Free boundary viscous flows at micro and mesolevel during liquid composites moulding process

Numerical simulation aspects, related to low Reynolds number free boundary viscous flows at micro and mesolevel during the resin impregnation stage of the liquid composite moulding process (LCM), are presented in this article. A free boundary program (FBP), developed by the authors, is used to track the movement of the resin front accurately by accounting for the surface tension effects at the boundary. Issues related to the global and local mass conservation (GMC and LMC) are identified and discussed. Unsuitable conditions for LMC and consequently GMC are uncovered at low capillary numbers, and hence a strategy for the numerical simulation of such flows is suggested. FBP encompasses a set of subroutines that are linked to modules in ANSYS. FBP can capture the void formation dynamics based on the analysis developed. We present resin impregnation dynamics in two dimensions. Extension to three dimensions is a subject for further research. Several examples are shown and efficiency of different stabilization techniques are compared. Copyright © 2004 John Wiley & Sons, Ltd.     

A hybrid finite-element–volume-of-fluid method for simulating free surface flows and interfaces

A numerical technique is developed for the simulation of free surface flows and interfaces. This technique combines the strength on the finite element method (FEM) in calculating the field variables for a deforming boundary and the versatility of the volume-of-fluid (VOF) technique in advection of the fluid interfaces. The advantage of the VOF technique is that it allows the simulation of interfaces with large deformations, including surface merging and breaking. However, its disadantage is that is solving the flow equations, it cannot resolve interfaces smaller than the cell size, since information on the subgrid scale is lost. Therefore the accuracy of the interface reconstruction and the treatment of the boundary conditions (i.e. viscous stresses and surface tension forces) become grid-size-dependent. On the other hand, the FEM with deforming interface mesh allows accurate implementation of the boundary conditions, but it cannot handle large surface deformations occurring in breaking and merging of liquid regions. Combining the two methods into a hybrid FEM-VOF method eliminates the major shortcomings of both. The outcome is a technique which can handle large surface deformations with accurate treatment of the boundary conditions. For illustration, two computational examples are presented, namely the instability and break-up of a capillary jet and the coalescence collision of two liquid drops.     

Viscous flows in corner regions: Singularities and hidden eigensolutions

Numerical issues arising in computations of viscous flows in corners formed by a liquid–fluid free surface and a solid boundary are considered. It is shown that on the solid a Dirichlet boundary condition, which removes multivaluedness of velocity in the ‘moving contact‐line problem’ and gives rise to a logarithmic singularity of pressure, requires a certain modification of the standard finite‐element method. This modification appears to be insufficient above a certain critical value of the corner angle where the numerical solution becomes mesh‐dependent. As shown, this is due to an eigensolution, which exists for all angles and becomes dominant for the supercritical ones. A method of incorporating the eigensolution into the numerical method is described that makes numerical results mesh‐independent again. Some implications of the unavoidable finiteness of the mesh size in practical applications of the finite‐element method in the context of the present problem are discussed. Copyright © 2009 John Wiley & Sons, Ltd.     

A marker‐and‐cell approach to free surface 2‐D multiphase flows

This work describes a methodology to simulate free surface incompressible multiphase flows. This novel methodology allows the simulation of multiphase flows with an arbitrary number of phases, each of them having different densities and viscosities. Surface and interfacial tension effects are also included. The numerical technique is based on the GENSMAC front‐tracking method. The velocity field is computed using a finite‐difference discretization of a modification of the Navier–Stokes equations. These equations together with the continuity equation are solved for the two‐dimensional multiphase flows, with different densities and viscosities in the different phases. The governing equations are solved on a regular Eulerian grid, and a Lagrangian mesh is employed to track free surfaces and interfaces. The method is validated by comparing numerical with analytic results for a number of simple problems; it was also employed to simulate complex problems for which no analytic solutions are available. The method presented in this paper has been shown to be robust and computationally efficient. Copyright © 2012 John Wiley & Sons, Ltd.     

