NASA Technical Reports Server (NTRS) 20030012929: A Comparison of Ffowcs Williams-Hawkings Solvers for Airframe Noise Applications

12 Pages·0.95 MB·English

by  NASA Technical Reports Server (NTRS)

AIAA 2002-2580
A Comparison of
Ffowcs Williams-Hawking;s Solvers
for Airframe Noise Applications
David P. Lockard
NASA Langley Research Center
Hampton, VA 23681
8th AIAA/CEAS Aeroacoustics Conference
June 17-19, 2002
Breckenridge, CO
For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics
1801Alexander Bell Drive, Suite 500, Reston, VA 20191-4344
AIAA-2002-2580
A COMPARISON OF FFOWCS WILLIAMS-HAWKINGS SOLVERS
FOR AIRFRAME NOISE APPLICATIONS
David R Lockard*
NASA Langley Research Center
Hampton, VA
This paper presents a comparison between two imple- Cartesian surface velocity components
Vi
mentations ofthe Ffowcs Williams and Hawkings equation x,y,z Cartesian observer coordinates
for airframe noise applications. Airframe systems are gen-
Greek:
erally moving at constant speed and not rotating, so these
mapped surface coordinates
conditions are used in the current investigation. Efficient OZl, OZ2
$ 4]-_ M2
and easily implemented forms of the equations applica-
ble to subsonic, rectilinear motion of all acoustic sources 5(f) Dirac delta function
are used. The assumptions allow the derivation of a sim- 5_j Kronecker delta
ple form ofthe equations inthe frequency-domain, and the P fluid density
time-domain method uses the restrictions onthe motion to T retarded or emission time, t - r/co
Source coordinates
reduce the work required to find the emission time. The
comparison between the frequency domain method and the Superscript:
retarded time formulation reveals some of the advantages ' perturbation quantity (e.g. J p- po)
of the different approaches. Both methods are still capable unit vector
of predicting the far-field noise from nonlinear near-field time derivative
flow quantities. Because of the large input data sets and
Subscript:
potentially large numbers of observer positions of inter-
est in three-dimensional problems, both codes utilize the emission or retarded quantity
o freestream quantity
message passing interface to divide the problem among
different processors. Example problems are used to demon- r inner product with _i
ret quantity evaluated at retarded time _-
strate the usefulness and efficiency of the two schemes.
,r_ inner product with _i
Nomenclature
co ambient speed of sound Introduction
f integration surface defined by f 0
Despite recent advances in computational aeroacous-
F_ dipole source terms
tics, numerical simulations that resolve wave propagation
H Heaviside function
from near-field sources to far-field observers are still pro-
hibitively expensive and often infeasible. Integral tech-
Mi local source Mach number vector, vile
niques that can predict the far-field signal based solely on
M Machnumber,I_l
near-field input are ameans to overcome this difficulty.
_i Outward directed unit normal vector
The Ffowcs Williams and Hawkings ] (FW-H) equation
p pressure
is a rearrangement of the exact continuity and Navier-
Q monopole source term
Stokes equations. The time histories of all the flow vari-
Qi components of vector in Eq.(6)
ables are needed, but no spatial derivatives are explicitly
required. The solution to the FW-H equation requires a
ri radiation vector, xi - (_
surface and avolume integral, but the solution is often well
r magnitude of radiation vector, Ir_l
approximated by the surface integral alone. Singer et al.2
t time
and Brentner and Farassat 3have shown that when the sur-
ui Cartesian fluid velocity components
face is inthe near field of asolid body, the FW-H approach
*Research Scientist, Senior Member, AIAA correctly filters out the part of the solution that does not
Copyright @ 2002 by the American Institute of Aeronantics and As- radiate as sound, whereas the Kirchhoff method produces
tronautics, Inc. No copyright is asserted in the United States under Tire erroneous results. Many applications of the FW-H and
17, U.S. Code. The U.S. Government has aroyalty-free license to exercise Kirchhoff methods can be found in the area of rotorcrafl
all rights under the copyright claimed herein for Governmental Purposes.
acoustics.4 7The FW-H method has typically been applied
All other rights are reserved by the copyright owner.
AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS PAPER
AIAA-2002-2580
by having the integration surface coincide with solid bod- by Co. A prime is used to denote a perturbation quantity
ies, but the method is still applicable when the surface is relative to the free-stream conditions denoted by the sub-
off the body and permeable. The codes developed in this script o. The Cartesian coordinates and time are xi and
work are valid for both cases. t, respectively. The usual convention, which is followed
For three-dimensional flows, the time-domain FW-H here, involves a quiescent ambient state with f prescribed
formulations developed by Farassat 8 are efficient and as a function of time so that it always surrounds a moving
amenable to numerical computations. Some simplifica- source region of interest. H(f) is the Heaviside function
tions are applied here based on restrictions of the surface which is unity for f > 0and zero for f < 0. The deriva-
motion, but the development follows that of Farassat very tive of the Heaviside function H' (f) 5(f) is the Dirac
closely. The frequency-domain version of the FW-H uses delta function, which is zero for f ¢ 0, but yields a finite
the form of the equations developed by Lockard 9for two- value when integrated over a region including f 0. The
dimensional problems. However, the derivation was done inviscid part, Pij pSij, of the compressive stress tensor
using Cartesian tensor notation, and is equally valid in Pij is used inthis work.
three-dimensions as long as the appropriate Green function Equation 1 is typically solved using a Green function
is employed. technique. The temporal and spatial convolution of the
The remainder of the paper describes the time- and free-space Green function with the source terms yields the
frequency-domain formulations in enough detail to show solution for p'. In the farfield, p' c_p'. The three-
the similarities and differences between the methods. A dimensional Green function 1°is
discussion of various parallelization strategies follows. Fi-
nally, two example problems are used to demonstrate the
47rr
utility and efficiency of the schemes. The last example
involves a noise calculation based on a large-scale CFD where ri xi - _; is the radiation vector, and r IriI.
computation of a landing gear. Because of the delta function, the solution to equation 1
can be manipulated into various forms, some of which are
Governing Equations amenable to numerical solution such as formulation 1A of
Farassat. 8 An alternative approach is to solve the FW-H
The FW-H equation can be written in differential form 1°
equation in the frequency domain. In this work we fol-
as
low the frequency-domain approach of Lockard 9 that is
restricted touniform, rectilinear motion, but other methods
C2__
can be used that still allow general motion. Inthe following
sections, the current implementations ofthese methods are
discussed.
OxiOxj TijH(f)
( )o( )
Time-Domain Method
(9 FiS(f) + QS(/) (1)
a:ci A concise, time-domain solution to equation 1thatis ap-
plicable tonondeforming, porous surfaces can be obtained
where
from the derivation of Farassat ]]using the variables
TU pu;uj + Pij - C2op,6;j, (2)
Qi (po - p)vi + pui, and
Fi Pij + pui(uj - vj) --, and (3)
Oxd
as proposed by Di Francescantonio. 6 The integral solution
Q poVi + p(ui - vi) axi" (4)
is given by
The contribution of the Lighthill stress tensor, T_j, to the
47rp'(xi, t) (7)
right-hand side is known as the quadrupole term. The
[
dipole term F_involves an unsteady force, and Q gives rise
ds
to a monopole-type contribution that can be thought of as
an unsteady mass addition. The function f 0 defines f 0 Ji'et
the surface outside of which the solution is desired. The
normalization IVfl 1is used for f. The total density
and pressure are given by pand p, respectively. The fluid f 0 ret
velocities are ui, while the vi represent the velocities of
the surface f. The Kronecker delta, 5ij, is unity for i j
/ VLj_'J(rJ_J'_J_C_(MJ'_J-M2)) I ]),
and zero otherwise. The ambient speed of sound is denoted
f 0 ret
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where p_ is the quadrupole term, which isneglected in this reuse, loop unrolling, and inlining of the interpolation rou-
work. Mi vi/Co isthe local Mach number vector, and the tine decreased the total run time by half.
