Non-uniform Grid Research Articles

A high-order implicit-explicit flux reconstruction lattice Boltzmann method for viscous incompressible flows

In this paper, a high-order implicit-explicit flux reconstruction lattice Boltzmann method (FRLBM) is proposed for simulating viscous incompressible flows. The discrete velocity Boltzmann equation (DVBE) is solved by using a high-order flux reconstruction scheme (FR) for spatial discretization and a high-order implicit-explicit Runge-Kutta scheme (IMEX RK) for time discretization in the generalized curvilinear coordinates with uniform and non-uniform grids to obtain high simulation efficiency. Numerical validations of the proposed method are implemented by simulating (a) Taylor-Green vortex problem, (b) doubly periodic shear layer flow, (c) lid-driven cavity flow, (d) cylindrical Couette flow and (e) flow over a circular cylinder. Numerical results demonstrate that the present method is stable and accurate even at high Reynolds number. Compared with the existing high-order off-lattice Boltzmann methods, the present method is more efficient. In addition, the present method holds the compactness of the standard LBM for parallel computing, but greatly improves the adaptability of complicated geometries.

Staggered Semi-Implicit Hybrid Finite Volume/Finite Element Schemes for Turbulent and Non-Newtonian Flows

This paper presents a new family of semi-implicit hybrid finite volume/finite element schemes on edge-based staggered meshes for the numerical solution of the incompressible Reynolds-Averaged Navier–Stokes (RANS) equations in combination with the k−ε turbulence model. The rheology for calculating the laminar viscosity coefficient under consideration in this work is the one of a non-Newtonian Herschel–Bulkley (power-law) fluid with yield stress, which includes the Bingham fluid and classical Newtonian fluids as special cases. For the spatial discretization, we use edge-based staggered unstructured simplex meshes, as well as staggered non-uniform Cartesian grids. In order to get a simple and computationally efficient algorithm, we apply an operator splitting technique, where the hyperbolic convective terms of the RANS equations are discretized explicitly at the aid of a Godunov-type finite volume scheme, while the viscous parabolic terms, the elliptic pressure terms and the stiff algebraic source terms of the k−ε model are discretized implicitly. For the discretization of the elliptic pressure Poisson equation, we use classical conforming P1 and Q1 finite elements on triangles and rectangles, respectively. The implicit discretization of the viscous terms is mandatory for non-Newtonian fluids, since the apparent viscosity can tend to infinity for fluids with yield stress and certain power-law fluids. It is carried out with P1 finite elements on triangular simplex meshes and with finite volumes on rectangles. For Cartesian grids and more general orthogonal unstructured meshes, we can prove that our new scheme can preserve the positivity of k and ε. This is achieved via a special implicit discretization of the stiff algebraic relaxation source terms, using a suitable combination of the discrete evolution equations for the logarithms of k and ε. The method is applied to some classical academic benchmark problems for non-Newtonian and turbulent flows in two space dimensions, comparing the obtained numerical results with available exact or numerical reference solutions. In all cases, an excellent agreement is observed.

An improved Picard iteration scheme for simulating unsaturated flow in porous media

The Richards equation is widely used in the simulation of unsaturated flow and related fields. In the numerical solution process, the finite volume method can be used to carry out numerical discretization, and the Picard method is used for iterative solution. However, in order to obtain a more reliable numerical solution, the space step size of the conventional uniform grid is often small. Especially under some unfavorable numerical conditions, such as infiltration into dry soil and layered soil with very different hydraulic conductivity, this will be very time-consuming and even slow to converge. Thus, combined with the non-uniform grid in the form of Chebyshev, an improved Picard method based on the non-uniform two-grid correction scheme (NTG-PI) is proposed to model 1D unsaturated flow in porous media. Through three examples of unsaturated flow under unfavorable conditions, the proposed scheme was verified. The results show that, compared with the conventional Picard method, NTG-PI can obtain higher numerical accuracy with a smaller number of nodes, and also has a faster convergence rate. This method can provide a certain reference for the numerical simulation of unsaturated flow.

