Showing papers on "Rayleigh number published in 1998"
TL;DR: In this article, the authors investigated the Coriolis effect on three-dimensional gravity-driven convection in a rotating porous layer heated from below, and showed that the higher the viscosity, the less stable the fluid.
Abstract: Linear stability and weak nonlinear theories are used to investigate analytically the Coriolis effect on three-dimensional gravity-driven convection in a rotating porous layer heated from below. Major differences as well as similarities with the corresponding problem in pure fluids (non-porous domains) are particularly highlighted. As such, it is found that, in contrast to the problem in pure fluids, overstable convection in porous media is not limited to a particular domain of Prandtl number values (in pure fluids the necessary condition is Pr<1). Moreover, it is also established that in the porous-media problem the critical wavenumber in the plane containing the streamlines for stationary convection is not identical to the critical wavenumber associated with convection without rotation, and is therefore not independent of rotation, a result which is quite distinct from the corresponding pure-fluids problem. Nevertheless it is evident that in porous media, just as in the case of pure fluids subject to rotation and heated from below, the viscosity at high rotation rates has a destabilizing effect on the onset of stationary convection, i.e. the higher the viscosity the less stable the fluid. Finite-amplitude results obtained by using a weak nonlinear analysis provide differential equations for the amplitude, corresponding to both stationary and overstable convection. These amplitude equations permit one to identify from the post-transient conditions that the fluid is subject to a pitchfork bifurcation in the stationary convection case and to a Hopf bifurcation associated with the overstable convection. Heat transfer results were evaluated from the amplitude solution and are presented in terms of Nusselt number for both stationary and overstable convection. They show that rotation has in general a retarding effect on convective heat transfer, except for a narrow region of small values of the parameter containing the Prandtl number where rotation enhances the heat transfer associated with overstable convection.
188 citations
TL;DR: In this article, the effect of the transverse magnetic field on flow field patterns and heat transfer processes in a tilted square cavity was numerically investigated, where the horizontal walls of the enclosure are assumed to be insulated while the vertical walls are kept isothermal.
Abstract: This paper numerically investigates the effect of the transverse magnetic field on flow field patterns and heat transfer processes in a tilted square cavity. The horizontal walls of the enclosure are assumed to be insulated while the vertical walls are kept isothermal. The power law control volume approach is developed to solve the conservation equations at Prandtl number of 0.71. Validation tests with existing data demonstrate the ability of the present scheme to produce accurate results. The effects of Grashof number, enclosure inclination angle, and Hartmann number are also investigated. The study covers the range of the Hartmann number from 0 to 100, the enclosure inclination angle from 0° to ‐90° with Grashof number of 104 and 106. The effect of the magnetic field is found to suppress the convection currents and heat transfer inside the cavity. This effect is significant for low inclination angles and high Grashof numbers. Additionally, it is noted that there is no variation of average Nusselt number with respect to inclination angle for high Hartmann number.
139 citations
TL;DR: In this article, a new turbulence model was proposed to describe stellar turbulent convection, which is able to reproduce the high Rayleigh number laboratory and direct numerical simulation data corresponding to medium-to-high values of the Peclet number (a measure of the efficiency of convection).
