Laminar-Turbulent Bifurcation Scenario in 3D Rayleigh-Benard Convection Problem

Abstract

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.

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

“…The linear stability was analyzed in 1916 by Lord Rayleigh himself [9]. The emergence of secondary stationary and oscillatory flows was later confirmed by many authors, e.g., [10][11][12]. Last two references contain bifurcation scenarios for 2D and 3D rectangular domains for laminar-turbulent transition ("frozen turbulence" in 2D case since stretching of vortex tubes is locked).…”

“…For viscous part of equations, we use finite element nodal method that is described for Poisson equation in Ref. [12]. Time integration is explicit (WENO)-implicit (FEM) method of the third order.…”

“…For troublesome problems, we apply either exponential or Cayley transformations. See [12,59] for more details on implementation features and application examples. In all calculations, unless specified otherwise, we use k ¼ 10 target eigenvalues with additional Krylov vectors totaling m ¼ 8, i.e., Krylov subspace dimension is dimðKÞ¼k Á m ¼ 80.…”