Nanofluidic thermal-fluid transport in a split-driven porous system working under a magnetic environment

PurposeThis study aims to focus on a thermo-fluid flow in a partially driven cavity (PDC) using Cu-water nanoliquid, magnetic field and porous substance. The cooling and sliding motion are applied on the upper half of the vertical walls and the bottom wall is heated. Thermal characteristics are explored to understand magnetohydrodynamic convection in a nanoliquid filled porous system from a fundamental viewpoint. The governing parameters involved to cater to the moving speed of the sidewalls and partial translation direction are the relative strength of thermal buoyancy, porous substance permeability, magnetic field intensity, nanoparticle suspension and orientation of the cavity.Design/methodology/approachThe coupled transport equations of the problem are solved using an in-house developed finite volume-based computing code. The staggered nonuniform grids along the x and y directions are used. The SIMPLE algorithm technique is considered for the iterative solution of the discretized equations with the convergence check of the continuity mass defect below 10–10.FindingsThe present study unveils that the heat transfer enhances at higher Ri with the increasing value of Re, irrespective of the presence of a porous substance or magnetic field or the concentration of nanofluid. Apart from different flow controlling parameters, the wall motions have a significant contribution to the formation of flow vortices and corresponding heat transfer. Orientation of the cavity significantly alters the transport process within the cavity. The upward wall velocity for both the sidewalls could be a better choice to enhance the high heat transfer (approximately 88.39% at Richardson and Reynolds numbers, respectively, 0.1 and 200).Research limitations/implicationsConsidering other multi-physical scenarios like porous layers, conducting block, microorganisms and the present investigation could be further extended to analyze a problem of complex flow physics.Practical implicationsIn this study, the concept of partially driven wall motion has been adopted under the Cu-water nanoliquid, magnetic field, porous substance and oblique enclosure. All the involved flow-controlling parameters have been experimented with under a wide parametric range and associated thermo-flow physics are analyzed in detail. This outcome of this study can be very significant for designing as well as controlling thermal devices.Originality/valueThe convective process in a partially driven cavity (PDC) with the porous medium has not been investigated in detail considering the multi-physical scenarios. Thus, the present effort is motivated to explore the thermal convection in such an oblique enclosure. The enclosure is heated at its bottom and has partially moving-wall cold walls. It consists of various multi-physical conditions like porous structure, magnetic field, Cu–H2O nanoliquid, etc. The system performance is addressed under different significant variables such as Richardson number, Reynolds number, Darcy number, Hartmann number, nanoliquid concentration and orientation of cavity.

In-depth isolation of highly permeable zones for reservoir conformance control

In-depth isolation of flooded-out reservoir zones is widely used in water injection and water production control in oilfields. This paper looks at the research results of a new technological solution – a method and mechanism of deep-penetrating insulation of highly permeable reservoir areas. Approximate mathematical models are used to describe oil-saturated reservoirs flooding with chemical compositions used as a displaced fluid. The mathematical model of the problem is reduced to a partial differential equation with initial boundary conditions. The article also presents the results of the numerical implementation of the transfer equations of concentrations of gel-forming components and a thickener. In a non-uniform grid domain, the problem at hand is replaced by a discrete problem using a combination of an explicit and implicit-difference scheme that increases the order of accuracy and is solved by the run-through method. Effective algorithms for solving the one-dimensional problem are obtained, taking into account the equations of adsorption and convective diffusion. The developed method increases the efficiency of the in-deep isolation treatment of flushed zones and can be applied to related reservoir flooding projects.

Stable numerical schemes for time-fractional diffusion equation with generalized memory kernel

The paper aims to develop the stable numerical schemes for generalized time-fractional diffusion equations (GTFDEs) with smooth and non-smooth solutions on the non-uniform grid. In time, the generalized Caputo derivative is discretized by a difference scheme of order (2−α) on a non-uniform grid where 0<α<1. Choosing the non-uniform meshes in the case of the smooth and non-smooth solution is also essential, so we graded the mesh in both cases separately. Stability and convergence for smooth as well as non-smooth solutions are obtained in L2-norm and L∞-norm respectively. Several numerical results are presented to show how the grading of meshes is essential. Also, numerical results validate the efficiency and effectiveness of proposed schemes and show how a non-uniform grid produces better results.

