Dynamics of large solid particles in bioconvective sedimentation

Settling of one or two large solid particles in a bioconvection flow induced by gyrotactic motile microorganisms is investigated using a 2D numerical model. The results of varying the initial positions of large particles on the bioconvection flow pattern are investigated. The Chimera method is utilized to generate subgrids around the moving particles. It is demonstrated that the introduction of a single large particle displaces bioconvection plume and changes its shape. The introduction of two particles on the same side of the bioconvection plume further displaces the plume while the introduction of two particles on opposite sides reduces this displacement. The influence of the bioconvection plume on the particles' settling paths and particles' settling velocities is investigated. Copyright © 2006 John Wiley & Sons, Ltd.     

A fast non‐Fickian particle‐tracking diffusion simulator and the effect of shear on the pollutant diffusion process

This paper presents a fast method for the generation of non‐Fickian particle paths within a particle‐tracking pollutant diffusion model based on a Fourier spectral representation of fractional Brownian motion (fBm), a generalization of ordinary Brownian motion. Correlated diffusive components in a particle‐tracking algorithm are modelled using fBm increments that have long‐range correlations over numerous spatial and/or temporal scales; hence producing non‐Fickian diffusion. A fast algorithm to generate fBm and its increment by using its power spectral density S(f) in a fast Fourier transform algorithm is given. A general equation for the scaling of fBm within a velocity flow field with simple linear shear is presented. An initial numerical study of the nature of fBm shear dispersion has been conducted by incorporating fBm increments into a non‐Fickian particle‐tracking algorithm. It is shown that the effect of simple (i.e. linear) shear on the diffusion process is to produce enhanced diffusive phenomena with the longitudinal spreading of the plume scaling with exponent ∼1+H, where H is the Hurst exponent used to describe fBm. Finally, a more complex shear zone at the entrance of a coastal bay model is investigated using both a traditional particle‐tracking method and the fBm‐based method. Copyright © 2000 John Wiley & Sons, Ltd.     

Improved MPS method with variable‐size particles

Using variable‐size particles in the moving particle semi‐implicit method (MPS) could lead to inaccurate predictions and/or numerical instability. In this paper, a variable‐size particle moving particle semi‐implicit method (VSP‐MPS) scheme is proposed for the MPS method to achieve more reliable simulations with variable‐size particles. To improve stability and accuracy, a new gradient model is developed based on a previously developed MPS scheme that requires no surface detection MPS. The dynamic particle coalescing and splitting algorithm is revised to achieve dynamic multi‐resolution. A cubic spline function with additional function is employed as the kernel function. The effectiveness of the VSP‐MPS method is demonstrated by three verification examples, that is, a hydrostatic pressure problem, a complicated free surface flow problem with large deformation, and a dynamic impact problem. The new VSP‐MPS scheme with variable‐size particles is found to have balanced efficiency and accuracy that is suitable for simulating large systems with complex flow patterns. Copyright © 2015 John Wiley & Sons, Ltd.     

Overstability Analysis of Thermo-bioconvection Saturating a Porous Medium in a Suspension of Gyrotactic Microorganisms

A linear stability analysis is performed to analyze bioconvection in a dilute suspension of gyrotactic microorganisms in horizontal shallow fluid layer cooling from below and saturated by a porous medium, in the rigid boundary case. It is established that due to cooling from below thermally stratified layer is stabilized, which opposes the development of bioconvection and the situations for oscillatory convection may take place. The stability criterion is obtained in terms of thermal Rayleigh number, bioconvection Rayleigh number, gyrotactic number, bioconvection Peclet number, measure of cell eccentricity, Prandtl number, and Lewis number. It is observed that the presence of porous medium results in decrease of the magnitude of critical bioconvection Rayleigh number in comparison with its non-existence; hence due to porous effect, the system becomes less stable.     

Dynamics of dual-particles settling under gravity

As a first step towards understanding particle–particle interaction in fluid flows, the motion of two spherical particles settling in close proximity under gravity in Newtonian fluids was investigated experimentally for particle Reynolds numbers ranging from 0.01 to 2000. It was observed that particles repel each other for Re>0.1 and that the separation distance of settling particles is Reynolds number dependent. At lower Reynolds numbers, i.e. for Re<0.1, particles settling under gravity do not separate.The orientation preference of two spherical particles was found to be Reynolds number dependent. At higher Reynolds numbers, the line connecting the centres of the two particles is always horizontal, regardless of the way the two particles are launched. At lower Reynolds numbers, however, the particle centreline tends to tilt to an arbitrary angle, even of the two particles are launched in the horizontal plane. Because of the tilt, a side migration of the two particles was found to exist. A linear theory was developed to estimate the side migration velocity. It was found that the maximum side migration velocity is approximately 6% of the vertical settling velocity, in good agreement with the experimental results.Counter-rotating spinning of the two particles was observed and measured in the range of Re=0–10. Using the linear model, it is possible to estimate the influence of the tilt angle on the rate of rotation at low Reynolds numbers. Dual particles settle faster than a single particle at small Reynolds numbers but not at higher Reynolds numbers, because of particle separation. The variation of particle settling velocity with Reynolds number is presented. An equation which can be used to estimate the influence of tilt angle on particle settling velocity at low Reynolds number is also derived.     

