The SPH approach to the process of container filling based on non-linear constitutive models

In this work,the transient free surface of container filling with non-linear constitutive equation’s fluids is numerically investigated by the smoothed particle hydrodynamics(SPH) method.Specifically,the filling process of a square container is considered for non-linear polymer fluids based on the Cross model.The validity of the presented SPH is first verified by solving the Newtonian fluid and OldroydB fluid jet.Various phenomena in the filling process are shown,including the jet buckling,jet thinning,splashing or spluttering,steady filling.Moreover,a new phenomenon of vortex whirling is more evidently observed for the Cross model fluid compared with the Newtonian fluid case.     

Modelling of Power-law Fluid Flow Through Porous Media Using Smoothed Particle Hydrodynamics

The flow of non-Newtonian fluids through two-dimensional porous media is analyzed at the pore scale using the smoothed particle hydrodynamics (SPH) method. A fully explicit projection method is used to simulate incompressible flow. This study focuses on a shear-thinning power-law model (n < 1), though the method is sufficiently general to include other stress-shear rate relationships. The capabilities of the proposed method are demonstrated by analyzing a Poiseuille problem at low Reynolds numbers. Two test cases are also solved to evaluate validity of Darcy’s law for power-law fluids and to investigate the effect of anisotropy at the pore scale. Results show that the proposed algorithm can accurately simulate non-Newtonian fluid flows in porous media.     

PTT黏弹性流体的光滑粒子动力学方法模拟

运用光滑粒子流体动力学(smoothed particle hydrodynamics, SPH)方法对基于PHan-Thien-Tanner (PTT)模型的黏弹性流动进行了数值模拟. 首先, 利用SPH方法模拟了基于PTT模型的平板 Poiseuille流, 通过与文献结果的比较, 验证了SPH方法模拟黏弹性流动的准确性和有效性; 随后, 基于PTT模型对黏弹性自由表面流-液滴碰撞问题进行了SPH模拟, 研 究了PTT模型中拉伸参数对碰撞过程的影响. 为了解决张力不稳定问题, 采用简化的 人工应力公式. 数值结果表明, SPH方法可有效而灵活地模拟黏弹自由表面流问题.     

A coupled WC-TL SPH method for simulation of hydroelastic problems

A coupled weakly compressible (WC) and total Lagrangian (TL) smoothed particle hydrodynamics (SPH) method is developed for simulating hydroelastic problems. The fluid phase is simulated using WCSPH method, while the structural dynamics are solved using TLSPH method. Fluid and solid components of the method are validated separately. A sloshing water tank problem is solved to test the WCSPH method while oscillation of a thin plate and large deformation of a cantilever beam are simulated to test the TLSPH method. After validating each component, the coupled WC-TL SPH scheme is used to simulate two benchmark hydroelastic problems. The first test case shows the evolution of water column with an elastic boundary gate, and the second one investigates the breaking water column impact on elastic structures. The agreement between WC-TL SPH results and literature data shows the ability of the proposed method in simulating hydroelastic phenomena.     

粘弹流体轴对称流动的格子Boltzmann方法

基于BGK碰撞模型，通过在迁移方程中引入作用力项，建立了粘弹流体的轴对称格子Boltzmann模型．通过Chapman-Enskog展开，获得了准确的柱坐标下轴对称宏观流动方程．采用双分布函数对运动方程和本构方程进行迭代求解，模拟分析了粘弹流体管道流动，获得了流场中的速度和构型张量的分布，通过与解析解进行比较，验证了模型的准确性．研究了作为粘弹流体流动基准问题的收敛流动，对涡旋位置进行了定量分析，将回转长度的计算结果与有限体积法进行了比较，两种数值结果十分吻合．研究结果表明，模型能够准确表征粘弹流体的轴对称流动，具有较广阔的应用前景．     

