Showing papers on "Streamlines, streaklines, and pathlines published in 1998"

Numerical simulation of electroosmotic flow

Abstract: We have developed a numerical scheme to simulate electroosmotic flows in complicated geometries. We studied the electroosmotic injection characteristics of a cross-channel device for capillary electrophoresis. We found that the desired rectangular shape of the sample plug at the intersection of the cross-channel can be obtained when the injection is carried out at high electric field intensities. The shape of the sample plug can also be controlled by applying an electric potential or a pressure at the side reservoirs. Flow induced from the side channels into the injection channel squeezes the streamlines at the intersection, thus giving a less distorted sample plug. Results of our simulations agree qualitatively with experimental observations.

Stratified flow over topography: the role of small-scale entrainment and mixing in flow establishment

Abstract: Stratified flow over topography is examined in the context of its establishment from rest. A key element of numerical and steady–state analytical solutions for large amplitude topographic flow is the splitting of streamlines, which then enclose a trapped wedge of mixed fluid above the rapidly moving deeper layer. Measurements have been acquired that illustrate the development of this wedge and the role played by small–scale instabilities and mixing formed initially by the acceleration of subcritical stratified flow over the obstacle crest. The volume of trapped fluid progressively increases with time, permitting the primary flow to descend beneath it over the lee face of the obstacle. Throughout the evolution of this flow, small–scale instability and consequent entrainment would seem to be a prime candidate for producing the weakly stratified wedge, thus allowing establishment of the downslope flow to take place. Velocity structure of instabilities within the entrainment zone is observed and the associated entrainment rate determined. The entrainment is sufficient to produce a slow downstream motion within the upper layer and a density step between the layers that decreases with downstream distance. The resulting internal hydraulic response is explained in terms of a theory that accommodates the spatially variable density difference across the sheared interface. The measurements described here were acquired in a coastal inlet subject to gradually changing tidal currents. It is proposed that the observed mechanism for flow establishment also has application to atmospheric flow over mountains.

Trajectory and entrainment of a round jet in crossflow

Abstract: This paper examines the trajectory and entrainment characteristics of a round jet in crossflow. A series of large eddy simulations was performed at Reynolds numbers of 1050 and 2100 and at jet to crossflow velocity ratios of 2.0 and 3.3. Trajectories, which are defined based on the mean streamlines on the centerplane, all collapse to a single curve far from the jet exit, and this curve can be represented with a power law fit. Within this power law region, entrainment of crossflow fluid is shown to be the primary mechanism by which the jet trajectory is determined. Upstream of the power law region, near the jet exit, jet trajectory varies from changes in pressure drag and from differences in the turbulence intensities in the incoming pipe flow.

Shallow‐water flow solver with non‐hydrostatic pressure: 2D vertical plane problems

Abstract: A numerical solution for shallow-water flow is developed based on the unsteady Reynolds-averaged Navier–Stokes equations without the conventional assumption of hydrostatic pressure. Instead, the non-hydrostatic pressure component may be added in regions where its influence is significant, notably where bed slope is not small and separation in a vertical plane may occur or where the free-surface slope is not small. The equations are solved in the σ-co-ordinate system with semi-implicit time stepping and the eddy viscosity is calculated using the standard k–ϵ turbulence model. Conventionally, boundary conditions at the bed for shallow-water models only include vertical diffusion terms using wall functions, but here they are extended to include horizontal diffusion terms which can be significant when bed slope is not small. This is consistent with the inclusion of non-hydrostatic pressure. The model is applied to the 2D vertical plane flow of a current over a trench for which experimental data and other numerical results are available for comparison. Computations with and without non-hydrostatic pressure are compared for the same trench and for trenches with smaller side slopes, to test the range of validity of the conventional hydrostatic pressure assumption. The model is then applied to flow over a 2D mound and again the slope of the mound is reduced to assess the validity of the hydrostatic pressure assumption. © 1998 John Wiley & Sons, Ltd.

