Numerical simulation of transient planar flow of a viscoelastic material with two moving free surfaces

In this study, we examine the numerical simulation of transient viscoelastic flows with two moving free surfaces. A modified Galerkin finite element method is implemented to the two-dimensional non-steady motion of the fluid of the Oldroyd-B type. The fluid is initially placed between two parallel plates and bounded by two straight free boundaries. In this Lagrangian finite element method, the spatial mesh deforms in time along with the moving free boundaries. The unknown shape of the free surfaces is determined with the flow field u, v, τ, p by the deformable finite element method, combined with a predictor-corrector scheme in an uncoupled fashion. The moving free surfaces and fluid motion of both Newtonian and non-Newtonian flows are investigated. The results include the influence of surface tension, fluid inertia and elasticity.     

Numerical simulation method for viscoelastic flows with free surfaces—fringe element generation method

To simulate filling flow in injection moulding for viscoelastic fluids, a numerical method, based on a finite element method and a finite volume method, has been developed for incompressible isothermal viscoelastic flow with moving free surfaces. The advantages of this method are, first, good applicability to arbitrarily shaped mould geometries and, second, accurate treatment for boundary conditions on the free surface. Typical filling flows are simulated, namely filling flow into a 1:4 expansion cavity with and without an obstacle. Numerical results predict the position of weld lines and air-traps. The method also indicates the effects of elongational flow on molecular orientation.     

Impact of the constitutive equation and singularity on the calculation of stick-slip flow: The modified upper-convected maxwell model (MUCM)

Calculations of viscoelastic flows using the upper-convected Maxwell (UCM) model in geometries which include sharp corners or moving and free liquid/fluid contact lines are known to be non-convergent with mesh refinement. A modified upper-convected Maxwell (MUCM) model is proposed which partially alleviates this difficulty. The MUCM model is derivable from network theory and allows the fluid relaxation time to decrease at increasing values of the trace of the stress tensor. The MUCM model yields stress fields that reduce to the a symptotic expressions for a Newtonian fluid near singularities at non-deformable boundaries. Calculations using a Galerkin finite element method are presented for the planar stick-slip problem of the flow between two no-slip surfaces joined to two shear-free surfaces. Results for fine meshes show the correct asymptotic behavior near the singularity for the MUCM model and converge to much higher values of the Deborah number than for the UCM model. However, the results for the MUCM model are still constrained by numerical instabilities related to approximating the stress behavior near the singularity.     

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 Kantorovich computational method for free surface gravity flows

Computing free surface gravity flows involves basically two coupled problems, namely, the location of the free surface position and the determination of the internal flow field (for assumed values H₀ and Q of the total head and discharge, respectively). Solution techniques are invariably based on iterative procedures, but those that iterate between the two coupled problems may become unstable. In this paper we present a computational method in which the coupling is kept throughout the process of iteration. This is achieved by converting the coupled problems (by means of the Kantorovich method) into the single problem of finding a set of streamlines, including that of the free surface. These streamlines are moved (iteratively) to satisfy the stationary conditions of the governing variational principle. The algorithm is very stable and converges rapidly. It is also easy to implement to solve various types of steady flows with a free surface under gravity.     

Surface tension driven flows in micro-gravity conditions

The importance of convective flows generated by surface tension gradients, in comparison with the ones generated by other driving forces, has been investigated in connection with space technological applications involving fluid processes. A theoretical model of the boundary conditions at the interface, considered free and diffusive, has been derived in general tensor form to allow for the use of non orthogonal curvilinear co-ordinates. For the study of flow fields contained in enclosures, these co-ordinates are more suitable to fit all teh boundaries, in particular near the contact angle between the interface and the solid walls, thus giving more accurate numerical solutions. A computational procedure to solve the complete set of bulk and surface equations is proposed and applied to a simplified two dimensional flow in a rectangular enclosure with a temperature gradient between the lateral walls. The numerical results show the importance of considering the interface to be deformable and diffusive for an accurate evaluation of the convective flow in the fluid bulk.     

Numerical computations of critical flow over a weir

Computing critical flows in hydraulics involves three problems in one: the internal flow problem, the location of the free surface and the determination of the critical flow rate. The subject can involve such difficulties as non-uniqueness, non-existence, ill-conditioning and catastrophes. This paper discusses the difficulties relating to computing critical flows over weirs. A new rapidly convergent method of determining the critical flow rate is presented and various results are shown using it with finite element discretization and with a new streamline shifting method. Numerical results are in good agreement with published data, both numerical and experimental.     

