AIAA 2000-0888
Boundary Conditions for CFD Simulations
of Supersonic Boundary-Layer Flow
through Discrete Holes
D. B. Benson* and T. I-P. Shih#
Department of Mechanical Engineering
Michigan State University, East Lansing, Michigan
D. O. Davis+ and B. P. Willis++
NASA - Glenn Research Center
Cleveland, Ohio
ABSTRACT
In CFD simulations of supersonic boundary-layer flow
with bleed through discrete holes, resolving the flow
through each bleed hole and the associated bleed plenum
can be extremely intensive computationally. In this study,
four outflow boundary condition (BCs) for the bleed holes
are developed and evaluated by comparing results obtained
by using the four bleed BCs with simulations that resolve
the holes and plenum (DNS results). The four BCs
developed are (1) DNS BC (specify normal velocity, W,
into hole based on DNS results and extrapolate all other
variables), (2) Avg W BC (same as DNS BC except
specify constant over each hole), (3) Cd BC (same as
DNS BC except W calculated by using a constant
discharge coefficient over each hole), and (4) W Profile
(same as Avg W BC except W has positive and negative
values).
Results obtained for choked bleed of a supersonic
boundary layer on a flat plate through six rows of normal
holes show that all four BCs developed produce accurate
solutions for the velocity profile downstream of the bleed
region when compared to the DNS results. With BC (2)
_____________
*
#
+
++
Graduate Student, Member AIAA.
Professor. Associate Fellow AIAA.
Research Engineer. Senior Member AIAA.
Hypersonic Tunnel Facility Manager, Plum Brook Station.
Member AIAA.
Copyright © 2000 by the American Institute of Aeronautics and
Astronautics, Inc. No copyright is asserted in the United States
under Title 17, U.S. Code. The U.S. Government has a royaltyfree license to exercise all rights under the copyright claimed
herein for government purposes. All other rights are reserved by
the copyright owners.
and (3), the barrier shock forms just downstream of each
bleed hole instead of inside of it. Despite this, there is no
flow separation between bleed holes. However, a very
small separation bubble forms downstream of the last row
of holes. With BC (2), a simulation was also conducted
with only one cell per bleed hole. With this coarse mesh,
the barrier shock about each bleed hole is smeared, but
predicted velocity profile downstream of the bleed region
compares well with the DNS result.
INTRODUCTION
Bleeding through rows of circular holes is an effective
and widely used method for controlling boundary-layer
profiles in aerodynamic and propulsion devices1. The
importance of bleed in controlling boundary-layer flows
has led a number of investigators to use both experimental
and CFD methods to study it. CFD studies of boundarylayer bleed can be classified into two groups. One group
models the bleed process by using boundary conditions
and/or a roughness model without resolving the flow
through the bleed holes2-7. The advantage of this approach
is that it is more efficient computationally, which enables a
complete inlet configuration to be simulated as was done
in Refs. 3 and 6.
The other group studies the bleed
process by resolving the flow through the bleed holes and
plenum9-24. The advantage of this approach is that it can
reveal the nature of the flow governing the bleed process.
The understanding gained by these studies can guide the
construction of boundary conditions and roughness models
used by the first group.25
So far, all boundary conditions developed for bleed
(e.g., Refs. 2-7) have treated the entire bleed region as a
porous surface so that information on hole geometry and
arrangement cannot be represented. As a result, many
details of the bleed process are lost in this treatment. In
particular, the “barrier” shocks inside the bleed holes and
their influences on boundary-layer control are not
accounted for in these models.
The objective of this study is to develop and evaluate
boundary conditions (BCs) for bleed, referred to here as
bleed BCs, which can represent the geometry and
arrangement of the bleed holes in order to account for
more physics of the bleed process. The focus is on choked
bleed of a supersonic boundary layer on a flat plate
through rows of normal holes arranged in a staggered
fashion without incident shock. The bleed BCs developed
in this study were evaluated by comparing with results
from CFD simulations that resolve the flow in the holes
and the plenum.
Lx
y
left symmetry
boundary
L
inflow
boundary
freestream boundary
flat plate
z
Ly
right symmetry
boundary
x
x
outflow
boundary
H
t
plenum
Hp
Lp
Fig. 1. Schematic of problem studied.
