Boundary-fitted Coordinate System Research Articles

Evaluation of Several Turbulence Models for Turbulent Flow in Concentric and Eccentric Annuli

Turbulent flow in concentric and eccentric annuli is numerically simulated as part of an investigation aimed at modeling drilled cuttings transport in wellbores. A numerical code is developed to solve the time-averaged momentum equation wherein the Reynolds stresses are modeled using the eddy viscosity approach. A nonorthogonal curvilinear, boundary-fitted coordinate system is used to facilitate the implementation of boundary conditions. Several turbulence models, including a one-layer mixing length model developed as part of this study, a two-layer mixing-length model, and a low Reynolds number, two-equation (k-τ) model are used to simulate turbulent flow in several concentric and eccentric annuli. Performance of these turbulence models is evaluated by comparing numerical predictions to experimental data obtained from several sources. Results show that the proposed one-layer mixing length model performs as well as the two-layer mixing length model and the two-equation model while avoiding some of the difficulties associated with the implementation of these models.

A three‐dimensional hydrodynamic model in curvilinear co‐ordinates with collocated grid

SUMMARY A three-dimensional hydrodynamic model has been developed for turbulent flows with free surface. In the horizontal x‐y-plane, a boundary-fitted curvilinear co-ordinate system is adopted, while in the vertical direction, a s-co-ordinate transformation is used to represent the free surface and bed topography or lower boundary. Using the finite volume method, the convection terms are discretized using Roe’s second-order-accurate scheme. The governing equations are solved in a collocated grid system by a fractional three-step implicit algorithm that has been developed to handle the velocity‐pressure‐depth coupling problem of free surface incompressible fluid flows. The present study is the extension of previous work to three-dimensional turbulent flows. The model has been applied to three test cases. Comparison with available data shows that the model developed is successful, and is valuable to engineering application. © 1998 John Wiley & Sons, Ltd.

Heat and fluid flow in pulsed current GTA weld pool

In this study the heat transfer, fluid flow and phase change of the weld pool in pulsed current gas tungsten arc (GTA) welding were investigated. Transporting phenomena from the welding arc to the base material surface, such as current density, heat flux, arc pressure and shear stress acting on the weld pool surface, were taken from the simulation results of the corresponding welding arc. Various driving forces for the weld pool convection were considered, namely self-induced electromagnetic, surface tension, buoyancy, and impinging plasma arc force. Furthermore, the effect of deformed free surface due to the arc pressure acting on the weld pool surface was considered. Heat and mass transfer equations, including the generalized Navier-Stokes equation and electric transport equation, were solved by a finite difference method. Because the fusion boundary has a curved and unknown shape during welding, a boundary-fitted coordinate system was adopted to precisely describe the boundary for the momentum equation. The numerical model for pulsed current welding was applied to AISI 304 stainless steel and compared with the results of the constant current.

Numerical Analysis of Developing Turbulent Flow in a 180.DEG. -Bent Tube with Roughened Wall.

A numerical analysis has been performed for three dimensional developing turbulent flow in a 180° bent tube with roughened wall by algebraic Reynolds stress model which is adopted to solve anisotropic turbulent structure precisely. To our knowlege, very little has been reported about the numerical investigations of bent tube with roughened wall. In numerical analysis, an algebraic Reynolds stress model in conjunction with boundary-fitted coordinate system is applied to 180° bent tube with roughened wall and it with smooth wall. As a result of this analysis, it is found that calculated results show comparatively a good agreement with the experimental data of time averaged velocity and secondary vectors. In bend tube with roughened wall, the location of maximum streamwise velocity moves more remarkably toward the outer wall than it with smooth wall and the intensity of secondary flow increases. These phenomena are realized by the present method. As for the comparison of turbulent energy, the present model has been found to simulate characteristic features of turbulent energy, but it has a tendency to overpredict its value. Moreover, it is elucidated by using calculated results that the decrease of averaged turbulent intensity over cross-section along streamwise is caused by the gradient of stream wise velocity with respect to bend angle.

Numerical Analysis of Turbulent Flow Developing in a Sinuous Duct of Square Cross Section.

