Computational Aeroacoustics

Transcription

1 Computational Aeroacoustics Simple Sources and Lighthill s Analogy Gwénaël Gabard Institute of Sound and Vibration Research University of Southampton, UK gabard@soton.ac.uk ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Simple Sources and Lighthill s Analogy 1 / 45

2 Introduction Simple sources: The effect of source motion relative to the observer and relative to the mean flow (convective amplification, Doppler shift). Different categories of physical sources of sound (mass sources, external forces, external stresses). Boundary forcing: the wavy wall Lighthill s analogy: Derivation and underlying assumptions. How is it used? Scaling laws. Noise from turbulence. ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Simple Sources and Lighthill s Analogy 2 / 45

3 Effect of source motion ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Simple Sources and Lighthill s Analogy 3 / 45

4 A moving point source We consider a moving point source with position x s(t) and strength s(t) in a uniform mean flow u 0. 1 c 2 0 D 2 0p Dt 2 2 p = s(t)δ[x x s(t)]. (1) The material derivative in the mean flow: D 0 Dt = t + u0. (x, t) observer (x s, τ) v s u 0 Γ s point source With this simple test case we can identify the effects of: flow velocity relative to the observer (convected propagation) source motion relative to the observer (Doppler shift) source motion relative to the flow (convective amplification) convected propagation (u 0 ) observer fluid Doppler shift (v s ) convective amplification (v s u 0 ) source ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Simple Sources and Lighthill s Analogy 4 / 45

5 Acoustic pulse in a uniform mean flow To describe an acoustic pulse propagating in a uniform mean flow we need the Green s function for the convected wave equation. It can be defined as 1 c 2 0 D 2 0G Dt 2 2 xg = δ(x y)δ(t τ). (2) We can use the Green s function of the standard wave equation to obtain: G(x, t y, τ) = G(x y, t τ) = δ(t τ r /c0) 4π r, (3) where r is the distance between the observer and the centre of the acoustic pulse: r = x y u 0(t τ). (4) ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Simple Sources and Lighthill s Analogy 5 / 45

6 Acoustic pulse in a uniform mean flow r = c 0 (t τ) (x, t) observer u 0 (y, τ) u 0 (t τ) point source The centre of the spherical wave front moves with the mean flow and is located at y + u 0(t τ). The wave front expands spherically from its centre at a velocity c 0 and so the spherical spreading is measured by r = c 0(t τ) (and not by x y ). The direction of the wave front seen by the observer at x is r/ r. ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Simple Sources and Lighthill s Analogy 6 / 45

7 A moving point source The general solution to this problem can be written (subsonic case): p(x, t) = s(τ) 4π r D. (5) The spherical spreading is based on the distance: r = x x s(τ) u 0(t τ). (6) The retarded time τ is obtained by solving (x, t) observer (x s, τ) v s u 0 Γ s point source The Doppler factor D is given by x x s(τ) u 0(t τ) c 0(t τ) = 0. (7) D = 1 vs(τ) u0 r c 0 r. (8) ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Simple Sources and Lighthill s Analogy 7 / 45

8 A moving point source If we now compare this expression p(x, t) = s[τ(x, t)] 4π r D with the sound field for a fixed source at x s and no mean flow we have p(x, t) = s(t x xs /c0) 4π x x s, (9). (10) The spherical spreading is adjusted using u 0(t τ) to include the convection effect of the mean flow. A Doppler factor associated with the source velocity relative to the mean flow is now present. The retarded time is different. It was directly given by t x x s /c 0 in the standard case. But now we have to solve x x s(τ) u 0(t τ) c 0(t τ) = 0, (11) which can be particularly difficult depending on the trajectory x s(τ). ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Simple Sources and Lighthill s Analogy 8 / 45

9 Convective amplification The Doppler factor can be written D = 1 M r cos(θ) (12) M r = v s u 0 /c 0 is the Mach number of the source velocity relative to the mean flow. If the source is moving with the mean flow then D = 1. θ is the angle between v s u 0 and the wave front direction r/ r (with no mean flow this reduces to the angle between the observer and the source velocity). The factor 1/ D changes the amplitude of the radiated sound and depends on the direction. The change in directivity introduced by the motion of the source relative to the ambient fluid is called the convective amplification: If cos(θ) is positive then the radiated sound is stronger (with no mean flow this is when the source is moving towards the observer). If cos(θ) is negative then the radiated sound is weaker (with no mean flow this is when the source is moving away from the observer). ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Simple Sources and Lighthill s Analogy 9 / 45

