Numerical simulations of two-dimensional (2D) turbulent thermal convection for inhomogeneous boundary condition are investigated using the lattice Boltzmann method (LBM). This study mainly appraises the temporal evolution and the scaling behavior of global quantities and of small-scale turbulence properties. The research results show that the flow is dominated by large-scale structures in the turbulence regime. Mushroom plumes emerge at both ends of each heat source, and smaller plumes increasingly rise. It is found that the gradient of root mean-square (rms) vertical velocities and the gradient of the rms temperature in the bottom boundary layer decrease with time evolution. It is further observed that the temporal evolution of the Kolmogorov scale, the kinetic-energy dissipation rates and thermal dissipation rates agree well with the theoretical predictions. It is also observed that there is a range of linear scaling in the 2nd-order structure functions of the velocity and temperature fluctuations and mixed velocity-temperature structure function.

A new sharp interface method with the combination of Ghost Fluid Method (GFM) and Cut Cell scheme is developed to study compressible multi-phase flows with clear interfaces. Straight-line cutting is applied on the cells passed by the interface. A new real-ghost mixing method is presented and applied around the cut cells to deal with very small cut cells. A cut face reconstruction method similar to volume of fluid is applied to deal with topological change problems. A high order Level Set (LS) method is applied to evolve the free interface, with the Level Set velocities from exact Riemann solver on the cut faces. Various 1D and 2D numerical examples are tested to show the robustness and ability of the present method in wide flow variable domains. This method benefits from cut cell on the sharp interface description, shows good conservation performance, and does not have the topological change difficulty of the full cut cell method presented in Chang, Deng & Theofanous, J. Comput. Phys., 242 (2013), pp. 946–990.

The Stokes axisymmetric flow of an incompressible viscous fluid past a micropolar fluid spheroid whose shape deviates slightly from that of a sphere is studied analytically. The boundary conditions used are the vanishing of the normal velocities, the continuity of the tangential velocities, continuity of shear stresses and spin-vorticity relation at the surface of the spheroid. The hydrodynamic drag force acting on the fluid spheroid is calculated. An exact solution of the problem is obtained to the first order in the small parameter characterizing the deformation. It is observed that due to increase spin parameter value, the drag coefficient decreases. Well known results are deduced and comparisons are made with classical viscous fluid and micropolar fluid.

In this paper, the magnetic field effects on natural convection of power-law nanofluids in rectangular enclosures are investigated numerically with the lattice Boltzmann method. The fluid in the cavity is a water-based nanofluid containing Cu nanoparticles and the investigations are carried out for different governing parameters including Hartmann number (0.0≤$Ha$≤20.0), Rayleigh number (104≤$Ra$≤106 ), power-law index (0.5≤$n$≤1.0), nanopartical volume fraction (0.0≤$ϕ$≤0.1) and aspect ratio (0.125≤$AR$≤8.0). The results reveal that the flow oscillations can be suppressed effectively by imposing an external magnetic field and the augmentation of Hartmann number and power-law index generally decreases the heat transfer rate. Additionally, it is observed that the average Nusselt number is increased with the increase of Rayleigh number and nanoparticle volume fraction. Moreover, the present results also indicate that there is a critical value for aspect ratio at which the impact on heat transfer is the most pronounced.

In this article a comparison study of the numerical methods for compressible two-phase flows is presented. Although many numerical methods have been developed in recent years to deal with the jump conditions at the fluid-fluid interfaces in compressible multiphase flows, there is a lack of a detailed comparison of these methods. With this regard, the transport five equation model, the modified ghost fluid method and the cut-cell method are investigated here as the typical methods in this field. A variety of numerical experiments are conducted to examine their performance in simulating inviscid compressible two-phase flows. Numerical experiments include Richtmyer-Meshkov instability, interaction between a shock and a rectangle $SF_6$ bubble, Rayleigh collapse of a cylindrical gas bubble in water and shock-induced bubble collapse, involving fluids with small or large density difference. Based on the numerical results, the performance of the method is assessed by the convergence order of the method with respect to interface position, mass conservation, interface resolution and computational efficiency.

