Body-fitted Coordinates Research Articles

NUMERICAL SIMULATION OF COUPLED FLUID FLOW AND SOLIDIFICATION IN A SINGLE ROLL CASTER

A mathematical model was developed to study fluid flow and solidification in a single roll casting system. Fluid flow coupled with energy equations were solved numerically using a finite volume method. A body fitted coordinate (BFC) transformation technique was adopted to handle the irregularly shaped domain. In order to couple the continuity and momentum equations, a SIMPLEC algorithm was used. A fixed grid enthalpy method was considered to model the solidification in the system. The turbulence effect on flow field was modelled using a low Re k- model. The model developed in this work was further run under various conditions to optimize the process parameters of the process.On a développé un modèle mathématique pour étudier l’écoulement de fluide et la solidification dans un système de coulée à cylindre unique. On a résous numériquement l’écoulement de fluide couplé aux équations d’énergie en utilisant une méthode de volume fini. On a adopté une technique de transformation de coordonnée ajustée au corps (BFC) pour manier le domaine de forme irrégulière. Afin de coupler la continuité et les équations de quantité de mouvement, on a utilisé un algorithme SIMPLEC. On a considéré une méthode d’enthalpie à grille fixe pour modéliser la solidification dans le système. On a modélisé l’effet de turbulence sur le champ d’écoulement en utilisant un modèle à faible Re k-e. On a également opéré le modèle, développé dans ce travail, sous des conditions variées pour optimiser les paramètres du procédé.

Three-dimensional coupled impeller-volute simulation of flow in centrifugal pump and performance prediction

A three-dimensional turbulent flow through an entire centrifugal pump is simulated using κ-e turbulence model modified by rotation and curvature, SIMPLEC method and body-fitted coordinate. The velocity and pressure fields are obtained for the pump under various working conditions, which is used to predict the head and hydraulic efficiency of the pump, and the results correspond well with the measured values. The calculation results indicate that the pressure is higher on the pressure side than that on the suction side of the blade; The relative velocity on the suction side gradually decreases from the impeller inlet to the outlet, while increases on the pressure side, it finally results in the lower relative velocity on the suction side and the higher one on the pressure side at the impeller outlet; The impeller flow field is asymmetric, i.e. the velocity and pressure fields are totally different among all channels in the impeller; In the volute, the static pressure gradually increases with the flow route, and a large pressure gratitude occurs in the tongue; Secondary flow exists in the rear part of the spiral.

Computation of flow and heat transfer in rotating two-pass rectangular channels (AR = 1:1, 1:2, and 1:4) with smooth walls by a Reynolds stress turbulence model

Numerical predictions of three-dimensional flow and heat transfer are presented for rotating two-pass smooth channels with three aspect ratios (AR = 1:1; 1:2; 1:4). Detailed predictions of mean velocity, mean temperature and Nusselt number for two Reynolds numbers ( Re = 10,000 and 100,000) were carried out. A total of fifteen calculations have been performed with various combinations of rotation number, Reynolds number, and coolant-to-wall density ratio. The rotation number and inlet coolant-to-wall density ratio varied from 0.0 to 0.28 and from 0.13 to 0.40, respectively. The focus of this study is to investigate the effect of the channel aspect ratio, the Reynolds number, and the coolant-to-wall density ratio on the nature of the flow and heat transfer. A multi-block Reynolds-averaged Navier–Stokes (RANS) method was employed in conjunction with a near-wall second-moment turbulence closure. In the present method, the convective transport equations for momentum, energy, and turbulence quantities are solved in curvilinear, body-fitted coordinates using the finite-analytic method.

Transient Photon Transport in Turbid Media with an Irregularly Shaped Foreign Object

A finite-volume method for the modeling of transient photon propagation in turbid media with an irregularly shaped foreign object is presented. The procedure can account for non-rectangular foreign object. Body-fitted coordinates are used to discretize the spatial domain. Absorbing, anisotropic scattering non-homogenous medium can be modeled. The procedure is validated using existing numerical results for a homogenous medium. The procedure produces physically possible trends for different optical properties of the foreign object.

Numerical solutions of a viscous uniform approach flow past square and diamond cylinders

Numerical results are presented for a uniform approach flow past square and diamond cylinders, with and without rounded corners, at Reynolds numbers of 250 and 1000. This unsteady viscous flow problem is formulated by the 2-D Navier–Stokes equations in vorticity and stream-function form on body-fitted coordinates and solved by a finite-difference method. Second-order Adams-Bashforth and central-difference schemes are used to discretize the vorticity transport equation while a third-order upwinding scheme is incorporated to represent the nonlinear convective terms. A grid generation technique is applied to provide an efficient mesh system for the flow. The elliptic partial differential equation for stream-function and vorticity in the transformed plane is solved by the multigrid iteration method. The Strouhal number and the average in-line force coefficients agree very well with the experimental and previous numerical values. The vortex structures and the evolution of vortex shedding are illustrated by vorticity contours. Rounding the corners of the square and diamond cylinders produced a noticeable decrease on the calculated drag and lift coefficients.

