RECENT IMPROVEMENTS IN THE NONEQUILIBRIUM VSL SCHEME FOR HYPERSONIC BLUNT-BODY FLOWS

BOA. Bhutta and CoHo Lewis VRA, Inc., Blacksburg, VA

29th Aerospace Sciences Meeting January 7-1 0, 1991/Reno, Nevada

Copyright @ VRA, Inca, 1991. All rights reserved

RECENT IMPROVEMENTS IN THE NONEQUILIBRIUM VSL SCHEME FOR HYPERSONIC BLl JNT-BODY FLOWS

Bilal A. Bhutta' a r k Clark I I. I,ewis2

VRA, Inc. Blacksburg, VA 24063

ABSTRACT

The nonequilibrium viscous shock-layer (VSI,) solution scheme is revisited to improve its sol- ution accuracy in the stagnation point region and also to minimize and control the errors in the conservation of elemental mass. The stagnation- point solution is improved by using a second- order expansion for the normal velocity and the elemental mass conservation is improved by di- rectly imposing the element conservation equations as solution constraints. These modifi- cations are such that the general structure and computational efficiency of the nonequilibrium VSL scheme is not affected. This revised none- quilibriurn VSL scheme is used to study the Mach 20 flow over a 7-deg sphere-cone vehicle under zero and 20-deg angle-of-attack conditions. Comparisons are made with the corresponding predictions of Navier-Stokes and Parabolized Navier-Stokes solution schemes. The results of these tests show that the noi~equilibrium blunt- body VSL scheme is indeed an accurate, fast and extremely efficient means for generating the blunt-body flowfield over spherical nosetips at small-to-large angles of attack.

NOMENCLATURE

A 1 1 A = angle of attack, a Ci - mass Fraction of the i-th species C FS = streamwise skin-friction coefficient CFW = crossflow skin-friction coeficient CP = nondimensional specific heat a t

constant pressure, Cp*/Cpt, I), = nondimensional binary difTusion

coefficient for the i-th species, Di* ~ * r n l ~ * r e f

L = mass fraction of the e-th element

h l k

1 ,e M n

%hl

N 13 NS I',P

1'1 11 I'INI7 I'r 1' w YW R I< B Re RIIO

scale factor, (1 + qnshl) nondimensional thermal conductivity, k*/pfrerCp*, Lewis number, pCpDi/k Mach number nondirnensional surface-normal coordinate n'/Rn* iirst-order term in the expansion of tlsh [Eqs. (5)] total number of elements total number of chemical species nondimensional pressure, P*/P*~(V*,)~ circumferential angle, 4 freestream static pressure, p , I'randtl number, pCp/k wall pressure, p, wall heat-transfer rate radial distance from body axis local body radius Reynolds number, (pVRn)/p density, p freestream density, p, nose radius nondimensional streamwise coor- dinate, s*/Rn* nondimensional temperature, rr*/T*ref dimensional reference temperature, (V*,)2/Cp*m nondimensional streamwise veloc- ity, u*/V*, total velocity nondimensional body-normal ve- locity, v*/V*, nondimensional crossflow velocity, w'/V', coordinate along body axis angle of attack

1 Chief Scienlist, senior Member AIAA

2 President, Associate Fellow AIAA

Superscript - -

Reynolds number parameter, [~* re~ l~*mV*ooRn ' l~~~ ctrcumferential angle measured from the windward side streamwise coordinate. s normalized body-normal coordi- nate, n/nsh crossflow coordinate, 4 nondimensional density, P * / P * ~ ~ nondimensional viscosity, p*/CL*ref

dimensional viscosity evaluated at T*rei

quantity scaled with respect to the corresponding shock valuc dimensional quantity

represents partial derivative freestream quantity reference quantity shock quantity wall quantity

INTRODUCTION

Over the last several years the prediction of high-altitude ( > 120kft) hypersonic reentry flows has become an area of significant interest and development. At such high altitudes, the charac- teristic reaction time is much longer than the characteristic flow time, and the vehicle is in the chemical nonequilibrium flow regime for most of the time.

Existing CFD schemes for predicting none- quilibrium hypersonic blunt-body flows can be broadly classified as (a) Boundary-Layer (BL) methods, (b) Viscous Shock-Layer (VSL) meth- ods, and (c) Navier-Stokes (NS) methods. I lypersonic reentry flows are, in general, charac- terized by low Reynolds numbers. Due to such typically low Reynolds number flows, the appli- cation of boundary-layer methods (such as the schcrnc of Dlottner et al.') encounters significant numerical difficulties (such as displacement- thickness interaction, streamline tracking, deter- mination of edge conditions, etc.). As far as nonequilibrium Navier-Stokes methods are con- cerned, over the last few years a number of such schemes have become a ~ a i l a b l e . ~ - I Iowevcr, such methods are typically very time consuming and not well suited for- practical design and anal- ysis studies.

On the other hand, the nonequilibrium VSI, blunt-body methods have shown great potential

for analyzing such nonequilibrium viscous reentry flows. Various studies over the last fifteen y c a r s U 4 have shown that the VSL scheme pro- vides a very efictive tool for accurately and effi- ciently prcdicting the flowfield in regions where thc flow is attached in the crossflow region. While the VSL schemes can not be used in re- gions of crossflow separation, even under large angle-of-attack conditions the flowfield in the nose rcgion typically remains attached in the crossflow direction. Consequently, the VSL schemes can still be used to provide the necessary starting solution to initiate the afterbody PNS solution schetne. Although reliable nonequilib- riurn Navier-Stokes schemes have recently be- come available, the nonequilibrium VSL blunt-hotly scheme can accurately predict the blunt-body flowfield for a small fraction of the computing time rcquired by these Navier-Stokes schcrncs.

Rcccntly, some comparisons between the predictions of the nonequilibrium VSL scheme of Swarninathm et aL7 and the nonequilibrium Navicr-Stokes predictions of Molvik and Merkle2 werc made by 13uclow et al.15 'I'he flowfield accu- racy of the corresponding perfect-gas and equilihrium-air VSL scheme has been recently confirmed hy TSdwards and Flores lhnd Ryan et al.17 by comparing with the predictions of the CNS (Compressible Navier-Stokes) code. The recent nonequilibrium comparisons of Buelow et al.ls were, however, not satisfactory. In this case, the Navier-Stokes and VSL methods were used to provide starting solutions for the PNS afterbody schemes and species profiles were compared at an axial location of 10 nose radii. The freestream flow corresponded to a Mach 20 reentry at an al- titude of 175 kft, and the geometry consisted of a 7" sphere-cone with a nose radius of 2 inch. This axisymrnctric case was based on an earlier 20-deg angle-of-attack case studied by Bhutta and I,ewis.I8 'I'hc rcsults of Duelow et al. '9howed that although thc wall pressure and shock-shape predictions of the nonequilibrium VSL and none- quilibriurn Nnvicr-Stokes schemes agreed quite well in thc blunt-body region, the species profiles at thc body end did not. Buelow et a1.15 accepted this wall-prcssurc and shock-shape agreement in the blunt-hotly region as a check of the blunt- body flow and, thus, incorrectly concluded that the disagreement at the body end was due to some error in the nonequilibrium PNS scheme of Ijhutta and I , c w i ~ ~ " ~ and Bhutta et In this papcr wc have studied this case in more detail, and our rcsults indicate that the difrerences ob- served in the species profiles at the body end are actually due to the differences in the flowfield

chemistry in the blunt-body region and not caused by the afterbody PNS calculations.

