Upwind Finite Volume Method Research Articles

An unstructured finite volume method for large-scale shallow flows using the fourth-order Adams scheme

In this paper, we introduce a new upwind finite volume method using unstructured grids for large-scale shallow flows. This method uses a high-order upwind scheme for the calculation of the numerical flux, and the fourth-order Adams method with a splitting approach for time integration. The process includes three stages: in the first and third steps the Coriolis term is integrated analytically, and in the second step the flux term is integrated numerically. Most upwind schemes perform well for gravity waves but they lead to a high level of damping or numerical oscillations for Rossby waves. The proposed method presents the advantage that it performs well for both gravity and Rossby waves. The use of fourth-order Adams method without any iteration on the corrector is enough to suppress the short-wave numerical noise without damping the long waves that are essential in the transport of energy Rossby waves, in large-scale oceanic and atmospheric flows.

Depth-Averaged Hydrodynamic Model for Gradual Breaching of Embankment Dams Attributable to Overtopping Considering Suspended Sediment Transport

AbstractA numerical model for simulation of embankment dams’ gradual breaching because of overtopping, named embankments breach software (EBS), is introduced in this paper. The model solves the equations for the depth-averaged flow (modified for shallow water flow on steep slopes), the k-e turbulence model, the sediment transport, and the bed elevation change in a coupled manner. An upwind cell center finite volume method is applied for developing the discrete form of the differential equations. A linearized scheme with second-order accuracy in space and explicit time integration on unstructured triangular meshes is utilized in this model. To simulate the flow over dry sloping surfaces of the embankment dam configuration, the developed model considers a powerful wetting and drying technique. Furthermore, the abilities of the model enhanced simulation of sudden bank failure of the breach channel because of the base erosion and transport of the failed solid material.

Residual distribution finite element method for convection-dominated problems

In this paper, we present a new stabilized finite element method for the simulation of nonlinear convection-dominated systems of PDEs. We seek to combine the robustness of upwind finite-volume methods for the discretization of strongly shocked flows with the ease and accuracy of C0 finite element reconstruction. This is achieved by exploiting the similarities between traditional SUPG method and finite volume residual distribution schemes. By constructing the finite element analog of finite-volume fluctuation-splitting schemes, we combine the ability to simulate PDEs with strongly discontinuous solutions of upwind finite-volume methods with the reliable accuracy of finite element reconstruction, which traditional finite-volume methods often lack on unstructured grids. Finally, we test the proposed algorithm with challenging problems drawn from hypersonic compressible flow and semiconductor device simulation.

A domain decomposition method for a two‐phase transport model of polymer electrolyte fuel cell containing micro‐porous layer

SUMMARYIn this paper, a nonlinear Dirichlet–Robin iteration‐by‐subdomain domain decomposition method is studied for a multidimensional, multiphysics, and multiphase model of polymer electrolyte fuel cell (PEFC) containing micro‐porous layer (MPL). Across the interface of gas diffusion layer and MPL in PEFC, it is well known that the capillary pressure is continuous, whereas liquid saturation is discontinuous, by which the liquid‐water removal in the porous electrodes can be significantly enhanced. We design a type of non‐overlapping domain decomposition method to deal with water transport in such multi‐layer diffusion media, where Kirchhoff transformation and its inverse techniques are employed to conquer the discontinuous and degenerate water diffusivity in the coexisting single‐phase and two‐phase regions. In addition, the conservation equations of mass, momentum, charge, and hydrogen and oxygen transport are also solved by the combined finite element–upwind finite volume method (FEM/FVM) to overcome the dominated convection effect in gas channels. Numerical simulations demonstrate that the presented techniques are effective in obtaining a fast and convergent nonlinear iteration for such a 3D PEFC model within around 50 steps, in contrast with the oscillatory and nonconvergent iteration conducted by standard FEM/FVM. A series of numerical convergence tests are also carried out to verify the efficiency and accuracy of the present numerical techniques. Copyright © 2012 John Wiley & Sons, Ltd.