Numerical Results of Crank-Nicolson and Implicit Schemes to Laplace Equation with Uniform and Non-Uniform Grids

AbstractIn this paper, we investigate the numerical results between Implicit and Crank-Nicolson method for Laplace equation. Based on the numerical results obtained, we get the conclusion that the absolute error of Crank-Nicolson method is smaller than the absolute error of Implicit method for uniform and non-uniform grids which both refer to the analytical solution of Laplace equation obtained by separable variable method.Keywords: Crank-Nicolson; Implicit; Laplace equation; separable variable method; uniform and non-uniform grids. AbstrakDalam makalah ini, kami menyelidiki hasil numerik antara etode Implisit dan Crank-Nicolson untuk persamaan Laplace. Berdasarkan hasil numerik yang diperoleh, kita mendapatkan kesimpulan bahwa kesalahan absolut metode Crank-Nicolson lebih kecil daripada kesalahan absolut metode Implisit untuk grid seragam dan tak-seragam yang keduanya mengacu pada solusi analitik persamaan Laplace yang diperoleh dengan metode separable.Kata kunci: Crank-Nicolson; Implisit; persamaan Laplace; metode variable terpisah; grid seragam dan tak-seragam.

An alternating-direction hybrid implicit-explicit finite-difference time-domain method for the Schrödinger equation

This paper proposes a novel hybrid FDTD method for solving the time-dependent Schrödinger equation, which is fundamental for modeling materials and designing nanoscale devices. The wave function is propagated on nonuniform grids by applying explicit updates in part of the grid and implicit updates elsewhere. The latter are based on the Alternating-Direction Implicit (ADI) scheme while the former are constructed with a central difference for the time derivative. A rigorous stability analysis proves that spatial steps can be selectively removed from the stability criterion thus combining the unconditional stability of the ADI scheme with fast explicit calculations. The scheme excels in its flexibility by efficiently discretizing and balancing explicit with implicit updates, as such expediting the computations. Moreover, it retains the linear complexity of explicit schemes with respect to space and time, making it especially scalable to numerically large problems. Several numerical experiments, including a laterally tunnel-coupled quantum wire and a nanowire double-barrier resonant-tunneling diode, show the validity of the scheme by demonstrating its high accuracy and decreased CPU time compared to traditional methods.

Statistical characteristics of turbulent mixing in spherical and cylindrical converging Richtmyer–Meshkov instabilities

In this paper, the Richtmyer–Meshkov instabilities in spherical and cylindrical converging geometries with a Mach number of approximately 1.5 are investigated by using the high resolution implicit large eddy simulation method, and the influence of the geometric effect on the turbulent mixing is investigated. The heavy fluid is sulphur hexafluoride (SF6), and the light fluid is nitrogen (N2). The shock wave converges from the heavy fluid into the light fluid. The Atwood number is 0.678. The total structured and uniform Cartesian grid node number in the main computational domain is 20483. In addition, to avoid the influence of boundary reflection, a sufficiently long sponge layer with 50 non-uniform coarse grids is added for each non-periodic boundary. Present numerical simulations have high and nonlinear initial perturbation levels, which rapidly lead to turbulent mixing in the mixing layers. Firstly, some physical-variable mean profiles, including mass fraction, Taylor Reynolds number, turbulent kinetic energy, enstrophy and helicity, are provided. Second, the mixing characteristics in the spherical and cylindrical turbulent mixing layers are investigated, such as molecular mixing fraction, efficiency Atwood number, turbulent mass-flux velocity and density self-correlation. Then, Reynolds stress and anisotropy are also investigated. Finally, the radial velocity, velocity divergence and enstrophy in the spherical and cylindrical turbulent mixing layers are studied using the method of conditional statistical analysis. Present numerical results show that the geometric effect has a great influence on the converging Richtmyer–Meshkov instability mixing layers.

