Aero-acoustics and noise
Sound can be understood as the pressure fluctuation in the medium. Acoustics is the study of sound propagation in the medium. AeroAcoustics deals with the study of sound propagation in air. With the stringent conditions imposed on the Aircraft industries for noise pollution, the focus now is shifting towards predicting the noise generated for a given aerodynamic flow. AeroAcoustics is an advanced field of fluid dynamics where in the flow scale is reloved to the acoustic levels. The first head-start in the field of AeroAcoustics is given by Sir James Lighthill when he presented an "Acoustic Analogy". With proper manipulation of the Euler equations, he derived a wave equation based on pressure as the fluctuating variable, and the flow variables contributing to the source of fluctuation. The resulting wave equation can then be integrated with the help of Green's Function, or can be integrated numerically. Thus, this equation can represent the sound propagation from a source in an ambient condition. With the success of acoustic analogy, many improvements were made on the derivation of the wave equation. Two common form of equation used in acoustic analogy are the Ffowcs Williams - Hawkins equation and the Kirchoff's Equation. Though Acoustic Analogy is able to solve the problem of noise prediction to a greater extent, the focus is now shifting towards direct computation, wherein the noise is computed directly by the flow solver. Ofcourse acoustic analogy is still applied in the far field propagation. But the near field sound generation is resolved to a grater extent. Large Eddy Simulation is widely used for these studies. DNS is still unreachable for problems of practical dimensions. The industries rather require a code that can provide them results in a day than a month. Hence, RANS based models (like JET3D by NASA) are also widely used in the industries. One of the main difficulties in Computational AeroAcoustics is the scale of the problem. Acoustics waves have a reasonably high velocity compared to the flow structures and at the same time, nearly 10 orders smaller in magnitude. Also, due to the propagation to long distances, the numerical scheme should be less dissipative and less dispersive. The CFD solvers have inherent dissipation to ensure stability. This makes most of the robust CFD solvers incapable of simulating acoustic flows. Advanced schemes such as Dispersion Relation Preserving (DRP) schemes, compact schemes etc. aim at less dispersive solution. Still, with the current computational capability, acoustic computation for a problem of practical interest is still out of reach.
Higher Order Schemes for Aero-acoustics
Acoustic problems are governed by the linearised Euler equation and it is known from wave propagation theory that the propagation characteristics of waves governed by a system of partial different equations are encoded in the dispersion relation.The dispersion relation of a system of equation can be used to determine the isotropy,group and phase velocities of all kinds of waves supported by the system of equations.With this idea in mind it is clear that we need a finite difference scheme which has almost similar dispersion relation to the original system of equations.It is well known that the first order schemes lead to excessive dissipation error and second order schemes have a lot of dispersion errors.This motivated the study to develop a class of finite difference schemes which can be suited to the modelling of wave propagation problems.This class of finite difference schemes are usually referred to as dispersion relation preserving schemes ( DRP Schemes ).
Construction of DRP Schemes
- Tam et. al , Tam,C.K.W and Webb,J.C. (1992), "Dispersion relation preserving Finite Difference Schemes for Computational Acoustics,” Journal of Computational Physics", Journal of Computational Physics, Vol. 107, pp 262–281.
- Lele, Lele, S. K. (1992), "Compact Finite Difference Schemes with Spectral-like Resolution,” Journal of Computational Physics", Journal of Computational Physics, Vol. 103, pp 16–42.