Prediction of pattern selection due to an interaction between longitudinal rolls and transverse modes in a flow through a rectangular channel heated from below

Abstract

Convection patterns in a flow through a horizontal channel that is heated from below are predicted on the basis of a weakly nonlinear theory. At a certain value of the Reynolds number and the Rayleigh number, the conduction state with steady shear flow becomes linearly unstable to both longitudinal rolls and transverse modes, simultaneously. The longitudinal rolls align along the streamwise direction whereas the transverse modes are periodic in the streamwise direction. Amplitude equations for the interaction between the longitudinal rolls and the transverse modes are derived in a consistent manner. Coefficients in the equations are determined numerically for a wide range of parameters. The longitudinal rolls are found to bifurcate supercritically. On the other hand, the transverse modes bifurcate subcritically or supercritically, depending on the Prandtl number, the aspect ratio of the channel, and the boundary conditions on the sidewalls. Stable convection patterns are classified in a parameter space. A mixed mode pattern, which is a mixture of the components of the longitudinal rolls and the transverse modes, is found to be stable for some sets of parameters.

Weakly nonlinear interaction of mixed convection patterns in porous media heated from below

“…A weakly nonlinear theory , which is valid near the double bifurcation point (Re * K , Ra * ), has been applied in order to derive coupled envelope equations for the marginally unstable T modes and L rolls. The procedure is similar to that presented by Kato and Fujimura [16], who obtained a coupled Landau equations describing the long-time behavior of the competition between T modes and L rolls in the Poiseuille-Rayleigh-Bénard problem. However, the situation near critical conditions is such, that a narrow spectrum of waves become unstable.…”

“…Substituting ∂/∂t → ∂/∂t + ε 2 ∂/∂τ , ∂/∂x → ∂/∂x + ε∂/∂X, according to (16) and inserting (14) as well as Pe = ε Pe into (7) yields at successive orders of ε,…”

“…It is a straightforward matter to determine the stability of the various equilibrium solutions to small perturbations (see for example [16]). Here we focus on the results and we omit the detail.…”

“…They applied a weakly nonlinear theory for perturbations which modulated both on time and space and obtained a coupled envelope equations. As pointed out by Kato and Fujimura [16], the disagreement between the predictions of [14] and [15] concerning the stability of a mixed mode comes from the difference of the coefficients of nonlinear interaction terms in their respective model. Therefore Kato and Fujimura [16] used the two time-scale analysis and derived rigorously a set of two coupled Landau equations, each of them describing the time evolution of L rolls and T modes respectively.…”

“…As pointed out by Kato and Fujimura [16], the disagreement between the predictions of [14] and [15] concerning the stability of a mixed mode comes from the difference of the coefficients of nonlinear interaction terms in their respective model. Therefore Kato and Fujimura [16] used the two time-scale analysis and derived rigorously a set of two coupled Landau equations, each of them describing the time evolution of L rolls and T modes respectively. Their reduced model is adequate to predict stable patterns for given sets of parameters but it does not take into account the modulation of the convective patterns which occur on a spatial scale.…”