One of the main objectives of this paper is to revisit the nonequilibrium blunt-body VSL scheme in order to improve the VSL predictions and resolve the observed disagreement between the nonequilibrium VSI,/PNS and Navier-Stokes predictions. In the absence of experimental data, an important measure of the numerical accuracy of a nonequilibrium blunt-body solution is the accuracy to which it satisfies global conservation of elemental mass. The VSL scheme historically has been formulated in a non-conservative form. Although the global conservation of mass is en- forced by adjusting the local shock standofl- distance, there is no direct means of controlling the non-conservative errors in the individual spe- cies conservation equations. These non- conservative errors in the species conservation equations are reflected in the global cor~scrvation of elemental mass. 'The results of this study show that, in some cases these errors in the global conservation of elemental mass can be largcr than the error in the global conservation of mass. It is important to note that these non-conservative errors are present in all nonequilibriurn VSI, schemes developed to date by various researchers (such as, Moss,5 Miner and L e ~ i s , ~ Swaminathan et al.,7 Shim et a1.,8 Kim et al.,9 Song and Lewis,Io Thompson et a].," Gupta et a1.,12 and Zoby et al.,13). In most instances, with these nonequilibrium VSL schemes the conservation of elemental mass may vary from good to satisfac- tory, depending upon the accuracy with which the global conservation of mass is satisfied. £low- ever, this is not automatically guaranteed for ev- ery case because of the errors due to (a) numerical inaccuracies, (b) nonconservative VSI, formu- lation, and (c) the fact that the conservation of element mass fraction is not directly imposed.

'I'he major focus of this paper is to present some recent itnprovements in the nonequilibrium VSI, blunt-body solution scheme. Briefly speak- ing, these improvements involve enhancements of the spherical stagnation-point solution and the reduction of the non-conservative numerical er- rors in element conservation by directly imposing the conservation equations of element mass. A direct irriposi tion of thcse element conservation equations does not afkct the global conservation of mass; however, it does improve the conserva- tion of mass of the individual elements. Detailed comparisons of these improved nonequilibrjum VSI, predictions have also been done with the corresponding predictions of the nonequilibrium Navier-Stokes scheme in terms of surface-

rneasurablc quantities as well as flowfield and species profiles.

SOLUTION SCHEME

'I'hc VSI, equations for a chemically reacting binary gas mixture were originally derived from the full Navier-Stokes equations by Davk21 Moss22 further extended the VSL scheme to study a five species ( 0 , 0 2 , N, Nz, and NO) gas model. Ilowever, these studies were restricted only to analytic bodics such as hyperboloids. Miner and Lewis6 extended the VSL scheme to study axisymmetric flows over non-analytic (sphere- cones) reentry vehicles using the seven species ( 0 , 0 2 , N, N2, NO, NO+, and e-) model of Blottner et al.1

Swaminathan et al.23 - 24 further extended the nonequilibrium VSL scheme to study the three- dimensional nonequilibrium viscous flows over multiconic recntry vehicles. To do so, they used the earlier approach of Waskiewicz and Lewis2" to couplc the first-order continuity and normal- momentum equations, resulting in increased sta- bility of the 3 - I l VSL solution scheme. Kim et al.9726 extended the nonequilibrium VSI, scheme to a general surface-orthogonal coordinate system to study the nonequilibrium flow over the space shuttle. All of these nonequilibrium VSL schemes used a scalar (uncoupled) solution of the species conservation equations in which the coupling of the production terms with other species is neg- Iccted, and each species conservation equation is solved independant of others.

'I'he governing VSL equations are derived from the steady Navier-Stokes equations for the axisymmetric flow of a multicomponent reacting gas mixture27 written in terms of the surface- normal coordinate system shown in Fig. 1. In this surface-normal coordinate system (Fig. 1) the 's' coordinate is tangent to the body in the streanwise direction, 'n' coordinate is normal to the surface, and the '4)' coordinate is measured from thc windward to the leeward pitch plane.

'I'hese governing Navier-Stokes equations are first normali7ed by variables which are of order one at thc body surface, and they are then again normalized by variables which are of order one at the outer bow shock. The Viscous Shock- Layer (VSI,) equations are then obtained from these Navicr-Stokes equations by neglecting terms highcr than second order [O(c2)]. Conse- quently, the VSL equations are uniformly sccond-ordcr accurate from the body to the shock. I;urthcrmore, the resulting nonequilib-

rium VSI, equations are also parabolic in the streamwise and crossflow directions.

In the normalized surface-normal coordinate system (t , q , c), the conservation equations for s-momentum, &momentum, energy and species mass-fraction are written in the standard parabolic form:

where W is the dependent variable. For the s- momentum equation the dependant variable is the streamwise velocity (u), for the &momentum equation the dependant variable is the crossflow velocity (w), for the energy equation the depen- dant variable is temperature (T), and for the spe- cies conservation equations the dependant variables are the species mass fractions (CJ.

The continuity and normal morncntum equations are first-order differential equations and, when solved independantly, pose numerical dificulties especially when surface discontinuitics (such as at the nose-afterbody tangent point) are involved. I Iowever, using the coupling approach dcvcloped by Waskiewicz and Lewis,25 these two first-order equations are coupled together to form a second-order system which can be solved using a tridiagonal solution procedure. This numerical coupling considerably enhances the overall nu- merical stability of the VSL solution scheme and makes it possible to treat geometric discontinui- ties.

The finite-difference algorithm used to solve these flowfield equations is based on the scalar tridiagonal approach developed by Murray and Lewis2* for perfect-gas flows and by Swaminathan et al.23 - 24 and Kim et al.9926 for finite-rate chemically-reacting flows.

All of the existing nonequilibriurn VSI, blunt- body schemess14 use Blottner's linearization a p p r ~ a c h ~ ? ~ to treat the rate-of-production term. In this linearization approach the production term appearing in a particular species conserva- tion equation is linearized such that only that species concentration appears as the unknown. Furthermore, the production terms in the energy equation are also linearized such that the tem- perature appears as the only u n k n ~ w n . ~ ~ ~ In this manner all equations can be solved uncoupled from each other. The detailed derivation of the conventional uncoupled VSL equations for a chemically reacting air mixture was given by Swaminathan et a1.25 24

Briefly speaking, the VSL solution begins on thc spherically blunted nose by first solving the govcrning equations along the stagnation streamline where the governing equations have a removable singularity, which is removed by using a scrics cxpansion of the flowfield variables around the stagnation streamline. Once the sol- ution along the stagnation streamline has been obtained, the governing axisymmetric VSL equations are solved using a 2-point backward- diffcrcnccd approximation for the streamwise de- rivativcs. At each marching step, the solution is iterated until it converges, and then the numerical solution steps downstream to obtain the solution at the next streamwise (s) station.

