V. V. Riabov and H. H. Legner- Numerical Study of Hypersonic Turbulent Flow About Segmented...

8/3/2019 V. V. Riabov and H. H. Legner- Numerical Study of Hypersonic Turbulent Flow About Segmented Projectiles

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Copyright 1998, American Institute of Aeronautics and Astronautics, Inc.

A98-32439 AIAA-98-2626NUM ERICAL STUDY OF HYPERSONIC TURBULENT FLOW ABOUT SEGMENTED PROJECTILES

V. V. Riabov*Daniel Webster College

Nashua, New Hampshire 03063-1300, U SAI. V. Yegorov1 and D. V. Ivanov1Central Aero-Hydrodynamics Institute

Zhukovsky-3, Moscow region, Russia ] 40160and

H. H. LegnersPhysical Sciences Inc.

Andover, Massachusetts 01810, U SA

The flow structure about a segmented projectile inhypersonic turbulent flow ha s been studied. The Reynolds-averaged Navier-Stokes equations were solved numericallyby application of the implicit monotonized scheme ofsecond-order accuracy, the modified New ton's method, andth e Christoffel-Schwarz grid-transformation technique. Astrong interference effect has been found. The effect can becharacterized by non-monotonous distributions of skinfriction, heat flux and pressure along the second segment ofth e projectile.

NomenclatureCf = local skin friction coefficientCp = pressure coefficientcp, c = specific heats at constant pressure and volumeD = diameter of the projectileE, G = flux -vectors in curvilinear coordinate system,

E q s . ( l ) , ( 2 )e = total energy per unit volume, p(c^T+(u~ ~ v~)/2)H = total enthalpy per unit volume, cpT + ( if -f v~)/2/? = node sizeJ = Jacobian of the coordinates transformation

*Adjunct Associate Professor, Department ofEngineering, M athematics and Sciences. M ember AJA A.TLeading Scientist, Aerothermody namics Division.:Research Scientist, Aerothermodynamics Division.Principal Research Scientist, Associate Fellow AJAA .Copyright > 1998 by the American Institute of

Aeronautics and Astronautics. Inc. All rights reserved.

k = Boltzmann's constantm molecular massPr = Prandtl numberp = pressureQ = vector of dependent variables, Eq. (1)q = heat flux vectorRe = Reynolds number, pjtj-/u,r = radius of the projectiles = distance along the body surfac e or along an ax isT = temperatureu = x -velocity componentV = velocity vectorv y-velocity componentx = Cartesian x-coordinatey = Cartesians-coordinate

1 Y = specific heat ratio, cjc^A = distance between the segmentsT J = curvilinear coordinateA = conductivity coefficientA, = eigenvalueu = viscosity coefficientE , = curvilinear coordinatep = density of fluidf k = regularization parameter, Eq . (17)T^ = viscous stress tensor

( f> = function, Eq. (11) - small parameter, Eq.(11)Subscriptsc - Cartesian coordinate systemw = wall value = freestream value

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IntroductionThe hypersonic projectile studies have been actively

revived with recent interest in electromagnetic launchers.Hypervelocity projectile aerophysics, includingaerodynamics and flight mechanics, ha s been reviewed byReinecke and Legner1. Aerothermodynamics of hypersonicnon-segmented projectiles has been studied experimentallyand numerically by Cayzac et al.2 Launch perturbationeffects in electromagnetic guns were evaluated by Seiler etal .3 Orphal an d Franzdn have found that th e terminaldamage of a hypervelocity projectile can be enhanced if theprojectile mass distribution is segmented along its axis.

In the present analysis, hypersonic turbulent flow abouttwo-segment projectile have been studied. The analysis oftwo-dimensional flow structure is based on numericalsolutions of the Reynolds-averaged Navier-Stokes equationsusing an implicit monotonized scheme of second-orderaccuracy (Total Variation Diminishing scheme) an dNewto n ' s method for solving the grid equations.5 Thetechnique has been successfully applied by Yegorov,Yegorova, and Riabov* and Yegorov et all in studies ofhypersonic viscous flow about bluff cylinders an d plateslocated one after another.

Th e influence of the Reynolds number and thegeometrical factor of interference between the segments,A h; on skin-friction, heat f lux , velocity, turbulenceparameter, pressure an d temperature in the flow field an dalong th e surfaces of the projectile has been studied.

