Showing papers on "Hele-Shaw flow published in 2006"

Edge of chaos in a parallel shear flow.

Abstract: We study the transition between laminar and turbulent states in a Galerkin representation of a parallel shear flow, where a stable laminar flow and a transient turbulent flow state coexist. The regions of initial conditions where the lifetimes show strong fluctuations and a sensitive dependence on initial conditions are separated from the ones with a smooth variation of lifetimes by an object in phase space which we call the ``edge of chaos.'' We describe techniques to identify and follow the edge, and our results indicate that the edge is a surface. For low Reynolds numbers we find that the surface coincides with the stable manifold of a periodic orbit, whereas at higher Reynolds numbers it is the stable set of a higher-dimensional chaotic object.

Comparison of flow resistance relations for debris flows using a one-dimensional finite element simulation model

Abstract: . This paper describes a one-dimensional finite element code for debris flows developed to model the flow within a steep channel and the stopping conditions on the fan. The code allows the systematic comparison of a wide variety of previously proposed one-phase flow resistance laws using the same finite element solution method. The one-dimensional depth-averaged equations of motion and the numerical model are explained. The model and implementation of the flow resistance relations was validated using published analytical results for the dam break case. Reasonable agreement for the front velocities and stopping location for a debris-flow event in the Kamikamihori torrent in Japan can be achieved with turbulent flow resistance relations including "stop" terms which allow the flow to come to rest on a gently sloping surface. While it is possible to match the overall bulk flow behavior using relatively simple flow resistance relations, they must be calibrated. A sensitivity analysis showed that the shape of the upstream input hydrograph does not much affect the flow conditions in the lower part of the flow path, whereas the event volume is much more important.

Mechanism of flow instability and transition to turbulence

Abstract: In this paper, a new mechanism of flow instability and turbulence transition is proposed for wall bounded shear flows. It is stated that the total energy gradient in the transverse direction and that in the streamwise direction of the main flow dominate the disturbance amplification or decay. Thus, they determine the critical condition of instability initiation and flow transition under given initial disturbance. A new dimensionless parameter K for characterizing flow instability is proposed which is expressed as the ratio of the energy gradients in the two directions for the flow without energy input or output. It is suggested that flow instability should first occur at the position of K max which may be the most dangerous position. This speculation is confirmed by Nishioka et al.'s experimental data. Comparison with experimental data for plane Poiseuille flow and pipe Poiseuille flow indicates that the proposed idea is really valid. It is found that the turbulence transition takes place at a critical value of K max of about 385 for both plane Poiseuille flow and pipe Poiseuille flow, below which no turbulence will occur regardless the disturbance. More studies show that the theory is also valid for plane Couette flows which holds a critical value of K max of about 370.

Measurements of Unsteady Wake Interference Between Tandem Cylinders

Abstract: A multi-phase, experimental study in the Basic Aerodynamics Research Tunnel at the NASA Langley Research Center has provided new insight into the unsteady flow interaction around cylinders in tandem arrangement Phase 1 of the study characterized the mean and unsteady near-field flow around two cylinders of equal diameter using 2-D Particle Image Velocimetry (PIV) and hot-wire anemometry These measurements were performed at a Reynolds number of 166 x 10(exp 5), based on cylinder diameter, and spacing-to-diameter ratios, L/D, of 1435 and 37 The current phase, Phase 2, augments this dataset by characterizing the surface flow on the same configurations using steady and unsteady pressure measurements and surface flow visualization Transition strips were applied to the front cylinder during both phases to produce a turbulent boundary layer upstream of the flow separation For these flow conditions and L/D ratios, surface pressures on both the front and rear cylinders show the effects of L/D on flow symmetry, pressure recovery, and the location of flow separation and attachment Mean streamlines and instantaneous vorticity obtained from the PIV data are used to explain the flow structure in the gap and near-wake regions and its relationship to the unsteady surface pressures The combination of off-body and surface measurements provides a comprehensive dataset to develop and validate computational techniques for predicting the unsteady flow field at higher Reynolds numbers

