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AD-A169 316 SINULATIONS OF SPECIAL INTERIOR BALLISTIC PHENONEM iNITH AND WITHOUT HEAT..(U) ARMY BALLISTIC RESEARCH LASABERDEEN PROVING GROUND HD R HEISER ET AL NAY B6

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US ARMYMATERI ELCOMMAND . -

00 TECHNICAL REPORT BRL-TR-2732

SIMULATIONS OF SPECIAL INTERIORBALLISTIC PHENOMENA WITH AND WITHOUT

HEAT TRANSFER TO GUN TUBE WALL

Rudi HeiserJames A. Schmitt

May 1986 J-L

APPROVED FOR FMUC RE.EASE; 00MUUOR UNLIMD.

US ARMY BALLISTIC RESEARCH LABORATORYABERDEEN PROVING GROUND, MARYLAND I:

••.-. . .-. .•. . ...-. ... . . . ,* . . . '

UNCLASS IF LED' CURITY CLASSIFICA't0N OF THIS PAGE 'Wlhen 17.. f~nered)

REPOT DCUMNTATON AGEREAD INSTRUCTIONSREPOR DOCUENTAION PGEBEFORE CO\1PI.ETIN, FCR.MI REPORT NUMBER Z. GOVT ACCESSION NOf 'ECIPIENT'S CATAj)G NUMBjER

Technical Report BRL-TR-2732 oq/ 3/4 To TL E (and S~btItle) P E O F R EPO RT S P E RI00 C .ER ED

Simulations of Special Interior Ballistic FinalPhenomena With and Without Heat Transfer to Sep 83 - Sep 84

the Gn Tue Wal f6. PERFORMING ORG. REPORT NMBER

7 AUTH-OR(a) 8. CONTRACT OR GRANT NUMBErli.;

Rudi Heiser*, James A. Schmitt

9 PERFORMING ORGANIZATION NAME AND ADDRESS 1C. PROGPAM ELEMENT. PROJEC7, -',,,AREA 'i WORK UNIT NUMSEPS

U.S. Army Ballistic Research laboratoryATTN: S LCBR- IB I L161102AH43AberdeenProvingGround._MD_21005-5066 ____ _____________

I CONTROLLING OFFICE NAME AND ADDOESS 12. REPORT DATEI.S Armv Ballistic Research Laboratory My18ATTN: S LCBR-DD-T

1 U B RO A EAberdeen Proving Ground, MD 21005-5066 7* IA1 MONITORING AGENCY NAME IS ArIORESSI differet Ion (>ot,oltinj office) ~SFSCURITY CLASS (.4f tftI raporf;

Unc lass ified

IS DCLSSFIATNDOWN-GRF AIN-G-SCHEDULE

* I16 DISTRIBUTION STATEMENT (of this. Report)

Approved for Public Release; Distribution Unlimited.

1.DISTRIBUTION STATEMENT 'of IIhe abstract entered In Block 20, If different from, Report)

IBLI 4 ijsspent a year at the Ballistic Research laboratory as avisiting scientist from Ernst-Mach Institut at Weil am Rhein in the Federal-Republic of Germany

19 6EY WO*RDS ony ip on fever.* Side If nei..earv and I en iy by block n be

flierir ~~lstts Groute £~~e rutiimensional Multiphase FlowHeat Transfer Turbulence ADI Finite Difference MethodNumerical Simulations One-Phase Flow Results

* 20. ASSTRACT rc-onthnwe an .r. e,,s ic it neaarv and IdentIfr by block -,nb.,)

The computer code DELTA uses a linearized Alternating Direction Implicit* (ALI) scheme to) provide a numerical approximation of the solution of thle

averaged two-phase (gas-solid) two-dimensional (axisymmetric) equaltionlsgoverning viscous interior ballistic flows within conventional uns. ino

further the understanding of phenomena affecting gun tube life as well ;i-gun performance, a heat transfer model, Which simlateS thle interaICtionlS Of

fluid dynamics and thL.rmal profile in the gun tube wall, hlas beeni

D AM 1A73 1413 EDITION OF 1 40Y 65 1%OBSOLE Tt LNCl.ASS IFl ;D 14

Z - 7 -

UNCIASS IF IEDSECURITY CLASSIFICATION 'V THIS PAGE(When Dais Entered)

incorporated in the DELTA code. The same linearized ADI r.iethod was utilized toobtain the numerical solution of the two-dimensional nonlinear heat conduction

equations in the gun tube. Our model of the heat transfer process couplescompletely and simultaneously all three controlling events without anyapproximations to the governing equations; that is, the axisymmetric viscousflow within the entire gun tube which naturally gives a precise definition of

the gas thermal boundary layer, the time-dependent balance of the heat fluxes

at the inner wall surface, and the two-dimensional temperature calculationwithin the gun tube wall. The nonlinear heat flux boundary conditions at the

inner and outer tube surfaces are linearized to be compatible with the solutionscheme.

a.

Results computed with DELTA are presented for two different types ofidealized one-phase interior ballistics flows. The first is a pure expansion

flow, and the other includes mass and heat sources. For each of the flows wepresent the effects of gas turbulence and the effects of heat conduction to thewall. Our results indicate that both effects are significant.-

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TABLE OF CONTENTS

Page

LIST OF ILLUSTRATIONS ............................................. 5

I. INTRODUCTION .......... . ...................................... 9

II. REVIEW OF THE MODELS IN DELTA ................................... 9

III. HEAT TRANSFER TO AND TEMPERATURE DISTRIBUTION IN THETUBE WALL... ................. ................................ 10

IV. TURBULENCE MODEL .............................................. 14

V. RESULTS...........................................................17

V 1. SUMMARY ........................................................ 61

REFERENCES ..................................................... 62

DISTRIBUTION LIST.............................................. 65

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LIST OF ILLUSTRATIONS

Figure Page

1. Standard Computational Mesh 19 x 49. Minimum Radial,;rid Size is 7.7 Pm .......................................... 18

2. Pressure Histories at the Center of the Breech and theProjectile Base for the Laminar Flow with Adiabatic Wallsin the Lagrange Gun (LG) and Real Gun (RG) Simulations ....... 19

3. Temperature Histories at the Center of the Breech and theProjectile Base for Laminar Flow with Adiabatic Walls forthe Lagrange Gun (LG) and Real Gun (RG) Simulations .......... 20

4. Projectile Velocity Histories for Laminar Flow withAdiabatic Walls in the Lagrange Gun (LC) and RealGun (RG) Simulations ......................................... 21

5. Projectile Displacement from the Breech for LaminarFlow with Adiabatic Walls in the Lagrange Gun (LG)and Real Gun (RG) Simulations ................................ 22

6. Lagrange Gun, Laminar Flow, Adiabatic Walls: SpatialDistribution of the Axial Gas Velocity at the Time ofMuzzle Clearance. Mesh is Uniformly Spaced in Axial

Direction and Nonuniformly Spaced in Radial Direction ........ 24

7. Lagrange Gun, Laminar Flow, Adiabatic Walls: SpatialDistribution of the Axial Gas Velocity at the Time ofMuzzle Clearance. Mesh is Nonuniformly Spaced inBoth Directions .............................................. 25

