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The evolution dynamics of wettability-induced jumping droplet is studied numerically by means of a high-density ratio based lattice Boltzmann method. The three phase moving contact line is modelled by a geometry based wetting formulation. We analyse the influence of pulse characteristics with respect to its nature, pulse width and double pulse actuation on the jumping dynamics. It is shown that these factors strongly affect the droplet lift-off mechanism. The influence of surrounding gas density and viscosity on the droplet behaviour is investigated systematically. It is found that the role of the ambient fluid becomes prominent during the recoiling and the lift-off phase of the jumping droplet. Finally, we demonstrate the applicability of wettability-induced droplet jumping in microfluidic devices where efficient transport of droplet is essential.

Three dimensional computations on successive droplet impingement on a solid surface are reported. A high-density ratio based lattice Boltzmann model is employed for the present two phase computations and a geometric-based contact angle formulation is used to model the moving contact line. The focus of the present work is on the impact behaviour and the interaction dynamics between the impinging droplets with an initial gap. The effects of trailing drop velocity, surface wettability, drop viscosity and surface tension on the time-dependent dynamics are investigated during the impingement process. Two different transient interaction modes are observed for different velocity ratios. With increase in advancing contact angle, it is observed that primary spreading increases in the case of in-phase mode of collision. Effect of eccentricity, through the offset between the droplets, is found to result in an asymmetric evolution of droplet morphologies. Passive particle tracers are seeded inside the droplets to visualize mixing process as a time sequence of closely interacting droplets during the impact with a solid surface.

A liquid drop impacting on a solid substrate is simulated to evaluate the effect of inhomogeneities in the wetting characteristics of the surface. A high density ratio multiphase solver based on phase-field lattice Boltzmann formulation has been employed. Influence of receding contact angle and wettability gradient is investigated to analyse the coupled dynamics of the impacting droplet on the solid surface. Simulations indicate that the receding contact angle is a key wetting parameter to determine the final physical outcome of the impact process. Interplay between tangential velocity component and receding contact angle results in interesting dynamical behaviours in the evolution of the droplet shape for oblique impact scenarios. Substrate inhomogeneities in the form of wettability gradient is found to induce a directional behaviour to the motion of the impacting droplet. The droplet motion is characterised by intense recoiling along the upstream part of the droplet, followed by secondary spreading of the downstream contact line.

We present a numerical study on the dynamic behaviour of two droplets impinging simultaneously onto a liquid film. A high density ratio based lattice Boltzmann method is employed for the present two-phase computations. Formation of a central uprising jet is observed due to the collision and coalescence of in between rims caused by the impacting droplets. Effect of separation gap between the droplets, film thickness, viscosity and gas density on the crown structure and jet behaviour are systematically studied. The influence of a moving wall with a liquid film on the jet behaviour is further investigated. It is shown that for a larger separation gap between the droplets, a delay in formation of central jet is observed while the spread length increases. For the thin films, the central jet height increases with the increase in film thickness, however a reversal in this behaviour is observed for the thicker films. The rate of increase of jet height and spread length is found to decrease with increase in gas density. Effect of wall velocity is found to enhance the upstream jet, while it suppresses the central and the downstream jets.

In this review, we provide an overview of the simulation techniques employed for modelling the flow of red blood cells (RBCs) in blood plasma. The scope of this review omits the fluid modelling aspect while focusing on other key components in the RBC-plasma model such as (1) describing the RBC deformation with shell-based and spring-based RBC models, (2) constitutive models for RBC aggregation based on bridging theory and depletion theory and (3) additional strategies required for completing the RBC-plasma flow model. These include topics such as modelling fluid-structure interaction with the immersed boundary method and boundary integral method, and updating the variations in multiphase fluid property through the employment of index field methods. Lastly, we summarily discuss the current state and aims of RBC modelling and suggest some research directions for the further development of this field of modelling.

Three dimensional simulations are performed to investigate the interaction dynamics between two drops impinging simultaneously on a dry surface. Of particular interest in this study is to understand the effects of impact velocity and surrounding gas density on droplet interactions. To simulate the droplet dynamics and morphologies, a computational framework based on the phase-field lattice Boltzmann formulation is employed for the two-phase flow computations involving high density ratio. Two different coalescence modes are identified when the impinging droplets have different impact speeds. When one of the droplet has a tangential impact velocity component, asymmetric ridge formation is observed. Influence of droplet impact angle on the interaction dynamics of the central ridge is further investigated. Traces of different fluid particles are seeded to analyse internal flow dynamics in oblique impact scenarios. Greater overlapping between the fluid particles is observed with increase in the impact angle. Finally, the present simulations indicate that the ambient gas density has a significant influence to determine the final outcome of the droplet interactions.

