Convection in a Porous Medium with Variable Gravity Field and Magnetic Field Effects

The problem of convection in a variable gravity field with magnetic field effect is studied using methods of linear instability theory and non-linear energy theory. Then, the accuracies of both the linear instability and global non-linear energy stability thresholds are tested using a three-dimensional simulation. The strong stabilizing effect of gravity field and magnetic field is shown. Moreover, the results support the assertion that the linear theory is very accurate in predicting the onset of convective motion, and thus regions of stability.     

Numerical simulation of incompressible flow driven by density variations during phase change1

A change in density during the solidification of alloys can be an important driving force for convection, especially at reduced levels of gravity. A model is presented that accounts for shrinkage during the directional solidification of dendritic binary alloys under the assumption that the densities of the liquid and solid phases are different but constant. This leads to a non‐homogeneous mass conservation equation, which is numerically treated in a finite element formulation with a variable penalty coefficient that can resolve the velocity field correctly in the all‐liquid region and in the mushy zone. The stability of the flow when shrinkage interacts with buoyancy flows at low gravity is examined. Copyright © 1999 John Wiley & Sons, Ltd.     

Experimental study of the gravity influence on the periodic thermocapillary convection around a bubble

The 3D periodic state, following the steady thermocapillary convection state, around an air bubble immersed in a low-Prandtl-number silicone oil layer under a heated wall was experimentally investigated on the ground and during parabolic flights. This oscillation was observed under reduced gravity conditions for the first time. Consequently, the initiation of this oscillation seems to be independent of gravity and so of buoyancy convection. The reduced and increased gravity conditions showed that the gravity level modifies the oscillation. Its frequency increases with the gravity level. The comparison with the results obtained on the ground shows the bubble aspect ratio is not a relevant parameter when the gravity varies. Received: 27 November 2000/Accepted: 25 May 2001     

Stationary Regimes of Electric Convection of a Low-Conductivity Fluid in the Case of Unipolar Charge Injection in a Constant Electric Field

Fluid Dynamics - Stationary nonlinear regimes of electric convection of a low-conductivity fluid in a horizontal capacitor in the gravity field and a constant electric field are studied in the case...     

Throughflow and g-jitter effects on binary fluid saturated porous medium

A non-autonomous complex Ginzburg-Landau equation (CGLE) for the finite amplitude of convection is derived, and a method is presented here to determine the amplitude of this convection with a weakly nonlinear thermal instability for an oscillatory mode under throughflow and gravity modulation. Only infinitesimal disturbances are considered. The disturbances in velocity, temperature, and solutal fields are treated by a perturbation expansion in powers of the amplitude of the applied gravity field. Throughflow can stabilize or destabilize the system for stress free and isothermal boundary conditions. The Nusselt and Sherwood numbers are obtained numerically to present the results of heat and mass transfer. It is found that throughflow and gravity modulation can be used alternately to heat and mass transfer. Further, oscillatory flow, rather than stationary flow, enhances heat and mass transfer.     

Stability of Solutal Convection in a Gravity Modulated Mushy Layer During the Solidification of Binary Alloys

The linear stability theory is used to investigate analytically the effects of gravity modulation on solutal convection in the mushy layer of solidifying binary alloys. The gravitational field consists of a constant part and a sinusoidally varying part, which is synonymous to a vertically oscillating mushy layer subjected to constant gravity. The linear stability results are presented for both the synchronous and subharmonic solutions. It is demonstrated that up to the transition point between the synchronous and subharmonic regions, increasing the frequency of vibration rapidly stabilizes the solutal convection. Beyond the transition point, further increases in the frequency tend to destabilize the solutal convection, but gradually. It is also demonstrated that the effect of increasing the ratio of the Stefan number and the solid composition (₀) is to destabilize the solutal convection.     

Stability of Convection in a Gravity Modulated Porous Layer Heated from Below

The linear stability theory is used to investigate analytically the effects of gravity modulation on convection in a homogenous porous layer heated from below. The gravitational field consists of a constant part and a sinusoidally varying part, which is tantamount to a vertically oscillating porous layer subjected to constant gravity. The linear stability results are presented for the specific case of low amplitude vibration for which it is shown that increasing the frequency of vibration stabilises the convection.     

