NASA/TMm1998-208403

Vortex/Body Interaction and Sound

Generation in Low-Speed Flow

Hsiao C. Kao

Lewis Research Center, Cleveland, Ohio

National Aeronautics and

Space Administration

Lewis Research Center

September 1998

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VORTEX/BODY INTERACTION AND SOUND GENERATION IN

LOW-SPEED FLOW

Hsiao C. Kao

National Aeronautics and Space Administration

Lewis Research Center

Cleveland, Ohio 44135

SUMMARY

The problem of sound generation by vortices interacting with an arbitrary body in a low-speed flow has been

investigated by the method of matched asymptotic expansions. For the purpose of this report, it is convenient to

divide the problem into three parts. In the first part the mechanism of the vortex/body interaction, which is essentially the inner solution in the inner region, is examined. The trajectories for a system of vortices rotating about their

centroid are found to undergo enormous changes after interaction; from this, some interesting properties emerged. In

the second part, the problem is formulated, the outer solution is found, matching is implemented, and solutions for

acoustic pressure are obtained. In the third part, Fourier integrals are evaluated and predicated results presented. An

examination of these results reveals the following: (a) the background noise can be either augmented or attenuated

by a body after interaction, (b) sound generated by vortex/body interaction obeys a scaling factor, (c) sound intensity

can be reduced substantially by positioning the vortex system in the "favorable" side of the body instead of the "unfavorable" side, and (d) acoustic radiation from vortex/bluff-body interaction is less than that from vortex/airfoil

interaction under most circumstances.

1. INTRODUCTION

Since the pioneer work of Lighthill, many theories and numerical methods for predicting aeroacoustics have

been put forward. Most of these theories involve volume or surface integrals to account for source distributions in

the turbulent near field. However, since a quantitative description of turbulent flows is difficult, perhaps we can,

from a practical point of view, justify exploring some simpler methods.

The aerodynamic aspect of sound production is known to be characterized by the formation of eddy motions

(see Powell's formulation of vortex sound in ref. 1 or 2). A simplified representation of these motions is the singular

distribution of vorticity, such as the discrete vortices moving in an inviscid fluid. The justification for this can be

found elsewhere (e.g., ref. 3); however, a brief description is provided here.

In a viscous fluid and at a relatively low Reynolds number, the velocity and vorticity fields are usually continuous. As the Reynolds number increases, the vorticity field becomes more concentrated and is often embedded in an

otherwise irrotational flow field. In taking the limit, these concentrated vorticity fields, which are governed by the

Biot-Savort law, may then be modeled by discrete vortices with given circulations. The discovery of large-scale

coherent structures in turbulent flows, especially in mixing layers, provides a further justification for this model.

Considerable numerical work has been done in the application of various forms of discrete vortex methods to simulate two-dimensional mixing layers (e.g., ref. 4).

Because of a great increase in computing speed, aeroacoustics computations based on the unsteady Euler or

Navier-Stokes equations can now be performed. However, obtaining accurate solutions depends on the resolution of

certain issues, such as the size of the computational domain, the outer boundary conditions, and such. Thus far, no

satisfactory solutions for these issues have been attained (ref. 5).

Notwithstanding these problems, computational aeroacoustics may still be the most promising method in the

long run. At present, the use of simpler methods without these problems may be justified. To this end, one chooses a

potential flow model with discrete vortices to represent the flow in the inner region. And it is known that "all acoustic motions in the vicinity of a singularity are solutions of Laplace's equation" (p. 40, ref. 6). To solve acoustic problems, the solution in the potential flow region has to be matched to the outer solution of the compressible flow

(acoustic field) by means of the method of matched asymptotic expansion. The acoustic equation to be used herein is

the acoustic equation for a moving medium instead of that for a stationary medium.

NASA/TM--1998-208403 1
Theproblem now has been separated into two parts: an inner potential flow region and an outer acoustic field.

This enables us to deal with complex geometries for the following two reasons. (a) There are a number of numerical

methods available for two-dimensional potential flow in the presence of arbitrary bodies. The method used here is

based on Martensen's formulation and is that favored mostly by British researchers (e.g., see ref. 7 or 8). The advantage of this method for the present problem is that a body is replaced by a distribution of surface vorticity elements.

Thus, the surface and the moving vortices can be treated similarly by the same Biot-Savart law. (b) The detailed

flow characteristics of the inner region in an acoustically compact region will not, in general, affect aeroacoustics.

Instead, only the gross properties of the potential flow influence the acoustic radiation. The evidence for this is that

after a Fourier transform has been taken, the acoustic equation becomes a Bessel equation whose solution for the outgoing wave is a Hankel function. Thus, the solution for sound radiation in the frequency domain is nearly determined.

The matching process merely determines the coefficients. For this reason, even for a body of very complex geometry,

once the inner solution is known, the pressure fluctuation in the far field in the frequency domain is established.

The idea of using asymptotic matching to study aeroacoustics in a low-speed flow has been implemented in the

past (e.g., refs. 9 to 15). In these studies if a body was present, it was usually of a simple shape, so the method of

conformal mapping was applicable. Here we propose to broaden this method to include an arbitrary body and to

adopt the acoustic equation in a moving medium.

Although the outer solution is known in the frequency domain, an inverse Fourier transform is still necessary to

bring it back to the physical domain. This has to be done numerically for a general case. There are at least two methods available. One is the method of the discrete Fourier transform, and the other is the method of convolution integral. The former is well known and its application to the present problem is relatively straightforward. The latter

involves divergent integrals, which were derived by Conlisk and coworkers (ref. 13 and related papers), but which

have not been evaluated numerically for far-field acoustic pressure fluctuations. Let us re-examine this method to

see why divergent integrals appear, and to evaluate these integrals with the help of Hadamard's (ref. 16) definition

of the final part.

This study examines sound generation by a vortex/body interaction. Initially a group of spinning vortices (see

fig. 1) is introduced at an upstream station; as they move downstream and pass near the surface, an interaction takes

place. This interaction causes the vortex trajectories to deviate from their original paths, alters the characteristics of

the background noise, and gives rise to sound radiation. We studied each of these aspects in some detail and present

the results herein.

The results of this investigation are presented in the following sections. In sections 2 and 3, respectively, we

derive the potential flow solutions in the inner region and discuss the related topics of vortex trajectories and their

interaction with the body. In section 4, a brief description of the formulation is given, which leads to the Fourier

transform of the inner solutions in section 5 and the outer solutions in the frequency domain in section 6. Section 7

considers the process of matching and the solution for the acoustic pressure. Sections 8 to 10 cover two methods for

evaluating the Fourier integrals and the test cases for the background noise. In section 11 results are presented and

discussed.

ds m

K

M

M o

N

p'

R

airfoil chord length

length of surface element at point Sm

influence coefficient

total number of surface elements

free-stream Mach number

total number of free vortices

fluctuating (acoustic) pressure

radial coordinate in outer region

SYMBOLS

NASA/TM--1998-208403 2
r I

s

t

Vo

U_V

U s

W

x,y

z=x+iy

Zct

F

_'m

0

O'

Om,O.

%

radial coordinate in inner region

position of surface element

time in moving frame

free-stream velocity

velocity components in x and y directions

surface velocity

complex potential

Cartesian coordinates

complex number

initial position of vortex centroid

radius of spinning vortices

strength of vortex

surface vorticity per unit length at point s,n

radial coordinate of surface element Sra

angular coordinate of observation point

angular coordinate in inner and outer regions

angular coordinate of surface element m or n

time in fixed frame

velocity potential

stream function

angular velocity of spinning vortices

angular frequency in Fourier transform

frequency of spinning vortices

Subscript

i

m,n

0

ith free vortex

surface element (point) m,n

initial position or free-stream condition

NASA/TM--1998-208403 3
2. POTENTIAL FLOW SOLUTIONS IN THE INNER REGION

As stated previously, an incompressible potential flow is assumed to exist in the inner region, which includes a

body as well as the moving rectilinear vortices with constant circulations concentrated at the singularities. The complex potential W for an isolated vortex located at z0 in a free space is

iF _ _

W -- _ + i_ : -_- log(z - Z0) (1)

where _ is the velocity potential; _ is the stream function; F is the circulation defined to be positive in the clockwise direction; z = x + iy; and _ = ._ + iy. This is the building block for the potential flow to be discussed.

