A comparative numerical analysis of linear and nonlinear aerodynamic sound generation by vortex disturbances in homentropic constant shear flows

Abstract

Aerodynamic sound generation in shear flows is investigated in the light of the breakthrough in hydrodynamics stability theory in the 1990s, where generic phenomena of non-normal shear flow systems were understood. By applying the thereby emerged short-time/non-modal approach, the sole linear mechanism of wave generation by vortices in shear flows was captured [G. D. Chagelishvili, A. Tevzadze, G. Bodo, and S. S. Moiseev, “Linear mechanism of wave emergence from vortices in smooth shear flows,” Phys. Rev. Lett. 79, 3178-3181 (1997); B. F. Farrell and P. J. Ioannou, “Transient and asymptotic growth of two-dimensional perturbations in viscous compressible shear flow,” Phys. Fluids 12, 3021-3028 (2000); N. A. Bakas, “Mechanism underlying transient growth of planar perturbations in unbounded compressible shear flow,” J. Fluid Mech. 639, 479-507 (2009); and G. Favraud and V. Pagneux, “Superadiabatic evolution of acoustic and vorticity perturbations in Couette flow,” Phys. Rev. E 89, 033012 (2014)]. Its source is the non-normality induced linear mode-coupling, which becomes efficient at moderate Mach numbers that is defined for each perturbation harmonic as the ratio of the shear rate to its characteristic frequency. Based on the results by the non-modal approach, we investigate a two-dimensional homentropic constant shear flow and focus on the dynamical characteristics in the wavenumber plane. This allows to separate from each other the participants of the dynamical processes — vortex and wave modes — and to estimate the efficacy of the process of linear wave-generation. This process is analyzed and visualized on the example of a packet of vortex modes, localized in both, spectral and physical, planes. Further, by employing direct numerical simulations, the wave generation by chaotically distributed vortex modes is analyzed and the involved linear and nonlinear processes are identified. The generated acoustic field is anisotropic in the wavenumber plane, which results in highly directional linear sound radiation, whereas the nonlinearly generated waves are almost omni-directional. As part of this analysis, we compare the effectiveness of the linear and nonlinear mechanisms of wave generation within the range of validity of the rapid distortion theory and show the dominance of the linear aerodynamic sound generation. Finally, topological differences between the linear source term of the acoustic analogy equation and of the anisotropic non-normality induced linear mechanism of wave generation are found.

On the optimal systems of subalgebras for the equations of hydrodynamic stability analysis of smooth shear flows and their group-invariant solutions

“…In general, however, the case ofω = 0 leads to the appearance of a delta function on the right hand side of (27). The same conclusion can be drawn by regarding (26), which has an integrating factor µ =ω. So, by integrating once the following form can be found:…”

“…If the PV is nonzero, the system admits vortex mode perturbations, whereas zero PV perturbations ensure solely wave mode perturbations. 23,26 Due to its importance in analyzing the linear dynamics in smooth shear flows, the PV is included in our analysis. Below, we present the different governing sets of ODEs and higher order formulations in single dependent variables.…”

“…By considering non-zero values of D, and, thus non-zero PV, non-propagating vortex, and propagating wave modes are included in the analysis, whereas zero PV restricts the analysis to wave modes. 23,26,69 In fact, the term inside the squared brackets of (27) is often referred to as the acoustic wave equation in the canonical modal framework 9,70,71 and can be solved in terms of confluent hypergeometric/Whittaker functions by a suitable variable transformation (see, e.g., Refs. 70 and 72 for a detailed discussion of the solution and Refs.…”

“…26 (the equations for the shearwise velocity and density perturbations are given in Equations (D2) and (D3) in Appendix D). For the reason of simplicity, it is preferable to analyze (36), which has the form of a wave equation.…”

Linear stability analysis of the homogeneous Couette flow in a 2D isentropic compressible fluid

“…However, at least to our knowledge, a complete non-modal mathematical analysis of the Couette flow was not considered previously in the literature. The linearization around the Couette flow in the 2D isentropic compressible Euler dynamics was instead considered in the physics literature, both from the numerical point of view and from the theoretical one, a highly incomplete list of papers includes [3,15,16,35,36] and references therein. In particular, the 2D inviscid problem (with an additional Coriolis forcing term) has been considered as a first model to understand the formation of spiral arms in a rotating disk galaxy by Goldreich and Lynden-Bell [31,32].…”