10 Convective amplification Example of convective amplification for M r = 0.2, 0.4, 0.6, Log 10 (1/ D ) ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Simple Sources and Lighthill s Analogy 10 / 45

11 Doppler shift To understand the Doppler shift it is easier to consider a time-harmonic source s(t) = e iωt. Then the solution is p(x) = exp[iωτ(x, t)] 4π r D. (13) If the frequency is high (geometrical acoustics) we can, at least locally, represent the solution as a plane wave with effective frequency ˆω and effective wavenumber ˆk: p(x) exp(iˆωt iˆk x) 4π r D, ˆω = ω τ t = D0 D ω, ˆk = ω xτ = k0 r D r, (14) where D 0 = 1 + (u 0 r)/(c 0 r ). ˆω is the effective angular frequency perceived by the observer. This change in frequency is the well-known Doppler shift. There is no Doppler shift (ˆω = ω) only when the source is stationary relative to the observer (v s = 0) or when it moves perpendicularly to the observer. ˆk is the effective wavenumber seen by the observer. This wavenumber is unchanged only when the source is moving with the ambient medium (v s = u 0). ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Simple Sources and Lighthill s Analogy 11 / 45

12 Example If we assume no mean flow and a source moving along the positive x axis then: The effective frequency is ˆω = ω/d which is the standard expression for the Doppler shift. The wavenumber is also modified by the same factor ˆk = k 0/D. The convective amplification is also present. 1 Larger wavelength (ˆk), smaller amplitude (1/ D ), lower frequency. 2 Smaller wavelength (ˆk), larger amplitude (1/ D ), higher frequency. 3 Standard wavelength (k 0), standard amplitude (D = 1), standard frequency. ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Simple Sources and Lighthill s Analogy 12 / 45

13 Example If we assume a fixed source in a uniform mean flow along the positive x direction then: The effective frequency is the same as the source (ˆω = ω) and there is no Doppler frequency shift. The wavelength is also modified by the same factor ˆk = k 0/D = k 0/D 0. The convective amplification is also present. 1 Smaller wavelength (ˆk), larger amplitude (1/ D 0 ), standard frequency. 2 Larger wavelength (ˆk), smaller amplitude (1/ D 0 ), standard frequency. 3 Standard wavelength (k 0), standard amplitude (D = 1), standard frequency. ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Simple Sources and Lighthill s Analogy 13 / 45

14 Supersonic flows Supersonic sources: When the mean flow and/or the source is/are supersonic the sound radiation changes significantly. The wave front can only reach within the Mach cone. For more details, refer to Chapter 14 by A.P. Dowling in Crighton et al. (1992). (x, t) observer r = c 0 (t τ) u 0 (y, τ) point source u 0 (t τ) ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Simple Sources and Lighthill s Analogy 14 / 45

15 Source mechanisms ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Simple Sources and Lighthill s Analogy 15 / 45

16 Different sources We can introduce various kinds of sources to drive the acoustic field: A distributed unsteady mass source m. An external, unsteady force f. External stresses T ij. Or an external, unsteady heat source h (e.g. combustion). These sources contributes to the linearised conservation of mass and momentum and to the thermodynamic equation of state: ρ 0 D 0u Dt D 0ρ Dt + ρ0 u = m, (mass conservation) (15) + p = f Tij x j, (momentum conservation) (16) p = c 2 0 ρ + ρ 0(γ 1)h, (equation of state) (17) ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Simple Sources and Lighthill s Analogy 16 / 45

17 Different sources We can rewrite all this into a convected wave equation with a uniform mean flow: 1 c 2 0 D 2 0p Dt 2 2 p = D0m + Dt ρ0(γ 1) D 2 0h c0 2 Dt f + 2 Tij x i x j. (18) The mass source acts as a monopole of strength D 0m/Dt. To induce an acoustic field the source should be unsteady when seen by an observer moving with the mean flow. The force field induces a dipole source distribution with direction aligned with f (note that a purely rotational force will not excite the sound field). The external stresses T ij act as quadrupoles. The heat source generates a monopole source distribution. ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Simple Sources and Lighthill s Analogy 17 / 45

18 Moving mass source We can follow the same analysis as above for an unsteady mass source q(t) by setting s(x, t) = D0 {q(t)δ[x xs(t)]}. (19) Dt For an arbitrary source trajectory x s(t), the far-field solution is given by [ 1 p(x, t) q (τ) + Γs(τ) ] r q(τ). (20) 4πD D r c 0 r D We can identify two mechanisms that contribute to the acoustic far field: The source unsteadiness is contributing as a monopole. The source acceleration is contributing as a dipole aligned with Γ s(τ). Compared to the basic source, the convective amplification is stronger ( 1/D 2 for the monopole and 1/D 3 for the dipole). ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Simple Sources and Lighthill s Analogy 18 / 45

