Finite-amplitude Thermal Convection Research Articles

Weakly nonlinear analysis of Rayleigh-Bénard convection in a non-Newtonian fluid between plates of finite conductivity: Influence of shear-thinning effects.

Finite-amplitude thermal convection in a shear-thinning fluid layer between two horizontal plates of finite thermal conductivity is considered. Weakly nonlinear analysis is adopted as a first approach to investigate nonlinear effects. The rheological behavior of the fluid is described by the Carreau model. As a first step, the critical conditions for the onset of convection are computed as a function of the ratio ξ of the thermal conductivity of the plates to the thermal conductivity of the fluid. In agreement with the literature, the critical Rayleigh number Ra(c) and the critical wave number k(c) decrease from 1708 to 720 and from 3.11 to 0, when ξ decreases from infinity to zero. In the second step, the critical value α(c) of the shear-thinning degree above which the bifurcation becomes subcritical is determined. It is shown that α(c) increases with decreasing ξ. The stability of rolls and squares is then investigated as a function of ξ and the rheological parameters. The limit value ξ(c), below which squares are stable, decreases with increasing shear-thinning effects. This is related to the fact that shear-thinning effects increase the nonlinear interactions between sets of rolls that constitute the square patterns [M. Bouteraa et al., J. Fluid Mech. 767, 696 (2015)]. For a significant deviation from the critical conditions, nonlinear convection terms and nonlinear viscous terms become stronger, leading to a further diminution of ξ(c). The dependency of the heat transfer on ξ and the rheological parameters is reported. It is consistent with the maximum heat transfer principle. Finally, the flow structure and the viscosity field are represented for weakly and highly conducting plates.

Retrograde equatorial surface flows generated by thermal convection confined under a stably stratified layer in a rapidly rotating spherical shell

Finite-amplitude thermal convection in a rapidly rotating spherical shell associated with a stably stratified layer placed near the outer surface is investigated. Systematic numerical experiments are performed with an Ekman number of E = 10−3, a Prandtl number of P = 1 and an inner/outer radius ratio of η = 0.4, and the existence of a strongly stratified upper layer is shown to enhance the generation of equatorial surface retrograde flows when the Rayleigh number is approximately ten times larger than the critical value. The existence of the stable layer causes the bottom of the stable layer to behave as a virtual boundary for the convective motion underneath. Its effective dynamic condition varies from the free-slip condition to the no-slip condition as the Rayleigh number increases. The Reynolds stress of the convective vortices beneath the stable layer is weakened and is dominated by the transport of the planetary angular momentum. As a result, the latitudinal temperature gradient produced at the bottom of the stable layer induces the equatorial retrograde flow through the thermal wind balance. This diffuses through the stable layer by viscosity and produces the equatorial surface retrograde flow.

Coriolis effect on convection for a low Prandtl number fluid

The problem of finite-amplitude thermal convection in a horizontal layer of a low Prandtl number heated from below and rotating about a vertical axis is studied. Linear stability and weak non-linear theories are used to investigate analytically the Coriolis effect on gravity-driven convection. The non-linear steady problem is solved by perturbation techniques, and the preferred mode of convection is determined by a stability analysis. Finite-amplitude results, obtained by using a weak amplitude, correspond to both stationary and oscillatory convections. These amplitude equations permit to identify from the post-transient conditions that the fluid is subject to Pitchfork bifurcation in the stationary convection and Hopf bifurcation in the oscillatory convection. Thereafter, in the small perturbations hypothesis, an amplitude solution is evaluated and drawn in time and space scales.