Three dimensional numerical simulations of viscoelastic flows through planar contractions

Abstract: We present in this paper a fully three dimensional (3D) convergent numerical study of planar viscoelastic contraction flows. A 3D finite volume method (FVM) with the primary variable elastic viscous split stress (EVSS) formulation is employed, and a very efficient 3D block solver coupled with block correction is developed to speed up the convergence rate. Full 3D simulations of viscoelastic flows in 4:1 planar abrupt contractions are carried out using experimental conditions. Upstream vortex patterns comparable with the existing flow visualisation observations are captured using the upper convected Maxwell model for a Boger fluid and the Phan-Thien–Tanner model for a shear thinning fluid. Comprehensive comparisons between numerical simulation results and data measured in the dynamic fields in a 4:1 planar abrupt contraction are made, and the results indicate that the experimental measurements can be quantitatively reproduced if the fluid is well characterised by an appropriate viscoelastic model. It is confirmed numerically that the shear thinning of the fluid reduces the intensity of the singularity of viscoelastic flow near the re-entrant corner. With the Oldroyd-B model, by extensive computations on successively refined meshes with the minimum dimensionless size being 0.16–0.014 on the contraction plane in 2D configuration, it is revealed that, although the asymptotic flow behavior near the re-entrant corner and the build-up of the overall pressure and extensional stresses as well as the kinematic behavior along the centreline are insensitive to mesh refinement, completely different vortex activities may be predicted if the mesh is not sufficiently fine. It is verified numerically that, depending on the flow inertia and rheological properties of fluids, both the lip vortex mechanism and the corner vortex mechanism may be responsible for the vortex activities of viscoelastic fluids in 4:1 planar contraction flow, and the elasticity number E and Mach number M of the flow can be used to determine the vortex mechanism approximately. It is clear that the development process of the vortex activities could be underestimated with 2D simplification, and overpredicted with the creeping flow assumption, particularly when Re>0.5. Therefore, in planar contraction flow analyses, numerical artifacts may be produced with a coarse mesh, and 2D flow simulation is only a good approximation to the fully 3D flow if the upstream aspect ratio W/H in the experiment is at least 10.

Modelling Flow and Oscillations in Collapsible Tubes

Abstract: Laboratory experiments designed to shed light on fluid flow through collapsible tubes, a problem with several physiological applications, invariably give rise to a wide variety of self-excited oscillations. The object of modelling is to provide scientific understanding of the complex dynamical system in question. This paper outlines some of the models that have been developed to describe the standard experiment, of flow along a finite length of elastic tube mounted at its ends on rigid tubes and contained in a chamber whose pressure can be independently varied. Lumped and one-dimensional models have been developed for the study of steady flow and its instability, and a variety of oscillation types are indeed predicted. However, such models cannot be rationally derived from the full governing equations, relying as they do on several crude, ad hoc assumptions such as that concerning the energy loss associated with flow separation at the time-dependent constriction during large-amplitude oscillations. A complete scientific description can be given, however, for a related two-dimensional configuration, of flow in a parallel-sided channel with a segment of one wall replaced by a membrane under longitudinal tension T. The flow and membrane displacement have been calculated successively by lubrication theory, Stokes-flow computation, steady Navier–Stokes computation and unsteady Navier–Stokes computation. For a given Reynolds number, Re, steady flow becomes unstable when T falls below a critical value (equivalently, when Re exceeds a critical value for fixed T), and the consequent oscillations reveal at least one period-doubling bifurcation as T is further reduced. The effect of wall inertia has also been investigated: it is negligible if the flowing fluid is water, but leads to an independent, high frequency flutter when it is air. The computations require very large computer resources, and a simpler model would be desirable. Investigation of the streamlines of the flow and the distribution of viscous energy dissipation reveals how the one-dimensional model might be improved; but such improvement is as yet incomplete.

Numerical simulation of the flow behind a rotary oscillating circular cylinder

Abstract: A numerical study was made of flow behind a circular cylinder in a uniform flow, where the cylinder was rotationally oscillated in time. The temporal behavior of vortex formation was scrutinized over broad ranges of the two externally specified parameters, i.e., the dimensionless rotary oscillating frequency (0.110⩽Sf⩽0.220) and the maximum angular amplitude of rotation (θmax=15°, 30°, and 60°). The Reynolds number (Re=U∞D/ν) was fixed at Re=110. A fractional-step method was utilized to solve the Navier–Stokes equations with a generalized coordinate system. The main emphasis was placed on the initial vortex formations by varying Sf and θmax. Instantaneous streamlines and pressure distributions were displayed to show the vortex formation patterns. The oscillatory forcing was in the vicinity of the lock-on range, which can be applied to flow feedback control afterwards. The vortex formation modes and relevant phase changes were characterized by measuring the lift coefficient (CL) and the time of negative ma...