Numerical simulation of immiscible two-phase flow in porous media

Abstract: Nonlinear evolution of viscous and gravitational instability in two-phase immiscible displacements is analyzed with a high-accuracy numerical method. We compare our results with linear stability theory and find good agreement at small times. The fundamental physical mechanisms of finger evolution and interaction are described in terms of the competing viscous, capillary, and gravitational forces. For the parameter range considered, immiscible viscous fingers are found to undergo considerably weak interaction as compared to miscible fingers. The wave number of nonlinear fingers decreases rapidly due to the shielding mechanism and scales uniformly as t−1 at long times. We have observed that even a small amount of density contrast can eliminate viscous fingers. The dominant feature for these flows is the gravity tongue, which develops a “ridge instability” when capillary and gravity effects are of similar magnitude.

Optical Stokes flow estimation: an imaging-based control approach

Abstract: We present an approach to particle image velocimetry based on optical flow estimation subject to physical constraints. Admissible flow fields are restricted to vector fields satifying the Stokes equation. The latter equation includes control variables that allow to control the optical flow so as to fit to the apparent velocities of particles in a given image pair. We show that when the real unknown flow observed through image measurements conforms to the physical assumption underlying the Stokes equation, the control variables allow for a physical interpretation in terms of pressure distribution and forces acting on the fluid. Although this physical interpretation is lost if the assumptions do not hold, our approach still allows for reliably estimating more general and highly non-rigid flows from image pairs and is able to outperform cross-correlation based techniques.

Experimental and Computational Investigation of Flow Development and Pressure Drop in a Rectangular Micro-channel

Abstract: Flow development and pressure drop were investigated both experimentally and computationally for adiabatic single-phase water flow in a single 222 μm wide, 694 μm deep, and 12 cm long rectangular micro-channel at Reynolds numbers ranging from 196 to 2215. The velocity field was measured using a micro-particle image velocimetry system. A three-dimensional computational model was constructed which provided a detailed description of liquid velocity in both the developing and fully developed regions. At high Reynolds numbers, sharp entrance effects produced pronounced vortices in the inlet region that had a profound influence on flow development downstream. The computational model showed very good predictions of the measured velocity field and pressure drop. These findings prove the conventional Navier-Stokes equation accurately predicts liquid flow in micro-channels, and is therefore a powerful tool for the design and analysis of micro-channel heat sinks intended for electronic cooling.

On the flow past a magnetic obstacle

Abstract: This paper analyses numerically the quasi-two-dimensional flow of an incompressible electrically conducting viscous fluid past a localized zone of applied magnetic field, denominated a magnetic obstacle. The applied field is produced by the superposition of two parallel magnetized square surfaces, uniformly polarized in the normal direction, embedded in the insulating walls that contain the flow. The area of these surfaces is only a small fraction of the total flow domain. By considering inertial effects in the analysis under the low magnetic Reynolds number approximation, it is shown that the flow past a magnetic obstacle may develop vortical structures and eventually instabilities similar to those observed in flows interacting with bluff bodies. In the small zone where the oncoming uniform flow encounters the non-negligible magnetic field, the induced electric currents interact with the field, producing a non-uniform Lorentz force that brakes the fluid and creates vorticity. The effect of boundary layers is introduced through a friction term. Due to the localization of the applied magnetic field, this term models either the Hartmann braking within the zone of high magnetic field strength or a Rayleigh friction in zones where the magnetic field is negligible. Finite difference numerical computations have been conducted for Reynolds numbers Re=100 and 200, and Hartmann numbers in the range 1 ≤ Ha ≤ 100 (Re and Ha are based on the side length of the magnetized square surfaces). Under these conditions, a wake is formed behind the obstacle. It may display two elongated streamwise vortices that remain steady as long as the Hartmann number does not exceed a critical value. Once this value is reached, the wake becomes unstable and a vortex shedding process similar to the one observed in the flow past bluff bodies is established. Similarities and differences with the flow around solid obstacles are discussed.

