Boundary-fitted Coordinate System Research Articles

Numerical Analysis of Three-Dimensional Turbulent Flow in a 180.DEG.-Bent Tube Using by an Algebraic Reynolds Stress Model.

A numerical analysis has been performed for three dimensional developing turbulent flow in a 180° bent tube with straight inlet and outlet sections using by an algebraic Reynolds stress model. To our knowledge, only very few numerical investigations exist in which the detailed comparison between calculated results and experimental data contained Reynolds stresses. In numerical analysis, an algebraic Reynolds stress model in conjunction with a boundary-fitted coordinate system is applied to a 180° bent tube in order to solve the anisotropic turbulent structure precisely. As a result of this analysis, it is found that the calculated results show a comparatively good agreement with the experimental data of the time averaged velocity and the secondary vectors in the bent tube and the straight outlet sections. For example, the location of the maximum streamwise velocity, which appears near the top or bottom wall in the bent tube, is predicted correctly by the present method. As for the comparison of Reynolds stresses, the present model has been found to simulate many characteristic features of streamwise normal stress and shear stresses in the bent tube satisfactorily, but has a tendency to underpredict its value. Judging from the comparison between the calculated and the experimental results, the algebraic Reynolds stress model is applicable to the developing turbulent flow in a bent tube which is known as a flow with a strong convective effect.

Numerical Analysis of Reynolds Stress for Turbulent Flow Developing in a Rectangular-Sectioned 90 Degree Bend.

A numerical study has been performed on turbulent flow developing in a 90 deg curved duct of retangular cross section and an aspect ratio of 6. The ratio of bend mean radius of curvature to hydraulic diameter is 2.04 and straight ducts of 7.5 and 25.5 width are attached to the inlet and outlet of the bend, respectively. In addition to this straight duct. two-dimensional lateral contraction of the wind tunnel with an area ratio of 6 is considered as the numerical domain as well as the experimental apparatus. Calculated results were compared with the experimental data measured for Reynolds stress components in detail. In this calculation, an algebraic Reynolds stress model was adopted in order to predict anisotropic turbulence precisely, and a boundary-fitted coordinate system was introduced as the method of coordinate transformation. As a result of this examination, the present method can predict normal and shear stresses distributions of the inlet straight duct but cannot reproduce the characteristic destabilizing effect and shows an increase in the turbulence intensity close to the outer wall in the curved duct. The discrepancy may be caused by use of the wall function as the boundary condition. In the outlet straight duct. the calculated value of normal and shear stresses are smaller than those of the experiments.

Numerical Analysis of Three-Dimensional Turbulent Flow in a 90.DEG. Bent Tube by Algebraic Reynolds Stress Model.

A numerical analysis has been performed for three dimensional developing turbulent flow in a 90° bent tube with straight inlet and outlet sections by an algebraic Reynolds stress model. To our knowledge, very little has been reported about detailed comparison between calculated results and experimental data containing Reynolds stresses. In calculation, an algebraic Reynolds stress model together with a boundary-fitted coordinate system is applied to a 90° bent tube in order to solve anisotropic turbulent flow precisely. The calculated results display comparatively good agreement with the experimental data of time averaged velocity and secondary vectors. In addition, the present method predicts as a characteristic feature that the intensity of secondary flow near the inner wall is increased immediately downstream from the bend outlet by the pressure gradient. With regard to comparison of Reynolds stresses, the present method is able to reproduce well the distributions of streamwise normal stress and shear stress defined streamwise and radial velocity fluctuation except for the shear stress defined streamwise and circumferential velocity fluctuation. The present calculation has been found to simulate many features of the developing flow in bent tube satisfactorily, but it has a tendency to underpredict the Reynolds stresses.

Numerical Analysis of Reynolds Stresses for Turbulent Flow Developing in S-Shaped Diffuser.

