Generation of large scale circulation in turbulent convection

In a horizontal layer of fluid heated below and cooled above, cellular convection with horizontal length scale comparable to the layer depth occurs in a certain range of Rayleigh numbers. As the Rayleigh number is increased, cellular flow disappears and is replaced by a random array of transient plumes. Upon further increase, these plumes drift in one direction near the bottom and in the opposite direction near the top of the layer with the axes of the plumes tilted in such a way that horizontal momentum is transported upwardsvia the Reynolds stress. With the onset of this large-scale horizontal flow, the largest scale of motion increases from a scale comparable to the layer depth to a scale comparable to the layer width. The conditions for the occurrence and the determination of the direction of this largescale circulation are described in this paper. A simple mathematical model is also described which, like the experiment, has the feature of spontaneous generation of a large-scale horizontal flow,u. A truncated Fourier representation of the streamfunction and temperature field is taken, which allows the possibility of a large-scale flow. The bifurcations of the resulting equations have been studied. The first bifurcation is from the conduction state to steady cellular convection. The second is from steady cellular convection to a steady tilted cell convection which has an Eulerian (but not Lagrangian) horizontal mean velocity,u. The third bifurcation is from steady tilted cells to an oscillatory flow, which may be described as consisting of transient, drifting, tilted plumes. The flow in this regime does have a net Lagrangian horizontal mean velocity. Embedded in this regime are also regions of hysteretic and other regions of chaotic flow. A qualitative similarity to the observed flows is noted.     

Experimental study of laminar natural convection heat transfer from a vertical cylinder to slush nitrogen under constant heat flux conditions

Natural convection heat transfer from a vertical cylinder immersed in slush and subcooled liquid nitrogen and subjected to constant heat fluxes was investigated in order to determine the relative merits of slush nitrogen (SlN₂) for immersion cooling. A glass dewar was used as a test vessel in which a cylindrical heater was mounted vertically, and heat transfer measurements were carried out for SlN₂ and subcooled liquid nitrogen (LN₂) in the laminar flow range. The results revealed advantages of SlN₂ over subcooled LN₂ in natural convection cooling. The local temperatures of the heated surface surrounded by solid nitrogen particles are measured to increase at much slower rates than in subcooled LN₂, which is due to the latent heat of fusion of solid nitrogen. Even after the solid nitrogen particles surrounding the heater are apparently depleted, the average heat transfer coefficients for SlN₂ are still found to be greater than those for LN₂ with the improvement in heat transfer being larger for lower Grashof number regime. Our analysis also indicates that solid nitrogen particles in close proximity to heated surface do not discourage local convection due to the porous nature of SlN₂, making the heat transfer in SlN₂ more effective than in the case of solid–liquid phase change of nitrogen involving melting and conduction processes.     

Specific features of wall free convection in a temperature-stratified medium

A flow in the near-wall region with free convection in a temperature-stratified medium is analyzed. A laminar flow in a plane square domain and laminar and turbulent flows near a plate are considered in the cases of linear and nonlinear stratifications. Different nonstationary stages of the flow and a local self-similar regime with criterion for its realization are distinguished. General characteristics for the laminar and turbulent regimes are formulated.     

Transition to chaos in Rayleigh-Bénard convection

The Boussinesq equations for the Rayleigh-Bénard problem have been solved by analytical and numerical methods. Two different sequences of Hopf bifurcations, leading from, stationary two-dimensional rolls to non-periodic motion, have been identified. For one of the sequences the first bifurcation results in transverse oscillations of the rolls. The next bifurcation gives quasiperiodic flow, and the sequence ends in chaotic motion after the third instability. The second route is characterized by waves in the periodic regime, travelling along the rolls. Thereafter two quasiperiodic regimes follow with two and three frequencies, respectively. Both types of sequences have been detected in the experiments reported by Gollub and Benson (1980). The regime of travelling waves is also analysed by a perturbation method.     

Nonlinear Convective Flows in a Laterally Heated Two-Layer System with a Temperature-Dependent Heat Release/Consumption at the Interface

Nonlinear convective flows developed under the joint action of buoyant and thermocapillary effects in a laterally heated two-layer system filling the closed cavity, have been investigated. The influence of a temperature-dependent interfacial heat release/consumption on nonlinear steady and oscillatory regimes, has been studied. It is shown that sufficiently strong temperature dependence of interfacial heat sinks and heat sources can change the sequence of bifurcations and lead to the development of specific oscillatory regimes in the system.     

Convective Flow of a Colloidal Suspension in a Vertical Slot Heated from Side Wall

Convective flows and the transport of nanoparticles are numerically investigated in the vertical slot filled with a colloidal suspension and heated from the side. The thermodiffusion and gravitational sedimentation of the nanoparticles are taken into account. Two different regimes of laminar flow are found. The intensity of the first regime is much lower than in molecular liquids (the magnitudes of the convective and diffusion fluxes have the same order). The second regime is more intensive. The transitions between these two regimes are investigated. It is shown that intensive convective flow completely mixes the colloidal suspension to a homogeneous state as a result of the long transient process.     

