conatus.problems.rayleigh_benard_evp¶

conatus.problems.rayleigh_benard_evp(*, N, Ra, Pr, k)[source]¶

Linear onset of Rayleigh–Benard convection: normal modes exp(i k x + s t) of the conduction state, same equations and units as rayleigh_benard() (z in [-1/2, 1/2]):

s u - Pr (D^2 - k^2) u + i k p = 0,
s w - Pr (D^2 - k^2) w + D p - Pr Ra T = 0,
s T - (D^2 - k^2) T - w = 0,          (dT0/dz = -1)
i k u + D w = 0,
u = w = T = 0 at z = -1/2 and z = +1/2.

Eigenvalue s = growth rate (real at onset). The wavenumber k is a Parameter, so ParametricEVP(r.problem, [Ra, k]) and critical(...) vary it without rebuilding (k > 0: k = 0 has no pressure gauge here). Chebyshev(z, N), dealias 1 (linear, constant coefficients). Taus as in the IVP: the second velocity tau of w is also lifted into the continuity equation (generalized tau).

Validation reference: Reid & Harris (1958): Ra_c = 1707.762 at k_c = 3.117 (tests/problems/test_problems_convection.py: critical() over k, N = 24).

Parameters:

N (int)
Ra (float)
Pr (float)
k (float)

Return type:

Ready