The primary and secondary instabilities of flow generated by an oscillating circular cylinder

Abstract: Two- and three-dimensional instabilities of the two-dimensional symmetric flow generated by a circular cylinder oscillating with simple harmonic motion in quiescent fluid, or, alternatively, by oscillatory flow past a stationary cylinder at low Keulegan-Carpenter and Stokes numbers are investigated via Floquet analysis and direct numerical simulation. Previous experimental visualization has found that the flows produced at low amplitudes and frequencies of motion can be grouped by their visual characteristics into a number of distinct regimes. At low values of Keulegan-Carpenter and Stokes numbers, the flow is two-dimensional and has a reflection symmetry about the axis of oscillation, in addition to a pair of spatio-temporal symmetries. This study isolates and classifies the symmetry-breaking instabilities from these two-dimensional basic states as functions of these control parameters. It is found that while the initial bifurcations produced by increasing the parameters can be to three-dimensional flows, much of the behaviour can be explained in terms of two-dimensional symmetry-breaking instabilities. These have two primary manifestations: at low Stokes numbers, the instability is synchronous with the imposed oscillation, and gives rise to a boomerang-shaped mode, while at higher Stokes numbers, the instability is quasi-periodic, with a well-defined second period, which becomes infinite as Stokes numbers are reduced along the marginal stability boundary, ‘freezing’ the quasi-periodic mode into a synchronous one. These two-dimensional modes are, with further small increase in control parameter, unstable to three-dimensional secondary instabilities, and these are the flows which have been reported in previous experimental studies. In contrast, the mode first reported by Honji (J. Fluid Mech. vol. 107, 1981, p. 509), which arises at high Stokes numbers, and lower Keulegan-Carpenter numbers than the two-dimensional quasi-periodic mode, has a three-dimensional primary instability arising directly from the symmetrical two-dimensional basic state.

Efficient mixing of viscoelastic fluids in a microchannel at low Reynolds number

Abstract: In this paper, we examined mixing of various two-fluid flows in a silicon/glass microchannel based on the competition of dominant forces in a flow field, namely viscous/elastic, viscous/viscous and viscous/inertial. Experiments were performed over a range of Deborah and Reynolds numbers (0.36 < De < 278, 0.005 < Re < 24.2). Fluorescent dye and microshperes were used to characterize the flow kinematics. Employing abrupt convergent/divergent channel geometry, we achieved efficient mixing of two-dissimilar viscoelastic fluids at very low Reynolds number. Enhanced mixing was achieved through elastically induced flow instability at negligible diffusion and inertial effects (i.e. enormous Peclet and Elasticity numbers). This viscoelastic mixing was achieved over a short effective mixing length and relatively fast flow velocities.

The linear stability of high-frequency oscillatory flow in a channel

Abstract: The linear stability of the Stokes layers generated between a pair of synchronously oscillating parallel plates is investigated. The disturbance equations were studied using Floquet theory and pseudospectral numerical methods used to solve the resulting system. Neutral curves for an extensive range of plate separations were obtained and when the plate separation is large compared to the Stokes layer thickness the linear stability properties of the Stokes layer in a semi-infinite fluid were recovered. A detailed analysis of the damping rates of disturbances to the basic flow provides a plausible explanation of why several previous studies of the problem have failed to detect any linear instability of the flow.To compare more faithfully with experimental work the techniques used for the channel problem were modified to allow the determination of neutral curves for axisymmetric disturbances to purely oscillatory flow in a circular pipe. Critical Reynolds numbers for the pipe flow tended to be smaller than their counterparts for the channel case but the smallest critical value was still almost twice the experimentally reported result.

