Investigation of the Relative Motion of a Viscous Fluid and a Porous Medium Using the Brinkman Equations

Exact solutions are obtained for the following three problems in which the Brinkman filtration equations are used: laminar fluid flow between parallel plane walls, one of which is rigid while the other is a plane layer of saturated porous medium, motion of a plane porous layer between parallel layers of viscous fluid, and laminar fluid flow in a cylindrical channel bounded by an annular porous layer.     

Viscous incompressible flow in a narrow channel with one free wall and fluid supply through a porous insert

The problem investigated relates the plane unsteady flow of a viscous incompressible fluid in a narrow channel one of whose walls is free and acted upon by a given load, while the other is rigidly fixed. The fluid enters the channel through a porous insert in the stationary wall. A model of the flow of a thin film of viscous incompressible fluid and Darcy's law for flow in a porous medium are used to find the distribution of fluid pressure and velocity in the channel and the porous insert in the two-dimensional formulation for fairly general boundary conditions in the case where the length of the porous insert exceeds the length of the free wall. In the particular case where the length of the porous insert is equal to the length of the free wall an exact stationary solution of the problem is obtained for a given value of the channel height. The stability of the equilibrium position of the free wall supported on a hydrodynamic fluid film is examined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 16–24, January–February, 1986.     

A sharp interface Cartesian grid method for viscous simulation of shocked particle-laden flows

A Cartesian grid-based sharp interface method is presented for viscous simulations of shocked particle-laden flows. The moving solid–fluid interfaces are represented using level sets. A moving least-squares reconstruction is developed to apply the no-slip boundary condition at solid–fluid interfaces and to supply viscous stresses to the fluid. The algorithms developed in this paper are benchmarked against similarity solutions for the boundary layer over a fixed flat plate and against numerical solutions for moving interface problems such as shock-induced lift-off of a cylinder in a channel. The framework is extended to 3D and applied to calculate low Reynolds number steady supersonic flow over a sphere. Viscous simulation of the interaction of a particle cloud with an incident planar shock is demonstrated; the average drag on the particles and the vorticity field in the cloud are compared to the inviscid case to elucidate the effects of viscosity on momentum transfer between the particle and fluid phases. The methods developed will be useful for obtaining accurate momentum and heat transfer closure models for macro-scale shocked particulate flow applications such as blast waves and dust explosions.     

On local solutions to non-Newtonian slow viscous flows

The eigenfunction expansion method is used to obtain local solutions to some non-Newtonian slow viscous flows. The forms of viscosity variation amenable to such analysis are restricted but do include power-law fluids. Power-law flow near a sharp corner between plane boundaries is analysed and results are obtained for the critical corner angle for eddy formation. Flows near a 90° corner with either a moving boundary or a finite flow rate at the corner are also considered. The “stick-slip” behaviour of a power-law fluid at a plane solid boundary is shown to obey a simple law.     

Motion of a circular cylinder in a viscous liquid between parallel plates

The two-dimensional motion of a cylinder in a viscous fluid between two parallel walls of a vertical channel is studied. It is found that when the cylinder moves very closely along one of the channel walls, it always rotates in the direction opposite to that of contact rolling along the nearest wall. When the cylinder is away from the walls, its rotation depends on the Reynolds number of the flow. In this study two numerical methods were used. One is for the unsteady motion of a sedimenting cylinder initially released from a position close to one of the channel walls, where the Navier-Stokes equations are solved for the fluid and Newton's equations of motion are solved for the rigid cylinder. The other method is for the steady flow in which a cylinder is fixed in a uniform flow field where the channel walls are sliding past the cylinder at the speed of the approaching flow, or equivalently a cylinder is moving with a constant velocity in a quiescent fluid. The flow field, the drag, the side force (lift), and the torque experienced by the cylinder are studied in detail. The effects of the cylinder location in the channel, the size of the channel relative to the cylinder diameter, and the Reynolds number of the flow are examined. In the limit when the cylinder is translating very closely along one of the walls, the flow in the gap between the cylinder and the wall is solved analytically using lubrication theory, and the numerical solution in the other region is used to piece together the whole flow field.This research was supported by NSF DMR91-20668 through the Laboratory for Research on the Structure of Matter at the University of Pennsylvania and from the Research Foundation of the University of Pennsylvania.     

Generalized Plain Couette Flow and Heat Transfer in a Composite Channel

An analytical study of fluid flow and heat transfer in a composite channel is presented. The channel walls are maintained at different constant temperatures in such a way that the temperatures do not allow for free convection. The upper plate is considered to be moving and the lower plate is fixed. The flow is modeled using Darcy–Lapwood–Brinkman equation. The viscous and Darcy dissipation terms are included in the energy equation. By applying suitable matching and boundary conditions, an exact solution has been obtained for the velocity and temperature distributions in the two regions of the composite channel. The effects of various parameters such as the porous medium parameter, viscosity ratio, height ratio, conductivity ratio, Eckert number, and Prandtl number on the velocity and temperature fields are presented graphically and discussed.     

