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International Journal of Thermal Sciences 49 (2010) 2067e2075

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International Journal of Thermal Sciences

journal homepage: www.elsevier .com/locate/ i j ts

Entropy generation in impinging flow confined by planar opposing jets

Sheng Chen*, Chuguang ZhengState Key Lab of Coal Combustion, Huazhong University of Science and Technology, Pockelstr.3, Wuhan 430074, China

a r t i c l e i n f o

Article history:Received 21 February 2010Received in revised form19 May 2010Accepted 31 May 2010Available online 6 July 2010

Keywords:Entropy generationSecond law analysisImpinging flowOpposing jetsLattice Boltzmann method

* Corresponding author.E-mail address: shengchen.hust@gmail.com (S. Ch

1290-0729/$ e see front matter � 2010 Elsevier Masdoi:10.1016/j.ijthermalsci.2010.05.024

a b s t r a c t

In this paper, entropy generation in impinging flow confined by planar opposing jets is investigatedsystematically for the first time. Different from previous works on entropy generation for practical flows,in this study the lattice Boltzmann method, which is more suitable for massive parallel computing, isused to solve the governing equations for flow field as well as the entropy generation equation, instead oftraditional numerical methods. The effects of the Reynolds number 10� Re� 500 and the distance ratiobetween opposing jets 2/5�W/L� 4/5 on entropy generation are revealed. It is found that the localentropy generation number is more sensitive to the variation of W/L than Re when Re> 50. The totalentropy generation number increases exponentially with Re but decreases as a power function of W/L. Inaddition, the entropy generation will receive significant influence from the damping traveling pressurewave during the transient state and the maximum emerges when the gas ejected from the top andbottom jets begins meeting and impinging.

� 2010 Elsevier Masson SAS. All rights reserved.

1. Introduction

Impinging flow confined by two planar opposing jets is popu-larly used in many industrial applications such as drying, mixing,absorption, dust collection, catalytic reaction, side dump combus-tion of gases, reaction injection molding, due to the high transfercoefficients obtainable in this type of flow [1e5]. To date numerousworks have been done to investigate the fundamental character-istics of such flow. Hosseinalipour and Mujumdar [6e8] developedthe first numerical model for two confined plane opposing jets withisothermal boundaries and adiabatic walls in steady laminar as wellas turbulent flows, respectively. An extensive parametric study wascarried out to study the geometric and hydrodynamic effects on theflow and heat transfer characteristics and effects of equal andunequal opposing jets on the flow and heat transfer characteristics.Later, Devahastin et al. [9,10] numerically studied the flow patternsand mixing characteristics of two dimensional confined impingingstreams for both the laminar and turbulent flow regime. Theyfound that both the inlet jet Reynolds number and the ratio ofthe height of the mixer exit channel to the width of an inlet jetinfluenced the distance to attain the well-mixed condition in theoutlet. This behavior was quite different from what was observedin the turbulent impinging streams. As the jet Reynolds numberincreased mixing was improved until some specific values of the

en).

son SAS. All rights reserved.

dimensionless distance were reached. These critical valuesdepended on both the operating conditions as well as the geometryof the system. Wang and Mujumdar [11,12] performed thenumerical studies on the effects of unequal opposed jets on theflow pattern and mixing characteristic in a three-dimensionalconfined turbulent opposing jets using low Reynolds number keeturbulence models. Results showed that better mixing is obtainedfor unequal jets than equal jets for the same total mass flow rate. Insuccession, Wang and his cooperators [5] studied the mixingcharacteristics and flow field of two-dimensional laminar confinedopposing streams for various temperature differences of theopposing jets. They found that the effects of temperature-depen-dent fluid properties on the mixing characteristics are dependentstrongly on the magnitude of temperature difference, flow condi-tions and geometric configurations. Recently, Chen et al. [2,13] andLiu et al. [14] investigated the flow characteristic of laminarconfined impinging streams with/without reaction using a newnumerical method e the lattice Boltzmann (LB) method. Morerecently, experimental and numerical study on stagnation pointoffset of turbulent opposed jets was carried by Li and his coworkers[3]. It is found that the position of stagnation point becomesinsensitive to the variety of exit velocity ratio. Only a few of themcan be cited here since the amount of the publications is too large.However, almost all of them are based on the first-law of thermo-dynamics as well as the fluid mechanics, and avoid complicatedanalyses for optimum design.