Three-dimensional modelling of GeSi growth in presence of axial and rotating magnetic fields

A three-dimensional numerical simulation has been performed to study the growth of Ge_0.98Si_0.02 by the Traveling Solvent Method. We attempted to suppress the buoyancy convection, in the Ge_0.98Si_0.02 melt zone, by applying axial and rotating magnetic fields. The effects of the applied magnetic field intensity, on the transport structures in the melt (flow and concentration fields, heat and mass transfer), have been investigated in detail. The steady-state full Navier–Stokes equations, as well as energy, mass species transport and continuity equations are numerically solved using the finite element method. By applying an axial magnetic field of various intensities (2, 10, and 22 mT), we found that as the axial magnetic field increases, the silicon distribution nearby the growth interface becomes more uniform. In the case of a rotating magnetic field, with different applied rotational speeds (2, 7 and 10 rpm), we found that such kind of magnetic field has a marked effect on the silicon concentration, which changes its shape from a convex one to a nearly flat shape as the magnetic field intensity increases. An alternative method to reduce or suppress buoyancy convection, in the melt zone, is the growing of the sample in a microgravity environment, with a gravity level of at least 10^?4 the earth normal gravity level; in this case the results revealed smooth and almost perfect straight concentration contours, due to the buoyancy convection weakness.     

热毛细对流的液桥几何位形及浮力效应

本文讨论重力对不同高度、直径比液桥的热毛细对流的影响。当液桥高度、直径比增大时,液桥中的等流函数线呈双涡结构,这种流动图样并不必然与热毛细振荡流相联系。在地面热毛细对流实验中模拟空间微重力情况,液桥高度需小于1.5mm。在微重力环境中,液桥内的流场和温度分布介于地面相同参数液桥的上部加热和下部加热两种结果之间。因此,可以用地面实验结果估计空间液桥的对流和热输运情况。     

Benard convection in a non-linear magnetic fluid under the influence of a non-vertical magnetic field

This work examines the convective instability of a horizontal layer of magnetohydrodynamic fluid of variable permeability when subjected to a non-vertical magnetic field. We use a model proposed by P. H. Roberts [9] in the context of neutron stars but the results obtained are aso relevant to the area of ferromagnetic fluids. The presence of the variable permeability has no effect on the development of instabilities through the mechanism of stationary convection but influences the threshold of overstable convection which is often the preferred mechanism in non-terrestrial applications. In the context of ferromagnetic fluids, both stationary and overstable instability can be expected to be realisable possibilities.     

Application of the Jacobi–Davidson method to accurate analysis of singular linear hydrodynamic stability problems

The aim of this paper is to show that the Jacobi–Davidson (JD) method is an accurate and robust method for solving large generalized algebraic eigenvalue problems with a singular second matrix. Such problems are routinely encountered in linear hydrodynamic stability analysis of flows that arise in various areas of continuum mechanics. As we use the Chebyshev collocation as a discretization method, the first matrix in the pencil is nonsymmetric, full rank, and ill‐conditioned. Because of the singularity of the second matrix, QZ and Arnoldi‐type algorithms may produce spurious eigenvalues. As a systematic remedy of this situation, we use two JD methods, corresponding to real and complex situations, to compute specific parts of the spectrum of the eigenvalue problems. Both methods overcome potentially severe problems associated with spurious unstable eigenvalues and are fairly stable with respect to the order of discretization. The real JD outperforms the shift‐and‐invert Arnoldi method with respect to the CPU time for large discretizations. Three specific flows are analyzed to advocate our statements, namely a multicomponent convection–diffusion in a porous medium, a thermal convection in a variable gravity field, and the so‐called Hadley flow. Copyright © 2012 John Wiley & Sons, Ltd.     