Coupling Eulerian‐Lagrangian method of air‐particle two‐phase flow with population balance equations to simulate the evolution of vehicle exhaust plume

In this paper, we present a new numerical scheme to describe the dynamic evolution of multiphase polydisperse systems in terms of time, space, and properties by coupling the Eulerian‐Lagrangian method for air‐particle two‐phase flow and population balance equations to describe particle property evolution due to microbehaviors (eg, aggregation, breakage, and growth). This coupling scheme was used to comprehensively simulate the two‐phase flow structure, particle size spectrum, particle number, and volume concentrations. These were characterized by a high‐resolution particle tracking using the Lagrangian approach and the high precision of moments of the particle size spectrum by solving the population balance equation with the quadrature method of moments. The algorithm of the coupling scheme was incorporated into the open source computational fluid dynamics software OpenFOAM to simulate the dynamic evolution of vehicle exhaust plume. The impacts of vehicle velocity, exhaust temperature, and aggregation efficiency on the distribution of auto exhaust particles in space and changes in their properties were analyzed. The results indicate that the particle number concentration, volume concentration, and average diameter of particles in the vehicle exhaust plume could be strongly affected by the plume structure and flow properties.     

Instability Analysis of Gyrotactic Microorganisms: A Combined Effect of High-Frequency Vertical Vibration and Porous Media

A linear stability analysis is carried out to predict the instability analysis in a dilute suspension of gyrotactic microorganisms in horizontal fluid-saturated porous layer influenced by high-frequency vertical vibration. The governing equations, describing the mean flow, are the time-averaged Boussinesq equations and the analytical solution of the problem has been obtained using Galerkin method. A secular relation involving bioconvection Rayleigh number and its vibrational analogs and other parameters have been established. The graphical interpretations for dependence of bioconvection Rayleigh–Darcy number and corresponding wave number, on gyrotactic number and bioconvection Péclet number in the presence of vibration are utilized to understand the problem.     

A hybrid Lagrangian–Eulerian particle‐level set method for numerical simulations of two‐fluid turbulent flows

A coupled Lagrangian interface‐tracking and Eulerian level set (LS) method is developed and implemented for numerical simulations of two‐fluid flows. In this method, the interface is identified based on the locations of notional particles and the geometrical information concerning the interface and fluid properties, such as density and viscosity, are obtained from the LS function. The LS function maintains a signed distance function without an auxiliary equation via the particle‐based Lagrangian re‐initialization technique. To assess the new hybrid method, numerical simulations of several ‘standard interface‐moving’ problems and two‐fluid laminar and turbulent flows are conducted. The numerical results are evaluated by monitoring the mass conservation, the turbulence energy spectral density function and the consistency between Eulerian and Lagrangian components. The results of our analysis indicate that the hybrid particle‐level set method can handle interfaces with complex shape change, and can accurately predict the interface values without any significant (unphysical) mass loss or gain, even in a turbulent flow. The results obtained for isotropic turbulence by the new particle‐level set method are validated by comparison with those obtained by the ‘zero Mach number’, variable‐density method. For the cases with small thermal/mass diffusivity, both methods are found to generate similar results. Analysis of the vorticity and energy equations indicates that the destabilization effect of turbulence and the stability effect of surface tension on the interface motion are strongly dependent on the density and viscosity ratios of the fluids. Copyright © 2007 John Wiley & Sons, Ltd.     

A numerical continuous model for the hydrodynamics of fluid particle systems

In order to understand the hydrodynamic interactions that can appear in a fluid particle motion, an original method based on the equations governing the motion of two immiscible fluids has been developed. These momentum equations are solved for both the fluid and solid phases. The solid phase is assumed to be a fluid phase with physical properties, such as its behaviour can be assimilated to that of pseudo‐rigid particles. The only unknowns are the velocity and the pressure defined in both phases. The unsteady two‐dimensional momentum equations are solved by using a staggered finite volume formulation and a projection method. The transport of each particle is solved by using a second‐order explicit scheme. The physical model and the numerical method are presented, and the method is validated through experimental measurements and numerical results concerning the flow around a circular cylinder. Good agreement is observed in most cases. The method is then applied to study the trajectory of one settling particle initially off‐centred between two parallel walls and the corresponding wake effects. Different particle trajectories related to particulate Reynolds numbers are presented and commented. A two‐body interaction problem is investigated too. This method allows the simulation of the transport of particles in a dilute suspension in reasonable time. One of the important features of this method is the computational cost that scales linearly with the number of particles. Copyright © 1999 John Wiley & Sons, Ltd.     

