Non-staggered Mesh Research Articles

The influence of prandtl number on flow characteristics of the fume from GMAW welding

GMAW has become one of the most popular welding methods due to its high productivity in the manufacturing industry. However, there has been agrowing concern about the health effects of the fume produced by this welding method, as it may flow into the breathing zone of the welding operator.In this study, the impact of the Pr number on the behavior of the fume flow produced by GMAW was studied numerically using Computational Fluid Dynamics (CFD) simulation. Navier-Stokes and energy equations in two-dimensional axisymmetric coordinates were used to establish the numerical model for the GMAW fume flow.A transient finite-volume method with non-staggered mesh was applied to solve the numerical model. The numerical simulations were run 2.0 ≤ Pr ≤ 15.0 with other parameters kept constant at Fr = 5.5, Re = 100, and H/X0 = 10. The time series of the spreading distance plotted at various Prnumbers show different types of distances at different stages of the fume flow, the initial maximum, the maximum, and the final distances. When the Pr number is increased from 2.0 to 15.0.the gap between the initial maximum and the maximum distances decreases while the gap between the initial maximum and the final distances decreases.A nonlinear relationship was observed when the initial maximum and final distance data were plotted against the Pr number. The best fit for the initial maximum and the final distance data were obtained using the power of -1/2 and -2/3.

Numerical assessment of finite volume schemes for incompressible viscous flows on non-staggered meshes

Numerical assessment of finite volume schemes for incompressible viscous flows on non-staggered meshes

Analyses of Effects of Cutting Parameters on Cutting Edge Temperature Using Inverse Heat Conduction Technique

During machining energy is transformed into heat due to plastic deformation of the workpiece surface and friction between tool and workpiece. High temperatures are generated in the region of the cutting edge, which have a very important influence on wear rate of the cutting tool and on tool life. This work proposes the estimation of heat flux at the chip-tool interface using inverse techniques. Factors which influence the temperature distribution at the AISI M32C high speed steel tool rake face during machining of a ABNT 12L14 steel workpiece were also investigated. The temperature distribution was predicted using finite volume elements. A transient 3D numerical code using irregular and nonstaggered mesh was developed to solve the nonlinear heat diffusion equation. To validate the software, experimental tests were made. The inverse problem was solved using the function specification method. Heat fluxes at the tool-workpiece interface were estimated using inverse problems techniques and experimental temperatures. Tests were performed to study the effect of cutting parameters on cutting edge temperature. The results were compared with those of the tool-work thermocouple technique and a fair agreement was obtained.

Simulation of Three-dimensional Complex Flows in Injection Molding Using Immersed Boundary Method

In this paper, an immersed boundary method (IBM) has been developed to simulate three-dimensional (3D) complex flows in the injection molding process, in which the irregular boundary of mould is treated by a level set function. The melt front (melt-air interface) is captured and treated using the coupled level set and volume of fluid (CLSVOF) method. The finite volume method on the nonstaggered meshes is implemented to solve the governing equations, and the melt filling process is simulated in a rectangular mould with both thick- and thin-wall sections. The numerical result shows good agreement with the available data. Finally, the melt filling processes in three different moulds, i.e., ring-shaped channels with concentric, eccentric and thin cylindrical insets, are investigated respectively. It is found that the so-called “race-tracking effect” and air traps phenomena could be observed successfully, which demonstrates the applicability of IBM to complex flows in injection molding.

A hybrid vertex-centered finite volume/element method for viscous incompressible flows on non-staggered unstructured meshes

This paper proposes a hybrid vertex-centered finite volume/finite element method for solution of the two dimensional (2D) incompressible Navier-Stokes equations on unstructured grids. An incremental pressure fractional step method is adopted to handle the velocity-pressure coupling. The velocity and the pressure are collocated at the node of the vertex-centered control volume which is formed by joining the centroid of cells sharing the common vertex. For the temporal integration of the momentum equations, an implicit second-order scheme is utilized to enhance the computational stability and eliminate the time step limit due to the diffusion term. The momentum equations are discretized by the vertex-centered finite volume method (FVM) and the pressure Poisson equation is solved by the Galerkin finite element method (FEM). The momentum interpolation is used to damp out the spurious pressure wiggles. The test case with analytical solutions demonstrates second-order accuracy of the current hybrid scheme in time and space for both velocity and pressure. The classic test cases, the lid-driven cavity flow, the skew cavity flow and the backward-facing step flow, show that numerical results are in good agreement with the published benchmark solutions.

