A formulation of boundary condition for flows with small disturbances is presented. The authors test their methodology in an axisymmetric jet flow calculation, using both the Navier-Stokes and Euler equations. Solutions in the far field are assumed to be oscillatory. If the oscillatory disturbances are small, the growth of the solution variables can be predicted by linear theory. Eigenfunctions of the linear theory are used explicitly in the formulation of the boundary conditions. This guarantees correct solutions at the boundary in the limit where the predictions of linear theory are valid

Hagstrom, Thomas

Hayder, M. Ehtesham

English

NASA Technical Reports Server

NASA Technical Memorandum 106930
ICOMP-95-10
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An Outflow Boundary Condition for
Aeroacoustic Computations
M. Ehtesham Hayder
Institute for Computational Mechanics in Propulsion
Lewis Research Center
Cleveland, Ohio
Thomas Hagstrom
Institute for Computational Mechanics in Propulsion
Lewis Research Center
Cleveland, Ohio
and The University of New Mexico
Albuquerque, New Mexico
Prepared for the
ASME Fluids Engineering Division Summer Meeting
cosponsored by ASME, JSME, and EALA
Hilton Head Island, South Carolina, August 13-18, 1995
National Aeronautics and
Space Administration
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AN OUTFLOW BOUNDARY CONDITION FOR AEROACOUSTIC COMPUTATIONS
M. Ehtesham Hayder
Institute for Computational Mechanics in Propulsion
Lewis Research Center
Cleveland, Ohio 44135
Thomas Hagstrom
Institute for Computational Mechanics in Propulsion
Lewis Research Center
Cleveland, Ohio 44135
and Department of Mathematics and Statistics
The University of New Mexico
Albuquerque, New Mexico 87131
ABSTRACT
We present a formulation of boundary condition for flows
with small disturbances. We test our methodology in an
axisymmetric jet flow calculation,_ using both the Navier-
Stokes and Euler equations. Solutions in the far field are
assumed to be oscillatory. If the oscillatory disturbances
are small, the growth of the solution variables can be pre-
dicted by linear theory. We use the eigenfixnctions of the
linear theory explicitly in our formulation of the boundary
conditions. This guarantees correct solutions at the bound-
art in the limit where the predictions of linear theory are
valid.
Keywords: Nourefiecting boundary condition, Com-
putational aeroacoustics, Jet flow computations
INTRODUCTION
Any attempt to directly compute the noise source from
the flow field demands high accuracy of the numerical meth-
ods, including the treatment of the boundary conditions.
Treatment of the outflow boundary for stable and accurate
flow simulations has attracted considerable attention [see
for example Engqulst and Majda (1977), Bayliss and Turkel
(1982), Scott and Hankey (1985), Hagstrom and Hariharan
(1988), Roe (1989), Giles (1990), Harihffr_m and Hagstrom
(1990), Thompson (1990), Tam and Webb (1993), Atkins
and Casper (1994)]. One usually idealizes a physical prob-
lem to formulate the conditions at the boundary. The ef-
fectiveness of the boundary condition is dependent on the
degree of validity of the idealized assumptions in the actual
flow situation. Approaches based on linear analysis, espe-
cially variations of the characteristic methods, are widely
used. Various investigators have derived essentially the
same asymptotic pressure boundary conditions [See Hay-
der and Turkel(1994) for a discussion of various works on
this boundary condition, an evaluation of its effectiveness,
and comparisons with other boundary conditions]. Hayder
and 2_trkel (1994) observed that the asymptotic pressure
boundary condition gave reasonable results. They how-
ever recommended a small exit region beyond the region
of interest. Their experiments indicated that Giles(1990)
and characteristic boundary conditions with a larger exit
layer also yields reasonable solutions. Because of the differ-
ence in the asymptotic forms of wave equations in two and
three dimensions, this boundary condition is slightly differ-
ent in three dimensions from two dimensions [see Hayder
and Turkel (1994)]. Hariharan and Hagstrom (1990) formu-
lated higher order forms of this boundary condition. As we
stated earlier, a boundary condition will give satisfactory re-
sults if the assumptions used to derive the condition closely
follow the actual flow situation.
In this paper we present a new approach to boundary
treatment based on the linear stability theory. The govern-
ing equations of the fluid flow are nonlinear. However, if
a mean flow is excited by a small disturbance, the linear
theory can be used to predict its growth. Also, the eigen-
functions given by the linear theory describe the profiles of
the disturbances after an initial adjustment region. This
phenomenon motivates our present effort to find a bound-
ary condition for a flow with small disturbances. "We as-
sume the profiles of the disturbances at the outflow can be
approximated by the eigenftmctions predicted by the linear
theory. The particular eigenfunctions chosen would gener-
ally correspond to the most unstable modes. However, any
eigenmodes could, in principle, be used. The latter may
be relevant for forced problems, where the excited modes
may be determined by the forcing. The boundary condition
that is developed here should be accurate for cases where
the linear theory accurately describes the disturbances at
the boundary and where these are dominated by a single,
kno_ mode. It may not be appropriate when the nonlin-
