A Defect Correction Method for the Retrieval of Acoustics Waves

For a given mathematical problem and a given approximate solution, the residue or defect may be defined as a quantity to measure how well the problem has been solved. Such information may then be used in a simplified version of the original mathematical problem to provide an appropriate correction quantity. The correction can then be applied to correct the approximate solution in order to obtain a better approximate solution to the original mathematical problem. Such idea has been around for a long time and in fact has been used in a number of different ways. A famous example of defect correction is the computation of a solution to the nonlinear equation f(x) = 0. Suppose x̄ is an approximate solution, then −f (x̄) is the defect. One possible version of the original problem is to define f̄(x) ≡ f ′(x̄)(x− x̄)+ f(x̄) = 0. In fact, if one replaces x−x̄ as v, then v is the correction which is obtained by solving f ′(x̄)v = −f (x̄) and an updated approximation can be obtained by evaluating x := x̄+v. Most defect correction are used in conjunction with discretisation methods and two-level multigrid methods [BS84]. This paper is not intended to give an overview of defect correction methods but to use the basic concept of a defect correction in conjunction with fluctuations in flow field variables for sound and noise retrieval. Recall that sound waves manifested as pressure fluctuations are typically several orders of magnitude smaller than the pressure variations in the flow field that account for flow acceleration. Furthermore, they propagate at the speed of sound in the medium, not as a transported fluid quantity. A decomposition of variables was first introduced in [DLP97] and has been further examined in [Dja98] to include three types of components. These components include (1) the mean flow, (2) flow perturbations or aerodynamic sources of sound, and (3) the acoustic perturbation. We have demonstrated the accurate computation of (1) and (2) in [DLP98]. Mathematically, the flow variable U may be written as ū+ u where ū denotes the mean flow and part of aerodynamic sources of sound and u denotes the remaining part of the aerodynamic sources of sound and the acoustic perturbation. While flow perturbation or aerodynamic sources of sound may be easier to recover, it is not true for the acoustic perturbation because of its comparatively small magnitude. In fact, the solutions of the Reynolds averaged Navier-Stokes equations reveal only a truncated part of the full physical quantities. This paper follows the basic principle of the defect correction as discussed above and applies the concept to the recovery of the propagating acoustic perturbation. The method relies on the use of a lower order partial differential equation defined on the same computational domain