Semi-implicit Finite Volume Research Articles

Studies on Double Diffusive Convection and Macro-segregation During Solidification of an Al-Alloy Based on Macro–Micro Model

In the present work, a study on the double diffusive convection and the macro-segregation during solidification of an Al-alloy (A356) is considered based on the macro–micro model. The model considers a volume average single-domain approach to represent all the variables and properties as continuum in the entire domain where the transport phenomena during solidification are represented by mass, momentum, energy and species conservation equations. The evolution of solid during solidification, in the model, is predicted based on the microscopic phenomena, i.e. considering the nucleation and the growth of the nuclei. A semi-implicit finite volume method is adopted to discretize the governing equations and the discretized linear simultaneous equations are solved based on the SIMPLER and the TDMA algorithms. The simulation involves prediction of temperature, velocity and species in the computation domain during solidification of the alloy. A parametric study is also considered.

A coastal ocean model of semi-implicit finite volume unstructured grid

A two-dimensional coastal ocean model based on unstructured C-grid is built, in which the momentum equation is discretized on the faces of each cell, and the continuity equation is discretized on the cell. The model is discretized by semi-implicit finite volume method, in that the free surface is semi-implicit and the bottom friction is implicit, thereby removing stability limitations associated with the surface gravity wave and friction. The remaining terms in the momentum equations are discretized explicitly by integral finite volume method and second-order Adams-Bashforth method. Tidal flow in the polar quadrant with known analytic solution is employed to test the proposed model. Finally, the performance of the present model to simulate tidal flow in a geometrically complex domain is examined by simulation of tidal currents in the Pearl River Estuary.

FVCOM2D: A Two Dimensional Semi-Implicit Finite Volume Free-Surface Ocean Model

A two dimensional semi-implicit finite volume free-surface ocean model(FVFOM2D) based on unstructured C-grid is built, in which the momentum equation is discretized on the faces of each cell, the continuity equation is discretized on the cell. The model is discretized by semi-implicit finite volume method, in that the free-surface is semi-implicit and the bottom friction is implicit, thereby removing stability limitations associated with the surface gravity wave and friction. The remaining terms in the momentum equations are discretized explicitly by integral finite volume method and second order Adams-Bashforth method. The performance of the present model to simulate tidal flow in a geometrically complex domain is examined by simulation of tidal currents in Pear River Estuary.

Finite volume solution of heat and moisture transfer through three-dimensional textile materials

This paper focuses on the numerical study of heat and moisture transfer in three-dimensional clothing assemblies, based on a multi-component and multiphase flow model, which includes heat/moisture convection and conduction/diffusion as well as phase change (in the form of the condensation/evaporation and vapor absorption by fiber). A class of flux type boundary conditions is used for the clothing assemblies. To maintain physical conservations, a splitting semi-implicit finite volume method on a structured hexahedron mesh is proposed for solving the system of nonlinear convection–diffusion–reaction equations, in which the calculation of liquid water content absorbed by fiber is decoupled from the rest of the computation. Four types of clothing assemblies are investigated and comparisons with experimental measurements are also presented.

A numerical study of air–vapor–heat transport through textile materials with a moving interface

This paper focuses on the numerical study of heat and moisture transfer in clothing assemblies, which is described by a multi-component and multiphase air–vapor–heat flow with a moving interface. A splitting semi-implicit finite volume method is applied for the system of nonlinear parabolic equations and an implicit Euler scheme is used for the interface equation. In terms of classical Dirichlet to Neumann map, the implicit system can be solved directly and no iteration is needed. Two types of clothing assemblies are investigated and the comparison with experimental measurements is also presented.

High order accurate semi-implicit WENO schemes for hyperbolic balance laws

In this paper we propose a family of well-balanced semi-implicit numerical schemes for hyperbolic conservation and balance laws. The basic idea of the proposed schemes lies in the combination of the finite volume WENO discretization with Roe’s solver and the strong stability preserving (SSP) time integration methods, which ensure the stability properties of the considered schemes [S. Gottlieb, C.-W. Shu, E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Rev. 43 (2001) 89–112]. While standard WENO schemes typically use explicit time integration methods, in this paper we are combining WENO spatial discretization with optimal SSP singly diagonally implicit (SDIRK) methods developed in [L. Ferracina, M.N. Spijker, Strong stability of singly diagonally implicit Runge–Kutta methods, Appl. Numer. Math. 58 (2008) 1675–1686]. In this way the implicit WENO numerical schemes are obtained. In order to reduce the computational effort, the implicit part of the numerical scheme is linearized in time by taking into account the complete WENO reconstruction procedure. With the proposed linearization the new semi-implicit finite volume WENO schemes are designed. A detailed numerical investigation of the proposed numerical schemes is presented in the paper. More precisely, schemes are tested on one-dimensional linear scalar equation and on non-linear conservation law systems. Furthermore, well-balanced semi-implicit WENO schemes for balance laws with geometrical source terms are defined. Such schemes are then applied to the open channel flow equations. We prove that the defined numerical schemes maintain steady state solution of still water. The application of the new schemes to different open channel flow examples is shown.

