Conservation of Quadratic Quantities

In this section, we check the conservations of the volume-integrated quadratic quantities. See the governing equations for the definitions of these quantities.

Configuration

First, we simulate for \(50\) time units to get a random flow. \(Ra\) is set to an extremely high value to mimic an inviscid condition. This is followed by another run for \(10\) time units, restarted from the previous simulation without the buoyancy force, so that the quadratic quantities should be conserved.

To see the effect of the time step size, I consider four different Courant numbers \(0.1\), \(0.2\), \(0.4\), \(0.8\).

Results

As derived in the governing equaations, the quadratic quantities should be constant over time. This is not the case in this project, since we adopt an explicit Runge-Kutta scheme to integrate the advective terms in time, which is dissipative (see Morinishi et al., J. Comput. Phys. (143), 1998, Coppola et al., Appl. Mech. Rev. (71), 2019). Thus these two quantities decrease monotonically over time, which is unfavourable but advantageous from a stability point of view.

The following graphs show the quadratic quantities as a function of time:

2D:

3D:

Here four different time step sizes are considered. Also, the decays of \(K\) and \(H\) (\(t = 50\) and \(60\) are compared) are displayed.

2D:

3D:

Here the third-order convergence is observed, which is also reported by the previous works (e.g., Morinishi et al., J. Comput. Phys. (143), 1998, Ham et al., J. Comput. Phys. (177), 2002 and Coppola et al., Appl. Mech. Rev. (71), 2019). This indicates that the decay, which is a numerical artifact, is properly converged.