1 AIAA-88-0419

AN EQUILIBRIUM

FOR HYPERSONIC FLOWS AIR NAVIER-STOKES CODE

D.K. OTA, S.R. CHAKRAVARTHY, AND J.C. DARLING

ROCKWELL INTERNATIONAL SCIENCE CENTER THOUSAND OAKS, CA.

AlAA 26th Aerospace Sciences Meeting January 11-14, 1988/Reno, Nevada

For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics 370 L’Enfant Promenade, S.W., Washington, D.C. 20024

A N EQUILIBRIUM AIR NAVIER-STOKES CODE FOR H Y P E R S O N I C FLOWS

Dale I<. Ota', Sukrimar R . Chahravarthy**, and

Jill C. Darling***

Rockwell International Scicricc Ccnter

Ahstr.ilci A nFw 2 D/axisymmetric turbulent NavierAtokcs

in species rqirations to takr into account tbp iionfqiiihb

rium chemistry. This is a major modification, and as a first stcp, the assumption of equilibrium chemistry can hr used.

cnrle has hrrn dcveloped with an cqnililxium air cquation This means we will assume that all air chcmistry is hap^

of state. This codr has been validated against threr hy- pening fast eiioiigh that thc air is i n cquilihriimi. This as^ pcrsonic cxpcriments. Thc cod? lias thc vcrsatility to he sumption is good for moderate hypcrsonir M i d i iiiimhcrs; run either in a spacc marching modc or a time-dependent liowcvrr, for very high hypcrsonic Mach nunihws, i i o I i ( . q i i i ~ mode. Features such as the high accuracy TVD (Total

~ ~ ~ i ~ ~ i ~ , ~ ~ j ~ ~ i ~ i ~ l , i ~ ~ ) fc,rln,llatio,, r,f ti,c! coilvcctjvr terms lilxi,izn chemistrp slioii ld hv i lsvd. Tlw I ly~~rrso~i i i . Ilyprr-

to avoid numerical oscillations, ,lSe of ~ i ~ ~ ~ , ~ ~ ~ solvcrs holoiil w r r k of Moss' s t ii M a c h nunihcr nf 19.08 s h o w that,

to col,struct the fluxes for following prop. t,lrc rqidihsium air ais)inipt,ion compar'rs ~4 with ~011x1

2Lgation properties, and multi-zone c a p a ~ ~ i ~ i t y for noncqidihritnn rcsiilt,s. T l r data wlrcrr thr comparison is

gcoInetrias exist as boforc from previous perfect gas ver. not, as good still givrc fairly good t,rcnds aiid lrvcls.

sions. Verification runs for three cases arc presented. The th rw cases arc a hypcrsonic ramp (2~-D), hypersonic inlet

( 2 1)): i l r l<l a Iiypcrsool>ic l~y1>crIjc~loicI (;ixisg*ilmc,tric). ~i~~

t.wo 2 D c i w s rrml>arc well with rsI>cri,,,,,,,tni <l;it,a arrr l rninotrir casc compares wrll with ~ ~ ~ ~ ~ ~ i ~ , ~ t ~ , t i o , ~ ~ , l

Ltilit,y of thr codr to hv rim cit1ii.r i n n

tirrrr dnpcndent. or spaco marcliing modc in v i ~ y p o w r f d , ullowing both fur attnchcd typc flows for spam ~narching aI1d fillly snrmatcd flow fnr iimr dcpcndmt (:omputations without Imviiig to iisc two different codcs.

The present codr uscs ;in cqdihriarri air cqi i i i t ior l of

statc. The nlcthod uscs CIITW fits of prrssiire, trmprriitiirr, and thc cocfficiants of viscosit.y and thcrrnal conductivity

dcveloprcl as functions of dnisitg and ititcrnal cncrgy.

Tlie cquilihrinm air code, l i k its pcrfcct gas p r d r ~

cpssor, still incorporatcs a very powcrful unified approach which allows for either a space~~marching modr or a full t,ime- depend< t computation. Many other features are

built into the code such as a zonal mcthodology for complex gcometries, multi -.grid sequencing for convergence speed-

up, and automatic singular point treatment. All these codcs arr part of the "USA" scries of codes (Unified Sohition Al- gorithms).

-

1.0 Introduction

Ilwmtly, &sign of hypersonic flight vehiclrs has be- cnme an issue of national importance. The dewlopment of computational c.odes which could assist in the dcsign of the overall vehicle in a hypersonic regime would he of grcat use.

Tlic flow rcgimr which necds to be computed is that of a

very high Mach nnmher. The high tcmpcraturcs associatcd wit11 these flight conditions can lead to vibrational cxcita- t'ion, clissociation, and ionization of thr air. Othcr rffcrts of tlie extremc hcating would be radiation, cicfnmmation of tlw siirfaccs, and surface catalysis for chemical rcnctions. Tlrrsc high tcmpcrature effects can have a significant in flu^.

C l l C c 011 th? typr of physics which will occur, and tlioreforr thFy iicc:d to he modelled c.ompntation;~Ily.

