AD

co

N CONTRACT REPORT BRL-CR-564 D TCELECTE

I MAR 5 97I

A MODEL FOR ANALYSIS OF SECONDARYCOMBUSTION IN GUN EXHAUST PLUMES

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GASL TR 276 BRL-CR-564

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______________________________RDT&E 162618 AH8011. TITLE (/ncludei, Security Classification)

A MODEL FOR ANALYSIS OF SECONDARY COMBUSTION IN GUN EXHAUST PLUMES

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16. SUPPLEMENTARY NOTATION

17. COSATI CODES 18. SUBJECT TERMS (Continue on reverse if necessary and identify by block number)

FIELD GROUP SUB-GROUP FMUZZLE R'ASH, . AEROTHERMOCHEMISTRY,19/, n4 MUZZLE "AST19 STEADY .TURBULENT.-MIXING '

19. ASTRACT (Continue on reverse if necessary a ridentify by block number)

',An analytical model of the unsteady turbulent mixing and chemical reaction at the pro-pellant-air interface in a muzzle blast flowfield has been developed. Particular attentionhas been given to the interplay between falling pressure and entrainment of variable-

temperature gases as a key element of the processes leading to secondary combustion of thepropellant in the gun exhaust plume. For the sake of computational economy, the model hasbeen restricted to the axis of symmetry, and a two-layer integral-method has been developed.Instructions for use-of the computer code are presented. [,

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GASL TR-276

TABLE OF CONTENTS

LIST OF TABLES. ...... ......................... iv

LIST OF FIGURES. ............................... v

1.0 INTRODUCTION .. ........................... 1

2.0 DISCUSSION OF THE MODEL .. ......... ........... 3

3.0 DESCRIPTION OF THE ANALYTICAL FORMULATION .. ............ 7

Assumptions and Basic Equations. ...... ........... 7

Integral Form of the Continuity and Energy Equations. .. ..... 11

Distribution Functions. .. .......... .......... 12

Species Equations. ..... ........... ........ 15

Final System of Energy and Species Equation .. .......... 16

Turbulence Modeling. ..... ........... ....... 21

Chemical Kinetics Modeling. .. ......... ......... 22

4.0 DESCRIPTION OF THE COMPUTER CODE .. ..... ........... 24

Overview. .. ......... ........... ....... 24

Input data. .......... ........... ..... 25

Output Data. ..... ........... ........... 28

ii

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5.0....................... .................. 31

RE.FERENCES. ................... .............. 34

DISTRIBUTION LIST .. ...... ...... ..... ...... ..... 45

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LIST OF TABLES

TABLE 1. Data Format for UMLAC Input File. ...... ........ 32

TABLE 2. Data Format for UMLAC Boundary Condition File. .. ...... 32

TABLE 3. Data Format for CREK Input File .. ..... ......... 33

Accesion For

NTIS CR jDTIC TABUnannoir'ced [Justificatwii

By ................... ... ...

Availability Codes

Avajii a,'d for A .Dist Sp~cial Ir~fNSP~cre

A,1 6

iv

~*- '. ~. ~.Y9M A ~

GASL TR-276

LIST OF FIGURES

Figure 1. Schematic of Muzzle Blast Flowfield ...... ............ 36

Figure 2. Schematic Showing Typical Streamline Traces and Entropy . 37

Figure 3. Typical Input Data File for UMLAC (Logical Unit 5) . . .. 38

Figure 4. Typical Boundary Condition Data File for UMLAC (Logical

Unit 9) ...... ..... ..... ...... .... 39

Figure 5. Typical Thermodynamic and Reaction Kinetics Data File for

UMLAC and CREK (Logical Unit 15) .... ............. ... 40

Figure 6. Typical Output Data from UMLAC (Logical Unit 6) . . . 41-46

V

r .1 ,11

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1.0 INTRODUCTION

Muzzle blast and flash are inherent to ejection of a chemically-propelledprojectile from a gun barrel. Blast is the result of release of high-pressuregases and the ensuing over-pressure of the surrounding atmosphere. Flash isthe radiation from the expelled propellant gases, which is frequently augmentedby afterburning of the hydrogen-rich propellent gases in the atmosphere. Blastand flash, as well as the dispersal of propellant gases and combustionproducts, represent environmental hazards to the gunner and other personnel ormachinery in the area of the gun platform. Additionally, they are functionallyundesirable in that they reveal the gun location. Consequently, considerable

effort has been devoted to development of mechanical devices which are intendedto suppress blast and flash or to deflect muzzle gases away from sensitiveareas (e.g., aircraft engine inlets). Although blast, flash and thedistribution of propellent exhaust gases are closely interrelated phenomena,muzzle devices which are successful in achieving one objective (e.g., exhaustgas diverters and recoil attenuation devices) are often counterproductive inother areas (i.e., flash suppression).

The coupling between muzzle blast and flash is evident in theamplification of blast overpressure by propellant afterburning, in theaugmentation of flash by muzzle brakes (which divert the exhaust gases), and inflash suppression by conical exhaust nozzles (which focus the exhaust gases).Blast and flash therefore clearly involve an interplay between mechanicaldesign of the gun and chemical composition of the propellant. A more thoroughdiscussion of the Yhenomenology and recent experience can be found in thestudies by Keller ,2 and Schmidt3 , as well as in the substantial body of priorwork cited therein.

MR, The present work concentrates on development of a model of muzzle flashwhich embodies the essential thermo-chemistry and fluid-dynamics of the processwhile retaining computational simplicity appropriate to exploratory studies andengineering analyses. The essential features of the phenomenon are consideredto be unsteady mixing of the propellant exhaust gases with shock-heated air andconcurrent chemical reaction in a flowfield dominated by an expanding blastwave. Emphasis has been placed on identification and representation of theprimary physical processes rather than detailed numerical analysis orcomprehensive comparative studies.

1 Keller, G.E., "Secondary Muzzle Flash and Blast of the British 81-mm,

L16A2, Mortar" ARBRL-MR-03117, July 1981.

E4.1 2 Keller, G.E., "The Effect of Propellant Composition on Secondary

Muzzle Blast Overpressure" ARBRL-MR-03270, April 1983.3 Schmidt, E.M., "Secondary Combustion in Gun Exhaust Flows" ARBRL-TR-

02373, October 1981.%

1

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The basic fluid-dynamical model draws on the muzzle blast analysis ofErdos and DelGuidice4 , which describes the unsteady flowfield along the gunaxis from the muzzle to the blast wave. This model assumes, however, that thepropellant gases are effectively inert after expulsion from the muzzle andneglects any mixing along their interface with the atmosphere. Both theseassumptions are removed in the present model. The chemical-kinetic mechanismsgoverning the reaction of the propellant gases with air, including the effectsof potassium additives, are drawn from the work of Yousefian5 6 . This workpays careful attention to the complex kinetics of the afterburning reactionsand the potential for flash suppression by chemical additives. However, thefluid-dynamics are modeled by steady-state mixing with constant boundaryconditions, which unfortunately ignores essential transients in pressure andvariations in boundary conditions on the mixing layer due to the blast wave.

1

The present work also draws on the study of secondary combustion bySchmidt3 . Schmidt recognizes the essentially unsteady nature of thephenomenon, but attempts to apply the unsteady conditions to empirical flashcriteria without modeling the rate of mixing or of reaction. His study is alsoimportant in that it identifies the mechanism by which muzzle brakes tend toaugment flash, namely by the generation of multiple shocks. The present workdoes not attempt to explicitly model muzzle devices, such as brakes. Howeverit does allow the effects of such devices to be modeled through the fluid-dynamic boundary conditions placed on the mixing layer, in the same fashion asSchmidt considers their effect on the empirical flash criteria.

Finally, the evaluation of available muzzle flash codes performed byKeller 7 is pointed out since it reinforces the requirement for a model such asproposed herein.

4 Erdos, J. and DelGuidice P., "Calculation of Muzzle Blast Flowfields"

AIAA Journal, Vol. 13, No. 8, August 1975 pp. 1048-1056.

5 Yousefian, V., "Muzzle Flash Onset" ARBRL-CR-00477, Feb. 1982.

6 Yousefian, V., "Muzzle Flash Onset: An Algebraic Criterion and

Further Validation of the Muzzle Exhaust Flow Field Model" AIRRL-CR-00506,March 1983.

