Finite difference solution of a Newtonian jet swell problem

A finite difference technique has been developed to study the Newtonian jet swell problem. The streamfunction and vorticity were used as dependent variables to describe the jet flow. The boundary-fitted co-ordinate transformation method was adopted to map the flow geometry into a rectangular domain. The standard finite difference method was then applied for solving the flow equations. The location of the jet free surface was updated by the kinematic boundary condition, and an adjustable parameter was included in the free-surface iteration. We could obtain numerical solutions for the Reynolds number as high as 100, and the differences between the present study and previous finite element simulations on the jet swell ratio are less than 5%.     

A finite difference technique for solving the Newtonian jet swell problem

A finite difference technique that incorporates a numerical mapping has been successfully applied to analyse both planar and axisymmetric Newtonian jets. A pressure gradient equation and a free-surface slope equation have been derived for free-surface iteration. The computation of pressure inside the jet surface using the pressure gradient equation is stable and accurate at high Reynolds numbers. The free-surface slope equation is needed for updating the free surface and is applicable for jets with strong surface tension effects. The present development can simulate the Newtonian jets for Reynolds numbers as high as 2000 and capillary number as low as 10^?5. Numerical predictions by the present technique are close to the results of previous finite element simulations.     

Modeling and simulation of the viscoelastic fluid mold filling process by level set method

A model for unifying a viscoelastic fluid and a Newtonian fluid is established, in which the governing equations for the viscoelastic fluid and the Newtonian fluid are successfully united into a system of generalized Navier–Stokes equations. A level set method is set up to solve the model for capturing the moving interface in the mold filling process. The physical governing equations are solved by the finite volume method on a non-staggered grid and the interpolation technique on the collocated grid is used for the pressure-velocity and the stress-velocity decoupling problems. The level set and its reinitialization equation are solved by the finite difference method, in which the spatial derivatives are discretized by the 5th-order Weighted Essentially Non-Oscillatory (WENO) scheme, and the temporal derivatives are discretized by the 3rd-order Total Variation Diminishing Runge–Kutta (TVD-R–K) scheme. The validity of the method is verified by some benchmark problems. Then a simulation of viscoelastic fluid mold filling process is pursued with the method. The moving interface and all the information of the physical quantities during the injection process are captured. The die swelling phenomenon is found in the simulation. The influences of elasticity and viscosity on the physical quantities such as stresses etc. in the mold filling process are analyzed. Numerical results show that elastic characteristics such as the stretch and die swelling etc. reinforce accordingly as Weissenberg number increases. Pressures increase continuously in the mold filling process and the pressure maintains the maximum value at the inlet. Injection velocity is proportional to injection pressure. A higher viscosity leads to a higher pressure distribution, that is, the pressure decreases as Reynolds number increases.     

Extrudate swell through an orifice die

The extrudate swell of a viscoelastic fluid through an orifice die is investigated by using a mixed finite element and a streamline integration method (FESIM), using a version of the K-BKZ model. The free surface calculation is based on a local mass conservation scheme and an approximate numerical treatment for the contact point movement of the free surface. The numerical results show a vortex growth and an increasing swelling ratio with the Weissenberg number. Convergence with mesh refinement is demonstrated, even at a high Weissenberg number of O(587), where the swelling ratio reaches a value of about 360%. In addition, it is found that the effective flow channel at the entrance region next to the orifice die is reduced due to the enhanced vortex growth, which may be a source of flow instability.     

Converged solutions of the Newtonian extrudate‐swell problem

Both the axisymmetric and the planar Newtonian extrudate‐swell problems are solved using the standard and the singular finite element methods. In the latter method, special elements that incorporate the radial form of the stress singularity are used around the exit of the die. The convergence of each of the two methods with mesh refinement is studied for various values of the Reynolds and the capillary numbers. The numerical results show that the singular finite elements perform well if coarse or moderately refined meshes are used, and appear to be superior to the standard finite elements only when the Reynolds number is low and the surface tension is not large. The standard finite elements perform better as the surface tension or the Reynolds number are increased. This implies that the effect of the stress singularity on the accuracy of the numerical solution in the neighborhood of the die exit becomes less significant when the Reynolds number is high or the surface tension is large. Copyright © 1999 John Wiley & Sons, Ltd.     

