Upwind Finite Volume Method Research Articles

A Finite Volume Method for a 2D Dam-Break Simulation on a Wet Bed Using a Modified HLLC Scheme

This study proposes a numerical model for depth-averaged Reynolds equations (shallow-water equations) to investigate a dam-break problem, based upon a two-dimensional (2D) second-order upwind cell-centre finite volume method. The transportation terms were modelled using a modified approximate HLLC Riemann solver with the first-order accuracy. The proposed 2D model was assessed and validated through experimental data and analytical solutions for several dam-break cases on a wet and dry bed. The results showed that the error values of the model are lower than those of existing numerical methods at different points. Our findings also revealed that the dimensionless error parameters decrease as the wave propagates downstream. In general, the new model can model the dam-break problem and captures the shock wave superbly.

The application of upwind meshless approximation finite volume method to the convection dominated problems on the unstructured and deformed meshes

The primary objective of this research is to develop a finite volume method that can incorporate the most widely recognized upwind schemes using meshless techniques on an unstructured mesh. The recently developed Meshless Approximation Finite Volume Method (MAFVM) is expected to provide a solution to the convection–diffusion equation with an extremely high Pélect number, regardless of the type or quality of mesh, even when dealing with highly distorted meshes. By incorporating the local compactly supported radial basis functions differential quadrature method (LCRBFDQ), a meshless approximation technique, the field variables, flux, and gradient are directly approximated using information from neighboring scattered nodes. Building upon this concept, we propose two meshless approximation approaches for computing the flux and gradient within the framework of the finite volume method. Elliptic type problem on regular mesh is solved using the proposed MAFVM by different flux computation methods. Parameters for MAFVM that provide accurate and robust results are suggested. Next standard linear and second order upwind schemes as well as the well-known total variation diminishing (TVD) methods are integrated into the proposed MAFVM, using the developed flux calculation method to solve convection–diffusion problem with very high Pélect number on standard or highly deformed mesh. Accurate and robust results can be observed which show that the current MAFVM can provide an alternative and better way to deal with strong convection problem on meshes of arbitrary types by the proposed meshless flux computation methods.

Numerical Investigation on the Effects of Wind and Shielding Conductor on the Ion Flow Fields of HVDC Transmission Lines

Ion flow field is an important aspect of high voltage direct current (HVDC) transmission lines. In this paper, we apply the upwind finite volume method for solving the ion flow field of three HVDC configurations to clarify the effect of the wind and the role of shielding conductors. For the monopolar configuration installation, the ground current distribution with underbuilt shield wires was studied numerically and experimentally. For the ±250 kV bipolar configuration, the calculated peak electric field and current density are verified with the values in a reference. The ±500 kV bipolar configuration is used to investigate the change in electric field and ion current within the same corridors of the existing HVAC lines. We analyze the ion flow field with and without the dedicated metallic return conductor (DMRC). In the absence of wind, the maximum of the electric field is lower than that of the HVAC lines and the current density is very low on the ground. In the presence of wind, the electric field and the current distribution become unsymmetrical on the ground level. The peak current density increases significantly on the downwind side. The HVDC line without DMRC has the electric field peak higher than that of the HVAC lines. With the higher wind shear coefficient, the peak of the electric field and the current density increases on the downwind side. Overall investigations summarize that DMRC can effectively reduce the peak of the electric field to be lower than that of the existing 500 kV HVAC lines under all wind conditions.

Developing the optimization control of pollution constrained by the convection diffusion equations

In this paper, an optimization control of pollution subject to the convection diffusion equations is developed and analyzed. The optimization objective function is to consider the pollution effects for minimizing the weighted derivation between simulated concentrations and allowable concentrations at observation sites and the costs of emission reductions, pollution taxes, and the cost related to layout of pipelines. The proposed optimal control problem is defined on the multiple observation and polluting points and is constrained by the convection diffusion equations. We analyze this optimization problem from a rigorous mathematical viewpoint and prove the existence of a solution of the pollution optimal control. We build the numerical optimal control approach by a second-order improved splitting upwind finite volume method to solve the convection diffusion equations and by the constrained difference evolution (DE) algorithm to compute the optimization procedure. Three numerical experiments are investigated, where the first one aims at searching the optimal disposal fluxes while minimizing concentration deviation and reduction cost, the other two are to minimize the whole economical costs that contain waste water treatment cost, pipeline layout cost and pollution taxation by finding the optimal effluent concentration of waste water. The developed optimal control is efficient and it can be applied in applications.

