Category Noise Sources in Turbulent Shear Flows

In Curle’s formulation (section 4.6) we restricted ourselves to fixed con­trol volumes. When considering sound produced by moving objects such as fan blades, it is more convenient to use a moving control volume. Ffowcs Williams and Hawkings (1969b) use generalized functions to take into ac­count the motion of the sources, the result being a generalization of Curle’s equation in which Doppler factors appear. In a further step Ffowcs Williams and Hawking [Goldstein (1976), Dowling and Ffowcs Williams (1983), Crighton et al. (1992)] introduce the boundaries of the control volume in the equation of motion, see next chapter. We now focus on the derivation

of the Doppler effect for point sources. A moving point source is described by:

where te = t – r and r = x – y. Using the properties of the delta function we get after spatial integration:

(107)

dT

(108)

with H(ti) = 0. In the present case we have:

x – xs (t)

Co

so that:

(110)

where Mr is the ratio of the source velocity component in the direction of the observer and the speed of sound. The sound field is given by:

(111)

(112)

For subsonic velocities there is only one root (t = te) of H(t) = 0. For a harmonically oscillating sound source with constant frequency ш, the fre­quency of the signal reaching the observer is:

dwte ш

dt 1 – Mr (te)

because d = (tc).

A further discussion of the Doppler Effect is provided in the next chap­ter, where it is shown that for supersonic Mach numbers, the sound source will have a strong radiation for directions such that Mr = 1. An example of such a radiation occurs when elastic bending waves in a plate propa­gate supersonically with respect to the surrounding fluid. As the velocity of propagation of bending waves increases with the frequency this occurs typically above a critical frequency fc, which is called the coincidence fre­quency. This explains why we hear a high pitch when we hit a glass window.

From equation (111) we observe that in addition to the change in fre­quency we have an effect of the source motion on the amplitude reaching the observer. This effect can be understood as a result of the change in ratio of source size to acoustic wave length. From equation (63) we know that with increasing Helmholtz numbers the radiate sound amplitude of a compact object increases. In the direction of motion of the source, the emit­ted acoustic wave length is shorter by a factor 1 — Mr, with an increased effective Helmholtz number as a result. In figure 5 we provide an intuitive interpretation of the Doppler shift in frequency.

Furthermore we note that for a moving object of volume V the sound source is q(x, t) = p0dVS(x — xs(t)). Hence we have:

It shows that due to the time dependency of the retarded time dte/dt an object of constant volume will radiate sound if its velocity varies. This is the so called thickness noise p’th, which is very important in aircraft fans. In the far field approximation for a rigid of volume V body moving at subsonic speed, we have:

Another example is the sound radiated by a moving point force:

f = F (t)S(x — xs(t)) (116)

which is given by:

In the far-field approximation we have:

F(te) .

4п|x – xs (te )||1 – Mr (te) | ‘

(118)

4.8 Influence of speed of sound gradient and of convective effects

Whenever a source of sound is compact we can separate the sound gen­eration from the wave propagation. Even with this simplification the wave propagation remains extremely complex.

In the presence of flow and gradients in the speed of sound, acoustic

waves display complex propagation behaviour [Dowling 1983, Pierce 1990, Rienstra 1999]. An example of this is the sound propagation in the at­mosphere. As a result of the non-uniformity of the temperature in the atmosphere waves are deflected from the straight path assumed in the ele­mentary solutions for uniform stagnant fluid. An example being that a gun shot or thunder heard at large distances can be repeated multiple times, which yields a roll sound. This is due to the fact that sound can reach our ears along multiple paths.

We now consider a very basic problem of a plane wave that is reflected at the interface (shear layer) between two uniform media a and b with each having uniform flow speeds respectively Ua = (Ua, 0, 0) and Ub = (Ub, 0, 0) and speeds of sound ca and cb respectively.

In presence of a uniform flow the plane wave solution (52) becomes:

Ux Un

co + n ■U,

with П = (cos в, sin в, 0) and к = un/(c0 + П ■ U).

We assume an incident wave with amplitude I and wave number ki = in region a. This induces a reflected wave with amplitude R and

and a transmitted wave with amplitude T and (Figure 6).

Cb+UT■Ub V & ’

At the interface x2 = 0 we have continuity of pressure so that for x2 =0 we have:

(120)

As this equation should hold for any value of the coordinate xi (along the shear layer) the exponents should be identical:

cos вІ

Ca + Ua cos ві

cos eR

Ca + Ua cos вП

cos вт

cb + Ub cos вт

The first equality of (121) implies that cos в1 = cos eR, so that the reflection angle is equal to the incidence angle eR = —вІ.

