Showing papers on "Hele-Shaw flow published in 1969"

Numerical Solution of a Uniform Flow over a Sphere at Intermediate Reynolds Numbers

Abstract: Numerical solutions of the transient uniform flow around a sphere are obtained. The transition takes place between an initial potential flow and a fully developed viscous field. The fluid is incompressible, homogeneous, and its flow is governed by the complete Navier‐Stokes equations. The range of Reynolds number studied is Re = 1–1000 where a recirculatory wake appears and the nonlinear terms are essential, that is, they cannot be neglected or approximated. The flow is assumed to be axisymmetric throughout this range. A time‐dependent stream function‐vorticity formulation is adopted. The solution is obtained by constructing a finite difference approximation to the vorticity transport equation on an expanding spherical polar grid system. Central differencing of second‐order accuracy both in time (Dufort‐Frankel) and space is utilized. Experiments with numerical stability show an appreciable deviation from linearized stability analysis due to the large gradients of vorticity in the field. Quantitative physical results are obtained. The geometrical parameters characterizing the recirculatory wake compare favorably with those recorded in physical experiments. The detailed distribution of the vorticity on the sphere agrees with results obtained via the steady‐state approach at Re = 10, 40, and 100. The computed drag coefficient CD agrees well with the standard drag curve over the range of Reynolds numbers investigated.

Numerical solution for the flow around a cylinder at Reynolds numbers of 40, 200 and 500

Abstract: Finite difference solutions for the time dependent equations of motion have been carried out in order to extend the range of available data on steady flow around a cylinder to larger Reynolds numbers. At the termination of the calculations for R = 40 and 200, the separation angle, the drag coefficient and the pressure and vorticity distributions around the surface of the cylinder were very close to their steady-state values. For R = 500 the separation angle and drag coefficient were very close to their steady-state values but the pressure distribution and vorticity distribution at the rear of the cylinder were still changing slightly. The results at R = 500 were found to be quite different from those at R = 200 so it is not clear how closely we approximated the steady solution for R → ∞. The forces on the cylinder due to viscous drag and due to pressure drag are found to be smaller for steady flow than for laboratory experiments where the wake is unsteady.

Nonlinear Flow in Porous Media

Abstract: Many investigators are concerned about the validity of the Forchheimer equation which represents the relationship between the velocity of flow and pressure gradient in porous media. A theoretical development of this equation through analysis of the dimensionless form of the Navier-Stokes equation is presented. It shows that energy losses at high-flow velocities in porous medium are a result of convective acceleration effects not turbulent effects. In addition, two dimensionless terms representing the flow behavior are defined and evaluated. It is shown that a constant could be used to represent the geometric properties of the medium and that a characteristic length representative of the flow exist. Both of these quantities are easily evaluated through hydraulic measurements of gradients and flow velocities. Experimental data from many sources were used to evaluate the theoretical results.

Time‐Dependent Viscous Flow over a Circular Cylinder

Abstract: A finite difference method is extended to high Reynolds number flow about circular cylinders with particular emphasis given to the quantitative description of fine flow features. The method is of the explicit type and includes a directional difference scheme for the nonlinear terms which enhances calculational stability at high Reynolds numbers. A cell structure is chosen which provides local cell dimensions consistent with the structure of solutions expected. Solutions are presented for a range of Reynolds numbers from 1 to 3 × 105 in which the flow is started impulsively from rest, and the development is studied up to the approach of the steady‐state or the limit cycle condition, whichever is appropriate to the particular Reynolds number.

Separation and flow reversal in swirling flows in circular ducts.

Abstract: The swirling motion of a laminar incompressible viscous flow in a circular duct is studied. The duct consists of two smoothly joined sections, one stationary and the other rotating with a constant angular velocity. A linearized analytical solution, valid for flows having small Reynolds numbers and large swirl ratios, is developed. Solutions for a wider range of Reynolds numbers and swirl ratios are obtained by numerically solving the discretized angular momentum and vorticity transport equations. The occurrence of flow reversal on the axis and near the tube wall is studied, in particular, and conditions for incipient flow reversal are established.