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 Numerical Study of Flow and Heat Transfer in Rotating Rectangular Channels AR=4 With 45 deg Rib Turbulators by Reynolds Stress Turbulence Model

Computations were performed to study three-dimensional turbulent flow and heat transfer in stationary and rotating 45 deg ribbed rectangular channels for which experimental heat transfer data were available. The channel aspect ratio (AR) is 4:1, the rib height-to-hydraulic diameter ratio e/Dh is 0.078 and the rib-pitch-to-height ratio P/e is 10. The rotation number and inlet coolant-to-wall density ratios, Δρ/ρ, were varied from 0.0 to 0.28 and from 0.122 to 0.40, respectively, while the Reynolds number was fixed at 10,000. Also, two channel orientations (β=90deg and 135 deg from the rotation direction) were investigated with focus on the high rotation and high density ratios effects on the heat transfer characteristics of the 135 deg orientation. These results show that, for high rotation and high density ratio, the rotation induced secondary flow overpowered the rib induced secondary flow and thus change significantly the heat transfer characteristics compared to the low rotation low density ratio case. A multi-block Reynolds-Averaged Navier-Stokes (RANS) method was employed in conjunction with a near-wall second-moment turbulence closure. In the present method, the convective transport equations for momentum, energy, and turbulence quantities are solved in curvilinear, body-fitted coordinates using the finite-analytic method.

Parallel computation and numerical visualization for non-uniform particles included in gas and liquid flows

A parallel computation method has been proposed for the mixing and segregation of granular mixture included in gas and liquid flows. In this method, a three-dimensional (3D) computational volume is decomposed into multiple sub-blocks and their geometries are represented by 3D body-fitted coordinates. The fluid-particle interactions are treated by two types of models: a two-way model for liquid-solid flows and a one-way model in case of gas-solid flows. The computations of the particle motions in the multiple sub-blocks are executed simultaneously on the basis of the distinct element method (DEM). Since a graphic process is also executed as one of the parallel jobs, the particle distributions can be visualized during the computations. The computational method was applied to the gas-solid flows consisting of different diameters and densities in the horizontal and inclined cylinders rotating around their axes. From the comparison with the experimental results, nearly uniform mixing and particle segregation are successfully predicted in the oscillating liquid flows. In addition, it has been indicated that the particle pathline is very effective to visualize and understand the flow patterns of the particles with different properties. The result of the computations for the liquid-solid flows demonstrated that the vertical segregation of the non-uniform particles is reasonably reproduced.

Numerical Modelling of Fluid Flow with Heat and Mass Transfer at the Surface of a Gas Bubble

A numerical modelling of steady axisymmetric viscous external flow and heat and mass transfer at the surface of an inviscid gas bubble has been carried out. A numerical method based on a velocity-pressure formulation combined with a finite-volume discretization of the Navier-Stokes related equations written in a non-orthogonal body-fitted coordinate (BFC) is employed. The method has been validated by calculating the fluid flow with heat and mass transfer around the surface of a rigid sphere and comparing with published data that are well regarded as benchmark solutions. The feasibility of the method for simulating a gas bubble problem is tested by calculating fields around an inviscid fluid sphere (or a pseudo-bubble) at similar conditions to the rigid sphere. On this basis, a deformable gas bubble at Sc = 500 and Pr = 2 is studied. The fields of flow, temperature, and concentration change with Reynolds and Weber numbers are simulated. It is found that the interactions between the fields and the deformation of the bubble interface are significant.

Stability of thermocapillary flows in non-cylindrical liquid bridges

The thermocapillary flow in liquid bridges is investigated numerically. In the limit of large mean surface tension the free-surface shape is independent of the flow and temperature fields and depends only on the volume of liquid and the hydrostatic pressure difference. When gravity acts parallel to the axis of the liquid bridge the shape is axisymmetric. A differential heating of the bounding circular disks then causes a steady two-dimensional thermocapillary flow which is calculated by a finite-difference method on body-fitted coordinates. The linear-stability problem for the basic flow is solved using azimuthal normal modes computed with the same discretization method. The dependence of the critical Reynolds number on the volume fraction, gravity level, Prandtl number, and aspect ratio is explained by analysing the energy budgets of the neutral modes. For small Prandtl numbers (Pr = 0.02) the critical Reynolds number exhibits a smooth minimum near volume fractions which approximately correspond to the volume of a cylindrical bridge. When the Prandtl number is large (Pr = 4) the intersection of two neutral curves results in a sharp peak of the critical Reynolds number. Since the instabilities for low and high Prandtl numbers are markedly different, the influence of gravity leads to a distinctly different behaviour. While the hydrostatic shape of the bridge is the most important effect of gravity on the critical point for low-Prandtl-number flows, buoyancy is the dominating factor for the stability of the flow in a gravity field when the Prandtl number is high.

