Showing papers on "Streamlines, streaklines, and pathlines published in 1992"

Kinematic simulation of homogeneous turbulence by unsteady random Fourier modes

Abstract: The velocity field of homogeneous isotropic turbulence is simulated by a large number (38–1200) of random Fourier modes varying in space and time over a large number (> 100) of realizations. They are chosen so that the flow field has certain properties, namely (i) it satisfies continuity, (ii) the two-point Eulerian spatial spectra have a known form (e.g. the Kolmogorov inertial subrange), (iii) the time dependence is modelled by dividing the turbulence into large- and small-scales eddies, and by assuming that the large eddies advect the small eddies which also decorrelate as they are advected, (iv) the amplitudes of the large- and small-scale Fourier modes are each statistically independent and each Gaussian. The structure of the velocity field is found to be similar to that computed by direct numerical simulation with the same spectrum, although this simulation underestimates the lengths of tubes of intense vorticity.Some new results and concepts have been obtained using this kinematic simulation: (a) for the inertial subrange (which cannot yet be simulated by other means) the simulation confirms the form of the Eulerian frequency spectrum , where e,U0,ω are the rate of energy dissipation per unit mass, large-scale r.m.s. velocity, and frequency. For isotropic Gaussian large-scale turbulence at very high Reynolds number, CE ≈ 0.78, which is close to the computed value of 0.82; (b) for an observer moving with the large eddies the ‘Eulerian—Lagrangian’ spectrum is ϕEL11 = CELeω−2, where CEL ≈ 0.73; (c) for an observer moving with a fluid particle the Lagrangian spectrum ϕL11 = CLeω−2, where CL ≈ 0.8, a value consistent with the atmospheric turbulence measurements by Hanna (1981) and approximately equal to CEL; (d) the mean-square relative displacement of a pair of particles 〈Δ2〉 tends to the Richardson (1926) and Obukhov (1941) form 〈Δ2〉 = GΔet3, provided that the subrange extends over four decades in energy, and a suitable origin is chosen for the time t. The constant GΔ is computed and is equal to 0.1 (which is close to Tatarski's 1960 estimate of 0.06); (e) difference statistics (i.e. displacement from the initial trajectory) of single particles are also calculated. The exact result that Y2 = GYet3 with GY = 2πCL is approximately confirmed (although it requires an even larger inertial subrange than that for 〈Δ2〉). It is found that the ratio [Rscr ]G = 2〈Y2〉/〈Δ2〉≈ 100, whereas in previous estimates [Rscr ]G≈ 1, because for much of the time pairs of particles move together around vortical regions and only separate for the proportion of the time (of O(fc)) they spend in straining regions where streamlines diverge. It is estimated that [Rscr ]G ≈ O(fc−3). Thus relative diffusion is both a ‘structural’ (or ‘topological’) process as well as an intermittent inverse cascade process determined by increasing eddy scales as the particles separate; (f) statistics of large-scale turbulence are also computed, including the Lagrangian timescale, the pressure spectra and correlations, and these agree with predictions of Batchelor (1951), Hinzc (1975) and George et al. (1984).

Constructing stream surfaces in steady 3D vector fields

Abstract: A streamline is a curve which is everywhere tangent to a fluid velocity field. A stream surface is the locus of an infinite set of such curves, rooted at every point along a continuous originating line segment, or rake. A stream surface may be approximated by the triangular tiling of adjacent pairs of integrated streamlines. Such a surface may be refined by repeatedly splitting the widest of the ribbons by the insertion of new curves.A more efficient method begins with a discretization of the rake. These particles are repeatedly advanced a short distance through the flow field. New polygons are appended to the downstream edge of the surface. The spacing of the particles is adjusted to maintain an adequate sampling across the width of the growing surface. This approach more efficiently accesses the flow field volume data and produces a better distribution of points over the two dimensions of the stream surface.

Two-Dimensional, Nonequilibrium, Wet-Steam Calculations for Nozzles and Turbine Cascades

Abstract: The paper describes a method of computing nonequilibrium, steady flows of wet steam in two- and quasi-three-dimensional turbine cascades. The mixture conservation equations are solved in an Eulerian reference frame using an inviscid time-marching method that includes the effects of the centrifugal and Coriolis acceleration terms in rotating blade rows. Nucleation and growth of water droplets are computed by integrating the relevant equations along true streamlines in a Lagrangian reference frame. Steam properties are computed using equations that display commercial steam table accuracy for pressures below 10 bar. Special procedures for grouping the range of droplet sizes present are described that allow an accurate representation of the droplet size distribution to be retained without requiring a large increase in CPU time. All types of wet-steam flow, including those involving secondary nucleation, can be computed. Examples are presented that display the sensitivity of the calculation procedure in computing nucleation affected by the shock- and expansion-wave structure in the region of a turbine blade trailing edge. Typical CPU time requirements for nonequilibrium solutions involving primary or secondary nucleations are about three times those for perfect gas calculations.

