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Developing an Accurate and Efficient Method
for Compressible Flow Simulations
- An Example of CFD in Aeronautics -
K. Fujii
The Institute of Space and Astronautical Science
Sagamihara, Japan
Abstract
Capability of the current CFD technology in the
aeronautical society is discussed with one of the rep-
resentative Navier-Stokes codes in Japan, as an exam-
ple. The code is named LANS3D and was developed
for the numerical simulation of high-Reynolds number
compressible flows. The algorithm used in this code
is briefly described First and then the applications for
aircraft simulation, cortical flow simulation and spacer
plane simulation are presented. These example show
how the code has been improved to satisfy two impor-
ta~t requirements of the computational fluid dynamics
(CFD) codes; efficiency and accuracy. Importance of
developing a supporting system on graphic worksta-
tions is kin ally discussed.
1. Introduction
Computational Fluid Dynamics (CFD) is begin-
ning to play an important role in the aeronautical
industry all over the world. People now realize that
CFD can be a new and effective design tool with the
aid of supercomputer. At the same time, CFD is be-
coming an important scientific tool for the fluid dy-
namics research. As high-speed supercomputers with
large memory become popular, the research now is
focused on the Navier-Stokes simulations for under-
standing fluid physics as well as for the future use as
a design tool in engineering.
In the application of the CFD to the aeronautical
problems, there are two important remarks to note.
First, representative Reynolds number is large com-
pared to many of the other CFD applications. Even
though viscous equations are solved, contribution of
viscous terms is small in most of the flow field. Thus,
computational algorithms that have been developed
for the Euler equations are mainly used. Physically,
viscous effect is confined to the thin layers near the
body surface and to evaluate viscous effect properly
with the Navier-Stokes simulations, grid spacing near
the body surface should be very small. For instance,
typical By for the transonic flow simulation is o(10~S-
10-6~. Thus, explicit time integration is not appropri-
ate and implicit time integration is indispensable.
At the same time, adding high-order artificial dissipa-
tion is required to keep the computation stable espe-
cially when the central difference is used for convective
terms.
Second, accurate prediction of aerodynamic co-
efficients is required. A few percent of the drag re-
duction is important for aircraft design, and the ef-
fect of Reynolds number and other parameters should
be accurately evaluated. Transition and turbulence
modeling are also important. Simulation results for
lower Reynolds-number flows can not be extrapolated
to high Reynolds numbers. In some applications such
as leading-edge separation flow field over a delta wing
as shown later, strong separation vortices become key
factor for the aerodynamic coefficients. Since the
Reynolds number is high, flow field away from the
body surface is considered to be rotational inviscid
in such applications. However, as is the case of an-
other applications, viscous effect near the body surface
should be properly evaluated because that is the key
factor to determine the location and the strength of
the separation vortices.
The present author has been engaged in devel-
oping an efficient and accurate method for the simu-
lation of complicated flows by solving the compress-
ible Navier-Stokes equations. In the initial stage of
the development, main effort was laid on improving
the efficiency. With the progress of supercomputer
capability, recent effort tends to be focused on the
improvement of the accuracy. In the present paper,
the method and the developed Navier-Stokes computer
code named LANS3D ~ PLANS' stands for the LU-ADI
Navier-Stokes code) is briefly described first. Some of
the application examples are then presented. First ex-
ample is the transonic flow over transport aircraft con-
figuration. Second example is the simulation of vortex
breakdown over a strake-delta wing conducted when
the author was a research associate at NASA Ames
Research Center. In this simulation, the importance
of the grid resolution and the accuracy of the numeri-
cal algorithm was realized and the computer code was
modified. Finally, recent application to the spaceplane
configuration is presented. The spacecraft design is so
critical that the simulation should be accurate for the
s
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complicated flow field. These examples will show how
the accuracy and efficiency of the code have been im-
proved and what problems are left to be improved.
