Viscoelastic Free Surface Flows Research Articles

Simulation of viscoelastic free-surface flows with the Particle Finite Element Method

Viscoelastic fluids are central in numerous applications from polymer manufacturing to the pharmaceutical industry and biological research. However, since analytical solutions are generally not available or too complex, it is common practice to study free-surface viscoelastic flows through numerical simulation techniques. This work proposes the use of the so-called particle finite element method (PFEM), a Lagrangian approach combining standard FEM techniques with a remeshing strategy. The PFEM is able to efficiently handle mesh distortion and to accurately track the free-surface evolution. Therefore, it is exploited in this work to deal with large displacements problems in the context of nonlinear viscoelasticity. An implementation of the Oldroyd-B constitutive model in the PFEM framework is here presented including details regarding how to deal with the transfer of the internal variables during remeshing events. Additionally, an innovative approach to impose unilateral Dirichlet boundary conditions ensuring optimal mass conservation is presented. The implementation is verified with two free-surface highly viscous benchmark flows: the impacting drop and the jet buckling problems. The results show perfect agreement with those obtained with other numerical techniques. The proposed framework opens the way for using PFEM in various applications, ranging from polymer extrusion to more sophisticated scenarios involving viscoelastic and viscoelasto-plastic constitutive laws.

SPH simulations of transient non-isothermal viscoelastic flows with free surfaces

• Transient non-isothermal viscoelastic free surface flows are simulated by SPH. • SPH discretization of the temperature equation is derived. • The impacting droplet and the filling process of circular disc with core are studied. • Both Oldroyd-B and UCM fluid models are considered. • The work provides references to handle non-isothermal viscoelastic flows in an entirely meshfree framework. This study presents a smoothed particle hydrodynamics (SPH) method for simulating transient non-isothermal viscoelastic flows with free surfaces. The basic equations governing the non-isothermal free surface flow of an Oldroyd-B fluid are considered and approximated by SPH. To model non-isothermal fluid flows, a working SPH discretization of the temperature equation is derived, and the temperature-dependent viscosity and relaxation time are represented by the time–temperature superposition principle. The proposed SPH method is first validated by solving the viscoelastic Poiseuille flow problem under isothermal and non-isothermal conditions and comparing the SPH solutions with the analytical solutions or those obtained by other methods. Then, the SPH method is extended to deal with the impact of a non-isothermal viscoelastic droplet with a cold wall and the non-isothermal filling process of a circular disc with a core. The convergence of the method is verified by four particle spacings of different levels of refinement. The effects of some important parameters, including the temperature dependency coefficient, reference temperature of the fluid, and Péclet number, on the flow behavior are analyzed in detail. All the results demonstrate that the proposed SPH method is capable of accurately and stably simulating transient non-isothermal viscoelastic free surface flows in an entirely meshfree framework.

Development of SPH for simulation of non-isothermal viscoelastic free surface flows with application to injection molding

In the present work, we focus on developing the smoothed particle hydrodynamics (SPH) method for simulation of non-isothermal viscoelastic free surface flows with application to the injection molding process. Specifically, the viscoelastic fluid is governed by the eXtended Pom-Pom (XPP) constitutive equation. The temperature-dependent viscosity and relaxation time are modeled using the Arrhenius relation. The governing equations are discretized by an improved SPH scheme, as presented by Xu and Yu, Appl. Math. Model. 73 (2019) pp. 715–731. We first verify the validity of the SPH method by solving the isothermal/non-isothermal Poiseuille flow of an XPP fluid and comparing the SPH solutions with the analytical solutions or finite volume method solutions. Then, we extend the method to 2D non-isothermal viscoelastic injection molding process of a C-shaped cavity. Some relevant physical information such as temperature, pressure, velocity, flow-induced stresses during the molding process are displayed. The convergence of the method is examined with several different particle sizes. The effects of the Péclet number, Reynolds number, Weissenberg number, viscous ratio, and XPP parameters, on the evolutions of the temperature and flow-induced stresses are analyzed in details. Results for 3D non-isothermal viscoelastic injection molding process of a door handle are lastly presented.