The behavior of a droplet impinging onto a solid substrate can be influenced significantly by the horizontal motion of the substrate. The coupled interactions between the moving wall and the impacting droplet may result in various outcomes, which may be different from the usual normal droplet impact on a stationary wall. In this paper, we present a method to suppress drop rebound on hydrophobic surfaces via transverse wall oscillations, normal to the impact direction. The numerical investigation shows that the suppression of droplet rebound has a direct relationship with the oscillation phase, amplitude, and frequency. For a particular range of oscillation frequencies and amplitudes, a lateral shifting of the droplet position is observed along the oscillating direction. While large oscillation amplitude favors the process of droplet deposition, a high frequency promotes droplet rebound from the oscillating wall. A linear trend in the transition region between deposition and rebound is observed from a scaled phase diagram of the oscillation amplitude versus frequency. We provide a systematic investigation of drop deposition and elucidate the mechanism of rebound suppression through the temporal evolution of the nonaxial kinetic energy and the velocity flow field.

A two-dimensional flow model, incorporating mass transport, has been developed to simulate flow in a microchannel bioreactor with a porous wall. A two-domain method was implemented which was based on finite volume method. For the porous-fluid interface, a stress jump condition was used with continuity of normal stress; and the mass interfacial conditions were continuities of mass and mass flux. Two parameters are defined to characterize the mass transports in the fluid and porous regions. The porous Damkohler number is the ratio of consumption to diffusion of the substrates in the porous medium. The fluid Damkohler number is the ratio of substrate consumption in the porous medium to substrate convection in the fluid region. The concentration results are found to be well correlated by the use of a reaction-convection distance parameter which incorporates the effects of axial distance, substrate consumption and convection.

The steady flow around and through a porous circular cylinder was studied numerically. The effects of the two important parameters, the Reynolds and Darcy numbers, on the flow were investigated in details. The recirculating wake existing downstream of the cylinder is found to either penetrate into or be completely detached from the cylinder. It is also found that, contrary to that of the solid cylinder, the recirculating wake develops downstream of or within the porous cylinder, but not from the surface of it. These new findings provide additional evidence to Leal’s conclusion (Leal LG. Vorticity transport and wake structure for bluff bodies at finite Reynolds number. Phys Fluids A 1989;1:124) that the appearance of recirculating wakes at finite Reynolds number is due to vorticity accumulation, but not a result of the same physical phenomena associated with separation in boundary layers in adverse pressure gradients. Also presented in the current study are the variation of the critical Reynolds number for the onset of a recirculating wake as a function of Darcy number and the variation of a newly defined parameter, the penetration depth, as a function of the Reynolds number and Darcy number.

A two-dimensional flow model, incorporating mass transport, has been developed to simulate a microchannel enzyme reactor with a porous wall. A two-domain approach based on the finite volume method was implemented. Two parameters are defined to characterize the mass transports in the fluid and porous regions: the porous Damkohler number and the fluid Damkohler number. For reactions close to first-order type (enzyme reactor), the concentration results are found to be well correlated by the use of a reaction–convection distance parameter which incorporates the effects of axial distance, substrate consumption and convection. The reactor efficiency reduces with reaction–convection distance parameter because of reduced reaction (or flux) due to the lower concentration. Increased fluid convection improves the efficiency but it is limited by the diffusion in the fluid region. The correlated results can find applications for the design of enzyme reactors with a porous wall.

Numerical simulations have been performed for the flow past and through a porous square cylinder over a wide range of Reynolds numbers (1 ⩽ Re ⩽ 50) and Darcy numbers (10−6 ⩽ Da ⩽ 10−1). It is found that the recirculating wake existing downstream of the cylinder is completely detached from the body under a certain range of parameters. The size and the location of the wake mainly depend on Re and Da. Specially, the wake may initially increase but then decrease in size with an increase in Re, and eventually disappear when the Reynolds number is sufficiently large. The variation of the critical Reynolds number for the onset of recirculating wake as a function of Darcy number is summarized. The results also show that in the range of Da investigated, different from that of the solid cylinder, the recirculating wake is initially developed downstream of the porous cylinder, but not on the surface of it.

In this paper, a hybrid method is proposed to study the flow-induced deformation of three-dimensional capsules. The capsules consist of Newtonian liquid drops enclosed by thin elastic membranes. In the proposed approach, the front-tracking method is coupled with the lattice Boltzmann method. The fluids inside and outside the capsule is treated as one fluid with varying physical properties, and is modeled by the lattice Boltzmann equation. The capsule membrane is explicitly tracked by the membrane nodes that are advected by the flow. The multi-block strategy of the lattice Boltzmann method is employed to refine the mesh near the capsule, which greatly increase the accuracy and efficiency of the three-dimensional computation. The capsule membrane is discretized into unstructured flat triangular elements, and a finite element model is incorporated to account for the membrane mechanics. With the present method, the transient deformation of initially spherical capsules with membrane following Neo-Hookean constitutive laws is simulated in shear flow, under various dimensionless shear rates and ratios of internal to surrounding liquid viscosities. The present results, including the Taylor shape parameter, the capsule inclination angle and the tank-treading frequency, agree well with previously published numerical results.

