Simplified Marker And Cell Research Articles

MECHANISM OF MODE SELECTION FOR TAYLOR VORTEX FLOW BETWEEN COAXIAL CONICAL ROTATING CYLINDERS

A combined numerical and experimental investigation was conducted to study mode selection in Taylor-vortex flow (TVF) between coaxial conical cylinders (truncated cones), where the inner cylinder rotates and the outer one is at rest. Simulations, using a finite-difference method with simplified marker and cell (SMAC) formulation, confirmed the same dependence of mode selection on the acceleration rate β of the inner conical cylinder as observed experimentally. Different TVF modes were obtained in the Reynolds number (Re) range studied. Three different modes corresponding to six, seven and eight pairs of steady Taylor vortices were obtained when the inner conical cylinder rotation speed was increased linearly at different acceleration rates. The transition diagram obtained by Re-β mapping successfully summarizes the various flow states and modes observed.

Numerical Simulation of Axisymmetric Free Surface Flows

This paper describes an extension of the GENSMAC code for solving two-dimen-sional free surface flows to axisymmetric flows. Like GENSMAC the technique is finite difference based and embodies, but considerably extends, the SMAC (simplified marker and cell) ideas. It incorporates adaptive time stepping and an accurate representation of the free surfaces while at the same time only uses surface particles to define the free surfaces, greatly increasing the computational speed; in addition, it employs a graphic interface with solid modeling techniques to provide enhanced three-dimensional visualization. Various simulations are undertaken to illustrate and validate typical flows. Both G. I. Taylor's viscous jet plunging into a fluid and a liquid drop splashing onto a fluid are simulated. Also, the important industrial application of container filling is illustrated. Finally, a comparison is made with the linear theory of standing waves and the code is validated by a numerical convergence study.

Parallel solution of three-dimensional Marangoni flow in liquid bridges

SUMMARY This paper describes the implementation and performances of a parallel solver for the direct numerical simulation of the three-dimensional and time-dependent Navier‐Stokes equations on distributedmemory, massively parallel computers. The feasibility of this approach to study Marangoni flow instability in half zone liquid bridges is examined. The results indicate that the incompressible, non-linear Navier‐Stokes problem, governing the Marangoni flows behavior, can effectively be parallelized on a distributed memory parallel machine by remapping the distributed data structure. The numerical code is based on a three-dimensional Simplified Marker and Cell (SMAC) primitive variable method applied to a staggered finite difference grid. Using this method, the problem is split into two problems, one parabolic and the other elliptic A parallel algorithm, explicit in time, is utilized to solve the parabolic equations. A parallel multisplitting kernel is introduced for the solution of the pseudo pressure elliptic equation, representing the most time-consuming part of the algorithm. A grid-partition strategy is used in the parallel implementations of both the parabolic equations and the multisplitting elliptic kernel. A Message Passing Interface (MPI) is coded for the boundary conditions; this protocol is portable to different systems supporting this interface for interprocessor communications. Numerical experiments illustrate good numerical properties and parallel efficiency. In particular, good scalability on a large number of processors can be achieved as long as the granularity of the parallel application is not too small. However, increasing the number of processors, the Speed-Up is ever smaller than the ideal linear Speed-Up. The communication timings indicate that complex practical calculations, such as the solutions of the Navier‐Stokes equations for the numerical simulation of the instability of Marangoni flows, can be expected to run on a massively parallel machine with good efficiency. Copyright © 1999 John Wiley & Sons, Ltd.

A simplified approach for the simulation of metal flow in a cylindrical sleeve diecasting cavity

Abstract A cylindrical sleeve is a basic geometry of diecasting shapes. This paper describes the computer simulation of metal flow in a cylindrical sleeve cavity of pressure diecasting, in the cylindrical coordinates system. The effects of the wall thickness and velocity profile between two bounding walls are taken into account. Through variable transformation, the governing equations can be solved by the SMAC (Simplified Marker and Cell) technique in a similar way as for the Cartesian coordinate system. Considerable simplification is realized for the calculation of metal flow in a cylindrical sleeve cavity. The rate of die cavity filling for different wall thickness is predicted.

Internal characteristics and numerical analysis of plunging breaker on a slope

Many investigations about the direct measurements of velocities to clarify the internal mechanism of the breaker have been carried out as a result of recent progress in the measuring techniques. This research attempts to clarify the breaking wave transformation system on a slope by an experiment and numerical analysis. In an experiment, the velocities in the surf zone were measured directly using an electromagnetic current meter, and the space distribution characteristic of the vorticity ω = (∂u/∂y − ∂u/∂x) and the skewness γ = (∂u/∂y + ∂u/∂x) were examined. Also, occurrence situations of the vortices at the time of water mass inrush were measured by video tape recorder (VTR) image processing. However, because the breaker is a violent phenomenon that is entrained with plentiful bubbles, the extent to which we can clarify breaker transformation in experiments is limited. Numerical simulations are substituted for experiments as a method to clarify breaker transformation. In numerical analysis, finite amplitude wave analysis based on the potential theory (non-viscous fluid) is possible before wave breaking; however, the analysis must take into account the viscous fluid after breaking. So, we use the Reynolds equations to develop a numerical simulation system of the breaker transformation on a sloping bottom. The numerical energy dissipation model of the breaker was compared to the experimental results, and a modified Simplified Marker and Cell (SMAC) method is presented. The internal characteristics of the breaker transformation are described using application examples.

