Showing papers on "Hele-Shaw flow published in 1975"

Stokes flow past a particle of arbitrary shape: a numerical method of solution

Abstract: The problem of determining the slow viscous flow of an unbounded fluid past a single solid particle is formulated exactly as a system of linear integral equations of the first kind for a distribution of Stokeslets over the particle surface. The unknown density of Stokeslets is identical with the surface-stress force and can be obtained numerically by reducing the integral equations to a system of linear algebraic equations. This appears to be an efficient way of determining solutions for several external flows past a particle, since it requires that the matrix of the algebraic system be inverted only once for a given particle.The technique was tested successfully against the analytic solutions for spheroids in uniform and simple shear flows, and was then applied to two problems involving the motion of finite circular cylinders: (i) a cylinder translating parallel to its axis, for which the local stress force distribution and the drag were determined; and (ii) the equivalent axis ratio of a freely suspended cylinder, which was calculated by determining the couple on a stationary cylinder placed symmetrically in two different simple shear flows. The numerical results were found to be consistent with the asymptotic analysis of Cox (1970, 1971) and in excellent agreement with the experiments of Anczurowski & Mason (1968), but not with those of Harris & Pittman (1975).

Hydromechanics of low-Reynolds-number flow. Part 2. Singularity method for Stokes flows

Abstract: The present study furthcr explores the fundamental singular solutions for Stokes flow that can be useful for constructing solutions over a wide range of free-stream profiles and body shapes. The primary singularity is the Stokeslet, which is associated with a singular point force embedded in a Stokes flow. From its derivatives other fundamental singularities can be obtained, including rotlets, stresslets, potential doublets and higher-order poles derived from them. For treating interior Stokes-flow problems new fundamental solutions are introduced; they include the Stokeson and its derivatives, called the roton and stresson. These fundamental singularities are employed here to construct exact solutions to a number of exterior and interior Stokes-flow problems for several specific body shapes translating and rotating in a viscous fluid which may itself be providing a primary flow. The different primary flows considered here include the uniform stream, shear flows, parabolic profiles and extensional flows (hyperbolic profiles), while the body shapcs cover prolate spheroids, spheres and circular cylinders. The salient features of these exact solutions (all obtained in closed form) regarding the types of singularities required for the construction of a solution in each specific case, their distribution densities and the range of validity of the solution, which may depend on the characteristic Reynolds numbers and governing geometrical parameters, are discussed.

Viscous incompressible flow between concentric rotating spheres. Part 3. Linear stability and experiments

Abstract: The stability of flow of a viscous incompressible fluid contained between a stationary outer sphere and rotating inner sphere is studied theoretically and experimentally. Previous theoretical results concerning the basic laminar flow (part 1) are compared with experimental results. Small and large Reynolds number results are compared with Stokes-flow and boundary-layer solutions. The effect of the radius ratio of the two spheres is demonstrated. A linearized theory of stability for the laminar flow is formulated in terms of toroidal and poloidal potentials; the differential equations governing these potentials are integrated numerically. It is found that the flow is subcritically unstable and that the observed instability occurs at a Reynolds number close to the critical value of the energy stability theory. Observations of other flow transitions, at higher values of the Reynolds number, are also described. The character of the stability of the spherical annulus flow is found to be strongly dependent on the radius ratio.

Hydromechanics of low-Reynolds-number flow. Part 3. Motion of a spheroidal particle in quadratic flows

Abstract: Exact solutions in closed form have been found using the singularity method for various quadratic flows of an unbounded incompressible viscous fluid at low Reynolds numbers past a prolate spheroid with an arbitrary orientation with respect to the fluid. The quadratic flows considered here include unidirectional paraboloidal flows, with either an elliptic or a hyperbolic velocity distribution, and stagnation-like quadratic flows as typical representations. The motion of a force-free spheroidal particle in a paraboloidal flow has been determined. It is shown that the spheroid rotates about three principal axes with angular velocities governed by a set of Jeffery orbital equations with the rate of shear evaluated at the centre of the spheroid. These angular velocities depend on the minor-to-major axis ratio of the spheroid and its instantaneous orientation, but are independent of its actual size. The spheroid also translates at a variable speed, depending on its orientation relative to the surrounding fluid, along a straight path parallel to the main flow direction without any side drift or migration. This ‘jerk’ motion obeys a trajectory equation which is size dependent.

Hydromagnetic flow past a porous flat plate with hall effects

Abstract: An investigation is made of the flow of a conducting liquid past an infinite porous flat plate taking Hall effects into account, the liquid being permeated by a transverse magnetic field. It is shown that asymptotic solution for the velocity and magnetic field exists both for suction or blowing at the plate. Further when the magnetic Reynolds number is very small, the flow pattern is remarkably similar to that for a non-conducting flow past a flat plate in a rotating frame.