求解带自由面渗流问题移动网格法的改进及其工程应用

采用有限元固定网格和移动网格相结合的方法,求解带自由面的坝体稳定渗流问题。本方法是针对一般坝体的结构,在有限元网格生成之后,又自动生成了网格移动所需的信息,实现了网格在迭代计算中随自由面的变化自动均匀移动;讨论了网格奇异的不同类型,提出了避免单元形状怪异的移动网格方法;在欧拉空间中分区记录坝体不同的渗透性能,在网格移动到新位置后,自动识别网格所在的物性空间,从而确定单元在新位置上的渗流系数。本文将该方法应用到吉林丰满混凝土重力坝防渗加固的实际工程当中,在考虑大坝原有防渗措施的前提下,利用数值分析,研究了防渗芯墙对大坝渗流性能的影响。计算结果表明,加芯墙后坝体自由面位置明显下移,湿水部分大为减少,坝基扬压力没有明显提升,下游出水点位置没有明显下降。计算过程表明,本文方法及措施具有很好的收敛性与稳定性,且收敛性与自由面初始位置的选择无关。     

Sigma transformation and ALE formulation for three‐dimensional free surface flows

In this paper we establish a link between the sigma transformation approach and the arbitrary Lagrangian–Eulerian (ALE) approach. For that purpose we introduce the ALE‐sigma (ALES) approach, which consists in an ALE interpretation of the sigma transformation. Taking advantage of this new approach, we propose a general ALES transformation, allowing for a great adaptability of the vertical discretization and therefore overcoming some drawbacks of the classical sigma transformation. Numerical results are presented, showing the advantages of this general coordinate system, as, for example, a better representation of horizontal stratifications. Copyright © 2008 John Wiley & Sons, Ltd.     

Use of lagrangian methods to describe drop deposition and distribution in horizontal gas—liquid annular flows

Drop distribution and deposition in horizontal gas—liquid annular flow is described by a diffusion model, which views the concentration field as the result of dispersion from a distribution of sources. Drops originating from a wall source are considered to diffuse in a field of homogeneous turbulence, while simultaneously being swept downward by the gravitational field. Deposition is assumed to be controlled by two mechanisms operating in parallel, and boundary conditions are derived which correctly satisfy conservation of mass. This analysis for an instantaneous source is shown to be equivalent to considering diffusion in a coordinate system moving with the settling velocity of the particles. The results are found to be useful for understanding droplet distribution and deposition.     

Objective flow classification parameters and their use in general steady flows

Three quantitative flow classification parameters have been studied in the context of Tanner and Huilgol’s suggestion of strong and weak flows. Seen in this context, the different types of streamlines possible for general 3-D flows furnish no indication with respect to the flow strength. This is in total contrast to 2-D flows, where the type of the streamline and the strength of the flow go hand in hand. Astarita’s [J Non-Newton Fluid Mech, 6:69–76, 1979] flow classification parameter takes care of this fact and, if properly generalized, can be applied to more general flows: Two other flow classification parameters also have their basis in homogeneous 2-D flows, but their generalization leads, for general flows, to nonuniqueness and other unacceptable results. For 3-D flows, none of the parameters can quantitatively be used in general, and additional parameters, with their basis outside the 2-D flow regime, seem to be called for.

Extension of domain‐free discretization method to simulate compressible flows over fixed and moving bodies

This paper is the first endeavour to present the local domain‐free discretization (DFD) method for the solution of compressible Navier–Stokes/Euler equations in conservative form. The discretization strategy of DFD is that for any complex geometry, there is no need to introduce coordinate transformation and the discrete form of governing equations at an interior point may involve some points outside the solution domain. The functional values at the exterior dependent points are updated at each time step to impose the wall boundary condition by the approximate form of solution near the boundary. Some points inside the solution domain are constructed for the approximate form of solution, and the flow variables at constructed points are evaluated by the linear interpolation on triangles. The numerical schemes used in DFD are the finite element Galerkin method for spatial discretization and the dual‐time scheme for temporal discretization. Some numerical results of compressible flows over fixed and moving bodies are presented to validate the local DFD method. Copyright © 2006 John Wiley & Sons, Ltd.     

