NASA Contractor Report 3331 

Viscous Theory of Surface 

Noise Interaction Phenomena 

John E. Yates 

CONTRACT NASl-15539 

SEPTEMBER 1980 

NASA 

CR 

3331 

C.1 
NASA Contractor Report 3331 

Viscous Theory of Surface 

Noise Interaction Phenomena 

John E. Yates 

Aeronautical Research Associates of Princeton, Inc. 

Princeton, New Jersey 

Prepared for . 

Langley Research Center 

under Contract NASl-15539 

National Aeronautics 

and Space Administration 

Scientific and Technical 

Information Branch 

1980 

II. 

III. 

IV. 

v. 

TABLE OF CONTENTS 

SUGARY ............................................ ..l 

INTRODUCTION ...................................... ...2 

GENERAL THEORY OF A CLASS OF LINEAR 

AEROACOUSTIC PROBLEMS ............................. ...6 

A. The Surface Interaction Problem ............... ...6 

B. A Self Consistent Energy Equation .............. ...8 

C. Mode Splitting and Boundary Conditions..........1 0 

D. The Associated Inviscid Boundary Value 

Problem.........................................l3 

E. An Integral Equation for Bernoulli Enthalpy 

(Uniqueness)....................................l4 

F. The Acoustic Potential..........................1 9 

G. Chebyshev Representation of the Surface 

Load and Acoustic Far Field.....................2 2 

H. Effect of Edge Bluntness........................2 4 

APPLICATIONS OF "INVISCID" THEORY...................2 4 

A. Zero Thickness Two-Dimensional Surface..........2 4 

B. Vortex Fly By a Sharp Edge......................3 2 

SUPPLEMENTARY MATERIAL..............................3 4 

A. Discussion of Trailing Edge Noise Models........3 4 

B. Summary of the Supplemental Report..............3 6 

CONCLUSIONS.........................................37 

REFERENCES..........................................39 

iii 
SUMMARY 

A generalized linear surface noise interaction problem 

is formulated. Noise production by an oscillating surface, 

turbulent or vertical interaction with a surface and scattering 

of sound by a surface are included in the generalized treatment. The direct effect of viscosity is included in the 

perturbation equations and the boundary conditions at the surface. An energy equation is derived to illustrate how the 

work or virtual work done at the interface is partitioned 

into the acoustic and vertical modes. 

The problem is re-expressed in terms of a near field 

incompressible viscous problem for the Bernoulli enthalpy and 

a problem for the compressible but inviscid acoustic potential. 

A viscous integral equation for the Bernoulli enthalpy.(or 

surface load) is derived. It is shown that viscosity will 

lead to a "unique" solution of the integral equation without 

the Kutta condition or other auxillary singularity requirement. The acoustic enthalpy is calculated by quadrature 

over the near field solution. 

The results of inviscid two-dimensional airfoil theory 

are used to discuss the interactive noise problem in the 

limit of high reduced frequency and small Helmholtz number. 

The acoustic spectrum is directly proportional to the surface 

load. The extreme limits of a full Kutta condition and no Kutta 

condition (actually no vorticity production) are considered. 

It is shown that in the case of vortex interaction with the surface the noise produced with the full Kutta condition is 3 dB 

less than the no Kutta condition result. Also, the spectrum with 

Kutta condition decays monotonically with frequency while the 

corresponding result without the Kutta condition decays in an 

oscillatory manner. It is suggested that the difference in the 

high frequency spectra could be detected experimentally. 

The results of a supplementary study of an oscillatory 

airfoil in a medium at rest are discussed. It is concluded 

that viscosity can be a controlling factor in analyses and experiments of surface noise interaction phenomena. It is further concluded that the effect of edge bluntness and viscosity 

must be included in the problem formulation to correctly calculate the interactive noise. 
I. INTRODUCTION 

In two previous reports (Refs. 1 and 2) the homentropic theory of aeroacoustics, based on a kinematic 

definition, of sound and the concept of Bernoulli enthalpy, 

was formulated in detail and applied to a variety of problems that illustrate the three basic questions of "free 

flow" aeroacoustics (i.e., in the absence of solid boundaries). In the spirit of Chu and Kovasznay (Ref. 3) the 

modal concept has been stressed in all of the previous 

work. The problem of how to decouple the acoustic and 

vertical modes (to the extent that it is possible) of 

energy transport is oneofthe most interesting questions in 

the field of aeroacoustics. 

For example, when the problem of noise production by a 

turbulent flow (e.g., a jet) is considered, it is customary 

to adopt the "Lighthill hypothesis". Because of differences 

in the basic definition of sound in the various aeroacoustic 

formulations and the level at which the Lighthill hypothesis 

is invoked, there have been serious arguments over what constitutes the "source" of sound. With the modal approach 

there is no "source" per se, and the application of the Lighthill hypothesis is tantamount to assuming a unilateral transfer of energy from the vertical mode to the acoustic mode. 

The only internal mechanism for acoustic feedback to the vortical mode is the coriolis coupling acceleration between the 

acoustic particle velocity and the vorticity of the primary 

flow (see Eq. (2.28)of Ref. 1). The coriolis mechanism is 

important in the problem of sound scattering by a steady vortical flow and the stimulated emission of sound from an unsteady vertical flow. Both of these problems were investigated in detail in Ref. 2. It was shown by explicit calculation that the broadband noise radiated from a discrete 

vortex array can be enhanced by an incident sound field. 

(For a concise summary of this work, see Ref. 4). 

The distinction between acoustic and vertical energy 

transport and the coriolis coupling mechanism between the two 

modes is reasonably well understood in the description of the 

internal dynamics of a fluid medium. However, there remains 

the important problem of how these modes interact in the 

presence of solid boundaries, and in particular when the 

solid boundary has "sharp" edges. The important feature of 

the surface interaction problem is that viscosity becomes a 

controlling factor in the mode coupling at the surface of 

the body. It is the objective of this report to shed some 

light on the importance of viscosity in a class of aeroacaustic problems that involves thin airfoil like surfaces, 

with edges. Two experiments due to Brooks (Ref. 5) and Brooks 

and Hodgson (Ref. 6) will be used to guide and test the 

development of the theory. 

2 
NOMENCLATURE 

a 

An 

C 

C* 

C 

C mn 

E 

gm 

G 

h 

h' 

LJJ 

J,(z) 

k 

i; 

Ko(z) 

1 

M 

free stream speed of sound 

spectral coefficients of the surface load, 

see Eq. (2.81) 

airfoil chord 

compressibility, see Eqs. (2.72) and (2.73) 

denotes chord of the surface S in the twodimensional problem 

see Eq. (2.83) 

energy density, see Eq. (2.9) 

surface deflection along z-axis 

surface force distribution, see Eq. (2.47) 

magnitude of the surface force, see Eq. (2.47) 

see Eq. (2.84) 

Green's function, see Eqs. (2.75) and (2.76) 

vertical distance of vortex from interactive 

surface 

perturbation enthalpy 

Bernoulli enthalpy 

unit vectors along x,y,z axes, see Fig. 1 

Bessel function 

wc/2v,, reduced frequency 

w/a 

modified Bessel function 

spanwise correlation length of the surface load 

v&, free stream Mach number 
M’ momentum flux, see Eq. (2.101 

n’ unit normal to a surface 

Aq 

Q 

R 

S 

sf 

S(a) 

t 

T,(x) 

