Inverse Problems in Unsteady Aerodynamics and Aeroacoustics

Home , Inverse problem

INVERSE PROBLEMS IN UNSTEADY AERODYNAMICS AND AEROACOUSTICS

y z

Sheryl M. Patrick and Ha z M. Atassi

Aerospace and Mechanical Engineering Department

University of Notre Dame

Notre Dame, Indiana

and

William K. Blake

David Taylor Mo del Basin

C.D./N.S.W.C.

Bethesda, Maryland

ABSTRACT duce uctuating pressure forces on the structural com-

p onents which leads to vibration and noise.

The feasibility of determining an unknown vortical

disturbance in an approach ow to a streamlined solid In order to either reduce or eliminate suchunwanted

b o dy from the radiated sound is studied. The problem e ects, the aero dynamics and aeroacoustics of the prob-

is treated in two separate parts. First, the inverse aero- lem need to be understo o d. This entails determining

dynamic problem, wherein the upstream disturbance which are the key parameters resulting in ecient or

is determined from unsteady surface pressure, is con- nonecient sources of vibration and noise and optimiz-

sidered. It is shown that the transverse comp onentof ing them for vibration and noise control.

an incoming vortical disturbance can b e uniquely pre-

To this end, the study of the aero dynamic prob-

dicted from the surface unsteady pressure. This result

lem asso ciated with impinging ow nonuniformities on

can b e extended for incoming turbulence in terms of the

a streamlined b o dy such as wings or blades, known as

surface pressure density sp ectrum. The second problem

the gust problem, has b een the fo cus of considerable in-

treated is the inverse acoustic problem, wherein the un-

terest in unsteady aero dynamics. The direct gust prob-

steady surface pressure is determined in terms of the

lem is to calculate the unsteady pressure distribution

radiated sound. It is shown that for a single frequency,

along the airfoil/blade surface knowing the upstream

the governing equation reduces to the Helmholtz equa-

disturbances and the shap e of the b o dy. The rst treat-

tion, and the solution to the inverse acoustic problem is

ment of this problem in incompressible owwas given

unique. However, the problem remains ill-p osed and ex-

by Sears 1941. A recent review on the sub ject was

hibits extreme sensitivity to noise in the far- eld data.

given byAtassi 1994.

The singular value decomp osition metho d and a simple

Another e ect asso ciated with the interaction of a

regularization metho d are used to solve the discretized

nonuniform ow and a solid body is the radiation of

system of equations resulting from the inverse acoustic

acoustic sound. Atlow Machnumb er, the direct acous-

problem. The simple regularization metho d is, however,

tic radiation problem can be treated using Lighthill's

more accurate.

acoustic analogy Lighthill, 1952 and the Ffowcs Wil-

liams-Hawkings equations Ffowcs Williams and Hawk-

INTRODUCTION

ings, 1969. In this treatment the unsteady pressure

Aircraft wings and propulsive system structural com-

along the airfoil/blade surface is equivalent to a dip ole

p onents often op erate in nonuniform ow conditions.

distribution. More recent analyses by Amiet 1976 and

The nonuniformities arise from atmospheric and inlet

Atassi et al. 1990 derive the far- eld sound directly

turbulence, viscous wakes, secondary ows, and rotor/

from the unsteady aero dynamic solutions. These treat-

stator interaction. Upstream ow nonuniformities pro-

ments consider dip ole and quadrup ole e ects. The lat-

ter b ecome signi cant as the mean- ow Machnumber

Fellow, Center for Applied Mathematics

and the reduced frequency increase and as the mean

Professor

airfoil/blade loading increases Atassi et al., 1993.

Research Scientist 1

ularization scheme. The Tikhonovscheme can b e used Another imp ortant asp ect of active control of vibra-

alone to obtain a solution or in conjunction with the tion and noise is sensing and analyzing data to predict

single value decomp osition to alleviate the dep endence their sources. This is the fo cus of the work presented

on very small singular values. in this pap er. Here, the feasibility of inverse aero dy-

namic and aeroacoustic problems is considered. The

The inverse problem discussed in this pap er, dif-

inverse problem allows for new metho ds of determining

fers from the inverse scattering problem investigated

the origins of unwanted vibration and noise. For the

by those mentioned ab ove. Here the incident eld is

aero dynamic problem, the pap er studies the feasibil-

unknown, although it is assumed to b e either vortical

ity of determining the upstream disturbances from the

or acoustic. The shap e of the b o dy is known, however

uctuating unsteady pressure on a moving streamlined

the unsteady pressure along the b o dy is not given. De-

body. The aeroacoustic problem concerns the feasibil-

termining the b oundary condition on the b o dy, i.e. the

ity of determining the surface unsteady pressure from

unsteady pressure, from far- eld measurements will b e

the radiated far eld. Ultimately b oth inverse problems

referred to as the inverse acoustic problem, while de-

can b e combined to analyze the upstream disturbances

termining the incident disturbance from the b oundary

from the radiated sound.