8. Lagrange Gun, Laminar Flow, Adiabatic Walls: SpatialDistribution of the Gas Pressure at the Time of Muzzle

Clearance .................................................... 26

9. Lagrange Gun, Laminar Flow, Adiabatic Walls: SpatialDistribution of the Gas Pressure at the Time ofMuzzle Clearance ............................................. 27

10. Lagrange Gun, Larminat Flow, Adiabatic Walls: SpatialDistribution of the Radial Gas Velocity at the Time ofMuzzle Clearance ............................................. 28

11. Lagrange Gun: Pressure Histories at the Center of the

Breech for Both Laminar and Turbulent Flows with AdiabaticWalls, and for Laminar Flow with Heat Transfer .............. 30

5

LIST OF ILLUSTRATIONS (CONT'D)

Figure Page

1. Lagrange Gun: Temperature Histories at the Center

dof the Breech for Both Laminar and Turbulent Flowswith Adiabatic Wails, and for Laminar Flow with Heat

Transfer ....................................................... 31

13. Lagrange Gun: Projectile Velocity Histories for Both

Laminar and Turbulent Flows with Adiabatic Walls and

for Laminar Flow with Transfer ................................. 32

14. Lagrange Gun, Turbulent Flow, Adiabatic Walls: SpatialDistribution of the Axial Gas Velocity at the Time ofMuzzle Clearance ............................................... 33

15. Lagrange Gun, Turbulent Flow, Adiabatic Walls: SpatialDistribution of the Radial Gas Velocity at the Time ofMuzzle Clearance ............................................... 34

16. Lagrange Gun, Adiabatic Walls: Radial Profiles of theAxial Gas Velocity for Both Laminar and Turbulent Flowat the Time of Muzzle Clearance at 0.25 m Away from

the Muzzle ..................................................... 35

17. Lagrarge Gun, Adiabatic Walls: Radial Profiles of theRadial Gas Velocity for Both Laminar and Turbulent Flowat the Time of Muzzle Clearance at 0.25 m Away fromthe Muzzle ..................................................... 36

18. Lagrange Gun, Adiabatic 1.alls: Radial Profiles of theGas Temperature for Both Laminar and Turbulent Flows at

the Time of Muzzle Clearance at 0.25 m Away fromthe Muzzle ..................................................... 37

19. Lagrange Gun, Laminar Flow, Heat Transfer: SpatialFrofile of the Axial Gas Velocity at the Time ofMuzzle Clearance ............................................... 38

20. Lagrange Gun, Laminai Vlov, Eeat Transfer: SpatialProfile of the Radial Gas Velocity at the Time ofMuzzle Clearance ............................................... 39

21. Lagrange Gun, Laminar Flow, Heat Trar.sfer: SpatialProfile of the Gas Pressure at the Time ofMuzzle Clearance ............................................... 40

22. Lagranpe Gun, Laminar Flow, Heat Transfer: SpatialProfile of the Gas Temperature at the Time of

Muzzle Clearance ............................................... 41

6

-? ;2:. ;-j-2' - - -. - .- -, .- ... ,. -,- - ... ,v , - . -- .- . * . . . . . . . .


Figure Page

23. Lagrange Gun, Laminar Flow: Radial Profiles of the AxialGas Velocity for Adiabatic and Heat Permeable Walls at the %Time of Muzzle Clearance 0.25 m Away from the Muzzle ......... 43

24. Lagrange Gun, Laminar Flow: Radial Profiles of the GasTemperature for Adiabatic and Heat Permeable Walls at theTime of Muzzle Clearance 0.25 m Away from the Muzzle ......... 44

25. Lagrange Gun, Laminar Flow: Radial Profiles of the RadialGas Velocity for Adiabatic and Heat Permeable Walls at the

Time of Muzzle Clearance 0.25 m Away from the Muzzle ......... 45

26. History of the Wall Surface Temperature for a LaminarExpansion Flow (LG) with Heat Transfer to the Tube Wallat the Locations 150 mm (inside the chamber) and 250 mm(inside the barrel) Away from the Breech ..................... 46

27. Real Gun, Laminar Flow, Adiabatic Walls: SpatialDistribution of the Axial Gas Velocity at 3.6 ms ............. 47

28. Real Gun, Laminar Flow, Adiabatic Walls: SpatialDistribution of the Gas Temperature at 3.6 ms ................ 48

29. Real Gun, Laminar Flow, Adiabatic Walls: SpatialDistribution of the Gas Pressure at 3.6 ms ................... 49

30. Real Gun, Laminar Flow, Adiabatic Walls: SpatialDistribution of the Axial Gas Velocity at MuzzleClearance (5.3 ms) ........................................... 50

31. Real Gun, Laminar Flow, Adiabatic Walls: SpatialDistribution of the Radial Gas Velocity at Muzzle

Clearance (5.3 ms) ........................................... 51

-2. ?eal Gun, Laminat Flow, Adiabatic Walls: SpatialPistributior of the Gas Temperature at Muzzle

Clearance (5.3 ms) ........................................... 52

33. Real Gun, Laminar Flow, Adiabatic Walls: SpatialDistribution of the Gas Pressure at Muzzle

Clearance (5.3 ms) ........................................... 53

34. Real Gun, Turbulent Flow, Adiabatic Walls: SpatialPistribution of the Axial Gas Velocity at 3.6 ms ............. 54

35. Real Gun, Turbulent Flow, Adiabatic Walls: SpatialDistribution of the Radial Gas Velocity at 3.6 ms ............ 55

7

.. - -. .. - . - -r z -- a -- -~.- . - .--. - '-- I - - 7


Figure Page

36. Real Gun, Turbulent Flow, Adiabatic Wells: Spatial

Distribution of the Las Temperature at 3.6 ns ................. 56

37. Real Cun, Turbulent Flow, Adiabatic Walls: SpatialDistribution of the Gas Pressure at 3.6 ms .................... 57

38. Real Gun, Turbulent Flow, Adiabatic Wails: Spatial

Distribution of the Axial Gas Velocity at Muzzle

Clearance (5.49 ms) ........................................... 58

39. Real Gun, Turbulent Flow, Adiabatic Walls: SpatialDistribution of the Radial Gas Velocity at Muzzle

Clearance (5.49 ms) ........................................... 59

40. Real Gun, Turbulent Flow, Adiabatic Walls: SpatialDistribution of the Gas Pressure at Muzzle

Clearance (5.49 ms) ........................................... 60

8

.°

,.__-

---- 4

,..

I. INTROIJCTION

Interior ballistics flows of conventional charges in gun tubes possesscomplex flow patterns. Their complexity is due to both the heterogeneousstructure of the charge and the rapidly changing flow conditions within a few -.

milliseconds. The fast rise in pressure and temperature caused by the burning

of the propellant initiates a turbulent, multidimensional, multiphase flow .

which is coupled with the accelerating projectile motion. A completemathematical model that describes all the physical phenomena occurring in an i

interior ballistics cycle is not presently available. However, s veral modelsthat simulate some of the phenomena exist or are being developed..

On the other hand, it is not possible for technical reasons to makedetailed experimental measurements of the complete interior ballistic cycle.