3 

UJx> 

v’l 

'd 

V a3 

v 

W 

denotes Fourier transform of variable q with 

respect to tS.me 

q(z = o-1 - q(z = O+>, jump in the dependent 

variable q across the plane z = 0, e.g., see 

Eq. (2.53) 

dissipation function, see Eq. (2.11) 

see Eq. (2.77) 

denotes a fixed or oscillating surface 

outer boundary of the volume V 

acoustic spectral function, see Eqs. (3.1) and 

(3.2) 

time 

Chebyshev polynomial of the first kind 

incompressible perturbation velocity 

Chebyshev polynomial of the second kind 

perturbation velocity field 

disturbance velocity field 

free stream velocity 

integration volume in energy integral 

upwash function see Eq. (2..4) 

Cartesian coordinates, see Fig. 1 

acoustic intensity, see Eqs. (3.36) and (3.37) 

kernel function, see Eqs. (2.65) and (2.66) 

integrated kernel, see Eq. (.2.85) 

total lift on two-dimensional surface, see Eqs. 

(3.16) and (3.10) 

normalized surface load distribution 

surface work or virtural work, see Eq. (2.13) 

4 
a 

62 

Y(X) 

r 

r. 

6(z) 

8 

K 

x,x* 

A 

V 

T 

9 

see Eqs. (3.28) and (3.34) 

see Eq. (3.3) 

1 - M2 

vortex strength distribution, see Eqs. (3.9) and (3.15) 

circulation, see Eq. (3.16) 

vortex strength, see Eq. (3.33) 

Dirac delta function 

far field directivity angle 

we/2a, Helmholtz number 

see Eq. (2.57) 

see Eq. (2.79) 

kinematic viscosity 

see Eq. (3.35) 

acoustic potential 

X hydrodynamic potential, see Eq. (3.7) 

w frequency in Fourier transform, see Eq. (2.42) 

z1 perturbation vorticity, see Eq. (2.21) 

curl vector curl operator 

div divergence 

grad vector gradient operator 

V2 

v: 

- 

I I 

Laplace operator 

surface Laplace operator, see Eq. (2.60) 

time average 

absolute value 

5 
GENERAL THEORY OF A CLASS 

%-LINEAR ~~Ro~cousT~c PROBLEMS 

A. The Surface Interaction Problem 

Below, a linear problem is formulated that will permit 

the investigation of three basic noise-surface interactions: 

1) An oscillating surface; 

2) An assumed vertical disturbance (e.g., turbulence) 

interacts with the surface (The Edge Noise Problem); 

and 

3) An assumed acoustic disturbance interacts with the 

surface (The Diffraction Problem). 

These problems are depicted schematically in Figure 1. The 

surface S may be finite or infinite and initially the suri 

face is supposed to be of zero thickness. Later, in Section 

III, the effect of finite edge geometry, is discussed. 

It is assumed that each of the three problems may be 

described with the theory of hornentropic small disturbances 

from a uniform main stream. The appropriate perturbation 

equations are: 

1 Dh' -- a2 Dt + div;' = 0 (2.1) 

where 

DGl 

- + gradh' = vV23' + Dt +grad (divfl) (2.2) 

D -= &+v a 

Dt -P (2.3) 

For the problem of an oscillating surface the boundary conditions on S are as follows: 

k’ . ff =w+ 

I 

on z = O+ in S 

dx2f=o (2.4) 

where f is the transverse (along the z-axis) deflection of 

the surface. The boundary conditions for the other two problems 

can also be expressed in the form of Eq. (2.4). Denote the 

incident velocity field due to 3 vertical or acoustic disturbance in the absence of S by v Then the interactive 

disturbance field must be such t R' 

on S; i.e., 

at it cancels the velocity 2, 
Noise Produced 

-_.._ 3. Incident Sound 

Interacts with S. 

Disturbance Interacts 

I. Direct Vibrati 

of Surface S. 

--.-. 

- - __ _. - . - 

__--- -- 

__-- ---_ 

Vortex Wake 

Produced 

Figure 1 - Three Types of Interactive Noise 

Production Mechanisms 
ii . v’? = w = -z . Gd 1 on z = Of in S 

i; x c1 = -ii x ;d (2.5) 

The boundary condition on the vertical, or z-component 

of velocity, is the usual boundary condition that is familiar 

in inviscid aerodynamic problems. The second boundary condition on the tangential surface perturbation velocity is necessary 

when viscous terms are retained in the momentum equation. The 

importance of the viscous terms and boundary conditions in 

establishing uniqueness of solution (without a Kutta condition) 

and in the partitioning of disturbance energy into the acoustic and vertical modes is the essential content of this report. 

Because of the linearity of the boundary value problem the 

solution can be obtained by superposition of two parts with 

the respective boundary conditions: 

iI . ft = w = -$ . 2, 1 on z = Oi in S 

dx?=o (2.6) 

and 

i; . fl= 0 1 on z = Of in S 

?i x f' = -2 x 2, (2.7) 

For all finite disturbance velocity fields Gd the solution 

of Eqs. (2.1) and (2.2) with the boundary conditions Eq. (2.7) 

is small of the order of the viscosity and will not be considered further in this report. The homogeneous viscous 

boundary condition in Eq. (2.4) or (2.6) is the main result 

that is needed to formulate The generalized linear aeroacoustic problem. Solve the perturbation equations lm 

and (2.2) subject to the boundary conditions (2.4) or (2.6) 

and the condition of outgoing decaying disturbances at infinity. 

B. A Self Consistent Energy Equation 

It is ins,tructive to derive an energy equation for second 

order perturbation quantities. In so doing the global partitioning of energy into acoustic and vertical parts will 

become evident. Multiply Eq. (2.1) by h', form the vector dot 

product of Eq. (2.2) with v' and add the results to obtain the 

local energy balance, 

g+divg Z-Q (2.8) 

8 
where 

Ed& + 

2a2 2 (2.9) 

Q= v1;'j2 + $J (div$')2 

For a statistically stationary disturbance velocity field, 

3d, the time average (denoted by an overbar) of Eq. (2.8) is 

aE vooax + divz = -v (2.12) 

Now integrate the last result over a large spherical volume, 

V, that encloses the surface S. On the outer boundary, S 

the effect of viscous transport is assumed to be negligib f e. ' 

The resultant global energy balance becomes 

v= 2k+ R, dxdy 

= s sf 

TdA + J Sf 

v,x l "EdA 

For the case of an oscillating airfoil the first example of 

the generalized problem, the quantity &is the rate at which 

work is done on the fluid by the oscillating surface. For 

the second and third examples the surface is rigid and does 

no work on the fluid. However, it is convenient to interpret 

ras the virtual work due to the surface motion (upwash) W 

that must be imposed to counter the normal component of the 

disturbance velocity field. The global energy equation (2.13) 

shows that the work done by the surface S is partitioned 

into sound radiation through the outer surface Sf, convective 

energy transport through Sf, and viscous dissipation throughout the volume V. The last two parts are actually different 

manifestations of the same physical process; i.e., the formation of vorticity at the surface S. If the surface Sf is 

sufficiently far from S, no convective transport will be detected and all of the vertical energy will be counted in the 

9 
viscous dissipation. If Sf is closer to S then convective 

transport of vertical energy will be a significant part of 

the energy balance. It is noted that similar statements can 

be made about the acoustic term in Eq. (2.13). Certainly 

if Sf is removed to infinity then all of the surface work is 

dissipated by viscosity. 

