Two-dimensional Shear Flow Research Articles

Stability of flow and kinetic energy dissipation in 2D annular shear flows of inelastic hard disk assemblies

We have used simulations of inelastic hard disks in two-dimensional shear flows to investigate the stability conditions of kinetic energy upon shearing through different circumstances. We study the characteristics of instability via the signals of kinetic energy dissipation and the statistics associated with this quantity. Our results reveal how the flow characteristics of hard-disk assembly can be modelled by a dynamical model for turbulence named as the GOY shell model. Our results clearly show that the behaviour of rapidly sheared hard-disk assemblies is similar to a fluid in highly chaotic state, so called turbulent state. The current results assist us in finding proper modelling ways in simulation of suspension flows, colloids and magnetic fluids.

A formulation for calculating the translational velocity of a vortex ring or pair

Cephalopods, among other marine animals, use jet propulsion for swimming. A simple actuator is designed to loosely mimic pulsatile jet formation in squid and jellyfish. The actuator is comprised of a cavity with an oscillating diaphragm and an exit orifice. Periodic oscillation of the diaphragm results in the formation of an array of vortex rings and eventually could generate a periodic pulsatile jet. A general formulation for calculating the velocity of a steadily translating vortical structure in two-dimensional and axi-symmetric shear flows is presented. This technique is based on taking the variational derivative of an energetic function at its critical point. This technique is general, applicable to vortices in liquid and gas media, with no limitation on the relative size of the vortex core. The technique is then implemented to estimate the translational velocity of a vortex ring in a Helmholtz vortex ring generator.

Numerical simulation of flow around a square cylinder in uniform-shear flow

Abstract This paper presents results obtained from a numerical simulation of a two-dimensional (2-D) incompressible linear shear flow over a square cylinder. Numerical simulations are performed, using the lattice Boltzmann method, in the ranges of 50⩽Re⩽200 and 0⩽K⩽0.5, where Re and K are the Reynolds number and the shear rate, respectively. The effect of the shear rate on the frequency of vortex shedding from the cylinder, and the lift and drag forces exerted on the cylinder are quantified together with the flow patterns around the cylinder. The present results show that vortex structure behind the cylinder is strongly dependant on both the shear rate and Reynolds number. When Re=50, a small K can disturb the steady state and cause an alternative vortex shedding with uneven intensity. In contrast, a large value of K will suppress the vortex shedding from the cylinder. When Re>50, the differences in the strength and size of vortices shed from the upper and lower sides of the cylinder become more pronounced as K increases. Vortex shedding disappears when K is larger than a critical value, which depends on Re. The flow patterns around the cylinder for different Re tend towards self-similarity with increasing K. The lift and drag forces exerted on the cylinder, in general, decrease with increasing K. Unlike a shear flow past a circular cylinder, the vortex shedding frequency past a square cylinder decreases with increasing the shear rate. A significant reduction of the drag force occurs in the range 0.15

The effect of confinement on the stability of two-dimensional shear flows

It has been shown recently that the instability of a two-dimensional wake increases when it is confined in the transverse direction by two flat plates. Confinement causes the transition from convective to absolute instability to occur at lower values of shear. This paper examines this effect comprehensively and concludes that it is due to the constructive interaction of modes with zero group velocity in the wake (or jet) and in the surrounding flow. Maximum instability occurs when the wavenumber of the fundamental mode in the wake (or jet) matches that of the fundamental mode in the surrounding flow. Other regions of high instability occur when the harmonics of one mode interact with the fundamental of the other. This effect is examined at density ratios from 0.001 to 1000. At each density ratio, the confinement which causes maximum absolute instability can be predicted. This study also shows that it is vital to examine the wavenumber of absolutely unstable modes in order to avoid over-predicting the absolute instability. In some situations this wavenumber is vanishingly small and the mode must be discounted on physical grounds.

Shear-driven magnetic reconnection in viscous resistive incompressible plasmas

Planar magnetic reconnection, driven by shear flows in viscous resistive incompressible plasmas, is analyzed. Time-dependent profiles of the flow velocity and the reconnecting magnetic field are computed by solving the magnetohydrodynamic equations for disturbances of a magnetostatic configuration. The resulting asymptotic profiles for large times are characterized by boundary layers, formed at the boundaries of the region, rather than by a localized reconnecting current sheet within the region. Analysis of steady viscoresistive solutions confirms that a smooth solution, reached by a dynamically evolving system, must be nonlocalized. The results imply that at least two-dimensional shear flows are required for driving magnetic reconnection in viscous resistive plasmas.

On pullback attractors for a class of two-dimensional turbulent shear flows

We consider a two-dimensional Navier–Stokes shear flow with time dependent boundary driving. We establish the existence of a unique global in time solution of the considered problem and then the existence of the pullback attractor for the associated evolutionary process. In the end, we estimate from above the dimension of the attractor in terms of given data. This research is motivated by a problem from lubrication theory.

