Practical evaluation of five partly discontinuous finite element pairs for the non‐conservative shallow water equations

This paper provides a comparison of five finite element pairs for the shallow water equations. We consider continuous, discontinuous and partially discontinuous finite element formulations that are supposed to provide second‐order spatial accuracy. All of them rely on the same weak formulation, using Riemann solver to evaluate interface integrals. We define several asymptotic limit cases of the shallow water equations within their space of parameters. The idea is to develop a comparison of these numerical schemes in several relevant regimes of the subcritical shallow water flow. Finally, a new pair, using non‐conforming linear elements for both velocities and elevation (P?P), is presented, giving optimal rates of convergence in all test cases. P?P₁ and P?P₁ mixed formulations lack convergence for inviscid flows. P?P₂ pair is more expensive but provides accurate results for all benchmarks. P?P provides an efficient option, except for inviscid Coriolis‐dominated flows, where a small lack of convergence is observed. Copyright © 2009 John Wiley & Sons, Ltd.     

Multivariant finite elements for three-dimensional simulation of viscous incompressible flows

Within multivariant elements, which have restricted degrees of freedom at some nodes, different velocity components have different variations. Shape functions for the multivariant elements Q P_o and R P_o are developed. With such shape functions the value of a velocity component within a multivariant element is shown to depend upon all the independent components of velocity at the nodes of the element. The use of the Q₁ P₀ element to simulate flows with discontinuous boundary conditions generated disturbance throughout the flow domain, giving erroneous pressure and velocity distributions. The Q P_o element restricted the disturbance due to such discontinuities to a small region near the singular points, whereas the P P_o element completely eliminated the fluctuations. Flows with discontinuous boundary conditions were simulated with reasonable accuracy by partially relaxing the no-slip condition on the Q₁ P_o elements near the singular points.     

The behaviour of several stress–velocity–pressure mixed finite elements for Newtonian flows

The Q₂/P₁, P/P₁, P₂/P₀ and Q₁/P₀ velocity–pressure mixed elements are extended to the stress–velocity–pressure formulation, using the same interpolants for stress and velocity, and tested in the 4-to-1 contraction problem for Stokes flow. The comparison shows significant differences among them, which are not present when the velocity–pressure formulation is used. To provide a better understanding of the phenomenon, several variants of the previous elements are introduced, obtained by either changing the pressure space or by enriching the stress space with bubble functions. The formulation exhibits a strong sensitivity to the first alternative, while the second produces only a minor effect. These observations are confirmed by a convergence test effected on a regular problem with the explicit analytical solution. Also, as a result of the whole comparison, the P/P/P₁ element looks promising for three-field calculations.     

Transition bifurcation branches in non-linear water waves

We are concerned with the numerical computation of progressive free surface gravity waves on a horizontal bed. They are regarded as families of bifurcation branches (λ,A)_Q of constant discharge Q. Numerically we determine two transition values Q₁ and Q₂ with corresponding transition bifurcation branches that classify waves into three disjoint branch sets B₁, B₂ and B₃. Their members are families of waves (λ,A)_Q satisfying the conditions 0<Q² ?Q, Q <Q² ?Q and Q <Q² <B/27, respectively. The bifurcation patterns are analysed in some detail from the computed bifurcation diagram, which shows that in B₁ bifurcation is to the left and the amplitude A increases as the wavelength λ decreases; in B₂ bifurcation is to the right and turning points are observed nearly at breaking point. In B₃ bifurcation is to the right and A increases monotonically with λ.     

Comparative study of Euler/Euler and Euler/Lagrange approaches simulating evaporation in a turbulent gas–liquid flow

This study deals with the Reynolds‐averaged Navier–Stokes simulation of evaporation in a turbulent gas–liquid flow in a three‐dimensional duct, focussing on the results obtained by a four‐equation turbulence model within the framework of the Euler/Euler approach for multiphase flow calculations: in addition to the two‐equation k?ε model describing the turbulence of the continuous (C) phase, the computational model employs transport equations for the turbulence kinetic energy of the disperse (D) phase and for the velocity covariance q=〈{u}^D{u}^C〉^D. In the present study, the evaporation model according to Abramzon and Sirignano (Int. J. Heat Mass Transfer 1989; 32 :1605–1618) has been extended by introducing an additional transport equation for a newly defined quantity ā, defined as the phase‐interface surface fraction. This allows the change in the drop diameter to be quantified in terms of a probability density function. The source term in the equation describing the dynamics of the volumetric fraction of the dispersed phase α^D is related to the evaporation time scale τ_Γ. The performance of the new model is evaluated by performing a comparative analysis of the results obtained by simulating a polydispersed spray in a three‐dimensional duct configuration with the results of the Euler/Lagrange calculations performed in parallel. Prior to these calculations, some selected (solid) particle‐laden flow configurations were computationally examined with respect to the validation of the background, four‐equation, eddy‐viscosity‐based turbulence model. Copyright © 2008 John Wiley & Sons, Ltd.     