conditions on the b o dy will b e termed the inverse gust

problem. In spite of the imp ortance of these inverse problems

to applications and, in particular, to control and sens-

The problem of determining unknown b oundary con-

ing, they have received much less attention than the

ditions for a given ob ject from far- eld measurements

direct problem. Like most inverse problems, these are

has b een extensively investigated in conjunction with

ill-p osed in the Hadamard sense; that is, a problem is

acoustic holographyVeronesi and Maynard, 1989, Kim

ill-p osed if one of the following criteria fails: a solution

and Lee, 1990, Sarkissian et al., 1993. These problems

exists, it is unique, it depends continuously on the data

di er from the present problem in that they do not con-

Hadamard, 1923. These issues will be considered in

sider a mean owvelo city.

regard to the present application.

Rayleigh, in 1896, showed that a source of sound

THE DIRECT PROBLEM

could not be determined uniquely from the far- eld

In order to gain a fundamental understanding of is-

sound Rayleigh, 1896. This statement is true for a

sues involved with our inverse problems, we restrict our-

general sound source whose propagation is governed by

selves to a at-plate airfoil in subsonic ow with vortical

the wave equation. For sound which is pro duced by

disturbances imp osed upstream. For the at-plate air-

a disturbance which is harmonic in time, however, the

foil, the mean ow remains uniform. The formulation

governing equation reduces to the Helmholtz equation

for the direct problem of a at-plate airfoil in a subsonic

and inverse solutions to the Helmholtz equation may

ow with convected vortical gusts is given in Atassi et

b e unique. The inversion of problems governed by the

al., 1990, where the unsteady pressure jump across the

Helmholtz equation has b een studied quite extensively

at plate is calculated using an integral equation and

in recent years. This recent interest can be traced to

then the scattered eld is calculated using Green's the-

an article by Kac 1966 entitled, \Can One Hear the

orem. The formulation is summarized here for clarity

Shap e of a Drum?". The emphasis in inverse acous-

and for reference.

tic scattering problems has b een on surface reconstruc-

tion. Many researchers have attempted to use the far-

The Gust Direct Problem

eld acoustics, pro duced by the interaction of a known

Assume the owisinviscid, and non-heat conduct-

acoustic wave and an unknown body, to reconstruct the

ing. When the upstream disturbance is small, i.e. the

b o dy surface This research has b een motivated by ap-

owisweakly rotational, the velo city, pressure and den-

plications in tomography, geophysics, optics and sonar.

sity can b e linearized ab out their mean values Atassi,

It has led to the establishmentofseveral theorems p er-

1994.

taining to the existence and uniqueness of solutions to

b oth direct and inverse scattering problems governed by

the Helmholtz equation. Researchers studying inverse

~ ~

acoustic scattering have proven that a radiating solu-

U ~x; t = U + ~u~x; t 1

tion to the Helmholtz equation can b e obtained uniquely

p~x; t = p ~x+p ~x; t 2

from the far- eld data, provided a radiating solution ex-

ists Colton and Kress, 1992.

~x; t = ~x+ ~x; t 3

Even though the theorems guarantee uniqueness

when a solution exists, the diculty remains in prac-

In the absence of upstream incident acoustic waves,

tice to nd this solution. In many cases, the formulation

the upstream ow can b e written as

leads to a Fredholm integral equation of the rst kind.

To solve this integral equation, usually, one has to invert

~ ~

an ill-conditioned matrix. Hence, existing metho ds for

U ~x; t=U i + ~u ~x i U t 4

1 1 1 1 1 1

solving very ill-conditioned matrices have b een further

where ~u represents the upstream rotational distur-

explored and new techniques have b een found Kress,

1989. The most widely used metho d found in the

bance in the ow and i is a unit vector. The axis and

literature is the singular value decomp osition metho d.

ow direction are shown in Figure 1.

Another metho d commonly used is the Tikhonov reg- 2 a

longitudinal 1

ik tk ~x

x ; 0;x =a e for 1

2 u

a2 1 3 2 1 k

Along the vortex sheet extending downstream from the

trailing edge, u is discontinuous, but 1 x a 2

p = 0 k 4 1 x

a 1

u = 0 x > 1 and x =0 10

2 4

a2 1 2 x 3

transverse

0 0 0

where 4p signi es p p .

