WENO Scheme Research Articles

A new third order finite volume weighted essentially non-oscillatory scheme on tetrahedral meshes

In this paper a third order finite volume weighted essentially non-oscillatory scheme is designed for solving hyperbolic conservation laws on tetrahedral meshes. Comparing with other finite volume WENO schemes designed on tetrahedral meshes, the crucial advantages of such new WENO scheme are its simplicity and compactness with the application of only six unequal size spatial stencils for reconstructing unequal degree polynomials in the WENO type spatial procedures, and easy choice of the positive linear weights without considering the topology of the meshes. The original innovation of such scheme is to use a quadratic polynomial defined on a big central spatial stencil for obtaining third order numerical approximation at any points inside the target tetrahedral cell in smooth region and switch to at least one of five linear polynomials defined on small biased/central spatial stencils for sustaining sharp shock transitions and keeping essentially non-oscillatory property simultaneously. By performing such new procedures in spatial reconstructions and adopting a third order TVD Runge–Kutta time discretization method for solving the ordinary differential equation (ODE), the new scheme's memory occupancy is decreased and the computing efficiency is increased. So it is suitable for large scale engineering requirements on tetrahedral meshes. Some numerical results are provided to illustrate the good performance of such scheme.

A new adaptive weighted essentially non-oscillatory WENO-θ scheme for hyperbolic conservation laws

A new adaptive weighted essentially non-oscillatory WENO- θ scheme in the context of finite difference is proposed. Depending on the smoothness of the large stencil used in the reconstruction of the numerical flux, a parameter θ is set adaptively to switch the scheme between a 5th-order upwind and 6th-order central discretization. A new indicator τ θ measuring the smoothness of the large stencil is chosen among two candidates which are devised based on the possible highest-order variations of the reconstruction polynomials in L 2 sense. In addition, a new set of smoothness indicators β ̃ k of the substencils is introduced. These are constructed in a central sense with respect to the Taylor expansions around the point x j . Numerical results show that the new scheme outperforms other comparing 6th-order WENO schemes in terms of improving the resolution at critical regions of nonsmooth problems as well as maintaining symmetry in the solutions.

A numerical framework for the direct simulation of dense particulate flow under explosive dispersal

In this paper, we present a Cartesian grid-based numerical framework for the direct simulation of dense particulate flow under explosive dispersal. This numerical framework is established through the integration of the following numerical techniques: (1) operator splitting for partitioned fluid–solid interaction in the time domain, (2) the second-order SSP Runge–Kutta method and third-order WENO scheme for temporal and spatial discretization of governing equations, (3) the front-tracking method for evolving phase interfaces, (4) a field function proposed for low-memory-cost multimaterial mesh generation and fast collision detection, (5) an immersed boundary method developed for treating arbitrarily irregular and changing boundaries, and (6) a deterministic multibody contact and collision model. Employing the developed framework, this paper further studies particle jet formation under explosive dispersal by considering the effects of particle properties, particulate payload morphologies, and burster pressures. By the simulation of the dispersal processes of dense particle systems driven by pressurized gas, in which the driver pressure reaches $$1.01325\times 10^{10}\,\,\mathrm {Pa}$$ ( $$10^5$$ times the ambient pressure) and particles are impulsively accelerated from stationary to a speed that is more than $$12000\,\,\mathrm {m/s}$$ within $$15\,\,\mathrm {\upmu s}$$ , it is demonstrated that the presented framework is able to effectively resolve coupled shock–shock, shock–particle, and particle–particle interactions in complex fluid–solid systems with shocked flow conditions, arbitrarily irregular particle shapes, and realistic multibody collisions.

Constrained moving least‐squares immersed boundary method for fluid‐structure interaction analysis

SummaryA numerical method is presented for the analysis of interactions of inviscid and compressible flows with arbitrarily shaped stationary or moving rigid solids. The fluid equations are solved on a fixed rectangular Cartesian grid by using a higher‐order finite difference method based on the fifth‐order WENO scheme. A constrained moving least‐squares sharp interface method is proposed to enforce the Neumann‐type boundary conditions on the fluid‐solid interface by using a penalty term, while the Dirichlet boundary conditions are directly enforced. The solution of the fluid flow and the solid motion equations is advanced in time by staggerly using, respectively, the third‐order Runge‐Kutta and the implicit Newmark integration schemes. The stability and the robustness of the proposed method have been demonstrated by analyzing 5 challenging problems. For these problems, the numerical results have been found to agree well with their analytical and numerical solutions available in the literature. Effects of the support domain size and values assigned to the penalty parameter on the stability and the accuracy of the present method are also discussed.