The second equality of (121) yields the modified Snelius law:

ca

cos вІ

cb

cos вт

Ub = (Ub,0,0)

Figure 6. Reflection and refraction of a plane wave at a flat shear layer x2 =0 separating two uniform flows.

or:

The maximum transmission angle is found for grazing incidence cos вІ = 1: In the particular case of ca = cb and Ub = 0 we find:

In high speed jets one does indeed observe a cone of silence along the axis of the jet, because the acoustic waves emitted along the main flow direction are bent away from the flow direction by the velocity gradient in the shear layers [Morfey (1978)].

The amplitude of the transmitted and reflected waves is calculated from the continuity of pressure at the interface I + R = T complemented by the continuity of particle displacement at the interface.

The analogy of Curle (1955) is the integral formulation (88) applied to Lighthill’s analogy (42) in terms of density fluctuations:

The observer is placed within the control volume V over which we carry out the integration. This equation is based on the assumption that at the listener’s position p’ = c2p’. We will further ignore the contribution from the external force field (f = 0). By means of partial integration we move

the space derivatives from the source terms towards the Green function:

Using the definition of the viscous stress tensor (26) and the momentum equation (9) we can write (102) in the form:

n

dG

dyi nidSy

Furthermore we neglect entropy fluctuations on the surface S.

By means of partial integration we move the time derivative in the sec­ond integral from the momentum flux to the Green’s function. Using the symmetry relations of the free field derivative with respect to space (85) and time derivatives (84), we find in the far field approximation (60):

In (104) we recognize the monopole sound production due to the volume flux leaving the surface (first integral), the dipole field generated by the force acting on the surfaces and the quadrupole field generated by fluctuations of the Reynolds stress tensor in the volume.

Using the superposition principle we obtain an integral formulation of the wave equation for free space conditions. We first consider the sound generated by a pulse from a point source. This implies a localization in time and space, obtained by using the delta function. The delta function S(t) is a generalized function defined by [Chrighton (1992)]:

TO

5(t)f (t)dt = f (0)

-TO

For any well behaving function f(t) and:

5(t)dt = 1

The delta function has no meaning outside an integral. The free-field Green function Go(x, ty, T) is the solution of the wave equation:

where S(x — y) = 5(x — y)5(x2 — y2)S(x3 — y3), for free-field boundary conditions and for the initial conditions:

Go(x, ty, T) = 0, t < t

corresponding to the causality condition that a wave cannot reach an ob­server before it has been emitted. In order to determine G0(x, ty, t) we use the Fourier transform G0 defined by:

and

Go(w, xy) = — Go(x, ty, T)exp(-iwt)dt.

2n.1^,

As we consider the field generated by a point source in free-field conditions we know that the Fourier transform of the Green function is given by:

л A

Go(u, xy) = — exp(—ikr)

where A is an amplitude which will be determined by using the properties of the delta function. We take the Fourier transform of the wave equation (71). Using the property (69) of the delta function:

2П f-oo S(t — T) exp(—i^t)dt =

(77)

= 2П f-o-r 6(t – T) exp(-iw(t – T) – ІШТ)d(t — T) = ЄХРІ2ПШТ}

we find:

— (k2 + V2)Go = ^Z! A. (78)

2П

We integrate this equation over a spherical volume V of radius R enclosing the source:

-f (k2 + V2)GodV = exp(~^T) . (79)

Jv

By taking the limit of a compact control volume kR << 1 and using the Gaussian Theorem we find:

which yields the amplitude A. Substituting A in (76) and transforming back to the time domain yields:

Go(x, ty, T )= ^(T4-rtg) (81)

where the emission (retarded) time te is defined by:

r

te = t——– . (82)

co

Because Green’s function in free-space only depends on the distance r and time difference (t — t), rather than on the source and observer’s coordinates (x, t) and (y, T) separately, it satisfies the important symmetry properties:

Equation (83) is the so-called reciprocity relation, which is also valid for Green’s functions in the presence of walls.

by using the superposition principle:

Substitution of (87) into (86) and using the definition (71) of Green’s function we can verify the validity of this solution.

In the presence of walls, we can still use the same free-field Green func­tion. However, now the solution of the wave equation will include surface integrals representing the effect of reflections of waves at the walls. Using Green’s theorem we have:

p'(x, t)= J—oo Uq(y, T )Go(x, ty, T )dVydr (88)

— I-o Is [p’vyGo — GoVyp’J ■ ndSydr. ( )

This integral formulation, in combination with Lighthill’s analogy, yields the integral formulation of Curle (1955). The control volume is chosen such that it encloses the observation point x. Note that in the literature the sign of the unit normal n is often chosen to be the opposite of the sign chosen here [Goldstein (1976), Dowling and Ffowcs Williams (1983)].

