A Two-Fluid Conditional Averaging Paradigm for the Theory ...and principles of theory and modeling,...

Research ArticleA Two-Fluid Conditional Averaging Paradigm for the Theoryand Modeling of Turbulent Premixed Combustion

Vladimir L Zimont

CRS4 Research Center POLARIS 09010 Pula (CA) Italy

Correspondence should be addressed to Vladimir L Zimont zimontcrs4it

Received 20 November 2018 Accepted 3 June 2019 Published 7 August 2019

Academic Editor Bruce Chehroudi

Copyright copy 2019 Vladimir L ZimontThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper extends a recent theoretical study that was previously presented in the form of a brief communication (Zimont CampF 1922018 221-223) in which we proposed a simple splitting method for the derivation of two-fluid conditionally averaged equations ofturbulent premixed combustion in the flamelet regime formulatedmore conveniently for applications involving unclosed equationswithout surface-averaged unknowns This two-fluid conditional averaging paradigm avoids the challenge in the Favre averagingparadigmofmodeling the countergradient scalar transport phenomenon and the unusually large velocity fluctuations in a turbulentpremixed flame It is a more suitable conceptual framework that is likely to be more convenient in the long run than the traditionalFavre averaging method In this article we further develop this paradigm and pay particular attention to the problem of modelingturbulent premixed combustion in the context of a two-fluid approachWe formulate and analyze the unclosed differential equationsin terms of the conditions of the Reynolds stresses 120591119894119895119906 120591119894119895119887 and the mean chemical source 120588119882 which are the only modelingunknowns required in our alternative conditionally averaged equations These equations are necessary for the development ofmodel differential equations for theReynolds stresses and the chemical source in the advancedmodeling and simulation of turbulentpremixed combustion We propose a simpler approach to modeling the conditional Reynolds stresses based on the use of the two-fluid conditional equations of the standard ldquo119896 minus 120576rdquo turbulence model which we formulate using the splitting method The mainproblem arising here is the appearance in these equations of unknown terms describing the exchange of the turbulent energy 119896and dissipation rate 120576 in the unburned and burned gases We propose an approximate way to avoid this problem We formulatea simple algebraic expression for the mean chemical source that follows from our previous theoretical analysis of the transientturbulent premixed flame in the intermediate asymptotic stage in which small-scale wrinkles in the instantaneous flame surfacereach statistical equilibrium while the large-scale wrinkles remain in statistical nonequilibrium

1 Introduction

We use the term lsquoparadigmrsquo in the title of this articleto emphasize that the two-fluid approach is a conceptualframework for analyzing and modeling turbulent premixedcombustion in the flamelet regime Paradigms in turbulentcombustion research with their basic conceptual viewpointsand principles of theory and modeling have been dis-cussed in the literature (see for example the paper entitledldquoParadigm in Turbulent Combustion Researchrdquo by Bilgeret al [1] which contains relevant citations) In this paperwe primarily analyze turbulent premixed combustion in thecontext of ldquoDamkohlerrsquos paradigmrdquo [1] which implies thatinstantaneous combustion takes place in a strongly wrinkled

laminar flame and we touch only briefly upon the thickenedflamelet regime BrayMoss and Libby proposed the progressvariable 119888 for use in the description of premixed combustionof a single quantity and developed BML formalism whichforms the basis of the BML model developed using theFavre ensemble averaging method [2] This more elaborateDamkohlerrsquos laminar flamelet paradigm can be referred toas the ldquoBML Favre averaging paradigmrdquo We remind readersthat this paradigm leads to the emergence of a funda-mental difficulty in the theory and modeling of turbulentpremixed combustion involving the adequate prediction of

the unknown mean flux of the progress variable 120588997888rarr119906 1015840101584011988810158401015840and stress tensor 120588997888rarr119906 10158401015840997888rarr119906 10158401015840 which appear in the unclosed

HindawiJournal of CombustionVolume 2019 Article ID 5036878 27 pageshttpsdoiorg10115520195036878

2 Journal of Combustion

Favre-averaged combustion and hydrodynamics equations

of the problem The point is that the scalar flux 120588997888rarr119906 1015840101584011988810158401015840 ispredominantly (but not always) countergradient (sometimescalled lsquocountergradient turbulent diffusionrsquo) and the stress

tensor120588997888rarr119906 10158401015840997888rarr119906 10158401015840 strongly depends on abnormally large velocityfluctuations observed in the turbulent premixed flame Thegas dynamic (nonturbulent) nature of these phenomena wasclearly explained by Pope in [3]

ldquoAs well as looking at the detail structure ofa turbulent premixed flame we can examinemean quantities Here too in comparison toother turbulent flows there are some unusualobservations the most striking being counter-gradient diffusion Within the flame there is amean flux of reactants due to the fluctuatingcomponent of the velocity field Contrary tonormal expectations and observations in otherflows it is found that this flux transports reac-tants up themean-reactants gradient away fromthe products (hence countergradient diffusion)A second notable phenomenon is the largeproduction of turbulent energy within the flameBehind the flame the velocity variance can be 20times its upstream value Both these phenomenaresult from the large density difference betweenreactants and products and from the pressurefield due to volume expansion [ ] From theEuler equations it is readily seen that a givenpressure gradient accelerates the light productsmore than the heavier reactants This mech-anism is responsible both for countergradientdiffusion and for turbulent energy productionrdquo

Many articles have been devoted to theoretical and exper-imental studies of these phenomena More advanced theo-retical and modeling approaches in the context of the BMLFavre averaging paradigm use the Favre-averaged unclosedequations in terms of the components of the scalar flux12058811990610158401015840119894 11988810158401015840 and stress tensor 12058811990610158401015840119894 11990610158401015840119895 and further approximationof the unknowns appears within turbulence theories (see forexample [4]) To justify the need for our study we emphasizethat the countergradient scalar flux and abnormally largevelocity fluctuations cannot be described adequately in thecontext of the BML Favre averaging paradigm (as we willillustrate below using as an example the results obtained in[4]) The reason for this is that these phenomena are causedby the large difference in the conditional mean velocities997888rarr119906 119906 and 997888rarr119906 119887 due to the different pressure-driven accelerationof the heavier unburned and lighter burned gases Thesevelocities which cannot be predicted by the Favre-averagedequations are at the same time described directly by the two-fluid conditional equations It would be appropriate here toquote a statement made by Forman Williams in his Hottellecture entitled ldquoThe Role of Theory in Combustion Sciencerdquo[5]

ldquoIt is relevant to distinguish between the scienceand the technology of the subject The march

of technology has never hesitated It uses sci-ence whenever possible but often especially incombustion forges ahead by trial and error orfortuitously by application of scientific miscon-ception but without scientific understandingrdquo

In this context the use of Favre averaging in themodelingof turbulent premixed combustion in the flamelet regimeis strictly speaking the ldquoapplication of scientific miscon-ceptionrdquo in contrast to two-fluid conditional averagingwhich is conceptually adequate Hence two fundamentalmodeling problems arise in the turbulent combustion theorydeveloped in the framework of the Favre averaging paradigmthe issue of adequate prediction in the premixed flame ofboth phenomena (ldquocountergradient turbulent diffusionrdquo andldquounusually large generation of turbulencerdquo) disappears in theframework of the two-fluid conditional average paradigm

As far as we know the pioneers of the theoreticaltwo-fluid approach to turbulent premixed combustion wereSpalding [6] and Weller [7] In [6] Spalding also analyzedsome previous papers related to the two-fluid approach In[7] Weller presented conditionally averaged two-fluid equa-tions that in fact were model equations The exact unclosedtwo-fluid conditionally averagedmass andmomentum equa-tions which served as a starting point for our study wereobtained in [8] This paper extends our recent theoreticaldevelopment of the two-fluid theory of turbulent premixedcombustion described in [9] in the format of a brief commu-nication in which we proposed a simple and physically clearsplitting method for the derivation of the unclosed two-fluidconditional equations of turbulent premixed combustionThis method made it possible to rederive the conditionalequations obtained in [8] and to understand the reason forthe appearance of surface-averaged variables in these theimpossibility of the well-founded modeling of which is oneof the obstacles to using this approach in applications inaddition this allowed us to indicate a way of excludingtheir appearance The only required modeling unknownsappearing in the original alternative two-fluid conditionalequations formulated in [9] are the conditional Reynolds

stress tensors 120591119906 = minus120588119906(997888rarr119906 1015840997888rarr119906 1015840)119906 and 120591119887 = minus120588119906(997888rarr119906 1015840997888rarr119906 1015840)119887 in theunburned and burned gases and the mean chemical source120588119882

This paper makes the following contributions(i) We describe in more detail the splitting method and

the derivation of the unclosed two-fluid conditionallyaveraged equations both those described in the liter-ature and the alternative versions

(ii) We obtain the unclosed equations in terms of theconditional moments (11990610158401198941199061015840119895)119906 and (11990610158401198941199061015840119895)119887 by splittingthe known unclosed equations in terms of the Favreaverage (12058811990610158401015840119894 11990610158401015840119895 ) and present the unclosed equationfor the mean chemical source 120588119882 which can be usedfor advanced modeling of the unknowns in the two-fluid conditionally averaged equations

(iii) We formulate simple models for estimating theunknown chemical source 120588119882 in the laminar and

Journal of Combustion 3

thickened flamelet combustion regimes based onour previous theoretical studies of transient turbulentpremixed flames in the intermediate asymptotic stagewhich were published elsewhere

(iv) We reformulate the equations for the classical ldquo119896 minus 120576rdquoturbulence model in terms of the conditional turbu-lent energy and dissipation rate using the slippingmethod and estimate the potentialities and problemsarising from the use of the obtained two-fluid ldquo119896 minus120576rdquo model to model turbulent characteristics and theconditional Reynolds stresses 120591119906 and 120591119887

We consider the two-fluid approach to the theory andmodel-ing of turbulent premixed combustion in the flamelet regimeas a promising alternative in the long run to traditionalmethods based on Favre ensemble averaging The two-fluidconditionally averaged equations adequately describe thehydrodynamic (due to the different pressure-driven accel-eration of the unburned and burned gases) and turbulent(using the corresponding two-fluid unclosed equation andcorresponding two-fluid turbulence model) effects and theirinteraction in the turbulent premixed flameThis allows us toeliminate the necessity of modeling the mean scalar flux andstress tensor which presents a challenge in the context of theFavre averaging framework

A strong argument in favor of the two-fluid approach isprovided by a review [10] of many studies of the structureof the instantaneous combustion zone in turbulent mixedflames which showed that combustion occurs in thinstrongly wrinkled flamelet sheets ldquoThin flamelets are foundto occur even when the Karlovitz number greatly exceedsunity The preheat zone average thickness is no larger thanthe laminar value in many studies while in some cases itis 2ndash4 times largerrdquo The two-fluid approach is applicablefor modeling not only premixed combustion in the laminarflamelet regime but also flames in the thickened flameletregime We draw the readerrsquos attention to the paper [11]quoted in [10] where the local flame structure of a premixedswirl-stabilized gas turbine burner has been investigated Weconsider the result obtained in [11] that the flamelet thermalthickness in the investigated highly turbulent lean premixedflames is close to the thermal thickness of the laminar flameas a significant argument in favor of the use of the two-fluidapproach

The two-fluid conditionally averaged unclosed equations

directly describe the conditional mean velocities997888rarr997888rarr119906 119906 and

997888rarr119906 119887the conditional mean pressures 119901119906 and 119901119887 the mean progressvariable 119888 and hence the probabilities of unburned gas 119875119906 =1minus119888 and burned gas119875119887 = 119888These can be used to calculate themean density 120588 and pressure 119901 the Favre-averaged progressvariable 119888 = 120588119888120588 the Reynolds- and Favre-averaged veloci-

ties997888rarr119906 and997888rarr119906 and themean scalar flux 120588997888rarr119906 1015840101584011988810158401015840 In conjunctionwith the corresponding combustion and turbulence modelsthese equations also describe the mean chemical source 120588119882the conditional Reynolds stresses 120591119906119894119895 = minus120588119906(11990610158401198941199061015840119895)119906 and 120591119887119894119895 =minus120588119906(11990610158401198941199061015840119895)119887 the Favre-averaged stresses 120591119894119895 = minus12058811990610158401015840119894 11990610158401015840119895 andsome other turbulent characteristics

In our study together with the conditional mean vari-ables we also use the Reynolds- and Favre-averaged oneswhich are useful in the interpretation of the results ofnumerical simulations and comparison with experimentaldata However the chemical source 120588119882 which dependson curtain statistical characteristics of randomly wrinkledinstantaneous flame is not a conditional mean characteristicThe source term 120588119882 represents the production rate per unitvolume of the product The rate per unit mass is a Favre-averaged characteristic = 120588119882120588 The unclosed equationfor the chemical source which is considered in this paper wasformulated in [12] in terms of We express the variables 120588and 997888rarr119906 appearing in this equation using 119888 997888rarr119906 119906 and 997888rarr119906 119887 whichare directly described by the two-fluid conditional equations

The paper is organized as followsSection 2 The two-fluid mathematical model and the

conditions of its applicability to premixed combustion in thelaminar and thickened flamelet regimes are described

Section 3 The potentialities and limitations of the Favreand two-fluid conditional averaging frameworks are dis-cussed and the aim of the study is formulated

Section 4 A splitting method for the derivation ofunclosed two-fluid conditional equations is presented itexplains the origin of surface-averaged unknowns anddemonstrates how to avoid them

Section 5 Two statistical concepts of premixed com-bustion in the flamelet regime are described in the formof a three-stage and a global one-stage process resultingin equations with and without surface-averaged unknownsrespectively

Section 6 An alternative system is presented for the two-fluid unclosed equations in which the only unknowns thatneed to bemodeled are the conditional Reynolds stresses andthe chemical source

Section 7 Unclosed equations are obtained for the condi-tional Reynolds stress using the splitting method

Section 8 Unclosed equations for the mean chemicalsource are presented and analyzed

Section 9 Equations for the conditional mean turbulentenergy and dissipation rate are obtained by splitting theFavre-averaged equations of the classical ldquo119896 minus 120576rdquo turbulencemodel

Section 10 The chemical source is modeled for transientflames in the laminar and thickened regimes

Section 11 A summary of this work and a conclusion arepresented

Appendix The hydrodynamichydraulic analytical the-ory of the countergradient scalar flux is described in detailand its applications for impinging and Bunsen flames andcriteria for transition of the scalar flux are discussed

The preliminary results of the current paper have beenpresented at three conferences (Zimont VL Gas Dynamicsin Turbulent Premixed Combustion Conditionally Aver-aged Unclosed Equations and Analytical Formulation ofthe Problem the 23rd International Colloquium on theDynamics of Explosions and Reactive System University ofCalifornia Irvine USA July 24-29 2011 pp 1-6 Zimont VLAn Alternative Approach to Modeling Turbulent Premixed


laminarflamelet

thickenedflamelet

turbulenteddies

(a) (b)

Figure 1 Laminar flamelet (a) and microturbulent (thickened)flamelet (b) combustion mechanisms

Combustion the Seventh Mediterranean Combustion Sym-posium Chia Laguna Cagliari Sardinia Italy September 11-15 2011 pp 1-11 Zimont VL An Approach in TurbulentPremixed Combustion Research Based on Conditional Aver-aging Joint Meeting of the British and Scandinavian-NordicSection of the Combustion Institute Cambridge UK March27-28 2014 pp 67-68)

2 The Two-Fluid Mathematical Model and ItsApplicability to Premixed Combustion

In this section we consider the conditions for applicability ofthe two-fluid mathematical model (in which it is postulatedthat instantaneous combustion takes place in a wrinkledflame of zero thickness) to combustion in the flamelet regime(where instantaneous combustion takes place in a stronglywrinkled laminar flame) and to combustion in the thickened(microturbulent) flamelet regime where eddies smaller thanthe laminar flame thickness penetrate and thicken the preheatzone of the instantaneous flame (Figure 1)

21The Two-FluidMathematical Model The two-fluidmath-ematical model corresponds to the limiting case in whichthe width of the instantaneous laminar flame 120575119871 is zero ieinstantaneous combustion takes place in a strongly wrinkledflame surface that propagates in a direction normal to itselfwith speed 119878119871 Thus the turbulent flame in the ldquoflamesurface regimerdquo consists of two fluids entirely unburned andcompletely burned gases

This model corresponds to a limiting case of the laminarflame in which the molecular transfer coefficient 120594 andchemical time 120591119888ℎ in the known expressions for the speed119878119871 asymp (120594120591119888ℎ)12 and width 120575119871 asymp (120594 sdot 120591119888ℎ)12 tend to zero120594 997888rarr 0 120591119888ℎ 997888rarr 0 while their ratio tends to a constant(120594120591119888ℎ) 997888rarr 119888119900119899119904119905 These conditions are as follows

lim119863119886997888rarrinfinRe

119905997888rarrinfin

(119863119886Re119905

) = 119888119900119899119904119905 (1a)



(120575119871120578 ) = 0 (1b)

where Re119905 = 1199061015840119871] is the turbulent Reynolds number (inwhich the kinematical viscosity coefficient ] and moleculartransfer coefficient 120594 are of the same order ] sim 120594) 119863119886 =120591119905120591119888ℎ is the Damkohler number (where 120591119905 = 1198711199061015840 isthe turbulent time) and 120578 = 119871Reminus34119905 is the Kolmogorovmicroscale (the size of the minimal eddies) Equation (1b)shows that close to the limit the size of the minimal eddies120578 remains much larger than the width of the flame 120575119871 Thisindicates that the influence of the stretch and curvature ofthe flame on its structure and speed caused by turbulence isnegligible ie in this limiting case the speed 119878119871 = 119888119900119899119904119905 isnot an assumption but an exact result

In this case the probability density function (PDF) of theprogress variable is bimodal 119901(119888) = 119875119906120575(119888)+119875119887120575(1minus119888) where119875119906 = 1 minus 119888 and 119875119887 = 119888 are the probabilities of unburned andburned gases respectively The symbol 120575 denotes the Diracgeneralized function and the values of the progress variableare 119888 = 0 for unburned and 119888 = 1 for burned gases

22 The Two-Fluid Mathematical Model and Premixed Com-bustion in the Laminar Flamelet Regime The laminar flameletregime arises in the case of developed turbulence (Re119905 gtgt 1)when the thickness of the laminar flame is much less thanthe size of the smallest eddies 120575119871 ltlt 120578 Since 119871 gtgt 120575119871in this case instantaneous combustion occurs in a highlywrinkled thin laminar flame and the application of thetwo-fluid mathematical model is justified We rewrite theinequality 120578120575119871 gtgt 1 using the expressions for 120575119871 120578 and thedefinition of the Damkohler number as follows

11986311988612Reminus14119905 gtgt 1 (2)

This inequality shows that in order for the instantaneousflame to be laminar the Damkohler number must be verylarge even in the case of smaller Reynolds numbers Forexample for values of Re119905 = 102 and Re119905 = 103 the LHSinequalities become 11986311988612 gtgt 3 and 11986311988612 gtgt 6 At smallerDamkohler numbers the laminar flamelet regime does notexist and the question of the applicability of the two-fluidapproach and the corresponding criteria need special consid-eration

23 The Two-Fluid Mathematical Model and Combustion inthe Microturbulent Flamelet Regime In the case of smallerDamkohler numbers for which the width of the laminarflame becomes larger than the sized of minimal turbulenteddies 120575119871 gt 120578 the smaller eddies penetrate the flameintensifying the transport processes and thickening the flameIf this thickening is not very large so that the instantaneousmicroturbulent flame remains strongly curved and the prob-ability of the intermediate states 0 lt 119888 lt 1 is small a two-fluid mathematical model will be applicable to this case Wenow obtain the necessary conditions using our theoreticalresults for the parameters of the thickened (microturbulent)flame as reported in [16] These results served as the basisfor the TFC turbulent premixed combustion model [17]which we use in this paper to model the mean chemicalsource appearing in the two-model conditionally averagedequations


The speed 119880119891 width 120575119891 and microturbulent transfercoefficient 120594119891 of the thickened flamelet are described by thefollowing expressions (Eq (16) in [16] and Eq (3) in [17])

119880119891 sim 1199061015840 sdot 119863119886minus12 (3a)

120575119891 sim 119871 sdot 119863119886minus32 (3b)

120594119891 sim 119863119905 sdot 119863119886minus2 (3c)

which were obtained on the assumption that the microtur-bulent transport inside the thickened flame is controlled byturbulence at statistical equilibriumwith a ldquo119896minus53rdquo spectrum

The speed of the turbulent flame is 119880119905 = 119880119891(1198781198780)119891where (1198781198780)119891 is the dimensionless area of the thickenedflamelet sheet Since (1198781198780)119891 gtgt 1 (meaning that the flameletsheet is strongly wrinkled) and 119880119905 sim 1199061015840 (the speed of thedeveloped turbulent premixed flame is of an equal orderof magnitude to the characteristic velocity fluctuation) itfollows that 119880119891 ltlt 1199061015840 From this and the inequality120575119891 gtgt 120578we can derive a criterion for the existence of a combustionregime in which the microturbulent flamelet sheet remainsrelatively thin (120575119891 ltlt 119871) and strongly wrinkled (119880119905 gtgt 119880119891)(Eqs (110) and (111) in [16] and Eq (7) in [17])

11986311988612 gtgt 1 gtgt 11986311988632Reminus34119905 (4)

We now modernize this criterion slightly keeping in mindthat thickening of the instantaneous flame takes place when120575119871120578 gt 1 ie when (120575119871120578)2 = 119870119886 gtgt 1 where 119870119886 is theKarlovitz number When the condition for strong wrinklingof the thickened flamelet sheet by large-scale eddies 119871120575119891 sim11986311988612 gtgt 1 is added the altered criterion is as follows

11986311988612 gtgt 1 gtgt 119870119886minus1 sim 119863119886 sdot Reminus12119905 (5)

These two criteria for the existence of a strongly wrinkledthickened flamelet sheet are similar in meaning and aresatisfied at large Reynolds and moderately large Damkohlernumbers

Example For = 10 Re119905 = 103 (4) and (5) become 3 gtgt 1 gtgt02 and 3 gtgt 1 gtgt 02 for 119863119886 = 30 Re119905 = 104 (4) and (5)become 55 gtgt 1 gtgt 0016 and 55 gtgt 1 gtgt 03

We now estimate the probability of the intermediatestates 0 lt 119888 lt 1 We assume that the speed and width ofthe turbulent flame are 119880119905 asymp 1199061015840 and 120575119905 asymp 10119871 and hence(1198781198780) = 119880119905119880119891 asymp 3 The ratio of the volume per unit areaof the flamelet sheet and the turbulent flame is (1198781198780)120575119891 asymp3119871 sdot 119863119886minus32 and asymp 10119871 giving a small characteristic valueof the probability of finding a flamelet sheet in the flame119875119891 asymp 3 sdot 119863119886minus32 asymp 10minus2

We see that the two-fluid approach is approximatelyapplicable to the thickened flamelet combustion regimewhenthe values of the Damkohler number are sufficiently large119863119886 gtgt 1 For smaller Damkohler numbers 119863119886 ge 1 atwhich the preheat zone becomes thicker and less wrinkledand especially for small numbers 119863119886 lt 1 direct applicationof the two-fluid approach becomes less suitable

In [16] a physical explanation was given for why thethickened flamelet which occurs when 120578 lt 120575119871 remains thinmeaning that there is no successive involvement of turbulenteddies of all sizes in the instantaneous flame and a distributedcombustion regime is achieved When its structure reachesstatistical equilibrium flamelet thickening reaches a limit inwhich ldquothe heat fluxes in the front because of heat conductionand convection and heat liberation because of chemicalreaction have one order of magnituderdquo (see Eq (17) in [16])

Even in the case of premixed flames subjected to extremelevels of turbulence studied in a recent paper [18] theinstantaneous reaction layer remains thin continuous andstrongly wrinkled

ldquoUnlike the preheat zones which grew exponen-tially with increasing values of 1199061015840119878119871 (up to sim10times their laminar value) the reaction layers inall 28 cases study here remained relatively thinnot exceeding 2 times their respective measuredlaminar values even though the turbulence level(1199061015840119878119871) increased by a factor of sim60 [ ] Thereaction layers are also observed to remaincontinuous that is local extinction events arerarely observedrdquo

These experimental data show that a distributed reactionzone is not observed and that accounting for pressure-driveneffects can be important for a proper description of theflame even in cases of very strong turbulence The reactionlayers in all 28 cases studied here remained relatively thinnot exceeding two times their respective measured laminarvalues even though the turbulence level (1199061015840119878119871) increasedby a factor of ldquosim60rdquo We notice that the preheat zoneswere investigated in [18] using laser induced fluorescenceof formaldehyde Obtained images of the preheat zonesdo not allow us to find the density gradient that is therelevant quantity for the estimation of a characteristic widthof the instantaneous flame The characteristic width can besignificantly smaller than what the images show keeping inmind the very small thickness of the instantaneous reactionzone The applicability of the two-fluid approximation tothe turbulent premixed flames subjected to extreme levels ofturbulence remains an open question

3 Unclosed Favre-Averaged andTwo-Fluid Conditional EquationsFormulation of the Problem

In this section we consider two systems of unclosed equa-tions for a turbulent premixed flame in the context ofthe Favre averaging and two-fluid conditional averagingframeworks compare their potentialities and limitations andformulate problems arising in the context of the two-fluidframework

31 Unclosed Favre-Averaged Equations and the Challengeof Countergradient Turbulent Diffusion The Favre-averagedcombustion and hydrodynamics equations for premixedcombustion are as follows120597120588119888120597119905 + nabla sdot (120588997888rarr119906119888) + nabla sdot 120588997888rarr119906 1015840101584011988810158401015840 = 120588119882 (6a)


120588 = 120588119906[1 + 119888 (120588119906120588119887 minus 1)] (6b)

120597120588997888rarr119906120597119905 + nabla sdot (120588997888rarr119906997888rarr119906) + nabla sdot 120588997888rarr119906 10158401015840997888rarr119906 10158401015840 = minusnabla119901 (6c)

120597120588120597119905 + nabla sdot 120588997888rarr119906 = 0 (6d)

The Favre-averaged value of a variable 119886 is 119886 = 120588119886120588 andits instantaneous value is 119886 = 119886 + 11988610158401015840 where the notation 119886identifies Reynolds averaging the mean density 120588 = 120588119906119875119906 +120588119887119875119887 and 120588119888 = 120588119888 = 120588119887119875119887 119875119906 and 119875119887 are the probabilitiesof the unburned and burned gases (119875119906 + 119875119887 = 1) The

term 120588997888rarr119906 1015840101584011988810158401015840 in the combustion equation (6a) is a scalar fluxfor which the 119894minuscomponent is 12058811990610158401015840119894 11988810158401015840 In the momentum

hydrodynamics equations (6c) 120588997888rarr119906 10158401015840997888rarr119906 10158401015840 is a tensor for whichthe 119894119895minuscomponent is 120588997888rarr119906 10158401015840119894 997888rarr119906 10158401015840119895 In (6b) for themean density 120588119906and 120588119887 are the densities of the unburned and burned gases

The variables defined for the system in (6a) (6b) (6c)and (6d) are the mean density 120588 and pressure 119901 the Favre-averaged progress variable 119888 and speed 997888rarr119906 The unknowns inthemodeling are the scalar flux (themean flux of the progress

variable) 120588997888rarr119906 1015840101584011988810158401015840 the stress tensor 120591 = minus120588997888rarr119906 10158401015840997888rarr119906 10158401015840 and themean chemical source (the mass consumption rate of theunburned gas per unit volume) which in the case of the flamesurface regime is 120588119882 = 120588119906119878119871Σ119891 where 119878119871 is the speed of theinstantaneous flame relative to the unburned gas and Σ119891 isthe flame surface density (the mean area of the instantaneousflame per unit volume) The mass equation (6d) does notcontain unknown terms

ldquoThe use of Favre averaging was initially introduced toreduce the system of transport equations to a form equivalentto the case of constant density flows and then using the sameturbulent closures for example gradient transport with eddydiffusivity modeling of second order transport correlationrdquoThe challenge of modeling the scalar flux and stress tensorin the turbulent premixed flame arises from the fact that theformer is unusual in the turbulent diffusion countergradientdirection and the latter is abnormally large for the turbulentflow velocity fluctuations described by the diagonal termsof the tensor As mentioned in the introduction the gasdynamics nature of these phenomena was clearly explainedby Pope in [3]

It is unlikely that the phenomena of the countergradientscalar flux and abnormally large velocity fluctuations in theturbulent premixed flame caused by the different pressure-driven acceleration of the unburned and burned gases canbe modeled adequately in the context of the Favre averagingparadigm even using an advanced approach based on attract-ing the unclosed differential equations for the scalar flux andstresses and even though these equations involving 12058811990610158401015840119894 11988810158401015840and 12058811990610158401015840119894 11990610158401015840119894 implicitly contain this hydrodynamic mecha-nismThepoint is that closure approximations for these equa-tions in the context of the Favre averaging framework wherethe unknowns are expressed in terms of the Favre-averaged

parameters defined by the equations of the problem do notprovide adequate modeling of the hydrodynamic pressure-driven effect This type of closure is performed in the samemanner as that commonly used in turbulence theory thescalar flux and stresses are presented in the model equationsas turbulent phenomena As an illustration of this shortcom-ing we will refer to the results of calculations of the scalarflux on the axis of the impinging premixed flame presentedin [19] in which a model theory of this type was developedfor the mean fluxes and stresses The results of the numericalsimulations performed in [4] (Figures 5 and 6 in [4]) show aqualitatively different behavior of the pressure profiles on theaxis of the flame In the flame close to the wall (Figure 5 in[4]) the pressure increases slightly across the flame due to thestrong influence of the wall on the pressure profile meaningthat in this case the scalar flux cannot be countergradientIn the free-standing impinging flame where influence of thewall is small the strong pressure drops across the flamewhere0 lt 119888 lt 1 (Figure 6 in [4]) can yield a countergradientscalar flux At the same time the numerically simulatedscalar fluxes presented in these figures are in both casescountergradient and close in value suggesting that the Favreaveraging paradigmdoes not provide a theoretically adequatedescription of the effects caused by the different pressure-driven acceleration of the unburned and burned gases

The reason for this drawback is that the closure of theseequations is performed in the context of the Favre averagingframework in which the unknowns are expressed in termsof the Favre-averaged parameters described by the equationsof the problem and does not adequately model the hydrody-namic pressure-driven effect described without modeling inthe context of a conditional averaging paradigmThis type ofclosure is performed in the same way as that commonly usedin the turbulence theory that is the scalar flux is interpretedas a turbulent phenomenon The closure used in [4] allowedthe authors to show agreement between their results (using acorresponding empirical constant) and known experimentaldata for the free-standing flame where the scalar flux wascountergradient (Figure 4 in [4]) However the results ofnumerical modeling with these empirical constants for theimpinging flame close to the wall were not physically feasibleeven in a qualitative sense

The results of numerical simulations of the impingingflame presented in [19] show a gradient scalar flux in theflame close to the wall and a predominantly countergradientflux in the free-standing flame In these simulations themean scalar flux was estimated as a sum of the terms thatdescribe the countergradient hydrodynamic contributiondue to the differences in pressure-driven acceleration of theheavier unburned and lighter burned gases and the gradientcontribution due to turbulent diffusion We discuss theseresults in the appendix

32 The Two-Fluid Equations and Modeling Challenges Theconservation equations for the unclosed two-fluid mass (7a)(7b) momentum (7c) (7d) and progress variable were for-mulated in [8] (Eqs (23)-(26a) and (26b)) and (17) as follows

120597 [120588119906 (1 minus 119888)]120597119905 + nabla sdot [120588119906 (1 minus 119888) 997888rarr119906 119906] = minus120588119906119878119871119906Σ119891 (7a)


120597 (120588119887119888)120597119905 + nabla sdot (120588119887119888997888rarr119906 119887) = 120588119887119878119871119887Σ119891 (7b)

120597 [120588119906 (1 minus 119888) 997888rarr119906 119906]120597119905 + nabla sdot [120588119906 (1 minus 119888) 997888rarr119906 119906997888rarr119906 119906)] + nabla

sdot [120588119906 (1 minus 119888) (997888rarr119906 1015840997888rarr119906 1015840)119906

] = minusnabla [(1 minus 119888) 119901119906]minus 997888rarr119906 119904119906120588119906119878119871119906Σ119891 + (119901997888rarr119899)

119904119906Σ119891

(7c)

120597 [120588119887119888997888rarr119906 119887]120597119905 + nabla sdot [120588119887119888997888rarr119906 119887997888rarr119906 119887)] + nablasdot [120588119887119888 (997888rarr119906 1015840997888rarr119906 1015840)

119887

] = minusnabla (119888 sdot 119901119887) + 997888rarr119906 119904119887120588119887119878119871119887Σ119891minus (119901997888rarr119899)

119904119887Σ119891

(7d)

120597119888120597119905 + 997888rarr119906 119904119906 sdot nabla119888 = (997888rarr119906 1015840119904119906997888rarr119899 1015840)119904119906 Σ119891 + 119878119871119906Σ119891 (7e)

where the indexes ldquo119906rdquo and ldquo119887rdquo refer to the unburdened andburned gases respectively 119878119871119906 and 119878119871119887 are the displacementspeeds of the instantaneous flame surface relative to theunburned and burned gases (119878119871119906 = 119878119871 using a commonnotation) Σ119891 is the flame surface density (FSD) the meanarea of the instantaneous flame per unit volume and 997888rarr119899is the unit normal vector on the flame surface toward theunburned side The mean progress variable 119888 is equal to theprobability of the burned gas 119875119887 119888 = 119888119906119875119906 + 119888119887119875119887 wherethe values of the progress variable are 119888119906 = 0 and 119888119887 = 1 ie119875119887 = 119888

In our analysis we will use the mean chemical source120588119882(997888rarr119909 119905) = Σ119891(997888rarr119909 119905) rather than the FSD Σ119891(997888rarr119909 119905) where = 120588119906119878119871119906 = 120588119887119878119871119887 is the mass flux per unit area of theinstantaneous flame We can therefore designate the RHSterms in (7a) and (7b) as minus120588119882 and +120588119882 the second RHSterms in (7c) and (7d) as as minus997888rarr119906 119904119906120588119882 and +997888rarr119906 119904119887120588119882 and thefinal RHS term in (7e) as 120588119882120588

The system in (3a) (3b) and (3c) describes the variables 119888997888rarr119906 119906997888rarr119906 119887119901119906 119901119887Theunknowns that need to bemodeled are the

conditional tensors (997888rarr119906 1015840997888rarr119906 1015840)119906 and (997888rarr119906 1015840997888rarr119906 1015840)119906 (or the conditionalReynolds stress tensors 120591119894119895119906 = minus120588119906(997888rarr119906 1015840997888rarr119906 1015840)119906 and 120591119894119895119887 =minus120588119887(997888rarr119906 1015840997888rarr119906 1015840)119906) themean chemical source 120588119882 and the surface-

averaged variables 997888rarr119906 119904119906 (119901997888rarr119899 )119904119906 997888rarr119906 119904119887 (119901997888rarr119899 )119904119887 and (997888rarr119906 1015840119904119906997888rarr119899 1015840)119904119906defined on the surfaces in the unburned and burned gasesadjacent to the instantaneous flame respectively

In contrast to the Favre averaging paradigm the mean

scalar flux 120588997888rarr119906 1015840101584011988810158401015840 does not need to be modeled as it isdescribed by the following closed expression

120588997888rarr119906 1015840101584011988810158401015840 = 120588119888 (1 minus 119888) (997888rarr119906 119887 minus 997888rarr119906 119906) (8)

where 120588 and 119888 in terms of the variable 119888 are as follows120588 = 120588119906 (1 minus 119888) + 120588119887119888 (9a)

119888 = 119888[119888 + (120588119906120588119887) (1 minus 119888)] (9b)

We can see that the phenomenon of the countergradientscalar flux in the turbulent premixed flame does need to bemodeled since 997888rarr119906 119906 997888rarr119906 119887 and 119888 are described by the unclosedsystem in (7a) (7b) (7c) (7d) and (7e)

The components of the tensor 120588997888rarr119906119906 are described by thefollowing expression

12058811990610158401015840119894 11990610158401015840119895 = 120588 (1 minus 119888) (11990610158401198941199061015840119895)119906 + 120588119888 (11990610158401198941199061015840119895)119887+ 120588119888 (119906119894119887 minus 119906119894119906) (119906119895119887 minus 119906119895119906)

(10)

Eq (10) shows that the ldquoabnormally large velocity fluctuationsin the turbulent premixed flamerdquo mentioned above whichare described by the diagonal terms of the tensor 120588997888rarr119906119906are controlled by the differences in the mean conditionalvelocities described directly by the unclosed system in (7a)(7b) (7c) (7d) and (7e)

We proceed on the premise that the conditional averagingframework is likely to be more convenient in the longrun than the Favre averaging framework since the two-fluid equations more adequately describe the pressure-drivenhydrodynamic processes in the premixed flame caused bya large difference in the densities of unburned and burnedgases Given the possibility of practical use of the two-fluidapproach we identified a critical obstacle in the fundamentaldifficulty of modeling of the surface mean variables ie theirexpression in terms of the variables described by the systemAt the same time we proceeded based on the possibility oftheoretical modeling of the unknown conditional Reynoldsstresses and mean chemical source by modifying existingmodeling approaches in the turbulence and combustiontheories for our case

Our formulation of the problem for this theoretical studyfollows from these considerations

33 Formulation of the Problem When we began to addressthis problem we noticed a connection between the structuresof the Favre-averaged mass and moment equations (6d) and(6c) and the two-fluid conditionally averaged mass andmoment equations (7a) (7b) and (7c) (7d) which influencedthe formulation of the problem By comparing the massequations (6d) and (7a) (7b) we can see that the LHS ofeach conditional mass equation for the unburned (7a) andburned (7b) gases has the structure of the Favre-averagedequation (6d) but is expressed in terms of the conditionalvariables The LHS of each conditional equation is weightedby the probabilities of the unburned (119875119906 = (1 minus 119888)) andburned (119875119887 = 119888) gas respectively Each equation has sinkand source terms on the RHS which are denoted as minus120588119882and +120588119882 where the chemical source 120588119882 is the same as inthe Favre-averaged equation (6a) A similar connection exists


between the structures of the momentum equations (6c) and(7c) (7d) where the sink and source represented by the twolast terms in (7c) (7d) express the impulse exchange betweenthe unburned and burned gases

In the derivation proposed by Lee and Huh [8] general-ized functionswere used in intermediate rather cumbersomemathematical manipulations and disappeared in the finalconditional equations The authors used zone averaging (asreflected in the title of the article) and did not actually relyon the two-fluid mathematical model Since cumbersomecalculations using an intermediate value of the progress vari-able that is nonexistent in the two-fluid mathematical model1 lt 119888lowast lt 1 led to the physically obvious result mentionedabove a simple method for obtaining these equations bysplitting the Favre-averaged mass and momentum equationsmust therefore exist

We therefore formulate the research problems consideredin this paper as follows

(1) A simple and physically insightful slipping methodfor the derivation of the two-fluid conditional meanequations which allows us to formulate alternativeunclosed two-fluid conditionally averaged momen-tum equations (using sound physical reasoning) thatdo not contain the surface-averaged unknowns toformulate unclosed two-fluid equations for the condi-tional Reynolds stresses and to reformulate the Favre-averaged equations of the ldquo119896minus 120576rdquo turbulence model interms of the mean conditional variables

(2) A derivation using an analysis of the splitting methodof the two-fluid conditional equation in terms of theconditional Reynolds stresses and their analysis

(3) An unclosed equation for the mean chemical sourcein the context of the two-fluid approach

(4) Practical approaches to modeling turbulence in theunburned and burned gases in the context of a two-fluid version of the standard ldquo119896minus120576rdquo turbulencemodeland the mean chemical source using a hypothesis ofstatistical equilibrium of the small-scale structures

In preparation for analyzing and resolving the listed tasksin further sections we aim to do the following

(i) To remind readers of the two-fluid one-dimensionalanalytical theory of the countergradient phenomenondeveloped by the present authors in [15] and to formu-late a criterion for the transition fromcountergradientto gradient scalar flux in the premixed flame

(ii) To illustrate these results using the examples ofimpinging and Bunsen flames considered in [19] and[15] respectively

Consideration of these results yields strong arguments in thefavor of the two-fluid conditional averaging framework Wepresent these materials in the appendix since although theyare not required in order to read the subsequent sections ofthis article they not only can help the reader to understandthe problem more deeply but also may be of interest in theirown right

4 A Splitting Method andAlternative Derivation of the KnownConditional Equations

In this section we consider the original derivation splittingmethod proposed in [9] and demonstrate its effectivenessby rederiving the known zone conditionally averaged massand momentum equations reported in [8] We notice thatthe derivation in [8] is rather cumbersome due to the use ofgeneralized functions and a particular value of the progressvariable 0 lt 119888lowast lt 1 in the intermediate mathematicalmanipulations which disappear from the final unclosedconditionally averaged equations Our derivation is moredirect and simple

In the next section we derive alternative conditionallyaveraged momentum equations using the splitting method

in which the only unknowns are (997888rarr119906 1015840997888rarr119906 1015840)119906 (997888rarr119906 1015840997888rarr119906 1015840)119887 and 120588119882

41 The Original Splitting Method The central idea of thisapproach is simple we split the Favre-ensemble-averagedmass (6d) and momentum (6c) equations into averaged two-fluid mass and momentum equations for the unburned andburned gases respectively using the obvious identity

120588119886 = 120588119906119886119906119875119906 + 120588119887119886119887119875119887 = 120588119906119886119906 (1 minus 119888) + 120588119887119886119887119888 (11)

(where 119886may be a constant scalar vector or tensor variable)We then use the conservation laws for the instantaneousflame to determine the sink and source terms appearing inthe two-fluid conditional mass and momentum equations

42 Conditionally Averaged Mass Equations To transform(6d) using (11) we use 119886 = 1 and 119886 = 997888rarr119906 This results in thefollowing equation

120597 [120588119906 (1 minus 119888)]120597119905 + nabla sdot [120588119906 (1 minus 119888) 997888rarr119906 119906]119906

+ 120597 (120588119887119888)120597119905 + nabla sdot (120588119887119888997888rarr119906 119887)119887

= 0(12)

where the expressions in the brackets 119906and 119887 refer toreactants and products respectively We can easily checkwhether the expression in the second braces is equal to 119887 =120588 To do this wemust eliminate997888rarr119906 from (1a) using (6a) andthe obvious expressions

120588119888 = 120588119888 = 120588119887119888 (13a)

997888rarr119906 = 997888rarr119906 119906 (1 minus 119888) + 997888rarr119906 119887119888 (13b)

Using (12) then gives 119906 = minus120588119882 and the conditional massequations are as follows

120597 [120588119906 (1 minus 119888)]120597119905 + nabla sdot [120588119906 (1 minus 119888) 997888rarr119906 119906] = minus120588119882 (14a)

120597 (120588119887119888)120597119905 + nabla sdot (120588119887119888997888rarr119906 119887) = 120588119882 (14b)


(14a) (14b) are identical to the two-fluid mass equations (23)and (24) in [8]

