An extended Chebyshev pseudo‐spectral benchmark for the 8:1 differentially heated cavity

Abstract

AbstractOur contribution to the benchmark is multifold. In addition to providing accurate unsteady simulations at the required Ra value of 3.4 × 105, we determine accurately the three first critical bifurcation points, investigate the supercritical regime, and study the differences between time‐averaged solutions and the corresponding base solution at Ra = 4× 105. We thereby establish the existence of, at least, 4 different branches of solutions and of 3 multiple unsteady periodic solutions for a Rayleigh value of 4 × 105. First appearance of quasi‐periodic flow is found at Ra about 5.0 × 105 and first appearance of chaotic solutions is found for 5.5 × 105 approximately. We investigate the differences between time‐averaged solutions and the corresponding base flow solution at Ra = 4 × 105. It is found that they exhibit symmetric feature and similar spatial distribution and that their amplitudes are proportional to the squared amplitude of periodic solutions.All these computations are carried out using 2D Chebyshev spatial approximations with spatial resolution up to 48 × 180. For the unsteady computations, a second order time stepping scheme is used, the incompressibility condition being strictly enforced through the use of an influence matrix technique, while for the accurate determination of the critical points, an original algorithm based on a combination of Newton, continuation and Arnoldi Krylov type methods was developed.Furthermore we investigate the stability of the 2D benchmark solution with respect to 3D periodic disturbances by using a spectral (Chebyshev and Fourier) time‐stepping (projection) code. In the 8:1 cavity we did not find any 3D instability before the onset of time dependence of 2D flows nor at the mandatory Ra of 3.4 × 105. 3D instabilities were observed for Ra about 4× 105 corresponding to a typical wavelength of two times the cavity width. Copyright © 2002 John Wiley & Sons, Ltd.