I h c to the normalization with respect to the variables which are order one a t the shock, the VSI, cquations depend on the shock slope nse6 and the strcarntvisc gradients of the flowfield var- iables bchind the shock. In the present VSL ap- proach, thc shock slope (n,h,g) over the solution domain is provided as input, and is obtained us- ing an appropriate inviscid solution scheme. The strcamwisc gradients of the flowfield behind the bow shock arc then obtained internally using the local shock slope and 2-point backward- differctlccd approximations. Although the shock slopc is kcpt lixed during the solution iterations, after each iteration the local shock standoff dis- tancc is updatcd by imposing the global conser- vation of mass. In general, an appropriate number ofglobal iterations can be done to further improve the input shock shape. However, no global itcrations were done in the present study.

Imposing the Conservation of Element Mass : In the prescnt approach, the mass-fraction distrib- ution of the N-F elements (E,) across the shock layer is obtained from the governing element conservation cquations. The VSL element con- scrvation cquations are obtaincd in a manner very similar to the species conservation equations. Evcn the form of these element conservation cquations is idcntical to the species conservation cquations, except for the absence of any pro- duction tcrms. Consequently, these element con- servation cquations are truly uncoupled from each othcr, and they are solved as such. Fur- thcrrnorc, since all the elemental mass fractions at a point must add up to unity, we only solve NE-I elcnicnt conservation equations, and the last clemcnt's mass fraction is determined from this conservation constraint.

The elcrnent conservation equations can also be writtcn in the following standard parabolic form

where Tl, is the mass fraction of the e-th element, and e = 1,2 ,..., NE-1.

For these element conservation equations the boundary conditions at the outer bow shock cor- respond to a frozen shock crossing [(L&, = (Ee)J. At the wall, the correct wall boundary condition for these element conserva- tion equations consist of the convection of ele- ments away from the wall and diffusion of the elements toward the wall. This convection- dimusion boundary condition is in fact a first- order dimerential equation written as

where (E,) - is the elemental mass fraction of the injectant gas.

Solution Improvements Along the Stagnation Streamline : The flowfield along the spherical stagnation streamline ( t = 0) involves a rernova- ble singularity. 'This singularity can be removed by expanding all quantities around the spherical stagnation streamline (5 = 0) in powers of t . In all existing VSL schemes MOSS,^^^^ Miner and I , e w i ~ , ~ and Murray and Lewis2R) first-order ex- pansions are used for all variables except pressure (p) and shock standom distance (nsh). Now the pressure field along this stagnation streamline is closely related to the normal velocity (v). Thus, in the present approach we have used a second- order expansion for the normal velocity (v). Thus, the flowfield variation around the stag- nation streamline is expressed as

The corresponding shock quantities appearing in the governing equations are expanded as

By using these expansions along the stag- nation streamline in the n-momentum equation, we can obtain the following improved expression for P2,v

+ PshlPsh2 plvl dvl

2 -1 Pshl a~

~ m i i i av2 - - av1 + v -)

P S ~ I ( 1 a?

whcre the terms enclosed by {) are the terms found in the works of MOSS,^,^^ Miner and Lewis6 and Murray and and the remaining terms are due to the second-order expansion o f v in Eqs. (4). The resulting second-order terms in the con- tinuity equation along the stagnation streamline ( ( = 0 ) can be combined in the following ex- pression for v2(q).

where v l , = V Z ~ = 0 for a no-blowing case.

'I'he solution scheme requires that for each solution iteration along the stagnation streamline, first the ~~(17) distribution is calculated using Eq. (7), and the resulting distribution is numerically ctifircntiatecl to obtain ~ 2 , ~ along the stagnation streamline. These v2 and ~ 2 , ~ distributions are then used to obtain the corresponding P2,q dis- tribution from I'q. (6), which is then numerically integrated liorn the shock to the body to obtain the required ji2 distribution along the stagnation streamline. 111 general, this improved stagnation point solution in not only more accurate, but it also results in more well-behaved pressure and

wall heat-transfer distributions in the stagnation region.

RESULTS AND DISCUSSION

In order to evaluate the accuracy of these en- hancements in the nonequilibrium VSI, scheme, we studied the Mach 20 flow over a sphere-cone configuration under zero and 20-deg angle-of- attack conditions at an altitude of 175 kft. The sphere-cone configuration considered consists of a 7-deg forecone with a 2-inch nose radius and an overall length of 20 inches (10 nose radii). The wall temperature was assumed to be 2000 R, and the vehicle surface was assumed to be fully cata- lytic. These test conditions are the same as those used by Buelow et al.I5 and are based on some test cases that we did in an earlier study.I8

It is important to note that, in our view, the main importance and usefulness of the VSL scheme is in generating the starting solutions for more accurate and robust afterbody solution schemes (such the Parabolized Navier-Stokes schemes). Although the VSI, solution schcme is indeed quite applicable in the afterbody region, there are some important issues which also need to be properly addressed in order to maintain solution accuracy under high-altitude (low Reynolds number) conditions. For example, un- der such conditions, if the input shock shape was generated using an inviscid approach (which is usually the case), it needs to be improved by per- forming the necessary global iterations. Needless to say, these global iterations may not be easy to do (as it turned out to be the case for the present test cases). In this study, no VSL global iter- ations were done, and the afterbody VSL sol- utions were only done in order to provide an additional set of predictions to be compared with the corresponding afterbody PNS predictions.

The results of these test cases are discussed below. Were possible, these nonequilibrium VSI, predictions are checked for accuracy by compar- ing with the corresponding predictions of the nonequilibrium Navier-Stokes solution scheme of Molvik and Merklq2 our 3-11 nonequilibrium Parabolized Navier-Stokes scherne,'hnd the Up- wind Parabolized Navier-Stokes schcme of Buelow et a].'"

fa) Zero Angle-of-Attack Test Case (Case 1) : For simplicity, the results of this zero angle-of-attack tes i cask a r e separated into (a) t h e blunt-body region and (b) the afterbody region.

Blunt-Body Flowfield Region : The main focus of the present study is the importance of accurately satisfvine the conservation of element mass in the none;uiibrium VSL scheme. It is important to note that in the VSI, solution scheme the global conservation of mass is dictated by the continuity equation and enforced in part by updating the shock-standolr distance based on an integrated forrn of the continuity equation (the shock slope remains unchanged during this process). As long as the input shock shape and the continuity equation are not changed, the global conservation of mass can not be affected. However, in the nonequilibrium case, the final error in the global conservation of mass is a n algebraic sum of indi- vidual errors in the conservation of element mass. 'I'hcse nonconservative errors can be of different signs and, thus, can be numerically much larger than thc final error in the global conservation of mass.

For cxi~tnple, for these Case 1 calculations the error in thc global conservation of mass at s/Rn = 1.6 was approximately + 6 % of the frcccctrcam mass flux. Without enforcing the ele- ment conservation equations, at the same lo- cation thc error in the global conservation of N atorns was -1- 1 1% and that of 0 atorns was -5% ( + 1 1 ?'h -5% = 6%). Ilowever, when the con- servation equations of N and 0 atoms were also enforced, a t the same location the error in the global conservation of N atoms was approxi- mately + 3.5% and that of 0 atoms was approx- imately + 2.5% ( + 3.5% + 2.5% = 6%). As can be seen, with the present approach of enforcing the element conservation equations, although the overall global conservation of mass error does not change (because the shock shape and the conti- nuity equation do not change), the individual er- rors in the conservation of element mass did improve. Figure 2 shows the mass-fraction of 0 atorns at this location with and without forcing the conservation of element mass. With the present approach the mass-fraction of 0 is almost constant at 0.23456 across the entire shock layer, whereas the previous VSL predictions were sig- nificantly lower in the peak temperature region.

