A simple high‐resolution advection scheme

A simple, robust, mass‐conserving numerical scheme for solving the linear advection equation is described. The scheme can estimate peak solution values accurately even in regions where spatial gradients are high. Such situations present a severe challenge to classical numerical algorithms. Attention is restricted to the case of pure advection in one and two dimensions since this is where past numerical problems have arisen. The authors' scheme is of the Godunov type and is second‐order in space and time. The required cell interface fluxes are obtained by MUSCL interpolation and the exact solution of a degenerate Riemann problem. Second‐order accuracy in time is achieved via a Runge–Kutta predictor–corrector sequence. The scheme is explicit and expressed in finite volume form for ease of implementation on a boundary‐conforming grid. Benchmark test problems in one and two dimensions are used to illustrate the high‐spatial accuracy of the method and its applicability to non‐uniform grids. Copyright © 2007 John Wiley & Sons, Ltd.     

Hybrid finite‐volume finite‐difference scheme for the solution of Boussinesq equations

A hybrid scheme composed of finite‐volume and finite‐difference methods is introduced for the solution of the Boussinesq equations. While the finite‐volume method with a Riemann solver is applied to the conservative part of the equations, the higher‐order Boussinesq terms are discretized using the finite‐difference scheme. Fourth‐order accuracy in space for the finite‐volume solution is achieved using the MUSCL‐TVD scheme. Within this, four limiters have been tested, of which van‐Leer limiter is found to be the most suitable. The Adams–Basforth third‐order predictor and Adams–Moulton fourth‐order corrector methods are used to obtain fourth‐order accuracy in time. A recently introduced surface gradient technique is employed for the treatment of the bottom slope. A new model ‘HYWAVE’, based on this hybrid solution, has been applied to a number of wave propagation examples, most of which are taken from previous studies. Examples include sinusoidal waves and bi‐chromatic wave propagation in deep water, sinusoidal wave propagation in shallow water and sinusoidal wave propagation from deep to shallow water demonstrating the linear shoaling properties of the model. Finally, sinusoidal wave propagation over a bar is simulated. The results are in good agreement with the theoretical expectations and published experimental results. Copyright © 2005 John Wiley & Sons, Ltd.     

Simulation of viscoelastic fluids in a 2D abrupt contraction by spectral element method

This study presents the vortex structure and numerical instability increase occurring when the level of elasticity is enhanced in inertial flows in planar contraction configuration for finitely extensible nonlinear elastic model by Peterlin (FENE‐P) fluid 1 . The re‐entrant corner effect on corner vortices is also considered. The calculations are performed using extended matrix logarithm formulation described in a previous paper: A. Jafari et al. A new extended matrix logarithm formulation for the simulation of viscoelastic fluids by spectral elements. Computer & Fluids 2010; 39 (9):1425–1438. In that reference, the proposed algorithm has been tested for simple geometry such as Poiseuille flow. In this study, we are interested in the capability of this algorithm for more complex geometry. This formulation helps to reach higher values of the Weissenberg number when compared with the classical one. Copyright © 2015 John Wiley & Sons, Ltd.     

Two‐dimensional flow past circular cylinders using finite volume lattice Boltzmann formulation

In this article, an extension to the total variation diminishing finite volume formulation of the lattice Boltzmann equation method on unstructured meshes was presented. The quadratic least squares procedure is used for the estimation of first‐order and second‐order spatial gradients of the particle distribution functions. The distribution functions were extrapolated quadratically to the virtual upwind node. The time integration was performed using the fourth‐order Runge–Kutta procedure. A grid convergence study was performed in order to demonstrate the order of accuracy of the present scheme. The formulation was validated for the benchmark two‐dimensional, laminar, and unsteady flow past a single circular cylinder. These computations were then investigated for the low Mach number simulations. Further validation was performed for flow past two circular cylinders arranged in tandem and side‐by‐side. Results of these simulations were extensively compared with the previous numerical data. Copyright © 2011 John Wiley & Sons, Ltd.     

