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A note on the interaction of unsteady flow with an acoustic liner
Journal of Sound and Vibration (1979) 63(3), 429-436
A NOTE ON THE INTERACTION OF UNSTEADY FLOW WITH AN ACOUSTIC LINER M. S.
HOWE
Bolt Beranek and Newman. Inc., 50 Moulton Street, Cambridge, Massachusetts (Received
19 August 1975, and in revised,#tirm 18 November
02138, U.S.A
1978)
The effects of a mean grazing flow on the energy exchanges involved in the interaction of a bias-flow acoustic liner with, respectively, incident sound and boundary layer turbulence arc contrasted. The analysis of model problems which make use of a line vortex to simulate large scale, unsteady boundary layer structures indicates that, whereas acoustic waves may be effectively attenuated, dissipation caused by “jetting” in the apertures of the liner can result in a net transfer of energy from the mean flow to the turbulence in the boundary layer.
1. INTRODUCTION
Certain differences may be expected between interactions of an absorbent liner with (i) incident sound and those with (ii) boundary layer turbulence. In particular, although acoustic energy may be efficiently absorbed by a liner, on the contrary boundary layer turbulence may generally be expected to be enhanced. This is because dissipative effects at the liner lead to a transfer of mean flow energy to the turbulent field, in essentially the same way that viscous dissipation in a boundary layer can result in the destabilization of TollmienSchlichting waves (c.f., e.g., [l, 21). This question may be examined analytically by considering an idealized liner consisting of an infinite, thin plate perforated by a uniform distribution of small circular apertures. Ffowcs Williams [3] has considered the effect of the apertures on the generation of aerodynamic sound by boundary layer turbulence in the absence of dissipation at the liner, but did not take account of the back-reaction of the liner on the turbulence. This aspect of the problem, being essentially non-linear, is difficult to model in any but the simplest of situations. One such model has been described by Purshouse [4], who used an idealized representation of a turbulent eddy in the form of a line vortex. Purshouse was concerned with a compliant but impermeable boundary, and showed that dissipation at the boundary resulted in the migration of the vortex towards the wall, an effect which implies that the kinetic energy of the disturbed flow gradually diminishes as the line vortex approaches its image in the plane [S, p. 2291. This result (that dissipation at the boundary leads to the decay of the turbulent field), although perfectly correct in the context in which it was considered by Purshouse, is likely to be reversed in practice, when account must also be taken of the presence of a mean grazing flow. This possibility is investigated, in what follows, by means of a re-examination of the Purshouse problem in terms of the perforated liner. Attention will be confined to a single dissipative mechanism: namely, that associated with “jetting” at the apertures. This is perhaps the dominant form of dissipation when there exists a mean flow through the liner (e.g., due to suction). Such “bias flow” liners are known to be particularly effective absorbers of sound (see, e.g., references [&S]) and accordingly the discussion will be framed in terms of this particular boundary condition. 429 0022-460X179/070429+08 SO2.00/0
@ 1979 Academic Press Inc. (London)
Limited
430
M. S. HOWE
The detailed characteristics of the idealized liner are described in section 2. The contrasting effects of the interaction of the liner with sound and with vertical flow are discussed in sections 3 and 4 respectively.
2. BOUNDARY CONDITIONS AT THE LINER Consider an infinite rigid plate which occupies the plane x2 = 0 of a rectangular coordinate system (xi, x2, XJ and is perforated with a uniform array of small circular apertures (N per unit area) each of radius R.The fluid in the upper region (x2 > 0) is in uniform low Mach number flow at speed Li parallel to the positive x,-axis (see Figure 1); in addition a steady pressure difference between the upper and lower surfaces of the plate produces a low Mach number bias flow through the liner into x2 < 0. Let (I denote the open-area ratio of the plate, and V( > 0) the mean bias velocity in the negative x,-direction in each aperture, so that the mean suction velocity is equal to oV
(a)
(b)
Figure 1. Unsteady flow produced by (a) an incident sound wave, (b) a line vortex aligned parallel to the x,-axis induces unsteady vortex shedding in the apertures of the perforated liner. V is the mean bias-flow velocity in each of the apertures.
In sections 3 and 4 problems will be considered in which it is assumed that above and below the liner the characteristic length scale of the disturbed flow is large compared with the aperture spacing and radius. The disturbance will be caused either by incident sound or localized vortex forcing, and the resulting motion induced in the apertures gives rise to the generation of vorticity (“jetting”) which is swept into x1 < 0 by the bias flow. The fluid motion in the vicinity of the plate may therefore be divided into two components. The first of these is a large scale potential flow which describes conditions at distances from the liner which exceed the characteristic spacing between the apertures; the second has a length scale - O(R), and includes small scale turbulence confined to regions close to the underside of the liner. Let the large scale flow be described by velocity potentials 0+ and O_, respectively, in x2 >( 0. Under the supposition that any relevant acoustic wavelengths greatly exceed the spacing of the apertures, the essentially incompressible nature of the aperture flows implies the continuity of normal velocity across the liner:
a
(x2= 0).
