Showing papers on "Hele-Shaw flow published in 1977"

Physical Fluid Dynamics

Abstract: Introduction Pipe and channel flow Flow past a circular cylinder Free convection between parallel walls Equations of motion Further basic ideas Dynamical similarity Low and high Reynolds numbers Some solutions of the viscous flow equations Inviscid flow Boundary layers, wakes, and jets Separation and attachment Lift Convection Stratified flow Flow in rotating fluids Instabilities Transition to turbulence in shear flows Turbulence Homogeneous isotropic turbulence Turbulent shear flows Convection in horizontal layers Double diffusive free convection Dynamical chaos Experimental methods Applications of fluid dynamics Notation Problems Hints and answers to problems Bibliography and references Index.

Prediction of the contributions to the Reynolds stress from bursting events in open-channel flows

Abstract: In this paper we intend to predict the magnitude of the contribution to the Reynolds stress of bursting events: ‘ejections’, ‘sweeps’, ‘inward interactions’ and ‘outward interactions’. We shall do this by making use of the conditional probability distribution of the Reynolds stress − uv, which can be derived by applying the cumulant-discard method to the Gram-Charlier probability distribution of the two variables u and v. The Reynolds-stress fluctuations in openchannel flows over smooth and rough beds are measured by dual-sensor hot-film anemometers, whose signals are conditionally sampled and sorted into the four quadrants of the u, v plane by using a high-speed digital data processing system.We shall verify that even the third-order conditional probability distribution of the Reynolds stress shows fairly good agreement with the experimental results and that the sequence of events in the bursting process, i.e. ejections, sweeps and interactions, is directly related to the turbulent energy budget in the form of turbulent diffusion. Also, we shall show that the roughness effect is marked in the area from the wall to the middle of the equilibrium region, and that sweeps appear to be more important than ejections as the roughness increases and as the distance from the wall decreases.

A finite volume method for transonic potential flow calculations

Abstract: It is proposed to solve the exact transonic potential flow equation on a mesh constructed from small volume elements, which can be conveniently packed around any reasonably smooth configuration. The calculation is performed on two sets of interlocking cells. The velocity and density are calculated in the primary cells, and a flux balance is then established in the secondary cells. The scheme is desymmetrized by the addition of artificial viscosity in the supersonic zone. Some results are included for a swept wing and a wing-cylinder combination.

Unsteady flows in a semi-infinite contracting or expanding pipe

Abstract: Physiological pumps produce flows by alternate contraction and expansion of the vessel. When muscles start to squeeze its wall the valve at the upstream end is closed and that at the downstream end is opened, and the fluid is pumped out in the downstream direction. These systems can be modelled by a semi-infinite pipe with one end closed by a compliant membrane which prevents only axial motion of the fluid, leaving radial motion completely unrestricted. In the present paper an exact similar solution of the Navier–Stokes equation for unsteady flow is a semi-infinite contracting or expanding circular pipe is calculated and reveals the following characteristics of this type of flow. In a contracting pipe the effects of viscosity are limited to a thin boundary layer attached to the wall, which becomes thinner for higher Reynolds numbers. In an expanding pipe the flow adjacent to the wall is highly retarded and eventually reverses at Reynolds numbers above a critical value. The pressure gradient along the axis of pipe is favourable for a contracting wall, while it is adverse for an expanding wall in most cases. These solutions are valid down to the state of a completely collapsed pipe, since the nonlinearity is retained in full. The results of the present theory may be applied to the unsteady flow produced by a certain class of forced contractions and expansions of a valved vein or a thin bronchial tube.

Stability of plane poiseuille flow of a highly elastic liquid

Abstract: The stability of fully developed pressure driven plane laminar flow of a Maxwell fluid has been studied using linear hydrodynamic stability theory. Elasticity is destabilizing in the inertial regime, but the flow is found to be stable to infinitesimal disturbances at low Reynolds numbers. This result contradicts previous calculations, which predicted a low Reynolds number flow instability at a critical recoverable shear of order unity. The previous calculations were carried out using less accurate numerical methods; the eigenvalue problem which must be solved is a delicate one, requiring sophisticated umerical techniques in order to avoid the calculation of spurious unstable modes. This work has direct bearing on the question of the mechanism of a low Reynolds number extrusion instability known as “melt fracture”. It is observed that the intensity of melt fracture increases with increasing die length for high density polyethylene, and it is therfore believed by some experimentalists that fully-developed die flow is unstable for this polymer above a critical recoverable shear. The analysis appears to be at variance with this interpretation of the experimental results.

