Upwind Finite Volume Research Articles

Uncertainty Quantification for Two‐Phase Flow in Heterogeneous Porous Media

AbstractThe simulation of flow and transport in porous media such as aquifers often involve dealing with complex heterogeneities. They are characterized by varying hydrogeological properties which differ strongly from the adjacent medium and often lead to significant changes of the flow behavior. However detailed information about the location of such heterogeneities is not always known. The deterministic models thus need to be extended stochastically to quantify uncertainties. As mathematical model we use the capillarity‐free fractional flow formulation for two immiscible and incompressible fluid phases in a two‐dimensional and partitioned domain. To cope with the randomly located heterogeneity interfaces we employ a stochastic Galerkin (SG) method [4]. The physical space of this system then is modelled by a central upwind finite volume scheme [5] in combination with mixed finite elements [7]. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A first‐order hyperbolic framework for large strain computational solid dynamics: An upwind cell centred Total Lagrangian scheme

SummaryThis paper builds on recent work developed by the authors for the numerical analysis of large strain solid dynamics, by introducing an upwind cell centred hexahedral finite volume framework implemented within the open source code OpenFOAM [http://www.openfoam.com/]. In Lee, Gil and Bonet (2013), a first‐order hyperbolic system of conservation laws was introduced in terms of the linear momentum and the deformation gradient tensor of the system, leading to excellent behaviour in two‐dimensional bending dominated nearly incompressible scenarios. The main aim of this paper is the extension of this algorithm into three dimensions, its tailor‐made implementation into OpenFOAM and the enhancement of the formulation with three key novelties. First, the introduction of two different strategies in order to ensure the satisfaction of the underlying involutions of the system, that is, that the deformation gradient tensor must be curl‐free throughout the deformation process. Second, the use of a discrete angular momentum projection algorithm and a monolithic Total Variation Diminishing Runge–Kutta time integrator combined in order to guarantee the conservation of angular momentum. Third, and for comparison purposes, an adapted Total Lagrangian version of the hyperelastic‐GLACE nodal scheme of Kluth and Després (2010) is presented. A series of challenging numerical examples are examined in order to assess the robustness and accuracy of the proposed algorithm, benchmarking it against an ample spectrum of alternative numerical strategies developed by the authors in recent publications. Copyright © 2016 John Wiley & Sons, Ltd.

Numerical simulation of the flow around a subsea pipeline with different protection methods

Hydrodynamic stress induced by marine currents subject subsea pipelines to failure vulnerability. Therefore, several methods have been established to protect such pipelines from hydrodynamic forces. The objective of this work is to investigate the performance of two different protection methods using computational fluid dynamics. A second-order accurate upwind finite volume computational fluid dynamics model was used to simulate isotropic turbulent flow around a subsea pipeline located on flat seabed. A comparison between four turbulence models revealed that both Menter’s shear stress transport k[Formula: see text] and the standard k[Formula: see text] models yield the best agreement with experimental measurements. Pipeline trenching and double-barrier protection methods were simulated with different geometrical characteristics. A comparison between those two methods was conducted and discussed. It is found that at small aspect ratios, the double-barrier method prevails over trenching in terms of its ability to isolate the pipe from the main current. While at large aspect ratio, trenching provides near-zero pressure coefficient along the pipe wall, which demonstrate its prevalence in protecting the pipeline.

