Unsteady Shear Flows Research Articles

Gaussian approximation for finitely extensible bead-spring chains with hydrodynamic interaction

The Gaussian Approximation, proposed originally by Öttinger [J. Chem. Phys., 90, 463–473 (1989)] to account for the influence of fluctuations in hydrodynamic interactions in Rouse chains, is adapted here to derive a new mean-field approximation for the FENE spring force. This FENE-PG force law approximately accounts for spring-force fluctuations, which are neglected in the widely used FENE-P approximation. The Gaussian Approximation for hydrodynamic interactions is combined with the FENE-P and FENE-PG spring force approximations to obtain approximate models for finitely extensible bead-spring chains with hydrodynamic interactions. The closed set of ordinary differential equations governing the evolution of the second moments of the configurational probability distribution in the approximate models is used to generate predictions of rheological properties in steady and unsteady shear and uniaxial extensional flows, which are found to be in good agreement with the exact results obtained with Brownian dynamics simulations. In particular, predictions of coil-stretch hysteresis are in quantitative agreement with simulations’ results.

Integrals of the Vorticity Equation. Part II: Special Two-Dimensional Flows

Abstract In Part I, a general integral of the 2D vorticity equation was obtained. This is a formal solution for the vorticity of a moving tube of air in a 2D unsteady stratified shear flow with friction. This formula is specialized here to various types of 2D flow. For steady inviscid flow, the integral reduces to an integral found by Moncrieff and Green if the flow is Boussinesq and to one obtained by Lilly if the flow is isentropic. For steady isentropic frictionless motion of clear air, several quantities that are invariant along streamlines are found. These invariants provide another way to obtain Lilly’s integral from the general integral.

Uniform shear flow past a circular cylinder

This paper solves the two-dimensional problem of the unsteady shear flow of a viscous incompressible fluid past a circular cylinder. The flow is calculated using two methods. The first takes the form of a double series solution where an expansion is carried out in powers of the time, t, and in powers of the parameter Open image in new window where R is the Reynolds number. This approximate analytical solution is valid for small times following the start of the motion and for large Reynolds numbers. The second method involves a spectral-finite difference procedure for numerically integrating the full Navier-Stokes equations expressed in terms of a stream function and vorticity. Our results demonstrate that for small times and moderately large R the two methods of solution are in good agreement. Results are presented for a wide range of Reynolds numbers. Comparisons with previous studies are also carried out and discussed.

Mesoscale constitutive modeling of magnetic dispersions: material functions for shear flows

We recently developed a constitutive model for magnetic dispersions by modeling the magnetic particles as rigid dumbbells dispersed in a solvent. The theory yielded a constitutive equation in which the stress tensor could be expressed as a function of the velocity gradient, an orientational order tensor, S , an average alignment vector, J , and any imposed external magnetic field, H . The constitutive equation is used here to predict material functions for steady shear flow (shear-rate dependent viscosity and first normal stress coefficient) as well as those for unsteady shear flows (stress growth upon inception of steady shear and small-amplitude oscillatory shear). The importance of effects of concentration, equilibrium nematic ordering in the dispersion, and anisotropy in the hydrodynamic drag are emphasized. Comparisons with available experimental data on viscosity for magnetic inks under steady shear flow and inception of steady shear flow show reasonably good agreement.

HYDRODYNAMIC FORCE EXERTING ON A SQURE PILLAR WITH ANGLE OF ATTACK BY A DAM BREAK FLOW

Hydrodynamic forces exerting on a squire pillar (length d) with angle of attack α, placed in the unsteady free surface shear flow, are investigated experimentally, being varied in the range of α=0°-45°. It is found that the assumption of hydrostatic pressure distribution at the front and back side of the pillar is not valid for the case of unsteady free surface shear flow and hydrodynamic force changes with the pressure difference Δh between the front and back side of the pillar. It is shown that Δh/d is dependent on the gradient dh/dx of water surface calculated from Ritter solution and Δh/d increases with |dh/dx|, and the drag coefficient Cd is smaller than that in uniform flows.