Finally it is pointed out that viscous dissipation cannot 

be discarded as an energy sink by the simple argument that 

the viscosity is small. It will be shown by explicit calculation in Section III and in the supplementary report that 

the vorticity is sufficiently singular to yield a finite integrated value for the dissipation even for vanishing small 

viscosity. 

C. Mode Splitting and Boundary Conditions 

The boundary value problem composed of equations (2.1), 

(2.2) and the generalized boundary conditions (2.4) or (2.6) 

is well posed. It is of interest, however, to recast the 

problem in terms of acoustic and vertical modes as follows: 

Let 

+ 

v’ = grad@ + 2, (2.14) 

where 6 is the acoustic notential and c' is an incomoressible 

velocity field. Substitute Eq. (2.14) into 

Dfl 

- + gradH = vV2z' Dt 

where 

is the Bernoulli enthalpy (or Pseudo Sound) 

the incompressible problem for 3'. Now use 

eliminate h' in Eq. (2.1). The result is a 

$; i.e., 

(2.2) to obtain 

(2.15) 

(2.16) 

associated with 

Eq. (2.16) to 

wave equation for 

(2.17) 

Note that, equations (2.15) and (2.17) could also be obtained 

by direct linearization of Eqs. (2.27) and (2.28) in Ref. 1. 

The acoustic mode is "driven" in the near field by the 

substantive rate of change of ?I . In the far field the 

10 
acoustic mode decays due to viscous dilitational damping. 

The near field hydrodynamic or vertical mode is not coupled 

linearly to the acoustic mode in the interior of the gas. 

However, mode coupling does occur due to the boundary conditions at the surface. (See Ref. 7 for further discussion of 

this point). 

The boundary conditions on z',?f are chosen to be the 

same as those for the complete perturbation velocity field; 

i.e., 

2 . E;’ =w 

I 

on z = Of in S (2.18) 

zx2f=o 

Then the boundary conditions for $ are homogeneous; i.e., 

z l grad+ = 0 1 

z x grad4 = 0 J on z = O- +in S (2.19) 

It is possible to apply the no-slip boundary conditions on 1 

3' because of the viscous term in Eq. (2.15). Similarly it 

is possible to apply no-slip conditions on the acoustic particle velocity as long as the viscous damping term is retained 

in Eq. (2.17). It is the presence of these terms that precludes the necessity for a Kutta condition or other singularity 

criteria to solve the relevant boundary value problem. 

A somewhat different insight into the role of viscosity 

and the coupling of the pressure and vorticity modes may be 

obtained as follows. Take the curl of Eq. (2.2) or (2.15) to 

obtain the vorticity diffusion equation 

DZ' -= Dt vV2Z' 

with 

* w’ = curlGf = curl2' 

(2.20) 

(2.21) 

Also take the substantive derivative of Eq. (2.1) and the divergence of (2.2) to obtain the damped wave equation for h'; 

i.e., 

11 
1 D'h' 4 v 

-- a2 -yyp 

V2Dh’ 

-73 Dt _ V2h' = 0 (2.22) 

As noted by Chu and Kovasznay (Ref. J).the vorticity and pressure modes decoutile completely in the interior of the gas. 

At the surface, the boundary conditions on 3' can be used to 

derive the relations between h' and 3'. First write Eq. (2.2) 

in the form, 

D? 

- + grad Dt 

h' + 4 V Dh' 

39-E =- ) 

vcurlz' (2.23) 

Because of the planar geometry of S and the no-slip boundary conditions, it follows that 

WZ =$ .;I = 0 (2.24) 

Now project Eq. (2.23) onto the surface S and use the 

boundary conditions on 3' to obtain the following boundary 

conditions. 

a 4 v Dh' 

az h'+j9Dt 

k'x grad 4 v Dh' 

h’ +ygTTE- > 

a2 

=Vaz on z = O+ in S (2.25) 

The intrinsic coupling of the pressure and vorticity modes 

is evident from these relations. The normal pressure gradient at the surface is balanced by the surface acceleration 

(real or virtual) and gradients of the surface vorticity. 

The second expression shows how vorticity is produced at a 

surface by pressure gradients along the surface. Near sharp 

edges that have large pressure gradients, the production of 

vorticity can be intense. It is also clear from this relation that to obtain a meaningful solution (noniconstant 

surface load) of the viscous boundary value problem, the product v(aw'f/az)must be finite even for a vanishingly small 

viscosity coefficient. 

very singular. 

The coupling between modes is thus 

12 
D. The Associated Inviscid Boundary Value Problem 

For each of the boundary value problems and energy relations derived thus far there is a corresponding "inviscid" 

relation. First of all, the perturbation enthalpy and 

velocity satisfy the relations 

1 Dh' -- a2 Dt +divf' =0 

Dfl -- Dt +gradh' -0 

(2.26) 

(2.27) 

with the boundary condition 

2 . 2’ = on z = Ot in S (2.28) 

and the radiation condition. The energy relation (2.13) 

becomes 

w-= 2A+ h'W dxdy 

= J Sf h'v;, dA + 

s sf 

VOS 1 l n'EdA 

(2.29) 

where the outer surface Sf can be taken to infinity since 

acoustic and vertical waves do not decay. The convective 

transport of energy in Eq. (2.29) is d ue to the singular vortex wake. It is the singular remnant of the effect of viscosity, and in general it is a significant part of the work 

done by the surface. Without the explicit effect of viscosity 

at the surface there is no way to determine the energy partitioning without recourse to an artificial uniqueness criteria 

like the Kutta condition. This will be demonstrated explicitly 

in the next section where the boundary value problem is cast 

in the form of a singular integral equation. 

The inviscid boundary value problems for 2, and $I are 

respectively: 

div :* = 0 (2.30) 

Dff 

- + grad%= 0 Dt 

13 
2 . ;t = W on 2 = O+ in S 

and 

with 

(2.34) 

1 D2$ -- a2 Dt2 

w 

az = 0 on z = Of in S 

The perturbation enthalpy 

h'=~-!?$ 

also satisfies the homogeneous wave equation 

(2.32) 

(2.33) 

1 D2h' 

SF - V2h' = 0 

and 

&= 

Dt 0 

It follows from (2.25) that 

ah' DW - =-- 

az Dt on z = Ok in S 

(2.35) 

(2.36) 

(2.37) 

(2.38) 

In each boundary value problem the boundary conditions on 

the tangential surface velocity components must be given up 

because the inviscid equations are of lower order,. Also, the 

inviscid problems do not have unique solutions and some 

auxiliary criterion (usually the Kutta condition) must be 

used to establish uniqueness. 