Comparison of difference scheme with customizable dissipative properties and WENO scheme in the case of one-dimensional gas and gas-particle dynamics problems

Subject of Research.The paper presents results of comparative testing of highly stable difference scheme with customizable dissipative properties and WENO5 scheme in the case of one-dimensional gas dynamics and mechanics of gas-suspensions problems. Method. The second order difference scheme is constructed by physical processes splitting into two phases. The first one uses the central differences with adaptive reconstruction of artificial viscosity and semi implicit approximation of the sources. TVD type reconstructions are used in the second phase. Main Results. For the representative collection of test problems, the proposed scheme has confirmed the efficiency and good quality numerical solutions at the level of the WENO5 scheme with substantially lower machine time cost. The wave structure in the decay of initial discontinuity in the non-equilibrium gas suspension and the convergence to exact self-similar solution are considered. Practical Relevance. A detailed resolution of structural peculiarities of gas flows and gas-particle mixtures allows reducing the amount of experimental development of new technologies and technical devices of coating, leakage control when using a trial particulate matter with particles of micro - and nano-sizes and other areas.

High-resolution simulation and parametric research on helicopter rotor vortex flowfield with TAMI control in hover

In order to investigate the effect of active jet control on the rotor vortex flowfield in hover, a high-accuracy numerical method is presented in this paper, and a novel computer code is developed to calculate rotor flowfield with tip air mass injection. In the present computational fluid dynamics method, the Navier–Stokes equation is selected as the governing equation. For the sake of reducing numerical diffusion in calculation, the upwind Roe scheme with a fifth-order WENO scheme is employed to spatial discretization. A dual-time method is utilized in time marching and the high-efficiency implicit LU-SGS scheme is applied to every pseudo time step. The Baldwin–Lomax turbulence model is employed for eddy viscosity modeling. A surface boundary condition, which may effectively simulate the effect of tip air mass injection, is introduced into the current method. The overset grid method, which can veritably simulate the rotor blade motion, is adopted to exchange the flowfield information between the rotor blade and background grids. By the established method, the rotor flowfield with jet control of different jet angles is simulated. Furthermore, parametric analyses of jet velocity are carried out at different tip March numbers and collective pitches. It is demonstrated that tip air mass injection can make the tip vortex rolled up and generated in advance compared with the clean rotor, which can change the position and strength of the vortex core greatly. It is also shown that the rotor aerodynamic characteristics, such as blade surface pressure and radial lift distributions, are more sensitive to tip air mass injection control at the condition with a lower Mach number or a lower collective pitch.

A stochastic Galerkin method for first-order quasilinear hyperbolic systems with uncertainty

This paper is concerned with generalized polynomial chaos (gPC) approximation for first-order quasilinear hyperbolic systems with uncertainty. The one-dimensional (1D) hyperbolic system is first symmetrized with the aid of left eigenvector matrix of the Jacobian matrix. Then the gPC stochastic Galerkin method is applied to derive a provably symmetrically hyperbolic equations for the gPC expansion coefficients. The resulting deterministic gPC Galerkin system is discretized by a path-conservative finite volume WENO scheme in space and a third-order total variation diminishing Runge–Kutta method in time. The method is further extended to two-dimensional (2D) quasilinear hyperbolic system with uncertainty, where the symmetric hyperbolicity of the one-dimensional gPC Galerkin system is carried over via an operator splitting technique. Several numerical experiments are conducted to demonstrate the accuracy and effectiveness of the proposed gPC stochastic Galerkin method.

Evolution of flame kernel in one eddy turnover of high-speed droplet laden shear layers

A high-speed droplet-laden reacting shear layer is modeled numerically to analyze the dynamics of flame kernel for better understanding ignition and extinction processes in spray combustion. The droplet-laden shear layer, separating a hot-air stream from a monodisperse spray-laden air stream, is modeled by the Eulerian-Lagrangian approach, in which the continuous phase is governed by the compressible Navier-Stokes equations together with species transport equations and the discrete droplets are tracked by the Lagrangian method. The convective flux terms of the conservative equations are solved by an adaptive central-upwind WENO scheme. Two-way coupling interactions consider exchanges of mass, momentum, and energy between the carrier-gas fluid and the liquid-fuel spray. The shear layer convective Mach number is specified as 0.4 in the present study, and hence, large scales of motion dominate both the fluid mixing and the droplet dispersion. The preferential concentration of droplets is observed. Auto-ignition kernels occur in the high-strain regions where sufficient fuel vapors distribute near the hot stream-boundary. The incipient chemical reaction consumes the available fuel vapors through a diffusion flame combined with a thin premixed flame, as a lean deflagration propagates across the mixing layer towards the spray. The present results unveil two different types of extinction behavior depending on the local strain rate and the available fuel for the reaction kernel. The latter type of extinction originates when the separated flame kernel is entrained into the core of an adjacent vortex but droplets cannot be entrained in the eddies to produce gaseous fuels for the chemical reaction due to the preferential segregation effect. The dynamics of ignition and extinction in the shear flows depend on the droplet dispersion, the droplet evaporation rate and the diffusion of fuel vapor.

