The adaptation of Crouzeix-Raviart finite element in the context of multi-scale finite element method (MsFEM) is studied and implemented on diffusion and advection-diffusion problems in perforated media. It is known that the approximation of boundary condition on coarse element edges when computing the multiscale basis functions critically influences the eventual accuracy of any MsFEM approaches. The weakly enforced continuity of Crouzeix-Raviart function space across element edges leads to a natural boundary condition for the multiscale basis functions which relaxes the sensitivity of our method to complex patterns of perforations. Another ingredient to our method is the application of bubble functions which is shown to be instrumental in maintaining high accuracy amid dense perforations. Additionally, the application of penalization method makes it possible to avoid complex unstructured domain and allows extensive use of simpler Cartesian meshes.

A mass-conserving lattice Boltzmann model based on the Bhatnagar-Gross-Krook (BGK) model is proposed for non-Newtonian fluid flows. The equilibrium distribution function includes the local shear rate related with the viscosity and a variable parameter changing with the shear rate. With the additional parameter, the relaxation time in the collision can be fixed invariable to the viscosity. Through the Chapman-Enskog analysis, the macroscopic equations can be recovered from the present mass-conserving model. Two flow problems are simulated to validate the present model with a local computing scheme for the shear rate, and good agreement with analytical solutions and/or other published results are obtained. The results also indicate that the present modified model is more applicable to practical non-Newtonian fluid flows owing to its better accuracy and more robustness than previous methods.

Homogeneous equilibrium model (HEM) has been widely used in cavitating flow simulations. The major feature of this model is that a single equation of state (EOS) is proposed to describe the thermal behavior of bubbly liquid, where both kinematic and thermal equilibrium is assumed between two phases. In this paper, the HEM was coupled with multi-relaxation-time lattice Boltzmann model (MRT-LBM) and the acoustic behavior was simulated. Two approaches were applied alternatively: adjusting speed of sound (Buick, J. Phys. A, 2006, 39:13807-13815) and setting real gas EOS. Both approaches result in high accuracy in acoustic speed predictions for different void (gas) volume of fractions. It is demonstrated that LBM could be successfully applied as a Navier-Stokes equation solver for industrial applications. However, further dissipation and dispersion analysis shows that Shan-Chen type approaches of LBM are deficient, especially in large wave-number region.

In this paper, the fluid flow and heat transfer around a circular cylinder are studied under various conditions (Reynolds number 10 < Re < 200; Prandtl number, 0.1 ≤ Pr ≤ 2). To solve the governing equations, we use the simplified thermal lattice Boltzmann model based on double-distribution function approach, and present a corresponding boundary treatment for both velocity and temperature fields. Extensive numerical results have been obtained to the flow and heat transfer behaviors. The vortices and temperature evolution processes indicate that the flow and temperature fields change synchronously, and the vortex shedding plays a determinant role in the heat transfer. Furthermore, the effects of Reynolds and Prandtl number on the flow and isothermal patterns and local and averaged Nusselt numbers are discussed in detail. Our simulations show that the local and averaged Nusselt numbers increase with the Reynolds and Prandtl numbers, irrespective of the flow regime. However, the minimum value of the local Nusselt number can shift from the rear point at the back of the cylinder with higher Prandtl number even in the steady flow regime, and the distribution of the local Nusselt number is almost monotonous from front stagnation point to rear stagnation point with lower Prandtl number in the unsteady flow regime.

Conducting lattice Boltzmann method on GPU has been proved to be an effective manner to gain a significant performance benefit, thus the GPU or multi-GPU based lattice Boltzmann method is considered as a promising and competent candidate in the study of large-scale complex fluid flows. In this work, a multi-GPU based lattice Boltzmann algorithm coupled with the sparse lattice representation and message passing interface is presented. Some numerical tests are also carried out, and the results show that a parallel efficiency close to 90% can be achieved on a single-node cluster equipped with four GPU cards. Then the proposed algorithm is adopted to study the hemodynamics of patient-specific cerebral aneurysm with stent implanted. It is found that the stent can apparently reduce the aneurysmal inflow and improve the hemodynamic environment. This work also shows that the lattice Boltzmann method running on the GPU platform is a powerful tool to study the fluid mechanism within the aneurysms and enable us to better understand the pathogenesis and treatment of cerebral aneurysms.