Assessment of the finite volume method applied to the v2 − f model

The objective of this paper is to assess the accuracy of low‐order finite volume (FV) methods applied to the v² ? f turbulence model of Durbin (Theoret. Comput. Fluid Dyn. 1991; 3 :1–13) in the near vicinity of solid walls. We are not (like many others) concerned with the stability of solvers ‐ the topic at hand is simply whether the mathematical properties of the v² ? f model can be captured by the given, widespread, numerical method. The v² ? f model is integrated all the way up to solid walls, where steep gradients in turbulence parameters are observed. The full resolution of wall gradients imposes quite high demands on the numerical schemes and it is not evident that common (second order) FV codes can fully cope with such demands. The v² ? f model is studied in a statistically one‐dimensional, fully developed channel flow where we compare FV schemes with a highly accurate spectral element reference implementation. For the FV method a higher‐order face interpolation scheme, using Lagrange interpolation polynomials up to arbitrary order, is described. It is concluded that a regular second‐order FV scheme cannot give an accurate representation of all model parameters, independent of mesh density. To match the spectral element solution an extended source treatment (we use three‐point Gauss–Lobatto quadrature), as well as a higher‐order discretization of diffusion is required. Furthermore, it is found that the location of the first internal node need to be well within y⁺=1. Copyright © 2009 John Wiley & Sons, Ltd.     

Consistent hybrid finite volume/element formulations: model and complex viscoelastic flows

The accuracy and consistency of a new cell‐vertex hybrid finite element/volume scheme are investigated for viscoelastic flows. Finite element (FE) discretization is employed for the momentum and continuity equation, with finite volume (FV) applied to the constitutive law for stress. Here, the interest is to explore the consequences of utilizing conventional cell‐vertex methodology for an Oldroyd‐B model and to demonstrate resulting drawbacks in the presence of complex source terms on structured and unstructured grids. Alternative strategies worthy of consideration are presented. It is demonstrated how high‐order accuracy may be achieved in steady state by respecting consistency in the formulation. Both FE and FV spatial discretizations are embedded in the scheme, with FV triangular sub‐cells referenced within parent triangular finite elements. Both model and complex flow problems are selected to quantify and assess accuracy, appealing to analysis and experimental validation. The test problem is that of steady sink flow, a pure extensional flow, which reflects some of the numerical difficulties involved in solving more generalized viscoelastic flows, where both source and flux terms may contribute equally to stress propagation. In addition, a complex transient filament‐stretching flow is chosen to compute the evolution of stress fields within liquid bridges. Shortcomings of the various stress upwinding schemes are discussed in this context, whilst dealing with such free‐surface type problems. Here, stress fluctuation distribution alone is advocated, and a Lax‐scheme is found to deliver accuracy and stability to the computational results, comparing well with the literature. Copyright © 2004 John Wiley & Sons, Ltd.     

An L2‐stable approximation of the Navier–Stokes convection operator for low‐order non‐conforming finite elements

We develop in this paper a discretization for the convection term in variable density unstationary Navier–Stokes equations, which applies to low‐order non‐conforming finite element approximations (the so‐called Crouzeix–Raviart or Rannacher–Turek elements). This discretization is built by a finite volume technique based on a dual mesh. It is shown to enjoy an L² stability property, which may be seen as a discrete counterpart of the kinetic energy conservation identity. In addition, numerical experiments confirm the robustness and the accuracy of this approximation; in particular, in L² norm, second‐order space convergence for the velocity and first‐order space convergence for the pressure are observed. Copyright © 2010 John Wiley & Sons, Ltd.     

Alternative subcell discretisations for viscoelastic flow: Velocity-gradient approximation