43 Conditionally Averaged Momentum Equations In a sim-ilar way to the analysis in the previous subsection we presentthe Favre-averaged momentum equations (6c) as a sum oftwo groups of terms which contain conditionally averagedparameters referring to the unburned and burned gases Toobtain these we insert into (6c) the expressions yielded by(11) with 119886 = 997888rarr119906 and 119886 = 997888rarr119906997888rarr119906 This results in the followingequation

120597 [120588119906 (1 minus 119888) 997888rarr119906 119906]

120597119905 + nabla sdot [120588119906 (1 minus 119888) 997888rarr119906 119906997888rarr119906 119906)] + nabla


]119906

+ 120597 (120588119887119888 119906119894119887)120597119905 + nabla

sdot (120588119887119888997888rarr119906 119887997888rarr119906 119887) + nabla sdot (120588119887119888 (997888rarr119906 1015840997888rarr119906 1015840)119887

)119887

= minusnabla [(1 minus 119888) 119901119906] minus nabla (119888119901119887)

(15)

where (997888rarr119906 1015840997888rarr119906 1015840)119906 (997888rarr119906 1015840997888rarr119906 1015840)119887 and 119901119906 119901119887 are conditionally aver-aged moments and pressures Splitting the equation in (15)yields the following two-fluid momentum equations



] = minusnabla [(1 minus 119888) 119901119906] + 997888rarr119865119906(16a)


119887

] = minusnabla (119888119901119887) + 997888rarr119865 119887(16b)

where 997888rarr119865119906 and997888rarr119865 119887 are equal in value and oppositely directed

(997888rarr119865119906+997888rarr119865 119887 = 0) terms that express the impulse sink and sourcedue to the mass exchange between the unburned and burnedgases and the inequality of the pressure on the different sidesof the instantaneous flame surface The equation 997888rarr119865119906 + 997888rarr119865 119887 =0 expresses the conservation of the averaged impulse on theinstantaneous flame surface The equation

997888rarr119906 119904119906120588119882 minus (119901997888rarr119899)119904119906Σ119891 = 997888rarr119906 119904119887120588119882 minus (119901997888rarr119899)

119904119887Σ119891 (17)

follows from the instantaneous impulse conservation law

997888rarr119906 119904119906 minus 119901119904119899997888rarr119899 = 997888rarr119906 119887119906 minus 119901119904119887997888rarr119899 (18)

where 997888rarr119899 is the unit vector normal to the instantaneous flamesurface As we shall see below the left-hand side and right-hand side of the equation (18) are equal respectively to 997888rarr119865119906

and 997888rarr119865 119887We represent the RHS of (6c) as follows

minus nabla119901 = minusnabla ((1 minus 119888) 119901119906) minus nabla (119888119901119887)+ (997888rarr119906 119904119887120588119882 minus (119901997888rarr119899)

119904119887Σ119891)

minus (997888rarr119906 119904119906120588119882 minus (119901997888rarr119899)119904119906Σ119891)

(19)

where the expression in the brackets is equal to zero inaccordancewith (17) After splitting the two-fluidmomentumequations (6c) with the RHS presented by (19) yields



] = minusnabla [(1 minus 119888) 119901119906]minus 997888rarr119906 119904119906120588119882 + (119901997888rarr119899)

119904119906Σ119891

(20a)


119887

] = minusnabla (119888119901119887) + 997888rarr119906 119904119887120588119882minus (119901997888rarr119899)

119904119887Σ119891

(20b)

which are identical to the conditional averaged momentumequations (25) and (26) in [8] that were derived usinggeneralized functions (see (7c) and (7d) above)

A comparison of (16a) (16b) (20a) and (20b) shows thatthe sink and source terms in (16a) and (16b) are as follows997888rarr119865119906 = minus997888rarr119906 119904119906120588119882 + (119901997888rarr119899)

119904119906Σ119891 (21a)

997888rarr119865 119887 = +997888rarr119906 119904119887120588119882 minus (119901997888rarr119899)119904119887Σ119891 (21b)

To avoid misunderstanding we should emphasize that thesplitting method uniquely determines the conditionally aver-aged mass equations (14a) and (14b) and momentum equa-tions (20a) and (20b)

The alternative conditional momentum equations with-out the surface-averaged unknowns which we formulate inthe next section are based on an original concept of a one-step statistical interpretation of instantaneous combustionand an alternative averaged impulse conservation law corre-sponding to this concept

5 Original Statistical Concept of PremixedCombustion in the Flame Surface Regime

The surface-averaged terms in (21a) and (21b) describe themomentum exchange between the unburned and burned


Instantaneous flame Adjacentsurfaces

Unburned gas Burned gas

u purarru u

u psurarru su

b pbrarru b

b psbrarru sb

Figure 2 Instantaneous parameters in the flow and on the adjacent surfaces

gases To avoid the use of the surface-averaged variable wemust express the momentum exchange between the gasesin terms of the conditional mean variables We do thiswithin the framework of the original statistical concept of aninstantaneous flame and we consider this below

51 Statistical Concepts of the One- and Three-Step Processesin Flame Surface Combustion We consider two statisticalconcepts of the combustion process in the turbulent pre-mixed flame in the flame surface regime In accordancewith the first concept the combustion is considered as aglobal one-step process in which unburned gas with parame-ters 120588119906 119901119906(997888rarr119909 119905) 997888rarr119906 119906(997888rarr119909 119905) transforms into burned gas withparameters 120588119887 119901119887(997888rarr119909 119905) 997888rarr119906 119887(997888rarr119909 119905) with a rate of transforma-tion equal to 120588119882(997888rarr119909 119905) This global one-step process can berepresented schematically as follows

120588119906 119901119906 997888rarr119906 119906 120588119882997904rArr 120588119887 119901119887 997888rarr119906 119887 (22)

In accordancewith the second concept the global processis split into three subprocesses these correspond to threesuccessive stages and are represented as follows

120588119906 119901119906 997888rarr119906 119906 997904rArr1 120588119906 119901119904119906 997888rarr119906 119904119906 997904rArr2 120588119887 119901119904119887 997888rarr119906 119904119887 997904rArr3 120588119887 119901119887 997888rarr119906 119887 (23)

The second stage corresponds to the stepwise transfor-mation of the unburned gas located on the surface adjacent(infinitely close) to the instantaneous flame to burned gasappearing on the infinitely close adjacent surface on the otherside of the flame In the first and third stages the change in theaveraged variables takes place in the unburned and burnedgases respectively

It is important to keep in mind that the averagedparameters 119901119906 997888rarr119906 119906 119901119904119906 997888rarr119906 119904119906 and 119901119887 997888rarr119906 119887 119901119904119887 997888rarr119906 119904119887 are definedat every point (997888rarr119909 119905) of the turbulent premixed flame whilethe instantaneous parameters 119901119906 997888rarr119906 119906 and 119901119887 997888rarr119906 119887 are definedin the unburned and burned gases respectively and theinstantaneous parameters 119901119904119906 997888rarr119906 119904119906 and 119901119904119887 997888rarr119906 119904119887 are definedon the corresponding surfaces adjacent to the instantaneousflame as shown in Figure 2

In the general case the conditional and surface-averagedvariables in the unburned and burned gases are not equalie 119901119906 = 119901119904119906 997888rarr119906 119906 = 997888rarr119906 119904119906 and 119901119887 = 119901119904119887 997888rarr119906 119904119887 = 997888rarr119906 119887The surface-averaged velocity 997888rarr119906 119904119887 of the products directlygenerated by the instantaneous flame is not necessarily equalto the conditional mean velocity of the burned gas 997888rarr119906 119887 Thusthe interaction of these ldquonewbornrdquo volumes of products withthe main flow of the products generates the pressure gradientin the rear zone adjacent to the instantaneous flame surfacegiving rise to the equalization of the velocities 997888rarr119906 119904119887 997904rArr 997888rarr119906 119887in the third stage Similarly the velocities 997888rarr119906 119906 and 997888rarr119906 119904119906 arenot necessarily equal due to the pressure gradient generatedby combustion in the front zone which yields the transition997888rarr119906 119906 997904rArr 997888rarr119906 119904119906

The flamelet sheet separates the gases with conditionalmean parameters 997888rarr119906 119906 119901119906 and 997888rarr119906 119887 119901119887 that is this sheetincludes the instantaneous flame surface and several adjacentlayers of unburned and burned gases in which the transitions997888rarr119906 119906 997904rArr 997888rarr119906 119904119906 and 997888rarr119906 119904119887 997904rArr 997888rarr119906 119887 take place In our mathematicalanalysis the width of this flamelet sheet is assumed to be zero

We draw the readerrsquos attention to an analogy between thetwo statistical concepts of the flamelet combustion describedabove and two classical theoretical concepts of a detonationwave

(i) In the Zelrsquodovich-Neumann-Doering (ZND) theory adetonation wave is a shock wave followed by a com-bustion front ie a global process of transformationof cold reactants into hot products that involves twostages compression of the unburned gas in the shockwave and its chemical transformation into burnedgas at the combustion front Thus the intermediateparameters of the compressed reactants appear in theequations of the ZND theory

(ii) In the Chapman-Jouguet (C-J) theory a detonationwave is considered to be a surface that divides theunburned and burned gases This surface includesthe shock wave and combustion front Hence theintermediate parameters of the gas do not appear inthe equations of the C-J theory which include only


the initial parameters of the reactants and the finalparameters of the products

Using this analogy we can say that the momentum equationscontaining intermediate surface-averaged variables (derivedby Lee and Huh in [8]) correspond conceptually to the ZNDtheory whereas our alternative momentum equations whichdo not contain intermediate surface-averaged variables (see(27a) and (27b) below) correspond to the C-J theory

52 Impulse Conservation Law Corresponding to the TwoConcepts of the Flame Sheet The impulse conservation law(17) that was used for formulation of the sink and sourceterms on the LHS of (20a) and (20b) corresponds to thesecond concept of the flamelet sheet It refers to the secondstage in (23) in which the unburned gas flow crossing theinstantaneous flame becomes the burned gas The problemof estimating the unknown surface-averaged parametersappearing in this case is reduced to one of modeling theeffects of the hydrodynamic mechanisms in the unburnedand burned gases that control the transformations of the gasparameters 997888rarr119906 119906 119901119906 997904rArr 997888rarr119906 119904119906 119901119904119906 and 997888rarr119906 119904119887 119901119904119887 997904rArr 997888rarr119906 119887 119901119887 inthe first and third stages

We formulate the impulse conservation law for this sheetas follows

997888rarr119906 119906120588119882 minus 119901119906997888rarr119899Σ119891 = 997888rarr119906 119887120588119882 minus 119901119887997888rarr119899Σ119891 (24)

where 997888rarr119899 (997888rarr119909 119905) is the averaged unit vector normal to theinstantaneous flame surface The terms 997888rarr119906 119906120588119882 and 997888rarr119906 119887120588119882describe the sink and source of the impulse per unit volumecaused by transformation of the unburned gas (with meanspeed 997888rarr119906 119906) into the burned gas (with mean speed 997888rarr119906 119887)while the terms minus119901119906997888rarr119899Σ119891 and minus119901119887997888rarr119899Σ119891 describe the sink andsource due to differences between the mean pressures 119901119906and 119901119887 in the unburned and burned gases Eq (24) doesnot contain a term describing the correlation between theinstantaneous pressures 119901119906 and 119901119887 and the unite vector 997888rarr119899 since these pressures refer to the unburned and burned gaseswhile the unit vector 997888rarr119899 shows the local orientation of theinstantaneous flame surface We remind the reader that thiscorrelation is significant in (17) since the pressures 119901119904119906 and119901119904119887 are defined on adjacent surfaces that are infinitely close tothe instantaneous flame with local orientations described bythe unit vector 997888rarr119899

In essence the originality of our theoretical analysisconsists of using a novel concept of a one-step conversion ofunburned gas with conditional mean parameters 997888rarr119906 119906 119901119906 intoburned gas with parameters 997888rarr119906 119887 119901119887 This allows us to avoidthe appearance of intermediate unknown surface-averagedparameters in the conditionally averaged momentum equa-tions

It seems that it is impossible to exclude surface-averagedparameters from the exact equation (17) meaning that thereis no mathematical equivalence between (17) in terms ofsurface-averaged parameters and (24) in terms of condi-tional mean parameters We therefore consider (24) as an

approximation till a possible more rigorous proof It shouldbe noted that (24) retains original physical meaning alsoin the case of the thickened flamelet regime in contrast to(17) where the surface-averaged variables that are definedon the surfaces adjacent to the instantaneous flame surfacemay cease to be strictly defined Below we shall validatethis equation against a result known in the literature fromdirect numerical simulation (DNS) of the one-dimensionalpremixed flame In answer to possible objections frompurists we note that full mathematical strictness of theunclosed equations is not obligatory since any turbulenceand combustion model that could be used for estimationof the unknown conditional Reynolds stresses and meanchemical source appearing in the unclosed equations wouldinevitably be approximate

In order to better understand the relationship betweenthese equations we use an analogy based on the result fromturbulence theory that gives an exact formula for the meandissipation rate 120576 = (120588]2)sum119894119895 (1205971199061015840119894120597119909119895 + 1205971199061015840119895120597119909119894)2 in termsof the velocity derivatives This result is controlled by small-scale turbulence the formula for the dissipation rate in strongturbulence (Re119905 gtgt 1) expressed in terms of the large-scale turbulent parameters 120576 = 11986011990610158403119871 (where119860 sim 1 isan empirical constant) does not follow from the previousformula but is a consequence of Kolmogorovrsquos hypothesisof statistical equilibrium in the small-scale structure ofturbulence Although the exact formula and its approximatetreatment cannot be mathematically equivalent the scientificcommunity consider Kolmogorovrsquos expression for the dissi-pation rate to be a theoretical result of turbulence mechanics

Analogously we consider the approximation in (24)which fortunately does not contain an empirical constant asan approximate impulse conservation law for the instanta-neous flame in the turbulent region at high Reynolds num-bers (ie developed turbulence) large Damkohler numbers119863119886 gtgt 1 (ie fast chemistry) and large density ratios120588119906120588119887 gtgt 1 (ie a strong pressure-driven effect) In contrastto the exact equation in (17) this equation is expressedin terms of the conditionally averaged parameters that aredescribed by the conditional mass equations (14a) and (14b)and formulated in the alternativemomentum equations (27a)and (27b) below

53 Accuracy of the Alternative Impulse Conservation LawWe have not attempted to estimate the accuracy of (24) withrespect to the global parameters of the turbulent flame Wedo not expect high accuracy at constant density 120588119906120588119887 =1 where both sides of (17) are identical At the same timethe conditional mean speeds in (24) are not equal in thegeneral case 997888rarr119906 119906 = 997888rarr119906 119887 and 119901119906 = 119901119887 (the former differencecontrols the scalar flux in (7a) and affects the stresses in (7b))Absent DNS results would indicate the extent to which thedifference in the velocities in the case of constant density iscompensated for by the inequality of the conditional meanpressures 119901119906 = 119901119887 The DNS results obtained in [20] fora steady-state planar premixed flame with density ratio 753and Damkohler number 181 demonstrate that in this casethe exact and approximate equations are fairly close In order


to estimate the terms in the exact impulse equation (17) the

pressure fluctuation correlations (1199011015840997888rarr119899 1015840)119904119906 and (1199011015840997888rarr119899 1015840)119904119906 areignored and the terms containing pressure are presented inthe forms (119901997888rarr119899 )119904119906Σ119891 asymp minus119901119904119906nabla119888 and (119901997888rarr119899 )119904119887Σ119891 asymp minus119901119904119887nabla119888The authors calculated these expressions which using ournotation become minus(997888rarr119906 119904119906 minus 997888rarr119906 119906)120588119882minus (119901119904119906 minus 119901119906)nabla119888 and (997888rarr119906 119904119887 minus997888rarr119906 119887)120588119882minus(119901119904119887minus119901119887)nabla119888These formulas approximately expressthe differences in the RHS and LHS terms of (17) and (24)respectively The graph for the former expression given inFigure 4 of [20] for the range 119888 = (01 minus 065) shows thatthe value of this expression is close to zero over the wholerange The graph in Figure 6 of [20] for the latter expressionpresented for the range 119888 = (03 minus 098) shows that the valueof this expression is nearly zero in the range 119888 = (03 minus 05)becomes negative and decreases relatively slowly in the range119888 = (05 minus 085) and then increases to zero in the range 119888 =(085 minus 098) In general these results support the conclusionthat the exact and approximate equations (17) and (23) areclose

We note that the authors of [20] proposed the use ofapproximation described above for the terms (119901997888rarr119899 )119904119906Σ119891 and

(119901997888rarr119899 )119904119887Σ119891 in modeling conditionally averaged momentumequations that do not contain unknown surface-averagedterms The pressure-related terms in these equations differfrom the corresponding terms in the momentum equationsthat are obtained in the next section using a statistical one-step model for transformation in the premixed flame of theunburned gas (with parameters 120588119906 997888rarr119906 119906 119901119906) into burned gas(with the parameters 120588119887 997888rarr119906 119887 119901119887)

It should be mentioned that this DNS has been per-formed for the turbulent premixed flame with a relativelylow Reynolds number We have no numerical data for thevalidation of (24) in the case of much higher Reynoldsnumber where the interaction between hot and cold volumescan be more significant and the difference in the conditionalmean velocities lower

6 Alternative Momentum Equations andComplete System of Unclosed Equations

In this section we obtain conditionally averaged momentumequations without unknown surface-averaged terms andformulate the complete system of unclosed equations as abasis for modeling as used in further sections

61 Formulation of the Alternative Conditional MomentumEquations To find 997888rarr119865119906 and

997888rarr119865 119887 in (16a) and (16b) which yieldthe alternative momentum equations we represent the RHSof (6a) as follows

minus nabla119901= minusnabla ((1 minus 119888) 119901119906) minus nabla (119888119901119887)+ (997888rarr119906 119887120588119882 minus 119901119887997888rarr119899Σ119891) minus (997888rarr119906 119906120588119882 minus 119901119906997888rarr119899Σ119891)

(25)

where the expression in the braces is equal to zero inaccordance with (24) After splitting the expressions for 997888rarr119865119906

and 997888rarr119865 119887 (25) yields997888rarr119865119906 = minus(997888rarr119906 119906120588119882 minus 119901119906997888rarr119899Σ119891)sin 119896

(26a)

997888rarr119865 119887 = (997888rarr119906 119887120588119882 minus 119901119887997888rarr119899Σ119891)119904119900119906119903119888119890

(26b)

Hence the desired alternative momentum equations are asfollows



] = minusnabla [(1 minus 119888) 119901119906]minus 997888rarr119906 119906120588119882 + 119901119906997888rarr119899Σ119891

(27a)


119887

] = minusnabla (119888119901119887) + 997888rarr119906 119887120588119882minus 119901119887997888rarr119899Σ119891

(27b)

We eliminate the variable Σ119891(997888rarr119909 119905) using the expressionΣ119891 = 120588119882 where = 120588119906119878119871 In order to eliminate the unit

vector 997888rarr119899 from (26a) and (26b) we use (24) to obtain 997888rarr119899 sim997888rarr119906 119906minus997888rarr119906 119887 ie the averaged unit vector997888rarr119899 is described by (28a)

997888rarr119899 = (997888rarr119906 119906 minus 997888rarr119906 119887)1003816100381610038161003816100381610038161003816997888rarr119906 119906 minus 997888rarr119906 119887

1003816100381610038161003816100381610038161003816 (28a)

119901119906 minus 119901119887 = 1003816100381610038161003816100381610038161003816997888rarr119906 119906 minus 997888rarr119906 1198871003816100381610038161003816100381610038161003816 (28b)

Combining (28a) with (24) and using the expression 120588119882 =Σ119891 result in (28b) which we will use below It followsfrom (28b) that the pressure difference becomes equal to thepressure drop across the laminar flameΔ119901 = 1205881199061198782119871(120588119906120588119887minus1)at the points of the turbulent flame where the vectors 997888rarr119906 119906 997888rarr119906 119887and 997888rarr119899 are collinear

The reader needs to keep two points in mind firstly weregard the unburned and burned gases as incompressiblefluids secondly the structures of the Favre-averaged equationand corresponding Reynolds-averaged equation of the sameproblem (which is associated with the constant densityapproximation) are identical In our case these are themomentum equation (6c) and the same equation in whichthe density is assumed to be constant

120588120597997888rarr119906120597119905 + 120588nabla sdot (997888rarr119906997888rarr119906) + 120588nabla sdot 997888rarr119906 1015840997888rarr119906 1015840 = minusnabla119901 (29)


Hence the constant density equation (29) also can beused for the formulation of two-fluid conditional equationsleading to (16a) and (16b) We use this consideration inSection 7 when deriving the unclosed two-fluid conditionalequations (33a) and (33b) for (11990610158401198941199061015840119895)119906 and (11990610158401198941199061015840119895)119887 proceedingfrom the constant density equation (32) for (11990610158401198941199061015840119895) This trickcan be useful when deriving other two-fluid conditionalequations

62 Complete System of Unclosed Equations for Turbulent Pre-mixed Combustion In this section we summarize the pre-vious results and represent the complete system of unclosedequations where the unknowns that need to be modeled arethe conditionally averaged Reynolds stresses and the meanchemical source (14a) (14b) (27a) (27b) (28a) and (28b)give the following system of unclosed equations


120597 (120588119887119888)120597119905 + nabla sdot (120588119887119888997888rarr119906 119887) = 120588119882 (30b)

120597 [120588119906 (1 minus 119888) 997888rarr119906 119906]120597119905 + nabla sdot [120588119906 (1 minus 119888) 997888rarr119906 119906997888rarr119906 119906)] minus nabla

sdot [(1 minus 119888) 120591119906] = minusnabla [(1 minus 119888) 119901119906]minus [[[997888rarr119906 119906 minus 119901119906

minus1 (997888rarr119906 119906 minus 997888rarr119906 119887)1003816100381610038161003816100381610038161003816997888rarr119906 119906 minus 997888rarr119906 1198871003816100381610038161003816100381610038161003816

]]]120588119882

(30c)

120597 [120588119887119888997888rarr119906 119887]120597119905 + nabla sdot [120588119887119888997888rarr119906 119887997888rarr119906 119887)] minus nablasdot [119888120591119887] = minusnabla (119888119901119887)+ [[[997888rarr119906 119887 minus 119901119887


]]]120588119882

(30d)

119901119906 minus 119901119887 = 1003816100381610038161003816100381610038161003816997888rarr119906 119906 minus 997888rarr119906 1198871003816100381610038161003816100381610038161003816 (30e)

The system includes three scalar equations (30a) (30b)and (30e) and two vector equations (30c) and (30d) Thesedescribe three scalar variables 119888 119901119906 and 119901119887 and two vectorvariables 997888rarr119906 119906 and 997888rarr119906 119887 The unknowns are the conditionalstress tensors 120591119906 and 120591119887 with components 120591119894119895119906 = minus120588119906(11990610158401198941199061015840119895)119906and 120591119894119895119887 = minus120588119887(11990610158401198941199061015840119895)119887 and the chemical source 120588119882

In the context of the two-fluid approach the terms thatdescribe the scalar flux and stress tensor do not appearin the equations this means that the challenge of model-ing the phenomena of the countergradient scalar flux andabnormal velocity fluctuations in the turbulent flame whichare artifacts of Favre ensemble averaging does not arise asmentioned above At the same time a quantitative estimate

of the mean scalar flux and stresses and the Reynolds- andFavre-averaged parameters may be necessary for physicalinterpretation of the results of numerical simulations andcomparison with experimental data These variables aredescribed by the following exact expressions

119888 = 119888[119888 + (120588119906120588119887) (1 minus 119888)] (31a)

997888119906 = 997888rarr119906 119906 (1 minus 119888) + 119906119887119888 (31b)

997888rarr119906 = 997888rarr119906 119906 (1 minus 119888) + 119906119887119888 (31c)

120588 = 120588119906[1 + 119888 (120588119906120588119887 minus 1)] (31d)

119901 = 119901119906 (1 minus 119888) + 119901119887119888 (31e)

12058811990610158401015840119894 11988810158401015840 = 120588119888 (1 minus 119888) (119906119894119887 minus 119906119894119906) (31f)


(31g)

(31a) (31b) (31c) (31d) (31e) (31f) and (31g) are notan inherent part of the system in (30a) (30b) (30c) (30d)and (30e) as they are not required for a solution of (30a)(30b) (30c) (30d) and (30e) for given values of 120591119894119895119906 120591119894119895119887119887and 120588119882 (31a) (31b) (31c) (31d) (31e) (31f) and (31g) areneeded to calculate 119888 997888rarr119906 120588 12058811990610158401015840119894 11990610158401015840119895 12058811990610158401015840119894 11988810158401015840 and 119901 using the

values of the variables 119888 119901119906 119901119887 997888rarr119906 119906 and 997888rarr119906 119887 described bythe system in (30a) (30b) (30c) (30d) and (30e) whichcan be used in the models for the unknown 120591119894119895119906 120591119894119895119887119887 and120588119882 and for the interpretation of numerical simulationsWe refer to the system that includes (30a) (30b) (30c)(30d) (30e) (31a) (31b) (31c) (31d) (31e) (31f) and (31g)as the complete equation system as it describes the mainReynolds- Favre- and conditionally averaged variables usedin applications We can add equations for example formulasfor 119879 120588119879101584010158402 11987910158402 and 119884119894 if the temperatures and speciesconcentrations 119884119894 in the unburned and burned gases areknown

We call attention to the absence from the system in(30a) (30b) (30c) (30d) and (30e) of a specific combustionbalance equation that is similar to (6a) Eqs (30a)ndash(30d) arehydrodynamic mass and momentum equations containingthe source 120588119882 meaning that the general problem cannot bebroken down into combustion and hydrodynamics subprob-lems This does not contradict the initial formulation of theproblem in which (6a) (6b) and (6c) (6d) correspond to thecombustion and hydrodynamic subproblems respectivelythe basic Favre-averaged combustion equation (6a) followsfrom the conditionally averaged mass equations (30a) (30b)and the expression (31f)

In the following two sections we present unclosed equa-tions for the unknown conditional Reynolds stresses and themean chemical source which can form the basis for advancedmodeling approaches In these sections we propose practical


approaches for the estimation of these unknowns using a two-fluid version of the standard ldquo119896 minus 120576rdquo turbulence model andsimple theoretical expressions describing the mean chemicalsource in the transient turbulent premixed flame observed inthe experiments

The obtained system of the two-fluid equations can benumerically simulated using as a model called ldquoEulerianMultifluid Modelingrdquo that is embedded in the commercialpackageAnsys Fluent 63 (httpswwwsharcnetcaSoftwareFluent6htmlugnode900htmsec-multiphase-eulerian)Fluentrsquos Eulerian multiphase model does not distinguishbetween fluid-fluid and fluid-solid (granular) multiphaseflows The Fluent solution can be summarized as follows(i) momentum and continuity equations are solved for eachphase (in our case for the unburned and burned gases)(ii) a single pressure is shared by all phases The lattercondition means that we must set 119901119906 = 119901119887 in the two-fluidmomentum equations (30c) and (30d) and omit (30e)which is superfluous in this case This assumption seemspermissible from a physical point of view

It is necessary to stress that we set 119901119906 = 119901119887 onlydue to peculiarities of this commercial package that isthis assumption is not caused by the essence of the two-fluid approach Note that assuming that conditional meanpressures are equal strictly speaking contradicts the basicidea of modeling the momentum interaction between thetwo fluids A consequence of this assumption is that the con-ditional equations (30c) and (30d) lose accuracy (althoughthe Favre-average equation (6d) remains exact while beingthe sum of these inaccurate equations) and therefore theconditional mean speeds as described by those equationsare not quite accurate However such inaccuracies may beinsignificantmdashasmay be indicated indirectly by a comparisonof the results which follow from the simple hydraulic two-fluid theory of the countergradient phenomenonmdashusing theassumption of equality of conditionally averaged pressureswith known experimental measurements of the counter-gradient scalar flux in the turbulent premixed flame seeAppendix A1

7 Unclosed Equations in Terms of Moments

In this section we obtain the two-fluid unclosed equations

for the unknown conditionalmoments (997888rarr119906 1015840997888rarr119906 1015840)119906 and (997888rarr119906 1015840997888rarr119906 1015840)119887We use the splittingmethod to formulate the exact equationswhich inevitably contain surface-averaged unknowns andapproximate alternative unclosed differential equations thatdo not contain these unknowns (this derivation is analogousto the use of the splittingmethod for the two-fluid conditionalmomentum equations)

We start with the constant density unclosed equations for(11990610158401198941199061015840119895) that are as follows [21] (molecular viscous terms areomitted)

120597 [120588 (11990610158401198941199061015840119895)]120597119905 + 120597 [120588119906119896 (11990610158401198941199061015840119895)]120597119909119896

= minus120597 [120588 (119906101584011989411990610158401198951199061015840119896)]120597119909119896minus [120588 (11990610158401198951199061015840119896) sdot 120597119906119894120597119909119896 + (11990610158401198941199061015840119896) sdot

120597119906119895120597119909119896]

minus [1199061015840119894 sdot 1205971199011015840

120597119909119895 + 1199061015840119895 sdot1205971199011015840120597119909119894]

(32)

Based on the remark at the end of Section 61 (32) leads tounclosed conditionally averaged equations for (11990610158401198941199061015840119895)119906 and(11990610158401198941199061015840119895)119887 as follows120597 [120588119906 (1 minus 119888) (11990610158401198941199061015840119895)119906]120597119905 + 120597 [120588119906 (1 minus 119888) 119906119896119906 (11990610158401198941199061015840119895)119906]120597119909119896= minus120597 [120588119906 (1 minus 119888) (119906101584011989411990610158401198951199061015840119896)119906]120597119909119896minus 120588119906 (1 minus 119888) [(11990610158401198951199061015840119896)119906 120597119906119894119906120597119909119896 + (11990610158401198941199061015840119896)119906

120597119906119895119906120597119909119896 ]

minus (1 minus 119888) [(1199061015840119895 sdot 1205971199011015840

120597119909119894)119906

+ (1199061015840119894 sdot 1205971199011015840

120597119909119895)119906

]minus 119876119906

119894119895

(33a)

120597 [120588119887119888 (11990610158401198941199061015840119895)119887]120597119905 + 120597 [120588119887119888 119906119896119887 (11990610158401198941199061015840119895)119887]120597119909119896= minus120597 [120588119887119888 (119906101584011989411990610158401198951199061015840119896)119887]120597119909119896minus 120588119887119888 [(11990610158401198951199061015840119896)119887 sdot 120597119906119894119887120597119909119896 + (11990610158401198941199061015840119896)119887 sdot

120597119906119895119887120597119909119896 ]

minus 119888 [(1199061015840119895 sdot 1205971199011015840

120597119909119894)119887

+ (1199061015840119894 sdot 1205971199011015840

120597119909119895)119887

] + 119876119887119894119895

(33b)

Here 119876119906119894119895 and 119876119887

119894119895 express the sink and source 119876119906119894119895 = 119876119887

119894119895

71 Exact Equations Containing Surface-Averaged UnknownsWe first derive exact expressions for 119876119906

119894119895 and 119876119887119894119895 in terms of

the surface-averaged parameters which need to be modeledand then avoid modeling by the formulation of approximateexpressions for the sink and source in terms of conditionalmean parameters

We begin the derivation of the exact expressions for 119876119906119894119895

and 119876119887119894119895 by formulating the following conservation law for

the 119894minus and 119895minus components of the impulse which connect


the parameters defined on the surfaces adjacent to theinstantaneous flame

1199061015840119894119904119906 minus (119901119904119906119899119894) + 119901119904119906119899119894 = 1199061015840119894119904119887 minus (119901119904119887119899119894) + 119901119904119887119899119894 (34a)

1199061015840119895119904119906 minus (119901119904119906119899119895) + 119901119904119906119899119895= 1199061015840119895119904119887 minus (119901119904119887119899119895) + 119901119904119887119899119895

(34b)

which follow from the instantaneous and averaged impulseequations for the 119894minus component

(119906119894119904119906 + 1199061015840119894119904119906) minus (119901119904119906119899119894)= (119906119894119904119887 + 1199061015840119894119904119887) minus (119901119904119887119899119894)

(35a)

119906119894119904119906 minus 119901119904119906119899119894 = 119906119894119904119887 minus 119901119904119887119899119894 (35b)

and similar equations for the 119895minus component By multiplyingthe RHS and RHS of (35a) and (35b) and averaging we obtainthe conservation law for the surface-averaged conditionalsecond moments (11990610158401198941199061015840119895)119904119906 and (11990610158401198941199061015840119895)119904119887 as follows

(11990610158401198941199061015840119895)119904119906 2 minus [(1199011199061015840119894119899119895)119904119906 + (1199011199061015840119895119899119894)119904119906] + [(1199012119899119894119899119895)119904119906 minus (119901119899119894)119904119906 (119901119899119895)119904119906]

= (11990610158401198941199061015840119895)119904119887 2 minus [(1199011199061015840119894119899119895)119904119887 + (1199011199061015840119895119899119894)119904119887] + [(1199012119899119894119899119895)119904119887 minus (119901119899119894)119904119906 (119901119899119895)119904119887]

(36)

When = 0 (ie there is no combustion) (34a) and (34b)become the identity 0 = 0 as all of the mean parametersdefined on both adjacent surfaces are equal For the case gt 0 we divide (36) by and then multiply it by Σ119891remembering that Σ119891 = 120588119882 we obtain the followingequation

(11990610158401198941199061015840119895)119904119906 120588119882 minus [(1199011199061015840119894119899119895)119904119906 + (1199011199061015840119895119899119894)119904119906] Σ119891+ [[(1199012119899119894119899119895)119904119906 minus (119901119899119894)119904119906 (119901119899119895)119904119906 ]

]Σ119891

= (11990610158401198941199061015840119895)119904119887 120588119882 minus [(1199011199061015840119894119899119895)119904119887 + (1199011199061015840119895119899119894)119904119887] Σ119891+ [[(1199012119899119894119899119895)119904119887 minus (119901119899119894)119904119906 (119901119899119895)119904119887 ]

]Σ119891

(37)

Hence the exact sink and source terms in (33a) and (33b) areas follows

119876119906119894119895 = (11990610158401198941199061015840119895)119904119906 120588119882

1

minus [(1199011199061015840119894119899119895)119904119906 + (1199011199061015840119895119899119894)119904119906] Σ1198912

+ [(1199012119899119894119899119895)119904119906 minus (119901119899119894)119904119906 (119901119899119895)119904119906 ] Σ1198913

(38a)

119876119887119894119895 = (11990610158401198941199061015840119895)119904119887 120588119882

1

minus [(1199011199061015840119894119899119895)119904119887 + (1199011199061015840119895119899119894)119904119887] Σ1198912

+ [(1199012119899119894119899119895)119904119887 minus (119901119899119894)119904119887 (119901119899119895)119904119887 ] Σ1198913

(38b)

The terms marked with lsquo1rsquo describe the sink and source dueto the transformation of the unburned gas into burned gas ata rate 120588119882 The terms marked with lsquo2rsquo which do not dependexplicitly on represent contributions to the sink and sourcedue to differences in the velocity fluctuations and pressureson the ldquocoldrdquo and ldquohotrdquo surfaces adjacent to the randominstantaneous flame with an orientation characterized by theunit vector 997888rarr119899 The terms marked with lsquo3rsquo depend explicitlyon and are caused by differences in the pressures on theisosurfaces All surface-averaged variables are unknowns

72 Alternative Two-Fluid Conditional Equations withoutSurface-Averaged Unknowns Eq (37) corresponds to thesecond stage in (23) Recall that the processes in stages 1 and3 do not make contributions to the sink and source terms119876119906119894119895 and 119876119887

119894119895 as these processes take place in the nonreactingunburned and burned gases meaning that (37) describes theglobal one-stage process denoted by (22) for the surface-averaged parameters To avoid the appearance of surface-averaged unknowns we formulate an approximate one-stageequation in terms of conditional averaged variables (in asimilar way to the conditional moment equations analyzedabove) as follows

(11990610158401198941199061015840119895)119906 120588119882 minus [(11990110158401199061015840119894)119906 119899119895 + (11990110158401199061015840119895)119906 119899119894] Σ119891+ [[11990110158402119906 (119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891

= (11990610158401198941199061015840119895)119887 120588119882 minus [(11990110158401199061015840119894)119887 119899119895 + (11990110158401199061015840119895)119887 119899119894] Σ119891+ [[11990110158402119887(119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891

(39)

Hence the approximate sink and source terms in (33a) and(33b) 119876119906

119894119895 and 119876119906119894119895 are as follows

119876119906119894119895 = (11990610158401198941199061015840119895)119906 120588119882 minus [(11990110158401199061015840119894)119906 119899119895 + (11990110158401199061015840119895)119906 119899119894] Σ119891

+ [[11990110158402119906 (119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891 = 119876119887

119894119895

= (11990610158401198941199061015840119895)119887 120588119882 minus [(11990110158401199061015840119894)119887 119899119895 + (11990110158401199061015840119895)119887 119899119894] Σ119891+ [[11990110158402119887(119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891

(40)

The conditional mean unknowns that need to be modeledare the terms in square brackets where the unit vector


997888rarr119899 is described by (28a) (120588119882 obeys the unclosed equationconsidered below and Σ119891 = 120588119882) In order to developmodel equations for the conditional Reynolds stresses theunknowns in (33a) (33b) and (40) must be expressed interms of the parameters described by the system (30a) (30b)(30c) (30d) and (30e)

8 Unclosed Equation in Terms of the MeanChemical Source

Themost commonly usedmethod for obtaining the unclosedequation describing themean chemical source is based on theequation 120588119882 = 120588119906119878119871Σ where the flame surface density (FSD)Σ (the mean area of the instantaneous flame per unit volume)obeys a surface-averaged unclosed Σminusequation We showedin [12] that this equation is inconsistent with other Favre-averaged equations (6a) (6b) (6c) and (6d) for the problemHere we show that the Σminusequation is also inconsistent withthe system of two-fluid conditional equations (30a) (30b)(30c) (30d) and (30e) In [12] we derived the Favre-averagedunclosed minus equation which directly describes the meanchemical source 120588119882 = 120588 and is consistent with the systemof Favre-averaged equations (6a) (6b) (6c) and (6d) Wereformulate this equation in terms of the conditional meanvariables so that it becomes compatible with the system oftwo-fluid conditional equations (30a) (30b) (30c) (30d) and(30e)

81 Incompatibility of the ΣminusEquation with Other EquationsThe flame surface density Σ is controlled by turbulencewhich generates wrinkles in the instantaneous flame andmoves with speed 119878119871 relative to the unburned gas whichsmoothes these wrinkles The FSD equations describing themean chemical source are as follows

120588119882 = 120588119906119878119871Σ (41a)

120597Σ120597119905 + nabla sdot (997888rarr119906 119904Σ) = [(nabla sdot 997888rarr119906)119904 minus (997888rarr119899997888rarr119899 nabla997888rarr119906)119904 Σ minus nablasdot [(119878997888rarr119899)

119904Σ] + (119878nabla sdot 997888rarr119899)

119904Σ

(41b)

where 997888rarr119899 = minusnabla119888(997888rarr119909 119905)|nabla119888(997888rarr119909 119905)| is the unit vector normal tothe instantaneous flame which points toward the reactantsand 997888rarr119899997888rarr119899 nabla997888rarr119906 = 119899119894119899119895120597119906119894120597119909119895 The ldquo119904rdquo symbol representssurface averaging This equation is defined on a particularisosurface inside the instantaneous flame travelingwith speed119878 While Σ is almost the same for all isosurfaces 120588119887 lt 120588 lt 120588119906the surface-averaged velocity 997888rarr119906 119904 and speed 119878 change acrossthe instantaneous flame on the ldquocoldrdquo isosurface

Although correct in itself the surface-averaged equation(41b) is inconsistent not only with the Favre-averaged equa-tions for the problem as shown in [12] but also with thetwo-fluid conditional equations (30a) (30b) (30c) (30d) and(30e) due to the appearance of differently averaged speeds997888rarr119906 119906 997888rarr119906 119887 and 997888rarr119906 119904 and other surface-averaged variables thatwould result in unnecessary and unjustified difficulties inmodeling

82 Consistent Unclosed Equation for the Mean ChemicalSource As mentioned in the introduction the mean chemi-cal source is not a conditionally averaged characteristic but astatistical characteristic of the traveling flame surface divid-ing the unburned and burned gases Hence the unclosedequation for the mean chemical source which is compat-ible with other two-fluid conditional equations inevitablycontains conditional mean variables referring to both theunburned and burned gases We propose to use the unclosedFavre-averaged equation in terms of for the turbulentpremixed combustion in the flamelet regime as derived in[12]

120597 (120588)120597119905 + nabla sdot (120588997888rarr119906) + nabla sdot (120588⟨997888rarr119906 1015840101584011988210158401015840⟩

119865)

+ 120588119906119878119871nabla sdot (997888rarr119899119882)= 120588 ⟨nabla119905 sdot 997888rarr119906⟩119865 + 120588⟨11988210158401015840 (nabla119905 sdot 997888rarr119906)10158401015840⟩

119865

+ 2120588119906119878119871119870119882

(42)

where = 120588119882120588 119870 = 05nabla sdot 997888rarr119899 is the curvature of theinstantaneous flame and the notations 119886 = ⟨119886⟩119865 and ⟨1198861015840101584011988710158401015840⟩119865indicate Favre averaging

The LHS of (42) contains one nonstationary term andthree transport terms describing convection by (i) meanvelocity transport (ii) turbulent diffusion-type transportdue to the correlation of velocity and combustion ratefluctuations and (iii) transport caused by the instantaneousflame movement The three terms on the RHS describethe effects of different physical mechanisms (sources andsinks) that control the Favre-averaged (i) the flamestretch due to the mean velocity field (ii) the flame stretchdue to the fluctuation component of the velocity and (iii)the effect of the wrinkled flame propagation on the sheetarea

In the case of the two-fluid mathematical model analyzedhere (ie for zero width of the instantaneous flame) there isno flame stretch as shown in Section 21 hence the first andsecond RHS terms must be omitted The unclosed equationfor the mean chemical source then becomes

120597 (120588119882)120597119905 + nabla sdot (120588119882997888rarr119906) + nabla sdot (120588997888rarr119906 1015840101584011988210158401015840) + 120588119906119878119871nablasdot (997888rarr119899119882) = 2120588119906119878119871119870119882

(43)

where the Favre-averaged velocity 997888rarr119906 and mean density 120588 areexpressed in terms of 997888rarr119906 119906 997888rarr119906 119887 120588119906 120588119887 and 119888 by (31a) (31b)and (31d) The unknowns that need to be modeled are in thethird and fourth terms on the LHS and the term on the RHSand should be expressed in terms of the variables describedby (30a) (30b) (30c) (30d) (30e) (31a) (31b) (31c) (31d)(31e) (31f) and (31g) Equation (43) was not presented in[12]


9 Practical Approaches to Modeling theConditional Reynolds Stresses

In this section we consider this relatively simple approach tomodeling the conditional Reynolds stresses 120591119894119895119906 = minus120588119906(11990610158401198941199061015840119895)119906and 120591119894119895119887 = minus120588119887(11990610158401198941199061015840119895)119887 in the context of the classical ldquo119896 minus 120576rdquoturbulencemodel [21] reformulated in terms of the two-fluidconditional variables using the splitting method We presentthe conditional mean turbulent stresses as follows

