Showing papers on "Hele-Shaw flow published in 1992"

Three-dimensional dynamics and transition to turbulence in the wake of bluff objects

Abstract: The wakes of bluff objects and in particular of circular cylinders are known to undergo a ‘fast’ transition, from a laminar two-dimensional state at Reynolds number 200 to a turbulent state at Reynolds number 400. The process has been documented in several experimental investigations, but the underlying physical mechanisms have remained largely unknown so far. In this paper, the transition process is investigated numerically, through direct simulation of the Navier—Stokes equations at representative Reynolds numbers, up to 500. A high-order time-accurate, mixed spectral/spectral element technique is used. It is shown that the wake first becomes three-dimensional, as a result of a secondary instability of the two-dimensional vortex street. This secondary instability appears at a Reynolds number close to 200. For slightly supercritical Reynolds numbers, a harmonic state develops, in which the flow oscillates at its fundamental frequency (Strouhal number) around a spanwise modulated time-average flow. In the near wake the modulation wavelength of the time-average flow is half of the spanwise wavelength of the perturbation flow, consistently with linear instability theory. The vortex filaments have a spanwise wavy shape in the near wake, and form rib-like structures further downstream. At higher Reynolds numbers the three-dimensional flow oscillation undergoes a period-doubling bifurcation, in which the flow alternates between two different states. Phase-space analysis of the flow shows that the basic limit cycle has branched into two connected limit cycles. In physical space the period doubling appears as the shedding of two distinct types of vortex filaments.Further increases of the Reynolds number result in a cascade of period-doubling bifurcations, which create a chaotic state in the flow at a Reynolds number of about 500. The flow is characterized by broadband power spectra, and the appearance of intermittent phenomena. It is concluded that the wake undergoes transition to turbulence following the period-doubling route.

Flow past a sphere with an oscillation in the free-stream velocity and unsteady drag at finite Reynolds number

Abstract: Unsteady flow over a stationary sphere with a small fluctuation in the free-stream velocity is considered at small Reynolds number, Re. A matched asymptotic solution is obtained for the frequency-dependent (or the acceleration-dependent) part of the unsteady flow at very small frequency, w, under the restriction St % Re 4 1, where St is the Strouhal number. The acceleration-dependent part of the unsteady drag is found to be proportional to St - w instead of the wi dependence predicted by Stokes’ solution. Consequently, the expression for the Basset history force is incorrect for large time even for very small Reynolds numbers. Present results compare well with the previous numerical results of Mei, Lawrence & Adrian (1991) using a finitedifference method for the same unsteady flow at small Reynolds number. Using the principle of causality, the present analytical results at small Re, the numerical results at finite Re for low frequency, and Stokes’ results for high frequency, a modified expression for the history force is proposed in the time domain. It is confirmed by comparing with the finite-difference results at arbitrary frequency through Fourier transformation. The modified history force has an integration kernel that decays as t-*, instead of t-i, at large time for both small and finite Reynolds numbers.

Turbulent flow between concentric rotating cylinders at large Reynolds number.

Abstract: Turbulent Taylor vortex flow is studied in experiments for Reynolds numbers ${10}^{3}$R${10}^{6}$. Simple scaling of the torque with Reynolds number is m/Inot observed for any range of R, although the characteristic time scales and the transport of passive scalars are found to scale with the global torque measurements. Above a nonhysteretic transition observed at R=1.3\ifmmode\times\else\texttimes\fi{}${10}^{4}$, the torque has a Reynolds number dependence similar to the drag observed in wall-bounded shear flows such as pipe flow and flow over a flat plate.

