A Hybrid Scheme Based on Finite Element/Volume Methods for Two Immiscible Fluid Flows

We have successfully extended our implicit hybrid finite element/volume (FE/FV) solver to flows involving two immiscible fluids. The solver is based on the segregated pressure correction or projection method on staggered unstructured hybrid meshes. An intermediate velocity field is first obtained by solving the momentum equations with the matrix-free implicit cell-centered FV method. The pressure Poisson equation is solved by the node-based Galerkin FE method for an auxiliary variable. The auxiliary variable is used to update the velocity field and the pressure field. The pressure field is carefully updated by taking into account the velocity divergence field. This updating strategy can be rigorously proven to be able to eliminate the unphysical pressure boundary layer and is crucial for the correct temporal convergence rate. Our current staggered-mesh scheme is distinct from other conventional ones in that we store the velocity components at cell centers and the auxiliary variable at vertices. The fluid interface is captured by solving an advection equation for the volume fraction of one of the fluids. The same matrix-free FV method, as the one used for momentum equations, is used to solve the advection equation. We will focus on the interface sharpening strategy to minimize the smearing of the interface over time. We have developed and implemented a global mass conservation algorithm that enforces the conservation of the mass for each fluid.

A Cip/Multi-Moment Finite Volume Method For Shallow Water Equations With Source Terms

A novel finite volume method has been presented to solve the shallow water equations. In addition to the volume-integrated average (VIA) for each mesh cell, the surface-integrated average (SIA) is also treated as the model variable and is independently predicted. The numerical reconstruction is conducted based on both the VIA and the SIA. Different approaches are used to update VIA and SIA separately. The SIA is updated by a semi-Lagrangian scheme in terms of the Riemann invariants of the shallow water equations, while the VIA is computed by a flux-based finite volume formulation and is thus exactly conserved. Numerical oscillation can be effectively avoided through the use of a non-oscillatory interpolation function. The numerical formulations for both SIA and VIA moments maintain exactly the balance between the fluxes and the source terms. 1D and 2D numerical formulations are validated with numerical experiments. Copyright (c) 2007 John Wiley & Sons, Ltd.

3D Finite Volume Scheme for Czochralski Single Crystal Growth

Czochralski (Cz) technique, which is used for growing single crystals, has dominated the production of single crystals for electronic applications. The Cz growth process involves multiple phases, moving interface and three-dimensional behavior. Much has been done to study these phenomena by means of numerical methods as well as experimental observations. A three-dimensional curvilinear finite volume based algorithm has been developed to model the Cz process. A body-fitted transformation based approach is adopted in conjunction with a multizone adaptive grid generation (MAGG) technique to accurately handle the three-dimensional problems of phase-change in irregular geometries with free and moving surfaces. The multizone adaptive model is used to perform a three-dimensional simulation of the Cz growth of silicon single crystals.Since the phase change interface are irregular in shape and they move in response to the solution, accurate treatment of these interfaces is important from numerical accuracy point of view. The multizone adaptive grid generation (MAGG) is the appropriate scheme for this purpose. Another challenge encountered is the moving and periodic boundary conditions, which is essential to the numerical solution of the governing equations. Special treatments are implemented to impose the periodic boundary condition in a particular direction and to determine the internal boundary position and shape varying with the combination of ambient physicochemical transport process and interfacial dynamics. As indicated above that the applications and processes characterized by multi-phase, moving interfaces and irregular shape render the associated physical phenomena three-dimensional and unsteady. Therefore a generalized 3D model rather than a 2D simulation, in which the governing equations are solved in a general non-orthogonal coordinate system, is constructed to describe and capture the features of the growth process. All this has been implemented and validated by using it to model the low pressure Cz growth of silicon. Accuracy of this scheme is demonstrated by agreement of simulation data with available experimental data. Using the quasi-steady state approximation, it is shown that the flow and temperature fields in the melt under certain operating conditions become asymmetric and unsteady even in the absence of extrinsic sources of asymmetry. Asymmetry in the flow and temperature fields, caused by high shear initiated phenomena, affects the interface shape in the azimuthal direction thus results in the thermal stress distribution in the vicinity, which has serious implications from crystal quality point of view.