THREE-DIMENSIONAL NUMERICAL MODEL FOR SIMULATING THE ACCUMULATION PROCESS OF IMMERSED TUBE TANK OF HMZ BRIDGE

In this paper, the development and implementation of a three-dimensional, numerical sediment transport model, which is based on staggered C-unstructured grids in the horizontal direction and Z-level grids in the vertical direction, is delineated. The three dimensional model is discretized by semi-implicit finite volume method, in that the free-surface and vertical diffusion are semi-implicit, thereby removing stability limitations associated with the surface gravity wave and vertical diffusion terms. The remaining terms in the momentum equations are discretized explicitly by integral method. The model is closed physically and mathematically using the Mellor and Yamada level-2.5 turbulent closure submodel. The numerical model is used for simulation accumulation process of immersed tube tank of HMZ (Hong Kong-Macau-Zhuhai) bridge. The model is calibrated and its performance extensively assessed against on-site experiment.

Transient flow simulation of municipal gas pipelines and networks using semi implicit finite volume method

An efficient transient flow simulation for gas pipelines and networks is performed. The proposed transient flow simulation is based on the transfer function models and finite volume method. The equivalent transfer functions of the nonlinear governing equations are derived for different boundary condition types. To verify the accuracy of the proposed simulation, the results obtained are compared with those of the experiments. The effect of the flow inertia is considered in this simulation with discretisation by TVD scheme. The accuracy of the proposed method is discussed for two test cases. It is shown that the proposed simulation has a sufficient accuracy.

Instability in leapfrog and forward–backward schemes: Part II: Numerical simulations of dam break

Following Sun’s approach [17], Shuman smoothing instead of conventional diffusion terms is used in a simple two-time step semi-implicit finite volume scheme to simulate dam break. When the Courant number is less than one, the absolute value of amplification factor of the 1D linearized shallow-water equations is 1 in this new scheme. Compared with the characteristic-based semi-Lagrangian schemes and the Riemann solver, this scheme produces excellent results of free water depth and speed of the shock. Numerical simulations show that the water inside the dam initially moves away radially until water almost depletes near the center; then the water moves back to the center and forms a vertical water column there. This paper proves that Shuman smoothing can be used not only in the linearized shallow-water equations discussed in Sun [17] but also in the nonlinear wave equations to control instability around shocks.

A Level Set Formulation of Geodesic Curvature Flow on Simplicial Surfaces

Curvature flow (planar geometric heat flow) has been extensively applied to image processing, computer vision, and material science. To extend the numerical schemes and algorithms of this flow on surfaces is very significant for corresponding motions of curves and images defined on surfaces. In this work, we are interested in the geodesic curvature flow over triangulated surfaces using a level set formulation. First, we present the geodesic curvature flow equation on general smooth manifolds based on an energy minimization of curves. The equation is then discretized by a semi-implicit finite volume method (FVM). For convenience of description, we call the discretized geodesic curvature flow as dGCF. The existence and uniqueness of dGCF are discussed. The regularization behavior of dGCF is also studied. Finally, we apply our dGCF to three problems: the closed-curve evolution on manifolds, the discrete scale-space construction, and the edge detection of images painted on triangulated surfaces. Our method works for compact triangular meshes of arbitrary geometry and topology, as long as there are no degenerate triangles. The implementation of the method is also simple.

Correlation for Nusselt number in pure magnetic convection ferrofluid flow in a square cavity by a numerical investigation

Magnetic convection heat transfer in a two-dimensional square cavity induced by magnetic field gradient is investigated numerically using a semi-implicit finite volume method. The side walls of the cavity are heated with different temperatures, the top and bottom walls are isolated, and a permanent magnet is located near the bottom wall. Thermal buoyancy-induced flow is neglected due to the nongravity condition on the plane of the cavity. Conditions for the different values of non-dimensional variables in a variety of ferrofluid properties and magnetic field parameters are studied. Based on this numerical analysis, a general correlation for the overall Nusselt number on the side walls is introduced for a wide range of effective parameters. Results showed that maximum error produced by use of this correlation is about 6 percent.