High-order upwind finite volume element method for first-order hyperbolic optimal control problems

We present a high-order upwind finite volume element method to solve optimal control problems governed by first-order hyperbolic equations. The method is efficient and easy for implementation. Both the semi-discrete error estimates and the fully discrete error estimates are derived. Optimal order error estimates in the sense of \(L_{2}\)-norm are obtained. Numerical examples are provided to confirm the effectiveness of the method and the theoretical results. doi:10.1017/S1446181116000031

A Roe-type scheme with low Mach number preconditioning for a two-phase compressible flow model with pressure relaxation

We describe two-phase compressible flows by a hyperbolic six-equation single-velocity two-phase flow model with stiff mechanical relaxation. In particular, we are interested in the simulation of liquid-gas mixtures such as cavitating flows. The model equations are numerically approximated via a fractional step algorithm, which alternates between the solution of the homogeneous hyperbolic portion of the system through Godunov-type finite volume schemes, and the solution of a system of ordinary differential equations that takes into account the pressure relaxation terms. When used in this algorithm, classical schemes such as Roe’s or HLLC prove to be very efficient to simulate the dynamics of transonic and supersonic flows. Unfortunately, these methods suffer from the well known difficulties of loss of accuracy and efficiency for low Mach number regimes encountered by upwind finite volume discretizations. This issue is particularly critical for liquid-gasmixtures due to the large and rapid variation in the flow of the acoustic impedance. To cure the problem of loss of accuracy at low Mach number, in this work we apply to our original Roe-type scheme for the two-phase flow model the Turkel’s preconditioning technique studied by Guillard–Viozat [Computers & Fluids, 28, 1999] for the Roe’s scheme for the classical Euler equations.We present numerical results for a two-dimensional liquid-gas channel flow test that show the effectiveness of the resulting Roe-Turkel method for the two-phase system.

HIGH-ORDER UPWIND FINITE VOLUME ELEMENT METHOD FOR FIRST-ORDER HYPERBOLIC OPTIMAL CONTROL PROBLEMS

We present a high-order upwind finite volume element method to solve optimal control problems governed by first-order hyperbolic equations. The method is efficient and easy for implementation. Both the semi-discrete error estimates and the fully discrete error estimates are derived. Optimal order error estimates in the sense of $L^{2}$-norm are obtained. Numerical examples are provided to confirm the effectiveness of the method and the theoretical results.

Semi-implicit finite volume schemes for a chemotaxis-growth model

This paper is concerned with finite volume approximations for a nonlinear parabolic–elliptic system for chemotaxis-growth in Rd, d=2,3. This model describes a process of pattern formation by some chemotactic biological individuals. We present two schemes which make use of a semi-implicit time discretization and an upwind finite volume approximation. For both schemes, we prove existence, uniqueness and nonnegativity of the approximate solutions under some conditions on the time step, and we show (for one of the schemes) that the numerical solution converges to a corresponding weak solution for the studied model. Numerical simulations are performed in two dimensional spaces to demonstrate the efficiency of the schemes to capture the pattern formations and to verify our theoretical results.

One-dimensional aggregation equation after blow up: Existence, uniqueness and numerical simulation

The nonlocal nonlinear aggregation equation in one space dimension is investigated. In the so-called attractive case smooth solutions blow up in finite time, so that weak measure solutions are introduced. The velocity involved in the equation becomes discontinuous, and a particular care has to be paid to its definition as well as the formulation of the corresponding flux. When this is done, the notion of duality solutions allows to obtain global in time existence and uniqueness for measure solutions. An upwind finite volume scheme is also analyzed, and the convergence towards the unique solution is proved. Numerical examples show the dynamics of the solutions after the blow up time.

An Exact Rescaling Velocity Method for some Kinetic Flocking Models

In this work, we discuss kinetic descriptions of flocking models, of the so-called Cucker-Smale and Motsch-Tadmor types. These models are given by Vlasov-type equations where the interactions taken into account are only given long-range bi-particles interaction potential. We introduce a new exact rescaling velocity method, inspired by a recent work of Filbet and Rey, allowing to observe numerically the flocking behavior of the solutions to these equations, without a need of remeshing or taking a very fine grid in the velocity space. To stabilize the exact method, we also introduce a modification of the classical upwind finite volume scheme which preserves the physical properties of the solution, such as momentum conservation.