The equilihrinrn air codr was verificd against three hy- personic test cases. The codr was first tcsted against a

2 ~ ~ D ramp at 15 degrcrs. This was a laminar rim at a Mach of 14.1. Next, the codr w&q tested against a 2 ~ ~ D

tllrhulent inlet with transition at a Mac11 number of 7.4. The code was then tested against a laminar axisylnmctric

i,yperholoid at a Mach nrinlhcr of 19.08.

2.0 Equations of Motion Tha Navier~~Stokes equations i n cart-sian coordinatrs

arc given in the following furin:

(2.1) .- The i n d completc model to incorporate into a rda ~ + a ( h + s l ) + 3 ( f 2 + 9 2 ) .~~ 0 w u i ~ l d hc to eliminate the perfect gas assumption, and 1"'t a f a7 aY

i ~ i t ,If this. thi. snl>wt ~ . .~~~ . ~ ~ . ~ -

(2.2) aq ai, aj2 31 a.r nY

v * Mmihm Ttdinical Staff, Mrmhcr AI..\..\ ** Managcr CFD Dcpt.> Mcinher AIAA - + - + - = O

"* Dcpt. of klathrmatirs, Cal Pol>- Sair Lnis Ohispo ill,!

1

In the above, p is pressure, p is density, and the Cartesian velocity components are u and v in the I and y directions, respectively. The total cncrgy per unit volume, e, is defined as p(e, + 0.5(u2 + v2)), where e , is the internal energy. For eqiiilihirurn air, the yressnreis assumed to he p = R l ( p , e i ) .

Thc viscous "flnxcs" 91 and 92 are given by

and the ternis rzz , T~~~ ryx , and r,, are given by

In the prwxling rquations, Re is the Reynolds number, p

is the cocffirieut of viscosity, li is the coefficient of ther-

nial conductivity, and T is the temperature, and quantities with subscripts r e f are for the reference states. The ref- erencc states typically would be chosen to he free^ stream conditions. Also, the referencc velocity has to satisfy the following condition for non-dimensionality,

The tcmperature and the coefficients of viscosity and thermal conductivity for equilibrium air arc assumed to be functions of density and internal energy like pressure, i.e.,

With the transformation of coordinates iniplied by T = R 2 ( p l e i ) , t 1 = %(P,e , ) ,K = R d ( p , e i ) .

r = t , C = F ( . , y , t ) , 0=7(z,y, t ) , (2.8)

Eq. 2.1 can be recast i n the conservation form givcn hy

where - q y = -

J ' 6. E - E l

f l = 74 + ,fl + Y f 2 J ,

J = a(C, v)/a(J, Y ) ( 2 11)

3.0 Implementa t ion of TVD Formulation Thp finitcvolume framework choscn to implerricnt th?

TVD formulation is presented. This is followcd by a d c ~ scription of the Riemann solver and thc TVD formulation. This is included to analyze the changes made by the as^

suinption of pressure, temperature, and thc coefficients of viscosity and thermal conductivity being arbitrary func~ tions of density and internal energy.

Associating the subscripts j, k with the (, q directions, a numerical approximation to Eq. 2.8 may be expressed in the semi-discrete conservation law form given by

- where f; + E, + E are numerical or representative fluxes

at the bounding sides of the cell for which discrete conserva- tion is considered, and q,,k is the representative conservrd quantity (the numerical approximation to q ) considerrd to be the centroidal value of the dependent variables times thr cell area. The half-integer subscripts denote cell sides and the integer subscripts the cell itself or its centroid. In the following, subscripts easily understood by implication will be dropped for brevity.

The semi-discrete conservation law given by Eq. 3.1 may be regarded as representing a finite volume discrvtiza~ tion if the following associations are made:

( 3 . 2 ) - q j , k 4 j . k A j . k

whcrc A is thc a rm of tlic crill under consideration;

In tlir ;hove, ~ i ~ , ~ arc tlir z, y componrntn of tlic rrprcs(:ri-

tativr normals to tlir stirfacc fonnrcl hy thc two points n , b

iiiiplic<l in nz,,(o., b) . Also, ( x 7 , y 7 ) j + , ~ , k + l / 2 are the z,y

~:omponwks of tlir nppropriatc ccll fiicc rapresentntive ve- locitics. Thcsr dcscrihr tho nint im nf the ccll f x r and will hp Z<TO for a stzitinnary grid. 111 thc following, th? not,ation n t is r w d t o descrihr t,hc reprcscntativr: ccll~ face normal vclocit,ie..;:

the numerical metrics (thc cell face normals). Thp inviscid numerical flux will now he dcfined

In the mcthodology being discussed, a Riemann solver

(upwind schcme) is coupled to a TVD approximation pro-

cedure in such a fashion as to eliminate esseiitially nll of the nnrncricd or spurious (unphysical) oscillations while, at tlir mnw time, achicving high accuracy. Tn describc this typr of discrctization, the underlying qnvind schemr usrd is first desrrihrd in terms of the corrrsponding approxiniatc Ricinann solver. This is followed by a drscriptim of h m v

high ;~c,,~:~cy and thr TVD p r ~ p ~ t y arc l nd t ii i . MWC (IC-

a11d i i i rcfcrcnrrs citcri thcrcin At rvery call intcrfnw vi + 112, Ict q:, , ~ , i l and q,yE+,12

