A study was completed of mathematically proper boundary conditions for unique numerical solution of internal, viscous, subsonic flows in the space shuttle main engine. The study has concentrated on well posed considerations, with emphasis on computational efficiency and numerically stable boundary condition statements. The method of implementing the established boundary conditions is applicable to a wide variety of finite difference and finite element codes, as demonstrated

Baker, A. J.

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lcomputational 
Mechanics 1 J  C Corporation 
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NASA/MSFC 
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Marshall Space Flight Center, AL 35812 
Reference: Contract NAS8-36650 
Subject: Final Technical Report 
Pursuant to the reporting requirements for the referenced 
contract, enclosed is the final Technical Report. 
Copies : 
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Information Faci lity(2 1. 
AP29-F( O),  AS24D(5), ATO1(1), CCOl/WOFFORD(l), EM13B-ll/J, 
(NASA-CR-178950) ANALYSIS CP E C U N D A B Y  N87-13062 
C C N D I T I C I S  EO35 SSEE S U E S C N I C  IliIEErNAL 
VISCOCS PLOW ANALYSIS ( C o m ~ u  taticjnal 
Hechanics C o n s u l t a n t s )  1 4  p CSCL 20D 
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TABLE OF CONTENTS 
Page 
i 
Technical Report 
ANALYSIS OF BOUNDARY CONDITIONS FOR SSME 
SUBSONIC INTERNAL VISCOUS FLOW ANALYSIS 
BY 
A. J. Baker 
Prepared for 
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION 
George C. Marshall Space Flight Center 
Huntsville, AL 35812 
Reference: Contract NAS8-36650 
Report No: COMCO TR 87-4.0 Date: Oct. 1986 
COMPUTATIONAL MECHANICS CORPORATION 
3601a Chapman Highway 
F2omille, m 37920 
. .  
I. sLEI;psRy . I  . ,  
A study has been cunpleted of mathematically proper boundary conditions for 
unique numerical solution of internal, viscous, subsonic flaws in the SSME, The 
study has concentrated on we1 1-posed considerations, with emphasis on 
canputational efficiency and numerically stable boundary condition statements. 
The method of implementing the established boundary conditions is applicable to 
a wide variety of finite difference and finite element d e s ,  as demonstrated. 
The results of this study are reported herein. 
11. " I Q l L D I S X E S I ~  
A. Introductsl 'on 
Over the past several years a focus at NASA Marshall Space Flight Center has 
been adaptation and application of canputational fluid dynamics (CFD) analysis 
techniques to f lowfield prediction in components of the SSME. Several 
"olynpiads" have been held, wherein purveyors of CFD codes have developed and 
compared solutions for model problem definition analyses to the turn-around 
duct-transfer duct SSME geanetry. The SSME geopnetry is defined to these codes 
via construction of meshes that possess boundary segments roughly coincident 
with solid walls and containing convenient flaw inlet and outlet planes. The 
numerical simulation of the associated f lowf ield is defined via appropriate 
specification of constraints on the (Navier-Stokes) conservation law system 
variables, e.g., velocity and pressure, over the entire boundary of the mesh. 
This study examined boundary condition specifications for the CFD models, 
with emphasis on mathematical we1 1-posedness with physical consistency. The 
SSME f lowf ield is characterized as complex turbulent three-dimensional and 
unsteady, at high Reynolds number and l aw  subsonic Mach number, le., essentially 
incompressible. Mathematically, the CFD algorithms/codes applied to this 
problem definition fall into two distinguishable categories. m e  family (GIM, 
nn-m v r n c . r n  
rfibh, v f i a 1 ,  ...) sets iip s~:iitiofi ss the * U Y S ~ S Z ~ Y  ~ * ~ = l ~ t i ~ n  ~f 2 h:~erhr\ l ic  
1 
. .  . . .  I .  
. I  * *  
conservation law system with the assmption of a ccmpressible fluid satisfying a 
polytropic gas law statement. Conversely, the second family (INS3D, PHOENIX, 
SIMPLE, FIDAP) specifically assumxi an incanpressible fluid, and directly seeks 
the steady-state solution without specifying a (physically significant) equation 
of state. The PHOENIX and SIMPLE algorithms seek the steady-state through a 
pressure relaxation procedure that explicitly requires pressure boundary 
carrdition specifications. FIDAP uses a finite element penalty methcd to totally 
replace the appearance of pressure. Alternatively, the INS3D theory employs a 
pseudo-canpressibility concept, yielding a hyperbolic conservation law-appearing 
statement for pressure that (only) requires approximation of the normal pressure 
gradient at boundaries. Thus, the mathematical boundary condition aspect of 
INS3D is mre analogous to that of GIM, etal., than PHOENIX, etd. 