Abstract: The description of stellar turbulent convection requires a minimum of five coupled, time-dependent, nonlocal, differential equations for the five variables: turbulent kinetic energy, turbulent potential energy, turbulent pressure, convective flux, and energy dissipation. Any fewer number of equations makes the model local. In this paper, we present the following results: 1. We derive the five coupled equations using a new turbulence model. The physical foundations and the turbulence statistics on which the model was tested are discussed. The model is able to reproduce the high Rayleigh number laboratory and direct numerical simulation data corresponding to medium-to-high values of the Peclet number (a measure of the efficiency of convection). 2. One of the major difficulties for any stellar convective model is the description of the low-efficiency, low Pe number region in which the physical timescale is no longer the turbulent timescale but the radiative one. No previous turbulence model has been able to incorporate these multiple timescales within the same framework properly. The present model does. 3. Overshooting is an unsolved problem in stellar structure. Its solution requires not only the above ingredients, but an additional one, a nonlocal model. This is because in the stably stratified region where ? - ?ad < 0, the only source of energy is diffusion, a nonlocal process. We discuss why the expressions used thus far to describe diffusion terms are inadequate. We then present a model that was successfully tested against LES data on the convective planetary boundary layer. 4. We analyze the nonlocal models of Gough and Xiong and discuss the approximations that are required to derive them from the full set of equations. 5. We discuss a model that relates the up/down drafts filling factors found by DNS/LES to the skewness of the velocity field which can be computed from the turbulence model. The results from DNS/LES and this model can thus be cross-checked. 6. We show that the stationary, local limit of the model reproduces recent local models (independently derived) which have been successfully tested against a variety of astrophysical data. 7. We discuss the fact that if the dissipation is described by a local model with a mixing length l (as done by all authors thus far), the remaining nonlocal equations exhibit divergences which preclude a physical solution to be found. OV results based on this method may be a coincidence since they are arrived at by fine tuning a coefficient. 8. The role of compressibility is discussed.
114 citations
TL;DR: In this article, a numerical investigation is made of natural convection of an incompressible fluid in a square cavity having a constant-temperature cold sidewall and an opposite hot sidewall with sinusoidally-varying temperature.
106 citations
TL;DR: In this paper, the onset of electrohydrodynamic motion associated with the imposition of an electric field across a thin layer of liquid has been investigated for the case in which the electrical conductivity varies linearly over the depth of the layer.
Abstract: The onset of electrohydrodynamic motion associated with the imposition of an electric field across a thin layer of liquid has been investigated for the case in which the electrical conductivity varies linearly over the depth of the layer. The variation of the conductivity is due to concentration gradients in the charge-carrying solutes and its spatiotemporal evolution is represented by a convective-diffusion equation. When the viscous relaxation time is long compared to the time for charge relaxation, the analysis reveals that the neutral stability curves for the layer can be characterized by three dimensionless parameters: Rae≡deE02Δσ/μKeffσ0, an electrical Rayleigh number; Δσ/σ0, the relative conductivity increment; and α, the transverse wave number of the disturbance. Here d is the thickness, e is the dielectric constant, and μ is the viscosity of the layer, E0 is the applied field strength at the lower conductivity boundary, and Keff is an effective diffusivity associated with the Brownian motion of the charge-carrying solutes. With stress-free boundaries, at which the electrical conductivity and current are prescribed, the critical Rae is 1.416×104 at a critical transverse wave number of 1.90 when Δσ/σ0 is 8. As Δσ/σ0 increases, the critical Rae increases and shifts to slightly shorter wavelength disturbances; the critical imposed field strength, however, passes through a minimum because the lower-conductivity boundary exerts a considerable stabilizing influence in the presence of steep conductivity gradients. For Δσ/σ0≲8, the critical Rayleigh number increases as Δσ/σ0 decreases and the layer is only sensitive to long wavelength disturbances (α<0.1) for Δσ/σ0 below 4. Similar trends were obtained for liquid layers with other boundary conditions; e.g., rigid boundaries and constant potential boundaries.
104 citations
TL;DR: In this article, a spectral-element time-stepping code is adapted to calculate bifurcation points and solution branches via Newton's method for a differentially heated binary mixture.
Abstract: Marangoni convection in a differentially heated binary mixture is studied numerically by continuation. The fluid is subject to the Soret effect and is contained in a two-dimensional small-aspect-ratio rectangular cavity with one undeformable free surface. Either or both of the temperature and concentration gradients may be destabilizing; all three possibilities are considered. A spectral-element time-stepping code is adapted to calculate bifurcation points and solution branches via Newton's method. Linear thresholds are compared to those obtained for a pure fluid. It is found that for large enough Soret coefficient, convection is initiated predominantly by solutal effects and leads to a single large roll. Computed bifurcation diagrams show a marked transition from a weakly convective Soret regime to a strongly convective Marangoni regime when the threshold for pure fluid thermal convection is passed. The presence of many secondary bifurcations means that the mode of convection at the onset of instability is often observed only over a small range of Marangoni number. In particular, two-roll states with up-flow at the centre succeed one-roll states via a well-defined sequence of bifurcations. When convection is oscillatory at onset, the limit cycle is quickly destroyed by a global (infinite-period) bifurcation leading to subcritical steady convection.