Some standard techniques as well as some new special techniques underdevelopment determine only specific quantities in real weapons or in

simulators under simplified flow conditions. Commonly measured quantities arethe gas pressure and projectile motion. Other quantities such as thetemperature distribution in the gas and in the gun tube wall, the velocitydistribution of the gas and solid particles inside or outside of boundarylayers, the particle distribution, the turbulence pattern, etc., cannot beaccurately determined by experiment. Thus, a need exists for modelling of theinterior ballistic cycle so that the dynamic development of these quantitiescan be studied, and their impact on ballistic problems can be evaluated.

A new computational capability for the investigation of interiorballistics flows is the DELTA code, which is under development at theBallistic Research laboratory. The purpose of this code is to addressparticular ballistics problems related to the boundary layer development, theheat transfer to the tube wall, the urbulence, and the time-dependentdistribution of additive particles. In the following, we shall give a shortdescription of the DELTA code and present computational results of aninvestigation of heat conduction and turbulence effects.

II. REVIEW OF THE MODELS IN DELTA

The flow which is modeled by DELTA is a multidimensional, two-phase flowinside a gun tube. Presently, the flow is assumed to be axisymmetric. At therear end, the so-called breech, the tube is closed by a stationary flat platewhile the front boundary is a moving flat-based projectile. The flow isassumed to be viscous and heat conducting, and it can be either laminar or

1"Fluid Dynamics Aspects of Internal Ballistics," AGARD Advisory ReportNo. 172, 1982.

9

turbulent. The wall may be either adiabatic or it may allow heat transferfrom the gas. Heat transfer is restricted to the tube wall (excluding thebreech). The core flow is fully coupled to the moving projectile, to theboundary layer development and, if desired, to the heat conduction in the tubewall. By fully coupled we mean that each of these phenomena can affect all

* others. For example, the boundary layer development can alter the details ofthe core flow. This would not be the case, for example, in a boundary layer

type model.

The mathematical model in DELTA (the balance equations for the gas-phase-and one solid phase), is based on an unsteady volume-averaged formulation.

The gas phase is described by averaged equations corresponding to the fullNavier-Stokes equations for a compressible fluid. The model is closed byaveraged coefficients of viscosity and heat conduction, an averaged viscousstress tensor, an averaged dissipation function and an averaged heatconduction function. Since interior ballistics flows usually produce high gaspressures, the Noble-Abel equation of state is used so that some real gaseffects can be included. The gas turbulence is represented by algebraicmixing length models. The solid phase is described by the averaged equationsfor arrays of incompressible particles which can undergo deformations. Thederivation of the equations is given in Ref. 2. The equations of the solution

algorithms are listed in Ref. 3.

The system of partial differential equations for the axisymmetric two-phase flow region is solved by a linearized Alternating Direction Implicit

(ADI) scheme. This scheme transforms the differential equations into a systemof linear algebraic equations. The corresponding matrix has a blocktridiagonal structure, allowing an efficient determination of the solution ateach new time-step. Details about the derivation of the scheme are presentedin Ref. 3.

Ill. HEAT TRANSFER TO AND T'1PFRATUKL DISTRIBUTION IN THE TUBE WALL

The heating of the gun tube wall caused by convection, heat conduction,

and radiation of the hot propellant gas enhancps the gun tube wear anderosion, and therefore, affects the lifetime of gun tubes. An experimentaldetermination of the inner wall surface temperature is quite difficult.C(omnon lv used thor nocouples ire )F limited use in interior ballistic

)

* C(;e~frs., , A.K.:., Scnmitt ,1 .A., "ree-Dimensional 'ot(elin, of .as-Corihustin7 Solid Two-Phase Flows, "Multi-Phase Flow and Heat TransterIII, Part B: Applications, T. N. Veziroglu and A.E. Berrles, editors,pp. 681-698, Elsevier Science Publishers, Amsterdam, 1984.

iSc"mi tt, Ll .A., " .uericai Algeorithr, n-or the "nlitidimensiona1, ,iiltimhase.VisCOuS f'quations of Interior Ballistics," Transactions of the Second ArmyConference on Applied Mathematics and Computing, ARO-Report 85-1,

pp. 649-691, 19F5.

1. C

applications because such applications require accurate measurements in veryshort time intervals, a close contact to the flow, and special thermalproperties of the gauges.

There are several mathematical models of the heat transfer to the gun

tube wall. They can be divided in four categories according to theircomplexity. The first type uses a very simplistic boundary layer calculationand a heat transfer correlation, e.g., Colburn's analogy, to obtain the heat

transfer. - Heat losses in the core flow are considered. The empirical heat

transfer correlations are derived for fully developed, steady, one-phase pipeflow. The main feature of this type of model is the emphasis on the

calculation of the core flow. In models of the second category the emphasisis on a decgiption of the boundary layer by using more general boundary layer

equations. The heat transfer to the tube wall again is described by

correlations. The boundary layer edge is not coupled with the computation ofthe core flow. Instead, one assumes for the latter values which represent

approximately the core flow. The third category makes use of general boundarylayer equations and of a balance of heat fluxes from the hot gas to the gun

tube wall at the inner wall surface.9 The conditions at the boundary layer

edge are comparable to those in the second category. Neither of these

approaches includes all the feedback mechanisms from the tube to the coreflow, and therefore, to the projectile motion. The fourth category of models

uses a fully coupled approach whereby the phenomena in the core flow aredirectly linked to the projectile motion, the boundary layer development, the

heat transfer to and the heat conduction in the tube wall, and vice versa.This is achieved by using a single system of equations everywhere in the gas

region and a coupled system of heat conduction equations in the tube wall.The solution of these sets automatically provides the boundary layer solutionin the boundary layer region, the core flow solution in the core flow, and all

the necessary coupling that naturally occurs in the flow. Although this type

4 Shelton, S., Bergles, A., Saha, P., "Study of Heat Transfer and Erosion inGun Barrels," Air Force Armament Laboratory, Eglin Air Force Base, Florida,

AFATL-TR-73-69, 1973.5 Gough, P., "'.odelin4 of Rigidized Gun Propelling Charges," Ballistic Research

Laboratory, Aberdeen Proving Ground, Maryland ARBRL-CR-00318, J983.

6 :els.n.C.,'.. Uard .7.. "Catlcurlion of Heat Trarsfer to the Gun Barrel

WaLt ," J. Ballistics 6 (3), pp. 1518-1524, 1982.

7 Barlett, E.P., Anderson, L.W., Kendall, L.:!., 'Time-Dependent Boundary Layerswith Application to Gun Barrel Heat Transfer," Proceedings 1972 Heat TransferFluid Mech. Institute, Stanford University, CA, 1972.

8 Buckingham, A.C. , "Modelin,- Prcpcl art. Combust ion IJteract in with n Eroding

Solid Surface," Lawrence Livermore laboratory, 1*CR[.-83727, 198n).

.... i , t er, .. "&nst.e;',\y tt',r1FP~ b1tind;, r' ., ' r- Pio;tvsis Am ied to

Gun arrei Wall Hleat Transfer," Int. .1. Heat 1,s Transler, Vol. 24, No. I _

pp. 1925-1935, 1981.

