Showing papers on "Hele-Shaw flow published in 1993"

A lattice Boltzmann model for multiphase fluid flows

Abstract: A lattice Boltzmann equation method for simulating multiphase immiscible fluid flows with variation of density and viscosity, based on the model proposed by Gunstensen et al. for two‐component immiscible fluids [Phys. Rev. A 43, 4320 (1991)] is developed. The numerical measurements of surface tension and viscosity agree well with theoretical predictions. Several basic numerical tests, including spinodal decomposition, two‐phase fluid flows in two‐dimensional channels, and two‐phase viscous fingering, are shown in agreement of experiments and analytical solutions.

The plane Symmetric sudden-expansion flow at low Reynolds numbers

Abstract: Detailed velocity measurements and numerical predictions are presented for the flow through a plane nominally two-dimensional duct with a Symmetric sudden expansion of area ratio 1:2. Both the experiments and the predictions confirm a symmetry-breaking bifurcation of the flow leading to one long and one short Separation zone for channel Reynolds numbers above 125, based on the upstream channel height and the maximum flow velocity upstream. With increasing Reynolds numbers above this value, the short separated region remains approximately constant in length whereas the long region increases in length.The experimental data were obtained using a one-component laser-Doppler anemometer at many Reynolds number values, with more extensive measurements being performed for the three Reynolds numbers 70, 300 and 610. Predictions were made using a finite volume method and an explicit quadratic Leith type of temporal discretization. In general, good agreement was found between measured and predicted velocity profiles for all Reynolds numbers investigated.

Rheological phenomena in focus

Abstract: Part 1 Introduction: What is rheology? Why flow visualization? On flow visualization techniques Material functions Dimensionless numbers Outline of the book. Part 2 General Phenomena Rod-climbing Weissenberg effects Post-extrusion effects Extensional viscosity effects Deformation-history effects. Part 3 Contraction and Expansion Flows: Flow through axisymmetric contractions Flow through planar contractions Flow through expansions. Part 4 Confined Flows: Flow over a hole Combined mixing and separating flow Flow in a channel obstructed by an antisymmetric array of obstacles Flow in a "T" geometry Flow past cylinders and sheres High Reynolds number flows Radial flow in a Hele-Shaw cell. Part 5 Rotating and Oscillating Flows Flow in cylindrical containers Flow caused by rotating solids of revolution Instabilities in rotating flows Flow modification in the two-roll mill Flow in the four-roll mill - the Uebler effect Flow generated by a vibrating cylinder. Part 6 Jet Breakup: Deformation and breakup of viscoelastic drops in planar extensional flows Rising bubble in a non-Newtonian liquid Drop entering a liquid Aggregation effects in suspensions of spheres in non-Newtonian liquids.

Mathematical Modeling of the Flow of Four Fluids in a Packed Bed.

Abstract: Macroscopic flow phenomena play important roles not only for improving productivity and energy efficiency but also for achieving the stable operation in metallurgical and chemical reactors. This review paper deals with flow phenomena of four fluids which are single-phase or multi-phase flow of gas, fine particles, liquid and packed particles. In some previous researches on the multi-phase flow, fundamental equations were derived for a continuous fluid phase and dispersed phases with different types of modeling. However, in this paper, continuous flow was assumed for each phase in the derivation of the equation of motion for obtaining the numerical solution. The model has been applied to the simulation of not only four phase flows but also one to three phase flows. Typical examples for application will be described for several processes.

Topology transitions and singularities in viscous flows.

