Free-energy-based lattice Boltzmann model for the simulation of multiphase flows with density contrast.

A free-energy-based phase-field lattice Boltzmann method is proposed in this work to simulate multiphase flows with density contrast. The present method is to improve the Zheng-Shu-Chew (ZSC) model [Zheng, Shu, and Chew, J. Comput. Phys. 218, 353 (2006)] for correct consideration of density contrast in the momentum equation. The original ZSC model uses the particle distribution function in the lattice Boltzmann equation (LBE) for the mean density and momentum, which cannot properly consider the effect of local density variation in the momentum equation. To correctly consider it, the particle distribution function in the LBE must be for the local density and momentum. However, when the LBE of such distribution function is solved, it will encounter a severe numerical instability. To overcome this difficulty, a transformation, which is similar to the one used in the Lee-Lin (LL) model [Lee and Lin, J. Comput. Phys. 206, 16 (2005)] is introduced in this work to change the particle distribution function for the local density and momentum into that for the mean density and momentum. As a result, the present model still uses the particle distribution function for the mean density and momentum, and in the meantime, considers the effect of local density variation in the LBE as a forcing term. Numerical examples demonstrate that both the present model and the LL model can correctly simulate multiphase flows with density contrast, and the present model has an obvious improvement over the ZSC model in terms of solution accuracy. In terms of computational time, the present model is less efficient than the ZSC model, but is much more efficient than the LL model.

Numerical investigation of transporting droplets by spatiotemporally controlling substrate wettability.

Droplet behaviors on substrates with wettability controlled both in space and in time are numerically investigated by using the lattice Boltzmann method. Several typical droplet responses are found under different designs of substrate wettability control. Special attention is drawn to the conditions under which rapid transport of droplets can be achieved. It is found that on alternating non-wetting-wetting units with proper non-wetting confining stripes, this objective can be realized when the frequency of wettability switch approximately matches that of the droplet to move across one unit. The variation of the "optimal" frequency with the size of the confining stripe is sought within certain ranges. The various types of droplet movement are analyzed by looking at the three-phase lines on the substrate, as well as the droplet shapes under different conditions. The results may provide useful implications for droplet manipulation in microfluidic devices.

Tank-treading, swinging, and tumbling of liquid-filled elastic capsules in shear flow.

The dynamic motion of three-dimensional (3D) capsules in a shear flow is investigated by direct numerical simulation. The capsules are modeled as Newtonian liquid droplets enclosed by elastic membranes, with or without considering the membrane-area incompressibility. The internal liquid of the capsules is the same as that outside. The dynamic motion of capsules with initially spherical and oblate spheroidal unstressed shapes is studied under various shear rates. The results show that spherical capsules deform to stationary configurations and then the membranes rotate around the liquid inside (steady tank-treading motion). Such a steady mode is not observed for oblate spheroidal capsules. It is shown that with the shear rate decreasing, the motion of oblate spheroidal capsules changes from the swinging mode (a capsule undergoes periodic shape deformation and inclination oscillation while its membrane is rotating around the liquid inside) to tumbling mode.

Transient deformation of elastic capsules in shear flow: effect of membrane bending stiffness.

The transient deformation of liquid capsules enclosed by elastic membranes with bending rigidity in shear flow has been 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. Bending moments, accompanied by transverse shear tensions, usually develop due to a preferred membrane configuration or its nonzero thickness. The present model can simulate flow induced deformation of capsules with arbitrary resting shapes (concerning the in-plane tension) and arbitrary configurations at which the bending energy has a global minimum (minimum bending-energy configurations). The deformation of capsules with initially circular, elliptical, and biconcave resting shapes was studied; the capsules' minimum bending-energy configurations were considered as either uniform-curvature shapes (like circle or flat plate) or their initially resting shapes. The results show that for capsules with minimum bending-energy configurations having uniform curvature (circle or flat plate), the membrane carries out tank-treading motion, and the steady deformed shapes become more rounded if the bending stiffness is increased. For elliptical and biconcave capsules with resting shapes as minimum bending-energy configurations, it is quite interesting to find that with the bending stiffness increasing, the capsules' motion changes from tank-treading mode to flipping mode, and resembles Jeffery's flipping mode at large bending stiffness.