The follawing sections examine CFD algorithm boundary condition issues fran 
the standpoint of, a) hyperbolic conservation law systems, and, b) pressure 
relaxation in an incanpressible f law specification. 
B. Hyperbolic &mservaticm Iaw Algorithm 
The conservation law system governing the kinematics, kinetics and 
thermodynamics of a viscous, heat-conducting fluid is generally termed the 
Navier-Stokes equations. The Cartesian tensor indicia1 fonn is, 
I 
ape) a 
Upe) = - + - (uJpe + u . p  - o..u. + q . )  = o 
at dx. J V I  J 
J (3) 
( 4 )  
2 
where P is dens i ty ,  Pui is  t h e  momentum vector, p is  pressure,  c l i j  i s  t h e  
Kronecker de l ta ,  e i s  s p e c i f i c  t o t a l  energy, and Y is  t h e  ra t io  of s p e c i f i c  
heats for a polytropic gas l a w  f luid.  The expression of const i tut ive properties 
of the f l u i d  is contained i n  the stress tensor ai ,  and heat f l u x  vector q 
simple f l u i d s  and laminar f law,  the accepted forms are, 
For 1. 
au. - + - 0 . .  
= '[ Re axj 
dT 
% 
= -A- 
'i 
(5) 
where the dynamic (molecular) viscosity P (TI, and the them1 conductibity k(T), 
are weak functions of temperature T, and the Reynolds n m k r  is Re=(PUL/u) ref. 
I n  t h e  l i m i t  of large Reynolds number, an i n v i s c i d ,  non-heat conducting 
assumption renders equations 5-6 ident ica l ly  zero. The resu l t ing  form of eqn 1- 
4 is tenned t h e  Eu le r  equat ions,  a homogeneous hyperbol ic  conservat ion l a w  
system. A l t e r n a t i v e l y ,  enforc ing  a statistical averaging procedure yields  a 
Reynolds-averaged Navier-Stokes system that introduces the concept of a Reynolds 
stress tensor and additional governing partial d i f f e ren t i a l  equations, eg., the 
two-equation turbulent kinet ic  energy-isotropic dissipation function system. 
I n  either instance the  generic form of the  governing equation system is, 
& df, 
+ s  = 0 ( 7 )  - +  2 ad ax. 
I 
where q conta ins  t h e  Navier-Stokes/Euler/Reynolds-averaged Navier-Stokes 
dependent var iable  set, f .  i s  t h e  corresponding f l u x  vector and s i s  a 
source/sink term, eg., 
3 
I I u . p k t k .  1 J 
The s ing le -po in t  c l o s u r e  equations for Q i j ,  q j ,  k. 3 and F: j are (cf., Baker, 
1986 
- - 
0 . .  = 0 . .  Y -pu' .u ' .  1 1  
- - 11 - - 
q, = q .  + p H u ' .  - u' 0 .  - U . O . .  J J 1 V 1 1J 
( 9 )  
e =  (q b )  5 
J ' J E  axi 
The E u l e r  equat ion form is contained i n  eqn 22 by d e l e t i o n  of k, E , kj,  c j, q,, 
ai, and s, and replacement of pH byp e + yp. 
A familiar a l t e rna t ive  form f o r  eqn 7 is established following imposition of 
= CE, n , 5 1 i s  a n y  
One pa r t i cu la r ly  useful form is to  a l ign  the 
j a c o o r d i n a t e  t r a n s f o r m a t i o n  xi = x i ( n j ) ,  where  
(curvi l inear)  coordinate system. 
coord ina te  E with t h e  d i r e c t i o n  of p r i n c i p a l  f low, whereupon eqn 7 can be 
where E, F and G are t h e  (Eu le r )  f l u x  vector components, E,, F, and G, are t h e  
const i tut ive closure model canponents (containing aij, q., kj and E , . ) ,  
expressed in terms of scalar canponents i n  the nj Coordinate system, ie,, 
Both are 3 3 
1 E =  - 
J 
4 
where J is the Jacobian of the coordinate transformation, and the convection 
velocity (U) vector contravariant scalar components parallel to the ( 5 ,  ,c )  
Ooorainate system are, 
u = $u + t u  + t Z W  
v = q,u + q u + qzw 
w = i,U + 5 u + $w 
Y 
Y 
Y 
(12) 
The constitutive scalar components E, Fv and Gv are formed in the similar 
manner. 