Fine structures for the solutions of the two-dimensional Riemann problems by high-order WENO schemes

The two-dimensional Riemann problem with polytropic gas is considered. By a restriction on the constant states of each quadrant of the computational domain such that there is only one planar centered wave connecting two adjacent quadrants, there are nineteen genuinely different initial configurations of the problem. The configurations are numerically simulated on a fine grid and compared by the 5th-order WENO-Z5, 6th-order WENO-źź6, and 7th-order WENO-Z7 schemes. The solutions are very well approximated with high resolution of waves interactions phenomena and different types of Mach shock reflections. Kelvin-Helmholtz instability-like secondary-scaled vortices along contact continuities are well resolved and visualized. Numerical solutions show that WENO-źź6 outperforms the comparing WENO-Z5 and WENO-Z7 in terms of shock capturing and small-scaled vortices resolution. A catalog of the numerical solutions of all nineteen configurations obtained from the WENO-źź6 scheme is listed. Thanks to their excellent resolution and sharp shock capturing, the numerical solutions presented in this work can be served as reference solutions for both future numerical and theoretical analyses of the 2D Riemann problem.

A hybrid method for improving the robustness of WENO scheme

In this paper, a hybrid scheme based on fifth-order finite difference WENO scheme is proposed for solving spurious oscillation in strong shock-capturing. In this scheme, we consider smoothness indicator as a measurement of gradient and construct a function of indicator, using which the new hybrid can switch between WENO scheme and Lax-Friedrichs (LxF) scheme. Numerical experiments demonstrate that the hybrid scheme dominates the accuracy of the solution in smooth region and non-oscillation can be promised around discontinuity. The new hybrid scheme could essentially preserve positive density and positive pressure. The idea of constructing the hybrid can be easily extended to other high resolution schemes (TVD, ROE etc.) coupled with WENO scheme. These ideas will be illustrated in further study.

Shock Capturing Shallow Water Model for Long Waves Generated by a Moving Atmospheric Pressure

ABSTRACT Choi, Y.-K. and Seo, S.-N., 2017. Shock capturing shallow water model for long waves generated by a moving atmospheric pressure. In: Lee, J.L.; Griffiths, T.; Lotan, A.; Suh, K.-S., and Lee, J. (eds.), The 2nd International Water Safety Symposium. Journal of Coastal Research, Special Issue No. 79, pp. 354–358. Coconut Creek (Florida), ISSN 0749-0208. The shock capturing numerical model based on nonlinear shallow water equations is developed to predict run-up as well as edge waves generated by a moving atmospheric pressure disturbance across a straight shoreline on a sloping beach. The HLL approximate Riemann solver with the fifth order accurate WENO scheme is used to compute numerical flux in spatial discretization, which yields better results in comparison with TVD-MUSCL type schemes. Numerical experiments are conducted for wave propagation by a moving atmospheric pressure disturbance normally approaching the coastline. When the atmospheric pressure disturbance suddenly acts on the water surface...

Model Gas-Dynamical Problems Possessing Cylindrical and Spherical Symmetry and Their Solution with the Aid of Weno Schemes

The solution of a number of problems of gas dynamics that are associated with cylindrically and spherically symmetric nonstationary compression of an inviscid gas is considered. Discretization of the Euler equations is carried out with the aid of the method of finite volumes and difference schemes of the WENO type. The results of numerical simulation are compared with exact solutions and with the results of the calculations available in the literature. The solutions of two- and three-dimensional problems possessing cylindrical and spherical symmetry are given in comparison with the solutions obtained in one-dimensional formulation. The calculations with the use of WENO schemes make it possible to obtain an exact and monotonic solution of the problem in the presence of both weak and strong gas-dynamical discontinuities.