On a classification scheme for flow fields

Abstract: There are some flows in which certain strain components grow exponentially in time, while there are other flows in which the components depend otherwise on the time. In this paper the former type are called strong flows and the latter weak. An examination of theJordan form of the matrix of the velocity gradient of a steady, homogeneous, isochoric flow is made, along with the eigenvalues of such a matrix, to discover when such a flow is strong or weak. It is shown that if the eigenvalues are all zero or if one is zero and the other two purely imaginary, then the flow is weak, with the remaining cases leading to strong flows.

Computation of three-dimensional, inviscid supersonic flows

Abstract: The paper sets forth in detail a method for the finite-difference computation of three-dimensional supersonic fields in an Eulerian mesh. First-, second-, and third-order finite difference schemes are examined. Attention is given to proper treatment of the impermeable and permeable boundaries encompassing the computational plane. Numerical results are presented for certain specific configurations: a conical wing-body combination, internal corner flow, a two-dimensional blunt body, an interfering shock problem, and three-dimensional inviscid supersonic flow past a shuttle-orbiter type vehicle.

The structure of three-dimensional separated flows in obstacle-boundary layer interactions

Abstract: : The subject is investigated with flow visualization techniques; the turbulent boundary layer on the wall of a continuous supersonic wind tunnel is used Sizeable separated flow regions can be studied since the wall width is 38cm and the boundary layer is typically 25cm thick The large scale of the experiment is required to resolve the fine details of the flow structure The flow visualization techniques are discussed The structure of the separated flow upstream of the obstacle is seen to change with relatively small changes in Reynolds number R; the number of vortices varies from 6 to 4 to 2 as R changes Data are presented for large and small protuberances, but the latter are emphasized

Experimental velocity profiles in laminar flow around spheres at intermediate Reynolds numbers

Abstract: Seeley, Hummel & Smith (1975) reported the results of experiments to study the dynamics of flow around spheres at intermediate Reynolds numbers using a nondisturbing flow-visualization technique. The flow patterns were recorded on cine photographs and the information stored was processed in order to obtain the velocity field. The position of fluid elements shown by the photochromic indicator traces were estimated by eye on a projection screen. In this paper, a new set of results based on the same films has been reduced and computed using the ‘POLLY’ film-reading system described by Esmail, Smith & Hummel (1976). Some numerical boundary-layer solutions are included to show the reliability of the data, and comparisons with the results previously reported by Seeley et al. (1975) are presented.J. W. Smith will be pleased to send a complete set of experimental data on request.

An experimental investigation of low Reynolds number secondary streaming effects associated with an oscillating viscous flow in a curved pipe

Abstract: This paper deals with nonlinear streaming effects in an oscillating fluid in a curved pipe. The secondary steady velocity field in the cross-sectional plane of the pipe is studied in detail. Our experimental results are compared with the theory of Lyne (1970; that part of his theory which is valid for Reynolds numbers Rs [Lt ] 1) and the theory of Zalosh & Nelson (1973). On the basis of these comparisons we conclude that the theories are in practice valid for higher Reynolds numbers Rs than was formally expected.

Stagnation-point viscous flow of an incompressible fluid between porous plates with uniform blowing

Abstract: The axially-symmetric laminar flow of an incompressible viscous fluid resulting from uniform injection through two parallel porous plates is analyzed. An exact numerical solution as well as asymptotic solutions for high and low Reynolds numbers are obtained. It is found that the velocity component normal to the porous plates is everywhere independent of radial position. This property of uniform accessibility may make this flow geometry a useful experimental tool analogous to the rotating disc. The analysis of high Peclet number mass transfer across the center plane of this geometry is presented as an example.

Non-parallel stability of boundary layer flows

Abstract: : The spatial stability of two-dimensional incompressible boundary-layer flows is analyzed by using the method of multiple scales. The analysis takes into account the streamwise variations of the mean flow, the disturbance amplitude, and the wavenumber. The theory is applied to the Blasius and the Falkner-Skan flows. For the Blasius flow, the non-parallel analytical results are in good agreement with the experimental data. The results show that the non-parallel effects increase as the pressure gradient decreases.

Secondary convection in a Hele Shaw cell

Abstract: A convective instability is produced by salt water diffusing onto the surface of a fresh-water layer in a Hele Shaw cell. Although the horizontal wavelength of the initial instability is small, an increase in the horizontal wavelength of the convective flow with time and depth is observed as the resulting two-dimensional convection develops. The phenomenon of wavelength variation is confirmed numerically, but quantitative observational and theoretical comparison is limited to small Rayleigh numbers. It is shown that perturbations in the density field cause horizontal pressure gradients, which in turn cause convective elements to combine.