103 citations
TL;DR: In this article, the authors present experimental observations of the convective flow produced by heating a horizontal layer of such viscoelastic fluids from below, in a long, narrow, annular geometry.
Abstract: Recent experiments on the response of individual, long DNA strands to externally-applied tension and fluid shear suggest that semi-dilute, aqueous suspensions of these molecules should behave as viscoelastic fluids with an elastic relaxation time of seconds and a viscosity comparable to that of water. I present experimental observations of the convective flow produced by heating a horizontal layer of such viscoelastic fluids from below, in a long, narrow, annular geometry. The convection patterns take the form of spatially-localized standing and travelling waves which exhibit small amplitudes and extremely long oscillation periods. The threshold Rayleigh numbers for the onset of oscillations are lower than the value measured for steady convection in a Newtonian fluid in the same apparatus and exhibit a decreasing trend with increasing elastic relaxation time. This behavior agrees with the predictions of theories of the linear instability of viscoelastic convection. However, the low frequencies are in gross quantitative disagreement with these analyses.
88 citations
TL;DR: In this article, the authors derived bounds on convective heat transport in a porous layer heated from below using the background field variational method based on the technique introduced by Hopf (1941).
Abstract: Bounds on convective heat transport in a porous layer heated from below are derived using the background field variational method (Constantin & Doering 1995a, b, 1996; Doering & Constantin 1992, 1994, 1996; Nicodemus, Holthaus & Grossmann 1997a) based on the technique introduced by Hopf (1941). We consider the infinite Prandtl–Darcy number model in three spatial dimensions, and additionally the finite Prandtl–Darcy number equations in two spatial dimensions, relevant for the related Hele-Shaw problem. The background field method is interpreted as a rigorous implementation of heuristic marginal stability concepts producing rigorous limits on the time-averaged convective heat transport, i.e. the Nusselt number Nu, as a function of the Rayleigh number Ra. The best upper bound derived here, although not uniformly optimal, matches the exact value of Nu up to and immediately above the onset of convection with asymptotic behaviour, Nu[les ]9/256Ra as Ra→∞, exhibiting the Howard–Malkus–Kolmogorov–Spiegel scaling anticipated by classical scaling and marginally stable boundary layer arguments. The relationship between these results and previous works of the same title (Busse & Joseph 1972; Gupta & Joseph 1973) is discussed.
82 citations
01 Jan 1998
TL;DR: In this article, a review of the literature on the internal structure, differentiation, convection, and tectonics of the icy satellites is presented, focusing on convection as a predominant endogenic process.
Abstract: Geodynamics concerns the internal structure, differentiation and convection, and tectonics of worlds. With respect to icy satellites there exists an excellent literature (e.g., Burns, 1986), and for the Earth a formidable body of new research results. In this review, I update some of the perspectives on how the icy satellites operate geodynamically, addressing the interplay between rheology, petrology, convection, and tectonics, and focusing on convection as a predominant endogenic process. Icy satellites, if they do undergo internal convection, are generally in the stagnant lid regime as defined by Solomatov, because the viscosity of water ice is strongly temperature-dependent. The Rayleigh number, a measure of the vigor of convection, for the actively convecting interior of an icy satellite is a very strong function of satellite radius (going at least as the sixth power). Convection was probable (if not vigorous) in all but the smallest middle-sized icy satellites early in solar system history. Today, vigorous convection only occurs in Ganymede, Callisto, and Titan, with weak convection occurring in Triton and Pluto. The pronounced polymorphism of the predominant ice, water ice, is expected to strongly modulate convective flow. The ice I-to-II transition should augment convective vigor, while both the ice I-to-III and II-to-V transitions should, by themselves, inhibit convective penetration. Convection within the larger icy satellites should be or have been layered. The negative activation volume for ice I ensures that convective flow in ice I is strongly coupled to the overlying icy lithosphere, which may in some circumstances generate sufficient stress in the lithosphere to induce brittle failure and surface tectonics.
