• Abstract

In this paper, a novel immersed boundary method is presented for simulating incompressible flows governed by Navier-Stokes equations. A virtual boundary is formed by the cell edges (for two-dimensional cases) in the vicinity of the immersed boundary. In the domain with the virtual boundary, the governing equations can be solved in the conventional way. Reconstructed velocity is imposed on the virtual boundary, which is determined via the interpolation along the direction normal to the wall and in conjunction with the no-slip condition for the actual boundary. For "freshly cleared nodes" on the virtual boundary encountered in moving-boundary problems, pressure at the previous time step is reconstructed by solving the local simplified momentum equation. In the test case for an analytical solution, the local accuracy of pressure is verified to be of the second order. In order to further validate the present method, various flows over the stationary and/or moving circular cylinder and NACA0012 airfoil have been simulated. The obtained results agree well with the available numerical or experimental data in the published literatures.

• Abstract

In this paper, we propose a novel Legendre neural network combined with the extreme learning machine algorithm to solve variable coefficients linear delay differential-algebraic equations with weak discontinuities. First, the solution interval is divided into multiple subintervals by weak discontinuity points. Then, Legendre neural network is used to eliminate the  hidden layer by expanding the input pattern using Legendre polynomials on each subinterval. Finally, the parameters of the neural network are obtained by training with the extreme learning machine. The numerical examples show that the proposed method can effectively deal with the difficulty of numerical simulation caused by the discontinuities.

• 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

In framework of the fictitious domain methods with immersed interfaces for the elasticity problem, the present contribution is to study and numerically validate the jump-integrated boundary conditions method with sharp interface for the vector elasticity system discretized by a proposed finite volume method. The main idea of the fictitious domain approach consists in embedding the original domain of study into a geometrically larger and simpler one called the fictitious domain. Here, we present a cell-centered finite volume method to discretize the fictitious domain problem. The proposed method is numerically validated for different test cases. This work can be considered as a first step before more challenging problems such as fluid-structure interactions or moving interface problems.

• Abstract

A convolutional neural network is developed for rapidly predicting

multiphase flow in heterogeneous porous media. Some direct numerical

methods can acquire accurate results of multiphase flow in porous media. However, once the geometry of the porous media changes, it takes much computational time to perform a new simulation. Here, a deep neural network model in the field of semantic segmentation is developed. It takes the two-dimensional microstructure of heterogeneous porous media as inputs and is able to predict corresponding multiphase flow fields (pressure and saturation fields). Compared to the direct lattice Boltzmann simulations, the inference time on new geometry of porous media can be reduced by several orders of magnitude. Our results show that the machine learning method is a good prediction tool in a wide range of porosity and heterogeneity. Besides, to better understand the inherent process, a visible explanation is presented on what our neural networks have learned.

• Abstract

In this paper, a fifth-order weighted essentially nonoscillatory scheme is presented for simulating dam-break flows in a finite difference framework. The new scheme is a convex combination of two quadratic polynomials with a fourth-degree polynomial in a classical WENO fashion. The distinguishing feature of the present method is that the same five-point information is used but smaller absolute truncation errors and the same accuracy order in the smooth region are obtained. The new nonlinear weights are presented by Taylor expansion of the smoothness indicators of the small stencils to sustain the optimal fifth-order accuracy. The linear advection equation, nonlinear scalar Burgers equation, and one- and two-dimensional Euler equations are used to validate the high-order accuracy and excellent resolution of the presented method. Finally, one- and two-dimensional Saint-Venant equations are tested by using the new fifth-order scheme to simulate a dam-break flow.

