Steady and oscillatory flows generated by Eckart streaming in the two-dimensional Rayleigh–Bénard configuration

Abstract

This paper presents a detailed analysis of the flows induced in a long two-dimensional cavity heated from below in the presence of streaming due to ultrasound acoustic waves emitted by a source. The problem is tackled by using performing spectral element codes, allowing continuation of steady solutions, bifurcation points and periodic cycles. For a given dimensionless source size, the governing parameters are the acoustic streaming parameter $A$ which modulates the acoustic force generating the Eckart streaming and the Rayleigh number ${\textit {Ra}}$ which quantifies the buoyant force responsible for the convection. The streaming flow, which goes to the right along the horizontal axis and returns along the lower and upper boundaries, influences the instability thresholds, which are first strongly stabilized above the pure Rayleigh–Bénard threshold ${\textit {Ra}}_0$ when $A$ is increased, before a destabilization to reach the pure streaming threshold $A_c$ at ${\textit {Ra}}=0$ . The steady multi-roll convective flow generated without streaming is replaced by periodic waves when $A$ is increased, forward waves for moderate $A$ and backward waves for large $A$ . The transition between these waves induces a specific dynamics involving steady flows, which has been elucidated. The waves also eventually disappear for a sufficient increase of the Rayleigh number, replaced by steady multi-roll flows hardly influenced by the streaming flow. A very rich dynamics is thus observed with the competition between the waves and the steady flows.