Finite element simulation of dip coating,I: Newtonian fluids

A finite element simulation of the dip coating process based on a discretization of the continuum with discontinuous pressure elements is presented. The algorithm computes the flow field from natural boundary conditions while an extra condition provided by the existence of free surface is employed to displace the meniscus location towards the actual position. The process is iterative and uses a pseudo-time stepping technique coupled to a cubic spline fitting of the free surface. Numerical predictions exhibit good agreement with experimental data for Newtonian fluids in the case of flat plate dip coating as well as in the case of wire dip coating.     

The addition of quasi-three-dimensional terms into a finite element method for transonic turbomachinery blade-to-blade flows

This paper describes the extension of a purely two-dimensional finite element method for the calculation of transonic turbomachinery blade-to-blade flows to include the quasi-three-dimensional terms. These terms account for the effect of variations in streamline radius, stream-tube height and blade rotation. By approximating the stream surface as a piecewise linear function, then using a local developed cone transformation on an element basis, the finite element equations are shown to remain of the same form as the two-dimensional equations. The numerical results presented demonstrate that the stream-tube height, streamline radius and blade rotation terms must be included if the prediction of the Mach number distribution around a gas turbine blade is to be calculated correctly.     

Development of new triangular elements for free surface flows

New finite elements have been developed to simulate steady and unsteady two-dimensional free surface flows. The depth-averaged velocity components with the free surface elevation have been used as independent variables in the model. The differences between the various elements presented lie in the choice of velocity approximation. The Newton–Raphson method has been used to solve the non-linear system of equations. Emphasis is put on bench-mark examples to assess the accuracy and efficiency of the elements. A simple stable new element tested herein shows promising advantages for industrial finite element codes.     

Time-dependent liquid metal flows with free convection and a deformable free surface

The finite element method is employed to investigate time-dependent liquid metal flows with free convection, free surfaces and Marangoni effects. The liquid circulates in a two-dimensional shallow trough with differentially heated vertical walls. The spatial formulation incorporates mixed Lagrangian approximations to the velocity, pressure, temperature and free surface position. The time integration is performed with the backward Euler and trapezoid rule methods with step size control. The Galerkin method is used to reduce the problem to a set of non-linear equations which are solved with the Newton–Raphson method. Calculations are performed for conditions relevant to the electron beam vaporization of refractory metals. The Prandtl number is 0·015 and Grashof number are in the transition range between laminar and turbulent flow. The results reveal the effects of flow intensity, surface tension gradients, mesh refinement and time integration strategy.     

Numerical simulation of the horizontal Bridgman growth. Part I: Two-dimensional flow

We study the generation of periodic velocity and temperature fields in a plane horizontal crucible of molten metal under the action of a horizontal temperature gradient. The geometry and the boundary conditions are similar to those encountered in the Bridgman growth process of semiconductor crystals, although the present paper is limited to two-dimensional flows. We use transient finite difference and finite element algorithms which lead to identical results. We demonstrate the oscillatory mechanism in two different geometries.     

Receptivity of a boundary layer to external sonic waves

The effect of external acoustic perturbations on a two-dimensional laminar boundary layer is studied within the framework of the asymptotic theory. Essentially nonparallel regimes of the basic flow when flows with a separation zone develop are investigated.     

A finite element analysis of a free surface drainage problem of two immiscible fluids

A sharp interface problem arising in the flow of two immiscible fluids, slag and molten metal in a blast furnace, is formulated using a two-dimensional model and solved numerically. This problem is a transient two-phase free or moving boundary problem, the slag surface and the slag–metal interface being the free boundaries. At each time step the hydraulic potential of each fluid satisfies the Laplace equation which is solved by the finite element method. The ordinary differential equations determining the motion of the free boundaries are treated using an implicit time-stepping scheme. The systems of linear equations obtained by discretization of the Laplace equations and the equations of motion of the free boundaries are incorporated into a large system of linear equations. At each time step the hydraulic potential in the interior domain and its derivatives on the free boundaries are obtained simultaneously by solving this linear system of equations. In addition, this solution directly gives the shape of the free boundaries at the next time step. The implicit scheme mentioned above enables us to get the solution without handling normal derivatives, which results in a good numerical solution of the present problem. A numerical example that simulates the flow in a blast furnace is given.     