Numerical prediction of vortex shedding behind a square cylinder

Abstract: It is common knowledge that flow around bluff bodies exhibits oscillatory behaviour. The aim of the present study is to compute the steady two-dimensional flow around a square cylinder at different Reynolds numbers and to determine the onset of unsteadiness through a linear stability analysis of the steady flow. Stability of the steady flow to small two-dimensional perturbations is analysed by computing the evolution of these perturbations. An analysis of various time-stepping techniques is carried out to select the most appropriate technique for predicting the growth of the perturbations and hence the stability of the flow. The critical Reynolds number is determined from the growth rate of the perturbations. Computations are then made for periodic unsteady flow at a Reynolds number above the critical value. The predicted Strouhal number agrees well with experimental data. Heat transfer from the cylinder is also studied for the unsteady laminar flow.

Stratified flow past a sphere

Abstract: The flow of a linearly stratified fluid past a sphere is considered experimentally in the Froude number Fi, Reynolds number Re, ranges 0.005 ≤ Fi ≤ 20 and 5 ≤ Re ≤ 10000. Flow visualization techniques and density measurements are used to describe the rich range of characteristic flow phenomena observed. These different flow patterns are mapped on a detailed Fi against Re flow regime diagram. In most instances the flow patterns were found to be very different from those observed in homogeneous fluids. Vortex shedding characteristics, for example, were found to be dramatically affected by the presence of stratification. Where possible, the results are compared with available analytical and numerical models.

Semi‐implicit and explicit finite element schemes for coupled fluid/thermal problems

Abstract: A comparative investigation, based on a series of numerical tests, of two purely explicit and one semi-implicit finite element methods used for incompressible flow computation is presented. The ‘segregated’ approach is followed and the equations of motion are considered sequentially. The fundamental concepts and characteristics of the formulations and the solution methodology used are described in technical detail. Various modifications to Chorin's projection algorithm are investigated, particularly with respect to their effects on stability and accuracy. The stability of the semi-implicit method is shown to be less restrictive when compared to the explicit methods as the Reynolds number increases. At large time steps the artificial viscosity is also reduced and higher accuracy is obtained. The performance of the methods discussed in this paper is illustrated by the numerical solutions obtained for the cavity flow and flow past a rearward-facing step problems at high Reynolds numbers, and free convection flow problem at high Rayleigh numbers. It is shown that the semi-implicit method needs fewer iterations than the explicit methods, and the accuracy of the present methods is guaranteed by comparison with the existing methods.

Potential flow and forces for incompressible viscous flow

Abstract: Forces on a finite body in an incompressible viscous flow are shown to be contributed by a potential flow and fluid elements of non-zero vorticity in a revealing formulation. The present study indicates that the potential flow play also a geometric role in determining the contribution of the fluid elements. Consideration is given to a solid body moving through a fluid, fluid accelerating past a solid body and a solid body which oscillates in a uniform stream. The effects of induced-mass and inertial forces appear naturally in the formulation and are separated from the contribution due to the surface vorticity and that due to the vorticity within the flow. Physical significance of the present analysis for vortical flows about a finite body is illustrated by examples, e.g. flow past a circular cylinder or an ellipsoid of revolution.

A hybrid method for computing the flow of viscoelastic fluids

Abstract: SUMMARY A hybrid method for computing the flow of viscoelastic and second-order fluids is presented. It combines the features of the finite difference technique and the shooting method. The method is accurate because it uses central differences. Its convergence is at least superlinear. The method is applied to obtain the solutions to three problems of flow of Walters’ B fluid (a) flow near a stagnation point, (b) flow over a stretching sheet and (c) flow near a rotating disk. Numerical results reveal some new characteristics of flows which are not easy to demonstrate using the perturbation technique.

Upward Vertical Two-Phase Flow Through an Annulus—Part II: Modeling Bubble, Slug, and Annular Flow

Abstract: This paper reports that mechanistic models have been developed for each of the existing two-phase flow patterns in an annulus, namely bubble flow, dispersed bubble flow, slug flow, and annular flow. These models are based on two-phase flow physical phenomena and incorporate annulus characteristics such as casing and tubing diameters and degree of eccentricity. The models also apply to the new predictive means for friction factor and Taylor bubble rise velocity. Given a set of flow conditions, the existing flow pattern in the system can be predicted. The developed models are applied next for predicting the flow behavior, including the average volumetric liquid holdup and the average total pressure gradient for the existing flow pattern. In general, good aggrement was observed between the experimental data and model predictions.

