Showing papers on "Streamlines, streaklines, and pathlines published in 1986"

Loss of evolution in the flow of viscoelastic fluids

Abstract: For the Maxwell model, it has been proved that an initially evolutionary flow remains so at any time: the property is transported along the streamlines. This is generally not the case for other differential models. A complete study is made of the local properties which are conserved by a Johnson-Segalman fluid flow. Evolution is not preserved, and counter examples are given where the evolutionary character is indeed lost. Two conservation properties are demonstrated and their relationship with the Second Law is also examined. The evolutionary character of a system is a physical requirement which should be satisfied by all models.

Hydrodynamic transitions to chaos in the convection of an anisotropic fluid

Abstract: We present the first spontaneous sequence of hydrodynamic instabilities due to the evolution of the convective flow from the rest state toward chaos in an anisotropic fluid: a nematic liquid crystal subjected to an AC electric field. The resulting structures are of decreasing symmetry as the control parameter is gradually increased. The study of the streamlines allows the interpretation of the entire sequence by the evolution of the convective flow up to the onset of chaos. We show that the first important step in the flow evolution is the build up of a second mode of rotation which is associated with a pinching instability of the vortex lines. A simple picture is then proposed in order to describe the essential steps of a possible route to chaos, and which could be extended to other convective systems On presente une suite ordonnee d'instabilites hydrodynamiques d'un ecoulement convectif, observees depuis l'etat de repos jusqu'au chaos dans un cristal liquide soumis a un champ electrique

A streamline element scheme for solving viscoelastic flow problems. Part I. Differential constitutive equations

Abstract: A finite element program for solving viscoelastic flow problems in plane and axisymmetric geometries is described. The program is based on a streamline element scheme (S.E.S.). One of the principal features of the program is that it employs an iterative process to treat elastic stresses as pseudo-body forces; during each iteration it generates updated streamline-based elements and integrates Maxwell-type constitutive equations along the streamlines forming the element boundaries. The performance of this method was satisfactory, considering both accuracy and efficiency. Results of two simple test problems and two complex flow problems are shown. These problems are simple shearing flow, uniaxial and biaxial elongation, extrudate swelling and flow around a sphere in a long cylinder. In the simple test problems the results were very accurate and were compared with exact solutions and inthe more complex flows the results were in close agreement with data from other programs based on different methods. Some limitations and the current state of development of the program are also discussed.

Rip-current generation near structures

Abstract: Data have been obtained for a wave-driven current system in a closed basin. Owing to interaction of the longshore current with the sidewall a strong rip current was generated. The velocity distribution over the depth of the rip current appeared to be more or less uniform. The current system has been modelled by means of a mathematical model. The effect of bottom topography, bottom friction, convection and turbulent viscosity on the current system has been investigated. The conclusions are that the rip current is dominated by convection and that the bottom topography plays a role in the convergence and divergence of the streamlines. The order of magnitude of the velocities is largely determined by the bottom-friction coefficient. The velocity field is modified by viscosity. First, turbulent viscosity entrains fluid in the longshore current and into the rip current, secondly it permits turbulent boundary layers and thirdly it is responsible for the existence of closed streamlines outside the breaker zone. Finally the model and the conclusions are extrapolated to prototype conditions.

Streamline Routing Through Fracture Junctions

Abstract: A series of laboratory tests was conducted to determine routing criteria for streamlines through fracture junctions. These tests showed that two criteria are all that is necessary to route streamlines through any two-dimensional junction under laminar flow conditions. These criteria are (1) that streamlines do not cross and (2) that flow along adjacent streamlines must be in the same direction. Using these two criteria, a unique distribution of streamlines can be determined for both continuous and discontinuous fracture junctions.

The limiting configuration of interfacial gravity waves

Abstract: Progressive gravity waves at the interface between two unbounded fluids are considered. The flow in each fluid is taken to be potential flow. The problem is converted into a set of integrodifferential equations, reduced to a set of algebraic equations by discretization, and solved by Newton’s method together with parameter variation. Meiron and Saffman’s [J. Fluid Mech. 129, 213 (1983)] calculations showing the existence of overhanging waves are confirmed. However, the present calculations do not support Saffman and Yuen’s [J. Fluid Mech. 123, 459 (1982)] conjecture that the waves are geometrically limited (i.e., that solutions exist until the interface intersects itself). It is proposed that along a solution branch starting with sinusoidal waves of small amplitude, one reaches solutions with vertical streamlines and then overhanging waves. Continuing on this branch, one returns to nonoverhanging waves and then back toward a wave with vertical streamlines. It is suggested that this succession of patterns ...

