Perturbation dynamics in viscous channel flows

Abstract

Plane viscous channel flows are perturbed and the ensuing initial-value problems are investigated in detail. Unlike traditional methods where travelling wave normal modes are assumed as solutions, this work offers a means whereby arbitrary initial input can be specified without having to resort to eigenfunction expansions. The full temporal behaviour, including both early-time transients and the long-time asymptotics, can be determined for any initial small-amplitude three-dimensional disturbance. The bases for the theoretical analysis are: (a) linearization of the governing equations; (b) Fourier decomposition in the spanwise and streamwise directions of the flow; and (c) direct numerical integration of the resulting partial differential equations. All of the stability criteria that are known for such flows can be reproduced. Also, optimal initial conditions measured in terms of the normalized energy growth can be determined in a straightforward manner and such optimal conditions clearly reflect transient growth data that are easily determined by a rational choice of a basis for the initial conditions. Although there can be significant transient growth for subcritical values of the Reynolds number, it does not appear possible that arbitrary initial conditions will lead to the exceptionally large transient amplitudes that have been determined by optimization of normal modes when used without regard to a particular initial-value problem. The approach is general and can be applied to other classes of problems where only a finite discrete spectrum exists (e.g. the Blasius boundary layer). Finally, results from the temporal theory are compared with the equivalent transient test case in the spatially evolving problem with the spatial results having been obtained using both a temporally and spatially accurate direct numerical simulation code.

Accretion disk instability revisited

“…Umurhan & Regev (2004) have performed numerical simulations of disk flow targeting the evolution of transient growth. In a related study (Criminale et al 1997), direct numerical solutions of the transient dynamics of pCf were able to reproduce all known stability data, but also found strong sensitivity to the form of the initial conditions, viz. : initial perturbations lacking normal vorticity gave inaccurate growth factors and unrealistic flow properties.…”

Direct Numerical Simulation of Stochastically Forced Laminar Plane Couette Flow: Peculiarities of Hydrodynamic Fluctuations

“…Actually, according to Refs. [13][14][15], the rigid boundaries affect the dynamics of perturbations at different l x independently. Analyzing the linear problem with Kelvin modes [16,17], one can say, that at the dynamics of initial (original) perturbations, boundaries induce ''secondary'' perturbations with the same l x to cancel the ''original'' perturbations on the boundaries [to satisfy no-slip boundary conditions, see Eq.…”

The Century of Turbulence Theory: The Main Achievements and Unsolved Problems

“…Since the available experimental and numerical estimates of 'intermittency corrections' are scattered and small, the reliable verification of Eqs. (18), (19) is apparently impossible at present. Let us consider, for example, the situation relating to the 'intermittency correction' £ 2 corresponding to the second-order structure function D 2 (r).…”

“…(17) may be compared with proposed by Barenblatt, Chorin and Goldenfeld Eqs. (18) and (19) which validity also must reflect some unknown symmetry features of flow structures determining the velocity differences. Moreover, Eq.…”

“…During the last twenty years many dozens of papers about such growth were published (papers [14][15][16][17][18][19][20] represent only a few examples of them), while much attention to this topic was also given in books [21,22] and a survey [23]. It was shown, in particular, that transient growth of nonmodal disturbances may exceed very much the growth of the linearly unstable wave modes.…”