NASA Technical Memorandum 102138 

AIAA-89-2669 

c 

High Speed Corner and Gap-Seal 

Computations Using an 

LU-SGS Scheme 

William J. Coirier 

Lewis Research Center 

Cleveland, Ohio 

fSASA-TB-lO2138) HIGE SPEED COBllBR AHD 1189-271 03 

CAP-SXAL COBPUTBTIORS USING hN LO-SGS SCEEHB 

(NASA., Levis Research Center) 16 pCSCL 200 

Unclas 

e3134 0217620 

Prepared for the 

25th Joint Propulsion Conference 

cosponsored by the AIAA, ASME, SAE, and ASEE 

Monterey , California, July 10- 12, 1989 
HIGH SPEED CORNER AND GAP SEAL COMPUTATIONS 

USING AN LU-SGS SCHEME 

* 

William J. Coirier t 

NASA Lewis Research Center 

Cleveland, Ohio 44135 

* 

Abstract 

The hybrid LU-SGS (Lower-Upper Symmetric 

Gauss-Seidel) algorithm has been added to a widely 

used series of 2D/3D Euler/Navier-Stokes solvers and 

has been demonstrated for a particular class of highspeed flows. A limited study has been conducted to 

compare the hybrid LU-SGS for approximate Newton iteration and diagonalized Beam-Warming (DBW) 

schemes on a work and convergence history basis. This 

study showed that for the cases considered, the hybrid 

LU-SGS algorithm is more efficient and easier to implement than the DBW scheme originally present in the 

code. The code has been validated for the hypersonic 

flow through two mutually perpendicular flat plates and 

then used to investigate the flow field in and around a 

simplified scramjet module gap seal configuration. Due 

to the similarities, the gap seal flow was compared to 

hypersonic corner flow at the same freestream conditions and Reynolds number. 

I. Introduction 

The hybrid LU-SGS (Lower-Upper Symmetric 

Gauss-Seidel) scheme has been added to the PARC 

2D/3D series of Euler/Navier-Stokes solvers. The 

PARC codes are based upon a central finite-difference 

scheme with fourth and second difference dissipative 

terms. The PARC codes originated as the AIR 2D/3D 

solvers by Pulliam (Ref. l), which used the AD1 algorithm presented by Beam and Warming (Ref. 2). 

To increase the computational efficiency and to reduce 

storage requirements, Pulliam (Ref. 3) diagonalized the 

block implicit operators of the original Beam-Warming 

AD1 algorithm, resulting in the ARC 2D/3D codes. 

Cooper (Ref. 4) further modified the ARC codes to 

be used in a production environment for solving the 

propulsion oriented problems at the Arnold Engineering Development Center, resulting in the PARC codes. 

The PARC codes are well validated and accepted and 

are in wide use by the aerodynamics community. 

I* 

t Aerospace Engineer, Computational Methods Br. 

Member AIAA. 

Copyright .? 1989 by the American Institute of Aeronautics 

and Astronautics. Inc. No copyright is asserted in the 

Iliiitccl State\ under Title 17. U.S Code. The U.S. Governmcni ha\ ;I ro~;ilry-trcc licci!x to cycrcisc all rights under 

tlic c(ip [right c.l;iitncd herein tor Govcrnmental purposes. 

A11 othcr right\ arc rexr\cd by the copyright owner. 

3 

Through the use of a unique method of grid patching 

and the segmented application of boundary conditions, 

the PARC codes are very useful in solving wide varieties 

of flow problems. A full description of the PARC codes 

and their implementation of the diagonalized BeamWarming algorithm can be found in Reference 4. 

In an attempt to further advance the practicality 

of these solvers, the present work investigates the addition of a relatively new algorithm to the codes. An 

LU implicit scheme, proposed by Jameson and Turkel 

(Ref. 5) and improved upon and demonstrated for the 

Euler equations by Jameson and Yoon (Ref. 6) was 

used in conjunction with a multigrid strategy for the 

Euler equations by Jameson and Yoon (Ref. 7). From 

this scheme, the LU-SGS (also referred to as the LUSSOR) scheme was developed and applied by Yoon and 

Jameson in References 8 and 9 for solutions to the Euler 

and Navier-Stokes equations. This scheme was shown 

to be very efficient and practical for the solution of 

the Navier-Stokes equations coupled in a fully implicit 

manner to the chemical transport equations by Shuen 

and Yoon (Ref. 10) in two dimensions and by Yu, Tsai 

and Shuen (Ref. 11) in three dimensions. Recently, 

this scheme was used by Park and Yoon (Ref. 12) to 

compute two dimensional flow in chemical and thermal 

non-equilibrium around a blunt body. This scheme has 

clear advantages when compared to others for implicitly 

solving fairly large systems of transport equations since 

it has the operational count and storage requirements 

of an explicit scheme, but still solves the equations implicitly and fully coupled. The following sections will 

show the derivation of the LU-SGS scheme, compare 

the LU-SGS and diagonalized Beam-Warming schemes 

in terms of convergence history on a work basis and use 

the LU-SGS scheme to analyze hypersonic corner and 

gap seal flowfields. 

