Showing papers on "Streamlines, streaklines, and pathlines published in 1990"

A general classification of three-dimensional flow fields

Abstract: The geometry of solution trajectories for three first‐order coupled linear differential equations can be related and classified using three matrix invariants. This provides a generalized approach to the classification of elementary three‐dimensional flow patterns defined by instantaneous streamlines for flow at and away from no‐slip boundaries for both compressible and incompressible flow. Although the attention of this paper is on the velocity field and its associated deformation tensor, the results are valid for any smooth three‐dimensional vector field. For example, there may be situations where it is appropriate to work in terms of the vorticity field or pressure gradient field. In any case, it is expected that the results presented here will be of use in the interpretation of complex flow field data.

On the three-dimensional instability of strained vortices

Abstract: The three‐dimensional (3‐D) instability of a two‐dimensional (2‐D) flow with elliptical streamlines has been proposed as a generic mechanism for the breakdown of many 2‐D flows. A physical interpretation for the mechanism is presented together with an analytical treatment of the problem. It is shown that the stability of an elliptical flow is governed by an Ince equation. An analytical representation for a localized solution is given and establishes a direct link with previous computations and experiments.

On a class of steady confined Stokes flows with chaotic streamlines

Abstract: The general incompressible flow uQ(x), quadratic in the space coordinates, and satisfying the condition uQ-n = 0 on a sphere r = 1, is considered. It is shown that this flow may be decomposed into the sum of three ingredients - a poloidal flow of Hill’s vortex structure, a quasi-rigid rotation, and a twist ingredient involving two parameters, the complete flow uQ(x) then involving essentially seven independent parameters. The flow, being quadratic, is a Stokes flow in the sphere. The streamline structure of the general flow is investigated, and the results illustrated with reference to a particular sub-family of ‘ stretch-twist-fold ’ (STF) flows that arise naturally in dynamo theory. When the flow is a small perturbation of a flow ul(x) with closed streamlines, the particle paths are constrained near surfaces defined by an ‘adiabatic invariant ’ associated with the perturbation field. When the flow u1 is dominated by its twist ingredient, the particles can migrate from one such surface to another, a phenomenon that is clearly evident in the computation of Poincar6 sections for the STF flow, and that we describe as ‘ trans-adiabatic drift ’. The migration occurs when the particles pass a neighbourhood of saddle points of the flow on r = 1, and leads to chaos in the streamline pattern in much the same way as the chaos that occurs near heteroclinic orbits of low-order dynamical systems. The flow is believed to be the first example of a steady Stokes flow in a bounded region exhibiting chaotic streamlines.

The role of breaking wavelets in air‐sea gas transfer

Abstract: Molecular diffusion sustains the flux of soluble gases on the water side of the air-sea interface. The “handover” of this flux to more efficient eddy mixing begins with the smallest eddies, of size l;, which interact with the surface diffusion boundary layer (DBL), of thickness δ. Owing to the discrepancy of the scales, δ ≪ l, the flow field on the δ scale consists of horizontal motions of a velocity constant with depth and varying horizontally on the l scale. The vertical velocity is proportional to the divergence of the horizontal flow and increases linearly with depth. An exact solution of the advection-diffusion equation for the simple model of divergent stagnation point flow shows the mass transfer coefficient (velocity) k to be proportional to (aD1/2) and DBL thickness δ to be proportional to (D/a1/2), where a is divergence, D diffusivity. Over a solid wall a similar model of Hiemenz flow yields a more complex relationship, also involving viscosity. These models reveal the mechanism by which the DBL is kept thin. The most intense surface divergences on a wind-blown sea surface are associated with rollers on breaking wavelets. Vorticity and divergence in the rollers are both proportional to u*2/v;, where u* is friction velocity and v is viscosity. The mass transfer coefficient resulting from divergences of this magnitude is then given by k = const u* Sc−1/2, where Sc is Schmidt number. Exact solutions of the advection-diffusion equation for model rollers reveal the details of the handover process. A thin DBL is maintained over divergences by the upward velocity. At convergences, narrow downward plumes convey DBL fluid into the turbulent interior. Flux lines (analogous to streamlines) are horizontal over divergences and dive down under convergences. Application to the sea surface requires a parameter quantifying the surface density of divergences. Laboratory data imply that a substantial fraction of the surface is covered by the divergences at higher wind speeds. However, in the open ocean straining by the large waves, and especially whitecapping, may significantly reduce the density of divergences and with it the area-average gas transfer rate. On the other hand, bubble and droplet production in whitecaps may diminish this effect or even reverse it.