Stress results from two-dimensional granular shear flow simulations using various collision models

Collision resolution is one of the key elements in a discrete element method algorithm for modeling granular flows. Several collision models have been proposed for this process. The hard-particle collision approach is typically used for dilute systems, or for those in which the assumption of binary and instantaneous particle-particle contact remains valid. As the solids fraction increases, however, multiple, enduring collisions can occur and a soft-particle approach is more appropriate for resolving the collision dynamics. In this work, the delineation between dilute and dense systems and the suitability of contact models are explored for a range of solid fractions. Stress results for two-dimensional shear flow simulations are compared using several collision models including an event-driven hard-particle model, a hysteretic spring soft-particle collision model following Walton and Braun [J. Rheol. 30, 949 (1986)], and a hybrid hard-particle-with-overlap model following Hopkins and Louge [Phys. Fluids A 3, 47 (1991)]. Results show that stresses are accurately predicted for a range of solids fractions, coefficients of restitution, and friction coefficients by both the hard-particle-with-overlap and soft-particle models so long as a sufficiently large loading stiffness is used for the soft-particle model. Additional results investigating the accuracy of the collision models and the amount of collisional overlap are presented as functions of the simulation time step and model parameters.

Onset of wetting failure in liquid–liquid systems

Some model problems are considered in order to investigate wetting failure in liquid–liquid systems. Three geometries are considered, two-dimensional two-phase shear flow, two-dimensional driven capillary rise, and both two- and three-dimensional two-phase driven cavity flow. In the first two cases, the two fluids are made equiviscous. The driven cavity flow is investigated for both equi- and non-equiviscous fluids. Three methods of analysis are used for the equiviscous case, an essentially exact Fourier series method, a quasi-parallel approximation and a phase-field model. The Fourier series validates the phase-field method in that they both give nearly identical results for onset of instability. At relatively large slip length divided by channel width (), the capillary number at which onset of wetting failure (entrainment of the receding fluid in the advancing) occurs is highly dependent on the type of flow. This dependence, however, appears to diminish rapidly as the slip length becomes smaller. The capillary number for the onset of instability is moderately dependent on gravity level.Three-dimensional phase-field calculations are then discussed that show wetting failure through tipstreaming and splitting instabilities. Spot checks indicate that the onset points of two- and three-dimensional instabilities are very close. It is hypothesized that tipstreaming can be understood in part as a quasi-two-dimensional phenomenon.

Manipulation of Single DNA Molecules via Lateral Focusing in a PDMS/Glass Microchannel

In a hybridized microchannel made of poly(dimethylsiloxane) (PDMS) and glass, nonuniform electroosmotic flow (EOF) was created when an electric field was applied. PDMS and glass pieces were bonded at room temperature after oxidization with oxygen plasma. After oxidization, the PDMS surface has a higher zeta (ζ) potential than that of glass, which results in faster EOF at that surface. When the channel has a large aspect ratio (width ≫ depth), the EOF can be considered as a two-dimensional gradient shear flow. By using a scanning confocal fluorescence microscope, we demonstrated that negatively charged DNA molecules were focused by the nonuniform EOF and the electric field into a thin layer at the glass surface of about one-fifth of the channel depth. This phenomenon was applied to selectively detect target DNA molecules using evanescent-field excitation. Streptavidin-conjugated quantum dots (QD) were used to selectively bind biotinylated DNA. After applying an electric field, free QD remained randomly distributed within the channel. Nonbiotinylated DNA molecules were focused onto the glass surface, but they were not labeled and thus nonfluorescent. Therefore, the increase in molecule count after application of the electric field can be solely attributed to QD−DNA complexes which were focused onto the glass surface. This method does not require the separation of excessive probes (QD) and can be operated continuously to achieve high throughput.

Fluid-Particle Interactions and Resuspension in Simple Shear Flow

The lattice Boltzmann method (LBM) is used to simulate the particle resuspension process in a two-dimensional simple shear flow. At first, the lift force on a single disk-shaped particle attached at the channel wall is computed. It is found that when the particle is allowed to freely move in the viscous fluid, the resulting lift force is smaller than that when the particle is constrained to be stationary. The bulk properties of fluids with particles suspended under various concentrations are numerically calculated. The results agree reasonably well with analytical results by Batchelor when the volume fraction is lower than 50%. The resuspension process of a group of particles (up to 500) is simulated at different particle-to-fluid density ratios. It is found that the height and shape of particle bed depend on the particle density ratio and flow conditions. Interactions of groups of particles as well as the final shape of the bedform of the particles were studied during this resuspension process. Finally, the pressure distribution and flow above the bedform of particles was examined. The results obtained agree well with those observed in naturally occurring bedforms of sediments.