Alternative method to construct equilibrium distribution functions in lattice-Boltzmann method simulation of inviscid compressible flows at high Mach number.

A method is proposed to construct an equilibrium density distribution function in the simulation of compressible flows at high Mach number by the lattice-Boltzmann method. In this method, the conventional Maxwellian distribution function is replaced by a circular function which is very simple and satisfies all needed statistical relations to recover the compressible Navier-Stokes equations. The circular function is then distributed to the lattice velocity directions by Lagrangian interpolation in such a way that all the needed statistical relations are exactly satisfied when the integral in the phase space is replaced by the summation in the context of the lattice-Boltzmann (LB) method. In this framework, the equilibrium distribution functions and the associated lattice velocity model can be derived naturally without assuming specific forms. Two LB models with adjustable specific heat ratio, respectively, for one-dimensional (1D) and two-dimensional (2D) compressible flows are shown in the paper. Some test cases of compressible flows with strong shock waves are simulated to validate the present approach. Excellent results are obtained. Note that in the simulation, the total variation diminishing (TVD) scheme was used to capture the discontinuity with coarse mesh.

Lattice Boltzmann interface capturing method for incompressible flows.

A lattice Boltzmann interface capturing method for incompressible flows is proposed in this paper. The interface is naturally captured by minimizing the free energy functional. It is easily implemented and does not require interface reconstruction as required by most of the traditional interface tracking methods such as the volume of fluid method. Moreover, the method does not require the isotropic property of the fourth order lattice tensor as do other lattice Boltzmann methods. Thus the D2Q5 (D2 means two dimensional, Q5 means five velocity model) discrete velocity model is applied in the method. The interface profile along the flat interface and coexistence curve can be given analytically. The proposed method is validated for some test cases, and compared to the volume of fluid and level set methods. Numerical results show that the present method performs very well and can generate very sharp interfaces.

Chaotic micromixers using two-layer crossing channels to exhibit fast mixing at low Reynolds numbers.

We report two chaotic micromixers that exhibit fast mixing at low Reynolds numbers in this paper. Passive mixers usually use the channel geometry to stir the fluids, and many previously reported designs rely on inertial effects which are only available at moderate Re. In this paper, we propose two chaotic micromixers using two-layer crossing channels. Both numerical and experimental studies show that the mixers are very efficient for fluid manipulation at low Reynolds numbers, such as stretching and splitting, folding and recombination, through which chaotic advection can be generated and the mixing is significantly promoted. More importantly, the generation of chaotic advection does not rely on the fluid inertial forces, so the mixers work well at very low Re. The mixers are benchmarked against a three-dimensional serpentine mixer. Results show that the latter is inefficient at Re = 0.2, while the new design exhibits rapid mixing at Re = 0.2 and at Re of O(10(-2)). The new mixer design will benefit various microfluidic systems.

Lattice kinetic scheme for the incompressible viscous thermal flows on arbitrary meshes.

A lattice kinetic scheme was developed for the incompressible viscous thermal flows on arbitrary meshes. The work was based on the lattice kinetic scheme proposed by Inamuro and the technique of Taylor series expansion- and least-square-based lattice Boltzmann method (TLLBM). Compared with the lattice Boltzmann method, the lattice kinetic scheme can save the computer memory since there is no need to store the density distributions. The implementation of the boundary condition is direct and just the same as the convectional Navier-Stokes solvers. By using the idea of TLLBM, the lattice kinetic scheme can be applied on arbitrary meshes, which makes this scheme suitable for practical applications. In order to validate this lattice kinetic scheme used on arbitrary meshes, numerical simulations of the natural convection in a square cavity and the natural convection in a concentric annulus between an outer square cylinder and an inner circular cylinder are carried out, and the results are compared very well with available data in the literature.