The transient deformation of two-dimensional non-circular and three-dimensional non-spherical capsules in simple shear flow is studied numerically, using the hybrid immersed boundary and multi-block lattice Boltzmann method recently proposed by the present authors. The capsules are modeled as Newtonian liquid drops enclosed by elastic membranes; the liquids inside and outside the capsule have the same physical properties. The present results show important different behaviors between two-dimensional and three-dimensional capsules in shear flow. For two-dimensional non-circular capsules without considering the membrane bending rigidity, or considering the bending with the minimum bending-energy configurations (shapes at which the bending-energy has a global minimum) being uniform-curvature shapes, the capsules will always achieve the steady tank treading motion (a capsule deforms to a steady shape with a steady inclination and the membrane rotates around the liquid inside). However, for three-dimensional non-spherical capsules without membrane bending rigidity, such a steady mode does not exist; with the shear rate decreasing, the three-dimensional capsules’ motion changes from swinging mode (a capsule undergoes periodic shape deformation and inclination oscillation while its membrane is rotating around the liquid inside) to flipping mode. The deformation of two-dimensional capsules, with their initial non-circular shapes taken as the minimum bending-energy configurations, is also considered. It is quite interesting to find such two-dimensional capsules behave qualitatively similar to three-dimensional capsules: there is a swinging-to-flipping transition induced by lowering the shear rate.

The flow through a channel partially filled with fibrous porous medium was analyzed to investigate the interfacial boundary conditions. The fibrous medium was modeled as a periodic array of circular cylinders, in a hexagonal arrangement, using the boundary element method. The area and volume average methods were applied to relate the pore scale to the representative elementary volume scale. The permeability of the modeled fibrous medium was calculated from the Darcy's law with the volume-averaged Darcy velocity. The slip coefficient, interfacial velocity, effective viscosity and shear jump coefficients at the interface were obtained with the averaged velocities at various permeabilities or Darcy numbers. Copyright © 2008 John Wiley & Sons, Ltd.

The effect of cell density on the fluid dynamics and oxygen transport in a micro-bioreactor with a tissue engineering scaffold was investigated. Variations of the permeability and porosity of the scaffold with cell density were estimated by a simple model. Based on the permeability and the porosity obtained, the flow and oxygen transport in a micro-bioreactor were simulated numerically. It was found that the flow rate passing through the scaffold is reduced with an increase in cell density. The oxygen concentration within the scaffold also decreases with an increase in cell density, due to lower oxygen convection caused by lower porous flow rate and higher oxygen consumption. The results suggest that, for high cell density culture, the oxygen concentration within the scaffold may be overestimated if the cell density effect is neglected.

A numerical method was developed for flows involving an interface between a homogenous fluid and a porous medium. The numerical method is based on the lattice Boltzmann method for incompressible flow. A generalized model, which includes Brinkman term, Forcheimmer term and nonlinear convective term, was used to govern the flow in the porous medium region. At the interface, a shear stress jump that includes the inertial effect was imposed for the lattice Boltzmann equation, together with a continuity of normal stress. The present method was implemented on three cases each of which has a porous medium partially occupying the flow region: channel flow, plug flow and lid-driven cavity flow. The present results agree well with the analytical and/or the finite-volume solutions.

PurposeIn this paper three‐dimensional numerical simulations were conducted for the periodically developed laminar flow in the sinusoid wavy fin‐and‐tube heat exchanger.Design/methodology/approachA novel CLEARER algorithm is adopted to guarantee the fully coupling between the pressure and velocity, and it can not only speed up the convergence rate, but also overcome the severe grid non‐orthogonality in the wavy fin‐and‐tube heat exchanger. The influence of wave amplitude, fin pitch, tube diameter and wave density on fluid flow and heat transfer characteristics is analyzed under different Reynolds numbers.FindingsThe numerical results show that with the increase of wave amplitude, tube diameter or wave density, both the friction factor and Nusselt number will increase, and the increase rate of friction factor is higher than that of Nusselt number. It is interesting to note that, at low Reynolds numbers the Nusselt number increases with the decrease of fin pitch, while at high Reynolds numbers, the Nusselt number increases with the increase of fin pitch.Originality/valueThe numerical results presented in this paper may provide some useful guidance in the design of the wavy fin‐and‐tube heat exchanger with large number of rows of tubes.

There have been a few recent numerical implementations of the stress-jump condition at the interface of conjugate flows, which couple the governing equations for flows in the porous and homogenous fluid domains. These previous demonstration cases were for two-dimensional, planar flows with simple geometries, for example, flow over a porous layer or flow through a porous plug. The present study implements the interfacial stress-jump condition for a non-planar flow with three velocity components, which is more realistic in terms of practical flow applications. The steady, laminar, Newtonian flow in a stirred micro-bioreactor with a porous scaffold inside was investigated. It is shown how to implement the interfacial jump condition on the radial, axial, and swirling velocity components. To avoid a full three-dimensional simulation, the flow is assumed to be independent of the azimuthal direction, which makes it an axisymmetric flow with a swirling velocity. The present interface treatment is suitable for non-flat surfaces, which is achieved by applying the finite volume method based on body-fitted and multi-block grids. The numerical simulations show that a vortex breakdown bubble, attached to the free surface, occurs above a certain Reynolds number. The presence of the porous scaffold delays the onset of vortex breakdown and confines it to a region above the scaffold. Copyright © 2008 John Wiley & Sons, Ltd.