'I'hc VSI, predictions of the flowfield and chemistry in the blunt-body region are compared in Figs. 3 thru 7 with the corresponding pred- ictions of the TIIFF (Navier-Stokes) code of Molvik and Merkle.2 These TIJFF calculations were donc by Buelow et aI.l5 and provided to us. 'l'hesc Case I VSI, calculations were done using 151 grid points between the body and the shock, whereas the 'I'IJIF calculations were done using 81 points between the body and the outer frccstrear~~ houndary. The VSL prediction of the

surface pressure in the blunt-body region is com- pared with the TUFF prediction in Fig. 3 and shows excellent agreement. The corresponding heat-transfer and skin-friction predictions from the 'I'UFF code were not available and, thus, could not be compared.

'The flowfield profiles along the stagnation streamline are compared in Figs. 4a thru 4c. The pressure profiles are compared in Fig. 4a and show excellent agreement. The corresponding density and temperature profiles are compared in Figs. 4b and 4c and also show excellent agree- ment between the VSL and TUFF predictions. These figures also show that the predicted VSL shock-standoff distance is in good agreement with the bow shock captured by the T U F F codc.

The O and Oz species profiles along the stag- nation streamline are compared in Figs. 5a and Sb, arid show excellent agreement. Thc N and N2 species profiles are compared in Figs. Sc and Sd, and show that the agreement is generally very good except that the pcak N concentration prc- dictcd by 'TUIT is approximately 20% higher and the corresponding minimum in the N2 profile is approximately 5% lower. The corrcsponding NO and NO' concentration profiles are also in exccl- lent agreement (Figs. Se and 5f), except that the TUFI; results predict the peak concentration of NO' to be approximately 20-25%0 more.

The flowfield and species profiles at s/Rn = 1 ST08 (x/Rn = 1 .O) are compared in Figs. 6 and 7. The pressure, density and temperature profiles from the VSL and TUFF calculations are compared in Figs. 6a, 6b and 6c, respectively. In general the agreement in the near-wall as well as outer flow region is excellent. IIowever, there are some small differences in the location of the bow shock and, thus, the conditions imrnediatcly be- hind the bow shock. It is fair to say that at this location, due to the tangent-point discontinuity, thc VSl, shock-standoF predictions may not bc very accurate, but a t the same time thc 'I'1JF17 results show the predicted bow shock to be slightly smeared. In any case, these difTerences are still small in magnitude and effect.

'The species profiles a t this location are com- pared in Figs. 7a thru 7f and show that the agreement between the VSL and TUFF pred- ictions is mostly excellent. There are some small differences (approximately 10-20%) in the pre- dicted extremas in the species profiles. Further- more, compared to the VSI, predictions, the TIJFF results show a slightly thinner layer of dissociated and ionized air, but this could also

simply be a reflection of the differences in the prcdictcd shock-standoff distance.

The computing times and grid used for this VSI, blunt-body solution are shown in Table 2, which shows that this VSL blunt-body calculation took only I I scc on Cray Y/MP. On the other hand, although the computing time required by thc 'I'IJFI; calculations is not available, Buelow et al.I5 havc reported that these TUFF calcu- lations took 1500 iterations to converge. I t is clear that for spherical nosetips, the present non- equilibrium VSI, scheme is an accurate and ef i - cient approach for generating the blunt-body flowfield.

Afterbody Flowfield Region : The VSI, solution for Case 1 conditions was also done over the en- tire blunt-body and afterbody region. These VSL predictions of thc surface-pressure and wall heat- transfer rate are cornpared in Figs. 8a and 8b with the corrcsponding predictions of our nonequilib- riurn VRAI'NS (Parabolized Navier-Stokes) codelQusing 150 points between the body and the shock. '!'he starting solution for this VRAPNS calculation was located at x = 1 Rn, and was gen- erated u ~ i n g thc present nonequilibrium VSI, blunt-body solution. These results show excellent agreement between the VSL and VRAPNS pred- ictions over the entire body length.

Ihclow et al.I5 have also done UPS (Upwind Parabolizcd Navier-Stokes) calculations for this case using the TUFF solution to start their afterbody I JPS code. Figures 9a thru 9d show the flowfield profiles a t the body end (x= 10 Rn) prcdictcd by the VSI,, VRAPNS and UPS codes. The pressure proliles are compared in Fig. 9a and show that the VSI, and VRAPNS predictions are in exccllcnt agreement with each other. There are some difircnces irnmcdiately behind the bow shock due to the larger shock-standoff distance predicted by thc VSL; however, these differences vanish very quickly. It should be pointed out that the VIIAI'NS schcme involves an implicit shock- fitting approach in which the shock shape is pre- dicted as part of the marching solution. It is interesting to note that the UPS predictions of the start of the captured bow shock wave are in ex- cellent agrccmcnt with the VRAPNS predictions. I Iowcvcr, the UPS prediction of the wall pressure is approximately 10Y0 lower than the corre- sponding VRAPNS and VSL predictions. It should also be noted that Figs. 3 and 6a show that the wall-pressure prediction of the TUFF codc (uscd to start the UPS code) is in excellent agreerrlcrlt with the blunt-body VSL prediction (used to start the VRAPNS code). Thus, it is our

view that these differences in the afterbody region are caused by the UPS predictions.

The density profiles are compared in Fig. 9b, and show that in the initial portion of the shock- layer the UPS, VRAPNS and VSL predictions are almost identical, whereas in the outer region the VRAPNS predictions are bounded by the VSI, and UPS results. However, these differences be- tween the VRAPNS and VSL are mainly due to the differences in conditions behind the shock. These VSL solutions were not globally iterated and, consequently, have a shock-layer thickness which is too thick and results in a weaker bow shock. The temperature profiles shown in Fig. 9c show that the VIUPNS and VSL predictions are almost identical over most of the shock layer, with small difirences in the outer region which are again due to the differences in the predicted shock-standoff distance. The UI'S predictions are in agreement with the VSI, and VRAPNS pred- ictions in the near-wall region; however, UPS predicts a 10- 1.5% higher peak temperature and much thinner high-temperature layer. It is inter- esting to note that the UI'S predictions of the \,ow shock are smeared over 10-20% of shock- layer thickness, and this may be a major reason for these differences. I'he streamwise velocity profiles at the body end are compared in Fig. 9d, which also shows excellent agreement between the VSL and VRAPNS predictions. For the most part, the UPS predictions are also in excellent agreement with the VSI, and VRAPNS pred- ictions, except for some slightly higher flow ve- locities near the middle of the shock-layer region. These higher UPS velocities are consistent with the thinner shock-layer thickness predicted by it.