II. Analysis 

The PARC2D/3D codes solve the Reynold's averaged Navier-Stokes equations in a generalized curvilinear coordinate system. The Navier-Stokes equations 

can be written in divergence form as 
where the flux variables q and a generalized inviscid flux vector, E, can be expressed as 

PC 

q=J-’ (i) , B=r’ (2) 

PWUC + &P 

(Et + P)% 

where J is the jacobian of the coordinate transformation and the contravariant velocities, uc, are defined 

in terms of the metric coefficients as 

21, = 6,u+6yv+6,w (3) 

(The viscous fluxes are omitted here for brevity, 

and can be found in many References, such as Reference 

19.) Following the procedure outlined in Ref. 10 and 

augmented with results shown in References 20 and 21, 

the hybrid LU-SGS scheme can be derived from the 

following. A prototype implicit scheme for solving (1) 

in delta form may be written as 

[I+ At(DtA + D,B + DtC)]Aq = -AtR (4) 

where De, D, and D< are difference operators that 

approximate a,, a,, and 8, and A, B and C are the flux 

jacobians 

(5) 

The PARC codes calculate the residual, R in (4), 

using central differences with added artificial dissipation in the manner due to Jameson (Ref. 23). If the 

flux jacobians in (4) are approximated so that the resulting jacobians can be split into ”+” and ”-” matrices, where the eigenvalues of the ”+” matrices are nonnegative and the eigenvalues of the ”-” matrices are 

non-positive, and then differenced according to their 

sign, (4) may be written as 

[I + At(VcAf + AtA- + V,Bt + A,B- (6) 

+V<Ct + Ace-)] = -AtR 

The choice of conditioning used to form the ap 

proximated jacobians is very important, and following 

the construction used in Reference 21, the approximated jacobians are formed as 

where 

TA,B,C = P~z~IAA,B,c I) (8) 

where P 2 1.0 and AA,B,C are the eigenvalues of 

the flux jacobians. Therefore, using the conditioning in 

(7), the unfactored scheme may be written as 

[I(l+At(TA+TB+Tc))+At(Aj-fl +Bi+l +CGl (9) 

-A+ ,-I - Bt-1 - CLl)]Aq = -AtR 

where (j,k,l) refers to the grid point index. (When 

an index is omitted it is assumed to be at j,k or 1). The 

hybrid LU-SGS scheme is then found by factoring (9) 

into L,D and U operators so that 

LD-lUAq = -AtR (10) 

(11) 

D-’ = I[1 A~(TA TB Tc)] (12) 

(13) 

L = I[1 4- At(rA + TB + Tc)] 

-At(A:-l + Bk+-l + CEl) 

u = I[1 + A~(TA + TB + Tc)] 

+At(Ai_tl+ K+l + CGl) 

All of the calculations performed here are only concerned with obtaining the steady state flowfield, so an 

approximate Newton iteration formulation is formed by 

dividing (10) by At and taking the limit as At approaches infinity, resulting in 

LUAq = -(rA 4- fg rc)R (14) 

L = [I(rA + TB + TC) - - Bt-1 - CLl] (15) 

U = [I(TA + TB + rc) + + Bi+l + CGl] (16) 

The solution of (14) is fully vectorieable on planes 

where j+k+l= constant and is efficient since it requires 

only scalar diagonal inversions. The approach is to first 

solve for an intermediate vector using (15) by sweeping in the direction of planes of increasing j+k+l and 

then solve for Aq using (16) by sweeping in the opposite direction. The efficiency in solving the chemical 

transport equations fully coupled to the Navier-S tokes 

equations has been demonstrated in References 10,11 

and 12 by using the LU-SGS scheme. The only modifications to the implicit operator are to account for the 

2 
15 

ti 

larger flux jacobians and the inclusion of the species 

production/destruction terms implicitly in the first operator, (15). Another advantage of this scheme is that 

when the approximate Newton iteration form is used, 

no choice of time step or Courant number is needed. 

This is very useful since for most problems it is not 

known a priori what time step or Courant mmber is 

needed to gain sufficient Convergence. 

111. LU-SGS and DBW Comparison 

Since the present work has replaced the original 

diagonalized Beam-Warming scheme (hereafter referred 

to as DBW) with this algorithm it was deemed prudent 

to make comparisons regarding the relative efficiencies 

of the two schemes. A limited study was made for inviscid 20 flows, and is summarized in the following sections. In all the comparisons, only the implicit operator 

was changed; the residual calculations were performed 

in the same manner with the same coefficients of artificial dissipation for both schemes. All calculations were 

performed on a CRAY-2 using the cft77 compiler. 

1II.a: M, =S Blunt Body 

The inviscid flow over the windward portion of a 

right circular cylinder at a free stream Mach number of 

3 was calculated using the LU-SGS and DBW schemes. 