Visualizing 3D flow

Abstract: We discuss volume line integral convolution (LIC) techniques for effectively visualizing 3D flow, including using visibility-impeding halos and efficient asymmetric filter kernels. Specifically, we suggest techniques for selectively emphasizing critical regions of interest in a flow; facilitating the accurate perception of the 3D depth and orientation of overlapping streamlines; efficiently incorporating an indication of orientation into a flow representation; and conveying additional information about related scalar quantities such as temperature or vorticity over a flow via subtle, continuous line width and color variations.

Spray jets in a cross-flow

Abstract: When droplets are expelled at a high velocity by a spray, a strong vertical air jet is induced throughout which the smallest droplets are dispersed (their Reynolds numbers associated with their relative motion being small). In our analysis we focus on the interaction between an external cross-flow and this spray jet. This interaction and the distances by which the spray jet and, over a longer distance, the large droplets are deflected are found to depend largely on the ratio of the cross-wind speed to the induced air speed U0/Uj. Using a multi-zone analysis we show that with a weak cross-flow (U0/Uj[les ]0.1), in the region immediately below the nozzle the spray entrains the external cross-flow and acts like a line sink; the streamlines close to the spray curve inwards to the centre, while further away the sink flow is weak and the streamlines follow the cross-wind. The external flow stagnates at a certain distance from the spray centreline which depends on U0/Uj. When U0/Uj[ges ]0.1 the cross-section of the spray jet and its velocity distribution change in the same way as a fluid jet in a cross-flow, whose inertia causes the deflection of the external flow around it and whose surface vorticity causes a pair of axial vortices on the downwind side of the spray. These vortices have a significant effect on the spray because they induce a back flow which reduces the tendency of the small droplets to leave the spray. When the cross-wind is strong (U0/Uj>0.3; U0[ges ]10 m s−1) the flow is too strong to be entrained; in this limit the main effect of the larger spray droplets is simply to resist the cross-flow which causes the cross-flow to slow down as it passes through the spray and to divert some of the cross-flow around the spray jet. Since the cross-flow now passes through the spray it carries the smallest droplets downwind.In this paper analytical models have been developed for all the practical ranges of the ratio of the jet speed to the cross-wind speed. This enables spray drift to be calculated. These models require very little computer time and can be run interactively. Spray droplet trajectories can be plotted straightforwardly for both axisymmetric and flat-fan sprays.

Hydromagnetic Natural Convection from AN Inclined Porous Square Enclosure with Heat Generation

Abstract: The problem of unsteady, laminar, two-dimensional hydromagnetic natural convection heat transfer in an inclined square enclosure filled with a fluid-saturated porous medium in the presence of a transverse magnetic field and fluid heat generation effects is studied numerically. The walls of the enclosure are maintained at constant temperatures. The flow in the porous region is modeled using the Brinkman-extended Darcy's law to account for the no-slip conditions at the walls. The control volume method is used to solve the governing balance equations for different values of the Darcy number, Hartmann number, and the inclination angle. Favorable comparisons with previously published work are performed. These comparisons confirmed the correctness of the numerical results. The obtained numerical results are presented graphically in terms of streamlines and isotherms as well as velocity and temperature profiles at midsections of the cavity to illustrate interesting features of the solution.