Adaptive method for the solution of 1D and 2D advection–diffusion equations used in environmental engineering

Abstract The paper concerns the numerical solution of one-dimensional (1D) and two-dimensional (2D) advection–diffusion equations. For the numerical solution of the 1D advection–diffusion equation, a method, originally proposed for the solution of the 1D pure advection equation, has been developed. A modified equation analysis carried out for the proposed method allowed increasing of the resulting solution accuracy and, consequently, to reduce the numerical dissipation and dispersion. This is achieved by proper choice of the involved weighting parameter being a function of the Courant number and the diffusive number. The method is adaptive because for uniform grid point and for uniform flow velocity, the weighting parameter takes a constant value, whereas for non-uniform grid and for varying flow velocity, its value varies in the region of solution. For the solution of the 2D transport equation, the dimensional decomposition in the form of Strang splitting technique is used. Consequently, this equation is reduced to a series of the 1D equations with regard to x- and y-directions which next are solved using the aforementioned method. The results of computational experiments compared with the exact solutions confirmed that the proposed approaches ensure high solution accuracy of the transport equations.

A three-dimensional atmospheric dispersion model for Mars

Atmospheric local-to-regional dispersion models are widely used on Earth to predict and study the effects of chemical species emitted into the atmosphere and to contextualize sparse data acquired at particular locations and/or times. However, to date, no local-to-regional dispersion models for Mars have been developed; only mesoscale/microscale meteorological models have some dispersion and chemical capabilities, but they do not offer the versatility of a dedicated atmospheric dispersion model when studying the dispersion of chemical species in the atmosphere, as it is performed on Earth. Here, a new three-dimensional local-to-regional-scale Eulerian atmospheric dispersion model for Mars (DISVERMAR) that can simulate emissions to the Martian atmosphere from particular locations or regions including chemical loss and predefined deposition rates, is presented. The model can deal with topography and non-uniform grids. As a case study, the model is applied to the simulation of methane spikes as detected by NASA’s Mars Science Laboratory (MSL); this choice is made given the strong interest in and controversy regarding the detection and variability of this chemical species on Mars.

Direct numerical simulation of natural convection in a square cavity at high Rayleigh numbers via the Lagrange interpolating polynomial scheme

Direct numerical simulation (DNS) is a powerful research technique to solve the Navier–Stokes equations based on turbulent flows. DNS delivers highly reliable physical solutions with extreme accuracy. Unfortunately, present hardware and computing technologies remain limited, making it difficult to implement DNS in practice. The recently proposed Lagrange interpolating polynomial (LIP) scheme is a discretization technique for the finite-volume method. The LIP scheme is easy to use with nonuniform grid simulations and can flexibly increase or decrease the order of accuracy. The objective of this study is to verify that the LIP scheme is suitable for DNS of natural convection in a square cavity at high Rayleigh numbers (Ra). Accordingly, code was developed in-house using a second-order LIP scheme with the finite-volume method and the semi-implicit method for pressure-linked equations (SIMPLE) algorithm. Simulations were performed at Ra values of 109 and 1.58 × 109. The solutions were then verified against experimental benchmark data and published numerical solutions. The maximum differences in the Nusselt numbers between the computed solutions and the experimental benchmark data and published numerical solutions are 19.23% and 5.51%, respectively. However, the comparison results indicate that most of the computed solutions are in good agreement with experimental benchmark data and published numerical solutions.

