The problem of turbulence arose more than hundred years ago to explain the nature of chaotic motion of the nonlinear continuous medium and to find ways for its description; so far it remains one of the most attractive and challenging problems of classical physics. Researchers of this problem have met with exclusive difficulties and there was an understanding of that the problem of turbulence always considered difficult, is actually extremely difficult. This problem is named by Clay Mathematics Institute as one of seven millennium mathematical problems [1] and it is also in the list of 18 most significant mathematical problems of XXI century formulated by S.Smale [2].The nature of turbulence -the disordered chaotic motion of a nonlinear continuous medium, the causes and mechanisms of chaos generation remain the main issue in the turbulence problem. Several models trying to explain the mechanisms of turbulence generation in nonlinear solid media were suggested at different time. Among such models the most known are Landau-Hopf and Ruelle-Takens models, explaining generation of turbulence by the infinite cascade of Andronov-Hopf bifurcations and, accordingly, by destruction of three-dimensional torus with generation of strange attractor. However, these models have not been justified by experiments with hydrodynamic turbulence.The universal unified mechanism of transition to dynamical chaos in all nonlinear dissipative systems of differential equations including autonomous and nonautonomous systems of ordinary and partial differential equations and differential equations with delay argument was theoretically and experimentally proven in number of recent papers by the authors [3][4][5][6][7][8][9][10][11][12]. The mechanism is developing by FSM (Feigenbaum-Sharkovskii-Magnitskii) scenario through subharmonic and homoclinic bifurcation cascades of stable cycles or stable two dimensional or many-dimensional tori.In this chapter we are presenting a consistent numerical solution method for 3D evolutionary Navier-Stokes equations with an arbitrary initial-boundary value problem posed. Then we

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.

We are considering two initial-boundary value problems for Rayleigh-Benard convection in Oberbeck-Boussinesq approximation for incompressible fluid in 3D-rectangular domain with 4:4:1 geometric ratio with periodicity in two directions and cubic domain with 1:1:1 ratio and zero velocity and temperature gradient boundary conditions. For this purpose, we use two numerical method: one is a Pseudo-Spectral-Galerkin method with trigonometric-Chebyshev polynomials and the other is finite element/volume method with WENO interpolation for advection term. Numerical methods are presented shortly and are benchmarked against known DNS data and against one another (for quasi-periodic domain problem). Then we perform stability analysis using analytical expression for main stationary solutions and eigenvalue numerical analysis by applying Implicitly Restarted Arnoldi (IRA) method. The IRA is used to perform linear stability analysis, find bifurcations of stationary points and analyze eigenvalues of monodromy matrices. Thus characteristic exponents of the system for time periodic solutions (limited cycles of various periods and resonance invariant tori) are computed. We show, numerically, the existence of multistable rotes to chaos through chaotic fractal attractors, full Feigenbaum-Sharkovski cascades and multidimensional torus attractors (Landau-Hopf scenario). The existence of these attractors is shown through analysis of phase subspaces projections, Poincare sections and eigenvalue analysis of numerically computed DNS data. These attractors burst into chaos with the increase of Rayleigh number either through resonance and phase-locking or through emergence of singular chaotic attractors.