Numerical model of wave-current interaction based on unstructured quadtree grid for nearshore coastal waters

According to the characteristics of complex topography, hydrology, sediment transport in coastal waters, the aim of this paper is to develop an implicit, depth-averaged 2-D numerical mode of coupling current and wave for coastal area. The hydrodynamic model is the two dimensional depth averaged non-linear shallow water equations with an unstructured non-staggered and multiple-level quadtree rectangular mesh with local refinement for important region. In this non-staggered system, primary variables u -, v -velocity, and water level are stored on the same set of grid points in order to simplify the program code, and fluxes at cell faces are determined using the Rhie and Chow’s momentum interpolation method to avoid spurious checkerboard oscillations. The discretized algebraic equations are solved iteratively using the GMRES method with ILUT preconditioning for speeding the computation. The Wave model is a two-dimensional spectral wave action model of taking into account the effect of wave breaking, shoaling, refraction, diffraction, wave-current interaction, bottom friction. The wave model readily provides the radiation stresses for the hydrodynamic model. The coupling model has been tested against measurement data for steady flow around a spur-dyke in a laboratory flume and tidal flows in Gironde Estuary, France and Grays Harbor, USA. The model reasonably well reproduces the temporal and spatial variations of water level, current speed and wave height observed in the measurements. It shows that the coupling current and wave model can simulate accurately the questions of tide and wave for complex domain, the modeling approach presented herein should be useful in engineering application and environmental protection for coastal area.

A Godunov-Type Scheme for Atmospheric Flows on Unstructured Grids: Euler and Navier-Stokes Equations

In recent years there has been a growing interest in using Godunov-type methods for atmospheric flow problems. Godunov's unique approach to numerical modeling of fluid flow is characterized by introducing physical reasoning in the development of the numerical scheme (van Leer, 1999). The construction of the scheme itself is based upon the physical phenomenon described by the equation sets. These finite volume discretizations are conservative and have the ability to resolve regions of steep gradients accurately, thus avoiding dispersion errors in the solution. Positivity of scalars (an important factor when considering the transport of microphysical quantities) is also guaranteed by applying the total variation diminishing condition appropriately. This paper describes the implementation of a Godunov-type finite volume scheme based on unstructured adaptive grids for simulating flows on the meso-, micro- and urban-scales. The Harten-Lax-van Leer-Contact (HLLC) approximate Riemann solver used to calculate the Godunov fluxes is described in detail. The higher-order spatial accuracy is achieved via gradient reconstruction techniques after van Leer and the total variation diminishing condition is enforced with the aid of slope-limiters. A multi-stage explicit Runge-Kutta time marching scheme is used for maintaining higher-order accuracy in time. The scheme is conservative and exhibits minimal numerical dispersion and diffusion. The subgrid scale diffusion in the model is parameterized via the Smagorinsky-Lilly turbulence closure. The scheme uses a non-staggered mesh arrangement of variables (all quantities are cell-centered) and requires no explicit filtering for stability. A comparison with exact solutions shows that the scheme can resolve the different types of wave structures admitted by the atmospheric flow equation set. A qualitative evaluation for an idealized test case of convection in a neutral atmosphere is also presented. The scheme was able to simulate the onset of Kelvin-Helmholtz type instability and shows promise in simulating atmospheric flows characterized by sharp gradients without using explicit filtering for numerical stability.

A numerical model for radiofrequency heating of sloshing ions in a mirror trap

A newly developed numerical model calculating the distribution and damping of radiofrequency fields by sloshing ions is presented. The model solves time-harmonic Maxwell's equations written in terms of the electric field. It uses a two-dimensional grid and a Fourier series in the third coordinate and is based on a non-staggered mesh not aligned along the steady magnetic field. The numerical stability of the scheme is discussed, and the convergence analysis is presented.

WENO schemes based on upwind and centred TVD fluxes

In this paper we propose to use second-order TVD fluxes, instead of first-order monotone fluxes, in the framework of finite-volume weighted essentially non-oscillatory (WENO) schemes. We call the new improved schemes the WENO-TVD schemes. They include both upwind and centred schemes on non-staggered meshes. Numerical results suggest that our schemes are superior to the original schemes used with first-order monotone fluxes. This is especially so for long time evolution problems containing both smooth and non-smooth features.

Numerical study of eccentric Couette?Taylor flows and effect of eccentricity on flow patterns

In this study, the differential quadrature (DQ) method was used to simulate the eccentric Couette–Taylor vortex flow in an annulus between two eccentric cylinders with rotating inner cylinder and stationary outer cylinder. An approach combining the SIMPLE (semi-implicit method for pressure-linked equations) and DQ discretization on a non-staggered mesh was proposed to solve the time-dependent, three-dimensional incompressible Navier–Stokes equations in the primitive variable form. The eccentric steady Couette–Taylor flow patterns were obtained from the solution of three-dimensional Navier–Stokes equations. The reported numerical results for steady Couette flow were compared with those from Chou [1], and San and Szeri [2]. Very good agreement was achieved. For steady eccentric Taylor vortex flow, detailed flow patterns were obtained and analyzed. The effect of eccentricity on the eccentric Taylor vortex flow pattern was also studied.

