Global Stability of Fluid Flows Despite Transient Growth of Energy

Verifying nonlinear stability of a laminar fluid flow against all perturbations is a central challenge in fluid dynamics. Past results rely on monotonic decrease of a perturbation energy or a similar quadratic generalized energy. None show stability for the many flows that seem to be stable despite these energies growing transiently. Here a broadly applicable method to verify global stability of such flows is presented. It uses polynomial optimization computations to construct nonquadratic Lyapunov functions that decrease monotonically. The method is used to verify global stability of 2D plane Couette flow at Reynolds numbers above the the energy stability threshold found by Orr in 1907 [The stability or instability of the steady motions of a perfect liquid and of a viscous liquid. Part II: A viscous liquid, Proc. R. Ir. Acad. Sect. A 27, 69 (1907)]. This is the first global stability result for any flow that surpasses the energy method.

Figure 1

Reynolds numbers (Re) at which laminar plane Couette flow is globally asymptotically stable against 2D perturbations of period $L$. Each symbol indicates values $(\mathrm{Re},L)$ where we verified stability using a quartic Lyapunov functional. Each functional depends explicitly on the flow’s projection onto 6, 8, 12, or 13 energy eigenmodes, and on the unprojected energy. Lines connect symbols to guide the eye. Orr’s energy stability threshold ${\mathrm{Re}}_E(L)$ is also shown.

Figure 2

Energy stability eigenmodes for $(\mathrm{Re},L)=(240,2)$. The top panel shows eigenvalues as a function of streamwise wave number $\alpha$. Shading indicates the minimum singular value of a boundary constraint matrix $\mathsf{B}$ (cf. the Supplemental Material [30]); black curves are zeros of this minimum, corresponding to eigenvalues. Eigenmodes consistent with $L=2$ occur at multiples of $\frac{2\pi }{L}$, marked by black dots. The mode with the $j$th largest eigenvalue among eigenmodes with wave number $\alpha =\frac{2\pi i}{L}$ is labeled $(i,j)$. Bottom panels show streamlines for selected modes. When $i\ne 0$ we include two modes, shifted by a quarter period in $x$, to span the relevant eigenspace. The 6-mode set consists of the (0, 0), (0, 1), (1, 1), and (1, 2) eigenmodes. The 8-mode set adds (2, 1), the 12-mode set adds (2, 2) and (3, 1), and the 13-mode set adds (0, 2).