Cellular Flow Field Research Articles

Error analysis of asymptotic solution of a heavy particle motion equation in fluid flows

For weakly inertial particles subjected to volumetric forces and Stokes drag force in fluid flows, we can solve the simplified particle motion equation using the perturbation method. This method allows us to obtain a recursive formula for the nth-order correction of the asymptotic solution of particle velocity. We verified the error of the asymptotic solution under two typical flow fields: a time-varying uniform flow field with a volumetric force field and a two-dimensional non-uniform cellular flow field. In the former, the relative error of the asymptotic solution of particle velocity and position increases with the Stokes number, and we provided a quantitative analysis of the results. In the latter, we verify and analyze the asymptotic solution from two perspectives: the behavior of a single particle and the collective behaviors of many particles. For asymptotic solutions with maximum velocity and position errors of less than 5%, we select the solution with the lowest order correction and designate it as the optimal asymptotic solution. The order of the optimal asymptotic solution increases with increasing Stokes numbers and motion durations. However, in most cases, for weakly inertial particles [St ∼ O(10−3)], and the time t* ∼ O(10), the first-order asymptotic solution can achieve accuracy, where both St and t* are defined using the flow field characteristic time, Tf = 4π s. The results validate the rationale behind utilizing first-order asymptotic solutions in the fast Eulerian method for turbulent dispersion of weakly inertial particles.

The dynamics of biofouled particles in vortical flows

When using mathematical models to predict the pathways of biofouled microplastic in the ocean, it is necessary to parametrise the impact of turbulence on their motions. In this paper, statistics on particle motion have been computed from simulations of small, spherical particles with time-dependent mass in cellular flow fields. The cellular flows are a prototype for Langmuir circulation and flows dominated by vortical motion. Upwelling regions lead to particle suspension and particles fall out at different times. The uncertainty of fallout time and a particle's vertical position is quantified across a range of parameters. A slight increase in settling velocities, for short times, is observed for particles with inertia due to clustering in fast downwelling regions for steady, background flow. For particles in time-dependent, chaotic flows, uncertainty is significantly reduced and we observe no significant increase in the average settling rates due to inertial effects.

The fluid dynamics of propagating fronts with solutal and thermal coupling

We numerically explore the propagation of reacting fronts in a shallow and horizontal layer of fluid. We focus on fronts that couple with the fluid due to density differences between the products and reactants and also due to heat release from the reaction. We explore fronts where this solutal and thermal coupling is cooperative or antagonistic. We quantify the fluid motion induced by the front and investigate the interactions of the front with the fluid as it propagates through quiescent, cellular and chaotic flow fields. The solutal coupling induces an extended convection roll that travels with the front, the thermal coupling due to heat release from the reaction generates a pair of convection rolls that travels with the front, and when both couplings are present there is a complex signature of these contributions. The details of the front dynamics depend significantly upon the interactions of the front-induced flow field with the fluid ahead of the front.

Computational geometric methods for preferential clustering of particle suspensions

A geometric numerical method for simulating suspensions of spherical and non-spherical particles with Stokes drag is proposed. The method combines divergence-free matrix-valued radial basis function interpolation of the fluid velocity field with a splitting method integrator that preserves the sum of the Lyapunov spectrum while mimicking the centrifuge effect of the exact solution. We discuss how breaking the divergence-free condition in the interpolation step can erroneously affect how the volume of the particulate phase evolves under numerical methods. The methods are tested on suspensions of 104 particles evolving in a discrete cellular flow field. The results are that the proposed geometric methods generate more accurate and cost-effective particle distributions compared to conventional methods.

Velocity and geometry of propagating fronts in complex convective flow fields.

We numerically study the propagation of reacting fronts through three-dimensional flow fields composed of convection rolls that include time-independent cellular flow, spatiotemporally chaotic flow, and weakly turbulent flow. We quantify the asymptotic front velocity and determine its scaling with system parameters including the local angle of the convection rolls relative to the direction of front propagation. For cellular flow fields, the orientation of the convection rolls has a significant effect upon the front velocity and the front geometry remains relatively smooth. However, for chaotic and weakly turbulent flow fields, the front velocity depends upon the geometric complexity of the wrinkled front interface and does not depend significantly upon the local orientation of the convection rolls. Using the box counting dimension we find that the front interface is fractal for chaotic and weakly turbulent flows with a dimension that increases with flow complexity.

