Showing papers on "Velocity gradient published in 2002"

Dynamics of singularity surfaces for compressible, viscous flows in two space dimensions

Abstract: We prove the global existence of solutions of the Navier-Stokes equations of compressible, barotropic flow in two space dimensions which exhibit convecting singularity curves. The fluid density and velocity gradient have jump discontinuities across these curves, exactly as predicted by the Rankine-Hugoniot conditions, and these jump discontinuities decay exponentially in time, more rapidly for smaller viscosities. The singularity curves remain C1+α despite the fact that the velocity fields which convect them are not continuously differentiable. © 2002 Wiley Periodicals, Inc.

Free surface flows of polymer solutions with models based on the conformation tensor

Abstract: A computational method is presented for analyzing free surface flows of polymer solutions with the conformation tensor. It combines methods of computing Newtonian free surface flows [J. Comp. Phys. 99 (1992) 39; V.F. deAlmeida, Gas–Liquid Counterflow Through Constricted Passages, Ph.D. thesis, University of Minnesota, Minneapolis, MN, 1995 (Available from UMI, Ann Arbor, MI, order number 9615160); J. Comp. Phys. 125 (1996) 83] and viscoelastic flows [J. Non-Newtonian Fluid Mech. 60 (1995) 27; J. Non-Newtonian Fluid Mech. 59 (1995) 215]. Modifications are introduced to compute a traceless velocity gradient, to impose inflow boundary conditions on the conformation tensor that are independent of the specific model adopted, and to include traction boundary conditions at free surfaces and open boundaries. A new method is presented for deriving and coding the entries of the analytical Jacobian for Newton’s method by keeping the derivatives of the finite element weighted residual equations with respect to the finite element basis functions in their natural vector and tensor forms, and then by mapping such vectors and tensors into the elemental Jacobian matrix. A new definition of extensional and shear flow is presented that is based on projecting the rate of strain tensor onto the principal basis defined by the conformation tensor. The method is validated with two benchmark problems: flow around a cylinder in a channel, and flow under the downstream section of a slot or knife coater. Regions of molecular stretch—determined by monitoring the eigenvalues of the conformation tensor—and molecular extension and shear rate—determined by projecting the rate of strain dyadic onto the eigenvectors of the conformation tensor—are shown in the flow around a cylinder of an Oldroyd-B liquid. The free surface coating flow between a moving rigid boundary and a parallel static solid boundary from which a free surface detaches is analyzed with several models of dilute and semidilute solutions of polymer of varying degree of stiffness based on the conformation tensor approach [J. Non-Newtonian Fluid Mech. 23 (1987) 271; A.N. Beris, B.J. Edwards, Thermodynamics of Flowing Systems with Internal Microstructure, 1st ed., Oxford University Press, Oxford, 1994; J. Rheol. 38 (1994) 769; M. Pasquali, Polymer Molecules in Free Surface Coating Flows, Ph.D. thesis, University of Minnesota, Minneapolis, MN, 2000 (Available from UMI, Ann Arbor, MI, order number 9963019)]. Although the boundary conditions at the static contact line introduce a singularity, that singularity does not affect the computation of flows at high Weissenberg number when a recirculation is present under the static boundary. In this case, a steep layer of molecular stretch develops under the free surface downstream of the stagnation point. Here the polymer aligns with its principal stretching axis parallel to the free surface. When the recirculation is absent, the singularity at the contact line strongly affects the computed velocity gradient, and the computations fail at moderate Weissenberg number irrespective of the polymer model and of whether the polymer is affecting the flow or not.

Flow modelling in compound channels : momentum transfer between main channel and prismatic or non-prismatic floodplains

