conatus.problems.pipe_evp

conatus.problems.pipe_evp(*, N, Re, m, k)[source]

Linear stability of Hagen–Poiseuille flow U0 = (1 - r^2) e_z, primitive variables.

Normal modes exp(i m phi + i k z + s t), k the axial wavenumber (alpha= is its deprecated v0.1 name):

s u + U0.grad(u) + u.grad(U0) + grad(p) - lap(u)/Re = 0,   div(u) = 0,   u = 0 at r = 1,

units of pipe() (Re = U_c R / nu). Eigenvalue s = growth rate + i frequency. Only the group plan.group(m, k) is meaningful: phi and z bases are ComplexFourier of sizes 2|m| + 4 and 4 (structural: they hold m and k), the z period is 2 pi / k. DiskBasis(r, N), dealias 1 (linear; U0 is exact). No-slip tau placed by add_bc(..., into=0).

Validation reference: Meseguer & Trefethen (2003), J. Comput. Phys. 186, 178, Table 1, Re = 1000: (m, k) = (1, 1) centre mode -0.0708640053 + 0.8467498288i, wall mode -0.0911426036 + 0.4691428789i (conjugates of the tabulated values: opposite sign convention of the frequency); (m, k) = (1, 0): -0.0146819706. Linearly stable at every Re. (tests/problems/test_problems_cylinders.py, N = 32.)

Parameters:

  • N (int)

  • Re (float)

  • m (int)

  • k (float)

Return type:

Ready