A horizontal layer of fluid heated from below is unstable. The warmer fluid at the bottom will be less dense than the cooler fluid at the top and it is this cooler top-heavy arrangement which is potentially unstable. At the onset of instability the flow evolves into a regular pattern which consists of many pairs of vortices with opposite signs, moving from the flow entrance to exit. This is the so-called Rayleigh-Bernard flow. The following is a numerical simulation of the Rayleigh-Bernard flow, carried out using a Spectral Element Method.
Parameters: Reynolds number Re=10, Richardson number Ri=150, Prandtl number Pr=2/3.
- Initial conditions: u_1(x,y,0)=6y(1-y), u_2(x,y,0)=0, T(x,y,0)=1-y+epsilon where epsilon is the temperature pertubation parameter (epsilon=10^{-2} is used in this calculation).
- Time discretization:
Backward differentiation/Runge-Kutta of 4th order semi-Lagrangian method.
- Space approximation:
P_N x P_{N-2} spectral element method.
- Outflow boundary conditions: Orlanski's type
Parameters: Reynolds number Re=1000, attack angle = 30 degree
- Initial conditions: u_1(x,y,0)=1, u_2(x,y,0)=0
- Time discretization:
Backward differentiation/Runge-Kutta of 4th order semi-Lagrangian method.
- Space approximation:
P_N x P_{N-2} spectral element method.
- Outflow boundary conditions: Orlanski's type