A multi-moment finite volume formulation for shallow water equations on unstructured mesh

A novel and accurate finite volume method has been presented to solve the shallow water equations on unstructured grid in plane geometry. In addition to the volume integrated average (VIA moment) for each mesh cell, the point values (PV moment) defined on cell boundary are also treated as the model variables. The volume integrated average is updated via a finite volume formulation, and thus is numerically conserved, while the point value is computed by a point-wise Riemann solver. The cell-wise local interpolation reconstruction is built based on both the VIA and the PV moments, which results in a scheme of almost third order accuracy. Efforts have also been made to formulate the source term of the bottom topography in a way to balance the numerical flux function to satisfy the so-called C-property. The proposed numerical model is validated by numerical tests in comparison with other methods reported in the literature. (C) 2010 Elsevier Inc. All rights reserved.

Perturbational Finite Volume Method For The Solution of 2-D Navier-Stokes Equations on Unstructured and Structured Colocated Meshes

Based on the first-order upwind and second-order central type of finite volume( UFV and CFV) scheme, upwind and central type of perturbation finite volume ( UPFV and CPFV) schemes of the Navier-Stokes equations were developed. In PFV method, the mass fluxes of across the cell faces of the control volume (CV) were expanded into power series of the grid spacing and the coefficients of the power series were determined by means of the conservation equation itself. The UPFV and CPFV scheme respectively uses the same nodes and expressions as those of the normal first-order upwind and second-order central scheme, which is apt to programming. The results of numerical experiments about the flow in a lid-driven cavity and the problem of transport of a scalar quantity in a known velocity field show that compared to the first-order UFV and second-order CFV schemes, upwind PFV scheme is higher accuracy and resolution, especially better robustness. The numerical computation to flow in a lid-driven cavity shows that the under-relaxation factor can be arbitrarily selected ranging from 0.3 to 0. 8 and convergence perform excellent with Reynolds number variation from 102 to 104.

Simulation of Free-Surface Flow in a Tank Using the Navier-Stokes Model and Unstructured Finite Volume Method

Modelling free-surface flow has very important applications in many engineering areas such as oil transportation and offshore structures. Current research focuses on the modelling of free surface flow in a tank by solving the Navier-Stokes equation. An unstructured finite volume method is used to discretize the governing equations. The free surface is tracked by dynamically adapting the mesh and making it always surface conforming. A mesh-smoothing scheme based on the spring analogy is also implemented to ensure mesh quality throughout the computaiton. Studies are performed on the sloshing response of a liquid in an elastic container subjected to various excitation frequencies. Further investigations are also carried out on the critical frequency that leads to large deformation of the tank walls. Another numerical simulation involves the free-surface flow past as submerged obstacle placed in the tank to show the flow separation and vortices. All these cases demonstrate the capability of this numerical method in modelling complicated practical problems.

A Multimoment Finite-Volume Shallow-Water Model On The Yin-Yang Overset Spherical Grid

A numerical model for shallow-water equations has been built and tested on the Yin-Yang overset spherical grid. A high-order multimoment finite-volume method is used for the spatial discretization in which two kinds of so-called moments of the physical field [i.e., the volume integrated average ( VIA) and the point value (PV)] are treated as the model variables and updated separately in time. In the present model, the PV is computed by the semi-implicit semi-Lagrangian formulation, whereas the VIA is predicted in time via a flux-based finite-volume method and is numerically conserved on each component grid. The concept of including an extra moment (i.e., the volume-integrated value) to enforce the numerical conservativeness provides a general methodology and applies to the existing semi-implicit semi-Lagrangian formulations. Based on both VIA and PV, the high-order interpolation reconstruction can only be done over a single grid cell, which then minimizes the overlapping zone between the Yin and Yang components and effectively reduces the numerical errors introduced in the interpolation required to communicate the data between the two components. The present model completely gets around the singularity and grid convergence in the polar regions of the conventional longitude-latitude grid. Being an issue demanding further investigation, the high-order interpolation across the overlapping region of the Yin-Yang grid in the current model does not rigorously guarantee the numerical conservativeness. Nevertheless, these numerical tests show that the global conservation error in the present model is negligibly small. The model has competitive accuracy and efficiency.