Control of Transient Free - Surface Flow

Abstract: A theory is presented to define methods of optimum gate operation for altering flow conditions in an open channel. The analytical procedure prescribes the motion of the control devices in the channel so the transient conditions are known and controlled during the period when the flow is being changed from one given situation to another desired flow condition. The method is applicable to cases of initiating flow, increasing flow, decreasing flow, or stopping flow in irrigation channels, power channels, forebay channels to pumping stations, etc. The theory is developed from the basic differential equations for unsteady flow in a prismatic channel, including frictional losses. Two examples are detailed in this study.

Thermal Entry for Low Reynolds Number Turbulent Flow

Abstract: Thermal entry problem solution for low Reynolds number turbulent gas flow based on Reynolds number dependent velocity profile

On the flow of viscous and elastico-viscous liquids through straight and curved pipes

Abstract: An experimental study is described which seeks to assess the effect of elasticity on the flow of elastico-viscous liquids as they are made to flow through curved pipes under a pressure gradient. Wherever possible, the experimental results are discussed in the light of existing theoretical predictions and good agreement between theory and experiment for viscous and elastico-viscous liquids is indicated. The general flow pattern for elastico-viscous liquids is shown to be similar to that for Newtonian liquids consisting of two spirals separated by the central plane. Elasticity has a significant effect on the pitch of these spirals. The flow rate in the case of laminar flow and in the transition region between laminar and turbulent flow is significantly enhanced by the presence of elasticity in the liquids, but the opposite is the case in turbulent flow at high Reynolds numbers. Many materials of industrial importance can be classified as elastico-viscous liquids in the sense that they flow under the action of applied stresses and yet have some of the properties usually associated with solids. In technological processes, these elastico-viscous liquids are often pumped through pipes which are usually straight but are sometimes curved. It is therefore of some practical importance to study the behaviour of elastico-viscous liquids as they are made to flow through straight and curved pipes under the action of a pressure gradient. The flow of viscous and elastico-viscous liquids through straight pipes is well documented, but very little consideration appears to have been given to the flow of elastico-viscous liquids through curved pipes. In the present paper, we shall indicate the various interesting features which have arisen from an experimental study of the flow of elastico-viscous liquids through straight and curved pipes of circular cross-section. Our work has been primarily motivated by technological requirements, but it may have applications in other spheres. For example, it may throw light on the flow of blood in the larger capillaries, since it is known that whole blood is non-Newtonian in its behaviour.

Hypersonic interactions near sharp leading edges.

Abstract: : Numerical solutions of a set of parabolic equations, uniformly valid throughout the entire continuum flow domain, were obtained for the axi-symmetric flow over slender sharp nose cones at zero angle of attack, the hypersonic interference flow between two finite length flat plates and the flow over a sharp flat plate of finite span. The regions of interest are the leading edge merged layer and the initial portion of the strong interaction flow. The analysis represents an extension of the theory first used successfully for the flow over two-dimensional sharp flat plates. Merged layer interference, the merging of streams at different pressures in the wake of the channel geometry, the three-dimensional flow and shock structure, as well as the significant differences between two-dimensional, and axi-symmetric or three-dimensional flow fields are discussed. Comparisons with available experimental data are good, as evidenced by surface pressure and heat transfer results, for the flow over cones. (Author)

Steady, Viscous Flow within a Circular Boundary

Abstract: The steady two‐dimensional flow of a viscous incompressible fluid within a circular vessel with prescribed wall velocity is studied both analytically and numerically in order to see how the flow pattern varies with the Reynolds number R. First, an algorithm to determine the Stokes' type successive approximation for any stage of approximation from the preceding one, when the wall velocity is given as a Fourier series, is established and carried out on a digital computer to the eighth approximation (correct to R8). Second, a finite‐difference method is applied to solve the Navier‐Stokes' equation numerically. Three typical cases of flow pattern are treated: flow without separation, flow consisting of two symmetric recirculating regions of semicircular form, and flow consisting of two unsymmetrical recirculating regions. It is found that the Stokes' type successive approximation is convergent for R below 32. It is also found that for larger Reynolds number the flow pattern agrees well with Batchelor's unifor...