Navier-Stokes Equations an d Boundary Conditions.The unsteady two-dimensional Navier- Stokes

equations in a curvilinear coordinate system ( , ? / ) , - v = xU",/7), v = v ( f , / ? ) , w here x. y are Cartesian coordinateshave a conservation form:8-"

..,, (2)where .1 = d(x,y)/d( 77) is a Jacobian of the

coordinates' transformation.Th e Cartesian vector components Qc, Ec, Gc, and Bc

fo r the two-dimensional Navier-Stokes equations have th efollowing form:

pp i /p vp(e+g2)Pgpc o

* R

0000

~"]p


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The heat flux vector q is calculated by formula:

Turbulent viscosity /*T and heat conductivitycoefficients X ^ are determined as follow:

Pr,

0i + -Pri= \/k; c o = e / A "

Th e mass-averaged Navier-Stokes equation system (1)for the compressible perfect gas model is completed by thestate equation as well as by two equations describing thedifferentional turbulent g- u > model offered by Coakley andHuang10 :

; C = 0 . 0 9

a =0.02=l-exp -a, r (c ,S 2divV) rVCl(C~J~r

fJL-c 'v} - c~ -' 23 co J" -(7)

ax av

o n c v . [ e nex dy

v~ v| CM ovox 6v

where p=0 for two-dimensional problem, and (3=1 foraxisymmetrical one. Other parameters are as fol low":

Cn=C1 2=l; C 22 = 0.833; C23 = 2.4

Also, it is assumed that the coefficient of viscosity iscalculated using a power law variation with temperature,/ / / / / = (T / T J 07 , and the Prandtl number is constant, Pr =0.7, and /V T = 0.9.

For further numerical analysis, new non-dimensionalparameters in Eqs. (1) - (6) were set up by norm alizing theCartesian coordinates to the characteristic length scale r(the radius of a cylinder), the Cartesian velocity components- to the upstream velocity , the pressure - to the doublevalue of the dynamic pressure in the upstream flow, andother parameters - to their values in the upstream f low.

To complete the finite-difference system of the Navier-Stokes equations, the following boundary conditions havebeen used. Theno-slip conditions (u - v = 0, g = 0, anddo)/dci = 0) , constant surface temperature (T = T J , an dextrapolations of a pressure from inner area nodes (with thecondition a p / ' d T J = 0) were posed on the body surface. Onthe outer surface of the computational area around the body,boundary conditions w ere w ritten in the form of Riemanninvariants and determined by the direction of theperturbation expansion9.

Th e Approximation of EquationsThe construction of a finite-difference scheme to solve

the Navier-Stokes equations (1) given in conservation lawsform is based upon an integro-interpolation method.9 Theutilization of an integro-interpolation method applied to thesolution of the Navier-Stokes equations gives finite-differences conservation laws analogies:

(9)

= 2

Here index n corresponds to time layer number; j, k -to node numbers along , and T), correspondingly; hph^-to the node sizes. The developed conservative finite-difference scheme is implicit, and this property of thescheme allows one to avoid any restrictions on the iterationtime-step caused by the instability of the ordinary differenceschemes in the solution of the stiff differential equations.

At semi-integer nodes, the convective components ofthe flux vectors E and G were approximated using a



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monotonized scheme of Godunov type.12 Th e eigenvaluesand eigenvectors at semi-integer nodes were calculated bythe Roe method13, for the approximate solution of theproblem of arbitrary discontinuity decay:

(10)1 C Q R -Here 3>(ALR) is a diagonal matrix w ith elements

parameters A t are the eigenvalues of the operator A =3E/5Q; an d RLR = R(Q,R) is a matrix with the columnsbeing the right-hand side eigenvectors of the operator A.The function ( f > ( A ) has the form:

(11)2e

The Roe method13 was utilized to compute eigenvaluesand eigenvectors of the A-operator in order to solveapproximately the Riemann problem of arbitrarydiscontinuity decay. Parameters 3>(A LR), R , R^ R"' werecalculated by the values of dependent variables, such as:

L R = -

(14)

Here the parameter c indicates the local speed ofsound.

