Lighthill Turbulence Stress Tensor

Aeroacoustics is a branch of acoustics that studies noise generation via either turbulent fluid motion or aerodynamic forces interacting with surfaces. Noise generation can also be associated with periodically varying flows. A notable example of this phenomenon is the Aeolian tones produced by wind blowing over fixed objects.

Although no complete scientific theory of the generation of noise by aerodynamic flows has been established, most practical aeroacoustic analysis relies upon the so-called ''aeroacoustic analogy'', proposed by Sir James Lighthill in the 1950s while at the University of Manchester. whereby the governing equations of motion of the fluid are coerced into a form reminiscent of the wave equation of "classical" (i.e. linear) acoustics in the left-hand side with the remaining terms as sources in the right-hand side.

History

The modern discipline of aeroacoustics can be said to have originated with the first publication of Lighthill in the early 1950s, when noise generation associated with the jet engine was beginning to be placed under scientific scrutiny.

Lighthill's equation

Lighthill rearranged the Navier–Stokes equations, which govern the flow of a compressible viscous fluid, into an inhomogeneous wave equation, thereby making a connection between fluid mechanics and acoustics. This is often called "Lighthill's analogy" because it presents a model for the acoustic field that is not, strictly speaking, based on the physics of flow-induced/generated noise, but rather on the analogy of how they might be represented through the governing equations of a compressible fluid.

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

:$\frac + \nabla\cdot\left(\rho\mathbf\right)=\frac + \rho\nabla\cdot\mathbf= 0,$

where $\rho$ and $\mathbf$ 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

:$\frac+ = -\nabla p+\nabla\cdot\sigma,$

where $p$ is the thermodynamic pressure, and $\sigma$ is the viscous (or traceless) part of the stress tensor from the Navier–Stokes equations.

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

:$\frac\left(\rho\mathbf\right) + \nabla\cdot(\rho\mathbf\otimes\mathbf) = -\nabla p + \nabla\cdot\sigma.$

Note that $\mathbf\otimes\mathbf$ is a tensor (see also tensor product). Differentiating the conservation of mass equation with respect to time, taking the divergence of the last equation and subtracting the latter from the former, we arrive at

:$\frac - \nabla^2 p + \nabla\cdot\nabla\cdot\sigma = \nabla\cdot\nabla\cdot(\rho\mathbf\otimes\mathbf).$

Subtracting $c_0^2\nabla^2\rho$, 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

:\frac-c^2_0\nabla^2\rho = \nabla\cdot\left nabla\cdot(\rho\mathbf\otimes\mathbf)-\nabla\cdot\sigma +\nabla p-c^2_0\nabla\rho\right

which is equivalent to

:\frac-c^2_0\nabla^2\rho=(\nabla\otimes\nabla) :\left rho\mathbf\otimes\mathbf - \sigma + (p-c^2_0\rho)\mathbb\right
where $\mathbb$ is the identity tensor, and $:$ denotes the (double) tensor contraction 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. $\rho\mathbf\otimes\mathbf-\sigma+(p-c^2_0\rho)\mathbb$, 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

:$\frac-c^2_0\nabla^2\rho=\frac,\quad (*)$

where

:$T_=\rho v_i v_j - \sigma_ + (p- c^2_0\rho)\delta_,$

and $\delta_$ is the Kronecker delta. Each of the acoustic source terms, i.e. terms in $T_$, may play a significant role in the generation of noise depending upon flow conditions considered. $\rho v_i v_j$ describes unsteady convection of flow (or Reynolds' Stress, developed by Osborne Reynolds), $\sigma_$ describes sound generated by viscosity, and $(p- c^2_0\rho)\delta_$ describes non-linear acoustic generation processes.

In practice, it is customary to neglect the effects of viscosity on 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 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 LifshitzL. D. Landau and E. M. Lifshitz, ''Fluid Mechanics'' 2ed., Course of Theoretical Physics vol. 6, Butterworth-Heinemann (1987) §75. 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 $\rho$ of the fluid. Furthermore, unlike Lighthill's equation, Landau and Lifshitz's equation is not exact; it is an approximation.

If one is to allow for approximations to be made, a simpler way (without necessarily assuming the fluid is incompressible) to obtain an approximation to Lighthill's equation is to assume that $p-p_0=c_0^2(\rho-\rho_0)$, where $\rho_0$ and $p_0$ are the (characteristic) density and pressure of the fluid in its equilibrium state. Then, upon substitution the assumed relation between pressure and density into $(*) \,$ we obtain the equation (for an inviscid fluid, σ = 0)

:$\frac\frac-\nabla^2p=\frac,\quad\text\quad\tilde_ = \rho v_i v_j.$

And for the case when the fluid is indeed incompressible, i.e. $\rho=\rho_0$ (for some positive constant $\rho_0$) everywhere, then we obtain exactly the equation given in Landau and Lifshitz, namely

:$\frac\frac-\nabla^2p=\rho_0\frac,\quad\text\quad\hat_ = v_i v_j.$

A similar approximation n the context of equation $(*)\,$ namely $T\approx\rho_0\hat T$, is suggested by Lighthill ee Eq. (7) in the latter paper

Of course, one might wonder whether we are justified in assuming that $p-p_0=c_0^2(\rho-\rho_0)$. The answer is affirmative, if the flow satisfies certain basic assumptions. In particular, if $\rho \ll \rho_0$ 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 $\rho$ 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 and Hamilton and Morfey.M. F. Hamilton and C. L. Morfey, "Model Equations," ''Nonlinear Acoustics'', eds. M. F. Hamilton and D. T. Blackstock, Academic Press (1998) chap. 3.

See also

* Acoustic theory
*Aeolian harp
*Computational aeroacoustics

References

Williams, J. E. Ffowcs, "The Acoustic Analogy—Thirty Years On" ''IMA J. Appl. Math.'' 32 (1984) pp. 113-124.

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

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

External links

* 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.
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* K. Naugolnykh and L. Ostrovsky, ''Nonlinear Wave Processes in Acoustics'', Cambridge Texts in Applied Mathematics vol. 9, Cambridge University Press (1998) chap. 1.
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* M. F. Hamilton and C. L. Morfey, "Model Equations," ''Nonlinear Acoustics'', eds. M. F. Hamilton and D. T. Blackstock, Academic Press (1998) chap. 3.
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