A Backwards-Tracking Lagrangian-Eulerian Method for Viscoelastic Two-Fluid Flows

A new Lagrangian–Eulerian method for the simulation of viscoelastic free surface flow is proposed. The approach is developed from a method in which the constitutive equation for viscoelastic stress is solved at Lagrangian nodes, which are convected by the flow, and interpolated to the Eulerian grid with radial basis functions. In the new method, a backwards-tracking methodology is employed, allowing for fixed locations for the Lagrangian nodes to be chosen a priori. The proposed method is also extended to the simulation of viscoelastic free surface flow with the volume of fluid method. No unstructured interpolation or node redistribution is required with the new approach. Furthermore, the total amount of Lagrangian nodes is significantly reduced when compared to the original Lagrangian–Eulerian method. Consequently, the method is more computationally efficient and robust. No additional stabilization technique, such as both-sides diffusion or reformulation of the constitutive equation, is necessary. A validation is performed with the analytic solution for transient and steady planar Poiseuille flow, with excellent results. Furthermore, the proposed method agrees well with numerical data from the literature for the viscoelastic die swell flow of an Oldroyd-B model. The capabilities to simulate viscoelastic free surface flow are also demonstrated through the simulation of a jet buckling case.

A least-squares particle model with other techniques for 2D viscoelastic fluid/free surface flow

In this work, a Lagrangian finite pointset model (FPM) is first developed to solve the 2D viscoelastic fluid governing equations, and then a corrected particle shifting technique (CPST) is introduced to eliminate the tensile instability in the long time simulations and added in the above FPM scheme (named as the FPMT). Subsequently, a coupled particle method (FPMT-SPH) is tentatively proposed to simulate the viscoelastic free surface flow, in which the FPMT method is used in the interior of fluid domain and the SPH method is adopted to treat the free surface near the boundary. The proposed FPMT method and FPMT-SPH method are motivated by: a) the spatial derivatives of the velocity and the viscoelastic stress are approximated and obtained by the weighted least squares method; b) the pressure is accurately solved by the application of projection method with a local iterative procedure; c) a corrected particle shifting technique is added to remedy the tensile instability (FPMT); d) an identification technique of free-surface particles is given in the FPMT-SPH method. The accuracy and the convergence of the proposed FPMT scheme for viscoelastic flow are first discussed by solving the planar flow based on the Oldroyd-B model, and compared with the analytical solutions. Secondly, the validity of the CPST is tested by several benchmarks, and compared with other numerical results. Thirdly, the viscoelastic lid-driven cavity flow at high Weissenberg number is simulated and compared with grid-based results, for further illustrating the robustness and the ability of the proposed FPMT method. Finally, the proposed coupled FPMT-SPH method is used to simulate the challenging free surface flow problem of a viscoelastic droplet impacting and spreading on rigid. All the numerical results show that the proposed particle method for the viscoelastic fluid or free surface flow is robust and reliable.

Macroscopic and mesoscopic simulation of viscoelastic free surface flow in gas‐assisted injection moulding process

AbstractIn this work, an improved simple coupled level‐set and volume of fluid (S‐CLSVOF) method is proposed to trace the moving interfaces in gas assisted injection moulding (GAIM) process based on the viscoelastic FENE‐P model. Firstly, the Kreisselmeier‐Steinhauser (KS) function constructed by means of Boolean operations is adopted as the shape level set (LS) function to precisely represent the complex mould cavities. Then, the benchmark problem of two‐dimensional deformation is used to verify the ability of the S‐CLSVOF method for capturing the moving interfaces. The stress birefringence is calculated and compared with the experiment result. Finally, the proposed method is further applied to the mould filling and gas penetration processes in GAIM. The macroscopic bubble appearance, temperature distribution, and the behaviour of the mesoscopic molecular orientation are shown and analyzed in detail. Due to the complexity of the gas‐liquid interaction, the phenomenon of asymmetrical gas flow is clearly observed in the gas penetration process. The influences of the macroscopic parameters on the macroscopic gas flows and mesoscopic molecular orientation are also discussed, such as insert positions, melt temperatures, and gas delay time. The numerical results illustrate that the coupled method can be applied to the multiscale numerical simulation of viscoelastic flows with complex free surfaces and provide significant guidance for the GAIM process.