A theoretical and experimental study of the velocity distribution and transition to turbulence in free oscillatory flow

Abstract: The free oscillatory flow of a viscous fluid in a U -shaped tube is considered. A theoretical analysis (in which an axial flow is assumed and the start-up of the column is taken into account) shows, depending on the value of the similarity parameter γ, various regimes of the flow. Measurements of the velocity distribution are made using hot-film velocity probes, operated with a constant-temperature anemometer, and visualizations of the flow are performed. Experimental results are in good agreement with theoretical ones when the flow is laminar, and show the possible existence of turbulent flows. Critical values, at which the flow is disturbed over a more or less extended range of the successive oscillations, are determined for the similarity parameters γ and h 0 / R.

Calculation of two-dimensional and axisymmetric bluff-body potential flow

Abstract: A numerical method incorporating some of the ideas underlying the wake source model of Parkinson & Jandali (1970) is presented for calculating the incompressible potential flow external to a bluff body and its wake. The effect of the wake is modelled by placing sources on the rear of the wetted surface of the body. Unlike Parkinson & Jandali's method, however, the body shapes that can be treated are not limited by the restrictions imposed by the use of conformal transformation. In the present method the wetted surface of the body is represented by a distribution of discrete vortices. Good agreement has been found between the pressure distributions predicted by the numerical method and the analytic expressions of Parkinson & Jandali for a ‘two-dimensional’ circular cylinder and flat plate. A flat plate at incidence and other asymmetric two-dimensional flows have also been treated. The method has been extended to axisymmetric bluff bodies and the results show good agreement with measured pressure distributions on a circular disk and a sphere.

Numerical Studies of Steady Flow in a Two-Dimensional Square Cavity at High Reynolds Numbers

Abstract: A simple iterative scheme making use of the local relaxation parameter is developed which allows the numerical computation of the Navier-Stokes equations at high Reynolds number by finite-difference methods; the second and fourth-order centered-difference approximations to the convective terms are used. The vortex structure in a square cavity is examined for Reynolds numbers 400, 700 and 1000. The effect on the vortex structure of the accuracy of the finite-difference approximation to the convective terms is also investigated. It is found that the accuracy of the convective terms plays an important role in determining the size of the invicid core, and the values of the stream function and the vorticity at the vortex center.

Forward marching procedure for separated boundary-layer flows

Abstract: A forward-marching procedure for separated boundary-layer flows which permits the rapid and accurate solution of flows of limited extent is presented. The streamwise convection of vorticity in the reversed flow region is neglected, and this approximation is incorporated into a previously developed (Carter, 1974) inverse boundary-layer procedure. The equations are solved by the Crank-Nicolson finite-difference scheme in which column iteration is carried out at each streamwise station. Instabilities encountered in the column iterations are removed by introducing timelike terms in the finite-difference equations. This provides both unconditional diagonal dominance and a column iterative scheme, found to be stable using the von Neumann stability analysis.

Turbulence measurements in smooth concentric annuli with small radius ratios

Abstract: The structure of turbulence of fully-developed flow through three concentric annuli with small radius ratios was investigated experimentally for a Reynolds number range Re = 2 × 104−2 × 105. Turbulence intensities were measured in three directions, and turbulent shear stresses in the radial and azimuthal direction, in annuli of radius ratios α = 0·02, 0·04 and 0·1, respectively. The results showed that the structure of turbulence for these asymmetric flows is not the same as that for symmetrical flows (tubes and parallel plates). The main difference between symmetrical and asymmetric flows is that, for the latter, the diffusion of turbulent energy plays an important role. This is the reason not only for the non-coincidence of the positions of zero shear stress and maximum velocity, but also for the failure of most turbulence models in calculating asymmetric flows.

A Comparative Study of Curved Flow and Density-Stratified Flow

Abstract: A semi-empirical theory, used to predict buoyancy effects in a density-stratified and shear-driven flow, is also applied to the case of a boundary layer with curvature. Curved flow data are available and interesting in their own right since it can be seen that the Reynolds stress is reduced to zero at a critical “curvature Richardson” number predicted reasonably well by the theory.

Analytical Solution for Real Cascade Equation in Steady State

Abstract: Steady state real cascades are analyzed. The method adopted is based on two difference equation representing the conservation of the total flow and the desired material flow, respectively. Introducing what is termed the “desired material flow cut” as a stage parameter corresponding to the “total flow cut” or “cut”, the flow relations of the desired material flow are reduced to the same form as that of the total flow rate. The flow rate is not subject to any restrictive assumption except the conservation of mass flow, so that the method can be applied to almost any type of real cascade with side flows and losses. Adoption of a suitable iterative calculation will permit this treatment to deal with cascades constituted of separators with large values of separation factor without resorting to approximation, because the “desired material flow cut” of each stage is described exactly by the separation factors of each stage. The difference equations are solved analytically, for which the determinants B(i,j) and D...