A mixed FE-BE method for coupled vibration of gravity dam-reservoir system with variable water depth

To solve the coupled vibration of a gravity dam-reservoir system with variable water depth by using a hybrid element method, the fluid region with variable water depth needs to be discretized by FE meshes. However, such a method asks for a great computational cost owing to the excessive unknowns, especially when the fluid region with variable water depth is relatively large. To overcome the shortcoming, a refined boundary element method is proposed to analyze the fluid field, in which only the discretization for the boundary of the variable depth region is required. But as a basis of this approach, it is necessary to construct a new Green's function corresponding to an infinite strip region. The problem is solved as the first step in this paper by employing Fridman's operator function theory, and then a mixed FE-BE formulation for analyzing the free vibration of the gravity damreservoir system is derived by means of the coupling conditions on the dam-reservoir interface. Finally, a numerical example is provided to illustrate a great improvement of the method developed herein over the hybrid element method. The project supported by the National Key Research Plan of China.     

Three-Dimensional Simulations for Convection Problem in Anisotropic Porous Media with Nonhomogeneous Porosity,Thermal Diffusivity,and Variable Gravity Effects

A convection problem in anisotropic and inhomogeneous porous media has been analyzed. In particular, the effect of variable permeability, thermal diffusivity, and variable gravity with respect to the vertical direction, has been studied. A linear and nonlinear stability analysis of the conduction solution has been performed. The validity of both the linear instability and global nonlinear energy stability thresholds are tested using a three- dimensional simulation. Our results show that the linear threshold accurately predicts on the onset of instability in the basic steady state. However, the required time to arrive at the basic steady state increases significantly as the Rayleigh number tends to the linear threshold.     

Nonlinear dynamics of hydrostatic internal gravity waves

Stratified hydrostatic fluids have linear internal gravity waves with different phase speeds and vertical profiles. Here a simplified set of partial differential equations (PDE) is derived to represent the nonlinear dynamics of waves with different vertical profiles. The equations are derived by projecting the full nonlinear equations onto the vertical modes of two gravity waves, and the resulting equations are thus referred to here as the two-mode shallow water equations (2MSWE). A key aspect of the nonlinearities of the 2MSWE is that they allow for interactions between a background wind shear and propagating waves. This is important in the tropical atmosphere where horizontally propagating gravity waves interact together with wind shear and have source terms due to convection. It is shown here that the 2MSWE have nonlinear internal bore solutions, and the behavior of the nonlinear waves is investigated for different background wind shears. When a background shear is included, there is an asymmetry between the east- and westward propagating waves. This could be an important effect for the large-scale organization of tropical convection, since the convection is often not isotropic but organized on large scales by waves. An idealized illustration of this asymmetry is given for a background shear from the westerly wind burst phase of the Madden–Julian oscillation; the potential for organized convection is increased to the west of the existing convection by the propagating nonlinear gravity waves, which agrees qualitatively with actual observations. The ideas here should be useful for other physical applications as well. Moreover, the 2MSWE have several interesting mathematical properties: they are a system of nonconservative PDE with a conserved energy, they are conditionally hyperbolic, and they are neither genuinely nonlinear nor linearly degenerate over all of state space. Theory and numerics are developed to illustrate these features, and these features are important in designing the numerical scheme. A numerical method is designed with simplicity and minimal computational cost as the main design principles. Numerical tests demonstrate that no catastrophic effects are introduced when hyperbolicity is lost, and the scheme can represent propagating discontinuities without introducing spurious oscillations.      

Vibrational Convection in the Hele-Shaw Cell. Theory and Experiment

The influence of high-frequency horizontal vibrations on convection in the Hele-Shaw cell located in a uniform gravity field is considered experimentally and theoretically. Nonlinear regimes of vibrational convection in the supercritical region are examined. It is shown that horizontal vibrations (directed toward the wide sides of the cell) decrease the threshold of quasi-equilibrium stability. Regions of existence of one- and two-vortex steady flows are found, and unsteady regular and random regimes of thermal vibrational convection are considered. New random regimes in the Hele-Shaw cell are found, which result from nonlinear interaction of the “lower” modes responsible for the formation of regular supercritical convective regimes. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 2, pp. 40–48, March–April, 2006.     

