Rayleigh-Benard Convection Problem Research Articles

Bénard–Taylor Convection in Temperature-Dependent Variable Viscosity Newtonian Liquids with Internal Heat Source

This study demonstrates a theoretical investigation of linear and weakly non-linear Rayleigh–Benard convection problem with rotation, variable viscosity and internal heat source. The linear stability analysis shows that an increase in the strength of rotation is to stabilize the system but an increasing internal heat generation and the thermorheological parameter destabilizes the system. The generalized Lorenz model derived for the problem is essentially fifth-order autonomous system of first ordered coupled differential equation with its coefficients depending on internal Rayleigh number, thermorheological parameter and Taylor number. The result on the parameter influence on critical Rayleigh number explains their influence on the Nusselt number as well. Quantification of heat transfer is made possible due to the numerical solution of the autonomous system. It is found that the effect of increasing the strength of rotation is found to have diminished heat transport but in the case of increasing internal Rayleigh number and thermorheological parameter enhances the same.

Boundary Layer Analysis for Navier-Slip Rayleigh–Bénard Convection: The Non-existence of an Ultimate State

We discuss the asymptotic behavior, at small viscosity and/or diffusivity, of the Rayleigh–Benard convection problem governed by the Boussinesq equations. The velocity vector field and the temperature are supplemented respectively with the Navier friction boundary conditions and the fixed flux boundary condition in a 3D periodic channel domain. By explicitly constructing the boundary layer correctors, which approximate the difference between the viscous/diffusive solutions and the corresponding limit solution, we validate the asymptotic expansions, and prove the vanishing viscosity and diffusivity limit with the optimal rate of convergence. Correctors in this setting include higher order diffusive effects than considered previously and accurately account for the interplay between the viscous and thermal layers. The boundary layer correctors satisfy a linear evolution equation indicating that for these boundary conditions, there is no turbulence in the boundary layer. The impact of this fact on the existence of an ‘ultimate state’ of turbulent convection is discussed, particularly in light of recent upper bounds on the heat transport that indicate such a state may exist in this setting.

Spectral method for vorticity-streamfunction equations, with application to Rayleigh-Benard convection

The spectral method with an explicit time integration scheme is implemented to solve the vorticity-streamfunction equations. In spectral method, functions are approximated with finite sums of periodic sine and cosine functions, which makes this method suitable for periodic problems. Numerical experiments for several cases are discussed. In Rayleigh-Benard convection problem, a horizontal layer of incompressible fluid is kept within high and low temperatures along the bottom and top boundaries, respectively. Triggered by a periodic initial condition, we simulate the occurrence of a regular pattern of convection cells, known as Benard cells.

Investigation on Instability of Rayleigh-Benard Convection Using Lattice Boltzmann Method with a Modified Boundary Condition

In this study, the effects of Prandtl number on the primary and secondary instability of the Rayleigh-Benard convection problem has been investigated using the lattice Boltzmann method. Two different cases as Pr=5.8 and 0.7 representing the fluid in liquid and gas conditions are examined. A body forces scheme of the lattice Boltzmann method was presented. Two types of boundary conditions in the presence of body forces are analyzed by the moment method and applied to a Poiseuille flow. Characteristic velocity was set in such a way that the compressibility effects are negligible. The calculations show that the increment of Prandtl number from 0.7 to 5.8 causes to create a secondary instability and onset of the oscillation in the flow field. Results show that at Pr=5.8, when the Rayleigh number is increased, a periodic solution appeared at Ra=48,000. It is observed that the dimensionless frequency ratio for Ra= 105 with Pr=5.8 is around 0.0065. The maximum Nusselt number for Ra = 105 with Pr=5.8 are estimated to be 5.4942.