Upwinding of Least Square Convection Estimate for Unstructured Grids.

A new method for estimating convection using a incompressible Navier-Stokes solver is presented. A least squares identification method was applied to obtain derivatives around a reference point. Higher order derivatives were also utilized in order to reduce artificial diffusivity caused by firstorder upwinding. An iterative method similar to SMAC (simplified marker and cell) was used for time marching, and least squares estimation was carried out to determine predictors of variables. Therefore, the estimation time can be neglected in the entire calculation. Flow around a cylinder and two-dimensional cavity flow were selected to examine the applicability of the current method. Static pressure distribution and velocity profiles agree fairly well with previously obtained data. However, it was found that artificial viscosity is not negligible, even if higher order derivatives were utilized.

Numerical Calculation of Viscoelastic Flows through Eccentric Abrupt Contraction.

Numerical simulations of viscoelastic flows through an eccentric four-to-one abrupt contraction are carried out using the Giesekus model. The SMAC (Simplified-Marker-and-Cell) method is used to analyze the three-dimensional flows. The velocity profiles along the path line passing through the center of the exit exhibit an overshoot near the entry section, and at high Weissenberg numbers an undershoot follows the overshoot. The magnitude of the stress along the same path line has a peak near the entry section, and its slow relaxation process indicates that a large downstream length is necessary for fully developed stress conditions to exist. The peak is lower than that for the flow through the concentric four-to-one abrupt contraction ; the decrease in the peak amplitude is understood to be due to the distortion of the path line in the eccentric geometry. A corner vortex, the height of which is a maximum at the widest corner, grows as the Weissenberg number increases. Furthermore, the tangential flow toward the widest section inside the vortex is determined.

A simplified marker and cell method for unsteady flows on non‐staggered grids

AbstractThe SMAC (simplified marker and cell) time‐advancing method for solving the unsteady incompressible Navier‐Stokes equations on non‐staggered grids is developed in generalized co‐ordinate systems. The primitive variable formulation uses Cartesian velocities and pressure, all defined at the centre of the control volume, as the dependent variables. A special elliptic flux correction at the faces of the finite volume is utilized in discretizing the continuity equation to suppress pressure oscillations. The test flows considered are a polar cavity flow starting from rest and the flow around a circular cylinder. The numerical results are compared with experimental results and results obtained by the well‐known SIMPLEC and PISO methods. The comparisons show that the elliptic flux correction technique works well in suppressing pressure oscillations and that the SMAC method is more efficient than the SIMPLEC and PISO methods for both steady and unsteady flows.

An Improvement of the Computational Efficiency of the SOLA Method.

The SOLA (solution algorithm) method is one of the most widely used methods for analyzing incompressible flows. However, this method often requires a number of iterations to obtain a converged solution for a practical problem on transient flow. Hence, a theoretical analysis of this method was carried out in the present study to clarify the reason why this method can give converged solutions. It was proven that the SOLA method is identical to the SMAC (simplified marker and cell) method for the interior cells of the computational domain. On the other hand, it is not identical to the SMAC method for the cells adjacent to the boundary cells. Hence, a modified SOLA method which improves computational efficiency was developed by taking into account boundary conditions in the iteration process. It was demonstrated that the modified SOLA method could analyze transient flow more efficiently than the original SOLA.

An Unsteady Implicit SMAC Scheme for Two-Dimensional Incompressible Navier-Stokes Equations.

A finite-difference method based on the SMAC (Simplified Marker and Cell) scheme for analyzing two-dimensional unsteady incompressible viscous flows is developed. The fundamental equations are the incompressible Navier-Stokes equations of contravariant velocities and the elliptic pressure equation in general curvilinear coordinates. With application of the Crank-Nicholson scheme, unsteady flow is calculated iteratively by the Newton method at each time step, and the elliptic pressure equation is solved by the Tschebyscheff SLOR method with alternating the computational directions. Therefore, the elliptic character of incompressible flow is well described. The present implicit scheme is stable under the proper boundary conditions, since spurious error and numerical instabilities can be suppressed by employing the staggered grid and upstream differences such as the modified QUICK scheme. Numerical results for two-dimensional flows through a decelerating cascade at high Reynolds numbers are shown. Some computed results of the surface pressure coefficient are in satisfactory agreement with the experimental data.

Numerical Solution Method for Incompressible Multidimensional Two-Fluid Model. A Method Suitable for the Evaluation of Constitutive Equations.