PurposeThe paper aims to report on the flow past a porous trapezoidal‐cylinder, in which the porous‐fluid interface was treated by implementing the stress jump boundary conditions.Design/methodology/approachThe numerical method was based on the finite‐volume method with body‐fitted and multi‐block grids. The Brinkman‐Forcheimmer extended model was used to govern the flow in the porous medium region. At its interface, a shear stress jump that includes the inertial effect was imposed, together with a continuity of normal stress.FindingsThe present model was validated by comparing with those for the flow around a solid circular cylinder. Results for flow around porous expanded trapezoidal cylinder are presented with flow configurations for different Darcy number, 10−2 to 10−7, porosity from 0.4 to 0.8, and Reynolds number 20 to 200. The flow develops from steady to unsteady periodic vortex shedding state. The first coefficient β has a more noticeable effect, whereas the second coefficient β1 has very small effect, even for Re = 200.Originality/valueThe effects of the porosity, Darcy number and Reynolds number on lift and drag coefficients, and the length of circulation zone or shedding period are studied.

An experimental set-up was introduced to study the dynamic behaviour of different types of check valves under pressure transient conditions. Three types of transient comparison methods were used, with similar results being obtained from all three methods. The experimental results show that the check valves with low inertia, assisted by springs or small travelling distance/angle, gave better performance under pressure transient conditions as compared to check valves without these features. Although different amounts of air entrainment were found to affect the experimental readings, the general characteristics of each check valve remain the same when compared between valves. This study can be applied to help in the choosing of suitable check valves for a particular pumping system.

The transient deformation of red blood cells (RBCs) in a shear flow is studied by a three-dimensional numerical model proposed by the present authors. The RBCs are approximated by ghost cells consisting of Newtonian liquid drops enclosed by Skalak membranes. The RBCs have an initially biconcave discoid resting shape, and the internal liquid is assumed to be the same to the fluid outside. The simulation is based on a hybrid method, in which the immersed boundary concept is introduced into the framework of the lattice Boltzmann method, and a finite element model is incorporated to obtain the forces acting on the nodes of the cell membrane which is discretized into flat triangular elements. The dynamic motion of RBCs is investigated in simple shear flow under a broad range of shear rates. At large shear rates, the present results show that the cells carry out a swinging motion, in which periodic inclination-oscillation and shape deformation superimpose on the membrane tank treading motion. With the shear rate decreasing, the swinging amplitude of the cell increases, and finally triggers a transition to tumbling motion.

The transient deformation of cylindrical liquid-filled capsules with elastic membranes is studied in simple shear flow at small and moderate Reynolds numbers. The aim of the study is to investigate the inertia effect on the deformation of elastic capsules and the flow structure around them. The simulation is based on a hybrid method which introduces the immersed boundary concept in the framework of the multi-block lattice Boltzmann model. The deformation of capsules with circular and elliptical resting shapes is studied; the membrane constitutive model is Hooke’s law. However, the present model can be implemented to capsules with arbitrary resting configurations and constitutive laws. The simulation results show that inertia has important effect on the transient deformation process, steady configuration and tank treading frequency of a capsule in simple shear flow; Inertia also significantly affects the flow structure and vorticity field around and inside a capsule.

The flow environment in the micro-bioreactor with a tissue engineering scaffold was numerically modeled. The finite volume method based on multi-block grid was applied to simulate the coupled flow and oxygen transport in both porous medium and homogenous fluid regions. At porous–fluid interface, a stress jump condition that includes both viscous and inertial effects was imposed. A parametric study was performed to investigate the effects of Reynolds, Darcy, and Damkohler numbers on the flow and oxygen concentration fields inside and outside the scaffold. The minimum oxygen concentration in the scaffold and its location under different conditions were summarized.

Vortex breakdown effects on oxygen concentration distributions in a micro-bioreactor were investigated by a finite volume model. The bioreactor consists of a cylindrical chamber with a rotating bottom wall. The minimum oxygen concentration and volumetric oxygen transfer coefficient were summarized at different Reynolds numbers, Damkohler numbers as well as bioreactor aspect ratios. It was found that vortex breakdown extends the axis region where oxygen concentration is higher. Also vortex breakdown tends to induce a uniform oxygen concentration distribution in the bioreactor. Thus, the occurrence of vortex breakdown is of benefit to animal cell culture. Compared with that in the bioreactor with a rigid lid, vortex breakdown in the bioreactor with free surface may be better for animal cell culture as the bubble has larger size and may occur apart from the axis.