The fine description of complex fluids can be carried out by describing the evolution of each individual constituent (e.g. each particle, each macromolecule, etc.). This procedure, despite its conceptual simplicity, involves many numerical issues, the most challenging one being that related to the computing time required to update the system configuration by describing all the interactions between the different individuals. Coarse grained approaches allow alleviating the just referred issue: the system is described by a distribution function providing the fraction of entities that at certain time and position have a particular conformation. Thus, mesoscale models involve many different coordinates, standard space and time, and different conformational coordinates whose number and nature depend on the particular system considered. Balance equation describing the evolution of such distribution function consists of an advection-diffusion partial differential equation defined in a high dimensional space. Standard mesh-based discretization techniques fail at solving high-dimensional models because of the curse of dimensionality. Recently the authors proposed an alternative route based on the use of separated representations. However, until now these approaches were unable to address the case of advection dominated models due to stabilization issues. In this paper this issue is revisited and efficient procedures for stabilizing the advection operators involved in the Boltzmann and Fokker-Planck equation within the PGD framework are proposed.

Finite-difference Lattice Boltzmann (LB) models are proposed for simulating gas flows in devices with microscale geometries. The models employ the roots of half-range Gauss-Hermite polynomials as discrete velocities. Unlike the standard LB velocity-space discretizations based on the roots of full-range Hermite polynomials, using the nodes of a quadrature defined in the half-space permits a consistent treatment of kinetic boundary conditions. The possibilities of the proposed LB models are illustrated by studying the one-dimensional Couette flow and the two-dimensional square driven cavity flow. Numerical and analytical results show an improved accuracy in finite Knudsen flows as compared with standard LB models.

Mixing processes of miscible viscous fluids in anisotropic porous media are studied through a lattice Boltzmann method in this paper. The results show that fluid mixing can be enhanced by the disorder and interfaces induced by viscous fingers. Meanwhile, the permeability anisotropy affects the developments of viscous fingers and the subsequent mixing behaviors significantly. Specifically, as the streamwise (longitudinal) permeability is larger than the spanwise (transverse) one, the mixing process is faster and stronger than that in the contrary case. Furthermore, the anisotropy can lead to different behaviors of the dissipation rates in streamwise and spanwise directions. Generally, the dissipation rate is dominated by the transverse concentration gradients when the longitudinal permeability is higher than the transverse one; on the contrary, as the transverse permeability is higher than the longitudinal one, the contributions to the dissipation rate from longitudinal and transverse concentration gradients are both significant.

The effective thermal conductivity of composite materials with thermal contact resistance at interfaces is studied by lattice Boltzmann modeling in this work. We modified the non-dimensional partial bounce-back scheme, proposed by Han et al. [Int. J. Thermal Sci., 2008. 47: 1276-1283], to introduce a real thermal contact resistance at interfaces into the thermal lattice Boltzmann framework by re-deriving the redistribution function of heat at the phase interfaces for a corrected dimensional formulation. The modified scheme was validated in several cases with good agreement between the simulation results and the corresponding theoretical solutions. Furthermore, we predicted the effective thermal conductivities of composite materials using this method where the contact thermal resistance was not negligible, and revealed the effects of particle volume fraction, thermal contact resistance and particle size. The results in this study may provide a useful support for materials design and structure optimization.

Newtonian and non-Newtonian dispersed phase droplet formation in non-Newtonian continuous phase in T-junction and cross junction microchannels are investigated by the immiscible lattice BGK model. The effects of the non-Newtonian fluid power-law exponent, viscosity and interfacial tension on the generation of the droplet are studied. The final droplet size, droplet generation frequency, and detachment point of the droplet change with the power-law exponent. The results reveal that it is necessary to take into account the non-Newtonian rheology instead of simple Newtonian fluid assumption in numerical simulations. The present analysis also demonstrates that the lattice Boltzmann method is of potential to investigate the non-Newtonian droplet generation in multiphase flow.

Despite its demonstrated importance in the deformation and dynamics of red blood cells, membrane viscosity has not received the same attention in computational models as elasticity and bending stiffness. Recent experiments on red blood cells indicated a power law response due to membrane viscosity. This is potentially much different from the solid viscoelastic models, such as Kelvin-Voigt and standard linear solid (SLS), currently used in computation to describe this aspect of the membrane. Within the context of a framework based on lattice Boltzmann and immersed boundary methods, we introduce SLS and power law models for membrane viscosity. We compare how the Kelvin-Voigt (as approximated by SLS) and power law models alter the deformation and dynamics of a spherical capsule in shear flows.

In this contribution, a new class of lattice Boltzmann schemes is introduced and studied. These schemes are presented in a framework that generalizes the multiple relaxation times method of d’Humières. They extend also the Geier’s cascaded method. The relaxation phase takes place in a moving frame involving a set of moments depending on a given relative velocity field. We establish with the Taylor expansion method that the equivalent partial differential equations are identical to the ones obtained with the multiple relaxation times method up to the second order accuracy. The method is then performed to derive the equivalent equations up to third order accuracy.