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LIGHTHILL'S EQUATION


Lighthill (1952) rearranged the Navier-Stokes equations, which govern the Flow of a Compressible Viscous Fluid , into an Inhomogeneous Wave Equation , thereby making an analogy between Fluid Mechanics and Acoustics .

The first equation of interest is the conservation of mass equation, which reads

: rac{\partial ho}{\partial t} +
abla\cdot\left( ho\mathbf{v} ight)= rac{D ho}{D t} + ho
abla\cdot\mathbf{v}= 0,

where ho and \mathbf{v} represent the density and velocity of the fluid, which depend on space and time, and D/Dt is the Substantial Derivative .

Next is the conservation of momentum equation, which is given by

:{ ho} rac{\partial \mathbf{v}}{\partial t}+{ ho(\mathbf{v}\cdot
abla)\mathbf{v}} = -
abla p+
abla\cdot\sigma,

where p is the thermodynamic Pressure , and \sigma is the viscous (or traceless) part of the Stress Tensor .

Now, multiplying the conservation of mass equation by \mathbf{v} and adding it to the conservation of momentum equation gives

: rac{\partial}{\partial t}\left( ho\mathbf{v} ight) +
abla\cdot( ho\mathbf{v}\otimes\mathbf{v}) = -
abla p +
abla\cdot\sigma.

Note that \mathbf{v}\otimes\mathbf{v} is a Tensor (see also Tensor Product ). Differentiating the conservation of mass equation with respect to time, taking the Divergence of the conservation of momentum equation and subtracting the latter from the former, we arrive at

: rac{\partial^2 ho}{\partial t^2} -
abla^2 p +
abla\cdot
abla\cdot\sigma =
abla\cdot
abla\cdot( ho\mathbf{v}\otimes\mathbf{v}).

Subtracting c_0^2
abla^2 ho, where c_0 is the Speed Of Sound in the medium in its equilibrium (or quiescent) state, from both sides of the last equation and rearranging it results in

: rac{\partial^2 ho}{\partial t^2}-c^2_0
abla^2 ho =
abla\cdot\left[
abla\cdot( ho\mathbf{v}\otimes\mathbf{v})-
abla\cdot\sigma +
abla p-c^2_0
abla ho ight],

which is equivalent to

: rac{\partial^2 ho}{\partial t^2}-c^2_0
abla^2 ho=(
abla\otimes
abla) :\left[ ho\mathbf{v}\otimes\mathbf{v} - \sigma + (p-c^2_0 ho)\mathbb{I} ight],
where \mathbb{I} is the operator.

The above equation is the celebrated Lighthill equation of aeroacoustics. It is a Wave Equation with a source term on the right-hand side, i.e. an inhomogeneous wave equation. The argument of the "double-divergence operator" on the right-hand side of last equation, i.e. ho\mathbf{v}\otimes\mathbf{v}-\sigma+(p-c^2_0 ho)\mathbb{I}, is the so-called ''Lighthill turbulence stress tensor for the acoustic field'', and it is commonly denoted by T.

Using Einstein Notation , Lighthill’s equation can be written as

: rac{\partial^2 ho}{\partial t^2}-c^2_0
  • )


where

:T_{ij}= ho v_i v_j - \sigma_{ij} + (p- c^2_0 ho)\delta_{ij},

and \delta_{ij} is the Kronecker Delta . Each of the acoustic source terms, i.e. terms in T_{ij}, may play a significant role in the generation of noise depending upon flow conditions considered.

In practice, it is customary to neglect the effects Viscosity of the fluid, i.e. one takes \sigma=0, because it is generally accepted that the effects of the latter on noise generation, in most situations, are orders of magnitude smaller than those due to the other terms. Lighthill (1952) provides an in-depth discussion of this matter.

In aeroacoustic studies, both theoretical and computational efforts are made to solve for the acoustic source terms in Lighthill's equation in order to make statements regarding the relevant aerodynamic noise generation mechanisms present.

Finally, it is important to realize that Lighthill's equation is exact in the sense that no approximations of any kind have been made in its derivation.


RELATED MODEL EQUATIONS

In their classical text on Fluid Mechanics , Landau and Lifshitz (1987) derive an aeroacoustic equation analogous to Lighthill's (i.e., an equation for sound generated by " Turbulent " fluid motion) but for the Incompressible Flow of an Inviscid fluid. The inhomogeneous wave equation that they obtain is for the ''pressure'' p rather than for the density ho of the fluid. Furthermore, unlike Lighthill's equation, Landau and Lifshitz's equation is not exact; it is an approximation.

  • ) \, we obtain the equation


: rac{1}{c_0^2} rac{\partial^2 p}{\partial t^2}-
abla^2p= rac{\partial^2 ilde{T}_{ij}}{\partial x_i \partial x_j},\quad ext{where}\quad ilde{T}_{ij} = ho v_i v_j.

And for the case when the fluid is indeed incompressible, i.e. ho= ho_0 (for some positive constant ho_0) everywhere, then we obtain exactly the equation given in Landau and Lifshitz (1987), namely

: rac{1}{c_0^2} rac{\partial^2 p}{\partial t^2}-
abla^2p= ho_0 rac{\partial^2\hat{T}_{ij}}{\partial x_i \partial x_j},\quad ext{where}\quad\hat{T}_{ij} = v_i v_j.


Of course, one might wonder whether we are justified in assuming that p-p_0=c_0^2( ho- ho_0). The answer is in affirmative, if the flow satisfies certain basic assumptions. In particular, if ho \ll ho_0 and p \ll p_0, then the assumed relation follows directly from the ''linear'' theory of sound waves (see, e.g., the Linearized Euler Equations and the Acoustic Wave Equation ). In fact, the approximate relation between p and ho that we assumed is just a Linear Approximation to the generic Barotropic Equation Of State of the fluid.

However, even after the above deliberations, it is still not clear whether one is justified in using an inherently ''linear'' relation to simplify a ''nonlinear'' wave equation. Nevertheless, it is a very common practice in Nonlinear Acoustics as the textbooks on the subject show: e.g., Naugolnykh and Ostrovsky (1998) and Hamilton and Morfey (1998).


REFERENCES

  • M. J. Lighthill, "On Sound Generated Aerodynamically. I. General Theory," ''Proc. R. Soc. Lond. A'' 211 (1952) pp. 564-587. This article on JSTOR .

  • M. J. Lighthill, "On Sound Generated Aerodynamically. II. Turbulence as a Source of Sound," ''Proc. R. Soc. Lond. A'' 222 (1954) pp. 1-32. This article on JSTOR .

  • L. D. Landau and E. M. Lifshitz, ''Fluid Mechanics'' 2ed., Course of Theoretical Physics vol. 6, Butterworth-Heinemann (1987) §75. ISBN 0750627670, Preview from Amazon .

  • K. Naugolnykh and L. Ostrovsky, ''Nonlinear Wave Processes in Acoustics'', Cambridge Texts in Applied Mathematics vol. 9, Cambridge University Press (1998) chap. 1. ISBN 052139984X, Preview from Google .

  • M. F. Hamilton and C. L. Morfey, "Model Equations," ''Nonlinear Acoustics'', eds. M. F. Hamilton and D. T. Blackstock, Academic Press (1998) chap. 3. ISBN 0123218608, Preview from Google .



EXTERNAL REFERENCES