The present study has carried out numerical analysis for turbulent flow developing in a sinuous duct with a square sectioned at Reynolds number of 4×104. The sinuous duct was formed from three 180 degree bends with 80 mm hydraulic diameter and 160 mm mean radius of curvature. Straight ducts of 80 and 30 hydraulic diameter long are attached to the inlet and outlet planes of the sinuous duct, respectively. In calculation, an algebraic Reynolds stress model was adopted in order to predict anisotropic turbulence precisely, and boundary-fitted coordinate system was introduced as the method of coordinate transformation. Calculated results were compared with the experimental data measured by hot-wire velocimeter. As a result of the comparison, it was found that the present method was able to reproduce the main features of streamwise mean-velocity and secondary flow although the agreement was certainly not perfect in all detail. As a great discrepancy, it is pointed out that the calculated intensity of the secondary flow near 90 degree of second and third bend shows greater value than the experimental data. Moreover, the pressure-driven, Reynolds stress-driven and centrifugal forces which generates the secondary flow, are evaluated quantitatively by using momentum transport equation. It is clarifyed that Reynolds stress-driven force also plays an important role near 90 degree of second and third sinuous duct.

EFFECT OF VARIOUS DRIVING FORCES ON HEAT AND MASS TRANSFER IN ARC WELDING

In this study the heat transfer and fluid flow of the molten pool in stationary gas tungsten arc welding using argon shielding gas were investigated. Transporting phenomena from the welding arc to the base material surface, such as current density, heat flux, arc pressure, and shear stress acting on the weld pool surface, were taken from the simulation results of the corresponding welding arc. Various driving forces for the weld pool convection were considered: self-induced electromagnetic, surface tension, buoyancy, and impinging plasma arc forces. Furthermore, the effect of surface depression due to the arc pressure acting on the molten pool surface was considered. Because the fusion boundary has a curved and unknown shape during welding, a boundary-fitted coordinate system was adopted to precisely describe the boundary for the momentum equation. The numerical model was applied to AISI304 stainless steel and compared with the experimental results.

Effects of density change and natural convection on the solidification process of a pure metal

The solidification of an initially superheated metal within a rectangular enclosure is studied numerically through the front-fixing method accounting for the solid-liquid density change and natural convection in the melt. The investigation focuses on the influence of both the volume contraction caused by solid-liquid density change and the natural convection effect in the melt. Difficulties associated with the complex time-dependent solid and liquid domains, the shapes of which are also a part of the solutions, are overcome by employing the boundary-fitted coordinate system. The computed results are validated and are shown in the forms of the transient position of the interface, the temperature distribution, the flow pattern, the Nusselt number and the solidification fraction, with respect to time.

Simulation of a free-surface and seepage face using boundary-fitted coordinate system method

The boundary-fitted coordinate (BFC) system method is applied to simulate steady groundwater seepage with a free-surface and seepage face using the finite-difference method. The BFC system method eliminates the difficulty of fitting finite-difference grids to a changeable free-surface which is not known a priori but will be obtained as part of a solution. Also, grid generation with this approach is simpler than with the finite-element method. At each iterative sweep, the changeable free-surface becomes a part of the boundary-fitted grid lines, making boundary condition implementation easy and accurate. An example problem demonstrating the simulation procedure and numerical results compares very well with the analytical solution.

SPECTRAL SIMULATION OF THERMOCAPILLARY CONVECTION WITH DEFORMABLE FREE SURFACE USING BOUNDARY-FITTED COORDINATES

A Chebyshev collocation spectral method has been developed to investigate thermocapillary and buoyancy-induced flow with a deformable free surface. The flow in a two-dimensional cavity is considered, and the equations of motion are solved in a generalized boundary-fitted coordinate (BFC) system to handle the free-surface deformation. The BFC grid is generated using a finite-difference Poisson solver. The metrics of transformation are evaluated using a spectral approximation for the mapping of the physical grid onto the computational grid. The transformed equations are solved using the influence matrix technique in order to produce a solenoidal flow field. Results from the two-dimensional calculations show that this method can be competitive with finite-element and spectral-element solutions previously reported in the literature.