80 citations
TL;DR: In this article, a new natural convection "benchmark problem" for validating CFD codes is defined, where a cubical air-filled cavity, tilted at 0, 45°, or 90°, has one pair of opposing faces at different temperatures, Th and Tc, respectively, the remaining faces having a linear variation from Tc to Th.
77 citations
TL;DR: In this article, it was shown that the normalized heat transport, the Nusselt number (Nu), obeys an Ra 1/5 powerlaw where Ra is the Rayleigh number based on the total temperature contrast along the bottom.
Abstract: Convection in a square container driven by non-uniform heating from below is investigated. Itis shown that the normalized heat transport, the Nusselt number (Nu), obeys an Ra 1/5 powerlaw where Ra is the Rayleigh number based on the total temperature contrast along the bottom.As the Prandtl number (Pr) is decreased from 10 to 1 (Ra>10 6 ), the stream function increasesrapidly while Nu decreases. For a given Ra, Nu is a strong function of the shape of the imposedtemperature distribution and increases as the lateral extent of the thermal contrast is reduced.It is also shown that the response of the system to a new temperature distribution is slow orrapid depending upon whether or not the interior temperature field is affected. These experimentsre-emphasize what is sometimes overlooked, namely that the thermohaline circulationis driven by the imposed horizontal (meridional density) contrast and not by the deep convectionper se. DOI: 10.1034/j.1600-0870.1998.t01-1-00006.x
TL;DR: In this article, a truncated Galerkin method was used to obtain a set of ordinary differential equations for the time evolution of the Galerikin amplitudes, subject to appropriate scaling, for free convection in porous media and the effect of the time derivative term in Darcy's equation.
TL;DR: In this article, the linear stability problem for rapidly rotating convection on an f-plane has been investigated and the existence of two distinct scales in the vertical structure of the critical eigenfunctions: a small length scale whose vertical wavenumber kz is comparable with the large horizontal wenumber k⊥ selected at onset, and a large scale modulation which forms an envelope on the order of the layer depth d.
Abstract: Investigation of the linear stability problem for rapidly rotating convection on an f-plane has revealed the existence of two distinct scales in the vertical structure of the critical eigenfunctions: a small length scale whose vertical wavenumber kz is comparable with the large horizontal wavenumber k⊥ selected at onset, and a large-scale modulation which forms an envelope on the order of the layer depth d. The small-scale structure in the vertical results from a geostrophic balance imposed by the Taylor–Proudman constraint. This primary balance forces rotational alignment and confines fluid motions to planes perpendicular to the rotation axis. For convective transport in the vertical this constraint must be relaxed. This is achieved by molecular dissipation which allows weak upward (downward) spiralling of hot (cold) fluid elements across the Taylor–Proudman planes and results in a large-scale vertical modulation of the Taylor columns.In the limit of fast rotation (i.e. large Taylor number) a multiple-scales analysis leads to the determination of a critical Rayleigh number as a function of wavenumber, roll orientation and the tilt angle of the f-plane. The corresponding critical eigenfunction represents the core solution; matching to passive Ekman boundary layers is required for a complete solution satisfying boundary conditions.An extension of this analysis, introduced by Bassom & Zhang (1994), is used to describe strongly nonlinear two-dimensional convection, characterized by significant departures of the mean thermal field from its conduction profile. The analysis requires the solution of a nonlinear eigenvalue problem for the Nusselt number (for steady convection) and the Nusselt number and oscillation frequency (for the overstable problem). The solutions of this problem are used to calculate horizontal and vertical heat fluxes, as well as Reynolds stresses, as functions of both the latitude and roll orientation in the horizontal, and these are used to calculate self-consistently north–south and east–west mean flows. These analytical predictions are in good agreement with the results of three-dimensional simulations reported by Hathaway & Somerville (1983).
