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Sound and vibration produced by an airfoil tip in boundary layer flow over an elastic plate
Journal of Sound and Vibration (1991) W(2),
SOUND
AND
VIBRATION
IN BOUNDARY
LAYER
229-245
PRODUCED FLOW
OVER
BY AN AIRFOIL AN ELASTIC
TIP
PLATE
M. S. HOWE BBN Laboratories,
(Received
10 Moulton Street, Cambridge, Massachusetts
02138, U.S.A.
10 July 1989, and in revised form 5 February 1990)
A theoretical model is analyzed to estimate the structural and acoustic noise produced when boundary layer turbulence impinges on the tip region of an airfoil, such as a rotor blade in a turbomachine. The airfoil has rectangular planform and its tip is immersed in the turbulent boundary layer on a wall modeled by a thin elastic plate. Numerical results are presented for a rigid airfoil adjacent to a steel plate in water. These indicate that the tip behaves as an acoustically bright source of sound, the intensity of which typically exceeds by 30 dB or more that which would be produced by the turbulence in the region of the tip when the airfoil is removed. Similarly, flexural motions induced in the wall (“structure-borne sound”) are shown to be substantially increased by the presence of the airfoil. This is important because structural waves may be scattered by surface discontinuities at remote points of the wall, resulting in an overall increase in the radiated sound.
1. INTRODUCTION The wall pressure fluctuations in turbulent boundary layer flow over a plane surface are characterized by the wall-pressure wavenumber frequency spectrum P(k, w), where the wavenumber vector k is conjugate to spatial co-ordinates in the plane of the wall, and w denotes the radian frequency. The spectrum is formally equal to the space-time Fourier transform of the wall-pressure correlation function, and is well defined for practical purposes provided that the largest length scales of interest are small compared to the length scale of variation of the mean properties of the boundary layer. For low Mach number flow over a homogeneous, flexible wall, the intensity of the sound generated by the turbulence is governed by the behavior of P(k, w) in the acoustic domain k = Ikl< w/c [l-4], where c is the speed of sound. In addition, flexural motions of the wall tend to be excited most favorably by pressure fluctuations in the adjacent subconvective region w/c
17 %03.00/O
,@) 1991
Academic
Press
Limited
230
M. S. HOWE
scattering of flexural wall motions that are themselves the product of diffraction of the convective domain by wall irregularities. Structural members that project from the wall into the boundary layer, or other control surfaces that are wetted by the turbulent flow, are also important sources of sound and vibration. Fins and stabilizers on marine vessels and buoyant test bodies, and struts, turning vanes and other flow control devices in wind tunnels, are frequently responsible for contamination of measurements of the low wavenumber and acoustic regions of the wall-pressure spectrum. Dowling [S] and Howe [9] have investigated a control surface in the form of a so-called large-eddy breakup device (“LEBU”), consisting of a thin, ribbon-like airfoil deployed transverse to the mean flow in the outer region of the boundary layer. Its purpose is to modify the turbulence structure in a manner that leads to a reduction in the wall skin friction. Decreases in skin friction of 20-40% on extensive sections of the wall have been reported in the literature (see, e.g., references [lo-121). According to reference [9], however, large pressure fluctuations established in the neighborhood of the LEBU in underwater applications are responsible for substantial increases in both the acoustic radiation and the structural noise generated by the flow. In this paper an investigation is made of the sound and vibration produced by a control surface in the form of a large airfoil, the tip of which is immersed in turbulent flow over a flexible wall. This may be regarded as an idealized model of a blade of a rotor in a turbomachine duct, although no account will be taken of mean blade motion, of the mutual interactions of neighboring blades, nor of the finite curvature of the duct wall. The objective is to estimate the increases in sound and vibration relative to the background radiations generated by the flow in the absence of the airfoil. The wall is modeled by a thin elastic plate, and detailed results are given for underwater applications, where the fluid-loading is high enough that the additional wall pressures and airfoil loadings generated by the turbulence-airfoil interaction would be expected to modify significantly the structural and acoustic noise. The analytical model is formulated and solved formally in section 2. In section 3 expressions are obtained for the acoustic pressure frequency spectrum of the sound generated by the airfoil, and for the spectrum of the power delivered to flexural motions of the wall. Numerical predictions are made (sections 4, 5) by use of data available in the literature to estimate the properties of the boundary layer turbulence.
2. THE ANALYTICAL 2.1.
THE
GOVERNING
MODEL AND ITS FORMAL
SOLUTION
EQUATIONS
Consider turbulent flow at low Mach number in the x,-direction of a rectangular co-ordinate system (x, , x2, x3). The fluid has mean density pO, sound speed c, main stream velocity U,, and occupies x3 > 5(x,, x2, t), where I denotes the flexural displacement of a thin plate, which in the undisturbed state coincides with the plane xj = 0. The region below the plate (x3< 5) is taken to be evacuated, and the flexural motions are small enough that 5 satisfies the linear, bending wave equation [13]
{D(a2/ax~+a2/ax~)‘+m a?/at”}l=
-p(x,,
x2,
0, f),
(2.1)
where p(x, t) is the pressure, and D, m are respectively the bending stiffness and specific mass density of the plate. A thin airfoil of rectangular planform and large aspect ratio is placed at zero angle of attack to the local mean flow (which is formally identified with the main stream velocity V,) with its spanwise dimension parallel to the x,-axis (normal to the undisturbed plane of the plate), as illustrated in Figure 1. The airfoil is modeled as a flat strip that occupies the region lx,1 < a, x2 = 0, tc < x3 < +OO; the chord 2a is of the
SOUND
Figure
1. Rectangular
airfoil
OF AIRFOIL
adjacent
TIP
IN BOlJNDARY
231
LAYER
to a flexible plate in low Mach number
turbulent
flow
same order or larger than the thickness 6 of the boundary layer, and the clearance t, between the tip of the airfoil and the plate is small compared to all other dimensions, but large enough for there to be no contact with the vibrating plate. When the properties of the boundary layer turbulence are known, or easily estimated, it is convenient to express the acoustic radiation and the flexural plate motions in terms of perturbations in the total enthalpy B. In homogeneous fluid in adiabatic motion, this is defined in terms of the pressure p and velocity v by B=
dp/p +;t?,
(2.2)
where p = p(p) is the local density. At high Reynolds number, and sufficiently low Mach number M = U,/c that refraction and scattering of sound by the flow can be neglected, B(x, t) may be regarded as the solution of the inhomogeneous wave equation [14], {d*/c*dt*-V”}B=div
(OAV),
0 = curl v.
(2.3)
in irrotational flow, where C$is a velocity potential satisfying v = VC$. At the wall v = (0,0, al/at), so that to leading order, the pressure may be replaced by poB(x,, x2, 0, t) on the right of the bending wave equation (2.1). Similarly, the high Reynolds number form of the momentum equation implies that a*l/at’ = -as/ax, at the wall. Thus equation (2.1) becomes B = -Q/dt
{D(a’/axf+a*/ax:)‘+
m a’lat’}
aB/ax,
= p,, a’B/dt’,
xj = 0.
(2.4)
232
M. S. HOWE
Equation (2.3) is to be solved subject to the boundary condition (2.4) and the radiation condition that B(x, 1) exhibits outgoing wave behavior at large distances (with the acoustic pressure p(x, t) - poB(x, t)). In obtaining the solution it will be assumed that the characteristic wavelengths of both the sound and bending waves on the plate are large relative to the chord of the airfoil. This condition permits a solution to be derived in analytic form, and is fulfilled over a wide range of frequencies in many underwater applications. 2.2.
APPROXIMATE
SOLUTION
OF THE
EQUATIONS
The analytical problem of determining B(x, t) in terms of the boundary layer vorticity and velocity distributions is analogous to that considered in reference [9] for the case of a LEBU in turbulent flow. The radiation produced by the turbulence-airfoil interaction may be attributed to a dipole source, the strength of which is equal to the unsteady force between the airfoil and fluid. This interaction can be described in terms of a Green function G(x, y; r - T), which is the solution of equation (2.3) when the term on the right is replaced by 6(x-y) ??(t - T). Application of the divergence theorem (in the manner described in reference [9]) permits the formal solution to be set in the form B(x,t)=-
(o~v)(y,~).dG(x,y;t-7)/ayd~ydT.
(2.5)
I
G(x, y; t-7) can be evaluated analytically in the low-frequency approximation in which the wavelengths of both bending and sound waves are large relative to the chord of the airfoil. The procedure is similar to that described in reference [9], and is outlined in the Appendix, where it is shown that x
G(x,y;
t-T)=-
XW3 .
I
-0c‘
L r(k)
(exp {iAk)b3-y311+ Nk, w) exp{iy(k)(x3+y3)l)
xexp{i[k.(x-Y)-w(t-~)]}d’kdw,
1xI+a,
(2.6)
where Y=
(Y,, y,+
The remaining r(k) =
V*(Y), O),
new quantities
sgn (w)(ki-
q*=Re{-iJ(z*-a*)}--yz,
z=y,+iy,.
on the right of equation
k’)“‘,
(2.6) are defined
k, = w/c
+i(k*-kg)“‘,
(acoustic
(2.7) as follows:
wavenumber), (2.8a-c)
R(k,w)={Dk4-mw2+ip,w2/y(k)}/{Dk4-mw”-ip0w2/y(k)}.
R(k, w) is the reflection coefficient for plane (propagating or evanescent) waves incident on the plate from xj > 0. In performing the integration with respect to k in equation (2.6), the singularity (pole) at the positive, real zero (k = K,, say, where K, > Ik,,j) of the denominator of expression (2.8~) is avoided by assigning to w a small positive imaginary component which is subsequently allowed to vanish. The pole characterizes flexural waves induced in the plate by a point source in the presence of the airfoil, and the formal limiting procedure ensures that these waves satisfy the radiation condition. 2.3.
LINEARIZATION
OF THE
AERODYNAMIC
SOURCE
TERM
To simplify expression (2.5) the vortex source OAV is linearized with respect turbulence velocity fluctuation. If U(x,) denotes the mean velocity distribution boundary layer, the leading order terms in expression (2.5)‘yield B(x, t) = -
u,(y,
7) &y(Y) 1
;+
WJ(Y,
7)
~b’,)
$
7>
G(x,
y;
f -
7)
d3y
d7,
to the in the
(2.9)
SOUND
OF
AIRFOIL
TIP
IN
BOUNDARY
LAYER
233
where r.+, w3 respectively denote the 3-component of the turbulence velocity u and the vorticity. In underwater applications one is frequently concerned with frequencies satisfying wS/ U, >>1, where 6 is the boundary layer thickness. In the region of intense boundary - 2*5u,/y, (u.+ being the friction velocity), whereas layer velocity fluctuations aU(y,)/ay, w3- u/y, and U - 20 u.+. The second term in the integrand of equation (2.9) is therefore typically an order of magnitude greater than the first, and in the leading approximation one can write B(x, t) = -
w,(y, T) U(y3)
dG (x, y; t ay,
T) d3y dr.