Three-wave resonant interaction involving unstable wave packets

In this work, we derive the evolution equations for spectrally narrow, resonantly interacting wave packets weakly unstable in terms of linear theory. We consider the following two cases. In the first case, the system is linearly unstable due to the weak coupling of two modes whose energies have opposite signs. As an example of such a system, we analyze the model of a three-layer two-dimensional shear flow, where both the density and vorticity of the undistorted flow have jumps at the layer interfaces and are uniform inside the layers. In the second case, we consider the linear instability of an individual mode. The instability of capillary‐gravity waves in a weakly supercritical flow described by the Kelvin‐Helmholtz model is an example of this instability. The evolution equations are different for these two cases. The equations are derived in the Hamiltonian formalism, which allows us to consider wave processes disregarding the features of a particular problem. We prove that bounded solutions of the evolution equations exist in both cases. In other words, waves unstable in terms of the linear theory can be stabilized due to their interaction with neutral waves.

Low-dimensional dynamical systems for a wake-type shear flow with vortex dislocations

Low-dimensional systems are constructed to investigate dynamics of vortex dislocations in a wake-type shear flow. High-resolution direct numerical simulations are employed to obtain flow snapshots from which the most energetic modes are extracted using proper orthogonal decomposition (POD). The first 10 modes are classified into two groups. One represents the general characteristics of two-dimensional wake-type shear flow, and the other is related to the three-dimensional properties or non-uniform characteristics along the span. Vortex dislocations are generated by these two kinds of coherent structures. The results from the first 20 three-dimensional POD modes show that the low-dimensional systems have captured the basic properties of the wake-type shear flow with vortex dislocation, such as two incommensurable frequencies and their beat frequency.

Closed-loop Lagrangian separation control in a bluff body shear flow model

We show how the location of Lagrangian coherent structures, such as unstable manifolds of Lagrangian separation points, can be controlled via feedback control in two-dimensional shear flows. Such control can be used, for instance, to guide fuel transport into designated regions of the flame in a combustor. Motivated by this example, we consider an unsteady vortex model for flow past a bluff body, and create unstable manifolds in this model at prescribed locations by applying control along the boundary. We find that oscillating the newly created unstable manifolds in 1:1 resonance with the von Kármán vortex shedding frequency enhances mixing in the wake significantly.

Intermittency of energy in rapid granular shear flows.

Hard-disk simulations are used for two-dimensional rapid granular shear flows of circular disks between two rotating cylinders. The intermittency effects associated with the rate of the energy dissipation of collisions are studied. The statistics of intermittent signals of energy dissipation reveals that a power law governs the dynamics of rapid shear granular flows. A dynamical system approach based on the Gledzer-Ohkitani-Yamada shell model of turbulence is employed to reproduce signals for energy dissipation that are statistically consistent with those from simulations. The results suggest that rapid granular flows can be analyzed by appropriate turbulent models.

超音速せん断流中の水素ガス初期燃焼への気体分子運動論的アプローチ｀＊１´ 第１報：混合拡散過程

Direct Simulation Monte Carlo (DSMC) method based on the Boltzmann equation, in-cooperating a new reactive inelastic molecular collision model, has been employed to analyze the fuel gas mixing in two-dimensional supersonic shear flow which originates from the meeting at a thin plate (lip) edge of two parallel flows, a supersonic air stream and a sonic hydrogen gas flow. In the present study, the influence of hydrogen gas temperature upon the mixing within the shear layer is clarified. In a companion paper (the 2nd report), the relationship between mixing and chemical reaction will be presented.

プラズマトーチ応用の超音速燃焼初期促進に関する気体分子運動論的解析 第１報：活性基の種類と供給形態による特徴

The effects of various radical species, supplied from a plasma torch ignitor upon initiating hydrogen-air chemical reaction in a two-dimensional supersonic shear flow have been investigated by a new numerical approach based on the molecular gas dynamics. Direct Simulation Monte Carlo (DSMC) method has been employed to derive the microscopic reaction phenomena from the Boltzmann equation with a reactive inelastic molecular collision model (IE Model). The plasma species and injection mode into air stream have been varied. The result shows that fuel hydrogen initial combustion can be significantly enhanced by plasma injection of atomic oxygen supplying in diffusive mode.