120591119894119895119906 = minus120588119906 (11990610158401198941199061015840119895)119906= ]119905119906 (120597119906119894119906120597119909119895 +

120597119906119895119906120597119909119894 ) minus (23) 120588119906119896119906120575119894119895

(44a)

120591119894119895119887 = minus120588119887 (11990610158401198941199061015840119895)119887= ]119905119887 (120597119906119894119887120597119909119895 +

120597119906119895119887120597119909119894 ) minus (23) 120588119906119896119887120575119894119895

(44b)

where ]119905119906 = 1198621205831198962119906120576119906 and ]119905119887 = 1198621205831198962119887120576119887 are the con-ditional turbulent viscosity coefficients (119862120583 is an empiricalcoefficient) and 120575119894119895 is the Kronecker delta function91 Two-Fluid Version of the ldquo119896 minus 120576rdquo Turbulence Model Theconditionally averaged turbulent energies and dissipationrates 119896119906 120576119906 119896119887 120576119887 can be described by conditionally aver-aged equations which we formulate by splitting the Favre-averaged equations of the standard ldquo119896 minus 120576rdquo turbulence model[21]

120597 (120588)120597119905 + 120597 (120588120572)120597119909120572 = 120591119894119895120597119895120597119909119894

+ 120597 [120588 (V119905120590119896) 120597120597119909119894]120597119909119894minus 120588120576

(45a)

120597 (120588120576)120597119905 + 120597 (120588119894120576)120597119909119894 = 1198621205761120591119894119895 (120576) 120597119906119895120597119909119894

minus 1198621205762120588(1205762 )+ 120597 [120588 (V119905120590120576) 120597120576120597119909119894 ] 120597119909119894

(45b)

where ]119905 = 119862120583(2120576) is the turbulent viscosity coefficient119862120583 1198621205761 1198621205762 120590119896 120590120576 are the empirical coefficients of thestandard ldquo119896minus120576rdquo turbulencemodel and 120591119894119895 are the componentsof the stress tensor

120591119894119895 = minus (12058811990610158401015840119894 11990610158401015840119895 ) = ]119905 ( 120597119894120597119909119895 +120597119895120597119909119894 ) minus (

23) 120588120575119894119895 (46)

These equations are as follows

120597 [120588 (1 minus 119888) 119896119906]120597119905 + 120597 [120588 (1 minus 119888) 119906120572119906119896119906]120597119909120572= 120588 (1 minus 119888) (120591119894119895119906120588119906) 120597119906119894119906120597119909119895 minus 120588 (1 minus 119888) 120576119906

+ 120597 [120588 (1 minus 119888) (V119905119906120590119896) 120597119896119906120597119909119894]120597119909119894 minus 119896119906120588119882+ 119870119906Σ119891

(47a)

120597 (120588119888119896119887)120597119905 + 120597 (120588119888119906119894119887119896119887)120597119909119894= 120588119888 (120591119894119895119887120588119887) 120597119906119894119887120597119909119895 minus 120588119888120576119887

+ 120597 [120588119888 (V119905119887120590119896) 120597119896119887120597119909119894]120597119909119894 + 119896119887120588119882 + 119870119887Σ119891

(47b)

120597 [120588 (1 minus 119888) 120576119906]120597119905 + 120597 [120588 (1 minus 119888) 119906119894119906120576119906]120597119909119894= 1198621205761120588 (1 minus 119888) (120576119906119896119906) (120591119894119895119906120588119906) 120597119906119894119906120597119909119895minus 1198621205762120588 (1 minus 119888) ( 1205762119906119896119906)

+ 120597 [120588 (1 minus 119888) (V119905119906120590119896) 120597119896119906120597119909120572]120597119909120572 minus 120576119906120588119882+ 119864119906Σ119891

(47c)

120597 (120588119888120576119887)120597119905 + 120597 (120588119888119906119894119887120576119887)120597119909119894= 1198621205761120588 (1 minus 119888) (120576119887119896119887) (120591119894119895119887120588119887) 120597119906119894119887120597119909119895minus 1198621205762120588119888( 1205762119887119896119887) +

120597 [120588119888 (V119905119906120590119896) 120597119896119887120597119909120572]120597119909120572+ 119864119887Σ119891 + 120576119887120588119882

(47d)

The underlined terms in the balance equations (47a) (47b)(47c) and (47d) obey the following conditions

minus119896119906120588119882 + 119870119906Σ119891 = +119896119887120588119882 + 119870119887Σ119891 (48a)

minus120576119906120588119882 + 119864119906Σ119891 = +120576119887120588119882 + 119864119887Σ119891 (48b)

The sink terms minus119896119906120588119882 minus120576119906120588119882 and source terms +119896119887120588119882+120576119887120588119882 are caused by transformation of the unburned gas


with turbulent parameters 119896119906 and 120576119906 into burned gas withturbulent parameters 119896119887 and 120576119887 The unknown exchangeterms which are symbolically represented as119870119906Σ119891 119870119887Σ119891 and119864119906Σ119891 119864119887Σ119891 describe the effects of the volume expansion ofthe gas due to combustion on the mean conditional kineticenergies and dissipation rates

92 A Simple Method to Avoid Modeling the UnknownExchange Terms To avoid modeling these exchange termswe took into account the fact that the influence of theexpansion due to combustion on turbulence of the unburnedgas is relatively small in comparison with its influence on tur-bulence of the burned gas due to effect of strong turbulizationof the gas when it crosses the wrinkled instantaneous flameOur proposed approach involves the following

(i) Omission of the terms 119870119906Σ119891 and 119864119906Σ119891 in (47a) and(47c) as negligible

(ii) Using the equations of the standard ldquo119896minus120576rdquo turbulencemodel instead of conditionally averaged equations(47b) and (47d)

(iii) Estimation of the conditional means 119896119887 and 120576119887 usingtwo algebraic expressions that involve 119896119906 119896119887 and120576 120576119906 120576119887 respectively

These simplifications lead to the following system of equa-tions

120597 [120588 (1 minus 119888) 119896119906]120597119905 + 120597 [120588 (1 minus 119888) 119906120572119906119896119906]120597119909120572= 120588 (1 minus 119888) (120591119894119895119906120588119906) 120597119906119894119906120597119909119895+ 120597 [120588 (1 minus 119888) (V119905119906120590119896) 120597119896119906120597119909119894]120597119909119894 minus 120588 (1 minus 119888) 120576119906minus 119896119906120588119882

(49a)

120597 [120588 (1 minus 119888) 120576119906]120597119905 + 120597 [120588 (1 minus 119888) 119906119894119906120576119906]120597119909119894= 1198621205761120588 (1 minus 119888) (120576119906119896119906) (120591119894119895119906120588119906) 120597119906119894119906120597119909119895+ 120597 [120588 (1 minus 119888) (V119905119906120590119896) 120597119896119906120597119909120572]120597119909120572 minus 1198621205762120588 (1 minus 119888)sdot ( 1205762119906119896119906) minus 120576119906120588119882

(49b)

120597 (120588)120597119905 + 120597 (120588120572)120597119909120572 = 120591119894119895120597119895120597119909119894+ 120597 [120588 (V119905120590119896) 120597120597119909119894]120597119909119894 minus 120588120576

(49c)

120597 (120588120576)120597119905 + 120597 (120588119894120576)120597119909119894 = 1198621205761120591119894119895 (120576) 120597119906119895120597119909119894+ 120597 [120588 (V119905120590120576) 120597120576120597119909119894 ] 120597119909119894 minus 1198621205762120588(1205762 )]

(49d)

= 119896119906 (1 minus 119888) + 119896119887119888 (49e)

120576 = 120576119906 (1 minus 119888) + 120576119887119888 (49f)

which allow us to avoid estimating the turbulization of theburned gas using the conditional turbulent energy 119896119887 anddissipation rate 120576119887 This turbulization in terms of and 120576 isdescribed to some extent by the Favre-averaged equations(49c) and (49d) and the effect on the values of the conditional119896119887 and 120576119887 is described by (49e) and (49f) It should beemphasized that the empirical constants in this modelingapproach are the same as in the standard ldquo119896 minus 120576rdquo turbulencemodel [21]

10 Simple Modeling of the MeanChemical Source

To model the mean chemical source 120588119882 we proceed fromthe following assumptions

(1) This source is the same in both the set of unclosedFavre-averaged equations (6a) (6b) (6c) and (6d)and the set of unclosed two-fluid conditional equa-tions (30a) (30b) (30c) (30d) and (30e)This meansthat we can use models obtained in the context ofthe Favre averaging framework and only reformulatethem in terms of the conditional mean variablesdescribed by the two-fluid equations

(2) The model for the chemical source depends on thecombustion regime We consider the laminar andthickened flamelet regimes since a distributed regimeis not observed in experiment In these regimesinstantaneous combustion takes place in thin andstrongly wrinkled flame sheets which can be consid-ered as random ones

(3) The chemical source model also depends on the stageof the turbulent flame Real turbulent flames aretransient meaning that complete statistical equilib-rium is not achieved in strongly wrinkled randomflamelet sheets We model the chemical source for atransient flame in the intermediate asymptotic stagewhere small-scale wrinkles in the flamelet sheet reachstatistical equilibrium while the large-scale wrinklesremain in nonequilibrium This concept allows us toobtain theoretical results that we will use to formulatevalid models of the chemical source

Following [16 17 22] we show that the flame in theintermediate asymptotic stage is characterized by a constantturbulent speed 119880119905 and increasing width 120575119905 We assume thatthe one-dimensional turbulent flame travels along the 119909minusaxisand that the wrinkled surface which shows the configuration


of the flamelet sheet is described by a random single-valuedfunction 119909 = ℎ(119910 119911 119905) with a continuous power spectraldensity (PSD) 119865(119896) The density is constant the turbulenceis uniform and stationary and the mean velocity of themedium is zero The speed of the flame is 119880119905 = 119878119871(1198601198600)where (1198601198600) gtgt 1 is the dimensionless mean area of theinstantaneous flame The area in terms of the spectrum isestimated as follows

( 1198601198600

) =1(1 + |nablaℎ|2)12 asymp

2|nablaℎ| asymp

3(|nablaℎ|2)12

=4(intinfin

01198962119865 (119896) 119889119896)12

(50)

The equality marked with lsquo1rsquo in (50) is an exact expression forthe area of the random surface described by the equation 119909 =ℎ(119910 119911 119905) The transition marked with lsquo2rsquo is valid for |nablaℎ|2 gtgt1 due to the assumption (1198601198600) gtgt 1 The transitionmarkedwith lsquo3rsquo is an estimate (an average of the absolute magnitudeof a random function with zero mathematical expectation isapproximately equal to the square root of its dispersion) Theequality marked with lsquo4rsquo is the exact expression of this root interms of the spectrum of the random surface 119865(119896) This gives

( 1198601198600

) asymp (intinfin01198962119865 (119896) 119889119896)12 (51a)

1205902119891 (119905) = (119909 minus 119909 (119905))2 = intinfin0119865 (119896 119905) 119889119896 (51b)

ie the small- and large-scale wrinkles (large and small wavenumbers 119896 in FSD) control the dimensionless mean area(1198601198600) and the dispersion 1205902119891 of the instantaneous flamerespectively

The assumption that the small-scale structures in thetransient turbulent flame are already statistical at equilibriumresults in a nearly constant speed of the flame 119880119905 =119878119871(1198601198600) Small-scale equilibrium is achieved when the rateof generation for small-scale wrinkles in the flamelet sheet bysmall-scale turbulent eddies with speed 1199061015840119890 that is equal to orless than the flamelet speed (1199061015840119890 le 119878119871 for the laminar flameletregime) and the rate of their consumption by the movingflamelet sheet surface are equal At the same time large-scalestructures are not yet at equilibrium resulting in an increasein the flame width In the case of strong turbulence when1199061015840 gtgt 119878119871 this increase is controlled by turbulent diffusionThe dispersion of the instantaneous flame 1205902119891 and the widthof the flame in this case are as follows

1205902119891 cong 2119863119905119905 (52a)

120575119905 sim (1199061015840119871119905)12 (52b)

where the turbulent diffusion coefficient119863119905 sim 1199061015840119871In [17 22] we referred to a flame with this type of

statistical structure as an ldquointermediate steady propagation(ISP) flamerdquo The kinematic equations describing a one-dimensional ISP flame in a motionless medium of constant

density and a three-dimensional ISP flame in a movingmedium are as follows

120597119888120597119905 = 11986311990512059721198881205971199092 + 119880119905 1003816100381610038161003816100381610038161003816100381612059711988812059711990910038161003816100381610038161003816100381610038161003816 (53a)

120597119888120597119905 + 997888rarr119906nabla119888 = nabla sdot (119863119905nabla119888) + 119880119905 |nabla119888| (53b)

The mean chemical sources in these equations are 119882 =119880119905|120597119888120597119909| and 119882 = 119880119905|nabla119888| In the case of variable densitythe chemical term in the corresponding Favre-averagedequations becomes

120588119882 = 120588 = 120588119906119880119905 |nabla119888| (54)

The problem is then reduced to obtaining theoreticalexpressions for the speed 119880119905 which are different for thelaminar and thickened flamelet combustion mechanismsThis problem was considered in [22] for the laminar flameletregime (using the flame surface mathematical model) and in[16 17] for the thickened (microturbulent) regime

101TheChemical Source for the Laminar Flamelet Regime Inthe flamelet regime in the case of constant density and strongturbulence (1199061015840 gtgt 119878119871) the speed and time interval of the ISPflame are described by the following expressions [22]

119880119905 sim (1199061015840119878119871)12 (55a)

120591119905 le 119905 ltlt (1199061015840119878119871)2 120591119905 (55b)

We use (55a) in modeling the chemical source in the casewhen the gas densities and the turbulent characteristics aredifferent for the unburned and burned gases We assume thatthe turbulent flame speed 119880119905 is controlled by the turbulenceof the unburned gas Hence the modal expression for thechemical source (120588119882)119871 used in the two-fluid conditionalequations becomes

(120588119882)119871= 119860119871120588119906radic11989612119906 119878119871 |nabla119888| (56)

where 119860119871 sim 1 is an empirical constant 119888 is expressed in (31a)in terms of 119888 as described by the two-fluid equations 119878119871 is thespeed of the laminar flame relative to the unburned gas and119896119906 = (32)11990610158401199062 is the turbulent energy in the unburned gasas described by the two-fluid model equations of turbulenceconsidered in Section 9

The formula (55a) is also invalid at large times 119905 gt(1199061015840119878119871)2120591119905 when wrinkles of all sizes reach statistical equi-librium A theoretical analysis of this steady-state flamepropagation was carried out in [22] using a hyperbolicequation that directly described the front edge of the flamecontrolling its propagation speed and showed that this speedwas 119880119905 = (11990610158402 + 1198782119871)12 Hence the flame speed at strongturbulence 1199061015840119878119871 gtgt 1 becomes 119880119905 cong 1199061015840 meaning thatthis analysis confirmed and clarified the known classical


Damkohler and Shelkin estimate 119880119905 sim 1199061015840 As this theoreticalresult indicates a paradoxical contradiction of the numericalexperimental data (since the speed of the turbulent flamedoes not depend on the physicochemical properties of thecombustible mixture) we referred to it in previous work asthe Damkohler-Shelkin paradox [22] We remind the readerthat the steady-state stage is unattainable in practice for realflames

Strictly speaking (55a) is also invalid for small times119905 le 120591119905 when the wrinkles of all sizes are at statisticaldisequilibrium Accurate simulations on these timescales aremainly important for spark-ignition engines We do notconsider this case here andmake the assumption that (56) andthe analogous equation (58) for the thickened flamelet regimeare valid at all times In this regard it should be noted thatthe statistical equilibrium of small-scale turbulent structuresis postulated in the ldquo119896 minus 120576rdquo turbulence model

102 The Chemical Source for the Thickened Flamelet RegimeThe possibility of applying the two-fluid approach in thecase of the thickened (microturbulent) flamelet regime wasanalyzed in Section 23 The conditions under which thethickened flamelet sheet remains thin and strongly wrinkledare described by the inequalities in (4) and (5)

The formula for the speed of the constant-pressure tran-sient ISP flame propagating in a uniform flow obtained in[16 17] (Eq (29) in [16] andEq (6) in [17]) and the inequalityfor the time interval for this transient flame are as follows

119880119905 = 119860119906101584011986311988614 = 11986011990610158403411987812119871 120594minus1411987114 (57a)

120591119905 lt 119905 ltlt 120591119905119863119886 (57b)

where 119860 is an empirical coefficient the Damkohler number119863119886 = 120591119905120591119888ℎ gtgt 1 the turbulent time 120591119905 = 1198711199061015840 and thechemical time 120591119888ℎ = 1205941198782119871

We use (57a) and assume that the turbulent flame speed119880119905 in the case of variable density is controlled by turbulence inthe unburned gas which is characterized by the conditionalturbulent energy 119896119906 and dissipation rate 120576119906 as describedby the turbulence model equations Thus we formulate themodel expression for the mean chemical source (120588119882)119898119905 forthe case of the microturbulent flamelet regime as follows

(120588119882)119898119905= 11986011989811990512058811990611989634119906 120576minus14119906 11987812119871 120594minus14119906 |nabla119888| (58)

where the empirical coefficients119860119898119905 and119860119871 appearing in themodel equations for the laminar andmicroturbulent flameletregimes have different values The molecular heat transfercoefficient 120594119906 and the speed of the laminar flame are thephysicochemical characteristics of the combustible mixturein this case

11 Conclusions

Widely used in applications RANS models of turbulent pre-mixed combustion are based on the use the Favre-averagedequations There is a large number of papers devoted todifferent aspects of the RANS approach some RANS models

implemented in commercial codes The totality of all resultsobtained in the context of the Favre averaging frameworkconstitute a certain technological tool widely used in manyindustries Nevertheless we are sure that the Favre averagingapproach is likely to be less convenient in the long runthan two-fluid conditional averaging approach for modellingturbulent premixed combustion in the flamelet regime Thereason is that from a conceptual point of view the latterapproach yields more adequate description of hydrodynamicand turbulent process in the unburned and burned gasesand their mutual interaction The challenge of modelingof the phenomena of the counter-gradient scalar flux andabnormally large velocity fluctuations in the premixed flamewhich in fact are treated in the context of Favre aver-age framework as turbulent phenomena (ldquocounter-gradientturbulent diffusionrdquo and ldquoabnormally strong turbulencerdquo)disappear in the context of two-fluid framework The reasonis that these phenomena are controlled by the difference ofthe conditional mean velocities which are described directly(without modeling) by the two-fluid conditional equations

At the same time the practical use of the two-fluidapproach faces significant difficulties This is due to theappearance in the two-fluid unclosed equations alongwith requiring modeling conditional Reynolds stresses alsosurface-averaged unknowns and the chemical source whichdepends on the statistical structure of the instantaneousflame-surface The unclosed equations become more cum-bersome when going beyond the two-fluid model is takeninto account the small but finite width of the instantaneousflame [23] which seems unjustified

We summarize the results of our affords to overcomethese difficulties and to propose possible way for resolvingarising modeling problem are as follows

(1) We formulate a system of two-fluid conditionallyaveraged unclosed equations (26a) and (26b) for tur-bulent premixed combustion in the flamelet regimewhere the only unknowns that need to bemodeled arethe conditional Reynolds stresses 120591119894119895119906 = minus120588119906(11990610158401198941199061015840119887)119906120591119894119895119887 = minus120588119887(11990610158401198941199061015840119887)119887 and the chemical source 120588119882 =120588119906119878119871Σ119891

(2) To avoid the appearance of unknown surface-averaged parameters which are inevitable in knownmathematically exact unclosed two-fluid equationswe propose the one-step statistical concept of theflame surface combustion regime in which everypoint of the turbulent flame undergoes directtransformation from unburned gas with parameters120588119906119901119906997888rarr119906 119906 to burned gas with parameters 120588119887119901119887997888rarr119906 119887

(3) We develop a simple splitting method (without usinggeneralized functions) that allows us to rederiveknown [8] equations containing surface-averagedterms and to formulate alternative conditional equa-tions that do not contain these unknown terms Thismethod is used for the formulation of equationsdescribing the conditional Reynolds stresses and forthe reformulation of the equations of the ldquo119896 minus 120576rdquo


turbulence model in the context of the two-fluidapproach

(4) We then formulate a system of unclosed equationsthat describes the conditional mean and Reynolds-and Favre-averaged parameters including the meanscalar flux and stress tensor which may be necessaryfor the interpretation of the results of numericalsimulations and their comparison with experimentaldata

(5) We touch on the problem of modeling the unknownReynolds stresses 120591119894119895119906 120591119894119895119887 and the mean chemicalsource 120588119882 as follows (i) we consider unclosedequations in terms of (11990610158401198941199061015840119887)119906 (11990610158401198941199061015840119887)119887 and whichmay be used in developing advanced models and(ii) we propose a simpler modeling approach inwhich the Reynolds stresses are estimated in thecontext of the modified ldquo119896 minus 120576rdquo turbulence modeland the chemical source is described by an algebraicexpression referring to a transient turbulent flame inthe intermediate asymptotic stage

(6) These unclosed two-fluid conditionally averagedequations and the approaches proposed here for theestimation of appearing unknowns can be treatedas an alternative paradigm for modeling turbulentpremixed combustion in the flamelet regime

Appendix

In this appendix we consider the phenomenon of thecountergradient scalar flux in the turbulent premixed flamecaused by the different pressure-driven acceleration of theheavier unburned and lighter burned gases An analysis ofthis hydrodynamic mechanism requires methods that falloutside the scope of the hypothesis-based approach usedabove We consider this important problem in the context ofour simple hydraulic two-fluid theory of the countergradientphenomenon

A The Countergradient Scalar Flux and ItsTransition to the Gradient Case

The hydrodynamic mechanism dominates in the turbulentpremixed flame when the fall in pressure across the flameΔ119901 = 1205881199061198802

119905 (120588119906120588119887minus1) is sufficiently largeThis takes place for alarge ratio of the densities 120588119906120588119887 gtgt 1 and strong turbulence1199061015840119878119871 gtgt 1 For estimation purposes we assume 120588119906120588119887 sim 101199061015840119878119871 sim 10 119880119905 sim 1199061015840 120575119905 sim (1199061015840119878119871)119871 sim 10119871 where 1199061015840and 119871 are the characteristic values of the velocity fluctuationsand the size of the turbulent eddies For estimation of thecharacteristic value 119902119901minus119889 of the pressure-driven componentof the scalar flux (1205881199061015840101584011988810158401015840)119901minus119889 using (8) we assume 119906119906 = 119880119905119906119887 = (120588119906120588119887)119880119905 sim 10119880119905 and 119888 sim 05 giving 119902119901minus119889 sim 2120588119880119905 Thecharacteristic value 119902119905119889 of the turbulent diffusion component(1205881199061015840101584011988810158401015840)119905119889 = minus120588119863119905119889119888119889119909 is 119902119905119889 sim 1205881199061015840119871120575119905 sim 10minus11205881199061015840Hence the ratio 119902119901minus119889119902119905119889 sim 20 meaning that the scalar fluxis controlled by the hydrodynamic effect caused by thermal

expansion The transition from countergradient to gradientscalar flux takes place when this ratio falls to 119902119901minus119889119902119905119889 sim 1In cases where 119902119901minus119889119902119905119889 ltlt 1 the pressure-driven effect issmall and the scalar flux is gradient

A1 Simple Hydrodynamic (Hydraulic) Two-Fluid Theory ofthe Countergradient Phenomenon In [15] we formulatedthe two-fluid mass and momentum equations assumingconservation of the total pressure of the unburned gas acrossthe flame brush which allowed us to find the mean scalarflux (1205881199061015840101584011988810158401015840)119901119889 in the turbulent premixed flame with knownspeed 119880119905 and the ratio 120588119906120588119887 In nondimensional form thesedimensionless equations in the coordinate system travelingwith the flame front are as follows using 119880119905 120588119906 and 1205881199061198802

119905 asquantities to normalize the velocity density andpressure [15]

(1 minus 119888) 119906119906 + 119888 119906119887Θ = 1 (A1a)

119901 + (1 minus 119888) 1199062119906 + 119888 1199062119887Θ = 119901minusinfin + 1 (A1b)

119901119906 + (12) 1199062119906 = 119901minusinfin + 12 (A1c)

whereΘ = 120588119906120588119887 and themean pressure119901 is given by119901 = (1minus119888)119901119906+119888119901119887 Assuming that the conditional mean pressures areequal ie 119901119906 = 119901119887(hydraulic approximation) (A1a) (A1b)and (A1c) reduce to the following set of equations

(1 minus 119888) 119906119906 + 119888 119906119887Θ = 1 (A2a)

(12 minus 119888) 1199062119906 +119888 1199062119887Θ = 12 (A2b)

which have the solution [15]

119906119887 = minus120573 + (minus4120572120574 + 1205732)12

2120572 119906119906 = 1 minus 119906119887119888Θ1 minus 119888 120572 = 119888Θ [ 05 minus 119888(1 minus 119888)2

119888Θ + 1] 120573 = minus2 119888Θ 05 minus 119888

(1 minus 119888)2 120574 = 05 minus 119888

(1 minus 119888)2 minus 05

(A3)

The dimensionless expressions for the scalar flux and pro-longed stress component are as follows

1205881199061015840101584011988810158401015840 = 120588119888 (1 minus 119888) (119906119887 minus 119906119906) (A4a)

1205881199061015840101584011990610158401015840 = 120588119888 (119906119887 minus 119906119906)2 (A4b)

Together with (A3) (31a) and (31d) yield the exact resultwithout empirical parametersThedimensionless conditional


0 025 05 075 10

005

01

015

02

025

03

035

04

045

05

u

c

uUt

c(a)

0 025 05 075 10

1

2

3

4

5

6

7

8

9

u uU

tu b

Ut

ubUt

uuUt

c(b)

Figure 3 Comparison of theoretical results for the dimensionless scalar flux and conditional mean velocities (solid curves) with Mossrsquosexperimental data [13] (the marks) for a fixed section of a turbulent Bunsen flame

mean velocities scalar flux and stress can be representedschematically by the following expressions

119906119906 = 1198911 (119888 Θ) (A5a)

119906119887 = 1198912 (119888 Θ) (A5b)

1205881199061015840101584011988810158401015840 = 1198913 (119888 Θ) (A5c)

1205881199061015840101584011990610158401015840 = 1198914 (119888 Θ) (A5d)

Here 1198911 1198912 1198913 and 1198914 are functions that follow directly fromthe previous equations

We consider this analysis to be an initial theoretical studyof the conditional velocities mean scalar flux and stressperformed in the context of the two-fluid approach

This hydrodynamics analysis is approximate for the fol-lowing reasons

(i) We ignore the effects of turbulence within theunburned and burned gases

(ii) We assume a constant total pressure in the unburnedgas

(iii) We assume that the pressure at each point in the flameis the same

In the main body of this paper we consider uncloseddifferential two-fluid conditional equations and proposedmodels for the estimation of unknowns that can be used tosolve the problem without these assumptions

A2 Application of the Two-Fluid Theory to a Bunsen FlameAlthough (4)ndash(7a) (7b) (7c) (7d) and (7e) were formulatedfor a one-dimensional steady-state flame they are applicableto a Bunsen flame since the contribution of the flame brushto the increase in the pressure drop across this turbulentflame is small in comparison with the pressure drop across

the flame caused by thermal expansion Figure 3 is takenfrom our previous work [15] and shows a comparison of thetheoretical results for the dimensionless scalar flux (the leftplot) and conditional velocities (the right plot) with Mossrsquosclassical experimental data [13] obtained from a fixed sectionof a Bunsen turbulent flame

Good agreement between theory and experiment sup-ports the conclusion that hydrodynamic effects dominatein this experiment ie the contribution of the gradientturbulent diffusion can be ignored The dashed curve in theleft-hand plot shows the upper estimate for the scalar flux thatgives (7a) using the assumption that the mean conditionalvelocities across the flame are constant and equal to 119906119906 = 119880119905and 119906119887 = Θ119880119905 (dashed lines in the right-hand plot)

These results show that the countergradient phenomenonand the velocity fluctuations in turbulent flames which areunusually large for turbulent flows are hydrodynamic innature This means that the scalar flux and stress tensorcannot be presented in gradient form ie the terms ldquocounter-gradient (negative) turbulent diffusionrdquo and ldquounusually largeturbulent velocity fluctuationsrdquo that are sometimes used forpremixed flames are misnomers

B Criterion for Distinguishing between theCountergradient and Gradient Scalar Flux

In the general case the mean conditional velocities 997888rarr119906 119906 and997888rarr119906 119887 are controlled by hydrodynamics and turbulence ie thetype of the scalar flux (gradient countergradient or neutral)which is governed by the balance between hydrodynamicand turbulent effects These interactive hydrodynamics andturbulent processes are accurately described in the contextof the two-fluid conditionally averaged differential equa-tions Here we approximate the scalar flux as the sum ofthe countergradient hydrodynamic contribution which we


describe in the context of the theory considered above andthe contribution of the gradient turbulent diffusion using theturbulent diffusion coefficient119863119905

1205881199061015840101584011988810158401015840 = (1205881199061015840101584011988810158401015840)119901minus119889

+ (1205881199061015840101584011988810158401015840)119905119889

= 1205881199061198801199051198913 (119888 Θ) minus 120588119863119905119889119888119889119909 (B1)

In order to explain the physical meaning of this assump-tion we note that in the case of constant density (1205881199061015840101584011988810158401015840 =12058811990610158401198881015840 and 119888 = 119888) there is no pressure-driven mechanism andthe scalar flux 12058811990610158401198881015840 and difference of the conditional meanvelocitiesΔ119906 = (119906119887minus119906119906) are controlled by turbulent diffusion(Δ119906 = Δ119906119905119889)

12058811990610158401198881015840 = 120588119888 (1 minus 119888) (119906119887 minus 119906119886) = minus120588119863119905119889119888119889119909 997904rArrΔ119906119905119889 = minus120588119863119905119889119888119889119909[119888 (1 minus 119888)]

(B2)

(B1) means that we represent the difference in theconditional mean velocities in the case of variable densityΔ119906 as a sum of the differences caused by the pressure-driveneffect Δ119906119901minus119889 and turbulent diffusion Δ119906119905119889

In (B1) we use elements of the two-fluid conditionalaveraging and Favre averaging frameworks when repre-senting a scalar flow as the sum of hydrodynamic andturbulent contributions and their estimates Equation (B1)is approximate since it does not include the effect on theflux 1205881199061015840101584011988810158401015840 of the interaction between hydrodynamic andturbulent processes which is automatically described by thetwo-fluid conditionally averaged equations

B1 Formulation of the Transition Criterion Eq (8) allows usto formulate original local and integral transition criteria

119870119897119900119888 =100381610038161003816100381610038161003816(1205881199061015840101584011988810158401015840)119901minus11988910038161003816100381610038161003816100381610038161003816100381610038161003816(1205881199061015840101584011988810158401015840)11990511988910038161003816100381610038161003816 =

10038161003816100381610038161205881199061198801199051198913 (119888 Θ)10038161003816100381610038161003816100381610038161003816120588 (119888)1198631199051198891198881198891199091003816100381610038161003816 (B3a)

119870int =100381610038161003816100381610038161003816⟨(1205881199061015840101584011988810158401015840)119901minus119889⟩10038161003816100381610038161003816100381610038161003816100381610038161003816⟨(1205881199061015840101584011988810158401015840)119905119889⟩10038161003816100381610038161003816

= (14) 120573 [Θ minus (2Θ minus 1)12] (1198801199051199061015840 )(120575119905119871 ) (B3b)

where ⟨(1205881199061015840101584011988810158401015840)119901minus119889⟩ and ⟨(1205881199061015840101584011988810158401015840)119905119889⟩ are the characteristicvalues (averaged over the flame section) for pressure-drivenhydrodynamic and turbulent diffusion contributions to thescalar flux and 120573 is a constant expected to be of orderunity The local criterion (B3a) which directly follows from(B1) shows that the scalar flux (1205881199061015840101584011988810158401015840) at points insidethe turbulent flame for which 119870119897119900119888 = 1 119870119897119900119888 lt 1and 119870119897119900119888 gt 1 is neutral gradient and countergradientrespectively Analogously the integral criterion (B3b) showsthat the scalar flux (1205881199061015840101584011988810158401015840) in the sections of the turbulent

flame where 119870int = 1 119870int lt 1 and 119870int gt 1 is neutralgradient and countergradient on average respectively Thelocal criterion (B3a) is uniquely defined in the context of thesimple theory considered above and its representation (B1)At the same time different estimates of the characteristicvalues of hydrodynamic and turbulent contributions mayappear in (B3b)

The assumptions that we used to obtain the integralcriterion (B3b) are as follows

(i) We assumed that the characteristic value of the meanflux ⟨(1205881199061015840101584011988810158401015840)119901minus119889⟩ corresponds to the flux describedby (A4a) with 119888∘ = 05 119888∘(1 minus 119888∘) = 14 and 120588 =120588(119888∘) although for simplification we estimated theconditional velocities 119906119906 and 119906119887 using dimensionlessequations (A2a) and (A2b) with 119888 = 1 or 119888 = 1Thisresults in the dimensionless velocities 119906119887 = Θ and119906119906 = [Θminus(2Θminus1)12] which somewhat overestimatethe difference in velocities described by (A2a) and(A2b) (see Figure 3(b))

(ii) The estimation of the dimension diffusion flux is⟨120588119863119905119889119888119889119909⟩ asymp 120588(119888∘)1199061015840119871(1120575119905)Comparing our and analogous criterion of transition119873119861 = (120591119878119871)(21205721199061015840) = 1 (where 120591 = (Θ minus 1) and 120572 (in

our case 120575119871119871 997888rarr 0) is a constant expected to be of orderunity) as proposed in [24] we notice that the latter was notbased on a comparison of the pressure-driven hydrodynamicand turbulent contributions to the scalar flux Hence thecriterion 119873119861 contains the parameters of the turbulent flame119880119905 and 120575119905 which affect the hydrodynamics and turbulenteffects respectively it therefore cannot predict for examplethe observed transition from the gradient scalar flux to thecountergradient flux along the Bunsen flame where 1199061015840 isnearly constant

B2 Qualitative Application of the Criterion to One-Dimensional and Bunsen Flames Wefirst apply these criteriato the one-dimensional steady-state flame where Θ gtgt 1The speed of the premixed flame with strong turbulence1199061015840 gtgt 119878119871 does not depend on 119878119871 and is equal to 119880119905 sim 1199061015840 inaccordance with the Damkohler qualitative estimation [25]which was confirmed and clarified theoretically (119880119905 cong 1199061015840) in[22] For the case of a typical stoichiometric mixture withΘ = 7 (119879119906 = 300119870 and 119879119887 = 2100119870) the integral criterion119870int (B3b) shows that a transition takes place (assuming in(B3b) that 120573 = 1) in the section where 120575119905119871 asymp 1 We believethat the width of this flame should be 120575119905119871 sim (5minus 10) ie thescalar flux is integrally countergradient At the same timein accordance with the local criterion (B3a) the scalar fluxmay be gradient in the vicinity of the front edge of the flamesince the difference in the velocities 119906119906 and 119906119887 is small in thisvicinity in accordance with the theory (see Figure 3(b))

The turbulent Bunsen flame is characterized by anincreasing width and at the same time a constant angle incli-nation to the mean flow ie nearly constant speed 119880119905 alongthe flameThis means that the gradient turbulent component|⟨(1205881199061015840101584011988810158401015840)119905119889⟩| in the integral criterion (B3b) is reduced


0 02 04 06 08 1

minus005

0

005

01

015

02

025

03

035Experiments

c

[kg

m2s]

u

c

(a)

0 02 04 06 08 1

minus005

0

005

01

015

02

025

03

035Calculations

c[k

gm

2s]

u

c

(b)

Figure 4 Comparison of experimental [14] (LHS) and theoretically estimated [15] (RHS) scalar flux in the fixed section of the turbulentBunsen flame

along the direction of the flame while the hydrodynamiccomponent |⟨(1205881199061015840101584011988810158401015840)119901minus119889⟩| remains nearly constant In thecase of a sufficiently large countergradient |⟨(1205881199061015840101584011988810158401015840)119901minus119889⟩|(intensive combustion 119880119905 sim 1199061015840 and a large ratio of thedensities 120588119906120588119887 gtgt 1) there will be a transition from gradientto countergradient scalar flux

In [22] we obtained the formula 119880119905 asymp (1199061015840119878119871)12 forthe speed of the transient flame under conditions of strongturbulence 1199061015840 gtgt 119878119871 with increasing width and constantspeed ie we must substitute 1198801199051199061015840 sim (1198781198711199061015840)12 in (B3b)The theoretical concept of this transient flame is based on theassumption of statistical equilibrium of small-scale wrinklesand at the same time disequilibrium of large-scale wrinkles inthe instantaneous turbulent flame at the intermediate asymp-totic stage We note that the theory for a one-dimensionalsteady-state flame is applicable to the Bunsen flame since theeffect of the increase in flame width on the pressure dropacross the flame is small In a typical case with Θ = 7 and1199061015840119878119871 = 5 the transition takes place in the section where120575119905119871 asymp 3 ie the transition can take place because the flamewidth can reach substantially larger values

C Gradient and Countergradient Scalar Fluxin Bunsen and Impinging Flames

Previous experimental studies of the scalar flux in the Bunsenflame usually report the results of cross-section measure-ments which are used in Figure 3 for example where thescalar flux measured in [13] was countergradient In [14]Bilger et al studied the Bunsen flame where the scalarflux in a fixed cross-section was countergradient neutral

or gradient depending on the turbulent parameters of theflow and the mixture composition In [15] we comparedthese experimental data with our theoretical analysis of thescalar flux in the Bunsen flame We briefly present thiscomparison for the readerrsquos convenience and also considerthe results of a study of the scalar flux in free-standingimpinging flames and those close to the wall In both caseswe use the representation (B1) in which the estimation of thehydrodynamic countergradient contribution is based on thetwo-fluid one-dimensional theory

We hope that these data provide an additional argumentin favor of the two-fluid conditional averaging frameworkdeveloped in later sections

C1 Scalar Flux Regimes in the Bunsen Flame In Figure 4taken from our paper [15] the left plot represent in thecoordinates 1205881199061015840101584011988810158401015840 = 119891(119888) the experimental data for thescalar flux in the fixed cross-section of the Bunsen flameobtained in [14] and in the right plot the results of ourestimations The velocity fluctuations 1199061015840 and ratios 1199061015840119878119871corresponding to different marks in the Figure 4 are asfollows 1199061015840 = 085119898119904 and 1199061015840119878119871 = 36 (rhombs) 1199061015840 =053119898119904 and 1199061015840119878119871 = 31 (squares) 1199061015840 = 083119898119904 and 1199061015840119878119871 =49 (circles) 1199061015840 = 079119898119904 and 1199061015840119878119871 = 88 (triangles)

In regimes with similar values of the velocity fluctuationsin the flow (1199061015840 = 079119898119904 1199061015840 = 083119898119904 and 1199061015840 = 085119898119904)the contributions of the gradient turbulent diffusion to thescalar flux were similar to each other while the contributionsof the countergradient hydrodynamic effect were differentThe latter increased with the values of the laminar flamespeed with variations in the mixture composition as follows119878119871 = 009119898119904 (1199061015840 = 079119898119904 1199061015840119878119871 = 88) 119878119871 = 017119898119904


0

002

0225 025 0275

1

1 1

02 03 04 026 027 028 029 03

026 027 028 029 03

X(m) x(m) X(m)

0225 025 0275x(m)

02 03 04X(m)

004

006

008

0101 03 05 07 09

012

minus006

minus004

minus002

0

08

06

04

02

0

004

002

006

minus006

minus004

minus002

0

002

004

006

0

002

004

006

008

01

012

minus0981

minus0984

minus0987

minus099

minus0993

minus0996

minus0999

３=04ＧＭ

３=04ＧＭ

３=2ＧＭ

０-０0０0

minus0987

minus0994

minus1001

minus1008

minus1015

minus1022

minus1029

０-０0０0

０-０0０0

５５0

５５0

1５５0

C

1

08

06

04

02

0

1

(Ｏ＝ＯＯ0)＄-＄Ｎ＝ＲＯＯ0

Ｏ＝ＯＯ0C

X(m)(Ｏ＝ＯＯ0)＄-＄Ｎ＝ＲＯＯ0

Ｏ＝ＯＯ0C

０-０0０0

５５0

C

３=04ＧＭ

３=2ＧＭ

３=2ＧＭ

Figure 5 Configurations and profiles of the parameters of the flame close to the wall (upper plots) and the free-standing flame (lowerplots)

(1199061015840 = 083119898119904 1199061015840119878119871 = 49) and 119878119871 = 024119898119904 (1199061015840 =085119898119904 1199061015840119878119871 = 36) The hydrodynamic effect is smallerfor the mixture with a smaller laminar flame speed 119878119871 dueto the lower turbulent speed 119880119905 and ratio Θ = 120588119906120588119887 andhence the smaller pressure drops across the turbulent flameΔ119901 = 1205881199061198802

119905 (Θ minus 1) and vice versa When 119878119871 = 009119898119904the turbulent diffusion component dominated and the scalarflux had a gradient direction when 119878119871 = 024119898119904 thehydrodynamic component dominated and the scalar fluxhada countergradient direction For an intermediate value of thespeed 119878119871 = 017119898119904 the contributions of the hydrodynamicsand turbulence effects were comparable resulting in near-neutral scalar flux In the fourth regime shown in Figure 4the velocity fluctuations were smaller (1199061015840 = 053119898119904) butthe laminar flame speed had the same intermediate value119878119871 = 017119898119904 (1199061015840119878119871 = 31) Hence both contributions weresmaller (the turbulent effect was smaller due to a smallervalue of 1199061015840 while the hydrodynamic effect was smaller due toa smaller value of119880119905 caused by smaller value of the speed 1199061015840)resulting in the near-neutral scalar flux in the experimentsand calculations shown in Figure 4

C2 Scalar Flux in Close-to-Wall and Free-Standing Imping-ing Flames Figure 5 (a reproduction of Figure 31 in [19])illustrates the potential of the two-fluid approach in thetheoretically grounded modeling of the scalar flux in theimpinging flame This figure shows the results of numericalsimulations of the two 2D flames in the channel (closeto the obstacle and free-standing impinging flames) Thesimulations were performed by Valirio Battaglia using thecommercial code ldquoFluentrdquo based on the TFC combustionmodel [17] (see [26] for more details)

The flow and turbulence parameters in both cases werethe same (119880 = 30119898119904 1199061015840 = 8119898119904 119871 = 054119888119898 119901 =1119886119905119898 119879 = 300119870 120588119906120588119887 = 7) although the combustiblemixtures were different Close-to-obstacle and free-standingflames had laminar flame speeds 119878119871 = 04119888119898119904 and 2119888119898119904respectively

The left-hand images show the geometry of the channeland obstacle (a 2Dplate) and the isosurfaces 119888 = 119888119900119899119904119905 whichshow the configuration of the flames The central picturesshow the dimensionlessmean pressure (119901minus1199010)119901 (solid line)mean velocity 1198801198800 (dotted line) and the progress variable