Abstract: Topological reconfigurations of the boundaries of thin fluid layers in Hele-Shaw flow are studied. A systematic treatment of the dynamics of the bounding interfaces is developed through an expansion in the aspect ratio of the layer, yielding nonlinear partial differential equations for the local thickness. For both density-stratified fluid layers and gravity-driven jets, numerical study of the dynamics at second order suggests strongly the collision of the interfaces in finite time. There are associated singularities both in the fluid velocity and in geometric properties of the interfaces

Two‐phase flow in a variable aperture fracture

Abstract: In this paper a dynamic two-dimensional two-phase flow model for a single variable aperture fracture is developed. The model is based on a finite volume implementation of the cubic law and the conservation of mass for each liquid. The two-phase fracture flow system is represented by incompressible parallel plate flow within two-dimensional subregions of constant aperture. The fluid phase distribution is represented by an explicit definition of the phase presence at each location within the domain. To achieve this definition, a phase distribution is assigned to each fracture subregion. Knowledge of the phase distribution allows calculation of interface capillary pressure based on the fracture aperture. One-dimensional analytic solutions for two-phase flow are developed and used to verify the model's behavior in one dimension. The model is verified against the Sandia Waste-Isolation Flow and Transport III model for the case of two-dimensional single-phase flow. Two-dimensional two-phase flow verification is performed qualitatively because no suitable analytic or physical model is currently available. Two-dimensional flow phenomena are investigated for variable aperture fractures generated using geostatistical methods. Results from these simulations illustrate the flow processes of phase isolation, pinching off of nonwetting phase globules, nonwetting phase refusal at the edges of tight regions, and downslope migration of a fluid countercurrent to flow of a less dense fluid.

Predicting equilibrium states with Reynolds stress closures in channel flow and homogeneous shear flow

Abstract: Turbulent channel flow and homogeneous shear flow have served as basic building block flows for the testing and calibration of Reynolds stress models. A direct theoretical connection is made between homogeneous shear flow in equilibrium and the log-layer of fully-developed turbulent channel flow. It is shown that if a second-order closure model is calibrated to yield good equilibrium values for homogeneous shear flow it will also yield good results for the log-layer of channel flow provided that the Rotta coefficient is not too far removed from one. Most of the commonly used second-order closure models introduce an ad hoc wall reflection term in order to mask deficient predictions for the log-layer of channel flow that arise either from an inaccurate calibration of homogeneous shear flow or from the use of a Rotta coefficient that is too large. Illustrative model calculations are presented to demonstrate this point which has important implications for turbulence modeling.

On the classification of solutions to the zero-surface-tension model for Hele-Shaw free boundary flows

Abstract: We discuss the classification of solutions to the zero-surface-tension model for Hele-Shaw flows in bounded and unbounded regions with suction and injection. We use results from the theory of univalent functions to derive estimates for certain geometric properties of the fluid region in the injection case.

A simultaneous solution for flow and fiber orientation in axisymmetric diverging radial flow

Abstract: The effect of the changing microstructure during the flow of fiber suspensions on the flow kinematics is studied. The suspension is assumed to consist of rigid cylindrical particles immersed in a highly viscous Newtonian fluid. Further, the suspension is modeled as an anisotropic fluid whose rheological properties are functions of the local microstructure. The effect of inertia is neglected during the flow of the suspension. The orientation of the particles is assumed to be governed by the flow field and the fiber—fiber interactions. The governing equations for the flow field and fiber orientation are coupled and are simultaneously solved in an axisymmetric radial-flow configuration. These solutions are compared to those obtained using the conventional decoupled approximation where the bulk flow field is assumed to be unaffected by the presence of the suspended particles.

Viscous flow normal to a flat plate at moderate Reynolds numbers

Abstract: An experimental and numerical investigation of the two-dimensional flow normal to a flat plate is described. In the experiments, the plate is started impulsively from rest in a channel for Reynolds numbers, based on the breadth of the plate, in the range 5 ≤ Re ≤ 20. Over this range of Re the flow remains symmetrical and stable and tends to a steady state but is shown to depend strongly on the ratio λ of the plate to channel breadth. The evolution of the experimental flow with time and Reynolds number is studied and the variation with λ in the range 0.05 ≤ λ ≤ 0.2 is investigated sufficiently to enable an estimate of properties of the flow as λ → 0 to be obtained for the steady-state flow. The numerical results are obtained for steady flow normal to a flat plate in an unbounded fluid for Reynolds numbers up to Re = 100. They supplement and extend results for this flow obtained for values of Re up to 20 by Hudson & Dennis (1985). The present solutions have been found using a vorticity-stream function formulation rather than the primitive-variable approach of Hudson & Dennis and provide an independent check on these results. A comparison of the theoretical results for Re ≤ 20 with the limit λ → 0 of the experimental results is, generally speaking, extremely satisfactory.