Miscible displacements in Hele-Shaw cells: two-dimensional base states and their linear stability

Abstract: Miscible fingering in a Hele-Shaw cell is studied by means of Stokes simulations and linear stability analysis. The two-dimensional simulations of miscible displacements in a gap indicate the existence of a quasi-steady state near the tip of the displacement front for sufficiently large Peclet numbers and viscosity ratios, in agreement with earlier work by other authors. The front thickness of this quasi-steady state is seen to scale with Pe -1/2 , while it depends only weakly on the viscosity ratio. The nature of the viscosity-concentration relationship is found to have a significant influence on the quasi-steady state. For the exponential relationship employed throughout most of the investigation, we find that the tip velocity increases with Pe for small viscosity ratios, while it decreases with Pe for large ratios. In contrast, for a linear viscosity-concentration relationship the tip velocity is seen to increase with Pe for all viscosity ratios. The simulation results suggest that in the limit of high Pe and large viscosity contrast, the width and tip velocity of the displacement front asymptote to the same values as their immiscible counterparts in the limit of large capillary numbers. In a subsequent step, the stability of the quasi-steady front to spanwise perturbations is examined, based on the three-dimensional Stokes equations. For all values of Pe, the maximum growth rate is found to increase monotonically with the viscosity ratio. The influence of Pe on the growth of the instability is non-uniform. For mild viscosity contrasts, a larger Pe is found to be destabilizing, while for large viscosity contrasts an increase in Pe has a slightly stabilizing influence. A close inspection of the instability eigenfunction reveals the presence of two sets of counter-rotating roll-like structures, with axes aligned in the cross-gap and streamwise directions, respectively. The former lead to the periodic acceleration and deceleration of the front, while the latter result in the widening and narrowing of the front. These roll-like structures are aligned in such a way that the front widens where it speeds up, and narrows where it slows down. The findings from the present stability analysis are discussed and compared with their Darcy counterparts, as well as with experimental data by other authors for miscible and immiscible flows.

Unsteady motion of two solid spheres in Stokes flow

Abstract: This study is concerned with the unsteady motion of two solid spherical particles in an unbounded incompressible Newtonian flow. The background flow is uniform and can be time dependent. In addition, the particle Reynolds numbers 2aVa∕ν and 2bVb∕ν, based on characteristic particles velocities Va and Vb, are assumed to remain small throughout the motion. Here, a and b denote the particle radii and ν is the kinematic viscosity of the fluid. Two approximate methods are employed in order to calculate the unsteady force exerted on each particle. In the first approach, a simplified method of reflections in combination with the point-force method is employed. In the second approach, a simplified method of reflections combined with Burger’s unsteady flow solution is considered. The forces due to the background flow and the disturbed flow created by the presence of particles are treated separately. The equation of motion for each particle is derived and some special cases are presented in detail including the moti...

Experimental investigation of three-dimensional flow instabilities in a rotating lid-driven cavity

Abstract: The swirling flow between a rotating lid and a stationary cylinder is studied experimentally. The flow is governed by two parameters: the ratio of container height to disk radius, h, and the Reynolds number, Re, based on the disk angular velocity, cylinder radius and kinematic viscosity of the working liquid. For the first time, the onset of three-dimensional flow behavior is measured by combining the high spatial resolution of particle image velocimetry and the temporal accuracy of laser Doppler anemometry. A detailed mapping of the transition scenario from steady and axisymmetric flow to unsteady and three-dimensional flow is investigated for 1 ≥ h ≥ 3.5. The flow is characterized by the development of azimuthal modes of different wave numbers. A range of different modes is detected and critical Reynolds numbers and associated frequencies are identified. The results are compared to the numerical stability analysis of Gelfgat et al. (J Fluid Mech 438:363–377, 2001). In most cases, the measured onset of three-dimensionality is in good agreement with the numerical results and disagreements can be explained by bifurcations not accounted for by the numerical stability analysis.

Flow rule, self-channelization, and levees in unconfined granular flows.

Abstract: Unconfined granular flows along an inclined plane are investigated experimentally. During a long transient, the flow gets confined by quasistatic banks but still spreads laterally towards a well-defined asymptotic state following a nontrivial process. Far enough from the banks a scaling for the depth averaged velocity is obtained, which extends the one obtained for homogeneous steady flows. Close to jamming it exhibits a crossover towards a nonlocal rheology. We show that the levees, commonly observed along the sides of the deposit upon interruption of the flow, disappear for long flow durations. We demonstrate that the morphology of the deposit builds up during the flow, in the form of an underlying static layer, which can be deduced from surface velocity profiles, by imposing the same flow rule everywhere in the flow.

