Gallery: fluids everywhere¶
One script per canonical case, written as in the paper that introduced it: the equation, physical
comments, explain(), and a result checked against a published value or a closed form: each
script ends with an assert against that value, with its tolerance, so it stops with an error if
the value is not reproduced. The pages below are generated from
examples/ at each build; the values are those recorded in examples/README.md.
Case |
Obtained |
Reference |
|---|---|---|
Weakly nonlinear theory derived from the IR: Landau and Ginzburg–Landau coefficients |
— |
— |
Rayleigh–Bénard onset in 3D: the nonlinear Boussinesq equations, linearized by the IR |
Ra = 1707.76178 at |
k |
Onset of rotating Rayleigh–Bénard convection on the f-plane (Chandrasekhar 1961, Ch. III) |
one pencil for all Ta (√Ta declared); free walls: Ra_c(Ta) = closed form to ≤ 2e-15 (Ta = 0…1e5); rigid: 1756.347 / 2151.341 / 4712.042 (Ta = 1e2, 1e3, 1e4) |
Chandrasekhar 1961, ch. III |
R_S/τ − R_T = 27π⁴/4 to ≤ 4e-16 (4 pairs) |
Stern 1960; Baines & Gill 1969, JFM 37 |
|
two-mode model: RPO “1” in the slice, T = 3.6415119771, θ = 0.0809697054; Lorenz homoclinic orbit as a connection, r = 13.9265574075 |
Budanur, Borrero-Echeverry & Cvitanović (arXiv:1411.3303), Table I: 3.64151221; Sparrow 1982: 13.926 |
|
Womersley flow in a plane channel: exact profile (LBVP and IVP) and Floquet stability (Hill) |
profile to 9e-16 (N = 24); RK443 order 2.85, SBDF2 1.95; Floquet ε = 0 = Orszag to 1e-9, O(ε²) deviation |
Womersley 1955; Orszag 1971, JFM 50, Table 3 |
Turing instability of the Brusselator (Prigogine & Lefever 1968) |
b_c to ≤ 9e-16 relative (3 sets) |
Prigogine & Lefever 1968; Peña & Pérez-García 2001, PRE 64 |
Complex Ginzburg–Landau equation: Benjamin–Feir–Newell and Eckhaus instabilities of plane waves |
complex form (conj() on the LHS, doubled system declared); BFN threshold: 1 + c₁c₃ < 1e-12 (Richardson k → 0); Eckhaus q_E² to 1e-7 |
Newell 1974; Aranson & Kramer 2002, RMP 74 |
Swift–Hohenberg: onset of rolls, weakly nonlinear amplitude, roll branch by continuation, 2D rolls |
threshold (r, q) on a single pencil (q a parameter, RealFourier); A(r = 0.1) = 0.3652433 (A² − 4r/3 → r²/144); periodic NLBVP (declared phase) = odd subspace to 1e-12; 2D IVP on the branch to 1e-9 |
Swift & Hohenberg 1977; Cross & Hohenberg 1993 |
Kelvin–Helmholtz instability of the tanh mixing layer (Michalke 1964), inviscid and viscous |
max ω_i = 0.189702 at α = 0.44492 (L = 20, N = 384); viscous → inviscid in O(1/Re) |
Michalke 1964, JFM 19, Table 1: 0.1897 at α = 0.4446 |
Non-modal stability of plane Poiseuille flow: transient growth and the continuous adjoint |
G_max = 4896.948 at t = 379.075 (Re = 5000, kz = 2.044; N = 48–96 identical to 1e-10); continuous vs discrete adjoint |
Butler & Farrell 1992; Reddy & Henningson 1993: 4897 at t = 379 |
wavemaker maximum S = 11.4856 at y = ±0.87387 (critical layer U = c_r); derivative of the pencil = Richardson finite difference to 5e-11; density (continuous adjoint) → pencil to 2e-12 (N = 80) |
no exact published value in a channel; closed forms of Chomaz (2005) for Ginzburg–Landau ( |
|
Stratified Kelvin–Helmholtz: Taylor–Goldstein equation for the tanh/tanh layer (Hazel 1972) |
whole line (tanh map, L = 2/α) + shifted complex path: J_n(1/2) = 1/4 to 2.5e-12 (N = 192) and J_n(0.3) = 0.21 to 3.8e-9 by Newton on c_i(J) = 0, without extrapolation, independent of L and δ; J = 0.3: every mode damped |
Hazel 1972, JFM 51 (R = 1); Miles 1961 / Howard 1961: J_c = 1/4 |
Where a script meets a capability that the library does not have, it says so and says what it does instead; nothing is worked around silently.