Numerical Analysis of Laminar‐Turbulent Bifurcation Scenarios in Kelvin‐Helmholtz and Rayleigh‐Taylor Instabilities for Compressible Flow

Abstract

In the chapter, we are focused on laminar-turbulent transition in compressible flows triggered by Kelvin-Helmholtz (KH) and Rayleigh-Taylor (RT) instabilities. Compressible flow equations in conservation form are considered. We bring forth the characteristic feature of supersonic flow from the dynamical system point of view. Namely, we show analytically and confirm numerically that the phase space is separated into independent subspaces by the systems of stationary shock waves. Floquet theory analysis is applied to the linearized problem using matrix-free implicitly restarted Arnoldi method. All numerical methods are designed for CPU and multiGPU architecture using MPI across GPUs. Some benchmark data and features of development are presented. We show that KH for symmetric 2D perturbations undergoes cycle bifurcation scenarios with many chaotic cycle threads, each thread being a Feigenbaum-Sharkovskiy-Magnitskii (FShM) cascade. With the break of the symmetry, a 3D instability develops rapidly, and the bifurcations includes Landau-Hopf scenario with computationally stable 4D torus. For each torus, there exist threads of cycles that can develop chaotic regimes, so the flow is more complicated and difficult to study. Thus, we present laminar-turbulent development of compressible RT and KH instabilities as the bifurcations scenarios.

Bifurcation Theory of Dynamical Chaos

“…Bases of FShM theory with reference to a wide class of nonlinear systems of partial differential equations are stated in Refs. [6][7][8][9]. This class includes systems of the equations of reaction-diffusion type, describing various autowave oscillatory processes in chemical, biological, social and economic systems, including the well-known brusselator equations; the equations of FitzHugh-Nagumo type, describing processes of chemical and biological turbulence in excitable media; the equations of Kuramoto-Tsuzuki (or Time Dependent Ginzburg-Landau) type, describing complex autooscillating processes after loss of stability of a thermodynamic branch in reaction-diffusion systems; the systems of Navier-Stokes equations, describing laminar-turbulent transitions in hydrodynamical and gazodynamical mediums.…”

“…The most famous among these models are: the Landau-Hopf model explaining turbulence by motion along an infinitedimensional torus generated by an infinite cascade of Andronov-Hopf bifurcations; and the Ruelle-Takens model, which explains turbulence by moving along a strange attractor generated by the destruction of a three-dimensional torus. In recent years, the author and his pupils have proved (see [8,9,[20][21][22]) that the universal bifurcation FShM mechanism for the transition to space-time chaos in nonlinear systems of partial differential equations through subharmonic cascades of bifurcations of stable cycles or two-dimensional and multidimensional tori also takes place in problems of laminar-turbulent transitions for Navier-Stokes equations…”

“…A numerically complete subharmonic cascade of bifurcations of stable two-dimensional tori is found up to a torus of period three in the famous Kolmogorov problem in two-dimensional and three-dimensional spatial cases [22]. The features of compressible flow and instabilities triggered by Kelvin-Helmholtz (KH) and Rayleigh-Taylor (RT) mechanisms are considered in Ref [9]. The Kelvin-Helmholtz instability is the instability of the shear layer, which is a tangential discontinuity for the inviscid liquid and which arises when there is a velocity difference at the interface of two liquids or when there is a velocity shift in one of the liquids.…”

“…Prigogine, that irregular attractors of complex nonlinear systems cannot be described by trajectory approach, that are systems of differential equations. And only in twenty-first century it has been proved and on numerous examples it was convincingly shown, that there is one universal bifurcation scenario of transition to chaos in nonlinear systems of mappings and differential equations: autonomous and nonautonomous, dissipative and conservative, ordinary, with private derivatives and with delay argument (see, for example, [1][2][3][4][5][6][7][8][9]). It is bifurcation Feigenbaum-Sharkovsky-Magnitskii (FShM) scenario, beginning with the Feigenbaum cascade of period-doubling bifurcations of stable cycles or tori and continuing from the Sharkovskii subharmonic cascade of bifurcations of stable cycles or tori of an arbitrary period up to the cycle or torus of the period three, and then proceeding to the Magnitskii homoclinic or heteroclinic cascade of bifurcations of stable cycles or tori.…”

“…The purpose of the present paper is once again to show on concrete new, not entered in [1][2][3][4][5][6][7][8][9], examples, that chaos in the system considered in Refs. [10,11], and also chaos in onedimensional unimodal mappings, dynamical chaos in systems of ordinary differential equations, diffusion chaos in systems of the equations with partial derivatives and chaos in Hamiltonian and conservative systems are generated by cascades of bifurcations under the FShM scenario.…”