Two distinct sources of vorticity must be included in this integral. The first, .zL,say, is contained in the turbulent flow impinging on the airfoil from upstream; the second is vorticity shed into the wake of the airfoil. The function 4p* in equations (2.6) and (2.7) for G governs the strength of the radiation produced by the interaction of the turbulence with the airfoil. This is henceforth referred to as the di$raction radiation. The contribution of acp*/ay, to the integral (210) is large when y is near the leading and trailing edges of the airfoil, so that most of the diffraction radiation is produced by vorticity close to the edges. Vorticity shed from the airfoil tends to oppose the production of sound at the trailing edge by the impinging vorticity (cf., [15]), and reduces the importance of the trailing edge as a source of sound and vibration. Consider the contribution to expression (2.10) from the impinging boundary layer vorticity R,, represented in the form a: ~3(K,w’;y,)exp{i(K.y-~‘r)}dZKdw’,K=(K,,K2,0). (2.11) MY, T) = I -II Using this and expressions (2.6), (2.7) in equation (2.10), and retaining only the contribution to the radiation from the interaction with the airfoil, one finds
x exp {i(k . x+K .y - wt)} d2k dw d2K d3y,
=-I a
47r
k2K,J,(K,a) yW)tKi2
Wy,)fi(i,(K,
(2.12a)
w; ~3)
x (exp {iy(k)lx3-y311+ R(k, w) exp {iy(k)(x3+.v3)l) x exp {i(k . x - tot)} d’k dw d’K dy,,
(2.12b)
where J, is a Bessel function. The suffix i designates the radiation produced by the impinging turbulence, and in passing from expression (2.12a) to expression (2.12b) the integrations with respect to y,, y2 (over the interval -03, ~0) have been performed by reference to results given in reference [ 161. When OS/ U, is large (i.e., for small-scale turbulence), the vorticity 0, may be regarded as effectively frozen during convect@ past the leading and trailing edges of the airfoil. In that case the Fourier transform R,(K, w; y3) is sharply peaked at K, = w/ U(y3), and the principal contribution to the integration with respect to w in expression (2.12b) is from a small interval containing o = U(y,)K, . For wa/ U, * OS/ CJ, >>1, it follows by
234
M. s. HOWE
use of the large argument asymptotic expansion of the Bessel function J, [16], that the integrand with respect to K, will then consist of two terms of the form (. 1 ev
{-iK,(a
+ U(y3)f)l,
(. ) exp {iK,(a
-
(2.13)
U(y3)t)l.
The phase factors characterize the times at which the dominant components of the diffraction radiation are produced, and (when wa/ U(y3) >>1) respectively correspond to radiation from the leading and trailing edges of the airfoil. In practice, there is destructive interference between the trailing edge component of expression (2.13) and the sound produced by the wake vorticity [15]. Indeed, for a two-dimensional airfoil in a uniform mean stream ( U = constant), the net radiation from the trailing edge is predicted to vanish identically when the wake vorticity convects at the same velocity U as the impinging vorticity. This is because the unsteady pressures exerted on the trailing edge region of the airfoil by the wake and by the impinging vorticity are equal and opposite, and the associated dipole source is therefore null. Evidently, the situation is more complicated when the convection velocity is sheared, as in boundary layer flow. However, when w6/ U, >>1, so that turbulent eddies are small relative to the length scale of variation of U(y3), interference between the trailing edge forces generated by the impinging vorticity and wake must still occur in an analogous fashion, and the net radiation from the trailing edge must also be very small, provided the local convection velocities are equal. It is assumed here that is the case, and so all quantities attributable to radiation from the trailing edge are formally discarded. This is done by replacing the Bessel function in equation (2.12b) by the leading term in its asymptotic expansion, and deleting the component with the second phase factor of expression (2.13). By this means one arrives at the following expression for the net radiation B(x, I) produced by the interaction of the airfoil with the boundary layer turbulence:
x (exp {idk)lx3-y311+R(k, w) exp {iy(k)(x3+y3)l) (2.14)
x exp {i( k . x - K,a - wt)} d’k dw d’K dy,.
3.
FREQUENCY
SPECTRA
OF
THE
SOUND
AND
THE
FLEXURAL
WAVE
POWER
3.1. THE DIFFRACTION RADIATION POWER The frequency spectra of the radiation will be determined by calculating the net acoustic and flexural wave powers Z7, say, produced by the interaction of the airfoil with the turbulence. This is given by [17] 17 = PO ~’((W)+ - (W-) I -1;
dx, d+,
(3.1)
where the angle brackets denote an ensemble average with respect to the boundary layer turbulence, and the suffixes +, - respectively denote evaluation at x1 = 6, x3 = +O; i.e., just “above” and “below” the boundary layer. The velocity u3 is expressed in terms of B by use of the relation dv3/a t = -dB/ax, . Since the radiation is governed by the low wavenumber components of the Fourier integral representations of B and Us, equation (2.14) may be used to evaluate expression (3.1). The velocity fluctuations in the boundary layer may be regarded as stationary random functions of time and position (x, , x2) parallel to the plate, and there accordingly exists
SOUND
OF AIRFOIL
a wavenumber-frequency spectrum component of vorticity, such that (&K,
w; y&&K’,
TIP
IN
&(K,
BOUNDARY
w; .Y~,y5)
w’; y;)) = %(K,
235
LAYER
= fi33(K,
a; y,, yj)W
o;
yj,
~3)
-K’)S(w
of
w’),
-
the
3-
(3.2)
where the asterisk represents the complex conjugate. Using this and expression (2.14) in equation (3.1) gives
x Wy3)
~(YW,,W,
w; ~3, Y;)
d’K
(3.3)
d*k dw dy3 d.4,
where C.C.denotes the complex conjugate of the preceding quantity. The term in large brackets on the right of equation (3.3) is non-zero only in the acoustic domain k < lb/ and at the flexural wavenumber k = K, (i.e., at the real pole of the reflection coefficient R( k, w)), where it has a S-function singularity. By considering the limiting behavior as Im (w) + +O, it is easily deduced that 4y(k)(Dk4-
l+ R(k, w) or(k)
+ C.C.=
mw’)’
w[(Dk4-mw*)*(k’-k;)+(p,,w’)‘]’
k < ikol
47volw I 6(Dk4(k*- k;)
k> Ikol ’
mw* -pow2/v’tk2-
k;)),
I
These terms correspond respectively to the powers LrA, LrF (n = l?,, + n,) the acoustic and bending wave modes, such that k;y(k)(Dk4[(Dk4l7,=
poaU&64
lwtk:Z’(w)
2
Ck*
where Z’(w) is dimensionless
3.2.
THE
The
ACOUSTIC
FREQUENCY
-
6.Dk4_m
radiating in
mw’)‘Z’(w)
mw*)‘(k*-
k;)+(p,w’)2]
(3.4)
d2k dw’
w’-p,w’/J(k*-k;))d’kdw,
(3Sa) (3Sb)
6)
and defined by
SPECTRUM
Q(W)
acoustic pressure frequency spectrum Q(w) at a far field point x satisfies (P’(X r)) =
uc 0(w) dw. I -X
(3.7)
@ is obtained from nA by introducing the transformation k=
k,(sin 0 cos C#J, sin 0 sin 4),
d*k = ki sin 19cos 6 dCJd&
k < Ikol,
(3.8)
in expression (3.5a) (so that y(k) = k, cos e), and by noting that the power dLrA, say, per unit frequency that radiates through the distant surface element 1x(’ sin 19dc? db specified by the spherical polar co-ordinates (1x1,8,~$) (see Figure I), is given by dnA = {@(w)/p,c}(xl'
sin 8 de d+.
(3.9)
It follows that CD(w) =
piaU&64kiZ’(
w )[ D( k, sin 0)” - mw ‘1’ co? + co? 8 27rlx]*{[D( k, sin 8)4 - mw’]‘( k,, cos e)‘+ (pow’)‘} ’
(3.10)
236
M. s. HOWE
where cos Cc,= sin 8 sin 4, and Cc,is the angle between the radiation direction x/lx1 and the x2-axis (normal to the airfoil). This result is rendered in dimensionless form by introducing the definitions K,,= (mw2/D)“4
(the in ZIUCUO bending wavenumber), E* = polkol/mK~= poc,d(l -2u)/d3p,c(l
CL= k@o,
-a),
(3.11)
where ps, c, respectively denote the density and (dilatational) sound speed of the material of the plate and 1.7is the Poisson ratio (see, e.g., reference [13]). One then has Z’(w)( k,6)*( mk,, cos B/p,,)‘[ 1 - cc4sin4 t9]’co? $
@to) p;UWVIxI)*
=
27r{1 + (mk, cos O/p,)*[ 1 - p4 sin4 131’)
’
(3.12)
This represents the field of an acoustic dipole whose axis is aligned with the x,-direction (normal to the plane of the airfoil); when mk, cos 13/p, >>1 the plate behaves as a rigid reflector of the dipole generated sound. 3.3. FLEXURAL WAVE This is defined by
FREQUENCY
SPECTRUM
p(W)
oc I&=
V(W) dw. I --oo
(3.13)
Performing the integration with respect to k in expression (3Sb), one finds
V(,) pJJ$ClCP =
~M(lICOIS)(SPOIm)Z’(w)A2 2p2[5A4-4(Q)*-11
’
(3.14)
where M = U,/ c, and A = I&./K,,, K, being the flexural wavenumber at frequency w, which is equal to the positive zero of the argument of the S-function in equation (3Sb), so that A satisfies: A4- 1= (E.J&J(A~-~~),
A > /_L.
(3.15)
4. MODELING THE TURBULENT BOUNDARY LAYER 4.1.
THE
VORTICITY
SPECTRUM
The most important contributions to the integral (3.6) defining Z’(o) are from the near wall region, say y,, yS.< 0.1 6, where the predominant turbulent fluctuations occur. To evaluate the integral it is necessary to specify the vorticity spectrum R,,(K, w; y,, yj). There is no analytic representation of this function based on deductive theory [ 1,2]. When 081 U, is large, however, the y,-correlation scale would be expected to be of order Q/W, where v* is the friction velocity, and for the purpose of computing the integral the following model can be introduced:
&AK, w; YS,Y;) = G,(K
w; y,)l[1 +(~/~*)*(Y~-YY;)~I”.
