Showing papers on "Hele-Shaw flow published in 1978"
678 citations
TL;DR: In this paper, the spectral distributions of the fluctuations in velocity are quantitatively related to the dimensions of the two unequal regions of flow recirculation, and it is shown that the intensity of fluctuating energy in these low Reynolds number flows can be larger than that in corresponding turbulent flows.
Abstract: Flow visualization and laser-Doppler anemometry have been used to provide a detailed description of the velocity characteristics of the asymmetric flows which form in symmetric, two-dimensional, plane, sudden-expansion geometries. The flow and geometry boundary conditions which give rise to asymmetric flow are indicated, and the reason for the phenomenon is shown to lie in disturbances generated at the edge of the expansion and amplified in the shear layers. The spectral distributions of the fluctuations in velocity are quantitatively related to the dimensions of the two unequal regions of flow recirculation. It is also shown that the intensity of fluctuating energy in these low Reynolds number flows can be larger than that in corresponding turbulent flows.
350 citations
TL;DR: In this paper, the series for steady fully developed laminar flow through a toroidal pipe of small curvature ratio has been extended by computer to 24 terms and it was shown that convergence is limited by a square root singularity on the negative axis of the square of the Dean number.
Abstract: Dean's series for steady fully developed laminar flow through a toroidal pipe of small curvature ratio has been extended by computer to 24 terms Analysis suggests that convergence is limited by a square-root singularity on the negative axis of the square of the Dean number An Euler transformation and extraction of the leading and secondary singularities at infinity render the series accurate for all Dean numbers For curvature ratios no greater than , experimental measurements of the laminar friction factor agree with the theory over a wide range of Dean numbers In particular, they confirm our conclusion that the friction in a loosely coiled pipe grows asymptotically as the one-quarter power of the Dean number based on mean flow speed This contradicts a number of incomplete boundary-layer analyses in the literature, which predict a square-root variation
141 citations
130 citations
TL;DR: In this paper, the linear stability of a flat Stokes layer is investigated and the results obtained show that, in the parameter range investigated, the flow is stable, and it is shown that the Orr-Sommerfield equation for this flow has a continuous spectrum of damped eigenvalues at all values of the Reynolds number.
Abstract: The linear stability of a flat Stokes layer is investigated. The results obtained show that, in the parameter range investigated, the flow is stable. It is shown that the Orr-Sommerfield equation for this flow has a continuous spectrum of damped eigenvalues at all values of the Reynolds number. In addition, a set of discrete eigenvalues exists for certain values of the Reynolds number. The eigenfunctions associated with this set are confined to the Stokes layer while those corresponding to the continuous spectrum persist outside the layer. The effect of introducing a second boundary a long way from the Stokes layer is also considered. It is shown that the least stable disturbance of this flow does not correspond to the least stable discrete eigenvalue of the infinite Stokes layer when this boundary tends to infinity.
116 citations
TL;DR: In this paper, the collocation technique was extended to handle a wide variety of non-axisymmetric creeping-motion problems with planar symmetry where the boundaries conform to more than a single orthogonal co-ordinate system.
Abstract: This paper describes how the collocation technique previously developed by the authors for treating both unbounded (Gluckman, Pfeffer & Weinbaum 1971; Leichtberg, Weinbaum, Pfeffer & Gluckman 1976) and bounded (Leichtberg, Pfeffer & Weinbaum 1976) multiparticle axisymmetric Stokes flows can be extended to handle a wide variety of non-axisymmetric creeping-motion problems with planar symmetry where the boundaries conform to more than a single orthogonal co-ordinate system. The present paper examines in detail the strong hydrodynamic interaction between two or more closely spaced identical spheres in a plane. The various two-sphere configurations provide a convenient means of carefully testing the accuracy and convergence of the numerical solution technique for three dimensional flow with known exact spherical bipolar solutions.The important difficulty encountered in applying the collocation technique to multi-particle non-axisymmetric flows is that the selection of boundary points is rather sensitive to the flow orientation. Despite this shortcoming one is able to obtain solutions for the quasi-steady particle velocities and drag for as many as 15 spheres in less than 30 s on an IBM 370/168 computer. The method not only gives accurate global results, but is able to predict the local fluid velocity and to resolve fine features of the flow such as the presence of separated regions of closed streamlines. Time-dependent numerical solutions are also presented for various three-sphere assemblages falling in a vertical plane. These solutions, in which the motion of each sphere is traced for several hundred diameters, are found to be in very good agreement with experimental measurements. The concluding section of the paper describes how the present collocation procedure can be extended to a number of important unsolved three-dimensional problems in Stokes flow with planar symmetry such as the arbitrary off-axis motion of a sphere in a circular cylinder or between parallel walls, or the motion of a neutrally buoyant particle at the entrance to a slit or pore.