7 Keller, G.E., "An Evaluation of Muzzle Flash Prediction Codes" ARBRL-MR-03318, Nov. 1983.

2@o4

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2.0 DISCUSSION OF THE MODEL

The essential features of a muzzle exhaust flowfield are sketched in

Figure (1). The propellant gases expand supersonically upon release from the

gun barrel. In a vacuum, the expansion process would approach a sphericalsource flow with it's origin at the muzzle. The propellant gases would form,in effect, an expanding spherical "balloon". In the atmosphere, the sphericalexpansion is contained by a finite back-pressure exerted by the surroundingair. This boundary condition leads to the barrel shocks which terminate thenear-spherical expansion and turn the propellant gases back to a near-axialflow direction. The expanding front of propellant gases still forms a"balloon" of sorts which drives a strong shock wave, i.e., a "blast wave", intothe atmosphere ahead of it. The pressure field behind the blast wave causesthe supersonic propellant gases to compress to a matching pressure acrossanother strong shock, termed the "Mach disc". The subsonic flowfield betweenthe Mach disc and the blast wave again resembles a spherical expansion, a factthat allowed Erdos and DelGuidice4 to model this region as a one-dimensionalunsteady flow with spherical symmetry.

The supersonic portion of the propellant expansion is quasi-steady; i.e.,it adjusts almost instantly to the time-varying conditions at the muzzle as the

propellant gases empty from the gun. The position of the Mach disc variescontinuously, but upstream of the disc the flowfield resembles a steady, under-expanded rocket plume. Mixing of the propellant gases with air will occuralong the lateral boundaries of the plume, and since the propellant gases aretypically rich in hydrogen and carbon monoxide, some afterburning in thismixing layer may be expected. However, since the air side of the layer isessentially quiescent while the propellant gases are moving at high supersonicspeeds, and since the total (stagnation) temperature of the air is much lowerthan that of the propellant gases, the maximum static temperature in the mixing

8.layer can be estimated by analogy with a hypersonic cold-wall boundary layerAccordingly, the maximum static temperature will be about one-quarter the total

temperature of the propellant gases, and this temperature will occur where thevelocity is about one-half the propellant velocity and the gas mixture willconsist of about one-half air and one-half propellant gases. The miy ire ratio

and temperature would therefore favor ignition, but the velocity may inhibitignition by allowing less time at this condition than the required ignitiondelay time. Characteristic gas residence time in this layer will be a smallfraction of the gun emptying time.

The subsonic portion of the propellant gas expansion i.e., the portionbetween the Mach disc and the propellant-air interface, is decidedly non-steady. Since the Mach number of the flow is low, the static temperature ofthe propellant gases approaches the total temperature. Furthermore, the

8 Dorrance, W.H., Viscous Hypersonic Flow, McGraw Hill Book Co., NewYork, 1962.

U%1

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velocity is low so the residence time in this layer will approach the gunemptying time. Consequently, afterburning is far more likely in the mixinglayer that develops at the propellant-air interface in this region (i.e., nearthe gun axis) than along the boundaries of the supersonic plume.

The static temperature of the propellant gas and of the air at theirinterface varies in time as the pressure drops. However, the entropy of thepropellant gas and of the air at any point in the flowfield defines thethermodynamic relationship between the temperature and pressure. The timevariation of entropy is governed by the energy equation which in the absence ofchemical reactions, heat conduction, diffusion, etc., simply states that theentropy is convected along streamlines (particle paths) except where thestreamlines cross shock waves. The entropies of the two sides of the interfaceare therefore determined by the initial conditions at release of the propellantgas. The blast wave begins as a strong shock wave and maintains its strengthas the cumulative amount of energy released by the escaping propellant gasesincreases. The strength of the blast wave begins to decay, however, as itacquires a spherical shape and grows in radius. The cumulative energy releasedby the propellant gases also begins to asymptote to a constant as the gunbarrel empties, and the blast wave decay rate transitions from the power lawassociated with a constant rate of energy deposition to a constant total energyrelease (i.e., a point explosion). The net result is that the entropy of theshock-heated air is initially about a constant near the propellant-airinterface and then steadily decreases with increasing distance from theinterface.

The entropy of the propellant gas, on the other hand, consists of theentropy generated by the propellant combustion process in the breech-end of thegun barrel plus the increment added as the gas passes across the plume Machdisc. The entropy layer on the propellant side of the interface therefore alsoexhibits a spatial variation associated with the initial increase in strengthof the Mach disc, but as the Mach disc later decays in strength and collapsesthe propellant-side entropy asymptotes to the combustion generated value. Thepropellant-side flowfield therefore exhibits a significant increase in entropywith increasing distance from the interface, a maximum entropy at some point,and a subsequent decrease and asymptotic approach to the entropy of the gasventing from the muzzle.

The significance of the propellant-side and air-side entropy layers isthat the shock-processed air is hot as it accumulates near the interface, andremains relatively warm as the pressure drops and volume expands. However, thehot propellant gas leaving the muzzle is further heated by the Mach disc, andas the pressure drops a temperature distribution resembling the entropy layerdevelops, namely: warm at the interface, hotter proceeding away from theinterface to a maximum temperature, and then cooling again and asymptoting tothe temperature of the gas venting through an isentropic subsonic expansion tothe atmosphere. Consequently, as the gases mix along the interface, the mixinglayer will initially entrain relatively hot propellant gas and air. (The term"relative" is used to denote the dependence on pressure.) As more propellant

GASL TR-276

gas is entrained, its relative temperature will increase, accelerating the rateat which afterburning reactions between the air and the unburned propellant gasproceed. The relative temperature of the entrained propellant gas will reach amaximum and then slowly decay toward an asymptotic final temperature.

Reactions between the unburned propellant gas and air are initiated in themixing layer at very high pressure, but as the pressure drops the molecularcollision frequency drops in direct proportion and the speed of the chain-forming ignition reactions may not be sufficient to sustain combustion.However, the ignition-sustaining reactions will be promoted by the entrainmentof increasingly hotter propellant gas, which will offset the effect of droppingpressure to some extent, since the reaction rates are typically exponentiallyproportional to temperature. If the ignition reactions become self-sustainingduring the period of entrainment of shock heated propellant gas, flash occurs.If they have not become self-sustaining by the time the entrained propellantgas has reached it's asymptotic decay period, there will be no flash due toafterburning.

Clearly then, the occurrence of flash is critically dependent on thecompeting mechanisms of falling pressure and entrainment of variabletemperature propellant gas which are superimposed on the mixing process. Thepresent model has been specifically formulated to address this phenemonology.

The role of chemical additives in the propellant is simply to interferewith the ignition process by tying up the free-radicals needed to sustaincombustion. The amotut of additive needed will depend on the transient fluid-mechanical characteristics of the specific muzzle flowfield. The present modelis also intended to provide a capability to rapidly assess the effectiveness ofchemical additives.

As the preceding discussion of the phenomology should indicate, thetransient characteristics of the flowfield, particularly the pressure impressed -on the mixing layer and the variation in temperature (i.e., entropy) of the Nentrained gas, are considered essential to evaluation of the chemical processes Ileading to flash. These processes occur on a time scale comparable to thecharacteristic time for the flowfield transients and must therefore be modeledas "finite rate" processes. Unfortunately, the cost of carrying out finite-rate chemical calculations can escalate very quickly. The fluid dynamics

-. require solution of three equations: conservation of mass, momentum andenergy. Finite-rate chemistry adds one more equation for each chemical speciesconsidered, and requires evaluation of complex reaction rate expressions for

'h every reaction in which the considered species participates. Consequentlythere is a strong fiscal motivation for keeping the number of spatial locationsat which the species equations must be solved to a minimum. The bare minimumis, of course, one point, e.g., the interface.

A two-layer model of the mixing layer has been adopted in which the

inviscid interface divides the two layers. The governing equations forconservation of mass and energy are analytically integrated from the interface

5

''p-

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outward to the "edge" or "front" of each layer, leading to a pair of urca:, qdifferential equations for the thermal thickness of each layer. The momer:r..equation is eliminated from present consideration by restricting attention"the axis of symmetry. The integral species conservation equation Sautomatically satisfied by assuming the thickness of the species diffus 0 rnlayer is equal to the thermal thickness.

V . Thus, the present model only requires solution of the governing equationsat the interface, plus the two equations for the thermal thickness of eachlayer.

A detailed description of the analytical formulation of the two-layermodel is presented in the following section.

0"

6:

4~o%

4.l

-4 ,4•. - . . . . - . - ° - . . .. . . -° .