A Mathematical Model for Transport and Growth of Microbes in Unsaturated Porous Soil

In this work, we develop a mathematical model for transport and growth of microbes by natural (rain) water infiltration and flow through unsaturated porous soil along the vertical direction under gravity and capillarity by coupling a system of advection diffusion equations (for concentration of microbes and their growth-limiting substrate) with the Richards equation. The model takes into consideration several major physical, chemical, and biological mechanisms. The resulting coupled system of PDEs together with their boundary conditions is highly nonlinear and complicated to solve analytically. We present both a partial analytic approach towards solving the nonlinear system and finding the main type of dynamics of microbes, and a full-scale numerical simulation. Following the auxiliary equation method for nonlinear reaction-diffusion equations, we obtain a closed form traveling wave solution for the Richards equation. Using the propagating front solution for the pressure head, we reduce the transport equation to an ODE along the moving frame and obtain an analytic solution for the history of bacteria concentration for a specific test case. To solve the system numerically, we employ upwind finite volume method for the transport equations and stabilized explicit Runge–Kutta–Legendre super-time-stepping scheme for the Richards equation. Finally, some numerical simulation results of an infiltration experiment are presented, providing a validation and backup to the analytic partial solutions for the transport and growth of bacteria in the soil, stressing the occurrence of front moving solitons in the nonlinear dynamics.

Numerical Analysis of Convection–Diffusion Using a Modified Upwind Approach in the Finite Volume Method

The main focus of this study was to develop a numerical scheme with new expressions for interface flux approximations based on the upwind approach in the finite volume method. Our new proposed numerical scheme is unconditionally stable with second-order accuracy in both space and time. The method is based on the second-order formulation for the temporal approximation, and an upwind approach of the finite volume method is used for spatial interface approximation. Some numerical experiments have been conducted to illustrate the performance of the new numerical scheme for a convection–diffusion problem. For the phenomena of convection dominance and diffusion dominance, we developed a comparative study of this new upwind finite volume method with an existing upwind form and central difference scheme of the finite volume method. The modified numerical scheme shows highly accurate results as compared to both numerical schemes.

Ion Flow Field Calculation Considering the Influence of Water Drops and Wind Flow

In this work, an effective calculation method of the ion flow field that considers the impact of wind flow and water drops is presented. To be explicit, the nominal electric field is solved by the charge simulation method (CSM) whilst the space charge density is calculated adopting a second order upwind finite volume method. In addition, a method that determines the roughness factor of a conductor surface is used to enhance calculation accuracy. The influence of various properties of the water drops exerting on an ion flow field is analyzed. Eventually, a practical experiment is conducted to verify the calculated result, and the effectiveness and reliability of this method are then proved via comparing calculated and measured results.

An implicit local time-stepping method based on cell reordering for multiphase flow in porous media

We discuss how to introduce local time-step refinements in a sequential implicit method for multiphase flow in porous media. Our approach relies heavily on causality-based optimal ordering, which implies that cells can be ordered according to total fluxes after the pressure field has been computed, leaving the transport problem as a sequence of ordinary differential equations, which can be solved cell-by-cell or block-by-block. The method is suitable for arbitrary local time steps and grids, is mass-conservative, and reduces to the standard implicit upwind finite-volume method in the case of equal time steps in adjacent cells. The method is validated by a series of numerical simulations. We discuss various strategies for selecting local time steps and demonstrate the efficiency of the method and several of these strategies by through a series of numerical examples.