Abstract: Flow modelling in a compound channel is a complex matter. Indeed, due to the smaller velocities in the floodplains than in the main channel, shear layers develop at the interfaces between these subsections, and the channel conveyance is affected by a momentum transfer corresponding to this shear layer, but also to possible geometrical changes in a non-prismatic reach. In this work, a one-dimensional approach, the Exchange Discharge Model (EDM), is proposed for such flows. The EDM accounts for the momentum transfer between channel subsections, estimated as proportional to the velocity gradient and to the discharges exchanged through the interface; where two main processes are identified : (1) the turbulent exchange, due to the shear-layer development; and (2) the geometrical transfer, due to cross-sectional changes. The EDM is successfully validated for discharge prediction, but also for water-profile computation, through comparison with existing laboratory and field measurements. The momentum transfer due to turbulent exchanges is then studied experimentally, theoretically and numerically. At first, new experimental data, obtained by using Particle Tracking Velocimetry techniques, are presented : the periodical vortex structures that develop in the shear layer are clearly identified and characterised. Secondly, a hydrodynamic linear stability analysis enables to predict quite successfully the wave length of some observed vortices. Lastly, an Unsteady-RANS numerical method is used to simulate the perturbation development. The estimated vortex wave lengths agree again with the measurements and the theoretical predictions, although vortices merging occurs in the simulation results, which was actually not observed experimentally. The velocity-profile prediction is found improved when the effect of vortices is considered, thanks to the corresponding additional shearing. The geometrical transfer is also investigated experimentally and numerically. Novel experiments are designed, with the measurements of the flow in a compound channel with symmetrically narrowing floodplains. The mass transfer and the evolution of the flow distribution along the channel length are clearly observed. A significant additional head loss due to this transfer is measured, in accordance with the EDM hypothesis. Measured water profiles are finally compared successfully with the EDM predictions. In addition to the EDM development and validation, the so-called Lateral Distribution Method (LDM) is also investigated and the significance of the secondary-currents models proposed by previous authors for this method is discussed. When considering the velocity-profile prediction, the effect of these helical secondary currents is again clearly highlighted, by using dispersion terms in the Saint-Venant equations. However, the actual physical meaning of the related dispersion coefficients remains uncertain. In addition, an extended LDM is also proposed and discussed for non-prismatic flow modelling, using the new narrowing-channel data set.

Measurement of the full shear-induced self-diffusion tensor of noncolloidal suspensions

Abstract: The full diffusion tensor of shear-induced self-diffusion has been measured experimentally for the first time. In addition to the well-known components in the velocity gradient, Dyy, and vorticity direction, Dzz, the coefficients Dxx and Dxy have been determined for concentrated suspensions of noncolloidal hard spheres as a function of particle volume fraction. Owing to the shear-induced nature of the phenomenon, these four coefficients are the only nonzero elements of the diffusion tensor. The newly determined diffusion quantities have been obtained by extending our correlation based technique [J. Fluid Mech. 375, 297 (1998); Phys. Rev. E 63, 021403 (2001)] with a method to subtract convective displacements due to the shear flow. The diffusion in the velocity direction, Dxx, is almost an order of magnitude larger than the other components and the only nonzero off-diagonal component, Dxy, is negative and small compared to the diagonal components of the diffusion tensor. In principle the applied technique is also feasible for measuring other anisotropic diffusion mechanisms, e.g., Brownian diffusion in steady shear flow.

Hydrodynamics study in a plane ultrafiltration module using an electrochemical method and particle image velocimetry visualization

Abstract: The wall shear stress is determined at the surface of a plane sheet of Plexiglas, taking the place of a membrane, using an electrochemical method. Several microelectrodes are mounted flush to this plane plate, and maps of shear stress are determined for two inlet and outlet configurations and three channel heights. The heterogeneity of the wall shear stress is observed for both configurations. Furthermore, the study of the turbulence features of the flow shows a decreasing fluctuating rate of velocity gradient when the channel height is decreased. The wall velocity gradients and turbulent intensity rates analysis are confirmed by a flow visualization using the particle image velocimetry method.

Theoretical interpretation of the toroidal rotation velocity observed in Alcator C-mod Ohmic H-mode discharges

