• Abstract

A particle-cluster treecode based on barycentric Lagrange interpolation

is presented for fast summation of hydrodynamic interactions through general Rotne-Prager-Yamakawa tensor in 3D. The interpolation nodes are taken to be Chebyshev points of the 2nd kind in each cluster.  The barycentric Lagrange interpolation is scale-invariant that promotes the treecode's efficiency. Numerical results show that the treecode CPU time scales like $\mathcal{O}(N \log N)$, where $N$ is the number of beads in the system.  The kernel-independent treecode is a relatively simple algorithm with low memory consumption, and this enables a straightforward OpenMP parallelization.

• Abstract

A multiple-relaxation-time (MRT) lattice Boltzmann model (LBM) is used to study the relative permeabilities in porous media. In many simulations in the literature, usually the periodic boundary condition at inlet and outlet and a uniform pressure gradient were applied to measure the relative permeabilities. However, it is not consistent with the pressure or velocity boundary conditions in the real experiments and may lead to unphysical results. Here using the convective outflow and constant velocity boundary conditions at outlet and inlet, respectively, we can simulate the real experimental setup. Meanwhile, the distribution of the two phases at the outlet can be resolved. The effects of wettability, initial saturation, viscosity ratio ($M\in(1,50)$), capillary number ($Ca \in(10^{-4},10^{-2})$) and micro two-phase distribution at the inlet on permeabilities are investigated comprehensively. It is found that generally speaking, the strong wetting, drainage, larger $Ca$, and larger $M$ may result in a larger relative permeability of the non-wetting phase. Different flow pattern, the lubrication effect of the wetting phase that attaches to the wall, and influence of stagnant pores may contribute to the feature. The study is helpful to further develop the LBM to simulate the real experimental process.

• Abstract

In this study, an improved single-relaxation-time multiphase lattice Boltzmann method (SRT-MLBM) is developed for simulating multiphase flows with both large density ratios and high Reynolds numbers. This model employs two distribution functions in lattice Boltzmann equation (LBE), with one tracking the interface between different fluids and the other calculating hydrodynamic properties. In the interface distribution function, a time derivative term is introduced to recover the Cahn-Hilliard equation. For flow field, a modified equilibrium particle distribution function is present to evolve the velocity and pressure field. The present method keeps simplicity of the conventional SRT-MLBM but enjoys good stability property in simulating multiphase. Apart from several benchmarks, the present model is validated by simulating various challenging multiphase flows, including two droplets impact on liquid film, droplet oblique splashing on a thin film and a drop impact on a moving liquid film. Numerical results show the reliability of present model for effectively simulating complex multiphase flows at density ratios of 1000 and high Reynolds numbers (up to 7000).

• Abstract

In hypersonic boundary layers, Mack mode is the most unstable mode and its secondary instability is a hot research topic on the laminar-turbulent transition. Understanding the mechanism of a secondary instability is very important to delay/promote turbulence generation. In this paper, we focus on the main routes of secondary instability to turbulence in hypersonic flows, including fundamental breakdown and subharmonic breakdown, especially the former one. Through the linear and non-linear stability analyse and secondary instability analysis at various flow temperature conditions, we are trying to find out the temperature effect on the secondary instability mechanism of Mack mode disturbances. The results point out that the fundamental mode always dominates the breakdown type when the saturated amplitude of the primary Mack mode is large enough. As the stagnation temperature increases, the maximum growth-rates of the fundamental mode and subharmonic mode both increase. Meanwhile, when the wall is cooling, the maximum growth-rates of the fundamental mode and subharmonic mode are both enlarged. In contrast, with the heating wall, the maximum growth-rates of the secondary instability both decrease.