(1)
UNSTEADY
FLOW/ACOUSTIC
LINER
431
INTERACTION
The dynamic effect of the liner can be expressed through the relation d@,/l3x, = Y(@+ - O_)
(2)
(x2 = 0),
where Y is the surface admittance (or conductivity) of the liner. In the absence of surface dissipation Y = 2NR, a result obtained by Rayleigh [9]. When dissipation occurs because of the generation of vorticity in the apertures, Y is complex and depends on the Strouhal number s = 20R/v (3) 01 being the radian frequency of a disturbance proportional
to emiwL.In this case one can se1
Y = 2NR(y - id),
(4)
where, for s 3 0, y and 6 are defined in terms of the modified Bessel functions I, and K, by PO1 v--id= l+(+ ne-“I,(s) - iK,(s) sinh(s))/s(+ne-” I,(s) + iK,(s) cash(s)). (51 For s < 0, y(s) = y( (s() and 6(s) = - 6() s(). It may easily be verified that y( / s I), 60 s / ) 3 0, and that the following asymptotic formulae are valid : Y z 2NR(l
-
i/s)
(s -+ LD),Y 1 -iNR~s/2
(s -+ 0).
(6)
The low Strouhal number limit can also be expressed in terms of the open-area ratio (7: Y Y -iNzR2mW
= -irm/V.
(7)
Observe that the defining relation (2) of the admittance is in terms of the normal velocity and the “potential difference” across the plate. In the absence of the grazing flow the linear relation p = -pod&& between the pressure p and the potential 4 (p, being the mean density), leads to an equivalent expression for the admittance in terms of the pressure jump across the plate. In the present case, however, the flux through the liner actually depends on jump in the total pressure p + p0v2/2 (v being the total velocity; see reference [lo] for further details), and therefore, by Bernoulli’s equation, on the potential difference. Recall that this difference is defined only to within an arbitrary constant; but such constant differences are of no practical significance, since the admittance Y(s) vanishes at s = 0.
3. ABSORPTION
OF ACOUSTIC
ENERGY
Consider a plane sound wave incident on the liner from .x2 > 0. The Mach number of the mean flow is assumed to be sufficiently small that convection of the sound by the flow may be neglected close to the plate-this approximation does not affect the nature of the results. but greatly simplifies the formulae. Introducing reflectio; and transmission coefficients 9 and ?-, respectively, and taking the propagation to be parallel to the (1,2)-plane. one has. inx, > 0, @+ = {,-ik,pZcosB + gpeikox~cosH; eiko(xlcinB-rt), 181
and, below the liner, in x2 < 0, Q
=
~--,-ik~x~cmB
+ ih,,(.xi
-inO
cl) (9)
where k, = o/c, c is the speed of sound, and 8 is the angle of incidence (see Figure l(a)). The coefficients 3’ and y are readily calculated from the surface conditions (1) and (2). When small terms involving the mean flow Mach number are discarded, the acoustic power I7 dissipated per unit surface area of the liner through the generation of turbulence is given by II/II,
= (1 - I%?]’- 15_12>E 2ik,cos8(Y
- Y*)/Jk, cos0
+ 2iY12.
(10)
432
M. S. HOWE
where n, is the intensity of the incident sound wave, and the asterisk denotes the complex conjugate. Using the representation (4) of the admittance Y one has = 8NRk,6
II/II,
cod/) k, co0 + 2iY[‘,
(11)
where, from the definition of 6, k,6 is non-negative for arbitrary real k, = o/c. Thus ZIjll, is also non-negative, confirming that acoustic energy is always absorbed by the liner. It is of interst to note the limiting form of the power dissipation ratio as the Strouhal number s -+ 0. Using the asymptotic formula (6) one finds -+ 4Mo cos0/[40 + M cos&j2
n/n,
(12)
where M = V/c is the Mach number of the aperture suction velocity. Thus the dissipation ratio ultimately becomes frequency-independent as the Strouhal number progressively diminishes, and varies linearly with the aperture suction velocity Vfor small Mach numbers M. These conclusions are consistent with the experimental results of Ingard and Ising [7].
4. INTERACTION
OF VORTICAL
FLOW WITH THE LINER
In the problem depicted in Figure l(b) a localized turbulent disturbance is modelled by a line vortex of circulation r which is located above the plate at @i(t), s*(t)) in the (1,2)-plane. Following Purshouse [4], the evolution of the unsteady motion resulting from the backreaction of the liner on the vortex may be determined by means of a perturbation expansion. The perturbation parameter is conveniently taken to be the open-area ratio CJ,and quantities will be calculated correct to O(a). First note that the kinetic energy of the (incompressible) vertical motion, 8, say, is given by b= (por2/47r) lnl s,(W)
(13)
+ O(o),
(see, e.g., references [4] and [5, p. 229]), where a is a length of the order of the radius of the vortex core. Thus the rate of change of total perturbation energy is
ad/at =
(pp/471)qtp,(t)
+ o(d),
(14)
where the migration velocity i,(t) of the vortex in the x,-direction is anticipated to be O(a), and the energy of the fluctuating flow increases/decreases according as the vortex recedes from/approaches the liner. The following more general form of the energy equation will also be required : &(~,,u~)=~(p~u~)
+ tJ&(p,u’)
= --~~ain~$
- div{u(p+
$p,,u’)},
(15)
2
1
where u = v - (U, 0, 0), and the grazing velocity U may depend on x2 when account is taken of mean shear. Integration of expression (15) over the fluid in x2 > 0 yields
a6 t=_,
ss m
dx
m
lo-
P#lUZ
dx, +
g 2
sm
(U,(P+ ~Ppou2)~x*=o dx1.