On Hele–Shaw flow of power-law fluids

Abstract: This paper reviews the governing equations for a plane Hele–Shaw flow of a power-law fluid. We find two closely related partial differential equations, one for the pressure and one for the stream function. Some mathematical results for these equations are presented, in particular some exact solutions and a representation theorem. The results are applied to Hele–Shaw flow. It is then possible to determine the flow near an arbitrary corner for any power-law fluid. Other examples are also given.

An advanced experimental investigation of quasi-two-dimensional shear flows

Abstract: Forced shear flows in a thin layer of an incompressible viscous fluid are studied experimentally. Streak photographs are used to obtain the stream function of vortical flow patterns arising after the primary shear flow loses stability. Various flow characteristics are determined and results are compared to the stability theory of quasi-two-dimensional flows. The applicability of the quasi-two-dimensional approximation is directly verified and the possibility of reconstruction of the driving force from the secondary flow pattern is demonstrated.

Spinodal decomposition in a Hele-Shaw cell

Abstract: As an experimentally realizable two-dimensional fluid system, we perform a cell-dynamical system simulation of a binary incompressible fluid that undergoes spinodal decomposition after a critical quench in a Hele-Shaw cell. Fluidity enhances fluctuations of interfaces, resulting in a form factor that continuously depends on the fluidity of the system. In this system, finite size coupled with the effect of incompressible flow is found to have a severe accelerating effect on the growth law.

Taking into account truncation problems and geomagnetic model accuracy in assessing computed flows at the core—mantle boundary

Abstract: SUMMARY Since the time Roberts & Scott (1965) first expressed the key ‘frozen flux’ hypothesis relating the secular variation of the geomagnetic field (SV) to the flow at the core surface, a large number of studies have been devoted to building maps of the flow and inferring its fundamental properties from magnetic observations at the Earth's surface. There are some well-known difficulties in carrying out these studies, such as the one linked to the non-uniqueness of the flow solution [if no additional constraint is imposed on the flow (Backus 1968)] which has been thoroughly investigated. In contrast little investigation has been made up to now to estimate the exact importance of other difficulties, although the different authors are usually well aware of their existence. In this paper we intend to make as systematic as possible a study of the limitations linked to the use of truncated spherical harmonic expansions in the computation of the flow. Our approach does not rely on other assumptions than the frozen flux, the insulating mantle and the large-scale flow assumptions along with some simple statistical assumptions concerning the flow and the Main Field. Our conclusions therefore apply to any (toroidal, steady or tangentially geostrophic) of the flow models that have already been produced; they can be summarized in the following way: first, because of the unavoidable truncation of the spherical harmonic expansion of the Main Field to degree 13, no information will ever be derived for the components of the flow with degree larger than 12; second, one may truncate the spherical harmonic expansion of the flow to degree 12 with only a small impact on the first degrees of the flow. Third, with the data available at the present day, the components of the flow with degree less than 5 are fairly well known whereas those with degree greater than 8 are absolutely unconstrained.

A Power-Law Formulation of Laminar Flow in Short Pipes

Abstract: In the quantification of air flow through penetrations in buildings, it is necessary to be able to characterize the flow without detailed knowledge of the geometry of the paths. At the conditions typical of buildings, the flow regime is partially developed laminar flow. This report develops a theoreticfal description of the hydrodynamic relationship based an a power-law representation between the air flow and applied pressure for laminar flow in short pipes