Our present interest is to study the interaction of vortices with a body in a stream and, therefore, the presence of

the body has to be taken into account. First, we consider Martensen's method of potential flow calculation for a

body of arbitrary shape without the moving vortices (for more details, see either of Martensen's papers, or ref. 7 or

8.) Since Martensen's method is not based on a transformation method, no images of vortices are necessary. The

body is defined by a string of surface vorticity elements separating the moving stream outside the body from a stationary medium inside the body. The complex potential is simply a summation of M surface vorticity elements of

undetermined strength, denoted as )'m' plus the uniform free stream:

M

w=_+i_ = _ i_,"as,, log(Z-Zra)+ Z

2_r

ra=l

(2)

Symbols here denote dimensionless quantities and are defined as follows:

w = z=r/i, r. =ym/ o, dsm=a ./i

where U-0 is the free-stream velocity; i is the reference length; Yrn is the surface vorticity per unit length located at

the point Zra on the surface of the body; and Agra is the length of the element (see fig. 2). (In subsequent calculations, the reference length for an airfoil is the chord and that for a rectangular block is the height.) The last term in

equation (2) is the complex potential for the uniform free stream assumed to be parallel to the x-axis. It follows that

dW M i_md$" " 1

--=u-iv= E +1

dz "=1 2_ z - zm

M _"ds" y - y"

u(x,Y)=m=lE 2/r (X_Xm)2+(y_y") 2

v(x, _ _ _'mdsm x- xm

Y)- - 2.,,,=1_ (x-x")2+(y-Y")2

(3)

The symbols u and v denote the velocity components in the x- and y-directions at a point x and y in the space induced by M surface vorticity elements. To determine the unknown _,n in equation (2), we apply equation (3) at the

surface elements sn and impose the Dirichlet condition that the intemal tangential velocity component is zero:

NASA/TM--1998-208403 4
21_

1__

ra=l,n_m

Z. ymdsmK(sn'sm) + Yn /l sn 4to Rn cos0n

K(sn,Sm) : (Yn - Ym)C°SOn -(Xn - Xm)SirlOn

(Xn-Xra)2+(yn-Yra) 2

n=l,2 ..... M

(4)

Here x n, Yn are the coordinates for Sn; R n is the radius of curvature at sn; On is the tangential angle at sn (see fig. 2);

and K(s n, sm) is the influence coefficient, which represents the tangential velocity component induced at sn by other

surface vorticity elements. The second term on the left side of the equation refers to the self-induced velocity of

vortex element )'ndsn.

Equation (4) is a system of M simultaneous equations for M unknowns Ym"The potential flow problem for the

inner region is now reduced to the solution of M linear simultaneous equations. Any standard method can be used to

solve it, for there is no ill-conditioning involved. We have used the Gaussian elimination method for every calculation performed so far.

The accuracy of this method has been assessed by Lewis (ref. 7), who compared its results with the known

exact solutions, including those for bodies with sharp comers. We have also done our own comparisons, with such

configurations as the NACA 0012 airfoil, and like Lewis, we concluded that we have a "high level of confidence" in

this method. (For a cusped airfoil the accuracy may deteriorate.) Plotted in figure 3 is the velocity distribution at the

forward comer of a sharp-edged rectangular block in a uniform flow. This is an indirect comparison since as the

length increases, the potential flow at the sharp comer approaches that of a step for which the exact solution exists.

The 2:1 ratio rectangle can almost simulate a step. Also plotted in figure 3 is the surface velocity distribution in the

forward face of the same rectangle but with round-edged corners. Both of these configurations will he used later for

predicting sound generation, and we will see that even slight rounding of the comers can reduce sound radiation.

Let us now consider the problem of vortex/body interaction. With the presence of free vortices in the field,

equation (4) assumes a somewhat different form:

1 M t_as

-2-- ) _tmdsmK(sn,Sm) + rnl

Air m=_,m_n _,_7r'l_n

l Z FiK(sn'zi)

= - cos 0. - _ ;=1

(5)

The last term on the right side of this equation accounts for the presence of the N free vortices, whose influence coefficient assumes the form

K(sn,Zi) = (Yn - yi)c°sOn -(xn - xi)sinOn

(Xn - xi) 2 +(Yn - Yi) 2

A two-dimensional vortex induces no velocity at its own center and moves with the local free-stream velocity

plus the velocities induced by the other vortices. With the presence of these free vortices Fi, equation (3) becomes

M N iFi 1

i]tmds m 1 _-1+ _".

iv j, = '_f,_._, 2tr zj -- Zm "--" 2re Zj -- Zi uj

m=l i=l,i:_j

where zj is the coordinate for vortex j, and uj, vj are the velocity components at zj. Here zi and Ym are functions of

time, but ds m, zm, and Fi are constants.

(6)

NASA/TM--1998-208403 5
Theevolutionofvortexj is governed at each time step by the following two first-order differential equations:

dxj = u,, dy./ = v/

dT _ d_

For a field with N vortices there are 2N such equations whose solutions give the trajectories for N vortices. At each

time step, we have to update Zm,by solving equation (5). A second-order method is used for computing the time

development from x to x + Ax:

(7)

yj(Z + AT)= yj ('t')+ l[3vj (T)--vj(r- AT)]AT

The time step Air is generally determined by comparing two solutions, one obtained with Az and the other with

AT/2. If these two solutions agree reasonably well, the time step AT is chosen; otherwise the process continues. In

general, A'r = 0.0125 to 0.025 was found to be acceptable. The initial conditions for equation (7) are the coordinates

of free vortices at locations sufficiently far upstream, where the interaction with the body is minimal.

The accuracy of this method for a single free vortex was assessed by comparing two trajectories past a circular

cylinder in an incompressible flow; one was obtained by the present method and the other was the exact solution

obtained by the image method. In a test run with Az= 0.1 and F= -1.0, four-figure accuracy was attained in the

vortex position, except very near the surface of the body, where the accuracy was reduced to three figures.

3. VORTEX TRAJECTORIES AND INTERACTIONS

With the necessary equations for a potential flow in place, we can now proceed with the solutions. First, a few

remarks axe in order.

Three conservation laws apply to a system of discrete vortices in a two-dimensional unbounded fluid at rest at

infinity; one of these states that the centroid of the system is stationary. If there are N equal, like-signed vortices

spaced equally around the circumference of a circle, they rotate about the centroid with a constant angular velocity.

In the present study N is equal to 4 or less. Emphasis will, however, be placed on a pair of spinning vortices, since

this is a classical example for acoustic radiation and there is an analytic solution for it. When such a system is placed

in a moving stream, the centroid moves with the local convective velocity.

The central topics of the present investigation are the vortex trajectory patterns and the acoustic radiation from

the interaction. Problems of a similar nature have been previously studied. For example, Hardin and Mason (ref. 17)

examined the blade/vortex interaction for a single vortex and the associated sound generation. Their results will be

used later for comparison with the present solutions. More recently, Panaras (ref. 18) investigated the vortex trajectories for many vortices and their interactions with several Joukowski airfoils. However, the relatively large number

of vortices used in the investigation may have obscured the process, since the vortices developed irregular patterns

prior to the interaction. As a result, the effect of the interaction on the vortex trajectories cannot be easily seen.

In order to see the effect of an interaction, it is preferable to limit the vortices to a small number and to arrange

them in such a way that the motion prior to interaction is regular; thus, the change in their trajectories can be discemed immediately. To this end, the present system consists of no more than four vortices, and it undergoes a circulax motion in a stationary medium. With the convective velocity included, the trajectory of each individual vortex

forms a wave pattern (figs. 4 to 9).

Hardin and Mason pointed out that vortex/blade interactions are highly nonlinear, which is also true here. So a

slight change in the initial conditions may result in distinctly different trajectories after interaction. For this reason a

number of plots are included here to show the different possibilities that an interaction can bring. In figure 4 three

trajectories by a pair of spinning vortices of equal and positive strength (clockwise rotation) are shown. Initially this

pair was stacked in an upstream station (fig. 1), whose vertical distances measured from the airfoil chord line are the

same, but the horizontal distances from the leading edge of the airfoil are different. With x 1= x 2 = -24.4, the fore

and aft trajectories are similar. However, as the initial position of the pair moves forward to -24.6 (20 percent of the

NASA/TM--1998-208403 6
chord),the trajectories after interaction (the aft trajectories) are no longer the same as the upstream trajectories (the

fore trajectories). The pair spins faster behind the airfoil. If the pair moves further upstream to -25.0, a pronounced

change in the aft trajectory pattern takes place, in which case the vortices depart from each other and, consequently,

the rotating speed becomes much smaller. This suggests that the background noise of a spinning vortex pair can be

attenuated by interaction with a body, which we will examine later. To understand this scenario, we cite a wellknown equation given in reference 19:

.O = _ (N - 1)F (8)

where _ is the angular velocity and N represents N equal, like-signed vortices F arranged uniformly about a circle

of radius _. According to this equation, the spin rate becomes smaller as the two vortices depart from each other,

and the associated sound decreases.

A closer examination of the trajectories reveals that here the aft trajectory patterns are periodic in the sense that

their evolution from a high to a low rotating rate repeats itself as the initial position of the vortex pair is repositioned

continuously forward or backward. The reason for this is that a vortex moves along a wavy path and its time of

arrival in the neighborhood of the airfoil is periodic.