earity in the flow is significant.
In Sections 2 and 3, we give, respectively, the governing
equations for our test problem and the derivation of our
new boundary condition. A discussion of the basic scheme
for our test problem is given in Section 4 and we present our
results in Section 5.
GOVERNING EQUATIONS
We compute the flow field of an axisymmetric jet to test
our new boundary conditions. We solve the Navier-Stokes
equations as given below
where
Q,+F=+G_=S
F_T
pu 2 -- _xx + P
puv - _zr
puH - uTxx - vTxr - aT_
V -_ r
puv - "r_r
pv 2 - T,.,- + p
pvH - u'rx,. - vm,.,. - aT,.
(o)0S= p - _oo
0
Q represents the solution variables, F and G are the fluxes
in the x and r directions respectively, S is the source term
that arises in the cylindrical polar coordinates, and rij are
the shear stresses.
DERIVATION OF THE BOUNDARY CONDITION
The governing equations for the fluid flows are the Navier
Stokes equations. For high Reynold number flows, the vis-
cous effect is small and one has to decide whether the bound-
ary treatment should be based on the Euler equations or
the Navier-Stokes equations. The difference between the
two approaches is not just the type of boundary conditions
but even the number of boundary conditions that need to
be given. For inviscid subsonic flow, one boundary condi-
tion needs to be specified at outflow corresponding to an
incoming acoustic wave. For supersonic flow no boundary
condition is required. For viscous flow the equations are no
longer hyperbolic but rather incompletely parabolic. For
inflow, four conditions need to be specified (in two dimen-
sions), while for outflow, three conditions need to be spec-
ified. In particular, the number of conditions does not de-
pend on whether the local flow is subsonic or supersonic.
Many codes use inviscid type boundary conditions. This
is based on the assumption that the flow in the far field
is essentially inviscid because of the high Reynolds number
and the lack of physical boundary layers. Hayder and Turkel
(1993) considered a framework for implementing the bound-
ary conditions, where the formulation is of a characteristic
type, but where viscous effects are also partially accounted
for. We will use that framework for our present study. At
subsonic outflow, we extrapolate three characteristic vari-
ables from the interior and impose one boundary condition.
This is done by solving the following set of equations.
pt - pcu_ = R1
pt + pcu_ = R2 (1)
p_ - c2p, = R3
Vt ._- R4
where R4 is determined by which variables are specified and
which are not. Whenever, the combination is not speci-
fied,/_/is just those spatial derivatives that come from the
Navier-Stokes equations. Thus, R/ contains viscous con-
tributions even though the basic format is based on invis-
cid characteristic theory. In implementing these differential
equations we convert them to conservation variables p , m
= p u , n = p v and E. Assuming an ideal gas we then have
u 2 + v 2
2
mt up_
%1,t --_
P P
nt vpt
P P
For subsonic outflow we calculate R2, R3, R4 from the
Navier-Stokes equations and set R1 as prescribed by the
given boundary condition. For supersonic flows, all the/_
at the outflow boundary can be calculated from the Navier-
Stokes equations or else by extrapolation of all the charac-
teristic variables from the interior.
In this work, we assume the solution variable Q at the
outflow behaves as
= +
where Q is the mean and QI is the oscillatory part of the
variable Q and
Q' = e_'_[C1 cos(a.x - wt) + C2 sin(a_x - wt)]
Thus at outflow,
v, = A coswt +V r Vi
p* Pr Pi
B sinwt
where w is the excitation frequency. Here, the vectors deter-
mining the structure of the disturbances are eigenfunctions
of the linear stability equations. They are used explicitly
in our boundary conditions. This form for the disturbances
should hold if the solution is well-approximated by linear
theory.
.R 1 Pt -pcut
w ( - APr sinwt + B Pi coswt)
- pew( -A u_ sin wt + B ui coswt)
Pt +pcut
w ( - APr sinwt + B Pi coswt)
+pcw( -A Ursin wt + B ui coswt)
pt - c2 pt
w ( - A Pr sinwt + B pi coswt)
- c2w( -A Pr sin wt + B pi coswt)
Let A = A w sinwt and B = B w coswt. Then
or
-p. - pcu_ Pi -4- pcu_ /
-Pr -t- c2 pr Pi - ¢2pi
= -_ p_ c2p_ -p_ - pcu_ R3
boundary condition [equation(2)]in other frameworks. For
example, the framework presented in Tam and Webb(1993)
uses the llnearizedEuler equations. One can implement our
condition by replacing the condition corresponding to the
incoming acoustic wave by equation (2). One may expect
to see some differenceswith the same boundary condition is
implemented in differentways.
We note that a similarconstruction could, in principle,be
carriedout at the inflow boundary. To do so, an eigenmode
corresponding to a left-moving wave should be identifiedand
itsamplitude relatedto the outgoing characteristicvariable,
RI. Finally,the incoming variablescould be specified,again
using the assumed form of the disturbance. We have not yet
explored the feasibilityof thisapproach.
BASIC SCHEME
We use a high order extension of the MacCormack Scheme
due to Gottlieb and Turkel(1976). It has a predictor and a
corrector stage and may be written as:
The predictor step with forward differences is
At "7"F '_ n
= Q?+ / ( - - (Fh - + ,',ts?
The corrector step with backward differences is
1
Q_+I = 2[(_ i + Q_
This scheme is second order in time and becomes fourth-
order accurate in the spatial derivatives when alternated
with symmetric variants. We define L1 as a one dimen-
sional operator with a forward difference in the predictor