2. Governing Equations and Numerical Algo-
r~thm
Compressible Navier-Stokes Equations
The basic equations under consideration are the
unsteady Navier-Stokes equations written for a body-
fitted coordinate system ((, 71, ()
TO + 06E + 0nF + 0
sipation term. Another modification is necessary in
the (-direction to evaluate implicit viscous terms. Ad-
ditional details of the derivation of the LU-ADI algo-
rithm are given in Ref. 1. In the right-hand side, ex-
plicit artificial dissipation is necessary. the dissipation
model was changed time to time, but currently, non- where
linear second-order and fourth-order mixed dissipation
model is implemented. The idea comes from the TVD
scheme and the model is called simplified TVD-type
dissipation. Again, the details are found in Ref. 1.
and
High-order upwind differencing has become pop-
ular in recently-developed Euler methods for com-
pressible inviscid flow simulations. This feature has
been extended in the straightforward manner for the
evaluation of convective terms of Navier-Stokes com-
putations. Lately, the matrix form of the dissipation
terms implicitly introduced by upwind methods was
studded t8,9] and, it was shown that such terms in the
upwind schemes such as Roe's flux difference splitting
became small in the viscous layers. Thus the use of
the proper upwind could improve the accuracy in the
viscous layer as well as make the discontinuities sharp.
Convective terms in the right-hand side of the
code was recently modified and the code includes the
option to choose either original central differencing
with artificial dissipation or flux difference splitting
with Roe's averaget10~. Higher-order extension of
flux difference splitting using the MUSCL approach
is found in Ref. 8, but is briefly described again.
When the convective terms are differenced with the
flux-difference splitting of Roe, the spatial derivatives
are written in the conservative form as a flux balance.
For instance in the (-direction,
~ ~ ~
ABED (Ej+~/2—Ej-il2) use
The numerical flux Ej+~/2 can be written as the
solution to an approximate Riemann problem and the
necessary metric terms are evaluated at the cell inter-
faces j+1/2.
Ej+il2 = 2 tE(Q~ ~ + E(QR)—AIR—I )]j+~/2' (6)
where E is the flux vector and A is the corre-
sponding Jacobian matrix computed using the Roe's
average state -. QL and QR are the state vari-
ables to the left and right of the half-cell interface.
These state variables are determined from the locally
one-dimensional non-oscillatory interpolations called
MUSCL approach. Primitive variables qua, U. V, W. p]T
are used for that purpose, and high-order accurate
monotone differencing is given by a one-parameter '` .
(q~)j+~/2 = qj + 4~1 —asps_ + (1 ~ KS)~+]j
(qR)j+~/2 = qj+l — 4~1 —asks+ ~ (1 + ~s)~_]j+l,
(~+)j = qj+l - qj, (~_)j = qj - qj_,
_ 2~_+6
S —
(,\+)2 + (a\ )2 + ~
(7)
s is the Van Albada's limiter and ~ is a small constant
to prevent zero division. For all the results here, third-
order accuracy corresponding to arc = 1/3 is used. Near
the boundary, the MUSCL interpolation goes back to
the first-order.
The flux evaluation for the central difference
method is written in the same form as Eq.~6) con-
sidering the artificial dissipation into account.
Ej+il2 = 2 fEj + Ej+,
- (~1 - ~52 - ~64V6~+~/2(Qj+i - Qj)]
where ~ is a kind of limiter function.
The first two terms Ej and Ej+i correspond to
the second-order central difference and the last term
corresponds to the dissipation model. The same is true
for Eq.~6~. The first two terms construct central dif-
ference and the last term is dissipative term implicitly
added by upwinding. Without limiter functions, first
two terms become second-order central difference for
first-order upwind method, and become fourth-order
central difference for third-order upwind method. The
point here is the form of the dissipation terms. In the
artificial dissipation model used with the central differ-
ence, magnitude of the coefficient is the same for all the
equations. Suppose we decompose the Euler equations
into a set of independent equations for the waves with
the characteristic speed u, u+c, u-c, the magnitude of
coefficient should be chosen to be large enough for any
of the waves. In the flux evaluation of the Roe's flux
difference splitting, this coefficient is in the form of ma-
trix A. If you rewrite this dissipation term, it is recog-
nized that dissipation depends on the strength of each
wave and the magnitude is automatically determined
for each wave by upwinding. Thus, artificial dissipa-
tion included in the high-resolution upwind methods
could be smaller than the dissipation used with the
central difference methods. It should also be noted
that Vatsa et al. demonstrated this dissipation terms
become automatically small in the viscous layers near
the body surfaced. On the other hand, some kind
of scaling to decrease the dissipation is necessary near
the body surface for the central difference approach.