Gasdynamic Aspects of Two-Phase Flow: Hyperbolicity, Wave Propagation Phenomena and Related Numerical Methods

Abstract: Preface 1 Introduction 2 Single-Phase Gas Flow 21 Euler equations forone-dimensional flow 22 Quasi-one-dimensional flow in ducts of variable cross section 23 Characteristic analysis of flow equations 24 Shockwaves 25 Flow through convergent-divergent nozzles 26 Shocktube 27 Multidimensional flow conditions References 3 Two-Fluid Model of Two-Phase Flow 31 Balance equations of two fluid model of two-phase flow 32 Single pressure two-fluid model 33 Remarks on interfacial transfer terms References 4 Simplified Two-Phase Flow Models 41 Homogeneous equilibrium model 411 Two-component two-phase flow 412 One-component two-phase flow 42 Homogeneous nonequilibrium two-phase flow 43 Wallis model References 5 A Hyperbolic Model for Two-Phase Flow 51 One-dimensional flow 511 Interfacial momentum coupling terms 512 Final form of conservation equations 513 Characteristic analysis- eigen values 514 Characteristic analysis - eigen vectors and splitting of coefficient matrix 515 Homogeneous flow conditions as a limiting case 516 Use of conservative variables 517 Quasi-one-dimensional flow through channels of variable cross section 52 Two-dimensional two-phase flow conditions 521 Basic flow equations for two-dimensional flow 522 Eigen values and split matrices 523 Conservative form of flow equations 53 Final remarks to the hyperbolic two-phase flow model References 6 Dispersion of Sound Waves 61 Acoustic approximation of flow equations 62 Dispersion analysis of gas-particle flows References 7 Numerical Methods for Hyperbolic Two-Phase Flow System Equations 71 Mathematical nature of two-phase flow equations 72 Overview on hyperbolic numerical methods 73 The Split Coefficient Matrix method 74 Godunov methods and Approximate Riemann solver 741 General Godunov approach 742 The linearized Riemann solver 743 The Roe solver 75 Flux Vector Splitting method References 8 Remarks on the Advanced Two-Phase Flow Module 81 Basic modeling approach 811 Balance equations of two-fluid model 812 Flow topology and interfacial area 813 Algebraicsourceterms 814 State properties 82 Numericalmethod 821 Conservative form of flow equations 822 Finite volume discretization 823 Second-order accuracy 824 Implicit time integration References 9 Numerical Results and Applications 91 Phase separation and voidwaves 911 Analytical model 912 Numerical results 92 U-tube oscillations 921 Analytical solution 922 Numerical results 93 Pressure wave propagation phenomena 931 Single-phase gas flow 932 Two-phase flow 94 Shocktube 941 Single-phase gas 942 Two-phase flow 95 Multidimensional wave propagation and explosion phenomena 951 Single-phase gas flow 952 Two-phase flow 96 Flow through convergent-divergent nozzles 961 The ASTA Rnozzle 962 Deichnozzle tests 963 Moby-Dick nozzle tests 97 Blow down phenomena 971 Edwards'pipe blow down 972 Canonexperiment 973 Two-vessel test case References 10 Summary and Concluding Remarks Appendices A Basic Flow Equations for Two-Fluid Model of Two-Phase Flow A1 Flow topology A11 Phase distribution function A12 Interfacial properties A13 Transport equation for interfacial area A2 Single-phase flow equations A3 Two-phase balance equations A31 Balance equation for mass A32 Balance equation for momentum A33 Balance equation for energy A34 Summary of two-phase balance equations B Characteristic Analysis of Flow Equations: Vectors and Matrices B1 Single-phase gas flow, one-dimensional conditions B2 Single-phase gas flow, two-dimensional conditions B3 Homogeneous non equilibrium two-phase flow B4 Wallis model B5 Hyperbolic two-phase flow model-one-dimensional conditions B6 Hyperbolic two-phase flow model-two-dimensional conditions Index