........................................................

of solution is most complex, it provides a flow description with the fewestassumptions and approximations. In tight of the scarcity of experimentalmeasurements with which to compare the calculations, we feel such an approachto be best. The DELTA code is an examole of a fourth category model.

The heat transfer model in DELTA consists of the equations governing theheat conduction in the tube wall, and the boundary conditions which couple thetemperature in the wall to the flow inside and outside the tube. The heatconduction in the tube wall is described by the two-dimensional, nonlinearaxisymmetric equation for the wall temperature Tw(tr,z):

pw(T) c(T) X= A (T W ww w t )z W IWw

3TZ

+ I 1 [r Xw(Tw) wr ,r wr .-.

The variables t, z and r denote the time, axial coordinate in the wall andradial coordinate in the wall, respectively. The specific heat cw and thethermal conductivity X of gun tube steel strongly depend on the

10temperature. By comparision, the density of the steel pw varies only alittle with temperature.

The most important boundary condition is at the inner tube wall surfacewhere the coupling of the flow region and the wall occurs. It consists of the . -

balance of heat fluxes with a radiation effect

, T 4 4 DTw+ I. o (T -T 'A-

g 9r g g w Drg

and of the temperature equilibrium equation

T = T-w

The variables o , , a, T and T denote the thermal conductivity of

the gas, the emissivity of the wall surface, the Stefan-Boltznann constant,the gas temperature, and the naximun gas temperature in a given cross-section(z = constant), respectively. We emphasize that in DELTA both canditions areused only at the inner tube wall surface, excluding the breech. The left handside of the first condition represents the heat flux on the gas side towardsthe wall, whereby the first term represents heat conduction, and the secondterm represents heat radiation. The right hand side gives the heat flux into .-the tube wall. For the boundary conditions at the outer tube wall, surface we

1 0 Aerospace Structural Metal Handbook, "Ferrous Alloys," 1973.

12

chose the simple engineering condition

w 3r h Tw T amb

where h is a heat transfer coefficient and Tamb is an outer ambienttemperature. A more sophisticated condition is not needed (at least for asingle shot weapon) because the heat usually does not reach the outer surfaceduring a ballistic cycle. Two additional boundary conditions are needed inaxial direction. At the projectile base, we set

Tw =Tamb,

across the wall thickness, that is, we assume that the projectile moves intoan area which is at ambient temperature. At the breech and at the projectile

base, an adiabatic condition

was assumed to be adequate.

The equations governing the temperature distribution in the tube wall aresolved using the same linearized ADI method as for the equations in the gasflow region; that is, the equations are linearized in time, and are splitalong coordinate directions. At each new time level, we first update thetemperature distribution in the wall, and then update the dependent variablesin the flow region. This is performed by the following sequence of sweepsalong coordinate directions: an axial sweep followed by a radial sweep in thewall, a radial sweep followed by an axial sweep in the gas region, and finallyan adjustment of the dependent variables along the inner wall surface to theflux boundary condition. We omit a discussion of the details of the numericalprocedure because they are discussed in Ref. 3. In the DELTA code, the

thermodynamic dependent variables in the gas region are the specific gasentropy (s) and the logarithm of the gas pressure (q). Therefore, the heatflux boundary condition must be reformulated in terms of s and q at the newinknown time level for the radial sweep in the gas region. To this end, wetransform the heat flux term on the right hand side of the condition via thechain rule as follows:

X X(T(s,q)) T(s,q) = X(T(s,q)) [T 3s + Trs r q r

,-,here T is the gas temperature, and T T denote the partial deriviatives off with respect to s and q, respectively, q(For simplicity, we dropped theindex g.) The linearization in time gives a relation between the unknown newtime level (n) and the known current time level (c)