Compact higher order discretization of 3D generalized convection diffusion equation with variable coefficients in nonuniform grids

A higher-order compact (HOC) discretization of generalized 3D convection-diffusion equation (CDE) in nonuniform grid is presented. Even in the presence of cross-derivative terms, the discretization uses only nineteen point stencil. Extension of this newly proposed discretization to semi-linear and convection-diffusion-reaction problems is seen to be straightforward and this inherent advantage is thoroughly exploited. The scheme being designed on a transformation free coordinate system is found to be efficient in capturing boundary layers and preserve the nonoscillatory property of the solution. The proposed method is tested using several benchmark linear and nonlinear problems from the literature. Additionally, problems with sharp gradients are solved. These diverse numerical examples demonstrate the accuracy and efficiency of the scheme proposed. Further, the numerical rate of convergence is seen to approach four confirming theoretical estimation.

A two-grid block-centered finite difference method for the nonlinear regularized long wave equation

In this paper, a Crank-Nicolson block-centered finite difference method is first developed and analyzed for the nonlinear regularized long wave equation. By using a cutoff technique, second-order convergences both in time and space are proved under a suitable time-space stepsize constraint condition. To further improve the computational efficiency, an efficient two-grid block-centered finite difference method is introduced and analyzed, in which a resulting small-scale nonlinear problem is first solved on a coarse grid space of size H, and then a resulting large-scale linear problem is solved on a fine grid space of size h. Under a rough time-space stepsize constraint condition Δt=o(H1/4), optimal-order error estimates for both the primal variable and its flux are derived on non-uniform spatial grids. Thus, the proposed method is competitive both in accuracy and efficiency compared with the fully nonlinear Crank-Nicolson block-centered finite difference scheme. Numerical experiments are presented to verify the theoretical analysis.

Approximation of Functions by Discrete Fourier Sums in Polynomials Orthogonal on a Nonuniform Grid with Jacobi Weight

Approximation of Functions by Discrete Fourier Sums in Polynomials Orthogonal on a Nonuniform Grid with Jacobi Weight

High order pseudo arc-length method for strong discontinuity of detonation wave

In this paper, in order to improve the resolution of capturing discontinuities, we introduce the pseudo arc-length parameter to make the mesh move to the discontinuities adaptively. By combining the high-precision WENO scheme with the pseudo arc-length algorithm, the advantages of both schemes can be shown, on the one hand, the solution has a higher convergence rate, on the other hand, it has a higher resolution for the region solution with larger physical variation . Because the traditional high-order scheme is based on Cartesian grid, and the grid in the pseudo arc-length numerical calculation is deformed. In view of the non-uniform grid and non-orthogonal deformation grid caused by the grid moving, the original deformed physical space is mapped to the uniform orthogonal arc-length calculation space by introducing coordinate transformation, and then the classical higher order scheme is used to solve the governing equations in the computational coordinate system. Through the comparison of some numerical examples and the analysis of numerical errors, it can be found that the pseudo arc-length algorithm is better than the finite volume method with fixed mesh. The high-order pseudo arc-length algorithm has a very high resolution to capture discontinuities, and the density of the grid near the discontinuities is very high. The adaptive grid movement weakens the singularity of the governing equation near the discontinuity, so the whole solution is smooth and the numerical oscillation is not obvious. This shows that the pseudo arc-length algorithm can overcome the shortcomings of high-order schemes which easily cause numerical oscillations. Finally, the chemical reaction flow problem is calculated. The results show that the high-order pseudo arc-length numerical algorithm with less mesh number has faster convergence rate and higher discontinuous resolution. Therefore, the high-order pseudo arc-length algorithm has obvious advantages in dealing with the strong discontinuity problem of explosion and shock.

Numerical study of the non-linear vibrations of electrically actuated curved micro-beams considering thermoelastic damping

Thermoelastic damping is a major source of intrinsic damping in vacuum operated Micro Electro Mechanical Systems (MEMS). In this article, the non-linear vibration of electrically actuated curved micro-beams is investigated, considering the effect of thermoelastic damping. The coupled thermoelasticity equations are derived on the basis of non-classical formulation, which is the modified couple stress theory and generalized heat conduction thermoelasticity equation. Afterwards, the order of the governing equations is reduced using the Generalized Differential Quadrature Method (GQDM). The validation of the derived model is achieved through comparison with existing published results. Then the effects of design parameters on snap-through buckling and pull-in instability have been investigated. Furthermore, it is observed that the GDQM with small number of nonuniform grid spacing performs accurate results without dealing with complicated shape functions and integrations. Finally the coupled thermo-mechanical modal analysis is carried out by FE simulation.