A mathematical model of gas tungsten arc welding considering the cathode and the free surface of the weld pool

A two-dimensional axisymmetric numerical model, including the influence of the cathode and the free surface of the weld pool, is developed to describe the heat transfer and fluid flow in gas tungsten arc (GTA) welding. In the model, a boundary-fitted coordinate system is adopted to precisely describe the cathode shape and deformed weld-pool surface. The current continuity equation has been solved with the combined arc plasma-cathode system, independent of the assumption of current density distribution on the cathode surface, which was essential in the previous studies of arc plasma. It has been shown that the temperature profile, the current, and the heat flux to the anode show good agreement with the experimental data. Moreover, the current and the heat-flux distributions may be affected by the shape of the cathode and the free surface of the weld pool.

NUMERICAL INVESTIGATION OF COUPLED TURBULENT FLOW, HEAT TRANSFER, AND MACROSCOPIC SOLIDIFICATION IN A VERTICAL TWIN-ROLL THIN-STRIP CASTER

A vertical twin-roll continuous thin-strip casting process for stainless steel has been mathematically modeled. The model takes into account the coupled turbulent flow, heat transfer, and macroscopic solidification aspects of the process. A low-Reynolds-number K-ε turbulence model was used to account for the turbulent effects. The transport equations for the wedge-shaped caster's cavity were solved using a boundary-fitted nanorthogonal coordinate system. A control-volume-based, iterative finite difference scheme on a staggered grid was used to solve the discretized transformed equations. The SIMPLE algorithm was employed to resolve the velocity-pressure coupling in the momentum equations. The parameters examined in this study include the Darcy coefficient and turbulent damping factor for the mushy region, the roll gap, and the inlet nozzle width. The effects of these process parameters on the turbulent flow and temperature fields, the turbulent eddy viscosity distributions, and the extents of the mushy and solidified regions were ascertained. A change of the Darcy coefficient above 800 did not show any significant effect on the results. The three types of liquid-fraction-dependent turbulent damping factor used for modeling the mushy region did not show a measurable effect on the solidified shell thickness but showed a noticeable influence on the velocity and temperature distributions in both liquid and mushy regions. For a fixed width of the nozzle, an increase in the roll gap increased the penetration depth of the plunging inlet jet but decreased the extent of the mushy region. The solidified shell profile was insensitive to the change of the roll gap. For a fixed roll gap, an increase in the width of the nozzle decreased the penetration depth of the nozzle but increased the extent of the mushy region, while the solidified shell thickness remained practically unaffected.

Numerical Modelling Of Wind-induced CirculationIn The Gulf Of Thermaikos Using A Non-orthogonalBoundary-fitted Coordinate System

Boundary-fitted coordinate systems are becoming increasingly popular in computational hydrodynamics since they allow an accurate representation of curved flow boundaries. This paper presents details of a two-dimensional, depth-averaged numerical model which utilises a non-orthogonal coordinate transformation to solve the shallow water equations. The model has been designed to predict wind-induced circulation patterns in semi-enclosed coastal seas of irregular shape. The scheme consists essentially of two modules: the first generates the non-orthogonal grid within the prescribed flow domain using an elliptic grid generation technique, whilst the second solves the transformed non-linear shallow water equations in a stable and nondiffusive manner. The curvilinear approach provides an accurate representation of the complex shape of natural flow domains and avoids the approximate 'staircase' perimeter which occurs in Cartesian finite-difference methods. The versatility of the model is demonstrated by simulating the wind-induced circulation patterns in the Gulf of Thermaikos, a semi-enclosed bay close to the City of Thessaloniki in Northern Greece.