E. An Integral Equation for B&noulli Enthalpy (Uniqueness) 

Consider the viscous boundary value problem for u" 

7f;i.e., and 

14 
div ff = 0 (2.39) 

Df' 

- + grad31 = vV2z' Dt (2.40) 

z . u’f =w 1 on z = 02 in S 

zx3=0 (2.41) 

It is convenient to work in the frequency domain. Take 

Fourier transforms of the above equations with respect to 

time and note for any dependent variable, q, that 

/ 

0) 

$d = 

e-iwt 

-03 q(t) dt 

Then 

with 

q(t) = & / -1 eiwt $"> dw 

div ;, = 0 

DG' 

- + grad% = vV2c' Dx 

d . ;I = E 1 on z = O- + in S 

iExG'=O 

and 

D 

= v a 

Dx coax +la 

(2.42) 

(2.43) 

(2.44) 

(2.45) 

(2.46) 

15 
The solution can be represented in terms of an unknown normal force distribution on the surface S; i.e., 

z= -2 shy> 6(z) (2.47) 

Add 2 to the right hand side of (2.44) and then take the 

divergence and curl to obtain 

and for the vorticity 

D;l 

- -vv2 Cl Dx = $ x gradz&(z) 

The solution of (2.48) is 

where the integration is over the surface S with 

Take the limit of (2.50) as z + 0 to obtain 

2: 

F 

= Limx = TZ 

z + o+ 2 

or 

,F = A?J = 7f!(x,y,o-1 - z(x,Y,o+) 

(2.48) 

(2.49) 

(2.50) 

(2.51) 

(2.52) 

(2.53) 

is the local jump in the Bernoulli enthalpy on the surface. 

For points off the surface A?J must vanish. 

The solution of (2.49) can be expressed in the form 

2 = d x grad& (2.54) 

16 

4. _-I 
where 

DQ 

Dx -vV2Q = ,F 6(z) (2.55) 

and it is readily shown that 

(2.56) 

where 

and 

A* = (A2 + iw/v)~ , A = g (2.57) 

curl;' = dV'Q aQ 

- grad 3z (2.58) 

From (2.441, with z # 0 

a+ 2 =+g.curl;’ 

DX az 

where 

Also 

= -vV:Q 

vi=& +a2 02 

(2.59) 

(2.60) 

(2.61) 

Thus 

D W 

- Dx =k J s _rCb~ 

l- exp[xl l (2 - Ti) - X*IZ - ZI] 

I x' -51 

d$ 

for z # 0 (2.62) 

Now solve (2.62) for l and take the limit as z + 0 to obtain 

17 
the following integral equation for x(z); i.e., 

(2.63) 

where the kernel is defined by the relation 

a3 s o exp(+iws/vo5) l- expIX(x - s> - A*lx' - Xsl) ds 

I ; - Tsl 

(2.64) 

The integral equation for the corresponding two-dimensional 

case can be obtained by integrating (2.64) over the spanwise 

coordinate. From a previous report (Ref. 8) the final expression is 

s L(<)~(x - 5) dS = - x(E) , x in C C 

(2.65) 

where 

X(X) = & c2-r exp(-iws/v,) 

0 

14x - 51 + exp{A(x - 5)) Ko(A*/x - 51 )]d5 

(2.66) 

and the integration is only over the chord C of the interactive surface S. 

The solution of (2.65) and a more general equation for 

airfoil type surfaces with finite thickness has been the subject of a recent development of "viscous thin airfoil theory". 

The results are summarized in a final report (Ref. 8) and an 

AGARD report (Ref. 9). For the moment the most important 

point to note is that the kernel function x(x) (Eq. (2.66)) 

has a logarithmic singularity for small argument no matter how 

small the coefficient of viscosity. The corresponding inviscid kernel of the incompressible Possio equation (obtained 

by dropping the second termin (2.66)has a Cauchy singularity. 

The viscous integral equation has a unique solution (See 

Ref. 8) while the inviscid equation has an eigensolution that 

18 
is associated with the circulation of the perturbation flow 

field (See Section IIIB). The strength of the eigensolution 

determines uniquely the amount of vertical energy that is 

shed into the wake. The eigensolution also contributes directly to the "source" term D?f /Dt in the acoustic problem 

(2.33) and (2.34) and so affects the radiated sound. Without viscosity there is no physical criterion to determine 

the eigensolution and it is customary to resort to the Kutta 

condition or principle of minimum singularity at the trailing edge to establish uniqueness. The solution of the 

viscous integral equation (2.66) has a square root singularity at a mathematically sharp leading or trailing edge. The 

singularities can be removed by slightly blunting the edges 

as shown in Section III and in that case there is no useful 

edge criteria whatsoever to establish uniqueness. The surface load, shed vorticity and noise must be obtained by 

solving the viscous boundary value problem. 

F. The Acoustic Potential 

Once H is known from the solution of the appropriate 

integral equation, there remains the problem of solving 

for the acoustic potential $; i.e., 

1 D*9 4 v V2D$ v24 = 1 D?-i -- a2Dt2 -San Dt , az Dt (2.67) 

with 

a@ =o 

az 

1 on z = Ok in S 

d x grad+ = 0 J (2.68) 

and the radiation condition. Actually, because of the dilatational damping term, it is sufficient to require decaying 

solutions of (2.67) at infinity. 

The interpretation of (2.67) and (2.68) is conceptually 

appealing. The origin of the interactive noise is the local 

substantive rate of change of 3. The surface itself is a 

passive element in the noise production process but it does 

cause compressible diffraction primarily through the boundary 

condition on the induced vertical component of velocity. Also, 

it is emphasized that by construction no vorticity is produced in the compressible part of the interaction process. 

A$1 vorticity is associated with the surface loading due to 

. It is this observation that permits an important simplification of the acoustic problem; i.e., the direct effect of 
viscosity can be omitted in the acoustic problem. Since 

there must be no compressible production of vorticity, the 

compressible viscous term is omitted in (2.67) and the viscous 

boundary condition in (2.68) is replaced by the condition that 

the acoustic potential be continuous across the plane z = 0 

for all points not in S; i.e., 

1 D2$ _ v2+ = 1 !?i -- a2 Dt2 a2 Dt (2.69) 

a$ 

3z = 0 on z = O? for (x,y> in S (2.70) 

$(z = o+) = Hz = O-) for (x,y> not in S (2.71) 

plus a radiation condition on the far field behavior of 9. 

Now take the Laplacian of (2.69) and introduce the compressibility variable c* such that 

v2$l = c* (2.72) 

Then, in the frequency domain 

1 D2s* 

gT---- - V2c” = 1 D vq-/ 

Dx2 - a2Dx - 

=---- 1 DE a6 

a2 Dx az (2.73) 

where (2.48) has been used to replace V27f. The solution for 

2." is 

1D a c*=-- a2Dxz J S z(t) G(x - I> dt (2.74) 

where 

(2.75) 

or 

G = 

exp(-i'i;/B2>(R - Mx) (2.76) 

4nR 

20 
with 

R = x2 + B2(y2 + i2) 

1 

% 

In the far field 

M= v-/a 

‘i; = w/a (2.77) 

C* g 1 -k .-?- G(z) 

a2 Dx az J S 

exp(i'i;A/B2) z(t) d? (2.78) 

with 

(2.79) 

From the continuity equation (2.1) it follows that the far 

field acoustic enthalpy is.the compressible continuation of 

H via the relation 

aG (Z) II; =-r exp(i!%B2) x(x) dz (2.80) 

Thus it appears that the far field pressure and the OASPL 

for example can be calculated from the incompressible 

problem without specifically solving for the acoustic potential. The actual acoustic potential and velocity and therefore the far field intensity must include the solution of 

(2.72) subject to the diffraction boundary conditions (2.70) 

and (2.71). 

