Scaling laws for the transient convective flow in a differentially and linearly heated rectangular cavity at Pr > 1

Abstract

The convective flow in a differentially heated cavity, with linear temperature profiles at two sidewalls, is investigated in the present study by a scaling analysis and direct numerical simulations (DNS). Scales for the thermal boundary layer and the subsequent intrusion are obtained through the scaling analysis. The velocity scale reveals that the characteristic velocity of the thermal boundary layer depends on both the streamwise position and the time after the initiation of the flow, which suggests a two-dimensional growth at the start-up stage, rather than the well-known one-dimensional growth of the thermal boundary layer induced by a constant temperature boundary condition. Furthermore, unlike the typical transition of the thermal boundary layer to a two-dimensional and steady stage that is characterized by the dying out of a “temperature overshoot” phenomenon, the thermal boundary layer under consideration enters a two-dimensional and steady stage smoothly, without the occurrence of the temperature overshoot. It is also found that, with the passage of time, whilst the characteristic velocity of the thermal boundary layer depends on the streamwise position, the thickness of the thermal boundary layer is streamwise position independent due to infinitesimally small time. Four possible flow regimes and corresponding scales for the unsteady intrusion flow underneath the cavity ceiling are finally obtained, which are two types of viscous-buoyancy dominated regimes and two types of inertial-buoyancy dominated ones. The important scales obtained in the present study are validated by corresponding DNS results.