Nonlinear stability analysis of plane Poiseuille flow by normal forms

In the subcritical interval of the Reynolds number 4320\leq R\leq R_c\equiv 5772, the Navier--Stokes equations of the two--dimensional plane Poiseuille flow are approximated by a 22--dimensional Galerkin representation formed from eigenfunctions of the Orr--Sommerfeld equation. The resulting dynamical system is brought into a generalized normal form which is characterized by a disposable parameter controlling the magnitude of denominators of the normal form transformation. As rigorously proved, the generalized normal form decouples into a low--dimensional dominant and a slaved subsystem. {}From the dominant system the critical amplitude is calculated as a function of the Reynolds number. As compared with the Landau method, which works down to R=5300, the phase velocity of the critical mode agrees within 1 per cent; the critical amplitude is reproduced similarly well except close to the critical point, where the maximal error is about 16 per cent. We also examine boundary conditions which partly differ from the usual ones.Comment: latex file; 4 Figures will be sent, on request, by airmail or by fax (e-mail address: rauh at beta.physik.uni-oldenburg.de

Simulation of Cavity Flow by the Lattice Boltzmann Method

A detailed analysis is presented to demonstrate the capabilities of the lattice Boltzmann method. Thorough comparisons with other numerical solutions for the two-dimensional, driven cavity flow show that the lattice Boltzmann method gives accurate results over a wide range of Reynolds numbers. Studies of errors and convergence rates are carried out. Compressibility effects are quantified for different maximum velocities, and parameter ranges are found for stable simulations. The paper's objective is to stimulate further work using this relatively new approach for applied engineering problems in transport phenomena utilizing parallel computers.Comment: Submitted to J. Comput. Physics, late