+ U

In the direct problem, the vortical disturbance is

known, i.e. 9 is given, and then the unsteady pressure

|a| flat plate wake

on the surface is found by solving the b oundary value

oblique

problem 6-10.

or a transverse gust the pressure jump on a at-

c F

wwas given by Sears 2 plate airfoil in incompressible o k

3 as:

1 x

Figure 1: Flat plate in ow with imp osed vortical gusts

ik t 0

a k S k e 4p x ;k =2 U

2 1 1 1 1 o 1

1+ x

From the splitting theorem Goldstein and Atassi,

where

1976, Atassi, 1994 the total unsteady disturbance ve-

lo city is split into an acoustic, irrotational part ~u , and

12 S k =

a vortical, rotational part ~u which is convected by

2 2

k k iH k H

1 1 1

1 0

the mean ow. Since the problem is linear, ~u can b e

broken down into its Fourier comp onents and a single

is the Sears function

Fourier comp onent can be considered without loss of

generality. So that

The Acoustic Direct Problem

i!tk ~x

Once the unsteady pressure on the at-plate is known,

i + ~ae + ~u ~x; t 5 U ~x; t=U

1 a 1

the scattered eld can be determined using a direct

calculation based on Green's theorem. The governing

where ~a =a ;a ;a is the amplitude vector and k =

1 2 3

equation for the far- eld pressure radiated from an air-

k ;k ;k is the wave number vector describing ~u .

1 2 3 1

foil in compressible ow with a three-dimensional gust

Since ~u is convected by the mean ow, ! = k U . In

1 1 1

ik x

3 3

such that the dep endence on x is given by e is

this formulation, lengths are normalized with resp ect

develop ed as follows:

to the half chord, c=2, velo cities with resp ect to U ,

and time with resp ect to c=2U . Thus the reduced

Taking the material derivative of 6, subtracting

frequency is k = !c=2U . ~u is the p otential ow dis-

the divergence of 7, and using the isentropic relation-

1 1 a

~ 0 2

ik tk ~x

, c is the sp eed of sound, the governing ship p = c

0 0

turbance that results from the interaction of ~ae

equation b ecomes

with the airfoil.

The linearized Euler's equations b ecome

D 1

2 0

5 p =0 13

c Dt

0 2 2 2

@ @ @

+ 5~u = 0 6

where 5 = + + . Let

2 2 2

@x @x @x

1 2 3

D ~u

0 a

k M

= 5p 7

K = 14

where

k x~

3 3

ik t+MK x~

1 1 1

p = P x ;x e 15

1 2

D @ @

= + 8

x~ = x 16

1 1

Dt @t @x

x~ = x 17

2 2

And from continuity ~a k =0. The b oundary conditions

x~ = x 18 are

3 3 3

p 0.2

and = with M = 1 M . Substituting 14-

c 0.18 total effect

to 13 gives the two-dimensional Helmholtz equa- 18 in 0.16 dipole effect

0.14 tion

0.12

2 2 |C | sin

~ r p' 0.1

5 + K P =0 19 0.08

0.06

2 2 2

@ @

~ 0.04

where 5 = + and

2 2

@ x~ @ x~ 2 1 0.02 0

2 -0.1 -0.06 -0.02 0.02 0.06 0.1

2 2

K = K 20 |C | cos

1 r p'

2 0.6 total effect

dipole effect

en by the pressure on

The b oundary conditions are giv 0.5

the body surface and a radiation condition in the far

0.4

eld. Atassi et al. 1990 develop ed the solution to

r |C | sin

h. The solution is 19 using a Green's function approac p'

0.3 Z

1 0.2

~ ~

@Gx~ jy~ 1

4P ~y dy~ 21 P x~= 1

1 0.1

2 @ y~

where -0.3 -0.2 -0.1 0 0.1 0.2 0.3 i

2 r |C | cos

~ ~ ~

~ p'

H K jy~ x~j 22 Gx~ jy~=

is the two-dimensional Green's function. Here 4P rep-

Figure 2: Far- eld pressure directivity. top M =

resents the transformed pressure jump across the at

:1;k =25:0, b ottom M = :8;k =1:0.

1 1

~ ~

plate, x~ is the observation p oint and y~ is the source

p oint.

In the rst case, M = :1;k =25;K =2:52, the lob e

It is clear from 21 that once the unsteady pres-

1 1

is due almost completely to the dip ole contribution. In

sure jump along the airfoil is known, the pressure in

the second case, M = :8;k =1:0;K =2:22, the lob e

the far eld is calculated immediately. This direct ap-

1 1

exists due to quadrup ole e ects.

proach captures the e ects that subsonic ow has on the

propagation of the unsteady pressure to the far eld.