A 101 X 101 evenly spaced grid was used, resulting in 

approximately 62 percent of the grid points being in the 

subsonic portion of the flow. Figure 1 shows the computed flowfield Mach number contours which clearly indicates the curved bow shock and the expansion along 

the body surface from the stagnation streamline. For 

the DBW calculations a spatially varying time step was 

used by holding the Courant number (CN) constant at 

all the grid points while p in (8) was linearly varied from 

it’s maximum, Pmaz, to 1.0 over the first 500 iterations 

for the LU-SGS calculations. Figure 2 compares the 

convergence histories of the LU-SGS and DBW calculations on a work basis, where one unit of work is equal 

to the cpu time for one iteration of the DBW scheme, 

Figure clearly shows the increasing rates of convergence 

with increasing CN for the DBW scheme and the relative insensitivity to Pmao of the LU-SGS scheme. At 

CN’s higher than 2, the oscillations present in the beginning stages of the iteration persisted over a longer 

range, retarding the convergence to less than that for 

the CN=2 calculation. This Figure shows that the two 

schemes converged at roughly the same rates, but only 

when the DBW was run at a proper Courant number. 

. for a range of Courant numbers and Pmaz values. The 

J 

For this case, the LU-SGS scheme was roughly 10 percent faster in time per iteration than the DBW scheme. 

1II.b: Transonic Nozzle 

A simplified transonic nozzle flowfield was calculated using the LU-SGS and DBW schemes. The geometry was constructed by matching two cubic polynomials at the throat region with the inlet and exit 

heights determined from 1-D theory to provide an inlet 

Mach number of 0.2 and an exit Mach number of 3.0. 

The total temperature and pressure were held constant 

at the inflow boundary and the static pressure was held 

constant at the exit. A 101 X 51 evenly spaced grid was 

used for this calculation. Figure 3 shows the computed 

Mach number contours and geometry of the nozzle, 

showing that the flow accelerates smoothly through the 

throat and then rapidly becomes supersonic, where an 

excessive ammount of turning causes a weak shock near 

the upper nosrle lip. As in the previous case, spatially 

varying time stepping was used for the DBW scheme 

by holding the Courant number constant at all the grid 

points, while /3 was varied linearly from a maximum to 

1.0 over 500 iterations. Figure 4 compares the LU-SGS 

and DBW calculations for a range of Courant numbers 

(CN) and values of pma,. As before, increasing the CN 

improved the convergence rate of the DBW scheme, but 

a limit was reached where beyond a CN=3.6, the DBW 

scheme diverged. Also, the LU-SGS was fairly insensitive to the p variation. For this case the LU-SGS 

was approximately 20 percent faster in time per iteration than the DBW scheme. The difference in efficiency 

when compared to the first case is attributable to the 

difference in the vector lengths (i.e.- grid sizes). Again, 

both schemes performed similarily, but only when the 

proper Courant number was used for the DBW scheme. 

1II.c: M, =2.5 Supersonic Inlet 

The inviscid flow through a generic supersonic inlet at a freestream Mach number of 2.5 was calculated 

using the two schemes. The computational grid was 

uniformly spaced in both coordinate directions for this 

200 (axially) by 75 (normally) grid point calculation. 

Figure 5 shows the computed pressure contours in the 

inlet, indicating the passing of the ramp shock out of 

the domain and the cowl generated shock attenuation 

and propagation into the inlet, terminating in a large 

sonic region. The DBW calculation used a spatially 

varying time step and the LU-SGS calculation used a 

@ that varied from 1.5 to 1.0 over the first 100 iterations. The comparison of the DBW (with a CN=5.0) 

3 
and the LU-SGS schemes on a work basis is shown in 

Figure 6. For this case, the LU-SGS scheme performed 

approximately 40 percent faster per iteration than the 

DBW scheme. The Figure shows that the two schemes 

converged roughly the same at the early stages of the 

iteration, but the LU-SGS scheme converged faster in 

the later stages. 

1II.d: M, =5.5 Supersonic Inlet 

This case is identical to the previous case, except 

the freestream Mach number was increased to M=3.5. 

The DBW scheme uses the same time stepping and 

Courant number as before, and the LU-SGS uses the 

same variation. Figure 7 shows the computed pressure contours in the inlet, showing that the ramp shock 

does not get ingested and the cowl shock is strong 

enough to to travel throughout the inlet. Figure 8 compares the two schemes convergence behavior for this 

case on a work basis. The Figure shows, as in the previous case, that the two schemes converge similarily 

in the early stages of the iterations while the LU-SGS 

scheme converges faster in the latter stages. As in the 

previous case, the LU-SGS scheme performed approximately 40 percent faster per iteration than the DBW 

scheme. 

1II.e: Summary of Convergence Study 

The results of this study comparing the convergence rates and efficiencies of the LU-SGS and DBW 

schemes show that the LU-SGS scheme is slightly more 

efficient on a work basis than the DBW scheme. Since 

the LU-SGS vectorization is performed on planes in 3D 

(lines in 2D) where j+k+l=constant the efficiency of 

the scheme is dependent upon the grid dimension since 

the vector lengths will be short in the plane corners 

and longer in the center of the planes. Although the 

convergence rates were nearly the same on an iteration 

basis, the DBW scheme required more user effort to 

find the best time step or Courant number, while for 

the cases considered here, the LU-SGS was fairly insensitive to "tuning" of the implicit dissipation. Since one 

often does not know what the "best" time stepping (or 

Courant number) is needed, this study indicates that 

the LU-SGS for approximate Newton iteration scheme 

may be easier to use. 