denotr the valrm of thc dcpcndcut varinblm ilrfinod just, to tlic right of and just t,o thv lcft of thr cell facr. 'Tlir Ricrnnnri Solvrr is a mechanism to dividc tlic flux diffcrrnrc bctvwn

and q, ;+ , / , ) into component p x t s associatd with cnch wwc ficld Thcsc ran in turn be dividcd into thoso that corrrs~~nrid t,o posi- tivi. and ncgativc wave speeds. When the numerical flux is computed at the cell face at m + 112, in the finite volume formulation, only the cell-face normals defined at 771 + 112 will he used in the terms cont r ihdng to that rcprescnta- tivc flux. The actual fluxes f, ,fl, when cvalnatcd with the metries cquated to cell-facr normals, cnii all lk, writtrn in the same functional form given hy

thmr ncighboriiig statcs (hctwem q,,, +

where the appropriate values of nrr ny arr used and N de- notes the set of t h o x normals. Using such notation, it is possible to present the necessary algebra wry concisely.

Let the Jacobian matrix of the flnx f with respect to the dependent wriahlcs q he denoted by af/& This Ja- cobian can also he called the coefficient matrix. Let the eigenvalues of the coefficient matrix bc drnotrd by A' and the corresponding left and right *igmvcctors by P' and r',

rcspectively. The matrix fnnncd by t h e . lcft. rigmvcctors as its rows i s then called the left rigrnvrctor nmtrix I, and the matrix of right eigenvectors comprising t,hr right eigenvec- tors as its columns is R. It is ronvcnicnt to choose an ortlrnnormal s c t of left and right rigrnvcctors for whicli L R = RL =: I , the identity mat.rix. In thc ahovc, t l x superscript i has bccn u s ~ d to denotr tlic association of the

i tli t:igenvalne with its corresponding rigenvcctor. Each eigenvalue is also a s s o c i a t d with its own waw field.

The underlying upwind scheme is hilsrd upon Roc's

approximate Riemann solrrr'. In this approach, ccll inter-

where VI = j or I;. This is slightly different than the perfect gas mrthod which uses cnthalpy instead of internal energy Oncc this is done, p,,+,/2, 9 a and consis- tent with prn+,/2 arid are computed using the curve fits. This method is consistent with Roe's averaging with a

"frozen" statr assumption In using this procedurc, thcrc is still ii littlr ineonsistrncy wit.11 the definition of the speed of sourid whvn conipared to the pcrfect gas formulation. This inconsistency does not seem to cause any problems numerically.

I'inowing ( u , u , p , p , 2, ~ ) , r l + l / z , we can compute the eigenvalncs and t,ho rnthonoirnal sct of left and right eigen-

v c c t i n s c,irrvspm~liug to a crll face. Tliesc uiay I I C dclu)tcd

P m+ I j z '

hy A:,,+,/, = x: , t+ ,p( ( l "L+I /2 1 N,r , .+ . l / z ) ,

f : , , + L / > = e : , , + , / , ( ~ , , , + , , , , ! ~ , , , + , , , ) : (3 .9 )

l.:,,+ 1 / 2 = C"+ 112 ((im+1/7 > Nm+ I / z 1 At each ccll facc, the positive and negative projections

of tlw eigcnvalucs may he drfined hy

(3.14)

with

(3.176)

P3 = PI(G t P -) (3.17d) P

and the matrix R of right eigenvectors is given by - To hclp Roc's Riemann solver avoid expansion shocks, only 0 1 - U a 2 + G 0 2 - P W t l ?

at sonic rarcfactions(X'(q,,Nmtllz) < o < Xi(qm+i ,Nm+i /2) ) , 1 - 1 -

c the corresponding positive and negative projections are r e R = - 2 '(.i" n, ~II~~ z C - + c y -+*, :' ] (3.1Sil)

% / 2 + 4 P e , q2 (3.18b) 0, = - t - + -

drfincd as U 6, - + f i g v . --ny "?I* y - C + , , Z = C C

(3.11) with (A'(u,.+r, lv,n+ljz) - A'(q , r I , Nm+1/2))

pc c 21.

Defining the contravariant velocity by

- U = n, t n,u t nyv , (3.12)

tlic cigcnvalrws arc giren by

u Thc rows of L and columtis of R correspond to the order <if

eigcnaalncs shown in E q 3.13. To co~istruct the cigcnvalucs

4

aJ/aq = R.\L (3 .19)

- Using tlic ;hove, thc nrnnrrical flux J , , i t l /2 is con

structed from

In the ahow, tlir mmpwssion parameter b is to be taken as

thr following fiirlction of the accuracy parameter 6 which will hc cxplain~~d shostly.

3 - 6 1 - 4

b = - (3.25)

The m i i m o r l slope-liinitrr operator is

u miiirnod[.r, y] = sign(x) max[0, min( 1 x 1 , ~ sign(z)}].

(3.26)

(3.27) whcrc

r:,, -.: , r ' (q ,,,. ( i \ : , , , ~ k l / 2 + r, ,,.., p ) / 2 ) . (3.2s)

.At maxiiiiii and minima. t l i r n i i n i ~ ~ n i i o p ~ r : ~ t o r rc tums a

%pro value and tlic Icft ; i d right st t l trs d u c t . to

- - q,,,+ i p ~ q'u

9.,,..J/2 = q r n (3.29)

t

resulting in a first-ordcr accurate srlicinc l o ~ d l y . The parameter C, defines schemes of varying accuracy.