GASL TR-276

3.0 DESCRIPTION OF THE ANALYTICAL FORMULATION

Assumptions and Basic Equations

As previously depicted in Figure (1), a mixing layer develops along theinterface between the propellant gas and the air. Within this mixing layer,hot propellant gas (shock-heated by the Mach disc) is brought into contact withhot oxygen (in the air which has been shock-heated by the blast wave). Themixing layer can be viewed as two sublayers: one which extends outward fromthe inviscid interface between the propellant gas and air and one which extendsinward from it. The inviscid interface provides a convenient frame ofreference for the mixing layer.

The present model is restricted to development of a planar mixing layer atthe gun axis. In other words, the radius of curvature of the inviscidinterface is assumed to be much larger than the thickness of the mixing layer.Furthermore, forces due to acceleration of the interface are neglected. Thelatter assumption is perhaps suspect at early times, but the interface locationbecomes virtually steady by the time secondary combustion is usually observed.The former assumption may also be questioned for later times when the mixinglayer becomes very thick. Nevertheless, the present model should adequatelymodel the phenomena of interest, and provide a basis for further modeling asneeded to refine the predictive capabililty.

The inviscid interface is therefore used as the base of a two-dimensional, rectangular coordinate system: x is the distance along theinterface and y is the distance normal to it.

Modeling of the diffusive processes poses a somewhat perplexing dilemma.Shadowgraphs of muzzle blast flowfields such as those taken by Schmidt andShear9 show highly turbulent structures in the mixing layers. However, thepropellant-air interface near the gun axis is not visible. No inviscidvelocity jump exists across the interface at the axis, and hence the usual 4mechanism for generating turbulent motion is absent at that point. Theinterface cannot support a pressure jump, and therefore the pressure gradientsalong the interface must be identical. Consequently, differences in thevelocity gradients must be associated with the differences in density acrossthe interface. Hence, velocity jumps that develop away from the axis due tothe pressure gradient are expected to be relatively small. Again, there seemsto be an absence of strong driving mechanisms for turbulence generation nearthe axis. By contrast, the inviscid velocity jumps across the lateralboundaries of the plume are large, and the observed turbulence in this area isclearly consistent with a shear-driven mechanism.

9 Schmidt, E. and Shear, D. "Flowfield About the Muzzle of an M16Rifle" BRL Report 1692, Jan., 1974. Also, AIAA Journal, Vol. 13, PP. 1088-1093, Aug., 1975.

7

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Acceleration of an interface between fluids of differing density in thedirection of the lower density fluid gives rise to the classical Rayleigh-Taylor instability of the interface. The observed broadening of the interfaceand turbulent mixing of driver and driven gases in a shock tube has beenattributed to this mechanismlO,'I '1 ,13 . The interface in a shock tube isproposed herein as an possible model for the muzzle flowfield interface at theaxis of symmetry.

In view of the lack of a well-defined basis for modeling the diffusivemechanism for the mixing layer that develops along the interface, three modelshave been postulated. The first is simply a laminar viscosity, thermalconductivity and diffusivity model. The second is a heuristic thermalconductivity model based on a classical eddy viscosity-type formulation. Thethird is a two-equation turbulent thermal energy model developed by analogywith the K-c model for the turbulent kinetic energy equation. In the lattertwo models, the effective Lewis and Prandtl numbers are assumed to be unity.

The propellant combustion products are described as a reacting mixture ofchemical species, each behaving as a perfect gas (in vibrational equilibrium).Similarly, air is described as a mixture of molecular and atomic nitrogen andoxygen, and nitric oxide (also in vibrational equilibrium). Fick's law, with asingle diffusion coefficient for all combinations of species, is used to-describe mass diffusion.

The thin-layer approximations to the Navier-Stokes equations will beassumed to apply, i.e.:

82/ay 2 >> a2/ax2 and p = p(x,t).

*The following set of governing equations thereby obtains (expressed here inconservative form):

10 Richtmeyer, R. D., "Taylor Instability in Shock Acceleration of

Compressible Fluids," Communications on Pure and Applied Mathematics, Vol.13,1960, pp. 297-319.

11 Levin, M. A., "Turbulent Mixing at the Contact Surface in a Driven

Shock-Wave," The Physics of Fluids, Vol. 13, No. 5, 1970, pp. 1166-1171.

12 Andronov, V. A. et al., "Turbulent Mixing at Contact Surface

Accelerated by Shock-Waves," Soviet Physics-JETP, Vol. 44, 1976, pp. 424-427.

13 Houas, L., Brun, R. and Hanana, M., "Experimental Investigation of

Shock-Interface Interactions," AIAA J, Vol. 24, No. 8, August 1986, pp. 1254-1255.

".0 *.0 " o .r -*8

U d -

GASL TR-276

a+ apu+ aV oat ax ay (1)

bu + a(pu2 + a(Puv) auat ax ay a y 5-y (2)

ap = 0ay (3)

+ auh + apvh a kTah + pat ax ay =y (p ay) at (4)

apa i +apua i aPVai - L (pD T 8aal

a t y ay + j (5)

with the following equations of state:

p - pRoTE(ai/MWi) (6)

h - Mai (hi + Ah?) (7)

where:

Thi c- f C (T) dT

0 (8)

and:C p . a i C , 9

Consideration will be further restricted to the axis (x - 0) where u = 0and a/ax - 0, reducing Equations (1) - (5) to:

at ay (10)

9

_ * . . .J..9~j * . . ..

GASL TR-276

3ph + h kT 8h d+ +at ay = ay Cp ay) dt (11)

apa i 8Pvai a- ( 8ai

-a- (PDr--;) + vi (12)

Determination of the chemical production term, *i, will employ themethod developed by Pratt14 . This, in turn, uses the thermodynamicrepresentation individual species developed by McBride, et.al. 15 Accordingly,a fifth-order polynomial is used in the present analysis for the sensibleenthalpy and specific heat of each species:

5hi = Z Cij TJ/j

j=1 (13)

5Cpi - E Cij TJ- (14)

j=1

The sixth coefficient defined by McBride, et.al. 15 , is identified herein as theheat of formation at AhT. These coefficients are additionally used to definethe equilibrium constant and either the forward or backward rate constant(whichever is not otherwise specified) from the Law of Mass Action.

Details of the technique for determination of the chemical production termVi are provided in Reference 14. For the present discussion it may be regardedas a "source" term for Equation (12) that is a function of the instantaneouspressure, enthalpy and mixture composition:

i f(p, h, aj) ; 1 , NS and i = 1, NS (15)

The CREK code described in Reference 14 is contained in the presentcomputermodel as an auxiliary program that integrates Equation (15) by theNewton-Raphson iteration technique described in Reference 14:

14 Pratt, D.T., "Calculation of Chemically Reacting Flows with Complex

Chemistry" in Studies in Convection, B.E. Launder, Ed., Academic Press, NY1977 15 McBride, B.J., Helms, S., Ehlers, J.G. and Gordon, S., "Thermodynamic

Properties to 6000°K for 210 Substances", NASA SP-30001, 1966.

10

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t+At

Aaireaction f LP dt (16)*t p

This integration is performed at constant pressure and enthalpy, but includes Uthe variation in temperature, density and composition due to reactions. (Thescheme admits both finite-rate reactions and fully-equilibrated reactions; onlythe finite-rate option is of present interest.) Blending of the change incomposition due to reactions with the change due to diffusion will be discussedlater in connection with the solution of Equation (12). However, it should be Irecognized that the changes in temperature and density obtained in this step ofthe solution are incomplete - the complete changes are obtained from thesolution of Equation (10) and (11). The influence of reactions entersimplicitly through the use of Equations (6) and (7). If, for example, Equation(11) had been written in terms of temperature rather than enthalpy, atemperature "source" term involving the product iAh9 would appear and thechange in temperature obtained concurrently with the solution of Equation (16)would be used to represent time-integration of this term.

Integral Form of the Continuity and Energy Equations

Equations (10) and (11) can be converted from a pair of partialdifferential equations in two independent variables (y and t) to a pair ofordinary differential equations with time (t) as the independent variable byanalytically integrating over y and representing the integrals in terms ofshape parameters that are functions of time. The limits of integration are theouter edge of the mixing layer on the propellant side, 62, and the outer edgeof the mixing layer on the air-side, 61. We choose to form a pair of equationsby integrating first from y = 0 to y = 61 and again from y = 0 to y = 62. Thesubscripts 0, 1 and 2 will be used hereafter to denote conditions at y = 0, 61and 62 respectively.