• Abstract

Singularity of Navier-Stokes equations is uncovered for the first time which explains the mechanism of transition of a smooth laminar flow to turbulence. It is found that when an inflection point is formed on the velocity profile in pressure driven flows, velocity discontinuity occurs at this point. Meanwhile, pressure pulse is produced at the discontinuity due to conservation of the total mechanical energy. This discontinuity makes the Navier-Stokes equations be singular and causes the flow to become indefinite. The analytical results show that the singularity of the Navier-Stokes equations is the cause of turbulent transition and the inherent mechanism of sustenance of fully developed turbulence. Since the velocity is not differentiable at the singularity, there exist no smooth and physically reasonable solutions of Navier-Stokes equations at high Reynolds number (beyond laminar flow). The negative spike of velocity and the pulse of pressure due to discontinuity have obtained agreement with experiments and simulations in literature qualitatively.

• Abstract

In this paper,  a linearized difference scheme is proposed for the Sine-Gordon equation (SGE) with a Caputo time  derivative of order $\alpha\in(1,2)$. Comparing with  the existing linearized difference schemes, the proposed numerical scheme is simpler and easier for theoretical analysis. The solvability, boundedness and convergence of the difference scheme are rigorously established in the $L_{\infty}$ norm. Finally, several numerical experiments are provided to support the theoretical results.

• Abstract

The prediction and control of the laminar-turbulent transition in three-dimensional boundary layers are crucial for the designs of vehicles, aircrafts, etc. Receptivity, the initial stage of transition, is the key to implement the prediction and control of the transition process. Former experimental results showed that, under a relatively low-level turbulence, the three-dimensional boundary-layer transition is mainly induced by the stationary cross-flow modes rather than the travelling ones. Near the leading edge, stationary cross-flow modes can be excited by three-dimensional localized roughness. And the receptive process is affected by both the size of roughness and the shape of leading edge which distorts the mean flow over the plate. Therefore, we perform direct numerical simulation to investigate the excitation of stationary cross-flow modes by three-dimensional localized wall roughness in swept-plate boundary layers with various elliptic leading edges. The effect of the leading-edge curvature on the induced stationary cross-flow modes is revealed. And the relations of the leading-edge curvatures with the amplitudes and dispersion relations of the stationary cross-flow modes are determined. Furthermore, the correlations between the receptivity coefficients and the geometries, locations and numbers of the roughness respectively are analyzed in different leading-edge curvatures. This research aims to complement the study of receptivity in the cross-flow boundary layer.

• Abstract

In this paper, we prove a lower convergence rate of the weak Galerkin finite element method for second order elliptic equations under a standard weak smoothness assumption. In all present literatures on weak Galerkin finite element methods for second order PDEs, the $H^2$ smoothness is compulsorily assumed for the real solution and hence a second order convergence is obtained. This lead to that the piecewise linear functions are excluded to construct finite element bases, although they behave very well in all numerical experiments. We intend to prove the $(1+s)$-order convergence rate under the $H^1$-smoothness assumption of the real solution and an additional $s>0$ regularity of the dual problem. Our strategy is that we firstly approximate the elliptic problem using the traditional finite element method with at least  $H^2$ smooth bases, and then we  apply the weak Galerkin method to approach this smooth approximating solution. Our result is an important supplementary for the weak Galerkin finite element method theory.

• Abstract

This paper investigates bifurcation and chaos of functionally graded carbon nanotube-reinforced composites (FG-CNTRCs) beam containing piezoelectric layer (PL) under combined electro-thermo-mechanical loads. We assumed that FG-CNTRC material properties were graded along thickness direction and determined them using mixtures' law. Governing equations of structures were derived according to the theory of Euler-Bernoulli beam, PL with thermal effects and von K\'{a}rm\'{a}n geometric nonlinearity. Next, the governing equations were transformed into second order nonlinear ordinary differential equations (SNODE) with cubic terms through Galerkin procedure and further into first order nonlinear ordinary differential equations (FNODE) through introducing additional state variables. Complex system dynamic behavior was qualitatively examined using fourth order Runge-Kutta method. The effects of different factors including applied voltage, volume fraction, temperature change, and distribution of carbon nanotubes (CNTs) on bifurcation and chaos of FG-CNTRC beams with PL were comprehensively studied.

