Formation of Structures of Multieddy Natural Convection in a Horizontal Layer Heated from Below

The Galerkin finite-element method is used to investigate the structures of supercritical thermogravitational convection in an elongated plane horizontal layer of a liquid with the Prandtl number Pr = 10. The solution of the problem with a constant heat-flux density preassigned at the lower boundary is compared to the solution of the problem for the case of boundaries at constant temperature. The results of comparative analysis of convective modes arising under different conditions at the boundaries of the layer and under different initial thermal conditions are given. The mathematical model is provided by two-dimensional unsteady-state equations for natural thermal convection in the Boussinesq approximation that are written in terms of the variables of speed eddy, stream function, and temperature.     

A nonlinear energy stability analysis of convection with temperature dependent viscosity

Summary We establish a nonlinear energy stability theory for convection, when the viscosity depends on the temperature. The linearized stability problem is shown to be intimately connected with the nonlinear one, and the nonlinear analysis requires an intricate use of embedding inequalities. Numerical results are given for the nonlinear stability threshold and these represent rigorous practical bounds which should prove useful in experimental investigations of sub-critical instabilities in convection in temperature-dependent viscosity fluids.     

The Helmholtz equation for convection in two-dimensional porous cavities with conducting boundaries

It is well-known that every two-dimensional porous cavity with a conducting and impermeable boundary is degenerate, as it has two different eigensolutions at the onset of convection. In this paper it is demonstrated that the eigenvalue problem obtained from a linear stability analysis may be reduced to a second-order problem governed by the Helmholtz equation, after separating out a Fourier component. This separated Fourier component implies a constant wavelength of disturbance at the onset of convection, although the phase remains arbitrary. The Helmholtz equation governs the critical Rayleigh number, and makes it independent of the orientation of the porous cavity. Finite-difference solutions of the eigenvalue problem for the onset of convection are presented for various geometries. Comparisons are made with the known solutions for a rectangle and a circle, and analytical solutions of the Helmholtz equation are given for many different domains.     

An operator-splitting algorithm for two-dimensional convection–dispersion–reaction problems

An operator-splitting algorithm for the two-dimensional convection–dispersion–reaction equation is developed. The flow domain is discretized into triangular elements which are fixed in time. The governing equation is split into three successive initial value problems: a pure convection problem, a pure dispersion problem and a pure reaction problem. For the pure convection problem, solutions are found by the method of characteristics. The solution algorithm involves tracing the characteristic lines backwards in time from a vertex of an element to an interior point. A cubic polynomial is used to interpolate the concentration and its derivatives on an element. For the pure dispersion problem, an explicit finite element algorithm is employed. Analytical solutions are obtained for the pure reaction problem. The treatment of the boundary conditions is also discussed. Several numerical examples are presented. Numerical results agree well with analytical solutions. Because cubic polynomials are used in the interpolation, very little numerical damping and oscillation are introduced, even for the pure convection problem.     

Steady free convection in viscoplastic liquids

The formulation and solution are given for the external problem of the steady free convection of a viscoplastic liquid.     

Magnetohydrodynamic,unsteady, one-dimensional viscous flows—II: Free convection and asymptotic behavior

The magnetohydrodynamic, unsteady free convection over an infinite vertical flat plate is dealt with, in the presence of a transverse magnetic field. The general solution is given for an ample class of boundary conditions. The conditions for the existence and the expression for the asymptotic solutions are discussed. As an application, the problem of sinusoidally changing boundary conditions at the wall is carried out.     

基于随机平均的非线性随机最优控制

首先建立非线性随机系统的最优控制问题,并介绍通过随机平均法导出平均系统、再由随机动态规划原理确定控制律的平均系统的非线性随机最优控制方法。然后,对于非线性随机系统的动态规划方程,提出应用随机平均法简化该方程、从而得到最优平均控制律的方法,并证明该最优平均控制律等价于平均系统的最优控制律。最后用一个例子说明方法及等价性,并指出在一定条件下,最优平均控制律将是动态规划方程的精确解。     

Active Control of Thermal Convection in a Rectangular Loop by Changing its Spatial Orientation

The problem of the automatic control of the fluid flow in a rectangular convective loop heated from below is studied theoretically and experimentally. The control is performed by using a feedback subsystem which changes the convection regimes by introducing small discrete changes in the spatial orientation of the loop with respect to gravity. We focus on effects that arise when the feedback controller operates with an unavoidable time delay, which is cause by the thermal inertia of the medium. The mathematical model of the phenomenon is developed. The dynamic regimes of the convection in the thermosyphon loop under control are studied. It is shown that the proposed control method can successfully stabilize not only a no-motion state of the fluid, but also time-dependent modes of convection including the irregular fluid flow at high values of the Rayleigh number. It is shown that the excessive gain of the proportional feedback can result in oscillations in the loop orientation exciting the unsteady convection modes. The comparison of the experimental data obtained for dielectric oil and dodecane with theory is given, and their good agreement is demonstrated.     