Viscoelastic simulations of stick‐slip and die‐swell flows

Numerical solutions of viscoelastic flows are demonstrated for a time marching, semi‐implicit Taylor–Galerkin/pressure‐correction algorithm. Steady solutions are sought for free boundary problems involving combinations of die‐swell and stick‐slip conditions. Flows with and without drag flow are investigated comparatively, so that the influence of the additional component of the drag flow may be analysed effectively. The influence of die‐swell is considered that has application to various industrial processes, such as wire coating. Solutions for two‐dimensional axisymmetric flows with an Oldroyd‐B model are presented that compare favourably with the literature. The study advances our prior fixed domain formulation with this algorithm, into the realm of free‐surface viscoelastic flows. The work involves streamline‐upwind/Petrov–Galerkin weighting and velocity gradient recovery techniques that are applied upon the constitutive equation. Free surface solution reprojection and a new pressure‐drop/mass balance scheme are proposed. Copyright © 2001 John Wiley & Sons, Ltd.     

Free surface tracking with polynomial reconstruction and error correction

A non‐linear method, PREC, for computation of the movement of a free surface is proposed here. The method is composed of three steps: identifying the free surface by using a non‐linear function from the volume fraction matrix, updating the volume fraction matrix using a volume projection method with error correction, and treatment of the results using overshooting or undershooting. Identification of the free surface includes using a polynomial function with 2, 4, or 8 coefficients for one‐, two‐, or three‐dimensional problems, respectively. The polynomial reconstruction involves non‐negligible numerical error. The second advection step includes a linear projection method in space and time. Advection of the volume fraction matrix is computed from the occupying volume of the mesh at the previous time step. At the new time step, the error at each grid point is assumed to be similar to the error at the previous time step and is used for correction. Overshooting or undershooting develops around the free surface mesh points due to the solution's finite time increment. The third step includes truncating the numerical overshooting or undershooting volumes, i.e. isotropic spreading of the excess fluid volumes. The PREC method is evaluated for a one‐dimensional flow case and several two‐dimensional simple flow cases with circular sections (cases include transition parallel to a coordinate, transition with an intersection angle to a coordinate, and rotation). The results from the present method are compared with analytical solutions and results from a donor‐cell VOF method. As a result of these comparisons, the PREC method is validated. Copyright © 2008 John Wiley & Sons, Ltd.     

Comparing analytical and numerical solution of nonlinear two and three‐dimensional hydrostatic flows

New test cases for frictionless, three‐dimensional hydrostatic flows have been derived from some known analytical solutions of the two‐dimensional shallow water equations. The flow domain is a paraboloid of revolution and the flow is determined by the initial conditions, the nonlinear advective terms, the Coriolis acceleration and by the hydrostatic pressure. Wetting and drying is also included. Some specific properties of the exact solutions are discussed under different hypothesis and relative importance of the forcing terms. These solutions are proposed for testing the stability, the accuracy and the efficiency of numerical models to be used for simulating environmental hydrostatic flows. The computed solutions obtained with a semi‐implicit finite difference—finite volume algorithm on unstructured grid are compared with the corresponding analytical solutions in both two and three space dimension. Excellent agreement are obtained for the velocity and for the resulting water surface elevation. Comparison of the computed inundation area also shows a good agreement with the analytical solution with degrading accuracy observed when the inundation area becomes relatively large and for long simulation time. Copyright © 2006 John Wiley & Sons, Ltd.     

A comparative study of truly incompressible and weakly compressible SPH methods for free surface incompressible flows