The species profiles at the body end (x= 10 Rn) are compared in Figs. 10a thru IOf. In gen- eral, the VSL and VRAPNS predictions are in excellent agreement with each other, except for some small (10-20%) differences in the predicted extremas of some species. The UPS predictions are in close agreement with these VSI, and VRAI'NS predictions in the near-wall region. In the outer shock-layer region the UPS predictions differ somewhat from these VSL and VRAI'NS predictions, but these difrerences are primarily due to the thinner shock-layer predicted by the IJI'S scheme.

It should be noted that the only diKerences in the nonequilibrium VRAPNS results of Ref. 18 and the present VRAPNS predictions are the dif- ferences in the starting VSI, blunt-body solution. Since the body length is quite short (in terms of nose radii), the starting solution does have a sig- nificant e f k t on the flowfield at the hody end.

I'hc computing times for these VSL and VRAI'NS calculations are shown in Table 2.

Jb) 20° Angle-of-Attack Test Case (Case 2) : These 20-dcg angle-of-attack (Case 2) calcu- lations werc done using VSL and VRAPNS sol- ution schcmcs. Briefly speaking, the VSL calculations were done using 9 crossflow planes and 51 points between the body and the shock. 'I'he corrcsponding VRAI'NS calculations were done using 3 1 crossflow planes and 50 points be- tween the hody and the shock. The grids used and the corresponding computing times for these Case 2 calculations are shown in Table 2.

In evaluating these VSL results it should be noted that when the crossflow separation begins on the lceside at large angles of attack, the VSL solution schcrnc (due to its parabolic crossflow naturc) can not march through this crossflow separatcd rcgion and, thus, drops these leeside solution plancs. Actually, this leeside VSL sol- ution rctnains attached for some short distance before it cncounters numerical difficulties and drops thcse troublcsorne leeside solution planes. For exatnplc, Tor thcse Case 2 calculations, the VRAI'NS calculations show that crossflow sepa- ration on the lceside starts somewhere between x = 2 Rn and x= 2.5 Rn. Although in this region the prcscnt VSL solution remains attached (maybe due to the coarse 9-plane grid), it is not expected to bc very accurate.

The surface-pressure distributions along the windward ( 4 = 0"), side (6 = 90") and leeward ( 4 = 180") planes are shown in Fig. I la, and the corresponding crossflow distributions at the body end are shown in Fig. 1 lb. These results show that, until thc start of the crossflow separated re- gion, the VSI, and VRAPNS predictions of the surface pressure are in excellent agreement with each other. 'I'he corresponding axial and cross- flow distributions of the wall heat-transfer rate are shown in Figs. 12a and 12b. There is excellent agreerncnt along the windward streamline, and the prcdictcd surface heating along the side (4=90°) is within 5-10% of each other. As mentioned before, this 3-11 VSL solution was not globally itcratcd and, thus, the predicted shock- layer thickness is slightly thicker than the corre- sponding VRAI'NS predictions. These observed differences in the surface heating, although small, are primarily due to this difference in the shock- layer thickness.

l'hc axial distribution of the streamwise skin- friction cocflicient is shown in Fig. 13, and fol- lows thc same trends as the wall heat-transfer ratc. 'I'hc crossflow distribution of the crossflow

skin-friction coefficient at the body end is shown in Fig. 14. This is one of the most sensitive quantities to predict accurately, and determines the extent of the crossflow separated region (CFWsO). These results show that up until the start of the crossflow separated region, the cross- flow skin-friciion coefficient predicted by VS1, and VRAPNS calculations is in good agreement. This is indeed refreshing to note the close agree- ment in these predictions, especially in view of the coarse VSL grid in the crossflow direction.

For completeness, the pressure and temper- ature contours at the body end using the VRAPNS predictions are shown in Figs. 15a and 15b. The corresponding contours of the species mass-fractions at the body end are shown in Figs. 16a thru 16f. These figures show that in all cases the predicted flowfield and species concentrations are quite well-behaved and smooth.

CONCLUSIONS

In this study we have enhanced the existing nonequilibrium VSL solution scheme to improvc its solution accuracy in the stagnation point re- gion and also minimize and control the errors in the conservation of elemental mass. 'I'he stagnation-point solution is improved by using a second-order expansion for the normal velocity, and the elemental mass conservation is improved by directly imposing the element conservation equations. The modifications are such that the general structure and computational efxciency of the nonequilibrium VSL scheme are not affected. 'The Mach 20 flow over a 7-deg sphere-cone ve- hicle is studied at zero and 20-deg angles of at- tack. Comparisons are made with the available predictions from Navier-Stokes and Parabolized Navier-Stokes solution schemes. The results of this study substantiate the following comments:

The solution accuracy of the nonequilibrium VSI, scheme is significantly enhanced by imposing the element conservation equations instead o r some of the species conservation equations.

The flowfield and chemistry predictions of the improved nonequilibrium blunt-body VSI, scheme are in excellent agreement with corresponding predictions of the nonequi- libriurn Navier-Stokes (TIJFF) code of Molvik and Merkle.2

In the afterbody region, the flowfield and chemistry predictions of this improved non- equilibrium VSI, scheme are in excellcnt

agreement with corresponding predictions of the nonequilibrium Parabolized Navier- Stokes schemes of Bhutta and LewisI8 and Duelow ct a1.15 In general, the agreement is better with the VRAPNS scheme of Bhutta and I , c ~ i s , ~ b h e r e a s the UPS scheme of Ruelow ct a1.I5 tends to predict a more dif- fused bow shock and, consequently, a thinner shock layer.

For short bodies (in terms of nose radii), the cflccts of the starting solution on the PNS aftcrhody solution schemes are not insignif- icant, and the accuracy of the blunt-body solution scheme used to generate this start- ing solution can not be ignored.

With solution enhancements, the nonequi- lihriutn blunt-body VSL scheme remains as an accurate, fast and extremely efficient nleans for generating the blunt-body flowlicld over spherical nosetips at small to largc angles of attack. These VSL blunt- body solutions can then be reliably used to start morc accurate afterbody PNS scheme.

ACKNOWLEDGEMENT

The work reported in this paper was supported in part by the NASA Lewis Research Center un- der contract number NAS3-25450. The encour- agement and cooperation provided by the contract monitor Dr. Tom Benson, Dr. Louis I'ovinelli and Mr. Dan Whipple during the course of this efrort are gratefully acknowledged. The authors would also like to thank Dr. J . Tannehill and his colleagues at Engineering Analysis, Inc., Ames, lowa, and Iowa State University, Ames, lowa, for their help in obtaining the detailed TIJIT and IJPS rcsults used in this study for comparisorl purposes.

REFERENCES

1 Dlottncr, IT.<;., Johnson, M., and Ellis, M., "('hernically Reacting Viscous Flow Program for Multi-Component Gas Mixtures," Report No. SC-RR-70-754, Sandia Laboratories, Albuquerque, NM, Dec. 1971.

Molvik, G.A., and Merkle, C.L., "A Set of Strongly-Coupled Upwind Algorithms for Computing Flows in Chemical Nonequilib- riurn," AIAA Paper 89-0199, Jan. 1989.

Candler, G., "On the Computation of Shock Shapes in Nonequilibrium I Iypersonic Flows," AIAA Paper 89-03 12, Jan. 1989.

Gnoffo, P.A., "Code Calibration in Support of the Aeroassist Flight Experiment," ~ o h a l of Spacecraft and Rockets, Vol. 27, March- April, 1990, pp. 131-142.

