M.F. Tomé

A finite difference technique for simulating unsteady viscoelastic free surface flows

This work is concerned with the development of a numerical method capable of simulating viscoelastic free surface flow of an Oldroyd-B fluid. The basic equations governing the flow of an Oldroyd-B fluid are considered. A novel formulation is developed for the computation of the non-Newtonian extra-stress components on rigid boundaries. The full free surface stress conditions are employed. The resulting governing equations are solved by a finite difference method on a staggered grid, influenced by the ideas of the marker-and-cell (MAC) method. Numerical results demonstrating the capabilities of this new technique are presented for a number of problems involving unsteady free surface flows.

A numerical technique for solving unsteady non-Newtonian free surface flows

Abstract A numerical method has been developed for solving two-dimensional generalized Newtonian fluid flow with multiple free surfaces. It is an extension of the GENSMAC code which solves the time-dependent Navier-Stokes equations for the primitive variables of velocity and pressure in an arbitrary domain. Like GENSMAC, it is a finite-difference technique, based on staggered grids, using (virtual) marker particles as a means of flow visualization. The code has been employed to solve three time-dependent problems: extrudate die swell, viscous jet buckling, and injection moulding in complex cavities. Both Newtonian and non-Newtonian results are displayed.

Numerical simulation of viscous flow : Buckling of planar jets

The phenomenon of viscous fluid buckling has a long and distinguished history, dating back to Taylor (1968). This paper is concerned with demonstrating that a numerical method, GENSMAC, is capable of simulating this physical instability. A table of the parameter values (e.g. the Reynolds number, the Froude number, inlet width, inlet velocity and aspect ratio) is provided giving details of when buckling occurs and when it does not. This allows the deduction of a possible buckling condition in terms of the Reynolds number and the ratio of height of the jet to the inlet width, modifying a previous hypothesis. Visualization of jet buckling is provided. This work has been motivated by the need of industry to understand jet filling of containers; jet buckling can lead to air entrapment and this is undesirable. Copyright

An experimental and numerical investigation of container filling with viscous liquids

This work is concerned with a study of container filling, with particular reference to the food industry. A computer code was developed and an experimental rig was built, the main purpose being to validate the software. The computational fluid dynamic (CFD) code, called GENSMAC, was specifically designed for relatively slow viscous flow and was capable of capturing multiple free surfaces. This paper focuses on the design of the experimental rig and how it functions. The visual output of the code is then compared with high-speed photographic shots of glucose syrup being jetted into a tub for a selected number of flow regimes

An implicit technique for solving 3D low Reynolds number moving free surface flows

This paper describes the development of an implicit finite difference method for solving transient three-dimensional incompressible free surface flows. To reduce the CPU time of explicit low-Reynolds number calculations, we have combined a projection method with an implicit technique for treating the pressure on the free surface. The projection method is employed to uncouple the velocity and the pressure fields, allowing each variable to be solved separately. We employ the normal stress condition on the free surface to derive an implicit technique for calculating the pressure at the free surface. Numerical results demonstrate that this modification is essential for the construction of methods that are more stable than those provided by discretizing the free surface explicitly. In addition, we show that the proposed method can be applied to viscoelastic fluids. Numerical results include the simulation of jet buckling and extrudate swell for Reynolds numbers in the range [0.01,0.5].

A Stable Semi-Implicit Method for Free Surface Flows

The present work is concerned with a semi-implicit modification of the GENSMAC method for solving the two-dimensional time-dependent incompressible Navier-Stokes equations in primitive varinbles formulation with a free surface. A projection method is employed to uncouple the velocity components and pressure, thus allowing the solution of each variable separately (a segregated approach). The viscous terms are treated by the implicit backward method in time and a centered second order method in space, and the nonlinear convection terms are explicitly approximated by the high order upwind variable-order nonoscillatory scheme method in space. The boundary conditions at the free surface couple the otherwise segregated velocity and pressure fields. The present work proposes a method that allows the segregated solution of free surface flow problems to be computed by semi-implicit schemes that preserve the stability conditions of the related coupled semi-implicit scheme. The numerical method is applied to both the simulation of free surface and to confined flows. The numerical results demonstrate that the present technique eliminates the parabolic stabiliy restriction required by the original explicit GENSMAC method, and also found in segregated semi-implicit methods with time-lagged boundary conditions. For low Reynolds number flows, the method is robust and very efficient when compared to the original GENSMAC method.