and a backward difference in the corrector. Its symmetric
variant L2 uses a backward difference in the predictor and
a forward difference in the corrector. For our computations,
the sweeps are arranged as
Q,_+I = LI_LI_Q_
Qn+2 = L2_L2_Q=+I
Further description of our implementations can be found
in Hayder et a1.(1993) and Mankbadi et a1.(1994).
R1 = -4(-p. + pcu.) +/} (p_ - pcui) (2)
Equation (2) is our new boundary condition and we use this
value of R1 in equation (1) for our numerical tests. We
would like to point out that one can implement our new
RESULTS
We test our new boundary condition for _n axisymmet-
ric jet flow calculation. Details of such calculations can be
found in Hayder et al. (1993) We note that the flow is
unstable, and hence provides a stiff test for any boundary
condition. Here, the initial axial velocity is specified as
1
_(r) = 2[(1 + u_) - (1 - ucc)tanh(4(r - 1))]
and the corresponding temperature is given by the Buse-
mann -Crocco integral of the energy equation:
T(r) = To + ---_M2(1 - _)(_ - uoo).
Here uoo = .25 and the jet center temperature is assumed
to be equal to the outer flow temperature, i.e., To = Too.
The jet Mach number is M=l.5 and Reynolds number based
on the jet radius is 364,000. We excite the inflow profile at
location r and time t as
W(r, t) = _V(r) + _Re(W' e _')
where W = (p,u,v,p) T, ITv is the mean and W t is the
eigenfunction of the linear stability equations correspond-
ing to the mean flow profile which has the most rapid
growth rate. For our numerical tests we used w = 1.08 and
e = 10 -6. Eigenfunctions (EF) are obtained by solving the
linear stability equations and also using our flow code. In
our implementation of the new boundary condition, we use
for the eigenfunctions the average of those obtained from
the linear stability calculations and those computed from
our flow code, i.e.,
E-/_Linear Theory "+ EFcode
E Fbc = 2
We show contours of vorticity magnitude for Navier Stokes
computations Figure 1. Similar computations with Euler
equations are shown in Figure 2. We used three compu-
tational domains. They are 60, 50 and 40 radii long. All
three computational domains are 5 radii wide in the trans-
verse direction. The results at the outflow boundary of the
shorter domains i.e., xl = 40 and 50 compare well with the
same quantities at the same locations in the long domains.
The present boundary condition gave satisfactory results.
Because of very high Reynolds number used in our compu-
tations, there is virtually no difference between our solutions
of the Euler and the Navier Stokes equations.
ACKNOWLEDGEMENT
The linear stability code used in this study was developed
by Dr. Lennart S. Hultgen. We gratefully acknowledge
his help using the code. The second author was partially
supported by NSF Grant DMS-9304406.
REFERENCES
Atldns, H. and Casper, J., (1994), "Nonrefiecting Bound-
ary Conditions for High-Order Methods", AIAA Journal,
Vol 32, No. 3, pp 512-518.
Bayliss, A. and Turkel, E., (1982), "Far Field Boundary
Condition for Compressible Flows", JournM o£ computa-
tionM Physics, Vol 48, pp 182-199.
Engqnist, B. and Majda, A.,(1977), "Absorbing Boundary
Conditions for the Numerical Simulation of Waves", Math-
ematics of Computations, Vol 31, No 139, 629-651.
Giles, M. G., (1990), "Nonreflecting Boundary Conditions
for Euler Equations Calculations," AIAA Journal, Vol 28,
pp 2050-2058.
Gottlieb, D. and Turkel, E,. (1976), "Dissipative Two-
Four Methods for Time Dependent Problems", Math.
Comp, Vol 30, pp 703-723.
Hagstrom, T. and Harlharan, S. I., (1988), "Accurate
Boundary Conditions for Exterior Problems in Gas Dynam-
ics", Mathematics of Computation, Vol 51, No 184, pp
581-592
Hariharan, S. I. and Hagstrom, T., (1990), "Far Field Ex-
pansion for Anisotropic Wave Equations" , Computational
Acoustics (IMACS), Vol 2, Lee, Cakmak and Vichnevetsky
(Editors), Elsvier Science Publisher, pp 283-294.
Hayder, M. E. and Turkel, E., (1994), "Boundary Con-
ditions for Jet Flow Computations", AIAA paper 94-2195,
June 1994.
Hayder, M. E. and Turkel, E., (1993), "High Order Accu-
rate Solutions of Viscous Problems", AIAA paper 93-3074,
NASA TM 106267, June 1993.
Hayder, M. E., Turkel, E. and Mankbadi, R. R.,(1993),
"Numerical Simulations of a High Much Number Jet Flow",
AIAA paper 93-0653.
Mankbadi, R. R., Hayder, M. E. and Povinelli, L. A.,
(1994), "The Structure of Supersonic Jet Flow and Its Ra-
diated Sound", AIAA Journal, Vol 32, No. 5, pp 897-906.
Roe, P. L., (1989), "Remote Boundary Conditions for
Unsteady Multidimensional Aerodynamic Computations",
Computer & Fluids, Vol 17, No. 1, 221-231
Scott, :l. N. and Hankey, W. L., (1985), "Boundary Con-
dition for Navier-Stokes Solutions of Unsteady Flow in a
Compressor Rotor", Three-Dimenslonal Flow Phenomena
in FIuid Machinery, Vol 32, ASME, New York.
Tam, C. K. W. and Webb, J. C., (1993), "Dispersion-
Relation- Preserving Finite Difference Schemes for Com-
putational Acoustics", Journal of Computational Physics,
107, pp 262-281.
Thompson, K. W. (1990), "Time Dependent Boundary
Conditions for Hyperbolic System, II', JournM of Compu-
tationM Physics, 89, 439-461.
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Figure 1: Solutions of the Navier-Stokes equations
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Figure 2b: Intermediate Computational Domain Solution (xl=50)
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Short Computational Domain Solution (xl:40)
MACH
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Figure 2: Solutions of the Euler equations