A numerical study of the effect of surface tension and noise on an expanding Hele-Shaw bubble

Abstract: In this paper, the dynamics of an interface under the influence of surface tension is studied numerically for flow in the Hele–Shaw cell, where the interface separates an expanding bubble of inviscid fluid from a displaced viscous fluid. Of special interest is the long–time behavior of the so‐called q‐pole initial data, whose motion is explicitly known and globally smooth for the zero surface tension flow. The numerical method is spectrally accurate and based upon a boundary integral formulation of the problem, together with a special choice for the frame of motion along the interface. In 64‐bit arithmetic, a transition from the formation of side branches to tip splitting is observed as the surface tension is decreased. The tip splitting occurs on a time scale that decreases with the surface tension. This is consistent with some experimental observations. However, by increasing the arithmetic precision to 128 bits, it is found that this transition occurs at a yet smaller surface tension. The tip splitting is associated with the growth of noise in the calculation at unstable scales allowed by the surface tension, and a simple linear model of this growth seems to agree well with the observed behavior. The robustness of the various observed structures to varying amounts of noise is also investigated numerically. It is found that the appearance of side branches seems to be the intrinsic effect of surface tension, and the time scales for their appearance increases as the surface tension decreases. These results suggest, with some qualification, that surface tension acts as a regular perturbation to evolution from this initial data, even for long times.

On the Origin of Streaks in Turbulent Shear Flows

Abstract: It is shown that the ideas of selective amplification and direct resonance, based on linear theory, do not provide a selection mechanism for the well-defined streak spacing of about 100 wall units (referred to as 100+ hereafter) observed in wall-bounded turbulent shear flows. For the direct resonance theory (Benney & Gustaysson, 1981; fang et al., 1986), it is shown that the streaks are created by the nonlinear self-interaction of the vertical velocity rather than of the directly forced vertical vorticity. It is then proposed that the selection mechanism must be inherently nonlinear and correspond to a self-sustaining process. The streak formation is only one stage of the complete mechanism and cannot be isolated from the rest of the process. The 100+ value should be considered as a critical Reynolds number for that self-sustaining mechanism. For the case of plane Poiseuille flow the 100+ criterion corresponds to a critical Reynolds number of 1250, based on the centerline velocity and the channel half-width, which is close to the usually quoted value of about 1000. In plane Couette flow, it corresponds to a critical Reynolds number of 625, based on the half velocity difference and the half-width.

Bridging between eddy-viscosity-type and second-order turbulence models through a two-scale turbulence theory.

Abstract: A gap between eddy-viscosity-type and second-order models is bridged using the results of a two-scale direct-interaction approximation developed for the study of turbulent shear flows. This work provides a method for incorporating the findings from models of eddy-viscosity-type into second-order models, and vice versa. Specifically, the effect of helicity controlling energy-cascade processes is incorporated into a second-order model. Then, a higher-order eddy-viscosity-type expression for the Reynolds stress is derived through the application of an iterative approximation to the second-order model. The latter result is tested in a turbulent rotating channel flow and its usefulness is confirmed. Effects of flow trajectory are also discussed in the context of the effect of an adverse pressure gradient on the isotropic eddy viscosity.