Outer acoustic streaming

Abstract: There is a general formulation by Nyborg that accounts for the streaming inside the viscous boundary layer on a solid surface in the presence of a sound wave [W. L. Nyborg, J. Acoust. Soc. Am. 30, 329 (1958)]. Using the streaming velocity at the edge of the viscous layer from Nyborg’s theory as a slip boundary condition, the streaming pattern at large outside the layer for various geometries is calculated. Compressibility of the first‐order wave motion is retained, such that its effect is reflected in the boundary condition for the secondary flow, although the latter is considered as incompressible. For the case of a cylinder or a sphere situated at a velocity antinode of a plane standing wave, it is found that the streamlines are closed loops as a consequence of compressibility. If the solid body is displaced from the antinode, the vortex pattern becomes asymmetric. A weak viscous drag acts on the object in the direction opposing the displacement. As a reaction, a weak net flow arises in the direction of...

Numerical Simulation of Natural Convection Heating of Canned Thick Viscous Liquid Food Products

Abstract: Sterilization of a thick viscous liquid food in a metal can sitting in an upright position and heated from the side wall (Tw= 394 K) only in a still retort was simulated. The liquid had temperature dependent viscosity but constant specific heat and thermal conductivity. Equations of mass, motion and energy conservation for an axisymmetric case were solved and plots of temperature, velocity and streamlines were provided for natural convection heating and isotherms compared with pure conduction contour plots. Results indicated that the natural convection moved the slowest heating point to the bottom center. The bottom of the can heated up slower than predicted by pure conduction heating. The magnitude of the axial velocity was found to be of the order of 10−5 m/sec which varied with time and position in the can.

Spatial organization of large structures in the turbulent far wake of a cylinder

Abstract: Using an array of x-probes aligned in the plane of mean shear in the turbulent far wake of a circular cylinder, instantaneous velocity vector patterns are obtained from which stream-function approximations and sectional streamlines are derived. Conditional patterns obtained using different methods for detecting the organised motion are essentially independent of the particular method used. The spatial arrangement of the organised motion about the flow centreline varies in a continuous manner between opposing and alternating modes. Results presented include conditional patterns for the opposing and alternating modes and the relative contributions made by each mode to the Reynolds stresses. A modified Rankine vortex kinematic model based as much as possible on experimental data and incorporating both modes, yields mean velocity and Reynolds stress distributions which agree well with experiment. A quasi-three-dimensional version of the model implies that large spanwise vortices and shear-aligned double rollers represent the same three-dimensional organised motion from two different viewpoints.

Simulation of Non-recirculating Flows of Dilute Fiber Suspensions

Abstract: An elementary continuum model is used in the simulation of the flow of dilute fiber suspensions in a Newtonian fluid. A pseudo-Newton-Raphson technique is developed in which the constitutive equation is solved via a stable and accurate integration along streamlines, while the remaining equations of motion are solved by a conventional finite element method. The numerical technique is used to simulate both planar extrusion and falling-ball rheometry. In planar extrusion, simulations preduct a strong dependence of extrudate swell on fiber concentration and aspect ratio. Simulations of falling-ball rheometry predict Newtonian behavior with an intrinsic viscosity that is insensitive to initial fiber orientation.

Vortex street impinging upon an elliptical leading edge

Abstract: The interaction of a Karman vortex street with an elliptical edge is investigated experimentally. Basic types of interaction, as a function of scale and transverse displacement of the incident vortex street, are revealed using flow visualization. Unsteady pressure fields induced by these interactions are measured by a phase-averaging technique and correlated with the visualized flow patterns for basic classes of interactions.For a generic vortex–edge interaction, measurements of the phase-averaged velocity field allow construction of streamlines and vorticity contours showing the details of the interaction, including distortion of the vortical structures near the edge. The pressure field is calculated from the measured velocity field and interpreted in relation to the vortical structures.Simulation of flow visualization using the measured velocity field demonstrates possible misinterpretations related to the underlying vorticity field.