A numerical analysis has been carried out for developing turbulent flow in a diffuser with S-shaped centerline at a Reynolds number of 4×104. The diffuser exit-to-inlet area ratio was 1.5 and the inler cross section was square, 40mm×40mm, and expanded on both curved surfaces to an enitplane section of 40mm×60mm. The S-shaped curvature was formed from two 22.5 degree bends of 280mm mean radius of curvature. Straight ducts 0.3m and 2m long are attached to the inlet and outlet planes of the S-shaped diffuser, respectively. In the calculation, an algebraic Reynolds stress model was adopted in order to predict anisotropic turbulence precisely, and a boundary-fitted coordinate system was introduced as the method of coordinate transformation. Special attention is paid to the comparison of Reynolds stresses between calculation results and experimental data. In addition, the difference in turbulent structure between S-shaped diffuser and S-shaped duct is presented by invariant analysis using the calculation results. The present method can be used accurate prediction of the characteristic features of normal and shear stresses. Invariant maps of the S-shaped diffuser show that the turbulent structure transforms anisotropic turbulent flow into a two-dimensional turbulent flow. Moreover, the calculation results suggent that the turbulence level of the S-shaped diffuser increases more actively than does that of the S-shaped duct.

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

A numerical analysis of developing turbulent flow in a diffuser with S-shaped centerline at a Reynolds number of 4×104 has been carried out. The diffuser exit-to-inlet area ratio was 1.5, the inlet cross section was square, 40mm×40mm, and was expanded on both curved surfaces to an exitplane section of 40mm×60mm. The S-shaped curvature was formed from two 22.5 degree bends of 280mm mean radius of curvature. Straight ducts 0.3m and 2m long were attached to the inlet and outlet planes of the S-shaped diffuser, respectively. In the calculation, an algebraic Reynolds stress model was adopted for predicting anisotropic turbulence precisely, and a boundary-fitted coordinate system was introduced for coordinate transformation. Calculation results for mean velocity were compared with the experimental data. In addition, the difference in flow between the S-shaped diffuser and the S-shaped duct is presented with calculation results. Using by the present method, we found that the flow at the inlet shows a core flow located near the inner wall of the first bend, and a region of low velocity fluid which has accumulated at the exit of the diffuser was observed. It was also found that the secondary flow patter of the S-shaped diffuser was similar to that of the S-shaped duct in the first bend but differed from it in the second bend.

Numerical Simulation of Turbulent Flow in a Channel with Large-Amplitude Wavy Surfaces.

The main purpose of this work is to calculate the developing turbulent structure for flow over large-amplitude wavy surfaces. In the calculation, and algebraic Reynolds stress model and a boundary-fitted coordinate system are applied to a turbulent flow in a channel containing wavy wall, in order to precisely solve the anisotropic turbulence. This turbulent flow is complicated, involving accelerated, separated and recirculating flows. The calculated results are compared with the experimental mean and the fluctuating velocities for the flow over a solid sinusoidal wave surface having a wavelength of 50.8mm and a wave amplitude of 5.08mm. For this flow, the experimental results show a large separated region. The present method predicts the locations of separation and reattachment without much discrepancy. The present model has been found to simulate characteristic features of the flow, but it has a tendency to show the different profiles near the wavy surface and to overestimate the value of the mean velocity. For the streamwise velocity fluctuation, the calculated results show similar profiles except at near-wall region. Moreover, a cross-sectional view shows that with this method, we can see how the secondary flow and the separated region develop. The calculated results show that a large variation of the secondary flow is generated close to the corner region of the wavy surface as a result of the pressure gradient.

Numerical Prediction of Developing Turbulent Structure Through a Square-to-Circular Transition Duct.