13..

- ~~~~~~.. .. . . . . . . . . . . . ........., ,. -. • l .. _ ' • . ... '

• .. ~ r_ 2- . -" ." -'

"% n (Tn c TC d T c 2X - ) = + (-) + - I At + 0 (t 2)

with

d T = d c ds c dq1 (3T c-T c

c X T 22)c dq + (T 22 T a)c ds

qq r sq ar dt ss r qs 3r -t

i The time-deriviatives we approximate by

dsn cdt _t

and

.43 t n cdt q -q.dt

With this approximation, the boundary condition provides a linear equation forsn and qn. The equation is compatible with the set of finite differenced andlinearized flow equations.

IV. TURBULENCE MODEL

In order to estimate the influence of turbulence on the flow pattern, weused two turbulence models. Both were equilibrium algebraic eddy viscosity

*models based on Prandtl's mixing length hypothesis. In these models, aturbulent eddy viscosity W and a turbulent thermal conductivity X are addedto the molecular viscosity 0 and thermal conductivity , respectvely,yielding the effective values

=+~"'eff = *

and

eff + X

which then are used in the laminar flow equations. The two models differ inthe underlying assumption that the boundary layer consists either of oneregion or is composed of two regions.

14

The one-layer model expresses the turbulent eddy viscosity by

2 1 w + u I1t = XZ - r 9z

where P is the local density, k is Prandtl's mixing length, w and u are thevelocity in axial and radial direction, respectively. In the DELTA testcalculations the velocity gradients were obtained from the solution of thegoverning differential equations. The mixing length was obtained lom acorrelation. For a steady incompressible flow in a tube Nikuradse

experimentally determined that

2= RI 0.14 - 0.08 (1 - Y)2 0.06 (1 - YL ) I.

where R is the tube radius and y is the distance from the tube wall. Usingthis correlation one assumes that it models the turbulence also in an unsteadyand compressible flow. We test this assumption by comparing the computedresults with those obtained using another turbulence model, a so-called two-layer model.

The two-layer model separates the boundary lay? j an inner and outerregion with different formulations for each region. The expressionfor the eddy viscosity in the inner region is

in

which is the same as the one-layer formulation. The difference is in thedefinition of the mixing length Z, which now is calculated by

k. k y D,

where k = 0.4 is the von Krm~n constant, y is the distance from the wall andD is the van Driest damping factor. The latter is given by

+ I+

AD = i e ,"

11 ." ic,. t in .- "houndar LavU-r Iheor ," '9criw- ii I • 9o6

12.,ubesin, .. *''"Numerica" Turbulence iode i i,,," ,\-S-86

SKusq OV , M .I. . ',it aS, J.' . . ,t-.,trs :tn, C.C., * l estigation of a '!'ree-Dimensional Shock Wave Separated Turbulent Boundary Layer," AIAA J., Vol.18 (1980), No. 12, pp. 1477.

15

where

Y+ y_ _ _Y = )W -Wt

w

and A = 26 is the van Driest constant. The subscript w indicates that thesubscripted quantities are to be evaluated at the wall surface (y-O). Thewall shear stress T is expressed by

w

w w y y=O.

In the outer layer we use for the eddy viscosity the correlation

0.0168 WPt ( -/ )-"

out [1 + 5.5 (y/6)6j.

where we denotes the axial edge velocity, 6 the kinematic boundary layerdisplacement thickness and 6 the boundary layer thickness. In our case we isthe maximum axial velocity in the cross-section z=constant, that is, thevelocity on the axis of symmetry. Hence the outer layer encompasses in thismodel the core flow. The complete two-layer eddy viscosity is given by

if y < Yc

t if y > Ycout :

where yc is the first point at which lit exceeds jt out

The turbulent thermal conductivity is calculated in both cases by

it . p •

t Pr t

where Prt = 0.9 is the turbulent Prandtl number and c is the specific heat atconstant pressure. P

16

, ,' , -.-. - -, . ' ' -.. .--.... .... -.. . --.. , .- ... . -. , . ', " ,. -,.. . ., - - ,., -. . .. .. - .. , .

..

V. RESULTS

Computational results are presented for two different types of interibrballistics flows. The first type simulates a pure gas expansion flow behind aprojectile moving in a constant cross-section tube. The tube is closed at ole

end by a stationary surface called the breech, and at the other end by the

movable flat based projectile. The initial states of the gas are uniform and

quiescent. Geometrical data, initial conditions as well as the thermodynamic

properties of the gas are li:ited in Table I. We designate this idealization

as the Lagrange gun.

TABLE I. LAGRANGE GUN PARAMETERS

Bore Diamneter 20 mm

Tube length 2.0 m

Chamber Length 0.175 mProjectile 'ass 120 g

Ratio of Specific Heats 1.271 3

Covolume 1.08 x10 m k"Molar Mass 23.8 g/mole

Initial Gas Pressure 300 MPaInitial Gas Temperature 3000 K

Initial Velocities 0 m/s

The flow in the Lagrange gun is very well suited for the study of several-

important features such as the performance of the numerical procedure, the

boundary layer development, the laminar and turbulent axisymmetric flow

patterns, and the heat transfer to the tube wall.

Since an expansion flow is quite removed from the phenomena occurrinnz

during a ballistic cycle, a second type of flow is simulated which hs tie--

dependent pressure and temperature profiles similar to a real weapon. It i".obtained by adding proper heat and mass to the one-phase flow via source

terms. An empirical burning law for pressure-dependent sources is used. Tlio-

sources move with the flow. We designate this idealization an the r -.

gun". The essential parameters for the real gun differ Lrom Tahle, I i,,Iv witLrespect to the initial conditions. The initial gas pressure is assiiried t, h,

a-mbient pressure (0.1 NPa) and the initial gas temperature to he amhi,'Ittemperature (293 K). In all cases involving the real 'ui simulation, theprojectile is released from its initial position when the pressure at t h,.projectile base reaches 30 MPa.

Because the DELTA code is based on an implicit finit 'hi,-loro , , ,

no stability condition restrict- the size of the tim, 'ro, . The ;,r,-, ,

results are all calctilated using a constant ti1ne,-st,,p of 1'' "

computational mesh consists of 49 uniformly or nonunifornlv pi-,d m',-h p

17

1I

in the axial direction and 19 nonuniformly spaced mesh points in the radialdirection. To obtain a finer spatial resolution in the boundary region, themesh points are concentrated near the wall. An example of a 19x49computational mesh, which is used in most of the computations, is shown inFigure 1. The smallest grid size in radial direction at the bore surface is7.7 im . The mesh for computing the heat conduction in the tube wall isgenerated in the same way with the same mesh distribution in axial directionas on the gas side, and a corresponding mesh concentration near the inner boresurface. Both the size of time-step and the number of grid points seem to bereasonable compromises between accuracy and computing time.

Some of the computed results for the Lagrange and real gun, respectively,

with laminar flows and adiabatic boundaries are compared in Figures 2-5. The

figures show the histories of the gas pressures and temperatures at the centerof the breech and projectile, and the velocities and displacements of theprojectile. The main differences are in the temporal distributions of allquantities shown, and between the final values of the muzzle velocities. Inthe real gun simulations, the pressure at the projectile base reaches 30 MPaat about 2.3 ms, at which time the projectile begins to accelerate down thetube.