Experiments on mixing in continuous chaotic flows

Abstract: We present the design and operation of a flow apparatus for investigations of mixing in time-periodic and spatially periodic chaotic flows. Uses are illustrated in terms of two devices operating in the Stokes regime: the partitioned-pipe mixer, a spatially periodic system consisting of sequences of flows in semicircular ducts, and the eccentric helical annular mixer, a time-periodic velocity field between eccentric cylinders with a superposed Poiseuille flow; other mixing flows can be implemented with relative ease. Fundamental differences between spatially periodic and time-periodic duct flows are readily apparent. Steady spatially periodic systems show segregated KAM-tubes coexisting with chaotic advection; such tubes are remarkably stable under a variety of experimental conditions. Time-periodic duct flows lead to complex streakline structures; since regular regions in the cross-sectional flow move through space, a streakline can find itself injected in a regular domain for some time then be trapped in a chaotic region, and so on, leading to ‘intermittent’ behaviour.

Visualizing three-dimensional flow with simulated streamlines and three-dimensional phase-contrast MR imaging.

Abstract: Three-dimensional (3D) velocity maps acquired with 3D phase-contrast magnetic resonance (MR) imaging contain information regarding complex motions that occur during imaging. A technique called simulated streamlines, which facilitates the display and comprehension of these velocity data, is presented. Single or multiple seed points may be identified within blood vessels of interest and tracked through the velocity field. The resulting trajectories are combined with a 3D MR angiogram and displayed with 3D volume visualization software. Mathematical analysis highlights potential applications and pitfalls of the technique, which was implemented both in phantoms and in vivo with excellent results. For example, single streamlines reveal helical flow patterns in aneurysms, and multiple streamlines seeded in the common carotid artery reveal branch filling-time relationships and slow filling of the carotid bulb. The technique is helpful in understanding these complex flow patterns.

Instability of bounded flows with elliptical streamlines

Abstract: In connection with the recent investigations of the instability of unbounded elliptical flows, some methods are discussed for the study of the instability of bounded flows. The stability of a ‘basic flow’ which is two-dimensional and rotating, with elliptical streamlines similar to the elliptical section of an experimentally studied cavity, is investigated in the framework of linear theory (for circular rotation, the flow discussed is stable). The regions of instability for three-dimensional disturbances are found in the plane of the parameters defining the geometry of the system (the height of the ellipsoidal cavity and the degree of ellipticity). It is shown that two types of instability exist, characterized by either monotone or oscillatory growth of the amplitudes of small disturbances.The influence of the Coriolis force field on this instability mechanism is also studied. Rotation of the system as a whole changes the regions of instability in parameter space characterizing the geometry of the cavity and the wavenumbers of unstable disturbances. As a result, the Coriolis force may stabilize or destabilize the basic flow for a given geometry.The instability of rotating density-stratified flow with elliptical streamlines is also considered.

A 3-D streamline tracking algorithm using dual stream functions

Abstract: A new methodology has been developed for constructing streamlines and particle paths in numerically generated fluid velocity fields. A graphical technique is used to convert the discretely defined flow within a cell into one represented by two three-dimensional stream functions. Streamlines are calculated by tracking constant values of each stream function, a process which corresponds to finding the intersection of two stream surfaces. The tracking process is mass conservative and does not use a time stepping method for integration, thus eliminating a computationally intensive part of traditional tracking algorithms. The method can be applied generally to any three-dimensional compressible or incompressible steady flow. Results presented here compare the performance of the new method to the most commonly used scheme and show that calculation times can be reduced by an order of magnitude.