Flow structure in motion of a spherical drop in a fluid medium at intermediate Reynolds numbers

Abstract: The problem of flow of a viscous fluid around a spherical drop has been examined for the limiting case of small and large Reynolds numbers in several investigations (see [1–3], for instance; there is a detailed review of various approximate solutions in [4]). For the intermediate range of Reynolds numbers (approximately 1≤Re≤100), where numerical integration of the complete Navier-Stokes equations is necessary, there are solutions of special cases of the problem —flow of air around a solid sphere [5–7], a gas bubble [8, 9], and water drops [10]. The present paper deals with flow around a spherical drop at intermediate Reynolds numbers up to Re=200 for arbitrary values of the ratio of dynamic viscosities Μ=Μ1/Μ2 inside and outside the drop. It is shown that a return flow can arise behind the drop in flow without separation. In such conditions the circulatory flow inside the drop breaks up. An approximate formula for the drag coefficient of the drop is given.

On cavity flow at high Reynolds numbers

Abstract: The flow in a square cavity is studied by solving the full Navier–Stokes and energy equations numerically, employing finite-difference techniques. Solutions are obtained over a wide range of Reynolds numbers from 0 to 50000. The solutions show that only at very high Reynolds numbers (Re [gt-or-equal, slanted] 30000) does the flow in the cavity completely correspond to that assumed by Batchelor's model for separated flows. The flow and thermal fields at such high Reynolds numbers clearly exhibit a boundary-layer character. For the first time, it is demonstrated that the downstream secondary eddy grows and decays in a manner similar to the upstream one. The upstream and downstream secondary eddies remain completely viscous throughout the range of Reynolds numbers of their existence. It is suggested that the behaviour of the secondary eddies may be characteristic of internal separated flows.

Thermal convection in a Hele-Shaw cell

Abstract: We derive the Rayleigh number RHS for thermal convection in a Hele-Shaw cell with gap width d and full width (gap plus walls) Y. For the state of marginal stability, the system of equations is found to be formally identical to that describing flow through a uniform porous medium, if d3/12Y is identified as the Hele-Shaw permeability. Thus Lapwood's (1948) thermal-instability analysis should apply, and the critical Rayleigh number should be 4π2 when the cell has impermeable isothermal boundaries.Baker's (1966) pH-indicator method for visualizing fluid flow has been adapted for use in a Hele-Shaw cell. In addition to revealing the convection pattern clearly, this technique proves to be an especially sensitive detector of incipient flow, and a highly accurate means of verifying the onset of convection. Our experiments confirm that the critical Hele-Shaw Rayleigh number is 40 ± 2, thereby validating our theoretically derived expression for the Rayleigh number. We also measure the vertical flow velocity wm and find that wm∝ (R2HS−402)½ closely fits our data for 40 < RHS < 140.

Hydrodynamic instability of viscous flow between rotating coaxial cylinders with fully developed axial flow

Abstract: A linear stability analysis is presented for flow between concentric cylinders when a fully developed axial flow is present. Small perturbations are assumed to be nonaxisymmetric. This leads to an eigenvalue problem with four eigenvalues: the critical Taylor number, an amplification factor and two wavenumbers. The presence of the tangential wavenumber permits prediction of the stability of spiral flow. This made it possible to model the flow more accurately and to extend the range of calculations to higher axial Reynolds numbers than had previously been attainable. Calculations were carried out for radius ratios from 0·95 to 0·1, Reynolds numbers as large as 300 and cases with co-rotation and counter-rotation of the cylinders.

The Secondary Flow of Newtonian Fluids in Cone‐and‐Plate Viscometers

Abstract: Secondary flow in cone‐and‐plate viscometers is studied by numerical integration of the equations of motion for steady incompressible flow of Newtonian fluids. Solutions over wide ranges of the two principal parameters, Reynolds number and gap angle, yield detailed information on the flow fields and elements of the rate of deformation tensor. Secondary flows are shown to cause large deviations in certain elements of the rate of deformation at Reynolds numbers more than an order of magnitude lower than those at which the torque is appreciably changed. Comparisons are given with prior analytical and experimental work.

