Pattern selection for Darcy-Bénard convection with local thermal nonequilibrium
Introduction
We shall consider how local thermal nonequilibrium (LTNE) effects affect the classical Darcy-Bénard problem, i.e. a uniform porous layer of constant thickness which is heated from below and cooled from above, and where the bounding temperatures are held constant in both time and space. Local thermal non-equilibrium is that special state whereby the use of a single heat transport equation is insufficient to model the microscopic heat transfer between the solid and fluid phases; in such cases two heat transport equations are used, one for each phase, and the system is closed by the addition of source/sink terms that are proportional to the temperature difference between the phases. This type of source/sink model was first introduced by Anzelius [1] and Schumann [2] in the 1920s and it continues to be used extensively to this day.
Three new parameters appear when local thermal nonequilibrium is being considered: H, which is a scaled interphase rate of heat transfer, γ, which is a porosity-weighted conductivity ratio and α, a thermal diffusivity ratio. Generally, local thermal equilibrium (LTE) corresponds to having sufficiently large values of either H or γ. It is often thought that a large conductivity contrast between the phases implies LTE, but the correctness of this assertiondepends on the magnitude of any externally imposed time scales; see [3]. Likewise, it is often assumed that LTE also arises when the flow is steady, but the works of Rees et al[4]. and Rees and Bassom [5] on the flushing of cold fluid using a hot fluid source, and the free convective boundary layer analyses of Rees and Pop [6] and Rees [7], show that LTNE effects persist for a wide range of values of both H and γ. Finally, we note that LTE is approached as the ratio between the microscopic and the macroscopic length scales tends to zero; the smaller microscales allow for a more rapid exchange of heat between the phases [8], [9]. But these matters are well-known and are now quite well-developed; see Nield and Bejan [10], Rees and Pop [11] and Kuznetsov [12].
In the general field of stability theory it is usual for studies to begin with the description of the basic state, followed by a consideration of the onset of instability by means of a linearised analysis. Then various types of nonlinear analysis are used, such as weakly nonlinear theory, energy stability theory and fully numerical simulations. But for the title topic, namely the effect of local thermal nonequilibrium on Darcy-Bénard convection, the very first publications presented some strongly nonlinear computations without the benefit of knowing the context provided by a linearised theory; see Combarnous [13] and Combarnous and Bories [14]. It was some time later that Banu and Rees [15] embarked on a comprehensive study of the onset problem. Like the classical Darcy-Bénard problem, there is an analytical expression for the critical Darcy-Rayleigh number, but a simple Newton-Raphson scheme is then required to minimise its value over thewavenumber.
A rather large number of papers have followed the appearance of Banu and Rees [15] by solving the linear instability problem for different variations of the Darcy-Bénard problem with LTNE. For example, Postelnicu and Rees [16], Postelnicu [17] and Malashetty et al[18]. considered how Brinkman effects alter the results of Banu and Rees [15]; generally, the addition of these effects serves to increase the critical Darcy-Rayleigh number because of the additional resistance to flow. Barletta and Rees [19] considered the effect of isoflux boundaries, and found that there is a region in (H, γ)-space where the critical wavenumber is zero, a result which is consistent with the LTE form of the stability problem discussed in §6.2 of Nield and Bejan [10]. However, in the remaining part of parameter space the critical wavenumber is nonzero. Nouri-Borujerdi et al. [20] studied the consequence of having internal heat generation within the layer, while a very recent work by Lagziri and Bezzazi [21] considers a case which effects a transition between cases studied by Banu and Rees [15] and Barletta and Rees [19] by employing thermal boundary conditions of the third kind. Celli et al. [22] have also considered the effect of an open upper surface. The work of Banu and Rees [15] has also been re-examined by Straughan [23] using an energy stability theory and this work confirms the absence of subcritical instabilities. Further papers may also be found which also include the effects of inclination, of anisotropy, rotation, unsteady heating and so on.
In the present paper our focus is solely on the use of a weakly nonlinear analysis of the cross-roll instability to determine whether rolls always form the preferred convective planform. The identity of the preferred pattern will inform the direction of further work on this topic.
Section snippets
Governing equations
The main interest of this study is to investigate the onset and subsequent development of Darcy-Bénard convection in a horizontal porous layer when the solid and fluid phases are not in local thermal equilibrium. Thus we consider unsteady three-dimensional convection when the local thermal non-equilibrium (LTNE) two-field model is valid, and where the lower bounding surface is held at the constant temperature, Thot, while the upper surface is held at the lower temperature, Tcold. The governing
Perturbation equations
It is seen readily that the full system, Eqs. (10)–(12), is satisfied by the solution,We may now concentrate on the behaviour of convection cells by perturbing about this basic state: letwhere the disturbances, P, Θ and Φ, are asymptotically small in magnitude. Our analysis of the cross-roll instability begins by making the assumption that the Darcy-Rayleigh number is slightly larger than the critical value for the onset of convection. More
Linear stability theory
The linear stability analysis corresponds to solving for the O(ϵ) terms in Eq. (24). It is this which forms the subject of the work of Banu and Rees [15] and the following will summarise and show how their work is modified when using the present pressure/temperature formulation.
The O(ϵ) terms may be shown to satisfy,These equations may be found to have the following roll
Weakly nonlinear analysis
The aim now is to determine whether or not the convective roll which has been described in §4 forms the stable planform of convection. We shall not present a study of the Eckhaus or sideband instability of rolls — such an analysis is essentially identical in form to that presented in Newell and Whitehead [24] for the Rayleigh-Bénard problem. In such an analysis the band of stable wavenumbers for slightly postcritical roll convection is centred on the critical wavenumber and is of of the
Conclusions
We have undertaken a weakly nonlinear analysis of the onset of convection in the classical Darcy-Bénard problem where local thermal non-equilibrium effects are significant. Whilst the Eckhaus and zigzag instabilities may also be considered, the symmetries of the configuration are such that the conclusions of Newell and Whitehead [24] for the Rayleigh-Bénard problem also apply here and therefore we have not covered this aspect. Rather, we have considered the cross-roll instability mechanism in
Declaration of Competing Interest
The authors have no conflict of interest.
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