IV: Corner and Gap Seal Flow Field Study 

Effort is currently underway to investigate the heat 

transfer and aerodynamic loads in and around a scramjet module gap seal configuration. The seal system 

must prevent high temperature and pressure gases from 

leaking through the gaps between the articulating engine panel walls and stationary splitter panels (Ref. 

13). This system is crucial to the design of hypersonic 

airbreathing propulsion systems and presents a significant challenge to structural designers to build a viable 

seal system. To assist the design effort, the present research complements the structural analysis underwdy 

in an attempt to better understand the fluid dynamic 

and heat transfer phenomena in and around the gap 

seal system. 

1V.a: Hypersonic Corner Validation 

To obtain some level of confidence in the modelling of the gap seal flow it is necessary to validate the 

3D Navier-Stokes solver with experimental data taken 

from a class of flows similar to the gap seal flow. Figure 9 shows the simplified geometry of a typical gap 

seal, showing that the geometry is similar to two mutually perpendicular flat plates (corner) aligned with the 

freestream flow with a "groove" situated at the plate 

intersections. Due to this similiarity, the 3D code has 

been validated against the experimental data of hypersonic corner flow taken by Cresci (Ref. 14). These tests 

were conducted at a Mach number of 11.8 over a range 

of Reynolds numbers of 0.15 X loe to 0.5 X 10' per 

foot. The data presented in Ref. 14 is axial pressures 

and heat transfer rates plotted against the hypersonic 

interaction parameter 

for different lateral locations measured from the 

corner. In addition, surface pressures, heat transfer 

rates and skin friction data are available (Ref. 15) as a 

function of lateral distance from the corner at various 

axial locations. Due to the simple geometry of this case, 

other researchers have successfuly used this data to validate space marching (Ref. 16) and Navier-Stokes (Ref. 

17) calculations. This data not only tests the prediction of the mutual interaction of the two plate boundary 

layers upon each other but also the self induced leading edge pressures caused by the boundary layer growth 

close to the leading edges. To test the predictions of the 

2D self induced leading edge shocks and heat transfer 

rates, the 2D and 3D codes were run and compared with 

the experimental data of Ref. 14 (taken at a lateral 

distance sufficiently far from the corner to ensure two 

dimensionality) and with the theoretical pressures deduced by Bertram (Ref. 18). The freestream conditions 

4 
li 

were set at the lower Reynolds number and stagnation 

condtions from the experiment (M,=11.8, Re=0.15 X 

1OB/foot and To =1700 Rankine) while the wall temperature was held fixed at =8.6543. Figures 10 and 

11 show the predicted and measured surface pressures 

and heat transfer rates versus x with the experimental 

and theoretical data (Ref. 18). A 36 X 41 grid was used 

with the grid clustered near the wall using a Robert's 

transformation (Ref. 19) with ,f3 = 1.01 and axially near 

the leading edge with 1.1. Five grid points were extended ahead of the leading edge to aid in correctly capturing the leading edge shock strength. The 3D code 

used a similar grid with 11 XY planes and symettric 

boundary condtions applied at the lateral boundaries. 

As seen in the Figure, the wall pressures were slightly 

over predicted while the heat transfer rates were predicted very well. 

The results from the 2D study were then used to 

guide the 3D corner flow computations. Two grid sizes 

were used in the study (36 X 41 X 41 and 36 X 61 X 

61) for plate lenghts of 8 inches and plate widths of 2.5 

inches at a Reynolds number of 0.15 X 10'. The grids 

were clustered normally using the Robert's transformation in both planes with ,f3 =1.01 and axially near the 

leading edge using p = 1.1 with five points extended 

ahead of the leading edge. The convergence histories 

for the both grids using the LU-SGS scheme are shown 

along with a coarse grid calculation using the DBW 

scheme in Figure 12. The DBW calculation was made 

at a constant Courant number of 2.5, and contrary to 

the 2D inviscid results, the LU-SGS was only 3 percent 

faster in time per iteration than the DBW scheme. This 

Figure shows that for this case the LU-SGS computations seemed to damp the initial transients more quickly 

than the DBW and that the convergence rate at the latter stages of the iterations was faster for the LU-SGS 

scheme. At the inflow plane the flux variables were held 

constant at their freestream values, constant temperature no-slip wall conditions were applied on the plates 

and grid extrapolation of the primitive variables was 

used at the outflow and lateral planes. Figures 13.a,bIc 

and d show the predicted and measured wall heat transfer rates measured laterally from the corner at different axial locations. Figures 14.a through e compare 

the predicted and measured wall pressures at different 

lateral locations plotted versus the hypersonic interaction parameter. Figure 15 shows spanwise surface pressure comparisons while Figure 16 shows the spanwise 

skin friction coefficients. Figures 13 a,b and d show remarkably good comparisons between the predicted and 

measured heat transfer rates, peak levels and peak locations, while Figure 13.c shows the correctly predicted 

level but incorrectly predicted peak location. The pressure comparisons show good agreement near the corner 