(16)
-m
The first term on the right represents the rate at which mean flow energy is converted into fluctuating (“turbulent”) flow by the Reynolds stress -p0ulu2 per unit distance in the x,-direction (out of the plane of the paper). The second integral is taken along the upper face of the liner. The dynamic surface condition (2) of section 2 is a function of the radian frequency o, in as much as Y z Y(s), s = 2wR/V. In order to calculate the motion of the line vortex it is
UNSTEADY
FLOW/ACOUSTIC
LINER
43.1
INTERACTION
therefore necessary to determine the effect of the liner on each Fourier component of the disturbance produced by the vortex. When the grazing velocity U is constant, the velocity potential in 0 < x2 < sl(r) due to the vortex can be set in the integral form cosh(kx,) e ik[xl-sl(t)l-lkIsz(t)
dk,
k
in the case in which there are no apertures in the plate. This result may be expressed as a Fourier decomposition in frequency by means of
f&t 4
cosh(kx,) ei(kr~-ot) k
dk do,
(181
where
f(k,U,)=& m
J
ei[wt-ksl(t)l
- Ik /sz(~) &.
(19)
--m
When the presence of the apertures in the liner is taken into account, O(a) quantities a(k. co) and P(k, co) can be introduced such that
cosh(kx2) +a
e-jklxl
k
1
f(k, co) eitkxl -W dk do
(0 < .
@_(x, t) =
,”
x2 <
s,(t)),
(20)
mJ
/leiklX2f(k, w) ei(kxl-ot) dk do
(x2 < 0),
(21)
J -m
and @* satisfy Laplace’s equation for incompressible, irrotational flow. The complex coefficients a and /I are calculated from the surface conditions (1) and (2), and one finds a = -/I = -Y/k((k)
+ 2Y).
(22)
Use of this expression in equation (20) determines, in principle, the velocity potential just above the liner, from which formulae for the motion of the vortex can be written down. Such formulae would implicitly involve the unknown location of the vortex, however, through the definition (19) of the Fourier coeflicient f(k, co). Accordingly one may adopt an iterative procedure in which f(k, co) is calculated in the zeroth approximation by setting s,(t) = U,t, Qt) = const., where u, = u + r/4?cs, ;
(23)
r/4ns, is the translational velocity of the vortex induced by its image in a rigid, unperforated plate, so that U, is the net translational velocity in the absence of perforations. In this case one has f(k, co) = 6(0 -
U,k) e-lkis2,
(24)
and, in the lowest order of approximation, Y(CJok)e-Ikixz
,-lklsz+ik(.x-LidI
k((kl + 2:‘(U,k))
where the second term in the curly brackets of the integrand is O(g).
dk,
125)
434
M. S. HOWE
The migration velocity S, of the vortex is equal to the sum of the suction velocity -01’ and the induced velocity of the core calculated from expression (25):
a sgn(k) Y(U,k) e- 21hlsl
‘1
s,(t) = -rrV - ;
_ -c
(I k / + iYq7g-d”.
(261
As the real and imaginary parts of Y(U,k) are respectively even and odd functions of k, it follows from equation (4) that one may also write (27) The integrand is non-negative for U, > 0, since 6 3 0 for k 3 0. Hence, when the circulation r of the vortex is positive/negative the effect of aperture dissipation is to cause the vortex to move towards/away from the liner. From the previous discussion, this implies that when the vortex is taken to characterize a large eddy in a boundary layer flow, for which the vorticity would be expected to be predominantly negative (directed into the paper in Figure l), the energy of the fluctuations has a tendency to increase. When there is no grazing flow (U = 0, so that U, = r/471s2 and sgn(b(U,k)) = sgn(r)), equation (27) reveals that S, is always negative, vertical energy being absorbed by the liner; this is the case discussed by Purshouse [4]. To illustrate this conclusion in more detail, consider the particular case in which the flow induced in the apertures of the liner by the vortex is of low Strouhal number s = 20R/V, where w = U,k. The effective range of integration in expression (27) satisfies 0 < ks, 5 1, and the low Strouhal number approximation therefore requires that (2U,/V)(R/s,) < 1, in which case the low frequency approximation (6) may be used in expression (27) to show that i,(t) = -al/
- (r/27cs,)o( U,/V)/(l
+ 4a2( U,/V)‘).