Abstract: It is shown that neoclassical theory explains quite well the origin of the co-current toroidal rotation velocity measured in the core of stationary Alcator C-Mod edge localized mode (ELM)-free Ohmic high confinement (H)-mode discharges. Both edge and core toroidal rotation velocity profiles are determined to a good approximation by the edge ion temperature and density pedestals, where the gradients are large and the plasma is in the high collisionality regime. Under these conditions, the predicted radial electric field profile is similar to those measured in the DIII-D tokamak whereas the usual expression for the poloidal velocity is modified by finite Larmor radius (FLR) effects. Over the entire plasma cross section, the expression of the toroidal velocity can approximately be cast as the product of a dimensionless non-local functional of the pedestal normalized profiles Ti(r)/Ti(rinf) and Ni(r)/Ni(rinf) with powers of the plasma density, temperature, safety factor and magnetic field at the pedestal inflexion point rinf provided the FLR related corrections are independent of the latter parameters. The collapse of the core toroidal rotation velocity when either an internal transport barrier forms (that leads to impurity accumulation), or the plasma experiences a transition from the H- to the low confinement (L)-mode, or ELMs appear, and the spin up at the L-H transition are also explained. In the edge region, power balance is consistent with the prediction from sub-neoclassical ion energy transport theory at high collisionality. The role of charge exchange neutrals is discussed and the critical density above which they are expected to noticeably slow down the rotation is estimated. The toroidal velocity gradient predicted by theory at the edge of the ELM-free Ohmic H-mode discharge mainly under study (qs = 3.4) is near the onset value for the Kelvin-Helmholtz (K-H) parallel velocity shear (PVS) instability; this result is very interesting since a transition from ELM-free to enhanced D? (EDA) H-modes occurs at q3.5-4; the PVS K-H instability appears to have the characteristics of the `quasi-coherent' mode that is present in all EDA plasmas, but not in ELM-free H-modes.

Layered droplet microstructures in sheared emulsions: finite-size effects.

Abstract: We investigate the influence of confinement on the steady state microstructure of emulsions sheared between parallel plates, in a regime where the average droplet dimension is comparable to the gap width between the confining walls. Utilizing droplet velocimetry, we find that the droplets can organize into discrete layers under the influence of shear. The number of layers decreases from two (at relatively higher shear rates) to one (at lower shear rates), as the drops grow slightly larger due to coalescence. We argue that the layering and overall composition profile may be controlled by the interplay of droplet collisions (which can cause separation of droplet centers in the velocity gradient direction), droplet migration toward the centerline (due to wall effects), and droplet packing constraints. We also study the effects of mixture composition on droplet microstructure, and summarize these results in the form of a morphology diagram in the parameter space of mass fraction and shear rate. We find that formation of strings of the suspended phase (reported earlier by our group in flow-visualization studies on confined emulsions) is observed over a broad composition window. We also find a stable (nontransient) morphology wherein the droplets are arranged in highly ordered pearl-necklace chain structures.

Stability of wall modes in fluid flow past a flexible surface

Abstract: The stability of wall modes in fluid flow past a flexible surface is analyzed using asymptotic and numerical methods. The fluid is Newtonian, while two different models are used to represent the flexible wall. In the first model, the flexible wall is modeled as a spring-backed, plate-membrane-type wall, while in the second model the flexible wall is considered to be an incompressible viscoelastic solid of finite thickness. In the limit of high Reynolds number (Re), the vorticity of the wall modes is confined to a region of thickness O( Re −1/3 ) in the fluid near the wall of the channel. An asymptotic analysis is carried out in the limit of high Reynolds number for Couette flow past a flexible surface, and the results show that wall modes are always stable in this limit if the plate-membrane wall executes motion purely normal to the surface. However, the flow is shown to be unstable in the limit of high Reynolds number when the wall can deform in the tangential direction. The asymptotic results for this case are in good agreement with the numerical solution of the complete governing stability equations. It is further shown using a scaling analysis that the high Reynolds number wall mode instability is independent of the details of the base flow velocity profile within the channel, and is dependent only on the velocity gradient of the base flow at the wall. A similar asymptotic analysis for flow past a viscoelastic medium of finite thickness indicates that the wall modes are unstable in the limit of high Reynolds number, thus showing that the wall mode instability is independent of the wall model used to represent the flexible wall. The asymptotic results for this case are in excellent agreement with a previous numerical study of Srivatsan and Kumaran.