Efficient algorithm on a nonstaggered mesh for simulating Rayleigh-Bénard convection in a box.

An efficient semi-implicit second-order-accurate finite-difference method is described for studying incompressible Rayleigh-Bénard convection in a box, with sidewalls that are periodic, thermally insulated, or thermally conducting. Operator-splitting and a projection method reduce the algorithm at each time step to the solution of four Helmholtz equations and one Poisson equation, and these are solved by fast direct methods. The method is numerically stable even though all field values are placed on a single nonstaggered mesh commensurate with the boundaries. The efficiency and accuracy of the method are characterized for several representative convection problems.

Development of a new simulation method of mold filling based on a body-fitted coordinate system

A new model has been developed for the simulation of mold filling in curved-shape mold cavities. In the present method, the SIMPLE scheme was adopted to solve the momentum transport and the VOF (Volume of Fluid) method to trace the free surface in mold filling processes. In order to improve the solution accuracy in modeling mold filling with curved-shape and thin-walled mold cavities, the body-fitted-coordinate (BFC) concept, which was known as the most effective method to predict a flow field in a curved-shape cavity, was adopted. The governing equations for fluid flow and heat transfer were transformed based on the BFC concept. The non-staggered mesh and the momentum interpolation method were used which are essential for the BFC method. In addition, the standard VOF method was modified for the treatment of free surface in the BFC system. The standard DAFA (donor and acceptor flux approximation) method was also revised as a suitable form to the BFC system. In order to verify the present SIMPLE-BFC-VOF method, several examples on mold filling problems having curved-shape mold cavities were simulated, and the results were compared with the experimental results and other simulation methods. It is concluded that the present method can be used as an effective simulation method for the simulation of mold filling in thin-walled and curved-shape mold cavities.

A Highly Accurate Technique for the Treatment of Flow Equations at the Polar Axis in Cylindrical Coordinates Using Series Expansions

Numerical methods for solving the flow equations in cylindrical or spherical coordinates should be able to capture the behavior of the exact solution near the regions where the particular form of the governing equations is singular. In this work we focus on the treatment of these numerical singularities for finite-difference methods by reinterpreting the regularity conditions developed in the context of pseudo-spectral methods. A generally applicable numerical method for treating the singularities present at the polar axis, when nonaxisymmetric flows are solved in cylindrical coordinates using highly accurate finite-difference schemes (e.g., Pade schemes) on nonstaggered grids, is presented. Governing equations for the flow at the polar axis are derived using series expansions near r=0. The only information needed to calculate the coefficients in these equations are the values of the flow variables and their radial derivatives at the previous iteration (or time) level. These derivatives, which are multivalued at the polar axis, are calculated without dropping the accuracy of the numerical method using a mapping of the flow domain from (0, R)*(0, 2π) to (− R, R)*(0, π), where R is the radius of the computational domain. This allows the radial derivatives to be evaluated using high-order differencing schemes (e.g., compact schemes) at points located on the polar axis. The accuracy of the method is checked by comparison with the theoretical solution corresponding to a circular compressible forced jet in the regime of linear growth. The proposed technique is illustrated by results from simulations of laminar-forced jets and turbulent compressible jets using large eddy simulation (LES) methods. In terms of the general robustness of the numerical method and smoothness of the solution close to the polar axis, the present results compare very favorably to similar calculations in which the equations are solved in Cartesian coordinates at the polar a xis, or in which the singularity is removed by employing a staggered mesh in the radial direction without a mesh point at r=0, following the method proposed recently by Mohseni and Colonius [1]. Extension of the method described here for incompressible flows or for any other set of equations that is solved on a nonstaggered mesh in cylindrical or spherical coordinates with finite-difference schemes of various levels of accuracy is immediate.

Analysis of hybrid algorithms for the Navier–Stokes equations with respect to hydrodynamic stability theory

AbstractHybrid three‐dimensional algorithms for the numerical integration of the incompressible Navier–Stokes equations are analyzed with respect to hydrodynamic stability in both linear and nonlinear fields. The computational schemes are mixed—spectral and finite differences—and are applied to the case of the channel flow driven by constant pressure gradient; time marching is handled with the fractional step method. Different formulations—fully explicit convective term, partially and fully implicit viscous term combined with uniform, stretched, staggered and non‐staggered meshes, x‐velocity splitted and non‐splitted in average and perturbation component – are analyzed by monitoring the evolution in time of both small and finite amplitude perturbations of the mean flow. The results in the linear field are compared with correspondent solutions of the Orr–Sommerfeld equation; in the nonlinear field, the comparison is made with results obtained by other authors. Copyright © 2002 John Wiley & Sons, Ltd.