(28)
The rate at which energy is transferred between the mean flow and the vortex is obtained by substitution of the migration velocity (27) into the vortex energy equation (14). It may be verified that this yields precisely the same result as that dven by the energy equation (16) when use is made of the potential function @+ defined by expression (25) to calculate the perturbation quantities. Since there is no mean shear, equation (16) becomes
(29) and it can be shown by direct calculation that the value of the integral is independent of x2 provided that 0 d x2 < s2. For x2 > s2 the integral vanishes identically. Thus the location of the vortex may be identified with a critical layer at which the convection velocity of vorticity in the flow exactly equals the phase velocity U, of hydrodynamic, evanescent wavelike disturbances produced by fluctuating flow in the apertures of the liner. At this layer these waves exchange energy with the vortex field (cf., reference [ll], section 4.6). Some idea of the practical significance of these theoretical conclusions may be obtained by taking the line vortex r to model a large scale boundary layer eddy, whose convection velocity is U, 5 U/2, say, so that r/47cs, z - U/2. It then follows from equations (14) and (28) that, in the low Strouhal number case, (o2/2)(U/aV)2 d& = np,U?s,oV ;It i 1 + a4( U/pV)2
1‘*. 1 (
(30)
UNSTEADY
FLOW/ACOUSTIC
LINER
INTERACTION
435
The growth of the eddy following its displacement s1 = I/t/2 parallel to the mean How is accordingly given in order of magnitude by an expression of the form dB al/ - = /l ...(-){--.gds, A U
(~Z/2)(WJV2
_ I
(31)
1 + dyupjT
where p is a quantity of order unity and A is a measure of the boundary layer width. E‘or a suction velocity equal to 10% of the mean stream velocity U (oV/U = @l), and an open area ratio of 40% (a = @4), the term on the right of equation (31) is equal to 0.12~14 and corresponds to an e-folding distance s, of the eddy of about 1OA. The same type of analysis indicates that the growth rates of smaller eddies, located much closer to the wall, are on the average negative; the amplification due to the second term on the right of equation (28) is more than compensated by the effect of the mean su&on term -al/. 4. I.
THE EFFECT
OF MEAN SHEAR
It might be argued that the absence of mean shear in the above discussion would limit the generality of the conclusions. An approximate treatment of the more general situation can be effected in the manner illustrated in Figure 2, in which the mean grazing velocity is specified by U(x,) = -Q(X,
- h)
= 0
(0 < X, < h),
where Q (< 0) is the vorticity of the mean conditions of the problem are unchanged.
Mean
(x1 > h)
flow,
(33)
assumed to be constant. The remaining
shear
flow
/
Figure 2. Idealized presence of a uniform
model used to calculate mean shear flow.
the interaction
of a line vortex
x,=6
I‘ with the bias-flow
liner in the
Proceeding with an iterative analysis of the type described above, and subsequently allowing the width b of the irrotational sublayer to vanish, one finds that the rate of change of perturbation energy is again given by equation (14), with the migration velocity defined by .
,f,(t) = -0v
-
s cc
sgn(k)Y(U,k)e-2ik1”~ + IkiTd”’ 27X _~ {(2 + nlv,~kJ)Y(U,k)
JL
(33)
436
M. S. HOWE
where U, = U(s,) + r/47+ As before, the second term on the right of this result. which gives the motion of the vortex due to dissipation at the boundary, is positive/negative according as the circulation r is negative/positive. In the energy equation (16) there is now a non-trivial contribution from the Reynolds stress term: i.e., from the first of the integrals on the right side, since ~U/&X, = -Q f 0. However, s2 is constant in the shear layer, and the integrated stress fmm~o~I~2 dx, m’ay be shown to be constant and non-zero in 0 < x2 < s2, and to be zero for x2 > s2. Similar remarks apply, as before, to the second term on the right of equation (16), so that in this more general case, both of the energy flux terms give rise to a transfer of energy between the mean and fluctuating flows at the critical layer x2 = s2.
5. CONCLUSION
Energy exchange mechanisms have been examined for a bias-flow acoustic liner at which dissipation occurs through the generation of vorticity (“turbulence”) at the apertures in the liner. It has been shown that in the presence of a mean grazing flow the liner always serves to attenuate sound, but that its action on turbulence can be different. For turbulent boundary layer flows of the type encountered in the jet pipe of an aeroengine, the dissipation at the boundary can be accompanied by an enhanced transfer of energy from the mean flow to the large scale boundary layer fluctuations. These conclusions, although derived for a specific modelling of the liner, would be expected to characterize the behaviour of all dissipative boundaries in the presence of a mean grazing flow.
ACKNOWLEDGMENT
This work was performed under contract from the NASA Langley Research Center.
REFERENCES
1. T. BROOKEBENJAMIN1960 Journal of Fluid Mechanics 9, 513-532. Effects of a flexible boundary on hydrodynamic stability. 2. T. BROOKEBENJAMIN1963 Journal of FZuid Mechanics 16,436-450. The threefold classification of unstable disturbances in flexible surfaces bounding inviscid flows. 3. J. E. FFOWCSWILLIAMS1972 Journal of Fluid Mechanics 51,737-749. The acoustics of turbulence near sound absorbent liners. 4. M. PURSHOUSE1976 Journal of Sound and Vibration 49, 423-436. On the damping of unsteady flow by compliant boundaries. 5. H. LAMB1932 Hydrodynamic. Cambridge University Press, sixth edition. 6. F. VON BARTHEL1958 Frequenz 12, l-11. Untersuchungen iiber nichtlineare Helmholtzresonatoren. 7. U. INGARDand H. ISING 1967 Journal of the Acoustical Society bf America 42, 6-17. Acoustic nonlinearity of an orifice. 8. P. D. DEANand B. J. TESTER1975 NASA CR-1 34998. Duct wall impedance control as an advanced concept in acoustic suppression. 9. LORD RAYLEIGH1945 Theory of Sound, Volume 2. New York: Dover. 10. M. S. HOWE 1979 (in Press). Proceedings of the Royal Society London. On the theory of unsteady high Reynolds number flow through a circular aperture. 11. M. J. LIGHTHILL1978 Waoes in Fluiak Cambridge University Press.
