Non-staggered Mesh Research Articles

NUMERICAL SIMULATION OF THREE-DIMENSIONAL INCOMPRESSIBLE FLOW BY A NEW FORMULATION

A new mathematical formulation, called the pseudovorticity–velocity formulation, of the three-dimensional incompressible Navier–Stokes equations is presented as an alternative to the vorticity–velocity approach. For the model lid-driven cavity flow problem in two and three dimensions, combined with an explicit mixed spectral /finite different numerical scheme the proposed formulation is found to be efficient and very accurate as compared with the results available in the literature. In particular, the simulation results demonstrate an attractive feature of the present formulation compared with the vorticity–velocity approach, namely that the divergence-free condition of the velocity field can always be achieved on a non-staggered mesh.

Ellipticity, accuracy, and convergence of the discrete Navier-Stokes equations

The introduction into the continuity equation of additional terms to recover grid-scale ellipticity, for the Navier-Stokes equations discretised on a non-staggered mesh, results in an increase in the discretisation error. The introduced error is a combination of the additional truncation error and a false source resulting from the inconsistent construction of the conservation equations used in the finite volume scheme considered. The false source error component is removed by constructing the conservation terms consistently, while the additional truncation error is shown to be of the same order as the leading order truncation error associated with the unmodified equations. A method of reducing the magnitude of the additional terms, thereby reducing the additional error, is considered. It is shown that although this does reduce the magnitude of the error it also reduces the ellipticity of the equations and leads to slower convergence.

Numerical prediction of phase separation in two-phase flow through T-junctions

The paper presents a predictive numerical method which solves the full, three-dimensional, two-fluid model equations for dispersed two-phase flow by control-volume discretization. The methodology incorporates general coordinates, indirect-addressing for easy mapping of non-rectangular domains and is based on the use of non-staggered meshes. Turbulence is modelled either by the standard k— ε turbulence model or by an extension of that model which accounts for void-fraction fluctuations. The method is applied to air-water bubbly flow in a rectangular cross-section T-junction for which experimental data is available. Comparisons of the predictions with measured velocities and phase separation ratios show good agreement. Contours of the volume-fraction reveal the presence of a pocket of high gas concentration at the entrance to the side arm, similar to that actually found in experiments. The effects of interfacial drag model, turbulence model and bubble diameter on the predictions are also investigated.

Calculation of a Confined Axisymmetric Laminar Diffusion Flame Using a Local Grid Refinement Technique

Abstract In this paper a laminar diffusion flame is predicted using a local grid refinement technique and non-staggered meshes. A single coarse grid covers the whole domain and local grid refinement is carried out in the regions of high gradients without changing the basic grid structure. The Navier-Stokes, the energy and the chemical species transport equations are solved and a finite rate chemical kinetics combustion model is employed. Transport equations for the species concentration are solved using a pseudo time splitting technique. The predictions are in good agreement with the experimental measurements for a confined laminar methane flame. The numerical method proved to be robust and accurate.

METHOD FOR CALCULATING INCOMPRESSIBLE VISCOUS FLOWS

A numerical method for solving incompressible viscous flow problems is introduced. The method uses colocated primitive variables on a nonstaggered finite-difference mesh and is applicable to both two- and three-dimensional flow. The principle of the method consists of introducing an amount of intermediate compressibility into a projection scheme to obtain a Helmholtz type expression for a pressure variable. Computational results for a driven cavity flow are presented.

Finite difference solutions of the Navier-Stokes equations on staggered and non-staggered grids

Finite difference schemes for the Navier-Stokes equations are considered, with particular attention to the discretization of the pressure gradient and divergence terms. It is demonstrated that the relation between these terms is the principal reason for the use of the conventional staggered mesh with SIMPLE and other schemes. The important factors in the differencing are that it should lead to an elliptic scheme that satisfies the integrability constraint. The SIMPLE scheme automatically satisfies these two constraints. A non-staggered mesh scheme that also satisfies these constraints has been developed. Comparisons between it and a SIMPLE scheme for natural convection in a cavity indicate that the schemes have equivalent accuracy. For the non-uniform mesh used in this problem the non-staggered scheme is more efficient. It is expected that this increase in efficiency would be greater for adaptive and multi-grid algorithms.

Development of Three-Dimensional Analysis Code for Two-Phase Flow Using Two-Fluid Model.

A numerical calculation method for two-phase flow behavior using a three-dimensional, two-fluid model has been developed. In this model, the control volume formulation and non-staggered mesh scheme were employed instead of the finite difference method and staggered mesh scheme which have been commonly used in numerical analyses for two-phase flow. Moreover, the porous medium approach was implemented for modeling of internal structures in the flow path in order to evaluate their effects with comparatively fewer meshes. Since the control volume formulation is based on the space integrated amount of the conserved quantities, the present method can satisfy the accuracy requirements of conservation equations based on the two-fluid model. In the non-staggered mesh scheme, pressure, density and void fraction are defined at the same location as phase velocities are defined. Calculation accuracy of this scheme can be higher than that of staggered mesh scheme because that does not require the interpolation of the void fraction between neighboring nodes at the location where phase velocities are defined. The present method has been applied to the analyses of two-dimensional two-phase flow behavior in a piping elbow and in a three-dimensional sudden expanded flow area, in order to confirm the calculation capability of phase separation and void fraction distribution caused by the density difference.

Numerical Computation of 3-D Free Surface Flows by N-S Solver and Detection of Sub-breaking

The Navier-Stokes equation is directly solved by the finite-difference method to simulate 3-dimensional viscous flow fields with free-surface around a Wigley model. The body-fitted coordinates, non-staggered mesh system and the third-order upstream differencings for the convection terms are used in the present method. The solution procedure is a time-marching calculation of the velocities, followed by a solution of the pressure field in which the pressure iterations are made to reach convergence. A closer examination of the numerical scheme such as grid schemes and computing domains are made.It is found that either a narrow domain or a coarse grid system suppresses the development of the free-surface elevation. The computational results around the bow agree well with the experimental results even though the Reynolds number is not the same, but the stern flow does not. This may be partially due to the difference in the Reynolds number.The criterion for the sub-breaking is applied to the computed results. It detects the appearance of sub-breaking waves in the course of computation. The results corresponds to the observed free-surface flows.

Numerical Analysis Method for Two-Dimensional Two-Fluid Model Using Control Volume Formulation

A numerical calculation technique for the two-dimensional two-fluid model has been developed. The control volume formulation and non-staggered mesh scheme are employed in order to confirm that solutions satisfy the conservation equations of the two-fluid model. Numerical instability due to the non-staggered mesh is overcome by considering additional flows induced from a local pressure gradient on control volume boundaries. The Godunov method and SIMPLE method are used to estimate the additional flow for compressible and incompressible two-phase flows, respectively. These methods are modified to take the density difference between the phases and the added mass effect into account. The present method has been applied to analyses of two-dimensional two-phase flow discharge from a pipe and phase separation due to gravity in a horizontal circular pipe in order to investigate the capability of dealing with fundamental phenomena of two-dimensional two-phase flow. The present technique enables a small scale compu...