Under subcell discretisation for viscoelastic flow, we have given further consideration to the compatibility of function spaces for stress/velocity-gradient approximation [see F. Belblidia, H. Matallah, B. Puangkird, M.F. Webster, Alternative subcell discretisations for viscoelastic flow: stress interpolation, J. Non-Newtonian Fluid Mech. 146 (2007) 59–78]. This has been conducted through the three scheme discretisations (quad-fe(par), fe(sc) and fe/fv(sc)). In this companion study, we have extended the application of an original implementation for velocity-gradient approximation, being of localised superconvergent recovered form, continuous and quadratic on the parent fe-triangular element. This has led to the consideration of both localised (pointwise) and global (Galerkin weighted-residual) approximations for velocity-gradient, highlighting some of their advantages and disadvantages. The global form is equivalent to the discontinuous elastico–viscous stress splitting (DEVSS-type) technique of Fortin and co-workers. Each representation, local or global, is based on linear/quadratic order upon parent or subcell element stencils. We consider Oldroyd modelling and the contraction flow benchmark, covering abrupt and rounded-corner planar geometries. The localised superconvergent quadratic velocity-gradient treatment affords strong stability and accuracy properties for the three scheme discretisations considered. Through associated analysis and iterative solution processes, we have successfully linked global approximations to their localised counterparts, depicting the inadequacy of inaccurate but stable versions through their corresponding solution features. These issues pervade all formulations, coupled or pressure-correction, and in focusing on velocity-gradient approximation, also apply universally to all discrete representations of stress. The inaccuracy of the global treatment can be somewhat repaired through an increase in (mass) iteration number. The efficiency of localised schemes (and associated properties) is particularly attractive over their global alternatives, being less restrictive to choice of spatial-order (higher-order). Conversely, global implementations are more restrictive in satisfaction of the space inclusion principle. Localised schemes come into their own when chosen to represent strongly localised solution features, such as arise in non-smooth flows. Analysis has also proved helpful in clarifying that space inclusion (extended LBB-condition) is a non-necessary convergence condition in the viscoelastic context.Overall, the localised-quadratic velocity-gradient treatment for both linear (subcell) and quadratic (parent) stress interpolation has achieved both stability and accuracy. Under DEVSS-type approximations (global), once function spaces for stress and velocity-gradients have been selected, this choice dictates the state of system consistency. Additionally, stability gains are recognised through the further application of strain-rate-stabilisation procedures.     

Further application of hybrid solution to another form of Boussinesq equations and comparisons

Recently, a new hybrid scheme is introduced for the solution of the Boussinesq equations. In this study, the hybrid scheme is used to solve another form of the Boussinesq equations. The hybrid solution is composed of finite‐volume and finite difference method. The finite‐volume method is applied to conservative part of the governing equations, whereas the higher order Boussinesq terms are discretized using the finite‐difference scheme. Fourth‐order accuracy is provided in both time and space. The solution is then applied to several test cases, which are taken from the previous studies. The results of this study are compared with experimental and theoretical results as well as those of the previous ones. The comparisons indicate that the Boussinesq equations solved here and in the previous study produce quite similar results. Copyright © 2006 John Wiley & Sons, Ltd.     

A comparative study of two fast nonlinear free‐surface water wave models

This paper presents a comparison in terms of accuracy and efficiency between two fully nonlinear potential flow solvers for the solution of gravity wave propagation. One model is based on the high‐order spectral (HOS) method, whereas the second model is the high‐order finite difference model OceanWave3D. Although both models solve the nonlinear potential flow problem, they make use of two different approaches. The HOS model uses a modal expansion in the vertical direction to collapse the numerical solution to the two‐dimensional horizontal plane. On the other hand, the finite difference model simply directly solves the three‐dimensional problem. Both models have been well validated on standard test cases and shown to exhibit attractive convergence properties and an optimal scaling of the computational effort with increasing problem size. These two models are compared for solution of a typical problem: propagation of highly nonlinear periodic waves on a finite constant‐depth domain. The HOS model is found to be more efficient than OceanWave3D with a difference dependent on the level of accuracy needed as well as the wave steepness. Also, the higher the order of the finite difference schemes used in OceanWave3D, the closer the results come to the HOS model. Copyright © 2011 John Wiley & Sons, Ltd.     

A non‐diffusive,divergence‐free,finite volume‐based double projection method on non‐staggered grids

Second‐order accurate projection methods for simulating time‐dependent incompressible flows on cell‐centred grids substantially belong to the class either of exact or approximate projections. In the exact method, the continuity constraint can be satisfied to machine‐accuracy but the divergence and Laplacian operators show a four‐dimension nullspace therefore spurious oscillating solutions can be introduced. In the approximate method, the continuity constraint is relaxed, the continuity equation being satisfied up to the magnitude of the local truncation error, but the compact Laplacian operator has only the constant mode. An original formulation for allowing the discrete continuity equation to be satisfied to machine‐accuracy, while using a finite volume based projection method, is illustrated. The procedure exploits the Helmholtz–Hodge decomposition theorem for deriving an additional velocity field that enforces the discrete continuity without altering the vorticity field. This is accomplished by solving a second elliptic field for a scalar field obtained by prescribing that its additional discrete gradients ensure discrete continuity based on the previously adopted linear interpolation of the velocity. The resulting numerical scheme is applied to several flow problems and is proved to be accurate, stable and efficient. This paper has to be considered as the companion of: 'F. M. Denaro, A 3D second‐order accurate projection‐based finite volume code on non‐staggered, non‐uniform structured grids with continuity preserving properties: application to buoyancy‐driven flows. IJNMF 2006; 52 (4):393–432. Now, we illustrate the details and the rigorous theoretical framework. Copyright © 2006 John Wiley & Sons, Ltd.     