Simplified thermal lattice Boltzmann model for incompressible thermal flows.

Considering the fact that the compression work done by the pressure and the viscous heat dissipation can be neglected for the incompressible flow, and its relationship with the gradient term in the evolution equation for the temperature in the thermal energy distribution model, a simplified thermal energy distribution model is proposed. This thermal model does not have any gradient term and is much easier to be implemented. This model is validated by the numerical simulation of the natural convection in a square cavity at a wide range of Rayleigh numbers. Numerical experiments showed that the simplified thermal model can keep the same order of accuracy as the thermal energy distribution model, but it requires much less computational effort.

Cavitation phenomena in mechanical heart valves: the role of squeeze flow velocity and contact area on cavitation initiation between two impinging rods.

In this study, the closing dynamics of two impinging rods were experimentally analyzed to simulate the cavitation phenomena associated with mechanical heart valve closure. The purpose of this study was to investigate the cavitation phenomena with respect to squeeze flow between two impinging surfaces and the parameter that influences cavitation inception. High-speed flow imaging was employed to visualize and identify regions of cavitation. The images obtained favored squeeze flow as an important mechanism in cavitation inception. A correlation study of the effects of impact velocities, contact areas and squeeze flow velocity on cavitation inception showed that increasing impact velocities results in an increase in the risk of cavitation. It was also shown that for similar impact velocities, regions near the point of impact were found to cavitate later for those with smaller contact areas. It was found that the decrease in contact areas and squeeze flow velocities would delay the onset and reduce the intensity of cavitation. It is also interesting to note that the squeeze flow velocity alone does not provide an indication if cavitation inception will occur. This is corroborated by the wide range of published critical squeeze flow velocity required for cavitation inception. It should be noted that the temporal acceleration of fluid, often neglected in the literature, can also play an important role on cavitation inception for unsteady flow phenomenon. This is especially true in mechanical heart valves, where for the same leaflet closing velocity, valves with a seat stop were observed to cavitate earlier. Based on these results, important inferences may be made to the design of mechanical heart valves with regards to cavitation inception.

Numerical simulation of natural convection in a concentric annulus between a square outer cylinder and a circular inner cylinder using the Taylor-series-expansion and least-squares-based lattice Boltzmann method.

In this paper, natural convective heat transfer in a horizontal concentric annulus between a square outer cylinder and a heated circular inner cylinder is numerically studied using the Taylor-series-expansion and least-squares-based lattice Boltzmann method (TLLBM). The TLLBM is used to extend the current thermal model to more practical applications. Since the TLLBM is basically a meshless approach and can be applied to any complex geometry, we can easily use it to solve the complex thermal problem accurately and effectively. The present method is validated by comparing its numerical results with available data in the literature, and very good agreement has been achieved.

Taylor-series expansion and least-squares-based lattice Boltzmann method: Two-dimensional formulation and its applications.

An explicit lattice Boltzmann method (LBM) is developed in this paper to simulate flows in an arbitrary geometry. The method is based on the standard LBM, Taylor-series expansion, and the least-squares approach. The final formulation is an algebraic form and essentially has no limitation on the mesh structure and lattice model. Theoretical analysis for the one-dimensional (1D) case showed that the version of the LBM could recover the Navier-Stokes equations with second order accuracy. A generalized hydrodynamic analysis is conducted to study the wave-number dependence of shear viscosity for the method. Numerical simulations of the 2D lid-driven flow in a square cavity and a polar cavity flow as well as the "no flow" simulation in a square cavity have been carried out. Favorable results were obtained and compared well with available data in the literature, indicating that the present method has good prospects in practical applications.

Least-squares-based lattice Boltzmann method: a meshless approach for simulation of flows with complex geometry.