Reconciliation of size–density bivariate distributions over a separating node

Data reconciliation considers the restoration of mass balance among the noise prone measured data by way of component adjustments for the various particle size or particle density classes or assays over the separating node. In this paper, the method of Lagrange multipliers has been extended to balance bivariate feed and product size-density distributions of coal particles split from a settling column. The settling suspension in the column was split into two product fractions at 40% height from the bottom after a minute settling of homogenized suspension at start. Reconciliation of data assists to estimate solid flow split of particles to the settled stream as well as helps to calculate the profiles of partition curves of the marginal particle size or particle density distributions. In general, Lagrange multiplier method with uniform weighting of its components may not guarantee a smooth partition surface and thus the reconciled data needs further refinement to establish the nature of the surface. In order to overcome this difficulty, a simple alternative method of reconciling bivariate size-density data using partition surface concept is explored in this paper.     

A kernel gradient free (KGF) SPH method

The finite particle method (FPM) is a modified SPH method with high order accuracy while retaining the advantages of SPH in modeling problems with free surfaces, moving interfaces, and large deformations. In both SPH and FPM, kernel gradient is necessary in kernel and particle approximation of a field function and its derivatives. In this paper, a new FPM is presented, which only involves kernel function itself in kernel and particle approximation. The kernel gradient is not necessary in the whole computation, and this approach is thus referred to as a kernel gradient free (KGF) SPH method. This is helpful when a kernel function is not differentiable or the resultant kernel gradient is not sufficiently smooth, and thus it is more general in selecting a kernel function. Moreover, different from the original FPM with an asymmetric corrective matrix, in the new FPM, the resultant corrective matrix is symmetric, and this is advantageous in particle approximations. A series of numerical examples have been conducted to show the efficiencies of KGF‐SPH including one‐dimensional mathematical tests of polynomial functions with equal or variable smoothing length and two‐dimensional incompressible fluid flow of shear cavity. It is found that KGF‐SPH is comparable with FPM in accuracy and is flexible as SPH. Copyright © 2015 John Wiley & Sons, Ltd.     

On the approximate zeroth and first‐order consistency in the presence of 2‐D irregular boundaries in SPH obtained by the virtual boundary particle methods

In this paper, a new method to impose 2‐D solid wall boundary conditions in smoothed particle hydrodynamics is presented. The wall is discretised by means of a set of virtual particles and is simulated by a local point symmetry approach. The extension of a previously published modified virtual boundary particle (MVBP) method guarantees that arbitrarily complex domains can be readily discretised guaranteeing approximate zeroth and first‐order consistency. To achieve this, three important new modifications are introduced: (i) the complete support is ensured not only for particles within one smoothing length distance, h, from the boundary but also for particles located at a distance greater than h but still within the support of the kernel; (ii) for a non‐uniform fluid particle distribution, the fictitious particles are generated with a uniform stencil (unlike the previous algorithms) that can maintain a uniform shear stress on a particle‐moving parallel to the wall in a steady flow; and (iii) the particle properties (density, mass and velocity) are defined using a local point of symmetry to satisfy the hydrostatic conditions and the Cauchy boundary condition for pressure. The extended MVBP model is demonstrated for cases including hydrostatic conditions for still water in a tank with a wedge and for curved boundaries, where significant improved behaviour is obtained in comparison with the conventional boundary techniques. Finally, the capability of the numerical scheme to simulate a dam break simulation is also shown. Copyright © 2015 John Wiley & Sons, Ltd.     

The sedimentation of polydisperse suspensions in vessels having inclined walls

A theory is presented for describing the sedimentation of polydisperse suspensions in two-dimensional channels having walls that are inclined to the vertical. The theory assumes that the flow is laminar and that the suspension consists of spherical beads having small particle Reynolds numbers. The suspension may consist of either $\text{N}$ distinct species of particles or of a continuum of particle sizes and densities. For the sake of simplicity, the analysis is mostly confined to the case in which the hindered settling velocity of each particle is given by its Stokes settling velocity multiplied by a function of the total local solids concentration. Under these conditions, results are developed that are useful for the design of either batch or continuous settling devices. Experimental observations were found to be in good agreement with the predictions of the present theory.     