• Abstract

Two adaptive techniques for choosing relaxation factor, namely, Minimal Residual Relaxation (MRR) and Orthogonal Projection Relaxation (OPR), on basic iterative methods for solving linear systems are proposed. Unlike classic relaxation, in which the optimal relaxation factor is generally difficult to find, in these proposed techniques, non-stationary relaxation factor based on minimal residual or orthogonal projection method is calculated adaptively in each relaxation step with acceptable cost for Jacobi, Gauss-Seidel or symmetric Gauss-Seidel iterative methods. In order to avoid the "stagnation" of the successive locally optimal relaxations, a recipe of inserting several basic iterations between every two adjacent relaxations is suggested and the resulting MRR$(m)$/OPR$(m)$ strategy is more stable and efficient (here $m$ denotes the number of basic iterations inserted). To solve linear systems with multiple right-hand sides efficiently, block-form relaxation strategies are proposed based on the MRR$(m)$ and OPR$(m)$. Numerical experiments show that the presented MRR$(m)$/OPR$(m)$ algorithm is more robust and effective than classic relaxation methods. It is also showed that the proposed block relaxation strategies can efficiently accelerate the solution of systems with multiple right-hand sides in terms of total solution time as well as number of iterations.

• Abstract

In this paper, we study an implicit-explicit scheme for the radiation hydrodynamics in the equilibrium diffusion limit and in the grey nonequilibrium diffusion limit. We extend a popular Godunov-type method, the MUSCL-Hancock scheme, to the convective part of the radiation hydrodynamics, while a cell centered finite volume scheme is used for the radiative heat transfer. Moreover, the implicit-explicit scheme is easier to implement. Numerical simulations show the character of the radiative shock wave and the accuracy of the scheme.

• Abstract

In this paper, we present a new semi-analytical method based on the B-spline approximation for solving 2D time-dependent convection-diffusion-reaction equations to model transfer in the anisotropic inhomogeneous medium. The mathematical model is expressed as the initial-boundary value problem for quasi-linear parabolic equation with the second order elliptic spatial operator with mixed derivatives and variable coefficients. The time-stepping Crank-Nicolson scheme transforms the original equation into a sequence of quasi-linear elliptic partial differential equations.  The approximate solution is sought as series over basis functions which are taken in the form of the tensor products of the  B-splines with centers distributed inside the solution domain. Due to the modification of spline basis, the final approximate solution satisfies the boundary conditions of the initial problem with any choice of the coefficients of the series. The numerical examples demonstrate the high accuracy of the proposed method in solving 2D convection-diffusion-reaction problems in single- and multi-connected domains.

• Abstract

A new Explicit Immersed Boundary method (IBM) is presented in this work by analyzing and simplifying the system of equations developed from the implicit boundary condition-enforced immersed boundary method. By this way, the requirement to solve the matrix system has been bypassed. It makes the solver be computationally less expensive, especially when large number of Lagrangian points are used to represent the solid boundary. The lattice Boltzmann Flux solver (LBFS) was chosen as the flow solver in this paper as it combines the advantages of both Lattice Boltzmann (LB) solver and Navier- Stokes solver. However, it should be indicated that the new IBM can be incorporated into any flow solver. Comprehensive validations demonstrate that the new explicit scheme bears comparable numerical accuracy as the previous implicit IBM when having a geometry with curvature. The new method is computationally much more efficient than the previous method, especially for moving boundary problems.

• Abstract

We mainly research the reduced-order of the classical natural boundary element (CNBE) method for the two-dimensional (2D) viscoelastic wave equation by means of proper orthogonal decomposition (POD) technique. For this purpose, we firstly establish the CNBE model and analyze the existence, stability, and errors for the CNBE solutions. We then build a highly efficient reduced-order extrapolating natural boundary element (HEROENBE) mode including few degrees of freedom but possessing sufficiently high accuracy for the 2D viscoelastic wave equation by the POD method and analyze the existence, stability, and errors of the HEROENBE solutions by the CNBE method. We finally employ some numerical experiments to verify that the numerical results are accorded with the theoretical ones so that the validity for the HEROENBE model is further verified.