A Schwarz Method for a Rayleigh–Bénard Problem

The aim of this work is to study a Schwarz domain decomposition numerical method for a stationary Rayleigh–Benard convection problem. The model equations are the stationary version of the incompressible Navier–Stokes equations coupled with a heat equation under Boussinesq approximation. The problem is defined in a rectangular domain. The nonlinear stationary problem is dealt with a Newton method. Each step in the Newton method is solved with a Schwarz domain decomposition method with the domain partitioned into several subdomains with appropriate interface conditions. Their convergence properties are studied theoretically in a simplified domain divided in two subdomains including two artificial parameters in the equations. The numerical resolution of the problem confirms the theoretical results. The convergence rate is less than one when overlap is considered. Convergence is achieved for large values of the aspect ratio, which are inabordable for the standard Legendre collocation method. Convergence is optimal for some values of the parameters. Other advantages of this methodology compared with standard methods are parallelization and high order.

Buoyancy Term Evolution in the Multi Relaxation Time Model of Lattice Boltzmann Method with Variable Thermal Conductivity Using a Modified Set of Boundary Conditions

During the last few years, a number of numerical boundary condition schemes have been used to study various aspects of the no-slip wall condition using the lattice Boltzmann method. In this paper, a modified boundary condition method is employed to simulate the no-slip wall condition in the presence of the body force term near the wall. These conditions are based on the idea of the bounce-back of the non-equilibrium distribution. The error associated with the modified model is smaller than those of other boundary condition models available in the literature. Additionally, various schemes to simulate body forces have been studied. Based on the numerical results, the model demonstrating minimum error has been reported. Finally, it has been shown that the present model is capable of simulating the effect of high nonlinearity in the heat transfer equation in the presence of a variable thermal conductivity. This has been accomplished by employing a multi relaxation time scheme to model a Rayleigh-Benard natural convection current in a 2-D domain with high Rayleigh numbers. Previous studies reported that the onset of oscillation occurs at Ra≈30,000 and Pr=6.0. By the modified boundary condition method which is used in this study, the oscillation is removed until at least Ra≈ 45,000 and Pr=6.0. The results show that applying scheme 3 for the current boundary condition yields the least amount of error compared to the semi-empirical correlation. The Rayleigh-Benard convection problem has been revisited in the presence of a variable thermal conductivity and the simulation results remain stable for flows with a large variation of thermal conductivity ( = 0.7) and Rayleigh numbers up to 1,000,000 and Pr=0.7.

Experimental and Numerical Investigation of a Rayleigh-Bénard Convection Affected by Coriolis Force

In this paper the influence of an impressed Coriolis force field on the configuration of a turbulent Rayleigh-Benard convection problem is investigated in an experimental and numerical study. The main purpose of both studies lie on the analysis of a possible stabilising effect of a Coriolis acceleration on the turbulent unsteady structures inside the fluid. The relative Coriolis acceleration which is caused in the atmosphere by the earth rotation is realised in the experimental study by a uniform-rotational movement of the setup in a large-scale centrifuge under hyper-gravity. The same conditions as in the atmosphere in the beginning of a twister or hurricane should be realised in the experiment. The investigated Rayleigh numbers lie between 2.33 × 106 ≤ Ra ≤ 4.32 × 107.

Маломодовое приближение в задаче Бенара-Рэлея для двухфазной смеси

Маломодовое приближение в задаче Бенара-Рэлея для двухфазной смеси

Gravitational Instability in a Ferromagnetic Fluid Saturated Porous Medium with Non-Classical Heat Conduction

The problem of Rayleigh-Benard convection in a ferromagnetic fluid saturated porous medium with the Maxwell-Cattaneo law is studied by the method of small perturbation. Modified Darcy-Brinkman model is used to describe the fluid motion. The horizontal porous layer is cooled from the upper boundary, while an isothermal boundary condition is imposed at the lower boundary. The non-classical Maxwell-Cattaneo heat flux law involves a wave type heat transport and does not suffer from the physically unacceptable drawback of infinite heat propagation speed. The resulting eigenvalue problem is solved exactly for simplified boundary conditions and the thresholds for the marginal stability are determined. Some of the known cases are derived as special cases. The influence of porous, magnetic, and non-magnetic parameters on the onset of ferroconvection has been analyzed. It is found that the Benard problem for a Maxwell-Cattaneo ferromagnetic fluid is always less stable than the classical ferroconvection problem. It is shown that the destabilizing influence of the Cattaneo number is not attenuated by that of magnetic forces and vice versa, and that the aspect ratio of the convection cells changes when the parameters involved in the study vary with the porous structure bringing out considerable influence.