DESCRIPTION OF PROBLEM
A schematic of the problem investigated is shown in
Fig. 1 (not drawn to scale). It involves bleeding a
supersonic turbulent boundary layer flow on a flat plate
into a plenum through six rows of normal circular holes
arranged in a staggered fashion. All holes have the same
diameter of D = 0.635 cm; the thickness of the flat plate is
t = D; and the plenum has a depth of Hp = 10D and length
of Lp = L-D/2. The streamwise and spanwise distances
between centers of successive holes are Lx = 2D and Ly =
D, respectively.
The distance between the inflow
boundary and the center of the first row holes is 5D. The
distance between the center of the last row holes and the
outflow boundary is 6D. The freestream boundary is at a
distance of H = 10D from the plate.
The working fluid is air with a constant specific-heats
ratio of  = 1.4. The boundary layer flow at 4.5D upstream
of the first row of bleed holes has a momentum thickness
of  = 0.198 cm. The freestream has a Mach number of
M  = 2.46, a static temperature of T = 132.56 K, and a
static pressure of P = 10700 Pa. The exit of the plenum
has an average back pressure of Pb = 0.25618 P. With
this back pressure, the bleed flow through the holes are
choked.
For the boundary-layer bleed problem just described,
two types of simulations were performed. One type,
referred to direct numerical simulation (DNS), resolves the
flow above the plate, in the holes, and in the plenum. For
this type of simulation, the domain is the region bounded
by the solid lines shown in Fig.1. Note that only “half” of
each bleed hole needed to be included in the domain
because of the symmetry in the spanwise direction. The
other type of simulations does not resolve the flow in the
holes and the plenum. They only resolve the flow above
the plate. In these simulations, the bleed flow through the
six rows of holes are accounted for by applying an outflow
boundary condition, referred to as the bleed BC, over each
bleed hole. The bleed BCs developed and evaluated are
described in the next section.
FORMULATION OF PROBLEM:
BLEED BCS
The problem shown in Fig. 1 and described in the
previous section is modeled by the ensemble-averaged
conservation equations of mass (continuity), momentum
(compressible Navier-Stokes), and total energy for a
thermally and calorically perfect gas with Sutherland's
model for thermal conductivity. Turbulence was modeled
by a low Reynolds number k- model known as the shearstress transport model of Menter26,27.
For the simulation that resolve the flow above the
plate, in the holes, and in the plenum, the boundary
conditions (BCs) used are as follows (Fig. 1). At the
inflow boundary above the plate, all flow variables were
specified at the freestream condition except those in the
region containing the boundary layer. In the boundary
layer, the compressible turbulent boundary-layer profile of
Huang, et al.28 was employed. At the freestream boundary,
all flow variables were specified at the freestream
conditions. At the outflow boundary, all flow variables
were extrapolated. The BCs imposed at the two symmetry
boundaries in the spanwise direction was zero derivatives
of the dependent variables except for the velocity
component normal to those boundaries which was set
equal to zero. At the exit of the plenum, a back pressure
(Pb) was imposed, and density and velocity were
extrapolated. At all solid surfaces, the no-slip condition,
adiabatic walls, and zero normal-pressure gradient were
imposed.
For simulations that only resolve the flow above the
plate, the BCs used at the inflow, outflow, freestream, and
solid boundaries are the same as those just described for
the DNS. Over each hole on the plate, a bleed BC is
imposed. Four bleed BCs were developed and evaluated,
and they are as follows:
(1) DNS BC. With this BC, the normal velocity
distribution in each hole is specified. The
velocity distribution imposed is taken from the
DNS results. Density (), pressure (P), and the
other two velocity components in each hole are
extrapolated by assuming zero normal
derivatives. The total energy is then computed by
using the equation of state. The purpose of this
BC is to examine the importance of the normal
velocity distribution in the hole. Another purpose
is to examine if all other variables can be
extrapolated .
(2) Avg W BC. With this BC, the normal velocity in
each hole is a constant over the entire hole. The
numerical value of this constant is the average
normal velocity in the hole. Other conditions are
the same as those for the DNS BC. Since the
normal velocity distribution is in general
unknown but the bleed rate is known or can be
modeled, this BC is intended to examine the
effects of neglecting the spatial variations in the
normal velocity distribution.