(4.1)
The exponent n is arbitrary, but is assumed to be large enough that the range of integration in equation (3.6) with respect to yi can be extended to (-co, OO),and U(yj) can be replaced by U(y3). fln[,,(K, w; yJ is the 33-component of the vorticity spectrum at distance y, from the wall. The integration with respect to y; in equation (3.6) now reduces to OCI (4.2) &(K, w; ~3, Y;) dyS = (B&&JI~*).~S~(K w; ~3). I -cc
SOUND
OF AIRFOIL TIP IN BOUNDARY
237
LAYER
The value of the coefficient /3,, is tabulated for integer values of n in Table 1. If n = 3 is taken as the minimum value that gives a sufficiently rapid decay in coherence in the wall-normal direction, it is seen that possible alternative choices of n are likely to modify predictions of the radiation spectra by perhaps 2 or 3 dB. This must be borne in mind in interpreting the final results. TABLE
1
Values of Pn n
P,
1
2
3
4
5
6
I
1.77
0.89
0.66
0.55
0.48
044
0.40
To model aj,(K, w; y3) it will be assumed that small-scale features of the turbulence are essentially frozen during their interaction with the airfoil, and that they may be adequately approximated (for OS/ U,> 1) in terms of the von KStrmitn spectrum for isotropic turbulence [18, Chapter 31. Discrepancies that might arise from the use of this approximation as opposed to alternative possible representations of fi:, are likely to be small, say, no larger than the variations of Table 1, since detailed differences between the various models should be averaged out by the integration in equation (3.6). Thus one can take
where q’ = ( Iu(y3)12) is the mean square turbulence velocity and T(x) is the Gamma function. The parameter b = b(y,) determines the correlation scale of the small-scale turbulence fluctuations, and the b-function arises because the turbulence is convecting at the local mean stream velocity U(y3). By using the results (4.1)-(4.3) in equation (3.6), it is found that (4.4)
Z’(w) = &(V*I Uc)3(w~I &)2z(w), where Z(w) =
55r(5/6) 36UW) I
F(x) =
4.2.
NUMERICAL
I
EVALUATION
m {q(y3)l~*}2b5(y3)F[wSb(y,)l { U(Y3)l QJ’
Io
t3 dt
OF
Wy3)l
dy3
(4.5a) (4.5b)
INTEGRALS
value of the integral (4.5a) would not be expected to depend significantly on the boundary layer Reynolds number R8 = U,S/ Y (where u denotes the kinematic viscosity). To estimate the variation of the integrand with y3 one can therefore use standard experimental data given by Hinze [18] at R, = 7.5 x 104. The parameter b in the von KBrman spectrum is related to the turbulence energy dissipation E and to the turbulence intensity q2 by [18, Chapter 31 The
b =O~152(q2/uz,)“‘/(~6/u_:).
(4.6)
The following approximations for q2/ui and b that are adequate for the evaluation of expression (4.5a) and are piecewise linear in z = y3/ 6, are readily derived from data in
M. S. HOWE
238
section 7.7 of reference [18]: b=
q2/v;=
0<2<0.2 0.2
l.Sz, 0.3 ( 2*1-2z,
12002, 12.56-55*56z, ( 7.78( 1 -z),
(4.7)
O
.
(4.8)
The integrand in equation (4.5a) is assumed to vanish for yJ> S(z > 1). The mean velocity U = U(yJ can be approximated by the logarithmic formula
A In {(v*/ KJ&Y~/SI+ B U(Y3)l UC.3 = Aln{(v,/Um)&l+B ’
(4.9)
where A = 2.5 and B = 7.0. The failure of this approximation in the region very close to the wall is of no practical significance because the remaining terms in the integrand of expression (4.5a) tend rapidly to zero as y3 + 0. An asymptotic representation of Z(w) for WC?/U, >>1 can be obtained by noting that F(x) - 1/4x”/’ when x is large. Numerical evaluation of expression (4.5a) then yields: (4.10)
Z(w) - 1*07/(&j/ uX)‘7’3.
The error involved in using this formula is less than 1 dB when wS/ U, exceeds 12.
5. NUMERICAL RESULTS In this section illustrative numerical predictions are presented of the diffraction radiation spectra when the airfoil is adjacent to a steel plate in water. The following values of physical quantities are used in the calculations: for steel, pc = 7700 kg/m3, c, = 6100 m/s, (+= 0.28; for water p,, = 1026 kg/m3, c = 1500 m/s
(5.1)
Hence E* = O-144 (see equation (3.11)). The boundary layer is assumed to be specified by v*. u, = 0.037,
M = l&j/c = 0.005,
where the value of v* is that appropriate (4.2).
Pn = p3 = 0.66,
(5.2)
for Rs = 7.5 x 104, and P,, is defined in equation
5.1. THE ACOUSTIC RADIATION The acoustic pressure frequency spectrum has been evaluated for the two values S/A = 1,5, where A denotes the thickness of the plate. The solid curves in Figure 2 depict the variation of 10 log,, {@(w)/p~Uh(S/I~l)~} (dB) with wS/ U,, where e(o) has the explicit representation
G(w) &JzJz(S/]x])2
p,,M2(v+/
=
8/pJ2[1 --(CLsin r9)‘]‘cos2 rL , ~GT{ 1 + (mk, cos f3/p0)‘[ 1 - (E.Lsin e)“]‘] (5.3)
IY,)~(wS/
U,)‘Z(w)(mk,cos
and the angles specifying the radiation direction have the nominal values 0 = + = 45”. The reduced intensity of radiation at the larger value 6/A = 5 occurs because mko cos B/p0
SOUND
OF
AIRFOIL
TIP
IN
BOUNDARY
LAYER
239
Figure 2. Frequency spectrum 10 log,, {@(w)/pbU&a(8/lsI)‘) (dB) of the sound produced by the interaction of the turbulence and the airfoil in the case of a steel plate in water and parameter values defined in expression (5.2) and for 0 = JI = 45”. The dashed curve is the estimated excess 10 log,, {@(w)/@&w)} (dB) of the radiation from the airfoil over that produced by turbulence contained in area 2a6 of the wall.
is smaller at fixed wS/ Um: except at very high frequencies, the plate is then acoustically “softer”. To illustrate the importance of the acoustic radiation from the airfoil relative to the background noise generated by the boundary layer turbulence, one can estimate the spectrum of the sound that would be produced by turbulence in the “tip region” if the airfoil were absent. The tip region is defined to be the boundary layer turbulence over a rectangular section of the wall of area 2~6 (the product of the airfoil chord and the boundary layer thickness). The prediction is derived from an empirical formula given by Sevik [3] for the low wavenumber and acoustic regions of the turbulence boundary layer wall pressure spectrum P(k, w). Sevik’s formula is strictly applicable when wS/ CJ, > 24, and yields [4] the following prediction for the acoustic pressure frequency spectrum boundary layer @LL(u), say, of sound radiated from a region of area A of a turbulent formed on a rigid wall: (5.4) Chase [2] has proposed an alternative formula for the turbulent boundary layer acoustic pressure frequency spectrum that is applicable for moderate values of wS/ U,. In the interval 24 < OS/ U, < 90, however, Chase’s predictions are typically lo-20 dB smaller than those given by expression (5.4), and decrease further at lower frequencies. According to Chase, estimates made with expression (5.4) should be interpreted with great care, since it has been deduced from experimental data from tests with buoyant bodies that may well have been contaminated by extraneous sources of sound, and from wind tunnel measurements of acoustic wall pressure fluctuations that probably include contributions from the adjacent low wavenumber, hydrodynamic domain of turbulent wall pressures. For the purpose of the present discussion, however, it is convenient to regard expression (5.4) as furnishing an upper bound on the level of the background noise against which the diffraction radiation must compete.
240
M. S. HOWE
The influence of wall compliance on the turbulence into expression (5.4) by replacing @LL by [9],
QBL(w)
=
generated sound is incorporated
%(~Nmkocos elpo)*[l-(CL sine)“l’ {l+(mk,cos
0/p0)2[1-(~
sin e)“]‘} ’
(5.5)
The dashed curve in Figure 2 represents the excess 10 log,,,{@(w)/@,,(w)} (dB) of the radiation from the airfoil over that produced by turbulence in the tip region estimated by expression (5.5). For 06/ U, < 24, Sevik’s spectrum (5.4) has been arbitrarily assumed to be flat. The excess radiation is independent of S/A, the direct radiation from the turbulence being typically 40 dB below the diffraction radiation. Note that the predicted acoustic spectrum of the diffraction radiation will be lifted or depressed by a fixed amount over all frequencies if a different value of /II” is taken in expression (5.2). For the range of exponents n in Table 1, however, the change will not exceed 3 dB, and this suggests that the predictions in Figure 2 are probably correct to within about 15 dB. Changes of this order do not materially affect the import of the above conclusions. 5.2.
FLEXURAL
VIBRATIONS
The solid curves in Figure 3 depict the variation of the spectrum 10 log,,{ Y (w)/PoU&a8*} (dB) of the flexural wave power generated by the airfoil for conditions (5.2) and S/A = 1,5, where (from expressions (3.14) and (4.4)),
(5.6) These results will be compared with the power spectrum ?PBL(w) of flexural waves produced by turbulence in the “tip region” (of area 2a8) of the wall in the absence of the airfoil. For a smooth wall, these waves are generated by the component of wall pressure at wavenumber k = K,(w), which lies in the subconvective domain of the wall pressure spectrum P(k, w). A representation of the strength of the bending waves is easily
Figure 3. Spectrum 10 of the turbulence and the (5.2). The dashed curve airfoil over that produced
log,,, { ly(u~)/p,U&aS’} (dB) of the flexural wave power produced by the interaction airfoil in the case of a steel plate in water and parameter values defined in expressions is the estimated excess IOlog,,{Y(~)/P-gL(w)} (dB) of the power radiated by the by turbulence pressures applied over area 206 of the wall.
SOUND
OF
AIRFOIL
TIP
IN
BOUNDARY
LAYER
241
obtained in terms of an empirical model of P(k, w) developed by Chase [2] for a rigid wall (the “blocked” pressure spectrum), which is based on an analysis of data from wind tunnel and buoyant body tests. By using this formula, it is shown in reference [9] (see section 5.1) that
The dashed curves in Figure 3 represent the excess 10 log,,,{!P(,)/PB,(,)} (dB) of the flexural wave power generated by the airfoil over the direct radiation from turbulence in the tip region. The efficiencies with which flexural motions are excited by both the airfoil and the boundary layer increase with 6/A, but the increase is smaller for the airfoil-generated waves. The diffraction radiation is typically 30 dB greater than the direct radiation from the tip region turbulence. In other words, the flexural wave power generated by the airfoil is equivalent to that produced by the turbulence wall pressure acting over an area -2a6 x lo3 of the wall. 5.3. APPROXIMATE FORMULAE When OS/ Uoc> 10 the asymptotic representation (4.10) of Z(w) can be used in expressions (5.3) and (5.6). Other terms in these formulae can also be simplified provided
-130.
(
I ( , ,,,,,
,
, , (, (,(,
,
, , , ,,,p (0)
Figure 4. Comparison of the full and approximate and S/A = 5. (a) Acoustic spectrum at 0 = Y =45”: -, power spectrum: -, equation (5.6); - - -, equation
formulae for the radiation spectra for conditions (5.2) equation (5.3); - - -, equation (5.9). (b) Flexural wave (5.10).
242
M.
S.
HOWE
that the frequency does not exceed the critical frequency w, at which the bending wave phase velocity in vacua is equal to the sound speed in the fluid, where w,. = c’d(m/D)
= cp,,/m&,.