Kinetic theory boundary conditions for discrete velocity gases

Abstract: Boundary conditions are investigated for a model gas composed of identical particles with velocities restricted to a given finite set of vectors. A possible model of the gas surface interaction is given and an H theorem for a gas in a vessel is obtained. Then, the steady Couette flow between two parallel plates is studied. The model with four coplanar velocities is used and the velocity set has arbitrary orientation. The exact kinetic solution is compared with the solutions of the associated Navier–Stokes problems.

Separation in a slow linear shear flow past a cylinder and a plane

Abstract: In a slow linear shear flow over a cylinder in contact with a plane, there is an infinite set of eddies within the cusps at the point of contact. If the cylinder is not in contact with the plane, there is a flux of fluid between the cylinder and the plane, no matter how small the gap. When the gap is approximately 0·685 times the cylinder radius or less, the flow separates from the boundaries. Single eddies form alternately on the plane and the cylinder. These interlace as the cylinder approaches the plane and force the fluid which flows through the gap to take a tortuous path. The expressions for the force and torque acting on the cylinder are also given.

Computation of the flow between two rotating coaxial disks

Abstract: A numerical investigation of the problem of rotating disks is made using the Newton–Raphson method. It is shown that the governing equations may exhibit one, three or five solutions. A physical interpretation of the calculated profiles will be presented. The results computed reveal that both Batchelor and Stewartson analysis yields for high Reynolds numbers results which are in agreement with our observations, i.e. the fluid may rotate as a rigid body or the main body of the fluid may be almost at rest, respectively. Occurrence of a two-cell situation at particular branches will be discussed.

Nonlinear stability of parallel flows with subcritical Reynolds numbers. Part 2. Stability of pipe Poiseuille flow to finite axisymmetric disturbances

Abstract: A theoretical study is presented of the spatial stability of flow in a circular pipe to small but finite axisymmetric disturbances. The disturbance is represented by a Fourier series with respect to time, and the truncated system of equations for the components up to the second-harmonic wave is derived under a rational assumption concerning the magnitudes of the Fourier components. The solution provides a relation between the damping rate and the amplitude of disturbance. Numerical calculations are carried out for Reynolds numbers R between 500 and 4000 and βR [les ] 5000, β being the non-dimensional frequency. The results indicate that the flow is stable to finite disturbances as well as to infinitesimal disturbances for all values of R and βR concerned.

Stability of time-periodic flows in a circular pipe

Abstract: The stability of time-periodic flows in a circular pipe is investigated. The disturbance is assumed to be axially symmetric and to have a small amplitude, so that the governing differential equation is linear. Calculations are carried out for the first ten modes for a range of values of the frequency of the primary motion, of the wavenumber of the disturbance, and of the Reynolds number of the primary flow. In the ranges of the parameters for which the calculations have been carried out, the flows are found to be stable and, as for Stokes flows (von Kerczek & Davis 1974), it is conjectured that the flows under study here are stable for all frequencies and all Reynolds numbers.

On axisymmetric stokes flow past a torus

Abstract: A method is presented for determining the exact solution of the Stokes equation for axisymmetric streaming flow past a torus. By solving directly for the velocity and pressure fields rather than introducing a stream function, the problem is reduced to the solution of a second order difference equation for a coefficient sequence. The force acting on the torus is evaluated for a number of values of the parameter defining the torus geometry.

Numerical solution of axisymmetric boattail flow fields with plume simulators

Abstract: Turbulent separating flows over axisymmetric afterbody-boattail configurations with solid sting plume simulators are computed with a time-dependent finite-difference method to solve the compressible Navier-Stokes equations. The Reynolds stress terms are replaced with a two-layer eddy viscosity model including a relaxation formula to model the nonequilibrium effects of the separated flow. The mesh is alined with the boattail body through an analytic transformation which accommodates a wide variety of boattail geometries. Numerical results for a series of boattail geometries over a wide range of Reynolds number (140,000 to 140 million) are presented and, when possible, compared with experimental data or independent numerical results.