PRECONDITIONED ITERATIVE METHOD FOR REACTIVE TRANSPORT WITH SORPTION IN POROUS MEDIA

This work deals with the numerical solution of a nonlinear degenerate parabolic equation arising in a model of reactive solute transport in porous media, including equilibrium sorption. The model is a simplified, yet representative, version of multicomponents reactive transport models. The numerical scheme is based on an operator splitting method, the advection and diffusion operators are solved separately using the upwind finite volume method and the mixed finite element method (MFEM) respectively. The discrete nonlinear system is solved by the Newton–Krylov method, where the linear system at each Newton step is itself solved by a Krylov type method, avoiding the storage of the full Jacobian matrix. A critical aspect of the method is an efficient matrix-free preconditioner. Our aim is, on the one hand to analyze the convergence of fixed-point algorithms. On the other hand we introduce preconditioning techniques for this system, respecting its block structure then we propose an alternative formulation based on the elimination of one of the unknowns. In both cases, we prove that the eigenvalues of the preconditioned Jacobian matrices are bounded independently of the mesh size, so that the number of outer Newton iterations, as well as the number of inner GMRES iterations, are independent of the mesh size. These results are illustrated by some numerical experiments comparing the performance of the methods.

Calculation of Ion Flow Field of Monopolar Transmission Line in Corona Cage Including the Effect of Wind

In this work, the ion flow field of a monopolar transmission line inside the corona cage of a square cross-section is iteratively calculated concerning the effects of wind. The electric field distribution is solved analytically using the charge simulation method (CSM). Meanwhile, the upwind finite volume method (UFVM) with 2nd order accuracy is presented for the distribution of space charge density. Additionally, a dual mesh grid is established in the calculation domain, the interlaced geometric construction of the mesh assures a quick and effective convergence rate. In the final part, a reduced-scaled experiment is designed to examine the feasibility and accuracy of this approach, electric field and ion current density on the bottom side are measured by field mills and Wilson plates. The data numerically computed fits well with that acquired by measurement.

An integrated algorithm for hypersonic fluid–thermal–structural analysis of aerodynamically heated cylindrical leading edge

The accurate and efficient prediction of the fluid–thermal–structural performance of thermal protection systems on hypersonic vehicles to protect against severe aerodynamic heating is attracting increasing worldwide attention. In this paper, a new integrated fluid-structural-thermal investigation based on the finite volume method is proposed to study the thermal behavior of aerodynamically heated cylinder leading edges in hypersonic flows. A unified integral equation system is developed based on the governing equations for the physical processes of aerodynamic heating and structural heat transfer, which is resolved by using an up-wind finite volume method and a new two-thermal-resistance model in one integrated, vectorized computer program. To demonstrate its capability and reliability, applications for steady/unsteady fluid–thermal–structural analysis are demonstrated on aerodynamically heated cylinder leading edges at Ma 6.47. The results show that the steady maximum temperature of the cylinder can reach approximately 648 K at the stagnation point, and the unsteady results are in good agreement with the experimental data and related references. Compared with the partitioned approach, the integrated method shows better computational stability with relatively small sensitivity to mesh scale and time step, reducing the computation time for the same unsteady case by approximately 50%. The present study indicates that the integrated approach has potential for significant improvements and efficiency in predicting long-endurance fluid-structural-thermal problems of hypersonic vehicles.

Optimal weighted upwind finite volume method for convection–diffusion equations in 2D

This paper develops and analyzes the optimal weighted upwind finite volume method (OWUFVM) for convection–diffusion equations on nonuniform rectangular meshes. The dual meshes in this method are associated with the local Peclet numbers, which are determined by the convection and diffusion coefficients and the mesh size. We first prove the stability and H 1 -norm error estimate. Then we prove a weak error estimate, which plays a key role in deriving the optimal second-order error estimate in the L 2 norm. The L 2 -norm error estimate is the most important result of this paper. We also briefly examine the discrete maximum principle. Numerical experiments are presented to illustrate the theoretical analysis.

