Particle trajectory calculations with a two‐step three‐time level semi‐Lagrangian scheme well suited for curved flows

This study proposes a new two‐step three‐time level semi‐Lagrangian scheme for calculation of particle trajectories. The scheme is intended to yield accurate determination of the particle departure position, particularly in the presence of significant flow curvature. Experiments were performed both for linear and non‐linear idealized advection problems, with different flow curvatures. Results for simulations with the proposed scheme, and with three other semi‐Lagrangian schemes, and with an Eulerian method are presented. In the linear advection problem the two‐step three‐time level scheme produced smaller root mean square errors and more accurate replication of the angular displacement of a Gaussian hill than the other schemes. In the non‐linear advection experiments the proposed scheme produced, in general, equal or better conservation of domain‐averaged quantities than the other semi‐Lagrangian schemes, especially at large Courant numbers. In idealized frontogenesis simulations the scheme performed equally or better than the other schemes in the representation of sharp gradients in a scalar field. The two‐step three‐time level scheme has some computational overhead as compared with the other three semi‐Lagrangian schemes. Nevertheless, the additional computational effort was shown to be worthwhile, due to the accuracy obtained by the scheme in the experiments with large time steps. The most remarkable feature of the scheme is its robustness, since it performs well both for small and large Courant numbers, in the presence of weak as well strong flow curvatures. Copyright © 2009 John Wiley & Sons, Ltd.     

Semi‐Lagrangian semi‐implicit time‐splitting scheme for the shallow water equations

Time‐splitting technique applied in the context of the semi‐Lagrangian semi‐implicit method allows the use of extended time steps mainly based on physical considerations and reduces the number of numerical operations at each time step such that it is approximately proportional to the number of the points of spatial grid. To control time growth of the additional truncation errors, the standard stabilizing correction method is modified with no penalty for accuracy and efficiency of the algorithm. A linear analysis shows that constructed scheme is stable for time steps up to 2h. Numerical integrations with actual atmospheric fields of pressure and wind confirm computational efficiency, extended stability and accuracy of the proposed scheme. Copyright © 2006 John Wiley & Sons, Ltd.     

Implementation of high‐order compact finite‐difference method to parabolized Navier–Stokes schemes

The numerical solution to the parabolized Navier–Stokes (PNS) and globally iterated PNS (IPNS) equations for accurate computation of hypersonic axisymmetric flowfields is obtained by using the fourth‐order compact finite‐difference method. The PNS and IPNS equations in the general curvilinear coordinates are solved by using the implicit finite‐difference algorithm of Beam and Warming type with a high‐order compact accuracy. A shock‐fitting procedure is utilized in both compact PNS and IPNS schemes to obtain accurate solutions in the vicinity of the shock. The main advantage of the present formulation is that the basic flow variables and their first and second derivatives are simultaneously computed with the fourth‐order accuracy. The computations are carried out for a benchmark case: hypersonic axisymmetric flow over a blunt cone at Mach 8. A sensitivity study is performed for the basic flowfield, including profiles and their derivatives obtained from the fourth‐order compact PNS and IPNS solutions, and the effects of grid size and numerical dissipation term used are discussed. The present results for the flowfield variables and also their derivatives are compared with those of other basic flow models to demonstrate the accuracy and efficiency of the proposed method. The present work represents the first known application of a high‐order compact finite‐difference method to the PNS schemes, which are computationally more efficient than Navier–Stokes solutions. Copyright © 2008 John Wiley & Sons, Ltd.     

Depth‐averaged non‐hydrostatic extension for shallow water equations with quadratic vertical pressure profile: equivalence to Boussinesq‐type equations

We reformulate the depth‐averaged non‐hydrostatic extension for shallow water equations to show equivalence with well‐known Boussinesq‐type equations. For this purpose, we introduce two scalars representing the vertical profile of the non‐hydrostatic pressure. A specific quadratic vertical profile yields equivalence to the Serre equations, for which only one scalar in the traditional equation system needs to be modified. Equivalence can also be demonstrated with other Boussinesq‐type equations from the literature when considering variable depth, but then the non‐hydrostatic extension involves mixed space–time derivatives. In case of constant bathymetries, the non‐hydrostatic extension is another way to circumvent mixed space–time derivatives arising in Boussinesq‐type equations. On the other hand, we show that there is no equivalence when using the traditionally assumed linear vertical pressure profile. Linear dispersion and asymptotic analysis as well as numerical test cases show the advantages of the quadratic compared with the linear vertical non‐hydrostatic pressure profile in the depth‐averaged non‐hydrostatic extension for shallow water equations. Copyright © 2017 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.     

