A fully conservative high‐order upwind multi‐moment method using moments in both upwind and downwind cells

We propose a fully conservative high‐order upwind multi‐moment method for the conservation equation. The proposed method is based on a third‐order polynomial interpolation function and semi‐Lagrangian formulation and is a variant of the constrained interpolation profile conservative semi‐Lagrangian scheme with third‐order polynomial function method. The third‐order interpolation function is constructed based on three constraints in the upwind cell (two boundary values and a cell average) and a constraint in the downwind cell (a cell center value). The proposed method shows fourth‐order accuracy in a benchmark problem (sine wave propagation). We also propose a less oscillatory formulation of the proposed method. The less oscillatory formulation can minimize numerical oscillations. These methods were validated through scalar transport problems, and compressible flow problems (shock tube and 2D explosion problems). Copyright © 2016 John Wiley & Sons, Ltd.     

A kinetic energy preserving nodal discontinuous Galerkin spectral element method

In this work, we discuss the construction of a skew‐symmetric discontinuous Galerkin (DG) collocation spectral element approximation for the compressible Euler equations. Starting from the skew‐symmetric formulation of Morinishi, we mimic the continuous derivations on a discrete level to find a formulation for the conserved variables. In contrast to finite difference methods, DG formulations naturally have inter‐domain surface flux contributions due to the discontinuous nature of the approximation space. Thus, throughout the derivations we accurately track the influence of the surface fluxes to arrive at a consistent formulation also for the surface terms. The resulting novel skew‐symmetric method differs from the standard DG scheme by additional volume terms. Those volume terms have a special structure and basically represent the discretization error of the different product rules. We use the summation‐by‐parts (SBP) property of the Gauss–Lobatto‐based DG operator and show that the novel formulation is exactly conservative for the mass, momentum, and energy. Finally, an analysis of the kinetic energy balance of the standard DG discretization shows that because of aliasing errors, a nonzero transport source term in the evolution of the discrete kinetic energy mean value may lead to an inconsistent increase or decrease in contrast to the skew‐symmetric formulation. Furthermore, we derive a suitable interface flux that guarantees kinetic energy preservation in combination with the skew‐symmetric DG formulation. As all derivations require only the SBP property of the Gauss–Lobatto‐based DG collocation spectral element method operator and that the mass matrix is diagonal, all results for the surface terms can be directly applied in the context of multi‐domain diagonal norm SBP finite difference methods. Numerical experiments are conducted to demonstrate the theoretical findings. Copyright © 2014 John Wiley & Sons, Ltd.     

An entropy fixed cell‐centered Lagrangian scheme

On the basis of the work [P.‐H. Maire, R. Abgrall, J. Breil, J. Ovadia, SIAM J. Sci. Comput. 29 (2007), 1781–1824], we present an entropy fixed cell‐centered Lagrangian scheme for solving the Euler equations of compressible gas dynamics. The scheme uses the fully Lagrangian form of the gas dynamics equations, in which the primary variables are cell‐centered. And using the nodal solver, we obtain the nodal viscous‐velocity, viscous‐pressures, antidissipation velocity, and antidissipation pressures of each node. The final nodal velocity is computed as a weighted sum of viscous‐velocity and antidissipation velocity, so do nodal pressures, whereas these weights are calculated through the total entropy conservation for isentropic flows. Consequently, the constructed scheme is conservative in mass, momentum, and energy; preserves entropy for isentropic flows, and satisfies a local entropy inequality for nonisentropic flows. One‐ and two‐dimensional numerical examples are presented to demonstrate theoretical analysis and performance of the scheme in terms of accuracy and robustness.Copyright © 2013 John Wiley & Sons, Ltd.     