Thus far, only the positive vortices placed above the chord line have been studied. If they are replaced by two

negative (counterclockwise) vortices with the other conditions unchanged, the fore and aft trajectories of all three

previously mentioned cases become similar. For example, figure 5 is a counterpart of figure 4(c), with identical

parameters except for the sense of the vortices. In reference 14, Sen quotes Howe's result that a vortex produces

sound mainly when it cuts across base streamlines near an edge. The velocity field induced by a negative vortex is in

the direction of the local (base) convective velocity and, therefore, produces less disturbance. By contrast, the velocity field induced by a positive vortex goes against the local convective velocity near the surface, so the vortices are

more likely to be repelled to a different orbit. On the basis of this intuitive notion, we would expect a pair of positive

vortices placed below the chord line to have the same effect as a pair of negative vortices above the chord line. This

expectation turned out to be correct and has been verified by calculations.

Three different trajectory patterns are shown in figure 6, one of which involves four vortices. They are included

here because these patterns will be used later to calculate acoustic pressures. Note that the bluff body (a 2:1 ratio

block) shown in figure 6(b) and the following figures is merely a distorted schematic of the real geometry. Since

there is a large disparity between the vertical and horizontal scale, a 2:1 block would be unrecognizable if plotted to

scale.

Apparently this intuitive notion for an airfoil is also applicable to the vortex/bluff-body interaction. For

example, see figure 7, where two sets of trajectories for a sharp-edged bluff body are plotted. Again, the initial conditions for parts (a) and (b) of figure 7 are the same; only the sense of the vortices is different.

Previously, we dealt exclusively with equal and like-signed vortices. If we ease this constraint a little to permit

vortices of unequal strengths, the centroid will be shifted to one side; other aspects of the trajectory pattern will,

however, remain essentially the same. For this reason, no results and graphics have been included.

Let us now shift our attention to the interactions involving three like-signed vortices of equal strength rotating at

a constant speed (fig. 1). In this case the forward trajectories are essentially three intertwined identical curves, but

the aft trajectories are a variety of vastly different patterns. Consequently, only a representative case is shown in

figure 8, which consists of two sets of curves, one for a sharp-edged body and the other for an equivalent roundedged body. The purpose is to demonstrate the influence of a corner. Apparently, rounding off the corners slightly

can make a difference in the aft pattern as well as in the acoustic radiation.

Usually, after a vortex/body interaction, the trajectories become less organized, as can be seen in the aforementioned figures. The reverse may also be true in the sense that after an interaction vortices may spin in a more orderly

fashion than before. This possibility is depicted in figure 9, which shows all three aft vortices rotating at nearly the

same speed.

4. FORMULATION

The problem to be considered consists of two regions: an inner and an outer region. The governing equations

for the former have been given. In this section the differential equations for the latter are to be derived. Since the

NASA/TM--1998-208403 7
acousticpressureadiatedfromasourcewill be distorted after traveling through a convective stream, it is preferable

to find solutions based on the acoustic equation in a moving medium instead of in the stationary medium used in

earlier investigations. This equation for a fixed frame of reference is given by

o32p, 2 , D2p,

, o3 p u2 __ _ O,

o_x2 l- _- m 0 Dr 2 -

D 0 o3

where - + (9)

DT oar 0x

The symbols p', r, x, y, M 0 are, respectively, the dimensionless quantities for the perturbed pressure, time, spatial coordinates, and Mach number. They are defined as follows:

-- - m

2 y U o

p'= -# r= v°-t, x = M0=--

_oUg ' l "-f ' Y= l ' c o

where _ois the density, iis the reference length, t is time, and E0 is the speed of sound. To make the analysis more

tractable, this equation is transformed from a fixed frame to a moving coordinate system (ref. 20) by defining

x' =x- r, y' =y, and t= r. Thus,

t92p, a._._pp 2 , _ _,,2 02p,

ae 2 0y,2 _v_0

(10)

This equation is of the same form as the classical acoustic equation. Therefore, all available methods for solving the

classical equation can now be put to use. Once the solution is known, it can be transformed back to the fixed frame

of reference.

Introducing two outer variables defined as R = Mor' and O' = tan -1 y'lx', where r' = (x '2 + y,2)1/2, we obtain

02P' F1 0 p' + 1 02p ' c92P' = 0 (11)

oTR 2 R OR R 2 o30'2 o3t'2

This is the final form of the governing equation for the outer region, subject to the radiation condition at infinity and

asymptotic matching near the body.

On the basis of the premise that steady flows generate no sound, the inner potential flow equations and the subsequent numerical expressions can be written in such a way that the unsteady part is separated from the steady part.

For the purpose of matching, we have to express the unsteady velocity potential in terms ofp' given by the linearized Bernoulli equation:

D_' c9O'

- (12) P'= P-Po - Dr Ot

The operator D/Dr represents the time rate of change of O'as seen by an observer moving with the free-stream

velocity. Note that the notation r denotes time in a fixed frame. If a moving frame is used, ,will become t and

D_TDr becomes cgclJTOt.

5. FOURIER TRANSFORM OF INNER SOLUTION

With the governing equations for the inner and outer regions both attained, we can proceed to solve low-speed

aeroacoustics problems. As briefly mentioned in section 1, the procedure is as follows. First the unsteady part of the

inner solution in the limit as r _ ,,ois determined, from which, by means of equation (12), the perturbation pressure

p'is found. Next, the Fourier transform of p" is obtained for matching. The partial differential equation (eq. (11 )), is

NASA/TM-- 1998 -208403 8
thentransformedviatheFouriertransformto an equivalent ordinary differential (Bessei) equation, whose solution

for the outgoing wave at infinity in the frequency domain is the Hankel function. The limiting form of this function

as R --->0 is matched to the inner solution to secure the unknown coefficients. Then in the last step, an inverse transform is performed to return the solution to the physical space.

Following these steps, let us proceed to determine the inner solution by first considering a pair of spinning vortices moving past a body. The complex potential for this system is similar to that of equation (2):

_? irmaSm log(z--Z,, iF 2. - )+z+ 13)

m=l

Most symbols in this equation have been previously defined. The terms in the last two sets of brackets represent the

positions of the spinning pair. The symbol Zct refers to the initial position of the vortex centroid and is a constant.

The quantity z + Zct is the position of the centroid at time z convected by the free stream in the absence of a body.

The symbols +zo are the local coordinates of vortices relative to the centroid (fig. 1). If the body is absent, this pair

rotate freely, are always aligned, and need only the quantities z + Zct +-z0 to denote their instantaneous positions.

However, in the presence of a body, these positions will be perturbed because of the interaction with the body; thus

we introduce z I and z2.

After excluding the steady part in equation (13), the relevant portion is the unsteady part IV' given by

z.ll+ (,

_ log z 1- log m=_ Z.'j Z_L k Z-- _-- Zct J Z-- q_-- Zct

Expanding this equation for Izm/z I << 1, [ (z0 + Zl)/(z - T- Zct ) I << 1, [ (Z 0 -- Z2)/(Z -- "_-- Zct) I << 1, and defining z = r

exp(i0) and zm = t_m exp(iOm), we have

W'= M i_mdsm F, .^ a m i(Om-O)] ir 2..,_L,og r +,_--_-_ +_-#

m=l

Zl z2 z0 1

(z-.-z.fJ+""

(14)

M

The first two terms in the first pair of brackets are zero, because Em=l _mdsm : 0. This is due to the fact that

Em=l M _'mdsm = _ (t9 _ / 3 s)ds = 0, provided that there is no circulation.1 (Ymcan also denote the surface tangential

velocity component; see ref. 7.) Note that only one second-order term in this equation is retained; others have been

dropped. This second-order term represents the unsteady motion of the spin and produces the background noise.

Other terms in this equation are first-order terms that are essentially zero until the vortices experience an interaction

with the body, and they will return to zero after the interaction is over.

The next step is to put equation (14) in a more convenient form and then to find p' by means of equation (12).

Introducing the polar coordinates of the following quantities,

Z - ¢: -- Zct = r' e iO' and Zo = fl eit°°t

where to0 = F/4x [32, we transform equation (14) to the form

M ),mdsm t_ra i(O m -0) iF x 1 + iy 1 + x 2 + iy 2 e-i_ iF f12

W'=_'+itF'=-iE 21r r e -2"-_ r' 27tr '2

m=l

e 2i(t°Ot-O') (15)

1If there is a strong vortex system in the proximity of a symmetric airfoil, the airfoil may experience a local angle of attack even if the free stream

is parallel to the chord line. This local effect, however, is neglected here.

NASA/TM--1998-208403 9
Inthisequationthesymbolsr and 0in the first term are the polar coordinates measured from the fixed frame, since

the body is stationary. However, the acoustic equation in the outer region is given in terms of r" and 0'. Thus, it is

necessary to express r and 0in terms of r', 0', and t before any matching can take place. Although this is possible,

we can demonstrate mathematically to first-order accuracy that r and 0 can be replaced by r' and/9' during the process of matching. The physical explanation for this is that oat'm/oat_e0 only when vortices are in the vicinity of the

body, where r and 0 are approximately equal to r' and 0'. Note also that by defining z0 = fl exp(ic00t ) we implicitly

assume a semblance between the fore and aft trajectory patterns; otherwise z o has to be defined differently.