The notations oi and ai haw hrrn iised to d~f inc slopr- limitrd r a l r i c s of tlir n paramrtrrs. If wrl rcplnri t h r w hy their nnliinited valurs, wr obtain schemrs whose truncation crrm in one dimensi. la1 steady~~stat? prohlrnis on uniform gsids can he analyzed. The choice of 0 = 1/3 rrsiilts i n a TVD srhrme based on an nnddying third ordcr schrmr. The choice of 6 = -1 results i n a Tl'D sclicnir 1,iinrd on the fully upwind second-order arriiratr formulation. Fronnn's schemr arisrs whrn 6 = 0

- - -

4.0 Equilibrium Air Equation of State

To incorporate the equilibrium air eqiiation of state into the code, pressure, temprratnrr, and roeficicnts of

viscosity and thermal conductivity ar? assumed to hc fiinc- tions of density and intrrnd mcrgy. Tliesc have alrratly

hren noted as functions R1, R2. R3, and R.j ii i tlir prcsm-

tation of the equation sct. This assumption is gmpral and any fluid's equation of state can be incorporatcd (othcr than cq~iilihriuni air) once the specific fluid's pressnre, trmpcra-

turc, and corfficients of viscosity and thixnial conduct,ivity are dcsrrihrd as a function of dcnsitj and intcrnal ciirrgy.

Tlic frinrtioiis of density itnd intcmal mcrgy for q u i - lihrium air prcss"orc and t r m p m a t ~ ~ r c t w d for this r o ~ i ~ p ~ ~ .

tation havr bcen dcveloprd by Sr in iwsm, T n n d i i l l , iind ~V~diniicnster8. These ciirvr fits an, valid fnr tciupcrzltmrs up t n 25,000 degrees I<clrin and dcnsitirs froill to io3 amagats. An amagnt is thr ratio of densiiy to the standard dcnsity of air at 1 atmosplicre.

The functions of density and internid mmgy for cqni-

lilrium air for thr coefficients of visrosity nnd thrrmnl mw

5

ductivity have also been devcloprd by Sriuivasim, Tanne hill, and Weilmuenstd. Thesc curvc fits havc a snialler range up validity, that is for tempcratures u p to 15,000 degrccs Kelvin and densities from to 10' arnagats.

Tlic general forrri of the curve fit for prcssurc is

p = p e ; [ a l + a 2 Y + a 3 Z t a . , Y Z + . , Y ~

+ U 6 Z 2 + a,Y2Z + <L*YZZ + QYJ

t ai& + ( a l l + a i l Y + a132

+ a 1 4 Y Z + U I 5 Y 2 + a,sZ2 + n,,YZZ

+ Q ~ ~ Y Z ~ + a19y3 + n20Z3)/(1

+ ezp[azl + azzY + n x Z + U N Y Z ] ) - 1]

(4.1)

with Y = loglo(p/1.292) and Z = 1og10(e,/78408.4). Tem- perature and the coefficients of viscosity and thermal con- ductivity have similar curve fit equations. Since the curve fits are explicit fnnctions of p and e , , the partial derivatives

g, g,:, and needed for the k f t arid right eigen- vectors used in the Riemann solver and Jacobians for the implicit left-hand side are easily formed The dcrivatives for the coefficients of viscosity and thermal conductivity are left out of the current formulation. This is consistent with the formulation used by Prabhu and Tannrhill'o.

Thcse curvc fits and thr modifications to tlre perfect gas code liavc been coded up in fully vectorizcd form. The equilibrium air computations typically take 2-3 times the time per step of the pcrfcct gas code. The equilibrium air

code seems to be as robust as tlic perfect gas code. The memory requiremnits of the code are increased by seven arrays. This can he a large penalty for a 3-D code, but for

a 2-D code it is not that bad.

5.0 Turbulence Model

The turbulence model incorporatcd into the present

code is the Baldwin-Lomax model" with separation treat- ment". This separation model is algebraic in nature, and thus does not add much computation time per step be- yond the conventional Baldwin-Lomax model. This model has been demonstated i n 2-D/axisymmet,ric problems to handle separation usually better than 2-~equation models. The separation model is not limited to algebraic models. It can be used with 1 or 2-equation inodcls as well as a

Rcynolds~ stress model. The Baldwin-Lomax model has

also been mudificd to include the option of choosing a re- gion in which to transition to turbulence using an expo- nential function. This is useful when transition is known to occur. Also a multiple-wall averaging of the eddy vis-

cosity has heen added. This is useful whm doing problems

in whicli twhrilence is bring gcneratrd from inore than one

wall, such as in a cavity

6.0 Numerical Method

A finite vo~uine code incorporating an implicit npwinii srhcmc and a Total Variation Diminishing [TVD) form,,.

lation of tlic convective terms has been uscd to solvr tlre

N a v i c ~ S t o h rq,mtions. Tlic TVD imple~ncritntion givcs

tlic oscillation frcc property for flow discontinuities which

nllows shocks to be cap tu rd very crisply. Roe's app~ox-

iniatr Rieinann solucr is used to form the numerical flux

which allows for signal propagation to be more accuratcly

modellc(l. Thr Baldwin-Lomax turbulence niodel with s e p

aration treatment is used whcn turbulent flow is computed.