Effect of axial flow on viscoelastic Taylor–Couette instability

Abstract: Viscoelastic flow instabilities can arise from gradients in elastic stresses in flows with curved streamlines. Circular Couette flow displays the prototypical instability of this type, when the azimuthal Weissenberg number Weθ is O(e−1/2), where e measures the streamline curvature. We consider here the effect of superimposed steady axial Couette or Poiseuille flow on this instability. For inertialess flow of an upper-convected Maxwell or Oldroyd-B fluid in the narrow gap limit (e[Lt ]1), the analysis predicts that the addition of a relatively weak steady axial Couette flow (axial Weissenberg number Wez=O(1)) can delay the onset of instability until Weθ is significantly higher than without axial flow. Weakly nonlinear analysis shows that these bifurcations are subcritical. The numerical results are consistent with a scaling analysis for Wez[Gt ]1, which shows that the critical azimuthal Weissenberg number for instability increases linearly with Wez. Non-axisymmetric disturbances are very strongly suppressed, becoming unstable only when e1/2Weθ= O(We2z). A similar, but smaller, stabilizing effect occurs if steady axial Poiseuille flow is added. In this case, however, the bifurcations are converted from subcritical to supercritical as Wez increases. The observed stabilization is due to the axial stresses introduced by the axial flow, which overshadow the destabilizing hoop stress. If only a weak (Wez[les ]1) steady axial flow is added, the flow is actually slightly destabilized. The analysis also elucidates new aspects of the stability problems for plane shear flows, including the exact structure of the modes in the continuous spectrum, and illustrates the connection between these problems and the viscoelastic circular Couette flow.

Flow around a cylinder: Shallow-water modeling with diffusion-dispersion

Abstract: The equations for two-dimensional (2D), horizontal flow are obtained by vertical depth-integration of the continuity and momentum equations. Diffusion-dispersion terms appear, containing turbulent and dispersion stresses. Turbulent stresses are expressed with the eddy-viscosity concept; dispersion stresses are evaluated using velocity distributions along and across the curved streamlines. The differential equations are solved using the MacCormack scheme. The numerical simulation was done for three runs; the expected alteration of the flow field around the cylinder is evident, notably the wake behind the cylinder. For comparison, two similar runs were performed in a laboratory channel. The velocity vectors upstream from the cylinder and along its sides are in reasonably good agreement. However, very close to the cylinder, the simulation sometimes underpredicts the velocities. Downstream from the cylinder, agreement is satisfactory both inside and outside the wake. The flow depths at the centerline both upstream and downstream from the cylinder are also in good agreement; however, along the cylinder circumference, the simulated flow depths are lower than the observed ones.

Unsteady separation past moving surfaces

Abstract: Unsteady boundary-layer development over moving walls in the limit of infinite Reynolds number is investigated using both the Eulerian and Lagrangian formulations. To illustrate general trends, two model problems are considered, namely the translating and rotating circular cylinder and a vortex convected in a uniform flow above an infinite flat plate. To enhance computational speed and accuracy for the Lagrangian formulation, a remeshing algorithm is developed. The calculated results show that unsteady separation is delayed with increasing wall speed and is eventually suppressed when the speed of the separation singularity approaches that of the local mainstream velocity. This suppression is also described analytically. Only 'upstream-slipping' separation is found to occur in the model problems. The changes in the topological features of the flow just prior to the separation that occur with increasing wall speed are discussed.

Numerical solution of fiber suspension flow through a complex channel

Abstract: Both numerical simulations and fiber orientation observations were carried out to investigate two-dimensional fiber orientations in a Newtonian flow through a 1:4 backward-facing step channel. Furthermore, the flows of suspensions with high aspect-ratio fibers in a Newtonian solvent through a 1:4 backward-facing step channel were computed rigorously by coupling the flow field with fiber orientation. In the numerical simulations, a statistical scheme (number of fibers N =1800) was used. In a recirculating flow, all fibers completely align along the streamlines for large aspect-ratio fibers ( r a =10 000), while complete alignment can be achieved, however, the preferred angle lies obliquely to the streamlines for small aspect-ratio fibers ( r a =5). On the other hand, in a main flow, the preferred angle lies obliquely to the streamlines in the central region of the channel, and furthermore, the fibers are less oriented and their preferred angles tilt away largely from the streamlines as the Reynolds number (Re) decreases. In the computations of flows of suspensions with high aspect-ratio fibers, co-linear alignment of fibers was used in a recirculating flow, while a complete alignment condition was adopted in a main flow. The relation between the fiber parameter φ μ/ η and vortex length L v * follows a linear trend and its gradient decreases as Re increases, i.e. when inertia dominates the flow, the effect of fiber additives on flow structure becomes insignificant in an expansion flow. In an expansion flow of fiber suspensions for the complete alignment case, the worst orientation of fibers can be predicted clearly as a band-like pattern both near the vortex boundary and in the upper region of the expanding part. It can be supposed that fluid anisotropy near the vortex boundary is a key factor for the vortex enhancement in a complex flow.