Nonlinear analysis of flow in an elastic tube (artery): steady streaming effects

Abstract: We analyse the nonlinear flow of a Newtonian fluid in an elastic tube when subjected to an oscillatory pressure gradient with motivation from the problem of blood flow in arteries. Two parameters: the unsteadiness, α = R 0 (ω/ν) ½ and the diameter variation, e = ( R max − R 0 )/ R 0 , are important in characterizing the flow problem. The diameter variation (e) is taken to be small so that the perturbation method is valid, and asymptotic solutions for two limiting cases of the steady-streaming Reynolds number, R s = (αe) 2 (either small or large), are derived. The results indicate that nonlinear convective acceleration induces finite mean pressure gradient and mean wall shear rate even when no mean flow occurs. The magnitude of this effect depends on the amplitude of the diameter variation and the flow rate waveforms and the phase angle difference between them, which can be related to the impedance (pressure/flow) phase angle. Changes in the impedance phase angle, which is indicative of the degree of wave reflection, can change the direction of the induced mean flow. It is also shown that the induced mean wall shear rate is proportional to α when α is large. In addition, it is observed that the steady flow structure in the core can be influenced by wave reflection. The streamlines in the core are always parallel to the tube wall when there is no reflection. However, with total reflection, the induced mean flow recirculates between the nodes and points of maximum amplitude in a closed streamline pattern. Implications of the steady-streaming phenomena for physiological flow applications are discussed in a concluding section.

Birefringent pipes: the steady flow of a dilute polymer solution near a stagnation point

Abstract: In steady flows at high Deborah numbers high polymer stresses are often concentrated within thin boundary layers along streamlines downstream of flow stagnation points where the polymer extension is large. The layers appear as birefringent lines in optical experiments. Detailed observations of the flow near a stagnation point have shown a complex sequence of birefringence structures, which appear as the flow rate increases, for polymer concentrations above some critical value. The first transition is from a solid birefringent line to a hollow birefringent cylinder or 'pipe'. In this paper we calculate the modification of the flow due to the presence of polymer for a FENE (finitely extensible non-linear elastic) dumbbell model with non-linear hydrodynamic friction, and demonstrate that the associated reduction in strain rate at the stagnation point can be sufficient to produce a pipe structure. The polymer concentrations required to produce this transition are found to be in qualitative agreement with experiment. We determine also the thickness of birefringent strands as a function of polymer concentration, molecular weight, flow rate and inertia. These results too are found to be in qualitative agreement with experiment. We show finally that for a FENE model with constant hydrodynamic friction birefringent strands are produced, but we do not find pipes at realistic values of the parameters.

Experimental studies to define the geometry of the flow convergence region. Laser Doppler particle tracking and color Doppler imaging.

Abstract: The color flow convergence method for calculating volume flow through regurgitant or forward flow restrictive orifices has gained significant interest and a number of in vitro studies have suggested that this method is accurate, even in pulsatile models. Clinical application of the method over a wide range of conditions will require improved understanding of the effect of orifice size, flow geometry, and flow rate on the flow convergence geometry. In this study, we performed laser particle tracking investigations to allow visualization of streamlines into stenotic orifices. These streamlines are theoretically perpendicular to the isovelocity surfaces used for flow convergence calculations. We compared those observations to color flow map, flow convergence images obtained with a Toshiba 160A for orifices 5 to 15 mm 2 with flow rates of 1.5 to 9.7 L/min. Our results show that for large orifices, low flow rates, and/or low pressure gradients, more oblique streamlines in the velocity of the orifice correspond to nonhemispherical, but more elliptical, flow convergence geometries. This can be corrected for by using lower Nyquist limits and calculating flow convergence at greater distances from the orifice. Under high flow and high gradient conditions, increased Nyquist limits and shorter aliasing radii are more suitable. Our studies yield insights into flow convergence geometry and yield corrective procedures to improve volume flow calculation.

Streamlines, vorticity lines, and vortices around three-dimensional bodies

Abstract: The properties of three-dimensional, steady, vortical flows are studied using both theoretical analysis and computed flowfields. The centerline of the vortex is described using minimum streamline curvature, maximum normalized helicity, and the edge of a separation sheet. Analysis indicates that several criteria must be met for these three descriptions of the centerline to agree. In certain regions of the flow, computed flowfields indicate that they do agree, that the velocity and vorticity fields are aligned at the centerline, and that extrema in the velocity magnitude, vorticity magnitude, pressure, and density occur at the centerline

An Engineering Aerodynamic Heating Method For Hypersonic Flow

Abstract: A capability to calculate surface heating rates has been incorporated in an approximate three-dimensional inviscid technique. Surface streamlines are calculated from the inviscid solution, and the axisymmetric analog is then used along with a set of approximate convective-heating equations to compute the surface heat transfer. The method is applied to blunted axisymmetric and three-dimensional ellipsoidal cones at angle of attack for the laminar flow of a perfect gas. The method is also applicable to turbulent and equilibrium-air conditions. The present technique predicts surface heating rates that compare favorably with experimental (ground-test and flight) data and numerical solutions of the Navier-Stokes (NS) and viscous shock-layer (VSL) equations. The new technique represents a significant improvement over current engineering aerothermal methods with only a modest increase in computational effort.