In this paper we describe numerical analysis to investigate the three-dimensional turbulent structure and fluid flow behavior in a square-to-circular transition duct. The hydraulic diameter, 40mm, was the same in the square inlet and round exit sections, resulting in a cross-sectional area reduction of 21.5 percent. The transition takes place over two hydraulic diameters. In the calculation, an algcbraic Reynolds stress model combined with a boundary-fitted coordinate system is applied to a square-to-circular transition duct in order to solve anisotropic turbulent flow precisely. The calculated results are compared with the experimental results available to examine the validity of the present method. As a result of calculation, it is found that the present method predicts streamwise and horizontal mean velocities well but the calculated distributions of the normal stresses show a discrepancy compared with the experiment. Although the flow was fully developed at the inlet to the transition duct, as stated in the experimental report, the experimental results of normal stresses differ greatly from those of flow, while calculated results are similar to those for fully developed flow. In this transition duct, two kinds of the secondary flow exist, i.e., one is the secondary flow of the second kind in the square duct and the other is that of the first kind in the transition duct induced by the pressure gradient. Moreover, the secondary flow of the transition duct disappears when the flow enters the duct with a circular cross section. The present method can predict these features quantitatively.

Numerical Analysis of Turbulent Flow Developing in a Rectangular-Sectioned 90 Degree Bend.

The numerical study has been performed on turbulent flow developing in a 90 degree curved duct of rectangular crosssection and an aspect ratio of 6. The ratio of bend mean radius of curvature to hydraulic diameter is 2.04 and straight ducts of 7.5 and 25.5 width are attached to the inlet and outlet of the bend, respectively. In addition to this straight duct, two-dimensional lateral contraction of the wind tunnel with an area ratio of 6 is considered as the numerical domain as well as the experimental apparatus. Calculated results were compared in detai with the experimental data measured for meanvelocity and Reynolds stress components. In this calculation, an algebraic Reynolds stress model was adopted in order to predict anisotropic turbulence precisely, and a boundary-fitted coordinate system was introduced as the method of coordinate transformation. As a result of the comparison with experimental data, it was found that calculated results showed good agreement with the experimental data of streamwise and transverse velocites except for that of the inner wall region of the outlet straight duct and suggested the existence of the Taylor-Gortler-type vortices on the outer wall by the distributions of secondary flow and wall shear stress.

Simulation Of Wind-Flow Patterns Around Bridge Deck Sections

The finite-difference procedure is used to solve the non-conservative form of the two-dimensional, unsteady, incompressible, Navier-Stokes flow equations, in a body-fitted curvilinear coordinate system. An implicit formulation scheme is employed, in order to avoid excessive computation time, resulting from stringent numerical stability restrictions associated with explicit schemes. The finite-difference mesh is generated using an automatic grid generation technique, based on the boundary-fitted curvilinear coordinate system. Solving the primitive variable form of the Navier-Stokes equations directly yields the desired flow parameters: the two velocity components and the pressure values. The flow velocities are then used in a stream function equation to compute the flow streamlines. Application to the bridge deck section of the Chesapeake and Delaware canal bridge shows good prediction of the wind flow patterns around such a bluff body geometry.

An operator splitting method with FDM and FEM for the convection- diffusion equation using boundary-fitted coordinate system

An operator splitting method combining finite difference method and finite element method is proposed in this paper by using boundary-fitted coordinate system. The governing equation is split into advection and diffusion equations and solved by finite difference method using boundary-fitted coordinate system and finite element method respectively. An example for which analytic solution is available is used to verified the proposed methods and the agreement is very good. Numerical results show that it is very efficient.

An Educational Two-Dimensional Interactive Dynamic Grid Generator

This paper describes IDGG, an Interactive Dynamic Grid Generator, for use as an educational tool by students studying computational fluid dynamics. The package is a Windows applications and runs on IBM PC, or compatible, computers. It is written in Pascal and built using object-oriented programming. The computer program allows the user to generate boundary-fitted curvilinear grids in any two-dimensional domain. The procedure adopted requires the user to perform the transformation step by step allowing him/her to easily grasp the concept of boundary-fitted coordinate systems. In addition, IDGG may be used by CFD researchers to display results graphically in the form of vector fields, contours, and two- and three-dimensional plots. The examples provided show the effectiveness of the package as a teaching aid.