Finite amplitude thermal convection with variable gravity

Finite amplitude thermal convection is studied in a horizontal layer of infinite Prandtl number fluid with a variable gravity. For the present study, gravity is restricted to vary quadratically with respect to the vertical variable. A perturbation technique based on a small parameter, which is a measure of the ratio of the vertical to horizontal dimensions of the convective cells, is employed to determine the finite amplitude steady solutions. These solutions are represented in terms of convective modes whose amplitudes can be either small or of order unity. Stability of these solutions is investigated with respect to three dimensional disturbances. A variable gravity function introduces two non‐dimensional parameters. For certain range of values of these two parameters, double or triple cellular structure in the vertical direction can be realized. Hexagonal patterns are preferred for sufficiently small amplitude of convection, while square patterns can become dominant for larger values of the convective amplitude. Variable gravity can also affect significantly the wavelength of the cellular pattern and the onset condition of the convective motion.

Effect of Surface Corrugation on Convection in a Three-Dimensional Finite Box of Fluid Saturated Porous Material

The problem of finite amplitude thermal convection in a three-dimensional finite box of fluid saturated porous material is investigated, when the lower boundary of the fluid is corrugated. The nonlinear problem of three-dimensional convection in the box for the values of the Rayleigh number close to the classical critical value and for small values of the amplitude of the corrugations is solved by a perturbation technique. The preferred mode of convection is determined by stability analysis. In the absence of corrugation three-dimensional modes of convection can be either stable or unstable depending on the values of the aspect ratios of the box, while two-dimensional rolls are always stable, provided that the box aspect ratios allow the existence of such modes of convection. In the presence of boundary corrugation with the appropriate form, different three-dimensional or two-dimensional modes of corrugation can be stable or unstable. For a rough boundary with local roughness sites, the location, size, and number of the roughness elements plus the wave numbers of the convection modes and the box aspect ratios can all play a role leading to either stable or unstable particular three- or two-dimensional flow patterns. For a wavy boundary, resonant wave-vector excitation can lead to the preference of stable two- or three-dimensional flow patterns whose wave vectors are in a subset of those due to the wavy boundary, while nonresonant wave-vector excitation can lead to the preference of stable flow patterns whose wave vectors are not generally in a subset of those due to the wavy boundary. Heat transported by convection can either be enhanced or be reduced by certain proper forms of the corrugations and by appropriate values of the box aspect ratios. Due to the surface corrugation highly subcritical modes of convection are stable, while highly supercritical modes of convection are unstable.

Convection in a layer heated from below with a nearly insulating boundary

The problem of two- and three-dimensional finite amplitude thermal convection in a fluid layer heated from below is analyzed in the case when one boundary has a much higher and the other boundary has a much lower thermal conductivity than the fluid. Both stress-free and no-slip boundary conditions for the velocity field are considered. It is shown that two-dimensional convection rolls are stable for Rayleigh numbers not too far above the critical value ${R}_{c}$ for the onset of convection. But at a Rayleigh number ${R}_{\mathrm{II}}$ less than twice the critical value they become unstable and are replaced by hexaroll convection. (Here ``hexarolls'' denotes a new planform intermediate to hexagons and rolls.) At an even higher value of $R$ time dependent convection is found.

Finite bandwidth, long wavelength convection with boundary imperfectons: near‐resonant wavelength excitation

Finite amplitude thermal convection with continuous finite bandwidth of long wavelength modes in a porous layer between two horizontal poorly conducting walls is studied when spatially nonuniform temperature is prescribed at the lower wall. The weakly nonlinear problem is solved by using multiple scales and perturbation techniques. The preferred long wavelength flow solutions are determined by a stability analysis. The case of near resonant wavelength excitation is considered to determine the non‐modal type of solutions. It is found that, under certain conditions on the form of the boundary imperfections, the preferred horizontal structure of the solutions is of the same spatial form as that of the total or some subset of the imperfection shape function. It is composed of a multi‐modal pattern with spatial variations over the fast variables and with non‐modal amplitudes, which vary over the slow variables. The preferred solutions have unusual properties and, in particular, exhibit ‘kinks’ in certain vertical planes which are parallel to the wave vectors of the boundary imperfections. Along certain vertical axes, where some of these vertical planes can intersect, the solutions exhibit multiple ‘kinks’.