The essential effect of viscosity, then, is to establish 

Vuniqueness" of solution of the surface load distribution 

in the integral equation (2.65) with the kernel function 

(2.66). Once the surface loading is determined, the shed 

vorticity and the acoustic far field are uniquely determined. 

The same statement can be made about the corresponding inviscid problem (See Section IID). However, the corresponding 

compressible or incompressible integral equation for the surface load has an eigensolution. The strength of the acoustic 

field and the vortex wake depend on the strength of the 

21 
eigensolution. Without a viscous integral equation to determine the surface reactive load there is no apriori physical criterion to determine the eigensolution. It is then 

customary to invoke the Kutta condition, principle of minimum singularity or modifications thereof to establish 

uniqueness. It is important to note that edge singularities 

are completely admissable in the linearized viscous or inviscid problem. Only by accounting for finite edge curvature 

can these singularities be eliminated. It is also true that 

the edge singularities pose no real problem in the calculation 

of the noise or vortex wake. It is much more important to 

calculate the surface loading correctly, both in magnitude 

and phase, than to worry about the appearance or absence of 

an edge singularity. These remarks, by the way, are independent of frequency. Even for very low frequency interactions 

where the noise and wake vorticity scale with the total lift, 

it is known (Ref. 8) that the Kutta condition will usually 

overestimate the lift. 

G. Chebyshev Representation of the Surface Load 

and Acoustic Far Field 

Consider the class of quasi two-dimensional problems 

where the surface load is calculated from (2.70) and the far 

field from (2.79) with the effective spanwise length scale 

e over which the load is correlated or non-zero. The solution of the integral equation (2.65) can be expressed in a 

series of Chebyshev polynomials; i.e., 

gx) = c n=O An 

5+2x/c) 

(2.81) 

[1 - (2x/+)2]% 

where c is the chord of the two-dimensibnal surface. The 

spectral coefficients A, may be calculated from the finite 

matrix representation of the integral equation; i.e., 

N 

c 

C, An = g, m=O,l,..,N (2.82) 

n=O 

with 

c, = 

(m+l) ' 

rr2 / 

cosn+d$ - cos 0) 

0 / 

' cosI(m+l)B)d@q(cose 

0 

(2.83) 

22 
y-G 

/ 

1 

-1 (l 

_ x2 p/2 u,(x)jJ(x) dx (2.84) 

and 

Xl(X) = 25Fv, 

/ 

x c%k)dE ,a0 (2.85) 

The numerical solution of (2.82) for the surface load distribution is the subject of Ref. 8. 

Now substitute the Chebyshev series (2.81) into (2.80) 

to obtain the acoustic far field in terms of the spectral 

coefficients of the surface load. The result is 

h’ = _ g .e . 

c/2 

ma / 

exp[(lir/B2)(x/R - M 116 

-c/2 

m 

c An T,( 26/c > dE 

n=O ( l- (2Ud2) l/2 

aG =-- . 

az 

$zAn/" expI(i~/B~) 

n=O 0 

(x/R - M)cos8) cosnede 

= 

iK l & 

8 (z/R) exp{(-iK/B2)(R - Mx)) 

co 

c A, I" J,I( K/B2)(x/R - Ml1 

n=O 

with the Helmholtz number 

(2.86) 

K = we/2a = i&2 (2.87) 

For the range of compact frequencies ( K << 1) only the 

23 
first term in the series of (2.86) contributes to the far 

field and the magnitude of hi is proportional to A, or the 

total interactive lift that develops on the surface. The 

value of A, may be calculated uniquely with the viscous 

theory as shown in Refs. 8 and 9. 

H. Effect of Edge Bluntness 

From extensive calculations of the solution of (2.82) 

in Ref. 9 the following important result was found. For a 

surface with zero thickness the edges are mathematically 

sharp. As a result, the solution of (2.82) converges to the 

inviscid solution with Kutta condition (See Section III) for 

any Reynolds number greater than about 500. The departure 

from inviscid behavior is of order l/(Re)% . The reason 

is that by introducing a mathematically sharp edge or zero 

length scale the linear viscous effect is grossly overexaggerated at the edges. By introducing a finite edge radius 

the viscous edge effect is diminished and viscous action over 

the entire surface becomes the controlling factor in establishing the surface loading. In all cases calculated (See Ref. 8) 

the surface loading is diminished by introducing finite edge 

geometry. Anexample of the reduction in the steady state 

lift curve slope on a surface at angle of attack is shown in 

Figure 2. The lift on an elliptic section is about 15% less 

than the lift on a Joukowski airfoil with cusped trailing 

edge. More extensive comparisons of unsteady results for 

surfaces with and without edge curvature may be found in Ref. 

8. For moderately low values of reduced frequency (wc/2v, = 

order of a few tenths) the real and imaginary parts of A,, Al, 

the two leading spectral coefficients in the surface load 

series (2.80) for an elliptic section are substantially 

different from those of flat plate theory. The changes are 

in quantitative agreement with measurements of Davis and 

Malcolm (Ref. 10). 

To perform meaningful acoustic calculations it is necessary to solve the surface load problem for relatively large 

values of reduced frequency 1 (wc/2v,) = 0(1/M) and larger 1. 

The numerical problems in evaluating the kernel functions 

with thickness and viscosity for high frequency have not been 

completely resolved and detailed results on this aspect of 

the problem will be reserved for a future study. 

III. APPLICATIONS OF "INVISCID" THEORY 

A. Zero Thickness Two-Dimensional Surface 

In the following development the family of solutionsofthe 

well known two-dimensional incompressible oscillating airfoil 

24 
Re = 1000 

I I I I I I I I I 

- 1.0 -.5 0 .5 1.0 

6 

Figure 2 - Degradation of the Steady State Lift Curve Slope (CL ) 

due to Trailing Edge Bluntness (E'); note the a 

airfoils sketched above the graph. 
problem (Ref. 11) is used to draw some general inferences 

about the generation of sound for the generalized surface 

interaction problems. Following the work of Yates and 

Houbolt (Ref. 12) and recent work of Howe(Ref. 13) the extreme limits. of a full Kutta condition and no Kutta condition 

are used to illustrate why it is important to complete the 

development of the viscous theory as formulated in the previous Sections. 