The compressible Newtonian extrudate swell problem

We solve the compressible Newtonian extrudate swell problem in order to investigate the effect of compressiblity on the shape of the extrudate. We employ a first-order equation of state relating the density to the pressure and use finite elements for the numerical solution of the problem. Our results show that the shape of the extrudate and the final extrudate swell ratio are not significanlty affected even at high compressibility values.     

Finite element computations of viscoelastic two-phase flows using local projection stabilization

A three-field local projection stabilized (LPS) finite element method is developed for computations of a three-dimensional axisymmetric buoyancy driven liquid drop rising in a liquid column where one of the liquid is viscoelastic. The two-phase flow is described by the time-dependent incompressible Navier-Stokes equations, whereas the viscoelasticity is modeled by the Giesekus constitutive equation in a time-dependent domain. The arbitrary Lagrangian-Eulerian (ALE) formulation with finite elements is used to solve the governing equations in the time-dependent domain. Interface-resolved moving meshes in ALE allows to incorporate the interfacial tension force and jumps in the material parameters accurately. A one-level LPS based on an enriched approximation space and a discontinuous projection space is used to stabilize the numerical scheme. A comprehensive numerical investigation is performed for a Newtonian drop rising in a viscoelastic fluid column and a viscoelastic drop rising in a Newtonian fluid column. The influence of the viscosity ratio, Newtonian solvent ratio, Giesekus mobility factor, and the Eötvös number on the drop dynamics are analyzed. The numerical study shows that beyond a critical Capillary number, a Newtonian drop rising in a viscoelastic fluid column experiences an extended trailing edge with a cusp-like shape and also exhibits a negative wake phenomena. However, a viscoelastic drop rising in a Newtonian fluid column develops an indentation around the rear stagnation point with a dimpled shape.     

Simulation of three-dimensional unsteady viscous free surface flow around a ship model

We present here a numerical method for solving the free surface flow around a ship at forward speed in calm water. The fluid is assumed to be Newtonian and the Reynolds-averaged Navier-Stokes equations are solved by a finite difference method. Modelization of turbulence is achieved by the algebraic model proposed by Baldwin and Lomax. Fully non-linear free surface conditions are satisfied in the model and a method to avoid the incompatibility between free surface conditions and no-slip conditions at the waterline is proposed. Numerical results obtained for a Wigley hull are compared with experimental results.     

一种在网格内部捕捉间断的Walsh函数有限体积方法

传统有限体积或有限元方法假定流动变量在单元内连续,间断仅限于控制体的交界面上,因此它们无法在控制体内部捕捉间断.本文摒弃控制体内流动变量连续的假设,将自身具有间断特点的Walsh基函数应用于有限体积方法,把控制体内的流场变量表示成间断基函数的组合形式.按照Walsh基函数在控制体内引入的间断数目和位置,将控制体单元虚分...     

基于格子Boltzmann两相模型的粘弹流体挤出胀大分析

挤出胀大的数值模拟是非牛顿流体研究中具有挑战性的问题．本文运用格子Boltzmann方法(LBM)分析Oldroyd-B和多阶松弛谱PTT粘弹流体的挤出胀大现象,采用颜色模型模拟出口处粘弹流体和空气的两相流动,通过重新标色获得两种流体的界面,并最终获得胀大的形状．Navier-Stokes方程和本构方程的求解采用双分布函数模型．将胀大的结果与解析解、实验解和单相自由面LBM结果进行了比较,发现格子Boltzmann两相模型结果与解析解和实验结果相吻合,相比于单相模型,收敛速度更快,解的稳定性更高．研究了流道尺寸对胀大率的影响,并对挤出胀大的内在机理进行了分析．     

Numerical simulation of the start-up of Bingham fluid flows in pipelines

This paper describes a mathematical model used to simulate the restart of an axial, compressible and transient flow of a Bingham fluid in circular or annular pipes. The model is based on the mass and momentum conservation equations plus a state equation that relates pressure to density. The viscous effect is modeled by employing a friction factor approach. The governing equations are discretized by using the finite volume method with a first-order upwind scheme, and the resulting non-linear algebraic equations are then solved iteratively. The model results were corroborated by an analytical solution for Newtonian flows. Additionally, the results were also in reasonable agreement with results reported in the literature. We also conducted sensitivity analyses with respect to Reynolds number, aspect ratio, gravity and the non-linear advective terms of the governing equations.     