Some comments on steady, laminar flow through twisted pipes

Abstract: Steady laminar flow through pipes of straight centre line and fixed cross-sectional geometry is consideredfor pipes in which the orientation of the cross-section changes slowly with distance along the axis of the pipe For small rates of twist, the (local) departure from (local) Poiseuille flow is small and it is shown that a part of the mathematical problem for this secondary flow is identical to that for the small, transverse displacement of a clamped elastic plate due to constant loading A detailed examination of the loss of flow rate (due to the twist) is given The special case of a pipe of elliptical cross-section is found to be analytically tractable (with the aid of a computer) and is considered in detail

Plane flow of a second-order fluid past submerged boundaries

Abstract: Considers the plane flow of a second-order fluid past submerged obstacles such as circular and elliptical cylinders in situations where inertia effects cannot be neglected. The effect of the short memory of the fluid upon the flow features is analyzed in detail. In particular, it is found that the viscoelasticity of the fluid reduces the drag coefficient for very low Reynolds numbers, while the opposite is true for large Reynolds numbers.

Stability of low Reynolds number flow with viscous heating

Abstract: The stability of fully developed plane Couette flow and pipe flow with viscous heating is studied at low Reynolds number for a Newtonian liquid with a temperature-dependent viscosity. The solution is obtained by a direct integration method of the eigenfunction equations, with eigenvalues located in the complex plane by means of the argument principle of complex variable theory. An instability will occur in plane Couette flow, but outside the parameter range which will be encountered in practice. There is no comparable instability in pipe flow. It can be concluded that a thermal mechanism does not cause the low Reynolds number instabilities observed in polymer processing operations.

Numerical solution of supersonic laminar flow over a three-dimensional compression corner

Abstract: A newly developed, rapid numerical scheme is extended to three dimensions to solve the complete Navier-Stokes equations for a supersonic, laminar flow over a compression corner with sidewall effects. The program is coded so that it can solve for a general curved ramp surface geometry such as found in inlets and fuselage-wing-flap junctions. A test case of Mach 3.0 flow is calculated. In regions where three-dimensional effects are small, good agreement is obtained between the present calculation and previous two-dimensional solutions. In other regions, the results show complex three-dimensional flow-field interactions including shock-shock and shock/boundary-layer interactions

Aerodynamic Effects of Inviscid Parallel Shear Flows

Abstract: A general theory of planar disturbances in inviscid parallel shear flows, analogous to thin-wing theory in potential flows, has been developed. Integral relations between surface pressure and deformation are obtained that are similar to, and can be solved by the same numerical methods as those of potential flow. Computed results are shown that illustrate the effects of a model turbulent boundary layer on various lifting and nonlifting surfaces, including an elastic panel in low supersonic flow and an airfoil control surface in subsonic flow.

Computation of transonic boattail flow with separation

Abstract: The relaxation procedure of South and Jameson for the full potential transonic flow equation was coupled to a modified Reshotko-Tucker integral boundary-layer technique with an empirical model for separated flow. The viscous and inviscid flows were solved iteratively until convergence was obtained. This iterative method was then applied to the subsonic and transonic flow over a series of axisymmetric circular-arc boattails with solid jet plume simulators. Comparisons of theoretical and experimental surface pressures and boattail drag are presented over a free-stream Mach numbers below 0.90. The qualitative variation of boattail drag with free-stream Mach number and boattail angle well into the region of transonic drag rise was correctly predicted; however, the absolute drag levels were significantly underpredicted. For separated flows, the empirical discriminating streamline model gives good results up to a free-stream Mach number of about 0.90 and allows reasonable predictions for shock-induced separation if the proper separation location and separation turning angle are known.

The equation of motion for a small aerosol in a continuum

Abstract: The motion of an aerosol can be described by a general force balance equation, independent of the detailed structure of the flow, provided that the interaction between the external flow field and the local flow induced by the aerosol is weak. A necessary and sufficient condition for the interaction to be weak is that the length scale of the aerosol is much less than that of the external flow. High and low Reynolds number regimes can be distinguished for the motion of an aerosol relative to the external flow. In some extreme conditions the equation of motion reduces to an algebraic equation for the aerosol velocity.