After Bejan showed that the exergy destruction in convectivefluid flow is due to heat transfer and viscous shear stresses [15],

y=W

x=L0

du/dx=0dv/dx=0p=p

0

0

du/dx=0dv/dx=0p=p

u=0,v=v

0

0u=0,v=v

Fig. 1. Schematic configuration and coordinate system of the flow field.

Nomenclature

c fluid particle speedHTI Heat Transfer IrreversibilityFFI Fluid Friction Irreversibilityu! fluid velocity vectore!k discrete velocity vectorFk source term in Eq. (4)gk distribution function for velocitygkeq equilibrium distribution function

W distance between jetsL length of jetRe Reynolds numberPr Prandtl numberBe Bejan numberEc Eckert numbercs speed of soundStotal total entropy generation numberS local entropy generation numberT0 reference temperaturep pressure

x! phase space

Greek symbolsDx grid spacingDt time stepz dimensionless timen kinematic viscositysu relaxation time for velocityr densityuk the weights for equilibriumck parameter in Eq. (8)4 irreversibility distribution ratiojfj magnitude of the strain rate tensor

Subscripts and superscriptsD thermalm viscousk discrete velocity direction_ averagea, b spatial indexU global

S. Chen, C. Zheng / International Journal of Thermal Sciences 49 (2010) 2067e20752068

a lot of numerical investigations for practical engineering flowbased on the second law analysis were carried out. But surprisingly,the open literature of entropy generation analysis on impingingflow is quite sparse. To the best knowledge of the present author,until now there is only five open publications discussing this topic.Ruocco [16] perhaps is the pioneer in this field. In Ref. [16] heconducted the numerical prediction on the distribution of entropygeneration in conjugate heat transfer from a finite thickness plateto a laminar confined impinging planar jet by using a finite volumeprocedure. It is found that the entropy generation rate increasedwith the thermal conductivity ratio. Shuja et al. analyzed theentropy generation in an impinging jet [17] and swirling jetimpingement on an adiabatic wall [18e20] for various flowconditions. In Ref. [17] the authors used the second law analysistechnology to evaluate the various turbulence models for a fluid jetimpinging on a heated wall. In the work, it is shown that the totalentropy generation attains higher values close to the stagnationregion and across the shear layer between the jet and thesurrounding fluid; The volumetric entropy generation due to heattransfer is slightly lower than that due to fluid friction; Theconstant volumetric entropy generation lines due to fluid frictionare similar in the far field for all turbulent models except forstandard keemodel; Low Reynolds number keemodel predicts theminimum volumetric entropy generation close to the stagnationregion whereas Reynolds stress models predict slightly high volu-metric entropy generation as the radial distance from the stagna-tion region extends. Then the same authors examined the effect ofthe nozzle exit velocity profile and the swirling velocity on the flowfield and entropy generation rate [18e20]. They found that the totalentropy generation increases with increasing swirl velocity for lowvelocity profile numbers. The Merit number improves for lowswirling velocity and high velocity profile numbers. However, thesecond law analysis of laminar impinging flow confined by planaropposing jets still remains an unsettled question so far.