100 citations
TL;DR: In this article, the theory for predicting flow pattern transition under transient flow conditions is developed and compared with experiment, which represents an extension of the methods presented by Taitel and Dukler (1976) for steady state flows.
Abstract: The theory for predicting flow pattern transition under transient flow conditions is developed and compared with experiment. This work represents an extension of the methods presented by Taitel and Dukler (1976) for steady state flows. Under transient conditions, flow pattern transitions can take place at flow rates substantially different than would occur under steady flow conditions. In addition, flow patterns can appear which would not be expected for a slow change in flow rates along that same path. Methods are presented for predicting the flow rates at which flow pattern transitions will take place during flow transients. The method also reveals when spurious flow patterns will appear.
87 citations
10 Jul 1978
76 citations
TL;DR: In this paper, it is shown that the flow is always stable to these disturbances, even in the case of infinitesimal disturbances of the type first studied by Gortler and Hammerlin.
Abstract: Experiments have shown that the two-dimensional flow near a forward stagnation line may be unstable to three-dimensional disturbances. The growing disturbance takes the form of secondary vortices, i.e. vortices more or less parallel to the original streamlines. The instability is usually confined to the boundary layer and the spacing of the secondary vortices is of the order of the boundary-layer thickness. This situation is analysed theoretically for the case of infinitesimal disturbances of the type first studied by Gortler and Hammerlin. These are disturbances periodic in the direction perpendicular to the plane of the flow, in the limit of infinite Reynolds number. It is shown that the flow is always stable to these disturbances.
62 citations
TL;DR: In this paper, the authors describe a sequence of two-dimensional numerical simulations of inflection point instability in a stably stratified shear flow near the ground, and investigate the growth inhibiting effect of the ground which is predicted by linear theory and the Reynolds number dependence of the process of growth to finite amplitude.
Abstract: We describe a sequence of two-dimensional numerical simulations of inflection point instability in a stably stratified shear flow near the ground. The fastest growing Kelvin-Helmholtz modes are studied in detail; in particular we investigate the growth inhibiting effect of the ground which is predicted by linear theory and the Reynolds number dependence of the process of growth to finite amplitude. We consider flows which are both above and below the critical Reynolds number (Re = 300) which has been reported by Woods (1969) to mark the boundary between flows which have turbulent final states and those which do not. A global energy budget reveals a fundamental difference in character of the finite amplitude billows in these two Reynolds number regimes. However, for relatively high Reynolds numbers (Re = 103) we do not find any explicit evidence for secondary instability. Above the transition Reynolds number the modified mean flow induced by wave growth is characterized by a splitting of the origi...
51 citations
TL;DR: In this article, an analogy is drawn between the end effect issue of concern here, called the "end effect", and the celebrated "Saint-Venant's Principle" of the theory of elasticity.
Abstract: One of the classic problems of laminar flow theory is the development of velocity profiles in the inlet regions of channels or pipes. Such entry flow problems have been investigated extensively, usually by approximate techniques. In a recent paper [4], Horgan & Wheeler have provided an alternative approach, based on an energy method for the stationary Navier-Stokes equations. In [4], concerned with laminar flow in a cylindrical pipe of arbitrary cross-section, an analogy is drawn between the end effect issue of concern here, called the “end effect”, and the celebrated “Saint-Venant's Principle” of the theory of elasticity.
