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

Abstract

SUMMARYOur contribution to the benchmark is multifold. In addition to providing accurate unsteady simulations at the required Ra value of 3:4 × 10 5 , we determine accurately the three ÿrst critical bifurcation points, investigate the supercritical regime, and study the di erences between time-averaged solutions and the corresponding base solution at Ra = 4 × 10 5 . We thereby establish the existence of, at least, 4 di erent branches of solutions and of 3 multiple unsteady periodic solutions for a Rayleigh value of 4 × 10 5 . First appearance of quasi-periodic ow is found at Ra about 5:0 × 10 5 and ÿrst appearance of chaotic solutions is found for 5:5 × 10 5 approximately. We investigate the di erences between time-averaged solutions and the corresponding base ow solution at Ra = 4 × 10 5 . 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 in uence 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 ÿnd any 3D instability before the onset of time dependence of 2D ows nor at the mandatory Ra of 3:4 × 10 5 . 3D instabilities were observed for Ra about 4 × 10 5 corresponding to a typical wavelength of two times the cavity width.

Periodic orbits in tall laterally heated rectangular cavities

“…The critical eigenfunctions at these points have also been compared, and can be seen in Ref. [15] and in Fig. 1b-d.…”

“…In order to check the code, the critical parameters for the appearance of oscillatory flows were compared with those given in Ref. [15], where primitive variables to formulate the problem and a code for the continuation of fixed points were used. Some of them are shown in Table I.…”

“…To facilitate the comparison, all the frequencies are expressed in thermal units taking into account that ω N S = √ RaP r ω XQ , where the subscript N S refers to the adimensionalization of the present paper and XQ to that of Ref. [15]. With any of the two resolutions shown, the relative errors of the critical Rayleigh numbers, Ra i c , i = 1, 2, 3, and frequencies ω i c , i = 1, 2, are below 0.01%, and only ω 3 c differs 0.5%.…”

“…Meth. Fluids (see [3,8,[12][13][14][15], among others). To compare the results several point data, temporal and spatial time averages, and fluctuations with respect to the mean values, of the velocity, vorticity, pressure differences and temperature were supplied by the contributors, in addition to the period, mean heat flux transport on the lateral boundaries, or a metric to measure the loss of the center-symmetry of the temperature field, although the full description of the solutions of this problem remained open.…”

Fast iterative solvers for buoyancy driven flow problems

“…Xin and Le Quéré [22], for the initial-value problem (1.1)-(1.5) uses semiimplicit time integration. This leads to a relatively simple linear algebra (Poisson or Stokes-type) problem at every time step, but there is a CFL stability restriction on the maximum time step size.…”