So along with capturing the radiation from the surface

THE INVERSE PROBLEM

dip oles, the e ects of di raction of the sound waves

Just as the direct problem is done in two steps con-

due to the sharp trailing edge and refraction due to

sisting of determining the unsteady pressure on the air-

the mean owinteraction with the radiating sound are

foil and then calculating the radiated sound, the inverse

captured. The contribution of the additional e ects,

problem must b e done in two steps. The rst calcula-

termed quadrup ole e ects, b ecomes signi cant as the

tion will determine the unsteady velo city, or gust, from

Machnumb er and the reduced frequency increase.

the unsteady pressure on the airfoil, and the second will

For years the b elief has b een that there is a single

determine the unsteady pressure on the airfoil from the

controlling parameter K , 14, in the direct aeroacous-

radiated sound.

tic problem. It is known that as K increases lob es ap-

p ear in the far- eld radiation patterns due to the non-

The Inverse Gust Problem

compact source e ects. What has not b een discussed

The inverse gust problem in this study is de ned

is that in some cases the lob es are due to dip ole e ects

as calculating the incident gust from knowledge of the

and in other cases they are due to quadrup ole e ects.

unsteady pressure distribution on the at plate. We

By isolating the dip ole term, the main contributors to

start by considering a single frequency gust. Such is

lob e formation b ecome identi able Atassi et al., 1990,

the case for rotating machinery. For instance, b ecause

Patrick et al. 1993. To isolate the dip ole term, take the

of the rotation of a prop eller, the acoustic signal of the

unsteady pressure distribution on the at plate exactly

harmonic nB will, in general, dep end on all p ossible

as the distribution used to calculate the total radiated

upstream mo des. For low sp eed prop ellers, we need to

sound and allow the waves to propagate as if there was

consider only the rst few mo des nB + s, where s =

no mean ow. Figure 2a-b show, b oth the total and the

0; 1; 2;::: Subramanian and Atassi, 1993.

dip ole, far- eld unsteady pressure magnitude directiv-

Consider the case of incompressible ow with im-

ity for two di erent combinations of Machnumb er and

p osed transverse gust. Equation 11 gives the expres-

reduced frequency whichhave similar K values. The

sion for the pressure jump across the at plate for this

at-plate is lo cated on the x-axis and just the top hemi-

case. It is clear that once the unsteady pressure on the

sphere is plotted since, for the at plate, the directivity

at plate is known, the amplitude of the second comp o-

is symmetric from top to b ottom. The unsteady pres-

nent of the vortical disturbance can b e found. In some

sure co ecient is normalized with resp ect to a U .

0 2 1 4

Equation 28 is the statistical equivalent of 11. In cases a can also be recovered. For instance, in the

this case once E is measured, a k a k is known. case of a prop eller turning in a nonuniform ow, k can

p 2 1 1 2

b e calculated for an airfoil section at radial lo cation r

The inversion shown in this section was "simple"

from the prop eller hub once k is known Atassi and

since the resp onse function can be written in closed

Scott, 1988. Then using the continuity relation that

form. Such a function do es not exist in closed form for

compressible ows, the three-dimensional rectangular

~a k =0, a can b e determined. In cases where k can-

1 2

wing, or wings with arbitrary geometries. For arbitrary

not b e found, a remains undetermined and the inverse

shap ed thin airfoils, Goldstein and Atassi 1976 and

solution is not unique. If the gust is three dimensional,

Atassi 1984 p erformed a p erturbation analysis to ar-

the inverse solution will not be unique either since a

rive at a resp onse function similar to S k . For com-

cannot b e determined.

pressible cases, the unsteady surface pressure must b e

For the case of turbulence, where the unsteady pres-

obtained from numerical solvers such as Gust3d Scott,

sure will not be p erfectly p erio dic, a continuous func-

1991 and CASGUST Atassi et al., 1993.

tion will describ e the unsteady pressure in the frequency

Necessary information for exp erimental application

domain. In this case, a statistical approach must be

of this technique is the number and lo cation of pres-

used to determine the second comp onent of the vortical

sure measurements on the body surface that will lead

disturbance.

to go o d reconstruction of the incoming disturbance. It

If weintro duce the Fourier transform,

seems that for the at-plate case, only one lo cation for

the unsteady pressure reading is necessary to determine

0 0 ik t

the amplitude of the second comp onent of the gust.

4p x; t= 4p^ x; k e dk 23

1 1

However, for accuracy a few lo cations should b e used.

This will not b e the case for other geometry airfoils.

and the pressure auto correlation

0 0

4p x; t 4 p x; t +

The Inverse Acoustic Problem

0 0

Theorems p ertaining to the existence and unique-

= lim 4p x; t 4 p x; t + dt24

T !1

ness of solutions to the inverse acoustic problem are

given by Colton and Kress 1992. A brief summary

for the two dimensional case is given here.