For the two-dimensional problem, the spectrum of the 

acoustic enthalpy (See Eq. (2.79)) can be expressed in the 

form 

where G is the asymptotic form of the three-dimensional Green's 

function (2.76) and 1 is some measure of the spanwise correlation of the surface disturbances. The airfoil semi-chord 

(c/2) is chosen as the unit of length and the spectral function S(a) is expressed in the form 

/ 

1 

S(a) = -1 exP(iaX) y(x) dx (3.2) 

where 

a = o.K/B2)(COSe - PI) (3.3) 

and y(x) is the normalized chordwise load distribution 

that must be calculated from the incompressible boundary 

value problem: 

div G = 0 

DZ 

Dx + gradP = 0 

3 l u’ =rony=O, -l<x<l (3.4) 

with 

D a Dx =ax +ik 

26 

(3.5) 

--..--.-. --. . ..-.. . . . . ,,,. , ,., 11111.1. _..I, I., I I. .-I I 111 
and 

9(x 1 = P(x,O-) - P(x,o+) (3.6) 

The solution of '(3.4) can be expressed in terms of a hydrodynamic potential X and a vortex sheet distribution y(x) 

that are related to&@ as follows: 

f= grad x for YZO (3.7) 

av au 

-- 

W=3Y ay = Y(X) 6(y) 

Y(X) = 

aAx 

w dy = ax 

(3.8) 

(3.9) 

(3.10) 

with 

AX = x(x,0-) - x(x,0+> (3.11) 

The potential is a harmonic function with the representation 

lo3 X =- J 

-1 

2TT -1 v(S)tan & dC (3.12) 

For x > 1 (in the wake) the following relations hold: 

ax> = 0 (3.13) 

AX =- Pexp{-ik(x - 1)) (3.14) 

Y(X) = - ikI'exp(-ik(x - 1)) (3.15) 

where f is the negative of the potential jump across the 

trailing edge. Also, I' is the circulation around the 

surface; i.e., 
y=or = Y- ii (3.16) 

y=o+ 

l d; = - Ax 

x=1 

x=1 

The value of r is unknown and in fact is completely 

indeterminate with inviscid theory. 

An integral equation for y(x) is obtained by applying 

the boundary condition (3.4) to the integral representation (3.12). The result is 

Y(S) m dE = g(x) (3.17) 

with 

m 

g(x) = -2-5jx) -1 F 

/ 1 

exP{-ik(S - 1)) dS 

(5 - xl 

where the slash through the integral denotes 

the Cauchy principle value. 

(3.18) 

The solution of (3.17) iS 

Y(X) = 

1 

TI(1 - x2+ L 

-r +A$!l l 

% 

(1 - s+ ds 

-1 s-x 

/ 

03 

1 

expf-ik(S - 1)) dS 

(5 - xl 

(1 - s+ 

(s - x> 

%ms) ds ] (3.19) 

a well known result (e.g., See Ref. 11). 

The spectral function S(a) from (3.2) can be expressed 

in terms of y(x) with the formula 

28 
S(a) = r $ eia + (5 - 1) J: eicLx y(x) dx 

(3.20) 

that is easily derived from (3.9) and (3.lO)by integration 

by parts. Now substitute (3.19)into (3.2O)to obtain the 

following representation of the acoustic spectrum: 

S(a) = r 1 - i Jo(c) + k eia 

K ) 

+ 4 (1 - i) 2 I" J,(a) /_: (1 - 5')' U,-,(S) WE) dS 

(3.21) 

where Jr,(a) is the ordinary Bessel function and U,(s) is the 

Chebyshev polynomial of the second kind. 

On the basis of the viscous analysis of a zero thickness 

surface, (Ref. 9) the value of the circulation r must be 

such that the inviscid flow at the trailing edge is smooth 

(the Kutta condition). In that case the expression in square 

brackets in (3.19) must vanish as x + 1‘. The resulting expression for r is, 

2 

rK = - J -1[(l++-s)]‘-%s) ds

ik eik 

(3.22) 

Kl(ik) + K,(ik) 1 

where K,, Kl are modified Bessel functions of the second kind. 

The subscript on r is used to indicate the Kutta condition. 

29 
On the other hand, if the surface has a finite thickness, 

the value of r is diminished in a way that must be calculated with the viscous theory. An extreme case is that 

when no vorticity is shed into the wake or I' = 0. One 

may refer to this example as a case of no Kutta condition. 

It is physically the more correct result for high frequency 

finite thickness surface interactions where there is insufficient time for viscosity to establish a finite circulation. Recall (Figure 2), however, that even at low frequency there are significant departures of the total lift 

from the result produced by the Kutta condition. 

The expression for the total lift on the surface is 

ikKl(ik)] -2ik / 1 

-1 

(1 - s2)%jms) ds 

(3.23) 

For k * 0 the lift is proportional to r. For high frequency and no Kutta condition the lift is given by 

/ 

1 

LNK = -2ik -1 (1 - s2)'_Wis)ds (3.24) 

On the other hand if the full Kutta condition is applied at 

high reduced frequency, the asymptotic form of the lift is 

given by the expression 

c(l+s)/(l -s:]'Hs) ds / 

1 

-2ik 

-1 Cl -s2)'_Wis)ds 

(3.25) 

The spectral form of the total lift is of interest 

because it is proportional to the far field acoustic spectrum 

in the compact limit. Consider the compact case where the 

reduced frequency is large and the Helmholtz number K in 

(3.21)is small or 

K << 1 << k (3.26) 

30 
The Mach number must, of course, be small for the validity 

of these relations. Then only the leading term in the series 

expression in (3.21) is important and s(c) is given by the 

asymptotic expression 

/ 

1 

S(a) = fik (n/2k)%exp(-is/4) -2ik -1 (1 - s2)'ps)ds 

(3.27) 

With the full Kutta condition and no Kutta condition, the 

asymptotic spectrum is proportional to LK and LNK,respectively. 

For convective disturbances (e.g.,a frozen turbulence 

pattern that interacts with the surface) the spectrum of the 

surface velocity is of the form 

(3.28) 

where 7T is some function of the frequency. In this case 

the integral in (3.27) can be evaluated exactly and the 

acoustic spectra and lift are given by the following expressions: 

sK = LK = -2 %(2n/k)' ei(k - JT'~) (3.29) 

'NK = LNK r -4 %(2n/k)' l 1 sin(k - n/4) (3.30) 

and for the spectral density functions, 

I2 = ILKI = 8rrlyo\2 l l/k (3.31) 

ILNKI 2 = 8~1Iyol~ l t sin2(k - v/4) (3.32) 

The interesting feature of these results is that with a 

Kutta condition the spectrum decays monotonically with k 

while the no Kutta spectrum has an oscillatory decay with 

local maxima and zeroes for particular values of k as 

31 
shown in Figure 3. .These results‘suggest that a simple 

experiment could be derived to measure the acoustic‘spectrum of a controlled convective or acoustic disturbance and 

determine by what degree the Kutta or no Kutta behavior is 

prevalent at high frequency. 

B. Vortex Fly By a Sharp Edge 

A specific example of the foregoing result is that of a 

single weak vortex convecting past a fixed surface. In that 

case the surface upwash is 

%+%,t) = *v- Eo (x -“,-t‘;$+ 01 h2 (3.33) 

where ro is the strength of the vortex and h is the 

vertical distance of the vortex from the plate. 

of '$fl is of the form (3.28) with 

The spectrum 

where 

T = 2h/c (3.35) 

and k is assumed to be positive. For convecting boundary 

layer turbulence, the parameter h is of the order of the 

boundary layer thickness, the spectrum is peaked and has a 

sharp cutoff near the reduced frequency, k = l/~, that is 

much greater than unity. Substitute (3.34) into (3.31) and 

(3.32) and estimate the acoustic intensity with the relation 

OD f= (2V,/C)2 $ / 

I&;1 2 dk 

0 

(3:36) 

The final result is 

y= & . $ . sin'8 ke-2k' F*dk 

(3.37) 

where 

r’* = 1 Kutta Condition 

32 
2ll 3a 

k 

Figure 3 - Spectrum of the Far Field Interactive Noise and 

Surface Load for an Airfoil Encountering a 

Sinusoidal Gust; k = WC/~V~. 
= 4sin2(k - n/4) No Kutta Condition (3.38) 

The integration over k is easily performed with and without 

the Kutta condition. The ratio of the acoustic intensity with 

and without the Kutta condition is 

&LEE= 1 

XNK (2 - 4r3)/(1 + .t212 

g l/2 for T << 1 (3.39) 

Thus the intensity is approximately 3dB less when the Kutta 

condition is applied. With a full Kutta condition convective 

vorticity is produced in the wake with an energy level that 

is the same order of magnitude as the radiated sound. In 

the supplement to this report a full viscous problem with no 

flow is considered and a similar result is obtained. The 

vertical energy in that case, however, is confined to the 

vicinity of the edges and is not convected into the far wake 

as above. An even more drastic result was obtained in the 

recent trailing edge noise study of Howe (Ref. 13). For a 

semi-infinite plate the edge noise due to convected turbulence 

is typically 1OdB greater when no Kutta condition is applied. 