Dispersion analysis of the least‐squares finite‐element shallow‐water system

The frequency or dispersion relation for the least‐squares mixed formulation of the shallow‐water equations is analysed. We consider the use of different approximation spaces corresponding to co‐located and staggered meshes, respectively. The study includes the effect of Coriolis, and the dispersion properties are compared analytically and graphically with those of the mixed Galerkin formulation. Numerical solutions of a test problem to simulate slow Rossby modes illustrate the theoretical results. Copyright © 2003 John Wiley & Sons, Ltd.     

Time‐linearized time‐harmonic 3‐D Navier–Stokes shock‐capturing schemes

In the present paper, a numerical method for the computation of time‐harmonic flows, using the time‐linearized compressible Reynolds‐averaged Navier–Stokes equations is developed and validated. The method is based on the linearization of the discretized nonlinear equations. The convective fluxes are discretized using an O(Δx) MUSCL scheme with van Leer flux‐vector‐splitting. Unsteady perturbations of the turbulent stresses are linearized using a frozen‐turbulence‐Reynolds‐number hypothesis, to approximate eddy‐viscosity perturbations. The resulting linear system is solved using a pseudo‐time‐marching implicit ADI‐AF (alternating‐directions‐implicit approximate‐factorization) procedure with local pseudo‐time‐steps, corresponding to a matrix‐successive‐underrelaxation procedure. The stability issues associated with the pseudo‐time‐marching solution of the time‐linearized Navier–Stokes equations are discussed. Comparison of computations with measurements and with time‐nonlinear computations for 3‐D shock‐wave oscillation in a square duct, for various back‐pressure fluctuation frequencies (180, 80, 20 and 10 Hz), assesses the shock‐capturing capability of the time‐linearized scheme. Copyright © 2007 John Wiley & Sons, Ltd.     

Nonisothermal two‐ and three‐dimensional flow simulations of inelastic and viscoelastic fluids by a finite‐volume method

This paper applies the finite‐volume method to computations of steady flows of viscous and viscoelastic incompressible fluids in complex two and three‐dimensional geometries. The materials adopted in the study obey different constitutive laws: Newtonian, purely viscous Carreau–Yasuda as also Upper‐Convected Maxwell and Phan‐Thien/Tanner differential models, with a Williams–Landel–Ferry (WLF) equation for temperature dependence. Specific analyses are made depending on the rheological model. A staggered grid is used for discretizing the equations and unknowns. Stockage possibilities allow us to solve problems involving a great number of degrees of freedom, up to 1 500 000 unknowns with a desk computer. In relation to the fluid properties, our numerical simulations provide flow characteristics for various 2D and 3D configurations and demonstrate the possibilities of the code to solve problems involving complex nonlinear constitutive equations with thermal effects. Copyright © 2009 John Wiley & Sons, Ltd.     

A two‐dimensional numerical scheme of dry/wet fronts for the Saint‐Venant system of shallow water equations

We propose a new two‐dimensional numerical scheme to solve the Saint‐Venant system of shallow water equations in the presence of partially flooded cells. Our method is well balanced, positivity preserving, and handles dry states. The latter is ensured by using the draining time step technique in the time integration process, which guarantees non‐negative water depths. Unlike previous schemes, our technique does not generate high velocities at the dry/wet boundaries, which are responsible for small time step sizes and slow simulation runs. We prove that the new scheme preserves ‘lake at rest’ steady states and guarantees the positivity of the computed fluid depth in the partially flooded cells. We test the new scheme, along with another recent scheme from the literature, against the analytical solution for a parabolic basin and show the improved simulation performance of the new scheme for two real‐world scenarios. Copyright © 2014 John Wiley & Sons, Ltd.     

High‐order 𝒞1 finite‐element interpolating schemes—Part II: Nonlinear semi‐Lagrangian shallow‐water models

The finite‐element, semi‐implicit, and semi‐Lagrangian methods are used on unstructured meshes to solve the nonlinear shallow‐water system. Several ??¹ approximation schemes are developed for an accurate treatment of the advection terms. The employed finite‐element discretization schemes are the P–P₁ and P₂–P₁ pairs. Triangular finite elements are attractive because of their flexibility for representing irregular boundaries and for local mesh refinement. By tracking the characteristics backward from both the interpolation and quadrature nodes and using ??¹ interpolating schemes, an accurate treatment of the nonlinear terms and, hence, of Rossby waves is obtained. Results of test problems to simulate slowly propagating Rossby modes illustrate the promise of the proposed approach in ocean modelling. Copyright © 2007 John Wiley & Sons, Ltd.     