An alternative approach is the use of a so-called tailored Green function [Dowling and Ffowcs Williams (1983)]. This is a Green function defined by the wave equation (71) and the same (locally reacting linear) boundary conditions as the acoustic field under consideration. In that case the surface integrals of (88) vanish. An example of such a Green function for the trailing edge of a plate will be discussed in later chapters, Part 2.

1.4 Monopole, dipole and quadrupole

We consider radiation of a spatially limited source-region under free field conditions. Whenever the source region (q(x, t) = 0) is compact, we can ne­glect variations in the retarded time te in the integral of equation (87).

Choosing the origin within the source region we get at distances large com­pared to the source region:

r = x — f* x (89)

so that we have:

P'(xX<> * XjX X — Xі dV

We call the integral fV q y, t — dVy the monopole strength of the source

region. Whenever the source is the divergence of a force field q(x, t) = —V-f integral (91) taken over a volume including the source region will vanish because the surface integral of the flux of the force field fS f • ndSy = 0 vanishes because f = 0 on the surface. The surface, including the control volume, is outside the source region so that the force is either uniform or zero. By partial integration and using the symmetry property (83) we can write the formal solution of the wave equation as:

p’(x, t) = — /-w fV(Vy • f(y, t))Go(x, ty, T)dVydr

= — /-^ fV f(y, r)VxGo(x, ty, T)dVydr.

As the integration over the source coordinates f does not interfere with the derivation by observer’s coordinates x we have:

p!(x, t) = —Vx •[ ( f(f, T )Go(x, ty, T )dVy dr. (93)

J-wJ V

For a compact source (kf << 1 and distances large compared to the dimension of the source region (x >> f), we have a dipole field:

p'(xt> *—Vx – (4ПХ X — f) ^ <94>

wher^ ^Jv f y, t — dVyj is the dipole strength.

An alternative way to find this expression is to consider the solution фі of the wave equation:

which leads to equation (93) because p'(x, t) = —V • ф.

While a monopole can be represented as a pulsating compact sphere, a dipole field is generated by a compact translating sphere. In a similar way we can obtain for the sound source found in the analogy of Lighthill:

In a compact source region this is a so-called quadrupole field.

An alternative approach to the multipole expansion of the source [Gold­stein (1976)] is to use a Taylor series expansion of the free space Green function around y = 0 in the general solution (87):

which, using the symmetry properties (85) of the Green’s function and the far field approximation, yields:

dVy і…

(100)

An intuitive interpretation of monopole, diopole and quadrupole on surface water waves is provided in Figure 4. Due to the oscillating momentum in the region between the two monopoles forming a dipole it is obvious that a dipole cannot exist without any force acting on the fluid. This force is needed to change the momentum. Thus, unsteady force induces a dipole radiation and a dipole radiation cannot exist without a force acting on the fluid.

For further reference we now consider the acoustic energy. Following the original approach of Kirchhoff, we start from the linearized mass and momentum equations:

Then we multiply the mass conservation law by p’/pQ and add the in-product of the momentum equation with the velocity v’, to find:

and the intensity I defined by:

f = p’v’ .

It should be noted that this derivation assumes that we did not neglect any relevant quadratic terms when using the linear approximation for the mass and momentum equation. This approach appears to be valid only for the case considered, i. e. of a uniform stagnant reference state [Morfey (1971), Landau and Lifchitz (1987), Pierce (1990), Myers (1991)].

Equation (66) clearly shows generating acoustic energy requires that a volume source should be placed at a position with a large acoustic pressure. A force needs an acoustic velocity to generate acoustic energy.

Considering a compact pulsating sphere near a rigid plane wave kh << 1 (Figure 3), we observe that due to reflection at the wall the amplitude of waves reaching an observer in the far field is roughly double the amplitude we would find in free space. Hence, the intensity is four times larger than in free space. However, the source only radiates into a half space, so that the time averaged power < P > generated by the source is doubled. This result can also be understood as a result of the doubling of the pressure fluctuations surrounding the source, due to reflection at the wall, which, following our energy corollary doubles the generated power. This implies

sphere placed near the wall: p’ 2 x 4nr2 <

PoC

that the radiated power is doubled compared to free field conditions. This example stresses the fact that the sound power does not only depend on the source but also on the surroundings of the source.