TL;DR: In this article, the force on a small sphere translating relative to a slow viscous flow was found to be O(Re ½ ) for two different fluid flows far from the sphere, namely pure rotation and pure shear.
Abstract: The force on a small sphere translating relative to a slow viscous flow is found to O(Re½) for two different fluid flows far from the sphere, namely pure rotation and pure shear. For pure rotation, the O(Re½) correction to the Stokes drag consists of an increase in the drag. For pure shear, the O(Re½) force contains a component perpendicular to the Stokes drag force.
TL;DR: In this article, an explicit representation for the unsteady motion on a transversely sheared mean flow is obtained which corresponds to the gustline motion on uniform mean flow, and the important features of this motion are discussed.
Abstract: An explicit representation for the unsteady motion on a transversely sheared mean flow is obtained which corresponds to the gustline motion on a uniform mean flow. The important features of this motion are discussed. It is shown that its velocity, pressure and vorticity are all induced by a certain disturbance field that is a linear combination of the vorticity and particle-displacement fields and is everywhere frozen in the mean flow. The general ideas are illustrated by considering the scattering of a gust by a half-plane embedded in a shear flow.
TL;DR: In this article, a method for solving boundary-layer equations with boundary conditions corresponding to two-dimensional plane and axisymmetric, laminar and turbulent internal flows is described and results are presented.
Abstract: A method for solving boundary-layer equations with boundary conditions corresponding to two-dimensional plane and axisymmetric, laminar and turbulent internal flows is described and results are presented. This method represents an extension of a procedure previously used with considerable success for an extensive range of two- and three-dimensional external flows, and is shown here to be equally successful when applied to internal flows. The efficient and accurate numerical scheme, which considers the pressure gradient as a nonlinear eigenvalue, is coupled with eddy-viscosity and eddy-conductivity assumptions for turbulent flow and is shown to represent the mean flow and heat transfer properties of flows including those with transition, provided the location of the onset of transition is known. An alternative method of considering the pressure gradient, the Mechul function approach, which offers the advantage of allowing solutions in the regions of flow separation, is also described.
TL;DR: In this article, it was shown that a stable secondary flow in the form of traveling waves bifurcates from the stationary flow at a certain Reynolds number, and that stationary flow is unstable above this number.
Abstract: The two cases of stationary Ekman boundary layer flow of an incompressible fluid near i) a plane boundary and ii) a free surface with constant shear are considered. It is proven that a stable secondary flow in the form of traveling waves bifurcates from the stationary flow at a certain Reynolds number, and that the stationary flow is unstable above this number. The values of the critical Reynolds number and of the numbers that characterize the traveling wave are computed and compared with experimental values.
TL;DR: In this paper, a simple momentum transfer model for Reynolds stresses is used to effect a solution for the secondary flow in a square duct, and the calculated secondary flows are similar in magnitude and direction to measurements reported in the literature.
Abstract: A requirement for calculating an accurate velocity distribution in noncircular conduits is a knowledge of the secondary flow distribution in these conduits. The cause of these secondary flows is considered in this paper and a simple momentum transfer model for Reynolds stresses is used to effect a solution for the secondary flow in a square duct. The calculated secondary flows are similar in magnitude and direction to measurements reported in the literature.
TL;DR: In this article, a separation of variables theory was developed for solving problems of Stokes flow in annular trenches bounded by horizontal parallel planes and concentric vertical cylinders, leading to a new set of eigenfunctions, adjoins eigen functions, biorthogonality conditions and an algorithm for the computation of the coefficients in an eigenfunction expansion of edge data prescribed on the horizontal boundaries.
Abstract: In this paper we develop a separation of variables theory for solving problems of Stokes flow in annular trenches bounded by horizontal parallel planes and concentric vertical cylinders. The theory leads to a new set of Stokes flow eigenfunctions, adjoins eigenfunctions, biorthogonality conditions and an algorithm for the computation of the coefficients in an eigenfunction expansion of edge data prescribed on the horizontal boundaries. To illustrate the algorithm we compute the motion and the shape of the free surface on a liquid in the space between two cylinders which are maintained at different temperatures. The solution is constructed as a domain perturbation of the rest state in powers of the temperature difference. At the lowest significant order the problem is reduced to a Stokes flow problem of the desired type with edge data prescribed on the horizontal boundaries.