(3) Cd BC. With this BC, the normal velocity, W, is
calculated by using a discharge coefficient; i.e.,
W = - CD P  , where CD is a constant over the
entire hole. Its value is chosen to ensure the
correct average W over the bleed hole based on
DNS result. Other conditions are the same as
those for the DNS BC. Since the bleed rate into a
hole can be modeled by a discharge coefficient,
the purpose of this BC s to examine the effects of
a constant discharge coefficient over the entire
hole.
(4) W Profile. With this BC, two different values of
constant normal velocity are imposed. In the
upstream 90% of the hole, the W velocity is a
constant negative, implying bleed into hole. In
the downstream 10% of the hole, the W velocity
is a constant positive, implying flow issuing from
the hole. Other conditions are the same as those
for the DNS BC. This BC is intended to model
the “barrier” shock inside each bleed hole.
One other bleed BC was attempted. In that bleed BC,
all flow variables at each bleed hole were specified with
the distributions taken from the DNS results. This BC is
not listed here because it was found to be unstable.
NUMERICAL METHOD OF SOLUTION
Solutions to the governing equations described in the
above section were obtained by using a cell-centered
finite-volume code called CFL3D.29,30 All inviscid terms
were approximated by the flux-difference splitting of
Fig. 3. Grid systems used when holes
and plenum are resolved.
Fig. 3. Close-up of grid system
when holes are resolved.
Roe31,32 (third-order accurate in the transformed domain)
with the slope limiter of Chakravarthy and Osher.33 All
diffusion terms were approximated conservatively by
differencing derivatives at cell faces. Since only steadystate solutions were of interest, time derivatives were
approximated by the Euler implicit formula. The system
of nonlinear equations that resulted from the
aforementioned approximations to the space- and timederivatives were analyzed by using a diagonalized
alternating-direction scheme34 with local time-stepping
(local Courant number always set to unity) and three-level
V-cycle multigrid.35,36
For the simulation that resolved the flow above the
plate,in the holes, and in the plenum, the grid system used
consists of 16 blocks of structured grids. Figures 1 and 2
show the grid system used in two planes. There are two HH grids for the region above the plate, a coarser grid for
the entire region above the plate (201 x 17 x 81) and a
finer embedded grid for the region about the bleed holes to
resolve the bleed process (449 x 33 x 49). For the plenum,
there are also two H-H grids, a coarser one for the
complete plenum (257 x 17 x 49) and a finer embedded
one for the region next to bleed holes (417 x 33 x 17) to
resolve the high-speed bled flow. For each bleed hole, two
grids are used, an O-H grid to get the geometry of the hole
correctly (73 x 49 x 25) and an H-H grid to remove the
centerline singularity associated with the O-H grid (73 x 9
x 17). The total number of grid points used is 2.054 x 10 6.
The grids in each of the bleed holes were overlapped. The
overlapped grids in the bleed holes were patched to the
grids above the plate and in the plenum, which are in turn
embedded in the coarser H-H grids.
For simulations that used bleed BCs, only the two HH grids above the plate are used with 1.003 x 106 grid
points, which is about half the number used for the DNS.
Though this is less, it is still a substantial number of grid
points. Simulations with the Avg W bleed BC was also
performed on an extremely coarse mesh with a one-block
grid system (23 x 3 x 81) in which there is only one grid
point in each bleed hole. The number of grid points in the
direction normal to the plate, however, remained the same
as before so that the boundary-layer profile is still resolved
accurately. The purpose of this grid coarsening study is to
examine the effects of grid on the velocity and pressure
profiles in the boundary layer downstream of the bleed
region.
RESULTS
Table 1 summarizes all of the simulations performed.
In that table, DNS refers to simulations that resolved the
flow above the plate, in the bleed holes, and in the plenum.
Bleed BC simulations resolved only the flow above the
plate. The fine grid refers to simulations performed with
2.054 x 106 grid point for DNS and 1.003 x 106 grid points
for the bleed BC simulations. The coarse mesh refers to
the grid with (23 x 3 x 81) or 5,589 grid points. The
results for these simulations are given in Figs. 4 to 10.
