Abstract
The origin and effect of each term in the equations of the viscous/acoustic splitting (VAS) method are studied in this paper. With its rigorous mathematical derivation, the VAS method could be an improved version of Lighthill’s acoustic analogy method (referred to here as the ILAA method). The ILAA method contains a new monopole acoustic source due to the time derivative of the incompressible pressure and a modified quadrupole acoustic source due to the increment in momentum flux and viscous stress from the compressible/incompressible flow. Based on the incompressible Navier–Stokes equations, the VAS method is also considered as an ILAA method in which the quadrupole acoustic source is due to the compressible momentum flux, compressible viscous stress and the incompressible pressure. A source decomposition method is proposed by which the sound waves are sorted by the source terms. We analyze the effects of each source in the VAS method and propose an improved VAS (IVAS) method by distinguishing between an unstable source and acoustic radiation sources. Using the IVAS method suppresses the pseudo-sound waves that are excited by the additional unstable source that is generated by the waves spreading in a non-uniform velocity field, and the radiating quadrupole source in the ILAA method (VAS method) is retained to the utmost extent.
Highlights
Computational aero-acoustics methods are used extensively to predict flow noise
∂ 2 ε ij ∂ xi ∂ xj Comparing Eq (13) to Eqs. (9)–(11), the equation of the viscous/acoustic splitting (VAS) method does not match that of the Lighthill’s acoustic analogy (LAA) method. We study this discrepancy in depth, paying particular attention to Eq (6), which we consider as the equation of an improved LAA (ILAA) method from which the acoustic source terms due to temporal and spatial pressure changing are retained
Rigorous mathematical deduction shows that the VAS method is equivalent to the ILAA method
Summary
Computational aero-acoustics methods are used extensively to predict flow noise. Lighthill’s acoustic analogy (LAA) method[1,2,3] is the most traditional computational aero-acoustics method, and many new hybrid methods have been developed to use more information about the flow field to predict flow noise. Hardin and Pope[4] proposed the original viscous/acoustic splitting (VAS) method for separating the variables of the compressible Navier–Stokes (N-S) equations into incompressible flow variables and perturbed acoustic variables, Shen[5,6] corrected and developed a new version of the VAS method. A perturbed viscous stress term is added by Sea[8] and Shen[9] to suppress the unstable pressure, what are called the perturbed compressible equations (PCE) method and the VAS method, respectively.
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