Hopf Bifurcation in the Double-Glazing Problem With Conducting Boundaries

Abstract

Oscillatory convection has been observed in recent experiments in a square, air-filled cavity with differentially heated sidewalls and conducting horizontal surfaces. We show that the onset of the oscillatory convection occurs at a Hopf bifurcation in the steady-state equations for free convection in the Boussinesq approximation. The location of the bifurcation point is found by solving an extended system of steadystate equations. The predicted critical Rayleigh number and frequency at the onset of oscillations are in excellent agreement with the values measured recently and with those of a time-dependent simulation. Four other Hopf bifurcation points are found near the critical point and their presence supports a conjectured resonance between traveling waves in the boundary layers and interior gravity waves in the stratified

Direct numerical simulation of a differentially heated cavity of aspect ratio 4 with Rayleigh numbers up to 1011 – Part I: Numerical methods and time-averaged flow

“…Due to the presence of high temperature areas at the bottom of the cavity, the configuration with perfectly conducting horizontal walls (linear distribution of temperature at the top and bottom walls) is more unstable than the configuration with adiabatic walls. The transition to unsteadiness of this configuration was studied by Winters [12], obtaining a critical number of Ra ¼ 2:109 Â 10 6 , later confirmed by Henkes [13]. For the square cavity with adiabatic horizontal walls, Le Quéré and Behnia [14] identified the critical number as Ra ¼ 1:82 AE 0:01 Â 10 8 and studied the time-dependent chaotic flows up to Ra ¼ 10 10 .…”

Linear stability analyses of natural convection flows in a differentially heated square cavity with conducting horizontal walls

“…There is therefore clearly the need for more suitable methods for carrying out linear stability analyses of flows in cavities. Winters 9 and Gelfgat and Tanasawa 10 have investigated stability analyses and determined the critical values in the case of a square cavity with conducting horizontal walls. Both studies relied on building explicitly the Jacobian of the nonlinear evolution operator and finding the corresponding spectrum of the eigenvalue problem.…”

“…Both studies relied on building explicitly the Jacobian of the nonlinear evolution operator and finding the corresponding spectrum of the eigenvalue problem. Winters 9 used the direct method for the determination of Hopf bifurcations proposed by Griewank and Reddien 11 which consists of solving the coupled system governing the base solution and the eigensystem. The coupled equations was solved by Newton's iteration via continuation technique: base solutions were obtained for several Rayleigh numbers and an inverse transformation of the Jacobian has been used to obtain initial guess of eigenvalues and eigenfunctions.…”

“…These methods thus seem limited to configurations in which the bifurcation takes place at relatively small values of the Reynolds or Rayleigh numbers so that the solution can be adequately approximated by relatively low spatial resolution. Winters 9 and Gelfgat and Tanasawa 10 have used a maximal spatial resolution of 31ϫ31. One must however be conscious that the numerical resolutions needed to approximate the base flow or the eigenmodes are not necessarily the same, since the eigenmodes may oscillate spatially more rapidly than the base flow, in particular in the case of Hopf bifurcations.…”

Boundary layer instability of the natural convection flow on a uniformly heated vertical plate

“…Direct numerical simulations of the natural convection cavity flow have been reported in [10][11][12][13][14][15][16][17][18][19][20][21][22][23] and of the unbounded plate with stratified ambient in [24][25][26]. The cavity studies were mainly concerned with the occurrence of global …”