7
3. Application Examples
3.1 Aircraft Simulation
- Toward Efficient Code-
Following the several trial computations to eval-
uate the code capability/4], the developed 3-D Navier-
Stokes code based on the LU-ADI algorithm was first
used for the simulation of an isolated wing named W-
14~23. This wing geometry was developed as a wing for
the transonic transport aircraft by Mitsubishi Heavy
Industries. The purpose of this study was to compare
the computed result with the experiment, and thus,
several computations were carried out for the angles
of attack varying from 0° to 7.5°. The Mach number
was fixed to be the design Mach number 0.82. Total
number of the grid points was 200,000. The angles
of attack assigned for the computations were slightly
modified from those of the experiment by MHI design-
ers, and the computers: Fujii and Obayashi, were not
informed of the experimental data in advance. The
flow was assumed to be fully turbulent and the so-
called Baldwin and Lomax turbulence model[11] was
used. The comparison of the surface Cp distribution
with the experiment for 2.46- deg. case is shown in
Fig. 1. The computed result shows the same tendency
as the experiment. The data agreement for both up-
per and lower surface is good at every station and, in
addition the pressure at the trailing edge is well pre-
dicted, which is important for the design process of a
new wing.
In early 1986, the code was modified for the sim-
ulation of wing-fuselage combinationt12~. Again, the
used geometry named W-18 was designed by the MHI
and seven angle of attacks (from 0.0° to 6.0°) that were
shifted a little from the experiment by the empirical
way were suggested to the computers who were not
informed of the experimental data. Figure 2 shows
the grid for the upper half volume of the computa-
tional domain. Since not only the viscous layers over
the wing but also the viscous layers over the fuselage
should be considered, the thin-layer approximation is
now adopted for two directions and thus, the basic
equations (1) are modified. The turbulence model was
also modified near the wing-fuselage junction. The
computational grid was generated by Takanashi us-
ing his modified conformal mapping technique, and
the number of the grid points was increased to about
800,000.
Figures 3a-3b show the surface pressure contour
plots for the 6.0-de" case. There occurs a large shock-
induced separation near the root section, and shock
wave exists even at the fuselage surface. The shock
wave has a strong spanwise curvature as in the case of
isolated wing, but remarkable difference exists near the
wing-fuselage junction. For an isolated wing, the shock
wave is always perpendicular to the center symmetry
plane. On the other hand, the shock wave curves for-
ward at the root section because of the large recircu-
lating region induced by the wing-fuselage interaction.
Computed off-body particle path traces are presented
in Fig. 4. The strong spiral is created and moved out-
board behind the shock wave over the wing. Near the
wing-fuselage junction, there exists a large recirculat-
ing region, where the vertical flow resembles the coiled
spring bent 90°. The vortex axis is perpendicular to
both the wing and the fuselage surface. The Cp dis-
tributions over the wing and the fuselage surfaces for
4.0-de" case are compared with the experiment in Figs.
5a and 5b respectively. The overall agreement is fairly
good, not only for the wing but also for the fuselage.
Note that the discrepancy in the pressure level at the
tip section can be explained by the elastic deformation
of the test model in the experiment. It is possible that
the tip section of the steel model was twisted by aero-
dynamic forces, since the aspect ratio of the wing is
very high. The discrepancy in the inboard region may
be due to the poor turbulence model. The details of
the computations are found in Ref. 12.
In 1988, along with the parametric study for
several wing-fuselage combinations, the flow simula-
tion over an almost complete aircraft was tried by
Takanashi et alt13~. In addition to the wing and fuse-
lage, vertical and horizontal tails were added. The
centerline symmetry plane was modified to include
the viscous effect of the vertical tail. The horizon-
tal tail wing were sandwiched between two chordwise
grid lines and the effect of viscous layers over the tail
surface was introduced there. Only the coarse grid
simulation using about 600,000 grid points was carried
out. The surface grid distribution and the computed
surface pressure contours are plotted in Fig. 6 and 7,
respectively. The Mach number is 0.6, the angle of
attack is 0 deg and the Reynolds number is 3.5x106
for this case. Reference 13 shows the details of the
computation.
All the computations above were carried out
within the specified time frame, as they should be for
the design purpose. The reliable solutions, thus, had
to be obtained within reasonable computer time and
the efficiency was the matter of concern. Initially, wing
simulation required roughly two hours on Fujitsu VP
supercomputer, and now requires 20 min. to 40 min.
of computer time. Wing-fuselage simulation could be
carried out within 2 to 3 hours. However, for the de-
sign purpose, twice as much should be considered for
safety. Checking the effect of parameters such as grid
spacing or artificial dissipation model is also necessary.
