Analysis of thermally induced rarefied gas flow in a square cavity: from near-free-molecular to near-continuum conditions

We investigate the gas flow induced in a closed, two-dimensional square cavity by imposed differences in the temperatures of its walls. The problem is studied over the entire range of gas rarefaction rates. Building upon the known free-molecular description, we derive an asymptotic solution at high yet finite Knudsen numbers. The latter is derived based on the Bhatnagar–Gross–Krook (BGK), Shakhov and full hard-sphere (HS) Boltzmann collision operator. These are complemented by direct simulation Monte Carlo and particle scheme calculations at arbitrary rarefaction rates, together with a thermodynamic ‘pure conduction’ continuum solution at vanishing italic Kn Kn $\textit{Kn}$ . The gas is found at an expectedly macroscopically quiescent state in both limiting states: the collisionless free-molecular limit, where heat is transported ballistically, and the leading-order Navier–Stokes–Fourier continuum limit, in which heat is transported by pure conduction. At large yet finite Knudsen numbers, however, even a small number of molecular collisions suffices to generate vortical flow, which magnitude vanishes as the free-molecular limit is approached. Analysis of the BGK asymptotic correction identifies the microscopic origin and rotation direction of this circulation. The asymptotic results are found in excellent agreement with the numerical calculations through upper O left parenthesis 1 right parenthesis O ( 1 ) $O(1)$ Knudsen numbers. The variation of the gas total kinetic energy with gas rarefaction is analysed, exhibiting a maximum at tilde upper O left parenthesis 10 Superscript negative 1 Baseline right parenthesis ∼ O ( 10 − 1 ) $\sim O(10^{-1})$ Knudsen numbers. The maximum location and magnitude are quantified. The quantitative differences between the HS, BGK and Shakhov model predictions are discussed.