Fluid mixing (stretching) by time periodic sequences for weak flows

Abstract: The stretching of material lines in time periodic sequences of ‘‘weak’’ flows (i.e., characterized by a linear stretching for long times) in which streamlines change at the end of each periodic unit is quantified in terms of a mixing efficiency. Two different flows and two different strain distributions give nearly identical results in terms of the mixing efficiency and indicate the existence of an optimal operating condition for maximum stretch. The results are relevant to fluid mixing in chaotic flows.

On the existence of localized rotational disturbances which propagate without change of structure in an inviscid fluid

Abstract: A wide class of solutions of the steady Euler equations, representing localized rotational disturbances imbedded in a uniform stream U, is inferred by considering the process of magnetic relaxation to analogous magnetostatic equilibria. These solutions, which may be regarded as generalizations of vortex rings, are characterized by their streamline topology, distinct topologies giving rise to distinct solutions. Particular attention is paid to the class of axisymmetric solutions described by Stokes stream function $(s, z). It is argued that the appropriate topological ‘invariant’ characterizing the flow is the function V($) representing the volume inside toroidal surfaces $ = const. in the region of closed streamlines where $ > 0. This function is described as the ‘signature’ of the flow, and it is shown that in a certain sense, flows with different signatures are topologically distinct. The approach yields a method by which flows of arbitrary signature V($) may in principle be found, and the corresponding vorticity wp = sF($) calculated.

The Functional Role of Wing Corrugations in Living Systems

Abstract: Questions concerning the functional role of spanwise wing corrugation in living systems are experimentally investigated. Attention was initially directed to this problem by observation of the irregular shape of many insect wings as well as other studies indicating higher lift on these wings. First, a flow visualization scheme was used to observe and photograph streamlines around two different wing sections. One of these, a sheet metal model with geometry matching that of a butterfly wing, was studied at a chord Reynolds number of 1500 and at a Reynolds number of 80 based on corrugation depth. A steady-state recirculation region near the model leading edge was found, and the separated flow region above this recirculation zone formed a laminar reattachment to the model. A second thicker wing was corrugated on the upper surface. Closed streamlines inside these upper surface corrugations were photographed at Reynolds numbers of 8000 and 3800 based on chord length, and 200 and 90 based on corrugation depth. Reductions in pressures on the corrugated upper wing surface relative to a smooth upper wing surface were then measured.

Topographic eddies in temporally varying oceanic flows

Abstract: In this paper we examine the behaviour of oceanic unsteady flow impinging on isolated topography by means of numerical simulation. The ocean model is quasigeostrophic and forced by an oscillatory mean flow. The fluid domain is of the channel type and open-boundary numerical conditions are used to represent downstream and upstream flow. In certain cases, vortex shedding, either cyclonic or anticyclonic, is observed in the lee of obstacles. Such shedding can be explained as the consequence of both an enhanced process of vorticity dissipation over the topography which locally affects the balance of potential vorticity on the advective timescale, and a periodic dominance of advective effects which sweep the fluid particles trapped on the seamount. For refined resolution and smallest viscosity the model will predict flows in which the shed eddies are coherent structures with closed streamlines. The model suggests a mechanism by which topographically generated eddies may be swept away from a seamount i...

Glancing interactions between single and intersecting oblique shock waves and a turbulent boundary layer

Abstract: The three-dimensional interactions of weak swept oblique shock and expansion waves and a turbulent boundary layer on a flat plate are investigated. Upstream influences in a single swept interaction are found to be consistent with a model of the flow involving shock/boundary-layer interaction characteristics. The model implies that there is more rapid thickening of the boundary layer close to the shock generator and this is seen to be consistent with surface streamline patterns. It is also found that a superposition principle, which is inherent in the triple-deck model of shock/boundary-layer interactions proposed by Lighthill, can be used to predict the pressure field and surface streamlines for the case of intersecting shock interactions and for the intersection of a shock with a weak expansion.

Buoyant Laminar Convection in a Vertical Cylindrical Annulus

Abstract: Solution to natural convective motion in a vertical cylindrical annulus of large aspect ratio is examined, with the inner wall subject to a constant heat flux. Similar transformations of the appropriately simplified Navier–Stokes model of the annular flow are sought. The issue of boundary condition limitations for the considered flow is resolved in terms of acceptable error bounds for valid solutions. Results are generated for a variety of outer wall boundary conditions over various ranges of the Rayleigh number, highlighting the effects on the patterns of streamlines and “heatlines.” Correlations presented for the Nusselt number are shown to be dependent on both the boundary conditions and the geometry.