and substitute 23 into 24 we get

Let

0 0

4p x; t 4 p x; t +

u ^x = b e 29

1 n

0 ik 0

n=1

= 4p^ x; k e 4 p^ x; k dk 25

1 1 1

where x^ = x=jxj, b e the scattered eld pattern de ned

iK r

as lim uxe r . If the co ecients in the expansion

where denotes conjugate. Then substituting 11 into

r !1

25 gives

satisfy the growth condition

0 0

4p x; t 4 p x; t +

2 2n

jb j < 1 30

n eK R

n=1

1 x 4

2 2

U =

0 1

2T 1+x

with some R>0, then the unique radiating solution of

the Helmholtz equation with scattered eld pattern u

a k a k S k S k e dk 26

2 1 1 1 1 1

If the one-dimensional energy sp ectrum is de ned as

in =4 n 2

Kre ; jxj >R b i H ux=

0 0

4p x; t 4 p x; t + = E x; k e dk

p 1 1

In other words if it is guaranteed that the far- eld

pattern is a solution to the Helmholtz equation, i.e. the

co ecients in the series satisfy the growth condition,

then using 26

then the near eld solution can b e given and it is unique.

2 1 x

2 2

Application to a Streamlined Bo dy . Referring to

a k S k a E x; k = U k S k

2 1 1 p 1 1 1

0 2 1

T 1+x

equation 19, to obtain the pressure jump across the

28 5

which shows the severe ill-p osedness. Since division by at plate from the scattered- eld data, a Fredholm inte-

extremely small singular values leads to large errors in gral equation of the rst kind must b e solved. This typ e

the solution, regularization techniques must be incor- of integral equation is notoriously dicult to solve [19].

p orated into the SVD. One technique that can b e used

is the sp ectral cut-o metho d. In this metho d, only the

Since we are interested in the relationship b etween

singular values which are greater than some sp eci ed

the unsteady surface pressure and the far- eld unsteady

value are allowed in the summation.

acoustic pressure, H in equation 19 may be re-

placed by its far- eld expansion. The integral equation

f u

to solve then has the form

A = v 38

jx~ j 8

Metho ds for cho osing the cut-o value are discussed in

iK jx~j=4

e f = P x~ 32

Kim and Lee, 1990, Sarkissian et al., 1993. Another

sin

metho d is due to Tikhonov 1963. This metho d in-

tro duces the parameter to alleviate dep endence on

iK cos y~

dy~ 33 4P ~y e =

1 1

singular values. The solution then takes the form

~ ~

where x~ =~r cos ; r~ sin . If a simple quadrature metho d

f u v 39 A =

i i

is used to solve the integral equation, by insp ection, one

i=1

can see how severely ill-conditioned the matrix gener-

ated from the kernel will b e. A p opular metho d for solv-

With p erfect far- eld information, generated from

ing such ill-conditioned problems is the singular value

an integral solver, a reconstruction of the pressure jump

decomposition, SVD, Baker, 1977, Kim and Lee, 1990,

along the at-plate was attempted using the SVD with

Sarkissian et al., 1993. After discretizing 33 using

b oth regularization techniques describ ed ab ove. The re-

a simple trap ezoidal quadrature metho d, the equation

construction is adequate in cases with low Machnum-

can b e written as

b ers and any reduced frequency. The reconstruction

is not as accurate for cases with high Mach numb ers.

To illustrate the various reconstruction techniques, two

MA = f 34

cases will b e used. The rst case: M = :1;k =25:0;

and the second case: M = :8;k =1:0; b oth have simi- M is m n with m n containing the information from

lar K values, 2.52 and 2.22 resp ectively. The unsteady the kernel, A is n 1 containing the discretized 4P ,

far- eld pressure magnitude directivity for these two and f is m 1 containing the far- eld information. The

cases are shown in Figure 2. The real and imaginary single value decomp osition consists of determining, the

matrices U , V and such that

parts of the far- eld unsteady pressure are plotted in

Figure 3. These complex far- eld pressure values are

the input for the inverse problem.

V 35 M = U

Figures 4 - 5 show the exact pressure jump across the

where is a m n diagonal matrix with the rst n diag-

airfoil and the reconstructions using the SVD metho d

onal terms containing the singular values, denoted here

with the sp ectral cut-o regularization technique to solve

as , and the others containing zero, U is an m m

equation 34. Here M is a 79 79 matrix resulting from

unitary matrix containing the left singular vectors in

discretizing the integral in 33 using the trap ezoidal

its columns and V is an n n unitary matrix contain-

quadrature rule. The real or imaginary part is shown

ing the right singular vectors in its columns. Here the

dep ending on which reconstruction has the most error.

bar over V denotes conjugate and the sup erscript t de-

Other choices of in the sp ectral cut-o metho d do not

notes transp ose. The singular values are the nonnega-

result in more accurate reconstructions. Figure 6 con-

tains the reconstruction when the Tikhonov metho d is

M M tive square ro ots of the eigenvalues of the matrix

used for the M = :8;k =1:0 case. This metho d do es

M Mx = x

i i

b etter than the sp ectral cut - o metho d but is still

inadequate.