Error estimates of the finite volume scheme for the nonlinear tensor-driven anisotropic diffusion

The paper deals with an error analysis of the semi-implicit diamond-cell finite volume scheme, introduced in [O. Drblíková, K. Mikula, Convergence analysis of finite volume scheme for nonlinear tensor anisotropic diffusion in image processing, SIAM J. Numer. Anal. 46 (1) (2007) 37–60], for solving the nonlinear tensor-driven anisotropic diffusion. First we present the finite volume scheme and its basic properties. Then the error estimate analysis is presented, where the piecewise constant approximation given by the finite volume scheme is compared with the weak solution to the problem. We proved that the error of the approximate solution in L 2 -norm is of order h, where h is a spatial resolution step under the natural relation k ≈ h 2 , where k is a time discretization step. The numerical results devoted to image processing applications are also given.

Numerical simulation of a stratigraphic model

A multi-lithology diffusive stratigraphic model is considered, which simulates at large scales in space and time the infill of sedimentary basins governed by the interaction between tectonics displacements, eustatic variations, sediment supply, and sediment transport laws. The model accounts for the mass conservation of each sediment lithology resulting in a mixed parabolic, hyperbolic system of partial differential equations (PDEs) for the lithology concentrations and the sediment thickness. It also takes into account a limit on the rock alteration velocity modeled as a unilaterality constraint. To obtain a robust, fast, and accurate simulation, fully and semi-implicit finite volume discre tization schemes are derived for which the existence of stable solutions is proved. Then, the set of nonlinear equations is solved using a Newton algorithm adapted to the unilaterality constraint, and preconditioning strategies are defined for the solution of the linear system at each Newton iteration. They are based on an algebraic approximate decoupling of the sediment thickness and the concentration variables as well as on a proper preconditioning of each variable. These algorithms are studied and compared in terms of robustness, scalability, and efficiency on two real basin test cases.

Convergence Analysis of Finite Volume Scheme for Nonlinear Tensor Anisotropic Diffusion in Image Processing

In this article we design the semiimplicit finite volume scheme for coherence enhancing diffusion in image processing and prove its convergence to the weak solution of the problem. The finite volume methods are natural tools for image processing applications since they use piecewise constant representation of approximate solutions similarly to the structure of digital images. They have been successfully applied in image processing, e.g., for solving the Perona-Malik equation or curvature-driven level set equations, where the nonlinearities are represented by a scalar function dependent on a solution gradient. Design of suitable finite volume schemes for tensor diffusion is a nontrivial task here we present the first such scheme with a convergence proof for the practical nonlinear model used in coherence-enhancing image smoothing. We provide basic information about this type of nonlinear diffusion including a construction of its diffusion tensor, and we derive a semiimplicit finite volume scheme for this nonlinear model with the help of covolume mesh. This method is well known as the diamond-cell method owing to the choice of covolume as a diamond-shaped polygon. Further, we prove a convergence of a discrete solution given by our scheme to the weak solution of the problem. The proof is based on Kolmogorov's compactness theorem and a bounding of a gradient in the tangential direction by using a gradient in the normal direction. Finally computational results illustrated in figures are discussed.

Steady Flow of Power Law Fluids across a Circular Cylinder

AbstractThe momentum equations describing the steady cross‐flow of power law fluids past an unconfined circular cylinder have been solved numerically using a semi‐implicit finite volume method. The numerical results highlighting the roles of Reynolds number and power law index on the global and detailed flow characteristics have been presented over wide ranges of conditions as 5 ≤ Re ≤ 40 and 0.6 ≤ n ≤ 2. The shear‐thinning behaviour (n < 1) of the fluid decreases the size of recirculation zone and also delays the separation; on the other hand, the shear‐thickening fluids (n > 1) show the opposite behaviour. Furthermore, while the wake size shows non‐monotonous variation with the power law index, but it does not seem to influence the values of drag coefficient. The stagnation pressure coefficient and drag coefficient also show a complex dependence on the power law index and Reynolds number. In addition, the pressure coefficient, vorticity and viscosity distributions on the surface of the cylinder have also been presented to gain further physical insights into the detailed flow kinematics.

