Theoretical Approach to the Onset of Convection in a Porous Layer

We examine the effect of viscous forces on the displacement of one fluid by a second, immiscible fluid along parallel layers of contrasting porosity, absolute permeability and relative permeability. Flow is characterized using five dimensionless numbers and the dimensionless storage efficiency, so results are directly applicable, regardless of scale, to geologic carbon storage. The storage efficiency is numerically equivalent to the recovery efficiency, applicable to hydrocarbon production. We quantify the shock-front velocities at the leading edge of the displacing phase using asymptotic flow solutions obtained in the limits of no crossflow and equilibrium crossflow. The shock-front velocities can be used to identify a fast layer and a slow layer, although in some cases the shock-front velocities are identical even though the layers have contrasting properties. Three crossflow regimes are identified and defined with respect to the fast and slow shock-front mobility ratios, using both theoretical predictions and confirmation from numerical flow simulations. Previous studies have identified only two crossflow regimes. Contrasts in porosity and relative permeability exert a significant influence on contrasts in the shock-front velocities and on storage efficiency, in addition to previously examined contrasts in absolute permeability. Previous studies concluded that the maximum storage efficiency is obtained for unit permeability ratio; this is true only if there are no contrasts in porosity and relative permeability. The impact of crossflow on storage efficiency depends on the mobility ratio evaluated across the fast shock-front and on the time at which the efficiency is measured.     

Decay of shear layers and vortex sheets

An incompressible fluid is contained in the domain between two stationary infinited parallel rigid plates. It is assumed that for shear flows, the shear stress in an element of the fluid depends linearly on history of the velocity gradient in that element. It is supposed that initially two steady shear layers exist in the fluid and are symmetrically disposed with respect to the mid-plane. The time-dependent velocity field which results from the removal of the forces maintaining this steady flow is calculated in the cases when the fluid is Newtonian and when it is Maxwellian. The limiting cases when the shear layers reside in an unbounded space of the fluid and when they further become vortex sheets are discussed.     

Steady Ostwald flow between two differentially rotating co-axial discs

The steady flow of a power-law (Ostwald) fluid between two differentially rotating, parallel, co-axial discs has been considered for both large and small Reynolds number. All edge effects of the discs are neglected, the discs rotate in the same sense and the distance between the two discs is much smaller than their radius. In the large Reynolds number case a similarity solution is sought. It is assumed that the flow consists of boundary layers on the discs, while the core rotates as a rigid body with speed intermediate of those of the discs. The boundary layer is thinner than in the equivalent Newtonian problem, and the decay of the boundary layers is found to be algebraic. This slow decay contrasts with the faster exponential decay in the Newtonian case. For the low Reynolds number problem, the ratio of the disc separation to radius was taken to be much smaller than the Reynolds number. This is, in effect, a lubrication-type problem. The velocity components are expressed as expansions in ascending powers of the Reynolds number. For both the large and small Reynolds number flow, the torque is calculated as a function of the disc speeds, for various values of the power-law index n.     

Wave propagation in alternating solid and fluid layers

The exact solution for harmonic plane wave propagation through periodically stratified media composed of alternating elastic solid and ideal fluid layers is derived for all frequencies and wavenumbers. The modal structure and the structure of the pass and stop bands is shown in the plane of slowness parallel to the stratification, s₁, and frequency, ω. In the low frequency limit, the slowness component perpendicular to the stratification, s₃, is found as a function of s₁, defining slowness surfaces in the s₁ and s₃ plane. At any angle of phase propagation through the medium (except perpendicular) there are two modes, a fast wave for which the motion in the solid and fluid layers is in phase and a slow wave for which the motion is 180° out of phase. An equivalent acoustic medium is found which has an anisotropic density tensor and a nonlocal constitutive relation between pressure and dilatation whose terms are fourth derivatives in time and in the space coordinate parallel to the stratification. This equivalent homogeneous anisotropic medium exhibits the indentical wave behavior as the layered solid fluid medium in the long wavelength limit as it has the same slowness surfaces.     