Finite difference Hermite WENO schemes for the Hamilton–Jacobi equations

In this paper, a new type of finite difference Hermite weighted essentially non-oscillatory (HWENO) schemes are constructed for solving Hamilton–Jacobi (HJ) equations. Point values of both the solution and its first derivatives are used in the HWENO reconstruction and evolved via time advancing. While the evolution of the solution is still through the classical numerical fluxes to ensure convergence to weak solutions, the evolution of the first derivatives of the solution is through a simple dimension-by-dimension non-conservative procedure to gain efficiency. The main advantages of this new scheme include its compactness in the spatial field and its simplicity in the reconstructions. Extensive numerical experiments in one and two dimensional cases are performed to verify the accuracy, high resolution and efficiency of this new scheme.

Dimension-by-dimension moment-based central Hermite WENO schemes for directly solving Hamilton-Jacobi equations

In this paper, a class of high-order central Hermite WENO (HWENO) schemes based on finite volume framework and staggered meshes is proposed for directly solving one- and two-dimensional Hamilton-Jacobi (HJ) equations. The methods involve the Lax-Wendroff type discretizations or the natural continuous extension of Runge-Kutta methods in time. This work can be regarded as an extension of central HWENO schemes for hyperbolic conservation laws (Tao et al. J. Comput. Phys. 318, 222–251, 2016) which combine the central scheme and the HWENO spatial reconstructions and therefore carry many features of both schemes. Generally, it is not straightforward to design a finite volume scheme to directly solve HJ equations and a key ingredient for directly solving such equations is the reconstruction of numerical Hamiltonians to guarantee the stability of methods. Benefited from the central strategy, our methods require no numerical Hamiltonians. Meanwhile, the zeroth-order and the first-order moments of the solution are involved in the spatial HWENO reconstructions which is more compact compared with WENO schemes. The reconstructions are implemented through a dimension-by-dimension strategy when the spatial dimension is higher than one. A collection of one- and two- dimensional numerical examples is performed to validate high resolution and robustness of the methods in approximating the solutions of HJ equations, which involve linear, nonlinear, smooth, non-smooth, convex or non-convex Hamiltonians.

Tribological study of an aerodynamic thrust bearing in the supersonic regime

Nowadays, aerodynamic air thrust bearing are mainly used over a large panel of turbo-machineries. These systems become increasingly faster and up to operate in supersonic regime. They have not reached a sufficient level of research in terms of high speed. The possibility of an aerodynamic thrust bearing operating in a supersonic regime is studied. First, the air film dynamic study for high Reynolds number is based on the “modified Reynolds” equations, which take into account the inertia terms, the viscosity’s variation in the film thickness, and the turbulence. It’s an extension of the traditional model used in lubrication called the generalized Reynolds equation. The results show that a depression occur at the location of the change of slope of the tapper flat geometry. The hypothesis of presence of shock or rarefaction waves shows that the pressure gradient in the film thickness may be no longer negligible. The modified Reynolds equation may be not enough to describe the problem. A new system is built to study these phenomena: the Navier-Stokes equation are adapted to the film’s geometry. The dynamic air film’s behavior study in supersonic regime requires a shock capturing scheme called WENO scheme (“Weighted Essentially Non Oscillatory”), mainly used in shock and turbulence studies. The numerical results give the film behavior modelling of a fixed air thrust bearing pad. The evolution of the quantities shows that shock wave can occur in a thin film.

On consistent boundary closures for compact finite-difference WENO schemes

The accuracy of compact finite-difference schemes can be degraded by inconsistent domain or box boundary treatments. A consistent higher-order boundary closure is especially important for block-structured Cartesian AMR solvers, where the computational domain is generally decomposed into a large number of boxes containing a relatively small number of grid points. At each box boundary, a consistent higher-order boundary closure needs to be applied to avoid a reduction of the formal order-of-accuracy of the numerical scheme. This paper presents such a boundary closure for the fifth-order accurate compact finite-difference WENO scheme by Ghosh and Baeder [1]. The accuracy of the new boundary closure is validated by employing the method of manufactured solutions. A comparison of the new compact boundary closure with the original explicit boundary closure demonstrates the improved accuracy for the new compact boundary closure, while the behavior of the scheme across discontinuities appears unaffected. The linear stability analysis results indicate that a linearly stable compact WENO boundary closure is achieved.