(Figures 14.a,b and c) but get worse at farther distances 

from the corner (Figures 14.d and e). The experimcntal data indicates that the plate pressure returns to the 

2D pressure at a distance of approximately 1.5 inches 

from the corner at x = 5.0, but the present calculations 

only show a very gradual lateral reduction in pressure 

from the peak corner values. This behavior is similar to 

the computed results in Ref. 17 in that the near corner 

pressures are computed well, but the spanwise pressure 

relief is underpredicted. 

W.b: Gap Seal and Corner Comparisons 

In an attempt to characterize the gap seal flow, 

comparisons were made between corner and gap seal 

flows at similar Reynolds and Mach numbers. Computations were made using the LU-SGS scheme for a 

Mach number of 10 at an altitude of 100,000 feet of 

the flow fields through a corner and a gap seal configuration. At this altitude the freestream pressure and 

temperature are 23.085 psf and 418.79 Rankine yielding a Reynolds number per foot of 1.0 X 10'. The 

length of the plates and gap seal configurations were 

taken to be 5 inches with widths of 1 inch, resulting in 

a Reynolds number based on plate length of 420,000. 

The gap seal "groove" (shown in Figure 9) was taken 

to be 1/8 by 1/8 inches and is considered to run along 

the entire length of the plates. The wall temperatures 

were held constant at 1200 Fahrenheit. The corner flow 

computation used a 36 X 51 X 51 grid with the same 

clustering as the previous corner case with 5 points extended ahead of the leading edge to help capture the 

leading edge shock strengths. The gap seal computation used a rectangular grid with clustering centered 

along the gap seal faces and toward the pIate walls as 

well as axial clustering near the leading edge with 5 

leading edge extension points. This computation used 

a base grid of 36 X 61 X 61 of which approximately 75 

percent of the grid points were active in the calculation. 

Figure 17 compares the convergence rates of the corner 

and gap seal computations using the LU-SGS scheme, 

showing that both cases converged monotonically and 

rapidly towards steady st at e. 

Figures 18.aIb and c compare the wall heat transfer coefficients along the gap seal floor and lateral wall 

with the corner only heat transfer rates for different 

axial locations. Examination of the Figures shows the 

peak heat transfer to the gap seal floor is reduced by 

approximately 4 times while the lateral wall heat transfer levels are approximately the same as the corner only 

levels. This reduction in heat transfer in the gap seal 

is due the the gap region filling up with low momentum and energy fluid caused by the viscous effects in 

5 
the gap. Figure 19 shows axial velocity contours near 

the end of the seal configuration in a plane perpendicular to the free stream velocity vector. This Figure 

shows the gap region filled with low velocity fluid and 

the existence of a vortical flow situated a short distance 

laterally from the gap seal. This vortical flow is caused 

by the rapid expelling of the flow from the gap seal 

to the outer lateral plate. Future computations will 

investigate whether this vortex will act as a barrier to 

keep the energetic outer flow from entering the gap seal 

region. 

V: Conclusions 

A hybrid LU-SGS scheme has been successfully implemented in the PARC2D/3D Navier-Stokes solvers 

and limited comparisons to the original algorithm in 

the code (diagonalized Beam-Warming ADI) have been 

made. The convergence comparisons indicated that the 

hybrid LU-SGS scheme for approximate Newton iteration is slightly more efficient on a work basis than the 

DBW scheme and requires less user effort to achieve 

an acceptable convergence rate. This is most likely due 

to a combination of the inherent implicit dissipation of 

the LU-SGS scheme and the approximate Newton iteration formulation, so that no time step specification is 

required. The scheme was demonstrated by validating 

the code against a M, =11.8 corner flow experiment 

and has been used to investigate the similarities between gap seal and corner flow geometries. The validation effort indicated that the code predicted the corner 

case heat transfer rates, peaks and peak locations very 

well while it predicted the wall pressures very well near 

the corner but poorly away from the corner. The gap 

seal/corner flow field comparisons indicated that the 

heat transfer rates to the floor of the gap seal were signifigantly reduced when compared to the corresponding corner only flow and that the lateral heat transfer peaks were displaced laterally by approximately the 

gap width distance. Future work will investigate the 

gap seal/corner flows in the limit of large and small 

gap aspect ratios as well as the inclusion of inert gas 

”blow by” cooling through the gap seal floor. 

References 

Pulliam, T.H., and Steger, J.L. “Implicit FiniteDifference Simulations of Three Dimensional Compressible Flow.” AIAA Journal, Vol. 18, No. 2, 

Feb. 1980, pp. 159-167. 