One‐dimensional solute transport in porous media with partial well‐to‐well recirculation: Application to field experiments

Abstract: A solute transport model incorporating well-to-well recirculation was developed to facilitate the interpretation of pilot-scale field experiments conducted for the evaluation of a test zone chosen for in situ restoration studies of contaminated aquifers, where flow was induced by recirculation of the extracted fluid. A semianalytical and an approximate analytical solution were derived to the one-dimensional advection-dispersion equation for a semi-infinite medium under local equilibrium conditions, with a flux-type inlet boundary condition accounting for solute recirculation between the extraction-injection well pair. Solutions were obtained by taking Laplace transforms to the equations with respect to time and space. The semianalytical solution is presented in Laplace domain and requires numerical inversion, while the approximate analytical solution is given in terms of a series of simple nested convolution integrals which are easily determined by numerical integration techniques. The applicability of the well-to-well recirculation model is limited to field situations where the actual flow field is one dimensional or where an induced flow field is obtained such that the streamlines in the neighborhood of the monitoring wells are nearly parallel. However, the model is fully applicable to studies of solute transport through packed columns with recirculation under controlled laboratory conditions. The model successfully simulated tracer breakthrough responses at a field solute transport study, where an induced flow field superimposed on the natural gradient within the confined aquifer was created by a well pair with extraction to injection rates of 10:1.4.

Oscillating flows over periodic ripples

Abstract: Oscillating flows over periodic ripples are of practical as well as scientific interest because of their relevance to beach processes. When either the ripples are sufficiently steep or the amplitude of ambient oscillations large, streamlines of a viscous flow are no longer parallel to the ripple surface. Circulation cells are formed which can help redistribute suspended sediments. Here we study theoretically these cells for a low-viscosity fluid such as pure water over rigid ripples. In particular we have calculated cells whose dimensions are as large as the ripple wavelength and therefore represent viscous effects far above the usual Stokes boundary layer. An idea of Stuart which was originated for stationary mean circulations around a cylinder is extended here. For large ambient amplitude, large oscillating vortices drifting with the ambient flow are found by seeking the stationary cells in a moving coordinate system.

Steady two-dimensional flow through a row of normal flat plates

Abstract: A numerical and experimental study is described for the two-dimensional steady flow through a uniform cascade of normal flat plates. The Navier–Stokes equations are written in terms of the stream function and vorticity and are solved using a second-order-accurate finite-difference scheme which is based on a modified procedure to preserve accuracy and iterative convergence at higher Reynolds numbers. The upstream and downstream boundary conditions are discussed and an asymptotic solution is employed both upstream and downstream. A frequently used method for dealing with corner singularities is shown to be inaccurate and a method for overcoming this problem is described. Numerical solutions have been obtained for blockage ratio of 50 % and Reynolds numbers in the range 0 [les ] R [les ] 500 and results for both the lengths of attached eddies and the drag coefficients are presented. The calculations indicate that the eddy length increases linearly with R , at least up to R = 500, and that the multiplicative constant is in very good agreement with the theoretical prediction of Smith (1985 a ), who considered a related problem. In the case of R = 0 the Navier–Stokes equations are solved using the finite-difference scheme and a modification of the boundary-element method which treats the corner singularities. The solutions obtained by the two methods are compared and the results are shown to be in good agreement. An experimental investigation has been performed at small and moderate values of the Reynolds numbers and there is excellent agreement with the numerical results both for flow streamlines and eddy lengths.

Thermal convection with large viscosity variation in an enclosure with localized heating

Abstract: The present study is undertaken in order to gain an understanding of convective transport in a magma chamber. We have chosen to represent the chamber by an enclosure with localized heating from below. Results of both laboratory experiments and computer modeling are reported. The experimental apparatus consists of a transparent enclosure with a square planform. An electrically heated strip, with a width equal to one-fourth of the length of a side of the enclosure, is centered on the lower inside surface of the enclosure. For the experiments reported here, the top of the fluid layer is maintained at a constant temperature and the depth of the layer is equal to the width of the heated strip. The large viscosity variation characteristic of magma convection is simulated by using corn syrup as the working fluid. Measured velocity and temperature distribution as well as overall heat transfer rates are presented. The experiment is numerically simulated through use of a finite element computer program. Numerically predicted streamlines, isotherms, and velocity distributions are presented for the transverse vertical midplane of the enclosure. Good agreement is demonstrated between predictions and measurements. 23 refs., 8 figs., 2 tabs.