On the nonlinear critical layer below a nonlinear unsteady surface wave

Abstract: A theory is presented for the development of the nonlinear critical layer below an unsteady free surface wave, of amplitude e, described by the Korteweg-de Vries (KdV) equation. The problem is formulated (via the Euler equations) for wave propagation over an arbitrary shear in a two-dimensional channel which contains a critical level. The equations are scaled so as to be valid in the far-field regime of the surface wave, appropriate to the existence of the KdV equation, i.e. long waves. The regions above and below the critical layer are solved (to O(e), as e → 0) and thence expanded in the neighbourhood of the critical layer itself. The symmetry of the critical layer solution, assuming that it exists, is then sufficient to determine the Burns integral for the linearized wave speed, and the relevant KdV equation. These turn out to be the classical results evaluated in terms of finite parts.The critical layer, of thickness O(e½), is analysed to O(e) and matched to the outer regions of the flow field. The initial configuration is taken to contain no closed streamlines, and so the vorticity can, presumably, be assigned from the undisturbed conditions at infinity. The initial surface profile must therefore contain a single peak, but by virtue of the KdV equation this can evolve into any number of solitons. Between consecutive pairs of peaks there will now appear regions of closed streamlines (cat's-eyes) with known vorticity. No recourse to a viscous argument is necessary to uniquely determine this vorticity. However, it is shown that the vorticity cannot be prescribed arbitrarily at all orders, initially: the long-wave assumption imposes a certain structure on the problem, and then the continuity of stream function and particle velocity fixes the vorticity. This agrees with the work of Varley & Blyth (1983) on the hydraulic equations. The vorticity inside the separating streamlines is obtained to O(e), but it is shown that for unsteady motion this asymptotic expansion is not uniformly valid as the bounding streamlines are approached. An alternative method, which exploits Varley & Blythe's approach, is used to confirm the correctness of our results away from these boundaries, and to indicate that a non-uniformity is present near the separating streamlines. Thus the model requires the inclusion of a vortex sheet; for steady flow a jump in vorticity is sufficient. The removal of the discontinuity by allowing a distortion of the main flow outside the critical layer is briefly discussed.Some results are presented for the formation of a single cat's-eye by using the exact 2-soliton solution of the KdV equation.

Extrudate Swell in Double‐Layer Flows

Abstract: A general‐purpose finite element program has been used to simulate the flow of two immiscible viscous fluids extruded concentrically from a capillary and from an annular die. The interface and free surfaces are found iteratively by taking into consideration that these surfaces are streamlines. The pressure discontinuities across the interface have been simulated numerically by double nodes to ensure continuity of the stresses and the velocities. A range of viscosity ratios is studied for several cases of inner/outer diameter ratios of the fluids. Extrudate swell calculations show that for inelastic Newtonian fluids, shrinkage occurs when the less viscous layer encapsulates the more viscous, which is also the preferential configuration in coextrusion from long dies. Swelling occurs when the more viscous fluid wets the capillary wall. These results are in general agreement with Tanner's inelastic theory of extrudate swell. In flow from annular dies, bending of the extrudate stream occurs toward the more viscous component due to the viscosity mismatch. The present results reveal some important trends that can be expected in actual coextrusion operations, depending upon viscosity and feed ratios.

Equal‐energy streamlines

Abstract: Energy streamlines show the direction of energy flow in a sound field: The intensity vector at any point is tangential to the streamline passing through that point. In earlier work, a procedure for computing energy streamlines was described [J. Acoust. Soc. Am. 78, 758–762 (1985)], but it did not give equally spaced streamlines. The latter have certain advantages, and a method for computing them is presented here. It is shown that in sound fields which vary in only two space dimensions, a scalar stream function exists, the contours of which are the energy streamlines. The stream function is obtained from the intensity function (or values) by an integration procedure. Examples of equally spaced streamlines are given for the point‐driven, fluid‐loaded plate, and for the baffled rigid piston.

Rotating flows along indented coastlines

Abstract: The problem of a steady, inviscid reduced-gracity rotating flow in a wedge around a sharp corner is examined. Using two or more corner solutions, plausible flow streamlines can be generated in more complicated domains, as long as no two corners are closer than the Rossby radius of deformation. This procedure is illustrated with two examples: 1) circulation in a channel mouth; and 2) flow around a square bump in a coastline.

Nonlinear stability of the Kelvin-Stuart cat's eyes flow

Abstract: Conditions which ensure the nonlinear stability of the Kelvin-Stuart cat's eyes solution for two dimensional ideal flow are given. The solution is periodic in the x direction and is bounded by two streamlines, which contain the separatrix, in the y-direction. The stability conditions are given explicitly in terms of the solution parameters and the domain size. The method is based on a technique originally developed by Arnold [1969].