A combined analytical–numerical method for treating corner singularities in viscous flow predictions

A combined analytical–numerical method based on a matching asymptotic algorithm is proposed for treating angular (sharp corner or wedge) singularities in the numerical solution of the Navier–Stokes equations. We adopt an asymptotic solution for the local flow around the angular points based on the Stokes flow approximation and a numerical solution for the global flow outside the singular regions using a finite‐volume method. The coefficients involved in the analytical solution are iteratively updated by matching both solutions in a small region where the Stokes flow approximation holds. Moreover, an error analysis is derived for this method, which serves as a guideline for the practical implementation. The present method is applied to treat the leading‐edge singularity of a semi‐infinite plate. The effect of various influencing factors related to the implementation are evaluated with the help of numerical experiments. The investigation showed that the accuracy of the numerical solution for the flow around the leading edge can be significantly improved with the present method. The results of the numerical experiments support the error analysis and show the desired properties of the new algorithm, i.e. accuracy, robustness and efficiency. Based on the numerical results for the leading‐edge singularity, the validity of various classical approximate models for the flow, such as the Stokes approximation, the inviscid flow model and the boundary layer theory of varying orders are examined. Although the methodology proposed was evaluated for the leading‐edge problem, it is generally applicable to all kinds of angular singularities and all kinds of finite‐discretization methods. Copyright © 2004 John Wiley & Sons, Ltd.     

A new weighted upwind finite volume element method based on non‐standard covolume for time‐dependent convection–diffusion problems

In modern numerical simulation of problems in energy resources and environmental science, it is important to develop efficient numerical methods for time‐dependent convection–diffusion problems. On the basis of nonstandard covolume grids, we propose a new kind of high‐order upwind finite volume element method for the problems. We first prove the stability and mass conservation in the discrete forms of the scheme. Optimal second‐order error estimate in L₂‐norm in spatial step is then proved strictly. The scheme is effective for avoiding numerical diffusion and nonphysical oscillations and has second‐order accuracy. Numerical experiments are given to verify the performance of the scheme. Copyright © 2013 John Wiley & Sons, Ltd.     

A monotone finite volume scheme for advection–diffusion equations on distorted meshes

A new monotone finite volume method with second‐order accuracy is presented for the steady‐state advection–diffusion equation. The method uses a nonlinear approximation for both diffusive and advective fluxes that guarantee the positivity of the numerical solution. The approximation of the diffusive flux is based on nonlinear two‐point approximation, and the approximation of the advective flux is based on the second‐order upwind method with proper slope limiter. The second‐order convergence rate for concentration and the monotonicity of the nonlinear finite volume method are verified with numerical experiments. Copyright © 2011 John Wiley & Sons, Ltd.     

MUSTA schemes for multi‐dimensional hyperbolic systems: analysis and improvements

We develop and analyse an improved version of the multi‐stage (MUSTA) approach to the construction of upwind Godunov‐type fluxes whereby the solution of the Riemann problem, approximate or exact, is not required. The new MUSTA schemes improve upon the original schemes in terms of monotonicity properties, accuracy and stability in multiple space dimensions. We incorporate the MUSTA technology into the framework of finite‐volume weighted essentially nonoscillatory schemes as applied to the Euler equations of compressible gas dynamics. The results demonstrate that our new schemes are good alternatives to current centred methods and to conventional upwind methods as applied to complicated hyperbolic systems for which the solution of the Riemann problem is costly or unknown. Copyright © 2005 John Wiley & Sons, Ltd.     