Nonequilibrium Reynolds stress for the dispersed phase of solid particles in turbulent flows

The nonequilibrium turbulent transport of dispersed-solid phase is investigated theoretically, and confirmed with simple numerical experiments. The nonequilibrium form of the particle Reynolds stress was obtained as a function of Stokes number, Stτ (=τp/τ), where τp is the particle relaxation time and τ the characteristic time scale of particle–turbulence interaction. The nonequilibrium memory effect could be explicitly expressed as a damped residual contribution of initial particle Reynolds stress. Major interest is given to the region of τ∼O(τL), where τL is the integral time scale associated with the Lagrangian autocorrelation of fluctuating velocities of carrier turbulence. In two-dimensional (2D) simple shear flows with stationary homogeneous turbulence, explicit expressions for the equilibrium and nonequilibrium particle Reynolds stress were obtained, and their characteristics were analyzed extensively for a wide range of particle relaxation time and various conditions of mean shearing of the carrier phase. The reliability of theoretical predictions was tested through a comparison with results of stochastic simulation of particle motion. It was shown that the particle inertia and the mean shearing of the carrier phase seriously affect the relaxation time of particle Reynolds stress required to reach its equilibrium state in such a way that the relaxation time to equilibrium increases in proportional to the increase of τp and this trend is augmented by increasing the mean shearing of the carrier phase. Finally, a nonequilibrium constitutive equation for the dispersed phase Reynolds stress was obtained in a form usable in the two-fluid Eulerian approach for two-phase turbulent flows.

Windowing, re-shaping and re-orientationinterrogation windows in particle image velocimetry for theinvestigation of shear flows

We present a procedure for analysing images obtained inparticle image velocimetry. This consists of weighting, re-shapingand re-orientating the interrogation windows to improve themeasurements in flow regions exhibiting velocity differences in spaceas in shear flows. The procedure is applied recursively together withthat used to determine the optimal window offset and does not requireany window deformation. Tests are performed on synthetic images indifferent conditions (on a two-dimensional wall shear flow providedin the web by the Visualisation Society of Japan) and on imagesacquired in the near field of a turbulent axisymmetric water jet (ata Reynolds number equal to 8000). Under these conditions, the newalgorithm substantially improves the results obtained with thestandard procedure. The use of an offset procedure with a variableiteration scheme (not simple halving) is also considered; although itintroduces some degree of arbitrariness, sometimes it can alsoimprove the quality of the results in comparison to the standardprocedures.

Coherent structures and self-consistent transport in a mean field Hamiltonian model.

A study of coherent structures and self-consistent transport is presented in the context of a Hamiltonian mean field, single wave model. The model describes the weakly nonlinear dynamics of marginally stable plasmas and fluids, and it is related to models of systems with long-range interactions in statistical mechanics. In plasma physics the model applies to the interaction of electron "holes" and electron "clumps," which are depletions and excesses of phase-space electron density with respect to a fixed background. In fluid dynamics the system describes the interaction of vortices with positive and negative circulation in a two-dimensional background shear flow. Numerical simulations in the finite-N and in the N--> infinity kinetic limit (where N is the number of particles) show the existence of coherent, rotating dipole states. We approximate the dipole as two "macroparticles" (one hole and one clump) and consider the N=2 limit of the model. We show that this limit has a family of symmetric, rotating integrable solutions described by a one-degree-of-freedom nontwist Hamiltonian. A perturbative solution of the nontwist Hamiltonian provides an accurate description of the mean field and rotation period of the dipole. The coherence of the dipole is explained in terms of a parametric resonance between the rotation frequency of the macroparticles and the oscillation frequency of the self-consistent mean field. This resonance creates islands of integrability that shield the dipole from regions of chaotic transport. For a class of initial conditions, the mean field exhibits an elliptic-hyperbolic bifurcation that leads to the filamentation, chaotic mixing and eventual destruction of the dipole. (c) 2002 American Institute of Physics.

The effects of continuous size distributions on the rapid flow of inelastic particles

Molecular-dynamics simulations are employed to investigate the stresses and granular energy in granular materials with Gaussian and lognormal size distributions. Specifically, smooth circular disks of uniform material density engaged in unbounded two-dimensional shear flow are simulated using an event-driven algorithm. Particle collisions are treated as hard-sphere collisions and all collisions have the same coefficient of restitution. The resulting stresses, when nondimensionalized with the root-mean-square diameter, are found to remain relatively constant as the widths of the particle size distributions are increased away from the monodisperse limit. As a consequence, the stresses predicted by monodisperse kinetic theory (using the root-mean-square diameter) are reasonably accurate in the Gaussian and lognormal systems studied herein. This width-independent nature of the total stresses is traced to an effective balancing of the stresses between the larger particles, which generate relatively high stresses, and smaller particles, which generate lower stresses. Moreover, similar to binary-sized systems, the granular energy in Gaussian and lognormal systems is found to be unequally distributed among the various sizes of particles, with large particles possessing more granular energy than their smaller counterparts (i.e., an equipartition of energy is not observed). This difference in granular energy between two particles increases with both inelasticity and the size difference.