On the implicit time integration of semi‐discrete viscous fluxes on unstructured dynamic meshes

Numerical simulations of viscous flow problems with complex moving and/or deforming boundaries commonly require the solution of the corresponding fluid equations of motion on unstructured dynamic meshes. In this paper, a systematic investigation of the importance of the choice of the mesh configuration for evaluating the viscous fluxes is performed when the semi‐discrete Navier–Stokes equations are time‐integrated using the popular second‐order implicit backward difference algorithm. The findings are illustrated with the simulation of a laminar viscous flow problem around an oscillating airfoil. Copyright © 1999 John Wiley & Sons, Ltd.     

Two-dimensional flows of a viscous binary fluid between moving solid boundaries

A class of exact solutions of hydrodynamic equations with additional Korteweg stresses is obtained which is characterized by a linear dependence of part of the velocity components on the space variable. In this class, exact solutions of two problems of binary fluid flow between moving flat solid boundaries was found. A family of particular exact solutions is obtained for the problem of viscous fluid flow between planes which approach or move away from each other according to a special law.     

Stochastic methods in dispersion theory

In this study we show how methods from the theory of stochastic processes can be applied to problems in dispersion theory.First, we show that Taylor dispersion with adsorbing boundaries is easily transformed into a new Taylor dispersion problem without adsorbing boundaries. The transformed problem can then be solved using any of the traditional methods used for Taylor dispersion.Secondly, we consider the dispersion of particles in a channel (between parallel plates) with one partially adsorbing surface and one perfectly reflecting boundary. We determine the exact law of the position of adsorption for an arbitrary channel flow in terms of an infinite series of iterated integrals of the flow field, which is assumed to be a function of the cross-channel coordinate only. We also consider the case of shear flow over an adsorbing plane, by taking the limit where one of the boundaries is taken to infinity     

Flow and heat transfer over a generalized stretching/shrinking wall problem—Exact solutions of the Navier-Stokes equations

In this paper, we investigate the steady momentum and heat transfer of a viscous fluid flow over a stretching/shrinking sheet. Exact solutions are presented for the Navier-Stokes equations. The new solutions provide a more general formulation including the linearly stretching and shrinking wall problems as well as the asymptotic suction velocity profiles over a moving plate. Interesting non-linear phenomena are observed in the current results including both exponentially decaying solution and algebraically decaying solution, multiple solutions with infinite number of solutions for the flow field, and velocity overshoot. The energy equation ignoring viscous dissipation is solved exactly and the effects of the mass transfer parameter, the Prandtl number, and the wall stretching/shrinking strength on the temperature profiles and wall heat flux are also presented and discussed. The exact solution of this general flow configuration is a rare case for the Navier-Stokes equation.     

Oscillations of a body filled with a viscous fluid

The oscillations of a physical pendulum containing a spherical cavity filled with an incompressible viscous liquid were discussed in [1]. In this paper we consider the mote general problem of the motion of an axially symmetric solid with a spherical cavity filled with an incompressible viscous fluid and moving about a fixed point. It is assumed that the center of the cavity and the fixed point lie on the axis of symmetry of the body.     

Stability of a pipeline section with an elastic support

We systematically study the stability of a pipeline section filled with a moving nonviscous fluid. The computational scheme of the pipeline is a rod one of whose ends is rigidly fixed and the other is elastically supported. For the problem parameters we take the fluid relative mass, the fluid flow rate, and the rigidity of the elastic support. We study the dynamic buckling frequencies and modes for various critical values of the parameters and the behavior of characteristic exponents on the complex plane. We also analyze the influence of the elastic support on the position of the stability region boundaries and on the type of buckling in the transition to a critical state.     

Formation of the fine structure of laminated stratified flows

The process of formation of the fine structure of stratified fluid flows is investigated in detail on the basis of exact solutions of the initial boundary value problems constructed using the Laplace transform. The problems of formation of plane-parallel flows generated in the neighborhood of a plane or a cylinder inclined to the horizontal and moving in translation are considered as model problems. The transformation of the properties of the solutions is studied by successively going over from the model of a homogeneous to that of an inhomogeneous fluid and to the model which takes diffusion of the stratifying admixture into account. It is shown that despite of the smallness of the parameters responsible for the stratification and diffusion effects their contribution with respect to other effects (inertial and viscous forces) is determined not only by the value of the scalar parameters (Froude and Schmidt numbers) but also to a substantial degree by the dynamic factors. Due to the great difference between the scalar parameters four time zones (three zones of small times, namely, viscous, buoyancy, and diffusion times, and the large-time zone) on whose boundaries the flow model changes are distinguished in the study. It is shown that the passage to the limit as t → ∞ is nonuniform for both plane and cylindrical geometry. Transition to steady-state models leads not only to nonuniformity of the double passage to the limit but frequently also to the need to reformulate the boundary conditions.     