超声速边界层/混合层组合流动的稳定性分析

利用可压缩线性稳定性理论研究了超声速混合层考虑壁面影响流动时的失稳特性. 基本流场选取了具有不同速度特征的2 股均匀来流,进入存在上下壁面的流道中. 混合层与边界层的距离为1~3 个边界层厚度,其中壁面取为绝热壁. 分析了该流动在超声速情况下的稳定性特征,同时还讨论了不同波角下的三维扰动波的演化特点,并与二维扰动波进行了比较和分析. 研究结果表明,在此流动情况下,边界层流动和混合层流动的稳定性特征同时存在,并互有影响,其流动稳定性特征既有别于单纯的平板边界层,也有别于单纯的平面混合层,呈现出了新的稳定性特征.      

超声速边界层/混合层组合流动的稳定性分析简

利用可压缩线性稳定性理论研究了超声速混合层考虑壁面影响流动时的失稳特性. 基本流场选取了具有不同速度特征的2 股均匀来流,进入存在上下壁面的流道中. 混合层与边界层的距离为1~3 个边界层厚度,其中壁面取为绝热壁. 分析了该流动在超声速情况下的稳定性特征,同时还讨论了不同波角下的三维扰动波的演化特点,并与二维扰动波进行了比较和分析. 研究结果表明,在此流动情况下,边界层流动和混合层流动的稳定性特征同时存在,并互有影响,其流动稳定性特征既有别于单纯的平板边界层,也有别于单纯的平面混合层,呈现出了新的稳定性特征.     

Singularities on wave fronts of slow waves in anisotropic fluid-saturated porous media

Energy focusing is found on the wave fronts of slow waves, which is a new propagation characteristic for slow waves in fluid-saturated porous materials. The material parameters, as well as the propagation directions, are chosen as the control parameters. Combined with the two axial variables, the influence of the anisotropy of the solid skeleton and pore fluid parameters on the propagation characteristic of slow waves in anisotropic fluid-saturated porous materials is discussed. The correspondence between the focusing on the wave fronts and the contours of zero Gaussian curvature on the slowness surface is explored. The development of the focusing patterns is investigated and the distinct trends in the energy flux focusing structures are revealed. This is helpful in further understanding the roles of the pore fluid in the damage of the fluid-saturated porous media.     

Turbulent penetrative convection with an internal heat source

An infinite fluid with a vertical cubic temperature profile in the absence of fluid motion is considered as a model for penetrative convection in which a central unstably stratified fluid layer is bounded above and below by stably stratified layers. Turbulence statistics from direct and large eddy numerical simulations for the mean temperature gradient, the velocity and temperature variances and the heat flux are presented for Rayleigh numbers R up to four orders of magnitude above critical. By means of a simplified second-moment closure, analytical scaling laws for the statistics are determined. For high Rayleigh numbers, the mean temperature gradient approaches zero in a central well-mixed layer, a reduced positive (stable) value in upper and lower partially mixed layers, and an unmixed value far above and below. The temperature variance is a factor of R^1/3 larger in the partially mixed layers compared to the well-mixed layer; the velocity variance and heat flux scales the same in both layers. Approximation of the three layers by a two layer model yields an estimate for the height of the mixed layer: the height decreases slowly with increasing Rayleigh number and at the highest Rayleigh number simulated is approximately 30% longer than the unstable layer in the absence of fluid motion.     

Slow steady flows of a conducting fluid at large Hartmann numbers

We consider slow steady flows of a conducting fluid at large values of the Hartmann number and small values of the magnetic Reynolds number in an inhomogeneous magnetic field. The general solution is obtained in explicit form for the basic portion (core) of the flow, where the inertia and viscous forces may be neglected. The boundary conditions which this solution must satisfy at the outer edges of the boundary layers which develop at the walls are considered. Possible types of discontinuity surfaces and other singularities in the flow core are examined. An exact solution is obtained for the problem of conducting fluid flow in a tube of arbitrary section in an inhomogeneous magnetic field.The content of this paper is a generalization of some results on flows in a homogeneous magnetic field, obtained in [1–8], to the case of arbitrary flows in an inhomogeneous magnetic field. The author's interest in the problems considered in this study was attracted by a report presented by Professor Shercliff at the Institute of Mechanics, Moscow State University, in May 1967, on the studies of English scientists on conducting fluid flows in a strong uniform magnetic field.     