Impact of mass lumping on gravity and Rossby waves in 2D finite‐element shallow‐water models

The goal of this study is to evaluate the effect of mass lumping on the dispersion properties of four finite‐element velocity/surface‐elevation pairs that are used to approximate the linear shallow‐water equations. For each pair, the dispersion relation, obtained using the mass lumping technique, is computed and analysed for both gravity and Rossby waves. The dispersion relations are compared with those obtained for the consistent schemes (without lumping) and the continuous case. The P₀?P₁, RT₀ and P?P₁ pairs are shown to preserve good dispersive properties when the mass matrix is lumped. Test problems to simulate fast gravity and slow Rossby waves are in good agreement with the analytical results. Copyright © 2008 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.     

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.     

Symbolic computation of flow in a rotating pipe

A perturbation solution of the fully developed flow through a pipe of circular cross-section, which rotates uniformly around an axis oriented perpendicularly to its own, is considered. The perturbation parameter is given by R = 2Ωa²/ν in terms of the angular velocity Ω, the pipe radius a and the kinematic viscosity ν of the fluid. The two coupled non-linear equations for the axial velocity ω and the streamfunction ? of the transverse (secondary) flow lead to an infinite system of linear equations. This system allows first the computation of a given order ?_n, n ? 1, of the perturbation expansion ? = ∑ Rⁿ?_n in terms of ω_n-1, the (n-1)-th order of the expansion ω = ∑ Rⁿω_n, and of the lower orders ?₁,…,?_{n ? 1}. Then it permits the computation of ω_n from ω₀,…,ω_{n ? 1} and ?₁,…,?;_n. The computation starts from the Hagen–Poiseuille flow ω₀, i.e. the perturbation is around this flow. The computations are performed analytically by computer, with the REDUCE and MAPLE systems. The essential elements for this are the appropriate co-ordinates: in the complex co-ordinates chosen the two-dimensional harmonic (Laplace, Δ) and biharmonic (Δ²) operators are ideally suited for (symbolic) quadratures. Symmetry considerations as well as analysis of the equations for ω_n, ?_n and of the boundary conditions lead to general (polynomial) formulae for these functions, with coeffcients to be determined. Their determination, order by order, implies, in complex co-ordinates, only (symbolic) differentiation and quadratures. The coefficients themselves are polynomials in the Reynolds number c of the (unperturbed) Hagen–Poiseuille flow. They are tabulated in the paper for the orders n ? 6 of the perturbation expansion.     

Reynolds‐stress modelling of M = 2.25 shock‐wave/turbulent boundary‐layer interaction

M = 2.25 shock‐wave/turbulent‐boundary‐layer interactions over a compression ramp for several angles (8, 13 and 18°) at Reynolds‐number Re=7 × 10³ were simulated with three low‐Reynolds second‐moment closures and a linear low‐Reynolds standard k–ε model. A detailed assessment of the turbulence closures by comparison with both mean‐flow and turbulent experimental quantities is presented. The Reynolds‐stress model which is wall‐topology free and which uses an optimized redistribution closure, is in good agreement with experimental data both for wall‐pressure and mean‐velocity profiles. Detailed analysis of three components of the Reynolds‐stress tensor (comparison with measurements and transport‐equation budgets) provides a critical evaluation of full Reynolds‐stress models for the separated supersonic compression ramp. The discrepancy observed in the shock‐wave foot region, between computations and measurements for the Reynolds‐stresses profiles, could be explained by considering the experimental shock‐wave oscillation and directions for future modelling work are indicated. Copyright © 2007 John Wiley & Sons, Ltd.     

Building generalized open boundary conditions for fluid dynamics problems

This paper deals with the design of an efficient open boundary condition (OBC) for fluid dynamics problems. Such problematics arise, for instance, when one solves a local model on a fine grid that is nested in a coarser one of greater extent. Usually, the local solution U^loc is computed from the coarse solution U^ext, thanks to an OBC formulated as , where B_h and B_H are discretizations of the same differential operator (B_h being defined on the fine grid and B_H on the coarse grid). In this paper, we show that such an OBC cannot lead to the exact solution, and we propose a generalized formulation , where g is a correction term. When B_h and B_H are discretizations of a transparent operator, g can be computed analytically, at least for simple equations. Otherwise, we propose to approximate g by a Richardson extrapolation procedure. Numerical test cases on a 1D Laplace equation and on a 1D shallow water system illustrate the improved efficiency of such a generalized OBC compared with usual ones. Copyright © 2012 John Wiley & Sons, Ltd.     