Non-darcian effects on double diffusive convection in a sparsely packed porous medium

Summary The linear and non-linear stability of double diffusive convection in a sparsely packed porous layer is studied using the Brinkman model. In the case of linear theory conditions for both simple and Hopf bifurcations are obtained. It is found that Hopf bifurcation always occurs at a lower value of the Rayleigh number than one obtained for simple bifurcation and noted that an increase in the value of viscosity ratio is to delay the onset of convection. Non-linear theory is studied in terms of a simplified model, which is exact to second order in the amplitude of the motion, and also using modified perturbation theory with the help of self-adjoint operator technique. It is observed that steady solutions may be either subcritical or supercritical depending on the choice of physical parameters. Nusselt numbers are calculated for various values of physical parameters and representative streamlines, isotherms and isohalines are presented.     

Convective heat transfer in air flow around a cylinder at low Reynolds numbers

The principal objective of the present work is to conduct investigations leading to a more complete explanation of heat-transfer processes on the external wall of a heated cylinder in laminar axial flow around it under high pressures. Investigations are aimed at determination of the limits of existence of mixed convection, explanation of the influence of free convection on the disturbances of heat transfer during laminar flow of a medium, and final explanation of intensification of heat-transfer processes occurring in a flow at high pressures. Published in Inzhenerno-Fizicheskii Zhurnal, Vol. 78, No. 6, pp. 163–169, November–December, 2005.     

A tertiary hopf bifurcation with applications to problems with symmetries

We develop explicit criteria for the occurrence of a tertiary Hopf bifurcation, and stability of the bifurcating orbits, in a special class of two-parameter systems of ordinary differential equations. We use these results to discuss tertiary Hopf and torus bifurcations in some bifurcation problems with symmetries such as steady-state-Hopf and Hopf-Hopf interaction problems. To analyse (and even detect) these bifurcations we use invariant coordinates and rescaling techniques     

Transient natural convection from a horizontal cylinder confined between vertical walls—a finite element solution

A general finite element computer code is developed for transient two dimensional natural convection heat transfer in a laminar regime. This code is used to study heat transfer from a horizontal isothermal cylinder confined between two vertical adiabatic walls. Results are presented for Ra = 1000, Pr = 0.7, and for different wall spacing to cylinder diameter ratios. The time variations of the local and average Nusselt numbers along with some transient and steady state velocity and temperature fields are demonstrated. The steady state results predict the existence of an optimum wall distance for the maximum average Nusselt number. The steady state and the transient Nusselt numbers are compared with the available experimental or theoretical results for a few cases. Their agreement is very good.     

A review of free,forced and mixed convection in a saturated porous annulus

A computational study of heat transfer in a liquid-saturated porous annulus with a heated inner wall and a cold outer wall is reported here. Results are presented for heat transfer rates from the inner cylinder for a wide range of parameters which characterize free, forced and mixed convective regimes of flow. Both horizontal and vertical annulii are included in the study. Heat transfer results have also been obtained from a non-Darcy model for flow and these are compared to the Darcy solutions. Non-Darcy effects are taken to arise from inertial and viscous effects in the fluid phase and the variation of porosity near the solid wall. The approach of heat transfer results of an annulus with a large radius ratio towards those of an isolated circular cylinder is discussed. Limited data on transient convection is also presented in this work.     

用Normal Form直接法研究壁板的热颤振与控制

建立了二维受热壁板在超音速气流中的颤振方程。运用分岔理论求得了系统的Hopf分岔点,应用Normal Form直接法计算得到系统Hopf分岔Normal Form系数。引入wash-out滤波器技术对壁板热颤振进行了主动控制,延迟系统Hopf分岔的产生而不改变分岔类型。最后采用数值模拟验证了理论分析。     

The effect of rotating cylinder on the heat transfer in a square cavity filled with porous medium

The heat transfer in a square cavity filled with clear fluid or porous medium is numerically investigated in the present study. To change the heat transfer in the cavity a rotating circular cylinder is placed at the centre of the cavity. The ratio of cylinder diameter to cavity height is chosen as 0.8. Depending on the angular velocity of the cylinder the convection phenomena inside the cavity becomes natural, mixed, and forced. To keep the number of data low the Grashof number, Gr, is set to 10⁶, while the parameter defining the convection regime in the cavity, Gr/Re², is changing from 0.0625 to 10². The Darcy number in the cavity is set to 10⁻², 10⁻³, and 10⁻⁴. Galerkin finite element method is used to solve the Navier–Stokes equations with Brinkman–Forcheimer extended Darcy’s law, and energy equation in 2-D non-dimensional form. The solution methodology is compared and validated with the literature for a similar problem, and good agreement is achieved. The results are presented in terms of Nusselt numbers, velocity profiles and temperature contours. The results show that rotation is more effective in the forced convection regime than in mixed and natural convection regimes, and at high spin velocities the heat transfer is almost independent from the Darcy number.     