Modeling and simulation of three‐dimensional extrusion swelling of viscoelastic fluids with PTT,Giesekus and FENE‐P constitutive models

The investigation of the extrusion swelling mechanism of viscoelastic fluids has both scientific and industrial interest. However, it has been traditionally difficult to afford theoretical and experimental researches to this problem. The numerical methodology based on the penalty finite element method with a decoupled algorithm is presented in the study to simulate three‐dimensional extrusion swelling of viscoelastic fluids flowing through out of a circular die. The rheological responses of viscoelastic fluids are described by using three kinds of differential constitutive models including the Phan‐Thien Tanner model, the Giesekus model, and the finite extensible nonlinear elastic dumbbell with a Peterlin closure approximation model. A streamface‐streamline method is introduced to adjust the swelling free surface. The calculation stability is improved by using the discrete elastic‐viscous split stress algorithm with the inconsistent streamline‐upwind scheme. The essential flow characteristics of viscoelastic fluids are predicted by using the proposed numerical method, and the mechanism of swelling phenomenon is further discussed.Copyright © 2013 John Wiley & Sons, Ltd.     

Numerical solution of the eXtended Pom-Pom model for viscoelastic free surface flows

In this paper we present a finite difference method for solving two-dimensional viscoelastic unsteady free surface flows governed by the single equation version of the eXtended Pom-Pom (XPP) model. The momentum equations are solved by a projection method which uncouples the velocity and pressure fields. We are interested in low Reynolds number flows and, to enhance the stability of the numerical method, an implicit technique for computing the pressure condition on the free surface is employed. This strategy is invoked to solve the governing equations within a Marker-and-Cell type approach while simultaneously calculating the correct normal stress condition on the free surface. The numerical code is validated by performing mesh refinement on a two-dimensional channel flow. Numerical results include an investigation of the influence of the parameters of the XPP equation on the extrudate swelling ratio and the simulation of the Barus effect for XPP fluids.     

Numerical study of concentrated fluid–particle suspension flow in a wavy channel

The particle migration effects and fluid–particle interactions occurring in the flow of highly concentrated fluid–particle suspension in a spatially modulated channel have been investigated numerically using a finite volume method. The mathematical model is based on the momentum and continuity equations for the suspension flow and a constitutive equation accounting for the effects of shear‐induced particle migration in concentrated suspensions. The model couples a Newtonian stress/shear rate relationship with a shear‐induced migration model of the suspended particles in which the local effective viscosity is dependent on the local volume fraction of solids. The numerical procedure employs finite volume method and the formulation is based on diffuse‐flux model. Semi‐implicit method for pressure linked equations has been used to solve the resulting governing equations along with appropriate boundary conditions. The numerical results are validated with the analytical expressions for concentrated suspension flow in a plane channel. The results demonstrate strong particle migration towards the centre of the channel and an increasing blunting of velocity profiles with increase in initial particle concentration. In the case of a stenosed channel, the particle concentration is lowest at the site of maximum constriction, whereas a strong accumulation of particles is observed in the recirculation zone downstream of the stenosis. The numerical procedure applied to investigate the effects of concentrated suspension flow in a wavy passage shows that the solid particles migrate from regions of high shear rate to low shear rate with low velocities and this phenomenon is strongly influenced by Reynolds numbers and initial particle concentration. Copyright © 2008 John Wiley & Sons, Ltd.     

聚合物熔体三维挤出胀大的数值模拟

采用有限元方法分析K-BKZ本构方程描述的聚合物熔体的三维挤出胀大.对于本构方程中偏应力张量的计算,首先给出质点的运动轨迹,分段求出局部的变形梯度张量,再求出整体的变形梯度、Cauchy-Green应变张量和 Finger应变张量,沿轨迹采用分段高斯积分计算应力.把应力作为方程的右端项,给出迭代方法,求解非线性方程组.并根据自由面处的边界条件,迭代得出出口处自由面的最终位置.对轴对称流道和矩形流道进行分析计算,并把结果与二维分析和实验结果进行了比较,显示方法是可行的.     