The limit of the Boussinesq system of Rayleigh-Bénard convection as the Prandtl number approaches infinity

In this paper, the initial layer problem and infinite Prandtl number limit of Rayleigh-Benard convection is studied by the asymptotic expansion methods of singular perturbation theory and the classical energy methods. For ill-prepared initial data, an exact approximating solution with expansions up to any order are given and the convergence rates O(ɛm+1/2) and the optimal convergence rates O(ɛm+1) are obtained respectively. This improves the result of J.G. SHI.

On the Stability Analysis of Rayleigh-Bénard Convection with Temperature Dependent Viscosity for General Boundary Conditions

The problem of Rayleigh-Benard convection with temperature dependent viscosity is investigated for all combinations of rigid and dynamically free boundaries. The principle of exchange of stabilities is shown to be valid for the problem which yields that the onset of convection is through stationary mode. Further, Rayleigh numbers for each possible combination of boundary conditions derived using Galerkin's technique. The effect of variable viscosity on the onset stationary convection for each case of boundary combination is computed numerically and depicted graphically. The present analysis yields that the positive values of temperature-dependent viscosity parameter has stabilizing effect on the onset of stationary convection while negative values have destabilizing effect. Further, various conclusions and results for the problem of Rayleigh-Benard convection with constant viscosity for different cases of boundary conditions have been worked out and are discussed in detail.

Numerical Method of Modeling Thermal Creeping Flow in Heterogeneous Medium

AbstractA hybrid method for modeling the creeping flow is proposed. In our method the so‐called marker‐in‐cell (MIC) and the Finite Element Method (FEM) algorithm are combined together to simulate the thermal creeping flow concerning heterogeneous medium deformation. In particular, the unknown parameters at the Euler mesh‐nodes are calculated using the FEM. The cell‐markers in each element carry the material composition and history variables during the flowing process. The momentum and continuity equation are solved in terms of the pressure‐stabilizing Petrov‐Galerkin method (PSPG) with the equal‐order interpolation of the velocity and pressure, and the energy equation is solved using the streamline upwind Petrov‐Galerkin method (SUPG). In the MIC algorithm, the bilinear interpolation corresponds to the interpolation function in the finite elements. The FEM and MIC algorithm are independent from each other. The data in these two processes communicate through the nodal points. In addition, the triangular element algorithm makes possible to solve the problems with irregular mesh‐grid in complex structures. Our computation program has been verified with the classical Rayleigh‐Benard convection problem. As an example, the descent of an irregular geometry block is calculated. The stability of numerical solution is also investigated.

On the relation between Rayleigh–Bénard convection and Lorenz system

Based on an extension of the Lorenz truncation scheme, a chaotic mathematical model is developed to provide a profile of the chaotic attractor associated with the Rayleigh–Benard convection problem in a plane fluid motion. The attractor of the Lorenz system is a cross-section of the attractor of the proposed model, in which solutions always exist in circles mirroring those appearing in the convection problem.