The main originality of the present work is to investigatesystematically the effects of the Reynolds number 10� Re� 500and the distance ratio between opposing jets 2/5�W/L� 4/5 onentropy generation for impinging flow confined by planar opposingjets for the first time. The above literature survey clearly shows thatthere is no study in the literature on this topic. The governingequations for flow field as well as the entropy generation equation

all are solved by a LB model recently developed by the presentauthor [2,13,21] instead of traditional numerical methods. Asshown in previous study of the present author, the LB methodpossesses significant advantages over traditional CFD methods forentropy generation analysis [21]: for example, the LB method ismore suitable for massive parallel computing of engineering flows.

2. Specification of the problem and governing equations

The configuration of planar opposing jets illustrated in previousworks conducted by the present author [2,13] is used in the presentstudy again since such configuration is a benchmark one popularlyused in advanced combustion technologies investigation [22e30].The problem domain and boundary conditions are summarized inFig. 1. The impinging flow is confined by two equal opposing jetswith length L. The distance between the planar opposing jets is W.Gas is uniformly ejected from the top and bottom jets and flowsoutward along the x-direction. As a primeval study, for simplicity,

S. Chen, C. Zheng / International Journal of Thermal Sciences 49 (2010) 2067e2075 2069

the flow is assumed isothermal in this work to avoid the influencecaused by temperature field.

The dimensionless continuity and momentum equations inCartesian coordinates are given as [2,13]

V$ u! ¼ 0 (1)

v u!vt

þ u!$V u! ¼ �Vpþ 1Re

D u! (2)

where u! ¼ ðu; vÞ is the velocity vector. Re is the Reynolds numberdefined as [13]

Re ¼ v0W2n

; (3)

where n is the kinematic viscosity and v0¼ 0.1 is the mean inletvelocity. p0¼1/3 is the reference pressure.

Fig. 2. Comparison of y-direction velocity v of the impinging flow along line x¼ 0.

3. Lattice Boltzmann model for impinging flow

Recently, the LB method has matured for simulating andmodeling complicated physical and chemical systems [31,32]. Theimplementation of a LB procedure is quite easy. Parallelization ofa LB model is natural since the relaxation is local and the perfor-mance increases nearly linearly with the number of CPUs. More-over, the LB moldes have been compared favourably with spectralmethods [33], artificial compressibility methods [34,35], finitevolume methods [36e38], finite difference methods [39e41],projection methods [42,43] and multigrid method [44,45], allquantitative results further validate excellent performance of theLB method not only in computational efficiency but also innumerical accuracy. Due to these advantages, the LB method hasbeen successfully used to simulate many problems, from laminarsingle phase flows to turbulent multiphase flows [31,32].

In the present study, the LB models proposed in our previousworks [2,13] are adopted to solve Eqs. (1) and (2). The evolutionequation for the flow field reads [2,13,21]

gkð x!þ c e!kDt; t þ DtÞ � gkð x!; tÞ¼ �s�1

u

hgkð x!; tÞ � gðeqÞk ð x!; tÞ

iþ DtFk (4)

where e!k is the discrete velocity direction. c¼Dx/Dt is the fluidparticle speed. Dx, Dt and su are the lattice grid spacing, the timestep and the dimensionless relaxation time for the flow field,respectively. The force term Fk must satisfy

Xk�0

Fk ¼ 0;Xk�0

Fk e!

k ¼ F!

(5)

where F!

is the external force. In this study F! ¼ 0.

The force term Fk is given as [21]

Fk ¼ uk

�1� 1

2su

��c e!k$ F

!c2s

þ ðc e!k$ v!Þðc e!k$ F

!Þc4s

� v!$ F!

c2s

�(6)

where cs is the speed of sound and the equilibrium velocity v! isdefined as

v! ¼Xk�0

c e!kgk þDt2

F!

(7)

gkð x!; tÞ is the distribution function at node x! and time t withvelocity e!k, and gðeqÞk ð x!; tÞ is the corresponding equilibriumdistribution. The equilibrium distribution in the present model isdefined by

gðeqÞk ¼ ckpþ sk (8)

where

sk ¼ uk

"c e!k$ v

!c2s

þ ðc e!k$ v!Þ2

2c4s� j v!j2

2c2s

#(9)

the parameter ck is determined by the moment constraints [13],which lead

ckjðks0Þ ¼ uk=c2s ;ckjðk¼0Þ ¼ ðu0 � 1Þ=c2s (10)

the values of uk for one-, two- and three-dimensional problems canbe found in Ref. [13].