A NOTE ON THE INTERACTION OF UNSTEADY FLOW WITH AN ACOUSTIC LINER M. S.
HOWE
Bolt Beranek and Newman. Inc., 50 Moulton Street, Cambridge, Massachusetts (Received
19 August 1975, and in revised,#tirm 18 November
02138, U.S.A
1978)
The effects of a mean grazing flow on the energy exchanges involved in the interaction of a bias-flow acoustic liner with, respectively, incident sound and boundary layer turbulence arc contrasted. The analysis of model problems which make use of a line vortex to simulate large scale, unsteady boundary layer structures indicates that, whereas acoustic waves may be effectively attenuated, dissipation caused by “jetting” in the apertures of the liner can result in a net transfer of energy from the mean flow to the turbulence in the boundary layer.
1. INTRODUCTION
Certain differences may be expected between interactions of an absorbent liner with (i) incident sound and those with (ii) boundary layer turbulence. In particular, although acoustic energy may be efficiently absorbed by a liner, on the contrary boundary layer turbulence may generally be expected to be enhanced. This is because dissipative effects at the liner lead to a transfer of mean flow energy to the turbulent field, in essentially the same way that viscous dissipation in a boundary layer can result in the destabilization of TollmienSchlichting waves (c.f., e.g., [l, 21). This question may be examined analytically by considering an idealized liner consisting of an infinite, thin plate perforated by a uniform distribution of small circular apertures. Ffowcs Williams [3] has considered the effect of the apertures on the generation of aerodynamic sound by boundary layer turbulence in the absence of dissipation at the liner, but did not take account of the back-reaction of the liner on the turbulence. This aspect of the problem, being essentially non-linear, is difficult to model in any but the simplest of situations. One such model has been described by Purshouse [4], who used an idealized representation of a turbulent eddy in the form of a line vortex. Purshouse was concerned with a compliant but impermeable boundary, and showed that dissipation at the boundary resulted in the migration of the vortex towards the wall, an effect which implies that the kinetic energy of the disturbed flow gradually diminishes as the line vortex approaches its image in the plane [S, p. 2291. This result (that dissipation at the boundary leads to the decay of the turbulent field), although perfectly correct in the context in which it was considered by Purshouse, is likely to be reversed in practice, when account must also be taken of the presence of a mean grazing flow. This possibility is investigated, in what follows, by means of a re-examination of the Purshouse problem in terms of the perforated liner. Attention will be confined to a single dissipative mechanism: namely, that associated with “jetting” at the apertures. This is perhaps the dominant form of dissipation when there exists a mean flow through the liner (e.g., due to suction). Such “bias flow” liners are known to be particularly effective absorbers of sound (see, e.g., references [&S]) and accordingly the discussion will be framed in terms of this particular boundary condition. 429 0022-460X179/070429+08 SO2.00/0
@ 1979 Academic Press Inc. (London)
Limited
430
M. S. HOWE
The detailed characteristics of the idealized liner are described in section 2. The contrasting effects of the interaction of the liner with sound and with vertical flow are discussed in sections 3 and 4 respectively.
2. BOUNDARY CONDITIONS AT THE LINER Consider an infinite rigid plate which occupies the plane x2 = 0 of a rectangular coordinate system (xi, x2, XJ and is perforated with a uniform array of small circular apertures (N per unit area) each of radius R.The fluid in the upper region (x2 > 0) is in uniform low Mach number flow at speed Li parallel to the positive x,-axis (see Figure 1); in addition a steady pressure difference between the upper and lower surfaces of the plate produces a low Mach number bias flow through the liner into x2 < 0. Let (I denote the open-area ratio of the plate, and V( > 0) the mean bias velocity in the negative x,-direction in each aperture, so that the mean suction velocity is equal to oV
(a)
(b)
Figure 1. Unsteady flow produced by (a) an incident sound wave, (b) a line vortex aligned parallel to the x,-axis induces unsteady vortex shedding in the apertures of the perforated liner. V is the mean bias-flow velocity in each of the apertures.
In sections 3 and 4 problems will be considered in which it is assumed that above and below the liner the characteristic length scale of the disturbed flow is large compared with the aperture spacing and radius. The disturbance will be caused either by incident sound or localized vortex forcing, and the resulting motion induced in the apertures gives rise to the generation of vorticity (“jetting”) which is swept into x1 < 0 by the bias flow. The fluid motion in the vicinity of the plate may therefore be divided into two components. The first of these is a large scale potential flow which describes conditions at distances from the liner which exceed the characteristic spacing between the apertures; the second has a length scale - O(R), and includes small scale turbulence confined to regions close to the underside of the liner. Let the large scale flow be described by velocity potentials 0+ and O_, respectively, in x2 >( 0. Under the supposition that any relevant acoustic wavelengths greatly exceed the spacing of the apertures, the essentially incompressible nature of the aperture flows implies the continuity of normal velocity across the liner:
a
(x2= 0).