Th e diffusion components of the flux vectors E and Gat the node side were approximated by the second ordercentral difference scheme:

This form satisfies the "entropy" condition (or thecriterion) in the choosing of a numerical solution with thecorrect physical properties.To increase the order of the finite-differenceapproximations up to the second order, the Monotone-Upstream-Scheme-for-Conservation-Laws principle of theminimum derivatives14 w as used to interpolate dependentvariables on the node side as follows:

QJ + 7 mninocKQ,

Th e function minmod(a,/>) has the form:

(12)

a , ab>Q. i a i < i A 'b, ab>0, \ a \ > \ b \ (13)

0. ab


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Solution of Nonlinear Differences EquationsThe nonlinear system of grid equations (F(X) = 0,

where X is a vector of unknown discrete functions) w assolved using the modified Newton's method:5-8-'7

(16)

Here D = dF/oX is the Jacobi matrix; k is the iterationnumber. In computations, the regularization parameter Tkwas calculated bv formula18:

(17)

where AXM is a vector of corrections. The iterationprocess is convergent with the second order of theconvergence rate and Tk- 1.

The iteration Jacobi matrix was found by employing theprocedure of finite increments of the residual vector of therequired grid functions. The approximation of the Navier-Stokes equations determinates the type of the Jacobi D-operator, namely, the rarefied structure of triangularmatrices and the initial 7x?-dense matrix coincide.

The system of linear algebraic equations obtained in anonlinearity iteration was solved by expanding the matrixinto a product of two triangular matrices L and U, where Lis the lower triangular matrix and U is the upper triangularmatrix, an d d f / d X - L*U. This operation w as preceded bythe analysis of the sparsity structure of matrices L and U. Inorder to reduce the total num ber of the arithmetic operationsand economize on RAM, the variables were numberedusing the generalized method of nested dissection 19 '20. Thistechnique w as successfully used many times incomputational experiments and prove d its effectiveness andreliability.9-11

Numerical Calculations and TestsTh e calculations were mainly performed on the

2 0 1 x 1 5 1 grid of the H-type. The size of the nodes w asautomatically reduced near the body surface" and in thevicinity of the symmetry axis.

The convergence and accuracy of the numericalsolutions were tested by carrying out a series of calculationsof hypersonic viscous flow about a cylinder (at Reynoldsnumber Re,r = 10 5 an d Mach number in upstream flow M,= 5) on the grids of different size. The analysis of theresults5"7-11 showed that the numerical solution of theproblem is convergent. Calculations were carried out at theWork Station RS6000/58H.

To improve the convergence rate of the iterationprocess, the new technique using the Jacobi matrix wasused. The method is based upon the usage of a truncated3x3-dense matrix. The com puting time (approximately 3h .20 min) of each variant was reduced by the factor of three.

ResultsThe flowfield around a two-segment projectile w as

calculated for a Mach number M a = 6 and for a Reynoldsnumber Re,, = 105, 106, an d 107. It was assumed that y =1.4, and the body surface is isothermal at TJT, = 1. Thedistance betw een the segments w as A/D = 1.2 an d 2.4.

Th e contours of constant values of local Mach numberM, temperature T /T m and velocity at Rem r - 10 6 and A ID =1.2 are shown in Figs. 1, 2, and 3, correspondingly. Th eflow structure changes significantly in the areas between thesegments an d behind the projectile. The zone between thesegments becomes totally subsonic and turbulent. As aresult, the hot-gas area near the down-stream segmentspreads fa r up-stream, up to the rear zone of the firstsegment. Behind the second segment, the wake areabecomes narrow and it is filled with ho t gas.

Th e distribution of the turbulence ^-parameter isshown in Fig. 4. The flow near the second segment isturbulent and extremely complex. Density contours in theflow between the segments are shown in Fig. 5. Theirpattem is significantly different from the zones withdifferent values of a turbulent ^-parameter.

Th e distributions of pressure and skin-frictioncoefficients, and heat flux along the surface of the firstsegment (the distance s is calculated from the nose) areshown in Figs. 6,7, and 8 correspondingly. T he developingrecirculation zone between the projectile segmentsinfluences these aerothermodynamic parameters on the rearsurface of the first body considerably. A t Rear = 107, theflow becomes turbulent. It results in a significant increase



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Copyright 1998, American Institute of Aerona utics and Astron autics, Inc.

of shear stress and heat flux. Th e size of the recirculationzone depends on both the distance between the segmentsand the Reynolds number. Th e influence of the first factoron the aerodynamic characteristics of the first segment islocalized in the rear area.