The viscous fluxes, E,E, are treated more or less con^

ventionally, using central difference approximations for all

second- derivative terms, except the cross-derivatives, which are handled with special directional differencing to

augment diagonal dominance. This treatment is discussed

i n Ref. 6.

W

Thr code allows for both space-marching using re-

laxation and time-dependent computations using approxi-

mate-factorization. The structure of the algorithm can be shown in general terms. Equation 3.1 is linearized about a

known state ?". v

where the index i is the iteration index. This iteration

index allows for relaxation iterations and global interrid iterations. The initial guess for $'=') is U". When the iter-

ation process is finished then *'] = ?"+'. The Jacobians

A and B are respectively and with the appropriate

super and subscripts.

For a time-factored scheme, thc left-hand side of

Eq. 6.1 is split into two factors.

[ I + Ar6j(Ai + B i ) ] [ I + A ~ s k ( A 2 + Bz)]'($'' ~ U') =

A ~ R H S ' (6 .2 )

For the space-marching scheme, the left-hand si& is

set up for Gauss-Seidel line relaxation.

[? + 6k(Az + B2)]'(Viti - 7j') = RHS' (6.3) v

with 7 = [A + ( A , + B I ) ~ ] ' .

Sincc this is a full rclaxatioii mcthod, ii time acciiratr relaxation ran he dune for finite timr stcps. Tn invoke

the spacr-marching schnnrt, :in "infinitc" time stcp (AT ̂-i IOzo) is input to canc.cl ont the time trrms in thr discretized equations. Earh sur ivr solution plaiir is srt to thc solii- tion plane just coniput,rrl to gnarmtcc that the downstrram niimerical flux is zcro to prevent any ripstream influence on the first relaxatinn swerp.

Y

For complex geometries, n zonal structure allows for casier gridding. Grid singularitics due to a fan grid, for example, are automatically takcn care of. For convergence speed-up, a multi p i d option has also heen inchiiled. Boundary conditions are done point -wise, thus allowing for user implementation of very complex boundary conditions in the simplest possible and most flcxiblc fashion.

7.0 Resul t s

The following test cases were chosen to validate the present equilibrium air code in the hypersonic flow regime. There are two 2-~D validation cases and one axisymmetric validation cilse. One of tlic 2%D cases is tnrbitlent.

7.1 Hypersonic Ramp

This case is a laniinnr hypersonic test cilsc. Thc 2 D ramp is nt an angle of 15 dcgrees. Sincc this case is onr of no separation, i t anahles 11s to comparc both the timc dcpcndent aiid space-marching methods.

- The grid for tlrc t ime dependent computation is shown

in Fig, 1. This is a 90x40 grid with the first point off the wall at 0.0001 meters. The freestreani flow conditions for this case are M , = 14.1, p , = 23.9 N / m Z , T, = 72.2 d e grees Kelvin, Tw.ll = 297 degrees Kelvin, Pr, = 0.72, and I ? e / i r ~ = 2.322 * IO5. This corresponds to thc experimental case of Holden and M o s ~ l l c ' ~ . This case cnnvergcd i n 700 steps using a variable time step strategy. This is a very high Mach number with strong shocks which lends itself well to the oscillation~~frec property of the TVD implementation of the present codc. Comparison of the pressure coefficient in Fig. 2 and the heat transfer cocfficicnt in Fig. 3 shows good agrcement with data. The pressurc contours are shown in Fig. 4. The pressur<: coefficient is dcfirird conventionally as

aiid the heat transfer coefficient normal to a surface is

This rase was next run in the space-marching mode

Thc same grid chmsity nf 40 gnints was nsrd i n thr normal

direction. The space marching s t q i was 1.5.r10-'m, result- ing in 610 marching planes. This i s t,hr samr code as be- fore except several input phramrtrrs have hcrn changed to makc thr cod? hranrli to the corrcrt snbrontiiics for s p a c e marching. Again thc prt-ssiirc and beat trimsfrr cocffirient

are in good agrcement with thv data as shmvn in Figs. 5 and F. The pressure contours arc prrsentrd i l l Fig. 7.

The noticeable difference brtwern tlir two sohitions is right at the corner of the ramp for thc h r i ~ t , tri$nsf<,r cot:f- ficieiit. The time dependent solution shows a kink right at the corner. This is an upstream viscous influence which is not seen in the space-marching solution sincr t,hcre cannot be any upstream influence in the spacc marching modc. This slight difference seems to cause thr heat transfer of the space-marchirig mode to he slightly highm up the ramp than the time-dependent sohition.

The space-marcliingmrthod had no problem in march-

ing through the leading edge. No special t.reatment was

necessary. be applied. Also, since this method is a space..

marching relaxation method, no special treatment is done for the subsonic region. The full NavieAtokes cqiiations are solved with each successive plane heing srt I ~ K , I to the previous marching plane just updated. This cliininates any signal propagation upstream, since the numerical flux will he exactly zero.