Crossflow of elastic liquids through arrays of cylinders

Abstract: The flow of a dilute solution of polyisobutylene in polybutene transverse to unidirectional arrays of cylinders has been investigated at Reynolds numbers less than 0.1. Two different arrays were used—a triangular pitch array and a rectangular pitch array. Both arrays have a porosity of 0.704, the same bed length and comprise identical cylinders. Steady state permeation experiments were run over a range of superficial velocities in both arrays, to study the onset of departure from Darcy's law. The rheology of the fluid was evaluated in shear before and after each set of runs. While departures from Darcy's law occurred in both arrays at similar values of Deborah number, mechanical degradation of the polymer solution was much more severe with the triangular pitch array than with the rectangular pitch array. Specifically, after several runs through the triangular array the relaxation time was halved while the change in viscosity was relatively minor; this reveals loss of the high molecular weight tail in the original polymer. This degradation was irrecoverable; no recovery was noted after two weeks. Measurements of molecular weight distribution on the same samples in Odell's laboratory confirm that the highest molecular weight components are degraded. Finite element simulations of Stokes flow were carried out for the two different geometries to determine extensional strain rates along the flow direction in several regions. This was followed by calculations of polymer chain deformation in these regions, with the nonlinear elastic dumbbell model. These calculations reveal that the maximum stretch rate in the triangular pitch array occurs along the streamline joining the stagnation points on adjacent cylinders; this leads to nearly complete extension of the polymer chain at a nominal Deborah number of 1 in the triangular array. However, in the rectangular pitch array, the maximum stretch rate occurs along streamlines considerably removed from the stagnation points, and the polymer chains are not extended along those streamlines up to a Deborah number of 1.

Natural Convection of Cold Water in a Vertical Annulus With Constant Heat Flux on the Inner Wall

Abstract: A numerical solution for steady laminar natural convection of cold water in a vertical annulus with a constant-heat-flux heated inner wall and an isothermally cooled outer wall is examined. Results are generated for flow in annuli with aspect ratio 0.5 {le} A {le} 8, the radius ratio varying between 1.2 and 10, and the density inversion parameter ranging from {minus}2 and 1 for 10{sup 3} {le} Ra{sup *} {le} 10{sup 6}. The heat and fluid flow structures of cold water are vividly visualized by means of contour maps of heatlines and streamlines. The results clearly indicate that the mixed boundary conditions considered can have a significant influence on the geometric dependent of heat transfer characteristics and fluid flow structures in comparison with those reported for isothermal boundary conditions. Multicellular flow behavior of cold water can arise in a tall annulus of A = 8.

Particle-image-velocimetry applied to thermocapillary convection

Abstract: Thermocapillary convection is studied experimentally using particle-image-velocimetry for flow visualization and analysis This method offers the advantage of measuring the entire flow field (velocity field, streamlines etc) in a selected plane within the fluid at a given instant of time in contrast to point by point methods like laser-Doppler-velocimetry (LDV) The paper describes the method and presents quantitative results for different Marangoni numbers

Heat Transfer Augmentation Through Wall-Shape-Induced Flow Destabilization

Abstract: Experiments on heat transfer augmentation in a rectangular cross-section water channel are reported The channel geometry is designed to excite normally damped Tollmein-Schlichting modes in order to enhance mixing In this experiment, a hydrodynamically fully developed flow encounters a test section where one channel boundary is a series of periodic, saw-tooth, transverse grooves Free shear layers span the groove openings, separating the main channel flow from the circulating vortices contained within each cavity The periodicity length of the grooves is equal to one-half of the expected wavelength of the most unstable mode The remaining channel walls are flat, and the channel has an aspect ratio of 10:1 Experiments are performed over the Reynolds number range of 300 to 15,000 Streakline flow visualization shows that the flow is steady at the entrance, but becomes oscillatory downstream of an onset location This location moves upstream with increasing Reynolds numbers Initially formed traveling waves are two dimensional with a wavelength equal to the predicted most unstable Tollmien-Schlichting mode Waves become three dimensional with increasing Reynolds number and distance from onset Some evidence of wave motion persists into the turbulent flow regime Heat transfer measurements along the smooth channel boundary opposite the grooved wallmore » show augmentation (65%) over the equivalent flat channel in the Reynolds number range 1,200 to 4,800 The degree of enhancement obtained is shown to depend on the channel Reynolds number, and increases with the distance from the onset location« less