Effect of crucible geometry on the flow pattern in the magnetic field applied Czochralski process

The effects of crucible geometry on the flow pattern and the oxygen concentration at the crystal-melt interface are studied numerically for the cusp type magnetic field applied Czochralski process. The numerically generated boundary-fitted orthogonal curvilinear coordinate systems are used with the finite difference method. The crucible geometry is found to have non-trivial effects on both the flow pattern and the oxygen concentration at the melt-crystal interface. Especially when the natural convection is dominant, the numerical results suggest that there exists a crucible shape which is advantageous over the conventional cylindrical shape in both the uniformity and the average level of oxygen concentration at the crystal-melt interface.

Natural convection in the annulus between concentric horizontal circular and square cylinders

Numerical solutions are presented for natural convection heat transfer between a heated horizontal cylinder placed concentrically inside a square enclosure. Three different aspect ratios (R/L 0.1, 0.2, and 0.3), and four different Rayleigh numbers (Ra = 10, 10, 10, and 10), are considered. The governing elliptic conservation equations are solved in a boundary-fitted coordinate system using a control volumebased numerical procedure. Results are displayed in the form of streamlines, isotherms, maximum stream function estimates, and local and average normalized Nusselt number values. At constant enclosure aspect ratio, the total heat transfer increases with increasing Rayleigh number. For constant Rayleigh number values, convection contribution to the total heat transfer decreases with increasing values of R/L. For convection-dominated flows, the average Nusselt number correlation is expressed as Nu = 0.92Ra(R/ L)*. Generated results are in good agreement with previously published experimental and numerical data.

Prediction of thermal stresses during vertical solidification of a pure metal with density change

A numerical methodology for the calculation of the thermal stresses in a solidifying body with volume contraction due to solid-liquid density change is developed and applied to the vertical solidification of pure aluminum. In this algorithm, the phase-change heat-transfer analysis is performed by a finite-volume method (FVM) and the thermal-stresses analysis in a solidifying body by a finite-element method (FEM). Difficulties associated with the 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. A hypoelastic-viscoplastic constitutive model and a rate form of the principle of virtual work are involved to model the stresses and the deformation. The effect of various cooling rates on the motion of the solid-liquid interface, on the temperatures and on the thermal stresses is presented and discussed.

Developing three-dimensional laminar mixed convection in a circular tube inserted with longitudinal strips

A finite-difference numerical study with a boundary-fitted coordinate system was made to investigate the three-dimensional, steady-state, laminar entrance flow of an axially uniformly heated horizontal tube with different adiabatic inserts. Parametric calculations have been performed to determine the nature and effect on thermally driven secondary flows of Reynolds number (100 ⩽ Re ⩽ 1000), Grashof number (10 3 ⩽ Gr ⩽ 10 6), aspect ratio (1 ⩽ AR ⩽ 8) and radius ratio (0 ⩽ R∗ ⩽ 1). Typical streamline patterns, isotherms and axial velocity along the flow direction are obtained. Local friction factor, Nusselt number and entrance length in the hydrodynamically and thermally developing, and fully developed regions are presented and discussed. Furthermore, flow visualization was made and compared with the corresponding numerical results. The agreement is good for Re = 500 at the two different downstream distances.

Numerical Analysis of Turbulent Structure Developing in Circular Pipe Containing One Rod Located Off-Center.

A numerical analysis has been performed on turbulent flow developing in a circular pipe containing one rod 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 to set the boundary condition. The Reynolds number in this calculation is the same as that in the experiment. Calculated results were compared with the experimental data available. The calculated profiles of secondary flow and Reynolds stresses for developing turbulent flow are shown 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. Moreover, the calculated result suggests that a pair of vortices are generated by anisotropic turbulence in the region between the off-center rod and circular pipe. As a characteristic feature, it is pointed out that the intensity of secondary flow is maximum before becoming a fully developed turbulent flow. The location at which it is maximum corresponds to that of the appearance of a merged boundary layer. In addition to these results, the production terms of secondary flow in the vorticity equation are evaluated using calculated results in order to provide further understanding of the generation of secondary flow.

