Nonlinear dynamics of two-dimensional convection in a vertically stratified slot with and without gravity modulation

Abstract

The convective flow in a vertical slot with differentially heated walls and vertical temperature gradient is considered for very large Rayleigh numbers. Gravity is taken to be vertical and to consist of both a mean and a harmonic modulation (‘jitter’) at a given frequency and amplitude. The time-dependent Boussinesq equations governing the two-dimensional convection are solved numerically. To this end an economic operator-splitting scheme is devised combined with internal iterations within a given time step. The approximation of the nonlinear terms is conservative and no scheme viscosity is present in the approximation. The flow is investigated for a range of Prandtl numbers from Pr = 1000 when fluid inertia is insignificant and only thermal inertia plays a role to Pr = 0.73 when both are significant and of the same order. The flow is governed by several parameters. In the absence of jitter, these are the Prandtl number, Pr, the Rayleigh number, Ra, and the dimensionless critical stratification, τ_B. Simulations are reported for Pr = 10³ and a range of τ_B and Ra, with emphasis on mode selection and finite-amplitude states. The presence of jitter adds two more parameters, i.e. the dimensionless jitter amplitude ε and frequency ω, rendering the flow susceptible to new modes of parametric instability at a critical amplitude ε_c. Stability maps of ε_c vs. ω are given for a range of ω. Finally we investigate the response of the system to jitter near the neutral curves of the various instability modes.