2D Buoyancy-driven Cavity

Overview

The 2D buoyancy-driven cavity, a differentially-heated square cavity, is a natural-convection benchmark for incompressible flow solvers with heat transfer. A square cavity of side $L=1\text{m}$ has a hot left wall and a cold right wall held at fixed temperatures, insulated (adiabatic) top and bottom walls, and gravity acting in the $-y$ direction. Buoyancy in the variable-density fluid drives a recirculating cell: warm, light fluid rises along the hot wall and cool, dense fluid sinks along the cold wall.

Unlike the small-temperature-difference (Boussinesq) limit, a temperature ratio $T_h/T_c=4$ is imposed here so that the full variable-density capability of the solver is exercised: density is computed from the ideal-gas law and the buoyancy body force is the complete $\rho \overrightarrow{g}$, not a linearized approximation.

Problem Setup

The hot and cold walls are held at $T_h=461.04\text{K}$ and $T_c=115.26\text{K}$ (so $T_h/T_c=4$); the top and bottom walls are adiabatic and all four are no-slip. Gravity is $g=9.81{\text{m/s}}^2$ in $-y$. Air is modeled as an ideal gas with $\gamma =1.4$, $R=287.058\text{J/(kg·K)}$; the dynamic viscosity $\mu =1.716\times {10}^{-5}\text{kg/(m·s)}$ and thermal conductivity $\kappa =0.02462\text{W/(m·K)}$ are constant, giving a laminar Prandtl number $Pr=0.7$.

The flow is governed by the Rayleigh number,

$$Ra=\frac{\Delta T{\rho }^{2}g{c}_p{L}^3}{\mu \kappa },\Delta T=\frac{2(T_h-T_c)}{T_h+T_c}=1.2.$$

The Rayleigh number is set by the operating density ${\rho }_0$ in the cavity, which fixes the constant thermodynamic pressure through the ideal-gas law; results are reported for $Ra={10}^3,{10}^5,{10}^6$. The flow is solved on a $257\times 257$ uniform structured grid. Velocities are normalized by the reference velocity $V_{ref}=\sqrt{FrgL}$ with Froude number $Fr=1.2$.

Quantities of Interest

The x-velocity profile along the vertical centerline ($x/L=0.5$), $u/V_{ref}$ versus $y/L$, is compared to the reference solution of Sockol (2003) for $Ra={10}^3,{10}^5,{10}^6$.