NASA Technical Reports Server (NTRS) 20030112851: Mean Flow Boundary Conditions for Computational Aeroacoustics

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by  NASA Technical Reports Server (NTRS)

L
AIAA-2003-3299
Mean Flow Boundary Conditions for Computational Aeroacoustics
R. Hixon*
Mechanical, Industrial, and Manufacturing Engineering Department
University of Toledo
Toledo, OH 43606
email: [email protected]
M. Nallasamy**
QSS Group, Inc.
Mail Stop 500-QSS
NASA Glenn Research Center
Cleveland, OH 44 135
S. Sawyer***
Mechanical Engineering Department
University of Akron
Akron,OH 44325
R. Dyson****
Mail Stop 54-3
NASA Glenn Research Center
Brook Park, OH 44135
Abstract
In this work, a new type of boundary condition for time-accurate Computational
Aeroacoustics solvers is described. This boundary condition is designed to complement
the existing nonreflective boundary conditions while ensuring that the correct mean flow
conditions are maintained throughout the flow calculation. Results are shown for a
loaded 2D cascade, started with various initial conditions.
(CAA) is concerned with the time-accurate
Introduction calculation of unsteady flow fields.
Theflow problems of interest can be
The field of Computational Aeroacoustics divided into two types: initial-value, and
-----------___________________________lo_n_g-- time unsteady problems. In an initial-
* Assistant Professor, Member AIAA value problem, the initial flow solution is
** Senior Member AIAA known exactly, and the computation is
***Assistant Professor, Member AIAA
focused on obtaining a time-accurate
**** Member AIAA
unsteady solutio throughout the
02003 by the American Institute of Aeronautics calculation. In a long-time unsteady
and Astronautics, Inc. All rights reserved. problem, the initial flow field is not
This is a preprint or reprint of a paper intended for presentation at a
conference. Because changes may be made before formal
publication, this is made available with the understanding that it will
not be cited or reproduced without the permission of the author.
correctly specified, and the solution must evolve be exactly specified. On the other hand, many
over time until it converges to the long-time realistic flow problems can be classified as long-
unsteady flow. time unsteady. Unlike an initial-value problem,
where the entire calculation must be highly
It is relatively rare to solve an initial-value flow accurate, the long-time unsteady flow calculation
problem, mainly because the initial solution must can be divided into two distinct phases.
instantaneous flow solution at the boundary;
In the first phase, an initial flow solution is instead, it acts on the time-averaged mean flow at
specified; since this is generally not the correct the boundary. In this way, the instantaneous
long-time solution, the flow evolves through a outgoing waves that do not affect the mean flow
transient phase before converging to a long-time will not be reflected.
unsteady solution. During the transient phase, the
goal is to converge the flow to the long-time Governing Equations and Numerical Method
unsteady solution rapidly; thus the transient
computation need not be highly accurate in time. In this work, the Euler equations are solved. The
2D nonlinear Euler equations may be written in
Once the flow has converged to the long-time Cartesian form as:
unsteady solution, the second phase of the
computation begins. Here, the desired unsteady
Q + E + F =O (1)
flow data is gathered; thus, a highly time-accurate t X Y
calculation is desired. The data gathered in this
phase is the desired output for the flow simulation. The NASA Glenn Research Center BASS code
was used to solve this The BASS
The boundary conditions used for each phase of code uses optimized explicit time marching
the calculation should be chosen to achieve the combined with high-accuracy finite-differences to
goals of each phase. Currently, nonreflective or accurately compute the unsteady flow. The code
damping boundary conditions are used for most is parallel, and uses a block-structured curvilinear
CAA calculations.' These conditions are designed grid to represent the physical flow domain. A
to allow outgoing disturbances to exit the constant-coefficient 10" order artificial dissipation
computational domain without generating either model6 is used to remove unresolved high-
real or spurious incoming disturbances. These frequency modes from the computed solution.