Comparison of Flow Models Used to Delineate Capture Zones of Wells

Abstract: Analytical, semianalytical, and numerical flow models of a leaky-confined fractured-carbonate aquifer were used with particle-tracking/stream-function programs to delineate traveltime-related capture zones of a municipal wellfield and to assess “model” errors imposed by requisite assumptions associated with each flow model. The analytical flow model uses the Hantush-Jacob equation describing two-dimensional transient drawdown surrounding a well in a leaky-confined aquifer with superposition of a nonuniform regional flow field. The semianalytical flow model employs the Thiem equation describing two-dimensional steady-state drawdown surrounding a well in a fully confined aquifer with superposition of a uniform regional flow field. The numerical flow model uses a finite-difference solution describing three-dimensional steady-state flow and incorporates five model layers, specified-flux boundary conditions, spatially variable recharge, and recharge from a lake. Goodness-of-fit statistics show that the semianalytical flow model does not calibrate as well to measured heads as do the analytical and numerical flow models. Model errors caused by the inability of the semianalytical flow model to account for vertical leakage and by the required use of a uniform regional flow field cause the size of the five-year capture zone to be overestimated and its shape to be too simplistic compared to those of the analytical and numerical flow models. Polar plots of reverse-tracked pathline endpoints suggest that model error also produces significant differences in pathline trajectories. The results show that in this type of hydrogeologic setting, it is necessary to use a flow model that accounts for vertical leakage and for the nonuniform character of the regional flow field.

A Numerical Study of the Shape of the Surface Separating Flow Into Branches in Microvascular Bifurcations

Abstract: The shape of the separating surface formed by the streamlines entering the branches of microvascular bifurcations plays a major role in determining the distribution of red blood cells and other blood constituents downstream from the bifurcation. Using the finite element method, we determined the shape of the surface through numerical solution of three dimensional Navier-Stokes equations for fluid flow at low Reynolds numbers in a T-type bifurcation of circular tubes. Calculations were done for a wide range of daughter branch to parent vessel diameter ratios and flow ratios. The effect of Reynolds number was also studied. Our numerical results are in good agreement with previously reported experimental data of Rong and Carr (Microvascular Research, Vol. 39, pp. 186-202, 1990). The numerical results of this study will be used to predict the concentration of blood constituents downstream from microvascular bifurcations providing that the inlet concentration profile is known.

An Experimental Study of Subcooled Film Boiling on a Vertical Surface—Hydrodynamic Aspects

Abstract: The aim of the present study is: (1) To determine the physical characteristics of the interfacial waves for different wall superheats and liquid subcoolings. (2) To determine the velocity field adjacent to the interface at different locations along the interface. Interface and liquid velocities near the leading edge of a vertical wall 6.3 cm wide and 10.3 cm high were measured during subcooled film boiling of water at 1 atm pressure. The interface and liquid velocities in the boundary layer adjacent to the interface were measured using the hydrogen bubble flow visualization method. Photographs taken from the front and side showed the existence of a finite vapor layer at the leading edge and the existence of ripples and large-amplitude waves (bulges) on the interface. The bulges and ripples did not slide on the interface but moved in unison with the interface. The wave amplitude and wavelength were also measured. For a given subcooling and wall superheat, the amplitude, the interfacial velocity, and the wavelength were found to attain an equilibrium value several millimeters downstream of the leading edge. The waves were highly nonlinear and the interface velocities, which are found to be governed by the wave amplitude, were much largermore » than those predicted from the smooth interface, laminar flow theory. Streamlines in the liquid were found to expand into the wave valleys. At the wave peaks the streamlines appeared to be clustered together and the measured interface velocity gradients were high. The overall picture is one of expansion in the wave valleys and contraction (of flow) at the wave peaks. The flow field in turn is found to affect the liquid side heat transfer in subcooled film boiling significantly.« less