GRID- - -- - - --.-

° It

* I I"

! iI i*I L I

0.000 0.035 0.070 0.105 0.140 0.1'5 ,AXIAL DISTANCE FROM BREECH, M

Fiure 1. Standard Computational Mesh 19 x 49. Minimum ~adia Crid Size is 7.7 lim.

18

-v- V

____TIMEHISTORY

*'BREECH (LG) ' BREECH (RG)

CD .....

BASE (RG) :

0 ..... ........ .......~~~.............. ....... ........................................ ............

C

0.0 1.0 2.0 3.0 4.0 5.0 6.0TIME,* MS

Figure 2. Pressure Histories at the Center of the Breech and theProjectile Base for the Laminar Flow with the AdiabaticWalls in the Lagrange Gun (LG) and the Real Gun (RG)Simulations.

19

TIME HISTORYBREECH (LG)

BASE AND BREECH (RG)~0

.. ... ............... ........... .........................

X IQ

BAEHoG:

0.0 1.0 2.0 3.0 4.0 5.0 6.0

TIMAE, MIS

Figure 3. Temperature Histories at the Center of th~e Breech and theProjectile Base for Laminar Flow with Adiabatic Wallsfor the Lagrange Gun (LG) and Real Gun (FG) Simulations.

20

* 2. * . * --. . . . . . . . . . . . . . . . . . k

6 TIMEHISTORY

-

-J

0 0

0.0 1.0 2.0 3.0 4.0 5.0 6.0

TI ME, MS

Figure 4. Projectile Velocity Histories for Laminar Flow withAdiabatic Walls in the Lagrange Gun (1LG) and RealGun (RC) Simulations.

21

0~TI MEH ISTORY

-4

o G RG

0.0 1.0 2.0 3.0 4-.0 5.0 6.0TIME, MS

Figure 5. Projectile Displacement from the Breech for LaminarFlow with Adiabatic Walls in the Lagrange Gun (LG)and Real Gun (RG) Simulations.

22

Detailed qualitative results for a Lagrange gun with a laminar flow andadiabatic walls are presented in Figures 6-10. Notice that the three-dimensional surfaces are shown from different directions in order to maketheir display clearer. The spatial distribution of the axial velocity isshown in Figures 6 and 7 at 3.75 ms. At this time, the projectile exits thegun tube with a muzzle velocity of 623 m/s. Figure 6 shows the axial velocityfield when a uniformly axially spaced mesh as in Figure 1 is used. Figure 7shows the same quantity but computed using a nonuniformly spaced axial mesh.The radial distribution of the mesh points are the same in both figures. Fora given axial position the axial velocity is constant across much of theradius of the tube (the core flow region), and decreases to zero only veryclose to the wall (the boundary layer region). The boundary layer is theresult of the no-slip condition (w=0) at the wall. The thickness of theboundary layer is approximately 0.2 to 0.3 mm. In the axial direction, thevelocity is distributed linearly in the core region between the zero value atthe breech and the muzzle velocity. The three-dimensional temperature andpressure distributions are given in Figures 8 and 9. Due to the adiabaticboundary condition at the wall surface the heat generated by the viscousforces near the tube wall cannot transfer to the tube wall, and the gastemperature rises towards the wall surface. Here again the boundary layer isonly 0.2 to 0.3 mm thick. The pressure, however, stays constant in radialdirection over the entire cross-section. The assumption in boundary layertheory that the radial pressure gradient is zero would be valid in thisexample. Figure 10 shows the 3-D graph of the radial gas velocity. Inapproximately the first 70% of the distance to the projectile, the radialvelocity is negative, i.e., the flow is directed towards the center line (C-L). Thereafter, it is positive and increases remarkably towards theprojectile base. Only very close to the projectile does the value of theradial velocity drop rapidly from its maximum value to zero. Of course, theradial velocity is small in comparison to the axial velocity since it isinduced only by molecular viscosity and heat conductivity. The results inFigures 2-10 agree very well with both a one-dimensional solution of the coreflow using the method of characterj tis, and the two-dimensional numericalcalculations of Heiser and Hensel.' 4 1

- ,., .,: :: ?A!, Ac:se.F . t risc' " Ie ,,hinenballistik, Teil I: Laminare Einphasenstrmung ohneWgrmeiibergang (AMI: An Axisymmetric Model of Interior Ballistics,

Part 1: Laminar One-Phase Flow without Heat Transfer)," Fraunhofer-Institut fNir Kurzzeitdynamik, Ernst-Mach-Institut, Abteilung furBallistik, Weil am Rhein, FRG, Report No. 4/80, 1980.

15*;eiser, R., :ensel, D., "Berechnung der GasstrSmunig in einem Waflenrohr

mit Hilfe des zweidimensionalen A l-Modells (Calculation of the Gas FlowInside a Gun Tube Using the Two-Dimensional AMI Model)." Fraunhofer-Institut fur Kurzzeitdynamik, Ernst-Mach-Institut, Abteiling f{r BalListik,Weil am Rhein, FRG, Report No. E 1/81, 198t.

23

*1

- .9. 0.

*1. Co

FigreE. Lagrange Gun, Laminar Flow, Adiabatic Walls: SpatialDistribution of the Axial Gas Velocitv at the Time ofMuzzle Clearance. Mesh is Uniformly Spaced in AxialDirection and Nonunifornilv Spaced in Radial Direction.

94

P,

4.00

Figure 7. Lagrange Gun, Lairinar Flow, Adiabatic Walls: Sl lt ia IDistribution of the Axial Gas Velocity at the TJnie ~Muzl Clearance. MUSh is NCn1uTIiforrnly Spqccd inBoth Direction.

77.

~pr

Figure 8. Lagrange Gun, Laminar Flow, Adiabatic Walls: SpatialDistribution of the Gas Pressure at the Time of MuzzleClearance.

26

Muzzle Cearacce

27

0%.0

0.

-rpm

Figure 10. Lagrange Gun, Laminar Flow, Adiabatic WISSpatial Distribution of the Radial (,as Velucit-%at the Time of MuzzlE Clearance.

282

For the Lagrange gun considered above the Reynolds number based on thetube's diameter and muzzle velocity is of the order of . Therefore, theassumption of a laminar flow might not be realistic, and we examined infurther calculations the effects of turbulence. In comparison to the laminarflow, several important differences were observed when the two algebrai"

turbulence models described in Section IV were applied. However, thedifference between the effects of the two turbulence models was found to beinsignificant. Therefore, we do not specify the type of model in thepresented results. First, one observes a difference in the projectileperformance between the laminar and turbulent type flows. This is shown inFigure 11 in terms of the pressure histories at the breech, in Figure 12 interms of the temperature histories at the breech, and in Figure 13 in terms ofthe projectile velocity histories. In the turbulent flow simulation, theprojectile muzzle velocity is about 50 m/s less than in the laminar flowsimulation. The axial velocity flow field at the time of muzzle clearance isshown in Figure 14. The velocity boundary layer is fully developed betweenthe center line and the tube wall. The axial velocity overshoots theprojectile velocity near the center line. This overshoot is related to theradial gas velocity (Figure 15) which, near the projectile base, is one tu .woorders of magnitude larger than in the case of a laminar flow (Figure 10). Inthis region, the greater radial gas flux toward the tube wall transports massaway from center-line which in turn can accelerate the axial flow. The radialvariation of the axial velocity (Figure 16), the radial velocity (Figure 17)and the temperature (Figure 18) taken 0.25 m upstream of the muzzle show somedetails of the differences between the laminar and turbulent flows at the timeof muzzle clearance. Boundary layer calculations in Ref. 7 show comparabletrends between laminar and turbulent flows. However, the flow patternscomputed by different types of turbulence models, e.g., non-algebraic models,may differ. An experiment corresponding to this idealized expansion flow isneeded to validate a turbulence model. Such experiments are being attemptedat the French-German Institute (ISL) in France and the Ernst-Mach-Institut(EMI) in Germany.