An upwind finite volume method for convection-diffusion equations on rectangular mesh

In this paper, we present an upwind finite volume method to solve the convection-diffusion equations with Dirichlet boundary on rectangular mesh. By utilizing the technique of element-by-element analysis, the stability of the method has been proved and the H1-norm error estimate is presented. Furthermore, we provide the proofs of the maximum principle and L∞-norm error estimate. Finally, some numerical experiments are provided to confirm our theoretical results.

A simple embedded discrete fracture–matrix model for a coupled flow and transport problem in porous media

Accurate simulation of fluid flow and transport in fractured porous media is a key challenge in subsurface reservoir engineering. Due to the high ratio between its length and width, fractures can be modeled as lower dimensional interfaces embedded in the porous rock. We apply a recently developed embedded finite element method (EFEM) for the Darcy problem. This method allows for general fracture geometry, and the fractures may cut the finite element mesh arbitrarily. We present here a velocity model for EFEM and couple the Darcy problem to a transport problem for a passive solute. The main novelties of this work are a locally conservative velocity approximation derived from the EFEM solution, and the development of a lowest order upwind finite volume method for the transport problem. This numerical model is compatible with EFEM in the sense that the same computational mesh may be applied, so that we retain the same flexibility with respect to fracture geometry and meshing. Hence, our coupled solution strategy represents a simple approach in terms of formulation, implementation and meshing. We demonstrate our model by some numerical examples on both synthetic and realistic problems, including a benchmark study for single-phase flow. Despite the simplicity of the method, the results are promising.

An integrated algorithm for hypersonic fluid-thermal-structural numerical simulation

In this paper, a fluid-structural-thermal integrated method is presented based on finite volume method. A unified integral equations system is developed as the control equations for physical process of aero-heating and structural heat transfer. The whole physical field is discretized by using an up-wind finite volume method. To demonstrate its capability, the numerical simulation of Mach 6.47 flow over stainless steel cylinder shows a good agreement with measured values, and this method dynamically simulates the objective physical processes. Thus, the integrated algorithm proves to be efficient and reliable.

A convergent staggered scheme for the variable density incompressible Navier-Stokes equations

In this paper, we analyze a scheme for the time-dependent variable density Navier-Stokes equations. The algorithm is implicit in time, and the space approximation is based on a low-order staggered non-conforming finite element, the so-called Rannacher-Turek element. The convection term in the momentum balance equation is discretized by a finite volume technique, in such a way that a solution obeys a discrete kinetic energy balance, and the mass balance is approximated by an upwind finite volume method. We first show that the scheme preserves the stability properties of the continuous problem (L ∞-estimate for the density, L ∞ (L 2)-and L 2 (H 1)-estimates for the velocity), which yields, by a topological degree technique, the existence of a solution. Then, invoking compactness arguments and passing to the limit in the scheme, we prove that any sequence of solutions (obtained with a sequence of discretizations the space and time step of which tend to zero) converges up to the extraction of a subsequence to a weak solution of the continuous problem.

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

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 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.

A Finite Volume Method for Modeling Shallow Flows with Wet-Dry Fronts on Adaptive Cartesian Grids

A second-order accurate, Godunov-type upwind finite volume method on dynamic refinement grids is developed in this paper for solving shallow-water equations. The advantage of this grid system is that no data structure is needed to store the neighbor information, since neighbors are directly specified by simple algebraic relationships. The key ingredient of the scheme is the use of the prebalanced shallow-water equations together with a simple but effective method to track the wet/dry fronts. In addition, a second-order spatial accuracy in space and time is achieved using a two-step unsplit MUSCL-Hancock method and a weighted surface-depth gradient method (WSDM) which considers the local Froude number is proposed for water depths reconstruction. The friction terms are solved by a semi-implicit scheme that can effectively prevent computational instability from small depths and does not invert the direction of velocity components. Several benchmark tests and a dam-break flooding simulation over real topography cases are used for model testing and validation. Results show that the proposed model is accurate and robust and has advantages when it is applied to simulate flow with local complex topographic features or flow conditions and thus has bright prospects of field-scale application.