conatus.problems.rayleigh_benard¶

conatus.problems.rayleigh_benard(*, Nx, Nz, Lx, Ra, Pr, dealias)[source]¶

2D Rayleigh–Benard convection between rigid, fixed-temperature plates (Boussinesq), IVP.

Equations (z in [-1/2, 1/2], x periodic of length Lx):

dt(u) - Pr lap(u) + grad(p) - Pr Ra T e_z = -u.grad(u),     div(u) = 0,
dt(T) - lap(T) = -u.grad(T),
u = 0 and T = +1/2 at z = -1/2 (hot bottom),  u = 0 and T = -1/2 at z = +1/2.

Units: length = gap d, time = thermal diffusion time d^2/kappa, temperature = the imposed difference; Ra = g alpha DeltaT d^3 / (nu kappa), Pr = nu / kappa. Conduction state T = -z, u = 0. Pressure by generalized tau, gauge tp with integ(p) = 0 (tutorial t02a).

Validation reference: Reid & Harris (1958): Ra_c = 1707.762, k_c = 3.117 (checked with rayleigh_benard_evp(), tests/problems/test_problems_convection.py). At Ra = 2500, k = pi, Pr = 1 the onset roll grows at sigma = 5.5425059 (tutorial t02a: IVP = EVP to 1e-5).

Parameters:

Nx (int)
Nz (int)
Lx (float)
Ra (float)
Pr (float)
dealias (float)

Return type:

Ready