119888 (dashed-dotted line) along the central axis The left-handimage shows the dimensionless scalar fluxes which in ournotation are the countergradient pressure-driven contribu-tion (1205881199061015840101584011988810158401015840)119901minus1198891205881199061198800 (solid line) the gradient turbulent dif-fusion contribution (1205881199061015840101584011988810158401015840)1199051198891205881199061198800 or (minus120588119906119863119905120597119888120597119909)1205881199061198800(dotted line) and the summary scalar flux (1205881199061015840101584011988810158401015840)1205881199061198800(bold solid line) It can be seen that the direction of the scalarflux in the flame close to the obstacle corresponds to gradientdiffusion while the scalar flux in the free-standing flame iscountergradient except for the zone adjacent to the leadingedge of the flame In [26] we discussed the reason for thedomination of turbulent diffusion in the head of the flame

From a theoretical viewpoint these results are reasonablesince they correlatewith the profile of the pressure in contrastto the results for the close-to-the-wall and free-standingimpinging flames mentioned in Section 31 and obtained in[4] in the context of the Favre averaging framework

Our approximate approach using (9a) and (9b) for theestimation of the scalar flux in the impinging flame wasvalidated in [27] against known experimental data for twoimpinging flames situated at different distances from the wall[28] In both flames the experimental and calculated scalarfluxes were countergradient but in the flame that was closerto the wall the level of the scalar flux was lower due to thesmaller pressure drop against the flame (Figure 3 in [27])

D Summary of Appendix

The results of the preliminary investigation of the two-fluidapproach considered in this appendix were based on anapproximate representation of the scalar flow in the form ofa sum of hydrodynamic and turbulent contributions and canhelp us to better understand the potentialities of the two-fluidapproach and the problems arising from this which facilitatesa deeper study of this issue Taking into account the basicadvantages of the two-fluid approach demonstrated aboveand bearing in mind the well-documented fact that in manypractical cases instantaneous combustion occurs in highlycurved thin layers we believe that the two-fluid conditionalaveraging framework shows significant promise formodelingapplications In particular it seems reasonable to reformulatethe TFC combustion model which is represented in thecommercial packages Fluent and CFX in the context of thetwo-fluid conditional averaging paradigm

Nomenclature

BML Bray-Moss-Libby119888 Progress variable119863119886 = 120591119905120591119888ℎ Damkohler number119870119886 = (120575119871120578)2 Karlovitz number 119896119906 119896119887 Favre- and conditionally averagedturbulent kinetic energy119871 Integral scale of turbulence997888rarr119899 Unit normal vector to instantaneousflame119901(119888) PDF of progress variable

PDF Probability density function

119875119906 119875119887 Probabilities of unburned andburned gases119901119906 119901119887 Conditional mean pressures inunburned and burned gases

Re119905 = 1199061015840119871] Turbulent Reynolds number119878119871 Speed of laminar flame1199061015840 = (11990610158402)12 Root-mean-square velocityfluctuation997888rarr119906 119904119906 119906119904119887 Surface-averaged speeds at the edgesof an instantaneous flame997888rarr119906 119906 119906119887 Conditional mean speeds ofunburned and burned gases119880119891 Thickened flamelet propagationspeed119880119905 Turbulent premixed flamepropagation speed

Greek Symbols

120575119891 Width of thickened flamelet120575119871 Width of laminar flame120575119905 Width of turbulent flame120576 120576119906 120576119887 Favre- and conditionally averageddissipation rates120578 = 119871Reminus34119905 Kolmogorov microscaleΘ = 120588119906120588119887 Ratio of densities of unburned andburned gases

] Kinematic viscosity coefficient120588 Density120588119906 120588119887 Density of unburned and burnedgases

120588997888rarr119906 1015840101584011988810158401015840 Mean scalar flux

120588997888rarr119906 10158401015840997888rarr119906 10158401015840 Mean stress tensor120588119882 Mean chemical sourceΣ119891 Flame surface density120591119888ℎ = 1205941198782119871 Chemical combustion time120591119894119895119906 120591119894119895119887 Conditional Reynolds stresses120591119905 = 1198711199061015840 Turbulent time120594 Molecular heat transfer coefficient

Data Availability

No data were used to support this study

Conflicts of Interest

The author declares no conflicts of interest

Acknowledgments

I have benefited greatly from the discussions about thistopic with Ken Bray (who encouraged me in due time todevelop this alternative approach) Denis Veynante (wholooked through a preliminary version of themanuscript) andmy former colleague Fernando Biagioli


References

[1] R W Bilger S B Pope K N C Bray and J F DriscollldquoParadigms in turbulent combustion researchrdquo Proceedings ofthe Combustion Institute vol 30 no 1 pp 21ndash42 2005

[2] K N C Bray P A Libby and J B Moss ldquoUnified modelingapproach for premixed turbulent combustionmdashpart i generalformulationrdquo Combustion and Flame vol 61 no 1 pp 87ndash1021985

[3] S B Pope ldquoTurbulent premixed flamesrdquoAnnual Review of FluidMechanics vol 19 no 1 pp 237ndash270 1987

[4] K N C Bray M Champion and P A Libby ldquoPremixedflames in stagnating turbulence part IV a new theory for theReynolds stresses and Reynolds fluxes applied to impingingflowsrdquo Combustion and Flame vol 120 no 1-2 pp 1ndash18 2000

[5] F A Williams ldquoThe role of theory in combustion sciencerdquoSymposium (International) on Combustion vol 24 no 1 pp 1ndash17 1992

[6] D B Spalding ldquoThe two-fluid model of turbulence applied tocombustion phenomenardquo AIAA Journal vol 24 no 6 pp 876ndash884 1986

[7] H G Weller ldquoThe developing of a new flame area combustionmodel using conditional averagingrdquo Thermo-Fluids SectionReport TF9307 Department of Mechanical EngineeringImperial College of Science Technology and Medicine 1993httpciteseerxistpsueduviewdocdownloaddoi=1011456303ampamprep=rep1ampamptype=pdf

[8] E Lee and K Y Huh ldquoZone conditional modeling of premixedturbulent flames at a highDamkohler numberrdquoCombustion andFlame vol 138 no 3 pp 211ndash224 2004

[9] V L Zimont ldquoA splitting method and alternative unclosed two-fluid conditional equations of turbulent premixed combustionwithout surface-averaged unknownsrdquo Combustion and Flamevol 192 pp 221ndash223 2018

[10] J F Driscoll ldquoTurbulent premixed combustion flamelet struc-ture and its effect on turbulent burning velocitiesrdquo Progress inEnergy and Combustion Science vol 34 no 1 pp 91ndash134 2008

[11] F Dinkelacker A Soika D Most et al ldquoStructure of locallyquenched highly turbulent lean premixed flamesrdquo Symposium(International) on Combustion vol 27 no 1 pp 857ndash865 1998

[12] V Zimont ldquoUnclosed Favre-averaged equation for the chemicalsource and an analytical formulation of the problem of turbu-lent premixed combustion in the flamelet regimerdquo Combustionand Flame vol 162 no 3 pp 874-875 2015

[13] J B Moss ldquoSimultaneous measurements of concentration andvelocity in an open premixed turbulent flamerdquo CombustionScience and Technology vol 22 no 3-4 pp 119ndash129 1980

[14] JH Frank P AKalt andRWBilger ldquoMeasurements of condi-tional velocities in turbulent premixed flames by simultaneousOH PLIF and PIVrdquo Combustion and Flame vol 116 no 1-2 pp220ndash232 1999

[15] V L Zimont and F Biagioli ldquoGradient counter-gradienttransport and their transition in turbulent premixed flamesrdquoCombustionTheory andModelling vol 6 no 1 pp 79ndash101 2002

[16] V L Zimont ldquoTheory of turbulent combustion of a homo-geneous fuel mixture at high reynolds numbersrdquo CombustionExplosion and Shock Waves vol 15 no 3 pp 305ndash311 1979

[17] V Zimont ldquoGas premixed combustion at high turbulence Tur-bulent flame closure combustionmodelrdquo ExperimentalThermaland Fluid Science vol 21 no 1-3 pp 179ndash186 2000

[18] A W Skiba T M Wabel C D Carter S D Hammack JE Temme and J F Driscoll ldquoPremixed flames subjected toextreme levels of turbulence part I Flame structure and a newmeasured regime diagramrdquoCombustion and Flame vol 189 pp407ndash432 2018

[19] V L Zimont ldquoKolmogorovs legacy and turbulent premixedcombustion modellingrdquo in New Developments in CombustionResearch W J Carey Ed p 93 Nova Science Publishers IncNew York NY USA 2006

[20] E Lee Y H Im and K Y Huh ldquoZone conditional analysis of afreely propagating one-dimensional turbulent premixed flamerdquoProceedings of the Combustion Institute vol 30 no 1 pp 851ndash857 2005

[21] D C Wilcox Turbulence Modeling for CFD DCW IndustriesInc Calif USA 3rd edition 2006

[22] V L Zimont ldquoDamkohler-Shelkin paradox in the theory ofturbulent flame propagation and a concept of the premixedflame at the intermediate asymptotic stagerdquo Flow Turbulenceand Combustion vol 97 no 3 pp 875ndash912 2016

[23] AN Lipatnikov ldquoConditionally averaged balance equations formodeling premixed turbulent combustion in flamelet regimerdquoCombustion and Flame vol 152 no 4 pp 529ndash547 2008

[24] D Veynante A Trouve K N Bray and T Mantel ldquoGradientand counter-gradient scalar transport in turbulent premixedflamesrdquo Journal of Fluid Mechanics vol 332 pp 263ndash293 1997

[25] G Damkohler ldquoDer einfluszligder turbulenz auf die flam-mengeschwindigkeit in gasgemischenrdquo Zeitschrift Fur Elektro-chemie und Angewandte Physikalische Chemie vol 46 no 11pp 601ndash626 1940

[26] V L Zimont and V Battaglia ldquoJoint RANSLES approach topremixed flame modelling in the context of the tfc combustionmodelrdquo Flow Turbulence and Combustion vol 77 no 1-4 pp305ndash331 2006

[27] F Biagioli and V L Zimont ldquoGasdynamics modeling ofcountergradient transport in open and impinging turbulentpremixed flamesrdquo Proceedings of the Combustion Institute vol29 no 2 pp 2087ndash2095 2002

[28] S C Li P A Libby and F Williams ldquoExperimental investiga-tion of a premixed flame in an impinging turbulent streamrdquoSymposium (International) on Combustion vol 25 no 1 pp1207ndash1214 1994

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Favre-averaged combustion and hydrodynamics equations

of the problem The point is that the scalar flux 120588997888rarr119906 1015840101584011988810158401015840 ispredominantly (but not always) countergradient (sometimescalled lsquocountergradient turbulent diffusionrsquo) and the stress

tensor120588997888rarr119906 10158401015840997888rarr119906 10158401015840 strongly depends on abnormally large velocityfluctuations observed in the turbulent premixed flame Thegas dynamic (nonturbulent) nature of these phenomena wasclearly explained by Pope in [3]

ldquoAs well as looking at the detail structure ofa turbulent premixed flame we can examinemean quantities Here too in comparison toother turbulent flows there are some unusualobservations the most striking being counter-gradient diffusion Within the flame there is amean flux of reactants due to the fluctuatingcomponent of the velocity field Contrary tonormal expectations and observations in otherflows it is found that this flux transports reac-tants up themean-reactants gradient away fromthe products (hence countergradient diffusion)A second notable phenomenon is the largeproduction of turbulent energy within the flameBehind the flame the velocity variance can be 20times its upstream value Both these phenomenaresult from the large density difference betweenreactants and products and from the pressurefield due to volume expansion [ ] From theEuler equations it is readily seen that a givenpressure gradient accelerates the light productsmore than the heavier reactants This mech-anism is responsible both for countergradientdiffusion and for turbulent energy productionrdquo

Many articles have been devoted to theoretical and exper-imental studies of these phenomena More advanced theo-retical and modeling approaches in the context of the BMLFavre averaging paradigm use the Favre-averaged unclosedequations in terms of the components of the scalar flux12058811990610158401015840119894 11988810158401015840 and stress tensor 12058811990610158401015840119894 11990610158401015840119895 and further approximationof the unknowns appears within turbulence theories (see forexample [4]) To justify the need for our study we emphasizethat the countergradient scalar flux and abnormally largevelocity fluctuations cannot be described adequately in thecontext of the BML Favre averaging paradigm (as we willillustrate below using as an example the results obtained in[4]) The reason for this is that these phenomena are causedby the large difference in the conditional mean velocities997888rarr119906 119906 and 997888rarr119906 119887 due to the different pressure-driven accelerationof the heavier unburned and lighter burned gases Thesevelocities which cannot be predicted by the Favre-averagedequations are at the same time described directly by the two-fluid conditional equations It would be appropriate here toquote a statement made by Forman Williams in his Hottellecture entitled ldquoThe Role of Theory in Combustion Sciencerdquo[5]

ldquoIt is relevant to distinguish between the scienceand the technology of the subject The march

of technology has never hesitated It uses sci-ence whenever possible but often especially incombustion forges ahead by trial and error orfortuitously by application of scientific miscon-ception but without scientific understandingrdquo

In this context the use of Favre averaging in themodelingof turbulent premixed combustion in the flamelet regimeis strictly speaking the ldquoapplication of scientific miscon-ceptionrdquo in contrast to two-fluid conditional averagingwhich is conceptually adequate Hence two fundamentalmodeling problems arise in the turbulent combustion theorydeveloped in the framework of the Favre averaging paradigmthe issue of adequate prediction in the premixed flame ofboth phenomena (ldquocountergradient turbulent diffusionrdquo andldquounusually large generation of turbulencerdquo) disappears in theframework of the two-fluid conditional average paradigm

As far as we know the pioneers of the theoreticaltwo-fluid approach to turbulent premixed combustion wereSpalding [6] and Weller [7] In [6] Spalding also analyzedsome previous papers related to the two-fluid approach In[7] Weller presented conditionally averaged two-fluid equa-tions that in fact were model equations The exact unclosedtwo-fluid conditionally averagedmass andmomentum equa-tions which served as a starting point for our study wereobtained in [8] This paper extends our recent theoreticaldevelopment of the two-fluid theory of turbulent premixedcombustion described in [9] in the format of a brief commu-nication in which we proposed a simple and physically clearsplitting method for the derivation of the unclosed two-fluidconditional equations of turbulent premixed combustionThis method made it possible to rederive the conditionalequations obtained in [8] and to understand the reason forthe appearance of surface-averaged variables in these theimpossibility of the well-founded modeling of which is oneof the obstacles to using this approach in applications inaddition this allowed us to indicate a way of excludingtheir appearance The only required modeling unknownsappearing in the original alternative two-fluid conditionalequations formulated in [9] are the conditional Reynolds

stress tensors 120591119906 = minus120588119906(997888rarr119906 1015840997888rarr119906 1015840)119906 and 120591119887 = minus120588119906(997888rarr119906 1015840997888rarr119906 1015840)119887 in theunburned and burned gases and the mean chemical source120588119882

This paper makes the following contributions(i) We describe in more detail the splitting method and

the derivation of the unclosed two-fluid conditionallyaveraged equations both those described in the liter-ature and the alternative versions

(ii) We obtain the unclosed equations in terms of theconditional moments (11990610158401198941199061015840119895)119906 and (11990610158401198941199061015840119895)119887 by splittingthe known unclosed equations in terms of the Favreaverage (12058811990610158401015840119894 11990610158401015840119895 ) and present the unclosed equationfor the mean chemical source 120588119882 which can be usedfor advanced modeling of the unknowns in the two-fluid conditionally averaged equations

(iii) We formulate simple models for estimating theunknown chemical source 120588119882 in the laminar and

























laminarflamelet

thickenedflamelet

turbulenteddies

(a) (b)









(119863119886Re119905

) = 119888119900119899119904119905 (1a)



(120575119871120578 ) = 0 (1b)




11986311988612Reminus14119905 gtgt 1 (2)





119880119891 sim 1199061015840 sdot 119863119886minus12 (3a)

120575119891 sim 119871 sdot 119863119886minus32 (3b)

120594119891 sim 119863119905 sdot 119863119886minus2 (3c)


















120588 = 120588119906[1 + 119888 (120588119906120588119887 minus 1)] (6b)


120597120588120597119905 + nabla sdot 120588997888rarr119906 = 0 (6d)


















119904119906Σ119891

(7c)


119887


119904119887Σ119891

(7d)











119888 = 119888[119888 + (120588119906120588119887) (1 minus 119888)] (9b)




(10)






















120588119886 = 120588119906119886119906119875119906 + 120588119887119886119887119875119887 = 120588119906119886119906 (1 minus 119888) + 120588119887119886119887119888 (11)




+ 120597 (120588119887119888)120597119905 + nabla sdot (120588119887119888997888rarr119906 119887)119887

= 0(12)


120588119888 = 120588119888 = 120588119887119888 (13a)




120597 (120588119887119888)120597119905 + nabla sdot (120588119887119888997888rarr119906 119887) = 120588119882 (14b)




120597 [120588119906 (1 minus 119888) 997888rarr119906 119906]



]119906

+ 120597 (120588119887119888 119906119894119887)120597119905 + nabla


)119887


(15)






119887





119904119887Σ119891 (17)






119904119887Σ119891)


(19)





119904119906Σ119891

(20a)


119887


119904119887Σ119891

(20b)



119904119906Σ119891 (21a)









u purarru u

u psurarru su

b pbrarru b

b psbrarru sb




120588119906 119901119906 997888rarr119906 119906 120588119882997904rArr 120588119887 119901119887 997888rarr119906 119887 (22)


































(25)



(26a)


(26b)





(27a)


119887


(27b)




1003816100381610038161003816100381610038161003816 (28a)









120597 (120588119887119888)120597119905 + nabla sdot (120588119887119888997888rarr119906 119887) = 120588119882 (30b)




]]]120588119882

(30c)



]]]120588119882

(30d)





119888 = 119888[119888 + (120588119906120588119887) (1 minus 119888)] (31a)

997888119906 = 997888rarr119906 119906 (1 minus 119888) + 119906119887119888 (31b)

997888rarr119906 = 997888rarr119906 119906 (1 minus 119888) + 119906119887119888 (31c)

120588 = 120588119906[1 + 119888 (120588119906120588119887 minus 1)] (31d)

119901 = 119901119906 (1 minus 119888) + 119901119887119888 (31e)

12058811990610158401015840119894 11988810158401015840 = 120588119888 (1 minus 119888) (119906119894119887 minus 119906119894119906) (31f)


(31g)













120597 [120588 (11990610158401198941199061015840119895)]120597119905 + 120597 [120588119906119896 (11990610158401198941199061015840119895)]120597119909119896


120597119906119895120597119909119896]

minus [1199061015840119894 sdot 1205971199011015840

120597119909119895 + 1199061015840119895 sdot1205971199011015840120597119909119894]

(32)


120597119906119895119906120597119909119896 ]

minus (1 minus 119888) [(1199061015840119895 sdot 1205971199011015840

120597119909119894)119906

+ (1199061015840119894 sdot 1205971199011015840

120597119909119895)119906

]minus 119876119906

119894119895

(33a)


120597119906119895119887120597119909119896 ]

minus 119888 [(1199061015840119895 sdot 1205971199011015840

120597119909119894)119887

+ (1199061015840119894 sdot 1205971199011015840

120597119909119895)119887

] + 119876119887119894119895

(33b)

Here 119876119906119894119895 and 119876119887


119894119895


119894119895 and 119876119887119894119895 in terms of







1199061015840119894119904119906 minus (119901119904119906119899119894) + 119901119904119906119899119894 = 1199061015840119894119904119887 minus (119901119904119887119899119894) + 119901119904119887119899119894 (34a)

1199061015840119895119904119906 minus (119901119904119906119899119895) + 119901119904119906119899119895= 1199061015840119895119904119887 minus (119901119904119887119899119895) + 119901119904119887119899119895

(34b)


(119906119894119904119906 + 1199061015840119894119904119906) minus (119901119904119906119899119894)= (119906119894119904119887 + 1199061015840119894119904119887) minus (119901119904119887119899119894)

(35a)

119906119894119904119906 minus 119901119904119906119899119894 = 119906119894119904119887 minus 119901119904119887119899119894 (35b)


(11990610158401198941199061015840119895)119904119906 2 minus [(1199011199061015840119894119899119895)119904119906 + (1199011199061015840119895119899119894)119904119906] + [(1199012119899119894119899119895)119904119906 minus (119901119899119894)119904119906 (119901119899119895)119904119906]

= (11990610158401198941199061015840119895)119904119887 2 minus [(1199011199061015840119894119899119895)119904119887 + (1199011199061015840119895119899119894)119904119887] + [(1199012119899119894119899119895)119904119887 minus (119901119899119894)119904119906 (119901119899119895)119904119887]

(36)


(11990610158401198941199061015840119895)119904119906 120588119882 minus [(1199011199061015840119894119899119895)119904119906 + (1199011199061015840119895119899119894)119904119906] Σ119891+ [[(1199012119899119894119899119895)119904119906 minus (119901119899119894)119904119906 (119901119899119895)119904119906 ]

]Σ119891

= (11990610158401198941199061015840119895)119904119887 120588119882 minus [(1199011199061015840119894119899119895)119904119887 + (1199011199061015840119895119899119894)119904119887] Σ119891+ [[(1199012119899119894119899119895)119904119887 minus (119901119899119894)119904119906 (119901119899119895)119904119887 ]

]Σ119891

(37)


119876119906119894119895 = (11990610158401198941199061015840119895)119904119906 120588119882

1

minus [(1199011199061015840119894119899119895)119904119906 + (1199011199061015840119895119899119894)119904119906] Σ1198912

+ [(1199012119899119894119899119895)119904119906 minus (119901119899119894)119904119906 (119901119899119895)119904119906 ] Σ1198913

(38a)

119876119887119894119895 = (11990610158401198941199061015840119895)119904119887 120588119882

1

minus [(1199011199061015840119894119899119895)119904119887 + (1199011199061015840119895119899119894)119904119887] Σ1198912

+ [(1199012119899119894119899119895)119904119887 minus (119901119899119894)119904119887 (119901119899119895)119904119887 ] Σ1198913

(38b)




(11990610158401198941199061015840119895)119906 120588119882 minus [(11990110158401199061015840119894)119906 119899119895 + (11990110158401199061015840119895)119906 119899119894] Σ119891+ [[11990110158402119906 (119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891

= (11990610158401198941199061015840119895)119887 120588119882 minus [(11990110158401199061015840119894)119887 119899119895 + (11990110158401199061015840119895)119887 119899119894] Σ119891+ [[11990110158402119887(119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891

(39)



119876119906119894119895 = (11990610158401198941199061015840119895)119906 120588119882 minus [(11990110158401199061015840119894)119906 119899119895 + (11990110158401199061015840119895)119906 119899119894] Σ119891

+ [[11990110158402119906 (119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891 = 119876119887

119894119895

= (11990610158401198941199061015840119895)119887 120588119882 minus [(11990110158401199061015840119894)119887 119899119895 + (11990110158401199061015840119895)119887 119899119894] Σ119891+ [[11990110158402119887(119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891

(40)







120588119882 = 120588119906119878119871Σ (41a)


119904Σ] + (119878nabla sdot 997888rarr119899)

119904Σ

(41b)





119865)


119865

+ 2120588119906119878119871119870119882

(42)





(43)





120591119894119895119906 = minus120588119906 (11990610158401198941199061015840119895)119906= ]119905119906 (120597119906119894119906120597119909119895 +

120597119906119895119906120597119909119894 ) minus (23) 120588119906119896119906120575119894119895

(44a)

120591119894119895119887 = minus120588119887 (11990610158401198941199061015840119895)119887= ]119905119887 (120597119906119894119887120597119909119895 +

120597119906119895119887120597119909119894 ) minus (23) 120588119906119896119887120575119894119895

(44b)


120597 (120588)120597119905 + 120597 (120588120572)120597119909120572 = 120591119894119895120597119895120597119909119894

+ 120597 [120588 (V119905120590119896) 120597120597119909119894]120597119909119894minus 120588120576

(45a)

120597 (120588120576)120597119905 + 120597 (120588119894120576)120597119909119894 = 1198621205761120591119894119895 (120576) 120597119906119895120597119909119894

minus 1198621205762120588(1205762 )+ 120597 [120588 (V119905120590120576) 120597120576120597119909119894 ] 120597119909119894

(45b)


120591119894119895 = minus (12058811990610158401015840119894 11990610158401015840119895 ) = ]119905 ( 120597119894120597119909119895 +120597119895120597119909119894 ) minus (

23) 120588120575119894119895 (46)



+ 120597 [120588 (1 minus 119888) (V119905119906120590119896) 120597119896119906120597119909119894]120597119909119894 minus 119896119906120588119882+ 119870119906Σ119891

(47a)

120597 (120588119888119896119887)120597119905 + 120597 (120588119888119906119894119887119896119887)120597119909119894= 120588119888 (120591119894119895119887120588119887) 120597119906119894119887120597119909119895 minus 120588119888120576119887

+ 120597 [120588119888 (V119905119887120590119896) 120597119896119887120597119909119894]120597119909119894 + 119896119887120588119882 + 119870119887Σ119891

(47b)


+ 120597 [120588 (1 minus 119888) (V119905119906120590119896) 120597119896119906120597119909120572]120597119909120572 minus 120576119906120588119882+ 119864119906Σ119891

(47c)

120597 (120588119888120576119887)120597119905 + 120597 (120588119888119906119894119887120576119887)120597119909119894= 1198621205761120588 (1 minus 119888) (120576119887119896119887) (120591119894119895119887120588119887) 120597119906119894119887120597119909119895minus 1198621205762120588119888( 1205762119887119896119887) +

120597 [120588119888 (V119905119906120590119896) 120597119896119887120597119909120572]120597119909120572+ 119864119887Σ119891 + 120576119887120588119882

(47d)


minus119896119906120588119882 + 119870119906Σ119891 = +119896119887120588119882 + 119870119887Σ119891 (48a)

minus120576119906120588119882 + 119864119906Σ119891 = +120576119887120588119882 + 119864119887Σ119891 (48b)










(49a)


(49b)

120597 (120588)120597119905 + 120597 (120588120572)120597119909120572 = 120591119894119895120597119895120597119909119894+ 120597 [120588 (V119905120590119896) 120597120597119909119894]120597119909119894 minus 120588120576

(49c)

120597 (120588120576)120597119905 + 120597 (120588119894120576)120597119909119894 = 1198621205761120591119894119895 (120576) 120597119906119895120597119909119894+ 120597 [120588 (V119905120590120576) 120597120576120597119909119894 ] 120597119909119894 minus 1198621205762120588(1205762 )]

(49d)

= 119896119906 (1 minus 119888) + 119896119887119888 (49e)

120576 = 120576119906 (1 minus 119888) + 120576119887119888 (49f)










( 1198601198600


2|nablaℎ| asymp

3(|nablaℎ|2)12

=4(intinfin

01198962119865 (119896) 119889119896)12

(50)


( 1198601198600

) asymp (intinfin01198962119865 (119896) 119889119896)12 (51a)

1205902119891 (119905) = (119909 minus 119909 (119905))2 = intinfin0119865 (119896 119905) 119889119896 (51b)



1205902119891 cong 2119863119905119905 (52a)

120575119905 sim (1199061015840119871119905)12 (52b)




120597119888120597119905 = 11986311990512059721198881205971199092 + 119880119905 1003816100381610038161003816100381610038161003816100381612059711988812059711990910038161003816100381610038161003816100381610038161003816 (53a)



120588119882 = 120588 = 120588119906119880119905 |nabla119888| (54)



119880119905 sim (1199061015840119878119871)12 (55a)

120591119905 le 119905 ltlt (1199061015840119878119871)2 120591119905 (55b)


(120588119882)119871= 119860119871120588119906radic11989612119906 119878119871 |nabla119888| (56)








119880119905 = 119860119906101584011986311988614 = 11986011990610158403411987812119871 120594minus1411987114 (57a)

120591119905 lt 119905 ltlt 120591119905119863119886 (57b)





11 Conclusions













Appendix








(1 minus 119888) 119906119906 + 119888 119906119887Θ = 1 (A1a)


119901119906 + (12) 1199062119906 = 119901minusinfin + 12 (A1c)


(1 minus 119888) 119906119906 + 119888 119906119887Θ = 1 (A2a)

(12 minus 119888) 1199062119906 +119888 1199062119887Θ = 12 (A2b)


119906119887 = minus120573 + (minus4120572120574 + 1205732)12


119888Θ + 1] 120573 = minus2 119888Θ 05 minus 119888

(1 minus 119888)2 120574 = 05 minus 119888

(1 minus 119888)2 minus 05

(A3)


1205881199061015840101584011988810158401015840 = 120588119888 (1 minus 119888) (119906119887 minus 119906119906) (A4a)

1205881199061015840101584011990610158401015840 = 120588119888 (119906119887 minus 119906119906)2 (A4b)



0 025 05 075 10

005

01

015

02

025

03

035

04

045

05

u

c

uUt

c(a)

0 025 05 075 10

1

2

3

4

5

6

7

8

9

u uU

tu b

Ut

ubUt

uuUt

c(b)



119906119906 = 1198911 (119888 Θ) (A5a)

119906119887 = 1198912 (119888 Θ) (A5b)

1205881199061015840101584011988810158401015840 = 1198913 (119888 Θ) (A5c)

1205881199061015840101584011990610158401015840 = 1198914 (119888 Θ) (A5d)
















1205881199061015840101584011988810158401015840 = (1205881199061015840101584011988810158401015840)119901minus119889

+ (1205881199061015840101584011988810158401015840)119905119889

= 1205881199061198801199051198913 (119888 Θ) minus 120588119863119905119889119888119889119909 (B1)



(B2)




119870119897119900119888 =100381610038161003816100381610038161003816(1205881199061015840101584011988810158401015840)119901minus11988910038161003816100381610038161003816100381610038161003816100381610038161003816(1205881199061015840101584011988810158401015840)11990511988910038161003816100381610038161003816 =

10038161003816100381610038161205881199061198801199051198913 (119888 Θ)10038161003816100381610038161003816100381610038161003816120588 (119888)1198631199051198891198881198891199091003816100381610038161003816 (B3a)

119870int =100381610038161003816100381610038161003816⟨(1205881199061015840101584011988810158401015840)119901minus119889⟩10038161003816100381610038161003816100381610038161003816100381610038161003816⟨(1205881199061015840101584011988810158401015840)119905119889⟩10038161003816100381610038161003816

= (14) 120573 [Θ minus (2Θ minus 1)12] (1198801199051199061015840 )(120575119905119871 ) (B3b)










0 02 04 06 08 1

minus005

0

005

01

015

02

025

03

035Experiments

c

[kg

m2s]

u

c

(a)

0 02 04 06 08 1

minus005

0

005

01

015

02

025

03

035Calculations

c[k

gm

2s]

u

c

(b)











0

002

0225 025 0275

1

1 1

02 03 04 026 027 028 029 03

026 027 028 029 03

X(m) x(m) X(m)

0225 025 0275x(m)

02 03 04X(m)

004

006

008

0101 03 05 07 09

012

minus006

minus004

minus002

0

08

06

04

02

0

004

002

006

minus006

minus004

minus002

0

002

004

006

0

002

004

006

008

01

012

minus0981

minus0984

minus0987

minus099

minus0993

minus0996

minus0999

３=04ＧＭ

３=04ＧＭ

３=2ＧＭ

０-０0０0

minus0987

minus0994

minus1001

minus1008

minus1015

minus1022

minus1029

０-０0０0

０-０0０0

５５0

５５0

1５５0

C

1

08

06

04

02

0

1


Ｏ＝ＯＯ0C


Ｏ＝ＯＯ0C

０-０0０0

５５0

C

３=04ＧＭ

３=2ＧＭ

３=2ＧＭ













Nomenclature





Greek Symbols





Data Availability




Acknowledgments



References































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


























laminarflamelet

thickenedflamelet

turbulenteddies

(a) (b)









(119863119886Re119905

) = 119888119900119899119904119905 (1a)



(120575119871120578 ) = 0 (1b)




11986311988612Reminus14119905 gtgt 1 (2)





119880119891 sim 1199061015840 sdot 119863119886minus12 (3a)

120575119891 sim 119871 sdot 119863119886minus32 (3b)

120594119891 sim 119863119905 sdot 119863119886minus2 (3c)


















120588 = 120588119906[1 + 119888 (120588119906120588119887 minus 1)] (6b)


120597120588120597119905 + nabla sdot 120588997888rarr119906 = 0 (6d)


















119904119906Σ119891

(7c)


119887


119904119887Σ119891

(7d)











119888 = 119888[119888 + (120588119906120588119887) (1 minus 119888)] (9b)




(10)






















120588119886 = 120588119906119886119906119875119906 + 120588119887119886119887119875119887 = 120588119906119886119906 (1 minus 119888) + 120588119887119886119887119888 (11)




+ 120597 (120588119887119888)120597119905 + nabla sdot (120588119887119888997888rarr119906 119887)119887

= 0(12)


120588119888 = 120588119888 = 120588119887119888 (13a)




120597 (120588119887119888)120597119905 + nabla sdot (120588119887119888997888rarr119906 119887) = 120588119882 (14b)




120597 [120588119906 (1 minus 119888) 997888rarr119906 119906]



]119906

+ 120597 (120588119887119888 119906119894119887)120597119905 + nabla


)119887


(15)






119887





119904119887Σ119891 (17)






119904119887Σ119891)


(19)





119904119906Σ119891

(20a)


119887


119904119887Σ119891

(20b)



119904119906Σ119891 (21a)









u purarru u

u psurarru su

b pbrarru b

b psbrarru sb




120588119906 119901119906 997888rarr119906 119906 120588119882997904rArr 120588119887 119901119887 997888rarr119906 119887 (22)


































(25)



(26a)


(26b)





(27a)


119887


(27b)




1003816100381610038161003816100381610038161003816 (28a)









120597 (120588119887119888)120597119905 + nabla sdot (120588119887119888997888rarr119906 119887) = 120588119882 (30b)




]]]120588119882

(30c)



]]]120588119882

(30d)





119888 = 119888[119888 + (120588119906120588119887) (1 minus 119888)] (31a)

997888119906 = 997888rarr119906 119906 (1 minus 119888) + 119906119887119888 (31b)

997888rarr119906 = 997888rarr119906 119906 (1 minus 119888) + 119906119887119888 (31c)

120588 = 120588119906[1 + 119888 (120588119906120588119887 minus 1)] (31d)

119901 = 119901119906 (1 minus 119888) + 119901119887119888 (31e)

12058811990610158401015840119894 11988810158401015840 = 120588119888 (1 minus 119888) (119906119894119887 minus 119906119894119906) (31f)


(31g)













120597 [120588 (11990610158401198941199061015840119895)]120597119905 + 120597 [120588119906119896 (11990610158401198941199061015840119895)]120597119909119896


120597119906119895120597119909119896]

minus [1199061015840119894 sdot 1205971199011015840

120597119909119895 + 1199061015840119895 sdot1205971199011015840120597119909119894]

(32)


120597119906119895119906120597119909119896 ]

minus (1 minus 119888) [(1199061015840119895 sdot 1205971199011015840

120597119909119894)119906

+ (1199061015840119894 sdot 1205971199011015840

120597119909119895)119906

]minus 119876119906

119894119895

(33a)


120597119906119895119887120597119909119896 ]

minus 119888 [(1199061015840119895 sdot 1205971199011015840

120597119909119894)119887

+ (1199061015840119894 sdot 1205971199011015840

120597119909119895)119887

] + 119876119887119894119895

(33b)

Here 119876119906119894119895 and 119876119887


119894119895


119894119895 and 119876119887119894119895 in terms of







1199061015840119894119904119906 minus (119901119904119906119899119894) + 119901119904119906119899119894 = 1199061015840119894119904119887 minus (119901119904119887119899119894) + 119901119904119887119899119894 (34a)

1199061015840119895119904119906 minus (119901119904119906119899119895) + 119901119904119906119899119895= 1199061015840119895119904119887 minus (119901119904119887119899119895) + 119901119904119887119899119895

(34b)


(119906119894119904119906 + 1199061015840119894119904119906) minus (119901119904119906119899119894)= (119906119894119904119887 + 1199061015840119894119904119887) minus (119901119904119887119899119894)

(35a)

119906119894119904119906 minus 119901119904119906119899119894 = 119906119894119904119887 minus 119901119904119887119899119894 (35b)


(11990610158401198941199061015840119895)119904119906 2 minus [(1199011199061015840119894119899119895)119904119906 + (1199011199061015840119895119899119894)119904119906] + [(1199012119899119894119899119895)119904119906 minus (119901119899119894)119904119906 (119901119899119895)119904119906]

= (11990610158401198941199061015840119895)119904119887 2 minus [(1199011199061015840119894119899119895)119904119887 + (1199011199061015840119895119899119894)119904119887] + [(1199012119899119894119899119895)119904119887 minus (119901119899119894)119904119906 (119901119899119895)119904119887]

(36)


(11990610158401198941199061015840119895)119904119906 120588119882 minus [(1199011199061015840119894119899119895)119904119906 + (1199011199061015840119895119899119894)119904119906] Σ119891+ [[(1199012119899119894119899119895)119904119906 minus (119901119899119894)119904119906 (119901119899119895)119904119906 ]

]Σ119891

= (11990610158401198941199061015840119895)119904119887 120588119882 minus [(1199011199061015840119894119899119895)119904119887 + (1199011199061015840119895119899119894)119904119887] Σ119891+ [[(1199012119899119894119899119895)119904119887 minus (119901119899119894)119904119906 (119901119899119895)119904119887 ]

]Σ119891

(37)


119876119906119894119895 = (11990610158401198941199061015840119895)119904119906 120588119882

1

minus [(1199011199061015840119894119899119895)119904119906 + (1199011199061015840119895119899119894)119904119906] Σ1198912

+ [(1199012119899119894119899119895)119904119906 minus (119901119899119894)119904119906 (119901119899119895)119904119906 ] Σ1198913

(38a)

119876119887119894119895 = (11990610158401198941199061015840119895)119904119887 120588119882

1

minus [(1199011199061015840119894119899119895)119904119887 + (1199011199061015840119895119899119894)119904119887] Σ1198912

+ [(1199012119899119894119899119895)119904119887 minus (119901119899119894)119904119887 (119901119899119895)119904119887 ] Σ1198913

(38b)




(11990610158401198941199061015840119895)119906 120588119882 minus [(11990110158401199061015840119894)119906 119899119895 + (11990110158401199061015840119895)119906 119899119894] Σ119891+ [[11990110158402119906 (119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891

= (11990610158401198941199061015840119895)119887 120588119882 minus [(11990110158401199061015840119894)119887 119899119895 + (11990110158401199061015840119895)119887 119899119894] Σ119891+ [[11990110158402119887(119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891

(39)



119876119906119894119895 = (11990610158401198941199061015840119895)119906 120588119882 minus [(11990110158401199061015840119894)119906 119899119895 + (11990110158401199061015840119895)119906 119899119894] Σ119891

+ [[11990110158402119906 (119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891 = 119876119887

119894119895

= (11990610158401198941199061015840119895)119887 120588119882 minus [(11990110158401199061015840119894)119887 119899119895 + (11990110158401199061015840119895)119887 119899119894] Σ119891+ [[11990110158402119887(119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891

(40)







120588119882 = 120588119906119878119871Σ (41a)


119904Σ] + (119878nabla sdot 997888rarr119899)

119904Σ

(41b)





119865)


119865

+ 2120588119906119878119871119870119882

(42)





(43)





120591119894119895119906 = minus120588119906 (11990610158401198941199061015840119895)119906= ]119905119906 (120597119906119894119906120597119909119895 +

120597119906119895119906120597119909119894 ) minus (23) 120588119906119896119906120575119894119895

(44a)

120591119894119895119887 = minus120588119887 (11990610158401198941199061015840119895)119887= ]119905119887 (120597119906119894119887120597119909119895 +

120597119906119895119887120597119909119894 ) minus (23) 120588119906119896119887120575119894119895

(44b)


120597 (120588)120597119905 + 120597 (120588120572)120597119909120572 = 120591119894119895120597119895120597119909119894

+ 120597 [120588 (V119905120590119896) 120597120597119909119894]120597119909119894minus 120588120576

(45a)

120597 (120588120576)120597119905 + 120597 (120588119894120576)120597119909119894 = 1198621205761120591119894119895 (120576) 120597119906119895120597119909119894

minus 1198621205762120588(1205762 )+ 120597 [120588 (V119905120590120576) 120597120576120597119909119894 ] 120597119909119894

(45b)


120591119894119895 = minus (12058811990610158401015840119894 11990610158401015840119895 ) = ]119905 ( 120597119894120597119909119895 +120597119895120597119909119894 ) minus (

23) 120588120575119894119895 (46)



+ 120597 [120588 (1 minus 119888) (V119905119906120590119896) 120597119896119906120597119909119894]120597119909119894 minus 119896119906120588119882+ 119870119906Σ119891

(47a)

120597 (120588119888119896119887)120597119905 + 120597 (120588119888119906119894119887119896119887)120597119909119894= 120588119888 (120591119894119895119887120588119887) 120597119906119894119887120597119909119895 minus 120588119888120576119887

+ 120597 [120588119888 (V119905119887120590119896) 120597119896119887120597119909119894]120597119909119894 + 119896119887120588119882 + 119870119887Σ119891

(47b)


+ 120597 [120588 (1 minus 119888) (V119905119906120590119896) 120597119896119906120597119909120572]120597119909120572 minus 120576119906120588119882+ 119864119906Σ119891

(47c)

120597 (120588119888120576119887)120597119905 + 120597 (120588119888119906119894119887120576119887)120597119909119894= 1198621205761120588 (1 minus 119888) (120576119887119896119887) (120591119894119895119887120588119887) 120597119906119894119887120597119909119895minus 1198621205762120588119888( 1205762119887119896119887) +

120597 [120588119888 (V119905119906120590119896) 120597119896119887120597119909120572]120597119909120572+ 119864119887Σ119891 + 120576119887120588119882

(47d)


minus119896119906120588119882 + 119870119906Σ119891 = +119896119887120588119882 + 119870119887Σ119891 (48a)

minus120576119906120588119882 + 119864119906Σ119891 = +120576119887120588119882 + 119864119887Σ119891 (48b)










(49a)


(49b)

120597 (120588)120597119905 + 120597 (120588120572)120597119909120572 = 120591119894119895120597119895120597119909119894+ 120597 [120588 (V119905120590119896) 120597120597119909119894]120597119909119894 minus 120588120576

(49c)

120597 (120588120576)120597119905 + 120597 (120588119894120576)120597119909119894 = 1198621205761120591119894119895 (120576) 120597119906119895120597119909119894+ 120597 [120588 (V119905120590120576) 120597120576120597119909119894 ] 120597119909119894 minus 1198621205762120588(1205762 )]

(49d)

= 119896119906 (1 minus 119888) + 119896119887119888 (49e)

120576 = 120576119906 (1 minus 119888) + 120576119887119888 (49f)










( 1198601198600


2|nablaℎ| asymp

3(|nablaℎ|2)12

=4(intinfin

01198962119865 (119896) 119889119896)12

(50)


( 1198601198600

) asymp (intinfin01198962119865 (119896) 119889119896)12 (51a)

1205902119891 (119905) = (119909 minus 119909 (119905))2 = intinfin0119865 (119896 119905) 119889119896 (51b)



1205902119891 cong 2119863119905119905 (52a)

120575119905 sim (1199061015840119871119905)12 (52b)