Approximation of Random Slow Manifolds and Settling of Inertial Particles Under Uncertainty

A method is provided for approximating random slow manifolds of a class of slow–fast stochastic dynamical systems. Thus approximate, low dimensional, reduced slow systems are obtained analytically in the case of sufficiently large time scale separation. To illustrate this dimension reduction procedure, the impact of random environmental fluctuations on the settling motion of inertial particles in a cellular flow field is examined. It is found that noise delays settling for some particles but enhances settling for others. A deterministic stable manifold is an agent to facilitate this phenomenon. Overall, noise appears to delay the settling in an averaged sense.

The motion of solid spherical particles falling in a cellular flow field at low Stokes number

We present a jointed experimental and numerical study examining the influence of vortical structures on the settling of solid spherical particles under the action of gravity at low Stokes numbers. The two-dimensional model experiment uses electroconvection to generate a two-dimensional array of controlled vortices which mimics a simplified vortical flow. Particle image-velocimetry and tracking are used to examine the motion of the particles within this vortical flow. Particle motion is compared to the predictions of a numerical simulation inspired by the model equation developed by Maxey [“The motion of small spherical particles in a cellular flow field,” Phys. Fluids 30, 1915 (1987)].

Front propagation in a chaotic flow field.

We investigate numerically the dynamics of a propagating front in the presence of a spatiotemporally chaotic flow field. The flow field is the three-dimensional time-dependent state of spiral defect chaos generated by Rayleigh-Bénard convection in a spatially extended domain. Using large-scale parallel numerical simulations, we simultaneously solve the Boussinesq equations and a reaction-advection-diffusion equation with a Fischer-Kolmogorov-Petrovskii-Piskunov reaction for the transport of the scalar species in a large-aspect-ratio cylindrical domain for experimentally accessible conditions. We explore the front dynamics and geometry in the low-Damköhler-number regime, where the effect of the flow field is significant. Our results show that the chaotic flow field enhances the front propagation when compared with a purely cellular flow field. We quantify this enhancement by computing the spreading rate of the reaction products for a range of parameters. We use our results to quantify the complexity of the three-dimensional front geometry for a range of chaotic flow conditions.

Cellular structure of detonation utilized in propulsion system

How to confine a detonation in a combustor is a key issue of detonation applications in propulsion systems. Based on achieving schemes, detonations applied in the combustor, including pulse detonation wave (PDW), oblique detonation wave (ODW) and rotating detonation wave (RDW), are different from that described by the classic CJ theory in fine structures and its self-sustaining mechanisms. In this work, the cellular structures and flow fields of ODW and RDW were obtained numerically, and the fundamental characteristics and self-sustaining mechanisms of the detonations were analyzed and discussed. ODW front consists of three parts: the ZND-like front, the single-headed triple point front and the dual-headed triple point front. Cellular structures of RDW are heterogeneous, and the cell size near the outer wall is smaller than that near the inner wall.

Numerical simulation of particles movement in cellular flows under the influence of magnetic forces

AbstractA numerical model of particle motion in fluid flow under the influence of hydrodynamic and magnetic forces is presented. As computational tool, a flow solver based on the Boundary Element Method is used. The Euler‐Lagrange formulation of multiphase flow is considered. In the case of a particle with a magnetic moment in a nonuniform external magnetic field, the Kelvin body force acts on a single particle. The derived Lagrangian particle tracking algorithm is used for simulation of dilute suspensions of particles in viscous flows taking into account gravity, buoyancy, drag, pressure gradient, added mass and magnetophoretic force. As a benchmark test case the magnetite particle motion in cellular flow field of water is computed with and without the action of the magnetic force. The effect of the Kelvin force on particle motion and separation from the main flow is studied for a predefined magnetic field and different values of magnetic flux density. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Numerical investigation of natural convection phenomena in a uniformly heated circular cylinder immersed in square enclosure filled with air at different vertical locations