A technique to remove the tensile instability in weakly compressible SPH

When smoothed particle hydrodynamics (SPH) is directly applied for the numerical simulations of transient viscoelastic free surface flows, a numerical problem called tensile instability arises. In this paper, we develop an optimized particle shifting technique to remove the tensile instability in SPH. The basic equations governing free surface flow of an Oldroyd-B fluid are considered, and approximated by an improved SPH scheme. This includes the implementations of the correction of kernel gradient and the introduction of Rusanov flux into the continuity equation. To verify the effectiveness of the optimized particle shifting technique in removing the tensile instability, the impacting drop, the injection molding of a C-shaped cavity, and the extrudate swell, are conducted. The numerical results obtained are compared with those simulated by other numerical methods. A comparison among different numerical techniques (e.g., the artificial stress) to remove the tensile instability is further performed. All numerical results agree well with the available data.

Simulation of the 3D viscoelastic free surface flow by a parallel corrected particle scheme**Project supported by the Natural Science Foundation of Jiangsu Province, China (Grant Nos. BK20130436 and BK20150436) and the Natural Science Foundation of the Higher Education Institutions of Jiangsu Province, China (Grant No. 15KJB110025).

In this work, the behavior of the three-dimensional (3D) jet coiling based on the viscoelastic Oldroyd-B model is investigated by a corrected particle scheme, which is named the smoothed particle hydrodynamics with corrected symmetric kernel gradient and shifting particle technique (SPH_CS_SP) method. The accuracy and stability of SPH_CS_SP method is first tested by solving Poiseuille flow and Taylor–Green flow. Then the capacity for the SPH_CS_SP method to solve the viscoelastic fluid is verified by the polymer flow through a periodic array of cylinders. Moreover, the convergence of the SPH_CS_SP method is also investigated. Finally, the proposed method is further applied to the 3D viscoelastic jet coiling problem, and the influences of macroscopic parameters on the jet coiling are discussed. The numerical results show that the SPH_CS_SP method has higher accuracy and better stability than the traditional SPH method and other corrected SPH method, and can improve the tensile instability.

Numerical simulation of flow-induced stresses in mold filling process using a three-dimensional two-phase flow model

The flow-induced stresses which arise during the filling stage have significant influences on the mechanical and optical properties of injection-molded products. The prediction of flow-induced stresses in mold-filling process is a challenging problem, which includes viscoelastic polymer melt free surface flow. Thus, this paper presents a three-dimensional (3D) non-isothermal two-phase flow model to quantitative predict the flow-induced stresses in polymer mold-filling process based on the XPP model. In this model, the governing equations for gas and viscoelastic melt are united into a system namely the generalized Navier–Stokes equations. In order to improve the stability of the numerical method in 3D, an enhanced treatment of stress solid boundaries is proposed. The stresses on solid boundaries are calculated using an effective second-order Runge–Kutta method. The model is successfully solved by using the finite volume method, and the melt front is accurately captured by level set method. The validity of the method is verified by two benchmark problems. Then the mold-filling process of a rectangular plate is studied numerically, obtaining the evolutions of flow-induced stresses at different levels along the gap-wise direction, which can not be simulated by 2.5D model. The predicted flow-induced birefringence is in agreement with the experimental result reported in the literature. Furthermore, the effects of injection velocity and some model parameters of XPP model on flow-induced stresses are numerically predicted. The numerical results show that the 3D simulation technology can effectively predict the flow-induced stresses and birefringence in non-isothermal viscoelastic polymer mold-filling process. The results can provide the theory foundation for improving the properties of products in actual production processes.

A new numerical framework to simulate viscoelastic free-surface flows with the finite-volume method

A new method for the simulation of 2D viscoelastic flow is presented. Numerical stability is obtained by the logarithmic-conformation change of variable, and a fully-implicit pure-streamfunction flow formulation, without use of any artificial diffusion. As opposed to other simulation results, our calculations predict a hydrodynamic instability in the 4:1 contraction geometry at a Weissenberg number of order 4. This new result is in qualitative agreement with the prediction of a non-linear subcritical elastic instability in Poiseuille flow. Our viscoelastic flow solver is coupled with a volume-of-fluid solver in order to predict free- surfaces in extrusion.