In a second series of calculations we investigated the significance ofheat transfer to the tube wall. The heating of the wall's surface haspractical implications because it influences the erosion of the tube. Ourheat transfer model together with the calculation of the heat conduction inthe wall couples the unsteady behavior in both media, and it is discussed inSection III. We now show results obtained for the laminar Lagrange gunexpansion flow with heat transfer from the gas to the tube wall. Initiallythe wall is assumed to be at ambient temperature. The thermal properties ofthe gun barrel are characterized by the barrel material density = 7.8 k/m,

thermal conductivity X = 43 W/(m K) and specific heat cw = 460 J^(kg K). TheWpressure and temperature histories at the breech, and projectile velocityhistory are given in Figures 11-13, and can be compared with the other twosimulations. The spatial profile of the axial velocity, radial velocity,pressure and temperature are plotted in Figures 19-22, respectively, it thetime of muzzle clearance. Comparing the results to those of the lTina;r flowwith adiabatic walls, we find that the velocity boundary layer as well a; thetemperature boundary layer are in the present case thinner, that the r ilial

29

IM

TIME HISTORY ____,

.... ...................................... .. ................................... . ............. ........ ......................... .......

C)

0: .0 1.0 2.0 3.0 4.0TIME, MS WAL

Figure 11. Lagrange Gun: Pressure Histories at the Centerof the Breech for Both Laminar and Turbulent Flows

with Adiabatic Walls, and for Laminar Flow with

Heat Transfer.

30

TI ME H I STORY

LAMINAR ADIABATIC WALL

'~ TURBULENT ADIABATIC WALL........................................ i............ ......* _ ... ................ ...................E-4

LAMINAR HEAT TRANSFER

0.0 L0 2.0 3.0 4.0TIME, MS

N7

Figure 12. Lagrange Gun: Temperature Histories at the Centerof the Breech fcr Both Laminar and -Turbulent Flowswith Adiabatic Walls, and for Laminar Flow withHeat Transfer.

31

0 111

TI ME H ISTORY

coLAMINAR ADIABATIC WALL

TURBULENT ADIABATIC WALL

LAMINAR HEAT TRANSFER0

-LJ

0.0 1L0 2.0 3.0 4.0TIME, MS

p1i,rk 13. Lagrange Gun: Projectile Velocity Histories for PDothLaminar and Turbulent Flows with Adiabatic Walls andfor Laminar Flow with Transfer.

3 2

711

J

Figure 14. Lagrange Gun, Turbulent [-low, Adiabatic nl-

Spatial Distribution of the Axial Cas Velocityat the Time of Muzzle Clearan)ce.

3 3 . .

Figure 15. Lagrange Gun, Turbulent Flow, Adiabatic lalls:Spatial Distribution of the Radial (as Vclocitv ait.the Time of Muzzle Clearance.

34

.

M.-

_ _ RAD IAL VIEW_________TURBULENT

.... ... ........ .............. .............. ... .......... ................... .. .............................. ......... .. .. .

S ........................ .... .............................. ... ............................. .. ..... .................. ........ ......... ....... ...... .... :

. i0.0 2.0 4.0 6.0 8.0 10.0

RADIAL DISTANCE FROM CENTER-LINE, MM

Figure 16. Lagrange Gun, Adiabatic Walls: Radial Profiles of theAxial Gas Velocity for Both Laminar and Turbulent Flow

at the Time of Muzzle Clearance at 0.23 m Away from

the Muzzle.

35

d

RAD IAL VI EW

/TURBULENT

.. .. . . . . ... ..

LAMINAR

0.0 2.0 i.0 6i.0 8.0 10.0

RADIAL DISTANCE FROM CENTER-LINE, MM

Fi uTt re 17. Lagrange Gun, Adiabatic Walls: Radial PrufileL of theRadial Cas Velocity for Both Laminar and TurbulentFlow at the Time of Muzzle Clearance at )..25 n- Aw'avfrom the Muzzle.

. . . .

§ ___RAD IAL VI EW

TURBULENT

o ________LAMINAR _____

0.0 2.0 4.0 6.0 8.0 10.0RADIAL DISTANCE FROM CENTER-LINE, MM

Figure 18. Lagrange Gun, Adiabatic Wails: Radial PrcI ile"-the Gas Tem~perature for Both Laminar aud lurbtiltiitFlows at the Time of Muzzle Clearance at C.253Away frov the Muzzle.

4 37

.....................

~~,qr~ W -- 7-r--J 7'

Ic

i Md

Muzzle0 Claane

&8

'-d4

NM

Figure 20. Lagrange Gun, Laminar Flow, Heat Transfer: SpatialProfile of the Radial Gas Velocity at the Time ofMuzzle Clearance.

39 1

1.2.

- 1 . lgrage G;un, Lrim r Flow, fleat Transfor:SptlProfile of the Gas Pressure at the Time of MuzzleClearance.

40

jis,

Figure 22. Lagrange Gun, I.amnj~r Flow, f!eat Transfer: Sp';~Profile of the Gas Temperature at the Tim. ofMuzzlc Clearance.

41

* velocity is about one order of magnitude higher, and that the radial velocity* is always directed toward the wall. For example, the velocity boundary layer

is only about 0.05 mm thick. Detailed comparisons across the boundary layerregions for axial velocities and temperatures are presented in Figures 23 and24, and across the tube cross-section for the radial velocities are shown inFigure 25. All these profiles are at 0.25 m from the muzzle at the time of

* muzzle clearance. Figure 26 shows the history of the wall surface temperatureat two different locations along the tube wall. One is taken at 150 mm awayfrom the breech. This wall point belongs to the chamber, and is heated upfrom the beginning of the flow cycle. The second wall point is 250 mm awayfrom the breech, and is heated up only after the projectile has passed it (at0.47 ms). The maximum wall temperature occurs early in the 4 ms cycle. Thelaminar gas layer close to the tube wall cools rapidly because the transportof heat in the tube wall by conduction is much faster than the transport of

heat by conduction and convection on the gas side towards the wall. To obtainthe heat transfer precisely from the gas to the wall, we need to compute theradial temperature gradient on both sides of the inner bore surface asaccurately as possible. The temperature boundary layer, however, is very thinas it is shown in Figures 22 and 24. This implies that very small grid sizesmust be used close to the wall. The smallest one in radial direction forFigure 26 is 0.73 um•

The final simulations are of one-phase flows in the real gun, i.e., of agas flow with mass and energy sources. The pressure and temperature historiesat the breech and projectile, and the projectile velocity and displacement

histories for the laminar flow simulation and for the real gun are given inFigures 2-5. Figures 27-40 are a series of three-dimensional profiles which

characterize both the laminar and turbulent flows inside an adiabatic tube.The group of figures associated with the laminar flow are at two times: thefirst at 3.6 ms when maximum pressure is achieved and the second at 5.3 mswhen the projectile exits the cube. For these calculations a nonuniform

radial grid is used with the minimum grid size of 0.73 jm because of thethinness of the boundary layer. We excluded the profile of the radialvelocity at 3.6 ms because the magnitude of this component was smaller than0.05 m/s. Three-dimensional profiles show the turbulent flow fields whenmaximum pressure occurs at 3.6 ms (Figures 34-37), and when the projectileexits the tube at 5.49 ms (Figures 38-40). As in the Lagrange gun, thedifferences are minimal between the simulations with different turbulencemodel, but significant between the laminar and turbulence simulations.