In this paper, the conjugate heat transfer in electronic cooling is numerically simulated with the newly proposed algorithm CLEARER on collocated grid. Because the solid heat source and substrate are isolated from the boundary, special attention is given to deal with the velocity and temperature in the solid region in the full field computation. The influence of openings on the substrate, heat source height and their distribution along the substrate on the maximum temperature and overall Nusselt number is investigated. The numerical results show that the openings on the substrate can enhance the heat transfer as well as increasing the heat source height, meanwhile, by arranging the heat sources coarsely in the front part and densely in the rear part of the substrate, the thermal performance can also be increased. Then the results are analyzed from the viewpoint of field synergy principle, and it is shown that the heat transfer improvement can all be attributed to the better synergy between the velocity field and temperature field, which may offer some guidance in the design of electronic devices.

An experiment setup was introduced to study dynamic behaviour of different types of check valves and the effects of air entrainment on the check valve performance under pressure transient condition. The experiment results show that the check valves with low inertia, assisted by springs or small traveling distance/angle gave better performance under pressure transient condition than check valves without these features. Air entrainment was found to affect both wave speed and reverse velocity. With the increase of the initial air void fraction in pipeline, the experiment results show that the wave speed was reduced, the reverse velocity was increased. The first peak pressure increased initially and then decreased with the increase of the initial air void fraction, the pressure surge periods were increased proportionally with air void fraction due to the greatly reduced wave speed. The study can be applied to help choosing suitable check valves for a particular pumping system.

The newly proposed CLEARER algorithm is extended to three-dimensional nonorthogonal curvilinear coordinates. This algorithm can not only guarantee full coupling between the pressure and velocity terms, and solution independence of underrelaxation factors, it can also satisfy the physical and geometric conservation laws, hence it owns high efficiency and more robustness than its companion SIMPLE-like algorithms. To ease implementation of the CLEARER algorithm, the irregular physical domain is transformed to a regular computational domain, and correspondingly, the governing equations are also transformed. The mathematical formulations in both the predictor step and the corrector step are provided in detail, as well as the solution procedure. The CLEARER algorithm is adopted to simulate numerically the fluid flow and heat transfer in a triangular wavy fin-and-tube heat exchanger, and the influence of eccentricity and transverse tube pitch on the friction factor and Nusselt number is investigated in detail. I...

Forced convection after a backward-facing step, with a porous floor segment, is investigated numerically using the SIMPEC method. The Brinkman-Forcheimmer extended model is used to govern the flow in the porous-medium region. At the interface, the flow boundary condition imposed is a shear stress jump, which includes the inertial effect, together with a continuity of normal stress. The thermal interfacial condition is continuities of temperature and heat flux. Results are presented for Reynolds number up to 800 and Darcy number up to 10−1. Also varied are the length and depth of the porous segment. Compared with the case with no porous floor, the local heat transfer is augmented after the porous floor. Within the porous floor, the heat transfer is reduced, but this may be offset by using a porous medium of higher conductivity than the fluid. To obtain good heat enhancement after the porous segment, it should approximately match the length of the recirculation region. The porous segment should have large p...

A numerical method was developed for flows involving an interface between a homogeneous fluid and a porous medium. It is based on the finite volume method with body-fitted and multi-block grids. The Brinkman–Forcheimmer extended model was used to govern the flow in the porous medium region. At its interface, the flow boundary condition imposed is a shear stress jump, which includes the inertial effect, together with a continuity of normal stress. The thermal boundary condition is continuity of temperature and heat flux. The forced convection through a porous insert over a backward-facing step is investigated. The results are presented with flow configurations for different Darcy numbers, 10−2 to 10−5, porosity from 0.2 to 0.8, Reynolds number from 10 to 800, and the ratio of insert length to channel height from 0.1 to 0.3. The heat transfer is improved by using porous insert. To enhance the heat transfer with minimal frictional losses, it is preferable to have a medium length of insert with medium Darcy number, and larger Reynolds number. The interfacial stress jump coefficients β and β1 were varied from −1 to 1, and within this range the average and local lower-wall Nusselt numbers are not sensitive to the parameters. Copyright © 2007 John Wiley & Sons, Ltd.

In this article the three-dimensional periodically developed incompressible flow in the triangular wavy fin-and-tube heat exchanger is numerically simulated. A novel CLEARER algorithm is adopted to guarantee the coupling between the pressure and velocity, and it can also overcome the severe grid non-orthogonality in the complex wavy fin-and-tube heat exchanger. The influence of wavy angle, fin pitch, tube diameter, and wavy density on pressure drop and heat transfer characteristics is provided under different Reynolds numbers. The numerical results show that with the increase of wavy angle, tube diameter, or wavy density both the friction factor and Colburn factor will increase, while the increase rate of friction factor is higher than that of the Colburn factor. Opposite to the wavy fin-and-tube heat exchanger with uniform inlet velocity distribution, the wavy fin-and-tube heat exchanger with periodically developed flow owns a higher Colburn factor at larger fin pitch. The influence of wavy angle, fin pi...