Development of a New Simulation Method of Mold Filling Based on a Body-fitted Coordinate System.

A new numerical model has been developed for the simulation of mold filling in curved-shape mold cavities. In the present method, the SIMPLE scheme was adopted to solve the momentum transport and the VOF (Volume of Fluid) method to trace the free surface in mold filling processes. In order to improve the solution accuracy in modeling mold filling with curved-shape and thin-walled mold cavities, the body-fitted-coordinate (BFC) concept, known as the most effective method to predict a flow field in a curved-shape cavity, was adopted. The governing equations for fluid flow were transformed based on the BFC concept. The non-staggered mesh and the momentum interpolation method were used which are essential for the BFC method. In addition, the standard VOF method was modified for the treatment of free surface in the BFC system. The standard DAFA (donor and acceptor flux approximation) method was also revised as a suitable form to the BFC system. In order to verify the present SIMPLE-BFC-VOF method, several examples on mold filling problems having curved-shape mold cavities were simulated, and the results were compared with the experimental results and other simulation methods. It is concluded that the present method can be used as an effective simulation method for the simulation of mold filling in thin-walled and curved-shape mold cavities.

INCOMPRESSIBLE FLOW AND HEAT TRANSFER COMPUTATIONS USING A CONTINUOUS PRESSURE EQUATION AND NONSTAGGERED GRIDS

A pressure-based algorithm for incompressible flows is presented. The algorithm employs a finite-volume discretization in general curvilinear coordinates on a nonstaggered mesh. This approach is derived from a finite-element algorithm, and is here extended to the finite-volume/finite-difference context. The algorithm can be classified as a SIMPLE-like sequential method, and is validated in two classical test cases: the lid-driven cavity and the differentially heated cavity problems. Good results, with no pressure checkerboarding, are achieved up to Reynolds numbers Re = 104 and Rayleigh numbers Ra = 108 , respectively.

Direct numerical simulation of transient performance in a rectangular closure with a non-staggered mesh scheme

Numerical simulations were obtained with a finite-volume scheme on a non-staggered mesh when hot or cold water enters into a rectangular closure and begins to develop duty cycles in it.

Development of a Two-Dimensional Hydrogen Detonation Analysis Code Devised to Reduce the Numerical Diffusion

Evaluation of thermal and mechanical loads on the containment vessel due to combustion of flammable gases is essential for addressing the important issues of safety analyses of nuclear power reactors under hypothesized severe accident conditions. In order to predict flammable gas detonation which is the most severe type of combustion, a two-dimensional numerial analysis code COMA has been developed using the explicit finite volume method and non-staggered mesh scheme. A new numerically stable transport calculation method has been devised to facilitate treatment of a large pressure discontinuity at the wave front with the minimum numerical diffusion. The test calculation of detonation propagation in a square cavity surrounded by walls showed that the calculated propagation velocity agreed well with the theoretical velocity by Chapman and Jouguet, and the wave reflection at rigid walls and interaction of detonation waves could be analyzed. The method was then validated through a comparison study with the hydrogen detonation tests in a cylindrical tube performed by NUPEC and BNL. The calculated results agreed with observed overall effects of hydrogen concentration and initial temperature on detonation wave propagation velocities.

Development of a Two-Dimensional Hydrogen Detonation Analysis Code Devised to Reduce the Numerical Diffusion

Evaluation of thermal and mechanical loads on the containment vessel due to combustion of flammable gases is essential for addressing the important issues of safety analyses of nuclear power reactors under hypothesized severe accident conditions. In order to predict flammable gas detonation which is the most severe type of combustion, a two-dimensional numerial analysis code COMA has been developed using the explicit finite volume method and non-staggered mesh scheme. A new numerically stable transport calculation method has been devised to facilitate treatment of a large pressure discontinuity at the wave front with the minimum numerical diffusion. The test calculation of detonation propagation in a square cavity surrounded by walls showed that the calculated propagation velocity agreed well with the theoretical velocity by Chapman and Jouguet, and the wave reflection at rigid walls and interaction of detonation waves could be analyzed. The method was then validated through a comparison study with the hy...

Finite difference solutions of vortex generation and shedding from a circular cylinder in oscillatory plus mean flow

A finite difference scheme using a zonal non-staggered mesh is employed to compute the two-dimensional flow around a circular cylinder in a combined unidirectional and oscillating flow. The computations were carried out for a Reynolds number range of up to 5000 and a Keulegan-Carpenter number range of up to 25. For such flows, the numerical solution successfully illustrated the eddy shedding and the impact of existing eddies on the flow and the shedding process.