Numerical Studies of Tritium and Helium-3 in the Thermocline

Abstract: The effects of circulation and lateral mixing on the distributions of tritium and heliurn-3 in the thermocline are investigated using a two-dimensional numerical model. The gyre circulation was approximated by a steady flow with closed streamlines and a western boundary current. Variations of the strength of the circulation and mixing showed that for either very high or very low mixing (relative to the strength of the circulations the distributions are analogous to a one-dimensional model. For very low mixing, the single dimension corresponds to the cross-stream dimension of the gyre. Mixing in the western boundary current increases the rate at which a tracer moves from the exterior streamlines the center of the circulation. The distributions and inventories of tritium and helium-3 are sensitive to the overall ventilation of the gyre and cannot easily resolve diffusive ventilation from direct ventilation. The results of simulations in a small parameter range are consistent with mappings of the di...

Large Magnetic Reynolds Number Dynamo Action in a Spatially Periodic Flow with Mean Motion

Abstract: We consider a problem of advection and diffusion of passive scalar and vector fields in a particular family of steady fluid flows. These flows are obtained by adding a small uniform velocity to a spatially periodic array of spiral eddies. The uniform flow, i/H, is taken to have the discrete form aH = e(M ,N ,0 )/(M 2 + N 2 )K 1, where M, N are relatively prime integers. The spatially periodic part, u may be expressed in terms of a streamfunction x]r9, u9 = (.........) = sin x sin y where K is a constant. The flow we study is therefore u = wH + w/. Our work is motivated by applications of dynamo theory and to classical diffusion of passive scalars. The above family of flows was chosen as typical of spatially periodic flow with non-zero mean velocity, ffH. The flows are comparatively simple because they are independent of z. Nevertheless the projection of the streamline pattern onto the plane z = 0 can be surprisingly complex, owing to the structure of u modulo the cell of periodicity of u9. This structure accounts for our special form of SH above, which makes the tangent of the angle of inclination of the uniform current a rational number. This uniform component breaks up the eddy pattern into closed eddies whose bounding streamlines begin and end at X-type stagnation points. The set of all such streamlines define the boundaries of the open channels, which fill the regions between the closed eddies and lie near the separatrices of u’. Then, for example, when is even the channel structure repeats under a shift ( , in the xy plane, leading to a periodicity in channel length of order L. Analogous results apply to the case L odd. This geometry raises interesting questions regarding the advection and diffusion of fields in the irrational limit, i.e. when M, N->■ oo, -*irrational. A basic result of this paper will be formal asymptotic expressions, for average physical quantities of interest, in the irrational limit. An asymptotic theory of advection-diffusion is exploited, based upon a separation into closed eddies, channels, and separatrix boundary layers. The fundamental assumption is that the dimensionless parameter R (a magnetic Reynolds number in the dynamo problem, a Peclet number in diffusion problems) is large, meaning that transport by molecular diffusion is nominally small compared with transport by advection. For large R, the X-type stagnation points trigger boundary layers, which for given MN, extend a distance of order L before repeating the structure. This leads to channel boundary layers of width L*R~*, compared with eddy boundary layer of width R~5 and channel widths of order eZT1, the eddies being separated by gaps of widths order e. In this setting the irrational limit is taken after the above asymptotic structure is isolated by the limit R-> oo. Our results consist of numerical studies for = eR? of order unity, and analytic asymptotic expressions derived under the condition > D. In the former, the eddy separation width is comparable with the eddy boundary layer width, so that we study the transition from transport dominated by boundary layers to transport dominated by channels. In the asymptotic theory for large the boundary-layer contributions may be neglected and the problem reduces to the analysis of channel geometry. Even here, the condition ft> Drestricts us to a countable set of mean flow orientations. The relation between solutions for these special orientations, and their immediate neighbours with irrational tangents, is discussed. Representative results for effective diffusion of a passive scalar field, and for mean induced electromotive force in an electrically conducting fluid (the a-effect) are presented. We also discuss the present examples in relation to the more complex problem of advection—diffusion by flows with chaotic lagrangian paths.