TL;DR: In this paper, the importance of degenerate critical points where the principal strain rate equals the rotation rate is discussed in relation to the birth of eddies and the effect that localized polymer chain extension can have in modifying various singular flows is examined.
Abstract: Flow singularities where the fluid velocity is zero are examined in terms of their relevance to polymer chain extension and flow instabilities. The singular flows are subdivided into two and three dimensional flows and flows that do and do not contain rotational components. Methods by which these various flows can be generated are reviewed. Theoretical considerations necessary to achieve flow induced polymer chain extension are briefly reviewed and the major conclusions presented in a simple form. The consequences of these conditions are then discussed in relation to the ability and manner in which chain extension can occur in various flows. The significance of degenerate and non degenerate two dimensional flows is examined in terms of flow stability and observed changes in the topology of flows; in particular the relevance of degenerate critical points where the principal strain rate equals the rotation rate is discussed in relation to the birth of eddies. Finally, the effect that localized polymer chain extension can have in modifying various singular flows is examined. When polymer is added to the flow a possible reduction in the persistent strain rate together with changes in the topology of flows are both considered.
TL;DR: In this paper, a technique has been developed and presented that can be used to compute secondary currents and, thus, the irregular three-dimensional structure of open channel flow in an alluvial channel.
Abstract: An open channel flow is usually three-dimensional consisting of the primary and secondary flows. The primary flow is defined herein as the flow component in the longitudinal direction of the channel, while the secondary flow includes the remaining flow components in the vertical and transverse directions of the channel. In an alluvial channel the channel cross sections and, thus, the three-dimensional flow structure vary irregularly or randomly along the flow. A technique has been developed and presented herein that can be used to compute secondary currents and, thus, the irregular three-dimensional structure of open channel flow. The technique uses analytical, hydrodynamic equations along with measured data of channel geometry and primary flow velocity distribution. Illustrative examples are presented with results of practical applications of the technique, describing the variability of the direction and magnitude of secondary currents with time, space, and discharge rate.
01 Jan 1978
TL;DR: Explicit, implicit, and characteristic finite-difference methods are applied to solve model equations representative of the compressible Navier-Stokes equations as mentioned in this paper, which has drastically reduced the computation time required to obtain viscous flow solutions.
Abstract: Explicit, implicit, and characteristic finite-difference methods are applied to solve model equations representative of the compressible Navier-Stokes equations. An approach is then formulated for solving the Navier-Stokes equation at high Reynolds numbers. The approach has drastically reduced the computation time required to obtain viscous flow solutions. Computational results for shock wave separated flows are presented.
01 May 1978
TL;DR: In this paper, the authors used a Hele-Shaw cell to simulate the motion of gas bubbles in a surrounding fluid under conditions of residual gravity in an earthbound workshop, and derived a formula for the translational velocity of the bubble relative to the liquid.
Abstract: In order zu simulate the motion of gas bubbles in a surrounding fluid under conditions of residual gravity in an earthbound workshop, a Hele-Shaw cell, i.e., a narrow rectangular channel filled with viscous liquid, can be utilized. Starting from the equations of creeping motion, the flow past a circular cylindrical bubble (as seen from above), located between the plates of the Hele-Shaw apparatus and surrounded by a viscous, incompressible and homogeneous fluid, has been investigated. A formula for the translational velocity of the bubble, relative to the liquid, has been derived. By employing the methods of conformal mapping noncircular (elliptical) bubble contours are taken into consideration also. The influence of surface tension at the liquid-gas interface has been dealt with approximately. Theoretical and experimental results show good agreement. The problem under study is related to the space processing of materials, especially degassing of molten matter.
01 Jul 1978
TL;DR: In this article, flow field measurements for a symmetrical NACA 64A010 airfoil section at transonic conditions were obtained for three angles of attack with the free-stream Mach number fixed at 0.8.