Comparisons of CIP,compact and CIP‐CSL2 schemes for reproducing internal solitary waves

Six different models were evaluated for reproducing internal solitary waves which occur and propagate in a stratified flow field with a sharp interface. Three stages were used to compute internal solitary waves in a stratified field: (1) first‐phase computation of momentum equations, (2) second‐phase computation of momentum equations, which corresponds to computing the Poisson's equation, and (3) density computation. The six models discussed in this paper consisted of combinations of four different schemes, a three‐point combined compact difference scheme (CCD), a normal central difference scheme (CDS), a cubic‐polynomial interpolation (CIP), and an exactly conservative semi‐Lagrangian scheme (CIP‐CSL2). The residual cutting method was used to solve the Poisson's equation. Three tests were used to confirm the validity of the computations using KdV theory; i.e. the incremental wave speed and amplitude of internal solitary waves, the maximum horizontal velocity and amplitude, and the wave form. In terms of the shape of an internal solitary wave, using CIP for momentum equations was found to provide better performance than CCD. These results suggest one of the most appropriate scheme for reproducing internal solitary waves may be one in which CIP is used for momentum equations and CCD to solve the Poisson's equation. Copyright © 2005 John Wiley & Sons, Ltd.     

Stability of a Crank–Nicolson pressure correction scheme based on staggered discretizations

In the context of LES of turbulent flows, the control of kinetic energy seems to be an essential requirement for a numerical scheme. Designing such an algorithm, that is, as less dissipative as possible while being simple, for the resolution of variable density Navier–Stokes equations is the aim of the present work. The developed numerical scheme, based on a pressure correction technique, uses a Crank–Nicolson time discretization and a staggered space discretization relying on the Rannacher–Turek finite element. For the inertia term in the momentum balance equation, we propose a finite volume discretization, for which we derive a discrete analogue of the continuous kinetic energy local conservation identity. Contrary to what was obtained for the backward Euler discretization, the dissipation defect term associated to the Crank–Nicolson scheme is second order in time. This behavior is evidenced by numerical simulations. Copyright © 2013 John Wiley & Sons, Ltd.     

A finite volume cell‐centered Lagrangian hydrodynamics approach for solids in general unstructured grids

A finite volume cell‐centered Lagrangian hydrodynamics approach, formulated in Cartesian frame, is presented for solving elasto‐plastic response of solids in general unstructured grids. Because solid materials can sustain significant shear deformation, evolution equations for stress and strain fields are solved in addition to mass, momentum, and energy conservation laws. The total stress is split into deviatoric shear stress and dilatational components. The dilatational response of the material is modeled using the Mie‐Grüneisen equation of state. A predicted trial elastic deviatoric stress state is evolved assuming a pure elastic deformation in accordance with the hypo‐elastic stress‐strain relation. The evolution equations are advanced in time by constructing vertex velocity and corner traction force vectors using multi‐dimensional Riemann solutions erected at mesh vertices. Conservation of momentum and total energy along with the increase in entropy principle are invoked for computing these quantities at the vertices. Final state of deviatoric stress is effected via radial return algorithm based on the J‐2 von Mises yield condition. The scheme presented in this work is second‐order accurate both in space and time. The suitability of the scheme is evinced by solving one‐ and two‐dimensional benchmark problems both in structured grids and in unstructured grids with polygonal cells. Copyright © 2013 John Wiley & Sons, Ltd.     

A frame invariant and maximum principle enforcing second‐order extension for cell‐centered ALE schemes based on local convex hull preservation

Two difficulties are clearly identified for high‐order extensions of ALE schemes for Euler equations: strict respect of the maximum principle and preservation of the Galilean invariance. We deal with these two issues in this paper. Our approach is closely related to the concepts of a posteriori limiting and convex hull spanning. We introduce the notion of local convex hull preservation schemes, which embodies these two concepts. We lean on this notion to propose a fully Galilean invariant ALE scheme. Moreover, we provide a new limiter (called Apitali for A Posteriori ITerAtive LImiter) for the remap step, enforcing the local convex hull preservation property. Copyright © 2014 John Wiley & Sons, Ltd.     