Laminar velocity profiles in developing flows using a laser doppler technique

Abstract: A laser doppler flowmeter is used to obtain velocity profiles in circular glass tubes downstream from a plug composed of brass screens. The apparatus for measuring point velocities of particles suspended in a fluid is described. Measurements were made very close to the plug at Reynolds numbers of 16.5, 47, and 274, and the flow development followed down the tube. The results show that the entrance flow development is dependent on the inlet profile shape and the Reynolds number for Reynolds numbers below 300. The flow development at Reynolds numbers less than 300 was considerably slower than that predicted by the boundary-layer solution of the equations of motion starting with a flat profile.

Nonexistence of axisymmetric separated flow.

Abstract: Axisymmetric separated flow in Chapman model shown nonexistent by flow visualization technique

Numerical integration of the navier-stokes equations in two dimensions.

Abstract: : A numerical method is described for solving the two-dimensional Navier-Stokes equations. Results for steady flow past a circular cylinder for Reynolds numbers up to 100, and past an elliptic cylinder for Reynolds numbers up to 200, are given. (Author)

The impingement of a circular jet normal to a flat surface with and without cross flow.

Abstract: : The flow field resulting from a circular jet impinging on a flat surface in a main cross-flowing stream was investigated. A cross-flowing stream produces considerable changes to the flow field of a jet impinging on a flat surface. Separation of the radial wall jet occurs along a horse-shoe shaped line with the formation of a separation bubble whose dimensions can be large compared to the jet nozzle diameter for large values of the jet to free stream velocity ratio. Because of the very high levels of turbulence encountered in the separation bubble, measurements with pressure or electrical probes were restricted to limited regions of the flow field. As a result, extensive use was made of flow visualization techniques. The results presented bring out some of the typical characteristics of the flow field investigated. An analysis is proposed to explain and predict some of these characteristics. (Author)

Validity of Stokes Theory for Accelerating Spheres

Abstract: Measured displacement-time relations for spheres gravitationally accelerating from rest through viscous fluids are compared with the displacement-time relations predicted by the Stokes flow solution. The Stokes flow solution, consisting of a viscous drag, an added mass effect, and a history-of-the-motion integral, agrees with experimental values over a range that is larger than expected. The total dimensionless distance for which the theoretical and measured values agree depends on the ratio of sphere density to fluid density. The instantaneous Reynolds number at the point of departure between measurements and theory depends on the Stokes theory terminal Reynolds number. For the experiments reported, the theory is valid to points for which the instantaneous Reynolds number is as high as 10,000.

Non-linear theory of unstable plane Poiseuille flow

Abstract: A theoretical study of plane Poiseuille flow is made using the full non-linear Navier-Stokes equations. The mathematical technique employed is to use a Fourier decomposition in the streamwise spatial variable, a Galerkin expansion in the lateral variable and numerical integration with respect to time. By retaining the non-linear terms, the limit cycle oscillations of an unstable (in a linear sense) flow are obtained. A brief investigation of the possibility of instability due to large (non-linear) disturbances is also made. The results are negative for the cases examined. Comparisons with results previously obtained by others from linear theory illustrate the accuracy and efficacy of the method.

Nondimensional Gradually-Varied Flow Profiles

Abstract: Chezy’s and Manning’s flow formulas are used to formulate flow equations for gradually-varied flow in wide rectangular channels. The flow equations are normalized by critical depth and critical slope, so that a parameter governing the flow is evident. This parameter is called normal-flow Froude number for flow on sustaining (mild, critical, and steep) slopes, and conceptual normal-flow Froude number for flow on adverse slopes. No third parameter governs flow on the horizontal slope. Critical depth, critical slope, and the normal-flow Froude number can be systematically found from nomographs for flow of a given discharge per unit width in a wide channel of given channel slope and roughness. Hydraulic-jump equation can also be normalized with the same normalizing quantities used in flow equations. Solutions of normalized flow and hydraulic-jump equations are graphically expressed, permitting direct use.

Correlation of physical and numerical experiments for incompressible laminar separated flows.