TL;DR: In this paper, the transition from two-dimensional thermoconvective steady flow to a time-dependent flow is considered for a liquid with a high Prandtl number (Pr=105) in a liquid bridge with a curved free surface.
Abstract: The transition from two-dimensional thermoconvective steady flow to a time-dependent flow is considered for a liquid with a high Prandtl number (Pr=105) in a liquid bridge with a curved free surface. Both thermocapillary and buoyancy mechanisms of convection are taken into account. The computer program developed for this simulation transforms the original nonrectangular physical domain into a rectangular computational domain. To solve the problem in body-fitted curvilinear coordinates, the time-dependent Navier–Stokes equations were approximated by central differences on a stretched mesh. For liquid bridges with a flat interface, the instability corresponding to an azimuthal wave number of m=0 is not found for the investigated range of Marangoni numbers. The instability corresponding to an m=0 is found for relatively low Marangoni numbers only in liquid bridges with a nonflat, free surface, and nonzero Rayleigh number. The steady state becomes unstable to axially running waves. It is shown that the onset ...
TL;DR: In this paper, the Galerkin and finite element methods are used to study the onset of the double-diffusive convective regime in a rectangular porous cavity, where two vertical walls of the cavity are subject to constant fluxes of heat and solute while the two horizontal ones are impermeable and adiabatic.
Abstract: The Galerkin and the finite element methods are used to study the onset of the double-diffusive convective regime in a rectangular porous cavity. The two vertical walls of the cavity are subject to constant fluxes of heat and solute while the two horizontal ones are impermeable and adiabatic. The analysis deals with the particular situation where the buoyancy forces induced by the thermal and solutal effects are opposing each other and of equal intensity. For this situation, a steady rest state solution corresponding to a purely diffusive regime is possible. To demonstrate whether the solution is stable or unstable, a linear stability analysis is carried out to describe the oscillatory and the stationary instability in terms of the Lewis number, Le, normalized porosity, e, and the enclosure aspect ratio, A. Using the Galerkin finite element method, it is shown that there exists a supercritical Rayleigh number, RsupTC, for the onset of the supercritical convection and an overstable Rayleigh number, RoverTC, at which overstability may arise. Furthermore, the overstable regime is shown to exist up to a critical Rayleigh number, RoscTC, at which the transition from the oscillatory to direct mode convection occurs. By using an analytical method based on the parallel flow approximation, the convective heat and mass transfer is studied. It is found that, below the supercritical Rayleigh number, RsupTC, there exists a subcritical Rayleigh number, RsubTC, at which a stable convective solution bifurcates from the rest state through finite-amplitude convection. In the range of the governing parameters considered in this study, a good agreement is observed between the analytical predictions and the finite element solution of the full governing equations. In addition, it is found that, for a given value of the governing parameters, the converged solution can be permanent or oscillatory, depending on the porous-medium porosity value, e.
TL;DR: In this paper, the Rayleigh number dependence of a stagnant-lid convection with a viscosity contrast of 106 was studied numerically in two and three dimensions in wide Cartesian domains.
Abstract: The strong dependence of the rheology of a fluid on temperature has a great impact on the style of thermally driven convection. When the viscosity contrast is sufficiently large, the viscosity of the coldest fluid at the top of a bottom-heated box is so high that this fluid layer becomes very stiff and a so-called cold “stagnant lid” develops on top of a hot convecting layer. Studying this style of convection is relevant for planetary mantles since the rheology of mantle material is likely to be very strongly temperature dependent. In this paper, the Rayleigh number dependence of stagnant-lid convection with a viscosity contrast of 106 is studied numerically in two and three dimensions in wide Cartesian domains. Like in constant-viscosity cases, the convection in the layer underneath the stagnant lid undergoes the typical transition from steady to time-dependent with the thinning of plumes and with the appearance of boundary layer instabilities as the Rayleigh number increases. A stagnant-lid style of con...