19 External force We can follow the same analysis for a force f(t) at a moving point x s(t) by setting s(x, t) = {f(t)δ[x x s(t)]}. (21) For an arbitrary source trajectory x s(t), the far-field solution is given by [ ( ) ] 1 r Γs(τ) p(x, t) 4πc 0D D r r f r f(τ) (τ) + c 0 r D. (22) We can identify two mechanisms that contribute to the acoustic far field: The source unsteadiness is contributing as a dipole aligned with f (τ). The source acceleration is contributing as a quadrupole aligned with f(τ) and Γ s(τ). Compared to the basic source, the convective amplification is stronger ( 1/D 2 for the dipole and 1/D 3 for the quadrupole). ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Simple Sources and Lighthill s Analogy 19 / 45

20 Interpretation These different sources can be understood in terms of external forces on a small fluid element: Monopole: Small expanding/contracting sphere. Zero net force. Contraction/expansion of the fluid element. Dipole: Oscillating piston. Non-zero net force. Translation of the fluid element. ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Simple Sources and Lighthill s Analogy 20 / 45

21 Interpretation Quadrupole: Zero net force. Shear stresses: lateral quadrupoles. Normal stresses: longitudinal quadrupoles. ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Simple Sources and Lighthill s Analogy 21 / 45

22 The wavy wall ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Simple Sources and Lighthill s Analogy 22 / 45

23 The wavy wall In 2D if we consider a half space (x 2 > 0) with a steady medium with sound speed c 0. At the wall x 2 = 0 the pressure is a travelling wave with frequency ω and velocity V c: x 2 wave fronts p(x 1, x 2 = 0) = A exp[iω(t x 1/V c)]. (23) θ V c x 1 The sound field generated in the fluid is a plane wave of the form: p(x 1, x 2) = A exp(iωt ik 1x 1 ik 2x 2), (24) where k 1 = ω/v c is the wavenumber of the wall oscillations. The transverse wavenumber k 2 in the fluid is given by: k 2 = ω c 0 1 1/M 2 c, (25) where M c = V c/c 0 is the Mach number of the wall oscillation velocity. ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Simple Sources and Lighthill s Analogy 23 / 45

24 The wavy wall Depending on the Mach number M c = V c/c 0 the transverse wavenumber k 2 = ω c 0 1 1/M 2 c (26) x 2 wave fronts can be either real or imaginary. θ V c x 1 If the wall oscillations are subsonic (M c < 1 or V c < c 0) then k 2 is imaginary. The wall oscillations fail to radiate efficiently into the fluid and they only induce evanescent waves decaying exponentially away from the wall. If the wall oscillations are supersonic (M c > 1 or V c > c 0) then k 2 is real. The wall oscillations generate a plane wave (a Mach wave) that radiates at an angle θ given by cos θ = 1/M c. Any wall oscillations can be decomposed as a sum of travelling waves (each with a different frequency ω and velocity V c). Only the supersonic components will be able to radiate sound to the far field. ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Simple Sources and Lighthill s Analogy 24 / 45

25 Mach waves The jet shear layer represents the boundary that separates the jet from the ambient medium. If the oscillations in the shear layer of the jet (large-scale structures, instability modes) are convected at a supersonic velocity V c relative to the ambient sound speed c 0, then Mach waves will be radiated. From Van Dyke (1982) p ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Simple Sources and Lighthill s Analogy 25 / 45

26 Wave packets Spatial modulation: oscillations that are growing and decaying. Even if the convection velocity V c is subsonic, part of the energy of the wave packet travels supersonically and will radiate noise. The presence of the envelope induces a very directive sound field. (Crighton and Huerre, 1990; Obrist, 2009; Reba et al., 2010; Cavalieri et al., 2011) V c wavepacket envelope travelling oscillations x 1 ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Simple Sources and Lighthill s Analogy 26 / 45

27 Lighthill s analogy ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Simple Sources and Lighthill s Analogy 27 / 45