superscripted caret (A)denotes a unit vector. The outward Once the kernel functions in equation 7have been deter-
directed unit normal vector is given by _zi. The subscript mined, Gaussian quadrature is used to perform the spatial
ret denotes evaluation at the retarded or emission time q-. integration over the surface. Because this integration is
For uniform, rectilinear motion, the normals are not func- performed repeatedly for each observer and retarded time,
tions of time, so finding Q,r_ Qi_zi and Qi_zi is sufficient parts of the integral that are independent of time and ob-
because a/at(@_) %a/at(@) + @a/at(%) _)i%. server position are precalculated and stored. The integra-
The dotabove avariable indicates atime derivative. Hence, tion is performed by linearly interpolating the data from
only four variables and their time derivatives are required the four corners of each cell using three-dimensional shape
to compute the kernel functions in equation 7. The time functions commonly employed in finite elements. 13Defin-
derivatives are computed using fourth-order, central differ- ing the mapped coordinates as (c_1,c_2),the elemental area,
ences. In the current application, the coordinates of the dS, is the magnitude of the normal as specified in equa-
surface and the flow field data are known at the grid points tion 8. The value of the elemental area at each Gaussian
of meshes on a series of surface patches that comprise the quadrature point is then stored for reuse. The values of the
complete, closed surface surrounding all sources. The nor- shape functions at each of the quadrature points are also
mals are determined using the grid metrics onthe patches, precalculated and stored. Typically, only four quadrature
which are computed using fourth-order, central differences. points are required over each cell. Four points will integrate
Because each surface patch has two independent indices, quadratic functions exactly. Although the function being
only two variables (ctl, ct2) are required to describe the integrated may be of higher order, a quadratic approxima-
variation on the mapped surface. The normal is computed tion is usually sufficient provided the surface shape is well
from the outer product behaved. Additional quadrature points and higher-order re-
constructions within each cell have not been found to be
very beneficial. Part of the reason may be the oscillatory
nature ofthe kernel function. The integration of alinear ap-
proximation of a sinusoid is often not much different from
Whether the normal points inward or outward is dependent the quadrature ofahigher-order fit. Grid refinement istypi-
on the ordering of the data. The current code reads a file cally much more influential onthe solution quality than the
which either specifies a source point within each surface order of the integration.
or the correct sign of the normal on each surface. When Because of the need to sample the source data at the re-
the surfaces come from CFD data, it is usually possible to tarded time, one cannot arbitrarily choose observer times.
deduce which direction the normal will point. However, a A separate code was written to determine the valid range of
visual inspection of the normals is always a good practice observer times based on a given range of input data. The
because incorrect signs are common and can cause errors Garrick triangle 12can be used to uniquely determine the
that are difficult to recognize. minimum and maximum reception times using equations 9
One of the more computationally intensive parts of a to solve for t given q-.
time-domain calculation is the determination of the re-
tarded time q-.For general motion, a root-finding technique Impenetrable Surfaces
must be employed. However, for uniform, rectilinear, sub- In many practical applications, the data for the FW-
sonic motion, the Garrick triangle ]2 uniquely determines
H solver is obtained on solid surfaces, and the equations
the emission time as
can be greatly simplified thereby decreasing the computa-
tional and memory requirements. The simplifications are
__ 1---M_ Mcos(O)+ _/1 - M_si.2(0) especially advantageous for the uniform, rectilinear motion
case. For impenetrable surface data, /9 vj and equations
r t - re(cid:0)Co (9)
6simplify to
where re is the distance between the source and the ob-
Qi poVi and Li P_i. (10)
server at the time of emission q-. The angle between the
surface velocity vector vi and the radiation vector ri is 0. Note that Q is independent of time. Furthermore, the nor-
The cos(0) can be determined from the inner product of the mals _zjare only a function of space. Hence, only the time
vectors. The variables Q,_, Li and their time derivatives history of the pressure and its time derivative are needed
must then be interpolated to the retarded time. This is an when the viscous terms are neglected in P_j. All of the
intensive part ofthe calculation because the interpolation is source terms are linear, yet the method has been shown to
performed for every grid point, retarded time, and observer give correct results when the surface data is in the nonlin-
position. Changing the indexing of data to increase cache ear near field.2 The failings of the Kirchhoff method for
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AIAA-2002-2580
problems with solid surfaces isclearly not due to its source equation 12becomes
terms being linear, but rather the assumptions used to de-
rive the equation.