Some considerations for high‐order ‘incremental remap’‐based transport schemes: edges,reconstructions, and area integration

The problem of two‐dimensional tracer advection on the sphere is extremely important in modeling of geophysical fluids and has been tackled using a variety of approaches. A class of popular approaches for tracer advection include ‘incremental remap’ or cell‐integrated semi‐Lagrangian‐type schemes. These schemes achieve high‐order accuracy without the need for multistage integration in time, are capable of large time steps, and tend to be more efficient than other high‐order transport schemes when applied to a large number of tracers over a single velocity field. In this paper, the simplified flux‐form implementation of the Conservative Semi‐LAgrangian Multi‐tracer scheme (CSLAM) is reformulated using quadratic curves to approximate the upstream flux volumes and Gaussian quadrature for integrating the edge flux. The high‐order treatment of edge fluxes is motivated because of poor accuracy of the CSLAM scheme in the presence of strong nonlinear shear, such as one might observe in the midlatitudes near an atmospheric jet. Without the quadratic treatment of upstream edges, we observe at most second‐order accuracy under convergence of grid resolution, which is returned to third‐order accuracy under the improved treatment. A shallow‐water barotropic instability also reveals clear evidence of grid imprinting without the quadratic correction. Consequently, these tests reveal a problem that might arise in tracer transport near nonlinearly sheared regions of the real atmosphere, particularly near cubed‐sphere panel edges. Although CSLAM is used as the foundation for this analysis, the conclusions of this paper are applicable to the general class of incremental remap schemes. Copyright © 2012 John Wiley & Sons, Ltd.     

A high‐order fully explicit flux‐form semi‐Lagrangian shallow‐water model

A new approach is proposed for constructing a fully explicit third‐order mass‐conservative semi‐Lagrangian scheme for simulating the shallow‐water equations on an equiangular cubed‐sphere grid. State variables are staggered with velocity components stored pointwise at nodal points and mass variables stored as element averages. In order to advance the state variables in time, we first apply an explicit multi‐step time‐stepping scheme to update the velocity components and then use a semi‐Lagrangian advection scheme to update the height field and tracer variables. This procedure is chosen to ensure consistency between dry air mass and tracers, which is particularly important in many atmospheric chemistry applications. The resulting scheme is shown to be competitive with many existing numerical methods on a suite of standard test cases and demonstrates slightly improved performance over other high‐order finite‐volume models. Copyright © 2014 John Wiley & Sons, Ltd.     

Applications of a higher‐order bounded numerical scheme to turbulent flows

This paper applies the higher‐order bounded numerical scheme Weighted Average Coefficients Ensuring Boundedness (WACEB) to simulate two‐ and three‐dimensional turbulent flows. In the scheme, a weighted average formulation is used for interpolating the variables at cell faces and the weighted average coefficients are determined from a normalized variable formulation and total variation diminishing (TVD) constraints to ensure the boundedness of the solution. The scheme is applied to two turbulent flow problems: (1) two‐dimensional turbulent flow around a blunt plate; and (2) three‐dimensional turbulent flow inside a mildly curved U‐bend. In the present study, turbulence is evaluated by using a low‐Reynolds number version of the k–ω model. For the flow simulation, the QUICK scheme is applied to the momentum equations while either the WACEB scheme (Method 1) or the UPWIND scheme (Method 2) is used for the turbulence equations. The present study shows that the WACEB scheme has at least second‐order accuracy while ensuring boundedness of the solutions. The present numerical study for a pure convection problem shows that the ‘TVD’ slope ranges from 2 to 4. For the turbulent recirculating flow, two different mixed procedures (Method 1 and Method 2) produce a substantial difference for the mean velocities as well as for the turbulence kinetic energy. Method 1 predicts better results than Method 2 does, comparing the analytical solution and the experimental data. For the turbulent flow inside the mildly curved U‐bend, although the predictions of velocity distributions with two procedures are very close, a noticeable difference of turbulence kinetic energy is exhibited. It is noticed that the discrepancy exists between numerical results and the experimental data. The reason is the limit of the two‐equation turbulence model to such complex turbulent flows with extra strain‐rates. Copyright © 2001 John Wiley & Sons, Ltd.     