The kinematic viscosity is determined by

n ¼ ðsu � 0:5Þc2sDt (11)

The velocity and pressure are given by

u! ¼Xk�0

c e!kgk þDt2

F!

(12)

p ¼ c2s1� u0

"Xks0

gk þ s0ð u!Þ#

(13)

4. Entropy generation

The local entropy generation number is given by [21]:

S ¼ ðVTÞ2þ4jfj2 (14)

where the irreversibility distribution ratio 4¼ BrT0 [21]. Br¼ PrEc isthe Brinkman number, where Ec is the Eckert number [21]. The firstterm in Eq. (14) reflects the entropy generation due to thermaldiffusion and the second due to viscous dissipation. Usually theyare referred to as Heat Transfer Irreversibility HTI and Fluid FrictionIrreversibility FFI, respectively. The dimensionless number, Bejannumber Be, is used to describe the ratio between them. In thepresent study, because the flow is isothermal therefore thecontribution of the first term in Eq. (14) vanishes, and Be¼ 0accordingly.

jfj is the magnitude of the strain rate tensor

0.4141960.405

0.390.3750.35

0.41 0.4

0.385 0.37

0.345

X

Y

0 1 2 3 4 50

0.5

1

1.5

2a

1.98609

6.06279

9.28694

9.74609

7.97548

3.94408

0.665742

9.8228910.666

10.0109

13.9205

27.6499

X

Y

0 1 2 3 4 50

0.5

1

1.5

2b

Fig. 4. Streamlines, pressure (a) and entropy generation number distribution (b) at Re¼ 50. (For interpretation of the references to color in this figure legend, the reader is referredto the web version of this article.).

0.413697

0 .41

0.395 83.0

0. 3 650.35

0.045

03.75

0.34

0.39

0.355

X

Y

0 1 2 3 4 50

0.5

1

1.5

2a

1.96271

6.047

9.35161

9.83444

8.00057

3.91354

0.654121

10.138

9.9248

10.854

14.2607

27.0611

28.8814

X

Y

0 1 2 3 4 50

0.5

1

1.5

2b

Fig. 5. Streamlines, pressure (a) and entropy generation number distribution (b) at Re¼ 500. (For interpretation of the references to color in this figure legend, the reader is referredto the web version of this article.).

0.418490.41

0 .39 5

0.345

0.365 0. 38

0 . 41 5

0.4

03. 85

03. 7

0.355 0.34

X

Y

0 1 2 3 4 50

0.5

1

1.5

2a

0.71455

4.0509

7.81565

9.349799.4599

9.01369

6.08065

2.12805

9.45999.59709

12.1997

14.282

21.1553

28.9185

X

Y

0 1 2 3 4 50

0.5

1

1.5

2b

Fig. 3. Streamlines, pressure (a) and entropy generation number distribution (b) at Re¼ 10. (For interpretation of the references to color in this figure legend, the reader is referredto the web version of this article.).

jfj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2fabfab

q(15)

where fab ¼ ðvaub þ vbuaÞ=2. For 2D problems [43]

jfj2 ¼ 2�vuvx

�2þ2

�vv

vy

�2þ�vuvy

�2þ2

�vuvy

��vv

vx

�þ�vv

vx

�2(16)

Apparently, it is inconvenient to calculate jfj (Eq. (15)) directly byconventional numerical methods due to its complex form of spatialderivative [43]. However, as shown in our previous study [21], in

the LB method, jfj can be calculated easily through computing themagnitude of the momentum fluxes Q

jfj ¼ 32suDt

jQ j (17)

and Q can be obtained by

Q ¼Xk

e!ka e!kb

�gk � geqk

�(18)

The total entropy generation number is defined as [15]

Fig. 6. Entropy generation number along line x¼ L/2 (a) and line y¼W/2 (b): Re¼ 10eblue solid line; Re¼ 50egreen dash line; Re¼ 500eblack dots (For interpretation of thereferences to color in this figure legend, the reader is referred to the web version of this article.).