(1)
UNSTEADY
FLOW/ACOUSTIC
LINER
431
INTERACTION
The dynamic effect of the liner can be expressed through the relation d@,/l3x, = Y(@+ - O_)
(2)
(x2 = 0),
where Y is the surface admittance (or conductivity) of the liner. In the absence of surface dissipation Y = 2NR, a result obtained by Rayleigh [9]. When dissipation occurs because of the generation of vorticity in the apertures, Y is complex and depends on the Strouhal number s = 20R/v (3) 01 being the radian frequency of a disturbance proportional
to emiwL.In this case one can se1
Y = 2NR(y - id),
(4)
where, for s 3 0, y and 6 are defined in terms of the modified Bessel functions I, and K, by PO1 v--id= l+(+ ne-“I,(s) - iK,(s) sinh(s))/s(+ne-” I,(s) + iK,(s) cash(s)). (51 For s < 0, y(s) = y( (s() and 6(s) = - 6() s(). It may easily be verified that y( / s I), 60 s / ) 3 0, and that the following asymptotic formulae are valid : Y z 2NR(l
-
i/s)
(s -+ LD),Y 1 -iNR~s/2
(s -+ 0).
(6)
The low Strouhal number limit can also be expressed in terms of the open-area ratio (7: Y Y -iNzR2mW
= -irm/V.
(7)
Observe that the defining relation (2) of the admittance is in terms of the normal velocity and the “potential difference” across the plate. In the absence of the grazing flow the linear relation p = -pod&& between the pressure p and the potential 4 (p, being the mean density), leads to an equivalent expression for the admittance in terms of the pressure jump across the plate. In the present case, however, the flux through the liner actually depends on jump in the total pressure p + p0v2/2 (v being the total velocity; see reference [lo] for further details), and therefore, by Bernoulli’s equation, on the potential difference. Recall that this difference is defined only to within an arbitrary constant; but such constant differences are of no practical significance, since the admittance Y(s) vanishes at s = 0.
3. ABSORPTION
OF ACOUSTIC
ENERGY
Consider a plane sound wave incident on the liner from .x2 > 0. The Mach number of the mean flow is assumed to be sufficiently small that convection of the sound by the flow may be neglected close to the plate-this approximation does not affect the nature of the results. but greatly simplifies the formulae. Introducing reflectio; and transmission coefficients 9 and ?-, respectively, and taking the propagation to be parallel to the (1,2)-plane. one has. inx, > 0, @+ = {,-ik,pZcosB + gpeikox~cosH; eiko(xlcinB-rt), 181
and, below the liner, in x2 < 0, Q
=
~--,-ik~x~cmB
+ ih,,(.xi
-inO
cl) (9)
where k, = o/c, c is the speed of sound, and 8 is the angle of incidence (see Figure l(a)). The coefficients 3’ and y are readily calculated from the surface conditions (1) and (2). When small terms involving the mean flow Mach number are discarded, the acoustic power I7 dissipated per unit surface area of the liner through the generation of turbulence is given by II/II,
= (1 - I%?]’- 15_12>E 2ik,cos8(Y
- Y*)/Jk, cos0
+ 2iY12.
(10)
432
M. S. HOWE
where n, is the intensity of the incident sound wave, and the asterisk denotes the complex conjugate. Using the representation (4) of the admittance Y one has = 8NRk,6
II/II,
cod/) k, co0 + 2iY[‘,
(11)
where, from the definition of 6, k,6 is non-negative for arbitrary real k, = o/c. Thus ZIjll, is also non-negative, confirming that acoustic energy is always absorbed by the liner. It is of interst to note the limiting form of the power dissipation ratio as the Strouhal number s -+ 0. Using the asymptotic formula (6) one finds -+ 4Mo cos0/[40 + M cos&j2
n/n,
(12)
where M = V/c is the Mach number of the aperture suction velocity. Thus the dissipation ratio ultimately becomes frequency-independent as the Strouhal number progressively diminishes, and varies linearly with the aperture suction velocity Vfor small Mach numbers M. These conclusions are consistent with the experimental results of Ingard and Ising [7].
4. INTERACTION
OF VORTICAL
FLOW WITH THE LINER
In the problem depicted in Figure l(b) a localized turbulent disturbance is modelled by a line vortex of circulation r which is located above the plate at @i(t), s*(t)) in the (1,2)-plane. Following Purshouse [4], the evolution of the unsteady motion resulting from the backreaction of the liner on the vortex may be determined by means of a perturbation expansion. The perturbation parameter is conveniently taken to be the open-area ratio CJ,and quantities will be calculated correct to O(a). First note that the kinetic energy of the (incompressible) vertical motion, 8, say, is given by b= (por2/47r) lnl s,(W)
(13)
+ O(o),
(see, e.g., references [4] and [5, p. 229]), where a is a length of the order of the radius of the vortex core. Thus the rate of change of total perturbation energy is
ad/at =
(pp/471)qtp,(t)
+ o(d),
(14)
where the migration velocity i,(t) of the vortex in the x,-direction is anticipated to be O(a), and the energy of the fluctuating flow increases/decreases according as the vortex recedes from/approaches the liner. The following more general form of the energy equation will also be required : &(~,,u~)=~(p~u~)
+ tJ&(p,u’)
= --~~ain~$
- div{u(p+
$p,,u’)},
(15)
2
1
where u = v - (U, 0, 0), and the grazing velocity U may depend on x2 when account is taken of mean shear. Integration of expression (15) over the fluid in x2 > 0 yields
a6 t=_,
ss m
dx
m
lo-
P#lUZ
dx, +
g 2
sm
(U,(P+ ~Ppou2)~x*=o dx1.