Illustrative examples of streaklines in unsteady vortices: Interpretational difficulties revisited

Abstract: As a preventive measure against false interpretations of the streakline pattern in unsteady flows, streaklines calculated for an isolated vortex and the infinite Karman vortex street are presented, which appear to create a visual impression of apparent growth and flow reversal, while in reality they do not exist.

Taylor dispersion in concentrated suspensions of rotating cylinders

Abstract: Laminar heat- or mass-transfer processes are theoretically investigated for two-dimensional spatially periodic suspensions of circular cylinders, each member of which rotates steadily about its own axis under the influence of an external couple. The novelty of the ensuing convective-diffusion phenomena derives from the absence of convective motion at the suspension lengthscale (the ‘macroscale’), despite its presence at the interstitial or particle lengthscale (the ‘microscale’). The latter fluid motion consists of a cellular vortex-like flow characterized by closed streamlines. These periodically closed streamlines give rise to a situation in which there exists no net flow at the macroscale. The resulting macroscale transport of heat or mass thus proceeds purely by conduction, the rate being characterized by a tensor diffusivity — dependent upon the angular velocity of the cylinders. Matched-asymptotic-expansion methods together with generalized Taylor dispersion theory are used to calculate this macroscale conductivity in the dual limit of large rotary Peclet numbers and small gap widths between adjacent cylinders. This prototype study illustrates the fact that the usual separation of transport processes into distinct convective and conductive contributions is not generally a scale-invariant concept; that is, microscale convec-tional contributions to the transport of heat or mass are not generally representable by corresponding macroscale convectipnal contributions to the transport. Possible applications of the analysis exist in the area of enhanced conduction rates in ferrofluids or other dipolar fluids rotating relative to a fixed external field (or conversely).

A method for determining corotating interaction regions in the equatorial plane of the heliosphere from solar wind velocity data

Abstract: An Euler-Lagrange coordinate transformation is applied to a computed solar wind velocity data set to obtain fluid streamlines which are also magnetic field lines. Plotting these lines for high-resolution simulation data provides detailed information about the time variation of corotating interaction regions in the equatorial plane of the heliosphere.

Linearly Stratified, Rotating Flow over Long Ridges in a Channel

Abstract: The flow of a rotating, linearly stratified fluid over a long symmetric ridge in a channel is investigated experimentally. The laboratory apparatus consists of a long channel of rectangular cross section. The upper bounding surface is a transparent, horizontal plane; the lower boundary is a horizontal flexible belt. The belt serves as a false bottom of the channel and is translated parallel to its long axis. Topographic features are mounted on the belt and are towed through a salt-stratified fluid which is otherwise at rest relative to a rotating observer; the channel rests on a rotating table whose rotation axis is vertical. The most important dimensionless parameters governing the motion are the Rossby and Ekman numbers, a stratification parameter defined as the ratio of the Brunt-Vaisala frequency to the Coriolis parameter, and the geometrical parameters defining the aspect ratio of the ridge, the ridge height to channel depth ratio and the ridge to channel width ratios. An analysis is presented that demonstrates the conditions under which centrifugal effects can be neglected in such laboratory experiments. The analysis also shows the conditions under which the laboratory flows should be a good approximation to the quasigeostrophic potential vorticity equations and attendant boundary conditions for the oceans and atmosphere. This analysis is made for a general three-dimensional topographic feature; i.e. it is not restricted to a long ridge. The laboratory system seems to be an excellent vehicle for modelling oceanic flows but does not properly reproduce a non-Boussinesq term in the atmospheric equation. An analysis for an infinitely long ridge is presented. The predictions so obtained are in good qualitative agreement with the experiments. The quantitative agreement, for the range of parameters considered however, is shown to be poor and this is attributed to neglecting the effects of the lateral bounding surfaces. The experiments demonstrate that for fixed rotation, stratification and free stream speed, the streakline deflection caused by the topography decays with height. For such experiments the flow in the lower levels for positive upward rotation deflects to the left before reaching the ridge, then continues to deflect to the left on the upwind side of the ridge before beginning a rightward drift slightly upstream of the mountain crest. This rightward drift continues on the downwind side of the ridge and well downstream of the ridge itself before reaching a maximum shift, from which a leftward drift again begins. Increased rotation, other parameters being held fixed, provides stronger horizontal streakline deflections. Stronger stratification, other parameters being fixed, leads to stronger downslope winds and possibly flow separation in the lee. Various characteristics of the flow field, such as the distance upstream to which substantial streakline curvature is observed, are measured as functions of the various system parameters; comparisons with the infinite ridge theory are also made. The downstream motion is accompanied by lee waves for the major portion of the parameter space examined. The amplitude of these waves is shown to decrease with increased rotation, other parameters being held fixed. Some non-rotating experiments are conducted and these are shown to be in good agreement with the model of Long (1955) and the measured wavelengths are found to be in good agreement with linear theory and laboratory measurements made by other investigators. Measurements supporting the theory of Queney (1947) are presented which show that the horizontal wavelengths of the lee waves decrease for increased rotation, other parameters being fixed. A flow regime map based on the observed structure of the vertical wave motion is developed and it is shown that for the range of parameters considered, rotation plays only a minor role, if any.