Semi-implicit finite volume scheme for image processing in 3D cylindrical geometry

Nowadays, 3D echocardiography is a well-known technique in medical diagnosis. Inexpensive echocardiographic acquisition devices are applied to scan 2D slices rotated along a prescribed direction. Then the discrete 3D image information is given on a cylindrical grid. Usually, this original discrete image intensity function is interpolated to a uniform rectangular grid and then numerical schemes for 3D image processing operations (e.g. nonlinear smoothing) in the uniform rectangular geometry are used. However, due to the generally large amount of noise present in echocardiographic images, the interpolation step can yield undesirable results. In this paper, we avoid this step and suggest a 3D finite volume method for image selective smoothing directly in the cylindrical image geometry. Specifically, we study a semi-implicit 3D cylindrical finite volume scheme for solving a Perona-Malik-type nonlinear diffusion equation and apply the scheme to 3D cylindrical echocardiographic images. The L ∞-stability and convergence of the scheme to the weak solution of the regularized Perona-Malik equation is proved.

Semi-implicit finite volume scheme for solving nonlinear diffusion equations in image processing

We propose and prove a convergence of the semi-implicit finite volume approximation scheme for the numerical solution of the modified (in the sense of Catte, Lions, Morel and Coll) Perona–Malik nonlinear image selective smoothing equation (called anisotropic diffusion in the image processing). The proof is based on $L_2$ a-priori estimates and Kolmogorov's compactness theorem. The implementation aspects and computational results are discussed.

3-D numerical analysis of natural convective liquid cooling of a 3×3 heater array in rectangular enclosures

The paper presents a three-dimensional (3-D) computational study of natural convection cooling on a 3-by-3 array of discrete heat sources flush-mounted on one vertical wall of a rectangular enclosure filled with various liquids ( Pr=5, 9, 25 and 130) and cooled by the opposite wall. The non-dimensional governing equations with appropriate boundary conditions are solved in the primitive variable form employing a semi-implicit finite volume formulation. The procedures are validated experimentally and numerically with published results. Computations are performed for a range of modified Rayleigh numbers from 10 4 to 10 8 while the enclosure aspect ratio varies from 1 to 20. The effects of modified Rayleigh number, enclosure aspect ratio and Prandtl number on heat transfer characteristics are investigated. The results reveal that the flow field is complex and the heat transfer from the discrete heaters is not uniform. The row-averaged Nusselt number increases with Rayleigh number in the power of 0.27. It also increases with enclosure aspect ratio in the range of 1–3 and attains the maximum when aspect ratio is greater than ∼3. The heater surface temperature is the highest at the top-row heaters regardless of the modified Rayleigh number and enclosure aspect ratio. The effects of Prandtl number are negligible in the range from 5 to 130.

SEMI-IMPLICIT FINITE VOLUME SHALLOW-WATER FLOW AND SOLUTE TRANSPORT SOLVER WITHk-ɛ TURBULENCE MODEL

A 3D semi-implicit finite volume scheme for shallow-water flow with the hydrostatic pressure assumption has been developed using the σ-co-ordinate system, incorporating a standard κ-e turbulence transport model and variable density solute transport with the Boussinesq approximation for the resulting horizontal pressure gradients. The mesh spacing in the vertical direction varies parabolically to give fine resolution near the bed and free surface to resolve high gradients of velocity, k and e. In this study, wall functions are used at the bed (defined by the bed roughness) and wind stress at the surface is not considered. Surface elevation gradient terms and vertical diffusion terms are handled implicitly and horizontal diffusion and source terms explicitly, including the Boussinesq pressure gradient term due to the horizontal density gradient. The advection terms are handled in explicit (conservative) form using linear upwind interpolation giving second-order accuracy. A fully coupled solution for the flow field is obtained by substituting for velocity in the depth-integrated continuity equation and solving for surface elevation using a conjugate gradient equation solver.

Efficient calculation of chemically reacting flow

A semi-implicit finite volume formulation is used to study flows with chemical reactions. In this formulation the source terms resulting from the chemical reactions are treated implicitly and the resulting system of partial differential equations is solved using two time-stepping schemes. The first is based on the Runge-Kutta method while the second is based on an Adams predictor-corrector method. Results show that improvements in computational efficiency depend to a large extent on the manner in which the source term is treated. Further, analysis and computation indicate that the Runge-Kutta method is more efficient than the Adams methods. Finally, an adaptive time stepping scheme is developed to study problems involving shock ignition. Calculations for a hydrogen-air system agree well with other methods.