Showing papers on "Hele-Shaw flow published in 1983"

Equation of motion for a small rigid sphere in a nonuniform flow

Abstract: The forces on a small rigid sphere in a nonuniform flow are considered from first prinicples in order to resolve the errors in Tchen’s equation and the subsequent modified versions that have since appeared. Forces from the undisturbed flow and the disturbance flow created by the presence of the sphere are treated separately. Proper account is taken of the effect of spatial variations of the undisturbed flow on both forces. In particular the appropriate Faxen correction for unsteady Stokes flow is derived and included as part of the consistent approximation for the equation of motion.

Small-scale structure of the Taylor–Green vortex

Abstract: The dynamics of both the inviscid and viscous Taylor–Green (TG) three-dimensional vortex flows are investigated. This flow is perhaps the simplest system in which one can study the generation of small scales by three-dimensional vortex stretching and the resulting turbulence. The problem is studied by both direct spectral numerical solution of the Navier–Stokes equations (with up to 256 3 modes) and by power-series analysis in time. The inviscid dynamics are strongly influenced by symmetries which confine the flow to an impermeable box with stress-free boundaries. There is an early stage during which the flow is strongly anisotropic with well-organized (laminar) small-scale excitation in the form of vortex sheets located near the walls of this box. The flow is smooth but has complex-space singularities within a distance $\hat{\delta}(t)$ of real (physical) space which give rise to an exponential tail in the energy spectrum. It is found that $\hat{\delta}(t)$ decreases exponentially in time to the limit of our resolution. Indirect evidence is presented that more violent vortex stretching takes place at later times, possibly leading to a real singularity ( $\hat{\delta}(t) = 0$ ) at a finite time. These direct integration results are consistent with new temporal power-series results that extend the Morf, Orszag & Frisch (1980) analysis from order t 44 to order t 80 . Still, convincing evidence for or against the existence of a real singularity will require even more sophisticated analysis. The viscous dynamics (decay) have been studied for Reynolds numbers R (based on an integral scale) up to 3000 and beyond the time t max at which the maximum energy dissipation is achieved. Early-time, high- R dynamics are essentially inviscid and laminar. The inviscidly formed vortex sheets are observed to roll up and are then subject to instabilities accompanied by reconnection processes which make the flow increasingly chaotic (turbulent) with extended high-vorticity patches appearing away from the impermeable walls. Near t max the small scales of the flow are nearly isotropic provided that R [gsim ] 1000. Various features characteristic of fully developed turbulence are observed near t max when R = 3000 and R λ = 110: a k − n inertial range in the energy spectrum is obtained with n ≈ 1.6–2.2 (in contrast with a much steeper spectrum at earlier times); th energy dissipation has considerable spatial intermittency; its spectrum has a k −1+μ inertial range with the codimension μ ≈ 0.3−0.7. Skewness and flatness results are also presented.

Secondary instability of wall-bounded shear flows

Abstract: The present analysis of a secondary instability in a wide class of wall-bounded parallel shear flows indicates that two-dimensional, finite amplitude waves are exponentially unstable to infinitessimal three-dimensional disturbances. The instability appears to be the prototype of transitional instability in such flows as Poiseuille flow, Couette flow, and flat plate boundary layers, in that it has the convective time scales observed in the typical transitions. The energetics and vorticity dynamics of the instability are discussed, and it is shown that the two-dimensional perturbation without directly providing energy to the disturbance. The three-dimensional instability requires that a threshold two-dimensional amplitude be achieved. It is found possible to identify experimental features of transitional spot structure with aspects of the nonlinear two-dimensional/linear three-dimensional instability.