Fully-Coupled Modeling of Shallow Water Flow and Pollutant Transport on Unstructured Grids

Understanding the space-time dynamics of pollutant transport remains an essential impediment to accurate prediction of impacts on the ecology of rivers and coastal areas and also for establishing efficient strategies for pollution control and environmental protection. Numerical models are a powerful tool to study the water flows and pollutant transport, and recently a new generation of models is being developed to simulate the coupled flow and pollutant transport in shallow water. In this paper, a two-dimensional fully-coupled model of shallow water flows and pollutant transport was developed using a triangular unstructured grid (TIN: triangular irregular network), which is also an important module of the PIHM-Hydro modeling system. The model is based on a cell-centered upwind finite volume method using the HLL approximate Riemann solver. A multidimensional linear reconstruction technique and multi-dimensional slope limiter was implemented to achieve a second-order spatial accuracy. In order to make the model efficient and stable, an explicit-implicit method was used in temporal discretization by an operator splitting technique. A test case of the pollutant transport in a square cavity is used to validate the model. Then the model was further applied to two pollutant transport scenarios: microscale pollutant transport following dam break and mesoscale pollutant transport driven by storm surge in Galveston Bay. The numerical results show that the model could accurately predict the flow dynamics and pollutant transport in extreme events such as a dam break and a storm surge. According to the prediction of the model, the storm surge caused by the Hurricane Ike significantly extended the polluted area.

Analysis of the upwind finite volume method for general initial- and boundary-value transport problems

This paper is devoted to the convergence analysis of the upwind finite volume scheme for the initial and boundary value problem associated to the linear transport equation in any dimension, on general unstructured meshes. We are particularly interested in the case where the initial and boundary data are in $L^\infty$ and the advection vector field $v$ has low regularity properties, namely $v\in L^1(]0,T[,(W^{1,1}(\O))^d)$, with suitable assumptions on its divergence. In this general framework, we prove uniform in time strong convergence in $L^p(\O)$ with $p<+\infty$, of the approximate solution towards the unique weak solution of the problem as well as the strong convergence of its trace. The proof relies, in particular, on the Friedrichs' commutator argument, which is classical in the renormalized solutions theory.

Two-dimensional numerical model for debris flows in the Jiangjia Gully, Yunnan Province

Debris flows are recurrent natural hazards in many mountainous regions. This paper presents a numerical study on the propagation of debris flows in natural erodible open channels, in which the bed erosion and sedimentation processes are important. Based on the Bingham fluid theory, a mathematical model of the two-dimensional non-constant debris flow is developed. The governing equations include the continuity and momentum conservation equations of debris flow, the sediment convection-diffusion equation, the bed erosion-deposition equation and the bed-sediment size gradation adjustment equation. The yield stress and shear stress components are included to describe the dynamic rheological properties. The upwind control-volume Finite Volume Method (FVM) is applied to discretize the convection terms. The improved SIMPLE algorithm with velocity-free-surface coupled correction is developed to solve the equations on non-orthogonal, quadrilateral grids. The model is applied to simulate a debris flow event in Jiangjia Gully, Yunnan Province and to predict the flow pattern and bed erosion-deposition processes. The results show the effectiveness of the proposed numercial model in debris flow simulation and potential hazard analysis.

Error estimate for the upwind finite volume method for the nonlinear scalar conservation law

In this paper we estimate the error of upwind first order finite volume schemes applied to scalar conservation laws. As a first step, we consider standard upwind and flux finite volume scheme discretization of a linear equation with space variable coefficients in conservation form. We prove that, in spite of their lack of consistency, both schemes lead to a first order error estimate. As a final step, we prove a similar estimate for the nonlinear case. Our proofs rely on the notion of geometric corrector, introduced in our previous paper by Bouche et al. (2005) [24] in the context of constant coefficient linear advection equations.