02 + k2- 2iM/k O0y_
Frequency-Domain Method
0 2
Several frequency-domain formulations of the FW-H
equation have been reported in the literature. 14 Here, the
method proposed by Lockard 9 for two-dimensional flows
will be extended to three-dimensions. Although this de- Oy_
velopment is restricted to surfaces in rectilinear motion at
constant speed, it is still useful for airframe noise, where Oy0_2Oyj (T/j(y,c_)H(f)). (15)
these assumptions are usually valid. The motion of the sur-
face is assumed to be governed by f f(x + Ut) where The wavenumber is defined by k c_/co, the Mach num-
the components of U are constant velocities describing the ber M U/co, and complex number i xfZT. Note that
motion of the surface. An application of the Galilean trans- the transform has been applied tothe groupings T/j, F/, and
formation from (x, t) to (y, t), Q because the equation is linear in these terms. However,
the desirable properties of the FW-H are maintained be-
Yi xi + Uit, t t,
cause all of the nonlinear products are included before the
0 0 0 0 0
transformation is applied. In a numerical implementation,
ax_ aye' at at + u/o7'7y__._ (11)
the products are formed first, and then a fast Fourier trans-
form (FFT) is applied. As a caution, the FFT must use the
to equation 1 leads to the convected wave equation
sign convention of equations 14, or the derivation must be
modified appropriately. The Green function for equation
02 + U/Uj Oy0iO2yj (12) 15 when M < 1can be obtained from a Prandtl-Glauert
transformation. Denoting the three-dimensional source co-
02 2 02
ordinates as (_,r/,4), and the observer position as (x,y,z) the
Green function is is
-1
OyiOyj G(x, y, z; _,q,4) _ exp( ik(d MT)/_ 2)
where, after the transformation, the F/and Q,r_become where
(x - g) cos c_cos ¢ + (y - q) sin c_
+ (z - 4)cos_ sin ¢,
- (x - g)sin ctcos 0+ (y - q) cos ct
Q,r, (p(ui+U/)-poU/)¢zi. (13)
+ (z - 4) sin c_sin ¢,
The Lighthill stress tensor T/j isunchanged, and f f(y) -(. - _)_i. ¢ + (_- ¢)¢o_¢,
is now only a function of the spatial coordinates. The sur- and
face velocities vi have been replaced by -U/, which can
be inferred from inspection of f(x + Ut) 0. Note d I_2 +/32(_2 + _2) (16)
that this implies that the mean flow is in the positive direc-
tion (or equivalently that the surface moves in the negative The angles are defined such that tan ¢ W/U, sin ct
direction) when U/ > 0. The vi used throughout the time- V/M, and M _/U2+ V2+ W2/co. The Prandtl-
domain formulation are of opposite sign and should not be Glauert factor is/3 -_- - M 2. The solution to equation
15for M < 1can now be written as
confused with the U/. Equation 12 is now in a convenient
form to perform the Fourier analysis. With application of
the Fourier transform pair H(f)C2op'(y,c_) - f Fi(_,c_) OG(y;_) ds
Oy_
f o
oo
5C{q(t)} q(c_) / q(t) exp(-ic_t)dt and - / ic_Q,_(_,c_)G(y;_) ds
oo f o
oo
bc S{q(c_)} q(t) 27 q(c_) exp(ic_t)dc_, (14) - / Tij(_,cv)H(f) O_G(y;_) d_. (17)
f>o
oo
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As has already been stated, the quadrupole contribution, server distances fixed, so this is not a major issue. How-
represented by the last term in equation 17,istypically ne- ever, comparisons with actual flight data should include the
glected because its influence is often small. Furthermore, Doppler effects.
the calculation is somewhat involved and expensive. Cer-
Impenetrable Surfaces
tain flows exist where the quadrupole cannot be ignored,
such as those containing significant refraction of waves by As was shown for the time-domain formulation, the
shear layers and wakes. As long as the integration sur- frequency-domain version can also be significantly simpli-
face is placed outside of all regions where Tij is large, fied when the input data is obtained on solid surfaces. For
the quadrupole contribution issubstantially included by the impenetrable surface data, uj -Uj and equations 13
surface sources even though the quadrupole integration is simplify to
not performed. This is also true for the time-domain for-
mulation. Q -poU_¢_i and F_ P¢_i. (18)
The frequency-domain solution process involves calcu- Note that Q is steady in time and has no impact on the
lating the surface normals and forming the products in Fi frequency domain solution. Hence, only the time history
and Q for all time at each point on the surface just as in of the pressure is needed. One only needs to determine
the time-domain version. However, the Fi and Q,r_func- the Fourier transform of the pressure and scale that result
tions are Fourier transformed, and the surface integrations by the appropriate normal to obtain the Fi terms. The
are performed for each frequency of interest instead of for savings in memory and computational requirements are so
each observer time. An inverse FFT can be used to recover
great that the solid surface formulation should be employed
the acoustic signal in the time domain. For truly periodic whenever possible. In the current implementation, differ-
problems one merely uses a single period of the flow data as ent versions of two subroutines are called depending on
input to the FW-H code. However, for more complicated, the case being run. Although some additional coding isre-
aperiodic flows, windowing the data isrequired. The win- quired, run times can be reduced by 60% and the memory
dowing should be applied to Fi and Q,r_after their mean load by over 70%.