Beam, R. and Warming, R.F. “An Implicit FiniteDifference Algorithm for Hyperbolic Systems in 

Conservation-Law Form.” Journal of Computational Physics, Vol. 22, No. 1, Sept. 1976, pp. 

(3) Pulliam, T.H. “Euler and Thin Layer NavierStokes Codes: ARC2D1ARC3D.” Notes for Computational Fluid Dynamics User’s Workshop, The 

University of Tennessee Space Institute, Tullahoma, Tennessee, (UTSI Publication E02-4005- 

023-84), March 12-16, 1984, pp. 15.1-15.85. 

(4) Cooper, G.K. “The PARC Code: Theory and Usage.” AEDC-TR- 87-24, October, 1987. 

(5) Jameson, A. and Turkel, E. “Implicit Schemes and 

LU Decompositions.” Mathematics of Computation, Vol. 37, No. 156, 1981, pp. 385-397. 

(6) Jameson, A. and Yoon, S., ”Lower-Upper Implicit 

Schemes with Multiple Grids for the Euler Equations.” AIAA Journal, Vol. 25, July 1987, pp. 929- 

935. 

(7) Jameson, A. and Yoon, S., ”Multigrid Solution 

of the Euler Equations using Implicit Schemes.”, 

AIAA Journal, Vol. 24, Nov. 1986, pp. 1737-1743. 

(8) Yoon, S. and Jameson, A., ”Lower-Upper Implicit 

Scheme for High-speed Inlet Analysis.”, AIAA 

Journal, Vol. 25, Aug. 1987, pp 1052-1053. 

(9) Yoon, S. and Jameson, A., “An LU-SSOR Scheme 

for the Euler and Navier-Stokes Equations.”, jAIAA paper 87-0600, January, 1987. 

(10) Shuen, J.S. and Yoon, S. “Numerical Study of 

Chemically Reacting Flows Using an LU Scheme.” 

AIAA paper 88-0436, January, 1988. 

(11) Yu, S.T., Tsai, Y.L. and Shuen, J.S. ”Three87-110. 

Dimensional Calculation of Supersonic Reacting 

Flows using an LU Scheme.” AIAA paper 89-0391, 

January, 1989. 

Park, C. and Yoon, S. ”Calculation of Real-Gas 

Effects on Blunt-Body Trim Angles.”, AIAA paper 

89-0685, Jan. 1989. 

Steinetz, B.M., DellaCorte, C. and Sirocky, P. ”On 

the Development of Hypersonic Engine Seals.” 

NASA TP-2854, Dec. 1988. 

Cresci, R.J. “Hypersonic Flow Along Two Intersecting Planes.”, AFOSR 66-0500, March 1966. 

Cresci, R.J., Rubin, S.G., Nardo, C.T. and Lin, 

T.C. ”Hypersonic Interaction along a Rectangular 

Corner.” AIAA Journal, Vol. 7, No. 12, p 2241 

2246. 

Buggeln, R.C., Kim, Y.N. and McDonald, H. 

”Computation of Multi-Dimensional Viscous Supersonic Flow.”, NASA CR-4021, Oct. 1986. 

Li, C.P. ”A Numerical Study of the Axial Corner 

Flow.”, AIAA paper 78-1146, July, 1978. 

Bertram, M.H. and Blackstock, T.A. ”Some Simple Solutions to the Problem of Predicting Bound6 
lIary - Layer Self-Induced Pressures.", NASA TND-798, April 1961. 

(19) Anderson, D.A., Tannehill, J.C. and Pletcher, 

R.H. "Computational Fluid Mechanics and Heat 

Transfer." McGraw-Hill Book Company, 1984. 

(20) Yoon, S. and Kwak, D. "Artificial Dissipation 

Models for Hypersonic Internal Flows." AIAA paper 88-3277, July, 1988. 

(21) Yoon, S. and Kwak, D. "Artificial Dissipation 

Models for Hypersonic External Flows." AIAA paper 88-3708, July, 1988. 

(22) Jameson, A., Schmidt, W. and Turkel, E. "Numerical Solutions of the Euler gquations by Finit Volume Methods Using Runge-Kutta Time-Stepping 

Schemes." AIAA paper 81-1259, 1981. 

7 
0 

-2 

-4 

h 

N 

_1 

v 

O - -6 01 

-1 0 

-0 

- 10 

-12 

Figure 1 M, = 3 Blunt Body: Mach Contours 

M=3 Blunt. Body 

LU-SGS 6 max-1 0 

_________ LU-SCS 6-mar-15 

________ LU-SCS ,E-mar-25 _____ DOY CN-IO 

- - - - DOW CN-05 

- - - DDY CN-20 

. -. 

'. 

I I 1 

1000 2000 3000 4000 

Work 

Figure 2 M,, = 3 Blunt Body: Convergence C'omparisons 

Figure 3 Transonic Norrle Case: Mach Contours 

Transonic Nozzle 

1.U-SCS b-mar-l 0 LU-SCS #-man-2 D 

0 

________ onw CN-20 _____ DllW CN-25 _ - _ - DUY CN-325 - _ - - DIJY CN-36 

-* t 

.. 0 400 800 1200 1600 2000 

Work 

Figure 4 Transonic Norrle Ce: Convergence Comparisons 
. 