Conservative 1D–2D coupled numerical strategies applied to river flooding: The Tiber (Rome)

Coupled 1D–2D numerical strategies are presented in this work for their application to fast computation of large rivers flooding. Both 1D and 2D models are built using explicit upwind finite volume schemes, able to deal with wetting-drying fronts. The topography representation is described via cross sections for the 1D model and with quadrilateral/triangular structured/unstructured meshes for the 2D model. The coupling strategies, free of hydraulic structures and tuning parameters, are firstly validated in a laboratory test dealing with a levee break and its flooding into a lateral plane. The numerical results are compared with a fully 2D model as well as with measurements in some gauge points giving satisfactory results. The simulation of a real flooding scenario in the Tiber river near the urban area of Rome (Italy) is then performed. A lateral coupling configuration is provided, in which the flood wave propagation in the main channel is simulated by means of a 1D model and the inundation of the riverside is simulated by means of a 2D model. On the other hand, a frontal coupling, in which the flood wave is simulated in a 1D model first and then it is propagated into a 2D model, is also performed. The flooding extension is almost well captured by all the schemes presented, being the 1D–2D lateral configuration the most confident with speed-ups of around 15x.

A new combined finite element-upwind finite volume method for convection-dominated diffusion problems

In this article, we develop a combined finite element-weighted upwind finite volume method for convection-dominated diffusion problems in two dimensions, which discretizes the diffusion term with the standard finite element scheme, and the convection and source terms with the weighted upwind finite volume scheme. The developed method leads to a totally new scheme for convection-dominated problems, which overcomes numerical oscillation, avoids numerical dispersion, and has high-order accuracy. Stability analyses of the scheme are given for the problems with constant coefficients. Numerical experiments are presented to illustrate the stability and optimal convergence of our proposed method.© 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2015

A Computational Modeling for Semi-Coupled Multiphase Flow in Atmospheric Icing Conditions

A two-dimensional second-order positivity-preserving finite volume upwind scheme is developed for a semi-coupled algorithm involving the air and droplet flow fields in the Eulerian frame in which shares the grid for each phase. Special emphasizes are placed on the computational modeling, which is induced from a strongly coupled algorithm, that satisfies the strict hyperbolicity and its numerical scheme based on the HLLC solver preserving the positivity to handle multiphase flow in the Eulerian frame, respectively. The proposed modeling associated with the semi-coupled algorithm including the Navier-Stokes and droplet equations takes into account different boundary conditions on the solid surface for each phase. The verification and validation studies show that the new scheme can solve the air and droplet flow fields in fairly good agreement with the exact analytical solutions and experimental data. In particular, it accurately predicted the maximum value of the droplet impingement intensity near the stagnation region and the droplet impingement area.

An upwind finite volume method on non-orthogonal quadrilateral meshes for the convection diffusion equation in porous media

The aim of this work is to solve the two-dimensional convection diffusion equation on non-rectangular grids formed only by quadrilaterals honoring the internal structures of a reservoir (preferential flow channels, faults, areas of high permeability contrast, changes in sediment type, etc.), taking into account different physical configurations of the porous medium. To take advantage of the good representation of the domain through these meshes, the finite volume method was used, which is conservative and facilitates the treatment of the boundary conditions. In this method, the convection diffusion equation is integrated on each quadrilateral (control volume) of the mesh, thus obtaining the integral form of the equation. The velocity value in the face of each quadrilateral is determined according to the direction of the flow (upwind scheme). After approximating the integrals involved and taking into account the boundary conditions, a discrete equation in each control volume showed up. Finally, a large sparse linear system is obtained, generally non-symmetric and ill-conditioned, which can be solved by iterative methods such as GMRES with incomplete LU preconditioning. Different scenarios were considered varying boundary conditions (Dirichlet and Neumann type), source term, and diffusion constant fluid velocity. The results are consistent with the physical interpretation of each configuration.

