A finite difference analysis of the extrudate swell problem

The extrudate swell phenomenon of a purely viscous fluid is analysed by solving simultaneously the Cauchy momentum equations along with the continuity equation by means of a finite difference method. The circular and planar jet flows of Newtonian and power-law fluids are simulated using a control volume finite difference method suggested by Patankar called SIMPLER (semi-implicit method for pressure-linked equations). This method uses the velocity components and pressure as the primitive variables and employs a staggered grid and control volume for each separate variable. The numerical results show good agreement with the analytical solution of the axisymmetric stick-slip problem and exhibit a Newtonian swelling ratio of 13.2% or 19.2% for a capillary or slit die respectively in accordance with previously reported experimental and numerical results. Shear thinning results in a decrease in swelling ratio, as does the introduction of gravity and surface tension.     

Extrudate swell for a round jet with large surface tension

The problem of extrudate swell of a viscoelastic fluid from a round pipe is studied by the method of domain perturbations. The perturbation problems are solved by a finite-element method through second-order in the flow rate parameter ∈ for small flow rates. The analysis extends the work of Sturges on swelling in two-dimensional channels to round capillary tubes. In perturbation studies for small ∈, the rheology of the fluid may be expressed by three parameters, the viscosity and the two constants α₁ and α₂ appearing at order two in the expansion of the extra stress around zero shear. Surface tension has an important influence on the shape of the jet at low speeds. The shape of the surface on a round jet depends on α₁ and α₂, in the plane jet only on α₁. The analysis predicts that no matter what the constitutive equation may be, the jet will first contract if the radius of the pipe is sufficiently small. The contraction takes place in a length less than $\text{1}\text{10}$ the diameter of the jet and is followed by a swell. The contraction is usually small and may be hard to observe. There are five different contributions to the jet shape at second-order but only the viscoelastic ones persist as the pipe radius goes to zero.     

Effect of the coflow stream on a plane wall jet

We propose in this work to study an isothermal and a non-isothermal laminar plane wall jet emerging in a coflow steam. The numerical solution of the governing equations was performed by a finite difference method. In this work, we are interested in the study of the influence of Grashof numbers on the wall jet emerging in a medium at rest. Further, we will examine the effect of the coflow stream on the behavior of the dynamic and thermal properties of the wall jet subjected to a constant temperature. A comparison with a simple wall jet is carried out. The results show that for a buoyant wall jet, two parameters can influence the flow: the inertial and buoyancy forces. The velocity effect indicates that the potential core length increases with the velocity ratio. We are also showed that when using a momentum length scale, the normalized longitudinal maximum velocity can reach an asymptotic curve at different velocity ratios.     

Delayed die swell

The experiments reported here establish that there is a general critical condition associated with die swell which we call delayed die swell. This condition is defined by a critical speed which is the area-averaged velocity, the extrusion velocity, at the exit of the pipe when the swell is first delayed. The delayed swell ratio and delay distance first increase for larger, post-critical values of the extrusion velocity; then the increases are terminated either by instabilities or by smoothing. The maximum post-critical velocity at the pipe exit was always greater than the shear wave speed measured on the shear-wave-speed meter. The post critical area averaged velocity at the position of maximum swell before termination was always less than the shear wave speed. There were always points in the region of swelling where the ratio of the local velocity to the shear wave speed, the viscoelastic Mach number, was unity. The swelling of the jet is a nonlinear phenomenon which we suggest is finally terminated either by instability or when the variations of the velocity, vorticity and stress field are reduced to zero by the inward propagation of shear waves from the free surface of the jet. This propagation is generated by discontinuous “initial” data along χ in which the prescribed values of velocity at the boundary change from no-slip in the pipe to no-shear in the jet. The measurements raise the possibility that the delay may be associated with a change of type from supercritical to subcritical flow.     

A marker-and-cell approach to viscoelastic free surface flows using the PTT model

This work is concerned with the development of a numerical method capable of simulating two-dimensional viscoelastic free surface flows governed by the non-linear constitutive equation PTT (Phan-Thien–Tanner). In particular, we are interested in flows possessing moving free surfaces. The fluid is modelled by a marker-and-cell type method and employs an accurate representation of the fluid surface. Boundary conditions are described in detail and the full free surface stress conditions are considered. The PTT equation is solved by a high order method which requires the calculation of the extra-stress tensor on the mesh contour. The equations describing the numerical technique are solved by the finite difference method on a staggered grid. In order to validate the numerical method fully developed flow in a two-dimensional channel was simulated and the numerical solutions were compared with known analytic solutions. Convergence results were obtained throughout by using mesh refinement. To demonstrate that complex free surface flows using the PTT model can be computed, extrudate swell and a jet flowing onto a rigid plate were simulated.     