Oldroyd-B黏弹性液滴弹跳行为的改进SPH模拟

基于光滑粒子流体动力学SPH(Smoothed Particle Hydrodynamics)方法对Oldroyd-B黏弹性液滴撞击固壁面产生的弹跳行为进行了模拟与分析。首先,为了解决SPH模拟黏弹性自由表面流出现的张力不稳定性问题,联合粒子迁移技术提出了一种改进SPH方法。然后,对Oldroyd-B黏弹性液滴撞击固壁面产生的铺展行为进行了改进SPH模拟,与文献结果的比较验证了方法的有效性。最后,通过降低Reynolds数捕捉到了液滴的弹跳行为;并在此基础上,分析了液滴黏度比、Weissenberg数和Reynolds数对液滴弹跳行为的影响。结果表明,改进SPH方法可有效地模拟黏弹性自由表面流问题;液滴黏度比、Weissenberg数和Reynolds数对液滴最大回弹高度均有显著的影响。     

A corrected symmetric SPH method to simulate viscoelastic free surface flows based on the PTT model

In this work, a corrected symmetric and periodic density reinitialized SPH (CSPDR‐SPH) method is proposed and extended to simulate the viscoelastic free surface flows based on the Phan–Thien–Tanner model. The improvements mainly lie in deriving a corrected symmetric kernel gradient, and combining it with a periodic density reinitialization procedure. In addition, a simple artificial viscosity and a simple artificial stress form are adopted. Thus, the CSPDR‐SPH method has higher accuracy and better stability than the SPH method, and conserves both linear and angular momentums. The consistency and convergence of the CSPDR‐SPH method are justified by approximating a function in one and two dimensions. The merits of CSPDR‐SPH method are demonstrated by several benchmarks. The simple flow in a two‐dimensional channel is investigated to show the capability of the CSPDR‐SPH method to simulate the viscoelastic free surface flow. Then the CSPDR‐SPH method is extended to simulate the impacting drop problem. Numerical results show that the CSPDR‐SPH method can precisely capture the viscoelastic free surface. The Reynolds number, Weissenberg number and elongation parameter have remarkable effect on the flows. Copyright © 2012 John Wiley & Sons, Ltd.     

An SPH multi‐fluid model based on quasi buoyancy for interface stabilization up to high density ratios and realistic wave speed ratios

We introduce a smoothed particle hydrodynamics (SPH) concept for the stabilization of the interface between 2 fluids. It is demonstrated that the change in the pressure gradient across the interface leads to a force imbalance. This force imbalance is attributed to the particle approximation implicit to SPH. To stabilize the interface, a pressure gradient correction is proposed. In this approach, the multi‐fluid pressure gradients are related to the (gravitational and fluid) accelerations. This leads to a quasi‐buoyancy correction for hydrostatic (stratified) flows, which is extended to nonhydrostatic flows. The result is a simple density correction that involves no parameters or coefficients. This correction is included as an extra term in the SPH momentum equation. The new concept for the stabilization of the interface is explored in 5 case studies and compared with other multi‐fluid models. The first case is the stagnant flow in a tank: The interface remains stable up to density ratios of 1:1000 (typical for water and air), in combination with artificial wave speed ratios up to 1:4. The second and third cases are the Rayleigh‐Taylor instability and the rising bubble, where a reasonable agreement between SPH and level‐set models is achieved. The fourth case is an air flow across a water surface up to density ratios of 1:100, artificial wave speed ratios of 1:4, and high air velocities. The fifth case is about the propagation of internal gravity waves up to density ratios of 1:100 and artificial wave speed ratios of 1:4. It is demonstrated that the quasi‐buoyancy model may be used to stabilize the interface between 2 fluids up to high density ratios, with real (low) viscosities and more realistic wave speed ratios than achieved by other weakly compressible SPH multi‐fluid models. Real wave speed ratios can be achieved as long as the fluid velocities are not very high. Although the wave speeds may be artificial in many cases, correct and realistic wave speed ratios are essential in the modelling of heat transfer between 2 fluids (eg, in engineering applications such as gas turbines).     

Viscoelastic flows studied by smoothed particle dynamics

A viscoelastic numerical scheme based on smoothed particle dynamics is presented. The concept goes a step beyond smoothed particle hydrodynamics (SPH) which is a grid-free Lagrangian method describing the flow by fluid-pseudo-particles. The relevant properties are interpolated directly on the resulting movable grid. In this work, the effect of viscoelasticity is incorporated into the ordinary conservation laws by a differential constitutive equation supplied for the stress tensor. In order to give confidence in the methodology we explicitly consider the non-stationary simple corotational Maxwell model in a channel geometry. Without further developments the scheme is applicable to ‘realistic’ models relevant for three-dimensional (3D) viscoelastic flows in complex geometries.     