7.2 Hypersonic Inlet This case is a turbulent hypersonic test cas?. The 2 ~ ~ D

inlet has a sharp wedge forebody. The leading cdgr aiiglc is

about 6.7 degrees. The cowl has a blunt leading edge. The frec'stream Mach number is 7.4, and the Rcynolds number

per meter is 8.86* IO6. The wall temperature is 303 dcgrcrs Kelvin, thr freestream pressure is 701.4 N / m 2 , and the freestream temperature is 67.9 degrees Kelvin. This exper- iment exhibited transition hoth on thc c.c.nterbody and the cowl surface. Transition occurred approximately at, 3' = 35 mi. on the centcrhody surface, and at z L 10F CIII. OIL the cowl surface. These conditions correspond to thr cxp& mental conditions run by Gnos and V';~t,son'~. Cornpnta- tions were done with the exact cowl lip geomrtry. Three computational miis were done: a laminar run, a turbulent run with transition at thc expcrimciitally observed x loca- tions, and a fully turhulcnt TUII. The transition points are

turned on rising exponential ramps. This case was run in the space-marching mode. The full geometry of the inlet and outer computational boundary can be seen in Fig. 8. No grid lines are shown at ;dl siiire it would just he tun dense. The grid used had 500 marching planes. Thr nor- mal direction had 100 grid points. The first point off the

7

wall is set at ,005 mi.

The local regions of tlic leading edge ramp of tlic wrtlgr

forebody and tlie hlunt Icading cdgc of thc cowl lip were

isolated i m < l miiltiplc rthaation passes wcrc rlone to corn

wrgr tlicsv solutions. Tlic innltiplr passes enable bettcr convergmrr o f the solntioir i n regions where tlie march- ing assumptions dn not hold - wcll. At lcading edge:"" of

the fnrclmdy wid tlir cowl, grarlicnts i n the niarching di- rection arc uot negligil~lc; tlicrcfore multiplr sweeps wcrc donc in thosr rcgions. This shows thc power of thc correut marching trchnique. Sinrc it is a full relaxation method of the Navier Stokcs cgnat,ioii, local flow regions where space-

marching will iiot give a good solution can be swept more than once to convrrge tlic solution locally. These options are fnlly c<mtr<,llcd by tlre user.

Tlic surfacc pressirc in tlii: inlet or1 tlic centerhudy and tlic cowl coinpare fairly w d l with tlir experimental data. These are shown in Figs. 9 and 10. The shock strengths are not quite correct; liowcver, d l shock locations are in better agrermmt.

Thc ~ K S S I I ~ C contours of the whole inlet (excluding tlie

forebody) arc shown iii Fig. 11. The local region of the cowl lip is isnlatrd and tlre details of the pressurc contours off thc cowl lil, c m he seen in Fig. 12.

Pitot p~wsurc pimfilcs and total temperature profiles arc BISO alrmvn for fnur stiramwise x locations (z = 104.14 cm, 5 = 109.22 cm, I = 114.3 cm, and x = 124.46 cm) in Figs. 13 20. Al l compare rcasonably well with tlie experi- mental data.

Tlic total temperaturc is defined conventionally as

(7.3)

(7.4a)

(7.46)

(7.4c)

(7.4dj

lor 2 1 1 > 1 ; , , Id

F , ( M ) = 1 r n l d F,(.M) = 1 (7.4,) V fo* .\I < 1.

Tlic, t l l i vc mns shcnv the rffect of thr t rx ldn ic r modd ancl tlir cfFc.i.t of niodrlling tha transition points. Thc &ita

swms clmsc t o th<, fidly rurl,nlmt rcsnlts at tirrics, Init sonw

tirucs i t i s clwx: t o the transitional results, and otlamviw thc laminar rrsults s r ~ i i i bcst to match tlic cxpcrimcntal rcstilts. This iurhiguity scems to support tlic commcnts in thr cxpcrimcntal p q x r that the flow is very transitioml If tlic turljulcncr mod<4 could more accnratcly captuw ill(,

c+Tccts d trmsitimal flow, mayhc the computatioid torr<:-

lation would l x better. Previous have h e n done with n prr-

fcct gas codr. Turljulencr was turncd on everywhere; no

transition modrlling was used. The ratio of specific hriits (7) was sct i L t 1.38. Thc change of specific heat was donr to taka c'cqidilxiinn'' effects into account . This seems to h r an r r r o n ~ o ~ ~ s assumption since when thc cquilibriim air

codc is used, the effective gamma never dcviatcs from 1.4

evcn i n locd rcgions of high pressures and temperatnrcs. 7.3 Hypersonic Hyperboloid

This case is a laminar axisymmetric hypersonic t cs t

dagrces. Tllc frccstrcam Mach number is 19.1, and the Reynolds ,>umber per meter is 1.2268 * IO6. The wall tcrn-

perature is 1500 degrees Kelvin, the freestream pressure is 19.757 N l n t 2 , and the frcestream temperature is 293.88 dcgrces Kelvin. This caw is interesting since it exhibits a strong c.quilit~riuni air cffect which shows up visibly in shuck location

case. Tlir hyperboloid has a total included angle of 45 v

This case was ruii in a timedependent mode. A grid of 9 0 ~ 4 0 points was used. The grid is shown in Fig. 21. Tlit first point off thc wall is set at ,001 cm. The stagnation

shock position is compared with a previous comput.rtioli'

i n Fig. 22. The perfect gas solution for this ru11 i s ais<,

iiicludcd in Fig. 22. The effect of the equilibrinm air ':C~)I;L-

tion of state is evident as the shock location ~noves cl~,st., to the wall tliau the pcrfcct gas solution I t c m be S C ~ I L t l l i L t

the agreement with thc second- order viscous shock Ijoui,&

ary layer computation is not exact; Iiowrvcr, i t is sir1liI;Lr.