The continuity equation, Equation (10), becomes:

d 6 i d6idtf pdy - pi + pivi = 0 (i = 1,2) (17)

0

where the boundary conditions v = 0 at y = 0 and a/ay = 0 at y = 6i have beenimposed and L'Hopital's rule has been observed.

V%11

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The energy equation, Equation (11), similarly becomes:

d 6i d6ia f phdy - pihi y-- + pivihi

0

k= 8h~ dd i= -Co (1h)o + !L (Pai) _ p !L(18CPO Bay' 0 d t dt (18)

The velocity vi can be eliminated by multiplying (17) by hi and substituting itinto (18), giving:

d di d 6i kTo ahd f phdy - hi a pdy -- - + 6 (i - 1,2) (19)

o cpo ayO idt

Evaluation if the integrals in Equation (19) requires representationof the integrands by some appropriate distribution functions satisfying theboundary conditions at y = 61 and 82- Additional accuracy can be obtained byalso satisfying the differential equations ((10 and (11)) at y - 0. Again,continuity can be substituted into the energy equation to eliminate (8v/ay)o ,yielding:

dho (kT h) )° + k (20)Po dt ay CPay 0 dt

Distribution Functions

Let f(y) represent any of the dependent variables. The boundaryconditions are:

a_ . 0 and f fi at y=-6 (21)ay

f = fo at y - 0 (22)

The following Fourier series is chosen to represent f(y):

f a + b cos in + c cos2nn (23)

12

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where n - (y - 62)/(61 - 62) (24)

This series automatically satisfies the derivative condition Iin (21). The coefficients a, b and c are defined by the remainingboundary conditions:

a =(-f 0 + I (f2 - fl) cos'rri + i (f2 + fl)cos 2vnri)/(cos 2nri0 1) (25)

b - (f2 -fl) (26)

c =(fo (f2 + fl) U ( 2 -fl) Cos Indj0 /(cos 2nnro 1) (27)

where:

10 62/(61 - 6)(28)

Note that a, b, c and no, are functions of time only. Their timederivatives are:

no (62'1 -6162)1(61 - 62)2(9

i= +- Co & (- + Cos wqo) + ...)/(cos 2wno 1) (30)

0- ( 2 il) (31)

00( + j(cos wrrl - 1) f1 + . . .)/(cos 2nrri - 1) (32)

It should also be noted that a, b, c are linear functions of fo, fl, f2, 61 and62:

it alio + a2i, + a3'2 + a491 + a5g2 (33)

6- b2 '1 + b3'2 (34)

M Clio + c2'1 + c3i2 + c4al + c562 (35)

13

GASL TR-276

The primary role of the distribution function is to permitevaluation of the integral of the function, viz:

61 b(61-62) C(61-62)f fdy - a6j IF si 10 2v sin 2vrri (36)0

and the derivative of the integral, viz:

6id fdy-a~i + 6iia+.(37

0

Evaluation of (37) is aided by the following definitions:

-Y, (a) - a (38)

Y'2(a, b, c, 61) - a6j -b 6 6 )sin iro - 2--LA..2sin 2 rr0 39

Y3 a 3a, 6) - -(61 62)(a2COSnno + a3cos2vrn0)6 (40)

I (a, b. c) -- (bsinwri0 c sin 27nri)/v (41)

where

62

+ i-622 (42)(61 -62)

Using these functions, Equation (37) becomes:

d fdy - y1(a)Si + Y2(&,6," 61) + I (a, b, C)(1 1 g 2)

0

+ y3(b, c, (62-62 )2 )81 + y,(a 2, a3, (6- 6 ) )62 (43)

14.

GASL TR-276

where a6,care given by Equations (33), (34) and (35), and

y2(, 6 ~,62) - 12(al, bl, cj, 6i) io +

+ Y2(a5, b5, C5 9 60)62 (44)

Accordingly, (43) can be written as:

6.dj f fdy - [yj(a)b , + [Y2(a]1, blo c1, 6i)1f0

0

" ['r2(a22 02, C2 6i5)f1

" [Y2(a3, b3, C3V 6J)]f2 (45)

" [IVa, b, C) + Y2(a4,, b4, C4 1 6i) + X3(b, c, 62 2)111(61 - 62)

+ [-I~a, b, C) + Y2(a5, b5, c5, 61) + X3(b, c. 12 -2)1'2(61 - 62)

Finally, the distribution function is also used to evaluate the first andsecond derivatives with respect to y at y =0:

=a (af) ~~= - (b sin ffn + 2c si n)(61..y)0 Taj 0 UY 0y, 61 62inr)(6

(b cos nno + 2 c cos inri) (47)

0y (61 - 6a2

Species Equations

The same integration steps as performed above could be applied to thespecies equations, Equation (12),which would yield a system of equationssimilar to Equation (19) but involving a separate limit of integration 6ij forthe two layers (1 -1,2) and each species Qj 1, NIS). - However, consistent

15%

P~~ N '1..

GASL TR-276

with the assumption of unit Prandtl and Lewis numbers, it may be assumed thateach species will diffuse to the "edge" of the mixing layer as defined by theconduction of heat across the layer, i.e., by the thermal thickness 6i .Equating these thicknesses makes the integral form of the species equationsredundant, and it becomes necessary to solve the species equation only at theinterface location y = 0. Each of the species mass fractions will be describedby a distribution function fi of the form given by Equation (23), viz:

fi= ai + bi cos nn + ci cos 2rr (i = 1, NS) (48)

where:

ai = ai (Mio0 ailai2,'Si, 62) (49)

bi b- (a- (50)1 'O i 2~

ci =c i (aioOil'ai2,61,62) (51)

and the coefficients ai, bi, c. have the same properties as displayed in %Equations (30) - (35) and (46), (47.

The species mass fractions at the interface are determined from Equation .

(12) evaluated at y = 0:

da.10 r8 ai

P - pDT + i o (52)o

Final System of Energy and Species Equation

Equations (19), (20) and (52), aided by (45), (46) and (47), form a,nhap atteit:ae ,,yed h eprtr at the

coupled linear system of first-order ordinary differential equations. Thecoupling occurs through the equations of state, Equation (6) and (7). Inparticular, the enthalpy at the interface, ho, yields the temperature at the ~ m

interface by iterative solution of (7). The coupling can be simplified, and theiteration eliminated by differentiating the equations of state and using thedensity, rather than enthalpy, as the dependent variable for the energyequation:

16• %

%V16'

• %.4, , "-I, b - , " , - , - - - - . ¢ . - - - , . - - . . . - , , . . . . . . .-

GASL TR-276

P/p - 6/p + T/T + R9 U (&i/MWi) (53)

and

6= &i(hi + Ah9) +jai(E CijTY.')i (54)

where

= IaiIM~ri)f-' (55)

Combining (20), (53) and (54) yields:

II 8 2) (a - -o) ziE (56)Pay2 0 io]

where:

C Z = aiCijTi (57)ii

Di A hi0 + E CijJ (58)j

Equation (56) now replaces (20), and the system (19), (52) and (56)can be expressed in matrix form as:

Aila, Ai2'2 + Ai3;0 + Ai,3+j&j - Bi (9

17

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where:

i = 1, NS +3 (number of equations) (60)

j = 1, NS (number of species) (61)

A i 0 (62)

A1 , 2 = 0 (63)

A1 ,3 = C (64)

A, = [+(- ( )CijTj (65)

B1 = ( + (1- PoAI,3) P - iD (66)c y2)o ayi

A2,1 = X1 (al) + I(al, a2, a3 ) + X 2(a8 , a2 , a3 , 61)

+ X3 (a 2 , a., 62/(61 - 62) 2 ) +. (67)

A2,2 -I(al, a2, a3 ) + ;2(al5 , a2

5, 5, 61)

+ X3 (a2 , a 3, -61/(61 - 62)2) (68)

A2 ,3 = X2(a11, a2

1, a31 61) (h0 + A1,3) + (69)

MW1a 2 a31 , 61)

A2,3+j Ml (£I- ii) CijTJ)'X2(al I a2 , a , )

1 (70)

- hlX 2 (al, b1, c1 , 61)

18

GASL TR-276

B2 = X2(all, a2 1, a31, 61) (-OAI 3 + 61) p(71)

_(I h 2 2 2-p a- 0- -plhlX 2(al , a2 , a 3 , &1)...