3.2 Vortex Breakdown Simulation
- Toward Accurate Code-
The flow over aircraft and missiles at moderate
8
to high angles of attack is characterized by the pres-
ence of large spiral vortices on the leeward side of the
body. These separation vortices induce low pressure
on the body upper surface, and this low pressure is
the predominant factor of the resulting aerodynamic
characteristic of the body. Research on such flow is
of great importance practically as well as physically
because understanding of the separated and vertical
flow fields may lead to the control of vortex behav-
ior and eventually to the enhancement of flight vehicle
performance.
To understand vertical flow field structure over
a strake-delta wing configuration, a series of computa-
tions was conducted when the author was a NRC re-
search associate at NASA Ames Research Centeri14~.
Two types of vortex breakdown - bubble and spiral
shaped - were successfully simulated and the differ-
ence of the flow structure of each breakdown was char-
acterized. However, the result at the same time in-
dicated still better grid resolution and reducing the
artificial dissipation are critically important for an ac-
curate simulation of vertical flow field. In all the air-
craft simulations shown above, central difference was
used for the convective terms. High-resolution upwind
difference might have less dissipation as noted in the
section 2, and was implemented as an option. Here,
some of the results t15] are presented and compared
with the result by central difference computation (note
that central difference always requires some artificial
dissipation model with it). The flow field is the sub-
sonic flow over a strake-delta wing. The freestream
Mach number is 0.3, and the Reynolds number based
on the root chord is 1.3x106 in the following computa-
tions.
In the first example, the total number of grid
points is about 850,000; 119 points in the chordwise
direction, 101 point circumferentially and 71 points
in the normal direction. Details of the grid genera-
tion and the grid distribution can be found in Ref.
14. Figure 8 shows the overall view of the spanwise
total pressure contour plots at several chordwise sta-
tions at the angle of attack 12 degrees. The contours
are plotted at 35% to 95 % chordwise stations with 10
% increase. At this angle of attack, there exist two
vortices over the upper surface of the wing; one ema-
nating from the strake leading edge and the other from
the main-wing leading edge. These two vortices merge
together over the main-wing surface because of the mu-
tual interaction. Both results indicate the existence of
two vortices over the wing surface and their interac-
tion. It seems that the merging of the two vortices
occurs more downstream in the upwind solution. The
corresponding particle path traces showing the vortex
trajectories are shown in Fig. 9 for the upwind result.
The computed vortex trajectories are presented in Fig.
9. Also presented are the experiment and the result
by the central difference computation. It is clear that
merging of two vortices occurs downstream in the up-
wind result, but still upstream than the experiment at
the same Reynolds number.
The same computation was carried out using
smaller number of grid points (about 120,000 in to-
tal). Compared to the previous grid, the number of
the grid points are decreased in all the directions. Fig-
ures 10a and 10b represent the total pressure contour
plots obtained by the upwind and central difference
computations, respectively. The contours are again
plotted at 35% to 95 % chordwise stations with 10 %
increase. The upwind result shown in Fig. 10a indi-
cates the existence of two vortices and their merging
process although the inner vortex is not as distinct as
the fine grid result. On the other hand, the central
difference result in Fig. 10b shows only one flattened
vortex instead of two vortices.
Final example is the result at 30 degrees. The
previous study in Ref. 15 showed that vortex break-
down takes place near the trailing edge both in the ex-
periment and in the computational results on the fine
grid. Here medium grid (previously mentioned grid
of about 120,000 points) computations are carried out
both with the central di~erencing and the upwind dif-
ferencing. The computed total pressure contour plots
are presented in Figs. Ha and lib. An abrupt in-
crease of the vortex-core is observed near the trailing
edge in the upwind result shown in Fig. lla. This in-
dicates that the vortex has undergone breakdown. In
fact, the plot of the streamwise velocity (although not
shown here) showed that there exists the reverse flow
region near the trailing edge. The central difference
result shown in Fig. lib, on the other hand, does not
show such a sudden change. Again, the resolution is
enhanced by the use of the present upwind scheme at
least on the grid used here (although a slight increase
of the number of grid points may introduce breakdown
phenomenon also in the central difference result).
