Numerical study of hypersonic turbulent flow about ...

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

A98-32439

AIAA-98-2626

NUMERICAL STUDY OF HYPERSONIC TURBULENT FLOW ABOUT SEGMENTED PROJECTILES V. V. Riabov* Daniel Webster College Nashua, New Hampshire 03063-1300, USA I. V. Yegorov1 and D. V. Ivanov1 Central Aero-Hydrodynamics Institute Zhukovsky-3, Moscow region, Russia ] 40160 and H. H. Legners Physical Sciences Inc. Andover, Massachusetts 01810, USA k m Pr p Q q Re r s T u V v x y 1 Y A TJ A A, u E, p fk T^ (f> £

The flow structure about a segmented projectile in hypersonic turbulent flow has been studied. The Reynoldsaveraged Navier-Stokes equations were solved numerically by application of the implicit monotonized scheme of second-order accuracy, the modified Newton's method, and the Christoffel-Schwarz grid-transformation technique. A strong interference effect has been found. The effect can be characterized by non-monotonous distributions of skin friction, heat flux and pressure along the second segment of the projectile. Nomenclature Cf = local skin friction coefficient Cp = pressure coefficient cp, c = specific heats at constant pressure and volume D = diameter of the projectile E, G = flux-vectors in curvilinear coordinate system, Eqs.(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 size J = Jacobian of the coordinates transformation

*Adjunct Associate Professor, Department of Engineering, Mathematics and Sciences. Member AJAA. T Leading Scientist, Aerothermodynamics 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.

= Boltzmann's constant — molecular mass = Prandtl number = pressure = vector of dependent variables, Eq. (1) = heat flux vector = Reynolds number, pjtj-/u«, = radius of the projectile = distance along the body surface or along an axis = temperature = x-velocity component = velocity vector —y-velocity component = Cartesian x-coordinate = Cartesians-coordinate = specific heat ratio, cjc^ = distance between the segments = curvilinear coordinate = conductivity coefficient = eigenvalue = viscosity coefficient = curvilinear coordinate = density of fluid = regularization parameter, Eq. (17) = viscous stress tensor = function, Eq. (11) - small parameter, Eq. (11)

Subscripts c - Cartesian coordinate system w = wall value «• = freestream value 427

American Institute of Aeronautics and Astronautics


Introduction

The hypersonic projectile studies have been actively revived with recent interest in electromagnetic launchers. Hypervelocity projectile aerophysics, including aerodynamics and flight mechanics, has been reviewed by Reinecke and Legner1. Aerothermodynamics of hypersonic non-segmented projectiles has been studied experimentally and numerically by Cayzac et al.2 Launch perturbation effects in electromagnetic guns were evaluated by Seiler et al.3 Orphal and Franzdn have found that the terminal damage of a hypervelocity projectile can be enhanced if the projectile mass distribution is segmented along its axis. In the present analysis, hypersonic turbulent flow about two-segment projectile have been studied. The analysis of two-dimensional flow structure is based on numerical solutions of the Reynolds-averaged Navier-Stokes equations using an implicit monotonized scheme of second-order

(2) ..,,

where .1 = d(x,y)/d( £ 77) is a Jacobian of the coordinates' transformation.

The Cartesian vector components Qc, Ec, Gc, and Bc for the two-dimensional Navier-Stokes equations have the following form:

pv

p(e+g2) Pg pco

accuracy (Total Variation Diminishing scheme) and Newton's method for solving the grid equations.5 The technique has been successfully applied by Yegorov,

* R

~"]p(ALR), R^, R^R"' were calculated by the values of dependent variables, such as:

(10)

CQR -

»LR = -

(14)

Here 3>(ALR) is a diagonal matrix with elements parameters At are the eigenvalues of the operator A = 3E/5Q; and RLR = R(Q,R) is a matrix with the columns being the right-hand side eigenvectors of the operator A. The function (f>(A) has the form: Here the parameter c indicates the local speed of sound.

The diffusion components of the flux vectors E and G at the node side were approximated by the second order central difference scheme:

(11)

2e

This form satisfies the "entropy" condition (or the criterion) in the choosing of a numerical solution with the correct physical properties. To increase the order of the finite-difference approximations up to the second order, the MonotoneUpstream-Scheme-for-Conservation-Laws principle of the minimum derivatives14 was used to interpolate dependent variables on the node side as follows:

QJ + 7 mninocKQ, (12)

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

a, ab>Q. i a i < i A '

b, ab>0, \ a \ > \ b \

(13)

0. ab 106. it has been found that the flow zone between the segments has become a fully recirculating subsonic flow. This effect is responsible for significant change of skin-friction and heatflux characteristics along the second-segment surface.