MOSS, J.N:, "Reacting Viscous Shock-Layer Solutions with Multicomponent Diffusion and Mass Injection," NASA 'I'R-R-411, June 1974.

6 Miner, E. W. and Lewis, C.1 I., "I Iypersonic Ionizing Air Viscous Shock-Layer Flows Over Nonanalytic Blunt Bodies," NASA CR-2550, May 1975.

Swarninathan, S., Kim, M.D., and Lewis, C.1 I., "Nonequilibrium Viscous Shock-Layer Flows Over Blunt Sphere-Cones at Angles of Attack," AIAA I'apcr No. 82-0825, June 1982.

"Shin, .l.l,., Moss, J.N., and Simmonds, A.l,., "Viscous Shock-Layer IIeating Analysis for the Shuttle Windward Plane with Surface 13- nitc Catalytic Recornbination Rates," AIAA Paper No. 82-0842, June 1982.

Kim, M.D., Swarninathan, S. and Lewis, C.1 I., "Three-Ilimensional Nonequilibrium Viscous Flow Computations Past the Space Shuttle," AIAA Paper No. 83-0487, Jan. 1983.

l o Song, D.J., and Lewis, C.1-I., "Hypersonic Finite-Rate Chemically Reacting Viscous Flows over an Ablating Carbon Surface," Journal of Spacecraft and Rockets, Vol. 23, Jan.-Feb. 1986, pp. 47-54.

Thompson, R.A., "Comparisons of Nonequi- librium Viscous Shock-Layer Solutions with Windward Surface Shuttle Heating Data," AIAA Paper No. 87-1473, June 1987.

l 2 (iupta, R.N., Lee, K.P., Moss, J.N., Zohy, E.V., and 'I'iwari, S.N., "Viscous Shock-Layer Analysis of Lone Slender Bodies," AIAA Pa- per No. 87-2487, Aug. 1987.

l 3 Zoby, E.V., Ixe, K.P., Gupta, R.N., Thompson, R.A., and Simmonds, A.l,., "Viscous Shock-Layer Solutions with None- quilibrium Chemistry for Ilypersonic Flows past Slender Bodies,'' Journal of Spacecraft and Rockets, Vol. 26, July-Aug. 1989, pp.

l 4 Ilhutta, HA., Song, D.J., and Lewis, C.II., "Nonequilibrium Viscous I Iypersonic Flows Over Ablating Teflon Surfaces," AIAA Paper No. 89-0314, Jan. 1989.

1 5 Buclow, P., levalts, J., and Tannehill, J., "Comparison of Three-Dimensional PNS Codes," A1 AA Paper No. 90- 1572, June 1990.

l 6 Iklwards, 'I'A., and Flores, J., "Computa- tional I;luid Dynamics Nose-to-Tail Capability : I Iypersonic Unsteady Navier-Stokes Code Validation," Journal of Spacecraft and Rock- ets, Vol. 27, March-April, 1990, pp. 123-130. -

l 7 Ryan, .J.S., Flores, J., and Chow, C.-Y., "De- velopment and Validation of a Navier-Stokes Code for 1 lypcrsonic External Flows," Journal of Spacecraft and Rockets, Vol. 27, March- April, 1090, pp. 160- 166.

18 Dhutta. B.A., and Lewis, C.II., "Three- Dimensional I Iypersonic Nonequilibrium Flows at 1,arge Angles o r Attack," AIAA Pa- per No. 88-2.568, June 1988.

l 9 Dhutta, D.A., and Lewis, I . , "Low Reynolds Number Flows Past Complex Multiconic Geometries," AIAA Paper No. 8.5-0362, .Jan. 1985.

2O Bhutta, B.A., Lewis, C.H., and Kautz 11, F A . , "A Fast Fully-Iterative Parabolized Navier- Stokes Scheme for Chemically-Reacting Re- entry 1710ws," AIAA Paper No. 85-0926, June 198.5.

21 Ikvis , R . , "IIypersonic Flow of a Chemically Reacting Binary Mixture Past a Blunt Body," AIAA Paper 70-805, July 1970.

22 MOSS, J.N., "Solutions for Reacting and Nonrcacting Viscous Shock Layers with Multicotnponent I>iflusion and Mass In- jection." PhD Dissertation, Virginia Polytechnic Institute and State University, Dlackshurg. VA, Oct. 1971.

23 Swaminathan, S., Kim, M.D., and Lewis, (:.I]., "Real Gas Flows over Complex Geom- etrics at Moderate Angles of ~ t t a c k , " Journal of Spacccrart and Rockets, Vol. 20, July-Aug. 1983, pp. 321-322; see also AIAA Paper

24 Swaminathan, S., Kim, M.D., and Lewis, I . , "'I'hrce-Dimensional Nonequilibrium Viscous Shock-Layer Flows over Complex Gcomctries," AIAA Journal, Vol. 22, June

1984, f p. 754-755; see also AIAA Paper 83-021 , Jan. 1983.

25 Waskiewicz, J.D., and Lewis, C.II., "Recent Developments in Viscous Shock-Layer The- ory," VPI&SU AERO-079, Virginia Polytechnic Institute and State University, Blacksburg, VA, March 1978.

2G Kim, M.D., Bhutta, B.A., and Lewis, C.H., "Three-Dimensional Effects upon Real Gas Flows Past the Space Shuttle," AIAA Paper 84-022.5, Jan. 1984.

2' Bird, R.R., Stewart, W.E. and Lightfoot, E.N., 17ransport Phenomena, John wiky and Sons, Inc., New York, NY, 1960.

28 Murray. A.L., and Lewis, C.H., "Three- I>imensional Fully Viscous Shock-Layer Flows Over Sphere-Cones at Iligh Altitudes and I Iigh Angles of Attack," VPI&SU AERO-078, Virginia Polytechnic Institute and State Uni- versity, Blacksburg, VA, Jan. 1978.

Table 1. Freestream conditions

- -- - - - - -

I I I I Quantity I I - I ==-==== I

I I Mach number 1 20.000 1 I Reynolds number ( 1.34E+4 1 1 Pressure (lbs/ft2) 1 1.102 1 I Density (slug/ft3) 1 1.32E-6 1 I Temperature (R) 1 485.151 1 I Velocity (ft/sec) 1 2.16E+4 1 I Angle of attack (deg) I I I Case 1 I O.OOOI I Case 2 1 20.0001 I Wall temperature(R) ( 2000.00 1 I Nose radius ( inch) I 2.00 1

Table 2. Computing Times

- ----------------------- - - I (a) (b) (c) I I Case Numerical x/Rn Grid CPU Time I I Scheme From-to N l x N 2 x N 3 (m:s) I I----------- -A-A------------

I Case 1 VRAVSL 0.0-1.00 17 x151 x 1 0: 10 I

0.0-10.0 5 0 x 1 5 1 ~ 1 0: 30 I

I VRAVSL 1.1-10.0 25 x151 x 1 0: 20

I I VRAPNS

0.0-2.50 20 x 50 x 9 0: 30 I

( Case 2 VRAVSL I I VRAVSL 0.0-10.0 35 x 50 x 9 1: 00 I I VRAPNS 1.2-10.0 22 x 50 x 31 3: 00 - ---- ------ -- I

------------------- Notes :

(a) VRAVSL = 3-D Viscous Shock-Layer solution VRAPNS = 3-D Parabolized Navier-Stokes solution

(b) Nl=number of streamwise marching steps N2=number of points between body and shock N3=number of equally spaced crossflow planes

(c) Equivalent computing time on Cray Y/MP with CFT77 compiler and auto vectorization.