120597119888120597119905 = 11986311990512059721198881205971199092 + 119880119905 1003816100381610038161003816100381610038161003816100381612059711988812059711990910038161003816100381610038161003816100381610038161003816 (53a)



120588119882 = 120588 = 120588119906119880119905 |nabla119888| (54)



119880119905 sim (1199061015840119878119871)12 (55a)

120591119905 le 119905 ltlt (1199061015840119878119871)2 120591119905 (55b)


(120588119882)119871= 119860119871120588119906radic11989612119906 119878119871 |nabla119888| (56)








119880119905 = 119860119906101584011986311988614 = 11986011990610158403411987812119871 120594minus1411987114 (57a)

120591119905 lt 119905 ltlt 120591119905119863119886 (57b)





11 Conclusions













Appendix








(1 minus 119888) 119906119906 + 119888 119906119887Θ = 1 (A1a)


119901119906 + (12) 1199062119906 = 119901minusinfin + 12 (A1c)


(1 minus 119888) 119906119906 + 119888 119906119887Θ = 1 (A2a)

(12 minus 119888) 1199062119906 +119888 1199062119887Θ = 12 (A2b)


119906119887 = minus120573 + (minus4120572120574 + 1205732)12


119888Θ + 1] 120573 = minus2 119888Θ 05 minus 119888

(1 minus 119888)2 120574 = 05 minus 119888

(1 minus 119888)2 minus 05

(A3)


1205881199061015840101584011988810158401015840 = 120588119888 (1 minus 119888) (119906119887 minus 119906119906) (A4a)

1205881199061015840101584011990610158401015840 = 120588119888 (119906119887 minus 119906119906)2 (A4b)



0 025 05 075 10

005

01

015

02

025

03

035

04

045

05

u

c

uUt

c(a)

0 025 05 075 10

1

2

3

4

5

6

7

8

9

u uU

tu b

Ut

ubUt

uuUt

c(b)



119906119906 = 1198911 (119888 Θ) (A5a)

119906119887 = 1198912 (119888 Θ) (A5b)

1205881199061015840101584011988810158401015840 = 1198913 (119888 Θ) (A5c)

1205881199061015840101584011990610158401015840 = 1198914 (119888 Θ) (A5d)
















1205881199061015840101584011988810158401015840 = (1205881199061015840101584011988810158401015840)119901minus119889

+ (1205881199061015840101584011988810158401015840)119905119889

= 1205881199061198801199051198913 (119888 Θ) minus 120588119863119905119889119888119889119909 (B1)



(B2)




119870119897119900119888 =100381610038161003816100381610038161003816(1205881199061015840101584011988810158401015840)119901minus11988910038161003816100381610038161003816100381610038161003816100381610038161003816(1205881199061015840101584011988810158401015840)11990511988910038161003816100381610038161003816 =

10038161003816100381610038161205881199061198801199051198913 (119888 Θ)10038161003816100381610038161003816100381610038161003816120588 (119888)1198631199051198891198881198891199091003816100381610038161003816 (B3a)

119870int =100381610038161003816100381610038161003816⟨(1205881199061015840101584011988810158401015840)119901minus119889⟩10038161003816100381610038161003816100381610038161003816100381610038161003816⟨(1205881199061015840101584011988810158401015840)119905119889⟩10038161003816100381610038161003816

= (14) 120573 [Θ minus (2Θ minus 1)12] (1198801199051199061015840 )(120575119905119871 ) (B3b)










0 02 04 06 08 1

minus005

0

005

01

015

02

025

03

035Experiments

c

[kg

m2s]

u

c

(a)

0 02 04 06 08 1

minus005

0

005

01

015

02

025

03

035Calculations

c[k

gm

2s]

u

c

(b)











0

002

0225 025 0275

1

1 1

02 03 04 026 027 028 029 03

026 027 028 029 03

X(m) x(m) X(m)

0225 025 0275x(m)

02 03 04X(m)

004

006

008

0101 03 05 07 09

012

minus006

minus004

minus002

0

08

06

04

02

0

004

002

006

minus006

minus004

minus002

0

002

004

006

0

002

004

006

008

01

012

minus0981

minus0984

minus0987

minus099

minus0993

minus0996

minus0999

３=04ＧＭ

３=04ＧＭ

３=2ＧＭ

０-０0０0

minus0987

minus0994

minus1001

minus1008

minus1015

minus1022

minus1029

０-０0０0

０-０0０0

５５0

５５0

1５５0

C

1

08

06

04

02

0

1


Ｏ＝ＯＯ0C


Ｏ＝ＯＯ0C

０-０0０0

５５0

C

３=04ＧＭ

３=2ＧＭ

３=2ＧＭ













Nomenclature





Greek Symbols





Data Availability




Acknowledgments



References































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




119880119891 sim 1199061015840 sdot 119863119886minus12 (3a)

120575119891 sim 119871 sdot 119863119886minus32 (3b)

120594119891 sim 119863119905 sdot 119863119886minus2 (3c)


















120588 = 120588119906[1 + 119888 (120588119906120588119887 minus 1)] (6b)


120597120588120597119905 + nabla sdot 120588997888rarr119906 = 0 (6d)


















119904119906Σ119891

(7c)


119887


119904119887Σ119891

(7d)











119888 = 119888[119888 + (120588119906120588119887) (1 minus 119888)] (9b)




(10)






















120588119886 = 120588119906119886119906119875119906 + 120588119887119886119887119875119887 = 120588119906119886119906 (1 minus 119888) + 120588119887119886119887119888 (11)




+ 120597 (120588119887119888)120597119905 + nabla sdot (120588119887119888997888rarr119906 119887)119887

= 0(12)


120588119888 = 120588119888 = 120588119887119888 (13a)




120597 (120588119887119888)120597119905 + nabla sdot (120588119887119888997888rarr119906 119887) = 120588119882 (14b)




120597 [120588119906 (1 minus 119888) 997888rarr119906 119906]



]119906

+ 120597 (120588119887119888 119906119894119887)120597119905 + nabla


)119887


(15)






119887





119904119887Σ119891 (17)






119904119887Σ119891)


(19)





119904119906Σ119891

(20a)


119887


119904119887Σ119891

(20b)



119904119906Σ119891 (21a)









u purarru u

u psurarru su

b pbrarru b

b psbrarru sb




120588119906 119901119906 997888rarr119906 119906 120588119882997904rArr 120588119887 119901119887 997888rarr119906 119887 (22)


































(25)



(26a)


(26b)





(27a)


119887


(27b)




1003816100381610038161003816100381610038161003816 (28a)









120597 (120588119887119888)120597119905 + nabla sdot (120588119887119888997888rarr119906 119887) = 120588119882 (30b)




]]]120588119882

(30c)



]]]120588119882

(30d)





119888 = 119888[119888 + (120588119906120588119887) (1 minus 119888)] (31a)

997888119906 = 997888rarr119906 119906 (1 minus 119888) + 119906119887119888 (31b)

997888rarr119906 = 997888rarr119906 119906 (1 minus 119888) + 119906119887119888 (31c)

120588 = 120588119906[1 + 119888 (120588119906120588119887 minus 1)] (31d)

119901 = 119901119906 (1 minus 119888) + 119901119887119888 (31e)

12058811990610158401015840119894 11988810158401015840 = 120588119888 (1 minus 119888) (119906119894119887 minus 119906119894119906) (31f)


(31g)













120597 [120588 (11990610158401198941199061015840119895)]120597119905 + 120597 [120588119906119896 (11990610158401198941199061015840119895)]120597119909119896


120597119906119895120597119909119896]

minus [1199061015840119894 sdot 1205971199011015840

120597119909119895 + 1199061015840119895 sdot1205971199011015840120597119909119894]

(32)


120597119906119895119906120597119909119896 ]

minus (1 minus 119888) [(1199061015840119895 sdot 1205971199011015840

120597119909119894)119906

+ (1199061015840119894 sdot 1205971199011015840

120597119909119895)119906

]minus 119876119906

119894119895

(33a)


120597119906119895119887120597119909119896 ]

minus 119888 [(1199061015840119895 sdot 1205971199011015840

120597119909119894)119887

+ (1199061015840119894 sdot 1205971199011015840

120597119909119895)119887

] + 119876119887119894119895

(33b)

Here 119876119906119894119895 and 119876119887


119894119895


119894119895 and 119876119887119894119895 in terms of







1199061015840119894119904119906 minus (119901119904119906119899119894) + 119901119904119906119899119894 = 1199061015840119894119904119887 minus (119901119904119887119899119894) + 119901119904119887119899119894 (34a)

1199061015840119895119904119906 minus (119901119904119906119899119895) + 119901119904119906119899119895= 1199061015840119895119904119887 minus (119901119904119887119899119895) + 119901119904119887119899119895

(34b)


(119906119894119904119906 + 1199061015840119894119904119906) minus (119901119904119906119899119894)= (119906119894119904119887 + 1199061015840119894119904119887) minus (119901119904119887119899119894)

(35a)

119906119894119904119906 minus 119901119904119906119899119894 = 119906119894119904119887 minus 119901119904119887119899119894 (35b)


(11990610158401198941199061015840119895)119904119906 2 minus [(1199011199061015840119894119899119895)119904119906 + (1199011199061015840119895119899119894)119904119906] + [(1199012119899119894119899119895)119904119906 minus (119901119899119894)119904119906 (119901119899119895)119904119906]

= (11990610158401198941199061015840119895)119904119887 2 minus [(1199011199061015840119894119899119895)119904119887 + (1199011199061015840119895119899119894)119904119887] + [(1199012119899119894119899119895)119904119887 minus (119901119899119894)119904119906 (119901119899119895)119904119887]

(36)


(11990610158401198941199061015840119895)119904119906 120588119882 minus [(1199011199061015840119894119899119895)119904119906 + (1199011199061015840119895119899119894)119904119906] Σ119891+ [[(1199012119899119894119899119895)119904119906 minus (119901119899119894)119904119906 (119901119899119895)119904119906 ]

]Σ119891

= (11990610158401198941199061015840119895)119904119887 120588119882 minus [(1199011199061015840119894119899119895)119904119887 + (1199011199061015840119895119899119894)119904119887] Σ119891+ [[(1199012119899119894119899119895)119904119887 minus (119901119899119894)119904119906 (119901119899119895)119904119887 ]

]Σ119891

(37)


119876119906119894119895 = (11990610158401198941199061015840119895)119904119906 120588119882

1

minus [(1199011199061015840119894119899119895)119904119906 + (1199011199061015840119895119899119894)119904119906] Σ1198912

+ [(1199012119899119894119899119895)119904119906 minus (119901119899119894)119904119906 (119901119899119895)119904119906 ] Σ1198913

(38a)

119876119887119894119895 = (11990610158401198941199061015840119895)119904119887 120588119882

1

minus [(1199011199061015840119894119899119895)119904119887 + (1199011199061015840119895119899119894)119904119887] Σ1198912

+ [(1199012119899119894119899119895)119904119887 minus (119901119899119894)119904119887 (119901119899119895)119904119887 ] Σ1198913

(38b)




(11990610158401198941199061015840119895)119906 120588119882 minus [(11990110158401199061015840119894)119906 119899119895 + (11990110158401199061015840119895)119906 119899119894] Σ119891+ [[11990110158402119906 (119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891

= (11990610158401198941199061015840119895)119887 120588119882 minus [(11990110158401199061015840119894)119887 119899119895 + (11990110158401199061015840119895)119887 119899119894] Σ119891+ [[11990110158402119887(119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891

(39)



119876119906119894119895 = (11990610158401198941199061015840119895)119906 120588119882 minus [(11990110158401199061015840119894)119906 119899119895 + (11990110158401199061015840119895)119906 119899119894] Σ119891

+ [[11990110158402119906 (119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891 = 119876119887

119894119895

= (11990610158401198941199061015840119895)119887 120588119882 minus [(11990110158401199061015840119894)119887 119899119895 + (11990110158401199061015840119895)119887 119899119894] Σ119891+ [[11990110158402119887(119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891

(40)







120588119882 = 120588119906119878119871Σ (41a)


119904Σ] + (119878nabla sdot 997888rarr119899)

119904Σ

(41b)





119865)


119865

+ 2120588119906119878119871119870119882

(42)





(43)





120591119894119895119906 = minus120588119906 (11990610158401198941199061015840119895)119906= ]119905119906 (120597119906119894119906120597119909119895 +

120597119906119895119906120597119909119894 ) minus (23) 120588119906119896119906120575119894119895

(44a)

120591119894119895119887 = minus120588119887 (11990610158401198941199061015840119895)119887= ]119905119887 (120597119906119894119887120597119909119895 +

120597119906119895119887120597119909119894 ) minus (23) 120588119906119896119887120575119894119895

(44b)


120597 (120588)120597119905 + 120597 (120588120572)120597119909120572 = 120591119894119895120597119895120597119909119894

+ 120597 [120588 (V119905120590119896) 120597120597119909119894]120597119909119894minus 120588120576

(45a)

120597 (120588120576)120597119905 + 120597 (120588119894120576)120597119909119894 = 1198621205761120591119894119895 (120576) 120597119906119895120597119909119894

minus 1198621205762120588(1205762 )+ 120597 [120588 (V119905120590120576) 120597120576120597119909119894 ] 120597119909119894

(45b)


120591119894119895 = minus (12058811990610158401015840119894 11990610158401015840119895 ) = ]119905 ( 120597119894120597119909119895 +120597119895120597119909119894 ) minus (

23) 120588120575119894119895 (46)



+ 120597 [120588 (1 minus 119888) (V119905119906120590119896) 120597119896119906120597119909119894]120597119909119894 minus 119896119906120588119882+ 119870119906Σ119891

(47a)

120597 (120588119888119896119887)120597119905 + 120597 (120588119888119906119894119887119896119887)120597119909119894= 120588119888 (120591119894119895119887120588119887) 120597119906119894119887120597119909119895 minus 120588119888120576119887

+ 120597 [120588119888 (V119905119887120590119896) 120597119896119887120597119909119894]120597119909119894 + 119896119887120588119882 + 119870119887Σ119891

(47b)


+ 120597 [120588 (1 minus 119888) (V119905119906120590119896) 120597119896119906120597119909120572]120597119909120572 minus 120576119906120588119882+ 119864119906Σ119891

(47c)

120597 (120588119888120576119887)120597119905 + 120597 (120588119888119906119894119887120576119887)120597119909119894= 1198621205761120588 (1 minus 119888) (120576119887119896119887) (120591119894119895119887120588119887) 120597119906119894119887120597119909119895minus 1198621205762120588119888( 1205762119887119896119887) +

120597 [120588119888 (V119905119906120590119896) 120597119896119887120597119909120572]120597119909120572+ 119864119887Σ119891 + 120576119887120588119882

(47d)


minus119896119906120588119882 + 119870119906Σ119891 = +119896119887120588119882 + 119870119887Σ119891 (48a)

minus120576119906120588119882 + 119864119906Σ119891 = +120576119887120588119882 + 119864119887Σ119891 (48b)










(49a)


(49b)

120597 (120588)120597119905 + 120597 (120588120572)120597119909120572 = 120591119894119895120597119895120597119909119894+ 120597 [120588 (V119905120590119896) 120597120597119909119894]120597119909119894 minus 120588120576

(49c)

120597 (120588120576)120597119905 + 120597 (120588119894120576)120597119909119894 = 1198621205761120591119894119895 (120576) 120597119906119895120597119909119894+ 120597 [120588 (V119905120590120576) 120597120576120597119909119894 ] 120597119909119894 minus 1198621205762120588(1205762 )]

(49d)

= 119896119906 (1 minus 119888) + 119896119887119888 (49e)

120576 = 120576119906 (1 minus 119888) + 120576119887119888 (49f)










( 1198601198600


2|nablaℎ| asymp

3(|nablaℎ|2)12

=4(intinfin

01198962119865 (119896) 119889119896)12

(50)


( 1198601198600

) asymp (intinfin01198962119865 (119896) 119889119896)12 (51a)

1205902119891 (119905) = (119909 minus 119909 (119905))2 = intinfin0119865 (119896 119905) 119889119896 (51b)



1205902119891 cong 2119863119905119905 (52a)

120575119905 sim (1199061015840119871119905)12 (52b)




120597119888120597119905 = 11986311990512059721198881205971199092 + 119880119905 1003816100381610038161003816100381610038161003816100381612059711988812059711990910038161003816100381610038161003816100381610038161003816 (53a)



120588119882 = 120588 = 120588119906119880119905 |nabla119888| (54)



119880119905 sim (1199061015840119878119871)12 (55a)

120591119905 le 119905 ltlt (1199061015840119878119871)2 120591119905 (55b)


(120588119882)119871= 119860119871120588119906radic11989612119906 119878119871 |nabla119888| (56)








119880119905 = 119860119906101584011986311988614 = 11986011990610158403411987812119871 120594minus1411987114 (57a)

120591119905 lt 119905 ltlt 120591119905119863119886 (57b)





11 Conclusions













Appendix








(1 minus 119888) 119906119906 + 119888 119906119887Θ = 1 (A1a)


119901119906 + (12) 1199062119906 = 119901minusinfin + 12 (A1c)


(1 minus 119888) 119906119906 + 119888 119906119887Θ = 1 (A2a)

(12 minus 119888) 1199062119906 +119888 1199062119887Θ = 12 (A2b)


119906119887 = minus120573 + (minus4120572120574 + 1205732)12


119888Θ + 1] 120573 = minus2 119888Θ 05 minus 119888

(1 minus 119888)2 120574 = 05 minus 119888

(1 minus 119888)2 minus 05

(A3)


1205881199061015840101584011988810158401015840 = 120588119888 (1 minus 119888) (119906119887 minus 119906119906) (A4a)

1205881199061015840101584011990610158401015840 = 120588119888 (119906119887 minus 119906119906)2 (A4b)



0 025 05 075 10

005

01

015

02

025

03

035

04

045

05

u

c

uUt

c(a)

0 025 05 075 10

1

2

3

4

5

6

7

8

9

u uU

tu b

Ut

ubUt

uuUt

c(b)



119906119906 = 1198911 (119888 Θ) (A5a)

119906119887 = 1198912 (119888 Θ) (A5b)

1205881199061015840101584011988810158401015840 = 1198913 (119888 Θ) (A5c)

1205881199061015840101584011990610158401015840 = 1198914 (119888 Θ) (A5d)
















1205881199061015840101584011988810158401015840 = (1205881199061015840101584011988810158401015840)119901minus119889

+ (1205881199061015840101584011988810158401015840)119905119889

= 1205881199061198801199051198913 (119888 Θ) minus 120588119863119905119889119888119889119909 (B1)



(B2)




119870119897119900119888 =100381610038161003816100381610038161003816(1205881199061015840101584011988810158401015840)119901minus11988910038161003816100381610038161003816100381610038161003816100381610038161003816(1205881199061015840101584011988810158401015840)11990511988910038161003816100381610038161003816 =

10038161003816100381610038161205881199061198801199051198913 (119888 Θ)10038161003816100381610038161003816100381610038161003816120588 (119888)1198631199051198891198881198891199091003816100381610038161003816 (B3a)

119870int =100381610038161003816100381610038161003816⟨(1205881199061015840101584011988810158401015840)119901minus119889⟩10038161003816100381610038161003816100381610038161003816100381610038161003816⟨(1205881199061015840101584011988810158401015840)119905119889⟩10038161003816100381610038161003816

= (14) 120573 [Θ minus (2Θ minus 1)12] (1198801199051199061015840 )(120575119905119871 ) (B3b)










0 02 04 06 08 1

minus005

0

005

01

015

02

025

03

035Experiments

c

[kg

m2s]

u

c

(a)

0 02 04 06 08 1

minus005

0

005

01

015

02

025

03

035Calculations

c[k

gm

2s]

u

c

(b)











0

002

0225 025 0275

1

1 1

02 03 04 026 027 028 029 03

026 027 028 029 03

X(m) x(m) X(m)

0225 025 0275x(m)

02 03 04X(m)

004

006

008

0101 03 05 07 09

012

minus006

minus004

minus002

0

08

06

04

02

0

004

002

006

minus006

minus004

minus002

0

002

004

006

0

002

004

006

008

01

012

minus0981

minus0984

minus0987

minus099

minus0993

minus0996

minus0999

３=04ＧＭ

３=04ＧＭ

３=2ＧＭ

０-０0０0

minus0987

minus0994

minus1001

minus1008

minus1015

minus1022

minus1029

０-０0０0

０-０0０0

５５0

５５0

1５５0

C

1

08

06

04

02

0

1


Ｏ＝ＯＯ0C


Ｏ＝ＯＯ0C

０-０0０0

５５0

C

３=04ＧＭ

３=2ＧＭ

３=2ＧＭ













Nomenclature





Greek Symbols





Data Availability




Acknowledgments



References































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



120588 = 120588119906[1 + 119888 (120588119906120588119887 minus 1)] (6b)


120597120588120597119905 + nabla sdot 120588997888rarr119906 = 0 (6d)


















119904119906Σ119891

(7c)


119887


119904119887Σ119891

(7d)











119888 = 119888[119888 + (120588119906120588119887) (1 minus 119888)] (9b)




(10)






















120588119886 = 120588119906119886119906119875119906 + 120588119887119886119887119875119887 = 120588119906119886119906 (1 minus 119888) + 120588119887119886119887119888 (11)




+ 120597 (120588119887119888)120597119905 + nabla sdot (120588119887119888997888rarr119906 119887)119887

= 0(12)


120588119888 = 120588119888 = 120588119887119888 (13a)




120597 (120588119887119888)120597119905 + nabla sdot (120588119887119888997888rarr119906 119887) = 120588119882 (14b)




120597 [120588119906 (1 minus 119888) 997888rarr119906 119906]



]119906

+ 120597 (120588119887119888 119906119894119887)120597119905 + nabla


)119887


(15)






119887





119904119887Σ119891 (17)






119904119887Σ119891)


(19)





119904119906Σ119891

(20a)


119887


119904119887Σ119891

(20b)



119904119906Σ119891 (21a)









u purarru u

u psurarru su

b pbrarru b

b psbrarru sb




120588119906 119901119906 997888rarr119906 119906 120588119882997904rArr 120588119887 119901119887 997888rarr119906 119887 (22)


































(25)



(26a)


(26b)





(27a)


119887


(27b)




1003816100381610038161003816100381610038161003816 (28a)









120597 (120588119887119888)120597119905 + nabla sdot (120588119887119888997888rarr119906 119887) = 120588119882 (30b)




]]]120588119882

(30c)



]]]120588119882

(30d)





119888 = 119888[119888 + (120588119906120588119887) (1 minus 119888)] (31a)

997888119906 = 997888rarr119906 119906 (1 minus 119888) + 119906119887119888 (31b)

997888rarr119906 = 997888rarr119906 119906 (1 minus 119888) + 119906119887119888 (31c)

120588 = 120588119906[1 + 119888 (120588119906120588119887 minus 1)] (31d)

119901 = 119901119906 (1 minus 119888) + 119901119887119888 (31e)

12058811990610158401015840119894 11988810158401015840 = 120588119888 (1 minus 119888) (119906119894119887 minus 119906119894119906) (31f)


(31g)













120597 [120588 (11990610158401198941199061015840119895)]120597119905 + 120597 [120588119906119896 (11990610158401198941199061015840119895)]120597119909119896


120597119906119895120597119909119896]

minus [1199061015840119894 sdot 1205971199011015840

120597119909119895 + 1199061015840119895 sdot1205971199011015840120597119909119894]

(32)


120597119906119895119906120597119909119896 ]

minus (1 minus 119888) [(1199061015840119895 sdot 1205971199011015840

120597119909119894)119906

+ (1199061015840119894 sdot 1205971199011015840

120597119909119895)119906

]minus 119876119906

119894119895

(33a)


120597119906119895119887120597119909119896 ]

minus 119888 [(1199061015840119895 sdot 1205971199011015840

120597119909119894)119887

+ (1199061015840119894 sdot 1205971199011015840

120597119909119895)119887

] + 119876119887119894119895

(33b)

Here 119876119906119894119895 and 119876119887


119894119895


119894119895 and 119876119887119894119895 in terms of







1199061015840119894119904119906 minus (119901119904119906119899119894) + 119901119904119906119899119894 = 1199061015840119894119904119887 minus (119901119904119887119899119894) + 119901119904119887119899119894 (34a)

1199061015840119895119904119906 minus (119901119904119906119899119895) + 119901119904119906119899119895= 1199061015840119895119904119887 minus (119901119904119887119899119895) + 119901119904119887119899119895

(34b)


(119906119894119904119906 + 1199061015840119894119904119906) minus (119901119904119906119899119894)= (119906119894119904119887 + 1199061015840119894119904119887) minus (119901119904119887119899119894)

(35a)

119906119894119904119906 minus 119901119904119906119899119894 = 119906119894119904119887 minus 119901119904119887119899119894 (35b)


(11990610158401198941199061015840119895)119904119906 2 minus [(1199011199061015840119894119899119895)119904119906 + (1199011199061015840119895119899119894)119904119906] + [(1199012119899119894119899119895)119904119906 minus (119901119899119894)119904119906 (119901119899119895)119904119906]

= (11990610158401198941199061015840119895)119904119887 2 minus [(1199011199061015840119894119899119895)119904119887 + (1199011199061015840119895119899119894)119904119887] + [(1199012119899119894119899119895)119904119887 minus (119901119899119894)119904119906 (119901119899119895)119904119887]

(36)


(11990610158401198941199061015840119895)119904119906 120588119882 minus [(1199011199061015840119894119899119895)119904119906 + (1199011199061015840119895119899119894)119904119906] Σ119891+ [[(1199012119899119894119899119895)119904119906 minus (119901119899119894)119904119906 (119901119899119895)119904119906 ]

]Σ119891

= (11990610158401198941199061015840119895)119904119887 120588119882 minus [(1199011199061015840119894119899119895)119904119887 + (1199011199061015840119895119899119894)119904119887] Σ119891+ [[(1199012119899119894119899119895)119904119887 minus (119901119899119894)119904119906 (119901119899119895)119904119887 ]

]Σ119891

(37)


119876119906119894119895 = (11990610158401198941199061015840119895)119904119906 120588119882

1

minus [(1199011199061015840119894119899119895)119904119906 + (1199011199061015840119895119899119894)119904119906] Σ1198912

+ [(1199012119899119894119899119895)119904119906 minus (119901119899119894)119904119906 (119901119899119895)119904119906 ] Σ1198913

(38a)

119876119887119894119895 = (11990610158401198941199061015840119895)119904119887 120588119882

1

minus [(1199011199061015840119894119899119895)119904119887 + (1199011199061015840119895119899119894)119904119887] Σ1198912

+ [(1199012119899119894119899119895)119904119887 minus (119901119899119894)119904119887 (119901119899119895)119904119887 ] Σ1198913

(38b)




(11990610158401198941199061015840119895)119906 120588119882 minus [(11990110158401199061015840119894)119906 119899119895 + (11990110158401199061015840119895)119906 119899119894] Σ119891+ [[11990110158402119906 (119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891

= (11990610158401198941199061015840119895)119887 120588119882 minus [(11990110158401199061015840119894)119887 119899119895 + (11990110158401199061015840119895)119887 119899119894] Σ119891+ [[11990110158402119887(119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891

(39)



119876119906119894119895 = (11990610158401198941199061015840119895)119906 120588119882 minus [(11990110158401199061015840119894)119906 119899119895 + (11990110158401199061015840119895)119906 119899119894] Σ119891

+ [[11990110158402119906 (119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891 = 119876119887

119894119895

= (11990610158401198941199061015840119895)119887 120588119882 minus [(11990110158401199061015840119894)119887 119899119895 + (11990110158401199061015840119895)119887 119899119894] Σ119891+ [[11990110158402119887(119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891

(40)







120588119882 = 120588119906119878119871Σ (41a)


119904Σ] + (119878nabla sdot 997888rarr119899)

119904Σ

(41b)





119865)


119865

+ 2120588119906119878119871119870119882

(42)





(43)





120591119894119895119906 = minus120588119906 (11990610158401198941199061015840119895)119906= ]119905119906 (120597119906119894119906120597119909119895 +

120597119906119895119906120597119909119894 ) minus (23) 120588119906119896119906120575119894119895

(44a)

120591119894119895119887 = minus120588119887 (11990610158401198941199061015840119895)119887= ]119905119887 (120597119906119894119887120597119909119895 +

120597119906119895119887120597119909119894 ) minus (23) 120588119906119896119887120575119894119895

(44b)


120597 (120588)120597119905 + 120597 (120588120572)120597119909120572 = 120591119894119895120597119895120597119909119894

+ 120597 [120588 (V119905120590119896) 120597120597119909119894]120597119909119894minus 120588120576

(45a)

120597 (120588120576)120597119905 + 120597 (120588119894120576)120597119909119894 = 1198621205761120591119894119895 (120576) 120597119906119895120597119909119894

minus 1198621205762120588(1205762 )+ 120597 [120588 (V119905120590120576) 120597120576120597119909119894 ] 120597119909119894

(45b)


120591119894119895 = minus (12058811990610158401015840119894 11990610158401015840119895 ) = ]119905 ( 120597119894120597119909119895 +120597119895120597119909119894 ) minus (

23) 120588120575119894119895 (46)



+ 120597 [120588 (1 minus 119888) (V119905119906120590119896) 120597119896119906120597119909119894]120597119909119894 minus 119896119906120588119882+ 119870119906Σ119891

(47a)

120597 (120588119888119896119887)120597119905 + 120597 (120588119888119906119894119887119896119887)120597119909119894= 120588119888 (120591119894119895119887120588119887) 120597119906119894119887120597119909119895 minus 120588119888120576119887

+ 120597 [120588119888 (V119905119887120590119896) 120597119896119887120597119909119894]120597119909119894 + 119896119887120588119882 + 119870119887Σ119891

(47b)


+ 120597 [120588 (1 minus 119888) (V119905119906120590119896) 120597119896119906120597119909120572]120597119909120572 minus 120576119906120588119882+ 119864119906Σ119891

(47c)

120597 (120588119888120576119887)120597119905 + 120597 (120588119888119906119894119887120576119887)120597119909119894= 1198621205761120588 (1 minus 119888) (120576119887119896119887) (120591119894119895119887120588119887) 120597119906119894119887120597119909119895minus 1198621205762120588119888( 1205762119887119896119887) +

120597 [120588119888 (V119905119906120590119896) 120597119896119887120597119909120572]120597119909120572+ 119864119887Σ119891 + 120576119887120588119882

(47d)


minus119896119906120588119882 + 119870119906Σ119891 = +119896119887120588119882 + 119870119887Σ119891 (48a)

minus120576119906120588119882 + 119864119906Σ119891 = +120576119887120588119882 + 119864119887Σ119891 (48b)










(49a)


(49b)

120597 (120588)120597119905 + 120597 (120588120572)120597119909120572 = 120591119894119895120597119895120597119909119894+ 120597 [120588 (V119905120590119896) 120597120597119909119894]120597119909119894 minus 120588120576

(49c)

120597 (120588120576)120597119905 + 120597 (120588119894120576)120597119909119894 = 1198621205761120591119894119895 (120576) 120597119906119895120597119909119894+ 120597 [120588 (V119905120590120576) 120597120576120597119909119894 ] 120597119909119894 minus 1198621205762120588(1205762 )]

(49d)

= 119896119906 (1 minus 119888) + 119896119887119888 (49e)

120576 = 120576119906 (1 minus 119888) + 120576119887119888 (49f)










( 1198601198600


2|nablaℎ| asymp

3(|nablaℎ|2)12

=4(intinfin

01198962119865 (119896) 119889119896)12

(50)


( 1198601198600

) asymp (intinfin01198962119865 (119896) 119889119896)12 (51a)

1205902119891 (119905) = (119909 minus 119909 (119905))2 = intinfin0119865 (119896 119905) 119889119896 (51b)



1205902119891 cong 2119863119905119905 (52a)

120575119905 sim (1199061015840119871119905)12 (52b)




120597119888120597119905 = 11986311990512059721198881205971199092 + 119880119905 1003816100381610038161003816100381610038161003816100381612059711988812059711990910038161003816100381610038161003816100381610038161003816 (53a)



120588119882 = 120588 = 120588119906119880119905 |nabla119888| (54)



119880119905 sim (1199061015840119878119871)12 (55a)

120591119905 le 119905 ltlt (1199061015840119878119871)2 120591119905 (55b)


(120588119882)119871= 119860119871120588119906radic11989612119906 119878119871 |nabla119888| (56)








119880119905 = 119860119906101584011986311988614 = 11986011990610158403411987812119871 120594minus1411987114 (57a)

120591119905 lt 119905 ltlt 120591119905119863119886 (57b)





11 Conclusions













Appendix








(1 minus 119888) 119906119906 + 119888 119906119887Θ = 1 (A1a)


119901119906 + (12) 1199062119906 = 119901minusinfin + 12 (A1c)


(1 minus 119888) 119906119906 + 119888 119906119887Θ = 1 (A2a)

(12 minus 119888) 1199062119906 +119888 1199062119887Θ = 12 (A2b)


119906119887 = minus120573 + (minus4120572120574 + 1205732)12


119888Θ + 1] 120573 = minus2 119888Θ 05 minus 119888

(1 minus 119888)2 120574 = 05 minus 119888

(1 minus 119888)2 minus 05

(A3)


1205881199061015840101584011988810158401015840 = 120588119888 (1 minus 119888) (119906119887 minus 119906119906) (A4a)

1205881199061015840101584011990610158401015840 = 120588119888 (119906119887 minus 119906119906)2 (A4b)



0 025 05 075 10

005

01

015

02

025

03

035

04

045

05

u

c

uUt

c(a)

0 025 05 075 10

1

2

3

4

5

6

7

8

9

u uU

tu b

Ut

ubUt

uuUt

c(b)



119906119906 = 1198911 (119888 Θ) (A5a)

119906119887 = 1198912 (119888 Θ) (A5b)

1205881199061015840101584011988810158401015840 = 1198913 (119888 Θ) (A5c)

1205881199061015840101584011990610158401015840 = 1198914 (119888 Θ) (A5d)
















1205881199061015840101584011988810158401015840 = (1205881199061015840101584011988810158401015840)119901minus119889

+ (1205881199061015840101584011988810158401015840)119905119889

= 1205881199061198801199051198913 (119888 Θ) minus 120588119863119905119889119888119889119909 (B1)



(B2)




119870119897119900119888 =100381610038161003816100381610038161003816(1205881199061015840101584011988810158401015840)119901minus11988910038161003816100381610038161003816100381610038161003816100381610038161003816(1205881199061015840101584011988810158401015840)11990511988910038161003816100381610038161003816 =

10038161003816100381610038161205881199061198801199051198913 (119888 Θ)10038161003816100381610038161003816100381610038161003816120588 (119888)1198631199051198891198881198891199091003816100381610038161003816 (B3a)

119870int =100381610038161003816100381610038161003816⟨(1205881199061015840101584011988810158401015840)119901minus119889⟩10038161003816100381610038161003816100381610038161003816100381610038161003816⟨(1205881199061015840101584011988810158401015840)119905119889⟩10038161003816100381610038161003816

= (14) 120573 [Θ minus (2Θ minus 1)12] (1198801199051199061015840 )(120575119905119871 ) (B3b)










0 02 04 06 08 1

minus005

0

005

01

015

02

025

03

035Experiments

c

[kg

m2s]

u

c

(a)

0 02 04 06 08 1

minus005

0

005

01

015

02

025

03

035Calculations

c[k

gm

2s]

u

c

(b)











0

002

0225 025 0275

1

1 1

02 03 04 026 027 028 029 03

026 027 028 029 03

X(m) x(m) X(m)

0225 025 0275x(m)

02 03 04X(m)

004

006

008

0101 03 05 07 09

012

minus006

minus004

minus002

0

08

06

04

02

0

004

002

006

minus006

minus004

minus002

0

002

004

006

0

002

004

006

008

01

012

minus0981

minus0984

minus0987

minus099

minus0993

minus0996

minus0999

３=04ＧＭ

３=04ＧＭ

３=2ＧＭ

０-０0０0

minus0987

minus0994

minus1001

minus1008

minus1015

minus1022

minus1029

０-０0０0

０-０0０0

５５0

５５0

1５５0

C

1

08

06

04

02

0

1


Ｏ＝ＯＯ0C


Ｏ＝ＯＯ0C

０-０0０0

５５0

C

３=04ＧＭ

３=2ＧＭ

３=2ＧＭ













Nomenclature





Greek Symbols





Data Availability




Acknowledgments



References































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







119904119906Σ119891

(7c)


119887


119904119887Σ119891

(7d)











119888 = 119888[119888 + (120588119906120588119887) (1 minus 119888)] (9b)




(10)






















120588119886 = 120588119906119886119906119875119906 + 120588119887119886119887119875119887 = 120588119906119886119906 (1 minus 119888) + 120588119887119886119887119888 (11)




+ 120597 (120588119887119888)120597119905 + nabla sdot (120588119887119888997888rarr119906 119887)119887

= 0(12)


120588119888 = 120588119888 = 120588119887119888 (13a)




120597 (120588119887119888)120597119905 + nabla sdot (120588119887119888997888rarr119906 119887) = 120588119882 (14b)




120597 [120588119906 (1 minus 119888) 997888rarr119906 119906]



]119906

+ 120597 (120588119887119888 119906119894119887)120597119905 + nabla


)119887


(15)






119887





119904119887Σ119891 (17)






119904119887Σ119891)


(19)





119904119906Σ119891

(20a)


119887


119904119887Σ119891

(20b)



119904119906Σ119891 (21a)









u purarru u

u psurarru su

b pbrarru b

b psbrarru sb




120588119906 119901119906 997888rarr119906 119906 120588119882997904rArr 120588119887 119901119887 997888rarr119906 119887 (22)


































(25)



(26a)


(26b)





(27a)


119887


(27b)




1003816100381610038161003816100381610038161003816 (28a)









120597 (120588119887119888)120597119905 + nabla sdot (120588119887119888997888rarr119906 119887) = 120588119882 (30b)




]]]120588119882

(30c)



]]]120588119882

(30d)





119888 = 119888[119888 + (120588119906120588119887) (1 minus 119888)] (31a)

997888119906 = 997888rarr119906 119906 (1 minus 119888) + 119906119887119888 (31b)

997888rarr119906 = 997888rarr119906 119906 (1 minus 119888) + 119906119887119888 (31c)

120588 = 120588119906[1 + 119888 (120588119906120588119887 minus 1)] (31d)

119901 = 119901119906 (1 minus 119888) + 119901119887119888 (31e)

12058811990610158401015840119894 11988810158401015840 = 120588119888 (1 minus 119888) (119906119894119887 minus 119906119894119906) (31f)


(31g)













120597 [120588 (11990610158401198941199061015840119895)]120597119905 + 120597 [120588119906119896 (11990610158401198941199061015840119895)]120597119909119896


120597119906119895120597119909119896]

minus [1199061015840119894 sdot 1205971199011015840

120597119909119895 + 1199061015840119895 sdot1205971199011015840120597119909119894]

(32)


120597119906119895119906120597119909119896 ]

minus (1 minus 119888) [(1199061015840119895 sdot 1205971199011015840

120597119909119894)119906

+ (1199061015840119894 sdot 1205971199011015840

120597119909119895)119906

]minus 119876119906

119894119895

(33a)


120597119906119895119887120597119909119896 ]

minus 119888 [(1199061015840119895 sdot 1205971199011015840

120597119909119894)119887

+ (1199061015840119894 sdot 1205971199011015840

120597119909119895)119887

] + 119876119887119894119895

(33b)

Here 119876119906119894119895 and 119876119887


119894119895


119894119895 and 119876119887119894119895 in terms of







1199061015840119894119904119906 minus (119901119904119906119899119894) + 119901119904119906119899119894 = 1199061015840119894119904119887 minus (119901119904119887119899119894) + 119901119904119887119899119894 (34a)

1199061015840119895119904119906 minus (119901119904119906119899119895) + 119901119904119906119899119895= 1199061015840119895119904119887 minus (119901119904119887119899119895) + 119901119904119887119899119895

(34b)


(119906119894119904119906 + 1199061015840119894119904119906) minus (119901119904119906119899119894)= (119906119894119904119887 + 1199061015840119894119904119887) minus (119901119904119887119899119894)

(35a)

119906119894119904119906 minus 119901119904119906119899119894 = 119906119894119904119887 minus 119901119904119887119899119894 (35b)


(11990610158401198941199061015840119895)119904119906 2 minus [(1199011199061015840119894119899119895)119904119906 + (1199011199061015840119895119899119894)119904119906] + [(1199012119899119894119899119895)119904119906 minus (119901119899119894)119904119906 (119901119899119895)119904119906]

= (11990610158401198941199061015840119895)119904119887 2 minus [(1199011199061015840119894119899119895)119904119887 + (1199011199061015840119895119899119894)119904119887] + [(1199012119899119894119899119895)119904119887 minus (119901119899119894)119904119906 (119901119899119895)119904119887]

(36)


(11990610158401198941199061015840119895)119904119906 120588119882 minus [(1199011199061015840119894119899119895)119904119906 + (1199011199061015840119895119899119894)119904119906] Σ119891+ [[(1199012119899119894119899119895)119904119906 minus (119901119899119894)119904119906 (119901119899119895)119904119906 ]

]Σ119891

= (11990610158401198941199061015840119895)119904119887 120588119882 minus [(1199011199061015840119894119899119895)119904119887 + (1199011199061015840119895119899119894)119904119887] Σ119891+ [[(1199012119899119894119899119895)119904119887 minus (119901119899119894)119904119906 (119901119899119895)119904119887 ]

]Σ119891

(37)


119876119906119894119895 = (11990610158401198941199061015840119895)119904119906 120588119882

1

minus [(1199011199061015840119894119899119895)119904119906 + (1199011199061015840119895119899119894)119904119906] Σ1198912

+ [(1199012119899119894119899119895)119904119906 minus (119901119899119894)119904119906 (119901119899119895)119904119906 ] Σ1198913

(38a)

119876119887119894119895 = (11990610158401198941199061015840119895)119904119887 120588119882

1

minus [(1199011199061015840119894119899119895)119904119887 + (1199011199061015840119895119899119894)119904119887] Σ1198912

+ [(1199012119899119894119899119895)119904119887 minus (119901119899119894)119904119887 (119901119899119895)119904119887 ] Σ1198913

(38b)




(11990610158401198941199061015840119895)119906 120588119882 minus [(11990110158401199061015840119894)119906 119899119895 + (11990110158401199061015840119895)119906 119899119894] Σ119891+ [[11990110158402119906 (119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891

= (11990610158401198941199061015840119895)119887 120588119882 minus [(11990110158401199061015840119894)119887 119899119895 + (11990110158401199061015840119895)119887 119899119894] Σ119891+ [[11990110158402119887(119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891

(39)



119876119906119894119895 = (11990610158401198941199061015840119895)119906 120588119882 minus [(11990110158401199061015840119894)119906 119899119895 + (11990110158401199061015840119895)119906 119899119894] Σ119891

+ [[11990110158402119906 (119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891 = 119876119887

119894119895

= (11990610158401198941199061015840119895)119887 120588119882 minus [(11990110158401199061015840119894)119887 119899119895 + (11990110158401199061015840119895)119887 119899119894] Σ119891+ [[11990110158402119887(119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891

(40)







120588119882 = 120588119906119878119871Σ (41a)


