Abstract
Abstract
This paper describes a simple spectral method of examining both modal and nonmodal stability of the laminar flow in cylindrical pipes. In particular, we present a rigorous method to find whether a perturbation of linearly stable laminar flow is also nonmodally stable, i.e., does not temporarily grow before its final exponential decay. We also discuss possible scenarios of intermittency in nonmodally unstable cases and the reason why stability persists for larger values of Reynolds number
R
if perturbations are very small.
Using an orthonormal basis of eigenfunctions of the Leray projection of Laplace operator we transform the Navier–Stokes equation (NSE) into a simple dynamical system, present a finite analytical expression for the elements of the matrix M of the linear approximation d
Δ
v/dt = M
Δ
v around its laminar solution and show that the physical meaning of these elements corresponds to energy flow to each perturbation mode from other modes and from the laminar flow. All the analytical results presented in this paper are mathematically rigorous consequences of NSE.
In sections 5 and 6 we also discuss some pilot numerical results that are tabulated in full in the supplementary material to this paper. These results support the assumption that the eigenvalues of M have negative real parts, that for large
R
matrix M is strongly nonnormal and that the nonmodal instability occurs for moderate values of
R
. The full numerical results as well as the source program and its detailed documentation are in the supplementary material of this paper.