Numerical simulation of delayed die swell

In a recent paper, Joseph et al. showed that, for a number of viscoelastic fluids, one can observe the phenomenon of delayed die swell beyond a critical extrusion velocity, or beyond a critical value of the viscoelastic Mach number. Giesekus had also observed that delayed die swell is a critical phenomenon.In the present paper, we find a set of material and flow parameters under which it is possible to simulate delayed die swell. For the viscoelastic flow calculation, we use the finite element algorithm with sub-elements for the stresses and streamline upwinding in the discretized constitutive equations. For the free surface, we use an implicit technique which allows us to implement Newton's method for solving the non-linear system of equations. The fluid is Oldroyd-B which, in the present problem, is a singular perturbation of the Maxwell fluid. The results show very little sensitivity to the size of the retardation time. We also show delayed die swell for a Giesekus fluid.This paper is dedicated to Professor Hanswalter Giesekus on the occasion of his retirement as Editor of Rheologica Acta.     

Computations for a jet impinging obliquely on a flat surface

A SIMPLE-C algorithm and Jones-Launder k-ε two-equation turbulence model are used to simulate a two-dimensional jet impinging obliquely on a flat surface. Both the continuity and momentum equations for the unsteady state are cast into suitable finite difference equations. The pressure, velocity, turbulent kinetic energy and turbulent energy dissipation rate distributions are solved and show good agreement with various experimental data. The calculations show that the flow field structure of the jet impinging obliquely on a flat surface is strongly affected by the oblique impingement angle. The maximum pressure zone of the obliquely impinging jet flow field moves towards the left as the oblique impingement angle is decreased.     

Prediction of extrudate swell in polymer melt extrusion using an Arbitrary Lagrangian Eulerian (ALE) based finite element method

Accurate prediction of extrudate (die) swell in polymer melt extrusion is important as this helps in appropriate die design for profile extrusion applications. Extrudate swell prediction has shown significant difficulties due to two key reasons. The first is the appropriate representation of the constitutive behavior of the polymer melt. The second is regarding the simulation of the free surface, which requires special techniques in the traditionally used Eulerian framework. In this paper we propose a method for simulation of extrudate swell using an Arbitrary Lagrangian Eulerian (ALE) technique based finite element formulation. The ALE technique provides advantages of both Lagrangian and Eulerian frameworks by allowing the computational mesh to move in an arbitrary manner, independent of the material motion. In the present method, a fractional-step ALE technique is employed in which the Lagrangian phase of material motion and convection arising out of mesh motion are decoupled. In the first step, the relevant flow and constitutive equations are solved in Lagrangian framework. The simpler representation of polymer constitutive equations in a Lagrangian framework avoids the difficulties associated with convective terms thereby resulting in a robust numerical formulation besides allowing for natural evolution of the free surface with the flow. In the second step, mesh is moved in ALE mode and the associated convection of the variables due to relative motion of the mesh is performed using a Godunov type scheme. While the mesh is fixed in space in the die region, the nodal points of the mesh on the extrudate free surface are allowed to move normal to flow direction with special rules to facilitate the simulation of swell. A differential exponential Phan Thien Tanner (PTT) model is used to represent the constitutive behavior of the melt. Using this method we simulate extrudate swell in planar and axisymmetric extrusion with abrupt contraction ahead of the die exit. This geometry allows the extrudate to have significant memory for shorter die lengths and acts as a good test for swell predictions. We demonstrate that our predictions of extrudate swell match well with reported experimental and numerical simulations.     