TL;DR: In this article, the authors visualized natural convection in 23 gas-liquid systems in a quiescent cell by Schlieren techniques and analyzed the relationship between the density effect of CO2 saturation and solvent density.
TL;DR: In this article, a finite element method is employed to simulate the convection-dominated melting of a phase change material (PCM) in a cylindrical-horizontal annulus heated isothermally from the inside wall.
TL;DR: In this article, a numerical solution for the transient natural convection flow over a vertical cylinder under the combined buoyancy effect of heat and mass transfer is presented, where velocity, temperature and concentration profiles, local and average skin-friction, Nusselt number and Sherwood number are shown graphically.
Abstract: A numerical solution for the transient natural convection flow over a vertical cylinder under the combined buoyancy effect of heat and mass transfer is presented. The velocity, temperature and concentration profiles, local and average skin-friction, Nusselt number and Sherwood number are shown graphically. It is observed that time taken to reach steady state increases with Schmidt number and decreases as combined buoyancy ratio parameter N increases. Stability and convergence of the finite difference scheme are established.
TL;DR: In this article, the effects of viscosity, magnetic field and buoyancy force on the unsteady free convection flow of an incompressible and electrically conducting fluid have been analyzed when the flow is generated by uniformly accelerated motion of a vertical plate subject to constant heat flux.
Abstract: The effects of viscosity, magnetic field and buoyancy force on the unsteady free convection flow of an incompressible and electrically conducting fluid have been analysed when the flow is generated by uniformly accelerated motion of a vertical plate subject to constant heat flux. The resulting boundary value problem has been solved exactly for the temperature and velocity variables. The fluid velocity and the skin friction have been computed for some saturated liquids and the effects of the fluid and external forces have been discussed. It is observed that the increase in velocity caused by the larger heat flux can be controlled by enhancing the magnetic field strength. The skin friction at the boundary increases for larger Prandtl and Hartmann numbers, and decreases with increasing Grashof number.
TL;DR: In this article, an experimental investigation of a steady state natural convection from uniformly heated helical coiled tubes oriented horizontally in air has been made, where average heat transfer coefficients are obtained for laminar naLural convection.
TL;DR: In this paper, the authors used a set of 32,000 density models with randomly perturbed input parameters to estimate the depth of the spinel to perovskite transformation on Mars.
Abstract: Previous convection models for Earth have shown that the endothermic phase transition from spinel to perovskite plus magnesiowustite at 660 km depth has a strong effect on mantle convection. For Mars, the depth of the main phase transitions is estimated using a set of 32,000 density models with randomly perturbed input parameters. It is found that, on average, the depth of the spinel to perovskite transformation is 1910 km. Thus a perovskite layer exists in the Martian mantle only if the core radius is less than 0.45 of the planetary radius. Recent results on the density of FeS at pressure and temperature conditions relevant to the Martiam core indicate that such a small core size is consistent with the cosmochemical estimate of around 14% sulphur content of the core. Numerical simulations of three-dimensional spherical convection demonstrate that the presence of the endothermic phase transition close to the core-mantle boundary is the crucial factor determining the style of convection. In addition to the endothermic phase boundary, the simulations assume an isoviscous mantle, which is heated both from below and within, a rigid upper boundary, and Rayleigh numbers of the order of 106. Under these conditions, it is shown that the natural planform of mantle convection exhibits only one or two strong mantle plumes. The number of plumes decreases with time during planetary evolution. For high Rayleigh numbers, the plumes start to oscillate about a mean position and pulsate in time. Strongly localized mantle upwellings are obtained even when the proportion of basal heating is only 10%. These results are robust in the sense that they do not critically depend on the parameters of the phase transition, the Rayleigh number, or the rate of basal heating. This planform of mantle convection is consistent with the volcanic history of Mars, where early volcanism was widespread, but later concentrated in two provinces, Tharsis and Elysium, and finally in Tharsis alone.