Free‐surface turbulence and mass transfer in a channel flow

Abstract: Free-surface turbulence in a fully developed, open-channel flow was measured for Reynolds numbers of 8,500–45,000. An analysis method of the 2-D divergence on the free surface has been developed to extract Hanratty's β values, or the velocity gradient into the free surface, from these measurements. Hanratty's β is the parameter that relates most directly to the turbulence effect on the liquid-film coefficient. Its measurement is a direct measurement of surface renewal. The spatial scales of β were 3 to 5 times smaller than those of the large upwelling events (boils) normally identified as surface renewal. The hypothesis is that the large upwelling events do not have the high-vorticity gradients associated with large β values. Instead, the locations of high-vorticity gradients on the free surface will also create the divergence required for high β values, occurring at the edges of a large upwelling event. Because the β frequency spectrum has properties to characterize the liquid-film coefficient, it was normalized to be determined from its maximum value, the wave number of this maximum value, and a shape factor used to scale the frequency. Measurements of the liquid-film coefficient from prior studies were also used to characterize the liquid-film coefficient by measured β values for this nonsheared surface. The larger β scales predominantly influence the liquid-film coefficient, in contrast to a previous study of a shear-free surface published by McCready et al. in 1986, where all β frequencies were equally important. Generally, higher frequency turbulence is more significant at a sheared water surface than at a water surface with minimal shear stress.

Global-in-time Hölder continuity of the velocity gradients for fluids with shear-dependent viscosities

Abstract: For an evolutionary nonlinear fluid model characterized by the viscosity being a decreasing function of the modulus of the symmetric velocity gradient we establish the global-in-time existence of the solution with the Holder continuous velocity gradients. Such a solution is unique in the class of weak solutions. We deal with the two dimensional space periodic problem.

Scaling the inner region of turbulent plane wall jets

Abstract: The present study reports refined laser Doppler anomemetry measurements in a turbulent plane wall jet. The Reynolds numbers based on jet exit conditions are in the range 7,500

Pressure Drops of Water Flow Through Micromachined Particle Filters

Abstract: When the particle is in the order of microns, flow through the small opening produces a large velocity gradient, leading to high viscous dissipation. Understanding the flow field is critical in determining the power requirement. We studied water flow through filters fabricated by micro-electro-mechanical system (MEMS) techniques

Turbulent gas-solid flows, part i: direct simulations and reynolds stress closures

Abstract: This work deals with direct numerical simulation (DNS) and development of a new Reynolds stress model (RSM) for description of gas-solid turbulent flows with low volume fraction and high density ratio. The coupling between the two phases is "two-way," which allows investigation of the effects of the mass loading ratio and the particle time constant on both phases. DNS is conducted of a homogeneous turbulent shear flow laden with monosize particles using a Eulerian-Lagrangian formulation. The RSM is based on a "two-fluid" methodology in which both the carrier phase and the dispersed phase are considered in the Eulerian frame. Closures are suggested for the unclosed terms (including the pressure- velocity gradient) which manifest the effects of two-way coupling. The final model predictions for all the components of the fluid, the particle, and fluid-particle Reynolds stresses are assessed via detailed comparisons against DNS data.

The formation of shear and density layers in stably stratified turbulent flows: linear processes

Abstract: The initial evolution of the momentum and buoyancy fluxes in a freely decaying, stably stratified homogeneous turbulent flow with r.m.s. velocity u′0 and integral lengthscale l0 is calculated using a weakly inhomogeneous and unsteady form of the rapid distortion theory (RDT) in order to study the growth of small temporal and spatial perturbations in the large-scale mean stratification N(z, t) and mean velocity profile u(z, t) (here N is the local Brunt–Vaisala frequency and u is the local velocity of the horizontal mean flow) when the ratio of buoyancy forces to inertial forces is large, i.e. Nl0/u′0[dbl greater-than sign]1. The lengthscale L of the perturbations in the mean profiles of stratification and shear is assumed to be large compared to l0 and the presence of a uniform background mean shear can be taken into account in the model provided that the inertial shear forces are still weaker than the buoyancy forces, i.e. when the Richardson number Ri = (N/[partial partial differential]zu)2[dbl greater-than sign]1 at each height. When a mean shear perturbation is introduced initially with no uniform background mean shear and uniform stratification, the analysis shows that the perturbations in the mean flow profile grow on a timescale of order N-1. When the mean density profile is perturbed initially in the absence of a background mean shear, layers with significant density gradient fluctuations grow on a timescale of order N−10 (where N0 is the order of magnitude of the initial Brunt–Vaisala frequency) without any associated mean velocity gradients in the layers. These results are in good agreement with the direct numerical simulations performed by Galmiche et al. (2002) and are consistent with the earlier physically based conjectures made by Phillips (1972) and Posmentier (1977). The model also shows that when there is a background mean shear in combination with perturbations in the mean stratification, negative shear stresses develop which cause the mean velocity gradient to grow in the density layers. The linear analysis for short times indicates that the scale on which the mean perturbations grow fastest is of order u′0/N0, which is consistent with the experiments of Park et al. (1994). We conclude that linear mechanisms are widely involved in the formation of shear and density layers in stratified flows as is observed in some laboratory experiments and geophysical flows, but note that the layers are also significantly influenced by nonlinear and dissipative processes at large times.