Table 1. Summary of Simulations*
Case #
1
2
3
4
5
6
*
DNS or Bleed BC
DNS
DNS-BC
Avg W BC
Cd BC
W Profile BC
Avg W BC
Fine or Coarse Grid
fine
fine
fine
fine
fine
coarse
For all cases, D = 0.635 cm, M  = 2.46, P = 10700
Pa, T = 132.56 K.
Figure 4 shows DNS results for the pressure contours
in the left symmetry plane and on the plate. It also shows
the velocity vectors at 10-5 m above the plate. From this
figure, the structure of the “barrier” shock formed about
each bleed hole can be seen. This figure also shows via
the velocity vectors that “barrier” shocks do not cause flow
separation. This is because the shocks are located in the
hole. This DNS simulation serves as a reference in
evaluating the bleed BCs developed in this study. Thus,
the term reference in Figs. 7 and 8 refer to the DNS results.
Figure 5 shows the pressure contours above the plate
for the DNS and the four bleed BCs developed by using
the same scales in the plotting. From this figure, it can be
seen that all bleed BCs developed produce the correct
qualitative features of the flow field, including the
formation of the “barrier” shocks. However, upon closer
inspection, there are differences. The most important
difference is that with the Avg W and the Cd BC, the
“barrier” shock forms just downstream of each bleed hole
instead of in it. This is because for these two bleed BCs,
all normal velocities have the same negative value.
Despite this change in shock location, no flow separation
formed between bleed holes in successive rows. But, a
very small separation bubble did form just downstream of
the last row of holes. This indicates that the row of holes
downstream of the “barrier” shock plays an important role
in accelerating flow through the “barrier” shock. Since
there were no flow separation between bleed holes, not
getting this part of the physics correct may be acceptable.
With the DNS BC and the W Profile BC, the “barrier”
shocks did form in each bleed hole so that no separation
took place between bleed holes in successive rows or after
the last row of bleed holes. For these two bleed BCs, the
normal velocity had both positive and negative values.
Figure 6 shows the variation in Mach number,
pressure, and normal velocity in the middle of the third
row holes. This figure shows that all of the bleed BCs
perform reasonably well when compared to the DNS
results. The greatest discrepancies always occur in the
structure of the “barrier”shock because of the shift in the
shock location.
Figure 7 shows the Mach number (velocity) and
pressure profiles near the plate 2D before and 1D after the
bleed holes. From this figure, it can be seen that all bleed
BCs are able to predict the downstream Mach number
profile with considerable accuracy when compared to the
DNS results. Note that bleed is able to energize the
boundary layer since the Mach number profile is fuller
downstream of the bleed region. The predicted pressure
profile downstream are also reasonable when compared to
the DNS results.
Figure 9 shows results obtained for Avg W BC for
two grid system, a fine grid with 1.003 x 10 6 grid points
and a coarse grid with 5,589 grid points. With the coarse
grid, there is only one grid point per bleed hole, which is
the fewest number of grid points that can be used and still
represent bleed hole arrangement. Figure 9 compares the
predicted “barrier” shocks. From this figure, it can be seen
that even with one grid point per bleed hole, “barrier”
shocks are still predicted though considerably weaker and
smeared. But, by being able to at least predict its presence,
porous-surface type bleed BCs can be improved. Figure
10 shows that the predicted Mach number and pressure
profiles downstream of the bleed region to compare
reasonably well when compared to DNS results. The
results of this coarse grid study indicate that it is possible
to construct a bleed BC by using very few grid points and
still be able to capture the key flow physics.
SUMMARY
Four bleed BCs were developed and evaluated. All
four bleed BCs developed were able to predict the
formation of the “barrier” shock. Though Avg W BC and
Cd BC predicted a slightly wrong location for the “barrier”
shock, the effect of this error on the flow is small. For the
Avg W BC, it was shown that “barrier” shocks can be
predicted by using only one grid point or cell per bleed
hole. Finally, all bleed BCs predicted accurately the Mach
number (velocity) and pressure profiles downstream of the
bleed region.
11.
12.
ACKNOWLEDGEMENT
13.
This work was supported by NASA grant NAG 3-2234
from NASA - Glenn Research Center. The computer time was
provided by the NASA’s NAS Facility. The authors are grateful
for this support.
14.
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