The conservation law system, eqn 7 or 10, is either hyperbolic or an 
initial-valued, inccmpletely elliptic boundary value problem, dependent upon the 
constitutive closure definitions (inviscid, viscous/turbulent). The solution 
domain il x t is a bounded subregion of a portion of the SSME duct region, and 
the boundary conditions on as2 , mathematically consistent with a well-posed 
problem, have been examined by Dutt(1985) following a dependent variable 
transformation to "entropy variables." The entropy transformation of the 
primitive (Euler) dependent variable set q = b , P ui, .pe), eqn 1-3, has been 
examined, cf., Harten(19811, Osher, et.al. (1984). For a family of strictly 
convex entropy functions, Mallet(1985) and others show that for the (Euler) 
extension to viscous and heat-conducting fluids, the sole ''useful" entropy 
function is the thermodynamic entropy p s. Hence, the transformation is, 
V(q) = - p s  = - p log (ph6 1 
and the entropy flux functions are 
fj = -pu#) = - mi(p s)  
The transformation to V(q) symnetrizes the conservation law statement, eqn 1-3, 
yielding a nonlinear energy estimate (for sufficiently smooth solutions to the 
mixed initial-boundary value problem) that corresponds to the Clausius-Duhem 
inequality (second law of thermodynamics). 
5 
For the (Euler equation)' hyperbolic conservation l a w  fom,  the appropriate 
number of boundary conditions on 00 i s  (Strickwerda, 1977): supersonic  in f low 
(5), subsonic in f low (41,  supersonic out f low (none), subsonic out f low (1). 
For t h e  Navier Stokes equations,  5 ( 4 )  boundary condi t ions  are required a t  
inflow (outflow). Dutt(1985) develops the set of "maximal dissipative" boundary 
condi t ions  for eqn 1-6 for t h e  (Navier-Stokes) boundary condi t ion s ta tement  
fom, 
(15) & - - + S q = b  as a t  
where R is a matrix of rank a t  most 4, Cis the coordinate normal to'>aQ, and for 
the Euler def ini t ion (E = 01, Sq=b is  a proper form f o r  the (unperturbed Euler) 
hyperbolic problem. The derived o u t f l o w  boundary condi t ions  are inne r  (dot) 
products of eqn 5-6 w i t h  the outward pointing uni t  n o m 1  vector fi ie., 1' 
- aiUl = bi , 1 < i < 3 - -  a. A 
q - f i  = o  I' j 
where U1 is the ve loc i ty  contravariant scalar cmponent parallel to  5 . Any SSME 
a p p l i c a t i o n  i n v o l v e s  o n l y  subsonic outflow, hence al(= aS 1 - > 0 and bl is a 
cons tan t  and a l l  o t h e r  components of ai and bi vanish as does t h e  normal heat 
flux. The derived i n f l w  boundary condition couples the in f lux  definit ion pU1 = 
bo w i t h  t he  more general form of eqn 16. 
PU1 = bo 
a i j  .fij - aiUi = bi 
- q..fi - a4T = b4 
, 1 - -  < i < 3 - 
i j  
In  ecp 17, the subscript bar on the contravariant ve loc i ty  vector Ui denotes the 
index i s  - not  summed, and ai and bi ( 0  1. i 1. 4 )  are cons tan t s  subject t o  
cons t r a in t s .  For t h e  SSME d e f i n i t i o n  of subsonic  in f low (M.(< 11, al = 0 = bl 
w h i l e  a2, a3 > b, /2 and a4 > 1 / 2  ( ?  - y + 2 ) b:o + E, where E corresponds t o  
the inverse Reynolds number, see eqn 5. Of sane interest ,  the last expression 
in ecp 17 is d i r e c t l y  amenable to physical interpretation; i n  expanded form, 
using eqn 6, 
6 
d l  -q. fi = k--- .fij = KW.fi = b4 + a4T = h (T-Tr) I' j ihc (18) 
Thus, a4 is interpreted as the boundary heat convection coefficient h, and Tr is 
the heat exchange reference temperature. 