It is recognized from these results that the
present upwind scheme has better resolution than the
conventional central difference scheme on the same
grid although grid resolution itself is, of course, an
important factor for an accurate flow simulation. Up-
wind scheme is more "vortex-preserving" than central
differencing scheme ~ with added dissipation ~ since it
has a lower level of dissipation. In the present upwind
scheme where the dissipation terms are constructed
in the matrix form, each characteristic wave has its
own minimum dissipation. On the other hand, cen-
tral difference scheme where dissipation terms are con-
structed in the scalar form, requires amount of dissi-
pation which is large enough for all the waves.
Of course, solutions of both central and upwind
difference schemes should converge to the solution of
9
the original partial differential equations as the compu-
tational grid is refined. The point here is that the accu-
racy estimation based on the idea of the Taylor expan-
sion is important but not good enough for the system
of nonlinear equations. What we need in numerical
schemes is the better representation of the properties
of original partial differential equations and, in that
sense, upwind difference scheme shows better result
than that of the central difference scheme for the grid
distributions feasible under the memory restriction of
the current supercomputers.
~~ ~ ~ A A ~— r
3.3 Spaceplane Simulation
- Accuracy and Efficiency Required-
Since February 1985, when U.S. President Rea-
gan announced the NASP project, there occurred a
strong acceleration on the research of orbiting or hy-
personic flight vehicle. The CFD is one of the im-
portant areas necessary for the development, and the
computations have been carried out for transonic to
hypersonic flow regime for the spaceplane configura-
tion proposed for the research at the NAL in Japan.
Following the comparison of the original central
di~erencing and the new upwind differencing that is
described abovet15], a series of simulations were con-
ducted. One of the result is presented in Fig. 12
where the computed surface density contours at the
Mach number 1.5, the Reynolds number 4 x106, and
the angle of attack 15 ° are plotted. The contours on
the wing surface and the fuselage surface indicate the
vertical flow generated above. The surface pressure
distributions at one cross section is compared with the
experiment in Fig. 13. The disagreement in the lower
surface Cp at the final station is due to the model sup-
port, and otherwise pretty good agreement was ob-
tained with the experiment. The details of the series
of computations and the effect of each element such as
tail fins will appear in Ref. 16.
3.4 Strategy for Complex Configurations
One of the important items to be solved in or-
der to apply the CFD to practical problems is how
to handle complex configurations. Although the so-
phisticated grid generation programs have been de-
veloped by many researchers, the application of the
flow solution codes is still restricted to relatively sim-
ple configurations, and a method is needed to make
the code applicable to the flow simulation over truly
complex configurations (say, complete aircraft with na-
celles, engines and so on). Another important item is
the problem of the grid resolution. As is seen in the
result shown above, even with the high-resolution up-
wind scheme, total number of grid points necessary
for accurate flow simulations becomes enormous. As
the number of the grid points becomes large, computer
time also becomes large and, as a result, supercomput-
ers with much large memory and faster speed would
be required.
Recent effort to solve these problems seems to be
in two directions. One is the use of unstructured mesh.
Using the unstructured mesh, handling complex geom-
etry is easier and such grid system is combined with
the finite element method or finite volume method for
arbitrary cell shapes. So far as the Euler equations
are concerned, such approach may be a good choice
since explicit time integration can be used. However,
for viscous flow simulations, implicit time integration
may be necessary and the problem how to apply the
implicit time integration and how to adopt turbulence
models should be solved. The other approach is the
zonal method. In this method, computational grid is
constructed for each element of the geometry. For in-
stance in aircraft simulation, each of the wing, fuselage
and engine has its own grid distributions. Although
modification of the existing finite difference codes to
the zonal codes is relatively easy, accurate and robust
interface method to transfer the data for each zonal
region should be developed. In this approach, num-
ber of the grid points is also easily increased locally.
One of the examplet174 is shown in Fig. 14. The flow
field and the solution algorithm are the same as Fig.
10b. Number of the grid points is globally decreased
but is locally increase in the vertical flow region, and
the result seems to be better than Fig. 10b although
the total number of the grid points is almost the same.
Currently the effort to develop the zonal interface algo-
rithm such as Chimera methodt18] is underway. With
such method, the zonal code can handle complex body
configuration, and the applicability of the code to the
practical problems is to be improved.