Theory and Applications of Viscous Fluid Flows

5R6. Theory and Applications of Viscous Fluid Flows. - RK Zeytounian (12 Rue Saint-Fiacre, Paris, 75002, France). Springer-Verlag, Berlin. 2004. 488 pp. ISBN 3-540-44013-5. $109.00.Reviewed by MF Platzer (Dept of Aeronaut and Astronaut, Naval Postgraduate Sch, Code AA/PL, Monterey CA 93943-5000).Starting with the derivation of the Navier-Stokes equations for viscous heat-conducting fluids the author proceeds to discuss various forms of these equations, including the special cases of compressible isentropic viscous flow of polytropic gases and viscous incompressible fluid flow. He then discusses the Orr-Sommerfeld theory for the plane Poiseuille flow as well as other basic flow cases, such as steady flow through an arbitrary cylinder, annular flow between concentric cylinders, Benard thermal convection flow, Benard-Marangoni flow induced by tangential gradients of variable surface tension, flow due to a rotating disc, and Rayleigh flow caused by an impulsively started flat plate. The next three chapters are devoted to the very large and very low Reynolds number limits and to the low Mach number incompressible limit. In the chapter on very large Reynolds number flow the author discusses the application of the method of matched asymptotic expansions to the two-dimensional steady flat-plate flow problem and delineates the relationship of the unsteady Navier-Stokes equations to the inviscid Euler, the Prandtl boundary layer, the one-dimensional gas dynamics and the Rayleigh compressible flow equations. He also discusses the triple deck concept, laminar flow separation on a circular cylinder, and the three-dimensional boundary layer equations. In the chapter on very low Reynolds numbers, the unsteady-state matched Stokes-Oseen solution for the flow past a sphere and the flow over an impulsively started circular cylinder are discussed, followed by a consideration of the Stokes and Oseen steady-state compressible flow equations and the asymptotic analysis for small Reynolds number flows on a rotating disc. In the next chapter on low Mach number incompressible limit, the author discusses subtleties involved in analyzing unsteady weakly compressible flows; flow in a bounded cavity and through large aspect ratio channels. He then provides further examples by analyzing the acoustic streaming effect caused by an oscillating circular cylinder, the incompressible flow past a rotating and translating cylinder, the Ekman and Stewartson layers on rotating cylinders, and the Benard-Marangoni thermo-capillary instability problem due to heating of a horizontal viscous liquid from below. Also presented are some aspects of non-adiabatic viscous atmospheric flows and a few other topics, such as the entrainment of a viscous fluid in a two-dimensional cavity and the laminar boundary layer separation phenomenon near the leading-edge region of an airfoil and on an impulsively started cylinder. In this regard, he emphasizes the need for the simultaneous solution of the boundary layer and inviscid flow equations in order to remove the singularity at the separation point, as implemented in the viscous-inviscid interaction procedures. The next two chapters are devoted to a discussion of the existence, regularity and uniqueness of solutions for the viscous incompressible and compressible flow equations and the stability theory of fluid motion. In particular, the Guiraud-Zeytounian asymptotic approach to nonlinear hydrodynamic stability is elucidated and applied to the Rayleigh-Benard convection problem, followed by an analysis of the Benard-Marangoni thermo-capillary instability problem and the Couette-Taylor viscous flow between two rotating cylinders. The final chapter of Theory and Applications of Viscous Fluid Flows presents the finite-dimensional dynamical systems approach to turbulence by reviewing the Landau-Hopf, Ruelle-Takens-Newhouse, Feigenbaum and Pomeau-Manneville transition scenarios to turbulence. The book is ended by giving examples of strange attractors occurring in various fluid flows, such as in viscous isobaric wave motions or in the flow of an incompressible but thermally conducting liquid down a vertical plane (the Benard-Marangoni problem for a free-falling vertical film). It is evident from this brief summary that the author’s emphasis is on the mathematical aspects of the viscous flow equations and their various asymptotic limit cases and analytical solution methods. His choice of topics and flow problems is meant to provide young researchers in fluid mechanics, applied mathematics and theoretical physics with an up-to-date presentation of some key problems in the analysis of viscous fluid flows. Although the author intentionally limited himself to a select few topics, teachers of advanced viscous flow courses and researchers in this field will welcome this book for its thorough review of current work and the listing of 1156 relevant papers. In my judgment, it meets the stated objective of bridging the gap between standard undergraduate texts in fluid mechanics and specialized monographs.

