Non-modal analysis

Transient growth of a linear operator, measured in the kinetic-energy norm.

A stable linear flow can amplify a perturbation by orders of magnitude before it decays, because its eigenvectors are not orthogonal. Non-modal analysis measures that amplification in a declared norm, here the kinetic energy \(E = \tfrac12\int |\boldsymbol u|^2\,\mathrm dy\) computed with its exact Gram matrix.

The operator

from conatus.expert import Y, max_growth, parallel_shear_flow, state_space

# Plane Poiseuille flow, streamwise-constant mode kx = 0, kz = 2, Re = 1000
flow = parallel_shear_flow(1 - Y**2, kx=0.0, kz=2.0, Re=1000.0, walls="no-slip")
desc = flow.discretize(48)                     # 48 Chebyshev coefficients per variable
ss = state_space(desc, infinite_tol=1e-10)     # energy-orthonormal generator
print(ss.explain())

parallel_shear_flow builds the linearized Navier–Stokes equations about \(U(y)\,\boldsymbol e_x\) in primitive variables for one pair of wavenumbers; the base flow is a symbolic expression in Y. state_space removes the infinite eigenvalues of this descriptor system and writes its generator \(\tilde A\) in energy-orthonormal coordinates, so that 2-norms are energy norms.

Optimal transient growth

\[ G(t) = \max_{q_0} \frac{E(t)}{E(0)} = \|e^{t\tilde A}\|_2^2 \]
G, t_opt = max_growth(ss, (1.0, 200.0), xtol=1e-6, n_scan=60)
print(f"G_max = {G:.2f} at t = {t_opt:.1f}")

A scan of n_scan times is followed by a bounded Brent search whose tolerance xtol you give. For a time-periodic base flow, transient growth uses the propagator of the time stepper instead (next section).

Any time-periodic base flow

Linearize the model about a base flow that depends on \(t\) and give its period. The gain over \([t_0, t_0 + T]\) and the Floquet exponents of one wavenumber then follow from the same operator:

  • The base flow \(U(y, t)\) is split into time harmonics \(U = \sum_m U_m(y)\, e^{2\pi i m t/P}\) (a discrete Fourier transform over 64 instants per period); explain() lists the harmonics kept and the size of the largest one dropped, and a base that is not smooth in \(t\) is refused.

  • optimal_gain integrates all unit initial states at once with the implicit–explicit scheme you name (RK111, RK222, RK443), the base-flow terms explicit at the exact stage times and \(dt = T/n_\text{steps}\) exactly. The gain is maximised over admissible initial states (those satisfying continuity and the boundary conditions) in the kinetic energy with its exact Gram matrix; g.seed and g.response are the optimal structures.

  • floquet(method="hill", Nh=..., targets=[...], tol=1e-12) solves the Hill problem of the same matrices (exponents nearest each target, residuals reported; tol, the Arnoldi stopping tolerance, is required); method="monodromy" takes the eigenvalues of the time-stepped monodromy matrix instead.

  • eigenvalues() is refused for a time-periodic base: it has no normal modes.

  • With real periodic axes (cn.periodic(L)), one wavenumber group holds the waves \((k_x, \pm k_z)\) together; complex=True separates them.

Expert form, with a linear problem per wave that you write yourself: conatus.growth.optimal_gain(build, base_harmonics, kx, kz, t0, T, steps, scheme, backend, fields=("u", "v", "w")), where build({"kx": ..., "kz": ...}) returns the IR problem and base_harmonics maps each base-flow field to {m: U_m} (the same input as hill_from_plan); floquet_exponents takes the same arguments (Floquet analysis).

What you choose

The norm (through the weights of the system), the resolution, the time window and the tolerance of the growth search. The threshold below which an eigenvalue counts as infinite is an argument (infinite_tol), reported with the gap that separates the two families.

See also: state_space, max_growth, the gallery case Non-modal stability of plane Poiseuille flow: transient growth and the continuous adjoint (Butler & Farrell 1992) and the mathematics in Non-modal analysis: transient growth.