Vortex simulation of unsteady shear flow induced by a vortex ring

A hybrid Eulerian–Lagrangian solver for unsteady shear flow induced by vortex ring/wall interactions is presented and discussed herein. Normal and oblique impacts of a vortex ring on a plane surface involve strong interactions between and external and the induced boundary layer, which undergoes separation and explosive growth, leading to the ejection of vorticity. For an oblique ring impact, the interaction is shown to be highly asymmetric. The development of the primary vortex ring and the pattern of the secondary vorticity generation can be seen clearly and the effects of the helical vortex lines around the primary ring also observed. The main features of the oblique impingement of a vortex ring on a wall are captured quite well and comparison is made with the experimental results. At the higher Reynolds number of 1000, the helical winding of vortex lines is however much more pronounced, and the secondary vortex structure has a thicker core. The results reveal detailed phenomena and provide physical insight into the flow mechanisms of vortex ring/wall interaction.

Brownian dynamics simulations of the stress and molecular configuration of polymers in exponential and linearly-ramped shear flow

The rheological and optical properties of dilute polymer solutions during the startup and subsequent relaxation in exponential shear flow are studied using Brownian dynamics simulations of freely draining, flexible bead-rod chains. We find that exponential shear flow is capable of effecting large molecular deformation for finite shear rates and strains, and furthermore, the polymer undergoes an initial stress and index of refraction relaxation that is much faster than a single exponential decay upon cessation of flow. When we compare the first normal stress difference ( τ 11– τ 22) to the corresponding components of the index of refraction tensor, n 11– n 22, we find that there is hysteresis when plotted over the course of a full startup and relaxation experiment. This is very similar to the hysteresis originally discovered by Doyle et al. [J. Non-Newtonian Fluid Mech. 76 (1998) 79–110] for uniaxial extensional flow. Moreover, the shear components of the stress and index of refraction tensors ( τ 12 and n 12) also exhibit hysteretic behavior. However, whereas the hysteresis loop for the 11–22 component traverses up the left branch during startup and relaxes on the right branch, that for the 12 component is in the completely opposite direction, i.e. the right branch corresponds to startup and the left branch relaxation. The presence of stress-index of refraction hysteresis clearly has important implications for the stress-optic rule. We calculate the stress-optic coefficient for the startup process, and find that the normal and shear components give the same value, but that it is strongly dependent on Wi. This result is consistent with that found by Doyle et al. during the startup of steady shear flow [Doyle and Shaqfeh, Dynamic simulation of freely-draining, flexible bead-rod chains: startup of extensional and shear flow, J. Non-Newtonian Fluid Mech. 76 (1998) 43–78]. We also simulated shear flows with a linearly ramped shear rate to compare to our results for exponential shear. We find that for comparable final shear rate and time, both exponential and linear ramping shear produce very similar hysteresis effects in both the shear and normal components suggesting that there may be many unsteady shear flow capable of creating large polymer stretch and stress-birefringence hysteresis. Finally, we compare our bead-rod chain simulations to the FENE dumbbell model, and we find that while the latter can qualitatively capture the normal stress hysteresis, it predicts negligible shear stress hysteresis at equivalent shear rates and shear strains.