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May 1995
4. TITLE AND SUBTITLE
An Outflow Boundary Condition for Aeroacoustic Computations
6. AUTHOR(S)
M. Ehtesham Hayder and Thomas Hagstrom
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)
National Aeronautics and Space Administration
Lewis Research Center
Cleveland, Ohio 44135-3191
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National Aeronautics and Space Administration
Washington, D.C. 20546-0001
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Technical Memorandum
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REPORT NUMBER
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10. SPONSORING/MONITORING
AGENCY REPORTNUMBER
NASA TM- 106930
ICOMP-95-10
11. SUPPLEMENTARY NOTES
Prepared for the ASME Fluids Engineering Division Summer Meeting cosponsored by ASME, JSME, and EALA, Hilton Head Island, South
Carolina, August 13-18, 1995. M. Ehtesharn Hayder, Institute for Computational Mechanics in Propulsion, NASA Lewis Research Center;,
Thomas Hagstrom, Institute for Computational Mechanics in Propulsion, NASA Lewis Research Center and Department of Mathematics and
Statistics, The University of New Mexico, Albuquerque, New Mexico 87131 (work funded under NASA Cooperative Agreement NCC3-370).
ICOMP Program Director, Louis A. Povinelli, organization code 2600, (216) 433-5818.
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13. ABSTRACT (Maximum 200 words)
We present a formulation of boundary condition for flows with small disturbances. We test our methodology in an
axisymmetric jet flow calculation, using both the Navier-Stokes and Euler equations. Solutions in the far field are
assumed to be oscillatory. If the oscillatory disturbances are small, the growth of the solution variables can be predicted
by linear theory. We use the eigenfunctions of the linear theory explicitly in our formulation of the boundary conditions.
This guarantees correct solutions at the boundary in the limit where the predictions of linear theory are valid.
14. SUBJECT TERMS
Nonreflecting boundary condition; Computational aeroacoustics; Jet flow computations
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An outflow boundary condition for aeroacoustic computations

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