Numerical experiments on Hele Shaw flow with a sharp interface

Abstract: The fingering instability of an interface between two immiscible fluids in a Hele Shaw cell is simulated numerically The algorithm used is based on a transcription of the equations of motion for the interface in which it formally becomes a generalized vortex sheet The evolution of this sheet is computed using a variant of the vortex-in-cell method The resulting scheme and code make it possible to follow the collective behaviour of many competing and interacting fingers well into the nonlinear, large-amplitude regime It is shown that in this regime the evolution is controlled essentially by just one dimensionless parameter, the ratio of fluid viscosities The effects of varying this parameter are studied and the results compared with experimental investigations Scaling properties of the average density profile across the evolving mixed layer between the two homogeneous fluid phases are investigated Many phenomena are observed that must be characterized as collective interactions and thus cannot be understood in terms of flows with just a single finger

Computation of supersonic viscous flows around pointed bodies at large incidence

Abstract: A recently reported parabolized Navier-Stokes method has been extended to compute turbulent supersonic flows around cones and an ogive-cylinder body at large incidence. The algebraic eddy-viscosity turbulence model contained in the code was modified to properly account for the large regions of cross-flow separation that occur in these flows. Extensive comparisons between computed results and experimentally measured flow fields are presented. The results show good agreement for viscous-layer profiles and details of the external leeward-side vortex structure at angles of attack up to three times the cone half angles. Details of the modified turbulence model are presented and discussed.

Numerical processing of flow-visualization pictures – measurement of two-dimensional vortex flow

Abstract: A new system has been developed for estimating experimentally some of the principal physical variables of fluid flows, through flow-visualization and image-processing techniques. Distributions of stream function, vorticity and pressure are calculated by this system with reasonable accuracy for two examples of two-dimensional flow: namely unsteady twin-vortex flow behind a circular cylinder accelerated impulsively to constant speed, and Karman vortices behind a circular cylinder moving at constant speed. A detailed explanation of the image-processing technique and the numerical calculation process is given first, and then some consideration is given to calculated results in these two types of flow. Comparison shows that some results of the unsteady twin-vortex experiment coincide well with those of previously published experimental investigations and theoretical calculations. Errors introduced at each stage of this system are estimated in some detail.

Numerical solution of viscous flows using integral equation methods

Abstract: A formulation of the boundary element method for the solution of non-zero Reynolds number incompressible flows in which the non-linear terms are lumped together to form a forcing function is presented. Solutions can be obtained at low to moderate Reynolds numbers. The method was tested using the flow of a fluid in a two-dimensional converging channel (Hamel flow) for which an exact solution is available. An axisymmetric formulation is demonstrated by examining the drag experienced by a sphere held stationary in uniform flow. Performance of the method was satisfactory. New results for an axisymmetric free jet at zero Reynolds number obtained using the boundary element method are also included. The method is ideal for this type of free-surface problem.

Flow between rotating disks. Part 1. Basic flow

Abstract: Laser-Doppler velocity measurements were obtained in water between finite rotating disks, with and without throughflow, in four cases: ω1 = ω2 = 0; ω2/ω1 = −1; ω2/ω1 = 0; ω2/ω1 = 1. The equilibrium flows are unique, and at mid-radius they show a high degree of independence from boundary conditions in r. With one disk rotating and the other stationary, this mid-radius ‘limiting flow’ is recognized as the Batchelor profile of infinite-disk theory. Other profiles, predicted by this theory to coexist with the Batchelor profile, were neither observed experimentally nor were they calculated numerically by the finite-disk solutions, obtained here via a Galerkin, B-spline formulation. Agreement on velocity between numerical results and experimental data is good at large values of the ratio RQ/Re, where RQ = Q/2πνs is the throughflow Reynolds number and Re = R22ω/ν is the rotational Reynolds number.

Numerical study of secondary flows and roll-cell instabilities in rotating channel flow

Abstract: A numerical study is conducted on the pressure-driven laminar flow of an incompressible viscous fluid through a rectangular channel subjected to a spanwise rotation. The full nonlinear time-dependent Navier–Stokes equations are solved by a finite-difference technique for various rotation rates and Reynolds numbers in the laminar regime. At weak rotation rates, a double-vortex secondary flow appears in the transverse planes of the channel. For more rapid rotation rates, an instability occurs in the form of longitudinal roll cells in the interior of the channel. Further increases in the rotation rate leads to a restabilization of the flow to a Taylor–Proudman regime. It is found that the roll-cell and Taylor–Proudman regimes lead to a substantial distortion of the axial-velocity profiles. The specific numerical results obtained are shown to be in excellent agreement with previously obtained experimental measurements and theoretical predictions.

