Table of Contents

## 2D Rayleigh-Bénard convection

Rayleigh-Bénard convection is observed in a thermal fluid subject to gravitation. The flow is constrained between a hot plate at the bottom and a cold plate at the top, and periodic in the horizontal direction. At low Rayleigh number, the fluid is motionless because of the symmetric domain layout. At a critical value of the Rayleigh number, this symmetry is broken, and two convection rolls are formed. The simulation is resolved by 200×100 lattice cells, the Prandtl number is Pr=1 and the time axis is discretized into intervals of $δ_t=0.001$. The following document explains how these values translate into lattice units of the simulation.

The simulation is executed on to grids, one for the fluid and one for the temperature. The coupling between the two is implemented by means of a linear buoyancy term, based on a Boussinesq approximation. This lattice Boltzmann model is described in a paper by Z. Guo e.a..

### Low Rayleigh number

At $Ra=10^4$, the system is beyond the critical Rayleigh number, and two convection rolls are visible. The image displays the temperature distribution in the steady state, after a sufficient amount of iteration steps.

Rayleigh-Bénard convection at Ra = $10^4$ The numerical values obtained from this thermal fluid simulation are very accurate. A typical benchmark criterion is the critical Rayleigh number $Ra_c$, at which the fluid switches from a rest state to a stationary state with two convection rolls. Although this is a difficult parameter to evaluate numerically, the thermal LB model quickly converges to the expected value, which is known from a stability analysis to be $Ra_c = 1708$. On a quite coarse grid of 100×50 cells, the LB simulation already yields an acceptable approximation of $Ra_c = 1721$.

### High Rayleigh number

As the Rayleigh number increases, the number of convection cells doubles, until a turbulent regime is reached. The following is a simulation at $Ra=10^8$, beyond the laminar regime of convection cells. One should be aware that the lattice resolution (200×100) is in all likelihood insufficient to represent the physics of the model accurately at such a high Rayleigh number. This animation should rather be taken as a proof of concept that lattice Boltzmann remains stable and yields a qualitative description of a high-Rayleigh flow in spite of the low lattice resolution. Both the still picture and the animations display the temperature distribution.

Animations: AVI file(2.8 Mb) / Animated Gif (3.6 Mb)

The same convective flow at Ra = $10^8$ community/gallery/rb_2d.txt · Last modified: 2011/12/20 17:02 (external edit)

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