TL;DR: In this article, a numerical scheme using the control volume approach is introduced when discretizing the governing flow and energy equations in a square cavity with differential top and bottom wall temperatures.
Abstract: A natural convection in a square cavity finds considerable interest in thermal engineering applications. However, the use of entropy generation concept enables to identify the optimum conditions for its practical application. Consequently, in the present study, natural convection in a square cavity with differential top and bottom wall temperatures is investigated. A numerical scheme using the control volume approach is introduced when discretizing the governing flow and energy equations. The study is extended to include the analysis of the entropy in the cavity. It is found that the local rise of temperature occurs at the right bottom of the cavity due to vertical circulation developed in the cavity. The entropy generation amplifies when circulation along the x-axis increases and, the entropy generation becomes minimum for a particular Rayleigh number. © 1998 John Wiley & Sons, Ltd.
TL;DR: In this paper, a numerical study of melting from a horizontal heated wall with vertically oriented fins embedded in the phase change material is presented, which is motivated by the need to improve the heat transfer rates during the charge and discharge cycles in latent heat thermal energy storage systems.
Abstract: A numerical study is reported of melting from a horizontal heated wall with vertically oriented fins embedded in the phase change material. This work is motivated by the need to improve the heat transfer rates during the charge and discharge cycles in latent heat thermal energy storage systems. A computational methodology based on a fixed‐grid enthalpy method is first presented for handling the complex problem of natural convection dominated melting from a finned wall. The model is validated with experimental data and next a parametric study is conducted to examine the effect of the heated wall (top or bottom), of the number of fins and of the Rayleigh number RaH on the melting process. Results show that melting is enhanced with a bottom finned heated wall and increasing Rayleigh number. They also indicate that, for a given Rayleigh number, the melting time is minimized for an optimal distance W between the fins. This optimal distance was correlated with W= a RaH + b for 2.10 × 106 ≤ RaH ≤ 8.57 × 106.
01 Apr 1998
TL;DR: In this article, a nonlinear analysis of the basic state gives the critical value of the control parameter for the primary instability onset, the nature of the most unstable modes and their growth rate above criticality.
Abstract: Introduction Instabilities in nonlinear systems driven far from equilibrium often consist of a transition from a motionless state to one varying periodically in space or time. Various examples, widely studied in the past, are Rayleigh – Benard convection, Couette–Taylor flow, waves in shear flows, instabilities of liquid crystals, oscillatory chemical reactions,…. The appearance of periodic structures in these systems driven externally by a forcing homogeneous in space or constant in time, corresponds to a bifurcation, characterized by one or several modes that become unstable as a control parameter is varied. Linear stability analysis of the basic state gives the critical value of the control parameter for the primary instability onset, the nature of the most unstable modes and their growth rate above criticality. Many examples have been studied for a long time and can be found for instance in the books of Chandrasekhar (1961) or Drazin and Reid (1981). However, linear stability analysis does not describe the saturation mechanism of the primary instability, and thus a nonlinear analysis should be performed to determine the selected pattern, its dynamics and in particular the secondary instabilities that occur as the control parameter is increased above criticality. Before considering these problems, we present some examples of the characteristic phenomena that occur above a pattern-forming instability onset. Example: the Faraday instability As a first example, consider a cylindrical vessel containing a liquid and its vapor (or any other gas), vertically vibrated at frequency ω e (see Fig. 1.1).
TL;DR: In this paper, the qualitative effect of a non-uniform basic temperature gradient on the linear stability analysis of the Rayleigh-Benard convection in an Eringen's micropolar fluid is studied numerically using a single-term Galerkin technique.
TL;DR: In this paper, the authors investigated the stability of a stably stratified layer of water which is heated above and cooled below and found that the cell size at onset is larger by a factor of √ 3 2 than in the classical Rayleigh-Benard convection problem.