A New Type of Modified WENO Schemes for Solving Hyperbolic Conservation Laws

In this paper we design a new type of the third order and fifth order modified weighted essentially nonoscillatory (MWENO) schemes in the finite difference framework for solving the hyperbolic conservation laws. These schemes adapt between the linear upwind scheme and the WENO scheme automatically by the usage of a new simple switching principle. The methodology to reconstruct numerical fluxes for the MWENO schemes is split into two parts: if all extreme points of the reconstruction polynomial for numerical flux in the big spatial stencil are located outside of the stencil, the numerical flux is approximated directly by the reconstruction polynomial, and the approximation is linear and of high order accuracy; otherwise, the WENO procedure in [G.-S. Jiang and C.-W. Shu, J. Comput. Phys., 126 (1996), pp. 202--228; C.-W. Shu, SIAM Review, 51 (2009), pp. 82--126] is applied to reconstruct the numerical flux. The main advantage of these new MWENO schemes is their robustness and efficiency compared to the classical WENO schemes specified in [G.-S. Jiang and C.-W. Shu, J. Comput. Phys., 126 (1996), pp. 202--228; C.-W. Shu, SIAM Review, 51 (2009), pp. 82--126.]. The MWENO schemes can be applied to compute some extreme test cases such as the Sedov blast wave, the Leblanc and the high Mach number astrophysical jet problems, etc., by using a normal CFL number without any further positivity preserving procedure for the purpose of controlling the concurrence of negative density and pressure. Extensive numerical results are provided to illustrate the good performance of the MWENO schemes.

Central Weighted ENO Schemes for Hyperbolic Conservation Laws on Fixed and Moving Unstructured Meshes

We present a novel arbitrary high order accurate central WENO spatial reconstruction procedure (CWENO) for the solution of nonlinear systems of hyperbolic conservation laws on fixed and moving unstructured simplex meshes in two and three space dimensions. Starting from the given cell averages of a function on a triangular or tetrahedral control volume and its neighbors, the nonlinear CWENO reconstruction yields a high order accurate and essentially non-oscillatory polynomial that is defined everywhere in the cell. Compared to other WENO schemes on unstructured meshes, the total stencil size is the minimum possible one, as in classical point-wise WENO schemes of Jiang and Shu. However, the linear weights can be chosen arbitrarily, which makes the practical implementation on general unstructured meshes particularly simple. We make use of the piecewise polynomials generated by the CWENO reconstruction operator inside the framework of fully discrete and high order accurate one-step ADER finite volume schemes on fixed Eulerian grids as well as on moving Arbitrary-Lagrangian-Eulerian (ALE) meshes. The computational efficiency of the high order finite volume schemes based on the new CWENO reconstruction is tested on several two- and three-dimensional systems of hyperbolic conservation laws and is found to be more efficient in terms of memory consumption and computational efficiency with respect to classical WENO reconstruction schemes on unstructured meshes. We also provide evidence that the new algorithm is suitable for implementation on massively parallel distributed memory supercomputers, showing two numerical examples run with more than one billion degrees of freedom in space. To our knowledge, at present these are the largest simulations ever run with unstructured WENO finite volume schemes.

Схема EBR-WENO для решения задач газовой динамики с разрывами на неструктурированных сетках

Abstract We develop the Edge-Based Reconstruction WENO scheme for solving Euler equations on unstructured meshes. It belongs to the class of edge-based schemes with quasi-1D reconstruction of variables. The scheme monotonization is provided by using a convex combination of three lower-order reconstructions of variables in a similar way as it is in the classical finite-difference WENO scheme. The new scheme damps oscillations near shocks on unstructured meshes and, due to its edge-based nature, requires rather low computational costs. The properties of the new scheme are demonstrated on several test problems.

3D Compressible Navier-Stokes Equations Solve Optical Properties of Forced Shear Layers

The optical properties of shear layers is a topic of research with applications in high power lasers and optical imaging systems. In recent years, it has been shown that large-scale structures are intrinsic features in a plane turbulent mixing layer. While there has been some research on the effects of turbulence on laser beams, there has been no study of the optical effects of large-scale structures as they exist in shear layer development, nor how these effects may be controlled. Since the susceptibility of a free shear layer to a forcing input increases as the location of the forcing moves closer to the trailing edge of the splitter plate. Disturbances can be imposed on shear layer using very small-amplitude displacements of the splitter-plate trailing edge in a direction perpendicular to the flow direction. We are studying the optical effects of large-scale structures in forced and unforced situations using periodic motion of a small flap at the trailing edge of the splitter plate. Three-dimensional Navier-Stokes equations in conservation law form are solved directly using the WENO scheme for the simulation of a spatially-developing free shear layer.