Non-modal analysis: transient growth

Shear flows are governed by non-normal operators: perturbations can grow by orders of magnitude before decaying, although every eigenmode decays. This growth is measured in a declared norm.

The norm

All quantities are measured in the kinetic energy of the perturbation,

\[ E = \tfrac12 \sum_i w_i \int |s_i|^2\, dy , \]

with declared weights \(w_i\) and the exact Gram matrix of the Chebyshev polynomials. With \(F\) a Cholesky factor, \(\lVert F x\rVert_2^2 = E(x)\), and every operator norm below is a 2-norm after this change of variables.

From the tau pencil to a generator

The discretised linearised Navier–Stokes equations form a pencil \(M x' + L x = 0\) with infinite eigenvalues (continuity, boundary rows, taus). An ordered generalised Schur decomposition separates the finite part,

\[ (-L) Z = Q S, \quad M Z = Q T, \qquad x = Z_1 q, \quad T_{11} q' = S_{11} q , \]

and a thin QR of \(F C Z_1\) (with \(C\) the map to the physical state) gives the generator \(\tilde A\) in coordinates orthonormal for the energy. Then

\[ \lVert e^{tA}\rVert_E = \lVert e^{t\tilde A}\rVert_2 , \]

without any truncated eigenvector expansion.

Quantities

  • Transient growth \(G(t) = \max_{x_0} E(t)/E(0) = \lVert e^{t\tilde A}\rVert_2^2\) and its optimal initial condition (leading right singular vector).

  • Numerical abscissa: the initial slope \(\max \lambda(\tilde A + \tilde A^H)/2\).

Time-periodic base flows

For a base flow periodic in time the propagator \(\Phi(t_0, T)\) is built by the same implicit–explicit schemes as the time stepper, applied to all unit initial states at once: the base-flow terms are explicit and evaluated at the exact stage times, the time step is \(T/n\) exactly, and the gain is the largest singular value of \(F\Phi F^{-1}\) restricted to admissible initial states (those satisfying continuity and the boundary conditions). Whether inadmissible inputs are allowed is an explicit choice.

References

  • L. N. Trefethen, A. E. Trefethen, S. C. Reddy and T. A. Driscoll, Hydrodynamic stability without eigenvalues, Science 261 (1993) 578–584.

  • K. M. Butler and B. F. Farrell, Three-dimensional optimal perturbations in viscous shear flow, Phys. Fluids A 4 (1992) 1637–1650.

  • S. C. Reddy and D. S. Henningson, Energy growth in viscous channel flows, J. Fluid Mech. 252 (1993) 209–238.

  • P. J. Schmid, Nonmodal stability theory, Annu. Rev. Fluid Mech. 39 (2007) 129–162.