conditions are ideal for either an initial-value
computation or the data-gathering phase of a long- The BASS code solves the Euler equations using
time unsteady computation, when the flow the nonconservative chain-rule formulation;
solution correctly represents the desired mean previous experience has indicated that the formal
flow. lack of conservation is offset by the increased
accuracy of the transformed equation^.^.^ The
However, these nonreflective boundary conditions chain-rule form of the Euler equations are:
have no mechanism for maintaining a desired
mean flow when implemented in a nonlinear flow Q,+ Et Q, Q,
+qf
solver. In the initial transient phase, large
disturbances may propagate through the boundary +ExE,+q E,
and exit the domain. These disturbances affect the
flow solution at the boundary, and the correct +E F,+q F,=O
Y Y
method to 'reset' the flow at the boundary is to
impose an incoming disturbance originating
For this work, the optimized low-storage RK56
outside of the computational domain.
scheme of Stanescu and Habashig was combined
with the prefactored sixth-order compact
Notice that in this work, the mean flow boundary
differencing scheme of Hixon".
condition (MFBC) does not act on the
Boundarv Condition Formulation
The mean flow boundary condition is
At the inflow and outflow boundaries of formulated in this way. First, the mean
the computational domain, boundary flow must be defined. The mean flow is
conditions must be specified. These simply the flow variables integrated over a
boundary conditions should, at a minimum, given length of time, T:
maintain the desired mean flow, be
nonreflective for transient waves that do f
not affect the mean flow, and introduce any QT= Qdt (4)
user-specified incoming disturbances. To
I-T
accomplish these goals, the time derivative
of the flow at the boundary is decomposed
Thus, the mean flow integration can be
into three parts:
performed using the same time integration
(et),,,,
( Next, the desired mean flow conditions
-
must be specified. For this problem, the
Ql)boundary- +
(Q,)incoming
desired conditions are specified in terms of
J 1
+ Q, nonreflective - the inflow stagnation temperature and
pressure, the inflow angle, and the outflow
Thus, there are three components to the Finally, the mean flow boundary condition
boundary conditions. The first component must be defined. TO accomplish this, the
(MFBC) maintains the desired mean flow. 1-D characteristics of Thomp~on’~a,r’e~
The second component (incoming) used, modified for an arbitrary rotation of
introduces any user-specified disturbances. the grid boundary:
The third component (nonreflective) is
nonreflective to transient waves. A A~ = (A p)- ?*(A p)
The nonreflective boundary condition used A A ~ = ( A V . ~ )
in this work was the Giles inflow and (5)
outflow conditions.” This boundary A A ~ (=A p)+pz(dV n)
a
condition has been previously tested on a
A A~ = (Ap )- pz(dV . n)
benchmark flat plate cascade with
acceptable result^.^
Here, the “n” vector is normal to the grid
The incoming gust specification boundary boundary, while the “m” vector is tangent
condition is also implemented in the BASS to the grid boundary.
code, though it was not used for this test.
This condition is simply designed to At each boundary point, a ’ target’mean
remove the incoming gust disturbances flow is defined; the object of the mean flow
from the Giles condition calculation, and boundary condition is to obtain the ’ target’
then add the time derivative of the gust mean flow. At the inflow boundary, the
components in at the inflow boundary. current mean flow error can be defined as:
This condition has been previously
validated.’**
-
AE=E --o(
larger currenl
Given that the error in the current mean flow is
known, then the incoming characteristics that will
correct this error can be calculated. At a subsonic where
inflow boundary, this results in a 4 x 4 matrix:
1 0 0 0
A Po
0 1 0 0
kl=
AT 0 0 1 0
[B] AA2 0 (7)
A A A h
3
A A 0
4
where
r In constrast to the inflow boundary, now the first
1 three characteristics are outgoing, and cannot be
I
modified.
" 0
Once the modifications to the characteristics are
known, the necessary modifications to the flow
variables may be calculated as:
I I
----
6(A1) 6(A2) &(Ai) 6(A4)
l o
0 0
l 1
(8) AQ=
Since the fourth characteristic is outgoing, it
cannot be modified.