Abstract: : Numerical and laboratory experiments for the incompressible viscous laminar flow over the two-dimensional simple back-step, blunt base, and 45 deg symmetric V-shaped cavity are presented. An explicit time-dependent finite difference scheme was used to obtain the numerical solution of the modelled flow field. A vorticity-stream function scheme, possessing conservative and transportive properties and utilizing upwind differencing for advection terms, yielded stable solutions for Reynolds numbers ranging from 0.1 to 1500. Experimental data obtained from hydrodynamic tow tank tests were compared with numerical results. Good qualitative and quantitative correlation was found. (Author)

Time‐Dependent Viscous Incompressible Flow past a Finite Flat Plate

Abstract: : The flow is started suddenly with a constant velocity and is parallel with the plate. This simple flow configuration is used to compare the accuracy and the speed of various numerical methods and to examine the effect of the far field boundary conditions and the variable mesh size on the flow. The governing Navier-Stokes equations are expressed in terms of vorticity and stream function. The (parabolic) vorticity equation is approximated by two finite difference methods. The (elliptic) stream function equation is approximated by two relaxation schemes. The details of the flow field around the flat plate are computed for Reynolds number equal to 4 and 993. One of the interesting results is that the velocity at the outer edge of the boundary layer exceeds that of the free stream and the maximum velocity overshoot is 9.5% at Re = 4 and 5.4% at Re - 993. The implicit alternating direction method is two to three times faster than the explicit method. (Author)

A Theoretical and Experimental Study of Confined Vortex Flow

Abstract: The interaction of a vortex with a stationary surface was studied both theoretically and experimentally. The flow field examined was that produced by radially inward flow through a pair of concentric rotating porous cylinders that were perpendicular to, and in contact with, a stationary flat plane. The complete Navier-Stokes equations were solved over a range of tangential Reynolds numbers from 0–300 and a range of radial Reynolds numbers from 0 to −13, the minus sign indicating radially inward flow. In order to facilitate the solution, the original equations were recast in terms of a dimensionless stream function, vorticity, and third variable related to the tangential velocity. The general validity of the numerical technique was demonstrated by the agreement between the theoretical and experimental results. Examination of the numerical results over a wide range of parameters showed that the entire flow field is very sensitive to the amount of radial flow, especially at the transition from zero radial flow to some finite value.

Low Reynolds Number Instability of an Incompressible Half‐Jet

Abstract: The instability of the free laminar boundary layer between parallel streams is investigated for an incompressible fluid. It is found that no minimum critical Reynolds number for instability exists based on the assumption of parallel undisturbed flow. However, when the parallel flow theory is corrected for the nonparallelism that actually exists at low Reynolds numbers, the minimum critical Reynolds number for instability becomes 10.2 with the wavenumber 0.135.

Investigations on the transonic flow around aerosoils

Abstract: The question of the stability of steady compressible two-dimensional potential flows with embedded regions of supersonic flow adjacent to a body is examined. The behaviour of unsteady disturbances is investigated and an essential difference between the stability of transonic one-dimensional and two-dimensional flow is shown to exist. It is shown that two-dimensional transonic potential flows are stable. An experimental investigation on quasi-elliptical aerofoil sections is discussed. The results of this investigation suggest that the theoretical potential flows can be approached arbitrarily close. It is concluded that the use of potential flow theory in the transonic region is equally acceptable as anywhere else in aerodynamics.

A modified method of integral relations approach to the blunt-body equilibrium air flow field, including comparisons with inverse solutions

Abstract: Numerical calculation of inviscid adiabatic flow field around blunt bodies at hypersonic speeds

Steady Flow Past Flat Plate in Channel

Abstract: An analysis is given of the application of inviscid fluid theory to the flow about a flat plate in a channel. An analogy is drawn, likening the plate to both an orifice and a wall. In this way the analytical results are related to flow past a sluice gate, or similar flat obstruction, normal to a stream, giving pressures, and drag and contraction coefficients over such a body. Experimental work carried out in a water tunnel illustrates the limits of application of the approach, showing where scale modelling needs to take particular account of the prototype situation.