Heat Transfer in Serpentine Flow Passages with Rotation

Abstract: Heat transfer characteristics of a three-pass serpentine flow passage with rotation are experimentally studied. The walls of the square flow passage are plated with thin stainless-steel foils through which electrical current is applied to generate heat. The local heat transfer performance on the four side walls of the three straight flow passages and two turning elbows are determined for both stationary and rotating cases. The through flow Reynolds, Rayleigh (centrifugal type), and rotation numbers are varied. It is revealed that three-dimensional flow structures cause the heat transfer rate at the bends to be substantially higher than at the straight flow passages. This mechanism is revealed by means of a flow visualization experiment for a nonrotating case. Along the first straight flow passage, the heat transfer rate is increased on the trailing surface but is reduced on the leading surface, due to the action of secondary streams induced by the Coriolis force

Perturbing Hele-Shaw flow with a small gap gradient.

Abstract: A controlled perturbation is introduced into the Saffman-Taylor flow problem by adding a gradient to the gap of a Hele-Shaw cell. The stability of the single-finger steady state was found to be strongly affected by such a perturbation. Compared with patterns in a standard Hele-Shaw cell, the single Saffman-Taylor finger was stabilized or destabilized according to the sign of the gap gradient. While a linear stability analysis shows that this perturbation should have a negligible effect on the early-stage pattern formation, the experimental data indicate that the characteristic length for the initial breakup of a flat interface has been changed by the perturbation.

Flow and temperature measurement of natural convection in a Hele-Shaw cell using a thermo-sensitive liquid-crystal tracer

Abstract: The temperature and flow field of natural convection in a Hele-Shaw cell is visualized by using a liquid-crystal tracer. The tracer photographs obtained by this method are compared with the interferograms of previous experiments using the same experimental setup, and the applicability of the present methods is validated. Quantitative data of the temperature and velocity were obtained by applying a colour-image-processing technique to the visualized images.

Miscible displacement in a Hele-Shaw cell

Abstract: We formulated a theory of simple mixtures of incompressible miscible liquids in terms of the mass averaged velocityu and the solenoidal volume averaged velocityW, We derived simplified equations for miscible displacement in a Hele-Shaw cell. We obtained a steady solution of these equations corresponding to displacement under gravity with prescribed values of concentration and mean normal stress at the inlet and exit of the cell. We studied the stability of this steady flow. This differs from previous works which treat the stability of unsteady miscible displacement using a quasi-static assumption and classical equations based on divu=0. In our problem, replacingu withW gives rise to a difference in the mean normal stress, which alters the pressure drop across the cell and changes the velocity of free fall. We found that the stability equations are the same in the two formulations, but the boundary conditions are slightly different; however the difference will be small if diffusion is slow or the thickness of the cell is small. The results show that steady miscible displacement in a Hele-Shaw cell is stable to long and short waves. Within certain ranges of parameters, the displacement of glycerin into water can be unstable. This instability is basically of a Rayleigh-Taylor type, regularized by diffusion. As the diffusion parameterS becomes smaller, the waves of disturbances become finer and are confined to an increasingly thin diffusion layer. Water displacing glycerin is always stable. This is due to the fact that the steady equilibrium profile is not steep enough to create a fingering instability.

Reacting and non-reacting flow simulation for film cooling in 2-D supersonic flows

Abstract: A computational fluid dynamics (CFD) model is developed for the analysis of reacting and non-reacting film cooling supersonic flows. The present study demonstrates the effectiveness of using an extended twoequation turbulence model with compressibility corrections for supersonic shear-layer simulations. An efficient finite-rate chemistry solution method is employed for effective simulation of stiff chemistry systems. Two-dimensional supersonic film cooling test cases with and without chemical reaction are investigated for the validation of the present model. The underlying fluid dynamics solver is a predictor plus multi-comector pressure based Navier-Stokes flow solver.