119904Σ] + (119878nabla sdot 997888rarr119899)

119904Σ

(41b)





119865)


119865

+ 2120588119906119878119871119870119882

(42)





(43)





120591119894119895119906 = minus120588119906 (11990610158401198941199061015840119895)119906= ]119905119906 (120597119906119894119906120597119909119895 +

120597119906119895119906120597119909119894 ) minus (23) 120588119906119896119906120575119894119895

(44a)

120591119894119895119887 = minus120588119887 (11990610158401198941199061015840119895)119887= ]119905119887 (120597119906119894119887120597119909119895 +

120597119906119895119887120597119909119894 ) minus (23) 120588119906119896119887120575119894119895

(44b)


120597 (120588)120597119905 + 120597 (120588120572)120597119909120572 = 120591119894119895120597119895120597119909119894

+ 120597 [120588 (V119905120590119896) 120597120597119909119894]120597119909119894minus 120588120576

(45a)

120597 (120588120576)120597119905 + 120597 (120588119894120576)120597119909119894 = 1198621205761120591119894119895 (120576) 120597119906119895120597119909119894

minus 1198621205762120588(1205762 )+ 120597 [120588 (V119905120590120576) 120597120576120597119909119894 ] 120597119909119894

(45b)


120591119894119895 = minus (12058811990610158401015840119894 11990610158401015840119895 ) = ]119905 ( 120597119894120597119909119895 +120597119895120597119909119894 ) minus (

23) 120588120575119894119895 (46)



+ 120597 [120588 (1 minus 119888) (V119905119906120590119896) 120597119896119906120597119909119894]120597119909119894 minus 119896119906120588119882+ 119870119906Σ119891

(47a)

120597 (120588119888119896119887)120597119905 + 120597 (120588119888119906119894119887119896119887)120597119909119894= 120588119888 (120591119894119895119887120588119887) 120597119906119894119887120597119909119895 minus 120588119888120576119887

+ 120597 [120588119888 (V119905119887120590119896) 120597119896119887120597119909119894]120597119909119894 + 119896119887120588119882 + 119870119887Σ119891

(47b)


+ 120597 [120588 (1 minus 119888) (V119905119906120590119896) 120597119896119906120597119909120572]120597119909120572 minus 120576119906120588119882+ 119864119906Σ119891

(47c)

120597 (120588119888120576119887)120597119905 + 120597 (120588119888119906119894119887120576119887)120597119909119894= 1198621205761120588 (1 minus 119888) (120576119887119896119887) (120591119894119895119887120588119887) 120597119906119894119887120597119909119895minus 1198621205762120588119888( 1205762119887119896119887) +

120597 [120588119888 (V119905119906120590119896) 120597119896119887120597119909120572]120597119909120572+ 119864119887Σ119891 + 120576119887120588119882

(47d)


minus119896119906120588119882 + 119870119906Σ119891 = +119896119887120588119882 + 119870119887Σ119891 (48a)

minus120576119906120588119882 + 119864119906Σ119891 = +120576119887120588119882 + 119864119887Σ119891 (48b)










(49a)


(49b)

120597 (120588)120597119905 + 120597 (120588120572)120597119909120572 = 120591119894119895120597119895120597119909119894+ 120597 [120588 (V119905120590119896) 120597120597119909119894]120597119909119894 minus 120588120576

(49c)

120597 (120588120576)120597119905 + 120597 (120588119894120576)120597119909119894 = 1198621205761120591119894119895 (120576) 120597119906119895120597119909119894+ 120597 [120588 (V119905120590120576) 120597120576120597119909119894 ] 120597119909119894 minus 1198621205762120588(1205762 )]

(49d)

= 119896119906 (1 minus 119888) + 119896119887119888 (49e)

120576 = 120576119906 (1 minus 119888) + 120576119887119888 (49f)










( 1198601198600


2|nablaℎ| asymp

3(|nablaℎ|2)12

=4(intinfin

01198962119865 (119896) 119889119896)12

(50)


( 1198601198600

) asymp (intinfin01198962119865 (119896) 119889119896)12 (51a)

1205902119891 (119905) = (119909 minus 119909 (119905))2 = intinfin0119865 (119896 119905) 119889119896 (51b)



1205902119891 cong 2119863119905119905 (52a)

120575119905 sim (1199061015840119871119905)12 (52b)




120597119888120597119905 = 11986311990512059721198881205971199092 + 119880119905 1003816100381610038161003816100381610038161003816100381612059711988812059711990910038161003816100381610038161003816100381610038161003816 (53a)



120588119882 = 120588 = 120588119906119880119905 |nabla119888| (54)



119880119905 sim (1199061015840119878119871)12 (55a)

120591119905 le 119905 ltlt (1199061015840119878119871)2 120591119905 (55b)


(120588119882)119871= 119860119871120588119906radic11989612119906 119878119871 |nabla119888| (56)








119880119905 = 119860119906101584011986311988614 = 11986011990610158403411987812119871 120594minus1411987114 (57a)

120591119905 lt 119905 ltlt 120591119905119863119886 (57b)





11 Conclusions













Appendix








(1 minus 119888) 119906119906 + 119888 119906119887Θ = 1 (A1a)


119901119906 + (12) 1199062119906 = 119901minusinfin + 12 (A1c)


(1 minus 119888) 119906119906 + 119888 119906119887Θ = 1 (A2a)

(12 minus 119888) 1199062119906 +119888 1199062119887Θ = 12 (A2b)


119906119887 = minus120573 + (minus4120572120574 + 1205732)12


119888Θ + 1] 120573 = minus2 119888Θ 05 minus 119888

(1 minus 119888)2 120574 = 05 minus 119888

(1 minus 119888)2 minus 05

(A3)


1205881199061015840101584011988810158401015840 = 120588119888 (1 minus 119888) (119906119887 minus 119906119906) (A4a)

1205881199061015840101584011990610158401015840 = 120588119888 (119906119887 minus 119906119906)2 (A4b)



0 025 05 075 10

005

01

015

02

025

03

035

04

045

05

u

c

uUt

c(a)

0 025 05 075 10

1

2

3

4

5

6

7

8

9

u uU

tu b

Ut

ubUt

uuUt

c(b)



119906119906 = 1198911 (119888 Θ) (A5a)

119906119887 = 1198912 (119888 Θ) (A5b)

1205881199061015840101584011988810158401015840 = 1198913 (119888 Θ) (A5c)

1205881199061015840101584011990610158401015840 = 1198914 (119888 Θ) (A5d)
















1205881199061015840101584011988810158401015840 = (1205881199061015840101584011988810158401015840)119901minus119889

+ (1205881199061015840101584011988810158401015840)119905119889

= 1205881199061198801199051198913 (119888 Θ) minus 120588119863119905119889119888119889119909 (B1)



(B2)




119870119897119900119888 =100381610038161003816100381610038161003816(1205881199061015840101584011988810158401015840)119901minus11988910038161003816100381610038161003816100381610038161003816100381610038161003816(1205881199061015840101584011988810158401015840)11990511988910038161003816100381610038161003816 =

10038161003816100381610038161205881199061198801199051198913 (119888 Θ)10038161003816100381610038161003816100381610038161003816120588 (119888)1198631199051198891198881198891199091003816100381610038161003816 (B3a)

119870int =100381610038161003816100381610038161003816⟨(1205881199061015840101584011988810158401015840)119901minus119889⟩10038161003816100381610038161003816100381610038161003816100381610038161003816⟨(1205881199061015840101584011988810158401015840)119905119889⟩10038161003816100381610038161003816

= (14) 120573 [Θ minus (2Θ minus 1)12] (1198801199051199061015840 )(120575119905119871 ) (B3b)










0 02 04 06 08 1

minus005

0

005

01

015

02

025

03

035Experiments

c

[kg

m2s]

u

c

(a)

0 02 04 06 08 1

minus005

0

005

01

015

02

025

03

035Calculations

c[k

gm

2s]

u

c

(b)











0

002

0225 025 0275

1

1 1

02 03 04 026 027 028 029 03

026 027 028 029 03

X(m) x(m) X(m)

0225 025 0275x(m)

02 03 04X(m)

004

006

008

0101 03 05 07 09

012

minus006

minus004

minus002

0

08

06

04

02

0

004

002

006

minus006

minus004

minus002

0

002

004

006

0

002

004

006

008

01

012

minus0981

minus0984

minus0987

minus099

minus0993

minus0996

minus0999

３=04ＧＭ

３=04ＧＭ

３=2ＧＭ

０-０0０0

minus0987

minus0994

minus1001

minus1008

minus1015

minus1022

minus1029

０-０0０0

０-０0０0

５５0

５５0

1５５0

C

1

08

06

04

02

0

1


Ｏ＝ＯＯ0C


Ｏ＝ＯＯ0C

０-０0０0

５５0

C

３=04ＧＭ

３=2ＧＭ

３=2ＧＭ













Nomenclature





Greek Symbols





Data Availability




Acknowledgments



References































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



















120588119886 = 120588119906119886119906119875119906 + 120588119887119886119887119875119887 = 120588119906119886119906 (1 minus 119888) + 120588119887119886119887119888 (11)




+ 120597 (120588119887119888)120597119905 + nabla sdot (120588119887119888997888rarr119906 119887)119887

= 0(12)


120588119888 = 120588119888 = 120588119887119888 (13a)




120597 (120588119887119888)120597119905 + nabla sdot (120588119887119888997888rarr119906 119887) = 120588119882 (14b)




120597 [120588119906 (1 minus 119888) 997888rarr119906 119906]



]119906

+ 120597 (120588119887119888 119906119894119887)120597119905 + nabla


)119887


(15)






119887





119904119887Σ119891 (17)






119904119887Σ119891)


(19)





119904119906Σ119891

(20a)


119887


119904119887Σ119891

(20b)



119904119906Σ119891 (21a)









u purarru u

u psurarru su

b pbrarru b

b psbrarru sb




120588119906 119901119906 997888rarr119906 119906 120588119882997904rArr 120588119887 119901119887 997888rarr119906 119887 (22)


































(25)



(26a)


(26b)





(27a)


119887


(27b)




1003816100381610038161003816100381610038161003816 (28a)









120597 (120588119887119888)120597119905 + nabla sdot (120588119887119888997888rarr119906 119887) = 120588119882 (30b)




]]]120588119882

(30c)



]]]120588119882

(30d)





119888 = 119888[119888 + (120588119906120588119887) (1 minus 119888)] (31a)

997888119906 = 997888rarr119906 119906 (1 minus 119888) + 119906119887119888 (31b)

997888rarr119906 = 997888rarr119906 119906 (1 minus 119888) + 119906119887119888 (31c)

120588 = 120588119906[1 + 119888 (120588119906120588119887 minus 1)] (31d)

119901 = 119901119906 (1 minus 119888) + 119901119887119888 (31e)

12058811990610158401015840119894 11988810158401015840 = 120588119888 (1 minus 119888) (119906119894119887 minus 119906119894119906) (31f)


(31g)













120597 [120588 (11990610158401198941199061015840119895)]120597119905 + 120597 [120588119906119896 (11990610158401198941199061015840119895)]120597119909119896


120597119906119895120597119909119896]

minus [1199061015840119894 sdot 1205971199011015840

120597119909119895 + 1199061015840119895 sdot1205971199011015840120597119909119894]

(32)


120597119906119895119906120597119909119896 ]

minus (1 minus 119888) [(1199061015840119895 sdot 1205971199011015840

120597119909119894)119906

+ (1199061015840119894 sdot 1205971199011015840

120597119909119895)119906

]minus 119876119906

119894119895

(33a)


120597119906119895119887120597119909119896 ]

minus 119888 [(1199061015840119895 sdot 1205971199011015840

120597119909119894)119887

+ (1199061015840119894 sdot 1205971199011015840

120597119909119895)119887

] + 119876119887119894119895

(33b)

Here 119876119906119894119895 and 119876119887


119894119895


119894119895 and 119876119887119894119895 in terms of







1199061015840119894119904119906 minus (119901119904119906119899119894) + 119901119904119906119899119894 = 1199061015840119894119904119887 minus (119901119904119887119899119894) + 119901119904119887119899119894 (34a)

1199061015840119895119904119906 minus (119901119904119906119899119895) + 119901119904119906119899119895= 1199061015840119895119904119887 minus (119901119904119887119899119895) + 119901119904119887119899119895

(34b)


(119906119894119904119906 + 1199061015840119894119904119906) minus (119901119904119906119899119894)= (119906119894119904119887 + 1199061015840119894119904119887) minus (119901119904119887119899119894)

(35a)

119906119894119904119906 minus 119901119904119906119899119894 = 119906119894119904119887 minus 119901119904119887119899119894 (35b)


(11990610158401198941199061015840119895)119904119906 2 minus [(1199011199061015840119894119899119895)119904119906 + (1199011199061015840119895119899119894)119904119906] + [(1199012119899119894119899119895)119904119906 minus (119901119899119894)119904119906 (119901119899119895)119904119906]

= (11990610158401198941199061015840119895)119904119887 2 minus [(1199011199061015840119894119899119895)119904119887 + (1199011199061015840119895119899119894)119904119887] + [(1199012119899119894119899119895)119904119887 minus (119901119899119894)119904119906 (119901119899119895)119904119887]

(36)


(11990610158401198941199061015840119895)119904119906 120588119882 minus [(1199011199061015840119894119899119895)119904119906 + (1199011199061015840119895119899119894)119904119906] Σ119891+ [[(1199012119899119894119899119895)119904119906 minus (119901119899119894)119904119906 (119901119899119895)119904119906 ]

]Σ119891

= (11990610158401198941199061015840119895)119904119887 120588119882 minus [(1199011199061015840119894119899119895)119904119887 + (1199011199061015840119895119899119894)119904119887] Σ119891+ [[(1199012119899119894119899119895)119904119887 minus (119901119899119894)119904119906 (119901119899119895)119904119887 ]

]Σ119891

(37)


119876119906119894119895 = (11990610158401198941199061015840119895)119904119906 120588119882

1

minus [(1199011199061015840119894119899119895)119904119906 + (1199011199061015840119895119899119894)119904119906] Σ1198912

+ [(1199012119899119894119899119895)119904119906 minus (119901119899119894)119904119906 (119901119899119895)119904119906 ] Σ1198913

(38a)

119876119887119894119895 = (11990610158401198941199061015840119895)119904119887 120588119882

1

minus [(1199011199061015840119894119899119895)119904119887 + (1199011199061015840119895119899119894)119904119887] Σ1198912

+ [(1199012119899119894119899119895)119904119887 minus (119901119899119894)119904119887 (119901119899119895)119904119887 ] Σ1198913

(38b)




(11990610158401198941199061015840119895)119906 120588119882 minus [(11990110158401199061015840119894)119906 119899119895 + (11990110158401199061015840119895)119906 119899119894] Σ119891+ [[11990110158402119906 (119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891

= (11990610158401198941199061015840119895)119887 120588119882 minus [(11990110158401199061015840119894)119887 119899119895 + (11990110158401199061015840119895)119887 119899119894] Σ119891+ [[11990110158402119887(119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891

(39)



119876119906119894119895 = (11990610158401198941199061015840119895)119906 120588119882 minus [(11990110158401199061015840119894)119906 119899119895 + (11990110158401199061015840119895)119906 119899119894] Σ119891

+ [[11990110158402119906 (119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891 = 119876119887

119894119895

= (11990610158401198941199061015840119895)119887 120588119882 minus [(11990110158401199061015840119894)119887 119899119895 + (11990110158401199061015840119895)119887 119899119894] Σ119891+ [[11990110158402119887(119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891

(40)







120588119882 = 120588119906119878119871Σ (41a)


119904Σ] + (119878nabla sdot 997888rarr119899)

119904Σ

(41b)





119865)


119865

+ 2120588119906119878119871119870119882

(42)





(43)





120591119894119895119906 = minus120588119906 (11990610158401198941199061015840119895)119906= ]119905119906 (120597119906119894119906120597119909119895 +

120597119906119895119906120597119909119894 ) minus (23) 120588119906119896119906120575119894119895

(44a)

120591119894119895119887 = minus120588119887 (11990610158401198941199061015840119895)119887= ]119905119887 (120597119906119894119887120597119909119895 +

120597119906119895119887120597119909119894 ) minus (23) 120588119906119896119887120575119894119895

(44b)


120597 (120588)120597119905 + 120597 (120588120572)120597119909120572 = 120591119894119895120597119895120597119909119894

+ 120597 [120588 (V119905120590119896) 120597120597119909119894]120597119909119894minus 120588120576

(45a)

120597 (120588120576)120597119905 + 120597 (120588119894120576)120597119909119894 = 1198621205761120591119894119895 (120576) 120597119906119895120597119909119894

minus 1198621205762120588(1205762 )+ 120597 [120588 (V119905120590120576) 120597120576120597119909119894 ] 120597119909119894

(45b)


120591119894119895 = minus (12058811990610158401015840119894 11990610158401015840119895 ) = ]119905 ( 120597119894120597119909119895 +120597119895120597119909119894 ) minus (

23) 120588120575119894119895 (46)



+ 120597 [120588 (1 minus 119888) (V119905119906120590119896) 120597119896119906120597119909119894]120597119909119894 minus 119896119906120588119882+ 119870119906Σ119891

(47a)

120597 (120588119888119896119887)120597119905 + 120597 (120588119888119906119894119887119896119887)120597119909119894= 120588119888 (120591119894119895119887120588119887) 120597119906119894119887120597119909119895 minus 120588119888120576119887

+ 120597 [120588119888 (V119905119887120590119896) 120597119896119887120597119909119894]120597119909119894 + 119896119887120588119882 + 119870119887Σ119891

(47b)


+ 120597 [120588 (1 minus 119888) (V119905119906120590119896) 120597119896119906120597119909120572]120597119909120572 minus 120576119906120588119882+ 119864119906Σ119891

(47c)

120597 (120588119888120576119887)120597119905 + 120597 (120588119888119906119894119887120576119887)120597119909119894= 1198621205761120588 (1 minus 119888) (120576119887119896119887) (120591119894119895119887120588119887) 120597119906119894119887120597119909119895minus 1198621205762120588119888( 1205762119887119896119887) +

120597 [120588119888 (V119905119906120590119896) 120597119896119887120597119909120572]120597119909120572+ 119864119887Σ119891 + 120576119887120588119882

(47d)


minus119896119906120588119882 + 119870119906Σ119891 = +119896119887120588119882 + 119870119887Σ119891 (48a)

minus120576119906120588119882 + 119864119906Σ119891 = +120576119887120588119882 + 119864119887Σ119891 (48b)










(49a)


(49b)

120597 (120588)120597119905 + 120597 (120588120572)120597119909120572 = 120591119894119895120597119895120597119909119894+ 120597 [120588 (V119905120590119896) 120597120597119909119894]120597119909119894 minus 120588120576

(49c)

120597 (120588120576)120597119905 + 120597 (120588119894120576)120597119909119894 = 1198621205761120591119894119895 (120576) 120597119906119895120597119909119894+ 120597 [120588 (V119905120590120576) 120597120576120597119909119894 ] 120597119909119894 minus 1198621205762120588(1205762 )]

(49d)

= 119896119906 (1 minus 119888) + 119896119887119888 (49e)

120576 = 120576119906 (1 minus 119888) + 120576119887119888 (49f)










( 1198601198600


2|nablaℎ| asymp

3(|nablaℎ|2)12

=4(intinfin

01198962119865 (119896) 119889119896)12

(50)


( 1198601198600

) asymp (intinfin01198962119865 (119896) 119889119896)12 (51a)

1205902119891 (119905) = (119909 minus 119909 (119905))2 = intinfin0119865 (119896 119905) 119889119896 (51b)



1205902119891 cong 2119863119905119905 (52a)

120575119905 sim (1199061015840119871119905)12 (52b)




120597119888120597119905 = 11986311990512059721198881205971199092 + 119880119905 1003816100381610038161003816100381610038161003816100381612059711988812059711990910038161003816100381610038161003816100381610038161003816 (53a)



120588119882 = 120588 = 120588119906119880119905 |nabla119888| (54)



119880119905 sim (1199061015840119878119871)12 (55a)

120591119905 le 119905 ltlt (1199061015840119878119871)2 120591119905 (55b)


(120588119882)119871= 119860119871120588119906radic11989612119906 119878119871 |nabla119888| (56)








119880119905 = 119860119906101584011986311988614 = 11986011990610158403411987812119871 120594minus1411987114 (57a)

120591119905 lt 119905 ltlt 120591119905119863119886 (57b)





11 Conclusions













Appendix








(1 minus 119888) 119906119906 + 119888 119906119887Θ = 1 (A1a)


119901119906 + (12) 1199062119906 = 119901minusinfin + 12 (A1c)


(1 minus 119888) 119906119906 + 119888 119906119887Θ = 1 (A2a)

(12 minus 119888) 1199062119906 +119888 1199062119887Θ = 12 (A2b)


119906119887 = minus120573 + (minus4120572120574 + 1205732)12


119888Θ + 1] 120573 = minus2 119888Θ 05 minus 119888

(1 minus 119888)2 120574 = 05 minus 119888

(1 minus 119888)2 minus 05

(A3)


1205881199061015840101584011988810158401015840 = 120588119888 (1 minus 119888) (119906119887 minus 119906119906) (A4a)

1205881199061015840101584011990610158401015840 = 120588119888 (119906119887 minus 119906119906)2 (A4b)



0 025 05 075 10

005

01

015

02

025

03

035

04

045

05

u

c

uUt

c(a)

0 025 05 075 10

1

2

3

4

5

6

7

8

9

u uU

tu b

Ut

ubUt

uuUt

c(b)



119906119906 = 1198911 (119888 Θ) (A5a)

119906119887 = 1198912 (119888 Θ) (A5b)

1205881199061015840101584011988810158401015840 = 1198913 (119888 Θ) (A5c)

1205881199061015840101584011990610158401015840 = 1198914 (119888 Θ) (A5d)
















1205881199061015840101584011988810158401015840 = (1205881199061015840101584011988810158401015840)119901minus119889

+ (1205881199061015840101584011988810158401015840)119905119889

= 1205881199061198801199051198913 (119888 Θ) minus 120588119863119905119889119888119889119909 (B1)



(B2)




119870119897119900119888 =100381610038161003816100381610038161003816(1205881199061015840101584011988810158401015840)119901minus11988910038161003816100381610038161003816100381610038161003816100381610038161003816(1205881199061015840101584011988810158401015840)11990511988910038161003816100381610038161003816 =

10038161003816100381610038161205881199061198801199051198913 (119888 Θ)10038161003816100381610038161003816100381610038161003816120588 (119888)1198631199051198891198881198891199091003816100381610038161003816 (B3a)

119870int =100381610038161003816100381610038161003816⟨(1205881199061015840101584011988810158401015840)119901minus119889⟩10038161003816100381610038161003816100381610038161003816100381610038161003816⟨(1205881199061015840101584011988810158401015840)119905119889⟩10038161003816100381610038161003816

= (14) 120573 [Θ minus (2Θ minus 1)12] (1198801199051199061015840 )(120575119905119871 ) (B3b)










0 02 04 06 08 1

minus005

0

005

01

015

02

025

03

035Experiments

c

[kg

m2s]

u

c

(a)

0 02 04 06 08 1

minus005

0

005

01

015

02

025

03

035Calculations

c[k

gm

2s]

u

c

(b)











0

002

0225 025 0275

1

1 1

02 03 04 026 027 028 029 03

026 027 028 029 03

X(m) x(m) X(m)

0225 025 0275x(m)

02 03 04X(m)

004

006

008

0101 03 05 07 09

012

minus006

minus004

minus002

0

08

06

04

02

0

004

002

006

minus006

minus004

minus002

0

002

004

006

0

002

004

006

008

01

012

minus0981

minus0984

minus0987

minus099

minus0993

minus0996

minus0999

３=04ＧＭ

３=04ＧＭ

３=2ＧＭ

０-０0０0

minus0987

minus0994

minus1001

minus1008

minus1015

minus1022

minus1029

０-０0０0

０-０0０0

５５0

５５0

1５５0

C

1

08

06

04

02

0

1


Ｏ＝ＯＯ0C


Ｏ＝ＯＯ0C

０-０0０0

５５0

C

３=04ＧＭ

３=2ＧＭ

３=2ＧＭ













Nomenclature





Greek Symbols





Data Availability




Acknowledgments



References































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





120597 [120588119906 (1 minus 119888) 997888rarr119906 119906]



]119906

+ 120597 (120588119887119888 119906119894119887)120597119905 + nabla


)119887


(15)






119887





119904119887Σ119891 (17)






119904119887Σ119891)


(19)





119904119906Σ119891

(20a)


119887


119904119887Σ119891

(20b)



119904119906Σ119891 (21a)









u purarru u

u psurarru su

b pbrarru b

b psbrarru sb




120588119906 119901119906 997888rarr119906 119906 120588119882997904rArr 120588119887 119901119887 997888rarr119906 119887 (22)


































(25)



(26a)


(26b)





(27a)


119887


(27b)




1003816100381610038161003816100381610038161003816 (28a)









120597 (120588119887119888)120597119905 + nabla sdot (120588119887119888997888rarr119906 119887) = 120588119882 (30b)




]]]120588119882

(30c)



]]]120588119882

(30d)





119888 = 119888[119888 + (120588119906120588119887) (1 minus 119888)] (31a)

997888119906 = 997888rarr119906 119906 (1 minus 119888) + 119906119887119888 (31b)

997888rarr119906 = 997888rarr119906 119906 (1 minus 119888) + 119906119887119888 (31c)

120588 = 120588119906[1 + 119888 (120588119906120588119887 minus 1)] (31d)

119901 = 119901119906 (1 minus 119888) + 119901119887119888 (31e)

12058811990610158401015840119894 11988810158401015840 = 120588119888 (1 minus 119888) (119906119894119887 minus 119906119894119906) (31f)


(31g)













120597 [120588 (11990610158401198941199061015840119895)]120597119905 + 120597 [120588119906119896 (11990610158401198941199061015840119895)]120597119909119896


120597119906119895120597119909119896]

minus [1199061015840119894 sdot 1205971199011015840

120597119909119895 + 1199061015840119895 sdot1205971199011015840120597119909119894]

(32)


120597119906119895119906120597119909119896 ]

minus (1 minus 119888) [(1199061015840119895 sdot 1205971199011015840

120597119909119894)119906

+ (1199061015840119894 sdot 1205971199011015840

120597119909119895)119906

]minus 119876119906

119894119895

(33a)


120597119906119895119887120597119909119896 ]

minus 119888 [(1199061015840119895 sdot 1205971199011015840

120597119909119894)119887

+ (1199061015840119894 sdot 1205971199011015840

120597119909119895)119887

] + 119876119887119894119895

(33b)

Here 119876119906119894119895 and 119876119887


119894119895


119894119895 and 119876119887119894119895 in terms of







1199061015840119894119904119906 minus (119901119904119906119899119894) + 119901119904119906119899119894 = 1199061015840119894119904119887 minus (119901119904119887119899119894) + 119901119904119887119899119894 (34a)

1199061015840119895119904119906 minus (119901119904119906119899119895) + 119901119904119906119899119895= 1199061015840119895119904119887 minus (119901119904119887119899119895) + 119901119904119887119899119895

(34b)


(119906119894119904119906 + 1199061015840119894119904119906) minus (119901119904119906119899119894)= (119906119894119904119887 + 1199061015840119894119904119887) minus (119901119904119887119899119894)

(35a)

119906119894119904119906 minus 119901119904119906119899119894 = 119906119894119904119887 minus 119901119904119887119899119894 (35b)


(11990610158401198941199061015840119895)119904119906 2 minus [(1199011199061015840119894119899119895)119904119906 + (1199011199061015840119895119899119894)119904119906] + [(1199012119899119894119899119895)119904119906 minus (119901119899119894)119904119906 (119901119899119895)119904119906]

= (11990610158401198941199061015840119895)119904119887 2 minus [(1199011199061015840119894119899119895)119904119887 + (1199011199061015840119895119899119894)119904119887] + [(1199012119899119894119899119895)119904119887 minus (119901119899119894)119904119906 (119901119899119895)119904119887]

(36)


(11990610158401198941199061015840119895)119904119906 120588119882 minus [(1199011199061015840119894119899119895)119904119906 + (1199011199061015840119895119899119894)119904119906] Σ119891+ [[(1199012119899119894119899119895)119904119906 minus (119901119899119894)119904119906 (119901119899119895)119904119906 ]

]Σ119891

= (11990610158401198941199061015840119895)119904119887 120588119882 minus [(1199011199061015840119894119899119895)119904119887 + (1199011199061015840119895119899119894)119904119887] Σ119891+ [[(1199012119899119894119899119895)119904119887 minus (119901119899119894)119904119906 (119901119899119895)119904119887 ]

]Σ119891

(37)


119876119906119894119895 = (11990610158401198941199061015840119895)119904119906 120588119882

1

minus [(1199011199061015840119894119899119895)119904119906 + (1199011199061015840119895119899119894)119904119906] Σ1198912

+ [(1199012119899119894119899119895)119904119906 minus (119901119899119894)119904119906 (119901119899119895)119904119906 ] Σ1198913

(38a)

119876119887119894119895 = (11990610158401198941199061015840119895)119904119887 120588119882

1

minus [(1199011199061015840119894119899119895)119904119887 + (1199011199061015840119895119899119894)119904119887] Σ1198912

+ [(1199012119899119894119899119895)119904119887 minus (119901119899119894)119904119887 (119901119899119895)119904119887 ] Σ1198913

(38b)




(11990610158401198941199061015840119895)119906 120588119882 minus [(11990110158401199061015840119894)119906 119899119895 + (11990110158401199061015840119895)119906 119899119894] Σ119891+ [[11990110158402119906 (119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891

= (11990610158401198941199061015840119895)119887 120588119882 minus [(11990110158401199061015840119894)119887 119899119895 + (11990110158401199061015840119895)119887 119899119894] Σ119891+ [[11990110158402119887(119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891

(39)



119876119906119894119895 = (11990610158401198941199061015840119895)119906 120588119882 minus [(11990110158401199061015840119894)119906 119899119895 + (11990110158401199061015840119895)119906 119899119894] Σ119891

+ [[11990110158402119906 (119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891 = 119876119887

119894119895

= (11990610158401198941199061015840119895)119887 120588119882 minus [(11990110158401199061015840119894)119887 119899119895 + (11990110158401199061015840119895)119887 119899119894] Σ119891+ [[11990110158402119887(119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891

(40)







120588119882 = 120588119906119878119871Σ (41a)


119904Σ] + (119878nabla sdot 997888rarr119899)

119904Σ

(41b)





119865)


119865

+ 2120588119906119878119871119870119882

(42)





(43)





120591119894119895119906 = minus120588119906 (11990610158401198941199061015840119895)119906= ]119905119906 (120597119906119894119906120597119909119895 +

120597119906119895119906120597119909119894 ) minus (23) 120588119906119896119906120575119894119895

(44a)

120591119894119895119887 = minus120588119887 (11990610158401198941199061015840119895)119887= ]119905119887 (120597119906119894119887120597119909119895 +

120597119906119895119887120597119909119894 ) minus (23) 120588119906119896119887120575119894119895

(44b)


120597 (120588)120597119905 + 120597 (120588120572)120597119909120572 = 120591119894119895120597119895120597119909119894

+ 120597 [120588 (V119905120590119896) 120597120597119909119894]120597119909119894minus 120588120576

(45a)

120597 (120588120576)120597119905 + 120597 (120588119894120576)120597119909119894 = 1198621205761120591119894119895 (120576) 120597119906119895120597119909119894

minus 1198621205762120588(1205762 )+ 120597 [120588 (V119905120590120576) 120597120576120597119909119894 ] 120597119909119894

(45b)


120591119894119895 = minus (12058811990610158401015840119894 11990610158401015840119895 ) = ]119905 ( 120597119894120597119909119895 +120597119895120597119909119894 ) minus (

23) 120588120575119894119895 (46)



+ 120597 [120588 (1 minus 119888) (V119905119906120590119896) 120597119896119906120597119909119894]120597119909119894 minus 119896119906120588119882+ 119870119906Σ119891

(47a)

120597 (120588119888119896119887)120597119905 + 120597 (120588119888119906119894119887119896119887)120597119909119894= 120588119888 (120591119894119895119887120588119887) 120597119906119894119887120597119909119895 minus 120588119888120576119887

+ 120597 [120588119888 (V119905119887120590119896) 120597119896119887120597119909119894]120597119909119894 + 119896119887120588119882 + 119870119887Σ119891

(47b)


+ 120597 [120588 (1 minus 119888) (V119905119906120590119896) 120597119896119906120597119909120572]120597119909120572 minus 120576119906120588119882+ 119864119906Σ119891

(47c)

120597 (120588119888120576119887)120597119905 + 120597 (120588119888119906119894119887120576119887)120597119909119894= 1198621205761120588 (1 minus 119888) (120576119887119896119887) (120591119894119895119887120588119887) 120597119906119894119887120597119909119895minus 1198621205762120588119888( 1205762119887119896119887) +

120597 [120588119888 (V119905119906120590119896) 120597119896119887120597119909120572]120597119909120572+ 119864119887Σ119891 + 120576119887120588119882

(47d)


minus119896119906120588119882 + 119870119906Σ119891 = +119896119887120588119882 + 119870119887Σ119891 (48a)

minus120576119906120588119882 + 119864119906Σ119891 = +120576119887120588119882 + 119864119887Σ119891 (48b)










(49a)


(49b)

120597 (120588)120597119905 + 120597 (120588120572)120597119909120572 = 120591119894119895120597119895120597119909119894+ 120597 [120588 (V119905120590119896) 120597120597119909119894]120597119909119894 minus 120588120576

(49c)

120597 (120588120576)120597119905 + 120597 (120588119894120576)120597119909119894 = 1198621205761120591119894119895 (120576) 120597119906119895120597119909119894+ 120597 [120588 (V119905120590120576) 120597120576120597119909119894 ] 120597119909119894 minus 1198621205762120588(1205762 )]

(49d)

= 119896119906 (1 minus 119888) + 119896119887119888 (49e)

120576 = 120576119906 (1 minus 119888) + 120576119887119888 (49f)










( 1198601198600


2|nablaℎ| asymp

3(|nablaℎ|2)12

=4(intinfin

01198962119865 (119896) 119889119896)12

(50)


( 1198601198600

) asymp (intinfin01198962119865 (119896) 119889119896)12 (51a)

1205902119891 (119905) = (119909 minus 119909 (119905))2 = intinfin0119865 (119896 119905) 119889119896 (51b)



1205902119891 cong 2119863119905119905 (52a)

120575119905 sim (1199061015840119871119905)12 (52b)




120597119888120597119905 = 11986311990512059721198881205971199092 + 119880119905 1003816100381610038161003816100381610038161003816100381612059711988812059711990910038161003816100381610038161003816100381610038161003816 (53a)



120588119882 = 120588 = 120588119906119880119905 |nabla119888| (54)



119880119905 sim (1199061015840119878119871)12 (55a)

120591119905 le 119905 ltlt (1199061015840119878119871)2 120591119905 (55b)


(120588119882)119871= 119860119871120588119906radic11989612119906 119878119871 |nabla119888| (56)








119880119905 = 119860119906101584011986311988614 = 11986011990610158403411987812119871 120594minus1411987114 (57a)

120591119905 lt 119905 ltlt 120591119905119863119886 (57b)





11 Conclusions













Appendix








(1 minus 119888) 119906119906 + 119888 119906119887Θ = 1 (A1a)


119901119906 + (12) 1199062119906 = 119901minusinfin + 12 (A1c)


(1 minus 119888) 119906119906 + 119888 119906119887Θ = 1 (A2a)

(12 minus 119888) 1199062119906 +119888 1199062119887Θ = 12 (A2b)


119906119887 = minus120573 + (minus4120572120574 + 1205732)12


119888Θ + 1] 120573 = minus2 119888Θ 05 minus 119888

(1 minus 119888)2 120574 = 05 minus 119888

(1 minus 119888)2 minus 05

(A3)


1205881199061015840101584011988810158401015840 = 120588119888 (1 minus 119888) (119906119887 minus 119906119906) (A4a)

1205881199061015840101584011990610158401015840 = 120588119888 (119906119887 minus 119906119906)2 (A4b)



0 025 05 075 10

005

01

015

02

025

03

035

04

045

05

u

c

uUt

c(a)

0 025 05 075 10

1

2

3

4

5

6

7

8

9

u uU

tu b

Ut

ubUt

uuUt

c(b)



119906119906 = 1198911 (119888 Θ) (A5a)

119906119887 = 1198912 (119888 Θ) (A5b)

1205881199061015840101584011988810158401015840 = 1198913 (119888 Θ) (A5c)

1205881199061015840101584011990610158401015840 = 1198914 (119888 Θ) (A5d)
















1205881199061015840101584011988810158401015840 = (1205881199061015840101584011988810158401015840)119901minus119889

+ (1205881199061015840101584011988810158401015840)119905119889

= 1205881199061198801199051198913 (119888 Θ) minus 120588119863119905119889119888119889119909 (B1)



(B2)




119870119897119900119888 =100381610038161003816100381610038161003816(1205881199061015840101584011988810158401015840)119901minus11988910038161003816100381610038161003816100381610038161003816100381610038161003816(1205881199061015840101584011988810158401015840)11990511988910038161003816100381610038161003816 =

10038161003816100381610038161205881199061198801199051198913 (119888 Θ)10038161003816100381610038161003816100381610038161003816120588 (119888)1198631199051198891198881198891199091003816100381610038161003816 (B3a)

119870int =100381610038161003816100381610038161003816⟨(1205881199061015840101584011988810158401015840)119901minus119889⟩10038161003816100381610038161003816100381610038161003816100381610038161003816⟨(1205881199061015840101584011988810158401015840)119905119889⟩10038161003816100381610038161003816

= (14) 120573 [Θ minus (2Θ minus 1)12] (1198801199051199061015840 )(120575119905119871 ) (B3b)










0 02 04 06 08 1

minus005

0

005

01

015

02

025

03

035Experiments

c

[kg

m2s]

u

c

(a)

0 02 04 06 08 1

minus005

0

005

01

015

02

025

03

035Calculations

c[k

gm

2s]

u

c

(b)











0

002

0225 025 0275

1

1 1

02 03 04 026 027 028 029 03

026 027 028 029 03

X(m) x(m) X(m)

0225 025 0275x(m)

02 03 04X(m)

004

006

008

0101 03 05 07 09

012

minus006

minus004

minus002

0

08

06

04

02

0

004

002

006

minus006

minus004

minus002

0

002

004

006

0

002

004

006

008

01

012

minus0981

minus0984

minus0987

minus099

minus0993

minus0996

minus0999

３=04ＧＭ

３=04ＧＭ

３=2ＧＭ

０-０0０0

minus0987

minus0994

minus1001

minus1008

minus1015

minus1022

minus1029

０-０0０0

０-０0０0

５５0

５５0

1５５0

C

1

08

06

04

02

0

1


Ｏ＝ＯＯ0C


Ｏ＝ＯＯ0C

０-０0０0

５５0

C

３=04ＧＭ

３=2ＧＭ

３=2ＧＭ













Nomenclature





Greek Symbols





Data Availability




Acknowledgments



References































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





u purarru u

u psurarru su

b pbrarru b

b psbrarru sb




120588119906 119901119906 997888rarr119906 119906 120588119882997904rArr 120588119887 119901119887 997888rarr119906 119887 (22)


































(25)



(26a)


(26b)





(27a)


119887


(27b)




1003816100381610038161003816100381610038161003816 (28a)









120597 (120588119887119888)120597119905 + nabla sdot (120588119887119888997888rarr119906 119887) = 120588119882 (30b)




]]]120588119882

(30c)



]]]120588119882

(30d)





119888 = 119888[119888 + (120588119906120588119887) (1 minus 119888)] (31a)

997888119906 = 997888rarr119906 119906 (1 minus 119888) + 119906119887119888 (31b)

997888rarr119906 = 997888rarr119906 119906 (1 minus 119888) + 119906119887119888 (31c)

120588 = 120588119906[1 + 119888 (120588119906120588119887 minus 1)] (31d)

119901 = 119901119906 (1 minus 119888) + 119901119887119888 (31e)

12058811990610158401015840119894 11988810158401015840 = 120588119888 (1 minus 119888) (119906119894119887 minus 119906119894119906) (31f)


(31g)













120597 [120588 (11990610158401198941199061015840119895)]120597119905 + 120597 [120588119906119896 (11990610158401198941199061015840119895)]120597119909119896


120597119906119895120597119909119896]

minus [1199061015840119894 sdot 1205971199011015840

120597119909119895 + 1199061015840119895 sdot1205971199011015840120597119909119894]

(32)


120597119906119895119906120597119909119896 ]

minus (1 minus 119888) [(1199061015840119895 sdot 1205971199011015840

120597119909119894)119906

+ (1199061015840119894 sdot 1205971199011015840

120597119909119895)119906

]minus 119876119906

119894119895

(33a)


120597119906119895119887120597119909119896 ]

minus 119888 [(1199061015840119895 sdot 1205971199011015840

120597119909119894)119887

+ (1199061015840119894 sdot 1205971199011015840

120597119909119895)119887

] + 119876119887119894119895

(33b)

Here 119876119906119894119895 and 119876119887


119894119895


119894119895 and 119876119887119894119895 in terms of







1199061015840119894119904119906 minus (119901119904119906119899119894) + 119901119904119906119899119894 = 1199061015840119894119904119887 minus (119901119904119887119899119894) + 119901119904119887119899119894 (34a)

1199061015840119895119904119906 minus (119901119904119906119899119895) + 119901119904119906119899119895= 1199061015840119895119904119887 minus (119901119904119887119899119895) + 119901119904119887119899119895

(34b)


(119906119894119904119906 + 1199061015840119894119904119906) minus (119901119904119906119899119894)= (119906119894119904119887 + 1199061015840119894119904119887) minus (119901119904119887119899119894)

(35a)

119906119894119904119906 minus 119901119904119906119899119894 = 119906119894119904119887 minus 119901119904119887119899119894 (35b)


(11990610158401198941199061015840119895)119904119906 2 minus [(1199011199061015840119894119899119895)119904119906 + (1199011199061015840119895119899119894)119904119906] + [(1199012119899119894119899119895)119904119906 minus (119901119899119894)119904119906 (119901119899119895)119904119906]

= (11990610158401198941199061015840119895)119904119887 2 minus [(1199011199061015840119894119899119895)119904119887 + (1199011199061015840119895119899119894)119904119887] + [(1199012119899119894119899119895)119904119887 minus (119901119899119894)119904119906 (119901119899119895)119904119887]

(36)


(11990610158401198941199061015840119895)119904119906 120588119882 minus [(1199011199061015840119894119899119895)119904119906 + (1199011199061015840119895119899119894)119904119906] Σ119891+ [[(1199012119899119894119899119895)119904119906 minus (119901119899119894)119904119906 (119901119899119895)119904119906 ]

]Σ119891

= (11990610158401198941199061015840119895)119904119887 120588119882 minus [(1199011199061015840119894119899119895)119904119887 + (1199011199061015840119895119899119894)119904119887] Σ119891+ [[(1199012119899119894119899119895)119904119887 minus (119901119899119894)119904119906 (119901119899119895)119904119887 ]

]Σ119891

(37)


119876119906119894119895 = (11990610158401198941199061015840119895)119904119906 120588119882