Shallow Water Model On Cubed-Sphere By Multi-Moment Finite Volume Method

A global numerical model for shallow water flows on the cubed-sphere grid is proposed in this paper. The model is constructed by using the constrained interpolation profile/multi-moment finite volume method (CIP/MM FVM). Two kinds of moments, i.e. the point value (PV) and the volume-integrated average (VIA) are defined and independently updated in the present model by different numerical formulations. The Lax-Friedrichs upwind splitting is used to update the PV moment in terms of a derivative Riemann problem, and a finite volume formulation derived by integrating the governing equations over each mesh element is used to predict the VIA moment. The cubed-sphere grid is applied to get around the polar singularity and to obtain uniform grid spacing for a spherical geometry. Highly localized reconstruction in CIP/MM FVM is well suited for the cubed-sphere grid, especially in dealing with the discontinuity in the coordinates between different patches. The mass conservation is completely achieved over the whole globe. The numerical model has been verified by Williamson's standard test set for shallow water equation model on sphere. The results reveal that the present model is competitive to most existing ones. (C) 2008 Elsevier Inc. All rights reserved.

3D Finite Volume Scheme for Czochralski Crystal Growth

A general three-dimensional model is developed for simulation of the growth process of silicon single crystals by Czochralski technique. The numerical scheme is based on the curvilinear non-orthogonal finite volume discretization. Numerical solutions show that the flow and temperature fields in the melt are asymmetric and unsteady for 8’’ silicon growth. The effects of rotation of crystal on the flow structure are studied. The rotation of crystal forms the Ekman layer in which the temperature gradient along solid/melt surface is small.

Perturbational finite volume scheme for the one-dimensional Navier-Stokes equations

Starting from the second-order finite volume scheme,though numerical value perturbation of the cell facial fluxes, the perturbational finite volume (PFV) scheme of the Navier-Stokes (NS) equations for compressible flow is developed in this paper. The central PFV scheme is used to compute the one-dimensional NS equations with shock wave.Numerical results show that the PFV scheme can obtain essentially non-oscillatory solution.

High Order Multi-Moment Constrained Finite Volume Method. Part I: Basic Formulation

A new high-order finite volume method based on local reconstruction is presented in this paper. The method, so-called the multi-moment constrained finite volume (MCV) method, uses the point values defined within single cell at equally spaced points as the model variables (or unknowns). The time evolution equations used to update the unknowns are derived from a set of constraint conditions imposed on multi kinds of moments, i.e. the cell-averaged value and the point-wise value of the state variable and its derivatives. The finite volume constraint on the cell-average guarantees the numerical conservativeness of the method. Most constraint conditions are imposed on the cell boundaries, where the numerical flux and its derivatives are solved as general Riemann problems. A multi-moment constrained Lagrange interpolation reconstruction for the demanded order of accuracy is constructed over single cell and converts the evolution equations of the moments to those of the unknowns. The presented method provides a general framework to construct efficient schemes of high orders. The basic formulations for hyperbolic conservation laws in 1- and 2D structured grids are detailed with the numerical results of widely used benchmark tests. (C) 2009 Elsevier Inc. All rights reserved.

A global shallow water model using high order multi-moment constrained finite volume method and icosahedral grid

A novel accurate numerical model for shallow water equations on sphere have been developed by implementing the high order multi-moment constrained finite volume (MCV) method on the icosahedral geodesic grid. High order reconstructions are conducted cell-wisely by making use of the point values as the unknowns distributed within each triangular cell element. The time evolution equations to update the unknowns are derived from a set of constrained conditions for two types of moments, i.e. the point values on the cell boundary edges and the cell-integrated average. The numerical conservation is rigorously guaranteed. in the present model, all unknowns or computational variables are point values and no numerical quadrature is involved, which particularly benefits the computational accuracy and efficiency in handling the spherical geometry, such as coordinate transformation and curved surface. Numerical formulations of third and fourth order accuracy are presented in detail. The proposed numerical model has been validated by widely used benchmark tests and competitive results are obtained. The present numerical framework provides a promising and practical base for further development of atmospheric and oceanic general circulation models. (C) 2009 Elsevier Inc. All rights reserved.