Stationary and oscillatory flow through coarse porous media

Abstract: Measurements in a U-tube tunnel were carried out to study flow through coarse granular material. Tests with stationary flow and tests with oscillatory flow were done to study the differences between both. The coefficients from the extended Forchheimer equation, which is supposed to describe non-stationary porous flow, were determined. It appeared that for oscillatory flow the turbulent resistance is larger than under stationary flow conditions. This additional resistance is depending on the flow-field, expressed by the Keulegan-Carpenter number. The contribution of the inertial resistance is depending on the flow field as well. Its contribution to the total resistance was rather limited. The influence of the non-stationary flow conditions have been implemented in the expressions for the turbulent resistance and the inertial resistance. Comparisons of the results from the stationary flow tests with other measurements show that the results correspond reasonably well. This is not the case for existing expressions for stationary flow. The existing formulae under-predict the a-values while they over-predict the b-values. Further research must be concentrated on the influence of parameters such as grading, the aspect ratio and shape. These dependencies can be determined under stationary flow conditions.

Shock Formation in Overexpanded Tip Leakage Flow

Abstract: Shock formation due to overexpansion of supersonic flow at the inlet to the tip clearance gap of a turbomachine has been studied. The flow was modeled of a water table using a sharp-edged rectangular channel. The flow exhibited an oblique hydraulic jump starting on the channel sidewall near the channel entrance. This flow was analyzed using hydraulic theory. The results suggest a model for the formation of the jump. The hydraulic analogy between free surface water flows and compressible gas flows is used to predict the location and strength of oblique shocks in analogous tip leakage flows. Features of the flow development are found to be similar to those of compressible flow in sharp-edged orifices

On pyroclastic flow emplacement

Abstract: In this note I investigate some theoretical characteristics of pyroclastic flow deposits, assuming that these flows are Bingham fluids, probably the simplest non-Newtonian fluids. Pyroclastic flows are modeled as laminar debris flows moving on an inclined plane, and their physics is discussed within the classical framework of lubrication theory. Using general hydrodynamics methods, I show that the arrestment and emplacement of pyroclastic flows may be seen as the time-asymptotic limit of their equations of motion. This limit is found to be a nonlinear ordinary differential equation, whose solution gives the shape of pyroclastic flow deposits. The model suggests that these flows stop when the supply of material from the source is depleted; deposit thickness is controlled principally by the flow yield stress τz, a parameter characteristic of Bingham fluids, while deposit length, a measure of flow mobility, depends on τz, on the source flux q0, and on the slope θ of the solid substrate. Even in this simple model, theoretical analysis shows a complex correlation between flow parameters and deposit profiles.

Quasiperiodicity and phase locking route to chaos in the 2-D oscillatory flow around a circular cylinder

Abstract: The two‐dimensional oscillatory flow around a circular cylinder is analyzed by means of the numerical approach described in Justesen [J Fluid Mech 222, 157 (1991)] For a fixed value of the ratio between the Stokes viscous thickness and the radius of the cylinder section, when the Reynolds number assumes low values, the flow is periodic and symmetric with respect to an axis aligned with the flow direction and crossing the axis of the cylinder An increase in the Reynolds number beyond a first critical value causes the flow to bifurcate: the velocity field loses its spatial symmetry even though it retains its time periodicity When the Reynolds number is larger than a second critical value, a new frequency appears in the flow This new frequency, which is much smaller than the frequency of the basic flow, increases for increasing values of the Reynolds number till a phase locking takes place A further increase in the Reynolds number leads the flow to a chaotic status The ‘‘quasiperiodicity and phase locking’’ route to chaos can be recognized

Periodic blocking in parallel shear or channel flow at low Reynolds number

Abstract: The two‐dimensional creeping flow disturbance due to a periodic array of wall‐attached barriers in either a parallel shear flow or a channel flow is determined by the use of a distribution of force singularities on each barrier. In each case, the Fredholm integral equation of the first kind, obtained by requiring no normal flow at a barrier, can be transformed into an infinite linear system of the second kind, whose coefficients involve, at most, infinite sums of elementary functions. The computed physical quantities include the displacement of the unbounded shear flow, the induced pressure gradient in the other flows, the maximum velocity in the symmetric flow through pairs of barriers, and the slip length derived by using the Maxwell slip‐flow approximation to model the effect of barriers on a wall.