一种三阶有限体积法及其在欠膨胀射流激波结构数值模拟中的应用

以喷管出口欠膨胀射流为研究对象,在Lagrange坐标系下建立欠膨胀射流二维积分形式的流动方程。通过在单元交接面处进行三阶ENO(essentially nonoscillatory)格式插值,构造得到一种适用于求解该方程的三阶ENO有限体积法。采用该格式对一维Sod激波管算例和喷管出口欠膨胀射流进行数值计算。计算结果表明,该方法具有高精度、基本无振荡的特点,能很好地捕捉包含激波、滑移线以及三波交点等复杂流场波系结构。计算得到的波系结构中马赫盘的位置与实验结果吻合很好,相对误差小于1.1%。     

Solution to the problem of a swirling jet from an annular orifice

The problem of the propagation of a laminar immersed fan jet with swirling was considered in [1–3]. In [1], the jet source scheme was used to find a self-similar solution for a weakly swirling jet. An attempt to solve by an integral method the analogous problem for a jet emanating from a slit of finite size was made in [2]. In [3], the equations of motion for a jet with arbitrary swirling were reduced under a number of assumptions to the equations that describe the flow of a flat immersed jet. This paper gives the numerical solution to the problem of the propagation of a radial jet emanating with arbitrary swirling from a slit of finite size and an analytic solution for the main section of the jet.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 49–54, March–April, 1991.     

Die swell measurements of second‐order fluids: numerical experiments

An analysis of the flow of a second‐order fluid is presented. Reference values for some variables are defined, and with these a non‐dimensional formulation of the governing equations. From this formulation, three dimensionless numbers appear; one is the Reynolds number, and two numbers that are called the first‐ and second‐dimensionless normal stress (NSD) coefficients. The equations of motion are solved by a finite element method using a commercially available program (Fidap), and the steady state converged solution was used to measure the die swell. The factors that influence die swell and that are studied in this work include: the die geometry for circular cross sectional dies, including tubular, converging, diverging, half‐converging/half‐tubular shapes; fluid characteristics such as Reynolds number and first‐ and second‐DNS coefficients (both positive and negative values); and flow rates, as determined by the maximum velocity in a parabolic velocity profile at the entrance to the die. The results suggest that shear and deformation histories of the fluid directly influence not only swell characteristics, but also convergence characteristics of the numerical simulation. © 1999 John Wiley & Sons, Ltd.     

SELF-INDUCED SLOSHING CAUSED BY AN UPWARD ROUND JET IMPINGING ON THE FREE SURFACE

Self-induced sloshing was found to occur in a cylindrical tank when an upward round jet on the tank axis created a surface swell at the impingement point. The conditions under which the sloshing occurred were investigated experimentally, together with the sloshing modes and frequencies. The sloshing was found to appear when the inlet velocity exceeded a certain value, which depends on the inlet-surface distance. The mode always had one diametrical node, and a lower mode appeared at increased inlet-surface distance and velocity. A feedback model was proposed in order to explain the energy supply mechanism to the sloshing, which was composed of a jet deflection process caused by a lateral pressure gradient and a pressure imbalance generation process caused by jet movement. The lateral pressure gradient deflects the jet, resulting in the displacement of the surface impingement point. When the jet moves, transformation of the jet-induced swell does not follow the movement immediately, generating an imbalance of pressure. The imbalance caused by the motion of the swell is considered to supply energy to the sloshing. The calculated jet deflection caused by the lateral pressure gradient was found to agree with the experimental findings. The measured lateral pressure gradient was calculated as the sum of the lateral pressures caused by the sloshing motion and by the jet impingement point movement. The calculated oscillating region in the water-depth jet-velocity map was found to agree fairly well with the experimental findings, which suggests the validity of the model.     

Finite element analysis of die swell of a highly elastic fluid

A mixed finite element method is applied to the die swell calculation of ana Oldroyd fluid B. The use of large entry lengths together with the presence of the retardation time in the constitutive equations allow us to reach values of the recoverable shear as high as four for the flow emerging from slit and circular dies, with swelling ratios of the order of 2. The numerical results are in good agreement with some available experimental data.     

The basic character of five two‐phase flow model equation sets

Three test problems were simulated using five different two‐phase flow model equation sets from the open literature. The test problems chosen were a fluidized bed, a batch settling, and a horizontal jet impingement on a vertical wall. These three problems demonstrate an important cross‐section of physical phenomena, such as fluidized bed voidage oscillations, phase separation, countercurrent flow, and jet formation. The dispersed flow regime is selected for all three problems. The study was performed to assess the basic character of the five‐field equation sets responding to the same initial and boundary conditions and using the same finite difference numerical scheme. The general performance of the five equation sets was found to be similar, even though one of them was ill posed as an initial‐value problem. Broad trends are the same and quantitative differences could be assessed by examining the fine structure of the results. None of the equation sets could be entirely rejected on the basis of producing physically impossible or unacceptable results. Copyright © 2000 John Wiley & Sons, Ltd.     