Local maximum principle satisfying high‐order non‐oscillatory schemes

The main contribution of this work is to classify the solution region including data extrema for which high‐order non‐oscillatory approximation can be achieved. It is performed in the framework of local maximum principle (LMP) and non‐conservative formulation. The representative uniformly second‐order accurate schemes are converted in to their non‐conservative form using the ratio of consecutive gradients. Using the local maximum principle, these non‐conservative schemes are analyzed for their non‐linear LMP/total variation diminishing stability bounds which classify the solution region where high‐order accuracy can be achieved. Based on the bounds, second‐order accurate hybrid numerical schemes are constructed using a shock detector. The presented numerical results show that these hybrid schemes preserve high accuracy at non‐sonic extrema without exhibiting any induced local oscillations or clipping error. Copyright © 2015 John Wiley & Sons, Ltd.     

Accuracy and stability of a set of free‐surface time‐domain boundary element models based on B‐splines

An analysis is given for the accuracy and stability of some perturbation‐based time‐domain boundary element models (BEMs) with B‐spline basis functions, solving hydrodynamic free‐surface problems, including forward speed effects. The spatial convergence rate is found as a function of the order of the B‐spline basis. It is shown that for all the models examined the mixed implicit–explicit Euler time integration scheme is correct to second order. Stability diagrams are found for models based on B‐splines of orders third through to sixth for two different time integration schemes. The stability analysis can be regarded as an extension of the analysis by Vada and Nakos [Vada T, Nakos DE. Time marching schemes for ship motion simulations. In Proceedings of the 8th International Workshop on Water Waves and Floating Bodies, St. John's, Newfoundland, Canada, 1993; 155–158] to include B‐splines of orders higher than three (piecewise quadratic polynomials) and to include finite water depth and a current at an oblique angle to the model grid. Copyright © 2000 John Wiley & Sons, Ltd.     

Slope limiting for discontinuous Galerkin approximations with a possibly non‐orthogonal Taylor basis

The use of high‐order polynomials in discontinuous Galerkin (DG) approximations to convection‐dominated transport problems tends to cause a violation of the maximum principle in regions where the derivatives of the solution are large. In this paper, we express the DG solution in terms of Taylor basis functions associated with the cell average and derivatives at the center of the cell. To control the (derivatives of the) discontinuous solution, the values at the vertices of each element are required to be bounded by the means. This constraint is enforced using a hierarchical vertex‐based slope limiter to constrain the coefficients of the Taylor polynomial in a conservative manner starting with the highest‐order terms. The loss of accuracy at smooth extrema is avoided by taking the maximum of the correction factors for derivatives of order p and higher. No free parameters, oscillation detectors, or troubled cell markers are involved. In the case of a non‐orthogonal Taylor basis, the same limiter is applied to the vector of discretized time derivatives before the multiplication by the off‐diagonal part of the consistent mass matrix. This strategy leads to a remarkable gain of accuracy, especially in the case of simplex meshes. A numerical study is performed for a 2D convection equation discretized with linear and quadratic finite elements. Copyright © 2012 John Wiley & Sons, Ltd.     

Comparative study of lattice‐Boltzmann and finite volume methods for the simulation of laminar flow through a 4:1 planar contraction

In the present paper, a comparative study of numerical solutions for Newtonian fluids based on the lattice‐Boltzmann method (LBM) and the classical finite volume method (FVM) is presented for the laminar flow through a 4:1 planar contraction at a Reynolds number of value one, Re=1. In this study, the stress field for LBM is directly obtained from the distribution function. The calculations of the stress based on the FVM‐data use the evaluations of velocity gradients with finite differences. The stress field for both LBM and FVM is expressed in the present study in terms of the shear stress and the first normal stress difference. The lateral and axial profiles of the velocity, the shear stress and the first normal stress difference for both methods are investigated. It is shown that the LBM results for the velocity and the stresses are in excellent agreement with the FVM results. Copyright © 2004 John Wiley & Sons, Ltd.     

A pressure‐based unstructured grid method for all‐speed flows

An all‐speed algorithm based on the SIMPLE pressure‐correction scheme and the ‘retarded‐density’ approach has been formulated and implemented within an unstructured grid, finite volume (FV) scheme for both incompressible and compressible flows, the latter involving interaction of shock waves. The collocated storage arrangement for all variables is adopted, and the checkerboard oscillations are eliminated by using a pressure‐weighted interpolation method, similar to that of Rhie and Chow [Numerical study of the turbulent flow past an airfoil with trailing edge separation. AIAA Journal 1983; 21 : 1525]. The solution accuracy is greatly enhanced when a higher‐order convection scheme combined with adaptive mesh refinement (AMR) are used. Copyright © 2000 John Wiley & Sons, Ltd.