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Aeroacoustics and Low Noise Design
Published in Ranjan Vepa, Electric Aircraft Dynamics, 2020
By considering non-zero base flows, one could extend the acoustic analogy to moving media. Lilley [41] extended Lighthill’s analogy to account for transversely sheared mean effects in the base flow. Lilley derived his analogy by retaining on the left-hand side of the analogy a third order, non-linear operator acting on the logarithm of the pressure. This non-linear operator could be approximated by the linear Pridmore-Brown operator [42]. The source terms corresponding to the linear Pridmore-Brown equation [43] appearing on the right-hand side could also be derived. The exact source terms on the right-hand side of Lilley’s analogy with the Pridmore-Brown operator are given in Colonius et al. [44], which also gives a simpler approximate form of the source terms, based on the work in Goldstein [45]. Moreover, Goldstein [46] also showed that for small fluctuations in the flow, the logarithm of pressure could be replaced by the pressure itself. Powell [47] formulated an aeroacoustic analogy which highlights the significance of vorticity as an acoustic source. In this formulation, the Lamb vector, which is the cross-product of the vorticity vector and the velocity vector, acts as the source. Powell’s analogy is an approximate version of Lighthill’s analogy where the independent variable is expressed in terms of the pressure and the right-hand side in terms of the vorticity vector. It can also be considered to be an approximation of Lilley’s analogy and naturally leads to the vortex theory of sound. To obtain Powell’s equation, it is further assumed that the fluid is incompressible inside the source region. The vortex sound theory reduces the source term only to the region where vorticity is not negligible which is typically smaller than the source region described by Lighthill’s analogy. Other extensions consider the acoustic analogy acting on different variables. These include Howe’s [48] and Doak’s analogies [49,50] that are applicable to highly compressible flows as well as other extensions which are only applicable to specific flows. Finally, Morfey and Wright [51] introduced the idea of developing non-linear acoustic analogies that can be used with methods of computational aero acoustics. Based on an aeroacoustic analogy proposed by Posson and Peake [52], Mathews [53] has modeled the pressure modes of a turbofan engine and obtained them numerically by approximating the modes using Chebyshev polynomials.
The timestep constraint in solving the gravitational wave equations sourced by hydromagnetic turbulence
Published in Geophysical & Astrophysical Fluid Dynamics, 2020
Alberto Roper Pol, Axel Brandenburg, Tina Kahniashvili, Arthur Kosowsky, Sayan Mandal
The direct analogy between sound generation and GW generation from isotropic homogeneous turbulence was exploited by Gogoberidze et al. (2007) and Kahniashvili et al. (2008). The aeroacoustic analogy allows approximated analytical description of turbulence generated GWs. In this work we focus on the numerical aspects of GW and MHD equations, and the generation of sound waves, as well as the aeroacoustic analogy for GW generation, is left for future studies. In a flat expanding universe, during the radiation-dominated epoch, using comoving spatial coordinates (which follow the expansion of the universe) and conformal time t (related to physical time through ), the GW equation (see appendix 1 for details) is given by (A.3). where refers to comoving spatial derivatives, is the scale factor in the Friedmann–Lemaître–Robertson–Walker (FLRW) model, which exhaustively describes the metric tensor in a spatially flat isotropic and homogeneous universe, are the scaled tensor-mode perturbations of the metric tensor, also called (scaled) strains, related to physical strains through , such that the spatial components of the metric tensor are , being the Kronecker delta; c is the speed of light, G is Newton's gravitational constant, and is the transverse and traceless (TT) projection of the comoving stress–energy tensor (see Grishchuk 1974; Deryagin et al.1986). Since (3) is the result of linearisation in unbounded space, we assume that the spatial average of vanishes. During the radiation-dominated epoch, evolves linearly with conformal time, as inferred from the Friedmann equations (Friedmann 1922) for a perfect fluid with relativistic equation of state . Hence, in (3) there is no damping term due to the expansion of the universe; see appendix 1 for details.