Turbulence modeling for complex hypersonic flows

Abstract: The paper presents results of calculations for a range of 2D turbulent hypersonic flows using two-equation models. The baseline models and the model corrections required for good hypersonic-flow predictions will be illustrated. Three experimental data sets were chosen for comparison. They are: (1) the hypersonic flare flows of Kussoy and Horstman, (2) a 2D hypersonic compression corner flow of Coleman and Stollery, and (3) the ogive-cylinder impinging shock-expansion flows of Kussoy and Horstman. Comparisons with the experimental data have shown that baseline models under-predict the extent of flow separation but over-predict the heat transfer rate near flow reattachment. Modifications to the models are described which remove the above-mentioned deficiencies. Although we have restricted the discussion only to the selected baseline models in this paper, the modifications proposed are universal and can in principle be transferred to any existing two-equation model formulation.

Numerical simulation of flows past periodic arrays of cylinders

Abstract: We present a detailed numerical investigation of three unsteady incompressible flow problems involving periodic arrays of staggered cylinders. The first problem is a uniperiodic flow with two cylinders in each cell of periodicity. The second problem is a biperiodic flow with two cylinders in each cell, and the last problem is a uniperiodic flow with ten cylinders. Both uniperiodic flows are periodic in the direction perpendicular to the main flow direction. In all three cases, the Reynolds number based on the cylinder diameter is 100, and initially the flow field has local symmetries with respect to the axes of the cylinders parallel to the main flow direction. Later on, these symmetries break, vortex shedding is initiated, and gradually the scale of the shedding increases until a temporally periodic flow field is reached. We furnish extensive flow data, including the vorticity and stream function fields at various instants during the temporal evolution of the flow field, time histories of the drag and lift coefficients, Strouhal number, initial and mean drag coefficients, amplitude of the drag and lift coefficient oscillations, and the phase relationships between the drag and lift oscillations associated with each cylinder. Our data confirms that, at this Reynolds number, there are no stable steady-state solutions with local symmetries. Of course, one can obtain such unphysical solutions by assuming symmetry conditions along the axes of the cylinders parallel to the main flow direction and taking half of the computational domain needed normally. In such cases, the “steady-state” flow fields obtained would be identical to the flow fields observed at the initial stages of our computations. However, we show that such flow fields do not represent the temporally periodic flow fields even in a time-averaged sense, because, in all three cases, the initial drag coefficients are different from the mean drag coefficients. Therefore, we conclude that stability studies involving periodic arrays of cylinders should be carried out, as it is done in this work, with the true implementation of the spatial periodicity.

Finite-time singularity formation in Hele-Shaw systems

Abstract: We investigate the behavior of the interface between two fluids in a two-dimensional flow driven by surface tension. The geometry is chosen so that one can apply a variant of the lubrication approximation and so that the more-viscous fluid will have a tendency to change its topology by separating into two masses. Simulations are used to show that, with appropriate initial and boundary conditions, this separation can occur in a finite time. We particularly focus our attention at the pinch point, i.e., the spacetime point at which the width of the viscous fluid first goes to zero. The lubrication approximation used contains a parameter ρ which measures the strength of the inertial forces

A novel hybrid numerical technique to model 3-D fountain flow in injection molding processes

Abstract: A novel hybrid two-dimensional (2-D)/three-dimensional (3-D) numerical technique is presented to model 3-D fountain flows in thin cavities as encountered in injection molding processes. At the fountain flow region, where all three velocity components are significant, the governing 3-D fluid flow equations are solved by using a pressure Poisson formulation. Behind the flow front, where out of plane flows are negligible, the 2-D Hele-Shaw formulation is employed, largely reducing the number of unknowns in comparison to a fully-three-dimensional formulation. Boundary fitted coordinate systems (BFCS) together with the finite difference method (FDM) are used to solve the governing equations on a non-staggered grid. The formulation is capable of handling non-linearities in the material behavior due to the shear-thinning characteristics of typical resin systems. Results are presented for isothermal flow of Newtonian and shear-thinning fluids through diverging and converging flow sections.