The claim
Let U(y) be the laminar profile between no-slip walls, U′ = 1 for plane Couette and U′ = −2y for plane Poiseuille flow, and let L be the linearized Navier–Stokes operator about it, with perturbations measured in the kinetic-energy norm. There are constants c, C > 0, independent of the Reynolds number R, such that:
- sup_t ‖e^{tL}‖ ≥ cR;
- ‖L⁻¹‖ ≥ cR², so the maximum resonance over real frequencies is at least cR²;
- at any R where L is eigenvalue-stable, some perturbation E with ‖E‖ ≤ CR⁻² gives L + E an eigenvalue with positive real part.
Derivation
- Decoupling. At streamwise wavenumber zero the linearized equations are ∂_t u + U′v = R⁻¹∇²u for the streamwise velocity, and ∂_t(v, w) = −∇p + R⁻¹∇²(v, w) with ∂_y v + ∂_z w = 0 for the cross-stream velocities. The (v, w) problem involves neither u nor U.
- Rescaling. Put t = Rτ and p = p̂/R. Then (v, w) as a function of τ solves a Stokes problem with no R in it. The streamwise velocity solves ∂_τ u = ∇²u − RU′v, so with u(0) = 0, u = Rû(τ) with û independent of R.
- Growth. In the energy norm, ‖q(Rτ)‖² = R²‖û(τ)‖² + ‖(v, w)(τ)‖². Take ‖(v, w)(0)‖ = 1, with v(0) nonzero where U′ ≠ 0. Then ∂_τ û(0) = −U′v(0) ≠ 0, so ‖û(τ*)‖ = c > 0 for some τ*, and ‖e^{tL}‖ ≥ cR at t = Rτ*.
- Resonance. Take steady forcing f = (0, f_v, f_w). Then (v, w) = RSf, with S the R-independent steady Stokes solve, and u = RΔ⁻¹(U′v) = R²Δ⁻¹U′Sf. For an f with Δ⁻¹U′Sf ≠ 0, ‖L⁻¹f‖ ≥ cR²‖f‖.
- Destabilization. For a closed operator, z is an eigenvalue of L + E for some E with ‖E‖ ≤ 1/‖(z − L)⁻¹‖. This is the pseudospectral characterization, used here, as in LIT-932, for the discretized operator. The resolvent is continuous in z, so ‖(δ − L)⁻¹‖ ≥ cR²/2 for small δ > 0. Some ‖E‖ ≤ 2/(cR²) then puts an eigenvalue at δ.
What it does not say
- That the growth is of order R and no more, or what the constants are. Table 1 of LIT-932, R/29.1, R/71.5, (R/8.12)² and (R/17.4)², comes from computation. That is THEORY-304’s, Proposed.
- Where the flows are eigenvalue-stable. All R for Couette flow and R < 5772 for Poiseuille flow is the source’s statement. The third bound is stated conditionally on it.
- Anything about transition. That transition happens through this growth plus nonlinear mixing is LIT-932’s conjecture, and is not filed.
- Anything in other norms. Non-normality is norm-relative (THEORY-323).