Aeroacoustics

Our point of departure for the basic theory of aeroacoustics is an exact restatement of the continuity and momentum equation for a compressible fluid, i.e. the Navier-Stokes equations², to the following form:

$$\underbrace{{\frac{1}{c^2}}{\frac{\partial ^{2} p ^{\prime }... ...%% \partial ^{2} \mu S_{ij} }{\partial x_{i}x_{j}}}_{Viscous}.$$ (1)

This equation is generally named after its inventor as the Lighthill equation (Lighthill, 1952, 1954).

The left-hand side of this equation can be recognized as the wave equation, in this case for the acoustic pressure perturbation $p^{\prime }$ which propagates with the speed of sound $c$. The equation given above shows that these propagating pressure fluctuations are due to various sources, which are the terms on the right-hand side. The first term on the the right-hand side denotes sound production due to unsteady mass injection and the second term the sound produced by external forces acting on the fluid. These two terms are known as the classical acoustic sources as for instance discussed by Rayleigh (1945). The third term of the Lighthill equation is the non-linear advection term, which is mainly responsible for the sound produced by the flow. This term is in particular large in a turbulent flow³. The fourth term represents the deviations from an isentropic state, including effect of non-constant speed of sound, convection and refraction of sound by temperature gradients. Basically all the acoustic effects associated with the internal energy $\rho C_{p}T$ of the flow are lumped in this term. It should be noted that in the Lighthill theory the equation for the internal energy is absent. The fifth term, in which $S_{ij}$ denotes the strain rate tensor, is due to viscous effects. These effects are in general very small except for the boundary-layers close to solid walls where the strain $S_{ij}$ can become very large. The first two sound sources are due to externally applied processes and fall by definition outside the scope of this research project. We will focus on the sound generated by the flow itself and in particular on the non-linear term, which for most practical circumstances is the most important contributor to flow noise. This is supported by considering the ratio of the non-linear inertial term to the viscous term given by

$${\frac{\rho u_{i}u_{j}}{2\mu S_{ij}}}\propto {\rm Re}$$

where Re is the Reynolds number. In most practical flows Re is much larger than one and this implies not only negligible influence of viscosity on the flow but also that the flow is turbulent flow. So we find that turbulence is a source for flow sound. The magnitude of the fourth term in the Lighthill equation depends strongly on the convection velocity $u$ of the fluid. In an Eulerian frame of reference, the acoustic perturbations are traveling with speed $u+c$. So the term $p^{\prime }-c^{2}\rho ^{\prime }$ is proportional to $u^{2}$ and if the Mach number of the flow:

$${\rm Ma}={\frac{u}{c}}$$

is small, this implies that $p^{\prime }-c^{2}\rho ^{\prime }$ is small so that the effect of source convection can be neglected. In the first stages of the proposed research we would therefore like to limit ourselves to cases with very low Mach numbers. Only in a later stage we aim to extend our study to the influence of the Mach number. Finally, it should be mentioned that the theory given in this section is far from complete, for instance we have ignored the effects of internal energy on the flow field completely (see for instance Morfey 1984, Goldstein 1974).