Sixth order compact finite difference schemes for Poisson interface problems with singular sources

Let Γ be a smooth curve inside a two-dimensional rectangular region Ω. In this paper, we consider the Poisson interface problem −∇2u=f in Ω∖Γ with Dirichlet boundary condition such that f is smooth in Ω∖Γ and the jump functions [u] and [∇u⋅n→] across Γ are smooth along Γ. This Poisson interface problem includes the weak solution of −∇2u=f+gδΓ in Ω as a special case. Because the source term f is possibly discontinuous across the interface curve Γ and contains a delta function singularity along the curve Γ, both the solution u of the Poisson interface problem and its flux ∇u⋅n→ are often discontinuous across the interface. To solve the Poisson interface problem with singular sources, in this paper we propose a sixth order compact finite difference scheme on uniform Cartesian grids. Our proposed compact finite difference scheme with explicitly given stencils extends the immersed interface method (IIM) to the highest possible accuracy order six for compact finite difference schemes on uniform Cartesian grids, but without the need to change coordinates into the local coordinates as in most papers on IIM in the literature. Also in contrast with most published papers on IIM, we explicitly provide the formulas for all involved stencils and therefore, our proposed scheme can be easily implemented and is of interest to practitioners dealing with Poisson interface problems. Note that the curve Γ splits Ω into two disjoint subregions Ω+ and Ω−. The coefficient matrix A in the resulting linear system Ax=b, following from the proposed scheme, is independent of any source term f, jump condition gδΓ, interface curve Γ and Dirichlet boundary conditions, while only b depends on these factors and is explicitly given, according to the configuration of the nine stencil points in Ω+ or Ω−. The constant coefficient matrix A facilitates the parallel implementation of the algorithm in case of a large size matrix and only requires the update of the right hand side vector b for different Poisson interface problems. Due to the flexibility and explicitness of the proposed scheme, it can be generalized to obtain the highest order compact finite difference scheme for non-uniform grids as well. We prove the order 6 convergence for the proposed scheme using the discrete maximum principle. Our numerical experiments confirm the sixth accuracy order of the proposed compact finite difference scheme on uniform meshes for the Poisson interface problems with various singular sources.

A robust node-shifting method for shape optimization of irregular gridshell structures

Evolutionary node shifting is an effective approach to the shape optimization of spatial structures for excellent mechanical performance. In this paper, a novel shape optimization algorithm is proposed, which is applicable to complex gridshell structures of irregular geometries and non-uniform grids. With the objective of maximizing the structural stiffness, the nodal coordinates are iteratively updated according to the sensitivity information. The perturbation displacements referring to the inverse hanging method are applied to the initial flat model. By analysis, we found that the irregular grids give rise to non-smooth gradient fields, which results in jagged surfaces. To normalize the non-smooth gradient fields and to prevent the optimization process from falling into local optima, a double filter scheme is introduced in the process of optimization. A variety of examples are presented to demonstrate that the proposed algorithm can effectively solve shape optimization problems of general gridshell structures.

Heat and mass transfer characteristics of double‐diffusive natural convection in a porous annulus: A higher‐order compact approach

AbstractIn this paper, we apply a reconstructed higher‐order compact scheme on nonuniform grids to the problem of double‐diffusive natural convection in a vertical porous annulus. The Darcy model is adopted for the fluid flow coupled with energy and mass equations. Computations are carried out in the regime , , , , and , where is the aspect ratio, is the radius ratio, is the Lewis number, is the buoyancy ratio, and is the thermal Rayleigh number. Extensive computations are carried out to investigate the influence of these parameters on the flow and a general pattern of heat and mass transfer characteristics emerges from the study. Particular attention has been paid to the transient flow behavior while investigating the influence of on the buoyancy ratio configuration.