An improved particle method for simulation of the non-isothermal viscoelastic fluid mold filling process

A non-isothermal molding filling process for a non-Newtonian viscoelastic fluid is investigated in this work. An extended smoothed particle hydrodynamics with diffusive term and kernel gradient correction (extended SPH_DTKGC) method is first presented to simulate the non-isothermal molding filling for the viscoelastic fluid based on the coupled eXtended Pom-Pom (XPP) and thermal power-law model. Meanwhile, a suitable discrete temperature scheme in the SPH frame is proposed. And a new way to treat the zig-zag boundary is given and expected to deal with arbitrary shaped rigid wall. Subsequently, the merits of the proposed boundary treatment are demonstrated by a spin-down problem and the filling process in a C-shaped mould. The ability of the extended SPH_DTKGC to simulate the viscoelastic free surface flow is verified by the several classical problems. Two benchmark problems are presented to display the reliability of the proposed temperature scheme. The non-isothermal filling process for a power-law fluid is performed to show the feasibility of the extended SPH_DTKGC to simulate the non-isothermal mold filling process. Then the extended SPH_DTKGC method combined with other improvements is applied to the non-isothermal filling processes based on the coupled XPP and power-law model. Especially, the influences of the parameters of the XPP model on the physical quantities, such as the first normal stress difference and the temperature field, are numerically predicted.

A New Method to Simulate Free Surface Flows for Viscoelastic Fluid

Free surface flows arise in a variety of engineering applications. To predict the dynamic characteristics of such problems, specific numerical methods are required to accurately capture the shape of free surface. This paper proposed a new method which combined the Arbitrary Lagrangian-Eulerian (ALE) technique with the Finite Volume Method (FVM) to simulate the time-dependent viscoelastic free surface flows. Based on an open source CFD toolbox called OpenFOAM, we designed an ALE-FVM free surface simulation platform. In the meantime, the die-swell flow had been investigated with our proposed platform to make a further analysis of free surface phenomenon. The results validated the correctness and effectiveness of the proposed method for free surface simulation in both Newtonian fluid and viscoelastic fluid.

A new model for shallow viscoelastic free-surface flows forced by gravity on rough inclined bottom

A thin-layer model for shallow viscoelastic free-surface gravity flows on slippery topographies around a flat plane has been derived recently in [Bouchut-Boyaval, M3AS (23) 2013]. We show here how the model can be modified for flows on rugous topographies varying around an inclined plane. The new reduced model extends the scope of one derived in [Bouchut-Boyaval, M3AS (23) 2013]. It is one particular thin-layer model for free-surface gravity flows among many ones that can be formally derived with a generic unifying procedure. Many rheologies and various shallow flow regimes have already been treated within a single unified framework in [Bouchut-Boyaval, HAL-ENPC (00833468) 2013]. The initial full model used here as a starting point is however a little different to one used in [Bouchut-Boyaval, HAL-ENPC (00833468) 2013], although the new thin-layer model is very similar to the one derived therein. Precisely, here, the bulk dissipation (due to e.g. viscosity) is neglected from the beginning, like in [Bouchut-Boyaval, M3AS (23) 2013]. Moreover, unlike in [Bouchut-Boyaval, HAL-ENPC (00833468) 2013], we perform here numerical simulations . The interest of the extension is illustrated in a physically interesting situation where new stationary solutions exist. To that aim, the Finite-Volume method proposed in [Bouchut-Boyaval, M3AS (23) 2013] needs to be modified, with an adequate discretization of the new source terms. Interestingly, we can also numerically exhibit an apparently new kind of “roll-wave” solution.

Modeling Die Swell of Second-Order Fluids Using Smoothed Particle Hydrodynamics

This work presents the development of a weakly compressible smoothed particle hydrodynamics (WCSPH) model for simulating two-dimensional transient viscoelastic free surface flow which has extensive applications in polymer processing industries. As an illustration for the capability of the model, the extrudate or die swell behaviors of second-order and Olyroyd-B polymeric fluids are studied. A systematic study has been carried out to compare constitutive models for second-order fluids available in literature in terms of their ability to capture the physics behind the swelling phenomenon. The effects of various process and rheological parameters on the die swell such as the extrusion velocity, normal stress coefficients, and Reynolds and Deborah numbers have also been investigated. The models developed here can predict both swelling and contraction of the extrudate successfully. The die swell of a second-order fluid was solved for a wide range of Deborah numbers and for two different Reynolds numbers. The numerical approach was validated through the solution of fully developed Newtonian and non-Newtonian viscoelastic flows in a two-dimensional channel as well as modeling the die swell of a Newtonian fluid. The results of these three benchmark problems were compared with analytic solutions and numerical results in literature when pertinent, and good agreements were obtained.