Fully coupled approach to modeling shallow water flow, sediment transport, and bed evolution in rivers

Our ability to predict complex environmental fluid flow and transport hinges on accurate and efficient simulations of multiple physical phenomenon operating simultaneously over a wide range of spatial and temporal scales, including overbank floods, coastal storm surge events, drying and wetting bed conditions, and simultaneous bed form evolution. This research implements a fully coupled strategy for solving shallow water hydrodynamics, sediment transport, and morphological bed evolution in rivers and floodplains (PIHM_Hydro) and applies the model to field and laboratory experiments that cover a wide range of spatial and temporal scales. The model uses a standard upwind finite volume method and Roe's approximate Riemann solver for unstructured grids. A multidimensional linear reconstruction and slope limiter are implemented, achieving second‐order spatial accuracy. Model efficiency and stability are treated using an explicit‐implicit method for temporal discretization with operator splitting. Laboratory‐and field‐scale experiments were compiled where coupled processes across a range of scales were observed and where higher‐order spatial and temporal accuracy might be needed for accurate and efficient solutions. These experiments demonstrate the ability of the fully coupled strategy in capturing dynamics of field‐scale flood waves and small‐scale drying‐wetting processes.

Type-VI and Type-V Shock-Shock Interactions on Double-Wedge Geometries Using AUSM+ on Unstructured Grid

The Euler equations are solved on unstructured triangular meshes for hypersonic flow over double-wedge geometries. The driving algorithm is an upwind biased cell centered finite volume method. AUSM+ method is used to split the fluxes. Edney (1968) studied the shock interactions by impinging an externally generated planar oblique shock on the bow shock generated by a cylinder. Depending upon the parametric conditions Edney classified the shock interactions in different types. Two interaction topologies, namely Type-VI and Type-V and the transition from Type-VI to Type-V are studied in details. Both six-shock and seven-shock configurations of Type-V interaction are presented.

Direct raycasting of unstructured cell-centered data by discontinuity Roe-average computation

In the field of computational fluid dynamics (CFD), the upwind finite volume method (FVM) is widely applied to solve 3D flows with discontinuity phenomena (e.g., shock waves). It produces unstructu...

Theoretical analysis of the upwind finite volume scheme on the counter-example of Peterson

When applied to the linear advection problem in dimension two, the upwind finite volume method is a non consistent scheme in the finite differences sense but a convergent scheme. According to our previous paper [Bouche et al. , SIAM J. Numer. Anal. 43 (2005) 578–603], a sufficient condition in order to complete the mathematical analysis of the finite volume scheme consists in obtaining an estimation of order p , less or equal to one, of a quantity that depends only on the mesh and on the advection velocity and that we called geometric corrector. In [Bouche et al. , Hermes Science publishing, London, UK (2005) 225–236], we prove that, on the mesh given by Peterson [SIAM J. Numer. Anal. 28 (1991) 133–140] and for a subtle alignment of the direction of transport parallel to the vertical boundary, the infinite norm of the geometric corrector only behaves like h 1/2 where h is a characteristic size of the mesh. This paper focuses on the case of an oblique incidence i.e. a transport direction that is not parallel to the boundary, still with the Peterson mesh. Using various mathematical technics, we explicitly compute an upper bound of the geometric corrector and we provide a probabilistic interpretation in terms of Markov processes. This bound is proved to behave like h , so that the order of convergence is one. Then the reduction of the order of convergence occurs only if the direction of advection is aligned with the boundary.