Th e pressure distribution between segments is shownin Fig. 9. At Re mr> 10 6, pressure increases in the turbulentflow near the front area of the second segment. This effectincreases with increasing the distance between the bodies.Th e distributions of pressure and skin-friction coefficients,and heat flux along the surface of the second segment areshown in Figs. 10,11, and 12 correspondingly. The extremevalues of these parameters occur on the front surface of thesegment. T he pressure coefficient and heat flux in this areaare larger than their values on the surface of the firstsegment by a factor of eight. The Reynolds numberinfluences significantly the aerodynamic characteristics(skin-friction and heat flux). Th e geometrical factor (adistance between the bodies) influences pressuredistribution in the front area of the second segment.

Th e flow behind the bodies remains turbulent at Re,rz10 6 (see Fig. 13). Th e pressure distribution pattern in thiscase is different from the pattern of the laminar flow at Re,r= 105. The increasing of the distance between the bodiesresults in increasing pressure in the rear local area behindthe second segment. Th e area width can be estimated as ID.

SummaryThe hypersonic turbulent flow parameters near a two-

segment projectile have been evaluated using an effectivenumerical algorithm The turbulence significantly influencesthe distributions of the pressure, skin friction and heat fluxalong the projectile surfaces as well as the flow parametersin the wakes behind the segments. At Re^ r > 106. it hasbeen found that the flow zone between the segments ha sbecome a fully recirculating subsonic flow. This effect isresponsible for significant change of skin-friction and heat-flux characteristics along the second-segment surface.

References'Reinecke, W. G., and Legner, H. H., "A Review of

Hypervelocity Projectile Aerophysics," A1AA Paper, No.95-1853, June 1995.

2Cayzac, R., Tassel, B., Carette, E., Champigny. P.. and

Bernhardt, J. M., "Aerothermodynamics of HypersonicProjectiles,"AIAA Paper, No. 95-1854, June 1995.

3Seiler, S., Legner, H. H ., Miller, M. G., and Reinecke,W ., "Impact of EM Effects on Hypervelocity ProjectileFlight," AIAA Paper, No. 95-1852, June 1995.

'Orphal, D. L., and Franzen, R. R., "PenetrationMechanics and Performance of Segmented Rods AgainstMetal Targets," International Journal of ImpactEngineering, Vol. 10 , 1990, pp . 427-438.

5Yegorov, I. V., Yegorova, M. V., Ivanov, D. V ., andRiabov, V. V., "Numerical Study of Hypersonic V iscousFlow About Plates Located Behind a Cylinder," AIAAPaper, No. 97-2573, June 1997.

6Yegorov, I. V., Yegorova, M. V ., and Riabov, V. V.,"Analysis of Hypersonic Viscous Flow About BluffCylinders Placed O ne After Another," AIAA Paper, No. 98-017 I .January 1998.

7Yegorov, I. V., Ivanov, D. V., and Obabko, A. ,"Newton Technique in Numerical Simulation of TurbulentFlows," 1997.

8Bashkin, V. A., Egorov, I. V., and Egorova, M. V.,"Supersonic Viscous Perfect Gas Flow Past a CircularCylinder," Fluid Dynamics, No. 6, Nov.-Dec. 1993, pp .833-838.

9Yegorov, I., and Zaitsev, 0., "Development ofEfficient Algorithms fo r Computational Fluid DynamicProblems," Proceedings of the Fifth InternationalSymposium on Computational Fluid Dynamics, Japan,Sendai, 1993, Vol. Ill, pp. 393-400.

10Coakley, T. J., and Huang, P. G., "TurbulenceModeling for High Speed Flows," AIAA Paper, No. 92-0436, January 1992.

"Ivanov, D. V., Obabko, A. V., and Yegorov, I. V.,"Simulation of Separated Flow s on the Base of DifferentialTurbulence Model," AIAA Paper, No. 97-1861, July 1997.

I2Godunov, S. K., "Finite Difference Method forNumerical Computation of Discontinuous Solutions of theEquations of Fluid'Dynamics," Matematicheskii Sbornik,Vol.47, No. 3,1959, pp . 271-306 (in Russian).

13Roe, P. L., "Approximate Riemann Solvers,Parameter Vectors, and Difference Scheme," Journal ofComputational Physics, Vol. 43, 1981, pp. 357-372.