values are subtracted. The subtraction has no effect on the
calculated noise because the derivatives of G all contain Parallel Implementation
aJ, and equation 17 shows that there is no contribution to Although one normally thinks of acoustic analogy com-
the noise at aJ 0 when the quadrupole term is neglected. putations as being extremely efficient, calculations involv-
The minimal amount oftime data typically available from a ing long time records and many observers can quickly be-
computational aeroacoustics calculation may lead to some come expensive. The cost of computing the time history
inaccuracies in the windowed FFT, but short time records at a single point using the FW-H may actually exceed that
are often just as much of an impediment for time-domain of a standard CFD method. However, the FW-H approach
formulations because information about the frequency con- allows the observer location to be anywhere outside of the
tent is usually desired. source region, whereas the CFD method must have grid
When the input to FW-H code is from a harmonic, lin- points from the source to the observer. The cost involved
earized Euler solver, Fi and Q,_ should also be linearized with all those intermediate points and the errors incurred
because the amplitudes from the linearized code may not in the long range propagation make standard CFD an in-
be physical. If they are too large, the nonlinearities in appropriate choice for most far-field noise computations.
the source terms can produce erroneous results. One must Still, when mapping out the directivity inthree-dimensions,
be careful when performing the linearization because the hundreds of observer locations may be required. In the re-
perturbation velocities ui are not necessarily small. For in- alistic problem of alanding gear given inthe examples sec-
stance, ona solid surface ui -Ui. Only aminor change tion, the FW-H computation of a single observer using the
in the code allows one to have a single code that is use- porous surface formulation requires seven minutes on an
ful for input from linear and nonlinear flow solvers. The SGI 250MHz R10K processor. The input record contains
frequency domain approach is particularly efficient forhar- 4096 time steps at 82,219 grid points, which consumes 1.4
monic data because only a single frequency needs to be GB of disk space. Mapping out a directivity is atime con-
calculated, and the FFT's donot need to be performed. suming process when performed serially. This motivated
A disadvantage of this particular frequency-domain for- the modification of the code to use the Message Passing
mulation is that the source and observer are always a fixed Interface (MPI) to perform parallel computations. Other
distance apart, and all Doppler effects are lost. In a time- researchers 16have used MPI in conjunction with acoustic
domain calculation, the distance between the observer and analogy methods.
the source can be changed for each time step to simulate FW-H solvers are ideal candidates for parallelization be-
a flyover condition. Most CFD computations and exper- cause the calculation of the signal at each observer is in-
iments are carried out in a laboratory frame with the ob- dependent, and the contributions from each portion of the
AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS PAPER
AIAA-2002-2580
data surface combine linearly. Even the computations at Test Examples
each time step or frequency are independent. Hence, one
has many choices of how to split the problem. Some initial
testing was done with different processors dealing with dif- Monopole inFlow
ferent observers. However, this requires that each processor As a first demonstration of the current implementations
have access to the entire data record for all the surface of the three-dimensional FW-H equation, the field from
patches. In the landing gear case, 181 patches comprise amonopole source is computed in the far field using the
the total surface. Either all of the nodes have to read all present technique. The source moves inthe -x direction at
the data, or one node must read it and broadcast it to all Mach 0.5. An equivalent flow involves afixed source atthe
the others. The total record for the landing gear is expected origin in auniform flow inthe +x direction. The complex
to be over 10GB on a medium mesh, and over 60 GB for potential forthe monopole is given byDowling and Ffowcs
a fine mesh. Either reading or passing that much data is Williams 15as
not reasonable. This same problem occurs when different
processors handle different frequencies or time steps. The ¢(x,y,z,t) A _1 expi(_t k(d M_)//32) (19)
other option is to divide the problem by surface patches, 47rd
and sum all of the contributions at the end. Each proces-
The variables needed in the FW-H equation are obtained
sor only needs to read the data for the particular patches
from the real parts ofp' -po (O¢/Ot + UoOO/Ox), u_
assigned to it. Hence, the data is only read once and only
O¢/Oxi, and pl pl/c_. Equation 19 is written in a
passed if the data isnot directly accessible byall the nodes.