State space approach to unsteady free convection flow of micropolar fluid

Summary The equations of unsteady free convection flow of a micropolar fluid are cast into a matrix form using the state space and Laplace-transform techniques. The results obtained are used to generate solutions in the Laplace-transform domain to a broad class of problems in free convection flow. The technique is applied to a heated vertical plate problem and to a problem pertaining to a plate under uniform heating. The inversion of Laplace-transform is performed using a numerical approach. Numerical results concerning velocity, microrotation and temperature are given and illustrated graphically for both problems.     

Short-Term Frequency Stability of the Rb87 Maser

Measurements of the short-term stability of the Rb87 maser are reported here. The measurements were made as a function of the maser power output and of the receiver cutoff frequency. The experimental data are compared to theoretical results obtained from an approximate theory. In this theory the transfer function of the maser for thermal noise is derived, and the spectral density of the phase fluctuations is calculated. An analytical expression for the "Allan variance" is also given. A comparison of the stability of the Rb87 maser with existing frequency standards shows its superiority for averaging times less than 1 s. We obtain ?f/f ? 1.3 × 10-3 ?-1. A stability of 5 × 10-12 for ? ?1000 s is also reported.     

Modelling convection in solidification processes using stabilized finite element techniques

Solidification of dendritic alloys is modelled using stabilized finite element techniques to study convection and macrosegregation driven by buoyancy and shrinkage. The adopted governing macroscopic conservation equations of momentum, energy and species transport are derived from their microscopic counterparts using the volume‐averaging method. A single domain model is considered with a fixed numerical grid and without boundary conditions applied explicitly on the freezing front. The mushy zone is modelled here as a porous medium with either an isotropic or an anisotropic permeability. The stabilized finite‐element scheme, previously developed by authors for modelling flows with phase change, is extended here to include effects of shrinkage, density changes and anisotropic permeability during solidification. The fluid flow scheme developed includes streamline‐upwind/Petrov–Galerkin (SUPG), pressure stabilizing/Petrov–Galerkin, Darcy stabilizing/Petrov–Galerkin and other stabilizing terms arising from changes in density in the mushy zone. For the energy and species equations a classical SUPG‐based finite element method is employed with minor modifications. The developed algorithms are first tested for a reference problem involving solidification of lead–tin alloy where the mushy zone is characterized by an isotropic permeability. Convergence studies are performed to validate the simulation results. Solidification of the same alloy in the absence of shrinkage is studied to observe differences in macrosegregation. Vertical solidification of a lead–tin alloy, where the mushy zone is characterized by an anisotropic permeability, is then simulated. The main aim here is to study convection and demonstrate formation of freckles and channels due to macrosegregation. The ability of stabilized finite element methods to model a wide variety of solidification problems with varying underlying phenomena in two and three dimensions is demonstrated through these examples. Copyright © 2005 John Wiley & Sons, Ltd.     

Calculation of convective streams in the liquid core of solidifying bodies of very simple shape

An analytical solution is given to the problem of distribution of velocity, temperature and heat flux in the liquid core of solidifying bodies of very simple shape, with natural convection and an arbitrary law of motion of the two-phase boundary.     

Effect of a non-constant magnetic field on natural convection in a horizontal porous layer heated from the bottom

An analytical and numerical investigation is conducted to study the effect of an electromagnetic field on natural convection in a horizontal shallow porous cavity filled with an electrically conducting fluid. The magnetic field is assumed to be induced by two long wires, carrying current, parallel to the horizontal boundaries of the system. A uniform heat flux is applied to the horizontal walls of the layer while the vertical walls are adiabatic. The governing parameters of the problem under study are the thermal Rayleigh number, Ra, Hartmann number, Ha, position of the electrical wires, d, current intensity ratio, r, and aspect ratio of cavity, A. An analytical solution, valid for a shallow layer (A ? 1), is derived on the basis of the parallel flow approximation. The critical Rayleigh number, Ra _c , for the onset of motion is derived in closed form in terms of the parameters of the problem. For finite-amplitude convection the heat and flow characteristics predicted by the analytical model are found to agree well with a numerical study of the full governing equations.     

Flow of a stream of unevenly heated liquid over a gas bubble at low marangoni numbers

The problem of the thermocapillary convection in an unevenly heated liquid near a gas bubble is solved analytically. Estimates are given for the velocity of drift and the shape of the bubble and the vortex boundary.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 32, No. 2, pp. 251–256, February, 1977.     