(5.8)
It follows from equation (3.11) that p =J(o/w,.). Thus, when w < w,. the term (p sin 0)’ can be discarded where it occurs in expression (5.3) (since sin 8 < 1 in all directions in which the radiation is finite), and one can take Q(w) p~U~a(~llxl)2c
0*17@,M2( v*/ U~)“(oS/ UJ5’3(
mko cos B/pO)’cos’ $ 7
{I+ (mb ~0s ~lkd*l 10 < d/
v, < w,6/ u,.
(5.9)
Similarly, the flexural wave power spectrum (5.6) can be taken in the simplified form U,)3M(wS/Um)-“‘3 T(w) _O~28P,(Sp0/m)‘(v,/ p0 ULaf3* EJl- ~*M(mlW(w~l &)I
10-cWCs/ u,,< WCS/ u,.
’ (5.10)
A comparison is made in Figures 4(a), (b) of these approximate formulae and the full representations (5.3) and (5.6) for conditions (5.2) and 6/A = 5. In this case w,S/ U, = 902. 6. CONCLUSIONS
The unsteady turbulent loads experienced by control surfaces are an important source of sound and vibration. In this paper an analysis has been presented of the production of structural and acoustic noise by an airfoil the tip of which is wetted by turbulence in boundary layer flow over a flexible wall. This is a first approximation to the problem of broadband noise generation by a rotor blade in a ducted flow or turbomachine. Strictly speaking, the analysis is applicable only for an airfoil with zero mean lift. For a lifting airfoil there will be an additional contribution to the noise from the interaction of the turbulence with the tip vortex. When wS/ I.& is large, however, the influence of this interaction on the airfoil loading would be confined to the trailing edge region, where its effect would tend to be countered by vortex shedding (in accordance with the unsteady Kutta condition). Thus, unless the airfoil becomes stalled, the influence of mean angle of attack and, in particular, of the tip vortex, will probably modify the present predictions by a few dB at the most. Numerical results for the case of heavy fluid loading, where the wall is modeled by a steel plate in water, reveal that the tip region of the airfoil generally constitutes an intense source of sound and flexural wall vibrations. The predicted acoustic radiation from the airfoil is typically 40-50 dB greater than that which would be generated by the turbulence in the tip region in the absence of the airfoil; similarly, the structure-borne sound produced by the airfoil is substantially greater (-30 dB, typically) than that generated by the flow in the tip region when the airfoil is removed. These estimates are based on the use of a particular empirical model of the boundary layer turbulence near the airfoil, but the possible error is estimated to be less than about *5 dB. The flexural wall modes generated by the airfoil can be an indirect source of additional acoustic radiation produced when structure-borne energy is scattered by wall discontinuities, at corners, ribs, structural supports, etc., at remote points of the wall. To eliminate secondary interactions of this kind it is necessary to attenuate the wall motions, for example, by cladding with a damping material, and predictions of the type discussed in this paper provide a starting point for the optimizing this procedure.
SOUND
OF AIRFOIL
TIP IN BOUNDARY
243
LAYER
ACKNOWLEDGMENT The research reported in this paper was supported by the Applied Hydrodynamics Research Program of the Office of Naval Research under Contract N00167-87-C-0021, administered by Mr James A. Fein.
REFERENCES 1. D. M. CHASE 1980 JournalofSound and Vibration 70,28-67. Modeling the wavevector-frequency spectrum of turbulent boundary layer wall pressure. 2. D. M. CHASE 1987 Sound and Vibration 112, 125-147. The character of the turbulent wall pressure spectrum at subconvective wavenumbers and a suggested comprehensive model. 3. M. M. SEVIK 1986 in Proceedings of the IUTAM Symposium on Aero- and Hydroacoustics, Lyon, 3-6 July, 1985. Berlin: Springer-Verlag. Topics in hydroacoustics. 4. M. S. HOWE 1988 Journal of the Acoustical Society of America (in press). Surface pressures and sound produced by low Mach number turbulent flow over smooth and rough walls. 5. D. G. CRIGHTON 1983 in Turbulence-induced Vibrations and Noise of Structures (editor M. M. Sevik). Long range acoustic scattering by surface inhomogeneities beneath a turbulent boundary layer. New York: ASME. 6. M. S. HOWE 1987 IMA Journal of Applied Mathematics 39, 99-120. Production of sound by turbulent flow over an embedded strut. 7. M. S. HOWE 1988 Journal of Sound and Vibration 125, 291-304. Diffraction of flow noise by a flexible seam. 8. A. P. DOWLING 1989 Journal of Fluid Mechanics 208,193-223. The effect of large-eddy breakup devices on flow noise. 9. M. S. HOWE 1989 Proceedings of the Royal Society London A424 461-486. Structural and acoustic noise generated by a large-eddy break-up device. 10. S. P. WILKINSON, J. B. ANDERS, B. S. LAZOS and D. M. BUSHNELL 1987 in Proceedings on Turbulent Drag Reduction by Passive Means (Royal Aeronautical Society) 1, l-32. Turbulent drag reduction research at NASA Langley-progress and plans. 11. A. BERTELRUD 1986 in Advances in Turbulence, 524-532. Manipulated turbulence structure in flight. New York: Springer-Verlag. 12. A. SAHLIN, A. V. JOHANSSON and P. H. ALFREDSSON 1988 Physics of Fluids 31,2814-2820. The possibility of drag reduction by outer layer manipulators in turbulent boundary layers. 13. L. CREMER, M. HECKL and E. E. UNGAR 1988 Structure-borne Sound (2nd edition). New York: Springer-Verlag. 14. M. S. HOWE 1975 Journal of Fluid Mechanics 71, 625-673. Contributions to the theory of aerodynamic sound, with application to excess jet noise and the theory of the flute. 15. M. S. HOWE 1988 Proceedings of the Royal Society, London A420, 503-523. Contributions to the theory of sound production by vortex-airfoil interaction, with application to vortices with finite axial velocity defect. 16. M. ABRAMOWITZ and I. A. STEGUN 1972 Handbook of Mathematical Functions (10th corrected edition). U.S. Department of Commerce, National Bureau of Standards, Applied Mathematics Series No. 55. 17. L. D. LANDAU and E. M. LIFSHITZ 1959 Fluid Mechanics. Oxford: Pergamon. 18. J. 0. HINZE 1975 Turbulence. New York: McGraw-Hill.
APPENDIX:
THE GREEN
FUNCTION
When the airfoil is absent the Green function Go, say, is given by equation (2.6) with Y = y. Let G = G, + G,, where G, accounts for the presence of the airfoil, and set G,(x,y;
t-T)=-
GQ(x, y; w) exp {-iw( t - T)} dw,
n = 0, s.
(Al)
According to the reciprocal theorem G’(x, y; w) = G’(y, x; w) [ 171, and the problem of determining the field G’(x, y; w) produced by a point source at y can be replaced by that of determining the field G’(y, x; w) as a function of y produced by a point source at x.
244
M.
S.
HOWE
In particular, one may consider situations in which 1x1+ co, and y is in the ncxglrborhood ofthe airfoil and plate, thereby concentrating attention on the component of G A-cs;ronsible for the radiation from turbulence sources in the boundary layer near the airl;>ii Now, ,X G&(x, y; w) =
--x
(w
{iAk)h
-.hll
+R(k w) exp {iy(k)(x3+y3)l)
em {ik - (X-Y)} r(k)
d2k
’
(‘42)
and the integral is dominated by the contributions from the acoustic domain k < Ikol and from the pole at k = K, when 1x(-, co. By hypothesis koa, Z&a <<1, so that when /xl+ co and y is in the boundary layer close to the airfoil, the integrand may be expanded in powers of k * y, giving to leading order GA= G&,+G&,
(A3)
where cr
Gdx,
Y;
WI=
(exp {iY(k)lx3-y3(}+R(k, --cc
G&(x,Y; 0) = -
I5
k * y(ev
w) exp {iy(k)(x,+y,)})
expiti)
X’d2k,
{iy(k)lx3-y311
--oc
expW - xl y(k)
+R(k ~1 exp{iY(k)(x3+.t41)
d2k
’
By construction, Gb satisfies the Fourier time transform of the boundary condition (2.4), with B replaced by G. In terms of the reciprocal variable y this becomes D(a’/ay:
+a’/ayj)‘aG’/ay3
= mw’aG’/ay,
- p,,dG’,
y, = 0.
(A9
The same condition must be satisfied separately by G: . One can now argue that, provided the clearance t, is smaller than all other relevant length scales, G:,(x,
Y; w) = -
+
X k2cP*(exp{iy(k)lx3-y,l} I --oi
R(k, w) ev {iy(k)(x3+y3)l)
exp {ik - x}
d2k
y(k)
may be taken as the leading order approximation to G,. In expression (A6) cp*, defined in equation (2.7), is a solution of Laplace’s equation that satisfies acp*/ayz = 1 on the rigid airfoil, which implies that a{ G;, + G:,}/ay, = 0 on the airfoil. When terms - 0{( k,a)‘} relative to unity are discarded, G:, is a solution of the time-harmonic acoustic wave equation that is significantly different from zero only for points y in the vicinity of the airfoil, where its characteristic length scale of variationO(a). When the derivatives of G: in expression (A5) are scaled on a, the inertia terms on the right side are respectively of orders (K,,u)~< (K,cI)~ (CC1) and (p,a/m)(K,a)4<(poa/m)(K,a)4 (cc 1) relative to the stiffness term on the left. Since SC*+ 0 for J(yf + yz) >>a, it follows that it is sufficient to require that acp*/dy3 = 0 on the plate y, = 0.
SOUND
Hence,
correct
to an error-
OF
AIRFOIL
O{(k,a)*},
TIP
IN
BOUNDARY
LAYER
245
one can take
G’= Gho+ G;, + G:,,
(A7)
near the airfoil (i.e., at positions closer than a characteristic wavelength from the airfoil). Equation (2.6) of the main text is a uniformly applicable approximation to G, such that G’ reduces to expression (A7) at source positions y close to the airfoil, and coincides with Gb (no airfoil) when y is far from the airfoil. The following remark may be made concerning the adopted functional form (2.7) of cp*. The Green function as derived above with the use of expression (2.7) is discontinuous across y, = 0 for ly,l< u, 0 < y, < t,, between the tip of the airfoil and the wail. This deficiency could in principle be rectified by use of an alternative (but naturally very much more complicated) potential function which satisfies d~*/ay, = 1 on the airfoil alone. It is unnecessary to do this, however, for the following reason. Within the confines of the linear perturbation theory of this paper, the convection of vorticity parallel to the wall within the space between the tip and the wall does not produce a significant fluctuation in airfoil lift except possibly for vorticity near the leading and trailing edges. To be sure, conditions there are homogeneous in the mean flow direction, and there is no mechanism for “scattering” high wavenumber turbulence energy into low wavenumber sound and mean lift fluctuations. All such effects occur predominantly at the leading and trailing edges of the airfoil (although the unsteady Kutta condition implies that the relative contribution of the trailing edge is small) where the representation (A7) is appropriate except right at the tip corners, where, however, the loading must tend rapidly to zero.