A finite volume solver for 1D shallow‐water equations applied to an actual river

This paper describes the numerical solution of the 1D shallow‐water equations by a finite volume scheme based on the Roe solver. In the first part, the 1D shallow‐water equations are presented. These equations model the free‐surface flows in a river. This set of equations is widely used for applications: dam‐break waves, reservoir emptying, flooding, etc. The main feature of these equations is the presence of a non‐conservative term in the momentum equation in the case of an actual river. In order to apply schemes well adapted to conservative equations, this term is split in two terms: a conservative one which is kept on the left‐hand side of the equation of momentum and the non‐conservative part is introduced as a source term on the right‐hand side. In the second section, we describe the scheme based on a Roe Solver for the homogeneous problem. Next, the numerical treatment of the source term which is the essential point of the numerical modelisation is described. The source term is split in two components: one is upwinded and the other is treated according to a centred discretization. By using this method for the discretization of the source term, one gets the right behaviour for steady flow. Finally, in the last part, the problem of validation is tackled. Most of the numerical tests have been defined for a working group about dam‐break wave simulation. A real dam‐break wave simulation will be shown. Copyright © 2002 John Wiley & Sons, Ltd.     

Direct numerical simulation of decaying two‐dimensional turbulence in a no‐slip square box using smoothed particle hydrodynamics

This paper explores the application of SPH to a DNS of decaying turbulence in a two‐dimensional no‐slip wall‐bounded domain. In this bounded domain, the inverse energy cascade, and a net torque exerted by the boundary, results in a spontaneous spin‐up of the fluid, leading to a typical end state of a large monopole vortex that fills the domain. The SPH simulations were compared against published results using a high‐accuracy pseudo‐spectral code. Ensemble averages of the kinetic energy, enstrophy and average vortex wavenumber compared well against the pseudo‐spectral results, as did the evolution of the total angular momentum of the fluid. However, although the pseudo‐spectral results emphasised the importance of the no‐slip boundaries as generators of long‐lived coherent vortices in the flow, no such generation was seen in the SPH results. Vorticity filaments produced at the boundary were always dissipated by the flow shortly after separating from the boundary layer. The kinetic energy spectrum of the SPH results was calculated using an SPH Fourier transform that operates directly on the disordered particles. The ensemble kinetic energy spectrum showed the expected k^?3 scaling over most of the inertial range. However, the spectrum flattened at smaller length scales (initially less than 7.5 particle spacings and growing in size over time), indicating an excess of small‐scale kinetic energy.Copyright © 2011 John Wiley & Sons, Ltd.     

A multi‐moment transport model on cubed‐sphere grid

A conservative, single‐cell‐based semi‐Lagrangian transport model is proposed in this paper. Using multi‐moment concept, an additional moment, i.e. volume‐integrated average (VIA), is treated as the model variable besides the point value (PV) updated in the traditional semi‐Lagrangian schemes. A quadratic interpolation function is constructed based on local degrees of freedom defined within each single cell. The PV moment is advanced by the semi‐Lagrangian formulation, whereas the VIA moment is updated by a finite volume formulation to rigorously ensure the numerical conservation. The numerical fluxes are computed from the PV moments defined along the boundary edges of the control volume. The scheme is extended to the spherical geometry through the application of the cubed‐sphere grid that eliminates the polar singularity in the conventional longitude/latitude coordinates by using the quasi‐uniform grid spacing covering the whole sphere. The single‐cell‐based scheme is well suited for the treatment of the connections between different patches. A simple quasi‐monotone limiter to the PV moment is applied to suppress non‐physical oscillations. The proposed scheme has been validated via representative benchmark tests and the performance is competitive to other existing transport schemes. Copyright © 2010 John Wiley & Sons, Ltd.     