laboratory frame where the flow is moving over a station-
ary source. The source terms in the FW-H equation are
Two paradigms were used to investigate the parallel im-
calculated from the flow variables evaluated over two pe-
plementation. First, a standard load-balancing approach
riods on the surface. For this case, M Uo/co 0.5,
was used. The size of each of the patches was read, and
_l/co 47r/46, A/(lco) 0.01 and the integration sur-
the largest patch assigned to the processor with the least face is a cube that extends from -51 to 51in all three coor-
points until all the patches were assigned. This strategy is
dinate directions. The reference length is1.Fifty uniformly
commonly employed in multiblock CFD codes. This ap-
spaced points are used on each side of the box. Figure l(a)
proach worked well when all ofthe processors were identi-
compares the directivity from the calculations to the ana-
cal and dedicated. However, because the code only passes
lytic solution in the z 0 plane. Figure l(b) makes a
a very minimal amount of data, it is typically run over a
similar comparison for the time history at (501, 0,0). The
standard network on the second processor of SGI Octane
agreement isexcellent, demonstrating that the formulations
workstations in the lab. These machines vary in speed,
are valid for problems with a uniform mean flow. Simi-
and sometimes both processors are in use when the job
lar agreement was found when each of the six sides of the
starts. Occasionally, one ofthe processors would take much
cube comprising the data surface were deformed to look
longer than all of the others to complete. To circumvent
like a Gaussian bell provided enough points were provided
this problem, the master-slave paradigm employed byLong
to adequately resolve the variation. The calculations were
and BrentneP 7was used. Here, one node does no work
performed in single precision, which appears sufficient as
other than to tell requesting nodes what patch they should
long as the acoustic signal on the surface is not less than
process. Again, the largest patches are assigned first so
five orders of magnitude smaller than the mean. A lack
that small patches are being used when the job nears com-
of smoothness ina time-domain formulation solution is an
pletion and the variability in processor speed is important.
indication of aprecision problem.
This paradigm worked extremely well and has resulted in a
nearly linear speed up onan SGI Origin. Although the mas- Landing Gear
ter node does not do any useful work, it needs to respond This example involves the calculation of the noise gener-
quickly to the requests from worker nodes to keep them ated by the unsteady flowfield surrounding alanding gear.
from becoming idle. The master's job can be assigned to An aerodynamic and acoustic analysis of a similar land-
a slow node or to a dual processor machine that also has a ing gear was performed by Souliez et al.18The near field
slave process. in this problem is highly nonlinear, and different parts
shed vorticity at different frequencies. Figure 2(a) shows
In a heterogeneous environment where one is trying to an instantaneous snapshot of the perturbation pressure on
use idle machines, the master-slave paradigm is preferred. all of the solid surfaces. The freestream Mach number
Furthermore, the load-balanced approach and master-slave is 0.2, and the gear is mounted on a fiat plate. The ref-
approach only affect a single subroutine in the code, so itis erence length is the diameter of the wheels which is 3.7
very easy to switch between the two depending onthe local in (0.09398 m). The input data for the acoustic calcula-
operating environment. tion is obtained from athree-dimensional, time-dependent
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high-frequency sources that are growing through most of
the time record. Some of the waves from these sources can
be seen on the door in Figure 2(a). Figure 2(b) shows the
time history of the pressure on the oleo in the contraction
just below the door as indicated bythe arrow inFigure 2(a).
The time history shows the complex, intermittent charac-
ter of the signal. The large amplitude oscillations occur
around one kHz (model scale) at varying intervals and ir-
regular durations. Riding on top of the signal are high
frequency oscillations generated by resonances in small,
triangular shaped spaces between the yokes and the door.
These spaces are found on the upper and lower yokes in
both the upstream and downstream junctures with the door.
Unfortunately, the signals took a long time to saturate, so
the calculation had to be run much longer than anticipated
to eliminate the transients. The upstream and downstream
cavities are slightly different in size, so the resonances oc-
cur at 20.5 kHz in the upstream cavity and at 25.4 kHz in
(a) Directivity at r' 50l the downstream one.