-p2b2X2(al 3, a2 , a33 6)

A3, 1 I (a,, a2, as) +I X2(a1'4, a2", a3'4, 62) + .. (72)

A2 ,1 =Xl(a,) - I (a,, a2, a3) + X2(als, a2 , a35 62) + ... (73)

A3 '3 = ) 2(a11 , a2

1 , a31 , 62) (ho + A1,3 ) + . .(74)

A33 = (m( - cdzj) CjTJ))X2(all a2 1, a31, 62)

j (75)

- h2X2(al, bl, cl, 62)

B3 =X 2(al1 a2 , a31 , 62) 0LA, 2

-(L h ) (;~lX(a,,,a2 2, a3 2, 62) ...cp ay 0 21

(;h)2X(a3 3 3,a2 ,a 3 ,62).

and:

A (77)

*A 3 -J,2 -0 (78)

A3 4 -3 ,3 - 0 (9

A3+J, -6jm (Kronecker delta) (80)

19

saA -'

'Il

GASL TR-276

N 1 kTa 2a.CT 2---)o + wj (81)3+j p. cpoY

where j = 1, NS and m = 1,NS.In view of the fact that A 3+j , 3+m = 68 m (i.e., 0 for j m m and 1 for j =

m) it is evident that the species equations are uncoupled and can be integratedindependently of the remainder of the system (over a single time step):

i = B3+i (82)

In this case the Bi's can be redefined as:

B.= Bi - ZA i = 1,2,3 (83)

Furthermore, since All = A1 2 = 0:

Po = B'I/A, 3 (84)

The B'2 and B'3 can be redefined again as:

B"2 = B'2 - A2,30 (85)

B"3 = B'3 - A3,3Po (86).B" 3

and the remaining 2 by 2 system is:

A2 ,1 61 + A2 ,262 = B"2 (87)

A3,1 61 + A3 ,262 = B"3 (88)

which is easily solved:

" A B(89)*61 A2 ,1A 3 ,2 - A3 ,1A2 ,2

20

%I

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B"3A2 1 - B"2 A3 1 (90)

62 A2 ,1A3 ,2 - A3 ,1A2 ,2

Thus the final system of differential equations consists of (81), (83),(87) and (88). These are integrated over a time step by a standard Runge-Kuttaprocedure.

Turbulence Modelin%

Three options are provided in the computer code for representation of thethermal conductivity kT and species diffusivity DT. It should be notedthat kT only appears in ratio to C and DT only appears in a product with p.Hence, for unit Prandtl and Lewis numbers:

kT/Cp f pDT - (91)

The first option is to use the laminar viscosity provided bySutherland's law:

T= S T3/2/ (T + SO (92)

where S1 and S, are onstants in the appropriate set of units (e.g., SI - 2.27x 10-8 ib sec/ft2/*R2 and S2 - 198.6°R).

The second option is to use a modified form of eddy viscosity; thefollowing is proposed here (based on analogy with accepted kinematic models):

= S3 ("' 61 (Pl - PO) + h2 62 P2 - Po)) (93)

where S3 is an empirical constant. (We have used S3 ..0143 lb sec/ft 2 in thecode.)

The third option is to use a model of the two-equation turbulentthermal energy equation. The model is derived by analogy with the widely-usedtwo-equation model of the turbulent kinetic energy equation. The two variables

in the present model are the mean square temperature fluctuation 5 2 and athermal dissipation coefficient, v. The thermal conductivity is related tothese variables by:

21

% % PI

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kT = CopC p /o (94)

These two variables are governed by differential equations developed from theturbulent thermal energy equation by analogy with the model equations for theturbulent kinetic energy equation:

De2 kT T 2 a kT ae 2 + PO _ 0 (95)

PA ckLT T2 a2

P2 ap

We have assumed that both 7 and v can be described by the samedistribution function used for the other variables, Equation (23),which,however, implies that their ratio is independent of y, and hence kT/Cp and pDTare likewise independent of y. It is therefore only necessary to evaluateEquation (93) and (94) at the interface, y = 0, to obtain the differentialequations for 002 and 00 :

2 kT aT )2 a2 92

Poo=C (y + ° -po~ 0(97)0 CP ay ay2

2

kT 02o21.)2 + L2 2- - 2 (98)$ - -(Cl-2ay + C2 2)o - C3 P0 - 0

P 8 dy eo 0

In this case, Equations (95) and (96) are added to the system of

equations to be integrated, and eo2 and *0 are added to the list of dependentvariables.

Chemical Kinetics Modelina

The final form of the species equation,Equation (82), can be writtenas:

22

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GASL TR-276

j i i

diffusion reaction (99)

where & and 6. are identifiable with the two terms inidiffusion . reaction

B3+i, Equation (81). Equation (97) can be integrated in a variety of ways,offering various advantages in speed, simplicity, stability, etc. We havefound the most stable procedure to be to evaluate &i first, by calling

reaction

the CREK code to integrate the chemical production rate over the time step At.Equation (82) is then integrated by the same Runge-Kutta scheme as thecontinuity and energy equations, yielding the combined effects of diffusion andreaction. The permissible step size At is governed by an error bound appliedto all equations integrated by the Runge-Kutta procedure. However, when therate of chemical reaction becomes very fast, the integration procedure in CREKcan be stable but yield changes in composition that are too large for theRunge-Kutta procedure to integrate accurately. Therefore, no more than a 180°Rchange in temperature due to reaction is permitted in any time step. If thislimit is exceeded the time step is halved and the evaluation of ai reaction isrepeated before proceeding to the Runge-Kutta integration.

The CREK code is very versatile, as well as accurate and stable. Thechemical constituents of the mixture are specified by their chemical names, thepolynomial coefficients defining their enthalpy, Ci., and the atomic weights ofthe elements making up the constituents. The kinetic rate mechanism is definedsymbolically by the names of the reactants and products in each reaction andeither the forward or backward rate constant in the form

ki - lOAiTBiexp (-Ci/T) (100)

All this data is specified in an input file and is therefore easily modified(without any recoding).

23

%-

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4.0 DESCRIPTION OF THE COMPUTER CODE

Overview

The preceding analysis, while complicated in derivation, results in afairly compact and efficient computer code. The Fortran code UMLAC (UnsteadyMixing Layer and Chemistry) executes on an HP 1000 minicomputer using the EMA(Extended Memory Area) capability of this machine. Although the calls to theCREK program are inefficient (the main program transfers data to CREK, thensuspends itself while CREK executes, (including a complete initialization of

the thermodynamic and kinetic rate data for each call,) CREK passes its resultsback to the main program, which then resumes execution,) the running times arenot unreasonable (i.e., 10's of minutes, typically.) A modest amount ofrecoding to include the CREK code as a true set of sub-routines and eliminatethe reinitialization would substantially accelerate the execution speed.

Input data is contained on three formatted ASCII files; one is read by themain program on logical unit 5, the second is read by the main program onlogical unit 9, and the third is read by both the main program and by CREK onlogical unit 15. The first file contains the initial conditions andintegration parameters for the case at hand. The second contains the boundarycondition data, i.e., pressure, inviscid flow densities on the air-side andpropellant-side of the interface, and species mass fractions on both sides, allas functions of time. The third contains the thermodynamic and kinetic ratedata; the thermodynamic data is read by both UMLAC and CREK.

In view of the importance of the inviscid entropy layer on the propellant-side of the interface pointed out in the earlier discussion, the programpermits up to 11 streamlines in the propellant-side flowfield to be specifiedas part of the boundary condition data. The local "edge" properties areobtained by linear interpolation in the y direction using the value of 62 atthe current time step, as indicated in Figure (2). Since the task of preparingthis massive amount of boundary data is quite formidable, a pre-processor codeDCREK has been written to provide an automated link between the inviscid muzzleblast flowfield code DAWNA16 and the present mixing layer code UMLAC. DCREKreads the pressure history along the interface from DAWNA and the densitiesalong each of the selected streamlines. (The density is isentropicallyadjusted to the interface pressure should a pressure difference exist betweenthe streamline and the interface). It also reads the initial chemicalcomposition for the streamlines, which can be equated to the equilibriumcomposition of the propellant gases at the muzzle exit, since that compositionremains effectively frozen through the subsequent supersonic expansion and

16 Ranlet, J. and Erdos, J. "Description of Fortran Program DAWNA for

Analysis of Muzzle Blast Flowfield" BRLCR-302, March 1975.