Numerical experiments on internal gravity waves in an accelerating shear flow

Abstract The influence of an accelerating shear flow on the propagation of an internal gravity wave in a continuously stratified fluid is studied by means of two-dimensional numerical simulations. These are motivated by earlier laboratory experiments [Thorpe, S.A. 1978b. On internal gravity waves in an accelerating shear flow, Vol. 88. J. Fluid Mech. pp. 623–639]. In these experiments the mean flow is an accelerated Couette flow and the mean density profile is linear. The laboratory experiments revealed the striking effect of the unsteady shear flow in the evolution of an internal gravity wave leading to the wave focusing in a region where the flow is extremum. This phenomenon is associated with the growth of small scale density fluctuations. As a result density overturns are sometimes observed. This behaviour is well reproduced by the numerical simulations. We provide insights on the flow dynamics in particular on the possible occurrence of wavebreaking. We show that the dynamics is characterized by two competitive mechanisms that is a damping of the wave and a local enhancement of its steepness leading sometimes to density overturns. The budget for the energy of the wave reveals that the initial damping of the wave results from wave-mean flow interactions. These interactions lead to the development of a fine scale vertical density structure which is associated with high vertical shear. We find that in some cases wavebreaking occurs as a result of shear instability. The value of the acceleration of the mean flow is very likely to influence the onset of the instability. The scaling laws of the wave evolution, in particular the rate of decrease of its energy, are determined. From these laws the lifetime of the wave is found as a function of the acceleration of the shear. It may be expected that, in the ocean, this development will result in the largest fluctuations derived from wave-flow interactions occurring where the mean flow in the wave direction is greatest. Waves travelling normal to a two-dimensional shear flow will be unchanged. Waves travelling parallel will be damped. This may have particular application at the continental shelf where flow, mainly parallel to the isobaths, will damp waves travelling along-slope, but allows waves travelling normal to the isobaths (e.g., directly across the shelf-break) to be transmitted without attenuation. Similar effects are expected for the evolution of a high frequency wave interacting with a lower frequency (e.g., near inertial) motion.

Rheology of Cetyltrimethylammonium p-Toluenesulfonate−Water System. 3. Nonlinear Viscoelasticity

The nonlinear viscoelastic behavior of the cetyltrimethylammonium p-toluenesulfonate (CTAT)−water system is investigated in steady and unsteady shear flow as a function of surfactant concentration and temperature. A rheo-optical study which includes measurements of dichroism, birefringence, and turbidity under flow at various shear rates is also discussed. The shear viscosity data in steady shear agree with the complex viscosity in the limit of low deformation rates. For moderate deformation rates, in the shear thinning region, the Cox-Merz rule is not followed. In all cases, a limiting stress or plateau stress was observed at shear rates that exceed one-half of the reciprocal of the main relaxation time [(2τd)-1]. At the stress plateau, the micellar solution most likely undergoes an isotropic-to-nematic phase transition induced by shear. However, our results do not conclusively exclude the possibility of a constitutive instability with respect to shear banding, in which simultaneous shear rates coexist u...

矩形断面まわりの流れにおける空力音の発生源の特性に関する数値流体解析

Numerical simulations of the flow around rectangular cross-sections are carried out to examinecharacteristics of aeroacoustic sources caused by the unsteady separated shear flows around the cross-sections.The aeroacoustic sources are identified by second derivative of pressure fluctuation in the incompressible flow around the cross-sections. It is shown that the locations of the aeroacoustic sources are not necessarily corresponding to those of high pressure fluctuation. Furthermore, it is found that reattachment of the separated shear flows to the solid surface near the trailing edges has significant effects on the intensity of the aeroacoustic sources.

Numerical modeling of three-dimensional shear flow in a cavity with moving lids

The results of a numerical calculation of the unsteady shear flow of a viscous incompressible fluid in a cavity with moving lids are given for various geometries and values of the Reynolds number. The paper describes the numerical method developed for solving the three-dimensional unsteady Navier-Stokes equations in a regular domain and also the direct method of parallelling the solution for a multiprocessor computer with distributed memory. Test results are compared with the results of other authors.