FIC/FEM Formulation with Matrix Stabilizing Terms for Incompressible Flows at Low and High Reynolds Numbers

Abstract: We present a general formulation for incompressible fluid flow analysis using the finite element method. The necessary stabilization for dealing with convective effects and the incompressibility condition are introduced via the Finite Calculus method using a matrix form of the stabilization parameters. This allows to model a wide range of fluid flow problems for low and high Reynolds numbers flows without introducing a turbulence model. Examples of application to the analysis of incompressible flows with moderate and large Reynolds numbers are presented.

Minimum Specific Energy and Critical Flow Conditions in Open Channels

Abstract: In open channels, the relationship between the specific energy and the flow depth exhibits a minimum, and the corresponding flow conditions are called critical flow conditions. Herein they are reanalyzed on the basis of the depth-averaged Bernoulli equation. At critical flow, there is only one possible flow depth, and a new analytical expression of that characteristic depth is developed for ideal-fluid flow situations with nonhydrostatic pressure distribution and nonuniform velocity distribution. The results are applied to relevant critical flow conditions: e.g., at the crest of a spillway. The finding may be applied to predict more accurately the discharge on weir and spillway crests.

Physics of flow instability and turbulent transition in shear flows

Abstract: Accepted 18 February, 2011 Turbulent transition is of great significance in modern sciences and industrial applications. The physics of flow instability and turbulent transition in shear flows is studied by analyzing the energy variation of fluid particles under the interaction of base flow with a disturbance. A simple model derived from physics is proposed to show that the flow instability under finite amplitude disturbance leads to turbulent transition. The proposed model is named as “energy gradient method”. It is demonstrated that it is the transverse energy gradient that leads to the disturbance amplification while the disturbance is damped by the energy loss due to viscosity along the streamline. The threshold of disturbance amplitude obtained is scaled with the Reynolds number by an exponent of -1, which is in good agreement with experiments in literature for pipe flow with injection disturbance. Experimental data for wall bounded parallel flows indicate that the critical value of the so called energy gradient parameter Kmax is same at turbulent transition (at least for pressure driven flows). The location of instability initiation accords well with the experiments for both pipe Poiseuille flow (r/R=0.58) and plane Poiseuille flow (y/h=0.58). It is also inferred from the proposed method that the transverse energy gradient can serve as the power for the self-sustaining process of wall bounded turbulence. Finally, the relation of “energy gradient method” to the classical “energy method” based on Rayleigh-Orr equation is also discussed.

Improved simulation of drop dynamics in a shear flow at low Reynolds and capillary number.

Abstract: The simulation of multicomponent fluids at low Reynolds number and low capillary number is of interest in a variety of applications such as the modeling of venule scale blood flow and microfluidics; however, such simulations are computationally demanding. An improved multicomponent lattice Boltzmann scheme, designed to represent interfaces in the continuum approximation, is presented and shown (i) significantly to reduce common algorithmic artifacts and (ii) to recover full Galilean invariance. The method is used to model drop dynamics in shear flow in two dimensions where it recovers correct results over a range of Reynolds and capillary number greater than that which may be addressed with previous methods.

Optical Stokes Flow Estimation: An Imaging-Based Control Approach

Abstract: We present an approach to particle image velocimetry based on optical flow estimation subject to physical constraints. Admissible flow fields are restricted to vector fields satifying the Stokes equation. The latter equation includes control variables that allow to control the optical flow so as to fit to the apparent velocities of particles in a given image pair. We show that when the real unknown flow observed through image measurements conforms to the physical assumption underlying the Stokes equation, the control variables allow for a physical interpretation in terms of pressure distribution and forces acting on the fluid. Although this physical interpretation is lost if the assumptions do not hold, our approach still allows for reliably estimating more general and highly non-rigid flows from image pairs and is able to outperform cross-correlation based techniques. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)