24

.94

GASL TR-276

recompression across the Mach disc. DCREK converts the pressure and density toan enthalpy history for each streamline using the initial composition, and thenperforms a finite-rate calculation along the streamline (preserving thepressure-enthalpy history). DCREK stores the pressure, composition and (post-reaction) density history for each streamline on an output file that becomesthe input data file for UMLAC. The same calculation is also performed on the

air-side, but only for the interface streamline. DCREK automaticallyinitializes the air-side composition to a mixture of 76.55% N2 and 23.45% 02(by mass).

Since the same thermodynamic and kinetic-rate data file is used by CREK inboth the DCREK and UMLAC codes, it should include the appropriate air

constituents and air reaction mechanisms as well as the propellant gasconstituents and mechanisms. For typical muzzle blast conditions the airconstituents are N2 , N, 02, 0 and NO, and the reactions among them are oxygendissociation and the Zeldovich mechanism for nitric oxide formation:

02 + M ; 0 + 0 + M (101)

N2 + 0 NO + N (102)

02 + N NO + 0 (103)

Input data

The initial conditions and option control data for UMLAC are given in4 + NS number of lines, where NS is the number of species. The format andvariables are shown in Table 1, and the variables are defined as follows:

Line 1: CI Initial value of time step, AtER Max integration error in any variable per time stepFCT Initial value ratio 81/(61 + 62)

Line 2: TINT Initial value of timeDELINT Initial value of total thickness, 61 + 62TFINAL Final time (at which run is terminated)DINF Reference length (e.g., gun barrel diameter)

THETIN Initial value of 7 (if used)~0

DHINT Initial value of o (if used)

Line 3: KINT Initial value of time step-counter (e.g., 0)

25

GASL TR-276

KEINAL Maximum value of time step counterLINT Print interval counterITURB Turbulence model flag: -1 for the two-equation TTE

model, 0 for the eddy viscosity (conductivity) model,+1 for laminar viscosity (Sutherland's law)

ICHEM Chemistry flag: 0 for frozen flow, 1 for finite-ratekinetics, 2 for fully equilibrated chemistry.

IWRITE Debug flag: 0 for normal printout, 1 forcomprehensive data dumps

Line 4: IALP Number of species, NSIELE Number of elements

Lines 4 + NS:MW(I) Molecular weight of ith speciesRNAME(I) Chemical name of ith speciesDUM Dummy variable; not used

In addition, certain semi-permanent data is stored in DATA Statementscontained in a BLOCK DATA subroutine. The variables and their currentlyassigned values are:

CO, Cl, C2, C3 - 0.09, 1.43, 0.77, 1.92

PINT W 2117. psf

RHOINF - 0.002376 slugs/ft3

GAMINF - 1.4

CPINF - 6006. ft2/sec 2/°R

The nondimensional constants are the values of C0 , CI, C2 and C3 appearingin the two-equation TTE turbulence model. The values of pressure,density,ratio of specific heats and specific heat at constant pressure correspond tostandard atmospheric conditions. Certain variables transferred into UMLAC fromDCREK may be non-dimensionalized with respect to these parameters, as well asthe reference length DINF appearing in Line 2 of the input data file. Thevalues of these parameters in entirely arbitrary, and they may be all definedas 1.0, for example, or may be defined in S.I units, etc. The only importantconsideration relative to their definition is that they are assigned consistentvalues in DCREK and UMLAC. (In this respect, values of 1.0 may be the bestchoice.)

26

GASL TR-276

The boundary conditions are given on a second input file, as described in*Table 2. In general, this file will be automatically generated by

Program DCREK and require no alterations by the user. However, it can bemanually prepared, if desired. The variables are defined as foilows:

Line 1: IALP Number of Species, NSNSL Number of streamlines on side #2 (max of 11)ISCT(1) Number of dummy entries in the data for the ith

streamline (corresponds to the time it takesbefore the ith streamline crosses the Mach disc).

Line 2: TFIT Time (at which the following values occur)

P PressureRH02(l) Density on side #2 of the interfaceRHOl Density on side #1 of the interfaceRH02(I) Density on ith streamline on side #2 (1-2, NSL)

Line 3: ALPHA2(l,I) Mass fractions of species #1 through NS on side#2 of the interface.

Line 4: ALPHAl(I) Mass fractions on side #1 of the interface.

Line 5: ALPHA2(J,I) Mass fractions on jth streamline on side #2.

The order of the species must match that used in lines 5 through 4 +NS on the input data file on logical unit 5 (Table 1) and that usedin the THERMO section of the CREK data file (Table 3). *

Line 6: ZSL(J) Axial distance from interface to the jth

streamline.

The thermodynamic and reaction kinetic data is on a third input file,which is described in Table 3. This file is read by the Program CREK which iscalled by UMLAC, and the identical file should be used by DCREK to generate the

boundary condition data. The variables in this file are defined as follows:

Line 1: ELEMENTS Key word identifying the type of data whichfollows

Line 2: NAME (I) Chemical name of the ith elementATWT(I) Atomic weight of the ith elementVAL (I) Valence of the ith element

Line 3: THERMO Key word identifying the type of data whichfollows

27

Lam

GASL TR-276

Line 4: NAME (I) Chemical name of ith speciesT1(I) Upper limit of temperature range for first set of

polynomial coefficientsT2(I) Upper limit of temperature range for second set

of polynomial coefficients

Line 5: C(I,J) 7 polynomial coefficients, as defined byReference (10), for the first temperature range,followed by 7 more for the second temperaturerange.

Line 6: MECHANISM Key work identifying type of data which follows

Line 7: D,E,F,G,H,I Chemical names of reactants and products. Mdenotes a third-body which can be any species.

A,B,C Terms in the rate constant given by k=1OATB exp(-C/T)

K K can be either a blank or one of the following 4character expressions: REVE denotes that reverserate data has been specified, CGS denotes thatthe data is not in S.I. units, COMM denotes acomment line (not interpreted as data). Anyother expression will be ignored.

Sample data sets for the input data, boundary condition data, and thermoand kinetic data files are presented in Figures (3), (4) and (5).

Output Data

The output from UMLAC is divided into three categories. The first is arepetition of input data (in card image format) to assist in checking for inputerrors. The record is a set of statements of calculated constants and initialconditions. The third, which comprises most of the output, is a statement ofthe solution and boundary conditions. A sample output is provided in Figure 6.

The first block of data is the repetition of the data read from LU # 5,and matches the description given in Table 1. The initial part of the outputdata shown in Figure 6 matches the input data shown in Figure 3.

Following this is a short summary of calculated scale factors used tonondimensionalize certain variables in the program, and a message indicatingthat the data file on LU # 9 (Figure 4) has been successfully read. Thereafteris a synopsis of the streamline data showing the initial time and postion atwhich each streamline begins. The warning statement at the end of this set ofdata cautions the user that if the program continues beyond 0.02898 seconds itwill use the boundary data at that time to the final time, 0.10 seconds in thisexample.

28

11,,

GASL TR-276

The set of calculated initial conditions for the species are given next,in the same format as used previously for the species input data on LU # 5. Inthis instance the first two columns of data now contain the mass fractions ofthe species on the air side of the interface (i.e. at y - 61) and on thepropellant side (i.e. at y 6 62). The latter values have been interpolatedfrom the streamline data at the initial time.

The solution data begins with a set of labels identifying the currentvalue of the time set counter, K. The following glossary identifies the outputvariables in the order they appear :

K Time step counter

TIME Current time (seconds)

DELTA 1 Thickness of air-side layer; 61 (ft)

DELTA 2 Thickness of propellant-side layer; 62 (ft)

THETA ** 2 Mean-squared temperature fluctuation; 7o

PHI Thermal energy dissipation parameter; *RHO Density at y = O;p o (non dimensional)

KTO Thermal conductivity at y 0

P Pressure (non dimensional)

RHI Density at y = 61; Pl

RH2 Density at y = 62; P2

TEMPI Temperature at y = 61, T1 (OR)

TEMP2 Temperature at y = 62, T2 (OR)

YZERO Location of 62 in streamline coordinate system

TEMP Temperature at y - 0, To (OR)

H Enthalpy at y - 0, h0 (ft2/sec 2)

ALPHA Array of species mass fractions at y - 0; ajo

(same order as given by the user in the input datafiles.)