Beyond the intermittency in the signal, another compli-
cation for obtaining spectral information about the solution
8E-05
is the predominantly low frequency content of the gear.
6E-05 Very long sampling times are needed to resolve these fre-
quencies. Although the time-domain code can exactly cal-
4E-05
culate the signal at the observer, the lack of long sampling
2E-05 times is a problem when the spectral content is needed.
Nonetheless, the comparison of the spectra at an observer
directly beneath the gear from the two FW-H codes is en-
couraging. The observer is 12l away from the gear. Figure
-2E-05
3(a) shows that the spectral content is nearly identical. In
-4E-05 both cases, a hamming window was applied to minimize
the effects of the aperiodic signal on the FFT's. The time
-6E-05
histories in Figure 3(b) are also similar. The curves are
-8E-05 offset because the frequency-domain calculation does not
0 20 40 60 include the effect of the zero or steady mode. All of the
Time
primary features of the signal can be observed in both re-
(b) Time history at (50/, 0,0) sults. Some discrepancy should be expected because the
Fig. 1 Solution comparisons for amonopole in a M 0.5 effect of the window is already included in the frequency
flow. Thepressure isnondimensionalized bypoC_ domain results because the window is applied to the input
data. However, the window is applied to the time-domain
CFD solution using the code CFL3D. 19,2°CFL3D was de- results after the calculation, so its effect is not seen in Fig-
veloped at NASA Langley Research Center to solve the ure 3(b).
three-dimensional, time-dependent, thin-layer Reynolds- Three different data records were used to investigate the
averaged Navier-Stokes (RANS) equations using a finite- influence of the duration and the particular time interval
volume formulation. The CFD data used in this work is used. Because of the complexity of the time histories on
discussed inmore detail inthe paper by Li et al.21 the gear, some variation should be expected, but drastic
The noise calculation involves 181 total patches com- changes would indicate an improper sample length or that
prising the data surface. All of the patches are on solid the flow has yet to eliminate transients. These sorts of vari-
surfaces, so only the pressure histories are needed. 147 ations were observed in many of the initial calculations.
patches are on the gear itself and 31 are on the plate above However, the comparison in Figure 4shows only minor dif-
the gear. So far, over 24,000 nondimensional time samples ferences between the three calculations. A time sample of
of//coAt 0.02 have been collected, but only half of the 8192 was used, which represents the time required for the
data is sufficiently free of transients to be useful for acous- flow topass by awheel 32times. The fullrecord was used,
tic calculations. The transient problem is exacerbated by then it was subdivided into two records of 4096 samples.
AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS PAPER
AIAA-2002-2580
1.0E-01
9.0E-02 -- Time-Domain
........................F..r.e.quency-Domain
8.0E-02
M 0.2
7.0E-02
6.0E-02
G.. 5.0E-02
_,- 4.0E-02
pIJ(OoC2_ 3.0E-02
i
2.0E-03
iiiiiiiiiiii1.0E-03 2.0E-02
0.0E+00
1.0E-02
-l.0E-03
-2.0E-03
ii
0.0E+001 10J 104
Frequency (Model Scale Hz)
(a) Perturbation pressure on solid surfaces (a) Spectra
e
0.7020 2.0E-05
1.5E-05
0.7000
1.0E-05
0.6980
_ 5.0E-06
@_ 0.6960 .'<
_'0.0E+00
0.6940
-5.0E-06
0.6920
-1.0E-05
0.6900 ,,,I,,,,I,,,,I,,,,I,,,,I,, ,I,,,,I,,,,I,,,,I
0 100 20T0ime 300 400 0 10 20 30 40Time 50 60 70 80 90
(b) Time history onthe oleo (b) Time history
Fig.2 CFD results for alanding gearinaM 0.2flow. Fig.3 FW-H results for anobserver directly belowalanding
gearin aM 0.2 flow.