For the outflow boundary, The development is
similar. Here, the matrix equation becomes:
Finally, the mean flow boundary condition is
written as:
In order to run the a time-marching flow
code, an initial flow condition must be
specified. For this work, seven initial
The value of CT used in this work was 1.0. conditions were used, in which the flow is
uniform throughout the computational
Initially, much higher values of CT were
domain. These seven initial conditions are
used, in an attempt to converge the mean
referred to as Tests 1-7, as defined in Table
flow solution more quickly. However, it
1. To validate the mean flow boundary
was found that instabilities occurred during
conditions further, an additional four tests
the first phase of the calculation, when very
were run in which the initial condition of
large transients were exiting the
Test 1 was used and the desired mean flow
boundaries. In future work, a variable CT
conditions were varied. These cases are
will be tried, where the value of CT is referred to as Tests 8-11, as defined in
initially low to allow the large transients to
Table 1:
leave the domain and then is increased in
order to more quickly converge the mean
Table 1: Flow Test Cases
flow solution.
Test Problem and Flow Initialization
The test problem chosen represents the i'
type of flow problem that the BASS code is 1 1 1 1 1 1 1 3 6 1 0 . 9 2
designed to solve. In this problem, the
mean flow about a loaded 2D cascade is
calculated. The cascade geometry is given
in Ref. (1 1). The mean flow conditions at 4 1 1 40 0.92
5
the inflow are:
-
p =1.0
-0
T =1.0 (13)
0 0.92
1 1 I I I
E= 36" 9 0.95 1 36 0.92
10 1 0.95 36 0.92
At the outflow,
11 1 1 36 0.85
-
p = 0.92 (14)
In the full problem, which is a benchmark
The first seven test cases given in Table 1
problem for the 4'h Computational
were run to test the ability of the MFBC to
Aeroacoustic Workshop on Benchmark
obtain the correct mean flow while starting
Problems, three simple-harmonic gusts are
from different initial conditions. For
introduced at the inflow boundary. For this
comparison, these cases were also run
work, the gusts are neglected; in future
without the mean flow boundary condition.
work, the gusts will be included in the
calculation.
Figures 1-7 show the convergence histones
of each mean flow variable at one point on
the inflow and outflow boundaries for each test Miller of the NASA Glenn Research Center for
case. Each figure compares the mean flow their contributions to this effort.
obtained both with and without the MFBC
enabled. It is clearly seen that the mean flow is References
not remaining constant through the run, and the
effect of the MFBC is to bring the computed flow 1) Hixon, R. ' Radiationand Wall Boundary
to the correct mean values. Also, the Conditions for Computational Aeroacoustics:
convergence to the mean flow as the run A Review' , International Journal of
progresses is shown. This is very important, as it Computational Fluid Dynamics, 2003.
indicates that the MFBC is only active during the 2) Hixon, R, Nallasamy, M., and Sawyer, S.,
initial transient portion of the code run. Parallelization Strategy for an Explicit
Computational Aeroacoustics Code' , AIAA
Another important point is that the MFBC was Paper 2002-2583, July 2002.
used in conjunction with the Giles nonreflective 3) Hixon, R., Nallasamy, M., and Sawyer, S.,
boundary condition; the results show the utility of Effectof Grid Singularities on the Solution
using both a mean flow and an instantaneous Accuracy of a CAA Code' AIAA Paper 2003-
nonreflective boundary condition. 0879, Jan. 2003.