Figure 5 M, = 2.5 Supersonic Inlet: Pressure Contours 

-' 

- 10 

M=2.S Supersonic Inlet 

LU-SCS: f?-max= 1.5 

DEW: CN=5.0 

0 400 800 1200 1600 2000 

Work 

Figure 6 M, = 2.5 S~personic Inlet: Convergence Comparisons 

M=3.5 Supersonic Inlet 

LU-SGS: f?-rnax=l.5 

DEW: CN=5.0 

-4 - 

-0 - h 

r4 

A 

0 

v 

7 

: -8- -I 

-10 - 

-12 - 

-14' , ! I I I I I I 

0 400 800 1200 1600 2000 

Work 

Figure 8 M, = 3.5 Supersonic Inlet: Convergence CompariSOnS 

Figure 7 M, = 3.5 Supersonic Inlet: Pressure Contours 

9 

Figure 9 Simplified Gap Sed Geometry 
35 

8 

11 

a 

\ 

A Experiment: Re = 0.125E6 

c] Experiment: Re = O300E6 

G Experiment: Re = 0.500E6 

00 

2 0 IO 

X 

Figure 10 Two-Dimensional Self Induced Lesding Edge Plate 

Pressures 

lo-' 

10-2 

c 

I/) 

10-3 

Heat Transfer: 2-D Data 

- ZD 31 x 41 5 pt I. Cxt 

__ ZD 31 x 41 -1 5 pt le ext 

p, Experiment Re = 0 125E6 0 Experiment Re = O300E6 

0 Experiment Re = 0 500E6 // 

A 

a A ,// 

lo-' 

X 

Figure 11 Two-Dimensional Heat "zansfer Rates 

10 

Hypersonic Corner Validation 

Figure 12 ThreoDhensiond Convergence Cornparisow 

lo-' 

I 

0 

lo-! 

Heat Transfer: x=8.3 

OExperiment: Re - O.15E6 

- 31 X 41 X 41 Grid 

________ 31 X 61 X 61 Grid 

00 05 I .o I5 20 25 

Z( in c hes) 

Figure 13 (a)Laterd Variation of Heat Transfer: x = 8.3 
. 

0.004 

I 

0 0.003 

0 001 

0 00: 

0 004 

I 

0 0 003 

0 002 

0 001 

- 

r 

1-{eat Transfer: x=i'.O 

3.5 

3.0 

25- 

8 

q 2.0 

a 

1.5 

__ 31 X 41 X 41 Grid 

_______ 31 X 81 X 61 Grid 

- 

- 

- 

- 

I\ 0 

! ', 

0 004 

I 

0 0 003 

0 I 2 3 4 5 0 

Z (Inches) 

- 

0 

Figure 13 (b)Lateral Variation of Heat Transfer: x = 7.0 

Heat Transfer: x=5.7 0 006 

- 31 X 41 X 41 Grid 

_______ 31 X 61 X 61 Grid 

Ooo5 1 

0 I 2 3 4 5 e 

Z (Inches) 

Figure 13 (c)Lateral Variation of Heat Transfer: x = 5.7 

Heat Transfer: x=4.6 

0°06....,, 

_______ 31 X 61 X 61 Grid 0.005 

0 002 

0 

0 

0 I 2 3 4 5 6 

Z (Inches) 

Figure 13 (d)Lateral Variation of Heat Transfer: x = 4.6 

Pressure: Z=O. 125 Inches 4.0 t 

0 

/' 