It should be pointed out however that the semi-infinite plate 

presents a much more singular problem because of the possibility of very weak upstream decay of the surface loading. 

IV. SUPPLEMENTARY MATERIAL 

,A. Discussion of Trailing Edge Noise Models 

During the earlier part of this program an inviscid dynamic model of discrete vortex fly by of a blunt (parabolic 

and rectangular) trailing edge was developed and calculations 

of the surface pressure and spectra near the trailing edge 

were made. The objective was to compare with the experimental results of Brooks and Hodgson (Ref. 6). The inviscid 

model accounts for two effects that are not included in the 

preceding analysis. First, the geometric shape of the 

edge (and the associated curvature of the mean flow) and 

second, the non-linear reaction of the convecting vortex to 

the surface boundary conditions. On the other hand, the 

important effect of vortex production and shedding due to 

viscosity is omitted. 

34 
The discrete vortex model was used in earlier work CRefs. 

1,2) to illustrate the basic coupling between the vorticity 

and acoustic modes. The model has been used by Hardin (Ref. 

14) to model the hydrodynamic origin of cavity noise. 

More recently Obermeier (Ref. 15) has-used the model to calculate the noise emminated due to vortex fly by of finite 

cylindrical bodies. Again, no vortex shedding is permitted 

in Obermeier's study and noise is calculated with the theory 

of Mohring (Ref. 16) that in practice differs very little 

from the theory of Powell (Ref. 17) as used by Hardin (Ref. 

14). 

While some of the basic velocity scaling laws and frequency parameters evolve from the inviscid discrete vortex 

model, the fundamental question of the trailing edge noise 

problem is side stepped completely. That is, how does viscosity control the vortex shedding and load alleviation near 

the edge? Howe, (Ref. 13) in a recent review of trailing 

edge noise theory has called attention to the Kutta condition as the fundamental uncertainty in the edge noise problem 

(See Section IIIB). If one adopts a potential theory point 

of view with regard to the trailing edge fluid mechanics, 

and, furthermore, if the edge is mathematically sharp, then 

the difficulty can indeed be reduced to an argument about the 

potential flow edge singularity or the Kutta condition. 

However, it is this authors point of view that "potential 

theory" as a fluid mechanical model is at its very worst near 

the trailing edge. Furthermore, with the slightest amount of 

edge blunting the edge singularity is eliminated and the 

Kutta condition becomes meaningless as a local vortex shedding 

criterion. More important is the fact that singularity 

arguments avoid rather than illuminate the basic physics of 

the problem. The main thesis of the present work is that 

viscosity is the missing piece of the physics and must be 

accounted for in the fluid mechanical calculation near the 

trailing edge. 

It was the realization of the importance of direct viscous effects in this study and in Ref. 8 that led to a redirection of the present effort from potential flow non-linear 

modeling to viscous flow linear modeling. Ideally, it would 

be desirable to calculate the complete nonlinear viscous fluid 

mechanics but that, of course, is an extremelydifficult problem at the present time. The present state of development of 

the linear viscous model is the following: The theory for 

the zero thickness two-dimensional surface is complete and 

some calculations have been made for low reduced frequency. 

The results are more directly applicable to unsteady airfoil 

work and are presented in Ref. 8. The evaluation of the YIScous kernel function and the load calculations for high frequency is not trivial but considerable progress has been made. 

35 
Based on the flat plate results for low frequency it was 

found, however, that direct viscous effects atthe trailing 

edge are so strong that the calculations converge to the 

inviscid results with Kutta condition for Reynolds number 

(based on the chord) greater than 500. 

The foregoing result means that when direct viscous 

effects are included in the linearized problem, there must 

not be any geometric length introduced that forces an infinitesimal or zero Reynolds number as for example at a 

mathematically sharp trailing edge. The correct formulation 

of the linearized viscous problem must include the finite 

trailing edge geometry, as discussed in Ref. 8. With a 

finite edge geometry the evaluation of the high frequency 

viscous function is even more difficult. To date only low 

frequency results have been obtained for oscillating rigid 

airfoils. The load calculations agree more closely with experimental results (Ref. 10) (both in magnitude and phase) 

than the flat plate calculations with Kutta condition. -Based 

on the low frequency results it is concluded that even 

greater departures from the flat plate theory will result as 

the frequency is increased. The fundamental linearized 

problem, then, is to solve for the viscous load distribution 

on a surface with finite geometry for frequencies well into 

the acoustic regime. The basic theory is available and it 

remains a computational problem to carry out this program. 

B. Summary of the Supplemental Report 

In Ref. 18, explicit viscous calculations have been made 

for an oscillating airfoil problem. The results are extremely 

important both for the conceptual understanding of noiseSurface interaction problems and for the quantitative calculation of interactive noise. The problem was motivated 

by the experiment of Brooks (Ref. 5). An airfoil section 

was vibrated at Helmholtz numbers of 1.3 and 2.04 in an 

anechoic chamber. The surface pressure and motion and the 

radiated pure tone sound were measured simultaneously. The 

surface measurements were also used in the "inviscid" Kirchhoff 

integral relation to calculate the radiated sound. The calculation overestimated the sound by 2 to 5 dB depending on 

direction. Also the measured chordwise surface pressure distribution has a flat behavior near the edges while aerodynamic 

theory predicts a square root decay of the edge loading. A 

more detailed description of the experiment may be found in 

(Ref. 5). 

The theoretical work in the supplementary report has 

shown that the measured surface load and sound field are 

mutually consistent if the effect of viscous dissipation at 

36 

&-. 
the edges is taken into account. The inviscid Kirchhoff integral has acoustic energy conservation built in so that all 

of the work done by the vibrating airfoil is propagated to 

the far field in the form of sound. When the same surface 

load is used in a viscous incompressible calculation the surface work appears in the near field (close to the edges) as 

kinetic energy associated with the vorticity formation due 

to viscosity. For a Helmholtz number of order unity and a 

near quadrupole distribution of the surface load it is shown 

that the viscous energy dissipation is nearly equal to that 

propagated as sound. 