Abstract: Flow-field measurements are presented for a symmetrical NACA 64A010 airfoil section at transonic conditions. Measurements were obtained for three angles of attack with the free-stream Mach number fixed at 0.8. The cases studied included a weak shock wave/boundary layer interaction, an interaction of medium strength with mild separation, and an interaction of sufficient strength to produce a shock-induced stall situation. Two nonintrusive optical techniques, laser velocimetry and holographic interferometry, were used to characterize the flows. The results include Mach number contours and flow angle distributions in the inviscid flow regions, and turbulent flow properties, including the turbulent Reynolds stresses, of the upper surface viscous layers, and of the near-wake. The turbulent flow measurements reveal that the turbulence fluctuations attain equilibrium with the local mean flow much faster than previously expected.
TL;DR: In this paper, the authors used a distorted coordinate system in which the locations of all shock waves do not change; the distortion is found as part of the solution. But the distortion was used only for transonic flows.
Abstract: The difficulty of treating the perturbation of transonic flow, during which shock waves change position, can be overcome by using a distorted coordinate system in which the locations of all shock waves do not change; the distortion is found as part of the solution. This device leads to a relation that allows a range of flows, with differing shock locations, to be related algebraically to two known 'calibration' flows. Results for flows around finite wings, including those with multiple, intersecting shock waves, are presented. A typical computing time for such examples is 0.3 sec on a CDC 7600 computer.
TL;DR: In this paper, the steady forced convection of a viscous fluid contained between two concentric spheres which are maintained at different temperatures and rotate about a common axis with different angular velocities is considered.
TL;DR: In this article, the incompressible time independent Navier-Stokes equations for two-dimensional laminar flow through arrays of parallel circular cylinders are solved numerically using a square mesh in the physical field.
TL;DR: In this paper, the unsteady flow near an infinite flat plate, which is oscillating harmonically in its own plane, is studied in the presence of a magnetic field subjected to suction or injection.
Abstract: The unsteady flow near an infinite flat plate, which is oscillating harmonically in its own plane, is studied in the presence of a magnetic field subjected to suction or injection. The magnetic field is perpendicular to the plate and the flow of viscous incompressible and electrically conducting fluid is regarded as being initially at rest. Exact solutions for velocity and skin-friction are obtained for the magnetic field fixed to the fluid or to the plate.
TL;DR: In this article, the slip flow of viscous fluid at low Reynolds numbers past a flat plate aligned with the flow is studied theoretically on the basis of Oseen-Stokes equations of motion, and an integral equation for the distribution of fundamental singularities representing the plate is derived and solved approximately in the vicinity of the edge and main portion of the plate.
Abstract: In this paper the slip flow of viscous fluid at low Reynolds numbers past a flat plate aligned with the flow is studied theoretically on the basis of Oseen-Stokes equations of motion. An integral equation for the distribution of fundamental singularities representing the plate is derived and solved approximately in the vicinity of the edge and main portion of the plate. A formula for the local skin friction is obtained and discussed numerically. It is also shown that the slippage of the flow gives rise to reduction of the drag force on the plate by an amount O ( K |ln K |), where K is the Knudsen number. The velocity change near the edge of the plate is of particular interest and is found to be logarithmically singular there.
TL;DR: In this paper, the particle motion and particle motion in a tube are solved numerically by coupling the Navier-Stokes equations with the dynamic equation of a particle, and the particulate flow is simulated from rest under a prescribed pressure drop.
Abstract: The fluid flow and particle motion in a tube are solved numerically by coupling the Navier-Stokes equations with the dynamic equation of a particle. The particulate flow is simulated from rest under a prescribed pressure drop. The symptotic steady state solution is in good agreement with the analytical solution of creeping flow and with the experimental data reported in the literature. The present computational analysis explores the Reynolds numbers effects on the flow processes and gives some insight into qualitative similarities of the kinematics and dynamics of particulate flows. The resistant coefficient, the apparent viscosity, the tube hemotocrit, and the distribution of viscous stresses and pressure on the particle and on the capillary wall are considered along with the curvilinear flow patterns.