The solution to 34 then is given by

Besides not pro ducing very accurate reconstructions

when the integral is discretized using a quadrature ap-

f u

proach, there is another practical problem with ap-

v 36 A =

proaching the inversion in this manner. For higher

6=0

frequencies, the number of p oints needed in the dis-

cretization of the integral can b e upwards of 100 or 200

If the singular values decay to zero very quickly then

to ensure good accuracy. This requires that there be

the problem is severely ill-p osed. If not then it is weakly

data for at least 200 p oints in the far eld in order

ill-p osed. For this problem the singular values decayas

to avoid having an underdetermined system to solve.

When using a numerical co de to generate the far- eld

eK r 1

data, as in this study, this p oses no problem, but in

2n 2n 6 0.5 Exact solution 0.15 Re(Cp') 0.4 Reconstruction: alpha = 1e-4 Im(Cp') 0.1 0.3 Reconstruction: alpha = 1e-7 0.2 0.05 0.1 Im ( C ) p' 0 C r p' 0 -0.1 -0.2 -0.05 -0.3 -0.4 -0.1 -0.5 -1.2 -0.8 -0.4 0 0.4 0.8 1.2 -0.15 l.e. t.e. 0 30 60 90 120 150 180 airfoil

0.6 Re(Cp')

6: Imaginary part of pressure, SVD metho d.

0.4 Im(Cp') Figure

M = :8;k =1:0. Discretization: quadrature. Regular-

0.2 1

ization: Tikhonov. r Cp' 0

-0.2

-0.4 practice when the far- eld information will come from

exp eriments this metho d cannot b e applied.

-0.6

Hence a collo cation technique is studied. The set of

-0.8

hosen such that two naturally arising

0 30 60 90 120 150 180 basis functions is c

conditions are met. One condition is that the unsteady

pressure will have a square ro ot singularity at the lead-

Figure 3: Real and imaginary parts of the far- eld un-

ing edge. The singularity is inherent to the fact that

the at plate has a sharp leading edge. The second

steady pressure. top M = :1;k = 25:0, b ottom

condition is the Kutta condition. Thus the following

M = :8;k =1:0.

expansion is assumed for P 1

0.5 Exact solution X

P ! =A cot A sinn! 40

0.4 Reconstruction: cut off = .0001 +

0 n

0.3 Reconstruction: cut off = .00001 2

n=1 0.2

0.1

where x~ = cos! . Substituting this into 33 and

Re ( C ) 1

p' 0

tegrating we get -0.1 in -0.2 -0.3

-0.4

~ ~ ~

f = A J K cos iJ K cos 41

0 1 -0.5 0 -1.2 -0.8 -0.4 0 0.4 0.8 1.2

l.e. t.e. 1 X

airfoil ~

J K cos

+ A i n 42

K cos

n=1

Figure 4: Real part of pressure, SVD metho d. M =

:1;k =25:0. Discretization: quadrature. Regulariza-

where J is the nth Bessel function of the rst kind.

tion: sp ectral cut-o .

Using the collo cation technique the matrix equation is

the same as in 34 but M now contains the Bessel

functions, and A represents the co ecients in the series

0.5 Exact solution

40. Our exp erience shows that a few terms in 40 will

0.4 Reconstruction: cut off = .0001

suce to give go o d accuracy for the unsteady pressure

0.3 Reconstruction: cut off = .00001

Hence this expansion has the advantage

0.2 along the plate.

umb er of exp erimental p oints required 0.1 of reducing the n Im ( C )

p' 0

in the far eld for the inverse problem.

-0.1

ve the prop erty -0.2 Notice that the Bessel functions ha

-0.3 that

-0.4

-0.5

n ez

-1.2 -0.8 -0.4 0 0.4 0.8 1.2 1

as n !1 43 J z

l.e. t.e. n

airfoil

Therefore the values of the elements in the columns

Figure 5: Imaginary part of pressure, SVD metho d.

of M decrease as the column number increases. So

M = :8;k =1:0. Discretization: quadrature. Regular-

this matrix is also ill-conditioned. But applying the

ization: sp ectral cut-o .

SVD metho d on this new linear system gives b etter 7 0.5 Exact solution 0.4 Reconstruction: alpha = 1e-4

0.3 Reconstruction: alpha = 1e-7 0.2 0.1 Im ( C ) p' 0 -0.1 -0.2 -0.3

0.5 Exact solution -0.4 0.4 Reconstruction: cut off = .0001 -0.5 -1.2 -0.8 -0.4 0 0.4 0.8 1.2 0.3 Reconstruction: cut off = .00001 l.e. t.e. airfoil 0.2 0.1 Re ( C )

p' 0

Figure 9: Imaginary part of pressure, SVD metho d.