The homogeneous scalar wave equation (49) satisfies the plane wave solution:

p’ = F (n ■ x — c0t) (50)

with П as the unit vector in the direction of propagation. This can easily be verified for П = (1, 0,0), in which case the wave equation (45) reduces to:

1 d [1] [2]p’ d 2p’

c2 dt2 dx2

Using the chain rule we can verify that p’ = F(x — c0t) is a solution. The function F(x) is determined by initial and boundary conditions. Also p’ = G(x + c0t) is a solution, representing a wave propagating in the opposite direction П = ( —1, 0, 0). For harmonic waves with a frequency f we can write this solution with the complex notation as:

p = A exp
iw	A exp І	wt — к ■ x	(52)

where A is the complex amplitude, к = (w/c0)n the wave vector and w = 2nf. Substitution of the plane wave solution into the momentum equation (45)with f = 0 yields:

u’ = —— П. (53)

Po co

Another elementary solution is obtained by considering spherical symmetric waves emanating from a point at source y. The pressure field is then only a function of time and of distance r = x — y between the source position f and the observer’s position x. The mass conservation law and momentum equation reduce to:

dP + po d (r2 =0

dt r2 dr dr j

where v’ is the fluid velocity in the radial direction. Eliminating the velocity and the density p’ = p’/c^ yields:
which is satisfied by the one-dimensional d’Alembert solution for the prod­uct of pressure p’ and distance r:

p’ = – F(r — cot) . (57)

r

By using this equation, we actually assume “free field” conditions. We assume that there are only outgoing waves and no incoming (or reflected) waves converging towards the source. For harmonic waves equation (57) becomes in complex notation:

A

p’ = — exp [i(ut — kr)] (58)

r

with k = ш/co. The corresponding radial velocity is found by substitution in the momentum equation:

We observe that for large distances compared to the wave length kr >> -, the solution can locally be approximated by a plane wave with: p’ = p0c0v’r. In this so-called “far field” approximation we have:

dp’ – dp’

dr c0 dt

In the opposite limit of near field kr << – the velocity varies quadrati – cally with the distance r, which is typical for the incompressible flow from a point volume source. Whenever characteristic flow dimensions are small compared to the wave length we can neglect wave propagation. Such a flow is called a “compact” flow.

Using these results (58-59) we can now consider the sound radiated by a pulsating sphere of radius

a = ao + a exp(iwt)

where a/a0 << – and Fa/c0 << -. Substituting (6-) into (59) and using (58) we find:

. л A exp(—ikao)F – , .

iwa = ———————- – + —— (62)

Pocoao V ikao J

and

This result shows that in the limit kaQ << 1 for a given volume flux ampli­tude 4^a2 wa, the amplitude of the radiated sound wave increases linearly with the frequency. At low frequency the pulsating sphere is compact and is a very inefficient source of sound. In the opposite limit kaQ >> 1 the radiated amplitude is independent of the frequency.

1.3 Wave equation

Taking the time derivative of (44), subtracting the divergence of (45) and using (46) and (47) in order to eliminate p’ and s’, we obtain the wave equation for pressure perturbations:

As can be seen from this equation, the unsteady heat production is a source of sound, which is due to the dilatation of the fluid. This is in line with our common experience that turbulent flames are noisy. Also an unsteady non-uniform force field appears to be a source of sound. This is the sound source when considering the whistling of a cylinder placed with its axis normal to a uniform flow. Due to hydrodynamic instability, the wave be­hind the cylinder breaks down into a vortex street of alternating rotation direction. This periodic vortex shedding induces an unsteady force of the flow on the cylinder. The reaction force from the cylinder on the fluid is the source of sound. The so-called Aelonian tone will be discussed in section 7.2.

The next sections will focus on wave propagation and hence assume that Qw = 0 and f = 0. We therefore consider solutions of the homogeneous wave equation of d’Alembert

As the flow is isentropic the equation of state (16) reduces to p’ = c?0p’.

The key idea of Lighthill’s analogy [Lighthill (1952-54)] is to derive a wave equation starting from the exact mass conservation equation (5) and the momentum equation (9):

and

Taking the time derivative of (5) and subtracting from it, the divergence of (9) we obtain the exact equation:

d2p d2pVjVj = d^p _ d2Tij dfi

dt2 dxldxj dx2 dxldxj dxl

1 d 2

which is quite meaningless. By adding Ф on both sides and rearranging the terms, making use of the fact that we chose c0 to be a constant, we can write (38) as a wave equation:

1 = d2pVjVj – Tij dfi + p_

Cq dt2 dx2 dxldxj dxl dt2 c2 P)

This equation is still exact and still generally meaningless. We could have chosen co to be a millimetre per century or equal to the speed of light. In order to have a meaningful equation we now assume that we consider sound production by a flow bounded by a fluid displaying small perturbations from a uniform stagnant state with speed of sound equal to c0 (Figure 2). We furthermore define the perturbations in the pressure p’ = p — p0 and density p’ = p — p0 as deviations from the state (p0,p0) of this reference uniform