The formation of shear and density layers in stably stratified turbulent flows: linear processes

Abstract: The initial evolution of the momentum and buoyancy fluxes in a freely decaying, stably stratified homogeneous turbulent flow with r.m.s. velocity u′0 and integral lengthscale l0 is calculated using a weakly inhomogeneous and unsteady form of the rapid distortion theory (RDT) in order to study the growth of small temporal and spatial perturbations in the large-scale mean stratification N(z, t) and mean velocity profile u(z, t) (here N is the local Brunt–Vaisala frequency and u is the local velocity of the horizontal mean flow) when the ratio of buoyancy forces to inertial forces is large, i.e. Nl0/u′0[dbl greater-than sign]1. The lengthscale L of the perturbations in the mean profiles of stratification and shear is assumed to be large compared to l0 and the presence of a uniform background mean shear can be taken into account in the model provided that the inertial shear forces are still weaker than the buoyancy forces, i.e. when the Richardson number Ri = (N/[partial partial differential]zu)2[dbl greater-than sign]1 at each height. When a mean shear perturbation is introduced initially with no uniform background mean shear and uniform stratification, the analysis shows that the perturbations in the mean flow profile grow on a timescale of order N-1. When the mean density profile is perturbed initially in the absence of a background mean shear, layers with significant density gradient fluctuations grow on a timescale of order N−10 (where N0 is the order of magnitude of the initial Brunt–Vaisala frequency) without any associated mean velocity gradients in the layers. These results are in good agreement with the direct numerical simulations performed by Galmiche et al. (2002) and are consistent with the earlier physically based conjectures made by Phillips (1972) and Posmentier (1977). The model also shows that when there is a background mean shear in combination with perturbations in the mean stratification, negative shear stresses develop which cause the mean velocity gradient to grow in the density layers. The linear analysis for short times indicates that the scale on which the mean perturbations grow fastest is of order u′0/N0, which is consistent with the experiments of Park et al. (1994). We conclude that linear mechanisms are widely involved in the formation of shear and density layers in stratified flows as is observed in some laboratory experiments and geophysical flows, but note that the layers are also significantly influenced by nonlinear and dissipative processes at large times.

Decorrelation-based blood flow velocity estimation: effect of spread of flow velocity, linear flow velocity gradients, and parabolic flow

Abstract: In recent years, a new method to measure transverse blood flow, based on the decorrelation of the radio frequency (RF) signals has been developed. In this paper, we investigated the influence of nonuniform flow on the velocity estimation. The decorrelation characteristics of transverse blood flow using an intravascular ultrasound (IVUS) array catheter are studied by means of computer modeling. Blood was simulated as a collection of randomly located point scatterers; moving this scattering medium transversally across the acoustical beam represented flow. First-order statistics were evaluated, and the signal-to-noise ratio from the signals were measured. The correlation coefficient method was used to present the results. Three velocity profiles were simulated: random spread of blood-flow velocity, linear blood-flow velocity gradient, and parabolic blood-flow. Radio frequency and envelope signals were used to calculate the decorrelation pattern. The results were compared to the mean decorrelation pattern for plug blood-flow. The RF signals decorrelation patterns were in good agreement with those obtained for plug blood flow. Envelope decorrelation patterns show a close agreement with the one for plug blood flow. For axial blood flow, there is a discrepancy between decorrelation patterns. The results presented here suggest that the decorrelation properties of an IVUS array catheter for measuring quantitative transverse blood flow probably will not be affected by different transverse blood-flow conditions.