Two-dimensional finite volume method for dam-break flow simulation

A numerical model based upon a second-order upwind cell-center finite volume method on unstructured triangular grids is developed for solving shallow water equations. The assumption of a small depth downstream instead of a dry bed situation changes the wave structure and the propagation speed of the front which leads to incorrect results. The use of Harten-Lax-vau Leer (HLL) allows handling of wet/dry treatment. By usage of the HLL approximate Riemann solver, also it make possible to handle discontinuous solutions. As the assumption of a very small depth downstream of the dam can change the nature of the dam break flow problem which leads to incorrect results, the HLL approximate Riemann solver is used for the computation of inviscid flux functions, which makes it possible to handle discontinuous solutions. A multidimensional slope-limiting technique is applied to achieve second-order spatial accuracy and to prevent spurious oscillations. To alleviate the problems associated with numerical instabilities due to small water depths near a wet/dry boundary, the friction source terms are treated in a fully implicit way. A third-order Runge–Kutta method is used for the time integration of semi-discrete equations. The developed numerical model has been applied to several test cases as well as to real flows. The tests are tested in two cases: oblique hydraulic jump and experimental dam break in converging-diverging flume. Numerical tests proved the robustness and accuracy of the model. The model has been applied for simulation of dam break analysis of Torogh in Iran. And finally the results have been used in preparing EAP (Emergency Action Plan).

New Numerical Techniques for a Three-Dimensional Liquid-Feed Direct Methanol Fuel Cell Model

In this paper, a three-dimensional two-phase transport model of a liquid-feed direct methanol fuel cell (DMFC) is studied, where flow, species, charge-transport, and energy equations are simultaneously addressed. Some new numerical techniques are specifically investigated in order to achieve fast and convergent numerical simulation. An updated source term of the water transport equation based on a combination of physical and numerical considerations, and a series of efficient numerical algorithms and discretizations by means of a combined finite element–upwind finite volume method, are designed and analyzed to assist in approaching this target. The three-dimensional numerical simulations demonstrate that the convergent physical solutions can be obtained within hundreds of steps, in contrast to the long-running and even nonconvergent nonlinear iterations operated by standard finite element/volume methods if no new numerical technique is adopted.

Turbine cascade design via multigrid-aided finite-difference progressive optimization

This paper proposes an efficient and robust procedure for the design optimization of turbomachinery cascades in inviscid and turbulent transonic flow conditions. It employs a progressive strategy, based on the simultaneous convergence of the design process and of all iterative solutions involved (flow analysis, gradient evaluation), also including the global refinement from a coarse to a sufficiently fine mesh. Cheap, flexible and easy-to-program Multigrid-Aided Finite Differences are employed for the computation of the sensitivity derivatives. The entire approach is combined with an upwind finite-volume method for the Euler and the Navier-Stokes equations on cell-vertex unstructured (triangular) grids, and validated versus the inverse design of a turbine cascade. The methodology turns out to be robust and highly efficient, the converged design optimization being obtained in a computational time equal to that required by 15 to 20 (depending on the application) multigrid flow analyses on the finest grid.

A multilevel multiscale mimetic (M 3) method for two-phase flows in porous media

We describe a multilevel multiscale mimetic (M 3) method for solving two-phase flow (water and oil) in a heterogeneous reservoir. The governing equations are the elliptic equation for the reservoir pressure and the hyperbolic equation for the water saturation. On each time step, we first solve the pressure equation and then use the computed flux in an explicit upwind finite volume method to update the saturation. To reduce the computational cost, the pressure equation is solved on a much coarser grid than the saturation equation. The coarse-grid pressure discretization captures the influence of multiple scales via the subgrid modeling technique for single-phase flow recently proposed in [Yu. A. Kuznetsov. Mixed finite element method for diffusion equations on polygonal meshes with mixed cells. J. Numer. Math., 14 (4) (2006) 305–315; V. Gvozdev. discretization of the diffusion and Maxwell equations on polyhedral meshes. Technical Report Ph.D. Thesis, University of Houston, 2007; Yu. Kuznetsov. Mixed finite element methods on polyhedral meshes for diffusion equations, in: Computational Modeling with PDEs in Science and Engineering, Springer-Verlag, Berlin, in press]. We extend significantly the applicability of this technique by developing a new robust and efficient method for estimating the flux coarsening parameters. Specifically, with this advance the M 3 method can handle full permeability tensors and general coarsening strategies, which may generate polygonal meshes on the coarse grid. These problem dependent coarsening parameters also play a critical role in the interpolation of the flux, and hence, in the advection of saturation for two-phase flow. Numerical experiments for two-phase flow in highly heterogeneous permeability fields, including layer 68 of the SPE Tenth Comparative Solution Project, demonstrate that the M 3 method retains good accuracy for high coarsening factors in both directions, up to 64 for the considered models. Moreover, we demonstrate that with a simple and efficient temporal updating strategy for the coarsening parameters, we achieve accuracy comparable to the fine-scale solution, but at a fraction of the cost.

