Optimal energy density growth in Hagen–Poiseuille flow

Abstract

Linear stability of incompressible flow in a circular pipe is considered. Use is made of a vector function formulation involving the radial velocity and radial vorticity only. Asymptotic as well as transient stability are investigated using eigenvalues and ε-pseudoeigenvalues, respectively. Energy stability is probed by establishing a link to the numerical range of the linear stability operator. Substantial transient growth followed by exponential decay has been found and parameter studies revealed that the maximum amplification of initial energy density is experienced by disturbances with no streamwise dependence and azimuthal wavenumber n = 1. It has also been found that the maximum in energy scales with the Reynolds number squared, as for other shear flows. The flow field of the optimal disturbance, exploiting the transient growth mechanism maximally, has been determined and followed in time. Optimal disturbances are in general characterized by a strong shear layer in the centre of the pipe and their overall structure has been found not to change significantly as time evolves. The presented linear transient growth mechanism which has its origin in the non-normality of the linearized Navier–Stokes operator, may provide a viable process for triggering finite-amplitude effects.

The temporal evolution of neutral modes in the impulsively started flow through a circular pipe and their connection to the nonlinear stability of Hagen–Poiseuille flow

“…The general conclusion of these studies is that HPF is stable to all linear disturbances and that the least-damped modes at high Reynolds number have azimuthal wavenumbers N = 0 (axisymmetric) and N = 1. This latter mode is also found to give the largest amplification through transient growth (Schmid & Henningson 1994).…”

On a solenoidal Fourier–Chebyshev spectral method for stability analysis of the Hagen–Poiseuille flow

“…For example, in Table 1, the convergence of the least stable eigenvalue has been tested for Re = 9600, n = 1 and k = 1, a case previously studied by other authors [14,20]. For Re = 3000, the spectra for different values of k and n have been computed in order to make comparisons with a first comprehensive linear stability analysis carried out in [22]. Our code provided spectral accuracy in all the computed cases.…”

Non-modal stability in sliding Couette flow

“…The numerical method for the general eigenvalue problem in the form Ax = ωBx has been described by Canuto et al [25] and Schmid and Henningson [10]. For more details on the stability problem in pipe flows, we refer the reader to the works by Schimid and Henningson [2] and by Khorrammi et al [23]. After we solve the eigenvalue problem, an eigenmode expansion method can be used to compute the transient energy growth and the response to external excitations.…”

“…The computational method 066318-5 is a standard procedure, and we refer the reader to Refs. [2] and [9].…”

“…It is well known that the Poiseuille flow in a circular pipe (Hagen-Poiseuille flow) is linear stable, i.e., no critical Reynolds number exists above which disturbances grow exponentially [2,3]. However, in most experiments transition in a circular pipe flow is observed at Re ≈ 2000 [4], whereas, in more tightly controlled conditions, laminar flow can be maintained until Re ≈ 12000.…”