An accurate numerical solution to the Saint-Venant-Hirano model for mixed-sediment morphodynamics in rivers

We present an accurate numerical approximation to the Saint-Venant-Hirano model for mixed-sediment morphodynamics in one space dimension. Our solution procedure originates from the fully-unsteady matrix-vector formulation developed in [54]. The principal part of the problem is solved by an explicit Finite Volume upwind method of the path-conservative type, by which all the variables are updated simultaneously in a coupled fashion. The solution to the principal part is embedded into a splitting procedure for the treatment of frictional source terms. The numerical scheme is extended to second-order accuracy and includes a bookkeeping procedure for handling the evolution of size stratification in the substrate. We develop a concept of balancedness for the vertical mass flux between the substrate and active layer under bed degradation, which prevents the occurrence of non-physical oscillations in the grainsize distribution of the substrate. We suitably modify the numerical scheme to respect this principle. We finally verify the accuracy in our solution to the equations, and its ability to reproduce one-dimensional morphodynamics due to streamwise and vertical sorting, using three test cases. In detail, (i) we empirically assess the balancedness of vertical mass fluxes under degradation; (ii) we study the convergence to the analytical linearised solution for the propagation of infinitesimal-amplitude waves [54], which is here employed for the first time to assess a mixed-sediment model; (iii) we reproduce Ribberink’s E8-E9 flume experiment [46].

A two-grid combined finite element-upwind finite volume method for a nonlinear convection-dominated diffusion reaction equation

In this paper, we first present a combined finite element-upwind finite volume method for a fully nonlinear convection-dominated diffusion reaction equation and derive its a priori optimal error estimate in H1-norm and sub-optimal error estimate in L2-norm for piecewise linear finite element combining with the first order upwind finite volume scheme. Then we study a type of two-grid method for the nonlinear convection-dominated transport equation together with the combined finite element-upwind finite volume method on the fine grid Th and the streamline diffusion finite element scheme on the coarse grid TH, which not only significantly reduces the computational cost on nonlinear iterations but also remains the numerical computation stabilized and the approximation accuracy unchanged. A priori error estimate of such designed two-grid method in H1-norm is proved to be O(h+H32), showing that the two-grid method achieves the optimal approximation as long as the mesh sizes satisfy h=O(H32). Finally, a numerical example is carried out to verify the accuracy and efficiency of the present numerical method.

Coupled nonlinear advection–diffusion–reaction system for prevention of groundwater contamination by modified upwind finite volume element method

The coupled advection-dominated diffusion reaction equations arising in the prevention of groundwater contamination problem are approximated by the modified upwind finite volume element method. In order to solve the resulting nonlinear system efficiently, we use a two-grid algorithm to decompose the nonlinear system into a small nonlinear system on a coarse grid with mesh size H and a linear system on a fine grid with mesh size h. It is shown that the approximation still achieves asymptotically optimal as long as the mesh sizes satisfy H=O(h1/3). Numerical examples are presented to illustrate efficiency and accuracy of the proposed method.

Oldroyd-B numerical solutions about a rotating sphere at low Reynolds number

This study investigates the numerical solution of a viscoelastic flow for an Oldroyd-B model, due to the rotation of a sphere about its diameter. Analysis of the elastic-viscous problem had been reported by Thomas and Walters (Q J Mech Appl Math 17:39–53, 1964), Walters and Savins (J Rheol 9:407–416, 1965) and Giesekus (Rheol Acta 9:30–38, 1970). In this respect, three different flow patterns (types 1–3) predicted by Thomas and Walters (Q J Mech Appl Math 17:39–53, 1964) have been successfully reproduced when using an Oldroyd-B fluid to represent a Boger fluid. Initially, solutions for the Oldroyd-B model were calibrated in the second-order regime against the analytical solution. Then, the work is extended to cover three different flows regimes (second-order regime, transitional and general flow) and two settings of polymeric solvent-fraction. Analysis based on the bounding sphere-radius, associated with type 2 flow, and through different flow regimes revealed that the distinctive symmetrical shape formed in the second-order regime was not preserved, but an elliptical shape was acquired. Moreover, for general and transitional flow regimes, a new and third vortex was identified in the polar region of the sphere. The adjustment of this feature between two different fluid compositions, with solutions for high-solvent and high-polymeric versions (low-high polymeric contributions), was contrasted. The second normal stress difference (N 2) on the field was increased as the m parameter developed across the different flow regimes. Different torque values for several m values were compared against the theory, demonstrating the expected linear behaviour. The numerical algorithm involved a hybrid sub-cell finite-element/finite volume discretisation (fe/fv), which solved the system of momentum-continuity-stress equations. It is based on a semi-implicit time-stepping Taylor-Galerkin/pressure-correction parent-cell finite element method for momentum continuity, whilst invoking a sub-cell cell-vertex fluctuation distribution finite volume scheme for the stress. The hyperbolic aspects of the constitutive equation were addressed discretely through finite volume upwind Fluctuation Distribution techniques and inhomogeneity calls upon Median Dual Cell approximation.

