Higher harmonic resonances in free convection between vertical parallel plates

Abstract

Nonlinear higher harmonic resonances have been investigated on a weakly nonlinear basis for free convection between vertical parallel plates with different uniform temperatures. It is pointed out that resonances of different classes take place in the convection: i.e. a resonance between stationary modes whose wavenumber ratio is 1:2, a resonance between a pair of travelling waves and a stationary mode with wavenumber ratio of 1 :1 :2, and a resonance between a travelling wave pair and two stationary modes with wavenumber ratio 1 :1 :2:4. Coupled amplitude equations for these resonating modes are derived and all the coefficients involved in the equations are determined numerically. Bifurcation diagrams are obtained based on the actual values of the coefficients.