SOUND
AND
VIBRATION
IN BOUNDARY
LAYER
229-245
PRODUCED FLOW
OVER
BY AN AIRFOIL AN ELASTIC
TIP
PLATE
M. S. HOWE BBN Laboratories,
(Received
10 Moulton Street, Cambridge, Massachusetts
02138, U.S.A.
10 July 1989, and in revised form 5 February 1990)
A theoretical model is analyzed to estimate the structural and acoustic noise produced when boundary layer turbulence impinges on the tip region of an airfoil, such as a rotor blade in a turbomachine. The airfoil has rectangular planform and its tip is immersed in the turbulent boundary layer on a wall modeled by a thin elastic plate. Numerical results are presented for a rigid airfoil adjacent to a steel plate in water. These indicate that the tip behaves as an acoustically bright source of sound, the intensity of which typically exceeds by 30 dB or more that which would be produced by the turbulence in the region of the tip when the airfoil is removed. Similarly, flexural motions induced in the wall (“structure-borne sound”) are shown to be substantially increased by the presence of the airfoil. This is important because structural waves may be scattered by surface discontinuities at remote points of the wall, resulting in an overall increase in the radiated sound.
1. INTRODUCTION The wall pressure fluctuations in turbulent boundary layer flow over a plane surface are characterized by the wall-pressure wavenumber frequency spectrum P(k, w), where the wavenumber vector k is conjugate to spatial co-ordinates in the plane of the wall, and w denotes the radian frequency. The spectrum is formally equal to the space-time Fourier transform of the wall-pressure correlation function, and is well defined for practical purposes provided that the largest length scales of interest are small compared to the length scale of variation of the mean properties of the boundary layer. For low Mach number flow over a homogeneous, flexible wall, the intensity of the sound generated by the turbulence is governed by the behavior of P(k, w) in the acoustic domain k = Ikl< w/c [l-4], where c is the speed of sound. In addition, flexural motions of the wall tend to be excited most favorably by pressure fluctuations in the adjacent subconvective region w/c
17 %03.00/O
,@) 1991
Academic
Press
Limited
230
M. S. HOWE
scattering of flexural wall motions that are themselves the product of diffraction of the convective domain by wall irregularities. Structural members that project from the wall into the boundary layer, or other control surfaces that are wetted by the turbulent flow, are also important sources of sound and vibration. Fins and stabilizers on marine vessels and buoyant test bodies, and struts, turning vanes and other flow control devices in wind tunnels, are frequently responsible for contamination of measurements of the low wavenumber and acoustic regions of the wall-pressure spectrum. Dowling [S] and Howe [9] have investigated a control surface in the form of a so-called large-eddy breakup device (“LEBU”), consisting of a thin, ribbon-like airfoil deployed transverse to the mean flow in the outer region of the boundary layer. Its purpose is to modify the turbulence structure in a manner that leads to a reduction in the wall skin friction. Decreases in skin friction of 20-40% on extensive sections of the wall have been reported in the literature (see, e.g., references [lo-121). According to reference [9], however, large pressure fluctuations established in the neighborhood of the LEBU in underwater applications are responsible for substantial increases in both the acoustic radiation and the structural noise generated by the flow. In this paper an investigation is made of the sound and vibration produced by a control surface in the form of a large airfoil, the tip of which is immersed in turbulent flow over a flexible wall. This may be regarded as an idealized model of a blade of a rotor in a turbomachine duct, although no account will be taken of mean blade motion, of the mutual interactions of neighboring blades, nor of the finite curvature of the duct wall. The objective is to estimate the increases in sound and vibration relative to the background radiations generated by the flow in the absence of the airfoil. The wall is modeled by a thin elastic plate, and detailed results are given for underwater applications, where the fluid-loading is high enough that the additional wall pressures and airfoil loadings generated by the turbulence-airfoil interaction would be expected to modify significantly the structural and acoustic noise. The analytical model is formulated and solved formally in section 2. In section 3 expressions are obtained for the acoustic pressure frequency spectrum of the sound generated by the airfoil, and for the spectrum of the power delivered to flexural motions of the wall. Numerical predictions are made (sections 4, 5) by use of data available in the literature to estimate the properties of the boundary layer turbulence.
2. THE ANALYTICAL 2.1.
THE
GOVERNING
MODEL AND ITS FORMAL
SOLUTION
EQUATIONS
Consider turbulent flow at low Mach number in the x,-direction of a rectangular co-ordinate system (x, , x2, x3). The fluid has mean density pO, sound speed c, main stream velocity U,, and occupies x3 > 5(x,, x2, t), where I denotes the flexural displacement of a thin plate, which in the undisturbed state coincides with the plane xj = 0. The region below the plate (x3< 5) is taken to be evacuated, and the flexural motions are small enough that 5 satisfies the linear, bending wave equation [13]
{D(a2/ax~+a2/ax~)‘+m a?/at”}l=
-p(x,,
x2,
0, f),
(2.1)
where p(x, t) is the pressure, and D, m are respectively the bending stiffness and specific mass density of the plate. A thin airfoil of rectangular planform and large aspect ratio is placed at zero angle of attack to the local mean flow (which is formally identified with the main stream velocity V,) with its spanwise dimension parallel to the x,-axis (normal to the undisturbed plane of the plate), as illustrated in Figure 1. The airfoil is modeled as a flat strip that occupies the region lx,1 < a, x2 = 0, tc < x3 < +OO; the chord 2a is of the
SOUND
Figure
1. Rectangular
airfoil
OF AIRFOIL
adjacent
TIP
IN BOlJNDARY
231
LAYER
to a flexible plate in low Mach number
turbulent
flow
same order or larger than the thickness 6 of the boundary layer, and the clearance t, between the tip of the airfoil and the plate is small compared to all other dimensions, but large enough for there to be no contact with the vibrating plate. When the properties of the boundary layer turbulence are known, or easily estimated, it is convenient to express the acoustic radiation and the flexural plate motions in terms of perturbations in the total enthalpy B. In homogeneous fluid in adiabatic motion, this is defined in terms of the pressure p and velocity v by B=
dp/p +;t?,
(2.2)
where p = p(p) is the local density. At high Reynolds number, and sufficiently low Mach number M = U,/c that refraction and scattering of sound by the flow can be neglected, B(x, t) may be regarded as the solution of the inhomogeneous wave equation [14], {d*/c*dt*-V”}B=div
(OAV),
0 = curl v.
(2.3)
in irrotational flow, where C$is a velocity potential satisfying v = VC$. At the wall v = (0,0, al/at), so that to leading order, the pressure may be replaced by poB(x,, x2, 0, t) on the right of the bending wave equation (2.1). Similarly, the high Reynolds number form of the momentum equation implies that a*l/at’ = -as/ax, at the wall. Thus equation (2.1) becomes B = -Q/dt
{D(a’/axf+a*/ax:)‘+
m a’lat’}
aB/ax,
= p,, a’B/dt’,
xj = 0.
(2.4)
232
M. S. HOWE
Equation (2.3) is to be solved subject to the boundary condition (2.4) and the radiation condition that B(x, 1) exhibits outgoing wave behavior at large distances (with the acoustic pressure p(x, t) - poB(x, t)). In obtaining the solution it will be assumed that the characteristic wavelengths of both the sound and bending waves on the plate are large relative to the chord of the airfoil. This condition permits a solution to be derived in analytic form, and is fulfilled over a wide range of frequencies in many underwater applications. 2.2.
APPROXIMATE
SOLUTION
OF THE
EQUATIONS
The analytical problem of determining B(x, t) in terms of the boundary layer vorticity and velocity distributions is analogous to that considered in reference [9] for the case of a LEBU in turbulent flow. The radiation produced by the turbulence-airfoil interaction may be attributed to a dipole source, the strength of which is equal to the unsteady force between the airfoil and fluid. This interaction can be described in terms of a Green function G(x, y; r - T), which is the solution of equation (2.3) when the term on the right is replaced by 6(x-y) ??(t - T). Application of the divergence theorem (in the manner described in reference [9]) permits the formal solution to be set in the form B(x,t)=-
(o~v)(y,~).dG(x,y;t-7)/ayd~ydT.
(2.5)
I
G(x, y; t-7) can be evaluated analytically in the low-frequency approximation in which the wavelengths of both bending and sound waves are large relative to the chord of the airfoil. The procedure is similar to that described in reference [9], and is outlined in the Appendix, where it is shown that x
G(x,y;
t-T)=-
XW3 .
I
-0c‘
L r(k)
(exp {iAk)b3-y311+ Nk, w) exp{iy(k)(x3+y3)l)
xexp{i[k.(x-Y)-w(t-~)]}d’kdw,
1xI+a,
(2.6)
where Y=
(Y,, y,+
The remaining r(k) =
V*(Y), O),
new quantities
sgn (w)(ki-
q*=Re{-iJ(z*-a*)}--yz,
z=y,+iy,.
on the right of equation
k’)“‘,
(2.6) are defined
k, = w/c
+i(k*-kg)“‘,
(acoustic
(2.7) as follows:
wavenumber), (2.8a-c)
R(k,w)={Dk4-mw2+ip,w2/y(k)}/{Dk4-mw”-ip0w2/y(k)}.
R(k, w) is the reflection coefficient for plane (propagating or evanescent) waves incident on the plate from xj > 0. In performing the integration with respect to k in equation (2.6), the singularity (pole) at the positive, real zero (k = K,, say, where K, > Ik,,j) of the denominator of expression (2.8~) is avoided by assigning to w a small positive imaginary component which is subsequently allowed to vanish. The pole characterizes flexural waves induced in the plate by a point source in the presence of the airfoil, and the formal limiting procedure ensures that these waves satisfy the radiation condition. 2.3.
LINEARIZATION
OF THE
AERODYNAMIC
SOURCE
TERM
To simplify expression (2.5) the vortex source OAV is linearized with respect turbulence velocity fluctuation. If U(x,) denotes the mean velocity distribution boundary layer, the leading order terms in expression (2.5)‘yield B(x, t) = -
u,(y,
7) &y(Y) 1
;+
WJ(Y,
7)
~b’,)
$
7>
G(x,
y;
f -
7)
d3y
d7,
to the in the
(2.9)
SOUND
OF
AIRFOIL
TIP
IN
BOUNDARY
LAYER
233
where r.+, w3 respectively denote the 3-component of the turbulence velocity u and the vorticity. In underwater applications one is frequently concerned with frequencies satisfying wS/ U, >>1, where 6 is the boundary layer thickness. In the region of intense boundary - 2*5u,/y, (u.+ being the friction velocity), whereas layer velocity fluctuations aU(y,)/ay, w3- u/y, and U - 20 u.+. The second term in the integrand of equation (2.9) is therefore typically an order of magnitude greater than the first, and in the leading approximation one can write B(x, t) = -
w,(y, T) U(y3)
dG (x, y; t ay,
T) d3y dr.