In this model, a numerical study of two dimensional steady natural convection is performed for a uniform heat source applied on the inner circular cylinder in a square air ( Pr = 0.7) filled enclosure in which all boundaries are assumed to be isothermal (at a constant low temperature). The developed mathematical model is governed by the coupled equations of continuity, momentum and energy and is solved by finite volume method. The effects of vertical cylinder locations and Rayleigh numbers on fluid flow and heat transfer performance are investigated. Rayleigh number is varied from 10 3 to 10 6 and the location of the inner cylinder is changed vertically along the centerline of the enclosure from − 0.25 L to 0.25 L upward and downward, respectively. It is found that at small Rayleigh numbers does not have much influence on the flow field while at high Rayleigh numbers have considerable effect on the flow pattern. In addition, the numerical solutions yield a two cellular flow field between the inner cylinder and the enclosure. Also, the total average Nusselt number behaves nonlinearly as a function of locations. Results are presented in terms of the streamlines, isotherms, local and average Nusselt numbers. Detailed results of the numerical has been compared with literature ones, and it gives a reliable agreement.

Dynamic Angular Segregation of Vesicles in Electrohydrodynamic Flows

We investigate a new type of behavior whereby small vesicles orbiting around a larger vesicle in a toroidal electrohydrodynamic flow undergo dynamic angular segregation. Application of a low frequency (approximately 50 Hz) electric field induces aggregation of adjacent unilamellar vesicles near the electrode, in a manner similar to that observed with rigid colloidal particles. For polydisperse vesicle suspensions, however, small vesicles (<10 microm) are often observed to "orbit" around larger vesicles (>20 microm) in a toroidal electrohydrodynamic flow field. While orbiting, the smaller vesicles gradually segregate into well-defined angular cross sections. Viewed from above, the vesicles appear to form dynamic "bands" at prescribed angles, separated by regions devoid of vesicles. We interpret the angular segregation in terms of induced dipolar interactions, and we propose a model based on point dipoles rotating in a cellular flow field. We demonstrate that the model yields a surprisingly diverse range of vesicle trajectories, including many that are qualitatively consistent with the experimental observations.

A rigid dumbbell settling under gravity in a periodic flow field

The motion of a rigid dumbbell settling under gravity in a stationary two-dimensional cellular flow field is investigated. The dumbbell is modeled as two identical beads connected by a rigid rod. Assuming that the bead radius is much shorter than the bead to bead distance, a simple model is obtained to describe the dynamical behavior of the dumbbell in terms of evolution of the dumbbell orientation and the velocity of the center of mass. The average velocity of the dumbbell in the gravity direction was found to depend on the dumbbell length, the inertia parameter and the terminal settling velocity in still fluid. For certain conditions the dumbbell remains suspended in the flow field. In one of the suspension regimes the dumbbell remains indefinitely attached to a fixed point. This behavior is related to the existence to saddle nodes in the cellular flow field.

Moving finite-size particles in a flow: A physical example of pitchfork bifurcations of tori

The motion of small, spherical particles of finite size in fluid flows at low Reynolds numbers is described by the strongly nonlinear Maxey-Riley equations. Due to the Stokes drag, the particle motion is dissipative, giving rise to the possibility of attractors in phase space. We investigate the case of an infinite cellular flow field with time-periodic forcing. The dynamics of this system are studied in a part of the parameter space. We focus particularly on the size of the particles, whose variations are most important in active physical processes, for example, for aggregation and fragmentation of particles. Depending on their size the particles will settle on different attractors in phase space in the long-term limit, corresponding to periodic, quasiperiodic, or chaotic motion. One of the invariant sets that can be observed in a large part of this parameter region is a quasiperiodic motion in the form of a torus. We identify some of the bifurcations that these tori undergo, as particle size and mass ratio relative to the fluid are varied. In this way we provide a physical example for sub- and supercritical pitchfork bifurcations of tori.

Influence of added mass on anomalous high rise velocity of light particles in cellular flow field: A note on the paper by Maxey (1987)

In this Brief Communication, we examine in detail the motion of light particles—or dense gas bubbles—rising under gravity in a cellular flow. We follow the methodology of Maxey [Phys. Fluids 30, 1915 (1987)], and we examine a range of parameters not fully discussed in the past, corresponding to particles lighter and yet more inertial than the surrounding fluid if the added mass effect is considered. We observe a nonmonotonic behavior of the particle rise velocity, which exhibits a maximum for density of the particles only slightly smaller than the density of the fluid. The maximum value of the average rise velocity corresponding to the “optimum” density is several times (more than 40) higher than those obtained for small density changes around the optimum. The occurrence of this maximum is due to the combined effects of particle inertia and fluid added mass, which segregate the rising particles into specific, fast pathlines where the local fluid velocity adds to the particle gravitational velocity. A plot covering particle rise velocity for this range—not covered by the analysis of Maxey—is provided.