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.     

Numerical Study of Turbulent Jet Ignition in a Lean Premixed Configuration

Direct numerical simulations (DNS) of a hot combustion product jet interacting with a lean premixed hydrogen-air coflow are conducted to fundamentally investigate turbulent jet ignition (TJI) in a three-dimensional configuration. TJI is an efficient method for initiating and controlling combustion in ultra-lean combustion systems. Fully compressible gas dynamics and species equations are solved with high order finite difference methods. The hydrogen-air reaction is simulated with a reliable detailed chemical kinetics mechanism. The physical processes involved in the TJI-assisted combustion are investigated by considering the flame heat release, temperature, species concentrations, vorticity, and Baroclinc torque. The complex turbulent flame and flow structures are delineated in three main: i) hot product jet, ii) burned-mixed, and iii) flame zones. In the TJI-assisted combustion, the flow structures and the flame features such as flame speed, temperature, and species distribution are found to be quite different than those in “standard” turbulent premixed combustion due to the existence of a high energy turbulent hot product jet. The flow structures and statistics are also found to be different than those normally seen in non-isothermal non-reacting jets.     

Finite‐element/level‐set/operator‐splitting (FELSOS) approach for computing two‐fluid unsteady flows with free moving interfaces

The present work is devoted to the study on unsteady flows of two immiscible viscous fluids separated by free moving interface. Our goal is to elaborate a unified strategy for numerical modelling of two‐fluid interfacial flows, having in mind possible interface topology changes (like merger or break‐up) and realistically wide ranges for physical parameters of the problem. The proposed computational approach essentially relies on three basic components: the finite element method for spatial approximation, the operator‐splitting for temporal discretization and the level‐set method for interface representation. We show that the finite element implementation of the level‐set approach brings some additional benefits as compared to the standard, finite difference level‐set realizations. In particular, the use of finite elements permits to localize the interface precisely, without introducing any artificial parameters like the interface thickness; it also allows to maintain the second‐order accuracy of the interface normal, curvature and mass conservation. The operator‐splitting makes it possible to separate all major difficulties of the problem and enables us to implement the equal‐order interpolation for the velocity and pressure. Diverse numerical examples including simulations of bubble dynamics, bifurcating jet flow and Rayleigh–Taylor instability are presented to validate the computational method. Copyright © 2004 John Wiley & Sons, Ltd.     

A numerical study of long rod impact onto a large target

The normal impact of a long rod onto a large target is studied using an Eulerian finite difference scheme. The impact velocity of 1.5 km s^?1 is chosen to be low enough for metal strength to be an important parameter characterising the impact. It is also sufficiently high for the rod to flow as a jet, which is consumed as it penetrates the target.The first numerical study neglects the material strength representation, so that the flow is inviscid. On impact, the flat face of the rod strikes the flat face of the target, and one dimensional analysis is used to check numerical predictions for the initial impact pressure and velocity. A steady state penetration is quickly achieved, at a velocity which is in agreement with theoretical predictions of jet flow. In the second numerical study, an elastic-perfectly plastic representation of material strength is included within the calculation. It is then found that the rod has to travel several rod diameters into the target before the penetration velocity falls from the one dimensional impact value to a steady state value. This implies that the resistance to flow increases with the depth into the target, and consequently the penetration achieved by a rod will be dependent on its diameter, as well as its length.     

The end correction for power-law fluids in the capillary rheometer

The finite element scheme developed by Nickell, Tanner and Caswell is used to compute the entry and exit losses for creeping flow of power-law fluids in a capillary rheometer. The predicted entry losses for a Newtonian fluid agree well with available experimental and theoretical results. The entry losses for inelastic power-law fluids increased with decreasing flow behaviour index and show an increasing deviation from available upper bound results as the flow behaviour index in the power-law decreases.The exit losses are found to be finite for inelastic power-law fluids and increase as the flow behaviour index decreases. The predicted die swell for Newtonian fluids agrees well with the available experimental data while the influence of shear thinning is to reduce the die swell.The end correction which is the sum of the entry and exit losses relative to twice the viscometric wall shear stress varies from 0.834 for n = 1 to 2.917 for n = 1/6. This figure reaches a very high value as n tends to zero. The experimental variation in the Couette correction factor in capillary rheometry is explained in terms of the shear thinning characteristics of the fluid. It is concluded that the exit flow is not viscometric, contrary to a common assumption.