Figure 2. Sound sources and listener in the analogy of Lighthill

stagnant reference fluid. As the reference state is constant and uniform we can write (39) as:

1 dV _ Sy = d2pvjVj – Tij dfi + (P_ _ Л (40)

c° dt2 dx2 dxidxj dxi dt2 c2 P

We will see (section 4) that this equation describes the propagation of acous­tic waves in the uniform stagnant fluid when the right hand side of the equation (40) is negligible. In regions where the right hand side is not neg­ligible, it describes the generation of sound. However, because the equation of Lighthill is a single exact equation for many unknowns, we will not obtain any result without approximations. Lighthill has shown that these approxi­mations can best be introduced into an integral formulation of (40). We will now consider basic acoustic wave propagation allowing to understand some elementary aspects of the problem and to derive the integral formulation.

An interesting aspect of the analogy is that the sound source we find depends on the choice of the acoustic variable. Until now we have chosen pressure fluctuations p’ to describe the acoustic field. We could also have followed a similar procedure to obtain a wave equation for the density fluc­tuations p’. Starting from (38) we now subtract from both sides of the equation the term c0V2p’ to find:

In principle equations (40) and (41) are identical. However the pressure formulation (40) is most convenient when considering sound production by combustion processes in which the time-dependent combustion yields time- dependent fluctuations in the entropy. In contrast, when considering a flow in spatially non-uniform fluids with large variations in speed of sound the density formulation (41) will be the most suitable. An example of this is the sound generation by turbulence in bubbly liquids (section 5.2). In this case the sound production appears to be dominated by the effect of differences in the speed of sound.

Equation (41) is often written for convenience in terms of the stress tensor of Lighthill :

dV _ 2 dV = d2 Tij df

dt2 0 dx2 dxidxj dxi

where the stress tensor Tij is defined by:

Tij = pviVj – Tij + (p’ – clp’) Sij

Sound production by flows occurs at relatively high Reynolds numbers. When considering wave propagation in air at audio frequencies, we can neglect friction and heat transfer over distances of the order of the wave length. Neglecting friction, heat transfer and heat production, the energy equation (11) becomes:

(28)

The momentum equation (7) reduces to the Euler equation:

In this isentropic flow approximation dQ = 0, so that it follows from the first law of thermodynamics:

The conservation laws are complemented by empirical constitutive equa­tions. For simplicity we assume that the fluid is locally in a state close to thermodynamic equilibrium, so that we can express the internal energy in terms of two other state variables:

e = ^, s)

where s is the entropy per unit of mass. Using the thermodynamic equation:

we get the equations of state:

The speed of sound c is defined by:

In most applications we will consider an ideal gas for which:

de = cv dT

with cv the specific heat capacity at constant volume. For an ideal gas this is a function of the temperature only. This further implies:

and

c =. [YP (20)

V P

with R = cp — cv the specific gas constant, 7 = cp/cv the Poisson ratio and cp is the specific heat capacity at constant pressure. By definition:

where the specific enthalpy is defined by:

Assuming local thermodynamic equilibrium, fluxes are linear functions of the flow variables. For the heat flux we use the law of Fourier:

q = —KVT,

where K is the heat conductivity. The viscous stress tensor is defined by:

with Sij the Kronecker delta, equal to unity for i = j and otherwise zero. The viscous stress tensor is described for a so-called Newtonian fluid in terms of the dynamic viscosity n and the bulk viscosity ц:

Tij 2n (Dij 3Dkk &ij^ + PDkk$ij (26)

1.2 Boundary conditions

The boundary conditions corresponding to the continuum assumption and the local thermodynamic equilibrium are, for a solid impermeable wall with velocity vw and temperature Tw : v = vw and T = Tw.

1.1 Conservation laws

The conservation of mass for an infinitesimal material element of density p and volume V is given in the continuum approximation by [Batchelor (1967), Landau and Lifchitz (1987), Kundu (1990)]:

In integral form this equation becomes:

in which, V is a fixed control volume delimited by the surface S with outer unit normal П (Figure 1). The second law of Newton applied to an infinites-

imal material element is:

where f is the density of a force field acting on the bulk of the fluid and P is the stress tensor representing the surface interaction between the particle and its surroundings. Using the definition of the convective derivative (2) and the mass conservation law (5) we obtain the conservation form of the momentum equation:

where e is the internal energy of the fluid per unit of mass, v = |v| ,q the heat flux and Qw the energy production per unit volume.