Taking the real part and applying equation (12), we obtain

p_=Zd)'rndsrn&rnsin(O,_Orn)+ Dx, +DX2]sinO,_(Dy, +Dy2_cosO,

ra=l o3r 2tr r' " " 2tr r Dr Dr / \ Dr Dr /

FrO0 t2

tr r" 2 cos 2(O90t - 0").

(16)

The first term represents the fluctuation of the surface pressure, the second term is due to the interaction, and the last

term represents the unsteady motion of the spinning pair. Notice that 7tinis differentiated with respect to r since the

body is fixed. Applying the Fourier transform pair

j_(R, to) = _f(R,t)e-iWtdt

to equation (16) gives the Fourier transform of p/:

and f(R,t) = _ (R, to)eitadto

--oo

t M0

27r L n L ,or Dr) n

Pi---_'_ E Ore) 1-M0 sin0" r ( Dxl + Dx2 ]e-i_dt cos0'

m=l n __o.

X 7 ( DyI+ Dy2 ]e -itatdt] /-'M2t°0fl2 {cos20'[tS(to-2mo)+t_(to+ 2tOo)]+isin20"[t_(to+ 2mo)-t_(to-2tOo)]}.

-bTj j n

(17)

where R = M0r'and t_ ( ) refers to the 6-function. The second-order term is proportional to IIR2, whereas the firstorder term is proportional to IlR.

6. OUTER SOLUTIONS IN THE FREQUENCY DOMAIN

In view of the inner solution (eq. (17)), the outer solution may be assumed to have the following form:

M

p_(R,O',t) = E pm(R't)sin( 0"- Ore) + Pl(R't)sinO' + p2(R't)c°sO" + p3(R't)sin 20" + P4(R, t)cos20'

m=l

Substituting this expression into equation (11) gives the following equations:

1 o3 ql l o32ql _ o32q2 1 o3 q2 4 o32q2

o32ql + R2 ql 0, -- +--- q2 - _ 0 dR 2 R OR ---_t 2 - O3R 2 R OR R2 =

where ql refers to Pro, Pl' or P2, and q2 refers to P3 or P4" These two equations can be transformed via the Fourier

transform to two equivalent Bessel equations:

(18)

NASA/TM-- 1998-208403 10
d2_ldR2 _IRddROIA(+ co2_.__l )ql^ =0 and d2_2+ld02 +(co2-_ 4). q2 =0 -- dR 2 R dR \

whose solutions for the outgoing waves at infinity are the Hankel functions of the second kind. We, therefore, obtain

M

P_) = Z AmH}2)(coR)sin(O'-Om) + AoH_2)(coR)sin 0' + BoH_2)(coR)c°s0' + C°H(2)(coR)sin20"

m=l +DoH2(2)(0_) cos 20, (19)

where A m, A o .... D O are the unknown coefficients to be determined by matching.

7. ASYMPTOTIC MATCHING

As mentioned briefly in Section 5, matching requires that the outer solution be put in the limiting form as R ---) 0

(ref. 21). With the aid of the formulas given in reference 22, we have

M . • • •

PO = Z 2i amsin(O,_Om) + 2i (Aosin0, + BocosO,)+_(Cosin20, + DOcos20, )

m = 1;ft.(fiR " _OJ_ /IXO /(

Equating this equation to equation (17) gives the unknown coefficients in equation (19):

CO -

Am -

mo _----

iFMoco s(Dxl+OX2_le-Oltdt

4 \ De: Dr J

--oo

B0

co [ (co+2coo)-*(co- 20,0)]

4

o32p, + 1 o3 p' + 1 o32p, o32p, _ 0 (11)

o3R 2 R o3R R 2 o30 '2 o3t '2

With the coefficients defined, we apply the Fourier inverse transform to equation (19) to give the acoustic pressure in the far field:

pP-- '=o=Edsm_msin(O'-Om) }o_t-l_2)(a_)eiaXd_ }_--_-ia_d_- @sine' 7coH_2)(o_)e'aXdco

8/_" m=l --_ --_ ¢_ --o.

DXl , Dx2 _

X i(--_.--_)e-ia_d_+_9-OcosO" icoH_2'(coR)ei_dco _( DyI + Dy2)e-i°_d_

-.o ___.o_.De 0 4 )

+ I-M_co3ofl2[J2(ZcooR)sin2(O ' +COot )- Y2(ZcooR)cos2(O" +coot)] (21)

NASA/TM-- 1998-208403 11
The last term, written in terms of Bessel functions of the first and the second kind, J2(2o90R) and Yz(2O90R),

refers to the background noise that will be heard by the observer constantly. All other terms represent the interaction

sound, which will be heard by the observer in the interval when the vortices are near the body.

The remaining task is to evaluate the Fourier integrals in equation (21). Two procedures are generally available.

One is to use the discrete Fourier transform, and the other is to rewrite the integrals as convolution integrals. These

convolution integrals are akin to the divergent integrals of the Hadamard type and have to be evaluated accordingly.

These two procedures will be discussed separately.

8. METHOD OF DISCRETE FOURIER TRANSFORM

Although the term O.i/-/1(2)(O9R) in the inverse integrals in equation (21) approaches o91/2 as o9-_ oo, the pulse

functions such as Oyrn/O_ are basically of the triangular wave forms whose Fourier transforms decay like 1/o92 at

infinity. Thus, the function as a whole is absolutely integrable, and the inverse integral exists. On this basis, the

existing methods for the discrete Fourier transform, such as those stated in reference 23, can be used. In order to

provide some assurance that this method was being applied properly, we repeated Crighton's solution numerically

(ref. 10) and found a close agreement with his results.

9. METHOD OF CONVOLUTION INTEGRAL

The procedure of writing the Fourier integrals in equation (21) in terms of convolution integrals was developed

by Conlisk and coworkers (see ref. 13 and related papers). To our knowledge, no actual evaluation of these integrals

has been made. Their concern has been with the near-field sound, which does not require evaluation. A close look at

their procedure suggests some questionable steps may have been taken. Such steps could be the reason that the

resulting integrals become divergent and are of the Hadamard type. However, the evidence that we have, on the

basis of actual evaluation, seems to indicate that these steps did not totally invalidate the outcome, except the emergence of divergent integrals. With the help of Hadamard's definition of the final part (ref. 16 or 24), the physically

meaningful part of the integral is extracted and the meaningless part is discarded. Perhaps for this reason, the acoustic pressure in the far field based on convolution integrals agrees reasonably well with some existing solutions under

certain circumstances. However, discrepancies begin to develop if the interaction becomes strong and vortices are

very close to the surface. This anomaly may be common to the methods using discrete vortex filaments with singular

centers, as was also mentioned by Hardin and Mason.

In order to make the explanation easier, let us repeat some of the steps in reference 13. Equation (21) contains

many terms, but they are all similar. It is, therefore, sufficient to pick only one term for illustration:

SDXl-io_._ iI-MoSDXl--S 2)(alR)eitO(t-,)do9 iI'Mo ogt_2)(o_)ei°_do9 --_e _a_:----_ --_d_ ttg4_ 8_r ....

(22)

The purpose of this step is to change the order of integration, which is permissible if the functions are squareintegrable (see p. 27 in ref. 25). However, as mentioned previously, o.i-/I(2)(O9R) ---) o91/2as o9---) oo.The squareoo (2) 2

integrability criterion _ mill (taR)l do9 < oo is therefore not satisfied.

Assuming this step is valid, we still have to write H(2)(ogR) in terms of the modified Bessel function KI( ) before we can consult a table. Equation (22) can now be reduced to

Dx I e-iO_ d_ -

il'Mo ttllt[2)(oAtC)ei_ do9 --_

81r --0. --.o

i/Mo 8Jr (__2 ]

--**--_

DXl d_ --_(°K

7

1( io9R)eit°(t-O do9

_ I_Mo R t--fRDx 1 d__ g 2 ]3/2 4It L De [(t_¢)2_

(23)

NASA/TM-- 1998-208403 12
Although we formally obtain a convolution integral, an improper use of the table has been made. For example, reference 26 states explicitly that the parameter in KI( ) must be real and positive. This constraint is violated in the above

equation. We nevertheless consider that this method has its merit as will be demonstrated later.

The convolution integral in equation (23) is similar to the acoustic potential function given in reference 13,

which has to be differentiated in order to obtain the acoustic pressure. However, special care is needed when a divergent integral is involved. For this reason p" is chosen as the dependent variable instead of the acoustic potential.

Divergent integrals similar to that in equation (23) have also been encountered in the linearized theory of supersonic flow (slender-body theory), which is known to be equivalent to the acoustic theory. Furthermore, the present

convolution integral is of the Hadamard type. Therefore, existing methods (ref. 16 or 24, for example) can be used to

define the final part. A brief description is given here to better illustrate a point that may affect the accuracy of some

numerical results.