The migration of a compound drop due to thermocapillarity

Abstract: The quasistatic thermocapillary motion of a compound drop in an unbounded fluid possessing a uniform temperature gradient is analyzed. For completeness, gravitational effects are included in the treatment. The general model is formulated, and the equations for the concentric case are solved using spherical polar coordinates, while the eccentric case is handled using bispherical coordinates. Results are given for the velocity of the drop as well as that of the droplet with respect to the drop, along with useful approximations. Illustrative results are presented graphically for the thermocapillary migration of a compound drop in the special case when the droplet is a gas bubble. In addition to the velocities of the drop and the bubble, representative isotherms and streamlines also are presented which display interesting qualitative features.

An application of topological analysis to studying the three-dimensional flow in cascades; part I—Topological rules for skin-friction lines and section streamlines

Abstract: Based on the working of Lighthill and Hunt et al., in the present paper the author has established the topological rules adapting to analysing the skin-friction lines and the section streamlines in cascades. These rules are (1) for a rotor cascade without shroud band, the total number of nodal points equals that the saddle points on the skin-friction line vector fields in each pitch range; (2) for an annular or straight cascade with no clearances at blade ends, the total number of saddle points is two more than that of nodal points on the skin-friction line fields in a pitch; (3) the total number of saddles in the secondary flow fields on cross-sections in cascade is one less than that of nodes; (4) in the section streamline vector fields on a meridian surface penetrating a flow passage, and on leading and trailing edge sections, the total number of nodes is equal to that of saddles; (5) on the streamline vector fields of a blade-to-blade surface, the total number of nodes is one less than that of saddles.

Swept wing ice accretion modeling

Abstract: An effort to develop a three-dimensional modeling method was initiated. This first step towards creation of a complete aircraft icing simulation code builds on previously developed methods for calculating three-dimensional flow fields and particle trajectories combined with a two-dimensional ice accretion calculation along coordinate locations corresponding to streamlines. This work is a demonstration of the types of calculations necessary to predict a three-dimensional ice accretion. Results of calculations using the 3-D method for a MS-317 swept wing geometry are projected onto a 2-D plane normal to the wing leading edge and compared to 2-D results for the same geometry. It is anticipated that many modifications will be made to this approach, however, this effort will lay the groundwork for future modeling efforts. Results indicate that the flow field over the surface and the particle trajectories differed for the two calculations. This led to lower collection efficiencies, convective heat transfer coefficients, freezing fractions, and ultimately ice accumulation for the 3-D calculation.

Chaotic advection of irrotational flows and of waves in fluids

Abstract: This paper treats the kinematics of particles advected passively by flow of an incompressible fluid. It is shown that for steady irrotational flow without circulation, and for many monochromatic waves in a fluid the particle paths are not chaotic, i.e. do not depend sensitively on initial conditions. However, if the flow is a time-periodic potential flow or a superposition of waves then the particle paths may be chaotic. This is shown by the application of the theory of Melnikov to the breakup of a heteroclinic orbit (which connects two stagnation points and may bound a region of closed streamlines) and the onset of chaos in two examples of two-dimensional flow. The first example is a simple unbounded irrotational flow comprising a steady flow with two stagnation points which has a time-periodic perturbation. The second example is of two Rossby waves with a mean zonal flow; the particle paths are examined geometrically and numerically, and consequences for pollutant dispersion are discussed in physical terms. Also the combination of the effects of chaotic advection and molecular diffusion on the transport of a solute are examined.

A Numerical Scheme for Determining Trajectories in Particle Models

Abstract: A scheme for modelling of trajectories of particles in Lagrangian advection/dispersion models is presented The second-order accurate method is numerically tractable and eliminates the tendency for particles to spiral outwards on curved streamlines The method is compared with Euler and Taylor series solutions