Convection induced by selective absorption of radiation: A laboratory model of conditional instability

When there is internal heating in a fluid layer, convection can occur even if the static state is one of stable stratification. We have been investigating through laboratory experiments such a stably stratified layer of water which is heated above and cooled below. The water contains in dilute solution thymol blue (a pH indicator), which normally colors the water orange. It turns yellow if the pH is low, blue if the pH is high. A small DC voltage is applied across the layer, by using the bottom boundary as the positive electrode, the top boundary as the negative electrode. The hydroxyl ions formed near the bottom boundary cause the orange fluid to turn blue. The fluid layer is uniformly and steadily illuminated from above with light from a sodium vapor lamp. This radiation travels with negligible absorption through the orange fluid but is strongly absorbed by the blue fluid. The resultant warming of the blue fluid can lead to convective instability, with the blue fluid rising into warm upper layers, which it would continue to penetrate as long as it remains blue and as long as the radiative heating is sufficient to exceed the higher ambient temperatures above. This radiative heating occurs only in the blue rising flow; the sinking fluid is orange and is not heated. We have found that with a strongly stably stratified layer, convective plumes are unable to penetrate far and they remain shallow. However, for a weakly stratified layer, plumes grow tall and furthermore collect into a large convective cluster which persists as a steady coherent structure. The present paper deals also with the formulation of the governing equations to include the fluid-state-dependent heat source. A linear stability analysis shows that the critical Rayleigh number for onset of motion is drastically reduced. Furthermore, the cell size at onset is larger by a factor of √32 than in the classical Rayleigh-Benard convection problem. However, the laboratory fluid cells were much further broadened (by a factor of 8 or 10) when they penetrated into the stably stratified fluid above. In this case, the rising region is narrow and the sinking region is broad, so that downward vertical velocities are correspondingly small. In this way, the downwards-forced warm fluid has time to cool by conduction to the cold boundary. Steady finite amplitude solutions and their stability are analyzed and it is shown that there is a parameter range in which finite amplitude hexagonal cells are stable.

An improved method of Nusselt number calculation

A novel method for calculating the Nusselt number, Nu, in a steady-state Rayleigh-Benard convection problem is presented, in which calculations are done for a square box with constant temperature, free-slip boundary conditions at the top and bottom, and a reflection symmetry along the side walls. The element heat flux is obtained by averaging over the entire element; element heat fluxes are then projected to the adjacent nodes. Compared with previous methods, the approach reduces the calculated depth variation in horizontally averaged flux by more than a factor of 10 and shows more rapid convergence of Nu as a function of grid size.

On Rayleigh-B�nard convection with rotation and magnetic field

A sufficient condition for the validity of the principle of exchange of stabilities in the Rayleigh-Benard convection problem under the simultaneous action of a uniform vertical rotation and magnetic field is derived. Furthermore the complex growth rate of an arbitrary oscillatory perturbation, neutral or unstable, is shown to lie inside a semicircle in the right half of the complexp-plane. These results are uniformly valid for all combinations of free insulating, rigid perfectly conducting and rigid insulating boundaries.

Horizontal Wave Number Dependence of Type II Solutions in Rayleigh?Benard Convection with Hexagonal Planform

The horizontal wave number dependence of the hexagonal planform solutions for the RayleighBenard convection problem, which have a nonzero vertical component of vorticity (type II solutions), has been established. Over the range of wave numbers which support cellular convection, comparisons between the thermal transport characteristics of these cyclonic type solutions and those traditionally obtained from nonlinear investigations of the single horizontal mode equations (type I solutions) have been made. From the numerical results obtained, it is found that the cell aspect ratio which maximizes the heat flux of type II solutions is larger than that for type I solutions, at quivalent parameter values, and that the value of the horizontal wave number giving maximum Nusselt number for type II solutions increases with Rayleigh number and decreases with Prandtl number.

Axisymmetric convection in the presence of a magnetic field

Abstract Numerical solutions have been calculated for the non-linear Rayleigh-Benard convection problem in a cylindrical geometry, including the effect of a magnetic field. The development of the field is governed by the induction equation and its back reaction on the motion is determined via the Lorentz force in the momentum equation. If the input field B 0 is weak a thin fluxrope is formed on the axis, and when B 0 is increased the amplified field reaches a maximum value as this region becomes stagnant. The equipartition value B e [tbnd] (μ o ρU2 )½ is easily exceeded. When Bo is stronger, heat transport is reduced and osciliations can occur; there are hysteresis effects, and erratic hehaviour is found for high magnetic Prandtl numbers. These results help to interpret ohserved flux concentrations on the Sun, and provide models for some subsidiary features of sunspots.