• Abstract

A novel upwind technique for the localized method of approximate particular solutions (LMAPS) is proposed to solve the convection-diffusion equations. An upwind approximation to the convective terms is implemented by choosing upwind interpolation stencils while the central interpolation stencils are used for the diffusive terms. The proposed upwind LMAPS scheme is also been compared with conventional LMAPS without upwind technique, to demonstrate its superiority in generating high accurate solutions than the latter. Numerical results show that the proposed upwind LMAPS has high accuracy and efficiency for a variety of convection-diffusion equations.

• Abstract

We present an efficient Galerkin-characteristic finite element method for the numerical solution of convection-diffusion problems in three space dimensions. The modified method of characteristics is used to discretize the convective term in a finite element framework. Different types of finite elements are implemented on three-dimensional unstructured meshes. To allocate the departure points we consider an efficient search-locate algorithm for three-dimensional domains. The crucial step of interpolation in the convection step is carried out using the basis functions of the tetrahedron element where the departure point is located. The resulting semi-discretized system is then solved using an implicit time-stepping scheme. The combined method is unconditionally stable such as no Courant-Friedrichs-Lewy condition is required for the selection of time steps in the simulations. The performance of the proposed Galerkin-characteristic finite element method is verified for the transport of a Gaussian sphere in a three-dimensional rotational flow. We also apply the method for simulation of a transport problem in a three-dimensional pipeline flow. In these test problems, the method demonstrates its ability to accurately capture the three-dimensional transport features.

• Abstract

In this paper, our purpose is to study the unconditional stability and convergence of characteristics finite element method (FEM) for the time-dependent viscoelastic Oldroyd  fluids motion equations. We deduce optimal error estimates in $L^2$ and $H^1$ norm. The analysis is based on an iterated time-discrete system, with which the error function is split into a temporal error and a spatial error. Finally, numerical results confirm the theoretical predictions.

• Abstract

In this paper, we use the radial basis function meshless method to solve the irregular region interface problem. The key idea is to construct radial basis functions corresponding to different regions divided by the interfaces. This method avoids the difficulty of mesh generation, and is efficient in the numerical simulation of partial differential equations in irregular domain with variable matrix coefficients. The numerical error is effectively reduced by using the direct method to handle the interface jump conditions. Numerical simulation results show that the radial basis function meshless numerical method can effectively deal with various kinds of interface problems with irregular domains and sharp-edged interfaces, including Poisson equations, heat conduction equations and wave equations.

• Abstract

Analytical study on nonlinear hydroelastic waves beneath a very large floating structure  or a thin ice sheet floating on deep water is presented. Adopting the special Cosserat theory of hyperelastic shells satisfying Kirchhoff's hypothesis to describe the floating sheet, we use the potential flow  theory  with the dynamic boundary condition expressing a balance among the hydrodynamic, surface tension, inertial, and elastic forces. For the case of incident progressive waves, the influences of different physical parameters on the hydroelastic waves are discussed with the aid of the homotopy analysis method. We compare the hydroelastic wave deflections based on nonlinear Plotnikov and Toland's  model  with those obtained by the corresponding linear Euler--Bernoulli model. It is found that the behaviors of both models are almost the same for small amplitudes, while the nonlinear plate deflections increase  greatly at large amplitudes. Further, the graphical comparisons are presented to show the behavior of the angular frequency versus wave amplitudes.

• Abstract

In this paper, a simple and robust multi-dimensional shock-capturing limiter is presented and applied to the high-order least square-based finite difference-finite volume (LSFD-FV) method for simulation of compressible inviscid flows on unstructured grids. The novel limiter considers all the spatial information of the neighbouring supporting cells of the reference cell instead of only the immediate neighbours. As a result, the local upper and lower bounds can be correctly defined based on the extrema at specific points within the supporting stencil and thereafter the numerical oscillations in the adjacent region to the discontinuities can be suppressed effectively. The main advantage of this limiting technique lies in its simple methodology and easy implementation. Furthermore, the multi-dimensional character and the point extrema-based methodology endow the proposed limiter with ability to solve multi-dimensional flow problems on unstructured grids. A series of smooth and non-smooth numerical examples in one, two and three dimensions on unstructured grids are tested. Numerical results obtained demonstrate the superior performance of the proposed limiter on simulating compressible inviscid flows.