Two distinct sources of vorticity must be included in this integral. The first, .zL,say, is contained in the turbulent flow impinging on the airfoil from upstream; the second is vorticity shed into the wake of the airfoil. The function 4p* in equations (2.6) and (2.7) for G governs the strength of the radiation produced by the interaction of the turbulence with the airfoil. This is henceforth referred to as the di$raction radiation. The contribution of acp*/ay, to the integral (210) is large when y is near the leading and trailing edges of the airfoil, so that most of the diffraction radiation is produced by vorticity close to the edges. Vorticity shed from the airfoil tends to oppose the production of sound at the trailing edge by the impinging vorticity (cf., [15]), and reduces the importance of the trailing edge as a source of sound and vibration. Consider the contribution to expression (2.10) from the impinging boundary layer vorticity R,, represented in the form a: ~3(K,w’;y,)exp{i(K.y-~‘r)}dZKdw’,K=(K,,K2,0). (2.11) MY, T) = I -II Using this and expressions (2.6), (2.7) in equation (2.10), and retaining only the contribution to the radiation from the interaction with the airfoil, one finds
x exp {i(k . x+K .y - wt)} d2k dw d2K d3y,
=-I a
47r
k2K,J,(K,a) yW)tKi2
Wy,)fi(i,(K,
(2.12a)
w; ~3)
x (exp {iy(k)lx3-y311+ R(k, w) exp {iy(k)(x3+.v3)l) x exp {i(k . x - tot)} d’k dw d’K dy,,
(2.12b)
where J, is a Bessel function. The suffix i designates the radiation produced by the impinging turbulence, and in passing from expression (2.12a) to expression (2.12b) the integrations with respect to y,, y2 (over the interval -03, ~0) have been performed by reference to results given in reference [ 161. When OS/ U, is large (i.e., for small-scale turbulence), the vorticity 0, may be regarded as effectively frozen during convect@ past the leading and trailing edges of the airfoil. In that case the Fourier transform R,(K, w; y3) is sharply peaked at K, = w/ U(y3), and the principal contribution to the integration with respect to w in expression (2.12b) is from a small interval containing o = U(y,)K, . For wa/ U, * OS/ CJ, >>1, it follows by
234
M. s. HOWE
use of the large argument asymptotic expansion of the Bessel function J, [16], that the integrand with respect to K, will then consist of two terms of the form (. 1 ev
{-iK,(a
+ U(y3)f)l,
(. ) exp {iK,(a
-
(2.13)
U(y3)t)l.
The phase factors characterize the times at which the dominant components of the diffraction radiation are produced, and (when wa/ U(y3) >>1) respectively correspond to radiation from the leading and trailing edges of the airfoil. In practice, there is destructive interference between the trailing edge component of expression (2.13) and the sound produced by the wake vorticity [15]. Indeed, for a two-dimensional airfoil in a uniform mean stream ( U = constant), the net radiation from the trailing edge is predicted to vanish identically when the wake vorticity convects at the same velocity U as the impinging vorticity. This is because the unsteady pressures exerted on the trailing edge region of the airfoil by the wake and by the impinging vorticity are equal and opposite, and the associated dipole source is therefore null. Evidently, the situation is more complicated when the convection velocity is sheared, as in boundary layer flow. However, when w6/ U, >>1, so that turbulent eddies are small relative to the length scale of variation of U(y3), interference between the trailing edge forces generated by the impinging vorticity and wake must still occur in an analogous fashion, and the net radiation from the trailing edge must also be very small, provided the local convection velocities are equal. It is assumed here that is the case, and so all quantities attributable to radiation from the trailing edge are formally discarded. This is done by replacing the Bessel function in equation (2.12b) by the leading term in its asymptotic expansion, and deleting the component with the second phase factor of expression (2.13). By this means one arrives at the following expression for the net radiation B(x, I) produced by the interaction of the airfoil with the boundary layer turbulence:
x (exp {idk)lx3-y311+R(k, w) exp {iy(k)(x3+y3)l) (2.14)
x exp {i( k . x - K,a - wt)} d’k dw d’K dy,.
3.
FREQUENCY
SPECTRA
OF
THE
SOUND
AND
THE
FLEXURAL
WAVE
POWER
3.1. THE DIFFRACTION RADIATION POWER The frequency spectra of the radiation will be determined by calculating the net acoustic and flexural wave powers Z7, say, produced by the interaction of the airfoil with the turbulence. This is given by [17] 17 = PO ~’((W)+ - (W-) I -1;
dx, d+,
(3.1)
where the angle brackets denote an ensemble average with respect to the boundary layer turbulence, and the suffixes +, - respectively denote evaluation at x1 = 6, x3 = +O; i.e., just “above” and “below” the boundary layer. The velocity u3 is expressed in terms of B by use of the relation dv3/a t = -dB/ax, . Since the radiation is governed by the low wavenumber components of the Fourier integral representations of B and Us, equation (2.14) may be used to evaluate expression (3.1). The velocity fluctuations in the boundary layer may be regarded as stationary random functions of time and position (x, , x2) parallel to the plate, and there accordingly exists
SOUND
OF AIRFOIL
a wavenumber-frequency spectrum component of vorticity, such that (&K,
w; y&&K’,
TIP
IN
&(K,
BOUNDARY
w; .Y~,y5)
w’; y;)) = %(K,
235
LAYER
= fi33(K,
a; y,, yj)W
o;
yj,
~3)
-K’)S(w
of
w’),
-
the
3-
(3.2)
where the asterisk represents the complex conjugate. Using this and expression (2.14) in equation (3.1) gives
x Wy3)
~(YW,,W,
w; ~3, Y;)
d’K
(3.3)
d*k dw dy3 d.4,
where C.C.denotes the complex conjugate of the preceding quantity. The term in large brackets on the right of equation (3.3) is non-zero only in the acoustic domain k < lb/ and at the flexural wavenumber k = K, (i.e., at the real pole of the reflection coefficient R( k, w)), where it has a S-function singularity. By considering the limiting behavior as Im (w) + +O, it is easily deduced that 4y(k)(Dk4-
l+ R(k, w) or(k)
+ C.C.=
mw’)’
w[(Dk4-mw*)*(k’-k;)+(p,,w’)‘]’
k < ikol
47volw I 6(Dk4(k*- k;)
k> Ikol ’
mw* -pow2/v’tk2-
k;)),
I
These terms correspond respectively to the powers LrA, LrF (n = l?,, + n,) the acoustic and bending wave modes, such that k;y(k)(Dk4[(Dk4l7,=
poaU&64
lwtk:Z’(w)
2
Ck*
where Z’(w) is dimensionless
3.2.
THE
The
ACOUSTIC
FREQUENCY
-
6.Dk4_m
radiating in
mw’)‘Z’(w)
mw*)‘(k*-
k;)+(p,w’)2]
(3.4)
d2k dw’
w’-p,w’/J(k*-k;))d’kdw,
(3Sa) (3Sb)
6)
and defined by
SPECTRUM
Q(W)
acoustic pressure frequency spectrum Q(w) at a far field point x satisfies (P’(X r)) =
uc 0(w) dw. I -X
(3.7)
@ is obtained from nA by introducing the transformation k=
k,(sin 0 cos C#J, sin 0 sin 4),
d*k = ki sin 19cos 6 dCJd&
k < Ikol,
(3.8)
in expression (3.5a) (so that y(k) = k, cos e), and by noting that the power dLrA, say, per unit frequency that radiates through the distant surface element 1x(’ sin 19dc? db specified by the spherical polar co-ordinates (1x1,8,~$) (see Figure I), is given by dnA = {@(w)/p,c}(xl'
sin 8 de d+.
(3.9)
It follows that CD(w) =
piaU&64kiZ’(
w )[ D( k, sin 0)” - mw ‘1’ co? + co? 8 27rlx]*{[D( k, sin 8)4 - mw’]‘( k,, cos e)‘+ (pow’)‘} ’
(3.10)
236
M. s. HOWE
where cos Cc,= sin 8 sin 4, and Cc,is the angle between the radiation direction x/lx1 and the x2-axis (normal to the airfoil). This result is rendered in dimensionless form by introducing the definitions K,,= (mw2/D)“4
(the in ZIUCUO bending wavenumber), E* = polkol/mK~= poc,d(l -2u)/d3p,c(l
CL= k@o,
-a),
(3.11)
where ps, c, respectively denote the density and (dilatational) sound speed of the material of the plate and 1.7is the Poisson ratio (see, e.g., reference [13]). One then has Z’(w)( k,6)*( mk,, cos B/p,,)‘[ 1 - cc4sin4 t9]’co? $
@to) p;UWVIxI)*
=
27r{1 + (mk, cos O/p,)*[ 1 - p4 sin4 131’)
’
(3.12)
This represents the field of an acoustic dipole whose axis is aligned with the x,-direction (normal to the plane of the airfoil); when mk, cos 13/p, >>1 the plate behaves as a rigid reflector of the dipole generated sound. 3.3. FLEXURAL WAVE This is defined by
FREQUENCY
SPECTRUM
p(W)
oc I&=
V(W) dw. I --oo
(3.13)
Performing the integration with respect to k in expression (3Sb), one finds
V(,) pJJ$ClCP =
~M(lICOIS)(SPOIm)Z’(w)A2 2p2[5A4-4(Q)*-11
’
(3.14)
where M = U,/ c, and A = I&./K,,, K, being the flexural wavenumber at frequency w, which is equal to the positive zero of the argument of the S-function in equation (3Sb), so that A satisfies: A4- 1= (E.J&J(A~-~~),
A > /_L.
(3.15)
4. MODELING THE TURBULENT BOUNDARY LAYER 4.1.
THE
VORTICITY
SPECTRUM
The most important contributions to the integral (3.6) defining Z’(o) are from the near wall region, say y,, yS.< 0.1 6, where the predominant turbulent fluctuations occur. To evaluate the integral it is necessary to specify the vorticity spectrum R,,(K, w; y,, yj). There is no analytic representation of this function based on deductive theory [ 1,2]. When 081 U, is large, however, the y,-correlation scale would be expected to be of order Q/W, where v* is the friction velocity, and for the purpose of computing the integral the following model can be introduced:
&AK, w; YS,Y;) = G,(K
w; y,)l[1 +(~/~*)*(Y~-YY;)~I”.