'"Kolgan, V. P., "Application of the Principle ofMinimum Value of Derivatives to the Construction ofFinite-Difference Schemes for C alculating DiscontinuousSolutions of Gas Dynamics," Uchenye Zapiski TsAGI, Vol.3, No. 6, 1972, pp . 68-77 (in Russian).

"Yegorov, I. V., and Ivanov, D. V., "Application of theComplete Implicit Monotonized Finite-DifferentialSchemes in Modeling Internal Plane Flows," Journal of

432American Institute of A eronautics and Astronautics


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Computational Mathematics and Mathematical P hysics,Vol. 36 , No. 12, 1996, pp. 91-107.

"Davis, R. T., "Numerical Methods for CoordinateGeneration Based on Schwarz-ChristoffelTransformations," AIAA Paper, No. 79-1463, June 1979.

'"Bashkin, V. A., Yegorov, I. V., and Ivanov, D. V.,"Application of Newton's Method to Calculation ofSupersonic Internal Separation Flows," Journal of AppliedMechanics an d Technical Physics, Vol.38, No.l, 1997, pp .30-42.

!SKarimov, T. Kh.,"Some Iteration Methods fo rSolving Nonlinear Equations in the Hilbert's Space,"DoldadyAkademiiNaukSSSR, Vol. 269, No. 5 ,1983 , pp .1038-1042 (in Russian).

19Lipton, R. X, Rose, D. X, and Tarjan, R. E.,"Generalized Nested Dissection," SLAM Journal ofNumerical Analysis, Vol. 16, No. 2, 1979, pp. 346-358.

^gorov, I. V., andZaitsev. 0. L., "On anApproach tothe Numerical Solution of the Two-Dimensional Navier-Stokes Equations by the Shock Capturing Method," Journalof Computational Mathematics and Mathematical Physics,Vol. 31, No. 2,1991, pp. 286-299.

Fig. 3 Velocity () contours near a two-segmentprojectile at Re, = 106, M , = 6.

Fig. 4 Turbulence ^-p arameter contours near atwo-segment projectile at R e, = 106, A f , = 6.

Fig. 1 Mach number contours near a two-segmentprojectile at Re, = 10 6, Af. = 6.

Fig. 2 Temp erature contours near a two-segmentprojectile at Re, = 10*, A f , = 6. Fig. 5 Density contours near a two-segmentprojectile at Re, = 106, Afm = 6.



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0.05

0.00 -

17

0 s/D 'Fig. 6 Pressure coefficient ( C f) distribution along thesurface of the first segment

qwxRe%

2 -

Re=105.4=1.2D

1 -

Re=107,4=1.2D

s/DFig. 8 Heat flux ( J distribution along the surface ofthe first segment

CfxR e1/2

4 -

2 -

0 -

Re=106,4=1.2D

0.00

Re=1D5,/i=1.2DRe=106,4=1.2D

-=_-- Re=1CT,4=2.4D L----- Re=10,4=2.4D //_._. Re=107,/i=2.4D

s/D

Fig. 7 Skin-friction coefficient ( C j ) distribution along Fig. 9 Pressure coefficient (C f) distribution between th ethe surface of the first segment segments of the projectile.



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0.2 -

0.1 -

0.0 -

Re=105. A=1.

Re=107,d=1-2DRe=105,4=2.4DRe=106,^=2.4DRe=107,4=2.4D

qwxRe

4 -

2 -

Re=105,4=1.2DRe=106, 4=1-2DRe=107,4=1.2D

- Re=106, A=2.4D

s/D s/ D12

Fig. 10 Pressure coefficient (Q distribution along the Fig. 12 Heat flux (#) distribution along the surface ofsurface of the second segment the second segment

Re=106,/\=1.2D

-2 -

0.00 -,

-0.01 -

-0.02 -

-0.03 -s/ D

Re=105, A=1.2D

Re=107,A=1-2D-- .-- R e = 1 0 5 , 4=--C--- Re=106,^=2.4D-..-.. Re=107,/i=2.4D

Fig. 11 Skin-friction coefficient (Q distribution along Fig. 14Pressure coefficient (Q distribution hi the wakethe surface of the second segment behind second segment.

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V. V. Riabov and H. H. Legner- Numerical Study of Hypersonic Turbulent Flow About Segmented...

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