42

RAD IAL VI EW________ ADIABATIC WALL

... .. ... .. ... ..

9.4 9.5 9.6 9.7 9.8 9.9 10.0RADIAL DISTANCE FROM CENTER-LI NE, MM

Figure 23. Lagrange Gun, Laminar Flow: Radial Profiles of theAxial Gas Velocity for Adiabatic and Feat PermeableWalls at the Time of Muzzle Clearance 0.25 m Awayfrom the Muzzle.

43

4.+

4,-

_RAD I AL VI EW i

_ ADIABATIC WALL

.......................... H EA T TRA N SFER............. ............ .

.. .. .. .. .. .. ................ .......... ............................ .......... ............... ........................................ .....................,

9.4 9.5 9.6 9.7 9.8 9.9 10.0

RADIAL DISTANCE FROM 'CENTER-LINE, MM

Figure 24. Lagrange Gun, Laminar Flow: Radial Profiles of OhL

Gas Temperature for Adiabatic and Heat Permeable WallF;at the Time of Muzzle Clearance 0.25 m Away fror theMuzzle.

44

", ," "" ". ,', . .' ' + : ." -+ , ,',_ . + "' .,, " , < _ . + .---+ +" , + .+ " ."'. ° ', " + '++ "+ + + , . .; -+ .• .-

...........- + l" . . ... .. .

o ____ RAD IALVI EW_ __

~6 ...........

lun) :HEAT TRANSFER

CS

0.0 ao 4.0 6.0 8.0 10.0RADIAL DISTANCE FROM CENTER-LI NE, MM

Figure 25. Lagrange Gun, Lampinar Flow: Radial Frofiles ofthe Radial Gas Velocity for Adiabatic and ReatPermeable Walls at the Time of Muzzle Claarance0.25 mn Away from the Muzzle.

45

LAMINAR EXPANSION FLOW WITH HEAT TRANSFER

1500-

d1=50 mm

LU

LL

500

d= DISTANCE FROM BREECH

0 1 2 3 4TIME, ms

Figture 26. History of the Wall Surface Temperature for a LaminarExpansion Flow (LG) with Heat Transfer to the TubeWall at the Locations 150 mm (inside the chamber) and250 mm (inside the barrel) Away from the Eteech.

46

- -- -. .

Figure 27. Real Gun, Laminar Flow, Adiabatic Walls:Spatial Distribution of the Axial GasVelocity at 3.6 m's.

47

*~~~~~~ ~~~~~ . S *.. ' ... S . . .. . S . S. . . . n

Figure 29. Real Gun, Laminar Flow., Aiabatic Walls:Spatial Distribution of the Gas Pressureat 3.6 mns.

49

Figure 31. Real Gun, Laminar Flow, Adiabatic Walls:Spatial Distribution of the Radial GasVelocity at Muzzle Clearance (5.3 ins).

51

........................................ *MM**.**.*.

lira

cve0c

Figure 32. Real Gun, Laminar Flow, Adiabatic Walls:

Spatial Distribution of the Gas Temperatureat Muzzle Clearance (5.3 ms).

52

- -"* ,."* - ;. J " " ' . ' :r " ' d-

.. . .... i , - _". ,, .. - . . . . . . . . . . . . . . . . ..: '_ " 7 "

117 ;7 IL.

Figure 33. Real ('IF , Il Hw Jitbatic alIs:Spatial Distiibution of the Gas Pressureat MIuzzlc Ci-carance (3.3 ins).

"Y" I. '* I~~- - -

4-A

to.

:''' Real Can, Turbulernt Flow, Adiabatic Walls:

Spatial Distribution of the AxialJ Gas

Ve'o)city aL '3.6 .

-~~~~~~~~~ ~~~~~~~~~ - .- .-C F .r r '.-7 rrrr~s ~ rrr ~ -N -

Figure 35. Real Gun, Turbulent Flow, Adiabatic Walls:Spatial Distribution of the Radial GasVelocity at 3.6 mns.

F i g ure 3(16. Real Gun, Turbul vit FIou , Adiabatic Wall~

Spatial Ditrihut ion of Hie Gas Teumpernt~rQat 3.6 mns.

M4~

t'N

Spa ia / i,' 1 H xt

3.V 6

Lc

&0p44p.

Figtire 38. Real Gun, Turbulent Flow, Adiabatic Walls:Spatial Distribution of the Axial GasVelocity at Muzzle Clearance (5.49 ins).

58

1.0.

1.2.

Figure 39. Real Gun, Turbulent Flow, Adiabatic Walls:Spatial Distribution of the Radial GasVelocity at Muzzle Clearance (5.49 mns).

59

0.18

Figur 40. Real Gun, Turbulent Flow:, Adiabatic Walls:Spatial Distribution of the Gas Pressure atMuzzle Clearance (5.49 ins).

60* ~ .. **... - .. * *" * % * j ~ '

VI. SUMMARY

Special interior ballistic phenomena are investigated by using the DiELTAcomputer code, which is designed to calculate the multidimensional, two-phaseflow behind an accelerating projectile inside a gun tube. Details of the heattransfer and turbulence submodels used in the simulations are given in this

report. The general mathematical model and numerical algorithm are describedin a companion paper, Ref. 3. Results are given for two types of idealized,one-phase interior ballistics gun simulations: a pure expansion flow, and a

flow with moving mass and heat sources. Comparisons are made between laminarand turbulent flows as well as between flows in an adiabatic tube and in atube that allows heat transfer. The comparisons show that the flow structure

and the motion of the projectile are both significantly influenced byturbulence and heat conduction, because of a strong coupling between the (-)reflow and boundary phenomena. A study of the effects of turbulence and heatconduction therefore can be best accomplished with a multi-dimensionalmathematical model, such as the DELTA code.

6i

61

REFERENCES

1. "Fluid Dynamics Aspects of Internal Ballistics," ACARD Advisory

Report No. 172, 1982.

2. Celmins, A.K.R., Schmitt, J.A., "Three-Dimensional Modeling Vof Gas-Combusting Solid Two-Phase Flows, "Multi-Phase Flowand Heat Transfer III, Part B: Applications, T. N. Veziroglu

and A.E. Bergles, editors, pp. 681-698, Elsevier Science

Publishers, Amsterdam, 1984

3. Schmitt, J.A., "A Numerical Algorithm for the Multidimensional,

Multiphase, Viscous Equations of Interior Ballistics,"Transactions of the Second Army Conference on Applied Mathematics

and Computing, ARO-Report 85-1, pp. 649-691, 1985

4. S.eiton, S., bergies, A., Sana, P., "Study of Heat Transfer and

Erosion in Gun Barrels," Air Force Armament Laboratory, EglinAir Force Base, Florida, AFATL-TR-73-69, 1973.

5. Cough, P., "Modeling of Rigidized Gun Propelling Charges,"

Ballistic Research laboratory, Aberdeen Proving Ground, MarylandARBRL-CR-00518, 1983.

6. Nelson, C.W., Ward, J.R., "Calculation of Heat Transfer to the GunBarrel Wall," J. Ballistics 6 (3), pp. 1518-1524, 1982.

7. Barlett, E.P., Anderson, L.W., Kendall, R.M., "Time-DependentBoundary Layers with Application to Gun Barrel Heat Transfer,"Proceedings 1972 Heat Transfer Fluid Mech. Institute, StanfordUniversity, CA, 1972.

8. Buckingham, A.C., "Modeling Propellant Combustion Interacting with

an Eroding Solid Surface," Lawrence Livermore Laboratory, UCRL-83727, 1980.

9. Adams, M.J., Krier, H., "Unsteady Internal Boundary Layer AnalysisApplied to Gun Barrel Wall Heat Transfer," Int. J. Heat Mass

Transfer, Vol. 24, No. 12, pp. 1925-1935, 1981.

If). Aerospace Structural 'Metal Handbook, "Ferrous Alloys," 1973.

11. Schlichting, H., "Boundary Layer Theory," McGraw-Hill, 1968.

12. Rubesin, M.W., "Numerical Turbulence Modeling," AGARD-LS-86.

13. Kussoy, M.I., Viegas, J.R., Horstman, C.C., "Investigation of aThree-Dimensional Shock Wave Separated Turbulent Boundary Layer,"AIAA J., Vol. 18 (1980), No. 12, pp. 1477.

63

14. i~ser , eii.cselu. "k L! Li Acixseusvrnz.etriscoes Acueii erInnenballistik, Tell 1: Laminare Einphasenstr6mung ohneWgrmedibergang (ANT: An Axisymmetric Model of InteriorBallistics, Part 1: Lamin~ar One-Phase Flow without HeatTransfer)," Fraunhofer-Institut fdr Kurzzeitdynamik, Ernst-Mach-Institut, Abteilung f~ir Ballistik, Well am Rhein, FRG, Report No.4/80, 1980.

15. ieiser, R., Miensel, , "Berechnung der G asstrbmung in elnemWaffenrohr mit Hilfe des zweidimensionalen AMI-Modells(Calculation of the Gas Flow Inside a Gun Tube Using the Two-Dimensional ANT Model)," Fraunhofer-Institut fir Kurzzeitdynamik,Ernst-Mach-Institut, Abteilung fUr Ballistik, Well am Rhein, FRG,Report No. E 1/81, 1981.

64

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