S. Chen, C. Zheng / International Journal of Thermal Sciences 49 (2010) 2067e2075 2071

Stotal ¼ZU

SvU (19)

where U means the global computational domain. What should beemphasized is that all quantities in the present study aredimensionless.

5. Numerical validation

In order to validate the present model, firstly we compare the y-direction velocity v obtained by the present model with thatobtained by the finite difference method (FDM) [13] for impingingflow with Re¼ 125, as shown in Fig. 2. It is obvious that the resultsobtained by the present model agree well with previous data [13].

6. Results and discussions

In the present study, we investigate the effects of the Reynoldsnumber and the distance ratio between opposing jets on entropygeneration for impinging flow confined by planar opposing jets bysetting 10� Re� 500 and 2/5�W/L� 4/5 since such configurationparameters are commonly found in practical applications [2,3,13].The behaviors of various results are considered by assuming one ofthese two parameters varying while the other is fixed. The irre-versibility distribution ratio 4¼ 1 in the present study. The gridresolution varies from 450�180 to 450� 360 according todifferent W/L. It has been demonstrated in previous works of thepresent author [2,13,22] that such grid resolution is fine enough toobtain grid-independent numerical solutions. The numerical code

Fig. 7. Total entropy generation number versus Re.

used here is described and validated in more detail in Refs.[13,21,22].

6.1. Effect of Reynolds number

To reveal the effect of the Reynolds number on entropy gener-ation distribution, in this subsection W/L¼ 2/5 is used to typify allcases in this study.

Figs. 3e5 illustrate the streamlines, pressure and entropygeneration number distributions at Re¼ 10, 50 and 500, respec-tively. For simplicity, the abscissa is normalized by x/(0.2L), and theordinate is normalized by y/(0.2L) for all figures in this paper. It isclear that the profiles are symmetric with respect to the domaincenter. The flow fields for different Re are almost same except slightdifferences between pressure distributions. Consequently, withoutsurprising, the plots of entropy generation number are also nearlysame, especially when Re> 50, which can be demonstrated byFig. 6, too. Fig. 6 shows the variations of S along the line x¼ L/2 andthe line y¼W/2 for Re¼ 10, 50 and 500. In Fig. 6, one can see thatthe profiles of S along the line x¼ L/2 and line y¼W/2 for Re¼ 50and Re¼ 500 almost overlap. According to this figure, it isstraightforward that S for Re¼ 10 is the smallest. The entropymainly generates near the outlets, namely the zones x< 1.2 andx> 3.8, especially near the corners, no matter what Re is. However,within the zone 1.2< x< 3.8, entropy generation is smaller and thecontours of entropy generation number form concentric toroid-coil-like curves with respect to the geometric centers of jets. Whenapproaching to the geometric centers of jets, entropy generationbecomes slighter and slighter. This is resulted from that thestreamlines can keep straight inside this zone but are stretched

Fig. 8. Total entropy generation number versus dimensionless time: Re¼ 10eblue dotline; Re¼ 500ered solid line (For interpretation of the references to color in this figurelegend, the reader is referred to the web version of this article.).

0.412959

0.41

0.395

0.35

0.365

0.38

0.04 5

0.3 9 0.375

0.355

0.41392

X

Y

0 1 2 3 4 50

0.5

1

1.5

2a

2.00445

6.07005

9.31618

9.80009

8.03844

3.95813

0.668718

9.88499 10.086510.7749

14.276

21.5802

16.4231

24.7261

X

Y

0 1 2 3 4 50

0.5

1

1.5

2b

Fig. 9. Streamlines, pressure (a) and entropy generation number distribution (b) at W/L¼ 2/5. (For interpretation of the references to color in this figure legend, the reader isreferred to the web version of this article.).