28 Lighthill s equation The starting point is the Navier-Stokes for a compressible viscous fluid: ρu i t ρ t + ρui x i = 0, (mass conservation) (27) + x j (ρu iu j + pδ ij τ ij) = 0, (momentum conservation) (28) where we have introduced the Kronecker symbol δ ij and the viscous stress tensor (µ is the shear viscosity) ( ) ui τ ij = µ + uj 2 u k δ ij. (29) x j x i 3 x k If we take the time derivative of (27) and the divergence of (28) and we can subtract the two to get 2 ρ t = 2 (ρu iu 2 j + pδ ij τ ij). (30) x i x j Finally we use the fact that 2 ρ/ t 2 = 2 ρ / t 2 where ρ = ρ ρ 0, and we subtract c ρ on either sides to obtain... ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Simple Sources and Lighthill s Analogy 28 / 45

29 Lighthill s equation The Lighthill (1952) equation: 2 ρ t 2 c2 0 2 ρ = 2 T ij x i x j, with T ij = ρu iu j + (p c 2 0 ρ )δ ij τ ij, (31) It is an exact rearrangement of the general Navier-Stokes equations. The Lighthill tensor T ij represents a distribution of quadrupole sources: ρu i u j is called the Reynolds stresses and represents non-linear effects p c0 2ρ is associated with the non-isentropic effects τ ij describes the effects of viscosity This is precisely what is left out when we model sound propagation. We choose to represent sound propagation with the standard wave equation and everything else is moved to the right-hand side and treated as sources. This is an acoustic analogy so these sources are (only) equivalent sources that will generate exactly the same sound field as if we had solved the full Navier-Stokes equations. Technically c 0 is arbitrary but to provide a meaningful model it is chosen as the sound speed in the fluid at rest. ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Simple Sources and Lighthill s Analogy 29 / 45

30 Using Lighthill s analogy A significant advantage of Lighthill s analogy is that the solution of the wave equation can be directly written using Green s formula: ρ (x, t) = 1 ( ) 2 T ij x y dy y, t 4πc0 2 y i y j c 0 x y. (32) We can use (32) to compute the density fluctuation ρ everywhere, provided that we know the sources terms. But ρ is also part of the sources... (x, t) far-field observer c 0 We have to split between a (small) source region (the near field) and the surrounding acoustic far field. (y, τ) source region ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Simple Sources and Lighthill s Analogy 30 / 45

31 Using Lighthill s analogy We rely on the basic property that acoustic waves are disturbances that can propagate far away from the flow features where they were generated. For instance for low Mach numbers the hydrodynamic velocity decays like 1/r 3 whereas acoustic waves decays like 1/r. The source terms are only significant in the source region and we can obtain the flow there (or a sufficient approximation) by other means (measurements, modelling, numerical simulations). In the acoustic far field we require that all the source terms are negligible so the only fluctuations are associated with acoustic waves. No feedback of sound on the flow field. ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Simple Sources and Lighthill s Analogy 31 / 45

32 Practical considerations In the geometric far field the general solution can be approximated as p (x, t) 1 ( 2 T ij y, t x ) + x y dy. (33) 4π x x i x j c 0 c 0 x Note that it is only in the far field that we can use p = c0 2 ρ. And we can also use time derivatives instead ( p 1 x ix j 2 T ij (x, t) 4πc0 2 x y, t x ) + x y dy. (34) x 2 t 2 c 0 c 0 x The use of time derivatives is often preferred as it is easier to implement and leads to more accurate predictions (Bastin et al., 1997). The source region often has to be truncated (Martínez-Lera and Schram, 2008). Absence of solid surfaces... ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Simple Sources and Lighthill s Analogy 32 / 45

33 Entropy sources A typical situation when entropy sources are significant is when there is an unsteady heat source in the fluid, such as the heat generated by combustion (a turbulent flame for instance). In that case we can write p c 2 0 ρ = ρ 0(γ 1)h, (35) where h is the unsteady heat added to the system. The far-field sound can then be written as ( p ρ0(γ 1) (x, t) 4πc0 2 x Q y, t x ) + x y dy, (36) t c 0 c 0 x where Q = h / t is the unsteady rate of heat release. This is a distribution of monopole sources. When present, and for low Mach number flows, this source tends to dominate the turbulence noise. (Crighton et al., 1992, Chapter 11 by A. Dowling) From Chen et al. in Comput. Sci. Discov. 2 (2009) ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Simple Sources and Lighthill s Analogy 33 / 45