TL;DR: In this article, a weakly nonlinear analysis was carried out to determine the relative stability of hexagon and roll patterns formed at the onset of bioconvection in a tilted chamber with its upper surface open to the atmosphere.
Abstract: Complex bioconvection patterns are observed when a suspension of the oxytactic bacterium Bacillus subtilis is placed in a chamber with its upper surface open to the atmosphere. The patterns form because the bacteria are denser than water and swim upwards (up an oxygen gradient) on average. This results in an unstable density distribution and an overturning instability. The pattern formation is dependent on depth and experiments in a tilted chamber have shown that as the depth increases the first patterns formed are hexagons in which the fluid flows down in the centre. The linear stability of this system was analysed by Hillesdon & Pedley (1996) who found that the system is unstable if the Rayleigh number r exceeds a critical value, which depends on the wavenumber k of the disturbance as well as on the values of other parameters. Hillesdon & Pedley found that the critical wavenumber k C could be either zero or non-zero, depending on the parameter values. In this paper we carry out a weakly nonlinear analysis to determine the relative stability of hexagon and roll patterns formed at the onset of bioconvection. The analysis is different in the two cases k c # 0 and k,. = 0. For the k,. 0 case (which appears to be more relevant experimentally) the model does predict down hexagons, but only for a certain range of parameter values. Hence the analysis allows us to refine previous parameter estimates. For the k,. = 0 case we carry out a two-dimensional analysis and derive an equation describing the evolution of the horizontal planform function.
TL;DR: In this article, the coupled streamfuction-temperature equations governing the Darcian flow and convection process in a fluid-saturated porous enclosure with an isothermal sinusoidal bottom sun face, has been numerically analyzed using a finite element method.
Abstract: The coupled streamfuction–temperature equations governing the Darcian flow and convection process in a fluid-saturated porous enclosure with an isothermal sinusoidal bottom sun face, has been numerically analyzed using a finite element method (FEM). No restrictions have been imposed on the geometrical non-linearity arising from the parameters like wave amplitude (a), number of waves per unit length (N), wave phase (Φ), aspect ratio (A) and also on the flow driving parameter Rayleigh number (Ra). The numerical simulations for varying values of Ra bring about interesting flow features, like the transformation of a unicellular flow to a multicellular flow. Both with increasing amplitude and increasing number of waves per unit length, owing to the shift in the separation and reattachment points, a row–column pattern of multicellular flow transforms to a simple row of multicellular flow. A cycle of n celluar and n+1 cellular flows, with the flow in adjacent cells in the opposite direction, periodically manifest with phase varying between 0 and 360°. The global heat transfer into the system has been found to decrease with increasing amplitude and increasing number of waves per unit length. Only marginal changes in the global heat flux are observed, either with increasing Ra or varying Φ. Effectively, sinusoidal bottom surface undulations of the isothermal wall of a porous enclosure reduces the heat transfer into the system. © 1998 John Wiley & Sons, Ltd.
TL;DR: In this article, a Mach-Zehnder interferometer was used to measure the laminar free convective heat transfer rate from an isothermal vertical surface adjacent to a set of aluminum Venetian blinds.
Abstract: A Mach-Zehnder interferometer was used to measure the laminar free convective heat transfer rate from an isothermal vertical surface adjacent to a set of aluminum Venetian blinds. Local and average heat transfer data were obtained at three different blind-to-plate spacings, and at four different blade angles. Data are presented up to a Rayleigh number of Ra =3 × 107, based on the distance from the leading edge. Flow visualization results that illustrate the cellular flow patterns between the blades are also presented. Overall, it was found that the flow field and local heat transfer distribution on the surface can be significantly affected by the presence a Venetian blind. When placed close to the surface, the blind causes a strong periodic variation in the local Nusselt number distribution. However, for all cases studied the average Nusselt number was within 13% of an isolated vertical flat plate at the same Rayleigh number.