Penetrative convection in thawing subsea permafrost

Penetrative convection is investigated in a porous medium bounded above by the ocean bed and below by the interface of the thawing permafrost ground. The thermal equation of state relating the density, temperature and salinity is assumed to be that of ocean water as proposed by the UNESCO formula. Employing the Boussinesq approximated Darcy-flow equations with such a realistic density formula in the buoyancy term, the problem of convective motion of brine is studied. Such convection flow is observed off the coast of Alaska. The field variables in question are the brine-velocity field, the temperature and the salinity, although we simplify the problem by imposing a temperature field that is linear in the depth variable. For this simplified system we study the continuous dependence of the velocity and salinity on the initial data, develop a linear instability analysis and, additionally, present a fully nonlinear three-dimensional stability analysis. This nonlinear analysis necessitates the introduction of a generalised energy (or Lyapunov function) due to the extra terms present in the realistic equation of state. Numerical results indicate that values of the critical Rayleigh numbers are smaller than when these extra terms are omitted. Received: March 10, 1997     

Galerkin finite element simulation of convection driven by rotation and gravitation

The performance of the Galerkin finite element method when applied to time-dependent convection involving rotation, self-gravitation and the normal gravity field in a horizontal cylinder is discussed in this paper. The governing equations, the parameters of the problem and our implementation of the numerical schemes are presented. The accuracy, spatial scale of resolution, flexibility and robustness of the resulting code show the element method as a valuable tool for research in this area or in related problems in astrophysical fluid dynamics. The numerical difficulties in large-amplitude flows are associated with an error-control scheme for time integration and the ‘short-time’ wiggles in transient Dirichlet problems. Coarse grids give the correct qualitative picture in most simulations, but the type of solution at short time (and hence grid refinement) presumably resolves the degeneracy in the azimuthal orientation of convection cells in flows driven by self-gravitation and (perhaps) centrifugal buoyancy. The final state in transient flows is a motionless isothermal fluid. However, residual motions can be nullified only in the limit of zero grid size in flows driven by centrifugal buoyancy (self-gravitation), while a fairly coarse grid is sufficient for this purpose in normal gravity-driven flows.     

Filtration convection in a high-frequency vibration field

The effect of high-frequency translational vibrations on the occurrence of filtration convection in a plane horizontal layer of a viscous incompressible liquid saturating a porous medium is studied. Constant temperature is maintained at the boundaries of the layer. It is established that for any vibration direction different from the vertical (transverse) direction, convection in gravity and thermal gravitational convection under both heating from above and heating from below can arise. In the case of reduced gravity, values of the vibration parameter that lead to transition to zero gravity are established. The results are obtained from an analysis of the averaged equations of filtration convection, derived for an arbitrary region. This work was presented at the joint X European and VI Russian Symposium on Physical Sciences in Microgravity (St. Petersburg, June 15–20, 1997). Rostov State university, Rostov-on-Don 344090. Rostov State Academy of Building, Rostov-on-Don 344022. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 3, pp. 22–29, May–June, 1999.     

竖直振动颗粒床对流机制的颗粒尺度实验研究

为了得到竖直振动颗粒床形成对流的运动模式及形成机制,本文通过高速摄影技术对竖直振动颗粒床进行了实验研究。实验发现,随着振动加速度的增加,对流环覆盖的粒子层数和强度明显增加。通过分析颗粒速度矢量图的演化,可以获得控制对流运动的各种应力波在在粒子床中传播的信息,发现应力波的强度和持续时间与振动加速度密切相关。通过实验发现,对流运动发端于重力波面上粒子从侧壁向粒子床中心的不可逆跃迁,这种横向对流的强度与重力波面的曲率密切相关,而持续的时间随粒子床振动周期的变化而变化。     

Energy methods for nonlinear stability in convection problems primarily related to geophysics

Applications of the theory of energy stability to problems in nonlinear convection are presented. Many topics are reviewed and examples of thermal, thermohaline and bio-convection are included. Specific sections address generalised energy methods, convection in a half-space, electrodynamic convection, surface tension driven convection, time-dependent flows such as gravity modulated convection and other topics.