A version of lattice Boltzmann method (LBM) is presented in this work, which is derived from the standard LBM by using Taylor series expansion and optimized by the least squares method. The method is basically meshless, and can be applied to any complex geometry and nonuniform grids. It can also be applied to different lattice models. The proposed method explicitly updates the distribution functions at mesh points by an algebraic formulation, in which the relevant coefficients are precomputed from the coordinates of mesh points. We have successfully applied this method to simulate many two-dimensional incompressible viscous flows. The numerical results are very accurate, and the computational time needed is much less as compared with other existing methods. In this paper, we mainly show the method.

Pulsatile flow studies of a porcine bioprosthetic aortic valve in vitro: PIV measurements and shear-induced blood damage.

A two-dimensional particle image tracking velocimetry (PIV) system has been used to map the velocity vector fields and Reynolds stresses in the immediate downstream vicinity of a porcine bioprosthetic heart valve at the aortic root region in vitro under pulsatile flow conditions. Measurements were performed at five different time steps of the systolic phase of the cardiac cycle. The velocity vector fields and Reynolds stress mappings at different time steps allowed us to chart a time history of the stress levels experienced by fluid particles as they move across the aortic root. This Lagrangian description of the stresses experienced by individual blood cells enabled us to estimate the propensity of shear-induced damage to platelets and red blood cells. Coupled with flow visualization techniques, the hydrodynamic consequences of introducing a porcine bioprosthetic heart valve into the aortic root was examined. Although the PIV measurements may lack the accuracy of single point measuring systems, the overall view of the flow in the aortic root region compensates for the shortcoming.

Steady flow dynamics of prosthetic aortic heart valves: a comparative evaluation with PIV techniques.

Particle Image Velocimetry (PIV), capable of providing full-field measurement of velocities and flow stresses, has become an invaluable tool in studying flow behaviour in prosthetic heart valves. This method was used to evaluate the performances of four prosthetic heart valves; a porcine bioprostheses, a caged ball valve, and two single leaflet tilting disc valves with different opening angles. Flow visualization techniques, combined with velocity vector fields and Reynolds stresses mappings in the aortic root obtained from PIV, and pressure measurements were used to give an overall picture of the flow field of the prosthetic heart valves under steady flow conditions. The porcine bioprostheses exhibited the highest pressure loss and Reynolds stresses of all the valves tested. This was mainly due to the reduction in orifice area caused by the valve mounting ring and the valve stents. For the tilting disc valves, a larger opening angle resulted in a smoother flow profile, and thus lower Reynolds stresses and pressure drops. The St. Vincent valve exhibited the lowest pressure drop and Reynolds stresses.

Pulmonary airway reopening: effects of non-Newtonian fluid viscosity.

This paper considers the effects of non-Newtonian lining-fluid viscosity, particularly shear thinning and yield stress, on the reopening of the airways. The airway was simulated by a very thin, circular polyethylene tube, which collapsed into a ribbon-like configuration. The non-Newtonian fluid viscosity was described by the power-law and Herschel-Buckley models. The speed of airway opening was determined under various opening pressures. These results were collapsed into dimensionless pressure-velocity relationships, based on an assumed shear rate gamma = U/(0.5 H), where U and H are the opening velocity and fluid film thickness, respectively. It was found that yield stress, like surface tension, increases the yield pressure and opening time. However, shear thinning reduces the opening time. An increased film thickness of the non-Newtonian lining fluid generally impedes airway reopening; a higher pressure is needed to initiate the airway reopening and a longer time is required to complete the opening process.

Steady flow velocity field and turbulent stress mappings downstream of a porcine bioprosthetic aortic valve in vitro.