Prcssnrc contours are shown in Fig. 23.

8.0 Conclusion

Tlic growing importance of having numerical r ~ & ~

which can hrlp designers in developing hypcrsoIlic: fl igi~t vrhiclw lrads us to modify perfect gas rotl<!s to Ilitrlciir t i l r

Ii?i".rsonic regime. The addition of fully rPit&lg rllrlnistry

W

is tlir final goal; howevrr, a good first step is to assmnc cqiii-

l ihr i im chemsitry. Thc prescni 2 D/nuisymmctrir Navicr Stokcs code with an rquilihirum air qimtion of s t n t c Iias

h e m shown to compai-c well with the cxpwimcntirl rcsiilts

of the hyprrmnic ramp, inlet, and thc roriput.atioiial rcsults of tlir hyprrl?oloid. Thc code has hecn modific,d with tlic iiii>nnption of pressurc, t,emprrature, axid t he mcfficicnts of

viscosity and thermal conductivity bcing functions of h i -

sity and internal energy. This is a geiieral formulation and

any fluid in equilihriiini can he I~an~llcd with thc prcscut codr. The curvc fit functions of Srinimwn, ct. al. were ixsed for implemeritiiig the equation of statr for cquililxium air.

‘This rlevchpment is part of ongoing work. YTCW vcr-

sioris of the 3 D tiirhnlent Navirr S tokc code h ; ~ w hreri clcvcloprd with the equilibrium air rapahility. A 2~ D mid 3 D finite rate chcinistry rapahility is XISO heing p u r s ~ ~ r d

References

[I] Moss, J.N., “Rciu:t,ing Viscous Shock layer Solutions with Multiromponcnt lliffiision aid Mass Injection”, NASA TR R- 411, NASA Lnngley Rmcarch Ccnter, Jiinc 1074.

121 Chakravarthy, S.R., and Oshcr, S., “A Ncw Class of

High Accuracy TVD Sclicmcs for IIypcrholic Conser- vation Laws”, AIAA Paper No. 85 0363. -

131 Chakravarthy, S.R., “Relaxation Methods for Unfac- tored Iinplicit Upwind Sclicmcs”, AIAA Pi~pcr No. 84 01F5.

141

[51

161

171

[SI

191 v

Chakravarthy, S.R., and Sacma I<GY., “ A n Eulcr SOIL dimensional Supersonic Flows with

Subsonic Pockets”, AIAA Paper No. W1703.

Chakravarthy S.R., “The Vnsatility and Reliability of

Eulcr Solvers based on High Accuracy TVD Formu- lations”, AIAA Paper No. 86 ~0243.

Chakravarthy, S.R., Saerna, I < . ~ Y., Goldhcrg, U.C., Gorski, JJ., a i d Osler, S., “Application of a New Class of High Accuracy TVD Schc~rics t u thc Navier Stokes Equations”, AIAA Paper No. 85- 0165,.

Roe, P.L., “Approximate Riemann Solvcrs, Parameter Vectors, and Diffcrcncc Schrmcs”, Journal of Comp. Physics, Vol. 43, p~’ . 357-372.

Srinivasan, S. , Tanirhill, d.C., and Wcilmuenster, K J . , “Sinlplificd Curve Fits for the Thermodynamic Propertics of F,qulihrinm Air”, ISU ERI Arncs-86401, Iowa Statr University, Jiinr 108F.

Stinivnsari, S.> Tantwhill, .I.C., and W<,ilmrienntrr, K.J. ; “SiniiMicil C i r v r Fits for rlir Transport Proper-

tics of Equilibriiiiii A i Y . ISL’ ERI .A111t.s 8840,i; Iowa Stat c Univrrsi ty, Scptcnihrr 19Si.

1101 l’rabliii, D.I<.: nnd Tannrhill; .l.C., “Knniwic;ii Solt~. thn of Spaw Shiittlr Orhitci Florvfielrl Iiicliirling R m l Gas Effccts”, Jonrnal of Spaw<raft, Vol. 23, X o 3. .Jmr 19SF, p p 2F4 272.

1111 Baldwin B.S., and L o ~ n a x , €I. , “Thin Layrr .4pprosi~

ination mid Algchmic Modd for Srparatcd Tur ldcnt Flows”, AIAA Papar Yo. 7 8 ~ 2.57.

1121 Goldhcrg, U.C., “Separated Flow Trcatment witli n

Kcw T d ~ u l c n c e Modcl”, AIAA Journal, Vol. 24, \lo.

10, Ort,ohcr 1986, pi’. 1711- 1713.

[13] Hdclc~ i , M.S., and Moselle, J.R., “Theoretical and Ex- pcrimcntal Studies of the Shock Wave-Boimdary Laycr Intcraction on Compression Surfaces in Hypersonic Flow”, ARL 70 ~0002, Cornell Aeronautical Lahora- tory, January 1070.