4. Supporting System Development
With the increase of the data obtained by the
CFD research, importance of visualization of the com-
puted results began to be recognized. To help people
to understand what happens in the flow field from the
obtained data, we have to visualize the flow and the
use of graphic workstations has an important role for
that purpose. Compared to the circumstances sev-
eral years ago, the level of the graphic software is im-
proved, and visualizing complex flow field in realistic
image now is not a difficult task. However, in most
cases, such softwares are used only to create nice and
beautiful pictures. That may be only important to ad-
vertise the CFD capability. What is really important
is the development of the software to help understand-
ing the result. Such softwares should be interactively
used without requiring researcher's effort and can dis-
play the plots they want quickly. Displaying beautiful
and real-image pictures does not have the first priority.
The computer programs that satisfy some of the
0
requirement mentioned above have been developed
at NASA Ames Research Center. They are called
' PLOT3D', 'GAS' and 'RIP' and each of them has
its own unique featuret19,20~. For instance in RIP,
particle trajectories are computed on the CRAY-2 su-
percomputer and the result is displayed on the IRIS
workstation. The graphic process is carried out inter-
actively on the IRIS display.
Graphic workstations are also useful for grid gen-
eration. Sometimes, grid generation process requires
more human time than flow computation itself. Inter-
active use of graphic workstation will reduce amount
of effort for the grid generation process.
One of the super Graphic Workstation has been
introduced at our laboratory lately and the preproces-
sor (grid generation) and postprocessor (flow visual-
ization programs) for Navier-Stokes and other solvers
having the features discussed in this paper are in the
process of being built up on that machinet214. Here
some of the display examples for the post processor are
presented. First, it has various function, not only con-
tour plotters but also particle path tracer, shock wave
detector and so on. Two or more flow functions can be
displayed not only on the same window but also on the
two or more windows respectively because displaying
two or more functions at the same time would help
understanding of the physical phenomenon. One of
these examples are shown in Figure 15. It shows pres-
sure contours of the flow field around a double delta
wing (discussed above) on the left window and total
pressure contours on the right window. The picture
are obtained in the following simple manner. Firstly,
the grid and flow data are read in. Then two windows
are opened. The left window is selected to display
the pressure contours and then the right window is se-
lected to display the total pressure contours. The third
window can be opened and on which another function
can be displayed if necessary. Also the transforma-
tions, which are the translation, the rotation and the
scaling, can interactively be done as will be described
later.
Next is the other way of using the multi-windows.
We use GWS because output list of enormous numbers
obtained by the CFD computation is too difficult to
understand. However, after displaying the computed
results on the screen we frequently find that we want
to extract the digital numbers from the result on the
screen. One example is the three-dimensional posi-
tions such as the center of a vortex. Figure 16 shows
an example of three-dimensional positioning using the
multi-windows. The left window is the front view and
the right window is the side view of the same objects.
Hair cursors move corresponding to the movement of
the mouse. If you want to know the position of the
vortex center of the third cross section, for instance, it
is achieved by setting the hair cursors crossed at the
center of vortex on the two windows as shown in Fig-
ure 16. Because the right window is perpendicular to
the left one, the three-dimensional position is uniquely
defined by the two set of the hair cursors.
The viewing of any windows can be modified by
mouse input while the viewing has its own default val-
ues. Users do not need to know the exact definitions
of viewing but move the mouse till the desired viewing
is achieved.
5. Summary
The computer code named LANS3D, one of the
representative Navier-Stokes codes in Japan, is taken
as an example and the capability of the current CFD
technology was discussed. This code was developed
for the numerical simulation of high-Reynolds num-
ber compressible flows. The algorithm used in this
code and how it has been improved so far explained
two important aspects of the computational fluid dy-
namics (CFD) codes: efficiency and accuracy. Some
of the application examples showed the capability of
the code for engineering problems. The code and its
modified versions have been extensively used by many
researchers. Reference 22 reviews such applications.
There are many problems (such as turbulence model,
unsteady effect) to be solved before making the CFD
codes an engineering tool, but imminent problem is
the treatment of complex configurations. To use the
best of the CFD, supporting system is important and
the graphic software development for fluid dynamic re-
search on the workstations should have attention.
Acknowledgement
The LANS3D code was mainly developed by the
first author of this paper and Dr. Shigeru Obayashi
(currently at NASA Ames Research Center) at the Na-
tional Aerospace Laboratory, Japan and many people
have helped improving the code capability and its ap-
plications. The authors express their special thanks
to Dr. Susumu Takanashi and Mr. Masahiro Yoshida
at the National Aerospace Laboratory and Ms. Kisa
Matsushima at Fujitsu Limited.
