Galerkin technique based on beam functions in application to the parametric instability of thermal convection in a vertical slot

Abstract

SUMMARYA Fourier-Galerkin spectral technique for solving coupled higher-order initial-boundary value problems is developed. Conjugated systems arising in thermoconvection that involve both equations of fourth and second spatial orders are considered. The set of so-called beam functions is used as basis together with the harmonic functions. The necessary formulas for expressing each basis system into series with respect to the other are derived.The convergence rate of the spectral solution series is thoroughly investigated and shown to be fifth-order algebraic for both linear and nonlinear problems. Though algebraic, the fifth-order rate of convergence is fully adequate for the generic problems under consideration, which makes the new technique a useful tool in numerical approaches to convective problems.An algorithm is created for the implementation of the method and the results are thoroughly tested and verified on different model examples. The spatial and temporal approximation of the scheme is tested. To further validate the scheme, a singular asymptotic expansion is derived for small values of the modulation frequency and amplitude and the numerical and analytic results are found to be in good agreement.The new technique is applied to the G-jitter flow, and the Floquet stability diagrams are produced. We obtain the expected alternating isochronous and subharmonic branches and find that stable motions are always isochronous while unstable motions can be either isochronous or subharmonic. The numerical investigation also leads to novel conclusions regarding the dependence of the amplitude of the solutions on some of the governing parameters.

Two-Gradient Convection in a Vertical Slot with Maxwell-Cattaneo Heat Conduction

“…Following [11,13], we consider natural convection in a vertical fluid with differentially heated lateral surfaces. The x-axis is across the slot and y is taken as the vertical coordinate.…”

“…which are termed the Cattaneo, Rayleigh, and Prandtl numbers, respectively. It can be shown that in terms of the stream function (see [11,13] and the literature cited therein), the ID equation of motion reads…”

“…Our basis functions are Lord Rayleigh's beam functions [15] and the trigonometric functions. The suitability of these two sets of functions, for problem of this kind, has been well-established (see [13] and the literature cited therein). All relevant formulas for the expansion of the various derivatives of the beam-functions into series with respect to the system, as well as the cross-expansion formulas, are given in the Appendix for the convenience of the reader.…”

“…The stability and accuracy of this scheme (Eq. (16)) are discussed in [13], where it is shown that the convergence rate of the spectral coefficients pi, dt is fifth-order algebraic for FHC; it is a straightforward task to extend those results to the MCHC case. For the g-jitter case, the main effect is that the introduction of time varying gravity can bring about a parametric instability.…”

“…To better understand the consequences of this material invariant (or frame indifferent) formulation, in this work, we study the flow of an incompressible Newtonian fluid in a vertical slot subject to both a vertical and a horizontal temperature gradient [9]. It should be noted that this flow was thoroughly investigated in [10,11,12,13] under FHC and with time dependent gravity (the so-called g-jitter flow). To highlight the role of the different terms in the objective rate, we at first investigate the limiting case of constant gravity using a truncated Galerkin expansions.…”