The time delay in electrochemical measurements of a finite-volume system

For a sphere electrode enclosed in finite-volume electrolyte, the measured current will deviate from the result predicted by the semi-infinite diffusion theory after some time. By random-walk simulation, we compared this time to the one needed for diffusion layer to reach electrolyte boundary, and revealed a clear signal delay of electrochemical current. Further we presented a quantitative description of this delay time. The simulation results suggested that the semi-infinite diffusion theory can even be applied when the theoretical diffusion layer grows to 1.28 electrolyte thicknesses, with an accuracy better than 0.5%. We attributed this time delay to the molecules' finite propagation velocity. Finally, we discussed how this delay can influence and facilitate the following electrochemical detection towards the nanometer and single-cell scale.

On dispersion-controlled principles for non-oscillatory shock-capturing schemes

The role of dispersions in the numerical solutions of hydrodynamic equation systems has been realized for long time. It is only during the last two decades that extensive studies on the dispersion-controlled dissipative (DCD) schemes were reported. The studies have demonstrated that this kind of the schemes is distinct from conventional dissipation-based schemes in which the dispersion term of the modified equation is not considered in scheme construction to avoid nonphysical oscillation occurring in shock wave simulations. The principle of the dispersion controlled aims at removing nonphysical oscillations by making use of dispersion characteristics instead of adding artificial viscosity to dissipate the oscillation as the conventional schemes do. Research progresses on the dispersion controlled principles are reviewed in this paper, including the exploration of the role of dispersions in numerical simulations, the development of the dispersion-controlled principles, efforts devoted to high-order dispersion-controlled dissipative schemes, the extension to both the finite volume and the finite element methods, scheme verification and solution validation, and comments on several aspects of the schemes from author's viewpoint.

Numerical-Value Perturbation Method with Application of Solving Convective-Diffusion Integral Equation

The convective--diffusion equation is of primary importance in such fields as fluid dynamics and heat transfer hi the numerical methods solving the convective-diffusion equation, the finite volume method can use conveniently diversified grids (structured and unstructured grids) and is suitable for very complex geometry The disadvantage of FV methods compared to the finite difference method is that FV-methods of order higher than second are more difficult to develop in three-dimensional cases. The second-order central scheme (2cs) offers a good compromise among accuracy, simplicity and efficiency, however, it will produce oscillatory solutions when the grid Reynolds numbers are large and then very fine grids are required to obtain accurate solution. The simplest first-order upwind (IUW) scheme satisfies the convective boundedness criteria, however. Its numerical diffusion is large. The power-law scheme, QMCK and second-order upwind (2UW) schemes are also often used in some commercial codes. Their numerical accurate are roughly consistent with that of ZCS. Therefore, it is meaningful to offer higher-accurate three point FV scheme. In this paper, the numerical-value perturbational method suggested by Zhi Gao is used to develop an upwind and mixed FV scheme using any higher-order interpolation and second-order integration approximations, which is called perturbational finite volume (PFV) scheme. The PFV scheme uses the least nodes similar to the standard three-point schemes, namely, the number of the nodes needed equals to unity plus the face-number of the control volume. For instanc6, in the two-dimensional (2-D) case, only four nodes for the triangle grids and five nodes for the Cartesian grids are utilized, respectively. The PFV scheme is applied on a number of 1-D problems, 2~Dand 3-D flow model equations. Comparing with other standard three-point schemes, The PFV scheme has much smaller numerical diffusion than the first-order upwind (IUW) scheme, its numerical accuracy are also higher than the second-order central scheme (2CS), the power-law scheme (PLS), the QUICK scheme and the second-order upwind(ZUW) scheme.