基于格子Boltzmann方法的挤出胀大的数值模拟

将单相格子Boltzmann方法(lattice Boltzmann method, LBM)引入到粘弹流体的瞬态挤出胀大的数值模拟中,建立了基于双分布函数的自由面粘弹性流动格子Boltzmann模型．分析得到的流道中流动速度分布和构型张量结果与理论解十分吻合．对粘弹流体瞬态挤出胀大过程进行了模拟,并分析了运动粘度比和剪切速率对挤出胀大率的影响,得到的胀大率结果与理论分析和其它模拟结果基本一致．表明给出的LBM可以捕捉挤出胀大的瞬态效应．     

Front‐tracking by the level‐set and the volume penalization methods in a two‐phase microfluidic network

Two‐phase immiscible fluids in a two‐dimensional micro‐channels network are considered. The incompressible Stokes equations are used to describe the Newtonian fluid flow, while the Oldroyd‐B rheological model is used to capture the viscoelastic behavior. In order to perform numerical simulations in a complex geometry like a micro‐channels network, the volume penalization method is implemented. To follow the interface between the two fluids, the level‐set method is used, and the dynamics of the contact line is modeled by Cox law. Numerical results show the ability of the method to simulate two‐phase flows and to follow properly the contact line between the two immiscible fluids. Finally, simulations with realistic parameters are performed to show the difference when a Newtonian fluid is pushed by a viscoelastic fluid instead of a Newtonian one. Copyright © 2015 John Wiley & Sons, Ltd.     

Numerical study of the axially symmetric motion of an incompressible viscous fluid in an annulus between two concentric rotating spheres

The research reported herein involved the study of the transient motion of a system consisting of an incompressible Newtonian fluid in an annulus between two concentric, rotating, rigid spheres. The primary purpose of the research was to study the use of a numerical method for analysing the transient motion that results from the interaction between the fluid in the annulus and the spheres which are started suddenly by the action of prescribed torques. The problems considered in this research included cases where: (a) one or both spheres rotate with prescribed constant angular velocities and (b) one sphere rotates due to the action of an applied constant or impulsive t?orque. In this research the coupled solid and fluid equations were solved numerically by employing the finite difference technique. With the approach adopted in this research, only the derivatives with respect to spatial variables were approximated with the use of the finite difference formulae. The steady state problem was also solved as a separate problem (for verification purposes), and the results were compared with those obtained from the solution of the transient problem. Newton's algorithm was employed to solve the algebraic equations which resulted from the steady state problem, and the Adams fourth-order predictor–corrector method was employed to solve the ordinary differential equations for the transient problem. Results were obtained for the streamfunction, circumferential function, angular velocity of the spheres and viscous torques acting on the spheres as a function of time for various values of the system dimensionless parameters.     

THREE-DIMENSIONAL EXTRUDATE SWELL: FORMULATION WITH THE STREAM TUBE METHOD AND NUMERICAL RESULTS FOR A NEWTONIAN FLUID

Stream tube analysis, already applied to two-dimensional extrudate swell problems involving rate and integral constitutive equations for incompressible fluids, is now considered in the problem of free surface determination in a three-dimensional flow situation. The method allows computation of the unknown free surface by considering only a ‘peripheral stream tube’ limited by the wall and the jet surface and an inner stream surface. Those boundary surfaces are determined by considering the conservation equations together with boundary condition equations, solved by the Levenberg/Marquatdt optimization algorithm. The method leads to a considerable reduction in the number of degrees of freedom and the storage area. As in a previous study in the two-dimensional case, singularity problems in the vicinity of the junction points between the wall and the free surface are avoided. However, the numerical method still allows evaluati on of stress peaks due to the singularity at the exit, as may be observed for results obtained with a Newtonian fluid in a duct of square cross-section.     

Computational Modelling of Non-Newtonian Effects on Flow in Channels with Moving Wall Indentations

Non-Newtonian effects in a channel with moving wall indentations are assessed numerically by a finite volume method for solving the unsteady incompressible Navier-Stokes equations and using a power-law model exhibiting shear thinning viscosity and Casson's model as the constitutive equations for the non-Newtonian fluid. The computations show that for a non-Newtonian fluid, there are differences in the velocity profiles and in the structure and size of the reversed flow regions as compared with the corresponding Newtonian fluid. The comparison of non-Newtonian and Newtonian wall shear stress reveals a slight decrease in the magnitude on the average for the non-Newtonian case, eventually resulting in the strength of the “wave train” being slightly weaker than those corresponding to a Newtonian fluid.