The comparison between the results for the full record and
the latter record isvery good. Somewhat more discrepancy plate in the CFD calculation coarsens very quickly away
is observed with the results for the first 4096 samples, but from the gear, sothe accuracy of high frequency signals is
it is still within the variation that should be expected for likely to be very poor over much of the plate. To investigate
such a complex flow. Even though there are not as many the impact of the accuracy on the plate, the FW-H solvers
samples available from the computation as one can obtain were run with the wall ignored and the gear surfaces mir-
from an experiment, the power spectral density should be rored about the plate. Figure 5shows the comparison with
calculated by averaging over the different samples to obtain results obtained by including and excluding the wall. At
a single answer, which should be stationary. lower frequencies, the case including the wall has much
One of the concerns with the CFD calculation was the higher levels than either the no wall or mirrored cases in-
accuracy ofthe solution on the wall above the landing gear. dicating that the signal on the wall is an important source
Although the plate doesn't produce any significant fluid of noise. The results in the figure are from the frequency-
mechanic fluctuations that would act as sound sources, it domain code, but very similar results were obtained from
does reflect acoustic signals, and some of the vorticity shed the time-domain code. Figure 5(b) zooms in on the high
by the side bars interacts with the wall. The grid on the frequency tones. For the tones, the wall and no wall re-
AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS PAPER
AIAA-2002-2580
1.0E-01 4 1.0E-01
ii Sample Interval
9.0E-02 ii -- 20155-24250 9.0E-02 -- AllSurfaces
M.............1..6..0..5..9..-.2..0..154 ........................N..o..Wall
8.0E-02 .... 16059-24250 8.0E-02 .... Mirrored
7.0E-02 7.0E-02
6.0E-02 6.0E-02
o.. 5.0E-02 o.. 5.0E-02
_- 4.0E-02 _- 4.0E-02
3.0E-02 3.0E-02
2.0E-02 2.0E-02
1.0E-02 1.0E-02
0.0E+001 0.0E+0_ 10 104
Frequency (Model Scale Hz) Frequency Model Scale Hz)
Fig. 4 FW-H results for anobserver directlybelow alanding (a)
gear ina3// 0.2 flow. Calculations from frequency-domain
code comparing effect of samplelength. 2.5E-01
sults are nearly identical which supports the idea that the
high-frequency signals on the wall are artificially damp- 2.0E-01 -- AllSurfaces
ened rapidly so that they have little impact on the acoustic ........................N..o..Wall
.... Mirrored
calculation. The mirrored result is lower at 20.5 kHz and
significantly higher at 25.4 kHz. The variability in the _D 15E-Ol II
effect of mirroring with frequency is likely caused by in- o_ 'f/_I1
terference effects which would be minimized for observers
farther from the gear. _, 1.0E-01 ] II
Because of the complexity of the landing-gear geometry
and the various locations of the sources, one would expect
the observer location to be important. Figure 6 compares 5.0E-02_ II 1
the spectra for observers at three locations. All of the ob- 0.0E+00
servers are 12/from the gear. The first location is directly 20400 20600 25000 f 25I500 26000
beneath the gear. The other two observers are at the same Frequency (Model Scale Hz)
vertical and streamwise locations as the wheels, but on op-
(b)
posite sides. The observer on the oleo side doesn't have
the door in the way to obscure some of the sources on the Fig.5 FW-H results for anobserver directly belowalanding
oleo and yokes. This ismost evident for the 25.4 kHz tone gearin a3// 0.2flow. Results from frequency-domain code
in Figure 6(b). Both side observers see much stronger tone comparing the effect ofincluding the wall and mirroring the
data.
signals indicating that the yoke resonances primarily radi-
ate horizontally. Part of the reason that the strength of the
Performance
tones observed underneath the gear is much less may be
because of blockage by the truck and wheels. The 20.5 In the landing gear calculations, all of the frequencies
kHz tone has the same amplitude for both side observers. were calculated in the frequency domain code, and an
The reason for the lack of influence of the door onthis tone equivalent number of time steps in the time domain code.
has not been investigated. At lower frequencies, the signal Table 1shows timing results for the serial runs with 4096
beneath the gear has a larger amplitude indicating that the samples and all 181 surfaces. These calculations involved
wheels and truck are best observed from below. The por- solid surface data, so the impenetrable surface simplifica-
tion of the signal around 1.4 kHz is completely absent for tions were employed. Run times are more than doubled if
the side observers. As expected, there isa significant direc- the surface isassumed to be penetrable. All of the frequen-
tivity associated with the gear, but the complex interplay cies were calculated by the frequency-domain code, and an
of source location, geometry, and observer position needs equivalent number of observer times in the time-domain
further investigation. code. When all of the data is computed, the frequency-
AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS PAPER