Prediction of Flow and Heat Transfer in Rotating Two-Pass Rectangular Channels With 45-deg Rib Turbulators

Abstract Numerical predictions of three-dimensional flow and heat transfer are presented for a rotating two-pass rectangular channel with 45-deg rib turbulators and channel aspect ratio of 2:1. The rib height-to-hydraulic diameter ratio e/Dh is 0.094 and the rib-pitch-to-height ratio P/e is 10. Two channel orientations are studied: β=90deg and 135 deg, corresponding to the mid-portion and the trailing edge regions of a turbine blade, respectively. The focus of this study is twofold; namely, to investigate the effect of the channel aspect ratio and the channel orientation on the nature of the flow and heat transfer enhancement. A multi-block Reynolds-averaged Navier-Stokes (RANS) method was employed in conjunction with a near-wall second-moment turbulence closure. In the present method, the convective transport equations for momentum, energy, and turbulence quantities are solved in curvilinear, body-fitted coordinates using the finite-analytic method. The numerical results compare reasonably well with experimental data for both stationary and rotating rectangular channels with rib turbulators at Reynolds number (Re) of 10,000, rotation number (Ro) of 0.11 and inlet coolant-to-wall density ratio (Δρ/ρ) of 0.115.

A numerical analysis of fluid flow, heat transfer and solidification in the bending-type square billet continuous casting process

A numerical modeling system was developed which can simulate the transport phenomena of a bending type square billet continuous casting process. Fluid flow and heat transfer were analyzed with a 3-dimensional finite volume method (FVM) with the aid of an effective heat capacity algorithm for the solidification. For a complex geometry of the bending type billet caster, a body-fitted-coordinate (BFC) system was employed. The bent structure of the caster allows a recirculating flow to develop in the upper and outer-radius region and the main stream to shift toward inner radius. This causes the thinner solid shell in the inner radius region than in the outer one. Besides standard operation conditions, we have analyzed the results when casting speed, caster shape, and tundish superheat changes. Lower casting speed makes the solid shell thicker by reducing heat flux from the mold. In the vertical caster, solid shell thickness are more uniform than that in the bending-type in entire region. When superheat increases by 5°C, solid shell thickness at the mold exit becomes thinner by 1 mm.

AN ALTERNATING DEPENDENT VARIABLES (ADV) METHOD FOR TREATING SLIP BOUNDARY CONDITIONS OF FREE SURFACE FLOWS WITH HEAT AND MASS TRANSFER

An alternating dependent variables (ADV) method is proposed to treat slip boundary conditions for interfacial flows of a nonspherical bubble in liquid when Cartesian velocity components are chosen as dependent variables in momentum equations written in nonorthogonal body-fitted coordinates. Cartesian interfacial velocities are solved alternatively at different segments of bubble profile, hence numerical instabilities due to a nearly infinite and zero slop of the profile are avoided. Numerical results indicate that the ADV is a suitable method for solving free surface flows of nonspherical bubbles with large curvature. On this basis, the interfacial characteristics of three types of nonspherical bubbles are presented.

Inverse radiation problem of source term in three-dimensional complicated geometric semitransparent media

A method is presented to identify the three-dimensional source term distribution in complicated geometric systems of known radiative properties from the knowledge of the radiative intensities exiting the boundaries. The inverse radiation problem is formulated as an optimization problem, and solved by the conjugate gradient method that minimizes the errors between the exit radiative intensities calculated and the experimental measurements. The analysis consists of the direct problem, the gradient equation, and the sensitivity problem. In this approach, the discrete ordinates method is employed to solve the direct and the sensitivity problems in general body-fitted coordinates. The effects of the measurement errors on the accuracy of the inverse analysis are investigated. The study shows that the three-dimensional source term distribution in complicated geometric systems can be estimated accurately, even with noisy data.

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.