Three-dimensional transient complex free surface flows: Numerical simulation of XPP fluid

In this paper we present a finite difference MAC-type approach for solving three-dimensional viscoelastic incompressible free surface flows governed by the eXtended Pom–Pom (XPP) model, considering a wide range of parameters. The numerical formulation presented in this work is an extension to three-dimensions of our implicit technique [Journal of Non-Newtonian Fluid Mechanics 166 (2011) 165–179] for solving two-dimensional viscoelastic free surface flows. To enhance the stability of the numerical method, we employ a combination of the projection method with an implicit technique for treating the pressure on the free surfaces. The differential constitutive equation of the fluid is solved using a second-order Runge–Kutta scheme. The numerical technique is validated by performing a mesh refinement study on a pipe flow, and the numerical results presented include the simulation of two complex viscoelastic free surface flows: extrudate-swell problem and jet buckling phenomenon.

Numerical simulation of 3D-unsteady viscoelastic free surface flows by improved smoothed particle hydrodynamics method

In this paper, a working smoothed particle hydrodynamics (SPH) method is introduced to solve three-dimensional (3D) transient viscoelastic flows with complex free surfaces. In order to alleviate the unphysical behavior of fracture and particle clustering in fluid stretching which is the so-called tensile instability, an artificial stress term is incorporated into the momentum equation. To facilitate the enforcement of 3D wall boundaries, a new boundary treatment technique, which can observably improve the computational efficiency, is proposed. The proposed SPH method is validated by solving the Hagen–Poiseuille flow of an Oldroyd-B fluid and comparing the SPH results with the available analytical solutions. Two challenging fluid flow problems, namely, a viscoelastic drop impacting on a rigid plate and jet buckling, are simulated to demonstrate the capability of the proposed SPH method in handing 3D viscoelastic free surface flows. Results for a Newtonian fluid are also shown for comparison. All numerical results obtained are in agreement with the available data.

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

SUMMARYIn 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.

Numerical prediction of three-dimensional time-dependent viscoelastic extrudate swell using differential and algebraic models

This study investigates the numerical simulation of three-dimensional time-dependent viscoelastic free surface flows using the Upper-Convected Maxwell (UCM) constitutive equation and an algebraic explicit model. This investigation was carried out to develop a simplified approach that can be applied to the extrudate swell problem. The relevant physics of this flow phenomenon is discussed in the paper and an algebraic model to predict the extrudate swell problem is presented. It is based on an explicit algebraic representation of the non-Newtonian extra-stress through a kinematic tensor formed with the scaled dyadic product of the velocity field. The elasticity of the fluid is governed by a single transport equation for a scalar quantity which has dimension of strain rate. Mass and momentum conservations, and the constitutive equation (UCM and algebraic model) were solved by a three-dimensional time-dependent finite difference method. The free surface of the fluid was modeled using a marker-and-cell approach. The algebraic model was validated by comparing the numerical predictions with analytic solutions for pipe flow. In comparison with the classical UCM model, one advantage of this approach is that computational workload is substantially reduced: the UCM model employs six differential equations while the algebraic model uses only one. The results showed stable flows with very large extrudate growths beyond those usually obtained with standard differential viscoelastic models.

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

AbstractIn 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.

The SPH method for simulating a viscoelastic drop impact and spreading on an inclined plate

In the present work, the phenomenon of an Oldroyd-B drop impact and spreading on an inclined rigid plate at low impact angles is simulated numerically using the smoothed particle hydrodynamics (SPH) method. In order to remove the unphysical phenomenon of fracture and particle clustering in fluid stretching which is the so-called tensile instability, an artificial stress term is employed which has been successfully proposed in simulations of elastic solids. Particularly, the effects of surface inclination and the different regimes of drop impact and spreading on an inclined surface are investigated. The numerical results show the capability of the proposed scheme in handing the unsteady viscoelastic free surface flows. All numerical results of using the SPH method are in agreement with the available data.