• Abstract

The forced-vibration response of a simply-supported isotropic thick-walled hollow elastic circular cylinder subjected to two-dimensional harmonic standing-wave excitations on its curved surfaces is studied within the framework of linear elastodynamics. Exact semi-analytical solutions for the steady-state displacement field of the cylinder are constructed using recently-published parametric solutions to the Navier-Lam\'{e} equation. Formal application of the standing-wave boundary conditions generates three parameter-dependent $6\times6$ linear systems, each of which can be numerically solved in order to determine the parametric response of the cylinder's displacement field under various conditions. The method of solution is direct and demonstrates a general approach that can be applied to solve many other elastodynamic forced-response problems involving isotropic elastic cylinders. As an application, and considering several examples, the obtained solution is used to compute the steady-state frequency response in a few specific low-order excitation cases. In each case, the solution generates a series of resonances that are in exact correspondence with a unique subset of the natural frequencies of the simply-supported cylinder. The considered problem is of general theoretical interest in structural mechanics and acoustics and more practically serves as a benchmark forced-vibration problem involving a thick-walled hollow elastic cylinder.

• Abstract

In this paper, we consider the optimization problem of identifying the pollution sources of convection-diffusion-reaction equations in a groundwater process. The optimization model is subject to a convection-diffusion-reaction equation with pumping point and pollution point sources. We develop a linked optimization and simulation approach combining with the Differential Evolution (DE) optimization algorithm to identify the pumping and injection rates from the data at the observation points. Numerical experiments are taken with injections of constant rates and  time-dependent variable rates at source points. The problem with one pumping point and two pollution source points is also studied. Numerical results show that the proposed method is efficient. The developed optimized identification approach can be extended to high-dimensional and more complex problems.

• 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 paper, the recently-developed singular boundary method is applied to address free boundary problems. This mesh-less numerical method is based on the use of the origin intensity factors with fundamental solutions. Three numerical examples and their results are compared with the results obtained using traditional methods. The comparisons indicate that the proposed scheme yields good results in determining the position of the free boundary.

• Abstract

In this paper, some two-grid finite-volume methods are constructed for solving the steady-state  Schrodinger equation. The method projects the original  coupled problem onto a coarser grid, on which it is less expensive to solve, and then prolongates the approximated coarse solution back to the fine grid, on which it is not much more difficult to solve the decoupled problem. We have shown, both theoretically and numerically,  that our schemes are more efficient and achieve asymptotically optimal accuracy as long as the mesh sizes satisfy $h=\mathcal{O}(H^2)$.

• 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

In this paper, we propose a finite element time-domain (FETD) method for the Maxwell's equations in chiral metamaterials (CMMs). The time-domain model equations are constructed by the auxiliary differential equations (ADEs) method. The source excitation method entitled total-field and scattered-field (TF/SF) decomposition technique is applied to FETD method for the first time in simulating the propagation of electromagnetic wave in CMMs, based on which a unified ADE-FETD-UPML-TF/SF scheme is proposed to simulate the wave in CMMs. The following properties of CMMs can be observed successfully from the numerical experiments based on our method, i.e., the ability of the polarization rotation, and the negative phase velocity. The amplitude of reflected wave can effectively be controlled by the physical parameters of CMMs.

• Abstract

An energy-preserving scheme is proposed for the coupled Gross-Pitaevskii equations.  The scheme is constructed by high order compact  method in the spatial direction and average vector field  method in the temporal direction, respectively. The  scheme is energy-preserving, stable, and of sixth order in space and of second order in time. Numerical experiments verify the theoretical results. The dynamic {behavior modeled} by the coupled Gross-Pitaevskii equations is also numerically investigated.

• 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 propose a new conservative gradient discretization method (GDM) for one-dimensional parabolic partial differential equations (PDEs). We use the implicit Euler method for the temporal discretization and conservative gradient discretization method  for spatial discretization. The method is based on a new cell-centered meshes, and it is locally conservative. It has smaller truncation error than the classical finite volume method on uniform meshes. We use the framework of the gradient discretization method  to analyze the stability and convergence. The numerical experiments show that the new method has second-order convergence. Moreover, it is more accurate than the classical finite volume method in flux error, $L^2$ error and $L^{\infty}$ error.

• 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.