Flow and heat transfer around a single row of circular cylinders

A numerical investigation of flow and heat transfer around a single row of circular cylinders was conducted using a boundary-fitted coordinate system. Numerical calculations for center-to-center distance between cylinders L/d=2.0, 2.5, 3.3, and ∞ were made of water flows in the Reynolds number range from 75 to 500. Numerical values of average Nusselt number for uniformly heated cylinders are in relatively good agreement with those obtained from experiments in water (Prandtl number Pr ≒ 8). The interaction of wake flows behind cylinders, observed in the experiments, was also found to occur with decreasing cylinder spacing L/d. © 1997 Scripta Technica, Inc. 25(3): 192–200, 1996

Numerical Analysis of Three-Dimensional Turbulent Structure and Secondary Flow in Trapezoidal Open-Channel Flows Using Algebraic Reynolds Stress Model.

An investigation of turbulence structure with the secondary flow of the second kind in a trapezoidal open channel is very important in basic hydraulics and river engineering. In this study, a numerical analysis has been carried out to develope turbulent flow in a trapezoidal open channel with three kinds of bank angle. An algebraic Reynolds stress model coupled with a boundary-fitted coordinate system was abopted in order to predict Reynolds stresses precisely. Calculated results were compared with the experimental data available. It has been pointed out from the experiment that velocity-dip, which means that the maximum velocity appears not at the free surface but rather just below it, is not observed in a trapezoidal open channel. The calculated results of streamwise velocity show that the maximum velocity exists at the free surface. For this reason, the present method predics this feature. Furthermore, the secondary vectors of the calculated result are in good agreement with those from the experiment, which proves the validity of the present method. Near the free surface, the secondary flow direction in a trapezoidal open channel is opposite to that in a rectangular open channel. The orign of this difference between the cross sections is examined by evaluatidng the production term of the streamwise vorticity equation.

Numerical Prediction of Three-Dimensional Turbulent Structure through a Circular-to-Rectangular Transition Duct.

Numerical analysis is used to investigate three-dimensional turbulent structure and fluid flow behavior in a circular-to-rectangular transition duct. Turbulent flow through a circular-to-rectangular duct is characterized by streamwise curvature and streamwise vorticity embedded in the boundary layer. In the calculation, an algebraic Reynolds stress model together with a boundary-fitted coordinate system is applied to to a circular-to rectangular duct in order to solve anisotropic turbulent flow precisely. The calculated results are compared with the experimental results to examine the validity of the present method. As a result of this calculation, it is found that the present method predicts well a secondary flow pattern which develops into a discrete vortex pair along the duct sidewalls. This secondary flow arises as a result of lateral skewing of the near-wall flow in the vicinity of the sidewall induced by transverse pressure gradients associated with wall curvature. Moreover, the calculated contours of six Reynolds stress components are shown for comparison with the experiment. The distortion of Reynolds stresses by the vortex pair is clearly observed as well as the primary flow. Although agreement is not perfect, the main features are reproduced by the present method using the algebraic Reynolds stress model.

Intimal thickening under shear in a carotid bifurcation—A numerical study

In the present study, a model for intimal thickening proposed by Friedman and his co-workers is employed to simulate the growth of intima in a carotid bifurcation with a steady flow. The computer code developed is a pressure-based, finite volume method in a boundary-fitted coordinate system. A model bifurcation is used to demonstrate the usefulness of the numerical tool. The predicted results are consistent with the available experimental observations. The localization of plaques is shown. It is found that thicker intima is formed in preferential regions near the junction and the carotid bulb due to relatively low wall shear stresses. The intimal thickening tends to reduce the size of low shear regions. It is also found that the new geometry resulting from the thickening yields a more even stress distribution over the bifurcation. The effect of the branching ratio on the distribution of intimal thickening is also studied. The computer code provides a potential tool for visualization of intimal thickening.