Forward roll coating flows of viscoelastic liquids

Roll coating is distinguished by the use of one or more gaps between rotating cylinders to meter and apply a liquid layer to a substrate. Except at low speed, the two-dimensional film splitting flow that occurs in forward roll coating is unstable; a three-dimensional steady flow sets in, resulting in more or less regular stripes in the machine direction. For Newtonian liquids, the stability of the two-dimensional flow is determined by the competition of capillary and viscous forces: the onset of meniscus nonuniformity is marked by a critical value of the capillary number. Although most of the liquids coated industrially are non-Newtonian polymeric solutions and dispersions, most of the theoretical analyses of film splitting flows relied on the Newtonian model. Non-Newtonian behavior can drastically change the nature of the flow near the free surface; when minute amounts of flexible polymer are present, the onset of the three-dimensional instability occurs at much lower speeds than in the Newtonian case.Forward roll coating flow is analyzed here with two differential constitutive models, the Oldroyd-B and the FENE-P equations. The results show that the elastic stresses change the flow near the film splitting meniscus by reducing and eventually eliminating the recirculation present at low capillary number. When the recirculation disappears, the difference of the tangential and normal stresses (i.e., the hoop stress) at the free surface becomes positive and grows dramatically with fluid elasticity, which explains how viscoelasticity destabilizes the flow in terms of the analysis of Graham [M.D. Graham, Interfacial hoop stress and instability of viscoelastic free surface flows, Phys. Fluids 15 (2003) 1702–1710].     

Flow simulation in the nip of a rigid forward roll coater

This paper shows the development of an efficient solution algorithm for the simulation of a forward roll coating flows with free surfaces. The technique is based on the method of successive approximation combined with a method for rapidly finding a good starting point, i.e. a good initial computational domain. Movement of the free surface from the initial to final positions is demonstrated using both kinematic and normal stress schemes. Adaptive domain decomposition at each pseudo time step is performed with no significant cost penalty. The flow field is computed using a finite element solution of the Navier–Stokes equation. The proposed scheme is shown to be flexible enough to accommodate different scenarios over the practical range of many applications, i.e. capillary numbers in the range of 0.01–300, and a wide range of gap settings and rotational speeds. A close match is found between simulation and experimental meniscus profiles as well as nip pressure profiles. Overall the technique is quite robust and is able to simulate a variety of coating flow situations without resorting to over‐simplifications. Copyright © 2005 John Wiley & Sons, Ltd.     

Finite element analysis of axisymmetric free jet impingement

A numerical algorithm to determine the impingement of an axisymmetric free jet upon a curved deflector is presented. The problem is considered within the potential flow theory with the allowance of gravity and surface tension effects. The primary dependent variable is the Stokes streamfunction, which is approximated through finite elements using the isoparametric Hermite Zienkiewicz element. To find the correct position of the free boundaries, a trial-and-error method is employed which amounts to solving a boundary value problem (BVP) for the Stokes streamfunction at each iteration step. An efficient method is proposed to solve this BVP. The algorithm to find the correct position of the free boundaries is tested by computing the impingement upon an infinite disc and a hemispherical deflector. To confirm the correctness of the solution, each problem has been solved using several different mesh gradings. A comparison between the Zienkiewicz and the other standard C⁰ finite elements is also given.     

A Lagrangian Panel Method in the Time Domain for Moving Free-surface Potential Flows

A Lagrangian-type panel method in the time domain is proposed for potential flows with a moving free surface. After a spatial semi-discretization with a low-order scheme, the instantaneous velocity-potential and normal displacement on the moving free surface are obtained by means of a time-marching scheme. The kinematic and dynamic boundary conditions at the free surface are non-linear restrictions over the related Ordinary Differential Equation (ODE) system and, in order to handle them, an alternative Steklov-Poincaré operator technique is proposed. The method is applied to sloshing like flow problems.     

A finite element model for free surface flows on fixed meshes

In this paper, we present a finite element model for free surface flows on fixed meshes. The main novelty of the approach, compared with typical fixed mesh finite element models for such flows, is that we take advantage of the particularities of free surface flow, instead of considering it a particular case of two‐phase flow. The fact that a given free surface implies a known boundary condition on the interface, allows us to solve the Navier–Stokes equations on the fluid domain uncoupled from the solution on the rest of the finite element mesh. This, together with the use of enhanced integration allows us to model low Froude number flows accurately, something that is not possible with typical two‐phase flow models applied to free surface flow. Copyright © 2007 John Wiley & Sons, Ltd.     

Improved method for implementing contact angle condition in simulation of liquid sloshing under microgravity

The finite element simulation of liquid sloshing under microgravity is focused on in this paper. For this class of flows, an important issue is to implement the contact angle boundary condition appropriately. A novel method of adding infinitely small free-surface mesh just adjacent to the contact line is proposed here, which coincides with the physical definition of the contact angle. This free-surface mesh has its orientation determined according to the contact angle, and the mean curvature at the contact line can be computed using this orientation. Hence, surface tension force can be computed and incorporated as an essential boundary condition into the pressure solve. This method is convenient to be applied in three-dimensional simulations, and the use of relatively coarse meshes near the contact line is allowed. In order to validate the proposed method, several numerical simulations of flows in two-dimensional circular and three-dimensional cylindrical and spherical containers are presented, including calculation of equilibrium positions of the free surfaces, unsteady flows of liquid reorientation, and large-amplitude nonlinear liquid sloshing under lateral excitation in microgravity. Parts of the numerical results are compared with the theoretical and published experimental results.