A single dumbbell falling under gravity in a cellular flow field

The authors of this paper have noticed that they made a mistake concerning the stability of the fixed points where the molecule is located horizontally. This results in figures 9 and 10 being incorrect. For full details and corrected figures, please see pdf.

A single dumbbell falling under gravity in a cellular flow field

We study the motion of a single polymer chain settling under gravity in an ensemble of periodic, cellular flow fields, which are steady in time. The molecule is an elastic dumbbell composed of two beads connected by a nonbendable Hookean spring. Each bead is subject to a Stokes drag and a Brownian force from the flow. In the absence of particle inertia, the molecule settles out at a rate which depends on three parameters: the particle velocity in a fluid at rest, Vg, the spring constant, B, and the diffusion coefficient, D. We investigate the dependence of the molecule settling velocity on B, for fixed Vg and D. It is found that this velocity strongly depends on B and it has a minimum value less than Vg. We also find that the molecule is temporarily trapped at fixed points for certain values of the parameters. We analyse one fixed point in detail and conclude that its stability is the main factor which contributes to slowing down the settling process.

Anomalous transport of particle tracers in multidimensional cellular flows.

Advection of tracers is studied numerically in time-dependent, two-dimensional cellular flows and a time-independent, three-dimensional cellular flow field. Tracers in these flows follow trajectories that are either periodic or chaotic and mimic correlated Lévy flights. The probability density function of displacements for particles in the ordered regions of the flow follows a classical Gaussian dispersion process. The particle trajectories in the chaotic regions of the flow exhibit anomalous diffusion and the probability density function of displacements is well modeled by a time-fractional diffusion equation of order alpha. The overall process of particle dispersion is found to be controlled mainly by the chaotic regions within the flow field. From the perspective of Lagrangian dynamics our results indicate that the advection of particles in flow fields prone to exhibit chaotic advection is a combination of both classical, i.e., Gaussian, behavior and anomalous, i.e., non-Gaussian, diffusion.

Three-dimensional simulation of flow and thermal field in a Czochralski melt using a block-structured finite-volume method

The three-dimensional and time-dependent turbulent flow field and heat transfer of the melt in a Czochralski crystal growth process were predicted using an efficient block-structured finite-volume Navier–Stokes solver. The present paper is a first step towards adopting a block-structured finite-volume method for simulating turbulent flow in a Czochralski melt. Different ways of creating high-quality block-structured grids for the crucible are described along with their advantages over structured grids. In order to study the cellular convection pattern in an industrial Czochralski melt, three-dimensional simulations of turbulent flow and thermal field were carried out on two grid levels in five blocks. Using a special form of discretization of the convective fluxes, the flow field was predicted without applying any turbulence model. The accuracy of the discretization of the convective terms and size of control volumes were shown to affect greatly the characteristics of the flow field. With fine grids, the predicted thermal field was seen to be in close agreement with the reported experimental observation and numerical prediction of melt surface thermal field as well as temperature fluctuations. The flow field was shown to transform through a number of intermediate stages such as spoke pattern, n-folded pattern with island, etc., until a stable cellular flow field is established with weak co-rotating waves at the surface. The mechanism of these waves is baroclinic instability.

Chaotic motion of nonspherical particles settling in a cellular flow field

Small, spheroidal particles undergo a tumbling motion as they settle out under gravity through a fluid flow. This motion is in response to the fluid forces that act on the particle causing it to rotate due to the local fluid vorticity and rate of strain. In this paper the nonlinear system of differential equations governing the particle motion in a spatially periodic, cellular flow field is considered. The motion of spherical particles is completely regular, and in some regions of the flow they may be permanently suspended. Using recent results on volume-preserving maps, it is shown that this behavior can persist for nonspherical particles. More usually the tumbling motion is chaotic. This is characterized by results on Poincar\'e sections of the motion and determination of Lyapunov exponents. The response to flow vorticity and rate of strain is analyzed for both full three-dimensional motion and restricted planar motion to determine the main factors governing the chaotic motion.