Buoyancy-Driven Flow in a Square Cavity
|Introduction||Solve the PDE Model|
|Multiphysics Model||Post-processing and Visualization|
Consider a two-dimensional flow within a square cavity of solid walls, where gravity is acting in the direction. The left and right boundaries have different temperature values at and , respectively. The top and bottom boundaries, however, are assumed to be thermally insulated.
As the fluid near the left boundary absorbs heat, it becomes less dense and rises due to thermal expansion. The surrounding cooler fluid then moves in to replace it, forming a convection current within the domain. This type of flow is known as the natural/free convection.
The following simulation models the temperature, pressure and velocity fields within the square cavity. Since a stronger convection flow results in a higher rate of the heat transfer, we can measure the level of the natural convection by calculating the heat flux at the boundary.
The symbols and corresponding units used here are summarized in the Nomenclature section.
Please refer to the information provided in "Heat Transfer" for a more general theoretical background for heat transfer analysis.
Since this problem considers more than one kind of physics, a multiphysics model is to be constructed. The heat equation is coupled to the Navier–Stokes equation for modeling heat transfer with the fluid flow.
For a steady-state heat transfer model without sources, the temperature distribution is described by the time-independent source-free heat equation (1). The heat convection by the fluid flow is modeled with the convective term. As such the velocity field coming for the fluid flow is set in the convective term of the heat equation:
The flow field is described by the steady-state Navier-Stokes equation (2). To account for the temperature effects on the fluid, a Boussinesq approximation  is used. This approximation neglects the variations in fluid properties except for the density differences induced by the temperature. In other words, the Boussinesq approximation assumes the buoyancy to be the only driving force acting on the fluid flow.
The top and bottom boundaries , are assumed to be perfectly insulated. The default thermally insulated boundary conditions are implicitly applied.
In the fluid dynamics model, the only boundary condition involved is the wall/no-slip boundary condition. At all four boundaries the flow velocity is set to zero to model the solid walls where the fluid has no-slip.
A stable solution can be found if the velocities are interpolated with a higher order than the pressure. NDSolve allows an interpolation order for each dependent variable to be specified. Here the velocities and and the temperature are set to be interpolated with second order and the pressure with first order.
Because the top and bottom boundaries are thermally insulated, the heat flux across the left boundary must equal to the one across the right boundary based on the energy conservation. Therefore, the heat flux can be calculated at either or .
To efficiently solve the updated heat transfer model, the previous result can be used as an initial guess for the new solution. Since the initial seeding based on the previous solution is already close to the final solution the nonlinear solver will converge in fewer steps, resulting in less time elapsed.
When the Rayleigh number increases, the driving force (buoyancy) of the convective flow increases compared to the resisting force (viscosity) of the fluid motion. A stronger convection flow is then formed.
The heat flux across the cavity has been increased from to , which implies a stronger convective heat transfer within the domain. This updated heat flux value also matches with the reference value presented in .