Fig. 10. Streamlines, pressure (a) and entropy generation number distribution (b) at W/L¼ 3/5. (For interpretation of the references to color in this figure legend, the reader isreferred to the web version of this article.).

S. Chen, C. Zheng / International Journal of Thermal Sciences 49 (2010) 2067e20752072

significantly outside. Velocity gradients are enhanced by stretchedstreamlines, so FFI increases. The streamlines are stretched mostsignificantly near the corners where the maximum of S appears.From Fig. 6(a) one can see that the minimum of S emerges at thegeometric centers of jets. The peak value of S at the line x¼ L/2appears at the geometric center of the domain. The peak value

Fig. 11. Streamlines, pressure (a) and entropy generation number distribution (b) at W/L¼referred to the web version of this article.).

increases with Re increasing although the increasing is slight whenRe> 50. The variation of S along the line x¼ L/2 is significant andlooks like a “mountain”. However, the variation of S along the liney¼W/2 is quite small except near the outlets, which makes thecurve look like a “riverbed”. Where 1< x< 4, the lines of S areparallel to the abscissa, no matter what value Re is. The peak value

4/5. (For interpretation of the references to color in this figure legend, the reader is

Fig. 12. Entropy generation number along line x¼ L/2 (a) and line y¼W/2 (b): W/L¼ 2/5eblue solid line; W/L¼ 3/5egreen dash line; W/L¼ 4/5eblack dots (For interpretation ofthe references to color in this figure legend, the reader is referred to the web version of this article.).

S. Chen, C. Zheng / International Journal of Thermal Sciences 49 (2010) 2067e2075 2073

of S at the line y¼W/2 appears at the outlets. Within a thin layernear the outlets, S increases quickly. Along the line y¼W/2 Sincreases with Re but the increase becomes very slowly whenRe> 50, agreeing with the conclusion obtained above.

Fig. 7 plots the variation of total entropy generation numberversus Re. It is clear that the total entropy generation number isa monotonic increasing function of Re. But when Re> 100, theincreasing is hardly observed. This curve can be approximated byan exponential function:

Stotal ¼ 114:34� 5:167� ð0:9497ÞRe (20)

Fig. 8 illustrates the variation of total entropy generation numberversus dimensionless time. For clarity, only two curves correspond-ing to Re¼ 10 and Re¼ 500 are shown. It is clear that the first tran-sient peak values of total entropy generation number for both casesemerge at the same time zz 10. The reason is that at time zz 10 thegas ejected from the top and bottom jets begins meeting andimpinging (please mention 0.5�W/v0¼1/0.1¼10). The first tran-sient peak value is also the maximum during the transient state andmuch bigger than what else. Moreover, the maximums for differentRe are almost the same. From the curve of Re¼ 10, it is obvious thatthere is a second transient peakvalueduring the transient state at zz800, when the disturbance caused by impinging spreads over thewhole domain. This phenomenon is not clear for Re¼ 500 becausewhen Re is bigger than 50, the flow field will receive significantinfluence of pressure wave bouncing back and forth between theboundaries before the wave completely being damped, as demon-strated in detail in our previous work [13], which will cause entropy

Fig. 13. Total entropy generation number versus W/L.

generation suddenly increasing. The sudden fluctuation of entropygeneration caused by the damping traveling pressure wave can befound clearly from the curve for Re¼ 500.

6.2. Effect of distance ratio

In order to show the effect of the distance ratio on entropygeneration distribution, in this subsection Re¼ 100 is used to typifyall cases in this study. As mentioned above, the variation of Sbecomes very slight when Re> 100.