(16)
-m
The first term on the right represents the rate at which mean flow energy is converted into fluctuating (“turbulent”) flow by the Reynolds stress -p0ulu2 per unit distance in the x,-direction (out of the plane of the paper). The second integral is taken along the upper face of the liner. The dynamic surface condition (2) of section 2 is a function of the radian frequency o, in as much as Y z Y(s), s = 2wR/V. In order to calculate the motion of the line vortex it is
UNSTEADY
FLOW/ACOUSTIC
LINER
43.1
INTERACTION
therefore necessary to determine the effect of the liner on each Fourier component of the disturbance produced by the vortex. When the grazing velocity U is constant, the velocity potential in 0 < x2 < sl(r) due to the vortex can be set in the integral form cosh(kx,) e ik[xl-sl(t)l-lkIsz(t)
dk,
k
in the case in which there are no apertures in the plate. This result may be expressed as a Fourier decomposition in frequency by means of
f&t 4
cosh(kx,) ei(kr~-ot) k
dk do,
(181
where
f(k,U,)=& m
J
ei[wt-ksl(t)l
- Ik /sz(~) &.
(19)
--m
When the presence of the apertures in the liner is taken into account, O(a) quantities a(k. co) and P(k, co) can be introduced such that
cosh(kx2) +a
e-jklxl
k
1
f(k, co) eitkxl -W dk do
(0 < .
@_(x, t) =
,”
x2 <
s,(t)),
(20)
mJ
/leiklX2f(k, w) ei(kxl-ot) dk do
(x2 < 0),
(21)
J -m
and @* satisfy Laplace’s equation for incompressible, irrotational flow. The complex coefficients a and /I are calculated from the surface conditions (1) and (2), and one finds a = -/I = -Y/k((k)
+ 2Y).
(22)
Use of this expression in equation (20) determines, in principle, the velocity potential just above the liner, from which formulae for the motion of the vortex can be written down. Such formulae would implicitly involve the unknown location of the vortex, however, through the definition (19) of the Fourier coeflicient f(k, co). Accordingly one may adopt an iterative procedure in which f(k, co) is calculated in the zeroth approximation by setting s,(t) = U,t, Qt) = const., where u, = u + r/4?cs, ;
(23)
r/4ns, is the translational velocity of the vortex induced by its image in a rigid, unperforated plate, so that U, is the net translational velocity in the absence of perforations. In this case one has f(k, co) = 6(0 -
U,k) e-lkis2,
(24)
and, in the lowest order of approximation, Y(CJok)e-Ikixz
,-lklsz+ik(.x-LidI
k((kl + 2:‘(U,k))
where the second term in the curly brackets of the integrand is O(g).
dk,
125)
434
M. S. HOWE
The migration velocity S, of the vortex is equal to the sum of the suction velocity -01’ and the induced velocity of the core calculated from expression (25):
a sgn(k) Y(U,k) e- 21hlsl
‘1
s,(t) = -rrV - ;
_ -c
(I k / + iYq7g-d”.
(261
As the real and imaginary parts of Y(U,k) are respectively even and odd functions of k, it follows from equation (4) that one may also write (27) The integrand is non-negative for U, > 0, since 6 3 0 for k 3 0. Hence, when the circulation r of the vortex is positive/negative the effect of aperture dissipation is to cause the vortex to move towards/away from the liner. From the previous discussion, this implies that when the vortex is taken to characterize a large eddy in a boundary layer flow, for which the vorticity would be expected to be predominantly negative (directed into the paper in Figure l), the energy of the fluctuations has a tendency to increase. When there is no grazing flow (U = 0, so that U, = r/471s2 and sgn(b(U,k)) = sgn(r)), equation (27) reveals that S, is always negative, vertical energy being absorbed by the liner; this is the case discussed by Purshouse [4]. To illustrate this conclusion in more detail, consider the particular case in which the flow induced in the apertures of the liner by the vortex is of low Strouhal number s = 20R/V, where w = U,k. The effective range of integration in expression (27) satisfies 0 < ks, 5 1, and the low Strouhal number approximation therefore requires that (2U,/V)(R/s,) < 1, in which case the low frequency approximation (6) may be used in expression (27) to show that i,(t) = -al/
- (r/27cs,)o( U,/V)/(l
+ 4a2( U,/V)‘).
