to deceptive conclusions (see, e.g., Ref. 16 and references herein), which do not coincide with experimental observations. 16,17 In the 1990s, the limiting nature of the classical modal approach was recognized and the nonnormal nature of non-uniform/shear flows was finally revealed and rigorously proven (see, e.g., Refs. 15,16,[18][19][20] and references herein)-a major breakthrough in the understanding of linear and nonlinear shear flow dynamics. In fact, the operators involved in the modal analysis of plane shear flows are non-normal, resulting in the non-orthogonality of the corresponding eigenfunctions, and hence strong interference phenomena among the eigenmodes. 19 The so-called non-modal approach, shifting the emphasis from the asymptotic to the short-time dynamics, became a well established alternative, taking into account the shortcomings of the modal approach. This resulted in the understanding and precise description of linear transient phenomena. In one branch of non-modal stability theory, the system's response to initial conditions is investigated, which is central to hydrodynamic stability theory. Here, the Kelvin mode approach, 21 stemming from the 1887 paper by Lord Kelvin, with a time-dependent shearwise wavenumber, has been extensively used for constant shear flows since the 1990s (see, e.g., . This mode represents the "simplest element" of constant shear flow physics 29 and has also been referred to as "flowing eigenfunctions" in expanding fluctuations. 30 In particular, there exist various terminologies for the Fourier mode with a time-dependent wave vector in the different areas of fluid mechanics, e.g., "Kelvin mode," "shear wave," "SH wave," "flowing eigenfunctions." Salhi and Cambon 31 gave a comprehensive survey of the method in the different communities (see Secs. I and II C).We make use of the Lie point symmetry analysis, as founded by the Norwegian mathematician Sophus Lie (1842-1899) and applied successfully in the area of fluid mechanics 32-48 to systematically derive invariant solutions in the context of linear stability of a two-dimensional (2-D) linearly sheared unbounded flow. The notion of invariant solutions was introduced by Sophus Lie, 49 and it denotes solutions whose representations are obtained with the help of any combinations of the admitted groups. This method identifies variable transformation by which new solutions can be generated from existing ones through the use of a differential operator, i.e., the generator of the group. The similarity analysis of systems of partial differential equations (PDEs) leading to group-invariant solution is systemically and well described in several textbooks, for instance, by Refs. 50-54. Compared to a simple dimensional analysis, this method is somehow superior. This is due to the fact that there is no need to analyze the equations and boundary/initial conditions at the same time in order to identify the desired similarity variables-non-dimensional combinations of variables. Further, it is possible to uncover types of similarity th...

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.

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We study wave packets with the small perturbation/gradient Mach number interacting with a smooth shear-layer in the linear regime of small amplitude perturbations. In particular, we investigate the temporal evolution of wave packets in shear-layers with locally curved regions of variable size using non-modal linear analysis and direct numerical simulations of the two-dimensional gas-dynamical equations. Depending on the wavenumber of the initially imposed wave packet, three different types of behavior are observed: (i) The wave packet passes through the shear-layer and constantly transfers energy back to the mean flow. (ii) It is turned around (or reflected) within the sheared region and extracts energy from the base flow. (iii) It is split into two oppositely propagating packages when reaching the upper boundary of the linearly sheared region. The conducted direct numerical simulations confirm that non-modal linear stability analysis is able to predict the wave packet dynamics, even in the presence of non-linearly sheared regions. In the light of existing studies in this area, we conclude that the sheared regions are responsible for the highly directed propagation of linearly generated acoustic waves when there is a dominating source, as it is the case for jet flows.