6

WINDWARD STREAMLINE

Fig. I . Coordinate system

Alt.=175 kft Mach=2O

7O sphere cone Rn=2 inch,

v LL! -

o TDFF Code (1x81~30 Grid)

I VRAVSL

LL ( 1 xl5l x17 Grid) z C(

a \

h

Fig. 3. Comparison of surface-pressure in the hlunt-11ody re- gion for Case I .

Alt. =I75 kft Mach=20 ALPEA=O 7O sphere cone

I

I

Q - Without Q

Element 8

(1x51 rid): 'Q

K Q e

With Element Conservation ( 1 xl51 Grid)#

/

Fig. 2. Effects of imposing the element ro~isersation equations on the mass fraction of oxygen atoms for Case I .

Alt. =I75 kft Mach=2O ALPHA=O 7O sphere cone Rn=2 inch

s/~n=O

o TOFF Code (81 points) - VRAVSL (151 points)

1 .f+O 1 .E+l 1 .E+2 1

Fig. 4a. Comparison of pressure profiles at s/Rn = 0 for Case 1.

Alt. =I 75 kft Mach=2O ALPHA=O 7O sphere cone Rn=2 inch

s/Rn=o

15,i o m F Code

(81 points) - VRAVSL (151 points)

1 .E+O 1 .E+1 1 .E+2 RHWRHOINF

Fig. 4h. Comparison of density profiles at s/Rn= 0 for Case I .

Alt.=175 kft Mach=20 ALPHA-0 O 7O sphere cone Rn=2 inch

a TUFF Code 15,2

(81 points) - VRAVSL 0 (151 points)

I

Fig. 5a. Comparison of 0 concentration profiles at s/Rn = 0 for Case 1.

Alt. =I75 kft Mach-20 ALPHA-0 O 7O sphere cone Rn=2 inch

s/~n=O

o !rUFF Code (81 points) - VRAVSL (151 points)

1 .Ed 1 .E+1 1 - .-

T/T I NF

Fig. 4c. Comparison of temperature profiles at s/Rn= O for Case I .

Alt. =I75 kft

Q 8

7O sphere cone

o Code 15,2

(81 points) - VRAVSL (1 51 points)

Fig. 5h. Comparison of Oz concentration profiles at s/Rn = O for Case I .

w 1 I I

Alt. -1 75 kf t Mach-20

N ALPHA= 0 O m 7O sphere cone d Rn=2 inch

s/Rn=O

o TUFF Code 15,2

Fig. 5c. Comparison of N concentration profiles at s/Rn = O for Case I .

Alt. =I75 kft Mach=2O ALPHA= 0 O 7O sphere cone Rn=2 inch

Z a \ L.

m Y a u

o TUFT Code 15,2 (81 points) - VRAVSL (151 points)

Fig. 5d. Comparison of NZ concentration profiles at ~ / R I I - 0 for Case I .

I I

Alt. =I75 kft Mach=2O ALpHA=o O 7- sphere cone Rn=2 inch

s/~n=O

o !lWPF Code (81 points) - VRAVSL

Z (151 points) a \ 0

m =f a u

.w CI (NO1

1 I 1

A l t . =I75 kf t Mach=20 ALPHA=O O 7O sphere cone ~ n = 2 inch

o TUFP Code 15,2

(81 points) - VRAVSL (151 points)

Fig. 5e. Comparison of NO concentration profiles at s/Rn= 0 for Case I .

Fig. Sf. Comparison of NO' concentration profiles at s/Rn= 0 for Case 1.

Alt. =I 75 kf t Mach=2O

8 ALPHA=O O 7O sphere cone

d Rn=2 inch

0 TUFF Code 15,2

(81 points) - VRAVSL (151 points)

1 .E+O 1 .E+1 1 .E+2 1

Fig. ha. Comparison of pressure profiles at s/Rn= l.2708 for Case I .

Alt. =I 75 kf t Mach=20 ALPHA= 0 7O sphere cone

d 4 Rn=2 inch

o T[IFF Code 15,2

(81 points) - YRAVSL (151 points)

1 .E+O 1 .E+1 1 T/TINF

Fig. 6c. Comparison of temperature profiles at s/Rn = 1.2708 for Case I .

ALT. =I 75 kft Mach=20 ALPHA=O O 7O sphere cone Rn=2 inch

s/~n=l. 5708

o TUFF Code 15,2

(81 points) - VRAVSL z (151 points)

a \ n

a Y a c.

1 .E+O 1, RHWRHOINF .

Fig. 6b. Comparison of density profiles at s/Rn= l.5708 for Case I .

8

Alt. =I75 kft Mach=20 ALPHA=O O

7O sphere cone Rn=2 inch

o TUW ~ o d e l ~ ' ~ (81 points) - VRAVSL

h (151 points)

Fig.7a. Comparison of 0 concentration profiles at s/Rn= 1.5708 for Case I .

I I

Alt. =I75 kft 8 Mach=2O Q ALPHA=O 0 8 7O sphere cone 8 - Rn=2 inch Q

- 15,2 d o TUFF Code Q

(81 points) - VRAVSL (151 points)

CI (021

Fig.711. Comparison of O2 concentration profiles at s/Rn = 1.5708 for Case I .

I 1 1

Alt. -1 75 kft Q Mach=20 Q

8 ALPHA=O O Q 7O sphere cone

d 1 Rn=2 inch Q Q

o TUFF Code 15,2 Q (81 points) a - VRAVSL (151 points)

.m

Fig. 7d. Comparison of Nt concentration profiles at s/Rn = 1.5708 for Case I.

e Alt. =I 75 kf t Mach=2O

8 ALPHA=O O 7- sphere cone d Rn=2 inch

o TUFF Code 15,2

(81 mints) - - B - VRAVSL

(151 points) \ - m 7 a Y

CI ( N l

F i g . 7 ~ . Comparison of N concentration profiles s/Rn = 1.5708 for Case I.

Alt. =I75 kft Mach=2O ALpHA=o O 7O sphere cone Rn-2 inch

o TUFF ode" (81 points) - VRAVSL (151 points

Fig. 7e. Comparison of NO cnncentration profiles at s/Rn= 1.5708 for Case 1 .

t Alt. =I 75 kf t Mach=2O

8 ALPHA=O O 7O sphere cone

d Rn=2 inch

o TUFF Code 15,2

(81 points) - VRAVSL (151 points)

Fig.7f. Comparison of NO+ concentration profiles at s/Rn= 1.5708 for Case I .

0 VRAVSL (151 points) - VRAPNS (1 50 points)

Fig. 8h. Comparison of wall heat-transfer rate in the afterludy region for Case I .

Alt. =I 75 kft Mach=20

7O sphere cone Rn=2 inch

0 VRAVSL (151 points) - VRAPNS ( 150 points

d + '?