The transient deformation of liquid capsules enclosed by elastic membranes in two-dimensional extensional flow is studied numerically, using an improved immersed boundary-lattice Boltzmann method. The purpose of the present study is to investigate the effect of interfacial bending stiffness on the deformation of such capsules, under the subcritical elasticity capillary number conditions. The present model can simulate flow-induced deformation of capsules with arbitrary resting shapes (concerning the in-plane tension) and bending-free configurations. The deformation of capsules with initially circular, elliptical, and biconcave resting shapes was investigated in the present study; the capsules' bending-free configurations were considered as either circular shapes or their initially resting shapes. The results show that for capsules with bending-free configuration as circles, membrane bending rigidity has significant rounding effect on the steady deformed profiles. For elliptical and biconcave capsules with resting shapes as the bending-free configurations, it is found that with the bending stiffness increasing, the capsules' steady shapes are more akin to their initial shapes.

A numerical investigation is carried out for steady, free convection inside a cavity filled with a porous medium. The cavity has vertical wavy walls which are isothermal. The top and bottom horizontal straight walls are kept adiabatic. The numerical method is based on the finite-volume method with body-fitted and nonorthogonal grids. A generalized model, which includes a Brinkman term, a Forcheimmer term, and a nonlinear convective term, is used. Studies are carried out for a range of wave ratio λ = 0 − 1.8, aspect ratio A = 1–5, Darcy number Da = 10−1–10−6, and Darcy-Rayleigh number Ra∗ = 10–105. Results are presented in the form of streamlines, isotherms, and local and average Nusselt numbers. The generalized model which considers viscous, inertia, and convective effects enables results to be obtained for a wider range of Darcy and Rayleigh numbers.

In this paper, the updating of the intermediate velocity in the correction stage in SIMPLE-like algorithms on staggered grid is discussed in detail, based on the discussion CLEARER algorithm that is extended from the non-staggered grid to the staggered grid. The performance of SIMPLER and CLEARER algorithms are compared using two numerical examples with reliable solutions. The results show that CLEARER can predict the numerical results as accurately as SIMPLER, and it can also enhance the convergence rate greatly under low under-relaxation factors. In some cases CLEARER algorithm can only need 27% of iteration number required by SIMPLER to reach the same convergence criterion.

The transient deformation of a liquid-filled elastic capsule, simulating a red blood cell, was studied in simple shear flow. The simulation is based on a hybrid method which introduces the immersed boundary concept in the framework of the multi-block lattice Boltzmann model. The method was validated by the study on deformation of an initially circular capsule with Hooke's membrane. Also studied were capsules with Skalak membrane of initially elliptical and biconcave shapes, which are more representative of a red blood cell. Membrane tank treading motion is observed. As the ratio between membrane dilation modulus and shear modulus increases, the capsule shows asymptotic behavior. For an initially elliptical capsule, it is found that the steady shape is independent of initial inclination angle. For an initially biconcave capsule, the tank treading frequency from two-dimensional modeling is comparable to that of real cells. Another interesting finding is that the tank treading velocity has not attained steady state when the capsule shape becomes steady; and at this state there is the internal vortex pair. The treading velocity continues to decrease and reaches a steady value when the internal vortex pair has developed into a single vortex.

In this article, an Improved SIMPLER (CLEARER) algorithm is formulated to solve the incompressible fluid flow and heat transfer on the nonstaggered, nonorthogonal curvilinear grid system. By virtue of a modified momentum interpolation method in calculating the interface contravariant velocity in both the predictor step and the corrector step, the coupling between pressure and velocity is fully guaranteed, and the conservation law is also satisfied. A second relaxation factor is introduced in the corrector step, of which the convergent solution is independent. By setting the second relaxation factor less than the underrelaxation factor for the velocity to some extent, both the convergence rate and robustness can be greatly enhanced. Meanwhile, the CLEARER algorithm can also overcome the severe grid nonorthogonality. With the simplified pressure-correction equation, the convergent solution can still be obtained even when the intersection angle among grid lines is as low as 1°, which may provide valuable gui...

In this article an Improved SIMPLER (CLEARER) algorithm is proposed to solve incompressible fluid flow and heat transfer problems. Numerical study shows with the CLEARER algorithm on a collocated grid, in the correction stage the velocities on the main nodes are overcorrected with the pressure correction, which lowers the convergence rate; hence a second relaxation factor is introduced to overcome this disadvantage. By setting this factor less than the underrelaxation factor for velocities, the convergence performance can be significantly enhanced; meanwhile, the robustness can also be increased. Four numerical examples with reliable solutions are computed to validate the CLEARER algorithm, and the results show that this algorithm can predict the numerical results accurately. Compared with the SIMPLER algorithm, CLEARER can enhance the convergence rate greatly, and in some cases it only needs as little as 17% of the iterations required by SIMPLER to reach the same convergence criterion.