Efficient computation of two‐dimensional steady free‐surface flows

We consider a family of steady free‐surface flow problems in two dimensions, concentrating on the effect of nonlinearity on the train of gravity waves that appear downstream of a disturbance. By exploiting standard complex variable techniques, these problems are formulated in terms of a coupled system of Bernoulli equation and an integral equation. When applying a numerical collocation scheme, the Jacobian for the system is dense, as the integral equation forces each of the algebraic equations to depend on each of the unknowns. We present here a strategy for overcoming this challenge, which leads to a numerical scheme that is much more efficient than what is normally used for these types of problems, allowing for many more grid points over the free surface. In particular, we provide a simple recipe for constructing a sparse approximation to the Jacobian that is used as a preconditioner in a Jacobian‐free Newton‐Krylov method for solving the nonlinear system. We use this approach to compute numerical results for a variety of prototype problems including flows past pressure distributions, a surface‐piercing object and bottom topographies.     

High‐order ADER‐WENO ALE schemes on unstructured triangular meshes—application of several node solvers to hydrodynamics and magnetohydrodynamics

In this paper, we present a class of high‐order accurate cell‐centered arbitrary Lagrangian–Eulerian (ALE) one‐step ADER weighted essentially non‐oscillatory (WENO) finite volume schemes for the solution of nonlinear hyperbolic conservation laws on two‐dimensional unstructured triangular meshes. High order of accuracy in space is achieved by a WENO reconstruction algorithm, while a local space–time Galerkin predictor allows the schemes to be high order accurate also in time by using an element‐local weak formulation of the governing PDE on moving meshes. The mesh motion can be computed by choosing among three different node solvers, which are for the first time compared with each other in this article: the node velocity may be obtained either (i) as an arithmetic average among the states surrounding the node, as suggested by Cheng and Shu, or (ii) as a solution of multiple one‐dimensional half‐Riemann problems around a vertex, as suggested by Maire, or (iii) by solving approximately a multidimensional Riemann problem around each vertex of the mesh using the genuinely multidimensional Harten–Lax–van Leer Riemann solver recently proposed by Balsara et al. Once the vertex velocity and thus the new node location have been determined by the node solver, the local mesh motion is then constructed by straight edges connecting the vertex positions at the old time level tⁿ with the new ones at the next time level t^{n + 1}. If necessary, a rezoning step can be introduced here to overcome mesh tangling or highly deformed elements. The final ALE finite volume scheme is based directly on a space–time conservation formulation of the governing PDE system, which therefore makes an additional remapping stage unnecessary, as the ALE fluxes already properly take into account the rezoned geometry. In this sense, our scheme falls into the category of direct ALE methods. Furthermore, the geometric conservation law is satisfied by the scheme by construction. We apply the high‐order algorithm presented in this paper to the Euler equations of compressible gas dynamics as well as to the ideal classical and relativistic magnetohydrodynamic equations. We show numerical convergence results up to fifth order of accuracy in space and time together with some classical numerical test problems for each hyperbolic system under consideration. Copyright © 2014 John Wiley & Sons, Ltd.     

A modified Galerkin/finite element method for the numerical solution of the Serre‐Green‐Naghdi system

A new modified Galerkin/finite element method is proposed for the numerical solution of the fully nonlinear shallow water wave equations. The new numerical method allows the use of low‐order Lagrange finite element spaces, despite the fact that the system contains third order spatial partial derivatives for the depth averaged velocity of the fluid. After studying the efficacy and the conservation properties of the new numerical method, we proceed with the validation of the new numerical model and boundary conditions by comparing the numerical solutions with laboratory experiments and with available theoretical asymptotic results. Copyright © 2016 John Wiley & Sons, Ltd.     