Semi-Lagrangian Algorithm for Two-Dimensional Advection-Diffusion Equation on Curvilinear Coordinate Meshes

A semi-Lagrangian method for the solution of the unsteady advection-diffusion equation in complex geometries is presented. The domain is discretized into a set of control volumes defined by an orthogonal, boundary-fitted coordinate system. A splitting strategy is employed that decouples the advective and diffusive processes. The advective update is realized by calculating back trajectories along characteristic lines and then, at the feet of these trajectories, the value of the transported field is calculated by interpolation. The interpolated values are calculated using a tensor product cubic spline function. The method is simple to program and produces results of high accuracy. Numerical errors are considerably lower than for conventional Eulerian schemes. In one-dimensional and rectangular two-dimensional domains, the present method gives comparable results to the well-known semi-Lagrangian scheme. However, the new method allows for greater flexibility in handling complex domains.

NUMERICAL SIMULATION OF TURBULENT FLOW AND HEAT TRANSFER IN THE WEDGE-SHAPED LIQUID METAL POOL OF A TWIN-ROLL CASTER

Controlled flow and heat transfer are important for the quality of a strip in a twin-roll continuous casting process. A numerical study was carried out to investigate the two-dimensional turbulent flow and heat transfer in the liquid stainless-steet-filted wedge-shaped cavity formed by the two counterrotating roils in a twin-roll continuous casting system. The turbulent characteristics of the flow were modeled using a low-Reynolds-number k-e turbulence model due to Launder and Sharma. The arbitrary nature of the computational domain was accounted for through the use of a nonorthogonal boundary-fitted coordinate system on a staggered grid. A control-volume-based finite difference scheme was used to solve the transformed transport equations. This study is primarily focused on elucidating the inlet superheat dissipation in the melt pool with the rolls being maintained at a constant liquidus temperature of the steel. A parametric study was carried out to ascertain the effect of the inlet superheat, the casting speed, and the roll gap at the nip of the rotating rolls on the flow and heat transfer characteristics. The velocity fields show two counterrotating recirculation zones in the upstream region. The local Nusselt number on the roll surface shows significant variations. The contours of temperature and turbulent viscosity show the complex nature of the turbulent transport phenomena to be expected in a twin-roll casting process

Pressure drop and heat transfer associated with flows moving laminarly in straight ducts of irregular, singly connected cross-sections

This paper presents a fast and dependable numerical procedure for the solution of the fully established velocity and temperature of low-Reynolds number flows inside straight ducts with irregular, singly connected cross sections. Relying on finite volume discretizations of the momentum and energy equations in a boundary-fitted coordinate system, the procedure has been applied with great success to a wide variety of ducts whose cross sections present different levels of difficulty. Numerical predictions for the pressure drop (friction factor) and the heat transfer rate (Nusselt number) for a sample of simple ducts are compared with benchmark results published by other researchers. Other numerical predictions for complex shapes where results are absent in the literature are compared among themselves using various degrees of refinement for the grids. In general, for a sub-family of ducts characterized by a geometric parameter, e.g. elliptic, moon, sine and rhombic, the data can be post-processed and correlation equations of special interest to design engineers can be easily deduced.

Boundary-fitted grid in landscape modeling

Landscape modeling requires the delineation of system boundaries and interior features. Quite often, these components are complex and difficult to accurately represent. A rectangular grid is used to represent the study and adjacent non-study areas in most cases. When the non-study area occupies a large portion of the grid, computer memory is wasted, and computational time increases. An elliptical grid generator for non-orthogonal curvilinear coordinates is used to generate a boundary-fitted grid for a landscape model. In a boundary-fitted grid coordinate system, one coordinate axis follows the landscape domain boundary and is non-orthogonal to the second axis. The boundary-fitted grid uses elliptic partial differential equations to distribute grid points inside the landscape domain. Although the boundary-fitted grid follows the domain boundary, the grid pattern and point allocation remain structured. Thus, a landscape model can use a boundary-fitted grid without changing the model’s data structure or the computational scheme. In this study, a boundary-fitted grid and a raster-based grid were applied to the Everglades Landscape Fire Model. Use of the boundary-fitted grid decreased model simulation time by about one fifth and computer storage by 58% relative to the raster-based grid. Also, the linear characteristics of interior geographical features such as rivers and airboat trails were preserved by the boundary-fitted grid, but not by the raster-based grid. This preservation provided a more reasonable base map for simulating ecological processes, such as fire across heterogenous landscapes.