34 Viscous sources According to Lighthill s analogy, viscosity also acts as a source of sound. We can rewrite the contribution from the viscous stresses as follows 2 τ ij x i x j = 4µ 3 2 Θ, with Θ = u, (37) where Θ is the rate of dilatation of the fluid. And the acoustic far field can be calculated using: p (x, t) µ ( 2 3πc0 2 x Θ y, t x ) + x y dy. (38) t 2 c 0 c 0 x The viscous sources can be described as a distribution of isotropic quadrupoles (they have the same directivity as monopoles) associated with the rate of expansion of the fluid. The viscous sources are often negligible (except in some special cases, e.g. the frequency is very high). In fact the term (4µ/3) 2 Θ represents sound absorption in a viscous fluid and could be considered as part of the propagation effects. (Obermeier, 1985; Morfey, 2003; Morfey et al., 2012) ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Simple Sources and Lighthill s Analogy 34 / 45

35 Radiating sources If we consider a time-harmonic source distribution s(x, ω), and apply the geometric far-field approximation, the solution can be written: p(x, t) = eiω(t x /c 0) ŝ 4π x ( k 0 x x, ω ). s(x, ω) x 2 θ x 1 x Retarded time t x /c 0. Spherical spreading. The only part of the source radiating toward the observer in the far field is the wavenumber k 0x/ x. ŝ(k, ω) k 2 The radiating wavenumber components are found on the sonic radius k = ω/c 0. The wave equation acts as a filter selecting only the part of the source on the sonic radius. θ k = k 0 x x k 1 ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Simple Sources and Lighthill s Analogy 35 / 45

36 Radiating sources The Lighthill source term contains a wide range of frequencies and wavenumber components, but only the sonic part will radiate to the far field. Consider the wavenumber components aligned with the observed (k = kx/ x ). Most of the energy is convected with the mean flow and is found along a line which is found for ω = ku c cos θ. A small part of the energy will radiate sound to the far field. It lies on the line ω = kc 0. ω sonic line convection line k ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Simple Sources and Lighthill s Analogy 36 / 45

37 Scaling laws We can estimate the sound amplitude and the total acoustic power by using dimensional analysis. To that end we introduce reference values: Source: Velocity U Length L Sound field: Velocity c 0 Length λ Density ρ 0 Compact source assumption: Density ρ 0 A source can be treated as a point source if it is much smaller than the acoustic wavelength: L λ. In that case the source and the sound field follow the same scaling for their reference time scales: L/U = λ/c 0 and we have Note that we do not consider the source motion. L λ = U c 0 = M 1. (39) ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Simple Sources and Lighthill s Analogy 37 / 45

38 Scaling laws For the non-linear source term we can use the following estimates T ij = ρu iu j ρ 0U 2 c 0 t λ = U L dy L 3 The acoustic density in the far field: ρ 1 x ix j 2 (x, t) 4πc0 4 x T x 2 t 2 ij (y,...) dy. (40) Note that x = x remains an independent parameter. Based on this dimensional analysis we get the following scaling law for the acoustic density, as well as for the acoustic intensity (using I c 3 0 ρ 2 /ρ 0) ρ ρ 0M 4 L x, I ρ0c3 0 M 8 L 2 x 2. (41) This is the well-known 8 th power law derived by Lighthill. Scaling law only provides trends but they can be used to identify the parameters driving aerodynamic noise generation. ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Simple Sources and Lighthill s Analogy 38 / 45

39 Scaling laws Distributed sources: The scaling laws we have obtained are only valid for compact sources. When the source distribution is not acoustically compact, its efficiency is reduced by the interference between different parts of the source region. Refer to Crighton (1975). ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Simple Sources and Lighthill s Analogy 39 / 45

40 Noise from turbulence For random sound source such as turbulence it is preferable to use a statistical description of the radiated sound field. Based on Lighthill s theory, one can obtain the following expression for the far-field sound intensity radiated by a region of turbulence: I(x) 1 x ix jx k x l 16π 2 ρ 0c0 5 x 2 x 4 where R ijkl is a correlation tensor: ( 4 R ijkl y, η, x η τ 4 c 0 x ) dydη, (42) R ijkl (y, η, τ) = T ij(y, t)t kl (y + η, t + τ). (43) One chooses a model for this correlation (typically a Gaussian) and defines several parameters such as length and time scales. These parameters are estimated from predictions of turbulence kinetic energy and dissipation rate for instance, typically using RANS simulations. Sound refraction by the mean flow is not included. (Tam and Auriault, 1999; Azarpeyvand and Self, 2009) ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Simple Sources and Lighthill s Analogy 40 / 45