Two-Equation Turbulence Model for Flow in Trenches

Abstract: Steady recirculating flow is described by means of a mathematical model based on the full unsteady Reynolds equations and the k-ϵ model for turbulence closure. The model is applied to the flow in a steep-sided trench perpendicular to the main flow direction. Inlet profiles are taken with reference to developed channel flow. For the wall boundary a local equilibrium is assumed, yielding among others a logarithmic behaviour for the mean flow velocity. The constants of the k-ϵ model are related to the roughness conditions. A sensitivity study is reported to identify the relative importance of the various constants and inlet conditions. Numerical results are compared with laboratory experiments.

Flow Characteristics in the Curved Rectangular Channels : Visualization of Secondary Flow

Abstract: In order to clarify the secondary flow pattern which appears in curved rectangular channels, flow visualization is made in a fully developed laminar flow. The channels have aspect ratios ranging from 0.5 to 2.5, curvature ratios ranging from 5 to 8 and a width of 20mm. From the experimental results the developing process of secondary flow vortices becomes clear. At a retarded layer near the outer concave wall, additional counter rotating pairs of vortices are observed. The critical Dean number takes the minimum and maximum values at the aspect ratios of about 1 and 2, respectively. The critical Dean number increases rapidly with decreasing aspect ratio when it is smaller than 1.

Numerical Study of Incompressible Slightly Viscous Flow Past Blunt Bodies and Airfoils

Abstract: A grid-free numerical method is used to simulate incompressible flow at high Reynolds numbers. The numerical method simulates the flow inside the boundary layer by vortex sheets and the flow outside this layer by vortex blobs. The algorithm produces a smooth transition between the sheets and the blobs.The accuracy of this hybrid numerical method is tested in several numerical experiments. In the first experiment, the algorithm is used to simulate slightly viscous flow past a circular cylinder. In the second experiment, the algorithm is used to simulate flow past a Joukowski airfoil at various angles of attack. In the latter case, the computations simulating the flow at the airfoil's trailing edge do not “blow-up”. In both experiments, the calculated flow and its functionals (such as lift and drag coefficients) are in good agreement with both theoretical results and wind tunnel experiments.

Wall-function boundary conditions in the solution of the Navier-Stokes equations for complex compressible flows

Abstract: To make computer codes for two-dimensional compressible flows more robust and economical, wall functions for these flows, under adiabatic conditions, have been developed and tested. These wall functions have been applied to three two-equation models of turbulence. The tests consist of comparisons of calculated and experimental results for transonic and supersonic flow over a flat plate and for two-dimensional and axisymmetrical transonic shock-wave/boundary-layer interaction flows with and without separation. The calculations are performed with an implicit algorithm that solves the Reynolds-averaged Navier-Stokes equations. It is shown that results obtained agree very well with the data for the complex compressible flows tested, provided criteria for use of the wall functions are followed. The expected savings in cost of the computations and improved robustness of the code were achieved.

Comparison Between Navier-Stokes and Thin-Layer Computations for Separated Supersonic Flow

Abstract: In the numerical simulation of high Reynolds-number flow, one can frequently supply only enough grid points to resolve the viscous terms in a thin layer. As a consequence, a body-or stream-aligned coordinate system is frequently used and viscous terms in this direction are discarded. It is argued that these terms cannot be resolved and computational efficiency is gained by their neglect. Dropping the streamwise viscous terms in this manner has been termed the thin-layer approximation. The thin-layer concept is an old one, and similar viscous terms are dropped, for example, in parabolized Navier-Stokes schemes. However, such schemes also make additional assumptions so that the equations can be marched in space, and such a restriction is not usually imposed on a thin-layer model. The thin-layer approximation can be justified in much the same way as the boundary-layer approximation; it requires, therefore, a body-or stream-aligned coordinate and a high Reynolds number. Unlike the boundary-layer approximation, the same equations are used throughout, so there is no matching problem. Furthermore, the normal momentum equation is not simplified and the convection terms are not one-sided differenced for marching. Consequently, the thin-layer equations are numerically well behaved at separation and require no special treatment there. Nevertheless, the thin-layer approximation receives criticism. It has been suggested that the approximation is invalid at separation and, more recently, that it is inadequate for unsteady transonic flow. Although previous comparisons between the thin-layer and Navier-Stokes equations have been made, these comparisons have not been adequately documented.