A finite volume approach for unsteady viscoelastic fluid flows

A finite volume, time‐marching for solving time‐dependent viscoelastic flow in two space dimensions for Oldroyd‐B and Phan Thien–Tanner fluids, is presented. A non‐uniform staggered grid system is used. The conservation and constitutive equations are solved using the finite volume method with an upwind scheme for the viscoelastic stresses and an hybrid scheme for the velocities. To calculate the pressure field, the semi‐implicit method for the pressure linked equation revised method is used. The discretized equations are solved sequentially, using the tridiagonal matrix algorithm solver with under‐relaxation. In both, the full approximation storage multigrid algorithm is used to speed up the convergence rate. Simulations of viscoelastic flows in four‐to‐one abrupt plane contraction are carried out. We will study the behaviour at the entrance corner of the four‐to‐one planar abrupt contraction. Using this solver, we show convergence up to a Weissenberg number We of 20 for the Oldroyd‐B model. No limiting Weissenberg number is observed even though a Phan Thien–Tanner model is used. Several numerical results are presented. Smooth and stable solutions are obtained for high Weissenberg number. Copyright © 2002 John Wiley & Sons, Ltd.     

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.     

A pressure-smoothing scheme for incompressible flow problems

A pressure-smoothing scheme for Stokes and Navier–Stokes flows of Newtonian fluids and for Stokes flow of Maxwell fluids is described. The stress deviator obtained from the calculated velocity field is substituted into the governing equilibrium equation. The resulting equation is then solved to obtain a new, smoothed pressure by a least square finite element method.     

非等温黏弹性复杂流动的改进SPH方法模拟

非等温黏弹性流体广泛存在于自然界和工业生产中,准确预测黏弹性流体的非等温流动机理和复杂流变特性有着重要的应用价值.文章提出一种改进的光滑粒子流体动力学(smoothed particle hydrodynamics,SPH)方法对非等温黏弹性复杂流动进行了数值模拟,其中流体的黏弹特性通过eXtended Pom-Pom本构模型来表征.为了提高模拟结果的精度,采用了一种核函数梯度的修正算法;为了灵活地施加边界条件,发展了边界粒子和虚拟粒子相联合的边界处理方法;为了消除流动过程中的拉伸不稳定性,施加了粒子迁移技术.运用改进SPH方法数值模拟了液滴撞击固壁和F型腔注塑成型问题,通过与Basilisk软件得到的结果进行比较验证了改进SPH方法求解非等温黏弹性流体的有效性.通过利用不同粒子初始间距进行计算,评价了改进SPH方法的数值收敛性.研究了非等温流动相较于等温流动的不同流动特征,深入分析了不同热流变参数对流动过程的影响.数值结果表明,文章提出的改进SPH方法可稳定、准确地描述非等温黏弹性复杂流动的传热机理、复杂流变特性和自由面变化特性.     

On the stability of plane parallel viscoelastic shear flows in the limit of infinite Weissenberg and Reynolds numbers

Elastic effects on the hydrodynamic instability of inviscid parallel shear flows are investigated through a linear stability analysis. We focus on the upper convected Maxwell model in the limit of infinite Weissenberg and Reynolds numbers. We study the effects of elasticity on the instability of a few classes of simple parallel flows, specifically plane Poiseuille and Couette flows, the hyperbolic-tangent shear layer and the Bickley jet.The equation for stability is derived and solved numerically using the spectral Chebyshev collocation method. This algorithm is computationally efficient and accurate in reproducing the eigenvalues. We consider flows bounded by walls as well as flows bounded by free surfaces. In the inviscid, nonelastic case all the flows we study are unstable for free surfaces. In the case of wall bounded flow, there are instabilities in the shear layer and Bickley jet flows. In all cases, the effect of elasticity is to reduce and ultimately suppress the inviscid instability.     

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.     