Numerical simulation of 3D backward facing step flows at various Reynolds numbers

The work deals with the numerical simulation of 3D turbulent flow over backward facing step in a narrow channel. The mathematical model is based on the RANS equations with an explicit algebraic Reynolds stress model (EARSM). The numerical method uses implicit finite volume upwind discretization. While the eddy viscosity models fail in predicting complex 3D flows, the EARSM model is shown to provide results which agree well with experimental PIV data. The reference experimental data provide the 3D flow field. The simulations are compared with experiment for 3 values of Reynolds number.

Dynamics Characterization of Missiles with Control Flaps Based on CFD

Numerical studies were carried out on the dynamics of the Basic Finner missile under supersonic velocity based on the three-dimensional unsteady N-S equation. Second-order upwind NND scheme finite volume was used to discretize the spatial derivative term of the flow-controlled equation group. The LU-SGS containing dual time steps was used to obtain higher efficiency and precision of time solution. A numerical model representing the relationship between the unsteady aerodynamic force/moment coefficients and state variables under single DOF (degree of freedom) pitching and rolling movements was built from the Etkin unsteady aerodynamic model. The numerical algorithm for solving damping-in-pitch derivatives and damping-in-roll derivatives by forced-harmonic analysis was proposed. The M910 target practice, discarding sabot with tracer (TPDS-T) was used as a validation example and proved that the numerical calculation gives the same result as the experiment. By simulating the forced pitching and rolling vibration of Basic Finner, the numerical identification succeeded in obtaining the damping-in-pitch derivative and damping-in-roll derivative to a higher precision than the calculated result obtained by the quasi-steady Euler equation. Besides, to address the longitudinal and horizontal cross coupling induced by the pitching movement, numerical identification was also carried out on the cross coupling derivative of Basic Finner when under forced pitching vibration.

An upwind finite volume method for incompressible inviscid free surface flows

An upwind finite volume method on curvilinear grids has been developed to simulate two-dimensional, inviscid, incompressible free surface flows. In this method, the level set method is used to capture the interface. The level set equation and the flow governing equations (Euler equations) are not solved separately, but incorporated into a system equation. The artificial compressibility method is adopted to solve the resulting hyperbolic equations. The MUSCL-Roe scheme, which is widely used in compressible flow computations, is employed to discretize the convective terms. A fully implicit dual-time stepping method with a highly efficient LU-SSOR algorithm as sub-iteration scheme is applied to advance the solution in time. In the present study, a re-initialization method on curvilinear grids for level set function is presented. An example of re-initialization for level set function on curvilinear grids is demonstrated and the resulting level set function is in good agreement with the exact distance function. Numerical results of a sloshing tank problem and a broken dam problem agree well with the theoretical and experimental data, respectively. The free surface flow over a submerged hydrofoil is also simulated using overset grids to demonstrate the capability of dealing with complex geometry. At last, the simulation of droplet splash problem is performed to show the ability of the present method in handling topological changes of the free surface.