Some Exactly Solvable Problems of the Radiation of Three–Dimensional Periodic Internal Waves

An exact solution of the problem of the generation of three–dimensional periodic internal waves in an exponentially stratified, viscous fluid is constructed in a linear approximation. The wave source is an arbitrary part of the surface of a vertical circular cylinder which moves in radial, azimuthal, and vertical directions. Solutions satisfying exact boundary conditions, describe both the beam of outgoing waves and wave boundary layers of two types: internal boundary layers, whose thickness depends on the buoyancy frequency and the geometry of the problem, and viscous boundary layers, which, as in a homogeneous fluid, are determined by kinematic viscosity and frequency. Asymptotic solutions are derived in explicit form for cylinders of large, intermediate, and small dimensions relative to the natural scales of the problem.     

The Compressible Viscous Surface-Internal Wave Problem: Nonlinear Rayleigh–Taylor Instability

This paper concerns the dynamics of two layers of compressible, barotropic, viscous fluid lying atop one another. The lower fluid is bounded below by a rigid bottom, and t he upper fluid is bounded above by a trivial fluid of constant pressure. This is a free boundary problem: the interfaces between the fluids and above the upper fluid are free to move. The fluids are acted on by gravity in the bulk, and at the free interfaces we consider both the case of surface tension and the case of no surface forces.We are concerned with the Rayleigh–Taylor instability when the upper fluid is heavier than the lower fluid along the equilibrium interface. When the surface tension at the free internal interface is below the critical value, we prove that the problem is nonlinear unstable.     

Two-phase flow modelling: the closure issue for a two-layer flow

The fully-developed, laminar flow of two fluid layers in a horizontal channel is studied by means of the height-averaged balance equations. The closure issue is addressed and the closure relations for the wall and interfacial shear stresses are given for some particular cases.     

Oscillatory Motion of a Viscous Fluid in Contact with a Flat Layer of a Porous Medium

Analytical solutions are obtained for two problems of transverse internal waves in a viscous fluid contacting with a flat layer of a fixed porous medium. In the first problem, the waves are considered which are caused by the motion of an infinite flat plate located on the fluid surface and performing harmonic oscillations in its plane. In the second problem, the waves are caused by periodic shear stresses applied to the free surface of the fluid. To describe the fluid motion in the porous medium, the unsteady Brinkman equation is used, and the motion of the fluid outside the porous medium is described by the Navier–Stokes equation. Examples of numerical calculations of the fluid velocity and filtration velocity profiles are presented. The existence of fluid layers with counter-directed velocities is revealed.     

An interfacial Gerstner-type trapped wave

Coastally trapped rotational interfacial waves are studied theoretically by using a Lagrangian formulation of fluid motion in a rotating ocean. The waves propagate along the interface between two immiscible inviscid incompressible fluid layers of finite depths and different densities, and are trapped at a straight wall due to the Coriolis force. For layers of finite depth, solutions are sought as series expansions after a small parameter. Comparison is made with the irrotational interfacial Kelvin wave. Both types of waves are identical to first order, having zero vorticity. The second order solution yields a relation between the vorticity and the velocity shear in the wave motion. Requiring that the mean motion in both layers is irrotational, then follows the well-known Stokes drift for interfacial Kelvin waves. On the other hand, if the mean forward drift is identically zero, we obtain the second order vorticity in the Gerstner-type wave. The solutions in both layers for the Gerstner-type interfacial wave are given analytically to second order. It is shown that small density differences and thin upper layers both act to yield a shape of the material interfacial with broader crests and sharper troughs. These effects also tend to make the particle trajectories at the interface in both layers become distorted ellipses which are flatter on the upper side. It is concluded that the effect of air excludes the possibility of observing the exact Gerstner wave in deep water.     