For Helmholtz numbers less than unity (into the compact 

regime) it is expected that even more energy will be transferred into the vorticity mode. Inviscid acoustic theory 

in the form of Kirchhoff integrals will in general overprediet the noise in the compact regime. In each application 

an estimate of the dissipation should be made with an incompressible viscous analysis like that in the supplementary 

report. A correction to the inviscid Kirchhoff calculation 

can then be made. For very small Helmholtz numbers it would 

appear that there is no alternative to a complete compressible 

viscous calculation. Furthermore, when there is a finite 

convection velocity the surface work transferred into vorticity is greatly enhanced and it is even more important to 

understand and calculate the effect of viscosity. 

v. CONCLUSIONS 

The most important conclusion of this study is that viscosity is an important missing physical element in aeroacoustic theories where interactions with a surface with edges is 

involved. Even in the absence of a mean flow there can be 

an appreciable viscous effect near the edge with an associated loss of energy to the acoustic mode. With a finite 

convection velocity, the magnitude of the circulation around 

the surface is determined uniquely by viscosity. The use of 

the Kutta condition will in general overestimate the circulation which means that more vorticity is shed into the wake and 

less noise is generated. There is no apriori reason why a 

potential flow singularity cannot occur at the trailing edge. 

It is much more important to estimate the magnitude of the 

load distribution correctly over the entire surface and this 

can be accomplished with the viscous theory. It is important 

that finite geometry and boundary layer displacement thickness effects be accounted for in the viscous calculation. 

The acoustic theory developed in this report and the associated 

report on viscous thin airfoil theory (Ref. 8) is the correct 

approach for carrying out the surface load calculation. The 

difficult practical step is the unsteady aerodynamic load calculation for high reduced frequency - a problem that has perplexed the unsteady aerodynamiscist for many years. 

37 
Considerable progress has been made during the course of this 

program but much work remains to be done before routine calculations can be made. 

38 
REFERENCES 

1. Yates, John E., "Application of the Bernoulli Enthalpy 

Concept to the Study of Vortex Noise and Jet Impingement Noise," NASA CR-2987, April, 1978. 

2. Yates, John E., "A Study of Scattering, Production, and 

Stimulated Emission of Sound by Vortex Flows,' NASA CR3139, May, 1979. 

3. Chu, B., and L.S.G. Kovasznay, "Nonlinear Interactions 

in a Viscous Heat-Conducting Compressible Gas," Journal 

of Fluid Mechanics, -~ vol. 3, 1957-58, pp. 494-514. 

4. Yates, J.E., "A Study of Production and Stimulated 

Emission of Sound by Vortex Flows," in Mechanics of Sound 

Generation in Flows, E.A. Muller, ed. Springer-Yezag, 

Berlin Heidelberg, 1979. 

5. Brooks, Thomas F., "An Experimental Evaluation of the 

Application of the Kirchhoff Formulation for Sound Radiation From An Oscillating Airfoil," NASA W-1048, 1977. 

6. Brooks, T.F. and T.H. Hodgson, "Prediction and Comparison 

of Trailing Edge Noise Using Measured Surface Pressures," 

AIAA Paper No. 80-0977, presented at the AIAA 6th Aeroacoustics Conference, Hartford, Conn., June 1980. 

7. Huerre, P. and K. Karamcheti, "Effects of Friction and 

Heat Conduction on Sound Propagation in Ducts,' JIAA 

TR-4, Stanford University, August 1976. 

8. Yates, J.E., "Viscous Thin Airfoil Theory," A.R.A.P. 

Tech. Memo No. 413, Ref. 1980. 

9. Yates, J.E., "Unsteady Viscous Thin Airfoil Theory,' 

AGARD Report No. 671, Sept. 1978. 

10. Davis, S.S. and G.N. Malcolm, "Experiments in Unsteady 

Transonic Flow, AIAA Paper No. 79-0769, presented at 

the AIAA/ASME/ASCE/AHS 20th Structures, Structural 

Dynamics and Materials Conferences, St. Louis, MO. 

April, 1979. 

11. Fung, Y.C., An Introduc.tion to The Theory of Aeroelasticty, -- John Wiley &Sons, Inc., New York, 1955. - 

12. Yates, J.E. and J.C. Houbolt, "On the Aerodynamic Forces 

of Oscillating Two-Dimensional Lifting Surfaces," AFOSRTR-71-1079, December 1970. 

39 
13. Howe, M.S., "A Review of the Theory of Trailing Edge 

Noise," NASA CR-3021, June 1978. 

14. Hardin, J.C., "Noise Calculation on the Basis of Vortex 

Flow Models," presented at ASME Synposium on Noise and 

Fluids Engineering, Atlanta, GA, Nov. 1977. 

15. Obermeier, F., "Effects of Solid Boundaries on Aerodynamic Sound Generation," Max-Planck-Institut Fur 

Stromungsforschung, Gb'ttingen, August 1979. 

16. Mb'hring, W., "On Vortex Sound at Low Mach Number," J. 

Fluid Mech., Vol. 85, 1978, p.685. - 

17. Powell, A., "Theory of Vortex Sound," Aerodynamic 

Noise, Sept. 1963, PP. 538-540. 

18. Yates, J.E., "The Importance of Viscosity in Experimental 

Applications of Kirchhoff-Type Integral Relations," AIAA 

Paper No. 80-0971, presented at the AIAA 6th Aeroacoustics 

Conference, Hartford, Conn.,June 1980. 

40 
~-- 

3. Recipient’s Catalog No. 

NASA CR-3331 

I- 4. Title end Subtitle 

VISCOUS THEORY OF SURFACE NOISE INTERACTION 

PHENOMENA ---iI 

-~ -- 

7. Autbr(sJ 

John E. Yates 

9. Performing Organization Name and Address 

Aeronautical Research Associates of Princeton, 

Inc. 11. Contract or Grant No. 

50 Washington Road, P.O. Box 2229 

Princeton. New Jersev 08540 1 12. Sponsoring Agency Name and Address 

National Aeronautics and Space Administration 

Washington, DC 20546 

1 

1 6. Supplementary Notes 

Langley Technical Monitor: Jay C. Hardin 

Final Report 

1 6. Abstract 

A viscous linear surface noise interaction problem is formulated that 

includes noise production by an oscillating surface, turbulent or 

vertical interaction with a surface, and scattering of sound by a 

surface. The importance of viscosity in establishing uniquesness of 

solution andpartitioningo.fenergy into acoustic and vertical modes 

is discussed. The results of inviscid two-dimensional airfoil theory 

are used to discuss the interactive noise problem in the limit of 

high reduced frequency and small Helmholtz number. It is shown that 

in.the case of vortex interaction with a surface, the noise produced 

with the full Kutta condition is 3 dB less than the no Kutta condition 

result. Also, the spectrum with Kutta condition decays monotonically 

with frequency while the corresponding result without the Kutta 

condition decays in an oscillatory manner. The results of a supplementary study of an airfoil oscillating in a medium at rest are discussed. 

It is concluded that viscosity can be a controlling factor in analyses 

and experiments of surface noise interaction phenomena. It is further concluded that the effect of edge bluntness as well as viscosity 

must be included in the problem formulation to correctly calculate 

the interactive noise. - 

li ‘. Kev Words (Suggested by Author(s)) 

Aeroacoustics 

Noise Surface Interactions 

Bernoulli enthalpy 

Surface Interactions 

__- 

16. Distribution Statement 

Unclassified-Unlimited 

Subject Category 71 

15 I. Security CIaoif. (of this report) 20. Security Classif. (of this pqe) 21. No. of Pqes 22. Rice 

Unclassified Unclassified 42 A03 

For sale by the National Technical Information Service, Springfield. Virgrnta 22161 

7