Numerical Solution of the Unsteady Navier-Stokes Equations for the Oscillatory Flow Over a Concave Body.

Abstract: An interesting oscillatory flow field over a concave shape, which has not been observed experimentally, has been calculated by numerically solving the unsteady Navier-Stokes equations.The unique characteristic of this flow is that it is not separated and yet is oscillatory in nature.All previously observed oscillatory flow fields contain separated regions. The general characteristics of the flow pattern have been described, together with the hypothesized flow mechanism which produces the oscillation. The flow field appears to be hydrodynamically unstable which results in an oscillatory variation of the flow field downstream of the forebody pressure minimum. Various changes in the afterbody shape do not alter the period of the oscillation and results in minor changes in the magnitude of the pressure oscillation. A number of numerical experiments have been performed and described, the results of which indicate that the oscillation is of physical origin. Further work is continuing to understand these flow fields and the underlying governing flow mechanisms. Experimental verification of this flow phenomenon is presently being pursued by Holden [1974] and a more detailed description of the flow will be forthcoming pending the results of this experimental study.

Experimental results for low Reynolds number flow between eccentric rotating spheres

Abstract: Experimental results concerning secondary flow and torque measurement for low Reynolds number flow in an eccentric spherical annulus are compared with previous theory. The results compare very well, especially for small eccentricities for which the perturbation theory is valid.

A generalized hyperbolic marching technique for three-dimensional supersonic flow with shocks

Abstract: The current paper introduces a generalized, nonorthogonal coordinate system for application to supersonic flows with such large angles between streamlines of the initial Cauchy data and the marching direction that the velocity component in the marching direction is actually subsonic. Finite-difference operators are developed to solve the three-dimensional, steady, inviscid conservation equations of fluid flow referenced to this general frame. Moreover, a new method of aligning the difference mesh to the bow shock wave is presented which eliminates both the necessity for differencing the free-stream flow properties and the spurious fluctuations that often arise in conservative variables when they are differenced across the discontinuity. Results are given for the case of flow past a conical surface.

Progress and Problems in Understanding Turbulent Shear Flows

Abstract: This discussion will be focussed on the plane turbulent mixing layer, featured in the illustration on the front of the program of this meeting. Another example of it is shown in Figure 1, while Figure 2 is a diagram which defines the basic parameters of this simplest of turbulent shear flows. Throughout this discussion the higher speed U1 will always be in the upper part of the diagram. The speed ratio U2/U1 will therefore always be less than unity, but the density ratio ρ2/ρ1 may be less than or greater than unity. In high-speed flow the Mach numbers M1 and M2 would also be parameters. In fact, our interest in this problem was initially addressed to the question of how the characteristics of the flow depend on the density ratio, it having been supposed by many investigators that this was the parameter governing the observed, strong variations of spreading angle in the shear layer at the edge of a supersonic flow (M1 > 1, M2 = 0).

Two-Phase Flow Through Normal Shock Wave

Abstract: The steady, one-dimensional flow of compressible gas containing particles through a normal shock wave is investigated. Following the derivation of a set of general equations, two extreme cases are treated, namely: 1 – Two-phase flow with high particle mass flow (so that their volume cannot be neglected) is first considered, and the final equilibrium conditions are analytically solved, whereas the relaxation zone is calculated numerically. The error in neglecting the particle volume is shown to be considerable in the calculation of the velocities. 2 – An approximate solution is secondly given for the case of low particle mass flow through a weak shock wave. Use is made here of small perturbations, for zeroth and first orders. The solution of this case describes the flow in the relaxation zone quite accurately, and shows its dependence on the physical properties of both gas and droplets.

Numerical Study of Steady Two-Dimensional Flow past a Square Cylinder in a Channel

Abstract: Finite-difference solutions of the Navier-Stokes equations are given in stream function-vorticity formulation for a series of Reynolds numbers and for varied positions of the cylinder in the channel. For flows in a doubly connected region, the value of the stream function on the body cannot always be given in advance. A new iteration scheme to solve such a problem is proposed for getting the proper solution by imposing a condition that the pressure is single-valued. The force and the moment of force acting on the cylinder are computed and their dependence on the Reynolds number as well as on the position of the cylinder is studied. The numerical errors caused by the existence of corners on the boundary are checked for Stokes flow by comparison with exact solutions, and their dependence on the mesh length is shown numerically.