A Class of Exact Solutions of Two-Dimensional Viscous Flow

Abstract: A viscous two-dimensional flow of shear layers superimposed on a stagnation-point flow is investigated This situation allows an exact solution of unsteady Navier-Stokes equation of an incompressible fluid in free space for a general initial condition The solution is exemplified for several sorts of initial condition One of them represents a flow in a balance between viscous diffusion and convective confinement of vorticity in the final asymptotic state Another shows a flow field of collision of two shear layers of opposite senses, which is forced to come into contact by the imposed flow, and this collision results in `pair annihilation' of the vortical layers The decay of the vortex strength of the layer shows a similarlity behaviour for different Reynolds numbers A comment is given about a possible dissipation mechanism in free flows

On angles of separation in Stokes flows

Abstract: A two-dimensional Stokes flow close to the line of contact of two touching cylinders or three-dimensional axisymmetric Stokes flow close to the point of contact of two touching bodies is shown in general to separate into infinite sets of eddies with angles of separation from the bodies which tend to 58.61° as the line or point of contact is approached. The flow near the vertex of a conical cusp is shown to be a system of nested toroidal vortices and the separation angles tend to 45.25° as the vertex is approached. Stokes flow between parallel planes or within a circular cylinder is shown in general to separate far from the generating disturbances with cellular eddy structure and separation angles which tend to 58.61° and 45.25° respectively. The mathematical equivalence of the various problems is established.

Sediment transport in stratified turbulent flow

Abstract: A mathematical model of sediment laden, density affected turbulent flows is presented. The finite element technique is used in conjunction with the Newton iterative method to solve the resulting partial differential equations. The model satisfactorily predicts velocity and concentration profiles for unidirectional open channel flows. It is concluded that the model can justifiably be extended to multidimensional flows.

The NCOREL computer program for 3D nonlinear supersonic potential flow computations

Abstract: An innovative computational technique (NCOREL) was established for the treatment of three dimensional supersonic flows. The method is nonlinear in that it solves the nonconservative finite difference analog of the full potential equation and can predict the formation of supercritical cross flow regions, embedded and bow shocks. The method implicitly computes a conical flow at the apex (R = 0) of a spherical coordinate system and uses a fully implicit marching technique to obtain three dimensional cross flow solutions. This implies that the radial Mach number must remain supersonic. The cross flow solutions are obtained by using type dependent transonic relaxation techniques with the type dependency linked to the character of the cross flow velocity (i.e., subsonic/supersonic). The spherical coordinate system and marching on spherical surfaces is ideally suited to the computation of wing flows at low supersonic Mach numbers due to the elimination of the subsonic axial Mach number problems that exist in other marching codes that utilize Cartesian transverse marching planes.

On certain aspects of three-dimensional instability of parallel flows

Abstract: The effect of small imperfections in shear flows on the development of finite-amplitude three-dimensional disturbances in the flow is examined. A model problem is studied, one in which the basic flow is plane Poiseuille flow in a channel and the small imperfection in the form of spanwise periodic variation of the basic flow is introduced from the channel boundaries. It is shown that this has a positive effect on the growth of larger Tollmien-Schlichting wave disturbances, which are in the form of standing waves in the spanwise direction. Equations for the amplitudes of these disturbances, based on Stuart-Watson-Eckhaus theory, are derived and over a range of Reynolds number, the regions in the wavenumber plane over which equilibrium solutions are possible are identified. The possibility of three-dimensional disturbances that are oscillatory in the streamwise direction but that may be growing exponentially in a spanwise direction is examined.

Simulation of Granular Flow

Abstract: A computer simulation of rapid granular flow has been developed to investigate the properties of granular materials. The system is designed to be very simple and efficient, so that simulated flow problems can be set up easily and run until a steady state flow is achieved. The particles are modeled as two-dimensional rigid disks with realistic coefficients of friction and restitution. Stochastic boundaries are used to eliminate wall effects, for the simulation of uniform flow. An advanced personal computer system is used to perform the computations and to display movies of the flow, for rapid comprehension of the local flow mechanics.