Radial stagnation flow on a rotating circular cylinder with uniform transpiration

Abstract: Radial stagnation flow of strain rate k impinging on a cylinder with uniform transpiration U0 and rotating at constant angular velocity ω is investigated. An exact reduction of the Navier-Stokes equations to a primary nonlinear equation for the meridional flow similar to that found by Wang [1] and a secondary linear equation for the azimuthal flow is obtained. The governing parameters are the stagnation-flow Reynolds number R = ka2/2v, the dimensionless transpiration S = U0/ka, and the dimensionless rotation rate ω = ω/k, where a is the cylinder radius and v is the kinematic viscosity of the fluid. The boundary-value problem is solved by numerical integration and by asymptotic analysis in certain limits. The results are succinctly summarized in plots of the axial and azimuthal shear-stress parameters as functions of R and S. Sample velocity profiles, meridional streamfunction plots, and projections of particle paths for both suction and blowing are given. An interesting double-layer structure in the azimuthal velocity profile, consisting of a removed free shear layer connected to a wall boundary layer, is observed at large values of blowing. This feature is consistent with results obtained from the asymptotic analysis.

Viscous oscillatory flow around a circular cylinder at low Keulegan-Carpenter numbers and frequency parameters

Abstract: SUMMARY The results of a numerical study of the viscous oscillating flow around a circular cylinder at low Keulegan‐ Carpenter numbers (KC) and frequency parameters (b) are presented in this paper. The finite element method was used for the solution of the Navier‐Stokes equations in the formulation where the streamfunction and vorticity are the field variables. The computation was conducted at Keulegan‐Carpenter numbers extending up to KCa 15 and frequency parameters ranging between ba 6 and 100. At low values of the Keulegan‐Carpenter number the flow remains symmetrical. As the Keulegan‐Carpenter number is increased over a certain value which depends also on the frequency parameter, asymmetries appear in the flow which are eventually amplified and lead finally to complex vortex-shedding patterns, some of which are markedly different from those observed at higher frequency parameters. The solution revealed that although for certain values of KC and b the shedding of vortices is periodic, there also exists a complicated flow regime in which the flow is not periodic but switches between different modes in consecutive cycles of flow oscillation. For the various flow cases examined, the traces of the hydrodynamic forces are presented and the hydrodynamic coefficients and RMS values of the inline force are compared with experimental evidence. # 1998 John Wiley & Sons, Ltd.

Elliptic instability in two-sided lid-driven cavity flow

Abstract: The recently discovered concentration of vorticity in slender vortex tubes in turbulent flow fields has motivated the investigation of a class of vortices with elliptical streamlines. As a prototype of this flow, long vortices confined in a rectangular cavity and driven by tangentially moting walls are studied. These vortices are characterized by a large rate of plane strain at the core. The quasi-two-dimensional flow is found to be unstable at small Reynolds numbers, if the eccentricity of the streamlines, i.e. the strain rate, is sufficiently large. The three-dimensional supercritical flow is found to be steady with a wavelength of the order of the vortex core diameter. The flow pattern appears in the form of rectangular cells that are very robust. Good agreement between experiment and numerical calculations is obtained. It is argued that the instability found is due to the elliptic instability mechanism.

Theoretical analysis of thermocapillary flow in cylindrical columns of high prandtl number fluids

Abstract: Steady and oscillatory thermocapillary flows of high Prandtl number fluids in the half-zone configuration are analyzed theoretically. Scaling analysis is performed to determine the velocity and length scales of the basic steady flow. The predicted scaling laws agree well with the numerically computed results. The physical mechanism of oscillations is then discussed. The deformation of free surface plays an important role for the onset of oscillations in that it alters the main thermocapillary driving force of the flow by changing the temperature field near the hot-corner region. This phenomenon triggers oscillation cycles in which the surface flow undergoes active and slow periods. Based on that concept a surface deformation parameter is derived by scaling analysis. The deformation parameter correlates available data for the onset of oscillations well

Steady-State Upscaling

Abstract: Steady-state upscaling is based on steady-state solutions to the two-phase flow equations We study features of steady-state solutions, and the applicability of upscaled flow functions based thereon in transient situations Also, efficient computation of steady-state solutions is addressed: The existence of a class of problems is proved, where two-phase streamlines (in the viscous limit) are identical to streamlines in the corresponding one-phase problem