In this paper, we consider the transform magnetic (TM) model of electromagnetic scattering in the cavity. By the Polynomial Preserving Recovery technique, we present superconvergence analysis for the vertex-edge-face type finite element. From the numerical example, we can see that the provided method is efficient and stable.

The vectorial form of the Wave Propagation Method (VWM), regarding the dispersion of harmonic plain (elasto-dynamic) waves within certain wave-guides, is developed for the vibration analysis of circular cylindrical shells. To obtain this goal, all plain waves are divided into positive-negative going wave vectors along with the shell axis. Based on the Flügge thin shell theory, the shell continuity as well as boundary conditions are well satisfied by introducing the propagation and reflection matrices. Furthermore, all elements of the reflection matrix are derived for certain classical supports. As an example, for demonstrating the feasibility of VWM in the shell vibration analysis, a circular cylindrical shell with two ended flexible support is adopted. The natural frequencies of the system as well as mode shapes are obtained using VWM. The aquired results are compared with those of the previous works and found in excellent agreement. It is also found that VWM could mathematically provide a reduced dimensional matrix (dominant matrix) to calculate the natural frequencies of the system. Accordingly, the proposed method can provide high computational efficiency and remarkable accuracy, simultaneously.

We present a convex-splitting scheme for the fourth order parabolic equation derived from a diffuse interface model with Peng-Robinson equation of state for pure substance. The semi-implicit scheme is proven to be uniquely solvable, mass conservative, unconditionally energy stable and $L^∞$ convergent with the order of $\mathcal{O}(∆t+h^2)$. The numerical results verify the effectiveness of the proposed algorithm and also show good agreement of the numerical solution with laboratory experimental results.

We propose a non-traditional finite element method with non-body-fitting grids to solve the matrix coefficient elliptic equations with imperfect contact in two dimensions, which has not been well-studied in the literature. Numerical experiments demonstrated the effectiveness of our method.

In this paper, based on the multi-symplectic formulations of the generalized fifth-order KdV equation and the averaged vector field method, two new energy-preserving methods are proposed, including a new local energy-preserving algorithm which is independent of the boundary conditions and a new global energy-preserving method. We prove that the proposed methods preserve the energy conservation laws exactly. Numerical experiments are carried out, which demonstrate that the numerical methods proposed in the paper preserve energy well.

In this work, element free Galerkin (EFG) method is posed for solving nonlinear, reaction-diffusion systems which are often employed in mathematical modeling in developmental biology. A predictor-corrector scheme is applied, to avoid directly solving coupled nonlinear systems. The EFG method employs the moving least squares (MLS) approximation to construct shape functions. This method uses only a set of nodal points and a geometrical description of the body to discretize the governing equation. No mesh in the classical sense is needed. However, a background mesh is used for integration purpose. Numerical solutions for two cases of interest, the Schnakenberg model and the Gierer-Meinhardt model, in various regions are presented to demonstrate the effects of various domain geometries on the resulting biological patterns.

Mass migration of cells (via wave motion) plays an important role in many biological processes, particularly chemotaxis. We study the existence of travelling wave solutions for a chemotaxis model on a microscopic scale. The interaction between nutrients and chemoattractants are considered. Unlike previous approaches, we allow for diffusion of substrates, degradation of chemoattractants and cell growth (constant and linear growth rate). We apply asymptotic methods to investigate the behaviour of the solutions when cells are highly sensitive to extracellular signalling. Explicit solutions are demonstrated, and their biological implications are presented. The results presented here extend and generalize known results.

Direct numerical simulations of the transition process from steady laminar to chaotic flow are considered in this study with the relatively new incompressible lattice Boltzmann equation. Numerically, a multiple relaxation time fully incompressible lattice Boltzmann equation is implemented in a 2D driven cavity. Spatial discretization is 2nd-order accurate, and the Kolmogorov length scale estimation based on Reynolds number ($Re$) dictates grid resolution. Initial simulations show the method to be accurate for steady laminar flows, while higher $Re$ simulations reveal periodic flow behavior consistent with an initial Hopf bifurcation at $Re$ 7,988. Non-repeating flow behavior is observed in the phase space trajectories above $Re$ 13,063, and is evidence of the transition to a chaotic flow regime. Finally, flows at Reynolds numbers above the chaotic transition point are simulated and found with statistical properties in good agreement with literature.