(28)
The rate at which energy is transferred between the mean flow and the vortex is obtained by substitution of the migration velocity (27) into the vortex energy equation (14). It may be verified that this yields precisely the same result as that dven by the energy equation (16) when use is made of the potential function @+ defined by expression (25) to calculate the perturbation quantities. Since there is no mean shear, equation (16) becomes
(29) and it can be shown by direct calculation that the value of the integral is independent of x2 provided that 0 d x2 < s2. For x2 > s2 the integral vanishes identically. Thus the location of the vortex may be identified with a critical layer at which the convection velocity of vorticity in the flow exactly equals the phase velocity U, of hydrodynamic, evanescent wavelike disturbances produced by fluctuating flow in the apertures of the liner. At this layer these waves exchange energy with the vortex field (cf., reference [ll], section 4.6). Some idea of the practical significance of these theoretical conclusions may be obtained by taking the line vortex r to model a large scale boundary layer eddy, whose convection velocity is U, 5 U/2, say, so that r/47cs, z - U/2. It then follows from equations (14) and (28) that, in the low Strouhal number case, (o2/2)(U/aV)2 d& = np,U?s,oV ;It i 1 + a4( U/pV)2
1‘*. 1 (
(30)
UNSTEADY
FLOW/ACOUSTIC
LINER
INTERACTION
435
The growth of the eddy following its displacement s1 = I/t/2 parallel to the mean How is accordingly given in order of magnitude by an expression of the form dB al/ - = /l ...(-){--.gds, A U
(~Z/2)(WJV2
_ I
(31)
1 + dyupjT
where p is a quantity of order unity and A is a measure of the boundary layer width. E‘or a suction velocity equal to 10% of the mean stream velocity U (oV/U = @l), and an open area ratio of 40% (a = @4), the term on the right of equation (31) is equal to 0.12~14 and corresponds to an e-folding distance s, of the eddy of about 1OA. The same type of analysis indicates that the growth rates of smaller eddies, located much closer to the wall, are on the average negative; the amplification due to the second term on the right of equation (28) is more than compensated by the effect of the mean su&on term -al/. 4. I.
THE EFFECT
OF MEAN SHEAR
It might be argued that the absence of mean shear in the above discussion would limit the generality of the conclusions. An approximate treatment of the more general situation can be effected in the manner illustrated in Figure 2, in which the mean grazing velocity is specified by U(x,) = -Q(X,
- h)
= 0
(0 < X, < h),
where Q (< 0) is the vorticity of the mean conditions of the problem are unchanged.
Mean
(x1 > h)
flow,
(33)
assumed to be constant. The remaining
shear
flow
/
Figure 2. Idealized presence of a uniform
model used to calculate mean shear flow.
the interaction
of a line vortex
x,=6
I‘ with the bias-flow
liner in the
Proceeding with an iterative analysis of the type described above, and subsequently allowing the width b of the irrotational sublayer to vanish, one finds that the rate of change of perturbation energy is again given by equation (14), with the migration velocity defined by .
,f,(t) = -0v
-
s cc
sgn(k)Y(U,k)e-2ik1”~ + IkiTd”’ 27X _~ {(2 + nlv,~kJ)Y(U,k)
JL
(33)
436
M. S. HOWE
where U, = U(s,) + r/47+ As before, the second term on the right of this result. which gives the motion of the vortex due to dissipation at the boundary, is positive/negative according as the circulation r is negative/positive. In the energy equation (16) there is now a non-trivial contribution from the Reynolds stress term: i.e., from the first of the integrals on the right side, since ~U/&X, = -Q f 0. However, s2 is constant in the shear layer, and the integrated stress fmm~o~I~2 dx, m’ay be shown to be constant and non-zero in 0 < x2 < s2, and to be zero for x2 > s2. Similar remarks apply, as before, to the second term on the right of equation (16), so that in this more general case, both of the energy flux terms give rise to a transfer of energy between the mean and fluctuating flows at the critical layer x2 = s2.
5. CONCLUSION
Energy exchange mechanisms have been examined for a bias-flow acoustic liner at which dissipation occurs through the generation of vorticity (“turbulence”) at the apertures in the liner. It has been shown that in the presence of a mean grazing flow the liner always serves to attenuate sound, but that its action on turbulence can be different. For turbulent boundary layer flows of the type encountered in the jet pipe of an aeroengine, the dissipation at the boundary can be accompanied by an enhanced transfer of energy from the mean flow to the large scale boundary layer fluctuations. These conclusions, although derived for a specific modelling of the liner, would be expected to characterize the behaviour of all dissipative boundaries in the presence of a mean grazing flow.
ACKNOWLEDGMENT
This work was performed under contract from the NASA Langley Research Center.
REFERENCES
1. T. BROOKEBENJAMIN1960 Journal of Fluid Mechanics 9, 513-532. Effects of a flexible boundary on hydrodynamic stability. 2. T. BROOKEBENJAMIN1963 Journal of FZuid Mechanics 16,436-450. The threefold classification of unstable disturbances in flexible surfaces bounding inviscid flows. 3. J. E. FFOWCSWILLIAMS1972 Journal of Fluid Mechanics 51,737-749. The acoustics of turbulence near sound absorbent liners. 4. M. PURSHOUSE1976 Journal of Sound and Vibration 49, 423-436. On the damping of unsteady flow by compliant boundaries. 5. H. LAMB1932 Hydrodynamic. Cambridge University Press, sixth edition. 6. F. VON BARTHEL1958 Frequenz 12, l-11. Untersuchungen iiber nichtlineare Helmholtzresonatoren. 7. U. INGARDand H. ISING 1967 Journal of the Acoustical Society bf America 42, 6-17. Acoustic nonlinearity of an orifice. 8. P. D. DEANand B. J. TESTER1975 NASA CR-1 34998. Duct wall impedance control as an advanced concept in acoustic suppression. 9. LORD RAYLEIGH1945 Theory of Sound, Volume 2. New York: Dover. 10. M. S. HOWE 1979 (in Press). Proceedings of the Royal Society London. On the theory of unsteady high Reynolds number flow through a circular aperture. 11. M. J. LIGHTHILL1978 Waoes in Fluiak Cambridge University Press.