Convective Mass Transport from Rectangular Cavities in Viscous Flow

Abstract: In this paper, we conduct a comprehensive numerical study of convective mass transport from rectangular cavities in low-Reynolds-number flows. The flow field is calculated by a high-order implementation of the boundary-integral method, while the convective diffusion equation is solved using the spectral-element method. Numerical convergence tests are presented showing the high precision of these algorithms. Physical results in the form of concentraton contours and local mass fluxes are presented for cavity aspect ratios from 1:1 to 4:1 and for P~clet numbers from 0 to 100,000. We investigate the effects of inlet flow profile and system boundaries on the velocity and concentration fields. The process of convective-transport from small cavities in a plane substrate is a problem of fundamental importance in a number of engineering applications. The diverse applications range from the cleansing of rough surfaces to mass-transfer from microscopic passages in biological systems. In the electrochemical area, the most notable examples are the evolution of corrosion pits and electrodeposition and electrochemical etching through passive masks. In these applications, the shape of the evolving surface is of paramount importance for the successful fabrication of the component. In eleetrodeposition, one seeks a uniform deposition of material in the cavity, without voids or overcoating of the mask. In etching, the goal is to maximize the degree of anisotropy in the evolving profile, producing deep cavities which do not undercut the mask. When the local mass-transfer rate controls the evolution of the surface shape, the fluid flow field becomes a major factor in the determination of the optimal design for the overall process. In forced convection, the convective-transport problem may be divided into two distinct parts: the investigation of the fluid flow field, and the solution for the concentration field once the flow has been specified. In a systematic study of these phenomena, we find it instructive to follow this division in our efforts. The problem of viscous flow in rectangular cavities is one of long standing interest in the fluid dynamics community. As early study owing to Pan and Acrivos I used finite-difference computations to model Stokes flow in a square cavity driven by translation of a rigid plate on the top surfa~ce. This work identified many of the essential features of cavity flows, including the presence of a large center eddy occupying the cavity and an infinite series of Moffatt ~ eddies in the corners. While this flow is quite interesting and has been adopted as a model test problem by many computational fluid dynamicists, the driven cavity does not accurately reflect the flow field in the problems of interest. Taneda 3 has conducted a detailed flow visualization study of Stokes flow over rectangular cavities showing how the size and position of eddies changed as a function ol the cavity aspect ratio. Finite-difference computations presented by Shen and Floryan 4 reproduced the flow fields observed by Taneda. Higdon ~ employed the boundary-integral

A time-accurate algorithm for chemical non-equilibrium viscous flows at all speeds

Abstract: A time-accurate, coupled solution procedure is described for the chemical nonequilibrium Navier-Stokes equations over a wide range of Mach numbers. This method employs the strong conservation form of the governing equations, but uses primitive variables as unknowns. Real gas properties and equilibrium chemistry are considered. Numerical tests include steady convergent-divergent nozzle flows with air dissociation/recombination chemistry, dump combustor flows with n-pentane-air chemistry, nonreacting flow in a model double annular combustor, and nonreacting unsteady driven cavity flows. Numerical results for both the steady and unsteady flows demonstrate the efficiency and robustness of the present algorithm for Mach numbers ranging from the incompressible limit to supersonic speeds.

Numerical and asymptotic methods for certain viscous free-surface flows

Abstract: This paper concerns the two-dimensional motion of a viscous liquid down a perturbed inclined plane under the influence of gravity, and the main goal is the prediction of the surface height as the fluid flows over the perturbations. The specific perturbations chosen for the present study were two humps stretching laterally across an otherwise uniform plane, with the flow being confined in the lateral direction by the walls of a channel. Theoretical predictions of the flow have been obtained by finite-element approximations to the Navier-Stokes equations and also by a variety of lubrication approximations. The predictions from the various models are compared with experimental measurements of the free-surface profiles. The principal aim of this study is the establishment and assessment of certain numerical and asymptotic models for the description of a class of free-surface flows, exemplified by the particular case of flow over a perturbed inclined plane. The laboratory experiments were made over a range of flow rates such that the Reynolds number, based on the volume flux per unit width and the kinematical viscosity of the fluid, ranged between 0.369 and 36.6. It was found that, at the smaller Reynolds numbers, a standard lubrication approximation provided a very good representation of the experimental measurements but, as the flow rate was increased, the standard model did not capture several important features of the flow. On the other hand, a lubrication approximation allowing for surface tension and inertial effects expanded the range of applicability of the basic theory by almost an order of magnitude, up to Reynolds numbers approaching 10. At larger flow rates, numerical solutions to the full equations of motion provided a description of the experimental results to within about 4% , up to a Reynolds number of 25, beyond which we were unable to obtain numerical solutions. It is not known why numerical solutions were not possible at larger flow rates, but it is possible that there is a bifurcation of the Navier-Stokes equations to a branch of unsteady motions near a Reynolds number of 25.