Investigation of oblique shock/boundary-layer bleed interaction

Abstract: The detailed flowfield characteristics in an oblique shock-wave/laminar-boundary-layer interaction with bleed were investigated. The numerical solution for the flowfield was obtained for the strong conservation-law form of the two-dimensional compressible Navier-Stokes equations using an implicit scheme. The computations mod- eled the flow in the interaction region and inside the bleed slot for an impinging oblique shock on a flat-plate boundary layer. The computed results for the streamlines and the pressure and Mach number contours inside the bleed slot indicate that the flow is choked in the slot, with a recirculation zone near the upstream slot corner. The bleed results in the interaction zone demonstrate that flow separation is controlled. The interaction length . is reduced and the downstream velocity profiles are more favorable than the separated flow results at the same shock strength without bleed. HE control of shock/boundary-layer interactions in inlets and nozzles and over vehicle surfaces is accomplished through bleed and/or blowing in the interaction zone. In the case of mixed compression supersonic inlets, the bleed system design is critical to the efficient and stable operation of the system. Hamed and Shang1 reviewed the existing experimen- tal data for shock-wave/boundary-layer interactions in super- sonic inlets and other related configurations. According to this survey, most of the experimental measurements in mixed compression supersonic inlets consisted of total pressure re- covery surveys at the engine face and static pressure distri- butions over the inner surfaces. In the few cases involving velocity profile measurements,2-3 the latter were obtained up- stream and downstream of the interactions. Comparisons of internal flow computational results3'5 with the experimental measurements in supersonic inlets2-3 revealed reasonable agreement between the computed and measured surface pres- sures upstream of the ramp bleed. However, discrepancies in the predicted shock locations and velocity profiles were ob- served downstream of shock/boundary-layer interactions with bleed. There is enough experimental evidence6'11 to indicate that local bleed can control flow separation in shock-wave/bound- ary-layer interactions. There are disagreements,1 however, among the different experimental studies regarding the effects of bleed hole size,7-8 and the location of the bleed holes in relation to the shock.6-9'11 The experimental data in these studies are not sufficient, however, to resolve these discrep- ancies. Strike and Rippy9 measured the surface pressure in the interaction zone of an oblique shock wave impinging a tur- bulent boundary layer over a flat plate, with suction. They determined that less suction is required to control separation, when applied upstream of the shock. Seebaugh and Childs11 investigated experimentally the axisymmetric flow in the in- teraction region of the boundary layer inside a duct. Contrary to the conclusions of Strike and Rippy,9 suction within the

Rayleigh-Bénard convection in a small box: spatial features and thermal dependence of the velocity field

Abstract: An experimental study of the spatial features of Rayleigh-Benard convection in a sall box is presented. Experiments are carried out in a rectangular cell (aspect ratios Γx = 2.03, Γy = 1.19) filled with silicone oil (Prandtl number, Pr = 130) for different Rayleigh numbers, Ra (up to Ra = 75Rac, Rac1707). The basic structure of the flow field for this range of Ra consists of two rolls with their axes parallel to the shorter horizontal side. Both senses of rotation for the rolls are observed, corresponding to the two branches of the bifurcation. Particle image velocimetry, with a 5 mW He-Ne laser as the illuminating source, is used to measure the velocity field in the midplane of the cell. From it the vorticity field (out of plane component) and two-dimensional streamlines are calculated. The flow has been measured to be three-dimensional, even for very low Ra, owing to the sidewall influence. The spatial features of the flow are shown to be dependent on both Ra and the sense of rotation of the rolls. Finally, a Fourier analysis of the velocity field is presented. The spatial and thermal dependences of the different Fourier terms are reported. The velocity field, in a first-order approximation, is quantitatively described.

Ageostrophic Pseudovorticity and Geostrophic C-Vector Forcing—A New Look at the Q Vector in Three Dimensions

Abstract: By combining the two Q-vector component equations with the third quasigeostrophic (QG) diagnostic equation (the vertical ageostrophic vorticity equation) a complete set of QG diagnostic equations is formed in a three-dimensional vector form with the ageostrophic pseudovorticity vector on the left-hand side and a newly defined geostrophic forcing vector (the C vector) on the right-hand side. The horizontal projection of the C vector is a rotated Q vector (by 90° to the right). The vertical C-vector component is proportional to the Gaussian curvature of the geopotential surface of constant pressure. Since C-vector streamlines can be viewed as ageostrophic pseudovortex lines, ageostrophic circulations can be easily inferred through three-dimensional “vorticity thinking,” which considers both the boundary effect and moist processes. The C vector is interpreted physically in terms of generation of Coriolis force curl and buoyancy curl due to the geostrophic advection alone. The basic techniques and po...

