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

Abstract

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...

On physically redundant and irrelevant features when applying Lie-group symmetry analysis to hydrodynamic stability analysis

“…In this study both types of transformation will be considered and applied concurrently, the change of reference frame as well as the change of representation, with the aim to reformulate a proposed change of representation in its optimal frame of reference. This will demonstrate that the new or more general invariant solutions proposed in the recent publication by Hau et al (2017), and in its precursor by Nold & Oberlack (2013), do not induce physically "new" or physically more "general" modes for unbounded linear shear flow as claimed, but only physically redundant ones, not going beyond the two different and already established representations of the modal and non-modal (Kelvin mode) † solution approaches.…”

“…Said differently, the aim in Hau et al (2017) and Nold & Oberlack (2013) is to discover and explore some alternative ways of describing the linearized dynamics of perturbed unbounded shear flow that should go beyond the classical modal and Kelvin mode approach. But the new representations obtained therein, by employing different combinations of symmetries of the underlying dynamical equations, are, not mathematically, but physically equivalent to the classical ones.…”

“…Hence, reformulating a proposed invariant solution into its optimal frame of reference, will give a method at hand, to identify whether the induced representation will lead to new or more † In Hau et al (2017), and for the first time in Chagelishvili et al (1994Chagelishvili et al ( , 1997, the Kelvin modes are also coined as "non-modal" to differentiate them from the modes of the usually applied temporal modal approach. Within a temporal framework to stability analysis (see e.g.…”

Symmetry analysis of an elastic beam with axial load

“…The Lie-method is also available in recent research addressing equations in mathematical physics [16][17][18]. In Ref.…”

“…[17]. Hau, et al presented a unifying solution framework for the linearized compressible equations for two-dimensional linearly sheared unbounded flows [18].…”