Velocity profiles and Reynolds stresses downstream of heart valve prostheses are vital parameters in the study of hemolysis and thrombus formation associated with these valves. These parameters have previously been evaluated using single-point measurement techniques such as laser Doppler anemometry (LDA). The purpose of this study is to map the velocity vector fields and Reynolds stresses downstream of a porcine bioprosthetic heart valve in the aortic root region with particle image velocimetry (PIV) techniques in vitro under steady flow conditions. PIV is essentially a multipoint measurement technique that allows full-field measurement of instantaneous velocity vectors in a flow field, thus allowing us to map the entire velocity or stress field over the aortic root (where single-point measurements are difficult). Coupled with flow visualization techniques, the hydrodynamic consequences of introducing a porcine bioprosthetic heart valve into the aortic root was examined, and compared with data obtained from an empty aortic root and an aortic root with the valve mounting ring alone. From our velocity and stress mappings, we found that the valve mounting ring effectively diminishes the central orifice area, giving rise to a higher central axial flow with strong recirculating regions and a corresponding large pressure drop. This in turn produces an intermixing zone between the central jet and recirculating region further downstream from the valve, which contributes to the high-stress zone measured. The development of the flow is further restricted by the valve stents, giving rise to stagnation regions and wakes. High-velocity gradients were also measured at the interface of the jet and recirculating region in the sinus cavity. The overall view of the velocity and stress mappings helps to identify regions of flow disturbances that otherwise may be lost with single-point measuring systems. Although the PIV measurements may lack the accuracy of single-point measuring systems, the overall view of the flow in the aortic root region compensates for the shortcoming.

Pressure/flow behaviour in collapsible tube subjected to forced downstream pressure fluctuations.

An experimental investigation has been made into the pressure/flow behaviour of a collapsible tube subjected to downstream pressure fluctuations. These downstream pressure waves are observed to be transmitted upstream beyond the point of collapse. The mean flow rate is not significantly affected by the amplitude or frequency of pressure fluctuations. However, the oscillatory flow amplitude is reduced at the higher frequency. The mean flow rate also remains independent of the mean driving pressure.

Particle image velocimetry in the investigation of flow past artificial heart valves.

Full-field measurement of instantaneous velocities in the flow field of artificial heart valves is vital as the flow is unsteady and turbulent. Particle image velocimetry (PIV) provides us the ability to do this as compared to other point measurement devices where the velocity is measured at a single point in space over time. In the development of a PIV system to investigate the flow field of artificial heart valves, many problems associated with the project arose and were subsequently resolved. Experience gained in the setting up of an environment conducive for PIV studies of artificial heart valves; from seed particle selection to refractive index matching, and the evolution of computer algorithms to satisfy the varied flow conditions in artificial heart valves are presented here. Velocity profiles and distributions are computed and drawn for a porcine tissue heart valve based on measurements with the PIV system developed.

Laser anemometry measurements of steady flow past aortic valve prostheses.

An experimental investigation was conducted in steady flow to examine the fluid dynamics performance of three prosthetic heart valves of 27 mm diameter: Starr-Edwards caged ball valve, Bjork-Shiley convexo-concave tilting disk valve, and St. Vincent tilting disk valve. It was found that the pressure loss across the St. Vincent valve is the least and is, in general, about 70 percent of that of the Starr-Edwards valve. The pressure recovery is completed about 4 diameters downstream. The velocity profiles for the ball valve reveal a large single reversed flow region behind the occluder while those for the tilting disks valves reveal two reversed flow regions immediately behind the occluders. Small regions of stasis are also found near the wall in the minor opening of Bjork-Shiley valve and in the major opening of St. Vincent valve. The maximum wall shear stresses of the three valves at a flow rate of 30 l/min are in the range 30-50 dyn/cm2 which can cause hemolysis of attached red blood cells. The corresponding maximum Reynolds normal stresses are in the range of 1600-3100 dyn/cm2. The Reynolds normal stresses decay quickly and return approximately to the upstream undisturbed level at about 4 diameters downstream while the wall shear stresses decay at a slower rate. The maximum Reynolds normal stresses occur at about 1 diameter downstream while the maximum wall shear stress is at about 2 diameters downstream. In general, the St. Vincent valve has better performance.(ABSTRACT TRUNCATED AT 250 WORDS)