Unsteady shear flow of a viscoplastic material

A viscoplastic, or yield-stress, liquid occupies the space between two infinite parallel plates. Initially the whole system is at rest. The lower plate is suddenly jerked into motion with given speed or shear stress, while the upper plate is kept fixed. The flow consists of two regions; (1) a lower sheared region bounded above by the yield surface, (2) an upper unyielded region bounded below by the yield surface. The yield surface propagates to the upper plate as time proceeds. We first consider the equivalent one plate problem of flow over a jerked plate, and find similarity solutions and small time asymptotic solutions for prescribed shear and speed cases respectively. These solutions are used as initial solutions for the two plate case. The motion of the yield surface and the time taken for the entire material to yield are investigated. The problems considered here are two dimensional representations of some control devices, for example the light duty clutch, which consists of two corotating, coaxial discs separated by a layer of electrorheological material. In this application it is useful to know the time taken for all the material to yield.

Evolution model of the vertical structure of the turbulent active layer of the sea

An one‐dimensional differential model of the vertical structure of the turbulent active layer of the sea based on the theory of the rotationally anisotropic turbulence is proposed. The present article consists of two parts. The first part gives the definition of the class of rotationally anisotropic turbulence (RAT) and presents the basics of the theory of RAT. In the second part the theory is used to describe the evolution of the vertical structure of the turbulent active layer of the ocean within the framework of the traditional model assumption about the horizontal homogeneity of all the fields under consideration. A set of submodels (including the analogue to the classical mixing length model) is derived and discussed. The model is used under different initial and boundary conditions. Depending on the actual conditions the model describes various effects: the appearance of the upper mixed layer with a steep thermocline below it, the switching between the convective and diffusive mixing regimes, the staircase‐like structure of the thermocline (caused by unsteady shear flow or by double‐diffusion instabilities), and the modulation of the velocity field by the period of the oscillation of the heat flux. The set of the model coefficients is determined on the basis of sea surface temperature data from the ocean weather station (OWS) Papa. The results of the following computations of temperature distribution show a reasonable similarity between the model results and the corresponding data from the OWS Papa and the Long‐Term Upper Ocean Study (LOTUS).

Boundary Element Method for Transient Viscoelastic Flow: The MPTT Model

This paper deals with the modelling of viscoelastic flows in transient regimes using the Modified Phan Tien and Tanner (MPTT) model. This model is the most recent one which can predict stress overshoot in unsteady shear flow and is based on the description of the liquid microstructure rather than empirical and mathematical developments. Here, the analysis of the squeeze film between parallel plates is presented. It is an alternative in quantifying and understanding the two main factors which may enhance the carrying capacity (inertia effect and viscoelastic behavior) situation which can occur, for instance, in the case of overloading in journal bearings. A boundary element method is used here to model such complex flow. The main factors have been extracted and the restrictions due to a rough estimate of the lubricant's flow are shown. Presented as a Society of Tribologlsts and Lubrication Engineers paper at the ASME/STLE Tribology Conference in Lahaina, Hawaii, October 16–20, 1994

Effects of shear and strain on temporal evolution of laminar diffusion flames

Closed-form analytical solutions that describe the evolution of diffusion flames in two-dimension al laminar flowfields due to an initial line source of fuel in an infinite oxidizer environment are derived. The velocity field is assumed to be divergence free and both steady and unsteady shear and plane stagnation flows are considered in this study. The flame surface is modeled as an infinitesimally thin flame sheet at which fuel and oxidizer are consumed at an infinitely fast rate; thus, chemical kinetic effects are ignored. Validity of the solutions are discussed. The flame shapes are observed to be elliptic. The flame front moves away from the source at early times; whereas at later times, the front collapses at the location of the fuel source. An expression for the time for complete consumption of the fuel is derived in each case, and it is shown that strain fields are more conducive to combustion. Nomenclature A = aspect ratio of the elliptical front a = shear rate b = strain rate Cf = mass concentration of fuel D = mass diffusivity e = base of natural logarithm Pe = Peclet number identical to L^/Dt^f Q = mixture fraction source strength Q = fuel source strength used in Eq. (1) rm = maximum radius of reaction front

Effect of a shear flow on the planform of thermal convection in a fluid of variable viscosity.