Measurements of heat transfer and flow structure in heated vertical channels

Abstract: Experiments are performed to study the buoyancy effects on the heat transfer and flow process in a finite, vertical, rectangular channel. One of the walls is insulated and the opposite wall is heated uniformly. Air flow with a uniform velocity profile is made to enter the channel. Both the cases for buoyancy-assisted flow and opposed flow are studied. The mean velocity is controlled so that the channel flow is either laminar or turbulent when the plate is not heated. Flow visualization and temperature fluctuation measurements are conducted and used to provide information on flow structure. For buoyancy opposed flow, the occurrence of flow reversal, separation of flow, and generation of vortices are observed, which cause oscillations of the mainstream and lead to fluctuations in temperature. Flow reversal occurs initially in the downstream region and extends gradually upstream as the buoyancy parameter GrIRe2 increases, which destabilizes the flow structure and enhances the heat transfer process in the region it traverses. The effect of buoyancy on the local and the average Nusselt number over the heated plate is measured and presented. Correlations of Nusselt number in terms of relevant nondimensional parameters are obtained for the Reynolds number varied from 600 to 2200 and the buoyancy parameter GrIRe2 from 0.7 to 95.

Parallel flow in Hele-Shaw cells

Abstract: We consider the parallel flow of two immiscible fluids in a Hele-Shaw cell. The evolution of disturbances on the fluid interfaces is studied both theoretically and experimentally in the large-capillary-number limit. It is shown that such interfaces support wave motion, the amplitude of which for long waves is governed by a set of KdV and Airy equations. The waves are dispersive provided that the fluids have unequal viscosities and that the space occupied by the inner fluid does not pertain to the Saffman-Taylor conditions (symmetric interfaces with half-width spacing). Experiments conducted in a long and narrow Hele-Shaw cell appear to validate the theory in both the symmetric and the non-symmetric cases.

Incorporation of 2D fluid flow into a pseudo-3D hydraulic fracturing simulator

Abstract: Pseudo 3D (P3D) hydraulic fracturing models often overpredict fracture height for a poorly contained fracture. This is caused partly by either the neglect of the fluid flow component in the vertical direction or a crude treatment of the 2D fluid flow in the fracture as 1D flow in the vertical direction in the fracture-height calculation. This paper presents a height-growth model that adopts a flow field more representative of the actual 2D flow in a fracture. In this model, the fracture is divided into two regions: an inner region where the flow direction is nearly horizontal, and an outer region where the flow field is approximated by a radial flow from an imaginary source. The governing equations for determining height growth rate and the numerical method for solving these equations are described.

Hele Shaw Flows with Time‐Dependent Free Boundaries Involving Injection Through Slits

Abstract: Consider the classical Hele Shaw situation with two parallel planes separated by a narrow gap. Suppose a Newtonian fluid to be injected into this gap through straight-line slits as well as, possibly, at isolated points. We are interested in predicting the plan-view of the resultant expanding blob, or blobs, of fluid. We show that the method of solution developed earlier for dealing with such problems when the injection is only at isolated points can be adapted to yield solutions in this more general situation.