Recent advances in the marker and cell method

SummaryIn this article recent advances in the MAC method will be reviewed. The MAC technique dates back to the early sixties at the Los Alamos Laboratories and this paper starts with a historical review, and then a summary of related techniques. Improvements since the early days of MAC (and the Simplified MAC-SMAC) include automatic time-stepping, the use of the conjugate gradient method to solve the Poisson equation for the corrected velocity potential, greater efficiency through stripping out all particles (markers) other than those near the free surface, more accurate approximations of the free surface boundary conditions, the addition of a bounded high accuracy upwinding for the convected terms (thereby being able to solve higher Reynolds number flows), and a (dynamic) flow visualization facility. This article will concentrate, in the main, on a three-dimensional version of the SMAC method. It will show how to approximate curved boundaries by considering one configurational example in detail; the same will also be done for the free surface. The article will avoid validation, but rather focus on many of the examples and applications that the MAC method can solve from turbulent flows to rheology. It will conclude with some speculative comments on the future direction of the methodology.

Surface tension implementation for Gensmac 2D

In the present work we describe a method which allows the incorporation of surface tension into the GENSMAC2D code. This is achieved on two scales. First on the scale of a cell, the surface tension effects are incorporated into the free surface boundary conditions through the computation of the capillary pressure. The required curvature is estimated by fitting a least square circle to the free surface using the tracking particles in the cell and in its close neighbors. On a sub-cell scale, short wavelength perturbations are filtered out using a local 4-point stencil which is mass conservative. An efficient implementation is obtained through a dual representation of the cell data, using both a matrix representation, for ease at identifying neighbouring cells, and also a tree data structure, which permits the representation of specific groups of cells with additional information pertaining to that group. The resulting code is shown to be robust, and to produce accurate results when compared with exact solutions of selected fluid dynamic problems involving surface tension.

Numerical solution of the Ericksen–Leslie dynamic equations for two-dimensional nematic liquid crystal flows

A finite difference method for solving nematic liquid crystal flows under the effect of a magnetic field is developed. The dynamic equations of nematic liquid crystals, given by the Ericksen–Leslie dynamic theory, are employed. These are expressed in terms of primitive variables and solved employing the ideas behind the GENSMAC methodology (Tome and McKee, 1994; Tome et al., 2002) [38,41]. These equations are nonlinear partial differential equations consisting of the mass conservation equation and the balance laws of linear and angular momentum. By employing fully developed flow assumptions an analytic solution for steady 2D-channel flow is found. The resulting numerical technique was then, in part, validated by comparing numerical solutions against this analytic solution. Convergence results are presented. To demonstrate the capabilities of the numerical method, the flow of a nematic liquid crystal through various complex geometries are then simulated. Results are obtained for L-shaped channels and planar 4:1 contraction for several values of Reynolds and Ericksen numbers.

Time-dependent flow in a penstock during head-gate closure

Abstract This paper is concerned with the study of the time-dependent water and air flow in a penstock caused by closing the head gate. Two approximate techniques are employed. The first approach uses simple global arguments based on mass and momentum conservation, and the time-dependent Bernoulli equation is used to derive evolution equations for the fluxes of water and air through the various components of the system. To complement this approach, the time-dependent Navier-Stokes equations were solved with a substantially reduced Reynolds number. Both approaches suggested that the pressure in the penstock remained above 1 2 atm, answering a question posed by Electricorp, New Zealand.