1

minus [(1199011199061015840119894119899119895)119904119906 + (1199011199061015840119895119899119894)119904119906] Σ1198912

+ [(1199012119899119894119899119895)119904119906 minus (119901119899119894)119904119906 (119901119899119895)119904119906 ] Σ1198913

(38a)

119876119887119894119895 = (11990610158401198941199061015840119895)119904119887 120588119882

1

minus [(1199011199061015840119894119899119895)119904119887 + (1199011199061015840119895119899119894)119904119887] Σ1198912

+ [(1199012119899119894119899119895)119904119887 minus (119901119899119894)119904119887 (119901119899119895)119904119887 ] Σ1198913

(38b)




(11990610158401198941199061015840119895)119906 120588119882 minus [(11990110158401199061015840119894)119906 119899119895 + (11990110158401199061015840119895)119906 119899119894] Σ119891+ [[11990110158402119906 (119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891

= (11990610158401198941199061015840119895)119887 120588119882 minus [(11990110158401199061015840119894)119887 119899119895 + (11990110158401199061015840119895)119887 119899119894] Σ119891+ [[11990110158402119887(119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891

(39)



119876119906119894119895 = (11990610158401198941199061015840119895)119906 120588119882 minus [(11990110158401199061015840119894)119906 119899119895 + (11990110158401199061015840119895)119906 119899119894] Σ119891

+ [[11990110158402119906 (119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891 = 119876119887

119894119895

= (11990610158401198941199061015840119895)119887 120588119882 minus [(11990110158401199061015840119894)119887 119899119895 + (11990110158401199061015840119895)119887 119899119894] Σ119891+ [[11990110158402119887(119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891

(40)







120588119882 = 120588119906119878119871Σ (41a)


119904Σ] + (119878nabla sdot 997888rarr119899)

119904Σ

(41b)





119865)


119865

+ 2120588119906119878119871119870119882

(42)





(43)





120591119894119895119906 = minus120588119906 (11990610158401198941199061015840119895)119906= ]119905119906 (120597119906119894119906120597119909119895 +

120597119906119895119906120597119909119894 ) minus (23) 120588119906119896119906120575119894119895

(44a)

120591119894119895119887 = minus120588119887 (11990610158401198941199061015840119895)119887= ]119905119887 (120597119906119894119887120597119909119895 +

120597119906119895119887120597119909119894 ) minus (23) 120588119906119896119887120575119894119895

(44b)


120597 (120588)120597119905 + 120597 (120588120572)120597119909120572 = 120591119894119895120597119895120597119909119894

+ 120597 [120588 (V119905120590119896) 120597120597119909119894]120597119909119894minus 120588120576

(45a)

120597 (120588120576)120597119905 + 120597 (120588119894120576)120597119909119894 = 1198621205761120591119894119895 (120576) 120597119906119895120597119909119894

minus 1198621205762120588(1205762 )+ 120597 [120588 (V119905120590120576) 120597120576120597119909119894 ] 120597119909119894

(45b)


120591119894119895 = minus (12058811990610158401015840119894 11990610158401015840119895 ) = ]119905 ( 120597119894120597119909119895 +120597119895120597119909119894 ) minus (

23) 120588120575119894119895 (46)



+ 120597 [120588 (1 minus 119888) (V119905119906120590119896) 120597119896119906120597119909119894]120597119909119894 minus 119896119906120588119882+ 119870119906Σ119891

(47a)

120597 (120588119888119896119887)120597119905 + 120597 (120588119888119906119894119887119896119887)120597119909119894= 120588119888 (120591119894119895119887120588119887) 120597119906119894119887120597119909119895 minus 120588119888120576119887

+ 120597 [120588119888 (V119905119887120590119896) 120597119896119887120597119909119894]120597119909119894 + 119896119887120588119882 + 119870119887Σ119891

(47b)


+ 120597 [120588 (1 minus 119888) (V119905119906120590119896) 120597119896119906120597119909120572]120597119909120572 minus 120576119906120588119882+ 119864119906Σ119891

(47c)

120597 (120588119888120576119887)120597119905 + 120597 (120588119888119906119894119887120576119887)120597119909119894= 1198621205761120588 (1 minus 119888) (120576119887119896119887) (120591119894119895119887120588119887) 120597119906119894119887120597119909119895minus 1198621205762120588119888( 1205762119887119896119887) +

120597 [120588119888 (V119905119906120590119896) 120597119896119887120597119909120572]120597119909120572+ 119864119887Σ119891 + 120576119887120588119882

(47d)


minus119896119906120588119882 + 119870119906Σ119891 = +119896119887120588119882 + 119870119887Σ119891 (48a)

minus120576119906120588119882 + 119864119906Σ119891 = +120576119887120588119882 + 119864119887Σ119891 (48b)










(49a)


(49b)

120597 (120588)120597119905 + 120597 (120588120572)120597119909120572 = 120591119894119895120597119895120597119909119894+ 120597 [120588 (V119905120590119896) 120597120597119909119894]120597119909119894 minus 120588120576

(49c)

120597 (120588120576)120597119905 + 120597 (120588119894120576)120597119909119894 = 1198621205761120591119894119895 (120576) 120597119906119895120597119909119894+ 120597 [120588 (V119905120590120576) 120597120576120597119909119894 ] 120597119909119894 minus 1198621205762120588(1205762 )]

(49d)

= 119896119906 (1 minus 119888) + 119896119887119888 (49e)

120576 = 120576119906 (1 minus 119888) + 120576119887119888 (49f)










( 1198601198600


2|nablaℎ| asymp

3(|nablaℎ|2)12

=4(intinfin

01198962119865 (119896) 119889119896)12

(50)


( 1198601198600

) asymp (intinfin01198962119865 (119896) 119889119896)12 (51a)

1205902119891 (119905) = (119909 minus 119909 (119905))2 = intinfin0119865 (119896 119905) 119889119896 (51b)



1205902119891 cong 2119863119905119905 (52a)

120575119905 sim (1199061015840119871119905)12 (52b)




120597119888120597119905 = 11986311990512059721198881205971199092 + 119880119905 1003816100381610038161003816100381610038161003816100381612059711988812059711990910038161003816100381610038161003816100381610038161003816 (53a)



120588119882 = 120588 = 120588119906119880119905 |nabla119888| (54)



119880119905 sim (1199061015840119878119871)12 (55a)

120591119905 le 119905 ltlt (1199061015840119878119871)2 120591119905 (55b)


(120588119882)119871= 119860119871120588119906radic11989612119906 119878119871 |nabla119888| (56)








119880119905 = 119860119906101584011986311988614 = 11986011990610158403411987812119871 120594minus1411987114 (57a)

120591119905 lt 119905 ltlt 120591119905119863119886 (57b)





11 Conclusions













Appendix








(1 minus 119888) 119906119906 + 119888 119906119887Θ = 1 (A1a)


119901119906 + (12) 1199062119906 = 119901minusinfin + 12 (A1c)


(1 minus 119888) 119906119906 + 119888 119906119887Θ = 1 (A2a)

(12 minus 119888) 1199062119906 +119888 1199062119887Θ = 12 (A2b)


119906119887 = minus120573 + (minus4120572120574 + 1205732)12


119888Θ + 1] 120573 = minus2 119888Θ 05 minus 119888

(1 minus 119888)2 120574 = 05 minus 119888

(1 minus 119888)2 minus 05

(A3)


1205881199061015840101584011988810158401015840 = 120588119888 (1 minus 119888) (119906119887 minus 119906119906) (A4a)

1205881199061015840101584011990610158401015840 = 120588119888 (119906119887 minus 119906119906)2 (A4b)



0 025 05 075 10

005

01

015

02

025

03

035

04

045

05

u

c

uUt

c(a)

0 025 05 075 10

1

2

3

4

5

6

7

8

9

u uU

tu b

Ut

ubUt

uuUt

c(b)



119906119906 = 1198911 (119888 Θ) (A5a)

119906119887 = 1198912 (119888 Θ) (A5b)

1205881199061015840101584011988810158401015840 = 1198913 (119888 Θ) (A5c)

1205881199061015840101584011990610158401015840 = 1198914 (119888 Θ) (A5d)
















1205881199061015840101584011988810158401015840 = (1205881199061015840101584011988810158401015840)119901minus119889

+ (1205881199061015840101584011988810158401015840)119905119889

= 1205881199061198801199051198913 (119888 Θ) minus 120588119863119905119889119888119889119909 (B1)



(B2)




119870119897119900119888 =100381610038161003816100381610038161003816(1205881199061015840101584011988810158401015840)119901minus11988910038161003816100381610038161003816100381610038161003816100381610038161003816(1205881199061015840101584011988810158401015840)11990511988910038161003816100381610038161003816 =

10038161003816100381610038161205881199061198801199051198913 (119888 Θ)10038161003816100381610038161003816100381610038161003816120588 (119888)1198631199051198891198881198891199091003816100381610038161003816 (B3a)

119870int =100381610038161003816100381610038161003816⟨(1205881199061015840101584011988810158401015840)119901minus119889⟩10038161003816100381610038161003816100381610038161003816100381610038161003816⟨(1205881199061015840101584011988810158401015840)119905119889⟩10038161003816100381610038161003816

= (14) 120573 [Θ minus (2Θ minus 1)12] (1198801199051199061015840 )(120575119905119871 ) (B3b)










0 02 04 06 08 1

minus005

0

005

01

015

02

025

03

035Experiments

c

[kg

m2s]

u

c

(a)

0 02 04 06 08 1

minus005

0

005

01

015

02

025

03

035Calculations

c[k

gm

2s]

u

c

(b)











0

002

0225 025 0275

1

1 1

02 03 04 026 027 028 029 03

026 027 028 029 03

X(m) x(m) X(m)

0225 025 0275x(m)

02 03 04X(m)

004

006

008

0101 03 05 07 09

012

minus006

minus004

minus002

0

08

06

04

02

0

004

002

006

minus006

minus004

minus002

0

002

004

006

0

002

004

006

008

01

012

minus0981

minus0984

minus0987

minus099

minus0993

minus0996

minus0999

３=04ＧＭ

３=04ＧＭ

３=2ＧＭ

０-０0０0

minus0987

minus0994

minus1001

minus1008

minus1015

minus1022

minus1029

０-０0０0

０-０0０0

５５0

５５0

1５５0

C

1

08

06

04

02

0

1


Ｏ＝ＯＯ0C


Ｏ＝ＯＯ0C

０-０0０0

５５0

C

３=04ＧＭ

３=2ＧＭ

３=2ＧＭ













Nomenclature





Greek Symbols





Data Availability




Acknowledgments



References































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


























(25)



(26a)


(26b)





(27a)


119887


(27b)




1003816100381610038161003816100381610038161003816 (28a)









120597 (120588119887119888)120597119905 + nabla sdot (120588119887119888997888rarr119906 119887) = 120588119882 (30b)




]]]120588119882

(30c)



]]]120588119882

(30d)





119888 = 119888[119888 + (120588119906120588119887) (1 minus 119888)] (31a)

997888119906 = 997888rarr119906 119906 (1 minus 119888) + 119906119887119888 (31b)

997888rarr119906 = 997888rarr119906 119906 (1 minus 119888) + 119906119887119888 (31c)

120588 = 120588119906[1 + 119888 (120588119906120588119887 minus 1)] (31d)

119901 = 119901119906 (1 minus 119888) + 119901119887119888 (31e)

12058811990610158401015840119894 11988810158401015840 = 120588119888 (1 minus 119888) (119906119894119887 minus 119906119894119906) (31f)


(31g)













120597 [120588 (11990610158401198941199061015840119895)]120597119905 + 120597 [120588119906119896 (11990610158401198941199061015840119895)]120597119909119896


120597119906119895120597119909119896]

minus [1199061015840119894 sdot 1205971199011015840

120597119909119895 + 1199061015840119895 sdot1205971199011015840120597119909119894]

(32)


120597119906119895119906120597119909119896 ]

minus (1 minus 119888) [(1199061015840119895 sdot 1205971199011015840

120597119909119894)119906

+ (1199061015840119894 sdot 1205971199011015840

120597119909119895)119906

]minus 119876119906

119894119895

(33a)


120597119906119895119887120597119909119896 ]

minus 119888 [(1199061015840119895 sdot 1205971199011015840

120597119909119894)119887

+ (1199061015840119894 sdot 1205971199011015840

120597119909119895)119887

] + 119876119887119894119895

(33b)

Here 119876119906119894119895 and 119876119887


119894119895


119894119895 and 119876119887119894119895 in terms of







1199061015840119894119904119906 minus (119901119904119906119899119894) + 119901119904119906119899119894 = 1199061015840119894119904119887 minus (119901119904119887119899119894) + 119901119904119887119899119894 (34a)

1199061015840119895119904119906 minus (119901119904119906119899119895) + 119901119904119906119899119895= 1199061015840119895119904119887 minus (119901119904119887119899119895) + 119901119904119887119899119895

(34b)


(119906119894119904119906 + 1199061015840119894119904119906) minus (119901119904119906119899119894)= (119906119894119904119887 + 1199061015840119894119904119887) minus (119901119904119887119899119894)

(35a)

119906119894119904119906 minus 119901119904119906119899119894 = 119906119894119904119887 minus 119901119904119887119899119894 (35b)


(11990610158401198941199061015840119895)119904119906 2 minus [(1199011199061015840119894119899119895)119904119906 + (1199011199061015840119895119899119894)119904119906] + [(1199012119899119894119899119895)119904119906 minus (119901119899119894)119904119906 (119901119899119895)119904119906]

= (11990610158401198941199061015840119895)119904119887 2 minus [(1199011199061015840119894119899119895)119904119887 + (1199011199061015840119895119899119894)119904119887] + [(1199012119899119894119899119895)119904119887 minus (119901119899119894)119904119906 (119901119899119895)119904119887]

(36)


(11990610158401198941199061015840119895)119904119906 120588119882 minus [(1199011199061015840119894119899119895)119904119906 + (1199011199061015840119895119899119894)119904119906] Σ119891+ [[(1199012119899119894119899119895)119904119906 minus (119901119899119894)119904119906 (119901119899119895)119904119906 ]

]Σ119891

= (11990610158401198941199061015840119895)119904119887 120588119882 minus [(1199011199061015840119894119899119895)119904119887 + (1199011199061015840119895119899119894)119904119887] Σ119891+ [[(1199012119899119894119899119895)119904119887 minus (119901119899119894)119904119906 (119901119899119895)119904119887 ]

]Σ119891

(37)


119876119906119894119895 = (11990610158401198941199061015840119895)119904119906 120588119882

1

minus [(1199011199061015840119894119899119895)119904119906 + (1199011199061015840119895119899119894)119904119906] Σ1198912

+ [(1199012119899119894119899119895)119904119906 minus (119901119899119894)119904119906 (119901119899119895)119904119906 ] Σ1198913

(38a)

119876119887119894119895 = (11990610158401198941199061015840119895)119904119887 120588119882

1

minus [(1199011199061015840119894119899119895)119904119887 + (1199011199061015840119895119899119894)119904119887] Σ1198912

+ [(1199012119899119894119899119895)119904119887 minus (119901119899119894)119904119887 (119901119899119895)119904119887 ] Σ1198913

(38b)




(11990610158401198941199061015840119895)119906 120588119882 minus [(11990110158401199061015840119894)119906 119899119895 + (11990110158401199061015840119895)119906 119899119894] Σ119891+ [[11990110158402119906 (119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891

= (11990610158401198941199061015840119895)119887 120588119882 minus [(11990110158401199061015840119894)119887 119899119895 + (11990110158401199061015840119895)119887 119899119894] Σ119891+ [[11990110158402119887(119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891

(39)



119876119906119894119895 = (11990610158401198941199061015840119895)119906 120588119882 minus [(11990110158401199061015840119894)119906 119899119895 + (11990110158401199061015840119895)119906 119899119894] Σ119891

+ [[11990110158402119906 (119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891 = 119876119887

119894119895

= (11990610158401198941199061015840119895)119887 120588119882 minus [(11990110158401199061015840119894)119887 119899119895 + (11990110158401199061015840119895)119887 119899119894] Σ119891+ [[11990110158402119887(119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891

(40)







120588119882 = 120588119906119878119871Σ (41a)


119904Σ] + (119878nabla sdot 997888rarr119899)

119904Σ

(41b)





119865)


119865

+ 2120588119906119878119871119870119882

(42)





(43)





120591119894119895119906 = minus120588119906 (11990610158401198941199061015840119895)119906= ]119905119906 (120597119906119894119906120597119909119895 +

120597119906119895119906120597119909119894 ) minus (23) 120588119906119896119906120575119894119895

(44a)

120591119894119895119887 = minus120588119887 (11990610158401198941199061015840119895)119887= ]119905119887 (120597119906119894119887120597119909119895 +

120597119906119895119887120597119909119894 ) minus (23) 120588119906119896119887120575119894119895

(44b)


120597 (120588)120597119905 + 120597 (120588120572)120597119909120572 = 120591119894119895120597119895120597119909119894

+ 120597 [120588 (V119905120590119896) 120597120597119909119894]120597119909119894minus 120588120576

(45a)

120597 (120588120576)120597119905 + 120597 (120588119894120576)120597119909119894 = 1198621205761120591119894119895 (120576) 120597119906119895120597119909119894

minus 1198621205762120588(1205762 )+ 120597 [120588 (V119905120590120576) 120597120576120597119909119894 ] 120597119909119894

(45b)


120591119894119895 = minus (12058811990610158401015840119894 11990610158401015840119895 ) = ]119905 ( 120597119894120597119909119895 +120597119895120597119909119894 ) minus (

23) 120588120575119894119895 (46)



+ 120597 [120588 (1 minus 119888) (V119905119906120590119896) 120597119896119906120597119909119894]120597119909119894 minus 119896119906120588119882+ 119870119906Σ119891

(47a)

120597 (120588119888119896119887)120597119905 + 120597 (120588119888119906119894119887119896119887)120597119909119894= 120588119888 (120591119894119895119887120588119887) 120597119906119894119887120597119909119895 minus 120588119888120576119887

+ 120597 [120588119888 (V119905119887120590119896) 120597119896119887120597119909119894]120597119909119894 + 119896119887120588119882 + 119870119887Σ119891

(47b)


+ 120597 [120588 (1 minus 119888) (V119905119906120590119896) 120597119896119906120597119909120572]120597119909120572 minus 120576119906120588119882+ 119864119906Σ119891

(47c)

120597 (120588119888120576119887)120597119905 + 120597 (120588119888119906119894119887120576119887)120597119909119894= 1198621205761120588 (1 minus 119888) (120576119887119896119887) (120591119894119895119887120588119887) 120597119906119894119887120597119909119895minus 1198621205762120588119888( 1205762119887119896119887) +

120597 [120588119888 (V119905119906120590119896) 120597119896119887120597119909120572]120597119909120572+ 119864119887Σ119891 + 120576119887120588119882

(47d)


minus119896119906120588119882 + 119870119906Σ119891 = +119896119887120588119882 + 119870119887Σ119891 (48a)

minus120576119906120588119882 + 119864119906Σ119891 = +120576119887120588119882 + 119864119887Σ119891 (48b)










(49a)


(49b)

120597 (120588)120597119905 + 120597 (120588120572)120597119909120572 = 120591119894119895120597119895120597119909119894+ 120597 [120588 (V119905120590119896) 120597120597119909119894]120597119909119894 minus 120588120576

(49c)

120597 (120588120576)120597119905 + 120597 (120588119894120576)120597119909119894 = 1198621205761120591119894119895 (120576) 120597119906119895120597119909119894+ 120597 [120588 (V119905120590120576) 120597120576120597119909119894 ] 120597119909119894 minus 1198621205762120588(1205762 )]

(49d)

= 119896119906 (1 minus 119888) + 119896119887119888 (49e)

120576 = 120576119906 (1 minus 119888) + 120576119887119888 (49f)










( 1198601198600


2|nablaℎ| asymp

3(|nablaℎ|2)12

=4(intinfin

01198962119865 (119896) 119889119896)12

(50)


( 1198601198600

) asymp (intinfin01198962119865 (119896) 119889119896)12 (51a)

1205902119891 (119905) = (119909 minus 119909 (119905))2 = intinfin0119865 (119896 119905) 119889119896 (51b)



1205902119891 cong 2119863119905119905 (52a)

120575119905 sim (1199061015840119871119905)12 (52b)




120597119888120597119905 = 11986311990512059721198881205971199092 + 119880119905 1003816100381610038161003816100381610038161003816100381612059711988812059711990910038161003816100381610038161003816100381610038161003816 (53a)



120588119882 = 120588 = 120588119906119880119905 |nabla119888| (54)



119880119905 sim (1199061015840119878119871)12 (55a)

120591119905 le 119905 ltlt (1199061015840119878119871)2 120591119905 (55b)


(120588119882)119871= 119860119871120588119906radic11989612119906 119878119871 |nabla119888| (56)








119880119905 = 119860119906101584011986311988614 = 11986011990610158403411987812119871 120594minus1411987114 (57a)

120591119905 lt 119905 ltlt 120591119905119863119886 (57b)





11 Conclusions













Appendix








(1 minus 119888) 119906119906 + 119888 119906119887Θ = 1 (A1a)


119901119906 + (12) 1199062119906 = 119901minusinfin + 12 (A1c)


(1 minus 119888) 119906119906 + 119888 119906119887Θ = 1 (A2a)

(12 minus 119888) 1199062119906 +119888 1199062119887Θ = 12 (A2b)


119906119887 = minus120573 + (minus4120572120574 + 1205732)12


119888Θ + 1] 120573 = minus2 119888Θ 05 minus 119888

(1 minus 119888)2 120574 = 05 minus 119888

(1 minus 119888)2 minus 05

(A3)


1205881199061015840101584011988810158401015840 = 120588119888 (1 minus 119888) (119906119887 minus 119906119906) (A4a)

1205881199061015840101584011990610158401015840 = 120588119888 (119906119887 minus 119906119906)2 (A4b)



0 025 05 075 10

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01

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c(a)

0 025 05 075 10

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Ut

ubUt

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c(b)



119906119906 = 1198911 (119888 Θ) (A5a)

119906119887 = 1198912 (119888 Θ) (A5b)

1205881199061015840101584011988810158401015840 = 1198913 (119888 Θ) (A5c)

1205881199061015840101584011990610158401015840 = 1198914 (119888 Θ) (A5d)
















1205881199061015840101584011988810158401015840 = (1205881199061015840101584011988810158401015840)119901minus119889

+ (1205881199061015840101584011988810158401015840)119905119889

= 1205881199061198801199051198913 (119888 Θ) minus 120588119863119905119889119888119889119909 (B1)



(B2)




119870119897119900119888 =100381610038161003816100381610038161003816(1205881199061015840101584011988810158401015840)119901minus11988910038161003816100381610038161003816100381610038161003816100381610038161003816(1205881199061015840101584011988810158401015840)11990511988910038161003816100381610038161003816 =

10038161003816100381610038161205881199061198801199051198913 (119888 Θ)10038161003816100381610038161003816100381610038161003816120588 (119888)1198631199051198891198881198891199091003816100381610038161003816 (B3a)

119870int =100381610038161003816100381610038161003816⟨(1205881199061015840101584011988810158401015840)119901minus119889⟩10038161003816100381610038161003816100381610038161003816100381610038161003816⟨(1205881199061015840101584011988810158401015840)119905119889⟩10038161003816100381610038161003816

= (14) 120573 [Θ minus (2Θ minus 1)12] (1198801199051199061015840 )(120575119905119871 ) (B3b)










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(a)

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0

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0225 025 0275

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X(m) x(m) X(m)

0225 025 0275x(m)

02 03 04X(m)

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minus004

minus002

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minus002

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minus0981

minus0984

minus0987

minus099

minus0993

minus0996

minus0999

３=04ＧＭ

３=04ＧＭ

３=2ＧＭ

０-０0０0

minus0987

minus0994

minus1001

minus1008

minus1015

minus1022

minus1029

０-０0０0

０-０0０0

５５0

５５0

1５５0

C

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Ｏ＝ＯＯ0C


Ｏ＝ＯＯ0C

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C

３=04ＧＭ

３=2ＧＭ

３=2ＧＭ













Nomenclature





Greek Symbols





Data Availability




Acknowledgments



References































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






120597 (120588119887119888)120597119905 + nabla sdot (120588119887119888997888rarr119906 119887) = 120588119882 (30b)




]]]120588119882

(30c)



]]]120588119882

(30d)





119888 = 119888[119888 + (120588119906120588119887) (1 minus 119888)] (31a)

997888119906 = 997888rarr119906 119906 (1 minus 119888) + 119906119887119888 (31b)

997888rarr119906 = 997888rarr119906 119906 (1 minus 119888) + 119906119887119888 (31c)

120588 = 120588119906[1 + 119888 (120588119906120588119887 minus 1)] (31d)

119901 = 119901119906 (1 minus 119888) + 119901119887119888 (31e)

12058811990610158401015840119894 11988810158401015840 = 120588119888 (1 minus 119888) (119906119894119887 minus 119906119894119906) (31f)


(31g)













120597 [120588 (11990610158401198941199061015840119895)]120597119905 + 120597 [120588119906119896 (11990610158401198941199061015840119895)]120597119909119896


120597119906119895120597119909119896]

minus [1199061015840119894 sdot 1205971199011015840

120597119909119895 + 1199061015840119895 sdot1205971199011015840120597119909119894]

(32)


120597119906119895119906120597119909119896 ]

minus (1 minus 119888) [(1199061015840119895 sdot 1205971199011015840

120597119909119894)119906

+ (1199061015840119894 sdot 1205971199011015840

120597119909119895)119906

]minus 119876119906

119894119895

(33a)


120597119906119895119887120597119909119896 ]

minus 119888 [(1199061015840119895 sdot 1205971199011015840

120597119909119894)119887

+ (1199061015840119894 sdot 1205971199011015840

120597119909119895)119887

] + 119876119887119894119895

(33b)

Here 119876119906119894119895 and 119876119887


119894119895


119894119895 and 119876119887119894119895 in terms of







1199061015840119894119904119906 minus (119901119904119906119899119894) + 119901119904119906119899119894 = 1199061015840119894119904119887 minus (119901119904119887119899119894) + 119901119904119887119899119894 (34a)

1199061015840119895119904119906 minus (119901119904119906119899119895) + 119901119904119906119899119895= 1199061015840119895119904119887 minus (119901119904119887119899119895) + 119901119904119887119899119895

(34b)


(119906119894119904119906 + 1199061015840119894119904119906) minus (119901119904119906119899119894)= (119906119894119904119887 + 1199061015840119894119904119887) minus (119901119904119887119899119894)

(35a)

119906119894119904119906 minus 119901119904119906119899119894 = 119906119894119904119887 minus 119901119904119887119899119894 (35b)


(11990610158401198941199061015840119895)119904119906 2 minus [(1199011199061015840119894119899119895)119904119906 + (1199011199061015840119895119899119894)119904119906] + [(1199012119899119894119899119895)119904119906 minus (119901119899119894)119904119906 (119901119899119895)119904119906]

= (11990610158401198941199061015840119895)119904119887 2 minus [(1199011199061015840119894119899119895)119904119887 + (1199011199061015840119895119899119894)119904119887] + [(1199012119899119894119899119895)119904119887 minus (119901119899119894)119904119906 (119901119899119895)119904119887]

(36)


(11990610158401198941199061015840119895)119904119906 120588119882 minus [(1199011199061015840119894119899119895)119904119906 + (1199011199061015840119895119899119894)119904119906] Σ119891+ [[(1199012119899119894119899119895)119904119906 minus (119901119899119894)119904119906 (119901119899119895)119904119906 ]

]Σ119891

= (11990610158401198941199061015840119895)119904119887 120588119882 minus [(1199011199061015840119894119899119895)119904119887 + (1199011199061015840119895119899119894)119904119887] Σ119891+ [[(1199012119899119894119899119895)119904119887 minus (119901119899119894)119904119906 (119901119899119895)119904119887 ]

]Σ119891

(37)


119876119906119894119895 = (11990610158401198941199061015840119895)119904119906 120588119882

1

minus [(1199011199061015840119894119899119895)119904119906 + (1199011199061015840119895119899119894)119904119906] Σ1198912

+ [(1199012119899119894119899119895)119904119906 minus (119901119899119894)119904119906 (119901119899119895)119904119906 ] Σ1198913

(38a)

119876119887119894119895 = (11990610158401198941199061015840119895)119904119887 120588119882

1

minus [(1199011199061015840119894119899119895)119904119887 + (1199011199061015840119895119899119894)119904119887] Σ1198912

+ [(1199012119899119894119899119895)119904119887 minus (119901119899119894)119904119887 (119901119899119895)119904119887 ] Σ1198913

(38b)




(11990610158401198941199061015840119895)119906 120588119882 minus [(11990110158401199061015840119894)119906 119899119895 + (11990110158401199061015840119895)119906 119899119894] Σ119891+ [[11990110158402119906 (119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891

= (11990610158401198941199061015840119895)119887 120588119882 minus [(11990110158401199061015840119894)119887 119899119895 + (11990110158401199061015840119895)119887 119899119894] Σ119891+ [[11990110158402119887(119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891

(39)



119876119906119894119895 = (11990610158401198941199061015840119895)119906 120588119882 minus [(11990110158401199061015840119894)119906 119899119895 + (11990110158401199061015840119895)119906 119899119894] Σ119891

+ [[11990110158402119906 (119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891 = 119876119887

119894119895

= (11990610158401198941199061015840119895)119887 120588119882 minus [(11990110158401199061015840119894)119887 119899119895 + (11990110158401199061015840119895)119887 119899119894] Σ119891+ [[11990110158402119887(119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891

(40)







120588119882 = 120588119906119878119871Σ (41a)


119904Σ] + (119878nabla sdot 997888rarr119899)

119904Σ

(41b)





119865)


119865

+ 2120588119906119878119871119870119882

(42)





(43)





120591119894119895119906 = minus120588119906 (11990610158401198941199061015840119895)119906= ]119905119906 (120597119906119894119906120597119909119895 +

120597119906119895119906120597119909119894 ) minus (23) 120588119906119896119906120575119894119895

(44a)

120591119894119895119887 = minus120588119887 (11990610158401198941199061015840119895)119887= ]119905119887 (120597119906119894119887120597119909119895 +

120597119906119895119887120597119909119894 ) minus (23) 120588119906119896119887120575119894119895

(44b)


120597 (120588)120597119905 + 120597 (120588120572)120597119909120572 = 120591119894119895120597119895120597119909119894

+ 120597 [120588 (V119905120590119896) 120597120597119909119894]120597119909119894minus 120588120576

(45a)

120597 (120588120576)120597119905 + 120597 (120588119894120576)120597119909119894 = 1198621205761120591119894119895 (120576) 120597119906119895120597119909119894

minus 1198621205762120588(1205762 )+ 120597 [120588 (V119905120590120576) 120597120576120597119909119894 ] 120597119909119894

(45b)


120591119894119895 = minus (12058811990610158401015840119894 11990610158401015840119895 ) = ]119905 ( 120597119894120597119909119895 +120597119895120597119909119894 ) minus (

23) 120588120575119894119895 (46)



+ 120597 [120588 (1 minus 119888) (V119905119906120590119896) 120597119896119906120597119909119894]120597119909119894 minus 119896119906120588119882+ 119870119906Σ119891

(47a)

120597 (120588119888119896119887)120597119905 + 120597 (120588119888119906119894119887119896119887)120597119909119894= 120588119888 (120591119894119895119887120588119887) 120597119906119894119887120597119909119895 minus 120588119888120576119887

+ 120597 [120588119888 (V119905119887120590119896) 120597119896119887120597119909119894]120597119909119894 + 119896119887120588119882 + 119870119887Σ119891

(47b)


+ 120597 [120588 (1 minus 119888) (V119905119906120590119896) 120597119896119906120597119909120572]120597119909120572 minus 120576119906120588119882+ 119864119906Σ119891

(47c)

120597 (120588119888120576119887)120597119905 + 120597 (120588119888119906119894119887120576119887)120597119909119894= 1198621205761120588 (1 minus 119888) (120576119887119896119887) (120591119894119895119887120588119887) 120597119906119894119887120597119909119895minus 1198621205762120588119888( 1205762119887119896119887) +

120597 [120588119888 (V119905119906120590119896) 120597119896119887120597119909120572]120597119909120572+ 119864119887Σ119891 + 120576119887120588119882

(47d)


minus119896119906120588119882 + 119870119906Σ119891 = +119896119887120588119882 + 119870119887Σ119891 (48a)

minus120576119906120588119882 + 119864119906Σ119891 = +120576119887120588119882 + 119864119887Σ119891 (48b)










(49a)


(49b)

120597 (120588)120597119905 + 120597 (120588120572)120597119909120572 = 120591119894119895120597119895120597119909119894+ 120597 [120588 (V119905120590119896) 120597120597119909119894]120597119909119894 minus 120588120576

(49c)

120597 (120588120576)120597119905 + 120597 (120588119894120576)120597119909119894 = 1198621205761120591119894119895 (120576) 120597119906119895120597119909119894+ 120597 [120588 (V119905120590120576) 120597120576120597119909119894 ] 120597119909119894 minus 1198621205762120588(1205762 )]

(49d)

= 119896119906 (1 minus 119888) + 119896119887119888 (49e)

120576 = 120576119906 (1 minus 119888) + 120576119887119888 (49f)










( 1198601198600


2|nablaℎ| asymp

3(|nablaℎ|2)12

=4(intinfin

01198962119865 (119896) 119889119896)12

(50)


( 1198601198600

) asymp (intinfin01198962119865 (119896) 119889119896)12 (51a)

1205902119891 (119905) = (119909 minus 119909 (119905))2 = intinfin0119865 (119896 119905) 119889119896 (51b)



1205902119891 cong 2119863119905119905 (52a)

120575119905 sim (1199061015840119871119905)12 (52b)




120597119888120597119905 = 11986311990512059721198881205971199092 + 119880119905 1003816100381610038161003816100381610038161003816100381612059711988812059711990910038161003816100381610038161003816100381610038161003816 (53a)



120588119882 = 120588 = 120588119906119880119905 |nabla119888| (54)



119880119905 sim (1199061015840119878119871)12 (55a)

120591119905 le 119905 ltlt (1199061015840119878119871)2 120591119905 (55b)


(120588119882)119871= 119860119871120588119906radic11989612119906 119878119871 |nabla119888| (56)








119880119905 = 119860119906101584011986311988614 = 11986011990610158403411987812119871 120594minus1411987114 (57a)

120591119905 lt 119905 ltlt 120591119905119863119886 (57b)





11 Conclusions













Appendix








(1 minus 119888) 119906119906 + 119888 119906119887Θ = 1 (A1a)


119901119906 + (12) 1199062119906 = 119901minusinfin + 12 (A1c)


(1 minus 119888) 119906119906 + 119888 119906119887Θ = 1 (A2a)

(12 minus 119888) 1199062119906 +119888 1199062119887Θ = 12 (A2b)


119906119887 = minus120573 + (minus4120572120574 + 1205732)12


119888Θ + 1] 120573 = minus2 119888Θ 05 minus 119888

(1 minus 119888)2 120574 = 05 minus 119888

(1 minus 119888)2 minus 05

(A3)


1205881199061015840101584011988810158401015840 = 120588119888 (1 minus 119888) (119906119887 minus 119906119906) (A4a)

1205881199061015840101584011990610158401015840 = 120588119888 (119906119887 minus 119906119906)2 (A4b)



0 025 05 075 10

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0 025 05 075 10

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tu b

Ut

ubUt

uuUt

c(b)



119906119906 = 1198911 (119888 Θ) (A5a)

119906119887 = 1198912 (119888 Θ) (A5b)

1205881199061015840101584011988810158401015840 = 1198913 (119888 Θ) (A5c)

1205881199061015840101584011990610158401015840 = 1198914 (119888 Θ) (A5d)
















1205881199061015840101584011988810158401015840 = (1205881199061015840101584011988810158401015840)119901minus119889

+ (1205881199061015840101584011988810158401015840)119905119889

= 1205881199061198801199051198913 (119888 Θ) minus 120588119863119905119889119888119889119909 (B1)



(B2)




119870119897119900119888 =100381610038161003816100381610038161003816(1205881199061015840101584011988810158401015840)119901minus11988910038161003816100381610038161003816100381610038161003816100381610038161003816(1205881199061015840101584011988810158401015840)11990511988910038161003816100381610038161003816 =

10038161003816100381610038161205881199061198801199051198913 (119888 Θ)10038161003816100381610038161003816100381610038161003816120588 (119888)1198631199051198891198881198891199091003816100381610038161003816 (B3a)

119870int =100381610038161003816100381610038161003816⟨(1205881199061015840101584011988810158401015840)119901minus119889⟩10038161003816100381610038161003816100381610038161003816100381610038161003816⟨(1205881199061015840101584011988810158401015840)119905119889⟩10038161003816100381610038161003816

= (14) 120573 [Θ minus (2Θ minus 1)12] (1198801199051199061015840 )(120575119905119871 ) (B3b)










0 02 04 06 08 1

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c

[kg

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0 02 04 06 08 1

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c[k

gm

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u

c

(b)











0

002

0225 025 0275

1

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02 03 04 026 027 028 029 03

026 027 028 029 03

X(m) x(m) X(m)

0225 025 0275x(m)

02 03 04X(m)

004

006

008

0101 03 05 07 09

012

minus006

minus004

minus002

0

08

06

04

02

0

004

002

006

minus006

minus004

minus002

0

002

004

006

0

002

004

006

008

01

012

minus0981

minus0984

minus0987

minus099

minus0993

minus0996

minus0999

３=04ＧＭ

３=04ＧＭ

３=2ＧＭ

０-０0０0

minus0987

minus0994

minus1001

minus1008

minus1015

minus1022

minus1029

０-０0０0

０-０0０0

５５0

５５0

1５５0

C

1

08

06

04

02

0

1


Ｏ＝ＯＯ0C


Ｏ＝ＯＯ0C

０-０0０0

５５0

C

３=04ＧＭ

３=2ＧＭ

３=2ＧＭ













Nomenclature





Greek Symbols





Data Availability




Acknowledgments



References































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia










120597 [120588 (11990610158401198941199061015840119895)]120597119905 + 120597 [120588119906119896 (11990610158401198941199061015840119895)]120597119909119896


120597119906119895120597119909119896]

minus [1199061015840119894 sdot 1205971199011015840

120597119909119895 + 1199061015840119895 sdot1205971199011015840120597119909119894]

(32)


120597119906119895119906120597119909119896 ]

minus (1 minus 119888) [(1199061015840119895 sdot 1205971199011015840

120597119909119894)119906

+ (1199061015840119894 sdot 1205971199011015840

120597119909119895)119906

]minus 119876119906

119894119895

(33a)


120597119906119895119887120597119909119896 ]

minus 119888 [(1199061015840119895 sdot 1205971199011015840

120597119909119894)119887

+ (1199061015840119894 sdot 1205971199011015840

120597119909119895)119887

] + 119876119887119894119895

(33b)

Here 119876119906119894119895 and 119876119887


119894119895


119894119895 and 119876119887119894119895 in terms of







1199061015840119894119904119906 minus (119901119904119906119899119894) + 119901119904119906119899119894 = 1199061015840119894119904119887 minus (119901119904119887119899119894) + 119901119904119887119899119894 (34a)

1199061015840119895119904119906 minus (119901119904119906119899119895) + 119901119904119906119899119895= 1199061015840119895119904119887 minus (119901119904119887119899119895) + 119901119904119887119899119895

(34b)


(119906119894119904119906 + 1199061015840119894119904119906) minus (119901119904119906119899119894)= (119906119894119904119887 + 1199061015840119894119904119887) minus (119901119904119887119899119894)

(35a)

119906119894119904119906 minus 119901119904119906119899119894 = 119906119894119904119887 minus 119901119904119887119899119894 (35b)


(11990610158401198941199061015840119895)119904119906 2 minus [(1199011199061015840119894119899119895)119904119906 + (1199011199061015840119895119899119894)119904119906] + [(1199012119899119894119899119895)119904119906 minus (119901119899119894)119904119906 (119901119899119895)119904119906]

= (11990610158401198941199061015840119895)119904119887 2 minus [(1199011199061015840119894119899119895)119904119887 + (1199011199061015840119895119899119894)119904119887] + [(1199012119899119894119899119895)119904119887 minus (119901119899119894)119904119906 (119901119899119895)119904119887]

(36)


(11990610158401198941199061015840119895)119904119906 120588119882 minus [(1199011199061015840119894119899119895)119904119906 + (1199011199061015840119895119899119894)119904119906] Σ119891+ [[(1199012119899119894119899119895)119904119906 minus (119901119899119894)119904119906 (119901119899119895)119904119906 ]

]Σ119891

= (11990610158401198941199061015840119895)119904119887 120588119882 minus [(1199011199061015840119894119899119895)119904119887 + (1199011199061015840119895119899119894)119904119887] Σ119891+ [[(1199012119899119894119899119895)119904119887 minus (119901119899119894)119904119906 (119901119899119895)119904119887 ]

]Σ119891

(37)


119876119906119894119895 = (11990610158401198941199061015840119895)119904119906 120588119882

1

minus [(1199011199061015840119894119899119895)119904119906 + (1199011199061015840119895119899119894)119904119906] Σ1198912

+ [(1199012119899119894119899119895)119904119906 minus (119901119899119894)119904119906 (119901119899119895)119904119906 ] Σ1198913

(38a)

119876119887119894119895 = (11990610158401198941199061015840119895)119904119887 120588119882

1

minus [(1199011199061015840119894119899119895)119904119887 + (1199011199061015840119895119899119894)119904119887] Σ1198912

+ [(1199012119899119894119899119895)119904119887 minus (119901119899119894)119904119887 (119901119899119895)119904119887 ] Σ1198913

(38b)




(11990610158401198941199061015840119895)119906 120588119882 minus [(11990110158401199061015840119894)119906 119899119895 + (11990110158401199061015840119895)119906 119899119894] Σ119891+ [[11990110158402119906 (119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891

= (11990610158401198941199061015840119895)119887 120588119882 minus [(11990110158401199061015840119894)119887 119899119895 + (11990110158401199061015840119895)119887 119899119894] Σ119891+ [[11990110158402119887(119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891

(39)



119876119906119894119895 = (11990610158401198941199061015840119895)119906 120588119882 minus [(11990110158401199061015840119894)119906 119899119895 + (11990110158401199061015840119895)119906 119899119894] Σ119891

+ [[11990110158402119906 (119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891 = 119876119887

119894119895

= (11990610158401198941199061015840119895)119887 120588119882 minus [(11990110158401199061015840119894)119887 119899119895 + (11990110158401199061015840119895)119887 119899119894] Σ119891+ [[11990110158402119887(119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891

(40)







120588119882 = 120588119906119878119871Σ (41a)


119904Σ] + (119878nabla sdot 997888rarr119899)

119904Σ

(41b)





119865)


119865

+ 2120588119906119878119871119870119882

(42)





(43)





120591119894119895119906 = minus120588119906 (11990610158401198941199061015840119895)119906= ]119905119906 (120597119906119894119906120597119909119895 +

120597119906119895119906120597119909119894 ) minus (23) 120588119906119896119906120575119894119895

(44a)

120591119894119895119887 = minus120588119887 (11990610158401198941199061015840119895)119887= ]119905119887 (120597119906119894119887120597119909119895 +