Image-guided streamline placement on curvilinear grid surfaces

Abstract: The success of using a streamline technique for visualizing a vector field usually depends largely on the choice of adequate seed points G Turk and D Banks (1996) developed an elegant technique for automatically placing seed points to achieve a uniform distribution of streamlines on a 2D vector field Their method uses an energy function calculated from the low-pass filtered streamline image to guide the optimization process of the streamline distribution This paper proposes a new technique for creating evenly distributed streamlines on 3D parametric surfaces found in curvilinear grids We make use of Turk and Banks's 2D algorithm by first mapping the vectors on a 3D surface into the computational space of the curvilinear grid To take into the consideration the mapping distortion caused by the uneven grid density in a curvilinear grid, a new energy function is designed and used for guiding the placement of streamlines in the computational space with desired local densities

Free convection heat transfer from an isothermal wavy surface in a porous enclosure

Abstract: The coupled streamfuction–temperature equations governing the Darcian flow and convection process in a fluid-saturated porous enclosure with an isothermal sinusoidal bottom sun face, has been numerically analyzed using a finite element method (FEM). No restrictions have been imposed on the geometrical non-linearity arising from the parameters like wave amplitude (a), number of waves per unit length (N), wave phase (Φ), aspect ratio (A) and also on the flow driving parameter Rayleigh number (Ra). The numerical simulations for varying values of Ra bring about interesting flow features, like the transformation of a unicellular flow to a multicellular flow. Both with increasing amplitude and increasing number of waves per unit length, owing to the shift in the separation and reattachment points, a row–column pattern of multicellular flow transforms to a simple row of multicellular flow. A cycle of n celluar and n+1 cellular flows, with the flow in adjacent cells in the opposite direction, periodically manifest with phase varying between 0 and 360°. The global heat transfer into the system has been found to decrease with increasing amplitude and increasing number of waves per unit length. Only marginal changes in the global heat flux are observed, either with increasing Ra or varying Φ. Effectively, sinusoidal bottom surface undulations of the isothermal wall of a porous enclosure reduces the heat transfer into the system. © 1998 John Wiley & Sons, Ltd.

On stagnation points and streamline topology in vortex flows

Abstract: The problem of locating stagnation points in the flow produced by a system of N interacting point vortices in two dimensions is considered. The general solution follows from an 1864 theorem by Siebeck, that the stagnation points are the foci of a certain plane curve of class N−1 that has all lines connecting vortices pairwise as tangents. The case N=3, for which Siebeck's curve is a conic, is considered in some detail. It is shown that the classification of the type of conic coincides with the known classification of regimes of motion for the three vortices. A similarity result for the triangular coordinates of the stagnation point in a flow produced by three vortices with sum of strengths zero is found. Using topological arguments the distinct streamline patterns for flow about three vortices are also determined. Partial results are given for two special sets of vortex strengths on the changes between these patterns as the motion evolves. The analysis requires a number of unfamiliar mathematical tools which are explained.

Oblique two-fluid stagnation-point flow

Abstract: Exact similarity solutions for the impingement of two viscous, immiscible oblique stagnation flows forming a flat interface are given. The problem is governed by three parameters: the ratios of density ρ = ρ1ρ2 and of viscosity μ = μ1μ2 of the two fluids and R = tanθ1tanθ2 where θ1 and θ2 are the asymptotic angles of the incident streamlines in each fluid layer. For given values of ρ, μ, and θ2, the compatible flows in the lower fluid, as measured by the strain rate ratio β = β1β2 of the two fluids and the asymptotic angle of incidence θ1, are found such that the interface remains horizontal in a uniform gravitational field. For ρ = 1, explicit solutions show that a family of co-current and counter-current shears supporting a flat interface exist for all finite, nonzero values of R. For ρ ≠ 1, the normal stress interfacial boundary conditions restricts the flow to a unique combination of asymptotic far-field shear and Hiemenz stagnation-point flow in each fluid layer. The displacement thicknesses in each layer are always positive when the fluid densities are not equal, but vanish simultaneously as ρ → 1. At each value of ρ the interfacial velocities increase with increasing viscosity ratio μ. As a generalization of the present oblique two-fluid stagnation-point flow problem, we discuss how the flat interface may be inclined with respect to the horizontal in a uniform gravitational field.