The effect of shear convection on diffusion measurements in liquid metals using the foton shear cell

The shear convection in the Foton shear cell was investigated quantitatively and interpreted as an additional diffusion-like mixing. Diffusion experiments were done under 1g and µg (Foton-M2 mission June 2005). Since the measured mean squarediffusiondepth (MSDD) is a linear function of time, the offset represents the additional mass transport (and the averaging effect due to the analysis by Atom Absorption Spectroscopy, AAS). The offset is 1–3% of the MSDD and can be used to eliminate this systematic error. The influence of the liquid’s viscosity on the shear convection is discussed, using a simple fluid dynamical model. The absence of buoyancy convection in 1g-experiments was shown by comparison with the µg-result.     

Free convection in hydromagnetic flows in a vertical wavy channel

A study is made of the free convection in hydromagnetic flows in a vertical wavy channel in the presence of heat source or sink. The governing equations for the hydromagnetic fluid flow and the heat transfer are solved subject to the relevant boundary conditions with the assumption that the solution consists of a mean part and a perturbed part. The zeroth-order, the first order and the total solution of the problem are numerically evaluated for various values of the magnetic parameter, heat source/sink parameter, wall-waviness parameter, and free convection parameter. The velocity and the temperature profiles are graphically represented for these parameters. The qualitative features of the hydromagnetic solution are discussed. A comparison is made between the hydromagnetic and the hydrodynamic solutions. The numerical values of the skin friction and the Nusselt number are tabulated for various parameters involved in the analysis. Special attention is given on the characteristic features of the flow, heat transfer, skin friction and the Nusselt numbers at the walls.     

Conjugate MHD flow past a flat plate

Summary A boundary layer solution for the conjugate forced convection flow of an electrically conducting fluid over a semi-infinite flat plate in the presence of a transverse magnetic field is presented. The governing nonsimilar partial differential equations are solved numerically using the Keller box method. Values of the temperature profiles of the plate are obtained for various values of the parameters entering the problem and are given in a table and shown on graphs.     

Rethinking convective quasi-equilibrium: observational constraints for stochastic convective schemes in climate models

Convective quasi-equilibrium (QE) has for several decades stood as a key postulate for parametrization of the impacts of moist convection at small scales upon the large-scale flow. Departures from QE have motivated stochastic convective parametrization, which in its early stages may be viewed as a sensitivity study. Introducing plausible stochastic terms to modify the existing convective parametrizations can have substantial impact, but, as for so many aspects of convective parametrization, the results are sensitive to details of the assumed processes. We present observational results aimed at helping to constrain convection schemes, with implications for each of conventional, stochastic or 'superparametrization' schemes. The original vision of QE due to Arakawa fares well as a leading approximation, but with a number of updates. Some, like the imperfect connection between the boundary layer and the free troposphere, and the importance of free-tropospheric moisture to buoyancy, are quantitatively important but lie within the framework of ensemble-average convection slaved to the large scale. Observations of critical phenomena associated with a continuous phase transition for precipitation as a function of water vapour and temperature suggest a more substantial revision. While the system's attraction to the critical point is predicted by QE, several fundamental properties of the transition, including high precipitation variance in the critical region, need to be added to the theory. Long-range correlations imply that this variance does not reduce quickly under spatial averaging; scaling associated with this spatial averaging has potential implications for superparametrization. Long tails of the distribution of water vapour create relatively frequent excursions above criticality with associated strong precipitation events.     

Combined convection in an annulus applied to a thermal storage problem

In a solar energy application involving thermal storage, one of the heat transfer situations is that of combined convection in vertical annuli for rather complex wall thermal boundary conditions. Predictive data of a high order of reliability are needed for incorporation within the suite of programs treating the whole problem. The program reported here treats the complete equations for combined free and forced convection in a vertical annulus. It allows for viscosity and density variation with temperature, and a variable heat flux or temperature at the walls. It was developed from a similar program for circular tubes. Comparisons are made with published data for velocity profiles and heat transfer performance. These are good, and show the step-wise energy balance method is necessary and valid. The strategy of generation of the required data is explained, together with sample output. These data are themselves analysed computationally; the performance equations agree with original predictions typically to within ± 11 per cent, with a standard deviation of around 2 per cent. The working fluid is water with 37 per cent ethylene glycol. Upward heated and downward cooled flows give aided combined convection. For the given design, laminar flow is predicted for the Reynolds number range 1800–2200.     

On the two-scales method for a class of integro-differential equations appearing in viscoelasticity

In this paper, we consider vibrations of a linear viscoelastic material, with long memory and short relaxation time (introduction of the small parameter ?). Related to this problem, an integro-differential equation is to be resolved. An approximate solution is constructed by using the two-scales method. It is shown that this formal approximation is valid on time intervals of order ?^?1. The method of averaging leads to the same first approximation.