0 

ooo * 

00 

__ 31 X 81 X 81 Grid 

- 31 X 41 X 41 Grid 

Experiment: Re = 0.125E6 

Experimenk Re = 0.300E6 

Experiment: Re = 0.500E6 

X 

Figure 14 (a)hid 'Variation of Wall Pressure: k0.125 

inches 

11 
Pressure: Z=0.25 Inches 40 

3.5 

3.0 

- 

- 

35- 

30- /" 25- 

8 

a 

a 

, 2.0 

.ISA0 

0 

0 

0 

0 

__ 31 X 61 X 61 Grid 

- 31 X 41 X 41 Grid 

A Experiment: Re = 0.125E6 

0 Experiment. Re = 0.300E6 

o Experiment. Re = 0.500E6 

- 

0 2 4 8 8 

X 

25 

E 

20- 

Q 

15- 

10- 

05 

Figure 14 (b)Arial Variation of Wall Pressure: 2=0.250 

inches 

- 

Pressure: 1,=0.3?5 Inches 

00 0 3 IO 

40 

35- 

30- 

00 

OA 

a 

2.5 

8 

a 

a 

, 2.0 

I5 

10- 

05- 

__ 31 X 61 X 61 Grid 

- 31 X 41 X 41 Grid 

A Experiment: Re = 0.125E6 

0 Experiment: Re = 0.300E6 

0 Experiment: Re = O500E6 

- 

- 

8 

20- 

a 

15- 

Figure 14 (c)Arial Variation of Wall Pressure: Z=0.375 

inches 

0 

A 

0 

Pressure: 2=0.500 Inches 

1.0 

0.5 

__ 31 X 61 X 61 Grid 

- 31 X 41 X 41 Grid 

A Experiment: Re - 0.125E6 

0 Experiment: Re = 0.300E6 

- 

- 

00 

__ 31 X 61 X 61 Grid 

- 31 X 41 X 41 Grid 

o Experiment: Re = O500E6 

A Experiment: Re - 0.125E6 

0 Experiment: Re = 0.300E6 051 

0 Experiment: Re - O500E6 

00 

x 

Figure 14 (d)Gd Variation of Wall Pressure: Z=0.500 

inches 

Pressure: Z= 1.000 Inches 40 

0 2.5 1 

Figure 14 (e)Adal Variation of Wall Pressure: Z=1.000 

inches 

12 
35- 

Spanwise Pressure: x=5.0 40 

0 Experiment: Re = 0.15E6 

- 31 X 41 X 41 Grid 

31 X 61 X 81 Grld 

I 

20- 

15- 

I 

8x 0 

.I 

00 05 10 15 20 25 

Z(inches) 

Figure 15 Lateral Pressure Distribution at x = 5.0 

Skin Friction: x=5.0 

Experiment: Re = 0.15E6 

- 31 X 41 X 41 Grid 

____ 31 X 81 X I31 Grid 

Z( inches) 

Hypersonic Corner/Gap Seal 

__ LU-SCS: Cap Seal Case 

- - - LU-SGS: Corner Case 

\ 

Figure 17 Convergence History: Corner and Gap Seal Comparison 

Heat Transfer: Y=O.X=2 inches 

- Corner Configuralion 

____ Cap-Seal Configuration 

10-3 

I 

0 

10-4 

I I 

0.0 0.4 0.8 

Y(inches) 
Heat Transfer: Y=O,X=4 inches 

- Corner Configuration 

____ Cap-Seal Configuration 

t 

t 

10-3 

I 

0 

10-4 

I I I I I 

0.0 0.4 0.8 

Y(inches1 

Figure 18 (b)Lateral Variation of Heat Transfer Rates, Corner/Gap seal: X=4.0 inches 

Heat Transfer: Y=O,X=5 inches 

____ Cap-Seal Configuration 

I 

0 

10-4 

10-5 

0.0 0.4 0.8 

Yfinches) 

Figure 18 (c)Laterd Variation of Heat Transfer Rates, Corner/Gsp seal: X=5.0 inches 

14 

Figure 19 Gap Seal Case: kial Velocity Component Contours at X=4.7 inches. 
National Aeronautics and Space Administration 

AIAA-89-2669 

4. Title and Subtitle 

Report Documentation Page 

5. Report Date 

NASA TM- 102 138 1. Report No. 

High Speed Corner and Gap-Seal Computations Using an LU-SGS Scheme 

7. Author@) 

I 2. Government Accession No. 

6. Performing Organization Code 

8. Performing Organization Report No. 

I 3. Recipient's Catalog No. 

9. Performing Organization Name and Address 

National Aeronautics and Space Administration 

Lewis Research Center 

Cleveland, Ohio 44135-3191 

505-80-2 1 

11. Contract or Grant No. 

13. Type of Report and Period Covered 

12. Sponsoring Agency Name and Address 

William J. Coirier 

Technical Memorandum 

E-4936 

10. Work Unit No. 

17. Key Words (Suggested by Author(s)) 

Three dimensional Navier-Stokes 

High speed flow 

LU-SGS algorithm 

Cornedgap-seal calculations 

18. Distribution Statement 

Unclassified -Unlimited 

Subject Category 34 

19. Security Classif. (of this report) 20. Security Classif. (of this page) 

Unclassified Unclassified 

National Aeronautics and Space Administration 

Washington, D.C. 20546-0001 

21. No of pages 22. Price' 

17 A03 

14. Sponsoring Agency Code 

15. Supplementary Notes 

Prepared for the 25th Joint Propulsion Conference cosponsored by the AIAA, ASME, SAE, and ASEE, 

Monterey, California, July 10-12, 1989. 

16. Abstract 

The hybrid LU-SGS (Lower-Upper Symmetric Gauss-Seidel) algorithm has been added to a widely used series of 

2D/3D Euler/Navier-Stokes solvers and has been demonstrated for a particular class of high-speed flows. A limited 

study has been conducted to compare the hybrid LU-SGS for approximate Newton iteration and diagonalized 

Beam-Warming (DBW) schemes on a work and convergence history basis. This study showed that for the cases 

considered, the hybrid LU-SGS algorithm is more efficient and easier to implement than the DBW scheme 

originally present in the code. The code has been validated for the hypersonic flow through two mutually perpendicular flat plates and then used to investigate the flow field in and around a simplified scramjet module gap seal 

configuration. Due to the similarities, the gap seal flow was compared to hypersonic comer flow at the same 

freestream conditions and Reynolds number.