Finite amplitude thermal convection with spatially modulated boundary temperatures

Finite amplitude thermal convection in a fluid layer between two horizontal walls with different fixed mean temperatures is considered when spatially modulated temperatures with amplitudes L 1 * and L u * are prescribed at the lower and upper walls, respectively. The nonlinear steady problem is solved by a perturbation technique, and the preferred mode of convection is determined by a stability analysis. In the case of a resonant wavelength excitation, regular or non-regular multi-modal pattern convection can be preferred for some ranges of L 1 * and L u *, provided the wave vectors for such patterns are contained in a certain subset of the wave vectors representing a linear combination of modulated upper and lower boundary temperatures. In the case of non-resonant wavelength excitation, a three (two) dimensional solution in the form of multi-modal (rolls) pattern convection can be preferred, even if the boundary modulations are one (two) or two (one) dimensional, provided the wavelengths of the modulations are not too small. Heat transported by convection can be enhanced by boundary modulations in some ranges of L 1 * and L u *.

Preferred pattern of convection in a porous layer with a spatially non-uniform boundary temperature

The problem of finite-amplitude thermal convection in a porous layer between two horizontal walls with different mean temperatures is considered when spatially non-uniform temperature with amplitude L* is prescribed at the lower wall. The nonlinear problem of three-dimensional convection for values of the Rayleigh number close to the classical critical value is solved by using a perturbation technique. Two cases are considered: the wavelength γ(b)n of the nth mode of the modulation is equal to or not equal to the critical wavelength γc for the onset of classical convection. The preferred mode of convection is determined by a stability analysis in which arbitrary infinitesimal disturbances are superimposed on the steady solutions. The most surprising results for the case γ(b)n = γc for all n are that regular or non-regular solutions in the form of multi-modal pattern convection can become preferred in some range of L*, provided the wave vectors of such pattern are contained in the set of wave vectors representing the spatially non-uniform boundary temperature. There can be critical value(s) L*c of L* below which the preferred flow pattern is different from the one for L* > L*c. The most surprising result for the case γ(b)n ≠ γc and γ(b)n ≡ γ(b) for all n is that some three-dimensional solution in the form of multi-modal convection can be preferred, even if the boundary modulation is one-dimensional, provided that the wavelength of the modulation is not too small. Here γ(b) is a constant independent of n.

Rolls versus squares in thermal convection of fluids with temperature-dependent viscosity

We consider finite-amplitude thermal convection, in a horizontal fluid layer. The viscosity of the fluid is dependent upon its temperature. Using a weakly nonlinear expansion procedure, we examine the stability of two-dimensional roll and three-dimensional square planforms, in order to determine which should be preferred in convection experiments. The analysis shows that the roll planform is preferred for low values of the ratio of the viscosities at the top and bottom boundaries, but the square planform is preferred for larger values of the ratio. At still larger values, subcritical convection is predicted. We also include the effects of boundaries having finite thermal conductivity, which enables favourable comparison to be made with experimental studies. A discrepancy between the present work and a previous study of this problem (Busse & Frick 1985) is discussed.

Nonlinear thermal convection with finite conducting boundaries

Finite-amplitude thermal convection in a horizontal layer with finite conducting boundaries is investigated. The nonlinear steady problem is solved by a perturbation technique, and the preferred mode of convection is determined by a stability analysis. Square cells are found to be the preferred form of convection in a semi-infinite three-dimensional region Ω in the (γb,γt, P)-space (γb and γt are the ratios of the thermal conductivities of the lower and upper boundaries to that of the fluid and P is the Prandtl number). Two-dimensional rolls are found to be the preferred convection pattern outside Ω. The dependence on γb, γt and P of the heat transported by convection is computed for the various solutions analysed in the paper.