Comparison of Methods Used to Delineate Capture Zones of Wells: 2. Stratified‐Drift Buried‐Valley Aquifer

Abstract: Analytical, semianalytical, and numerical models of the flow system in a stratified-drift buried-vailey aquifer were used with particle-tracking/stream-function programs to delineate traveltime-related capture zones of a municipal wellfield and to assess conceptual errors imposed by requisite assumptions associated with each flow model. The analytical flow model uses the Theis equation describing two-dimensional transient drawdown surrounding a well in a fully confined aquifer and superposition of a nonuniform regional flow field. The semianalytical flow model implicitly uses the Thiem equation describing two-dimensional steady-state drawdown surrounding a well in a fully confined aquifer and superposition of a uniform regional flow field. The numerical flow model uses a three-dimensional steady-state finite-difference solution and incorporates four model layers, specified-flux and head-dependent flux boundary conditions, spatially variable recharge, and head-dependent leakage to and from streams. The predictive ability of the flow models is based on comparison of mean absolute errors and root mean squared errors between measured and simulated heads as well as comparison of pathline distributions defining the one-year capture zones. The comparisons show that the uniform flow field and uniform transmissivity required of the semianalytical flow model cause it to be less accurate than the analytical and numerical flow models. The inability of the analytical flow model to incorporate spatial variations in transmissivity causes it to be less accurate than the numerical flow model. These conceptual errors associated with use of the analytical and semianalytical flow models in this hydrogeologic setting cause their one-year capture zones to be less reasonable than those from the numerical flow model.

Method for visualizing streamlines around hypersonic vehicles by using electrical discharge

Abstract: Floryan, J. M., and Dallmann, U., "Flow over a Leading Edge with Distributed Surface Roughness," Journal of Fluid Mechanics, Vol. 216, Dec. 1990, pp. 629-656. Stuart, J. T., "The Viscous Flow near a Stagnation Point When External Stream Has Uniform Vorticity," Journal of the Aero/Space Sciences, Vol. 26, No. 2, 1959, pp. 124-125. Tamada, K., *'Two-Dimensional Stagnation-Point Flow Impinging Obliquely on a Plane Wall," Journal of the Physical Society of Japan, Vol. 46, No. 1, 1979, pp. 310-311. Dorrepaal, J. M., "An Exact Solution of the Navier-Stokes Equation Which Describes Nonorthogonal Stagnation Point Flow in Two Dimensions," Journal of Fluid Mechanics, Vol. 163, Feb. 1986, pp. 141-147. Hall, P., Malik, M. R., and Poll, I. A., "On the Stability of Infinite Swept Attachment Line Boundary Layer," Proceedings of the Royal Society of London, Series A, Vol. A395, No. 1809, Oct. 1984 pp. 229-245.

On the simulation of viscoelastic flow past a sphere using spectral methods

Abstract: The flow past a sphere in an infinite expanse of viscoelastic fluid is simulated numerically using a pseudospectral technique. The problem is formulated using primitive variables. A time-splitting scheme is used that ensures that the velocity field is divergence-free at the end of each time step. The flow domain is treated without truncation using rational Chebyshev polynomials as basis functions in the radial direction. The drag on the sphere is computed for a range of values of the non-dimensional elasticity number We . A quantitative comparison of the drag coefficient with that obtained by a perturbation analysis is given. As We is initially increased, a reduction in the drag from its Newtonian value is observed. For larger values of We , drag enhancement is detected. In addition, for small values of We the streamlines are shifted slightly downstream in agreement with experimental observations. The numerical algorithm is described in detail and the numerical results compared qualitatively with experiments.

Chaos Associated with Fluid Inertia

Abstract: The low Reynolds number flow between two concentric steadily rotating spheres is considered. The pattern of streamlines is explained in terms of an adiabatic invariant. It is shown that, when the two rotation vectors are not parallel, the streamlines become chaotic when the Reynolds number Re is increased and the onset of global chaos occurs near Re = 20.

Numerical simulation of viscoelastic flow in two-dimensional L-shaped channels

Abstract: Numerical simulation of the flow of a White–Metzner type liquid in two‐dimensional L‐shaped channels has been carried out using the finite difference technique. Results are compared with experimental data obtained in previous work. Numerical predictions are in agreement with the experiment for velocity and pressure distributions. Further considerations are given to streamlines and stress component profiles. A streamline shift toward the outer wall accompanied by reverse flow occurs in the upstream vicinity of the re‐entrant corner. It is shown that the fluid is subjected to stretching near the re‐entrant corner, where reversal of flow occurs. Effects of inertia and elasticity are addressed by presenting the size of the reverse flow region. Elasticity has the effect of increasing its size, which is opposite to an inertial effect.