It is well known that if the variation with temperature of viscosity is taken into account, the onset of convection in an infinite layer of fluid that is heated from below takes the form of hexagonal convection cells. It is also well known that in a sufficiently strong shear flow, rolls aligned with the flow direction are the preferred mode of convection. Here the process by which a steady or unsteady shear flow de- stroys the usual hexagonal cell pattern is investigated. The critical size of the shear flow that enables rolls to be established at the onset of convection is determined for the case when the stress is prescribed at the boundaries.

Exact solution of the system of equations of the second-order kinetic moments for a two-scale homoenergetic affine monatomic gas flow

The exact solution of the system of equations for the second-order kinetic moments (stresses) is investigated for a flow with two microscales when an unsteady shear flow is superimposed on a one-dimensional unsteady gas flow.

Vorticity transport in modeling three-dimensional unsteady shear flows

This Letter examines how intrinsic three-dimensionality influences the flow structure of unsteady shear flows, and models its effects by considering spanwise averages of vorticity fluctuations. In particular, uniform flow past a half-cylinder is considered: the two-dimensional wake is very different from its three-dimensional counterpart and the evolution equation for the spanwise spatially averaged vorticity is analyzed. It is shown that the vector flux which transports spanwise vorticity due to spatial vorticity fluctuations is of considerable magnitude (large in comparison with molecular diffusion) and is far from co-linear with the gradient of the spanwise averaged vorticity. Modeling this flux using classical eddy viscosity ideas would therefore seem inappropriate.

Finite-amplitude convection in the presence of an unsteady shear flow

The effect of an unsteady shear flow on the planform of convection in a Boussinesq fluid heated from below is investigated. In the absence of the shear flow it is well-known, if non-Boussinesq effects can be neglected, that convection begins in the form of a supercritical bifurcation to rolls. Subcritical convection in the form of say hexagons can be induced by non-Boussinesq behaviour which destroys the symmetry of the basic state. Here it is found that the symmetry breaking effects associated with an unsteady shear flow are not sufficient to cause subcritical convection so the problem reduces to the determination of how the orientations of roll cells are modified by an unsteady shear flow. Recently Kelly & Hu (1993) showed that such a flow has a significant stabilizing effect on the linear stability problem and that, for a wide range of Prandtl numbers, the effect is most pronounced in the low-frequency limit. In the present calculation it is shown that the stabilizing effects found by Kelly & Hu (1993) do survive for most frequencies when nonlinear effects and imperfections are taken into account. However a critical size of the frequency is identified below which the Kelly & Hu (1993) conclusions no longer carry through into the nonlinear regime. For frequencies of size comparable with this critical size it is shown that the convection pattern changes in time. The cell pattern is found to be extremely complicated and straight rolls exist only for part of a period.

Open Boundary Conditions in the Simulation of Subsonic Turbulent Shear Flows

Dealing with open boundaries in the computer simulation of unsteady subsonic shear flows presents challenging problems. In practice, only a portion of the flow can be investigated and we must ensure that the presence of artificial boundaries does not pollute the solution in a significant way. One difficulty is related to the basic solution of the physical flow equations and involves choosing appropriate nonlocal open boundary conditions at the outflow boundaries which will adequately bound the computational domain while providing information about the virtual flow behavior outside. A second difficulty is related to the discretized computational problem, for which additional numerical boundary conditions that are consistent with the unsteady flow equations at the boundaries are required for closure. Recent approaches based on characteristic analysis and their practical implementation are discussed. Specific examples are used to illustrate the implementation of state-of-the-art approaches to open boundary conditions and, in particular, the potential sensitivity of subsonic free shear flows to the actual choice of open boundary conditions. This sensitivity is an intrinsic feature of the flows being studied rather than an artifact of the computations. Ideal free shear flows do not exist; actual flow realizations are defined by the numerical or laboratory boundary conditions in the experiments.