Usingf(_) to represent DxllD _, we may write equation (23) as

t-R f(_)d_ t--jR f(_)_ f(t- R) t-R

_ __t(,-+)-R ]

The first integral on the right side of the equation can be shown to be an ordinary improper integral and may be

evaluated accordingly. The second integral is a divergent integral and has to be determined by using Hadamard's

definition of the final part. We now write the first integral to be a sum of two parts with cr denoting a small quantity

t_R f(_)_ f(t- R) t-R-a t-R

j r,_....__,2_2d¢= _ f_(.___)--f(t-R)de+

___L(t-¢) -R ] ___ [(t-C) 2- t-R-tY

(24)

The first part is a regular integral and can be evaluated by a conventional method; the second part, after expanding

f(_) about the point t - R and retaining the first-order term, becomes

t-R t-R

f(_) - f(t - R) (t - R- _) d_

(t - _ - R)3/2(t- _ + R) 3/2

which has a closed-form solution. Our experience seems to indicate that trin equation (24) has to be chosen in such

a way that it is comparable to the size of A_. If it is too small, large local spikes may develop. Even with some precaution, this anomaly is still there and can be seen in figure 10.

10. BACKGROUND NOISE

It is perhaps worthwhile at this stage to present some evidence that the method of convolution integrals is

capable of yielding reasonable results, at least under some conditions. To this end a pair of spinning vortices convected in a free space is chosen as the first test case. An analytic solution to this problem is known (ref. 9) and is, in

fact, the last term in equation (21). The process leading to an analytic solution involves the use of the &function. If

we forgo this practice, we can still proceed, but in this case convolution integrals will emerge. Taking similar steps

as in equations (22) to (24), we have

I t-R t-R COS 2to0_ d_}

p'=-3_-FM2R2tOofl 2 sin 2t°°¢ de + cos20' _ ---- 5/2

sin7r [ 20'___ [(,__)---2--R---_15/2 -- [(,__)2_R-'21

(25)

NASA/TM--1998-208403 13
Thisexpression is similar to equation (23), except that the denominator in the integrand is raised to 5/2 power

instead of 3/2 power, which occurs because the acoustic pressure for a spinning pair is a second-order effect.

Figure 10 compares two sets ofp'. Although the frequency and the amplitude given by equation (25) are both

somewhat larger than the analytic quantities, the agreement is not really bad, especially in view of the fact that the

solution given in equation (25) involves two divergent integrals. In the present formulation, which uses the convective acoustic equation, the Doppler effect on acoustic radiation should appear by itself. Although this property cannot be seen easily in figure 10, it will be discernible at a higher Mach number for example, at M o > 0.3.

The acoustic pressure depicted in figure 10 may be assumed to represent the background noise in the problem of

a vortex/body interaction, if the fore and aft trajectories are similar. If they are different, the characteristics of the

background noise after interaction will be altered, that is, have a different frequency and amplitude. In this case we

have to choose two sets of average values to approximate the fore pattern from _,o to x 1 and the aft pattern from

x 2 to _. (The interval between x I and x 2 is the interaction zone, which is neglected as far as the background noise is

concerned.) For illustration, the step modulated inputflt) = U(t) cos toot is used to represent a cosine wave in the

interval from 0 to _,, where U(t) is the unit step function. The Fourier transform off(t) is

0)o/+ +0)0/]+i0)

(See p. 41 of ref. 25.) This expression is seen to be equivalent to D o in equation (20). However, since it has a singular term, a special treatment is required in the inverse transform. For this reason, the method of convolution integrals

is preferred.

Again following similar steps as in equations (22) to (24), we obtain

• 3/'M2R 2

p = - sin 20'

47r

,_2t}R U(-_ - Xl) sin 20)i_ 'sRU(_-x2)sin 20)_I 3/-M2 R2

0)IP' L [_t__)__-ff2_-_ d_+0)3f12__** _-___R2] 5/2 d¢I-_ 41r cos20'

(26)

where 0)1 and fll denote the average angular velocity and the radius of rotation in the forward portion of the trajectory, and 0)3 and t3 denote the same for the aft portion. The symbol -x I is the endpoint of the forward portion,

whereas x2 is the starting point of the aft portion (see fig. 4(c)). The symbol U( ) refers to the unit step function

shifted by the amount -x 1 or x 2 to account for the semi-infinite length. If the segment becomes infinitely long, the

unit step function becomes unity throughout and equation (26) reduces to equation (25).

To demonstrate the applicability of equation (26), we extricated the required information from the calculated

trajectories in figure 4(c) to evaluate p', which is plotted in figure 11. Notice that in this case the background noise

after an interaction has been almost totally attenuated. Because of the linearity property, this noise signature can be

superimposed on the acoustic radiation from an interaction to present a complete acoustic field. This is shown in the

following section. However, because of the scale difference, the background noise is not discernible. For this reason,

the body in figure 11 is removed and the vertical scale is enlarged.

11. RESULTS AND DISCUSSION

In the foregoing sections we discussed two methods to evaluate the Fourier integrals. The method of convolution integrals is rather complex and lacks mathematical rigor, yet it is an interesting approach and may provide simpler solutions on occasion, such as in the previous example. Furthermore, two more cases will be given to

demonstrate its applicability to sound generation, provided that the interaction is moderate. For a strong interaction

overpredictions are liable to occur. Thus, the discrete Fourier transform (DFT) is our primary method.

NASA/TM-- 1998-208403 14
Asmentionedearlier,HardinandMasoninvestigatedtheproblemofsoundgenerationbyavortexandpresentedtheircalculatedresultsforaJoukowski airfoil.Usingtheinitialandboundaryconditionsindicatedinreference17,wesimplifiedequation(21)forasinglevortexandsolveditbymeansoftheDFrtosecureasetof

corresponding acousticpressures. Figure12showsthiscomparison. Thecharacteristic lengthinreference17isthe

spanlength.However, nosuchlengthisavailableinthepresentinvestigation. Consequently, onlytherelativemagnitudescanbecompared. Theoverallshapeappearstobesimilar,thoughdetailsaredifferent.AlthoughHardinand

Masonanalyzedbothlift andnonliftingairfoils,onlythenonliftingcaseiscompared, sincetheangleofattackis

assumedtobezerointhisstudy.

Observingthetrajectoriesinfigure4,wecouldassumethathedegreeofinteractionbecomesprogressively

strongerfromparts(a)to(c)offigure4.Weintendtoshowthatasfarassoundgenerationisconcerned, thisassumptioniscorrect.ForthispurposeweusedtheDFr tosolveequation(21).Thereis,however, adisparityinthe

fore and aft trajectories in parts (b) and (c) of figure 4; the last term in equation (21) for these two cases has to be

replaced by equation (26). The results are plotted in figure 13, which shows that the background noise is relatively

low, especially in figure 13(c). To discern the background noise in figure 13(c), refer to figure 11, which has been

magnified. These three examples illustrate that the background noise is strongly affected by the dynamics of interaction, which can augment or attenuate its intensity. In the case of figure 13(c), the interaction almost silenced it.

Let us now digress to test the accuracy of the method of convolution integrals by repeating the calculation for

the case in figure 13(a). The results are shown in figure 14(a). The characteristics of the curves in these two figures

are approximately the same, although the intensities are dissimilar to various degrees. We take a more lenient view

of this method and consider this kind of accordance to be satisfactory.

Returning to the method of DFT, let us consider the case in figure 5. The initial and boundary conditions for this

case are identical to those in figure 13(c), except that F1= F2 = -0.15. Simply changing the sign of the vortices

greatly diminishes the pressure fluctuation in figure 14(b), and the first peak is now positive instead of negative. As

mentioned earlier, this outcome is borne out by Howe's result. In other words, sound generation by vortex/body

interaction depends on the sense of rotation. The effect is antisymmetric with respect to the chord line. For example,

if the vortex in figure 5 is shifted to a corresponding position below the chord line, the trajectory will be the same as

in figure 4(c), and the sound radiation will be identical to that in figure 13(c).

A possible application of this property is that if a design change is feasible, we can position the vortices on the

favorable side of the body to substantially reduce the noise level. Even for a single vortex, the direction of rotation

can make some difference. For example, the predictions by Hardin and Mason in figure 5 of reference 17 show the

different levels of sound radiation for Yo / E = -0.2 and +0.2, even at zero angle of attack.

A pair of fast-spinning vortices would be expected to produce a higher level of sound radiation than would a

pair of slow-spinning vortices. In order to substantiate this assumption, the initial positions of the vortices in figure

4(b) was modified slightly by reducing the radius of rotation, thereby making them spin faster, as depicted in figure

6(a). The corresponding acoustic radiation calculated by the DFT is plotted in figure 15(a); it is considerably higher

than that in figure 13(b).

