Rayleigh–Bénard Convection

The rbc case models turbulent Rayleigh–Bénard convection in a three-dimensional domain using the Boussinesq approximation. The fluid is heated from below and cooled from above, while the lateral directions are periodic. The resulting buoyancy forces generate rising hot plumes, descending cold plumes, and large-scale convective circulation.

Computational domain

The computational domain has dimensions , where is the domain height, with extents

(1)

The domain is discretized using hexahedral elements in the , , and directions, respectively, for a total of 2,048 elements. The mesh is uniform in the two periodic lateral directions and stretched in the vertical direction to increase resolution near the thermal and viscous boundary layers.

Governing equations

The nondimensional momentum and temperature equations are

(2)(3)

where is the Rayleigh number and is the Prandtl number. Gravity and buoyancy act in the vertical direction.

Boundary and initial conditions

Periodic boundary conditions are imposed in the and directions. No-slip, fixed-temperature conditions are imposed on the horizontal walls,

(4)

The initial conductive temperature profile is

(5)

Figure 1 shows an instantaneous temperature field, illustrating hot fluid rising from the lower wall and cold fluid descending from the upper wall.

Instantaneous temperature contours showing thermal plumes in Rayleigh--Bénard convection

Figure 1: Instantaneous temperature contours for Rayleigh–Bénard convection.

Figure 2 shows the corresponding instantaneous velocity-magnitude field generated by the buoyancy-driven convective motion.

Instantaneous velocity-magnitude contours for Rayleigh--Bénard convection

Figure 2: Instantaneous velocity-magnitude contours for Rayleigh–Bénard convection.

Verification criterion

The simulation restarts from a statistically developed flow field and advances the solution for four nondimensional time units using a seventh-order spectral-element discretization. During this interval, the instantaneous vertical convective heat flux is accumulated in time. The time-averaged field is then integrated over the domain to compute the volumetric Nusselt number,

(6)

where the overbar denotes a time average and denotes a volume average. The computed value is compared with the reference value reported by Togni et al. (Togni et al., 2015). The simulation results are considered to adequately match the reference data when

(7)

References

  1. Riccardo Togni, Andrea Cimarelli, and Elisabetta De Angelis. Physical and scale-by-scale analysis of rayleigh–bénard convection. Journal of Fluid Mechanics, 782:380–404, 2015.[Export]