CFD DEVELOPMENT AND SIMULATION OF FLOW FIELDS IN VENTILATED SPACES WITH MULTIPLE OCCUPANTS

Simulation of flow fields in ventilated spaces with multiple occupants (cows) was conducted using a computational fluid dynamics (CFD) technique. Airflow in a ventilated space is affected by the locations of the inlet and outlet, inlet or room shape, and positions of the occupants. Three simulation test cases considered are: (1) four cows standing in a row with one end wall simulating as an inlet, and the opposite end wall simulating as an outlet; (2) four cows standing in a row with the two end walls simulating as inlets, and an outlet located at the center of the ceiling; and (3) ten cows placed at random with inlet and outlet arrangements the same as in Case (2). The body–fitted coordinate (BFC) system is adopted to characterize the shape of a cow, and the governing equations in the Cartesian coordinate system are transformed into the BFC system. To generate the grid under the BFC system, partial differential equations and algebraic interpolation methods were used. The finite volume method is applied to transform the partial differential equations into algebraic ones, and the SIMPLE (Semi–Implicit Method for Pressure–Linked Equation) algorithm is extended to the BFC system to solve the flow field. The effects of obstructions (cows), and the positions of cows, inlets, and outlet on flow distribution were clearly demonstrated by velocity vectors, velocity profiles, and streamline plots generated. The flow field was more evenly distributed when cows were positioned at random than when in a row.

Inverse radiation problem in three-dimensional complicated geometric systems with opaque boundaries

An inverse radiation analysis is presented to identify the three-dimensional source term distribution in complicated geometric systems of known radiative properties from the knowledge of the exit radiative intensities. The inverse radiation problem is formulated as an optimization problem, and solved by the conjugate gradient method that minimizes the errors between the calculated exit radiative intensities and the experimental data. The measured data are simulated by adding random errors to the exact solution of the direct problem. The analysis consists of the direct problem, the gradient equation, and the sensitivity problem. In this approach, the discrete ordinates method is employed to solve the direct and the sensitivity problems in general body-fitted coordinates. The effects of the measurement errors, single scattering albedo, and scattering asymmetry parameter on the accuracy of the inverse analysis are investigated. The study shows that the three-dimensional source term distribution in complicated geometric systems with opaque and diffuse-gray boundaries can be estimated accurately for the exact and noisy data.

Numerical visualization of free surface oscillation predicted with arbitrary Lagrangian-Eulerian method

A numerical prediction method has been proposed to predict non-linear free surface oscillation in a three-dimensional container. The fluid motions are numerically predicted with Navier-Stokes equations discretized in a Lagrangian scheme with sufficient numerical accuracy. The profile of a free surface is precisely represented with three-dimensional body-fitted coordinates (BFC), which are regenerated in each computational step on the basis of the arbitrary Lagrangian-Eulerian (ALE) formulation. The computational method was applied to non-linear sloshings and transitions from sloshing to swirling motions. The predicted free surface motions were visualized as sequential image files and animations to understand their dynamic futures.

A Radiation Model for the Numerical Simulation of Coal-Fired Furnaces Using Body-Fitted Grids

A model for predicting radiative heal transfer in coal-fired furnaces is presented. The radiative transfer equation is modelled by the discrete ordinates method for body-fitted coordinates using the S4-approximation. The gas absorption coefficient is determined by a weighled-sum-of-grey-gases model, the radiative properties of coal and ash particles are derived from the specific area and a mean efficiency factor of the particle cloud. The scattering phase function is modelled by the Delta-Edding-ton approximation. The entire radiation model is adopted for vector and parallel computers guaranteeing numerical efficiency. It is tested at an idealized squared combustion chamber with black walls filled by an emitting, absorbing medium with constant absorption coefficient. The computations are carried out on several curvilinear non-orthogonal grids using different boundary conditions. On all grids the computed radiative source terms and wall heat fluxes are in very good agreement with the analytic solution. Furthermore, the model is validated at a coal-fired lest facility. Apart from the near burner region the computed temperature distribution only slightly deviates from the measured values.

Flow and Heat Transfer in a Rotating Square Channel With 45 deg Angled Ribs by Reynolds Stress Turbulence Model

Numerical predictions of three-dimensional flow and heat transfer are presented for a rotating square channel with 45 deg angled ribs as tested by Johnson et al. (1994). The rib height-to-hydraulic diameter ratio e/Dh is 0.1 and the rib pitch-to-height ratio (P/e) is 10. The cross section of the ribs has rounded edges and corners. The computation results are compared with the experimental data of Johnson et al. (1994) at a Reynolds number (Re) of 25,000, inlet coolant-to-wall density ratio Δρ/ρ of 0.13, and three rotation numbers (Ro) of 0.0, 0.12, and 0.24. A multiblock numerical method has been employed with a near-wall second-moment turbulence closure model. In the present method, the convective transport equations for momentum, energy, and turbulence quantities are solved in curvilinear, body-fitted coordinates using the finite-analytic method. Pressure is computed using a hybrid SIMPLER/PISO approach, which satisfies the continuity of mass and momentum simultaneously at every time step. The second-moment solutions show that the secondary flows induced by the angled ribs, rotating buoyancy, and Coriolis forces produced strong nonisotropic turbulent stresses and heat fluxes that significantly affected flow fields and surface heat transfer coefficients. The present near-wall second-moment closure model provided an improved flow and heat transfer prediction.