(4.1)
The exponent n is arbitrary, but is assumed to be large enough that the range of integration in equation (3.6) with respect to yi can be extended to (-co, OO),and U(yj) can be replaced by U(y3). fln[,,(K, w; yJ is the 33-component of the vorticity spectrum at distance y, from the wall. The integration with respect to y; in equation (3.6) now reduces to OCI (4.2) &(K, w; ~3, Y;) dyS = (B&&JI~*).~S~(K w; ~3). I -cc
SOUND
OF AIRFOIL TIP IN BOUNDARY
237
LAYER
The value of the coefficient /3,, is tabulated for integer values of n in Table 1. If n = 3 is taken as the minimum value that gives a sufficiently rapid decay in coherence in the wall-normal direction, it is seen that possible alternative choices of n are likely to modify predictions of the radiation spectra by perhaps 2 or 3 dB. This must be borne in mind in interpreting the final results. TABLE
1
Values of Pn n
P,
1
2
3
4
5
6
I
1.77
0.89
0.66
0.55
0.48
044
0.40
To model aj,(K, w; y3) it will be assumed that small-scale features of the turbulence are essentially frozen during their interaction with the airfoil, and that they may be adequately approximated (for OS/ U,> 1) in terms of the von KStrmitn spectrum for isotropic turbulence [18, Chapter 31. Discrepancies that might arise from the use of this approximation as opposed to alternative possible representations of fi:, are likely to be small, say, no larger than the variations of Table 1, since detailed differences between the various models should be averaged out by the integration in equation (3.6). Thus one can take
where q’ = ( Iu(y3)12) is the mean square turbulence velocity and T(x) is the Gamma function. The parameter b = b(y,) determines the correlation scale of the small-scale turbulence fluctuations, and the b-function arises because the turbulence is convecting at the local mean stream velocity U(y3). By using the results (4.1)-(4.3) in equation (3.6), it is found that (4.4)
Z’(w) = &(V*I Uc)3(w~I &)2z(w), where Z(w) =
55r(5/6) 36UW) I
F(x) =
4.2.
NUMERICAL
I
EVALUATION
m {q(y3)l~*}2b5(y3)F[wSb(y,)l { U(Y3)l QJ’
Io
t3 dt
OF
Wy3)l
dy3
(4.5a) (4.5b)
INTEGRALS
value of the integral (4.5a) would not be expected to depend significantly on the boundary layer Reynolds number R8 = U,S/ Y (where u denotes the kinematic viscosity). To estimate the variation of the integrand with y3 one can therefore use standard experimental data given by Hinze [18] at R, = 7.5 x 104. The parameter b in the von KBrman spectrum is related to the turbulence energy dissipation E and to the turbulence intensity q2 by [18, Chapter 31 The
b =O~152(q2/uz,)“‘/(~6/u_:).
(4.6)
The following approximations for q2/ui and b that are adequate for the evaluation of expression (4.5a) and are piecewise linear in z = y3/ 6, are readily derived from data in
M. S. HOWE
238
section 7.7 of reference [18]: b=
q2/v;=
0<2<0.2 0.2
l.Sz, 0.3 ( 2*1-2z,
12002, 12.56-55*56z, ( 7.78( 1 -z),
(4.7)
O
.
(4.8)
The integrand in equation (4.5a) is assumed to vanish for yJ> S(z > 1). The mean velocity U = U(yJ can be approximated by the logarithmic formula
A In {(v*/ KJ&Y~/SI+ B U(Y3)l UC.3 = Aln{(v,/Um)&l+B ’
(4.9)
where A = 2.5 and B = 7.0. The failure of this approximation in the region very close to the wall is of no practical significance because the remaining terms in the integrand of expression (4.5a) tend rapidly to zero as y3 + 0. An asymptotic representation of Z(w) for WC?/U, >>1 can be obtained by noting that F(x) - 1/4x”/’ when x is large. Numerical evaluation of expression (4.5a) then yields: (4.10)
Z(w) - 1*07/(&j/ uX)‘7’3.
The error involved in using this formula is less than 1 dB when wS/ U, exceeds 12.
5. NUMERICAL RESULTS In this section illustrative numerical predictions are presented of the diffraction radiation spectra when the airfoil is adjacent to a steel plate in water. The following values of physical quantities are used in the calculations: for steel, pc = 7700 kg/m3, c, = 6100 m/s, (+= 0.28; for water p,, = 1026 kg/m3, c = 1500 m/s
(5.1)
Hence E* = O-144 (see equation (3.11)). The boundary layer is assumed to be specified by v*. u, = 0.037,
M = l&j/c = 0.005,
where the value of v* is that appropriate (4.2).
Pn = p3 = 0.66,
(5.2)
for Rs = 7.5 x 104, and P,, is defined in equation
5.1. THE ACOUSTIC RADIATION The acoustic pressure frequency spectrum has been evaluated for the two values S/A = 1,5, where A denotes the thickness of the plate. The solid curves in Figure 2 depict the variation of 10 log,, {@(w)/p~Uh(S/I~l)~} (dB) with wS/ U,, where e(o) has the explicit representation
G(w) &JzJz(S/]x])2
p,,M2(v+/
=
8/pJ2[1 --(CLsin r9)‘]‘cos2 rL , ~GT{ 1 + (mk, cos f3/p0)‘[ 1 - (E.Lsin e)“]‘] (5.3)
IY,)~(wS/
U,)‘Z(w)(mk,cos
and the angles specifying the radiation direction have the nominal values 0 = + = 45”. The reduced intensity of radiation at the larger value 6/A = 5 occurs because mko cos B/p0
SOUND
OF
AIRFOIL
TIP
IN
BOUNDARY
LAYER
239
Figure 2. Frequency spectrum 10 log,, {@(w)/pbU&a(8/lsI)‘) (dB) of the sound produced by the interaction of the turbulence and the airfoil in the case of a steel plate in water and parameter values defined in expression (5.2) and for 0 = JI = 45”. The dashed curve is the estimated excess 10 log,, {@(w)/@&w)} (dB) of the radiation from the airfoil over that produced by turbulence contained in area 2a6 of the wall.
is smaller at fixed wS/ Um: except at very high frequencies, the plate is then acoustically “softer”. To illustrate the importance of the acoustic radiation from the airfoil relative to the background noise generated by the boundary layer turbulence, one can estimate the spectrum of the sound that would be produced by turbulence in the “tip region” if the airfoil were absent. The tip region is defined to be the boundary layer turbulence over a rectangular section of the wall of area 2~6 (the product of the airfoil chord and the boundary layer thickness). The prediction is derived from an empirical formula given by Sevik [3] for the low wavenumber and acoustic regions of the turbulence boundary layer wall pressure spectrum P(k, w). Sevik’s formula is strictly applicable when wS/ CJ, > 24, and yields [4] the following prediction for the acoustic pressure frequency spectrum boundary layer @LL(u), say, of sound radiated from a region of area A of a turbulent formed on a rigid wall: (5.4) Chase [2] has proposed an alternative formula for the turbulent boundary layer acoustic pressure frequency spectrum that is applicable for moderate values of wS/ U,. In the interval 24 < OS/ U, < 90, however, Chase’s predictions are typically lo-20 dB smaller than those given by expression (5.4), and decrease further at lower frequencies. According to Chase, estimates made with expression (5.4) should be interpreted with great care, since it has been deduced from experimental data from tests with buoyant bodies that may well have been contaminated by extraneous sources of sound, and from wind tunnel measurements of acoustic wall pressure fluctuations that probably include contributions from the adjacent low wavenumber, hydrodynamic domain of turbulent wall pressures. For the purpose of the present discussion, however, it is convenient to regard expression (5.4) as furnishing an upper bound on the level of the background noise against which the diffraction radiation must compete.
240
M. S. HOWE
The influence of wall compliance on the turbulence into expression (5.4) by replacing @LL by [9],
QBL(w)
=
generated sound is incorporated
%(~Nmkocos elpo)*[l-(CL sine)“l’ {l+(mk,cos
0/p0)2[1-(~
sin e)“]‘} ’
(5.5)
The dashed curve in Figure 2 represents the excess 10 log,,,{@(w)/@,,(w)} (dB) of the radiation from the airfoil over that produced by turbulence in the tip region estimated by expression (5.5). For 06/ U, < 24, Sevik’s spectrum (5.4) has been arbitrarily assumed to be flat. The excess radiation is independent of S/A, the direct radiation from the turbulence being typically 40 dB below the diffraction radiation. Note that the predicted acoustic spectrum of the diffraction radiation will be lifted or depressed by a fixed amount over all frequencies if a different value of /II” is taken in expression (5.2). For the range of exponents n in Table 1, however, the change will not exceed 3 dB, and this suggests that the predictions in Figure 2 are probably correct to within about 15 dB. Changes of this order do not materially affect the import of the above conclusions. 5.2.
FLEXURAL
VIBRATIONS
The solid curves in Figure 3 depict the variation of the spectrum 10 log,,{ Y (w)/PoU&a8*} (dB) of the flexural wave power generated by the airfoil for conditions (5.2) and S/A = 1,5, where (from expressions (3.14) and (4.4)),
(5.6) These results will be compared with the power spectrum ?PBL(w) of flexural waves produced by turbulence in the “tip region” (of area 2a8) of the wall in the absence of the airfoil. For a smooth wall, these waves are generated by the component of wall pressure at wavenumber k = K,(w), which lies in the subconvective domain of the wall pressure spectrum P(k, w). A representation of the strength of the bending waves is easily
Figure 3. Spectrum 10 of the turbulence and the (5.2). The dashed curve airfoil over that produced
log,,, { ly(u~)/p,U&aS’} (dB) of the flexural wave power produced by the interaction airfoil in the case of a steel plate in water and parameter values defined in expressions is the estimated excess IOlog,,{Y(~)/P-gL(w)} (dB) of the power radiated by the by turbulence pressures applied over area 206 of the wall.
SOUND
OF
AIRFOIL
TIP
IN
BOUNDARY
LAYER
241
obtained in terms of an empirical model of P(k, w) developed by Chase [2] for a rigid wall (the “blocked” pressure spectrum), which is based on an analysis of data from wind tunnel and buoyant body tests. By using this formula, it is shown in reference [9] (see section 5.1) that
The dashed curves in Figure 3 represent the excess 10 log,,,{!P(,)/PB,(,)} (dB) of the flexural wave power generated by the airfoil over the direct radiation from turbulence in the tip region. The efficiencies with which flexural motions are excited by both the airfoil and the boundary layer increase with 6/A, but the increase is smaller for the airfoil-generated waves. The diffraction radiation is typically 30 dB greater than the direct radiation from the tip region turbulence. In other words, the flexural wave power generated by the airfoil is equivalent to that produced by the turbulence wall pressure acting over an area -2a6 x lo3 of the wall. 5.3. APPROXIMATE FORMULAE When OS/ Uoc> 10 the asymptotic representation (4.10) of Z(w) can be used in expressions (5.3) and (5.6). Other terms in these formulae can also be simplified provided
-130.
(
I ( , ,,,,,
,
, , (, (,(,
,
, , , ,,,p (0)
Figure 4. Comparison of the full and approximate and S/A = 5. (a) Acoustic spectrum at 0 = Y =45”: -, power spectrum: -, equation (5.6); - - -, equation
formulae for the radiation spectra for conditions (5.2) equation (5.3); - - -, equation (5.9). (b) Flexural wave (5.10).
242
M.
S.
HOWE
that the frequency does not exceed the critical frequency w, at which the bending wave phase velocity in vacua is equal to the sound speed in the fluid, where w,. = c’d(m/D)
= cp,,/m&,.