References
1. Obayashi, S., Matsushima, K, Fujii, K. and
Kuwahara, K., 'MImprovements in Efficiency
and Reliability for Navier-Stokes Computations
Using the LU-ADI Factorization Algorithm,
AIAA Paper 86-338", Jan., 1986.
2. Fujii, K. and Obayashi, S., "Navier-Stokes Simu-
lations of liansonic Flows over a Practical Wing
Configuration," AIAA J., Vol. 25, No. 3, March,
1987 pp. 369-370, also AIAA Paper 86-0513.
3. Obayashi, S. and Kuwahara, K., "An Approx-
imate LU Factorization Method for the Com-
11
pressible Navier-Stokes Equations, Journal of
Computational Physics, Vol. 63, 1986, pp. 157-
167.
4. Obayashi, S. and Fujii, K., "Computation of
Thre - Dimensional Viscous liansonic Flows
with the LU Factored Scheme," AIAA Paper 85-
1510, July, 1985.
5. Jameson, A., "Successes and Challenges in Com-
putational Aerodynamics," AIAA Paper
87-1184, June, 1987.
6. Pulliam, T. H. and Steger, J. L., "Implicit Fi-
nite Difference Simulations of Three-Dimensional
Compressible Flow," AIAA Journal, Vol. 18, No.
2, Feb. 1980, pp. 159-167.
7. Steger, J. L. and Warming, R. F., "Flux Vector
Splitting of the Inviscid Gas-Dynamic Equations
with Application to Finite-Difference Methods,"
Journal of Computational Physics, Vol. 40, 1981,
pp. 263-293.
8. Vatsa, V. N., Thomas, J. L. and Wedan, B.
W., "Navier-Stokes Computations of Prolate
Spheroids at Angle of Attack," AIAA Paper 87-
2627, August, 1987.
9. Van Leer, B., Thomas, J. L., Roe, P. L. and
Newsome, R. W., 'MA Comparison of Numerical
Flux Formulas for the Euler and Navier-Stokes
Equations," AIAA Paper 87-1104CP, June, 1987.
10. Roe, P. L., "Finite-Volume Methods for the
Compressible Navier-Stokes Equations, Proc.
Int. Conf. Num. Methods for Laminar and lur-
bulent Flows," July, 1987.
11. Baldwin, B. S. and Lomax, H., "Thin Layer Ap-
proximation and Algebraic Model for Separated
Turbulent Flows," AIAA Paper 78-257, Jan.,
1978.
12. Fujii, K and Obayashi, S., "Navier-Stokes Sim-
ulations of liansonic Flows over Wing-Fuselage
Combination," AIAA J., Vol. 25, No. 12, Dec.,
1987, pp. 1587-1596, also AIAA Paper 86-1831.
13. Takanashi, S., Obayashi, S., Matsushima, K. and
Fujii, K., "Numerical Simulation of Compressible
Flows around Practical Aircraft Configurations,"
AIAA Paper 87-2410CP, August, 1987.
14. Fujii, K. and Obayashi, S., "Use of High-
Resolution Upwind Scheme for Vortical Flow
Simulations," AIAA Paper 89-1955, June, 1989.
15. Fujii, K. and Schiff, L. B., "Numerical Sim-
ulations of Vortical Flows over a Strake-delta
Wing," AIAA Paper 87-1987, 1987, to appear
as AIAA Journal in 1989.
16. Matsushima, K.,-Takanashi, S. and Fujii, K.,
"Navier-Stokes Computations of the Supersonic
Flows about a Spac~plane," AIAA Paper 89-
3402, August, 1989.
17. Fujii, K., ' 'A Method to Increase the Accuracy
of Vortical Flow Simulations," AIAA Paper 88-
2562CP, 1988.
18. Buning, P. G., Chin, I. T., Obayashi, S., Rizk, Y.
M. and Steger, J. L., "Numerical Simulation of
the Integrated Space Shuttle Vehicle in Ascent,"
AIAA Paper 88-4359, 1988.
19. Lasinski, T., et. al., "Flow Visualization of CFD
Using Graphics Workstations," AIAA Paper 87-
1180, 1987.
20. Watson, V., et. al., "Use of Computer Graph-
ics for Visualization of Flow Fields," presented
at AIAA Aerospace Engineering Conference and
Show, February 17-19, 1987.