Until now, the point of observation was assumed to be situated at 0 = 75 ° from the positive x-axis (the chord

line behind the airfoil). It would be interesting to know the sound radiation received by an observer at the same distance from the center but who is located ahead of the airfoil and slightly above the chord line (0 = 170°). In this

case, as displayed in figure 15(b), the intensity is reduced in comparison to figure 13(c), the signals are bunched

together, and there is a time delay in receiving the signal. Such characteristics are what we would expect.

Let us now direct our attention to bluff bodies. Contrary to our intuition, it seems that sound radiation from

vortex/bluff-body interactions is almost always lower than that from the vortex/airfoil interaction. The explanation

lies perhaps in the flow pattern in the nose region of a bluff body, where the fluid, after impacting a flat surface,

branches off into two streams and skirts the corners to be away from the body. As a result, the vortex never gets a

chance to be very close to the surface, and there is less interaction. This seems to be true even for the almost head-on

collision shown in figure 7(a). By contrast, a pair of spinning vortices can come very near the airfoil and practically

"scrub" the surface, thereby giving rise to a strong interaction. However, it is uncertain whether the discrete vortex

method is still valid, as vortices move into the boundary layer and the viscous effect becomes important. Further

investigations may be needed.'With an airfoil, the trailing edge may also intensify the interaction.

In order to clarify some of these intuitive explanations, in figure 16(a) we plotted acoustic pressure fluctuations

for the trajectory depicted in figure 7(a). As can be seen, this pattern changed enormously after interaction. In the

case for the airfoil, such a pattern would usually indicate high intensity radiation. Yet the acoustic pressure in

figure 16(a) is only moderate. (Note here that the background noise is not plotted in this figure and will not be plotted hereafter since its absence will not hinder our future discussion.)

NASA/TM-- 1998 -208403 15
Asa further test of the method of convolution integral, another example is provided. This method was used to

recalculate the acoustic pressure in figure 16(a). The results (see fig. 16(b)) show that the details are again at variance with figure 16(a), but the intensity is comparable.

As in figure 13, we can increase the intensity of the acoustic pressure by manipulating the vortex pair's horizontal distance from the leading edge. With a proper shift, the acoustic pressure can be increased considerably (see

fig. 17(a)), but it is still significantly less than that in figure 13(c). To lower the intensity further, simply round off

the comers slightly as in figure 3(b) or figure 8(b), and the pressure fluctuation (fig. 17(b)) lessens substantially. By

shifting the vortex pair somewhat upward and away from the centerline, the serrate peaks in figure t7 give way to

the smooth curve in figure 18. This transition from a serrate configuration to a smooth curve also occurs in a vortex/

airfoil interaction.

Finally, let us investigate the cases of three and four spinning vortices. In order to explain the scaling effect of

interaction, it is more convenient to pick the system of four vortices (fig. 6(c)). As in the previous cases, these are

equal and like-signed vortices that are spaced uniformly around a circle (fig. 1) and spin at a constant rate in a stationary medium (eq. (8)). Now we can remove two trailing vortices to form a pair of stacked vortices, as in previous

dispositions, thereby making the rotation radius smaller. The acoustic radiation from the interaction of this pair with

the airfoil is plotted as a solid line in figure 19(b). Next, the two previously removed vortices are restored to form a

quadruple. This system, however, spins at a rate 1.5 times faster than the rate of the pair, so the acoustic pressure

becomes substantially higher as shown in figure 19(b). (The motion of four vortices in the present configuration is

known to be stable (ref. 19), and there is no computational evidence of instability after interaction.)

It is proposed that the increase in acoustic pressure with the quadruple (fig. 19Co)) is almost entirely due to the

increase in rotational speed and nearly independent of the number of vortices involved. To illustrate this scaling

factor, let us reduce the vortex strength for the quadruple from 0.15 to 0.1, so that the rate of spinning matches that

of the pair, and recompute the acoustic pressure. This is represented by the dashed line in figure 19(a), which differs

only moderately from the solid line. This scaling effect should also be valid for a system of three vortices and for

bodies other than an airfoil; both of these cases are demonstrated next.

Knowing the trajectory pattern in figure 6(b) and following a procedure similar to that just discussed, we plotted

the solid lines in figure 20(a) to represent the acoustic pressure after the third trailing vortex is removed (fig. 1).

Restoring this vortex to the system produces radiation that is much more intense (see fig. 20(b)). If we then reduce

the vortex strength in a similar manner to make the two rotating speeds identical, we obtain the dashed lines shown

in figure 20(a). These two sets of lines agree fairy well, so the scaling factor is again validated.

It is pertinent to make another remark here. The scaling factor is expected to hold true for bluff bodies under

most circumstances; however, it may not be valid for strong vortex/airfoil interactions because one or more vortices

may scrub the surface. If the number of vortices is reduced, the likelihood of a close encounter is different.

Although some indirect comparisons have been made herein, there is a genuine need for detailed comparisons

with experimental data. The challenge to developing an experiment that simulates a two-dimensional flow may be

alleviated if three-dimensional problems can be solved. This is feasible, especially if the body is relatively simple

and the self-induced velocity on the curved vortex filament is calculated approximately. See, for example,

reference 27.

12. CONCLUDING REMARKS

A procedure to solve aeroacoustics problems due to vortex/body interactions in low-speed flow has been presented. An attempt was first made to understand the dynamics of a vortex/body interaction. It so happened that this

is a rather complex process that led to some interesting properties, which were discussed in some detail. In this

report only cases of zero angle of attack were investigated. However, extension to cases of nonzero angles of attack

is relatively straightforward.

The method of asymptotic matching enabled us to separate the problem into two parts: an inner potential flow

region and an outer acoustics region. The main advantage of this is the efficacy of dealing with a complex geometry

because only the potential flow sees the geometry, and there are efficient numerical methods for solving flow problems of an arbitrary body. Matching was applied merely to transmit the gross properties of the inner region to the

acoustic field. In fact the solution of acoustic pressure in the frequency domain is known; only the unknown coefficients had to be determined by matching.

NASA/TM--1998-208403 16
Afairamountofefforthasbeendevotedtodelineatingtheproblemsassociatedwiththemethodof convolution

integral and to demonstrate the usage of Hadamard' s definition of the final part. The evaluation of the convolution

integral is not essential to solving acoustics problems, but it is of some interest in unraveling the vexing properties of

a divergent integral.

The evidence from examining calculated sound generation seems to indicate that sound radiation from a vortex/

bluff-body interaction is actually less than that from a corresponding vortex/airfoil interaction. Although this may

appear to be paradoxical, there is an explanation for this property. We found a scaling factor that governs the sound

generated by the vortex interaction and a mechanism to attenuate the background noise. In addition, we proposed a

possible application that can lower sound production by simply positioning vortices on the "correct" side of the body

instead of the "wrong" side.

REFERENCES

1. Powell, A.: Theory of Vortex Sound. Acoust. Soc. Am. J., vol. 36, Jan. 1964, pp. 177-195.

2. Blake, W.K.: Mechanics of Flow-Induced Sound and Vibration. Academic Press, Orlando, FL, 1986.

3. Saffman, P.G.: Vortex Dynamics, Cambridge University Press, Cambridge, England, 1992.

4. Leonard, A.: Vortex Methods for Flow Simulation. J. Comp. Phy., vol. 37, Oct. 1980, pp. 289-335.

5. Tam, C.K.W.: Computational Aeroacoustics: Issues and Methods. AIAA J., vol. 33, no. 10, Oct. 1995,

pp. 1788-1796.

6. Dowling, A.P.; and Ffowcs Williams, J.E.: Sound and Sources of Sound, Ellis Horwood, Chichester, England,

1983.

7. Lewis, R.I.: Surface Vorticity Modeling of Separated Flows From Two-Dimensional Bluff Bodies of Arbitrary

Shape. J. Mech. Eng. Sci., vol. 23, Feb. 1981, pp. 1-12.

8. Gostelow, J.P.: Cascade Aerodynamics. Pergamon Press, Oxford, England, 1984.

9. Mueller, E.A.; and Obermeier, F.: The Spinning Vortices as a Source of Sound. Fluid Dynamics of Rotor and

Fan Supported Aircraft at Subsonic Speeds, AGARD CP-22, 1967, pp. 1-8.

10. Crighton, D.G.: Radiation From Vortex Filament Method Near a Half Plane. J. Fluid Mech., vol. 51, Jan. 1972,

pp. 357-362.

11. Davis, S.S.: Theory of Discrete Vortex Noise. AIAA J., vol. 13, no. 3, Mar. 1975, pp. 375-380.

12. Hardin, J.C.: Noise Radiation From the Side Edges of Flaps. AIAA J., vol. 18, 1980, pp. 549-552.

13. Conlisk, A.T.; and Veley, D.: The Generation of Noise in Impinging Vortex Motion Past a Step. Phy. Fluids,

vol. 28, Oct. 1985, pp. 3004-3012.