A comparative study of high‐order variable‐property segregated algorithms for unsteady low Mach number flows

In this paper, five different algorithms are presented for the simulation of low Mach flows with large temperature variations, based on second‐order central‐difference or fourth‐order compact spatial discretization and a pressure projection‐type method. A semi‐implicit three‐step Runge–Kutta/Crank–Nicolson or second‐order iterative scheme is used for time integration. The different algorithms solve the coupled set of governing scalar equations in a decoupled segregate manner. In the first algorithm, a temperature equation is solved and density is calculated from the equation of state, while the second algorithm advances the density using the differential form of the equation of state. The third algorithm solves the continuity equation and the fourth algorithm solves both the continuity and enthalpy equation in conservative form. An iterative decoupled algorithm is also proposed, which allows the computation of the fully coupled solution. All five algorithms solve the momentum equation in conservative form and use a constant‐ or variable‐coefficient Poisson equation for the pressure. The efficiency of the fourth‐order compact scheme and the performances of the decoupling algorithms are demonstrated in three flow problems with large temperature variations: non‐Boussinesq natural convection, channel flow instability, flame–vortex interaction. Copyright © 2010 John Wiley & Sons, Ltd.     

Monotonicity in high‐order curvilinear finite element arbitrary Lagrangian–Eulerian remap

The remap phase in arbitrary Lagrangian–Eulerian (ALE) hydrodynamics involves the transfer of field quantities defined on a post‐Lagrangian mesh to some new mesh, usually generated by a mesh optimization algorithm. This problem is often posed in terms of transporting (or advecting) some state variable from the old mesh to the new mesh over a fictitious time interval. It is imperative that this remap process be monotonic, that is, not generate any new extrema in the field variables. It is well known that the only linear methods that are guaranteed to be monotonic for such problems are first‐order accurate; however, much work has been performed in developing non‐linear methods, which blend both high and low (first) order solutions to achieve monotonicity and preserve high‐order accuracy when the field is sufficiently smooth. In this paper, we present a set of methods for enforcing monotonicity targeting high‐order discontinuous Galerkin methods for advection equations in the context of high‐order curvilinear ALE hydrodynamics. Published 2014. This article is a U.S. Government work and is in the public domain in the USA.     

Combined swept region and intersection‐based single‐material remapping method

A typical arbitrary Lagrangian–Eulerian algorithm consists of a Lagrangian step, where the computational mesh moves with the fluid flow; a rezoning step, where the computational mesh is smoothed and repaired in case it gets too distorted; and a remapping step, where all fluid quantities are conservatively interpolated on this new mesh. In single‐material simulations, the remapping process can be represented in a flux form, with fluxes approximated by integrating a reconstructed function over intersections of neighboring computational cells on the original and rezoned computational mesh. This algorithm is complex and computationally demanding – Therefore, a simpler approach that utilizes regions swept by the cell edges during rezoning is often used in practice. However, it has been observed that such simplification can lead to distortion of the solution symmetry, especially when the mesh movement is not bound by such symmetry. For this reason, we propose an algorithm combining both approaches in a similar way that was proposed for multi‐material remapping (two‐step hybrid remap). Intersections and exact integration are employed only in certain parts of the computational mesh, marked by a switching function – Various different criteria are presented in this paper. The swept‐based method is used elsewhere in areas that are not marked. This way, our algorithm can retain the beneficial symmetry‐preserving capabilities of intersection‐based remapping while keeping the overall computational cost moderate.     