Numerical Analysis of Developing Turbulent Flow in S-Shaped Duct.

A numerical analysis has been carried out of developing turbulent flow in S-shaped duct with a square cross section at a Reynolds number of 4×104. The S-duct was formed from two 22.5 degree bends with 40 mm hydraulic diameter and 280 mm mean radius of curvature. Straight ducts with hydraulic diameters of 7.5 and 50 are attached to the inlet and outlet planes of the S-duct, respectively. In calculation, an algebraic Reynolds stress model was adopted in order to predict anisotropic turbulence precisely, and boundary-fitted coordinate system was introduced as the method of coordinate transformation. Calculated results were compared with the experimental data obtained using a laser Doppler velocimeter. As a result of the calculation, it was predicted without conflict with the experimental data that the flow at the inlet shows a "core" flow located nearer the inner wall of the first bend and a region of low-velocity fluid accumulated at the exit of the S-duct is observed. Moreover, the transition from the secondary flow of the first kind to that of the second kind is examined in the outlet straight duct. It was found that after reaching a maximum value near the S-duct outlet, the secondary flow decays and gradually transforms to the secondary flow of the second kind at a hydraulic diameter of about 18 from the bend outlet.

Numerical Analysis of Developing Turbulent Structure in S-Shaped Duct.

Numerical simulation has been performed for developing turbulent flow in S-shaped duct with a square section at a Reynolds number of 4×104. The S-duct was formed from two 22.5 degree bends with 40 mm hydraulic diameter and 280 mm mean radius of curvature. Straight ducts with hydraulic diameters of 7.5 and 50 are attached to the inlet and outlet planes of the S-duct, respectively. In calculation, an algebraic Reynolds stress model was adopted in order to predict anisotropic turbulent flow precisely, and a boundary-fitted coordinate system was introduced for coordinate transformation. The calculated results were compared with the experimental data using a laser Doppler velocimeter. In particular, the distributions of Reynolds stresses are compared in this report. As a result of comparison with the experimental data, it was found that the present method can accurately predict contours of streamwise normal stress which show the active generation of turbulence near the outer wall of the second bend. Moreover, it was clarified by estimating the production term that the high value of streamwise normal stress near the outer wall of the second bend is responsible for the velocity gradient of streamwise mean velocity with respect to the radial direction. The profiles of other normal stresses and shear stress are also predicted using the anisotropic turbulent model without great discrepancy.

Numerical Analysis of Compressible Flow Using Pressure-Correction Method.

A new pressure-correction method is developed in order to calculate the flow field involving both subsonic and supersonic flows. The method is applied to the turbulent flow computation code, in which governing equations including the standard k-eturbulence model are solved in the physical component tensor form with a boundary-fitted coordinate system. The MUSCL-type TVD scheme is also used not only for the estimation of the convection term of the transport equations but also for the estimation of the mass residual of the pressure correction equation. In order to examine the accuracy of the present pressure-correction method, two-dimensional supersonic flow through convergent-divergent nozzles and two-dimensional transonic flow through a channel with the shock/turbulent boundary-layer interaction are computed. The computed results are compared with the experimental data and good agreement between them is obtained.

対向偏心直円管を有する円管路内の乱流構造解析

A numerical analysis has been performed on developing turbulent flow in a circular pipe containing two rods located off-center. In calculation, an algebraic stress model was adopted in order to predict Reynolds stresses precisely, and the boundary-fitted coordinate system was introduced as the method of the coordinate transformation. Calculated results were compared with the experimental data available. The calculated profiles of secondary flow and Reynolds stresses for developing turbulent flow were investigated in detail. As a result of this calculation, it was found that the calculated distributions of the streamwise velocity and friction velocity are predicted well within 3% error range. Although the secondary flow along the wall of the outer circular pipe is not observed in the experimental data, the calculated result suggested that its secondary flow is generated by anisotropic turbulent in the region between off-center rod and circular pipe. The generation mechanism for its secondary flow was presented by evaluating the production terms in vorticity equation quantitatively. But more active research is necessary to make clear whether the predicted secondary flow exists or not.