4) Nallasamy, M., Hixon, R., Sawyer, S., Dyson,
The final four tests (Tests 8-1 1) were designed to R., and Koch, L., ' AParallel Simulation of
investigate the ability of the MFBC to obtain Rotor Wake - Stator Interaction Noise' AIAA
different desired mean flows while starting from Paper 2003-3134, May 2003.
the same initial condition. Figure 8 illustrates the 5) Sawyer, S., Nallasamy, M., Hixon, R., Dyson,
time history of the four runs, and the final mean R., and Koch, D., ComputationaAeroacoustic
flows are those given in Table 1. Prediction of Discrete-Frequency Noise
Generated by a Rotor-Stator Interaction' AIAA
Conclusions and Future Directions Paper 2003-3268, May 2003.
6) Kennedy, C. A. and Carpenter, M. H., ' Several
In this work, a mean flow boundary condition for New Numerical Methods for Compressible
time-marching unsteady calculations was Shear-Layer Simulations' ,Applied Numerical
developed. The initial tests showed the Mathematics, Vol. 14, 1994, pp. 397-433.
effectiveness of this boundary condition; the mean 7) Hixon, R., Shih, S.-H., and Mankbadi, R. R.,
flow converged to the correct conditions 'Evaluation of Boundary Conditions for the
regardless of the initial flow condition. This will Gust- Cascade Problem', Journal of
be a useful addition to any nonlinear unsteady Propulsion and Power, Vol. 16, No. 1, 2000,
code using nonreflective boundary conditions, pp. 72-78.
since these nonreflective boundary conditions do 8) Hixon, R., Mankbadi, R. R., and Scott, J. R.,
not automatically maintain the desired mean flow. ' Validation of a High-Order Prefactored
Compact Code on Nonlinear Flows with
In future, the user-input unsteady gust will be Complex Geometries' AIAA Paper 200 1- 11 03,
added to the flow, and a full unsteady nonlinear Jan. 200 1.
calculation will be performed. This will provide 9) Stanescu, D. and Habashi, W. G., 2N-Storage
the final test for the mean flow boundary Low Dissipation and Dispersion Runge- Kutta
conditions. Schemes for Computational Acoustics' ,
Journal of Computational Physics, Vol. 143,
Acknowledpments NO. 2, 1998, p. 674-681.
10)Hixon, R., PrefactorecGmall-Stencil Compact
This work was funded by the NASA Glenn Schemes' Journal of Computational Physics,
Research Center under the Quiet Aircraft Vol. 165,2000, p. 522-541.
Technology program. The authors would like to 1l )www.math.fsu.edu/caa4
thank Dr. Edmane Envia and Dr. Christopher 12)Giles, M. B., ' Nonreflecting Boundary
Conditions for Euler Equation Fimres
Calculations' ,AIM Journal, Vol. 28,
NO. 12, 1990, pp. 2050-2058.
13)Thompson, K. W., ' TimeDependent
11
Boundary Conditions for Hyperbolic
i'
Systems' , Journal of Computational
imp-
Physics, Vol. 68, No. 1, 1987, pp. 1-24. d3
1 . 5
14) Thompson, K. W., ' TimeDependent
Boundary Conditions for Hyperbolic 0 9s
0 91
Systems 11' Journal of Computational e
Physics, Vol. 89, 1990, pp. 439-461. 11 05 a9
Ossi
08 .-
015
O ~ J O ~ O ~ ~ O U A M ~ O
I- IVnS
Figure 1: Mean Flow Test Case 1
Figure 2: Mean Flow Test Case 2
I,' 1 7 I ' I ' I ' I ' I ' I ' 11.1
Figure 3: Mean Flow Test Case 3
.
uy5 B
B
OY <~
OSSC
0 IO 20 30 40 M W 'Iu I0 ?U M 40 50 60 70
LLnC IMC
Figure 4: Mean Flow Test Case 4
Figure 7: Mean Flow Test Case 7
50 1 1 1 1 , 1 , 1 , ( , 1 , , , , , , , , 3
Figure 5: Mean Flow Test Case 5
Figure 8: Mean Flow Test Cases 8-11
0%
r
r03 VIF'Y !ji
5
0 811
0 20 40 60 ID 10 40 t4 80~00"
Figure 6: Mean Flow Test Case 6