Equations 15-18, i n  concer t  w i t h  Strickwerda's constraints, encanpass the 
range of boundary conditions, that are mathematically well-posed and n m r i c a l l y  
stable, for SSME f lm  CFD simulations formulated as approximate procedures for 
solving the hyperbolic conservation l a w  Euler extension to  Navier-Stokes. Since 
SSME f lawf ie lds  are uniformly subsonic, then only one exit D i r i c h l e t  boundary 
condition is allowed, taken as the static pressure. Up to four i n l e t  D i r i c h l e t  
boundary conditions are mathematically permitted; hawever, eqn 17 suggests that 
a l l  bu t  t he  mass i n f l u x  be replaced by Neumann cons t r a in t s .  Fur ther ,  i f  t h e  
mass in f lux  is specified, then the i n l e t  pressure may not be specified (unless a 
region of supersonic  f l o w  e x i s t s  between i n l e t  and o u t l e t ) .  Conversely,  a 
D i r i c h l e t  s p e c i f i c a t i o n  of i n l e t  to ta l  pressure could be made, whereupon t h e  
mass f l u x  w i l l  become determined by t h e  f lowpath t o t a l  p re s su re  drop. The 
repor ted  SSME s imula t ions  us ing  PAGE and VAST have g e n e r a l l y  employed t h e  
former, while the INS3D simulations have general ly  used the latter. 
C. Inoooopressible Navier-Stokes Algorithns 
As noted a t  t h e  end of Section A, t h e  a l t e r n a t i v e  SSME s imula t ion  CFD 
cons t ruc t ion  class u t i l i z e s  incompressibi l i ty  d i r e c t l y  t o  recast the  Navier- 
Stokes system i n t o  a non-hyperbolic conservation l a w  form. In these procedures, 
eg., PHOENIX, SIMPIS and FIDAP, the pressure dis t r ibut ion is derived ind i r ec t ly  
fran the ve loc i ty  constraint of divergence-freeness, hence no equation of state 
is (need be) assumed t o  e x i s t .  Thus, no ( D i r i c h l e t )  p re s su re  boundary 
conditions are needed or appropriate i n  defining the CFD simulation, although it 
i s  w e l l  publ ished t h a t  PHOENIX and SIMPLE employ a "pressure co r rec t ion  
equation" to  achieve convergence t o  a numerical  approximation of divergence- 
freeness. This is not s t r i c t l y  exact, as w i l l  be developed. 
7 
The mentioned class of incunpressible Navier-Stokes algorithms can be viewed 
i n  a unified manner as decisions made i n  evaluating a Taylor series on t he  time 
e v o l u t i o n  of t h e  v e l o c i t y  f i e l d  EI, where boldface  d e f i n e s  a v e c t o r  f i e l d .  
Assuning knawledge of the  solution a t  t i m e  G, where tn+l = & + A t ,  we have, 
f+'= - + A t $  + ... (19) 
The incompressible  form of eqn 2, p l u s  eqn 5, p rovides  t h e  expression f o r  t h e  
time-derivative i n  eqn 19, hence, 
- &'= f - A t  [@*V@ - - +Vp - Re-1V2#] - + ... (20)  
The basic CFD algorithm theoretical  choice lies i n  select ion of Vp i n  eqn 
A f i n i t e  element p e n a l t y  a lgori thm r e p l a c e s  t h e  v a r i a b l e  p w i t h  t h e  20. 
approximation t o  continuity, 
- 
p = - xv .u_ (21)  
.. 
where x is a large 0(106) constant. The superscript t i l d e  denotes that u is an 
approximation t o  a divergence-free velocity f ie ld .  Final ly ,  eqn 21 is evaluated 
a t  the  new t i m e  tn", hence eqn 20-21 is an implicit expression. 
The basic theory f o r  the  SIMPLE-class of incunpressible Navier-Stokes CFD 
algorithms is s i m i l a r l y  developed d i r e c t l y  from eqn 20. If :Vpn is  employed, 
then the solution 3' does not sa t i s fy  the  continuity equation. Hence, define 
a new pressure en+' that produces (assumption) a divergence-free ve loc i ty  f i e l d  
- Writing eqn 20 f o r  both pressures, taking the sun and subtracting yields,  
vx ($+I - p1, = 0 (22)  
Thus, t h e  d i s t i n c t i o n  between t h e  predicted and t h e  con t inu i ty - sa t i s fy ing  
ve loc i ty  f i e l d s  a t  tn+' can a t  most be the  gradient of a scalar f i e l d  0, ie., 
(23) u"+l - 2"+1 = -0 - - 
Subtracting eqn 22 i n t o  the  incanpressible form of eqn 1 yie lds  
v 2 0  = - v *&+I - (24)  
The boundary condition for eqn 23, for the  harmonic function 0, is obtained f ran 
q? 22 as 
v0.a = (31+1 - - $+l) c .a (25) 
where 6 is the uni t  outwarding pointing vector normal t o  the  solution danain 0. 