Bulk and interfacial shear thinning of immiscible polymers

Abstract: Nonequilibrium molecular-dynamics simulations are used to study the shear-thinning behavior of immiscible symmetric polymer blends. The phase-separated polymers are subjected to a simple shear flow imposed by moving a wall parallel to the fluid-fluid interface. The viscosity begins to shear thin at much lower rates in the bulk than at the interface. The entire shear-rate dependence of the interfacial viscosity is consistent with a shorter effective chain length s(*) that also describes the width of the interface. This s(*) is independent of chain length N and is a function only of the degree of immiscibility of the two polymers. Changes in polymer conformation are studied as a function of position and shear rate. Shear thinning correlates more closely with a decrease in the component of the radius of gyration along the velocity gradient than with elongation along the flow. At the interface, this contraction of chains is independent of N and consistent with the bulk behavior for chains of length s(*). The distribution of conformational changes along chains is also studied. Central regions begin to stretch at a shear rate that decreases with increasing N, while shear induced changes at the ends of chains are independent of N.

Size separation of supermicrometer particles in asymmetrical flow field-flow fractionation. Flow conditions for rapid elution.

Abstract: The performance of lift-hyperlayer asymmetrical flow field-flow fractionation using rapid elution conditions was tested through the separation of standard polystyrene latex particles of diameters from 2 to 20 mum. Optimization of flowrates was studied not only in order to obtain efficient and rapid separation; but also to work under conditions of various shape and steepness of the axial flow velocity gradient. Using extreme flow conditions, the five widely spaced particle sizes, 20.5-, 15.0-, 9.7-, 5.0-, and 2.0-mum diameter, could be resolved in 6 min, whereas for the narrower size range of 20.5-5.0 mum, 1 min was enough. The size selectivity in the size range 9.7-2.0 mum was studied as a function of flowrates and particle size and was found to be constant. A particle trapping device made it possible to separate particles of sizes >10 mum; which has previously proven to be difficult in asymmetrical channels.

Sidewall quenching of laminar premixed flames propagating along the single wall surface

Abstract: To clarify the characteristics of the sidewall flame quenching layer, experiments were performed using a combustion vessel. The results indicate that the thickness of the quenching layer is approximately constant until the flames propagate a certain distance from the leading edge downstream of the quenching wall: however, after that, the thickness rapidly increases with flame propagation. When the flame propagates further, the thickness of the quenching layer again remains constant, maintaining a high value. Regarding this phenomenon, it is hypothesized that when a flame propagates in the velocity boundary layer formed on the wall surface, the flames are stretched by the velocity gradient in the velocity boundary layer, which affects flame quenching: this sidewall quenching phenomenon was analyzed. The results indicate that the burning velocity of the flame propagating in the velocity boundary layer decreases due to flame stretch. Namely, the equation for the quenching Peclet number, P e Q =S·d q 3 /α =constant, holds. Here, S is the burning velocity under the influence of flame stretch, d q ′ is the thickness of the quenching layer, and α is the thermal diffusivity of the mixture. Further, when the flames suffer the cooling effect of the wall, as the flame thickness is greater than that of adiabatic flames, it is estimated that the thickness of the preheat zone also increases. Therefore, the influence of the flame stretch is greater than for an adiabatic flame. In this way, the wall surface affects flame quenching not only by exerting a quenching effect on the flame due to its cooling effect but also by stretching the flame by the velocity gradient in the velocity boundary layer formed on the surface of the wall.

Exact Solution of the Problem on a Six‐Constant Jeffreys Model of Fluid in a Plane Channel

Abstract: An exact analytic solution of the problem of a generalized viscoelastic Jeffreys fluid flow in a plane channel under the action of a pressure gradient is found. The velocity profiles are obtained in a parametric form with a velocity gradient taken as a parameter. The critical values of the pressure gradient are determined, which, when exceeded, lead to weak tangential discontinuities in the longitudinal velocity profile. When the pressure gradient changes smoothly over some range of parameters, a hysteresis loop emerges on the graph of the flow rate versus the pressure gradient.

Temporal behavior of flow in rotating cavities

Abstract: Temporal flow behaviors in a heated rotating cavity with an axial throughflow and sealed annulus are explored. Generally, for both systems, the ratio of the flow angular velocities to the system boundaries are found, even close to boundaries, not equal to unity. Both systems exhibit similar temporal velocity gradient component distributions. Most unsteadiness corresponds to regions of high tangential velocity gradients. As a result of the flow drift, these spatial gradients are converted to temporal gradients. For the present flows, computations suggest judicious coordinate system angular velocity choices may reduce demands on both convective term treatments and computer resources.