120597119906119895119887120597119909119894 ) minus (23) 120588119906119896119887120575119894119895

(44b)


120597 (120588)120597119905 + 120597 (120588120572)120597119909120572 = 120591119894119895120597119895120597119909119894

+ 120597 [120588 (V119905120590119896) 120597120597119909119894]120597119909119894minus 120588120576

(45a)

120597 (120588120576)120597119905 + 120597 (120588119894120576)120597119909119894 = 1198621205761120591119894119895 (120576) 120597119906119895120597119909119894

minus 1198621205762120588(1205762 )+ 120597 [120588 (V119905120590120576) 120597120576120597119909119894 ] 120597119909119894

(45b)


120591119894119895 = minus (12058811990610158401015840119894 11990610158401015840119895 ) = ]119905 ( 120597119894120597119909119895 +120597119895120597119909119894 ) minus (

23) 120588120575119894119895 (46)



+ 120597 [120588 (1 minus 119888) (V119905119906120590119896) 120597119896119906120597119909119894]120597119909119894 minus 119896119906120588119882+ 119870119906Σ119891

(47a)

120597 (120588119888119896119887)120597119905 + 120597 (120588119888119906119894119887119896119887)120597119909119894= 120588119888 (120591119894119895119887120588119887) 120597119906119894119887120597119909119895 minus 120588119888120576119887

+ 120597 [120588119888 (V119905119887120590119896) 120597119896119887120597119909119894]120597119909119894 + 119896119887120588119882 + 119870119887Σ119891

(47b)


+ 120597 [120588 (1 minus 119888) (V119905119906120590119896) 120597119896119906120597119909120572]120597119909120572 minus 120576119906120588119882+ 119864119906Σ119891

(47c)

120597 (120588119888120576119887)120597119905 + 120597 (120588119888119906119894119887120576119887)120597119909119894= 1198621205761120588 (1 minus 119888) (120576119887119896119887) (120591119894119895119887120588119887) 120597119906119894119887120597119909119895minus 1198621205762120588119888( 1205762119887119896119887) +

120597 [120588119888 (V119905119906120590119896) 120597119896119887120597119909120572]120597119909120572+ 119864119887Σ119891 + 120576119887120588119882

(47d)


minus119896119906120588119882 + 119870119906Σ119891 = +119896119887120588119882 + 119870119887Σ119891 (48a)

minus120576119906120588119882 + 119864119906Σ119891 = +120576119887120588119882 + 119864119887Σ119891 (48b)










(49a)


(49b)

120597 (120588)120597119905 + 120597 (120588120572)120597119909120572 = 120591119894119895120597119895120597119909119894+ 120597 [120588 (V119905120590119896) 120597120597119909119894]120597119909119894 minus 120588120576

(49c)

120597 (120588120576)120597119905 + 120597 (120588119894120576)120597119909119894 = 1198621205761120591119894119895 (120576) 120597119906119895120597119909119894+ 120597 [120588 (V119905120590120576) 120597120576120597119909119894 ] 120597119909119894 minus 1198621205762120588(1205762 )]

(49d)

= 119896119906 (1 minus 119888) + 119896119887119888 (49e)

120576 = 120576119906 (1 minus 119888) + 120576119887119888 (49f)










( 1198601198600


2|nablaℎ| asymp

3(|nablaℎ|2)12

=4(intinfin

01198962119865 (119896) 119889119896)12

(50)


( 1198601198600

) asymp (intinfin01198962119865 (119896) 119889119896)12 (51a)

1205902119891 (119905) = (119909 minus 119909 (119905))2 = intinfin0119865 (119896 119905) 119889119896 (51b)



1205902119891 cong 2119863119905119905 (52a)

120575119905 sim (1199061015840119871119905)12 (52b)




120597119888120597119905 = 11986311990512059721198881205971199092 + 119880119905 1003816100381610038161003816100381610038161003816100381612059711988812059711990910038161003816100381610038161003816100381610038161003816 (53a)



120588119882 = 120588 = 120588119906119880119905 |nabla119888| (54)



119880119905 sim (1199061015840119878119871)12 (55a)

120591119905 le 119905 ltlt (1199061015840119878119871)2 120591119905 (55b)


(120588119882)119871= 119860119871120588119906radic11989612119906 119878119871 |nabla119888| (56)








119880119905 = 119860119906101584011986311988614 = 11986011990610158403411987812119871 120594minus1411987114 (57a)

120591119905 lt 119905 ltlt 120591119905119863119886 (57b)





11 Conclusions













Appendix








(1 minus 119888) 119906119906 + 119888 119906119887Θ = 1 (A1a)


119901119906 + (12) 1199062119906 = 119901minusinfin + 12 (A1c)


(1 minus 119888) 119906119906 + 119888 119906119887Θ = 1 (A2a)

(12 minus 119888) 1199062119906 +119888 1199062119887Θ = 12 (A2b)


119906119887 = minus120573 + (minus4120572120574 + 1205732)12


119888Θ + 1] 120573 = minus2 119888Θ 05 minus 119888

(1 minus 119888)2 120574 = 05 minus 119888

(1 minus 119888)2 minus 05

(A3)


1205881199061015840101584011988810158401015840 = 120588119888 (1 minus 119888) (119906119887 minus 119906119906) (A4a)

1205881199061015840101584011990610158401015840 = 120588119888 (119906119887 minus 119906119906)2 (A4b)



0 025 05 075 10

005

01

015

02

025

03

035

04

045

05

u

c

uUt

c(a)

0 025 05 075 10

1

2

3

4

5

6

7

8

9

u uU

tu b

Ut

ubUt

uuUt

c(b)



119906119906 = 1198911 (119888 Θ) (A5a)

119906119887 = 1198912 (119888 Θ) (A5b)

1205881199061015840101584011988810158401015840 = 1198913 (119888 Θ) (A5c)

1205881199061015840101584011990610158401015840 = 1198914 (119888 Θ) (A5d)
















1205881199061015840101584011988810158401015840 = (1205881199061015840101584011988810158401015840)119901minus119889

+ (1205881199061015840101584011988810158401015840)119905119889

= 1205881199061198801199051198913 (119888 Θ) minus 120588119863119905119889119888119889119909 (B1)



(B2)




119870119897119900119888 =100381610038161003816100381610038161003816(1205881199061015840101584011988810158401015840)119901minus11988910038161003816100381610038161003816100381610038161003816100381610038161003816(1205881199061015840101584011988810158401015840)11990511988910038161003816100381610038161003816 =

10038161003816100381610038161205881199061198801199051198913 (119888 Θ)10038161003816100381610038161003816100381610038161003816120588 (119888)1198631199051198891198881198891199091003816100381610038161003816 (B3a)

119870int =100381610038161003816100381610038161003816⟨(1205881199061015840101584011988810158401015840)119901minus119889⟩10038161003816100381610038161003816100381610038161003816100381610038161003816⟨(1205881199061015840101584011988810158401015840)119905119889⟩10038161003816100381610038161003816

= (14) 120573 [Θ minus (2Θ minus 1)12] (1198801199051199061015840 )(120575119905119871 ) (B3b)










0 02 04 06 08 1

minus005

0

005

01

015

02

025

03

035Experiments

c

[kg

m2s]

u

c

(a)

0 02 04 06 08 1

minus005

0

005

01

015

02

025

03

035Calculations

c[k

gm

2s]

u

c

(b)











0

002

0225 025 0275

1

1 1

02 03 04 026 027 028 029 03

026 027 028 029 03

X(m) x(m) X(m)

0225 025 0275x(m)

02 03 04X(m)

004

006

008

0101 03 05 07 09

012

minus006

minus004

minus002

0

08

06

04

02

0

004

002

006

minus006

minus004

minus002

0

002

004

006

0

002

004

006

008

01

012

minus0981

minus0984

minus0987

minus099

minus0993

minus0996

minus0999

３=04ＧＭ

３=04ＧＭ

３=2ＧＭ

０-０0０0

minus0987

minus0994

minus1001

minus1008

minus1015

minus1022

minus1029

０-０0０0

０-０0０0

５５0

５５0

1５５0

C

1

08

06

04

02

0

1


Ｏ＝ＯＯ0C


Ｏ＝ＯＯ0C

０-０0０0

５５0

C

３=04ＧＭ

３=2ＧＭ

３=2ＧＭ













Nomenclature





Greek Symbols





Data Availability




Acknowledgments



References































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




1199061015840119894119904119906 minus (119901119904119906119899119894) + 119901119904119906119899119894 = 1199061015840119894119904119887 minus (119901119904119887119899119894) + 119901119904119887119899119894 (34a)

1199061015840119895119904119906 minus (119901119904119906119899119895) + 119901119904119906119899119895= 1199061015840119895119904119887 minus (119901119904119887119899119895) + 119901119904119887119899119895

(34b)


(119906119894119904119906 + 1199061015840119894119904119906) minus (119901119904119906119899119894)= (119906119894119904119887 + 1199061015840119894119904119887) minus (119901119904119887119899119894)

(35a)

119906119894119904119906 minus 119901119904119906119899119894 = 119906119894119904119887 minus 119901119904119887119899119894 (35b)


(11990610158401198941199061015840119895)119904119906 2 minus [(1199011199061015840119894119899119895)119904119906 + (1199011199061015840119895119899119894)119904119906] + [(1199012119899119894119899119895)119904119906 minus (119901119899119894)119904119906 (119901119899119895)119904119906]

= (11990610158401198941199061015840119895)119904119887 2 minus [(1199011199061015840119894119899119895)119904119887 + (1199011199061015840119895119899119894)119904119887] + [(1199012119899119894119899119895)119904119887 minus (119901119899119894)119904119906 (119901119899119895)119904119887]

(36)


(11990610158401198941199061015840119895)119904119906 120588119882 minus [(1199011199061015840119894119899119895)119904119906 + (1199011199061015840119895119899119894)119904119906] Σ119891+ [[(1199012119899119894119899119895)119904119906 minus (119901119899119894)119904119906 (119901119899119895)119904119906 ]

]Σ119891

= (11990610158401198941199061015840119895)119904119887 120588119882 minus [(1199011199061015840119894119899119895)119904119887 + (1199011199061015840119895119899119894)119904119887] Σ119891+ [[(1199012119899119894119899119895)119904119887 minus (119901119899119894)119904119906 (119901119899119895)119904119887 ]

]Σ119891

(37)


119876119906119894119895 = (11990610158401198941199061015840119895)119904119906 120588119882

1

minus [(1199011199061015840119894119899119895)119904119906 + (1199011199061015840119895119899119894)119904119906] Σ1198912

+ [(1199012119899119894119899119895)119904119906 minus (119901119899119894)119904119906 (119901119899119895)119904119906 ] Σ1198913

(38a)

119876119887119894119895 = (11990610158401198941199061015840119895)119904119887 120588119882

1

minus [(1199011199061015840119894119899119895)119904119887 + (1199011199061015840119895119899119894)119904119887] Σ1198912

+ [(1199012119899119894119899119895)119904119887 minus (119901119899119894)119904119887 (119901119899119895)119904119887 ] Σ1198913

(38b)




(11990610158401198941199061015840119895)119906 120588119882 minus [(11990110158401199061015840119894)119906 119899119895 + (11990110158401199061015840119895)119906 119899119894] Σ119891+ [[11990110158402119906 (119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891

= (11990610158401198941199061015840119895)119887 120588119882 minus [(11990110158401199061015840119894)119887 119899119895 + (11990110158401199061015840119895)119887 119899119894] Σ119891+ [[11990110158402119887(119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891

(39)



119876119906119894119895 = (11990610158401198941199061015840119895)119906 120588119882 minus [(11990110158401199061015840119894)119906 119899119895 + (11990110158401199061015840119895)119906 119899119894] Σ119891

+ [[11990110158402119906 (119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891 = 119876119887

119894119895

= (11990610158401198941199061015840119895)119887 120588119882 minus [(11990110158401199061015840119894)119887 119899119895 + (11990110158401199061015840119895)119887 119899119894] Σ119891+ [[11990110158402119887(119899119894119899119895 + 11989910158401198941198991015840119895) ]

]Σ119891

(40)







120588119882 = 120588119906119878119871Σ (41a)


119904Σ] + (119878nabla sdot 997888rarr119899)

119904Σ

(41b)





119865)


119865

+ 2120588119906119878119871119870119882

(42)





(43)





120591119894119895119906 = minus120588119906 (11990610158401198941199061015840119895)119906= ]119905119906 (120597119906119894119906120597119909119895 +

120597119906119895119906120597119909119894 ) minus (23) 120588119906119896119906120575119894119895

(44a)

120591119894119895119887 = minus120588119887 (11990610158401198941199061015840119895)119887= ]119905119887 (120597119906119894119887120597119909119895 +

120597119906119895119887120597119909119894 ) minus (23) 120588119906119896119887120575119894119895

(44b)


120597 (120588)120597119905 + 120597 (120588120572)120597119909120572 = 120591119894119895120597119895120597119909119894

+ 120597 [120588 (V119905120590119896) 120597120597119909119894]120597119909119894minus 120588120576

(45a)

120597 (120588120576)120597119905 + 120597 (120588119894120576)120597119909119894 = 1198621205761120591119894119895 (120576) 120597119906119895120597119909119894

minus 1198621205762120588(1205762 )+ 120597 [120588 (V119905120590120576) 120597120576120597119909119894 ] 120597119909119894

(45b)


120591119894119895 = minus (12058811990610158401015840119894 11990610158401015840119895 ) = ]119905 ( 120597119894120597119909119895 +120597119895120597119909119894 ) minus (

23) 120588120575119894119895 (46)



+ 120597 [120588 (1 minus 119888) (V119905119906120590119896) 120597119896119906120597119909119894]120597119909119894 minus 119896119906120588119882+ 119870119906Σ119891

(47a)

120597 (120588119888119896119887)120597119905 + 120597 (120588119888119906119894119887119896119887)120597119909119894= 120588119888 (120591119894119895119887120588119887) 120597119906119894119887120597119909119895 minus 120588119888120576119887

+ 120597 [120588119888 (V119905119887120590119896) 120597119896119887120597119909119894]120597119909119894 + 119896119887120588119882 + 119870119887Σ119891

(47b)


+ 120597 [120588 (1 minus 119888) (V119905119906120590119896) 120597119896119906120597119909120572]120597119909120572 minus 120576119906120588119882+ 119864119906Σ119891

(47c)

120597 (120588119888120576119887)120597119905 + 120597 (120588119888119906119894119887120576119887)120597119909119894= 1198621205761120588 (1 minus 119888) (120576119887119896119887) (120591119894119895119887120588119887) 120597119906119894119887120597119909119895minus 1198621205762120588119888( 1205762119887119896119887) +

120597 [120588119888 (V119905119906120590119896) 120597119896119887120597119909120572]120597119909120572+ 119864119887Σ119891 + 120576119887120588119882

(47d)


minus119896119906120588119882 + 119870119906Σ119891 = +119896119887120588119882 + 119870119887Σ119891 (48a)

minus120576119906120588119882 + 119864119906Σ119891 = +120576119887120588119882 + 119864119887Σ119891 (48b)










(49a)


(49b)

120597 (120588)120597119905 + 120597 (120588120572)120597119909120572 = 120591119894119895120597119895120597119909119894+ 120597 [120588 (V119905120590119896) 120597120597119909119894]120597119909119894 minus 120588120576

(49c)

120597 (120588120576)120597119905 + 120597 (120588119894120576)120597119909119894 = 1198621205761120591119894119895 (120576) 120597119906119895120597119909119894+ 120597 [120588 (V119905120590120576) 120597120576120597119909119894 ] 120597119909119894 minus 1198621205762120588(1205762 )]

(49d)

= 119896119906 (1 minus 119888) + 119896119887119888 (49e)

120576 = 120576119906 (1 minus 119888) + 120576119887119888 (49f)










( 1198601198600


2|nablaℎ| asymp

3(|nablaℎ|2)12

=4(intinfin

01198962119865 (119896) 119889119896)12

(50)


( 1198601198600

) asymp (intinfin01198962119865 (119896) 119889119896)12 (51a)

1205902119891 (119905) = (119909 minus 119909 (119905))2 = intinfin0119865 (119896 119905) 119889119896 (51b)



1205902119891 cong 2119863119905119905 (52a)

120575119905 sim (1199061015840119871119905)12 (52b)




120597119888120597119905 = 11986311990512059721198881205971199092 + 119880119905 1003816100381610038161003816100381610038161003816100381612059711988812059711990910038161003816100381610038161003816100381610038161003816 (53a)



120588119882 = 120588 = 120588119906119880119905 |nabla119888| (54)



119880119905 sim (1199061015840119878119871)12 (55a)

120591119905 le 119905 ltlt (1199061015840119878119871)2 120591119905 (55b)


(120588119882)119871= 119860119871120588119906radic11989612119906 119878119871 |nabla119888| (56)








119880119905 = 119860119906101584011986311988614 = 11986011990610158403411987812119871 120594minus1411987114 (57a)

120591119905 lt 119905 ltlt 120591119905119863119886 (57b)





11 Conclusions













Appendix








(1 minus 119888) 119906119906 + 119888 119906119887Θ = 1 (A1a)


119901119906 + (12) 1199062119906 = 119901minusinfin + 12 (A1c)


(1 minus 119888) 119906119906 + 119888 119906119887Θ = 1 (A2a)

(12 minus 119888) 1199062119906 +119888 1199062119887Θ = 12 (A2b)


119906119887 = minus120573 + (minus4120572120574 + 1205732)12


119888Θ + 1] 120573 = minus2 119888Θ 05 minus 119888

(1 minus 119888)2 120574 = 05 minus 119888

(1 minus 119888)2 minus 05

(A3)


1205881199061015840101584011988810158401015840 = 120588119888 (1 minus 119888) (119906119887 minus 119906119906) (A4a)

1205881199061015840101584011990610158401015840 = 120588119888 (119906119887 minus 119906119906)2 (A4b)



0 025 05 075 10

005

01

015

02

025

03

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04

045

05

u

c

uUt

c(a)

0 025 05 075 10

1

2

3

4

5

6

7

8

9

u uU

tu b

Ut

ubUt

uuUt

c(b)



119906119906 = 1198911 (119888 Θ) (A5a)

119906119887 = 1198912 (119888 Θ) (A5b)

1205881199061015840101584011988810158401015840 = 1198913 (119888 Θ) (A5c)

1205881199061015840101584011990610158401015840 = 1198914 (119888 Θ) (A5d)
















1205881199061015840101584011988810158401015840 = (1205881199061015840101584011988810158401015840)119901minus119889

+ (1205881199061015840101584011988810158401015840)119905119889

= 1205881199061198801199051198913 (119888 Θ) minus 120588119863119905119889119888119889119909 (B1)



(B2)




119870119897119900119888 =100381610038161003816100381610038161003816(1205881199061015840101584011988810158401015840)119901minus11988910038161003816100381610038161003816100381610038161003816100381610038161003816(1205881199061015840101584011988810158401015840)11990511988910038161003816100381610038161003816 =

10038161003816100381610038161205881199061198801199051198913 (119888 Θ)10038161003816100381610038161003816100381610038161003816120588 (119888)1198631199051198891198881198891199091003816100381610038161003816 (B3a)

119870int =100381610038161003816100381610038161003816⟨(1205881199061015840101584011988810158401015840)119901minus119889⟩10038161003816100381610038161003816100381610038161003816100381610038161003816⟨(1205881199061015840101584011988810158401015840)119905119889⟩10038161003816100381610038161003816

= (14) 120573 [Θ minus (2Θ minus 1)12] (1198801199051199061015840 )(120575119905119871 ) (B3b)










0 02 04 06 08 1

minus005

0

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[kg

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u

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(a)

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c[k

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u

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0

002

0225 025 0275

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X(m) x(m) X(m)

0225 025 0275x(m)

02 03 04X(m)

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minus006

minus004

minus002

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004

002

006

minus006

minus004

minus002

0

002

004

006

0

002

004

006

008

01

012

minus0981

minus0984

minus0987

minus099

minus0993

minus0996

minus0999

３=04ＧＭ

３=04ＧＭ

３=2ＧＭ

０-０0０0

minus0987

minus0994

minus1001

minus1008

minus1015

minus1022

minus1029

０-０0０0

０-０0０0

５５0

５５0

1５５0

C

1

08

06

04

02

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Ｏ＝ＯＯ0C


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０-０0０0

５５0

C

３=04ＧＭ

３=2ＧＭ

３=2ＧＭ













Nomenclature





Greek Symbols





Data Availability




Acknowledgments



References































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







120588119882 = 120588119906119878119871Σ (41a)


119904Σ] + (119878nabla sdot 997888rarr119899)

119904Σ

(41b)





119865)


119865

+ 2120588119906119878119871119870119882

(42)





(43)





120591119894119895119906 = minus120588119906 (11990610158401198941199061015840119895)119906= ]119905119906 (120597119906119894119906120597119909119895 +

120597119906119895119906120597119909119894 ) minus (23) 120588119906119896119906120575119894119895

(44a)

120591119894119895119887 = minus120588119887 (11990610158401198941199061015840119895)119887= ]119905119887 (120597119906119894119887120597119909119895 +

120597119906119895119887120597119909119894 ) minus (23) 120588119906119896119887120575119894119895

(44b)


120597 (120588)120597119905 + 120597 (120588120572)120597119909120572 = 120591119894119895120597119895120597119909119894

+ 120597 [120588 (V119905120590119896) 120597120597119909119894]120597119909119894minus 120588120576

(45a)

120597 (120588120576)120597119905 + 120597 (120588119894120576)120597119909119894 = 1198621205761120591119894119895 (120576) 120597119906119895120597119909119894

minus 1198621205762120588(1205762 )+ 120597 [120588 (V119905120590120576) 120597120576120597119909119894 ] 120597119909119894

(45b)


120591119894119895 = minus (12058811990610158401015840119894 11990610158401015840119895 ) = ]119905 ( 120597119894120597119909119895 +120597119895120597119909119894 ) minus (

23) 120588120575119894119895 (46)



+ 120597 [120588 (1 minus 119888) (V119905119906120590119896) 120597119896119906120597119909119894]120597119909119894 minus 119896119906120588119882+ 119870119906Σ119891

(47a)

120597 (120588119888119896119887)120597119905 + 120597 (120588119888119906119894119887119896119887)120597119909119894= 120588119888 (120591119894119895119887120588119887) 120597119906119894119887120597119909119895 minus 120588119888120576119887

+ 120597 [120588119888 (V119905119887120590119896) 120597119896119887120597119909119894]120597119909119894 + 119896119887120588119882 + 119870119887Σ119891

(47b)


+ 120597 [120588 (1 minus 119888) (V119905119906120590119896) 120597119896119906120597119909120572]120597119909120572 minus 120576119906120588119882+ 119864119906Σ119891

(47c)

120597 (120588119888120576119887)120597119905 + 120597 (120588119888119906119894119887120576119887)120597119909119894= 1198621205761120588 (1 minus 119888) (120576119887119896119887) (120591119894119895119887120588119887) 120597119906119894119887120597119909119895minus 1198621205762120588119888( 1205762119887119896119887) +

120597 [120588119888 (V119905119906120590119896) 120597119896119887120597119909120572]120597119909120572+ 119864119887Σ119891 + 120576119887120588119882

(47d)


minus119896119906120588119882 + 119870119906Σ119891 = +119896119887120588119882 + 119870119887Σ119891 (48a)

minus120576119906120588119882 + 119864119906Σ119891 = +120576119887120588119882 + 119864119887Σ119891 (48b)










(49a)


(49b)

120597 (120588)120597119905 + 120597 (120588120572)120597119909120572 = 120591119894119895120597119895120597119909119894+ 120597 [120588 (V119905120590119896) 120597120597119909119894]120597119909119894 minus 120588120576

(49c)

120597 (120588120576)120597119905 + 120597 (120588119894120576)120597119909119894 = 1198621205761120591119894119895 (120576) 120597119906119895120597119909119894+ 120597 [120588 (V119905120590120576) 120597120576120597119909119894 ] 120597119909119894 minus 1198621205762120588(1205762 )]

(49d)

= 119896119906 (1 minus 119888) + 119896119887119888 (49e)

120576 = 120576119906 (1 minus 119888) + 120576119887119888 (49f)










( 1198601198600


2|nablaℎ| asymp

3(|nablaℎ|2)12

=4(intinfin

01198962119865 (119896) 119889119896)12

(50)


( 1198601198600

) asymp (intinfin01198962119865 (119896) 119889119896)12 (51a)

1205902119891 (119905) = (119909 minus 119909 (119905))2 = intinfin0119865 (119896 119905) 119889119896 (51b)



1205902119891 cong 2119863119905119905 (52a)

120575119905 sim (1199061015840119871119905)12 (52b)




120597119888120597119905 = 11986311990512059721198881205971199092 + 119880119905 1003816100381610038161003816100381610038161003816100381612059711988812059711990910038161003816100381610038161003816100381610038161003816 (53a)



120588119882 = 120588 = 120588119906119880119905 |nabla119888| (54)



119880119905 sim (1199061015840119878119871)12 (55a)

120591119905 le 119905 ltlt (1199061015840119878119871)2 120591119905 (55b)


(120588119882)119871= 119860119871120588119906radic11989612119906 119878119871 |nabla119888| (56)








119880119905 = 119860119906101584011986311988614 = 11986011990610158403411987812119871 120594minus1411987114 (57a)

120591119905 lt 119905 ltlt 120591119905119863119886 (57b)





11 Conclusions













Appendix








(1 minus 119888) 119906119906 + 119888 119906119887Θ = 1 (A1a)


119901119906 + (12) 1199062119906 = 119901minusinfin + 12 (A1c)


(1 minus 119888) 119906119906 + 119888 119906119887Θ = 1 (A2a)

(12 minus 119888) 1199062119906 +119888 1199062119887Θ = 12 (A2b)


119906119887 = minus120573 + (minus4120572120574 + 1205732)12


119888Θ + 1] 120573 = minus2 119888Θ 05 minus 119888

(1 minus 119888)2 120574 = 05 minus 119888

(1 minus 119888)2 minus 05

(A3)


1205881199061015840101584011988810158401015840 = 120588119888 (1 minus 119888) (119906119887 minus 119906119906) (A4a)

1205881199061015840101584011990610158401015840 = 120588119888 (119906119887 minus 119906119906)2 (A4b)



0 025 05 075 10

005

01

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c

uUt

c(a)

0 025 05 075 10

1

2

3

4

5

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u uU

tu b

Ut

ubUt

uuUt

c(b)



119906119906 = 1198911 (119888 Θ) (A5a)

119906119887 = 1198912 (119888 Θ) (A5b)

1205881199061015840101584011988810158401015840 = 1198913 (119888 Θ) (A5c)

1205881199061015840101584011990610158401015840 = 1198914 (119888 Θ) (A5d)
















1205881199061015840101584011988810158401015840 = (1205881199061015840101584011988810158401015840)119901minus119889

+ (1205881199061015840101584011988810158401015840)119905119889

= 1205881199061198801199051198913 (119888 Θ) minus 120588119863119905119889119888119889119909 (B1)



(B2)




119870119897119900119888 =100381610038161003816100381610038161003816(1205881199061015840101584011988810158401015840)119901minus11988910038161003816100381610038161003816100381610038161003816100381610038161003816(1205881199061015840101584011988810158401015840)11990511988910038161003816100381610038161003816 =

10038161003816100381610038161205881199061198801199051198913 (119888 Θ)10038161003816100381610038161003816100381610038161003816120588 (119888)1198631199051198891198881198891199091003816100381610038161003816 (B3a)

119870int =100381610038161003816100381610038161003816⟨(1205881199061015840101584011988810158401015840)119901minus119889⟩10038161003816100381610038161003816100381610038161003816100381610038161003816⟨(1205881199061015840101584011988810158401015840)119905119889⟩10038161003816100381610038161003816

= (14) 120573 [Θ minus (2Θ minus 1)12] (1198801199051199061015840 )(120575119905119871 ) (B3b)










0 02 04 06 08 1

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[kg

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c[k

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u

c

(b)











0

002

0225 025 0275

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026 027 028 029 03

X(m) x(m) X(m)

0225 025 0275x(m)

02 03 04X(m)

004

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008

0101 03 05 07 09

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minus004

minus002

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minus004

minus002

0

002

004

006

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004

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minus0981

minus0984

minus0987

minus099

minus0993

minus0996

minus0999

３=04ＧＭ

３=04ＧＭ

３=2ＧＭ

０-０0０0

minus0987

minus0994

minus1001

minus1008

minus1015

minus1022

minus1029

０-０0０0

０-０0０0

５５0

５５0

1５５0

C

1

08

06

04

02

0

1


Ｏ＝ＯＯ0C


Ｏ＝ＯＯ0C

０-０0０0

５５0

C

３=04ＧＭ

３=2ＧＭ

３=2ＧＭ













Nomenclature





Greek Symbols





Data Availability




Acknowledgments



References































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





120591119894119895119906 = minus120588119906 (11990610158401198941199061015840119895)119906= ]119905119906 (120597119906119894119906120597119909119895 +

120597119906119895119906120597119909119894 ) minus (23) 120588119906119896119906120575119894119895

(44a)

120591119894119895119887 = minus120588119887 (11990610158401198941199061015840119895)119887= ]119905119887 (120597119906119894119887120597119909119895 +

120597119906119895119887120597119909119894 ) minus (23) 120588119906119896119887120575119894119895

(44b)


120597 (120588)120597119905 + 120597 (120588120572)120597119909120572 = 120591119894119895120597119895120597119909119894

+ 120597 [120588 (V119905120590119896) 120597120597119909119894]120597119909119894minus 120588120576

(45a)

120597 (120588120576)120597119905 + 120597 (120588119894120576)120597119909119894 = 1198621205761120591119894119895 (120576) 120597119906119895120597119909119894

minus 1198621205762120588(1205762 )+ 120597 [120588 (V119905120590120576) 120597120576120597119909119894 ] 120597119909119894

(45b)


120591119894119895 = minus (12058811990610158401015840119894 11990610158401015840119895 ) = ]119905 ( 120597119894120597119909119895 +120597119895120597119909119894 ) minus (

23) 120588120575119894119895 (46)



+ 120597 [120588 (1 minus 119888) (V119905119906120590119896) 120597119896119906120597119909119894]120597119909119894 minus 119896119906120588119882+ 119870119906Σ119891

(47a)

120597 (120588119888119896119887)120597119905 + 120597 (120588119888119906119894119887119896119887)120597119909119894= 120588119888 (120591119894119895119887120588119887) 120597119906119894119887120597119909119895 minus 120588119888120576119887

+ 120597 [120588119888 (V119905119887120590119896) 120597119896119887120597119909119894]120597119909119894 + 119896119887120588119882 + 119870119887Σ119891

(47b)


+ 120597 [120588 (1 minus 119888) (V119905119906120590119896) 120597119896119906120597119909120572]120597119909120572 minus 120576119906120588119882+ 119864119906Σ119891

(47c)

120597 (120588119888120576119887)120597119905 + 120597 (120588119888119906119894119887120576119887)120597119909119894= 1198621205761120588 (1 minus 119888) (120576119887119896119887) (120591119894119895119887120588119887) 120597119906119894119887120597119909119895minus 1198621205762120588119888( 1205762119887119896119887) +

120597 [120588119888 (V119905119906120590119896) 120597119896119887120597119909120572]120597119909120572+ 119864119887Σ119891 + 120576119887120588119882

(47d)


minus119896119906120588119882 + 119870119906Σ119891 = +119896119887120588119882 + 119870119887Σ119891 (48a)

minus120576119906120588119882 + 119864119906Σ119891 = +120576119887120588119882 + 119864119887Σ119891 (48b)










(49a)


(49b)

120597 (120588)120597119905 + 120597 (120588120572)120597119909120572 = 120591119894119895120597119895120597119909119894+ 120597 [120588 (V119905120590119896) 120597120597119909119894]120597119909119894 minus 120588120576

(49c)

120597 (120588120576)120597119905 + 120597 (120588119894120576)120597119909119894 = 1198621205761120591119894119895 (120576) 120597119906119895120597119909119894+ 120597 [120588 (V119905120590120576) 120597120576120597119909119894 ] 120597119909119894 minus 1198621205762120588(1205762 )]

(49d)

= 119896119906 (1 minus 119888) + 119896119887119888 (49e)

120576 = 120576119906 (1 minus 119888) + 120576119887119888 (49f)










( 1198601198600


2|nablaℎ| asymp

3(|nablaℎ|2)12

=4(intinfin

01198962119865 (119896) 119889119896)12

(50)


( 1198601198600

) asymp (intinfin01198962119865 (119896) 119889119896)12 (51a)

1205902119891 (119905) = (119909 minus 119909 (119905))2 = intinfin0119865 (119896 119905) 119889119896 (51b)



1205902119891 cong 2119863119905119905 (52a)

120575119905 sim (1199061015840119871119905)12 (52b)




120597119888120597119905 = 11986311990512059721198881205971199092 + 119880119905 1003816100381610038161003816100381610038161003816100381612059711988812059711990910038161003816100381610038161003816100381610038161003816 (53a)



120588119882 = 120588 = 120588119906119880119905 |nabla119888| (54)



119880119905 sim (1199061015840119878119871)12 (55a)

120591119905 le 119905 ltlt (1199061015840119878119871)2 120591119905 (55b)


(120588119882)119871= 119860119871120588119906radic11989612119906 119878119871 |nabla119888| (56)








119880119905 = 119860119906101584011986311988614 = 11986011990610158403411987812119871 120594minus1411987114 (57a)

120591119905 lt 119905 ltlt 120591119905119863119886 (57b)





11 Conclusions













Appendix








(1 minus 119888) 119906119906 + 119888 119906119887Θ = 1 (A1a)


119901119906 + (12) 1199062119906 = 119901minusinfin + 12 (A1c)


(1 minus 119888) 119906119906 + 119888 119906119887Θ = 1 (A2a)

(12 minus 119888) 1199062119906 +119888 1199062119887Θ = 12 (A2b)


119906119887 = minus120573 + (minus4120572120574 + 1205732)12


119888Θ + 1] 120573 = minus2 119888Θ 05 minus 119888

(1 minus 119888)2 120574 = 05 minus 119888

(1 minus 119888)2 minus 05

(A3)


1205881199061015840101584011988810158401015840 = 120588119888 (1 minus 119888) (119906119887 minus 119906119906) (A4a)

1205881199061015840101584011990610158401015840 = 120588119888 (119906119887 minus 119906119906)2 (A4b)



0 025 05 075 10

005

01

015

02

025

03

035

04

045

05

u

c

uUt

c(a)

0 025 05 075 10

1

2

3

4

5

6

7

8

9

u uU

tu b

Ut

ubUt

uuUt

c(b)



119906119906 = 1198911 (119888 Θ) (A5a)

119906119887 = 1198912 (119888 Θ) (A5b)

1205881199061015840101584011988810158401015840 = 1198913 (119888 Θ) (A5c)

1205881199061015840101584011990610158401015840 = 1198914 (119888 Θ) (A5d)
















1205881199061015840101584011988810158401015840 = (1205881199061015840101584011988810158401015840)119901minus119889

+ (1205881199061015840101584011988810158401015840)119905119889

= 1205881199061198801199051198913 (119888 Θ) minus 120588119863119905119889119888119889119909 (B1)



(B2)




119870119897119900119888 =100381610038161003816100381610038161003816(1205881199061015840101584011988810158401015840)119901minus11988910038161003816100381610038161003816100381610038161003816100381610038161003816(1205881199061015840101584011988810158401015840)11990511988910038161003816100381610038161003816 =

10038161003816100381610038161205881199061198801199051198913 (119888 Θ)10038161003816100381610038161003816100381610038161003816120588 (119888)1198631199051198891198881198891199091003816100381610038161003816 (B3a)

119870int =100381610038161003816100381610038161003816⟨(1205881199061015840101584011988810158401015840)119901minus119889⟩10038161003816100381610038161003816100381610038161003816100381610038161003816⟨(1205881199061015840101584011988810158401015840)119905119889⟩10038161003816100381610038161003816

= (14) 120573 [Θ minus (2Θ minus 1)12] (1198801199051199061015840 )(120575119905119871 ) (B3b)










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minus005

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005

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[kg

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u

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(a)

0 02 04 06 08 1

minus005

0

005

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02

025

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035Calculations

c[k

gm

2s]

u

c

(b)











0

002

0225 025 0275

1

1 1

02 03 04 026 027 028 029 03

026 027 028 029 03

X(m) x(m) X(m)

0225 025 0275x(m)

02 03 04X(m)

004

006

008

0101 03 05 07 09

012

minus006

minus004

minus002

0

08

06

04

02

0

004

002

006

minus006

minus004

minus002

0

002

004

006

0

002

004

006

008

01

012

minus0981

minus0984

minus0987

minus099

minus0993

minus0996

minus0999

３=04ＧＭ

３=04ＧＭ

３=2ＧＭ

０-０0０0

minus0987

minus0994

minus1001

minus1008

minus1015

minus1022

minus1029

０-０0０0

０-０0０0

５５0

５５0

1５５0

C

1

08

06

04

02

0

1


Ｏ＝ＯＯ0C


Ｏ＝ＯＯ0C

０-０0０0

５５0

C

３=04ＧＭ

３=2ＧＭ

３=2ＧＭ













Nomenclature





Greek Symbols





Data Availability




Acknowledgments



References































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia










(49a)


(49b)

120597 (120588)120597119905 + 120597 (120588120572)120597119909120572 = 120591119894119895120597119895120597119909119894+ 120597 [120588 (V119905120590119896) 120597120597119909119894]120597119909119894 minus 120588120576

(49c)

120597 (120588120576)120597119905 + 120597 (120588119894120576)120597119909119894 = 1198621205761120591119894119895 (120576) 120597119906119895120597119909119894+ 120597 [120588 (V119905120590120576) 120597120576120597119909119894 ] 120597119909119894 minus 1198621205762120588(1205762 )]

(49d)

= 119896119906 (1 minus 119888) + 119896119887119888 (49e)

120576 = 120576119906 (1 minus 119888) + 120576119887119888 (49f)










( 1198601198600


2|nablaℎ| asymp

3(|nablaℎ|2)12

=4(intinfin

01198962119865 (119896) 119889119896)12

(50)


( 1198601198600

) asymp (intinfin01198962119865 (119896) 119889119896)12 (51a)

1205902119891 (119905) = (119909 minus 119909 (119905))2 = intinfin0119865 (119896 119905) 119889119896 (51b)



1205902119891 cong 2119863119905119905 (52a)

120575119905 sim (1199061015840119871119905)12 (52b)




120597119888120597119905 = 11986311990512059721198881205971199092 + 119880119905 1003816100381610038161003816100381610038161003816100381612059711988812059711990910038161003816100381610038161003816100381610038161003816 (53a)



120588119882 = 120588 = 120588119906119880119905 |nabla119888| (54)



119880119905 sim (1199061015840119878119871)12 (55a)

120591119905 le 119905 ltlt (1199061015840119878119871)2 120591119905 (55b)


(120588119882)119871= 119860119871120588119906radic11989612119906 119878119871 |nabla119888| (56)








119880119905 = 119860119906101584011986311988614 = 11986011990610158403411987812119871 120594minus1411987114 (57a)

120591119905 lt 119905 ltlt 120591119905119863119886 (57b)





11 Conclusions













Appendix








(1 minus 119888) 119906119906 + 119888 119906119887Θ = 1 (A1a)


119901119906 + (12) 1199062119906 = 119901minusinfin + 12 (A1c)


(1 minus 119888) 119906119906 + 119888 119906119887Θ = 1 (A2a)

(12 minus 119888) 1199062119906 +119888 1199062119887Θ = 12 (A2b)


119906119887 = minus120573 + (minus4120572120574 + 1205732)12


119888Θ + 1] 120573 = minus2 119888Θ 05 minus 119888

(1 minus 119888)2 120574 = 05 minus 119888

(1 minus 119888)2 minus 05

(A3)


1205881199061015840101584011988810158401015840 = 120588119888 (1 minus 119888) (119906119887 minus 119906119906) (A4a)

1205881199061015840101584011990610158401015840 = 120588119888 (119906119887 minus 119906119906)2 (A4b)



0 025 05 075 10

005

01

015

02

025

03

035

04

045

05

u

c

uUt

c(a)

0 025 05 075 10

1

2

3

4

5

6

7

8

9

u uU

tu b

Ut

ubUt

uuUt

c(b)



119906119906 = 1198911 (119888 Θ) (A5a)

119906119887 = 1198912 (119888 Θ) (A5b)

1205881199061015840101584011988810158401015840 = 1198913 (119888 Θ) (A5c)

1205881199061015840101584011990610158401015840 = 1198914 (119888 Θ) (A5d)
















1205881199061015840101584011988810158401015840 = (1205881199061015840101584011988810158401015840)119901minus119889

+ (1205881199061015840101584011988810158401015840)119905119889

= 1205881199061198801199051198913 (119888 Θ) minus 120588119863119905119889119888119889119909 (B1)



(B2)




119870119897119900119888 =100381610038161003816100381610038161003816(1205881199061015840101584011988810158401015840)119901minus11988910038161003816100381610038161003816100381610038161003816100381610038161003816(1205881199061015840101584011988810158401015840)11990511988910038161003816100381610038161003816 =

10038161003816100381610038161205881199061198801199051198913 (119888 Θ)10038161003816100381610038161003816100381610038161003816120588 (119888)1198631199051198891198881198891199091003816100381610038161003816 (B3a)

119870int =100381610038161003816100381610038161003816⟨(1205881199061015840101584011988810158401015840)119901minus119889⟩10038161003816100381610038161003816100381610038161003816100381610038161003816⟨(1205881199061015840101584011988810158401015840)119905119889⟩10038161003816100381610038161003816

= (14) 120573 [Θ minus (2Θ minus 1)12] (1198801199051199061015840 )(120575119905119871 ) (B3b)










0 02 04 06 08 1

minus005

0

005

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015

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035Experiments

c

[kg

m2s]

u

c

(a)

0 02 04 06 08 1

minus005

0

005

01

015

02

025

03

035Calculations

c[k

gm

2s]

u

c

(b)











0

002

0225 025 0275

1

1 1

02 03 04 026 027 028 029 03

026 027 028 029 03

X(m) x(m) X(m)

0225 025 0275x(m)

02 03 04X(m)

004

006

008

0101 03 05 07 09

012

minus006

minus004

minus002

0

08

06

04

02

0

004

002

006

minus006

minus004

minus002

0

002

004

006

0

002

004

006

008

01

012

minus0981

minus0984

minus0987

minus099

minus0993

minus0996

minus0999

３=04ＧＭ

３=04ＧＭ

３=2ＧＭ

０-０0０0

minus0987

minus0994

minus1001

minus1008

minus1015

minus1022

minus1029

０-０0０0

０-０0０0

５５0

５５0

1５５0

C

1

08

06

04

02

0

1


Ｏ＝ＯＯ0C


Ｏ＝ＯＯ0C

０-０0０0

５５0

C

３=04ＧＭ

３=2ＧＭ

３=2ＧＭ













Nomenclature





Greek Symbols





Data Availability




Acknowledgments



References































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


A Two-Fluid Conditional Averaging Paradigm for the Theory ...and principles of theory and modeling,...

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Transcript of A Two-Fluid Conditional Averaging Paradigm for the Theory ...and principles of theory and modeling,...