Hessian‐based model reduction: large‐scale inversion and prediction

Hessian‐based model reduction was previously proposed as an approach in deriving reduced models for the solution of large‐scale linear inverse problems by targeting accuracy in observation outputs. A control‐theoretic view of Hessian‐based model reduction that hinges on the equality between the Hessian and the transient observability gramian of the underlying linear system is presented. The model reduction strategy is applied to a large‐scale ( degrees of freedom) three‐dimensional contaminant transport problem in an urban environment, an application that requires real‐time computation. In addition to the inversion accuracy, the ability of reduced models of varying dimension to make predictions of the contaminant evolution beyond the time horizon of observations is studied. Results indicate that the reduced models have a factor speedup in computing time for the same level of accuracy. Copyright © 2012 John Wiley & Sons, Ltd.     

Axisymmetric boundary integral formulation for a two‐fluid system

A 3D axisymmetric Galerkin boundary integral formulation for potential flow is employed to model two fluids of different densities, one fluid enclosed inside the other. The interface variables are the velocity potential and the normal velocity, and they can be solved for separately, the second linear system being symmetric. The algorithm is validated by comparing with the analytic solutions for a static interior spherical drop over a range of values for the relative densities of exterior and interior fluids and various boundary conditions. For time‐dependent simulations utilizing a level set method for the interface tracking, the accuracy has been checked by comparing against the known oscillation frequency of the sphere. Pinch‐off profiles corresponding to an initial two‐lobe geometry drop and D = 6 are also presented. Published in 2011 by John Wiley & Sons, Ltd.     

Performance of very‐high‐order upwind schemes for DNS of compressible wall‐turbulence

The purpose of the present paper is to evaluate very‐high‐order upwind schemes for the direct numerical simulation (DNS ) of compressible wall‐turbulence. We study upwind‐biased (UW ) and weighted essentially nonoscillatory (WENO ) schemes of increasingly higher order‐of‐accuracy (J. Comp. Phys. 2000; 160 :405–452), extended up to WENO 17 (AIAA Paper 2009‐1612, 2009). Analysis of the advection–diffusion equation, both as Δx→0 (consistency), and for fixed finite cell‐Reynolds‐number Re_Δx (grid‐resolution), indicates that the very‐high‐order upwind schemes have satisfactory resolution in terms of points‐per‐wavelength (PPW ). Computational results for compressible channel flow (Re∈[180, 230]; M?_CL ∈[0.35, 1.5]) are examined to assess the influence of the spatial order of accuracy and the computational grid‐resolution on predicted turbulence statistics, by comparison with existing compressible and incompressible DNS databases. Despite the use of baseline O(Δt²) time‐integration and O(Δx²) discretization of the viscous terms, comparative studies of various orders‐of‐accuracy for the convective terms demonstrate that very‐high‐order upwind schemes can reproduce all the DNS details obtained by pseudospectral schemes, on computational grids of only slightly higher density. Copyright © 2009 John Wiley & Sons, Ltd.     

A stabilized finite element method for transient Navier–Stokes equations based on two local Gauss integrations

On the basis of two local Gauss integrations, a stabilized finite element method for transient Navier–Stokes equations is presented, which is defined by the lowest equal‐order conforming finite element subspace such as (or ) elements. The best algorithmic feature of our method is using two local Gauss integrations to replace projection operator. The diffusion term in these equations is discretized by using finite element method, and the temporal differentiation and advection terms are treated by characteristic schemes. Moreover, we present some numerical simulations to demonstrate the effectiveness, good stability, and accuracy properties of our method. Especially, the rate of convergence study tells us that the stability still keeps well when the Reynolds number is increasing. Copyright © 2011 John Wiley & Sons, Ltd.     

A consistent and fast weakly compressible smoothed particle hydrodynamics with a new wall boundary condition

A modified weakly compressible smoothed particle hydrodynamics (WCSPH) is presented, which utilizes consistent discretization schemes for spatial derivatives in the flow equations. Here, each SPH particle is considered as a computational point that represents a specific part of the fluid. To overcome non‐physical oscillations that usually arise in standard WCSPH, we modified the mass conservation equation by using a numerical filter. This modification is based on the difference between two discretization schemes used for the term . Furthermore, a new implementation of wall boundary condition in SPH is introduced. This condition is imposed on the pressure of wall boundary particles to ensure that the acceleration of each boundary particle in normal direction to the wall is zero. Thus, no penetration through walls will occur. To examine the performance of the modified method, we solved a series of two‐dimensional incompressible internal flow benchmark problems. By comparing the result with analytical solutions and the results of the standard WCSPH, we show that the use of consistent schemes in conjunction with the proposed numerical filter improves both accuracy and speed of the numerical method. Copyright © 2011 John Wiley & Sons, Ltd.     