LL" Z

'? I 1 1 I

-ol. 00 2I.m 6'. 00 1.60 1L.O WRN

Fig. 8a. Comparison of surface-pressure in the afterhndy re- gion for Case I .

I 1 I

Alt. =I75 kft Mach120 ALPHA= 0 O 7- sphere cone Rn=2 inch

(1 51 points) - VRAPNS

P/PINF

Fig. 9a. Comparison of pressure profiles at x/Rn = I0 for Case 1 .

Alt. =I75 kft Mach=20 ALPHA-0 O 7O sphere cone ~ n = 2 inch

.- UPS (81 points) - VRAPNS (150 points)

0 VRAVSL (1 51 points)

RHO/RHO I NF

Fig. Yb. Comparison of density profiles at x/Rn= 10 for Case

Alt. 1 1 75 kf t Mach=2O ALPHA-0 O 7O sphere cone

-- UPS 15 (81 points) - VRAPNS ( 1 50 points)

0 VRAvsL u (1 51 points) h

Fig. Yd. Comparison of streamwise velocity profiles at x/Rn = I0 for Case I.

Alt. =I75 kft Mach=20 ALPHA=O 7O sphere cone Rn=2 inch

(1 50 points)

(151 points)

W 16.0 T/T I NF

Fig. Yc. Comparison of temperature profiles at x/Rn = 10 fnr Case I.

Alt. =I 75 kft Mach=20 ALPBA=o O 7- sphere cone Rn=2 inch i

-1 -+- UPS 15

(81 points) - VRAPNS ( 1 50 points i

0 VRAVSL (151 points)

Fig. 101. Comparison of 0 concentration profiles at x/Rn= 10 for Case 1.

Alt.=175 kft Mach=20 AI.pHA=o O 7O sphere cone Rn=2 inch

-+- UPS (81 points) - VRAPNS (1 50 points)

0 VRAVSL (1 51 points)

Fig. 1011. Comprrison of O2 concentration profiles at x/Rn= 10 for Case I .

C Alt. -1 75 kft Mach=20 ALPHA=O O ? 7. sphere cone Rn=2 inch

1 -+- UPS 15

(81 points) - VRAPNS Z (1 50 points) z 0 VRAVSL 3 3 (1 51 points)

Fig. IOd. Comparison of N2 concentration profiles at x/Rn= I 0 for Case I .

Alt. =I75 kft Mach=2O

. AI.PHA=O~ 7- sphere cone Rn=2 inch

- VRAPNS B ( 1 50 points \ n

0 VRAVSL

m (1 51 points)

Y a

.I2 CI ( N I

Fig. 10c. Comparison of N concentration profiles at s / R n = I0 for Case 1.

Alt. =I75 kft Mach=2O ALPHA=O O 7O sphere cone Rn=2 inch

-+- UPS 15

(81 points) - VRAPNS ( 1 50 points 1

0 VRAVSL (1 51 points)

CI (NO1

Fig. foe. Comparison of NO concentration profiles at x/Rn= 10 for Case 1.

Alt. =I75 kft Hach=20 ALPHA=O " 7" sphere cone Rn=2 inch

-+- UPS 15

(81 points) - VRAPNS ( 1 50 points

0 VRAVSL (1 51 points)

Fig. lOf. Comparison of NO+ concentration profiles at x/Rn = 10 for Case I .

Alt. =I75 kft Mach=2O ALPHA=20 O

7" sphere cone Rn=2 inch

- - - M - a - \ -

0+ - VRAPNS (31x50 Grid w I I I

-0.00 &.a, sb.00 1k.w 180.0 PHI ( E G I

1 I I

. Alt.=175 kft Mach=20 ALPHA=20 7O sphere cone Rn=2 inch

t o VRAVSL ( 9x51 Grid) - VRAPNS (31x50 Grid) 0,

Fig. 1 la. Comparison of the axial distribution of surface pres- sure for Case 2.

Alt. =I75 kft Mach=20 ALpHA=20 O

7O sphere cone Rn=2 inch A

o VRAVSL ( 9x51 Grid) - VRAPNS (31x50 Grid)

WRN

Fig. l lh. Comparison of the crossflow distrihution of surface Fig. 12a. Comparison of the axial distrihution of wall heat- pressure at x/Rn= 11) for Case 2. transfer rate for Case 2.

Alt. =I75 kft Mach-20 ALpErA=20 O 7O sphere cone - Rn=2 inch

n

Ln - \

o VRAVSL ( 9x51 Grid) 9 - VRAPNS (31x50 Grid) LL! -0.00 4k.00 9b.00 llbS.OO 180.0

PHI (DEGI

- I Alt. =I75 kf t o VRAVSL Mach=20 (9x51 Grid) ALPHA=20 O 7O sphere cone (31x50 Grid) Rn=2 inch

PHI=O O -wNs I

"6. w d . 6 ~ s'.m 7'. 60 1b.o WRN

Fig. 12h. Comparison of the crossflow distribution of wall lleat- F~E. 13. Comparison of the axial distrihution of streaniwis~ transfer rate at x /Rn= 10 for Case 2. skin-friction for Case 2.

8

Alt.=175 kft Mach=2O ALpHA=20° 7O sphere cone - d m = 2 inch

0

-

0 VRAVSL ( 9x51 Grid) - VRAPNS (31x50 Grid)

I I I

4k.m 9b.w h6.w lw.0 PHI (DEGI

Alt.=175 kft Mach=20 ALPHA=~O O

7O sphere- cone

Rn=2 inch

31x50 Grid ( W N S )

Fig. 14. Comparison of the crossf lo~ distrihtion of crossflow . Fig. 1%. Pressure contours at x /Rn= 1 0 for Case 2. skin-friction at x /Rn= I0 for Case 2.

21

Alt. =I 75 kf t Mach=2O ALPEA=20 O

7O sphere- cone

Rn=2 inch

Alt. =I75 kf t Mach=20 ALPHA=20° 7O sphere-

cone Rn=2 inch

Fig. 1 3 . Temperature contours at x /Rn= 10 for Case 2. Fig. 16a. 0 concentration contours at x /Rn= I0 for Caw 2.

Alt. =I75 kft Mach=20 A L P E ~ A = ~ O ~ 7O sphere-

cone Rn=2 inch

Alt. =I75 kft Mach=2O ALPHA= 2 0 O 7O sphere-

cone Rn=2 inch

31x50 Grid ( VRAPNS )

Fig. l6h. 0 2 concentration contours at x/Rn= I f ) for <'me 2. Fig. 16c. N concentration contours at x/Rn= 10 for <'ase 2.

Alt. =I 75 kf t Mach=2O ALPHA=20 7O sphere-

cone Rn=2 inch

31x50 Grid ( W N S 1

Fig. Ihd. NZ concentration contours at x/Rn= 10 for Case 2.

Alt.=175 kft Mach=2O ALPHA=20 O 7O sphere-

cone Rn=2 inch

31x50 Grid ( W N S )

Fig. 16e. NO concentration contours at x/Rn= I0 for Case 2.

Alt. =I75 kft Mach=2O ALPHA=20 O 7O sphere-

cone Rn=2 inch

Fig. 16f. NO' concentration contours at x/Rn = 10 for Case 2.