Solving problems in fluid mechanics

Abstract: 1 Dimensional Analysis 2 Dynamical Similarity Problems 3 Vortex Motion And Radial Flow 4 Streamlines And Stream Function 5 Gases At Rest 6 Flow Of Gases 7 Viscous Flow 8 Turbulent Flow 9 Flow Round Totally Immersed Bodies 10 Waterhammer And Pressure Transients 11 Non-Uniform Flow In Channels 12 Impulse And Reaction Turbines 13 Centrifugal Pumps 14 Reciprocating Pumps 15 Computational Fluid Dynamics Index

An Inverse (Design) Problem Solution Method for the Blade Cascade Flow on Streamsurface of Revolution

Abstract: On the basis of the fundamental equations of aerothermodynamics a method for solving the inverse (design) problem of blade cascade flow on the blade-to-blade streamsurface of revolution is suggested in the present paper. For this kind of inverse problem the inlet and outlet flow angles, the aerothermodynamic parameters at the inlet, and the other constraint conditions are given. Two approaches are proposed in the present paper: the suction-pressure-surface alternative calculation method (SSAC) and the prescribed streamline method (PSLM). In the first method the metric tensor (blade channel width) is obtained by alternately fixing either the suction or pressure side and by revising the geometric form of the other side from one iteration to the next. The first step of the second method is to give the geometric form of one of the streamlines. The velocity distribution or the mass flow rate per unit area on that given streamline is estimated approximately by satisfying the blade thickness distribution requirement. The stream function in the blade cascade channel is calculated by assuming initial suction and pressure surfaces and solving the governing differential equations. Then, the distribution of metric tensor on the given streamline is specified by the stream function definition. It is evident that the square root of the metric tensor is a circumferential width of the blade cascade channel for the special nonorthogonal coordinate system adopted in the present paper. The iteration procedure for calculating the stream function is repeated until the convergence criterion of the metric tensor is reached. A comparison between the solutions with and without consideration of viscous effects is also made in the present paper.

High-latitude convection on open- and closed-field lines for large IMF B/sub y/. Technical report

Abstract: Certain S3-3 electric-field observations show a single convetion cell engulfing the northern polar cap. The flow direction is that for a positive IMF B/sub y/ component. The particle data indicate that nearly half the duskside sunward flow occurs on closed field lines, whereas the dawnside flow is entirely on open field lines. The authors interpret this in terms of an IMF B/sub y/-induced deformation in the polar-cap boundary, where the deformation moves with the convective flow. Thus, convection streamlines cross the deformed polar-cap boundary, but no flow crosses the boundary because it is carried by the flow. Since southern-hemisphere convection is expected to occur with the opposite sense of rotation, it is predicted that closed-field lines will be forced to tilt azimuthally. On the nightside, the tilt produces a y component of the magnetic field in the same direction as the IMF for either sign of IMF B/sub y/. Our interpretation is consistent with observations of a greater y component in the plasma sheet than the tail lobes, which are difficult to understand in terms of the common explanation of IMF penetration. Alternatives to this interpretation are also discussed.

Analytical Solution of Oseen's Equation for Creeping Flow around a Spherical Gas-bubble in Liquid

Abstract: The first-order solution to Oseen's equation is determined for liquid flow around a spherical gas-bubble at low Reynolds numbers under the assumption that the velocities and the stress are continuous at the interface between gas and liquid. This solution leads to new drag-coefficients of a spherical gas-bubble rising in quiescent liquid as Cd = 16 (1+ Re/8)/Re, where Re is Reynolds number related to the diameter of a spherical gas-bubble. The measurements of hydro-dynamic drags experienced by small gas-bubbles in quiescent liquid at the Reynolds number between 0.03 and 1.0 show good agreement with the new drag formula introduced in the present paper. The visualized streamlines around a spherical gas-bubble at the Reynolds numbers below 0.3 agree well with the streamlines predicted by the first-order solution to Oseen's equation.