14. Sen R.: Vortex-Oscillation Model of Airfoil Side-Edge Noise. AIAA J., vol. 35, no. 3, Mar. 1997, pp. 441--449.

15. Hu, F.Q.; Martin, J.E.; and Hussaini, M.Y.: On Computing Sound Radiation of Temporally Evolving Mixing

Layer by Vortex Method and Matched Asymptotic Expansions. AIAA Paper 96-0875, 1996.

16. Hadamard, J.S.: Lectures on Cauchy's Problem in Linear Partial Differential Equations. Yale University Press,

New York, NY, 1923.

17. Hardin, J.C.; and Mason, J.P.: A New Look at Sound Generation by Blade/Vortex Interaction. J. Vib., Acoust.,

Stress and Reliab. In Design, vol. 107, Apr. 1985, pp. 224-228.

18. Panaras, A.G.: Numerical Modeling of the Vortex/Airfoil Interaction. AIAA J., vol. 25, Jan. 1987, pp. 5-11.

19. Acheson, D.J.: Elementary Fluid Dynamics. Oxford University Press, New York, NY, 1990.

20. Goldstein, M.E.: Aeroacoustics, McGraw-Hill, New York, NY, 1976.

21. Van Dyke, M.D:: Perturbation Methods in Fluid Mechanics. The Parabolic Press, Stanford, CA, 1975.

22. Abramowitz, M.; and Stegun, I.A.: Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables. Dover Publications, New York, NY, 1965.

23. Brigham, E O.: The Fast Fourier Transform. Prentice-Hall, Englewood Cliffs, NJ, 1974.

24. Ward, G.N.: Linearized Theory of Steady High-Speed Flow, Cambridge University Press, Cambridge, England,

1955.

25. Papoulis, A.: The Fourier Integral and Its Applications. McGraw-Hill, New York, NY, 1962.

26. Ryzhik, I.M.; and Gradstein, I.S.: Tables of Series, Products, and Integrals. VEB Deutscher Vedag der

Wissenschaften, Berlin, 1963.

27. Affes, H.; and Conlisk, A.T.: Model for Rotor Tip Vortex-Airframe Interaction. AIAA J., vol. 31, no. 12, Dec.

1993, pp. 2263-2273.

NASA/TM--1998-208403 17
iy

T

F

wO_

/" Zo •

/, " ,

I IL/zct I I

ZO' Z O'

• "_,zo'"

zo" I I

J

• zo,'e. .# ,,

,, ,, ,, ZO

X

Figure 1.--Initial configurations of two, three, and four spinning vortices.

Figure 2.--Surface vorticity elements on a twodimensional body.

NASA/TM--1998-208403 18
7 b

6 i

5--

4-

3--

2 -

1 -

U0

¢N

I

0.0

Method

Step Exact

• Sharp-edged block Numerical

o Round-edged block Numerical

• •

1 I 1 I

0.5 1.0 1.5 2.0

Nondimensional distance along surface

I I 1

2:1 ratio 2:1 ratio

Figure 3.--Comparison of surface velocity distributions of a potential flow over a step and

over two rectangular bluff bodies.

NASA/TM-- 1998- 208403 19
(a)

1.0

0.8

0.6

0.4

0.2

y 0.0

I I I I I I I t

-6 -4 -2 0 2 4 6 8

x

Figure 4.mTrajectodes of two spinning vortices passing an NACA 0012 airfoil at zero angle of

attack. For all cases, F = 0.15. Initial positions of vortices: (a) z 0 = -24.4 + iy 0 and z' 0 = -24.4 +

iy'0; (b) z 0 = -24.6 + iy 0 and z' 0 = -24.6 + iY'O; and (c) z 0 = -25.0 + iy 0 and z' 0 = -25.0 + iy' O, where

Y0 = 0.29 and Y'0 = 0.12.

(c)

1.0

0.8

0.6

0.4

0.2

y 0.0

-0.2

-0.4

-0.6-

-0.8-

-1.0

-8

m

m

I I I I I I I I

-6 -4 -2 0 2 4 6 8

x

Figure 5.mTrajectodes of two spinning vortices passing an NACA 0012 airfoil at zero angle of attack.

Initial positions are the same as in figure 4(c): z 0 = -25.0 + 0.29i and z' 0 = -25.0 + 0.12/, except

['=-0.15.

NASA/TM--1998-208403 20
C_

(a)

(b)

0.0

-0.2

-0.4

-0.6

-0.8

-1.0

1°I

0.8

0.6

0.4

0.2

(c)

I I I I I

-5 0 5 10 15

X

Figure 6.--Trajectories of spinning vortices at zero angle of attack. For all cases, [' = 0.15. (a) Two

vortices passing an NACA 0012 airfoil. Initial positions: z 0 = -24.65 + 22/and z'0 = -24.65 + 0.1 2/.

(b) Three vortices passing a sharp-edged block. Initial positions: z 0 = -25.7 + 0.41i; z' 0 = -25.7 +

0.24i, and z"0 = -25.5528 + 0,325/. (c) Four spinning vortices passing an NACA 0012 airfoil. Initial

positions: z 0 = -25.72 + 0.41i; z' 0 = -25.72 + 0.24/; z"0 = -25.55 + 0.24/; and z'"0 = -25.55 + 0.41i.

Y

02I

0.0

-0.2

-0.4

-0.6

-0.8

-1.0

(a)

1.0

0.8

0.6

0.4

I I I I

(b)

-5 0 5 10 15

X

Figure 7.--Trajectories of two spinning vortices passing a sharp-edged block at zero angle of attack.

Initial positions of vortices are the same for both cases: z 0 = -25.7 + 0.41i and z' 0 = -25.7 + 0.24/.

(a)r = 0.15.Co)r = -0.15.

NASA/TM--1998-208403 21
(a)

1.0

0.8

0.6

0.4

0.2

y 0.0

-0.2

-0.4

-0.6

-0.8 (b)

-1.0 r I I l

-5 0 5 10 15

X

Figure 8,mTrajectories of three spinning vortices at zero angle of attack. Initial conditions are identical for both cases: F = 0.15; z 0 = -25.55 + 0.41i; z'o = -25.55 + 0.24/; and z" 0 = -25.4028 + 0.325/.

(a) Sharp-edged block. (b) Equivalent round-edged block.

Y

1.0

0.8

0.6

0.4

0.2

0.0 I

-0.2

-0.4

-0.6

-0.8

-1.0 I I I

I I

-5 0 5 10

X

15

Figure 9.--Vortex trajectories rectified by a round-edged bluff body at zero angle of attack. Initial

conditions: r = 0.15; z 0 = -25.55 + 0.41i; z' 0 = -25.55 + 0.24/; and z" 0 = -25.19 + 0.325/.

NASA/TM-- 1998- 208403 22
(a)

p' 0.0

1.0xl 0 -4

-1.0 I I I I I I (b) l

0 5 10 15 20 25 30 35

T

Figure 10._Compadson of sound generation by two convected spinning vortices. (M 0 = 0.2; dt = 0.025;

point of observation: R = 9.0, 6 = 75 ° from positive x-axis.) (a) Analytic solution. (b) Method of convolution integral.

1.0xl 0 -4

p' 0.0

-1.0 I I I I I I I

0 5 10 15 20 25 30 35

T

Figure 11 .DBackground noise due to a pair of spinning vortices attenuated by an NACA 0012 airfoil

(same as the background noise in fig. 13(c) but plotted on enlarged scale).

NASA/TM-- 1998-208403 23
15 -

II

¢,j

0

"W 0

0

U

¢1

.g

I0

t,'-

0

E

N -10

10 -

5 -

(a)

-15

0

,._- ._oK-= o

/

_-o/c-=-o.2

I I I I I

2 3 4 5 6

0oUe

I

7

II

e3

O.

O

:=

-,s

O

_e

t-

.o

¢-.

G)

a

15

10

5

0

-5

i

_.r- Po/e = o

/

..F- .y0/_ = .-0.3

/

k --

(b)

-10 I I I 1 I l I

0 1 2 3 4 5 6 7

Oo?/e

Figure 12.--Comparison of two methods for determining sound generation by a vortex interacting with Joukowski airfoil at zero angle of attack. Observer is directly above airfoil (e = 90°).

(a) Hardin and Mason's results. (b) Results from discrete Fourier transform.

NASA/TM--1998-208403 24
p!

0.4x10-2

0.3

0.2

0.1

0.0

-0.1

-0.2

-0.3

-0.4

B

I I

l p!

0.4xl 0-2

0.3

0.2

0.1

0.0

-0.1

-0.2

-0.3-

-0.4

0

I I I tal,_,l I I I I I (b) ..I

0 5 10 15 20 25 30 5 10 15 20 25 30

'7 '7

0.6xl 0-2

r'-

0.4

0.2

p' -0.0

-0.2 -

-0.4

--0.6

0 5 10 15 20 25 30

"r

Figure 13.--Sound generation by a pair of spinning vortices interacting with an NACA 0012 airfoil. (For all cases,

F = 0.15, M 0 = 0.2, dt = 0.0125, and point of observation is R = 9.0, 6 = 75 ° from positive x-axis.) Initial positions

of vortices are the same as in figures 4(a) to (c): (a) z0 = -24.4 +/Y0 and z' 0 = -24.4 + iy'o; (b) z 0 = -24.6 + iy 0 and

z' 0 = -24.6 + iY'O; and (c) z 0 = -25.0 + iy 0 and z' 0 = -25.0 + iy' O, where Y0 = 0.29 and Y'0 = 0.12.