A numerical study of constitutive models endowed with Pom-Pom molecular attributes

The branched polymer melts are modeled respectively in this investigation by the existing XPP and PTT–XPP models, along with the proposed S-MDCPP (Single/Simplified Modified Double Convected Pom-Pom) model developed on the basis of the existing MDCPP model. A pressure stabilized mass equation is formulated with the finite increment calculus (FIC) process to restrain and further eliminate spurious oscillations of pressure field due to the incompressibility of fluids. The discrete elastic viscous stress splitting (DEVSS) technique is employed, in order to retain an elliptic contribution in the weak form of the momentum equation. An inconsistent streamline-upwind (SU) method is applied to spatially discretize the constitutive equations. The mass, momentum conservation and constitutive equations are discretized and solved by the iterative stabilized fractional step algorithm along with the Crank–Nicolson implicit difference scheme. Thus the finite elements with equal low-order interpolation approximations for velocity–pressure–stress variables can be devised to numerically simulate the viscoelastic contraction flows for branched LDPE melts. The influences of the three viscoelastic constitutive models and the branched arms at the end of the Pom-Pom molecule on the rheological behaviors occurring in this complex flow are discussed. The numerical results demonstrate that the proposed S-MDCPP model is capable of reproducing some properties similar to those predicted by the XPP model in high shear flow and, on the other hand, reproducing some properties similar to those predicted by the PTT–XPP model in high elongational flow. Furthermore, the proposed S-MDCPP model is capable of well identifying the macromolecule topological structures of branched polymer melts.     

A numerical study on nonlinear dynamics of three-dimensional time-depended viscoelastic Taylor-Couette flow

In this paper, three-dimensional viscoelastic Taylor-Couette instability between concentric rotating cylinders is studied numerically. The aim is to investigate and provide additional insight about the formation of time-dependent secondary flows in viscoelastic fluids between rotating cylinders. Here, the Giesekus model is used as the constitutive equation. The governing equations are solved using the finite volume method (FVM) and the PISO algorithm is employed for pressure correction. The effects of elasticity number, viscosity ratio, and mobility factor on various instability modes (especially high order ones) are investigated numerically and the origin of Taylor-Couette instability in Giesekus fluids is studied using the order of magnitude technique. The created instability is simulated for large values of fluid elasticity and high orders of nonlinearity. Also, the effect of elastic properties of fluid on the time-dependent secondary flows such as wave family and traveling wave and also on the critical conditions are studied in detail.     

Two-dimensional viscoelastic flows: experimentation and modeling

Extensive experimental data on the birefringence in converging and diverging flows of a polymeric melt have been obtained. The birefringence and pressure drop measurements were carried out in working cells of planar geometry having different contraction angles and contraction ratios. For investigation of diverging or abrupt expansion flow, the direction of flow in the cells was reversed. The theoretical predictions are based upon the Leonov constitutive equation and a finite element scheme with streamwise integration.In contrast to Newtonian and second-order fluids, viscoelastic fluids at high shear rates show significant differences in pressure drop and birefringence (i.e. stresses) in converging and diverging flows. For a constant flow rate, the pressure drop is higher and the birefringence smaller in diverging flows than in converging flows. This difference increases with increasing flow rate. Further, for the same contraction ratio but different contraction angles, the birefringence maximum increases considerably with contraction angle. In addition, an increase in contraction ratio has the same effect.The viscoelastic constitutive equation of Leonov has been shown to describe all the above viscoelastic effects observed in the experiments. In general, a reasonable agreement between theory and experiment has been obtained, which shows the usefulness of the Leonov model in describing actual flows.     

黏弹性板的大挠度蠕变屈曲

研究了具有初始小挠度受轴向压载黏弹性板的蠕变屈曲问题，在建立控制方程时，利用了von Karman非线性应变-位移关系，并考虑了初始挠度，用标准线性固体模型描述材料的黏弹性特性，在求解非线性积分方程时，利用梯形公式计算记忆积分式，将非线性积分方程化为非线性代数方程进行数值求解，得到了结构的蠕变变形过程，又将问题退化到小挠度情况进行研究，得到了挠度随时间扩展的解析解，分析了瞬时失稳临界载荷、持久临界载荷的物理意义，讨论了考虑几何非线性对黏弹性板蠕变屈曲的影响。