An adaptive boundary element approach to transient free surface flow as applied to injection molding

An adaptive (Lagrangian) boundary element approach is proposed for the general two‐dimensional simulation of confined moving‐boundary flow of viscous incompressible fluids. Only the quasi‐steady creeping (Stokes) flow of a Newtonian fluid is examined. The method is stable as it includes remeshing capabilities of the deforming moving boundary, and thus it can handle large deformations. An algorithm is developed for mesh refinement of the deforming moving‐boundary mesh. Several flow problems are presented to illustrate the utility of the approach, with particular emphasis on cavity filling and viscous fingering, as applied to conventional and gas‐assisted injection molding. The accuracy of the method is assessed through the problem of jet flow and the transient fountain flow between two flat plates. Copyright © 2000 John Wiley & Sons, Ltd.     

A hybrid immersed‐boundary and multi‐block lattice Boltzmann method for simulating fluid and moving‐boundaries interactions

A numerical method is developed for modelling the interactions between incompressible viscous fluid and moving boundaries. The principle of this method is introducing the immersed‐boundary concept in the framework of the lattice Boltzmann method, and improving the accuracy and efficiency of the simulation by refining the mesh near moving boundaries. Besides elastic boundary with a constitutive law, the method can also efficiently simulate solid moving‐boundary interacting with fluid by employing the direct forcing technique. The method is validated by the simulations of flow past a circular cylinder, two cylinders moving with respect to each other and flow around a hovering wing. The versatility of the method is demonstrated by the numerical studies including elastic filament flapping in the wake of a cylinder and fish‐like bodies swimming in quiescent fluid. Copyright © 2006 John Wiley & Sons, Ltd.     

Unitive analysis schemes on problems of multiple moving boundaries with 3-D liquid-solid multiple nonlinear coupling for uplift of anchored liquid storage tanks

I.Intr0ducti0nAnchored1iquidstoragetanksareextensivelyusedinbusiness,nuclearpowerplants,spaceflightandpetrochemicalindustry.Theystoragepetrochemicalproductions0fover9Opercentintheworld.Theanchoredliquidstoragetanksarecontinuelydamagedduringearthquakestoha…     

Structure of wall flow through a channel with a granular bed

The problem of viscous incompressible fluid flow through a plane channel with one linear and one sinusoidal boundary is considered. Using the representation of the system of Stokes equations in terms of the stream function in a region including a single periodic element, we obtain a boundary-value problem for the biharmonic operator. Its solution is found by the mixed Galerkin method - the straight line method. The near-degenerate matrix of unknown coefficients was calculated on a computer. The velocity vector component, pressure and streamline fields are found as functions of the curvature of the boundary. The features of the flow structure resulting from the asymmetry of the walls are established. The distortion of the pore space required to refine the dependence of the permeability on the structure is introduced. The results are of interest for analyzing the wall effect of increased flow velocity in a channel with a fixed granular bed.Ufa. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 9–13, November–December, 1994.     

DYNAMICAL ARTIFICIAL BOUNDARY FOR FLUID MEDIUM IN WAVE MOTION PROBLEMS

无限域流体介质的波动辐射效应是影响海域工程动力反应的重要因素,人工边界是实现此类开放系统近场波动问题数值分析的有效方法.基于位移格式的流体波动理论推导开放域流体介质的人工边界,分别给出一维、二维和三维空间中平面波、柱面波和球面波的流体介质动力人工边界条件,其中一维平面波动人工边界为经典的黏性边界,二维柱面波、三维球面波的人工边界处节点应力与节点速度和加速度成正比,可等效为由阻尼与质量系统构成的人工边界条件.讨论相应的数值模拟技术,给出流体介质动力人工边界在ANSYS软件平台的具体实现方法.近场流体介质动力反应问题的算例表明,所发展的流体动力人工边界对于轴对称波动与非轴对称波动在近场有限域截断处的透射吸收作用的模拟计算精度均较为良好,说明此流体介质人工边界具有较高的可靠性与实用性.所发展的流体介质动力人工边界可较为方便地与大型商用有限元软件结合,可为包括海域地形和海岛在内的海域工程的动力分析提供一定的方法借鉴.     

Compressible, viscous fluid dynamics (review). Part 1

The article is the first part of a survey of problems in compressible, viscous fluid dynamics as related to the dynamics of rigid and elastic bodies in a compressible, viscous fluid in the linearized formulation. The formulation of basic problems is discussed, along with a method of solution based on general solutions of the Navier-Stokes equations in vector and scalar form in dynamical problems. Forced harmonic vibrations of rigid bodies in rest and moving compressible, viscous fluids are discussed. Publications relevant to the stated problems are analyzed. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 1, pp. 25–52, January, 2000. Detailed information about the author can be found in the journalPrikladnaya Mekhanika, Volume 35, No. 1, pp. 104–108 (1999).