Surface roughness exists in most microfluidic devices due to the microfabrication technique or particle adhesion. The present study has developed a numerical model based on Finite Volume Method to simulate the fluid flow and mass transfer in a flat-plate microchannel bioreactor with an array of rough elements uniformly placed on the bottom wall. Both semicircle and triangle roughness are considered to include more shapes of roughness elements. A monolayer of cells is assumed to attach to the base of the channels and consumes species from culture medium. The results show that the roughness size ratio (α) and the roughness distribution ratio (β) have direct and significant effects on fluid flow and mass transfer. The dimensionless parameters Peclet number (Pe) and Damkohler number (Da) can also influence mass transfer greatly. Although the two types of roughness have similar effects, at the same condition, the triangle roughness has larger effect on shear stress by showing higher dimensionless values at the channel base; the semicircle roughness has larger effect on mass transfer by showing lower dimensionless minimum base concentration [Formula: see text] and higher dimensionless absorption rate (Δj%). However, it is important to ensure the lower maximum shear stress and the adequate minimum species concentration for cell growth in rough channels. Hence, if the maximum shear stress and minimum concentration in rough channels can satisfy the critical conditions for cell growth, rough channels would be better than smooth channels because of their lower shear stress at the flat-bed part and higher mass transfer efficiency. The results would provide guidance on the flow and perfusion requirements to avoid shear stress damage and solute depletion or toxicity during cell culture.

Microchannel bioreactors have been used in many studies to investigate the fluid microenvironment around cells. In the present study, a three-dimensional numerical model was developed to simulate the fluid flow and mass transfer in a 3D flat-plate rectangular microchannel bioreactor with a monolayer of cells adherent to the bottom. The numerical results of mass transfer were well correlated in terms of two dimensionless parameters: Sherwood number at base (Sh) and effective distance (κ). The generalized results show the mass transfer characteristic in the flat-plate microchannel bioreactors, which may find applications in culturing cells in these bioreactors.

A numerical model was developed for flows involving an interface between a homogenous fluid and a porous medium. The numerical model is based on the finite volume method with body-fitted and multi-block grids. The Darcy-Forchheimer extended model is used to govern the flow in the porous medium region. At its interface, a shear stress jump was imposed, together with a continuity of normal stress. Furthermore, the effect of the jump condition on the diffusive flux is considered, additional to that on the convective part which has been usually considered. Numerical results of three flow configurations are presented. The modeling is suitable for problems which have complex interface boundary conditions coupling between two flow domains.

A numerical method is developed for modelling the interactions between incompressible viscous fluid and moving boundaries. The principle of this method is introducing the immersed-boundary concept in the framework of the lattice Boltzmann method, and improving the accuracy and efficiency of the simulation by refining the mesh near moving boundaries. Besides elastic boundary with a constitutive law, the method can also efficiently simulate solid moving-boundary interacting with fluid by employing the direct forcing technique. The method is validated by the simulations of flow past a circular cylinder, two cylinders moving with respect to each other and flow around a hovering wing. The versatility of the method is demonstrated by the numerical studies including elastic filament flapping in the wake of a cylinder and fish-like bodies swimming in quiescent fluid. Copyright © 2006 John Wiley & Sons, Ltd.

A 3D numerical model was developed to study the flow field and oxygen transport in a micro-bioreactor with a rotating magnetic bar on the bottom to mix the culture medium. The Reynolds number (Re) was kept in the range of 100-716 to ensure laminar environment for animal cell culture. The volumetric oxygen transfer coefficient (k(L)a) was determined from the oxygen concentration distribution. It was found that the effect of the cell consumption on k(L)a could be negligible. A correlation was proposed to predict the liquid-phase oxygen transfer coefficient (k(Lm)) as a function of Re. The overall oxygen transfer coefficient (k(L)) was obtained by the two-resistance model. Another correlation, within an error of 15%, was proposed to estimate the minimum oxygen concentration to avoid cell hypoxia. By combination of the correlations, the maximum cell density, which the present micro-bioreactor could support, was predicted to be in the order of 10(12) cells m(-3). The results are comparable with typical values reported for animal cell growth in mechanically stirred bioreactors.

In this paper the fluid flow and heat transfer characteristics of mixed convection in three-dimensional rectangular channel with four heat sources are investigated numerically. The SIMPLEC algorithm is applied to deal with the coupling between pressure and velocity, and a new high-order stability-guaranteed second-order difference (SGSD) scheme is adopted to discretize the convection term. The influence of four parameters is studied: Richardson number, heat source distribution, channel height and inclination angle. The numerical results are analysed from the viewpoint of the field synergy principle, which says that the enhanced convective heat transfer is related not only to the velocity field and temperature field, but also to the synergy between them. It is found that the effects of the four parameters on the thermal performance can all be explained with the field synergy principle. To obtain better electronic cooling, the synergy between the velocity and temperature gradient should be increased when other conditions are unchanged. Copyright © 2006 John Wiley & Sons, Ltd.