1141 Gnos, A.V., and Watson, E.C., “Investigation of Flow Fields within Large Scale Hypersonic Inlet Models”, NASA TN D 7150, NASA Ames Research Ccntar, April 1973.

1151 Kunik, W.G., Benson, T.J., Ng, W., and Taylor, A,, “Two . and Thrrc Dimensional Viscous Computations of a Hypcrsonic Inlet Flow”, AIAA Paper No. 87 0283.

9

0.2!

0.2c

0.15

CP

0.1c

0.500E-0:

P<l*Oirn PRESSURE COEFFICIENT EQUILIBRIUM AIR M, = 14.1 6 = 15O

L

04 P,

0.20 0.34 0.48 0.62 0.76 0.90

X

Figiirc 2. Prcssnrc cocfficient for egnililxiim air

0.750E-0:

0.6OOE-Oi

0.45M-02

ch

0.300E-02

0.150E-02

0 c

HEAT TRANSFER COEFFICIENT 1511W6

EQUILIBRIUM AIR M, = 14.1 6 = 1 5 O

- T I --- -,-. 0 3 4 0 4 8 0 5 2 0 7 6 0 9 0

X

X

Fignrc 4. Prcssurc contours for q u i l i l r i t m i air. IC.O.1~ 0.25

PRESSURE COEFFICIENT EQUILIBRIUM AIR M,= 14.1 6=15O

0 . 2 0 ~ I SPACE-MARCHING

0.15-

CP

0 . 1 0 ~

0 . 0 5 -

0 0.14 0.33 0.52 0.71 0.90 0

X - Figure 5. Prcssiire coefficicrit for cquilibrinni air-

0.75 x

0.45 x 10.’

ch

0.30 x 10.’

0.15 x 10.’

space niarching

0

X

Figure 6 . Heat transfer coefficient for cqnililxianr air^

space marching.

-

-0.05 0.14. 0 33 0.52 0.71 0.90

x ICmI

Fignrr 8. Out,line of P8 inlet computational grid

1 ° L 1 0 100 106 112 1 1 8 124 130

X bml

Figure 10. Cowl surface pressure

24 I

0 - ___ 80 92 104 116 128 GO

X Icml

I pigure 11~. pressure contours-laminar.

24 1 111212.

I !

80 0 1 ~ ~ ~r ----

92 104 116 128 140

X (Cml

11h. Pressure contours-turhulcnt with transition

BCI>lll 2 4 '

0 80 92 104 118 128 140

x icm1

~ i ~ ~ , ~ ~ 11~ . pressure contour5 fully tiubulcnt

11

I ~~

18 3 I 18 3 .

!

!

18.1 ,,-- ,

- > E I E I

; I 17.91 17.91

! I P8 INLET ~ PRESSURE CONTOURS ! P 8 INLET PRESSURE CONTOURS

M, 7.4 17.71 M, ~~ 7 .4 R e / m ~ 8 86 x 106 R d m 8.86 x l o 6 LAMINAR FULL TURBULENT

i 1 7 7 i 1 7 5 _~.__/ ~ 17 5 , ~~ ~, ~ ~~. ~~~ ~~ . - ~~ ~

01.4 81.6 81.8 02.0 81.0 81.2 81 .4 81.6 s'l.8 8 2 0 x (Cm/ x Icm)

I 8 1 0

Figure 12a. Presswe contours near cowl ~ i p - ~ a m i n a ~ .

1 8 5 1 %LA1128

~ i ~ ~ , ~ ~ 12c. Pressure contours near cowl lip fully tntbulcnt.

, P8 INLET PRESSURE CONTOURS

1 7 7 ; M, ~ 7 .4 R d m i 8.86 x l o 6 TURBULENT WITH TRANSITION

~~~ -, 17.51 , ~ ~ ~~~

81.0 81.2 81.4 81.6 8; 8 82.0 x 1cmi

pigrlrc 12b. Prrssiirc contours nCar cowl li11 trlrbulent with transitiom

w

12

PP

Fignrc 15. Pitot prcssrirc profilrs at x=100.22 cm

F - >

14.4

1 3 2 -

12.0

~

- 0 2 0 4 0 6 0 6 1 0

Tt

Figure 16. Total tcmpcraturc profilcs 2Lt n=lO9.22 mi.

16.8 1

I AY--

12.0 ;"I". 0 0.02 0.04 0.06 0.08

PP

Figure 17. Pitot prcssurc profiles at x=114.30 cm

13

dd O Z L O 9600 Z L O O 8 V O O V Z O O

7- -7

0 0

3 Z L

E CL

b U t

< - : - 9 SL

3 9 1

3 8 1

In: 2.00 I

~7.~-

STAGNATION LINE PRESSURE Mm = 19.1 Re = 1.22x106im

- EQUlLlBRtUM AIR ----- PERFECT 6AS 1.20

0.400

0.000 0.000 100 200 300 400

P

Fignn: 22. Stagnation line shock location

Y

0.0001 m I I 28.8 31.3 33.8 36 .3 38.8

Figure 23. Pressure colltollrs.

3

15