(5.8)
It follows from equation (3.11) that p =J(o/w,.). Thus, when w < w,. the term (p sin 0)’ can be discarded where it occurs in expression (5.3) (since sin 8 < 1 in all directions in which the radiation is finite), and one can take Q(w) p~U~a(~llxl)2c
0*17@,M2( v*/ U~)“(oS/ UJ5’3(
mko cos B/pO)’cos’ $ 7
{I+ (mb ~0s ~lkd*l 10 < d/
v, < w,6/ u,.
(5.9)
Similarly, the flexural wave power spectrum (5.6) can be taken in the simplified form U,)3M(wS/Um)-“‘3 T(w) _O~28P,(Sp0/m)‘(v,/ p0 ULaf3* EJl- ~*M(mlW(w~l &)I
10-cWCs/ u,,< WCS/ u,.
’ (5.10)
A comparison is made in Figures 4(a), (b) of these approximate formulae and the full representations (5.3) and (5.6) for conditions (5.2) and 6/A = 5. In this case w,S/ U, = 902. 6. CONCLUSIONS
The unsteady turbulent loads experienced by control surfaces are an important source of sound and vibration. In this paper an analysis has been presented of the production of structural and acoustic noise by an airfoil the tip of which is wetted by turbulence in boundary layer flow over a flexible wall. This is a first approximation to the problem of broadband noise generation by a rotor blade in a ducted flow or turbomachine. Strictly speaking, the analysis is applicable only for an airfoil with zero mean lift. For a lifting airfoil there will be an additional contribution to the noise from the interaction of the turbulence with the tip vortex. When wS/ I.& is large, however, the influence of this interaction on the airfoil loading would be confined to the trailing edge region, where its effect would tend to be countered by vortex shedding (in accordance with the unsteady Kutta condition). Thus, unless the airfoil becomes stalled, the influence of mean angle of attack and, in particular, of the tip vortex, will probably modify the present predictions by a few dB at the most. Numerical results for the case of heavy fluid loading, where the wall is modeled by a steel plate in water, reveal that the tip region of the airfoil generally constitutes an intense source of sound and flexural wall vibrations. The predicted acoustic radiation from the airfoil is typically 40-50 dB greater than that which would be generated by the turbulence in the tip region in the absence of the airfoil; similarly, the structure-borne sound produced by the airfoil is substantially greater (-30 dB, typically) than that generated by the flow in the tip region when the airfoil is removed. These estimates are based on the use of a particular empirical model of the boundary layer turbulence near the airfoil, but the possible error is estimated to be less than about *5 dB. The flexural wall modes generated by the airfoil can be an indirect source of additional acoustic radiation produced when structure-borne energy is scattered by wall discontinuities, at corners, ribs, structural supports, etc., at remote points of the wall. To eliminate secondary interactions of this kind it is necessary to attenuate the wall motions, for example, by cladding with a damping material, and predictions of the type discussed in this paper provide a starting point for the optimizing this procedure.
SOUND
OF AIRFOIL
TIP IN BOUNDARY
243
LAYER
ACKNOWLEDGMENT The research reported in this paper was supported by the Applied Hydrodynamics Research Program of the Office of Naval Research under Contract N00167-87-C-0021, administered by Mr James A. Fein.
REFERENCES 1. D. M. CHASE 1980 JournalofSound and Vibration 70,28-67. Modeling the wavevector-frequency spectrum of turbulent boundary layer wall pressure. 2. D. M. CHASE 1987 Sound and Vibration 112, 125-147. The character of the turbulent wall pressure spectrum at subconvective wavenumbers and a suggested comprehensive model. 3. M. M. SEVIK 1986 in Proceedings of the IUTAM Symposium on Aero- and Hydroacoustics, Lyon, 3-6 July, 1985. Berlin: Springer-Verlag. Topics in hydroacoustics. 4. M. S. HOWE 1988 Journal of the Acoustical Society of America (in press). Surface pressures and sound produced by low Mach number turbulent flow over smooth and rough walls. 5. D. G. CRIGHTON 1983 in Turbulence-induced Vibrations and Noise of Structures (editor M. M. Sevik). Long range acoustic scattering by surface inhomogeneities beneath a turbulent boundary layer. New York: ASME. 6. M. S. HOWE 1987 IMA Journal of Applied Mathematics 39, 99-120. Production of sound by turbulent flow over an embedded strut. 7. M. S. HOWE 1988 Journal of Sound and Vibration 125, 291-304. Diffraction of flow noise by a flexible seam. 8. A. P. DOWLING 1989 Journal of Fluid Mechanics 208,193-223. The effect of large-eddy breakup devices on flow noise. 9. M. S. HOWE 1989 Proceedings of the Royal Society London A424 461-486. Structural and acoustic noise generated by a large-eddy break-up device. 10. S. P. WILKINSON, J. B. ANDERS, B. S. LAZOS and D. M. BUSHNELL 1987 in Proceedings on Turbulent Drag Reduction by Passive Means (Royal Aeronautical Society) 1, l-32. Turbulent drag reduction research at NASA Langley-progress and plans. 11. A. BERTELRUD 1986 in Advances in Turbulence, 524-532. Manipulated turbulence structure in flight. New York: Springer-Verlag. 12. A. SAHLIN, A. V. JOHANSSON and P. H. ALFREDSSON 1988 Physics of Fluids 31,2814-2820. The possibility of drag reduction by outer layer manipulators in turbulent boundary layers. 13. L. CREMER, M. HECKL and E. E. UNGAR 1988 Structure-borne Sound (2nd edition). New York: Springer-Verlag. 14. M. S. HOWE 1975 Journal of Fluid Mechanics 71, 625-673. Contributions to the theory of aerodynamic sound, with application to excess jet noise and the theory of the flute. 15. M. S. HOWE 1988 Proceedings of the Royal Society, London A420, 503-523. Contributions to the theory of sound production by vortex-airfoil interaction, with application to vortices with finite axial velocity defect. 16. M. ABRAMOWITZ and I. A. STEGUN 1972 Handbook of Mathematical Functions (10th corrected edition). U.S. Department of Commerce, National Bureau of Standards, Applied Mathematics Series No. 55. 17. L. D. LANDAU and E. M. LIFSHITZ 1959 Fluid Mechanics. Oxford: Pergamon. 18. J. 0. HINZE 1975 Turbulence. New York: McGraw-Hill.
APPENDIX:
THE GREEN
FUNCTION
When the airfoil is absent the Green function Go, say, is given by equation (2.6) with Y = y. Let G = G, + G,, where G, accounts for the presence of the airfoil, and set G,(x,y;
t-T)=-
GQ(x, y; w) exp {-iw( t - T)} dw,
n = 0, s.
(Al)
According to the reciprocal theorem G’(x, y; w) = G’(y, x; w) [ 171, and the problem of determining the field G’(x, y; w) produced by a point source at y can be replaced by that of determining the field G’(y, x; w) as a function of y produced by a point source at x.
244
M.
S.
HOWE
In particular, one may consider situations in which 1x1+ co, and y is in the ncxglrborhood ofthe airfoil and plate, thereby concentrating attention on the component of G A-cs;ronsible for the radiation from turbulence sources in the boundary layer near the airl;>ii Now, ,X G&(x, y; w) =
--x
(w
{iAk)h
-.hll
+R(k w) exp {iy(k)(x3+y3)l)
em {ik - (X-Y)} r(k)
d2k
’
(‘42)
and the integral is dominated by the contributions from the acoustic domain k < Ikol and from the pole at k = K, when 1x(-, co. By hypothesis koa, Z&a <<1, so that when /xl+ co and y is in the boundary layer close to the airfoil, the integrand may be expanded in powers of k * y, giving to leading order GA= G&,+G&,
(A3)
where cr
Gdx,
Y;
WI=
(exp {iY(k)lx3-y3(}+R(k, --cc
G&(x,Y; 0) = -
I5
k * y(ev
w) exp {iy(k)(x,+y,)})
expiti)
X’d2k,
{iy(k)lx3-y311
--oc
expW - xl y(k)
+R(k ~1 exp{iY(k)(x3+.t41)
d2k
’
By construction, Gb satisfies the Fourier time transform of the boundary condition (2.4), with B replaced by G. In terms of the reciprocal variable y this becomes D(a’/ay:
+a’/ayj)‘aG’/ay3
= mw’aG’/ay,
- p,,dG’,
y, = 0.
(A9
The same condition must be satisfied separately by G: . One can now argue that, provided the clearance t, is smaller than all other relevant length scales, G:,(x,
Y; w) = -
+
X k2cP*(exp{iy(k)lx3-y,l} I --oi
R(k, w) ev {iy(k)(x3+y3)l)
exp {ik - x}
d2k
y(k)
may be taken as the leading order approximation to G,. In expression (A6) cp*, defined in equation (2.7), is a solution of Laplace’s equation that satisfies acp*/ayz = 1 on the rigid airfoil, which implies that a{ G;, + G:,}/ay, = 0 on the airfoil. When terms - 0{( k,a)‘} relative to unity are discarded, G:, is a solution of the time-harmonic acoustic wave equation that is significantly different from zero only for points y in the vicinity of the airfoil, where its characteristic length scale of variationO(a). When the derivatives of G: in expression (A5) are scaled on a, the inertia terms on the right side are respectively of orders (K,,u)~< (K,cI)~ (CC1) and (p,a/m)(K,a)4<(poa/m)(K,a)4 (cc 1) relative to the stiffness term on the left. Since SC*+ 0 for J(yf + yz) >>a, it follows that it is sufficient to require that acp*/dy3 = 0 on the plate y, = 0.
SOUND
Hence,
correct
to an error-
OF
AIRFOIL
O{(k,a)*},
TIP
IN
BOUNDARY
LAYER
245
one can take
G’= Gho+ G;, + G:,,
(A7)
near the airfoil (i.e., at positions closer than a characteristic wavelength from the airfoil). Equation (2.6) of the main text is a uniformly applicable approximation to G, such that G’ reduces to expression (A7) at source positions y close to the airfoil, and coincides with Gb (no airfoil) when y is far from the airfoil. The following remark may be made concerning the adopted functional form (2.7) of cp*. The Green function as derived above with the use of expression (2.7) is discontinuous across y, = 0 for ly,l< u, 0 < y, < t,, between the tip of the airfoil and the wail. This deficiency could in principle be rectified by use of an alternative (but naturally very much more complicated) potential function which satisfies d~*/ay, = 1 on the airfoil alone. It is unnecessary to do this, however, for the following reason. Within the confines of the linear perturbation theory of this paper, the convection of vorticity parallel to the wall within the space between the tip and the wall does not produce a significant fluctuation in airfoil lift except possibly for vorticity near the leading and trailing edges. To be sure, conditions there are homogeneous in the mean flow direction, and there is no mechanism for “scattering” high wavenumber turbulence energy into low wavenumber sound and mean lift fluctuations. All such effects occur predominantly at the leading and trailing edges of the airfoil (although the unsteady Kutta condition implies that the relative contribution of the trailing edge is small) where the representation (A7) is appropriate except right at the tip corners, where, however, the loading must tend rapidly to zero.