-0.1

M = :8;k =1:0. Discretization: collo cation. Regular-

-0.2 1

-0.3

ization: Tikhonov. -0.4 -0.5 -1.2 -0.8 -0.4 0 0.4 0.8 1.2

l.e. t.e.

as illustrated in Figures 7 - 8. M used to

airfoil results

obtain the results presented ab oveisa79 14 matrix

which corresp onds to using 79 p oints in the far- eld

Figure 7: Real part of pressure, SVD metho d. M =

and 14 terms in the series representation of P given

:1;k =25:0. Discretization: collo cation. Regulariza-

in 40. Also, the sp ectral cut-o regularization tech-

tion: sp ectral cut-o .

nique is used. For comparison, Figure 9 shows results

obtained for M = :8;k =1:0 using the Tikhonov reg-

ularization metho d.

The b est reconstructions found using the SVD

metho d still oscillate ab out the exact solution. There

is another metho d which allows p erfect inversion when

given p erfect far- eld information.

The metho d is a simple regularization technique and

is linked quite closely to the SVD. The collo cation tech-

nique is used as describ ed ab ove to obtain the matrix

M . Then multiplying b oth sides of 34 by the adjoint

of M gives

t t

M MA = M f 44

so that

t t

=M M M f 45 0.5 Exact solution A

0.4 Reconstruction: cut off = .0001

diculty in applying this metho d is determining

0.3 Reconstruction: cut off = .00001 The

w many columns M will have. As mentioned earlier, 0.2 ho

0.1

the values of the elements in the columns of M decrease Im ( C )

p' 0

as the column number increases. Therefore including

-0.1

to o many columns prohibits the inversion of M . As -0.2 M

-0.3

mentioned b efore, the values of A in 40 decrease as n

-0.4

n increases, implying that the columns corresp onding

-0.5

the higher values of n can be excluded. In order -1.2 -0.8 -0.4 0 0.4 0.8 1.2 to

l.e. t.e.

determine how many columns will be included in

airfoil to

M M are calculated. the matrix M , the eigenvalues of

Recall these are exactly the square of the singular values

Figure 8: Imaginary part of pressure, SVD metho d.

of M . The number of eigenvalues with value greater

M = :8;k =1:0. Discretization: collo cation. Regular-

than a certain cut-o value, is the numb er of columns

ization: sp ectral cut-o .

that will be retained in the matrix M . And, in 40,

A = 0 when n the numb er of columns. Notice that

this corresp onds to the sp ectral-cut o regularization

technique. The exact solutions given in Figures 4 -9

are obtained when this metho d is used.

Because of the ill-conditioned nature of the matri-

M M , inversions are very sensitive to er- ces M and

rors in the far- eld data. For instance, the inversions 8

0.5 Exact solution

allow for good reconstruction. The linear system ob-

0.4

tained from the quadrature cannot b e solved accurately

0.3 Reconstruction from truncated data

y either the single value decomp osition with two dif-

0.2 b

t regularization techniques imb edded or the sim- 0.1 feren

Re ( Cp') hnique.

0 ple regularization tec

-0.1

was then used to discretize the inte-

-0.2 Collo cation

equation. The basis for the collo cation is con-

-0.3 gral

using knowledge of the direct problem. A -0.4 structed

-0.5

square ro ot singularity is imp osed at the leading edge -1.2 -0.8 -0.4 0 0.4 0.8 1.2

l.e. t.e. and the Kutta condition is imp osed at the trailing edge.

airfoil

The square ro ot singularity is inherent to the problem

0.5 Exact solution

the Kutta condition is an essential added condi-

0.4 but

We conjecture that the absence of the Kutta con- 0.3 Reconstruction from truncated data tion.

0.2

dition in the quadrature metho d is the reason for the

0.1

The matrix pro duced using Re ( C ) incorrect reconstructions.

p' 0

the collo cation technique is also ill-conditioned, so the -0.1

-0.2 SVD metho d and the simple regularization metho d are

again on this linear system. The solutions for

-0.3 tested

P are much b etter when using the collo cation tech-

-0.4

-0.5

But reconstructions by the singular value de- -1.2 -0.8 -0.4 0 0.4 0.8 1.2 nique.

l.e. t.e. w oscillations around the exact

airfoil comp osition metho d sho

solution for some cases. These oscillations p ersist for

all regularization parameters.