Equations of Fluid'Dynamics," Matematicheskii Sbornik, Vol.47, No. 3,1959, pp. 271-306 (in Russian). 13 Roe, P. L., "Approximate Riemann Solvers, Parameter Vectors, and Difference Scheme," Journal of Computational Physics, Vol. 43, 1981, pp. 357-372. '"Kolgan, V. P., "Application of the Principle of Minimum Value of Derivatives to the Construction of Finite-Difference Schemes for Calculating Discontinuous

References 'Reinecke, W. G., and Legner, H. H., "A Review of Hypervelocity Projectile Aerophysics," A1AA Paper, No. 95-1853, June 1995. 2 Cayzac, R., Tassel, B., Carette, E., Champigny. P.. and

Solutions 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 the Complete Implicit Monotonized Finite-Differential Schemes in Modeling Internal Plane Flows," Journal of

432 American Institute of Aeronautics and Astronautics


Computational Mathematics and Mathematical Physics,

Vol. 36, No. 12, 1996, pp. 91-107. "Davis, R. T., "Numerical Methods for Coordinate Generation Based on Schwarz-Christoffel Transformations," AIAA Paper, No. 79-1463, June 1979. '"Bashkin, V. A., Yegorov, I. V., and Ivanov, D. V., "Application of Newton's Method to Calculation of Supersonic Internal Separation Flows," Journal of Applied Mechanics and Technical Physics, Vol.38, No.l, 1997, pp. 30-42. !S Karimov, T. Kh., "Some Iteration Methods for Solving Nonlinear Equations in the Hilbert's Space," DoldadyAkademiiNaukSSSR, Vol. 269, No. 5,1983, pp. 1038-1042 (in Russian). 19 Lipton, R. X, Rose, D. X, and Tarjan, R. E., "Generalized Nested Dissection," SLAM Journal of Numerical Analysis, Vol. 16, No. 2, 1979, pp. 346-358. ^gorov, I. V., andZaitsev. 0. L., "On an Approach to the Numerical Solution of the Two-Dimensional NavierStokes Equations by the Shock Capturing Method," Journal of Computational Mathematics and Mathematical Physics,

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

Fig. 4 Turbulence ^-parameter contours near a two-segment projectile at Re, = 106, Af«, = 6.

Vol. 31, No. 2,1991, pp. 286-299.

Fig. 1 Mach number contours near a two-segment projectile at Re, = 106, Af. = 6.

Fig. 2 Temperature contours near a two-segment projectile at Re, = 10*, Af«, = 6.

Fig. 5 Density contours near a two-segment projectile at Re, = 106, Afm = 6.



qwxRe%

Re=105.4=1.2D

0.05 Re=107,4=1.2D

2 -

0.00 -

1 -

17

0

s/D

'

s/D Fig. 8 Heat flux (^J distribution along the surface of the first segment

Fig. 6 Pressure coefficient (Cf) distribution along the surface of the first segment

Re=1D5,/i=1.2D 1/2

Re=106,4=1.2D

Re=106,4=1.2D

CfxRe

4 -

-•=_--

Re=1CT,4=2.4D

-----

Re=10°,4=2.4D

_._.

£L

//

7

Re=10 ,/i=2.4D

2 -

0 0.00

s/D Fig. 7 Skin-friction coefficient (Cj) distribution along the surface of the first segment

Fig. 9 Pressure coefficient (Cf) distribution between the segments of the projectile.



q w xRe Re=105,4=1.2D

Re=105. A=1.

Re=106, 4=1-2D

4 -

0.2 -

Re=107,4=1.2D

Re=10 7 ,d=1-2D Re=105,4=2.4D Re=106,^=2.4D

0.1 -

-

Re=10 7 ,4=2.4D

Re=106, A=2.4D

2 -

0.0 -

1

s/D

s/D Fig. 10 Pressure coefficient (Q distribution along the surface of the second segment

2

Fig. 12 Heat flux (#„) distribution along the surface of the second segment

Re=10 6 ,/\=1.2D

0.00 -,

-0.01 Re=105, A=1.2D

Re=10 7 ,A=1-2D

-0.02 -

-2 -

--^.--

Re=10 5 , 4=

--C---

Re=10 6 ,^=2.4D

-.£.-..

Re=107,/i=2.4D

-0.03 -

s/D Fig. 11 Skin-friction coefficient (Q distribution along the surface of the second segment

Fig. 14 Pressure coefficient (Q distribution hi the wake behind second segment.

435 Amsrican Institute of Aeronautics and Astronautics