DPDT dp/dt

29 1

GASL TR-276

DRHIDT (dph/dt) at y = 01

DRH2DT (dph/dt) at y -f 02

ALPHA1 Mass fractions at y - 61; il

DALPIDT (dai/dt) at y - 6,

ALPHA2 Mass fractions at y = 62; ai2

DALP2DT (dai/dt) at y - 62

Following each solution statement is a set of 3 integer numbers thatrelate to the integration algorithm. The normal values are 4, 1 and 1. Alsoshown here are the current values of the (nondimensional) time step and thevalues that will be attempted on the next step.

The solution data will be repeated until either K > KFINAL or TIME >TFINAL.

The presence of muzzle flash is characterized by the occurrence of a rapidincrease in the interface temperature (TEMP), which otherwise drops off as the

pressure (P) decays and tends to remain bracketed by the bounding temperatures(TEMPI and TEMP2). The species composition at the interface (ALPHA) providesadditional diagnostic information regarding the chemical react;ons leading toor suppressing flash. In general, the user is advised to examine the timevariations of pressure (P) and temperature (TEMP) to determine if flash occurs

in a particular case, and then examine the time histories of the species(ALPHA), the thicknesses of the two layers (DELTAI and DELTA2), and thebounding temperatures (TEMPI and TEMP2) for diagnostic information.

30

GASL TR-276

5.0 SUMMARY

A model of one-dimensional, unsteady turbulent mixing and chemicalreaction along the interface between the propellant gas expelled from a gunmuzzle and the blast-heated air has been developed. The model incorporates allthe essential aspects of the unsteady flowfield processes, but thecomputational burden is economized considerably by employing a two-layerintegral-method for solving the governing gas-dynamic equations. Derivation ofthe integral-method of analysis and of the resulting system of coupled, firstorder ordinary differential equations has been presented in detail. Assumptionof unit Lewis and Prandtl numbers and of a single definition of the "edge" ofthe mixing layer for enthalpy and for species diffusion nearly uncouples theenergy equation from the species conservation equation, and requires that thespecies equation be solved only at the interface (at no small savings incomputer time).

The general requirements for operating the computer code and the specificinput and output data formats and a glossary of Fortran variables have beenpresented.

'p.

31

! 4'.

r-"S

%'

GASL TR-276

TABLE 1

DATA FORMAT FOR UMLAC INPUT FILE(Read on Logical Unit 5)

Line Format Variables

1 3E 10. CI, ER, FCT2 6E10. TINT, DELINT, TFINAL, DINF, THETIN, DHINT3 615 KINT, KFINAL, LINT, ITURB, ICHEM, IWRITE4 215 IALP, IELE5 3E10.,A4, DUM, DUM, MW(1), DUM, RNAME (1)

5X,A46 it'

4 + IALP " " " " "

TABLE 2

DATA FORMAT FOR UMIAC BOUNDARY CONDITION FILE(Read on Logical Unit 9)

Line Format Variables

1 1215 IALP, NSL, (ISCT(I), 1=2, NSL)I** 1 + (IALP-1)/6 lines are skipped

2 6E12 TFIT, P, RHO2(1), RHOI, (RHO2(1), 1=2, NSL)3 6E12 (ALPHA2(1,I), I - IALP)4 6E12 ALPHAI(I), I-I, IALP

' 5 6E12 (ALPHA2(J,I), I=1,IALP)Line 5 is repeated for J=2, NSL m

6 6E12 (ZSL(I), I=1,NSL)e Lines 2 through 6 are repeated up to 91 times. The READ loop will

terminate upon encountering an end-of-file mark. *

32 &

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TABLE 3

DATA FORMAT FOR CREK INPUT FILE(Read on logical unit 15)

Line Format Variables

1 12A4 ELEMENTS, a key name2 A2, 7X, 2F10 NAME (I), ATWT(I), VAL(I)

"" Line 2 is repeated for each element in the chemical system. A blankline signals the end of the elemental data *3 12A4 THERMO. a key name4 A12, 27X, 2F10 NAME (I), Tl(I), T2(I)5 5E15 (C1 (I,J), J=1,5)6 5E15 (C1 (1,J), J=6,7) (C2(I,J), J=1,3)7 5E15 (C2 (l,J), 3=4,7)

* Lines 4-7 are repeated for each species in the chemical system. A

blank line signals the end of the chemical data. Note that the number, namesand order of the species must agree with the data specified in other inputfiles. *8 12A4 MECHANISM, a key name9 12A4, 3F8.3, 2A4 D,E,F,GH,I,A,B,C,K

* Line 9 is repeated for each reaction. It can be used to specify eitherthe forward or backward rate of a reaction, in which case the opposite rate iscalculated from the equilibrium constant, or it can be used to specify bothforward and backward rates (2 lines per reaction). A blank line signals theend of the kinetics data.

a.,~33

-L _a m " '-

-,1

GASL TR-276

REFERENCES

1. Keller, G.E., "Secondary Muzzle Flash and Blast of the British 81-mm,L16A2, Mortar" ARBRL-MR-03117, July 1981.

2. Keller, G.E., "The Effect of Propellant Composition on Secondary MuzzleBlast Overpressure" ARBRL-MR-03270, April 1983.

3. Schmidt, E.M., "Secondary Combustion in Gun Exhaust Flows" ARBRL-TR-02373,

October 1981.

4. Erdos, J. and DelGuidice P., "Calculation of Muzzle Blast Flowfields" AIAAJournal, Vol. 13, No. 8, August 1975 pp. 1048-1056.

5. Yousefian, V., "Muzzle Flash Onset" ARBRL-CR-00477, Feb. 1982.

6. Yousefian, V., "Muzzle Flash Onset: An Algebraic Criterion and FurtherValidation of the Muzzle Exhaust Flow Field Model" ARBRL-CR-00506, March 1983.

7. Keller, G.E., "An Evaluation of Muzzle Flash Prediction Codes" ARBRL-MR-

03318, Nov. 1983.

8. Dorrance, W.H., Viscous Hypersonic Flow, McGraw Hill Book Co., New York,1982.

9. Schmidt, E. and Shear, D. "Flowfield About the Muzzle of an M16 Rifle"BRL Report 1692, Jan., 1974. Also, AIAA Journal, Vol. 13, PP. 1088-1093, Aug.,1975.

10. Richtmeyer, R. D., "Taylor Instability in Shock Acceleration ofCompressible Fluids," Communications on Pure and Applied Mathematics, Vol.

13,1960, pp. 297-319.

11. Levin, M. A., "Turbulent Mixing at the Contact Surface in a Driven Shock-Wave," The Physics of Fluids, Vol. 13, No. 5, 1970, pp. 1166-1171.

12. Andronov, V. A. et al., "Turbulent Mixing at Contact Surface Acceleratedby Shock-Waves," Soviet Physics-JETP, Vol. 44, 1976, pp. 424-427.

13. Houas, L., Brun, R. and Hanana, M., "Experimental Investigation of Shock-Interface Interactions," AIAA J, Vol. 24, No. 8, August 1986, pp. 1254-1255.

14. Pratt, D.T., "Calculation of Chemically Reacting Flows with ComplexChemistry" in Studies in Convection, B.E. Launder, Ed., Academic Press, NYU 1977.

34

U:.,6ft-

GASL TR-276

15. McBride, B.J., Heims, S., Ehlers, J.G. and Gordon, S., "ThermodynamicProperties to 6000*K for 210 Substances", NASA SP-30001, 1966.

16. Ranlet, J. and Erdos, J. "Description of Fortran Program DAWNA forAnalysis of Muzzle Blast Flowfield" BRLCR-302, March 1975.