Boundary-layer analysis of propagating mode II cracks in porous elastic media

The steady advance of semi-infinite cracks in porous elastic media is examined in the limiting cases of very slow and very rapid propagation. The low-speed limit is the elastic solution representing fully-drained behavior, with no boundary layers present. In the high-speed limit, two boundary layers are identified: one along the crack face and another centered at the crack tip. The zero-order outer solution is the elastic solution representing fully-undrained behavior, with both boundary layers acting as zones of transition to the effectively drained conditions at the crack itself. Closedform expressions are derived for the stresses and pore pressure within the crack-tip boundary layer. The true stress intensity factor is found to be reduced by an amount related to the disparity between the drained and undrained Poisson's ratios of the medium.     

Two fluid flow and heat transfer in an inclined channel containing porous and fluid layer

Convective flow and heat transfer in an inclined channel bounded by two rigid plates held at constant different temperatures with one region filled with porous matrix saturated with a viscous fluid and another region with a clear viscous fluid different from the fluid in first region is studied analytically. The coupled nonlinear governing equations are solved using regular perturbation method. It is found that the presence of porous matrix in one of the region reduces the velocity and temperature. Results have been presented for a wide range of governing parameters such as Grashof number, porous parameter, angle of inclination, ratio of heights of the two layers and also the ratio of viscosities.     

Numerical prediction of turbulent thermocapillary convection in superposed fluid layers with a free interface

The present paper introduces a new numerical method for predicting the characteristics of thermocapillary turbulent convection in a differentially-heated rectangular cavity with two superposed and immiscible fluid layers. The unsteady Reynolds form of the Navier–Stokes equations and energy equation are solved by using the control volume approach on a staggered grid system using SIMPLE algorithm. The turbulence quantities are predicted by applying the standard k–ε turbulence model. The level set formulation is applied for predicting the topological changes of the interface separating the two fluid layers and to provide an accurate and robust modeling of the interfacial normal and tangential stresses. The computational results obtained showed good agreement when compared with the previous experimental, numerical and analytical benchmark data for different validation cases in both laminar and turbulent regimes. The present numerical method is then applied to predict the velocity and temperature distribution in two immiscible liquid layers with undeformable interface for a wide range of Marangoni numbers. The laminar-turbulent transition is demonstrated by obtaining the turbulence features at high interfacial temperature gradient which is characterized by high Marangoni number. The effect of increasing Marangoni number on the interface dynamics in turbulent regime is also investigated.     

有限厚度流体层界面运动Rayleigh-Taylor不稳定性的数值模拟

采用二阶ＴＶＤ格式及ＬｅｖｅｌＳｅｔ方法计算了二维可压缩有限厚度流体层Ｒａｙｌｅｉｇｈ Ｔａｙｌｏｒ流体不稳定性。计算结果与Ｔａｙｌｏｒ的线性解和Ｏｔ的薄层非线性解析解符合很好。     

Numerical prediction of turbulent thermocapillary convection in superposed fluid layers with a free interface

The present paper introduces a new numerical method for predicting the characteristics of thermocapillary turbulent convection in a differentially-heated rectangular cavity with two superposed and immiscible fluid layers. The unsteady Reynolds form of the Navier–Stokes equations and energy equation are solved by using the control volume approach on a staggered grid system using SIMPLE algorithm. The turbulence quantities are predicted by applying the standard k–ε turbulence model. The level set formulation is applied for predicting the topological changes of the interface separating the two fluid layers and to provide an accurate and robust modeling of the interfacial normal and tangential stresses. The computational results obtained showed good agreement when compared with the previous experimental, numerical and analytical benchmark data for different validation cases in both laminar and turbulent regimes. The present numerical method is then applied to predict the velocity and temperature distribution in two immiscible liquid layers with undeformable interface for a wide range of Marangoni numbers. The laminar-turbulent transition is demonstrated by obtaining the turbulence features at high interfacial temperature gradient which is characterized by high Marangoni number. The effect of increasing Marangoni number on the interface dynamics in turbulent regime is also investigated.