A boundary element analysis of multiply connected three‐dimensional cavity mixing flow of polymer solutions

The emergence of non‐linear dynamics in cavity mixing is examined using the boundary element method (BEM). The method is implemented for the simulation of three‐dimensional transient creeping flow of Newtonian or linear viscoelastic fluids of the Jeffreys type. A boundary only formulation in the time domain is proposed for viscoelastic flow. Special emphasis is placed on cavity flow involving multiply connected moving domains. The BEM becomes particularly suited for this case, when part of the boundary (stirrer or rotor) is moving, and the remaining outer part (cavity) is at rest. In contrast to conventional volume methods, the BEM is shown to be much easier to implement since the kinematics of the elements bounding the fluid is known (imposed). It is found that, for a simple cavity flow induced by a rotating vane at constant angular velocity, the tractions at the vane tip and cavity face exhibit non‐linear periodic dynamical behaviour with time for fluids obeying linear constitutive equations. Copyright © 1999 John Wiley & Sons, Ltd.     

Numerical study on heavy rigid particle motion in a plane wake flow by spectral element method

Experimental particle dispersion patterns in a plane wake flow at a high Reynolds number have been predicted numerically by discrete vortex method (Phys. Fluids A 1992; 4 :2244–2251; Int. J. Multiphase Flow 2000; 26 :1583–1607). To address the particle motion at a moderate Reynolds number, spectral element method is employed to provide an instantaneous wake flow field for particle dynamics equations, which are solved to make a detail classification of the patterns in relation to the Stokes and Froude numbers. It is found that particle motion features only depend on the Stokes number at a high Froude number and depend on both numbers at a low Froude number. A ratio of the Stokes number to squared Froude number is introduced and threshold values of this parameter are evaluated that delineate the different regions of particle behavior. The parameter describes approximately the gravitational settling velocity divided by the characteristic velocity of wake flow. In order to present effects of particle density but preserve rigid sphere, hollow sphere particle dynamics in the plane wake flow is investigated. The evolution of hollow particle motion patterns for the increase of equivalent particle density corresponds to that of solid particle motion patterns for the decrease of particle size. Although the thresholds change a little, the parameter can still make a good qualitative classification of particle motion patterns as the inner diameter changes. Copyright © 2008 John Wiley & Sons, Ltd.     

Reconciliation of size-density bivariate distributions over a separating node

Data reconciliation considers the restoration of mass balance among the noise prone measured data by way of component adjustments for the various particle size or particle density classes or assays over the separating node. In this paper, the method of Lagrange multipliers has been extended to balance bivariate feed and product size-density distributions of coal particles split from a settling column. The settling suspension in the column was split into two product fractions at 40% height from the bottom after a minute settling of homogenized suspension at start. Reconciliation of data assists to estimate solid flow split of particles to the settled stream as well as helps to calculate the profiles of partition curves of the marginal particle size or particle density distributions. In general, Lagrange multiplier method with uniform weighting of its components may not guarantee a smooth partition surface and thus the reconciled data needs further refinement to establish the nature of the surface. In order to overcome this difficulty, a simple alternative method of reconciling bivariate size-density data using partition surface concept is explored in this paper.     

Simulation of incompressible viscous flows around moving objects by a variant of immersed boundary‐lattice Boltzmann method

A variant of immersed boundary‐lattice Boltzmann method (IB‐LBM) is presented in this paper to simulate incompressible viscous flows around moving objects. As compared with the conventional IB‐LBM where the force density is computed explicitly by Hook's law or the direct forcing method and the non‐slip condition is only approximately satisfied, in the present work, the force density term is considered as the velocity correction which is determined by enforcing the non‐slip condition at the boundary. The lift and drag forces on the moving object can be easily calculated via the velocity correction on the boundary points. The capability of the present method for moving objects is well demonstrated through its application to simulate flows around a moving circular cylinder, a rotationally oscillating cylinder, and an elliptic flapping wing. Furthermore, the simulation of flows around a flapping flexible airfoil is carried out to exhibit the ability of the present method for implementing the elastic boundary condition. It was found that under certain conditions, the flapping flexible airfoil can generate larger propulsive force than the flapping rigid airfoil. Copyright © 2009 John Wiley & Sons, Ltd.     

The Onset of Bioconvection in a Horizontal Porous-Medium Layer

In this note the problem of the onset of bioconvection in a horizontal layer occupied by a saturated porous medium is analyzed. Gyrotactic effects are incorporated in the analysis. The Darcy flow model is employed, and it is assumed that the bioconvection Péclet number is not greater than unity. Critical values of the bioconvection Rayleigh number and the corresponding critical Rayleigh number are obtained for various values of the bioconvection Péclet number, the gyrotaxis number and the cell eccentricity.