Figure 10: M = :1;k =25:0. Error in real part of re-

It is imp ortant to note that all the di erent solutions

construction when input data is truncated 1e-5. top

for P , obtained using the SVD metho d with di erent

using the SVD metho d. Discretization: collo cation.

regularization techniques and di erent regularization

Regularization: sp ectral cut-o . b ottom using the

parameters, repro duce the far eld with 10 precision

when they are plugged backinto 34. It seems, then,

simple regularization metho d. Discretization: collo ca-

that the solutions obtained for P using these meth-

tion.

o ds are all valid and one might b e tempted to conclude

that the di erences b etween the reconstructed solution

and exact solution are due to evanescent waves. This

shown in Figures 4 - 9 are obtained when the values

cannot b e true.

of and P in the far- eld are given with precision of

P determined using the simple regularization tech-

10 . Noisy far- eld data is mo delled by truncating

nique agrees completely with the exact solution for every

the values to 10 precision. Figure 10 shows how this

case. This leads to the reasoning that the discrepancy

small error manifests itself in a signi cant oscillation

between the reconstructed P and the exact P using

in the reconstructed surface pressure for the case of

the SVD metho d cannot b e evanescentwaves but inad-

M = :1;k = 25:0. Figure 10top shows the error

equacies in the scheme. If there are evanescentwaves

in the real part of the unsteady pressure jump when

present then no metho d would be able to reconstruct

the SVD metho d with cut - o .0001 is used and Figure

the surface pressure exactly from the far- eld informa-

10b ottom the error in the reconstruction of the real

tion.

part of the pressure jump when the simple regulariza-

tion technique is used.

The inadequacies in the SVD metho d must b e un-

dersto o d b etter. Also a b etter understanding of the

role of the regularization parameters is needed. For the

Summary and Notes . For the inverse acoustic prob-

case M = :8;k =1:0 where K =2:22 the reconstruc-

1 1

lem, we are assuming that the information in the far-

tion oscillates, even in the b est case when the Tikhonov

eld is exactly the radiated sound pro duced by the un-

regularization metho d is used with =10 . This con-

steady pressure distribution along the at plate. So the

trasts the case M = :1;k =25:0;K =2:52, for which

theorems summarized earlier imply that the unsteady

1 1

b oth the SVD and the simple regularization techniques

pressure on the at plate can be uniquely determined

gave adequate results. Since b oth cases have the same

from the far- eld information. In order to treat the

value for K ,we conjecture that the inadequacy of the

problem numerically it must b e discretized. This means

SVD metho d for high Machnumb er is also essentially

the problem remains il l-posed however, due to the dis-

numerical.

crete dep endence of the solution on the far- eld infor-

mation. The ill-p osedness manifests itself in a severely

As exp ected for ill-p osed problems, small errors in

ill-conditioned linear system, where small p erturbations

the far- eld data lead to large error in the inverse so-

in the input will create large errors in the solutions. So

lution. Noisy data was simulated by truncating from

8 5

diculties lie in the development of a solution pro ce-

10 precision to 10 . Solutions obtained using the

dure.

SVD metho d show more sensitivity to the noise than

solutions obtained from the simple regularization tech-

Straight forward discretization of the integral equa-

nique. This was unexp ected. This issue must b e con-

tion using the trap ezoidal quadrature rule, do es not 9

sidered, and hop efully a solution pro cedure can b e de- the solution in resp onse to small changes in the data.

signed that will be able to lter out p erturbations in

For real geometry problems, additional complexity

the far- eld data. In practice, measurements may be

is exp ected. For a real lifting surface such as an air-

valid to only one signi cant digit, and never to 6 or 7

foil or a prop eller blade, the mean owis nonuniform

which is the necessary accuracy for go o d reconstruction

and only in the far eld do es the governing equation

using the schemes outlined in this pap er.

reduce to a Helmholtz equation. In the near and mid

The matrices used for all the examples in this pa- elds the governing wave equation has nonconstant co-

per were built assuming 79 evenly spaced far- eld mea- ecients and an inhomogeneous source term which de-

surements. The reconstructions work in the same man- p end on the nonuniform mean ow. Physically, these

ner when less far- eld p oints are used if the collo cation nonconstant co ecients account for nonuniform refrac-

technique is used. As few as 10 p oints can b e used for tion of the radiated sound by the mean ow. This addi-

smaller K cases. As K increases so do es the number tional complexitymay lead to diculties in solving the

1 1

of p oints necessary in the far eld. This is physically inverse problem uniquely for real geometry problems.

consistent, since as K increases, so do es the number The knowledge b eing gained by studying the at-plate

of lob es in the scattered eld pattern. To represent airfoil case should give insights into the real geometry

the far- eld pattern accurately when there are several problems.

lob es, more measurements are necessary. The place-

ment of these measurements do es not seem to matter if

ACKNOWLEDGEMENTS

the data is p erfect. When the data is noisy, though, the

The researchwas supp orted by the Oce of Naval

numb er and lo cation of the measurements may help in

Research Grant No. N00014-92-J-1165 and monitored

reducing errors in the solution and, therefore, must b e

by Mr. James A. Fein. Sheryl M. Patrickwould liketo

considered more carefully.

thank

the Clare Bo oth Luce Foundation for their partial

CONCLUSIONS

supp ort.

The inverse gust problem, de ned as determining

the incoming vortical gust from the unsteady pressure

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