A Multistep Finite Volume Method with Penalty for Thermal Convection Problems in Two or Three Dimensions

The author considers a thermal convection problem with infinite Prandtl number in two or three dimensions. The mathematical model of such problem is described as an initial boundary value problem made up of three partial differential equations. One equation of the convection-dominated diffusion type for the temperature, and another two of the Stokes type for the normalized velocity and pressure. The approximate solution is obtained by a penalty finite volume method for the Stokes equation and a multistep upwind finite volume method for the convection-diffusion equation. Under suitable smoothness of the exact solution, error estimates in some discrete norms are derived.

Turbine cascade design via multigrid-aided finite-difference progressive optimization

This paper proposes an efficient and robust procedure for the design optimization of turbomachinery cascades in inviscid and turbulent transonic flow conditions. It employs a progressive strategy, based on the simultaneous convergence of the design process and of all iterative solutions involved (flow analysis, gradient evaluation), also including the global refinement from a coarse to a sufficiently fine mesh. Cheap, flexible and easy-to-program Multigrid-Aided Finite Differences are employed for the computation of the sensitivity derivatives. The entire approach is combined with an upwind finite-volume method for the Euler and the Navier-Stokes equations on cell-vertex unstructured (triangular) grids, and validated versus the inverse design of a turbine cascade. The methodology turns out to be robust and highly efficient, the converged design optimization being obtained in a computational time equal to that required by 15 to 20 (depending on the application) multigrid flow analyses on the finest grid.

On curl-preserving finite volume discretizations for shallow water equations

The preservation of intrinsic or inherent constraints, like divergence-conditions, has gained increasing interest in numerical simulations of various physical evolution equations. In Torrilhon and Fey, SIAM J. Numer. Anal. (42/4) 2004, a general framework is presented how to incorporate the preservation of a discrete constraint into upwind finite volume methods. This paper applies this framework to the wave equation system and the system of shallow water equations. For the wave equation a curl-preservation for the momentum variable is present and easily identified. The preservation in case of the shallow water system is more involved due to the presence of convection. It leads to the vorticity evolution as generalized curl-constraint. The mechanisms of vorticity generation are discussed. For the numerical discretization special curl-preserving flux distributions are discussed and their incorporation into a finite volume scheme described. This leads to numerical discretizations which are exactly curl-preserving for a specific class of discrete curl-operators. The numerical experiments for the wave equation show a significant improvement of the new method against classical schemes. The extension of the curl-free numerical discretization to the shallow water case is possible after isolating the pressure flux. Simulation examples demonstrate the influence of the modification. The vortex structure is more clearly resolved.

A conservative 2D model of inundation flow with solute transport over dry bed

In this paper, a transient 2D coupled vertically averaged flow/transport model is presented. The model deals with all kind of bed geometries and guarantees global conservation and positive values of both water level and solute concentration in the transient solution. The model is based on an upwind finite volume method, using Roe's approximate Riemann solver. A specific modification of the Riemann solver is proposed to overcome the generation of negative values of depth and concentration, that can appear as a consequence of existing wetting/drying and solute advance fronts over variable bed levels, or by the generation of new ones when dry areas appear. The numerical stability constraints of the explicit model are stated incorporating the influence of the flow velocity, the bed variations and the possible appearance of dry cells. Faced to the important restriction that this new stability condition can impose on the time step size, a different strategy to allow stability using a maximum time step, and in consequence a minimum computational cost is presented.