In this paper, the performance of the incompressible SPH (ISPH) method and an improved weakly compressible SPH (IWCSPH) method for free surface incompressible flows are compared and analyzed. In both methods, the Navier–Stokes equations are solved, and no artificial viscosity is used. The ISPH algorithm in this paper is based on the classical SPH projection method with common treatments on solid boundaries and free surfaces. The IWCSPH model includes some advanced corrective algorithms in density approximation and solid boundary treatment (SBT). In density approximation, the moving least squares (MLS) approach is applied to re‐initialize density every several steps to obtain smoother and more stable pressure fields. An improved coupled dynamic SBT algorithm is implemented to obtain stable pressure values near solid wall areas and, thus, to minimize possible numerical oscillations brought in by the solid boundaries. Three representative numerical examples, including a benchmark test for hydrostatic pressure, a dam breaking problem and a liquid sloshing problem, are comparatively analyzed with ISPH and IWCSPH. It is demonstrated that the present IWCSPH is more attractive than ISPH in modeling free surface incompressible flows as it is more accurate and more stable with comparable or even less computational efforts. Copyright © 2013 John Wiley & Sons, Ltd.     

Deformation and breakup of viscoelastic drops in planar extensional flows

A computer-controlled four-roll mill was used to investigate the deformation and break-up of polymeric drops in the well-characterized flow of an immiscible Newtonian fluid. Aqueous polymer solutions ranging in concentration from 160 ppm to 3% by weight were examined. For zero-shear-rate viscosity ratios greater than order 1, the deformation of the drops closely followed that of Newtonian fluids, irrespective of the droplet material. However, drops with viscosity ratios less than order 1 had significantly smaller critical deformations and the critical capillary number was found to be substantially smaller. Two modes of drop break-up were discovered that differed substantially from that observed for Newtonian drops in the inclusion of cusped ends and tip streaming.     

An adaptive numerical scheme for solving incompressible 2‐phase and free‐surface flows

In this paper, we present a numerical scheme for solving 2‐phase or free‐surface flows. Here, the interface/free surface is modeled using the level‐set formulation, and the underlying mesh is adapted at each iteration of the flow solver. This adaptation allows us to obtain a precise approximation for the interface/free‐surface location. In addition, it enables us to solve the time‐discretized fluid equation only in the fluid domain in the case of free‐surface problems. Fluids here are considered incompressible. Therefore, their motion is described by the incompressible Navier‐Stokes equation, which is temporally discretized using the method of characteristics and is solved at each time iteration by a first‐order Lagrange‐Galerkin method. The level‐set function representing the interface/free surface satisfies an advection equation that is also solved using the method of characteristics. The algorithm is completed by some intermediate steps like the construction of a convenient initial level‐set function (redistancing) as well as the construction of a convenient flow for the level‐set advection equation. Numerical results are presented for both bifluid and free‐surface problems.     

Dynamics of the outflow and its effect on the hydraulics of two-layer exchange flows in a channel

This paper reports that an experimental study is conducted to examine the dynamics of the outflow in two-layer exchange flows in a channel connecting between two water bodies with a small density difference. The experiments reveal the generation of Kelvin-Helmholtz (KH) instabilities within the hydraulically sub-critical flow region of the channel. During maximal exchange, those KH instabilities develops into large-amplitude KH waves as they escape the channel exit into the reservoir. The propagation speed of those waves, their generation frequency and their amplitudes are studied. The dynamics of the outflow and these waves are directly linked to the hydraulic conditions of the exchange flow within the channel.     

A staggered conservative scheme for every Froude number in rapidly varied shallow water flows

This paper proposes a numerical technique that in essence is based upon the classical staggered grids and implicit numerical integration schemes, but that can be applied to problems that include rapidly varied flows as well. Rapidly varied flows occur, for instance, in hydraulic jumps and bores. Inundation of dry land implies sudden flow transitions due to obstacles such as road banks. Near such transitions the grid resolution is often low compared to the gradients of the bathymetry. In combination with the local invalidity of the hydrostatic pressure assumption, conservation properties become crucial. The scheme described here, combines the efficiency of staggered grids with conservation properties so as to ensure accurate results for rapidly varied flows, as well as in expansions as in contractions. In flow expansions, a numerical approximation is applied that is consistent with the momentum principle. In flow contractions, a numerical approximation is applied that is consistent with the Bernoulli equation. Both approximations are consistent with the shallow water equations, so under sufficiently smooth conditions they converge to the same solution. The resulting method is very efficient for the simulation of large‐scale inundations. Copyright © 2003 John Wiley & Sons, Ltd.