Solution of the shallow‐water equations using an adaptive moving mesh method

This paper extends an adaptive moving mesh method to multi‐dimensional shallow water equations (SWE) with source terms. The algorithm is composed of two independent parts: the SWEs evolution and the mesh redistribution. The first part is a high‐resolution kinetic flux‐vector splitting (KFVS) method combined with the surface gradient method for initial data reconstruction, and the second part is based on an iteration procedure. In each iteration, meshes are first redistributed by a variational principle and then the underlying numerical solutions are updated by a conservative‐interpolation formula on the resulting new mesh. Several test problems in one‐ and two‐dimensions with a general geometry are computed using the proposed moving mesh algorithm. The computations demonstrate that the algorithm is efficient for solving problems with bore waves and their interactions. The solutions with higher resolution can be obtained by using a KFVS scheme for the SWEs with a much smaller number of grid points than the uniform mesh approach, although we do not treat technically the bed slope source terms in order to balance the source terms and flux gradients. Copyright © 2004 John Wiley & Sons, Ltd.     

A two‐step Taylor‐characteristic‐based Galerkin method for incompressible flows and its application to flow over triangular cylinder with different incidence angles

An alternative characteristic‐based scheme, the two‐step Taylor‐characteristic‐based Galerkin method is developed based on the introduction of multi‐step temporal Taylor series expansion up to second order along the characteristic of the momentum equation. Contrary to the classical characteristic‐based split (CBS) method, the current characteristic‐based method does not require splitting the momentum equation, and segregate the calculation of the pressure from that of the velocity by using the momentum–pressure Poisson equation method. Some benchmark problems are used to examine the effectiveness of the proposed algorithm and to compare with the original CBS method, and the results show that the proposed method has preferable accuracy with less numerical dissipation. We further applied the method to the numerical simulation of flow around equilateral triangular cylinder with different incidence angles in free stream. In this numerical investigation, the flow simulations are carried out in the low Reynolds number range. Instantaneous streamlines around the cylinder are used as a means to visualize the wake region behind, and they clearly show the flow pattern around the cylinder in time. The influence of incidence angle on flow characteristic parameters such as Strouhal number, Drag and Lift coefficients are discussed quantitatively. Copyright © 2009 John Wiley & Sons, Ltd.     

Numerical study of wave propagation in compressible two‐phase flow

We propose a new model and a solution method for two‐phase two‐fluid compressible flows. The model involves six equations obtained from conservation principles applied to a one‐dimensional flow of gas and liquid mixture completed by additional closure governing equations. The model is valid for pure fluids as well as for fluid mixtures. The system of partial differential equations with source terms is hyperbolic and has conservative form. Hyperbolicity is obtained using the principles of extended thermodynamics. Features of the model include the existence of real eigenvalues and a complete set of independent eigenvectors. Its numerical solution poses several difficulties. The model possesses a large number of acoustic and convective waves and it is not easy to upwind all of these accurately and simply. In this paper we use relatively modern shock‐capturing methods of a centred‐type such as the total variation diminishing (TVD) slope limiter centre (SLIC) scheme which solve these problems in a simple way and with good accuracy. Several numerical test problems are displayed in order to highlight the efficiency of the study we propose. The scheme provides reliable results, is able to compute strong shock waves and deals with complex equations of state. Copyright © 2006 John Wiley & Sons, Ltd.     

One‐dimensional simulation of supercritical flow at a confluence by means of a nonlinear junction model applied with the RKDG2 method

We investigate the one‐dimensional computation of supercritical open‐channel flows at a combining junction. In such situations, the network system is composed of channel segments arranged in a branching configuration, with individual channel segments connected at a junction. Therefore, two important issues have to be addressed: (a) the numerical solution in branches, and (b) the internal boundary conditions treatment at the junction. Going from the advantageous literature supports of RKDG methods to a particular investigation for a supercritical benchmark, the second‐order Runge–Kutta discontinuous Galerkin (RKDG2) scheme is selected to compute the water flow in branches. For the internal boundary handling, we propose a new approach by incorporating the nonlinear model derived from the conservation of the momentum through the junction. The nonlinear junction model was evaluated against available experiments and then applied to compute the junction internal boundary treatment for steady and unsteady flow applications. Finally, a combining flow problem is defined and simulated by the proposed framework and results are illustrated for many choices of junction angles. Copyright © 2007 John Wiley & Sons, Ltd.