Unsteady mixed convection flow of a thermomicropolar fluid on a long thin vertical cylinder

The unsteady laminar mixed convection boundary layer flow of a thermomicropolar fluid over a long thin vertical cylinder has been studied when the free stream velocity varies with time. The coupled nonlinear partial differential equations with three independent variables governing the flow have been solved numerically using an implicit finite difference scheme in combination with the quasilinearization technique. The results show that the buoyancy, curvature and suction parameters, in general, enhance the skin friction, heat transfer and gradient of microrotation, but the effect of injection is just opposite. The skin friction and heat transfer for the micropolar fluid are considerably less than those for the Newtonian fluids. The effect of microrotation parameter is appreciable only on the microrotation gradient. The effect of the Prandtl number is appreciable on the skin friction, heat transfer and gradient of microtation.     

Numerical simulation of unsteady MHD natural convection of CNT-water nanofluid in square cavity heated sinusoidally from below

ABSTRACT

Numerical unsteady natural convection flow in a square cavity in the presence of uniform magnetic field is investigated. The cavity is filled with CNT-water nanofluid and heated from below with sinusoidal temperature distribution. This study has been conducted for the certain pertinent parameters of Rayleigh number Ra, Hartmann number Ha and the CNT volume fraction from Φ?=?0 to 0.12. The results indicate, that for large values of Ha, increasing Φ results in an increase of the normalized average Nusselt number. The rate corresponding to CNT- volume fraction nanoparticles of 10% ensures an enhancement about 50% of the heat transfer rate compared to the standard case. The flow undergoes a period bifurcation at a Rayleigh number beyond the value 10⁶. A critical Rayleigh number Ra_c of 1.077?×?10⁶ is then computed. Finally, a correlation giving the average Nusselt number as a function of the frequency of the periodic regime is established.     

Control of Rayleigh-Bénard Convection in a Fluid Layer with Internal Heat Generation

The control of the onset of convection in a horizontal fluid layer with internal heat generation is studied. The horizontal boundaries of the system are cooled isothermally. The stability of the fluid layer is investigated on the basis of the linear stability theory and the resulting eigenvalues problem is solved numerically. Upon using a feedback proportional control, the heating power of the system is modulated in order to counteract any deviations of the temperature of the fluid from its conductive value. As a result, it is possible to postpone (or advance) significantly the onset of motion. The optimal positions of the thermal sensors can be predicted on the basis of the linear stability theory. The linear stability analysis also reveals the possible existence of Hopf’s bifurcations at the onset of motion. This type of bifurcation can be delayed using differential controllers. Two-dimensional numerical simulations of the full governing equations are carried out and found to agree well with the prediction of the linear stability theory.     

弦-梁耦合系统的动力学行为分析

本文研究了弦-梁耦合系统在初始平衡解处的稳定性与分岔情况,给出了特征值随阻尼参数的变化情况,并利用稳定性分析和特征值分析等解析方法,得到了初始平衡解、周期解和拟周期解的稳定边界以及导致Hopf分岔和2维胎面等分岔解的临界分岔曲线。最后,利用数值模拟方法研究了弦-梁耦合系统的稳定性与分岔情况。     

Transient free convection flow past an impulsively started isothermal vertical cylinder with mass flux

This paper concern with the laminar flows, which arise in fluids due to the interaction of the force of gravity and density differences, caused by temperature differences and material or phase constitution for both air and water. A solution of laminar boundary layer equations has been obtained for the transient free convective flow past an impulsively started semi-infinite isothermal vertical cylinder with uniform mass flux. The solutions of the dimensionless, unsteady, coupled and non-linear governing partial differential equations are obtained by a more accurate, unconditionally stable and fast converging implicit finite difference scheme. The results show many interesting effects on velocity, temperature and concentration profiles and local as well as average shear stress, rate of heat and mass transfer. It is observed that there is a rise in the velocity due to the presence of mass diffusion.     

Dynamics, bifurcations and chaos in coupled lasers

Experiments and numerical modelling on two different class B lasers that are subjected to external optical light injection are presented. This presentation includes ways of measuring the changes in the laser output, how to numerically describe the systems and how to construct diagrams of the dynamical states in the plane frequency detuning between lasers and injection strength. The scenarios for the semiconductor laser include an area of frequency locking and islands of chaotic behaviour embedded in and mixed with periodic doubling regimes. Using a rate equation model, the largest Lyapunov exponent is calculated as a measure of the stability of equilibriums and the amount of chaos in chaotic regimes. In the solid-state laser case, different dynamical regions were clearly observed. The found boundaries were identified experimentally, using an identification method, and numerically, from bifurcation analysis, as Hopf, saddle-node, period-doubling and torus bifurcations.