SIMPLE‐type preconditioners for cell‐centered,colocated finite volume discretization of incompressible Reynolds‐averaged Navier–Stokes equations

This paper contains a comparison of four SIMPLE‐type methods used as solver and as preconditioner for the iterative solution of the (Reynolds‐averaged) Navier–Stokes equations, discretized with a finite volume method for cell‐centered, colocated variables on unstructured grids. A matrix‐free implementation is presented, and special attention is given to the treatment of the stabilization matrix to maintain a compact stencil suitable for unstructured grids. We find SIMPLER preconditioning to be robust and efficient for academic test cases and industrial test cases. Compared with the classical SIMPLE solver, SIMPLER preconditioning reduces the number of nonlinear iterations by a factor 5–20 and the CPU time by a factor 2–5 depending on the case. The flow around a ship hull at Reynolds number 2E9, for example, on a grid with cell aspect ratio up to 1:1E6, can be computed in 3 instead of 15 h.Copyright © 2012 John Wiley & Sons, Ltd.     

Adaptive Q‐tree Godunov‐type scheme for shallow water equations

This paper presents details of a second‐order accurate, Godunov‐type numerical model of the two‐dimensional shallow water equations (SWEs) written in matrix form and discretized using finite volumes. Roe's flux function is used for the convection terms and a non‐linear limiter is applied to prevent unwanted spurious oscillations. A new mathematical formulation is presented, which inherently balances flux gradient and source terms. It is, therefore, suitable for cases where the bathymetry is non‐uniform, unlike other formulations given in the literature based on Roe's approximate Riemann solver. The model is based on hierarchical quadtree (Q‐tree) grids, which adapt to inherent flow parameters, such as magnitude of the free surface gradient and depth‐averaged vorticity. Validation tests include wind‐induced circulation in a dish‐shaped basin, two‐dimensional frictionless rectangular and circular dam‐breaks, an oblique hydraulic jump, and jet‐forced flow in a circular reservoir. Copyright © 2001 John Wiley & Sons, Ltd.     

Multigrid convergence of inviscid fixed‐ and rotary‐wing flows

The affect of multigrid acceleration implemented within an upwind‐biased Euler method is presented, and applied to fixed‐wing and rotary‐wing flows. The convergence of fixed‐ and rotary‐wing computations is shown to be vastly different, and multigrid is shown to be less effective for rotary‐wing flows. The flow about a hovering rotor suffers from very slow convergence of the inner blade region, where the flow is effectively incompressible. Furthermore, the vortical wake must develop over several turns before convergence is achieved, whereas for fixed‐wing computations the far‐field grid and solution have little significance. Results are presented for single mesh and two, three, four, and five level multigrid, and using five levels a reduction in required CPU time of over 80 per cent is demonstrated for rotary‐wing computations, but 94 per cent for fixed‐wing computations. It is found that a simple V‐cycle is the most effective, smoothing in the decreasing mesh density direction only, with a relaxed trilinear prolongation operator. Copyright © 2002 John Wiley & Sons, Ltd.     

One‐shot pseudo‐time method for aerodynamic shape optimization using the Navier–Stokes equations

This paper presents a numerical method for aerodynamic shape optimization problems in compressible viscous flow. It is based on simultaneous pseudo‐time stepping in which stationary states are obtained by solving the pseudo‐stationary system of equations representing the state, costate and design equations. The main advantages of this method are that it blends in nicely with previously existing pseudo‐time‐stepping methods for the state and the costate equations, that it requires no additional globalization in the design space, and that a preconditioner can be used for convergence acceleration which stems from the reduced SQP methods. For design examples of 2D problems, the overall cost of computation can be reduced to less than 2 times the forward simulation runs. Copyright © 2011 John Wiley & Sons, Ltd.     

A dual‐weighted trust‐region adaptive POD 4D‐VAR applied to a finite‐element shallow‐water equations model

We consider a limited‐area finite‐element discretization of the shallow‐water equations model. Our purpose in this paper is to solve an inverse problem for the above model controlling its initial conditions in presence of observations being assimilated in a time interval (window of assimilation). We then attempt to obtain a reduced‐order model of the above inverse problem, based on proper orthogonal decomposition (POD), referred to as POD 4‐D VAR. Different approaches of POD implementation of the reduced inverse problem are compared, including a dual‐weighed method for snapshot selection coupled with a trust‐region POD approach. Numerical results obtained point to an improved accuracy in all metrics tested when dual‐weighing choice of snapshots is combined with POD adaptivity of the trust‐region type. Results of ad‐hoc adaptivity of the POD 4‐D VAR turn out to yield less accurate results than trust‐region POD when compared with high‐fidelity model. Directions of future research are finally outlined. Copyright © 2009 John Wiley & Sons, Ltd.