Once @ is determined, using an appropriate (CFD) algorithm fo r  eqn 23-24, then 
the "corrected pressure" f i e l d  is 
A t  s teady-s ta te  convergence, eqn 23 becomes homogeneous, hence I O  h I 0 for  a 
(s ingle)  D i r i c h l e t  boundary specification i n  concert w i t h  eqn 24. Thus i n  the 
l i m i t ,  5 is the pressure f i e l d  that coexists w i t h  a canputed approximation t o  
the divergence-free ve loc i ty  f i e l d  u h . - 
Viewing eqn 20, 22-25, there is no admissible pressure boundary condition 
specification for the SIMPLE-class, CFD incompressible  Navier-Stokes algo- 
rithms. Equations 16-17 remain appropriate for t h e  remaining variables,  and the 
mass f lowrate (eqn 17a) mus t  be spec i f ied  t o  create a SSME problem statement.  
The auxiliary variable 0 carries the remaining boundary condition specification, 
and eqn 24 is hanogeneous Neumann everywhere that the dis t inct ion between # and 
u -h must vanish,  eg., i n l e t ,  so l id  (no-sl ip)  w a l l s ,  symmetry p lanes ,  etc. The 
sole D i r i c h l e t  constraint  f o r  @ therefore can only be applied a t  a location on 
 
the mass e f f lux  boundary segment of 00. AssLsning the CFD i t e ra t ion  sequence is 
convergent,  eqn 25 y i e l d s  t h e  corresponding s ta t ic  p res su re  d i s t r i b u t i o n  t o  
within an a rb i t ra ry  constant, wh ich  can be specified (for example) to  match 9 t o  
an i n l e t  o r  an o u t l e t  pressure leve l .  
9 
The literature contains numerous results documenting the robustness of the 
CFD algorithm boundary conditions developed in the preceeding sections. The 
ocmpressible hyperbolic conservation law formulation is exhaustively examined in 
Dutt(1985); Chang and Kwak(1984) document the total pressure specification 
option for the INS3D algorithm. To our knowleddge, the interpretation of 
SIMPLE-type algorithms developed in Section C is not CQrmon knowledge. The @ 
construction for incompressible parabolic Navier-Stokes algorithms is well 
. 
established; Baker(1983, Ch. 6-7) fully docments the range of application of 
the solution statement given as eqn 23-24. 
. 
10 
Baker, A.J. (19831, F in i t e  Element Cauputational Fluid Mechanics, 
Hemisphere publishing Corp., New York. 
Chang, J.L.C., Dao, S.C., and Kwak, D.(1984), Progress  i n  Space 
Shut t le  Main Engine Code Development and HGM Analyses, i n  
Cmputational Fluid Dynarmcs, Proceedings of A Workshop held a t  
MSFD, Huntsville, AL. 
Chang, J.L.C., and Kwak, D.(1984), On t h e  Method of Pseudo 
Ccnpressiaility for Numerically Solving Incompressible Flows, 
Tkch. Paper AIM-84-0252. 
Dutt, P.(1985), Stable Baundary carrditions aml Differenoe schenes f a r  
Navier-Stokes Equations, Tech. Report NASA ICASE 85-37. 
Ehgleman, M.S.(1984), FIDAP, F lu id  Dynamics Ana lys i s  Program, 
tJ=- '01 Manual, Control Data R e p o r t  No. 84003650. 
Harten, A. (19811, On t h e  Synmetric Form of Systems of Conservation 
Laws with Tech. Report, NASA ICASE 81-34. 
Kwak, D., and Chang, J.L.C.(1984), An Incampressible Navier-Stokes 
Flaw solver i n  '-1 Curvi l inear  (hordma - Wsystens 
Using Primitive V a r i a b l e s ,  Tech. Paper W-84-0253. 
Mallet, M. (19851, A Finite Elenerrt Algorithm far CCmpltatioMl Fluid 
DyMnics, PhD Dissertation, Stanford University, Dept. C i v i l  Ehq. 
Osher, S. and Chakravarthy, S. (19841, High Resolution Schemes and the  
Cbditim, SIAM J. Nunn. Anal., 21, 955-984. 
Patanker,  S.V.(1980), Numerical H e a t  T rans fe r  and F l u i d  Flow, 
Hemisphere Publishing Corp., New York. 
Strickwerda, J.C.(1977), I n i t i a l  Boundary V a l u e  Problems for 
Inanpletely Parabolic Byaticms, Cam. Pure Appl .  Mech., 30, 797- 
822. 
11 


Analysis of boundary conditions for SSME subsonic internal viscous flow analysis

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