The transition from roll to square-cell solutions in Rayleigh–Bénard convection

We consider three-dimensional finite-amplitude thermal convection in a fluid layer with boundaries of finite conductivity. Busse & Riahi (1980) and Proctor (1981) showed that the preferred planform of convection in such a system is a square-cell tesselation provided that the boundaries are much poorer conductors than the fluid, in contrast to the roll solutions which are obtained for perfectly conducting boundaries. We determine here the conductivity of the boundaries at which the preferred planform changes from roll to square-cell type. We show that, for low-Prandtl-number fluids (e.g. mercury), square-cell solutions are realized only when the boundaries are almost insulating; while, for high-Prandtl-number fluids (e.g. silicone oils), square-cell solutions are stable when the boundaries have conductivity comparable to that of the fluid.

Nonlinear convection in a porous layer with finite conducting boundaries

The problem of finite-amplitude thermal convection in a porous layer with finite conducting boundaries is investigated. The nonlinear problem of three-dimensional convection is solved by expanding the dependent variables in terms of powers of the amplitude of convection. The preferred mode of convection is determined by a stability analysis in which arbitrary infinitesimal disturbances are superimposed on the steady solutions. Square-flow-pattern convection is found to be preferred in a bounded region [Gcy ] in the (γ b , γ t )-space, where γ b and γ t are the ratios of the thermal conductivities of the lower and upper boundaries to that of the fluid. Two-dimensional rolls are found to be the preferred pattern outside [Gcy ]. The qualitative features of the convection problem appear to be essentially symmetric with respect to γ b and γ t . The dependence of the heat transported by convection on γ b and γ t is computed for the various solutions analysed in the paper.

Planform selection by finite-amplitude thermal convection between poorly conducting slabs

Three-dimensional finite-amplitude thermal convection in a fluid layer is considered in the case where the boundaries of the layer are much poorer conductors than the fluid. It can be shown that if the conductive heat flux through the layer is not too large, the horizontal scale of motion is much greater than the layer depth. Then a ‘shallow water theory’ approximation leads to a nonlinear evolution equation for the leading-order temperature perturbation, which can be analysed in terms of a variational principle. It is proved that the preferred planform of convection is a square cell tesselation, as found in a rather more restricted parameter range by Busse & Riahi (1980), in contrast to the roll solutions that obtain for perfectly conducting boundaries. It is also shown that the preferred wavelength of convection increases slowly with amplitude.

Thermal evolution of the earth

Models to study the thermal evolution of the earth have been developed based on a calculation of finite-amplitude thermal convection in an incompressible and viscous mantle with temperature and pressure-dependent physical parameters. The lateral dependences of the variables are resolved through their spherical harmonic representations, while the radial and time dependences are calculated using numerical procedures. The major emphasis of this paper is to illustrate the effects of different factors on the thermal evolution of the earth rather than to present a definite earth model. The main conclusions achieved in this study are: 1. (1) A significant portion of the present-day temperature in the mantle and surface heat flux is due to initial heat of the earth resulting from accretion and core formation. 2. (2) Since the effective Rayleigh number of the earth's mantle is high, solid-state convection is found to be sufficiently efficient to maintain an almost adiabatic temperature gradient within the convecting region, while large temperature gradients exist across the thermal boundary layers (core—mantle boundary) and below the lithosphere. 3. (3) The convection is oscillatory with avalanche-type properties. This is due to the slow development of instability in the thermal boundary layer near the surface which causes the episodic behavior. The periods of the oscillations are about 50–250 m.y. The oscillation in the mantle also induces periodic features in the surface heat flux and the thickness of the lithosphere. Mantle viscosity structure plays an important role in the cooling rate and nature of oscillations. 4. (4) The mobility of the lithosphere is an important factor controlling the thermal state of the earth's interior. A mobile lithosphere will cool the interior faster.

