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LIGHTHILL'S EQUATION


Lighthill essentially rearranged the Navier Stokes equations of Viscous Fluid Flow into an Inhomogeneous Wave Equation therefore making an acoustic analogy with Fluid Mechanics . For non-relativistic velocities, the exact governing equation for the conservation of mass is given by:

: 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 is the density of the fluid.

The exact equation for the conservation of momentum is given by:

:{ ho} rac{\partial \mathbf{v}}{\partial t}+{ ho\mathbf{v}
abla\mathbf{v}}+
abla p=0

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( ho\mathbf{v}\mathbf{v})+{
abla p}=0

Note that \mathbf{v}\mathbf{v} is a Tensor . Differentiating the conservation of mass with respect to time and differentiating the conservation of momentum with respect to space and subtracting gives

:
abla^2 p+ rac{\partial^2 ho}{\partial t^2}=
abla\cdot\left(
abla p+ rac{\partial}{\partial t}( ho\mathbf{v}) ight)

Combining both these expressions gives Lighthill's equation

: rac{\partial^2 ho}{\partial t^2}-c^2_0
abla^2 ho=
abla\cdot\left( rac{\partial}{\partial t}( ho\mathbf{v})+
abla( ho\mathbf{v}\mathbf{v})- rac{\partial}{\partial t}( ho\mathbf{v}) ight)

which simplifies to become

: rac{\partial^2 ho}{\partial t^2}-c^2_0
abla^2 ho=
abla^2( ho\mathbf{v}\mathbf{v})

The above expression is the famous Lighthill's equation of aeroacoustics. This equation is the same as the well-known wave equation except for the source term on the right-hand side. The product \mathbf{v}\mathbf{v} represents the so-called Lighthill’s stress tensor, commonly expressed as T_{ij}. Lighthill’s equation can be written using indicial tensor notation as

: rac{\partial^2 ho}{\partial t^2}-c^2_0
abla^2 ho= rac{\partial^2T_{ij}}{\partial x_i \partial x_j}

where T_{ij} is given by

:T_{ij}= ho u_i u_j+(p- ho c^2_0)\delta_{ij}+\sigma_{ij}

and \delta_{ij} is the Kronecker Delta .

Each of these acoustic source terms may play a significant role in the generation of noise depending upon flow conditions considered. It is generally accepted however, that the term \sigma_{ij} representing the effects of Viscosity on noise generation is orders of magnitude less than the other terms and can be consequently neglected in most situations. In most aeroacoustic studies, both theoretical and computational efforts are made to solve for the acoustic source terms in Lighthill's equation to make statments regarding the relevant aerodynamic noise generation mechanisms present.


EXTERNAL REFERENCES