Fujiwhara interaction of tropical cyclone scale vortices using a weighted residual collocation method

The fundamental interaction between tropical cyclones was investigated through a series of water tank experiements by Fujiwhara  6 - 8 . However, a complete understanding of tropical cyclones remains an open research challenge although there have been numerous investigations through measurments with aircrafts/satellites, as well as with numerical simulations. This article presents a computational model for simulating the interaction between cyclones. The proposed numerical method is presented briefly, where the time integration is performed by projecting the discrete system onto a Krylov subspace. The method filters the large scale fluid dynamics using a multiresolution approximation, and the unresolved dynamics are modeled with a Smagorinsky type subgrid scale parameterization scheme. Numerical experiments with Fujiwhara interactions are considered to verify modeling accuracy. An excellent agreement between the present simulation and a reference simulation at has been demonstrated. At , the kinetic energy of cyclones is seen consolidated into larger scales with concurrent enstrophy cascade – suggesting a steady increase of energy containing scales – a phenomena that is typical in two‐dimensional turbulence theory. The primary results of this article suggest a novel avenue for addressing some of the computational challenges of mesoscale atmospheric circulations. Copyright © 2015 John Wiley & Sons, Ltd.     

An efficient and stable finite element solver of higher order in space and time for nonstationary incompressible flow

In this paper, we present fully implicit continuous Galerkin–Petrov (cGP) and discontinuous Galerkin (dG) time‐stepping schemes for incompressible flow problems which are, in contrast to standard approaches like for instance the Crank–Nicolson scheme, of higher order in time. In particular, we analyze numerically the higher order dG(1) and cGP(2) methods, which are super convergent of third, resp., fourth order in time, whereas for the space discretization, the well‐known LBB‐stable finite element pair of third‐order accuracy is used. The discretized systems of nonlinear equations are treated by using the Newton method, and the associated linear subproblems are solved by means of a monolithic (geometrical) multigrid method with a blockwise Vanka‐like smoother treating all components simultaneously. We perform nonstationary simulations (in 2D) for two benchmarking configurations to analyze the temporal accuracy and efficiency of the presented time discretization schemes w.r.t. CPU and numerical costs. As a first test problem, we consider a classical ‘flow around cylinder’ benchmark. Here, we concentrate on the nonstationary behavior of the flow patterns with periodic oscillations and examine the ability of the different time discretization schemes to capture the dynamics of the flow. As a second test case, we consider the nonstationary ‘flow through a Venturi pipe’. The objective of this simulation is to control the instantaneous and mean flux through this device. Copyright © 2013 John Wiley & Sons, Ltd.     

Stability and consistency of nonhydrostatic free‐surface models using the semi‐implicit θ‐method

The θ‐method is a popular semi‐implicit finite‐difference method for simulating free‐surface flows. Problem stiffness, arising because of the presence of both fast and slow timescale processes, is easily handled by the θ‐method. In most ocean, coastal, and estuary modeling applications, stiffness is caused by fast surface gravity wave timescales imposed on slower timescales of baroclinic variability. The method is well known to be unconditionally stable for shallow water (hydrostatic) models when , where θ is the implicitness parameter. In this paper, we demonstrate that the method is also unconditionally stable for nonhydrostatic models, when for both pressure projection and pressure correction methods. However, the methods result in artificial damping of the barotropic mode. In addition to investigating stability, we also estimate the form of artificial damping induced by both the free surface and nonhydrostatic pressure solution methods. Finally, this analysis may be used to estimate the damping or growth associated with a particular wavenumber and the overall order of accuracy of the discretization. Copyright © 2012 John Wiley & Sons, Ltd.     

Fast Ewald summation for Stokesian particle suspensions

We present a numerical method for suspensions of spheroids of arbitrary aspect ratio, which sediment under gravity. The method is based on a periodized boundary integral formulation using the Stokes double layer potential. The resulting discrete system is solved iteratively using generalized minimal residual accelerated by the spectral Ewald method, which reduces the computational complexity to , where N is the number of points used to discretize the particle surfaces. We develop predictive error estimates, which can be used to optimize the choice of parameters in the Ewald summation. Numerical tests show that the method is well conditioned and provides good accuracy when validated against reference solutions. Copyright © 2014 John Wiley & Sons, Ltd.