The present work shows that vortex breakdown may also occur in a bioreactor for animal cell or tissue culture. The aim is to study the effect of vortex breakdown on the fluid environment for cell growth, particularly hydrodynamic stress and mass transfer. A numerical simulation of the fluid flow and oxygen transfer in a cylindrical bioreactor is presented, in which a rotating base wall is used to generate the medium flow and mixing. The steady and laminar, axisymmetric Navier-Stokes and species equations are solved by the numerical model based on finite volume method. The hydrodynamic stress and oxygen transfer conditions are systematically studied by varying the two key parameters which determine the flow behavior: bioreactor aspect ratio H/R and a rotation Reynolds number Re=ΩR2/ν. It is found that the oxygen concentration at the attached breakdown vortex center is significantly higher than that at the main recirculation center but the hydrodynamic stress level is almost similar in the two regions. The results would provide guidance on choosing the proper operating parameters for cell or tissue culture.

Surface roughness exists in most microfluidic devices due to the microfabrication technique or particle adhesion. In this study, a numerical model based on Finite Volume Method has been developed to simulate the mass transfer in a flat-plate microchannel bioreactor with semi-circular protrusions uniformly distributed on the bottom. The results show that the mass transfer in rough channel is enhanced, as shown by lower minimum species concentration in the rough channel compared with that in smooth channel. Non-dimensional parameters such as Peclet number (Pe), Damkohler number (Da) and the roughness size ratio (β) can influence the effect of roughness greatly. However, it is important to ensure that the minimum species concentration in the rough channel is adequate for cell growth. The results would provide guidance on the perfusion requirements to avoid solute depletion or toxicity during cell culture.

A numerical investigation on the characteristics of transitional turbulent flow over series bell-shape stenoses for a physiological pulsatile flow is presented in the present study. The flow behaviours for the physiological pulsatile flow are studied by considering the effects of the Reynolds number, Womersley number, constriction ratio and spacing ratio of the stenoses on the pulsatile turbulent flow fields. Especially, the mutual influences between the double stenoses under different flow conditions are considered. The numerical results show that the variation of these flow parameters puts significant impacts on the flow developments in the arteries with series stenoses. The double stenoses lead to the higher peak turbulence disturbance and the greater area with comparatively high turbulence intensity distal to the stenoses in comparison with the single stenosis. The analysis shows that for the physiological pulsatile flow, the downstream stenosis usually does not have perceptible influences on the upstream flow fields. Copyright © 2004 John Wiley & Sons, Ltd.

A detailed analysis on the characteristics of transitional turbulent flow over a bell-shape stenosis for a physiological pulsatile flow is presented. The comparison of the numerical solutions to three types of pulsatile flows, including a physiological flow, an equivalent pulsatile flow and a simple pulsatile flow, are made in this work. Then the effects of the Reynolds number, Womersley number and constriction ratio of stenosis on the pulsatile turbulent flow fields for the physiological flow are considered. The comparison of the three pulsatile flows shows that the flow characteristics cannot be properly estimated if an equivalent or simple pulsatile inflow is used instead of actual physiological one in the study of the pulsatile flows through arterial stenosis. The equivalent or simple pulsatile inflow can lead to higher disturbance intensity in the vicinity of the stenosis than the physiological inflow. For a physiological flow, the recirculation zones with high disturbance intensity occur mainly in the distal of the stenosis. The larger Reynolds number and severer constriction ratio may result in more complex flow field and cause some important flow variables to increase dramatically near stenosis. The higher Womersley number leads to a larger phase lag between the imposed flow rate changes and the final converged flow field in one cycle. The turbulence intensity decreases with the increase of Womersley number for the same Reynolds number.

The flow fields in the neighbourhoods of series vascular stenoses are studied numerically for the Reynolds numbers from 100 to 4000, diameter constriction ratios of 0.2–0.6 and spacing ratios of 1, 2, 3, 4 and ∞. In this study, it has been further verified that in the laminar flow region, the numerical predictions by k–ω turbulence model matched those by the laminar-flow modelling very well. This suggests that the k–ω turbulence model is capable of the prediction of the laminar flow as well as the prediction of the turbulent stenotic flow with good accuracy. The extent of the spreading of the recirculation region from the first stenosis and its effects on the flow field downstream of the second stenosis depend on the stenosis spacing ratio, constriction ratio and the Reynolds number. For c1 = 0.5 with c2 ≤ c1, the peak value of wall vorticity generated by the second stenosis is always less than that generated by the first stenosis. However, the maximum centreline velocity and turbulence intensity at the second stenosis are higher than those at the first stenosis. In contrast, for c1 = 0.5 with c2 = 0.6, the maximum values at the second stenosis are much higher than those at the first stenosis whether for centreline velocity and turbulence intensity or for wall vorticity. The peak values of the wall vorticity and the centreline disturbance intensity both grow up with the Reynolds number increasing. The present study shows that the more stenoses can result in a lower critical Reynolds number that means an earlier occurrence of turbulence for the stenotic flows. Copyright © 2003 John Wiley & Sons, Ltd.