NASA/TM-- 1998 -208403 25
0.4x10 -2

0.3

0.2

0.1

p' 0.0

--0.1

--0.2

-0.3

-0.4

0 5 10 15 20 25 30

'T

0.4xl 0-2

0.3

0.2

0.1

p' 0.0

-0.1

-0.2

-0.3

-0.4

0

I ] I I I (b) l

5 10 15 20 25 30

qr

Figure 14.-- Sound generation by a pair of spinning

vortices interacting with an NACA 0012 airfoil.

(For both cases M 0 = 0.2; point of observation:

R = 9.0, 0 = 75 ° from positive x-axis.) Initial positions

are the same as in figure 4(c): z 0 = -25.0 + 0.29/ and

z' 0 = -25.0 + 0.12/. (a) Convolution integral, F = 0.15,

dt = 0.025. (b) Discrete Fourier transform, F = -0.15,

dt = 0.0125.

NASA/TM-- 1998-208403 26
p!

p!

0.6xl 0 -2

0.4-

0.2-

0.0

-0.2

-0.4-

-0.6

0

(a)

I I I [ I I

5 10 15 20 25 30

't"

0.6xl 0-2

m

0.4

0.2

0.0

-0.2 -

-0.4 -

-0.6

0

(b)

I I I I I I

5 10 15 20 25 30

¢

Figure 15._Sound generation by a pair of spinning

vortices interacting with an NACA 0012 airfoil.

(For both cases M 0 = 0.2; dt = 0.0125; point of

observation: R = 9.0.) (a) Faster spinning vortices

with initial positions the same as in figure 6(a):

z0 = -24.65 + 22/and z'0 = -24.65 + 0.12/, 0 = 75°

from positive x-axis. (b) Recalculation of sound

generation for the same initial positions, except

0 = 170 o.

NASA/TM--1998-208403 27
0.4xl 0-2

pl

0.3

0.2

0.1

0.0

-0.1

-0.2

-0.3

-0.4

0

I I I I I (a) l

5 10 15 20 25 30

3"

0.4xl 0-2

B

0.3-

0.2-

0.1 -

p' 0,0

-0.1 -

-0.2-

-0.3 -

-0.4

0

t i l I I (b)l

5 10 15 20 25 30

'T

Figure 16._Sound generation corresponding to trajectories in figure 7(a) for a pair of spinning vortices.

(For both cases M 0 = 0.2; point of observation:

R = 9.0, 8 = 75° from positive x-axis.) (a) Discrete

Fourier transform, dt = 0.0125. (b) Convolution

integral, dt = 0.025.

NASA/TM-- 1998- 208403 28
p!

0.4xl 0-2

0.3-

0.2-

0.1 -

0.0

-0.1 -

-0.2-

-0.3-

-0.4

0

I I I I

5 10 15 20

T

(a)

I I

25 30

0.4xl 0-2

0.3

0.2

0.1

p' 0.0

-0.1

-0.2

-0.3

-0.4

0

\

I

5

I

10

I

15

1"

I

20

I (b)l

25 30

Figure 17._Sound radiation by two spinning vortices

interacting with rectangular blocks. Initial positions:

U = 0.15, z0 = -25.0 + 0.41i and z'0 = -25.0 + 0.24/.

(For both cases M 0 = 0.2; dt = 0.0125; point of

observation: R = 9.0, 0 = 75° from positive x-axis.)

(a) Sharp-edged block. (b) Equivalent round-edged

block.

NASA/TM-- 1998- 208403 29
0.4xl 0 -2

0.3 "-

0.2-

0.1 -

p' 0.0

-0.1

-0.2

-0.3 _

-0.4

0

I I I I I I

5 10 15 20 25 30

T

Figure 18._Sound generation by two spinning vortices skirting a sharp-edged block. Initial positions:

r = 0.15, z 0 = -25.0 + 0.52/and z'0 = -25.0 + 0.35/.

(M 0 = 0.2; opt= 0.0125; point of observation: R = 9.0,

e = 75 ° from positive x-axis.)

NASA/TM--1998-208403 30
pl

pJ

0.6xl 0-2

0.4

0.2

0.0

--0.2 --

-0.4

0

F = 0.15, two vortices

..... F = 0.1, four vortices

ff

d

I

5

I

10

I

15

,r

(a) I I I

20 25 30

0.6xl 0-2

0.4

0,2 --

0.0

-0,2

-0.4 I I I I [ (b) l

0 5 10 15 20 25 30

"r

Figure 19.--Scaling factor for acoustic radiation from

two and four spinning vortices interacting with an

NACA 0012 airfoil. (For both cases M 0 = 0.2;

dt = 0.0125; point of observation: R = 9.0, 0 = 75°

from positive x-axis.) (a) Two vortices and four

vortices. (b) Four vortices corresponding to the

initial positions in figure 6(c): F = 0.15, z0 = -25.72 +

0.41i; z' 0 = -25.72 + 0.24/; z°'0 = -25.55 + 0.24/; and

z"'0 = -25.55 + 0.41i.

NASA/TM-- 1998- 208403 31
iO I

0.6xl 0-2

0.4-

0.2-

0.0

-0.2-

-0.4 I

0 5

0.6xl 0-2

r = 0.15, two vortices

r = 0.1, three vortices

i i i i (a)_

10 15 20 25 30

pl

0.4

0.2

0.0

-0.2

-0.4 I I I 1 I ()bl

0 5 10 15 20 25 30

T

Figure 20.-- Scaling factor for acoustic radiation from

two and three spinning vortices interacting with a

sharp-edged block. (For both cases M 0 = 0.2;

dt = 0.0125; point of observation: R = 9.0, 0 = 75°

from positive x-axis.) (a) Two vortices and three

vortices. (b) Three vortices corresponding to the

initial positions in figure 6(b): I' = 0.15, z0 = -25.7 +

0.41i; z'0 = -25.7 + 0.24/, and z"0 = -25.5528 + 0.325/.

NASA/TM--1998-208403 32

REPORT DOCUMENTATION PAGE FormApproved

OMB No. 0704-0188

! Public reporting burden for this collection of information is estimated to averege 1 hour per response, including the time for reviewing inslructions, searching existing data sources,

gsthedng and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this

collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorele for information Operations end Reports, 1215 Jefferson

Davis Highway, Suite 1204, Arlington, VA 22202-4302, end to the Office of Management and Budget, Paperwork Reduction Project (0704-0188), Washington, DC 20503.

1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED

September 1998 Technical Memorandum

4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

Vortex/Body Interaction and Sound Generation in Low-Speed Flow

6. AUTHOR(S)

Hsiao C. Kao

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Lewis Research Center

Cleveland, Ohio 44135-3191

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

WU-522-31-23--00

8. PERFORMING ORGANIZATION

REPORT NUMBER

E-11235

10. SPONSORING/MONITORING

AGENCY REPORT NUMBER

NASA TM--1998-208403

11. SUPPLEMENTARY NOTES

Responsible person, Hsiao C. Kao, organization code 5860, (216) 433-5866.

12e. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified - Unlimited

Subject Categories: 02 and 71 Distribution: Nonstandard

This publication is available from the NASA Center for AeroSpace Information, (301) 621--0390

12b. DISTRIBUTION CODE

13. ABSTRACT (Maximum 200 words)

The problem of sound generation by vortices interacting with an arbitrary body in a low-speed flow has been investigated

by the method of matched asymptotic expansions. For the purpose of this report, it is convenient to divide the problem

into three parts. In the first part the mechanism of the vortex/body interaction, which is essentially the inner solution in the

inner region, is examined. The trajectories for a system of vortices rotating about their centroid are found to undergo

enormous changes after interaction; from this, some interesting properties emerged. In the second part, the problem is

formulated, the outer solution is found, matching is implemented, and solutions for acoustic pressure are obtained. In the

third part, Fourier integrals are evaluated and predicated results presented. An examination of these results reveals the

following: (a) the background noise can be either augmented or attenuated by a body after interaction, (b) sound generated by vortex/body interaction obeys a scaling factor, (c) sound intensity can be reduced substantially by positioning the

vortex system in the "favorable" side of the body instead of the "unfavorable" side, and (d) acoustic radiation from vortex/

bluff-body interaction is less than that from vortex/airfoil interaction under most circumstances.

14. SUBJECT TERMS

Vortex/body interaction; Sound generation; Asymptotic matching;

Discrete Fourier transform

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OF REPORT

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A03

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