41 Noise from turbulence Based on Ffowcs Williams expression, the sound intensity scales as follows: I(θ) ρ2 J UJ 3 MJ 5 L 2 ρ 0 D 5 x, with D = [(1 2 Mc cos θ)2 + m 2 ] 1/2. (44) D is a generalised Doppler factor that includes the fact that the sources in the jet (the turbulent eddies) have finite sizes. The Mach number m is associated with the time scale of the turbulence (m is much smaller than M c). In this model, the convective amplification is governed by a factor 1/D 5. For supersonic convection velocity (V c > c 0) Mach waves are radiated at an angle θ = cos 1 (1/M c). In that direction we get I(x) ρ2 JU 3 J ρ 0 L 2 x 2 (45) which is proportional to the jet mechanical power ( ρ J U 3 J ). Chapter 11 by J.E. Ffowcs Williams in Crighton et al. (1992) ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Simple Sources and Lighthill s Analogy 41 / 45

42 Source (in)efficiency The mechanical power available in a jet can be estimated by Aρ J U 3 J /2 so we can estimate the efficiency of the jet as a source of sound: This is only valid for low Mach numbers. P sound P flow M 5. (46) This shows that aerodynamic noise generation is a very inefficient process and only a small fraction of the flow energy is converted into acoustic energy. This also shows that the acoustic waves represent a minute by-product of the flow and that, if not careful, an approximation (numerical error, or simplifying assumption for a theoretical model) can be disastrous and lead to errors several orders of magnitude larger than the actual sound field amplitude. For supersonic jets the source efficiency is much higher but still remains of the order of 1%. ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Simple Sources and Lighthill s Analogy 42 / 45

43 Further reading The effect of source motion is also discussed in Chapter 14 by A.P. Dowling in Crighton et al. (1992). You can find detailed discussions on Lighthill s analogy in the review paper by Crighton (1975) and the books by Goldstein (1976), Dowling and Ffowcs Williams (1983), Crighton et al. (1992) and Howe (2003). ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Simple Sources and Lighthill s Analogy 43 / 45

44 References I Azarpeyvand, M and R. Self (2009). Improved jet noise modeling using a new time-scale. In: The Journal of the Acoustical Society of America 126.3, pp Bastin, F., P. Lafon and S. Candel (1997). Computation of jet mixing noise due to coherent structures: the plane jet case. In: Journal of Fluid Mechanics 335, pp Cavalieri, A. V. et al. (2011). Jittering wave-packet models for subsonic jet noise. In: Journal of Sound and Vibration , pp Crighton, D. (1975). Basic principles of aerodynamic noise generation. In: Progress in Aerospace Sciences 16.1, pp Crighton, D. and P. Huerre (1990). Shear-layer pressure fluctuations and superdirective acoustic sources. In: Journal of Fluid Mechanics 220, pp Crighton, D. et al. (1992). Modern methods in analytical acoustics. London: Springer-Verlag. Dowling, A. and J. Ffowcs Williams (1983). Sound and sources of sound. Ellis Horwood Limited. Goldstein, M. (1976). Aeroacoustics. McGraw-Hill. Howe, M. S. (2003). Theory of vortex sound. Vol. 33. Cambridge University Press. Lighthill, M. (1952). On sound generated aerodynamically I. General theory. In: Proc. Roy. Soc. London 211, pp Martínez-Lera, P. and C. Schram (2008). Correction techniques for the truncation of the source field in acoustic analogies. In: The Journal of the Acoustical Society of America 124.6, pp Morfey, C. (2003). The role of viscosity in aerodynamic sound generation. In: International Journal of Aeroacoustics 2.3, pp Morfey, C., S. Sorokin and G Gabard (2012). The effects of viscosity on sound radiation near solid surfaces. In: Journal of Fluid Mechanics 690, pp ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Simple Sources and Lighthill s Analogy 44 / 45

45 References II Obermeier, F. (1985). Aerodynamic sound generation caused by viscous processes. In: Journal of Sound and Vibration 99.1, pp Obrist, D. (2009). Directivity of acoustic emissions from wave packets to the far field. In: Journal of Fluid Mechanics 640, pp Reba, R., S. Narayanan and T. Colonius (2010). Wave-packet models for large-scale mixing noise. In: International Journal of Aeroacoustics 9.4, pp Tam, C. and L. Auriault (1999). Jet mixing noise from fine-scale turbulence. In: AIAA Journal 37.2, pp Van Dyke, M. (1982). An Album of fluid Motion. Parabolic Press. ISVR, University of Southampton, UK ERCOFTAC Computational Aeroacoustics Simple Sources and Lighthill s Analogy 45 / 45