AIAA 97-0609



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 royalty-free 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 boundary-layer 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.

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 γ = 1.4. The boundary layer flow at 4.5D upstream of the first row of bleed holes has a momentum thickness of θ = 0.198 cm. The freestream has a Mach number of [pic] = 2.46, a static temperature of [pic] = 132.56 K, and a static pressure of [pic] = 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-ω model known as the shear-stress 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 (ρ), 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[pic], 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 steady-state 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 time-derivatives 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 H-H 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 106. 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 H-H 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 # DNS or Bleed BC Fine or Coarse Grid

1 DNS fine

2 DNS-BC fine

3 Avg W BC fine

4 Cd BC fine

5 W Profile BC fine

6 Avg W BC coarse

* For all cases, D = 0.635 cm, [pic] = 2.46, [pic]= 10700 Pa, [pic] = 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 106 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.

ACKNOWLEDGEMENT

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.

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right symmetry boundary

t

flat plate

plenum

y

Lx

x

z

x

Ly

Lp

L

Hp

H

outflow

boundary

inflow

boundary

freestream boundary

left symmetry boundary

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