A number of accurate constitutive equations are required for reliable prediction of multidimensional two-phase flow. However, few constitutive equations have been established due to the lack of (1) simple numerical methods for solving governing equations of multidimensional two-phase flow and (2) reliable experimental data on multidimensional two-phase flow. In the present study, a simple and efficient numerical method for solving a multidimensional incompressible two-fluid model, which is based on the simplified marker and cell (SMAC) method, was proposed to facilitate the development of constitutive equations. This method enables us to calculate multidimensional two-phase transient phenomena with a personal computer. Details of the present method and a comparison of measured and predicted two-dimensional flow patterns of two-dimensional bubbly flow are described.

Numerical Simulation of Glass-Blowing Process.

The objective of this work is to understand the effects of several parameters on the blowing process involved in glass container production. The process was numerically simulated by the Simplified Marker And Cell (SMAC) method, assuming that glass was an incompressible viscous fluid. It was found that the glass-blowing process was strongly affected by the initial temperature and the viscous stress on the glass surface. It was proven, however, that the blowing process was little affected by the energy dissipation due to viscosity and the heat transfer to the mold.

Comparison of the SMAC, PISO and iterative time-advancing schemes for unsteady flows

Calculations of unsteady flows using a simplified marker-and-cell (SMAC), a pressure-implicit splitting of operators (PISO) and an iterative time-advancing (ITA) scheme are presented. A partial differential equation for incremental pressure is used in each time-advancing scheme. Example flows considered are a polar cavity flow starting from rest and self-sustained oscillatory flows over a circular and a square cylinder. For a large time-step size, the SMAC and ITA schemes are more strongly convergent and yield more accurate results than the PISO scheme. The SMAC scheme is the most efficient computationally. For a small time-step size, the three time-advancing schemes yield equally accurate Strouhal numbers. The capability of each time-advancing scheme to accurately resolve unsteady flows is attributed to the use of a new pressure correction algorithm that can strongly enforce the conservation of mass. The numerical results show that the low frequency of the vortex shedding is caused by the growth time of each vortex shed into the wake region.

An Improvement of the Computational Efficiency of the SOLA Method.

The SOLA (solution algorithm) method is one of the most widely used methods for analyzing incompressible viscous flows. However, this method often requires a number of iterations to obtain a converged solution for a practical problem on transient flow. Hence, a theoretical analysis of this method was carried out in the present study to clarify the reason why this method can give converged solutions. It was proved that the SOLA method is identical to the SMAC (simplified marker and cell) method in the interior cells of the computational domain. On the other hand, it is not identical to the SMAC at the cells adjacent to the boundary cells. Hence, a modified SOLA method which improves the computational efficiency was developed by taking into account boundary conditions in the iteration process. It was demonstrated that the modified SOLA could analyze transient flow more efficiently than the original SOLA.

Numerical Analysis of a Flow in a Three-Dimensional Cubic Cavity.

Incompressible viscous flow in a three-dimensional cubic cavity is studied by the finite difference method. The marker and cell (MAC), the simplified MAC (SMAC), and the highly simplified MAC (HSMAC) methods are applied to calculate the pressure field. All spatial derivatives in the Navier-Stokes equation are approximated by central difference, and the Euler forward scheme is used to integrate time-dependent terms. The staggered mesh system is employed in this calculation. As a result, the difference of the solution methods for the pressure in three-dimensional calculation affects pressure distribution and velocity distribution in a cubic cavity, which is different from the effect by two-dimensional calculation.

Modification of the smac method in two dimensions

The Marker-and-Cell (MAC) method, proposed by Harlow and Welch,' is an efficient numerical solution technique for investigating the dynamics of an incompressible fluid. However, the method requires the solution of the Poisson equation for pressure distribution. This disadvantage is removed by the Simplified MAC technique in which the pressure need not be calculated. The basic idea is to separate each calculation cycle into two parts 1. The tentative velocity field 1.1, is calculated by using an arbitrary pressure field. Correct velocity boundary conditions ensure that this tentative velocity field reproduces the vorticity in a correct way.3 However, the mass concentration equation is not satisfied. 2. The potential function C#I is introduced so that the final velocity determined by w = \.?, +grad C#I preserves the vorticity and satisfies the incompressibility condition. The potential function C#I is calculated from the Poisson equation for mass conservation. In two dimensions it is more interesting to use the stream function 4 instead of the potential. Assume the momentum equations without the pressure gradient to be

Numerical method for almost three-dimensional incompressible fluid flow and a simple internal obstacle treatment

Abstract An extension of the Simplified Marker and Cell (SMAC) method for the numerical solution of incompressible flows is used to investigate transient flows that are almost three dimensional. The equations of motion are two dimensional but contain functions that account for the effects of a third dimension. The method that results from this extension may also be used for the simple incorporation of internal obstacles in two-dimensional flows.