Finite-amplitude thermal convection and geostrophic flow in a rotating magnetic system

An electrically conducting Boussinesq fluid is confined between two rigid horizontal planes. The fluid is permeated by a strong uniform horizontal magnetic field and the entire system rotates rapidly about a vertical axis. By heating the fluid from below and cooling it from above the system becomes unstable to small perturbations when the adverse temperature gradient becomes sufficiently large. Attention is restricted to small values of the Ekman number E and the ratio q of the thermal and magnetic diffusivities (see (1.2) and (1.3) below). In this parameter range marginal convection is steady and its character depends on the relative sizes of the Coriolis and Lorentz forces as measured by the parameter λ (see (1.1) below). When λ [ges ] 2/3½, motion consists of a single roll, whose axis is perpendicular to the applied magnetic field. On the other hand, when λ < 2/3½, two distinct rolls are possible: the axis of each roll lies oblique but with equal angle to the applied magnetic field. Only the latter case is discussed here.Once the Rayleigh number R exceeds its critical value Rc only one of the two sets of single rolls remains stable, while its squared amplitude increases linearly with R – Rc. For certain values of the parameters λ and τ (see (1.6) below) a second critical value may be isolated at which the system becomes unstable to unidirectional geostrophic flow perturbations aligned with the applied magnetic field. The instability sets in as either a steady or oscillatory shear flow dependent on the values taken by λ and τ. In both cases, after the secondary instability sets in, the roll amplitude remains largely insensitive to further increase in the Rayleigh number with the consequence that the geostrophic flow is stabilized. The amplitude of the shear, on the other hand, increases with R, adjusting its magnitude to ensure stability of the convection rolls.

Numerical approaches to solid-state convection in planetary bodies

Numerical methods suitable for the calculation of finite-amplitude thermal convection in planetary interiors are reviewed. Three methods of approach are described and compared; namely, finite-difference, finite-element and spectral procedures. Application of each approach to three-dimensional solid-state convection with complex rheology is examined. Finite-difference methods have enjoyed some success to date. However, future developments are likely to favor finite-element methods when rheology and boundaries are complex, and spectral methods when accuracy is paramount and rheology and boundaries are simple.

Non-linear and finite-amplitude thermal convection in a heterogeneous terrestrial planet

Summary. Equations governing non-linear and finite-amplitude convection in a heterogeneous planetary interior are developed. Using spherical harmonic expressions of variables, together with Green's function of Laplacian operator in a spherical coordinate, the equations are reduced to one-dimensional integro-differential equations and their numerical solu6ons are obtained by a finite-difference scheme. The theory is then applied to several lunar models and the following conclusions are obtained. (1) The mean temperatures and velocities of convecting zones of variable viscosity models are higher than those of constant viscosity ones. This is due to the development of lithospheres with 400-500 km thicknesses in the former models, which reduce heat loss considerably. (2) Molten regions are continuous shells in variable viscosity models whereas they become discontinuous and localized in a constant viscosity model. The continuous molten shells decrease lateral variations of tempera- ture significantly and tend to stabilize convection. (3) Lateral variations of viscosity have negligible effects on the thermal evolution of the models considered.

カーブした垂直温度分布をもち,しかもシアのある流れをもつ流体中の熱対流の有限振幅論1.理論

カーブした垂直温度分布をもち,しかもシアのある流れをもつ流体中の熱対流の有限振幅論1.理論

Viscous coupling in a spherical body of temperature-dependent viscosity

The finite-amplitude thermal convection in a spherical body with temperaturedependent viscosity is considered with emphasis on the coupling of different convection modes. An axisymmetric convection is assumed, where the lateral dependence of the variables are resolved through their harmonic presentations whereas the radial and timedependence of the variables are determined by a finite difference method. In the case of a constant or radially-dependent viscosity model the coupling is due to the non-linear term in the energy equation. The former model tends to cool throughout the body whereas the latter model provides a thicker lithosphere and higher mean temperature below. In a laterally heterogeneous viscosity model all of the convection modes couple in a complicated manner. But the coupling does not change the temperature distribution significantly in the case of a small, moon-type, planetary body.