21. Tamura Y. and Fujii, K., "Use of Graphic Work-
stations for Computational Fluid Dynamics," to
be presented at the International Symposium on
Computational Fluid Dynamics-Nagoya, to be
held at Nagoya, Aug., 1989.
22. Fujii, K., "Capability of Current Supercomputers
for the Computational Fluid Dynamics," to be
presented at the 1st International Conference on
Applications of Supercomputers in Engineering,
to be held at Southampton, U.K. Sept., 1989.
2
°^ EXPERIMENl
PRESENT RESULT
-1.0 ~
_ ~ L I
.o -1
Cp o _
.u
1.0 _
-1.0
-.5
Cp o
.5
_
1.O
) it,
~ I ~6
~ ' ~
SEMI-SPAN STATION ~ 15.0%
_ ~ . ~ _ · · ~
. ~
- ~/-1 ~
~ 60.0Yo
0 .2 .4 .6 .8 1.0
x/c
Wo
37.0%
. . . . .
_ _.
85.0%
. . , . . ,
.2 .4 .6 .8 1.0
X/C
Fig. 1 Comparison of the computed Cp distributions with the experimental data at several
spanw~se stations
: Moo = 0.82, ct = 2.46°, and Re = 2.0 x 106.
Fig. 2 Overall view of the discretized region of the grids (upper half of the computational
volume).
13
a) overall view
b) close-up view
by,
"Ye
Fig. 3 Computed surface pressure contour plots
: Moo = 0.82, ~ = 6.00°, and Re = 1.67 x 106
AT
Fig. 4 Computed off-body particle path trace
: Moo = 0.82, ~ = 6.00°, and Re = 1.67 x 106.
14
or, deg
O EXPERIMENT 3.65
PRESENT RESUl-T 4.00
-1.2 r
-.8
-.4
Cp O
.4
8
1.2 _
-1.2
-~8
-.4
Cp O
.4
.8
1.2
_ ~0Oo
_ ,:
_ ~
_ SEMI SPAN
STATION ~ 13.4%
_ ~ ..
1P~
_ ~
_ ~
60.0X
0 .5
x/c
_ - ~~°
_ ~
22.6%
~ , ,
r Wo ~
~ '~
74.6%
. . · ~ .
1.0 0 .5 1.0 0
x/c
85.0X
.5 1.0
xlc
~ 95.0%
. .
0 .5 1.0
x/c
Fig. 5a Chordwise Cp distributions over a wing surface at several span-wise stations
: Moo = 0.82, ~ = 4.00°, and Re = 1.67 x 106.
-1t
-.5 ~
~ . o°
P
0f
.SI . , ,
-lr
Cp
-.5 1
Ol
.5 _
55o
o o
1
~ 0 0
113° ~\
~f
C~o
l
.35 .40 .45 .50 .55 35
1 39o
~,5~_
. I · . I
_ .40 .45 .50 .55 .35
X/L XIL
-O
~, 27.57
—1 ~ `~55.0
~_484.32
~j`~,3
I ~ ' ~
~139
180 158
G
180
XIL
a,de'
EXPERIMENT 3.65
~r
.55
Fig. 5b Chordwise Cp distributions over a fuselage surface at several circumferential stations
: Moo = 0.82, cx = 4.00°, and Re = 1.67 x 106.
15
Fig. 6 Body geometry and the surface grid for the wing-fuselage-tail combination.
Fig. 7 Computed pressure contour plots for the wing-fuselage-tail combination
: Moo = 0.60, ~ = 0.0°, and Re = 3.47 x 106.
16
PMAX = 0.9900
P Ml N = 0.8500
:< UP = 0.0050
\~
upwind difference result
PMAX = 0.9900
PMIN = 0.8500
\~, UP = 0.0050
\ ~
central difference result
Fig. 8 Computed total pressure contour plots at several chordwise stations
: Moo = 0.30, ~ = 12.0°, and Re = 1.3 x 106.
central difference
upwind result
result
Fig. 9 Computed vortex position compared with experiment
: Moo = 0.30, ax = 12.0°, and Re = 1.3 x 106.
17
INNER PRIMARY
VORTEX /;
OUTER PRIMARY
VO RTEX
\
EXPERIMENT BY
BR ENN ENSTU H L & H UMM E L:
Re = 1.3 X 106