Figs. 9e11 illustrate the streamlines, pressure and entropygeneration number distributions at W/L¼ 2/5, 3/5 and 4/5,respectively. Although the influence of distance ratio on flow field issmall, its influence on entropy generation is obvious. Withincreasing of W/L, the concentric toroid-coil-like curves withrespect to the geometric centers of jets, which represent thecontours of entropy generation number, spread outwards. Conse-quently, the area in which entropy generates intensively becomessmaller and smaller. Although the maximum of S still appears nearthe corners, its value becomes smaller against W/L increasing.

This conclusion is also supported by Fig. 12, in which the vari-ations of S along the line x¼ L/2 and line y¼W/2 for different W/Lare shown. The “mountains”, which mean the variations of S alongthe line x¼ L/2 for different W/L, in Fig. 12(a) become flatter andflatter withW/L increasing. The same phenomena also can be foundfor the “riverbeds”, which represent the variations of S along theline y¼W/2 for different W/L (Fig. 12(b)). The reason is that for

Fig. 14. Total entropy generation number versus dimensionless time: W/L¼ 2/5ebluedot line; W/L¼ 4/5ered solid line (For interpretation of the references to color in thisfigure legend, the reader is referred to the web version of this article.).

S. Chen, C. Zheng / International Journal of Thermal Sciences 49 (2010) 2067e20752074

bigger W/L, the flow receives smaller influence of stretching byboundaries. Consequently FFI decreases against W/L increasing.

Fig. 13 plots the variation of total entropy generation numberversus W/L. It is clear that the total entropy generation number is sa monotonic decreasing function of W/L, agreeing with theconclusion abstracted above. This curve can be approximated bya power function

Stotal ¼ 22:325� ðW=LÞ�1:769 (21)

which means the decreasing of total entropy generation numberbecomes slowly against W/L increasing.

Fig. 14 illustrates the variation of total entropy generationnumber versus dimensionless time. For clarity, only two curvescorresponding to W/L¼ 2/5 and W/L¼ 4/5 are shown. It is obviousthat the first transient peak values of total entropy generationnumber for both cases emerge at the same time z z 10, too. Asexplained above, at that time the gas ejected from the top andbottom jets begins meeting and impinging. The first transient peakvalue is also the maximum during the transient state and muchbigger than what else, which same as that in Fig. 8. Moreover, themaximums for different W/L are almost the same, too. From thisfigure, we also can observe the sudden fluctuation of entropygeneration caused by the damping traveling pressure wave. Forbigger W/L (for example W/L¼ 4/5), the traveling pressure wave isdamped quickly. The second transient peak value during the tran-sient state mentioned above can be observed from the curve forW/L¼ 2/5, although it is not clear. As explained above, when Re isbigger than 50, the fluctuation caused by the traveling pressurewave hampers our observation. But forW/L¼ 4/5, this peak value isoverlaid by the sudden fluctuation caused by the traveling pressurewave.

7. Conclusion

In this paper, entropy generation in impinging flow confined byplanar opposing jets is reported for the first time. The latticeBoltzmann model proposed in our previous work is used to solvethe governing equations for flow field as well as the entropygeneration equation, instead of traditional numerical methods.

The effects of the Reynolds number 10� Re� 500 and thedistance ratio between opposing jets 2/5�W/L� 4/5 on entropygeneration are revealed. It is found that the local entropy genera-tion number is more sensitive to the variation ofW/L than RewhenRe> 50. The total entropy generation number increases exponen-tially with Re but decreases as a power function ofW/L. And we findthat the entropy generation will receive significant influence fromthe damping traveling pressure wave during the transient state.The maximum entropy generation during the transient stateappears when the gas ejected from the top and bottom jets beginsmeeting and impinging.

Acknowledgments

This work was supported by the State Key Development Pro-gramme for Basic Research of China (Grant No. 2010CB227004-01),and the National Natural Science Foundation of China (Grant No.50936001 and 50721005).

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