35

GASL TR-276

.4j

U. I00

II-

.j1

NN

LL

4i

'U:

cm)

36

GASL TR-276

a. A. . r

T TIZ

TI, ,

/*2Tl/I

I

"-'-- '0 PSL - PROPELLANT STREAMLINE

.$- ASL - AIR STREAMLINE

AXIAL DISTANCE

FIGURE 2. SCHEMATIC SHOWING TYPICALSTREAMLINE TRACES AND ENTROPYDISTRIBUTION

37

__.v'V' *.

GASL TR-276

1.E-5 1.OE-3 0.5.00052 .0200 0.1 .5085 2.704E+01 1.020E+08

0 9000 01 -0 1 011 5

.0000E+00 .2955E+00 28.0150 CO

.OOOOE+O0 .1785E+00 44.0099 C02

.OOOOE-05 .2939E-05 1.0080 H

.0000E-01 .1257E-01 2.0159 H2

.0000E+00 .1711E+00 18.0153 H20

.7655E+00 .3268E+00 28.0134 N2

.00DOE-O8 .9712E-08 15.9994 0

.OOOOE-05 .6403E-05 17.0074 OH

.2345E-00 .1355E-08 31.9988 02

.OOOOE-02 .1450E-02 39.1000 K

.OOOOE-01 .1408E-01 56.0100 KOH

FIGURE 3. TYPICAL INPUT DATA FILE FOR UMLAC (LOGICAL UNIT 5)I

38

GASI TR-276

c.4

'-4 E-&-

NP4 N-

N P4 cnoz 0

N00 0 CDQ0000 0 0 0 00 00 00z 0 0 0 I 0 0 0C 0000E-0Q 0 0 0 0 0D 0 0 0 0 0 0 0 D0

0 C 0 0 0 0 0 0 0 0 0 c 0 0

'0 00 N &M N4 N N N N N1 N N4 N N1 00 00

On0= 0 " coc0 C1 0 N r0 C1 t "0N0N0 0 0 00r-"20 cc 0 0 -

co~0 ONCmocO~oNNoNOoNONONoOOONOO 00

0% 040 m Cn 00000000000000000000000 040%a 00 (~C U 0 0 0 0 0 0 0 0 0 0 0 0 00

'0 0 M0 0 4 T -,-4NNNNNNN-NNN -T NNN-NNT - -T0 0 00

~-4 0 -4M 0-4M 4M 4C4M 4M-4 C - M-0 0 -0S00 00-440000000000000000000000oc 00 p

4+ + smm a. 4 40 W-4

-To OrsO00000000000000000000000 %00NO mon o o o o o ~ o o o o o -40 4N 0 440 % % 0 0% CM% %CC%%0O'%0 00 '40 E-4

LA 0%0 %OO N N N N N N N N N N O 00

C4 -4 0 r 7 nw0w0w00 n0 n0 nc n0 L0 L u0 in co 0 CD 40 00% 0 00C 00.-400 000000000 00000000 HO 0 -N% 4+ + I4 + I I I I IT T 0 00 E II6I~ 4 4-4

(C90 'OIAOOOOOOOOOOOOOOOOOOOOOOO %000% 0 NWOOOOOOOOOOOOOOOOOOOOO0 00%00 0OW c O~ 0 %00%00%00'00%00%00%00'00'0 C090

N NO4 0 W0 in- 4W. - 4W-4 W -4 -W 4 W -4W 0 0 %00U

.0 N'%,- 4- 0 - Q %0,-4%-4 4%0-40 0 %0 4 %0 00 .0 >

0- 0 C40 00 00 00 0 0000 in0L0 Q0 00 00 0 0 0 0 0 0000Q 000D

-4 enN~ 0 4 0 0 00 0 0 0 0 0 0 0 CD 0D-4 40 0 0 ~ O 0 0 0 0 0 0 0 0 0 - 0 -400

MOO0-4 -40Q0-4 N-4N-4 N- N -4 N-4 N-4N"-4N-4 N-4 N0% 0 NOO

00 00CD0000-t000 00000000000000 00wo -0 NO 0 C

QOOO0N0 N-4 0 -4 a -4 0%-%-4 0 -4 a-4 C% -4 % -40M-4 %0% Q0 I 0 0. CD0 00 000 0 -404 00-40-4 40 -040 -400-4 00 0 r0 0.

P34

39

GASL TR-276

ELEMENSC 12.01115 4.000000H 1.007970 1.0000000 15. 999400-2 .000000N 14.006700 3.0K 39.100000 1.0

THIERMOCO J 9/65C 1.0 1.00 0.00 0.G 300.000 5000.0000.29840689E 01 0. 148913871-02-0.578996781-06 0. 103645761-09-0.693534991-14-0.14245227E 05 0.634791471 01 0.37100916E 01-0.161909641-02 0.369235841-05-0.20319673E-08 0.23953344E-12-0.143563091 05 0.295553401 01C02 J1 9/65C 1.0 2.00 0.00 0.G 300.000 5000.0000.446080401 01 0.309817171-02-0.123925661-05 0.227413231-09-0.155259481-13-0.48961438E 05-0.986359781 00 0.240077881 01 0.87350905E-02-0.660703611-050.20021860E-08 0.632740391-15-0.483775201 05 0.969514471 01R J 9/65H 1.00 0.00 0.00 0.0 300.000 5000.000

0.25000000E 01 0.0 0.0 0.0 0.00.254716251 05-0.460117581 00 0.250000001 01 0.0 0.00.0 0.0 0.25471625Z 05-0.46011758E 00

H2 J 3/61H 2.0 0.0 0.0 0.0 300.000 5000.0000.310018831 01 0.511194581-03 0.526442041-07-0.349099641-10 0.369453411-14-0.877380131 03-0.19629412E 01 0.305744461 01 0.267651981-02-0.580991492-050.552103431-08-0.181227261-11-0.98890430E 03-0.22997046E 01H20 J 3/61H 2.0 1.00 0.00 0.G 300.000 5000.0000.27167616E 01 0.294513701-02-0.80224368E-06 0.102266811-09-0.48472104E-14

-0.29905820E 05 0.66305666E 01 0.407012751 01-0.110844991-02 0.415211801-05-0.29637404E-08 0.80702101E-12-0.30279719E 05-0.322700381 00N2 J 9/65N 2.0 0.0 0.0 0.0 300.000 5000.0000.289631941 01 0.15154863E-02-0.57235275E-06 0.998073851-10-0.652235361-14-0.905861821 03 0.616151431 01 0.36748257E 01-0.12081496E-02 0.232401001-OS-0.63217520E-09-0.225772531-12-0.106115871 04 0.235804181 010 J 6/620 1.00 0.00 0.00 0.0 300.000 5000.0000.25420580E 01-0.27550603E-04-0.31028029E-08 0.455106701-11-0.436804941-150.292308011 05 0.49203072E 01 0.29b642831 01-0.163816641-02 0.24210303E-05

-0.16028432E-08 0.38906964E-12 0.29147641E 05 0.296399311 01OH J 3/660 1.H 1.00 0.00 0.0 300.000 5000.0000.29106417E 01 0.95931627E-03-0.194417001-06 0.13756646E-10 0.14224542E-150.39353811E 04 0.544234281 01 0.38175931E 01-0.10778855E-02 0.96830354E-060.18713971E-09-0.22571089E-12 0.3(412820E 04 0.49370009E 0002 J 9/650 2.0 0.0 0.0 0.0 300.000 5000.0000.36219521E, 01 0.73618256103-0.19652219106 0.36201556E-10-0:28945623E114

-0.67635071E-08 0.21555977E-11-0.104752251 04 0.43052769E 01K 3 K 1. 0.0 0.0 0.0 300.000 5000.0000.2561E 01-0.1373E -03 .11601 -06-0.5159E -10 0.1180E -13-0.76581 03 0.4699E 01 0.24031 +01 0.6656E -03-0.1599E -050. 16151 -08-.58439 -12-0.73491 03 0.54401 01KOH J K 1.0 I.H 1. 0.0 300.000 5000.0000.5643E 01 0.1243E -02-0.34241 -06 0.4206E -10-0.1816E -14-0.4035E 05 -0.4072E 01 0.3663E 01 0.12681 -01-0.23481 -040.20081 -07-0.63571 -11-0.40111 05 0.46431 01

MECHANISM -YOSEFIAN RATE CONSTANTS FOR M4UZZLE BLASTCo OH C02 H 4.227 1.3 -330.CO 02 C02 0 9.403 0.0 24000.CO 0 M C02 M 9.4046 0.0 2200.H 02 01 0 11.16 0.0 8250.0 H2 OH H 7.2569 1.0 4480.OR 0R 1120 0 9.780 0.0 550.OH H2 H H120 6.06 1.3 1825.0 H M OH M 12.559 -1. 0.0 4

0 0 M 02 M 8.0366 0.0 -900.H 01 M H20 M 16.559 -2. 0.0

H H M H2 M 12.0366 -1.0 0.0H KOH H20 K 10.035 0.0 1000.I

K OH M KOH M 14.7353 -1. 0.0

FIGURE 5. TYPICAL THERMODYNAMIC AND REACTION KINETICS DATA FILE FOR UMLACAND CREK (LOGICAL UNIT 15)

40 *.

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'4419ML L

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