In order to demonstrate the capabilities of GENSMAC3D we present a number of calculations performed by the FreeFlow3D code as follows:

Three-Dimensional Free Surface Flows

1. Numerical Simulation of container filling. We present a calculation which simulates the filling behaviour of the trapezoidal container. The following input data were employed: The nondimensional parameters were: $Re = 7.5$, $Fr = 2.9146$. The scaling parameters were: velocity reference value $U = 1.0$m/s (fluid velocity at the nozzle), length reference value $L = 12.0$mm (jet width in the $x-$direction), and kinematic viscosity $1.6^{-3}$m$^{2}$s$^{-1}$. Gravity was assumed to be acting in the $z$-direction with $g = -9.81$ms$^{-2}$. The mesh used in this example was $80\times 120 \times 70$ cells ($\Delta x = \Delta y = \Delta z = 1.0$ mm). The Freeflow3D code solved this problem with the above input data. Figure \ref{trapezoidalcontainer} displays a series of snapshots taken during this run. Numerical Simulation of container filling (square container): In this example we simulate the filling of a square container of dimensions 5 cm x 5cm with a inlet on the top side. The scaling parameters used in this example were: U = 1 m/s (inlet velocity), L = 6 mm (inlet size) and nu = 0.005 m^2/s. Figure 1 displays several snapshots taken from this example.

2. Jet Buckling of Axisymmetric Jets (Newtonian) When a cylindrical jet flows onto a rigid surface a phenomenon known as jet buckling can occur if the Reynolds number is smaller than a prescribed value. This flow has attracted the attention of a number of researchers and has been studied both experimentally and numerically by Cruickshank and Munson \cite{cruickshank2} and Cruickshank \cite{cruickshank}. They presented results, both experimental and theoretical, for Newtonian jets. From their study, they obtained estimates for when jet buckling occurs. These are based on the Reynolds number and the aspect ratio $H/D$, where $H$ is the height of the inlet to the rigid plate and $D$ is the jet diamenter. For an axisymmetric jet, they found that if the conditions $Re < 1.2$ and $H/D > 7.0$ were satisfied then the jet would buckle. In order to simulate this problem, we considered a cylindrical thin jet of diameter $D$ = 5 mm issuing from an inlet situated at a height $H$ = 5 cm above the plate (so that $H/D = 10$) flowing onto a rigid plate. A uniform input velocity of $U = 1$ ms$^{-1}$ was set at the inlet, and the fluid viscosity $\nu = 0.02$ m$^{2}$s$^{-1}$ was used. Surface tension was neglected. The scaling paramenters were $U, D, \nu$ so that $Re = 0.25$ and $1/F_{r}^{2} = 0.049$. A mesh size of $\Delta x = \Delta y = \Delta z = 0.005$ mm ($100\times 100 \times 100$) was employed. Figure 2 displays the fluid flow configuration at different times. A simulation showing the buckling of a planar jet was also performed. The results of this simulation are shown in Figure 2a.

3. Splashing Drop (Newtonian) A spherical drop of fluid of diameter $D = 10$mm is given an initial velocity of $U = 1$ms$^{-1}$ and released from a height of 4cm above a square container containing a quiescent fluid. The value of the viscosity was $\nu = 10^{-6}$m$^{2}$s$^{-1}$ so that the Reynolds number $Re = UD/\nu = 1,000$. The mesh used in this example was $100\times 100 \times 100$ cells ($\Delta x = \Delta y = \Delta z = 1.0$ mm). The initial indentation may be observed with the formation of a travelling wave. High pressure beneath this indentation causes fluid to travel upwards (in the form of a splash). As the ``splash'' reaches its peak we can observe the surface waves being reflected from the sides of the container. In this problem surface tension is insignificant and was neglected. Figure 3 displays several snapshots taken from this run.

4. Hydraulic Jump The Freeflow3D code has been applied to simulate a circular hydraulic jump. We considered a cylindrical jet of a viscous fluid flowing rapidly onto a rigid horizontal surface. The nondimensional parameters were: $Re = 400$, $Fr = 1.785$. The scaling parameters were: inflow velocity $U = 0.5$m/s, inflow diameter $D = 8.0$mm and kinematic viscosity $\nu = 10^{-5}$m$^{2}$s$^{-1}$. Gravity was considered to be acting in the $z$-direction with $g = -9.81$ms$^{-2}$. The mesh used in this example was $100\times 100 \times 100$ cells ($\Delta x = \Delta y = \Delta z = 1.0$ mm). The Freeflow3D code solved this problem with the above input data. Figure \ref{hydraulicjum} exhibits snapshots of this run at selected times. Surface tension has been neglected. Figure 4 displays several snapshots taken from this run.

5. Jet Buckling of Viscoelastic Jets - Oldroyd-B Model. To illustrate the effect of viscoelasticity on the buckling phenomenon we applied Freeflow3D to simulate thin jets flowing onto a rigid plate. We considered a rigid plate of dimensions 6cm x 6cm and an axisymmetric nozzle situated 12cm above the rigid plate. The nozzle radius was R=3mm and the velocity of the jet issuing from the nozzle was 1 ms^{-1}. Two simulations were performed: one run with a Newtonian jet and another run using a viscoelastic jet modelled by the Oldroyd-B constitutive equation. The fluid properties were rho = 1,000 kg/m^{3}, mu_{0} = 0.004615 Pa.s, lambda_{1} = 0.006s, lambda_{2} = 0.0006$s. Therefore we had Re = rho U R /mu_{0} = 1.3 and We = lambda_{1} U/R = 1. A mesh size of 60 x 60 x 120 cells was employed in both simulations. We anticipated that the Newtonian jet will not buckle as the restriction Re < 1.2 is not satisfied . The results are displayed in Figure 5 showing the jet flowing onto a rigid plate at different times. The Newtonian jet flows radially without any sign of the buckling instability while the jet modelled by the Oldroyd-B constitutive equation does undergo slight buckling. The reason why the Oldroyd-B jet buckles may be because of the extensional viscosity which increased after the jet impinged on the rigid plate. illustrates several snapshots taken from this run.

6. Die-Swell of Axisymmetric Viscoelastic Jets - Oldroyd-B We also applied the Freeflow3D code to the simulation of the extrudate swell of an Oldroyd-B fluid. The time-dependent flow of an axisymmetric jet flowing inside a tube and then extruded in air was considered. The no-slip boundary condition is imposed on the tube walls, while fully developed flow is assumed at the tube entrance. On the fluid free surface, the full stress conditions are applied. To simulate this problem the following input data were employed: tube radius $R = 1$cm, tube length $L = 3R$, $\Delta x = \Delta y = \Delta z = 0.1$cm ($20 \times 100 \times 100$ cells). The fluid was characterised by $\nu_{0} = 0.010$ m$^{2}$s$^{-1}$, $\lambda_{1} = 0.01$s. The scaling parameters were $R$, $U=1$, $\nu_{0}$ and $\lambda_{1}$, giving $Re = U\, R/\nu_{0} = 1$ and $We = \lambda_{1}\, U/R = 1$. To demonstrate that the code can deal with the Oldroyd-B model, we used these input data and performed three simulations. In the first simulation the value of $\lambda_{2} = 0.9\lambda_{1}$ was used and in the second simulation we used $\lambda_{2} = 0.70\lambda_{1}$ while in the third simulation we chose $\lambda_{2} = 0.5\lambda_{1}$. Recall that the effective Weissenberg number for the Oldroyd-B model is given by (see Yoo and Na \cite{yoo1991}) $We_{\bf effect} = \left(1-\frac{\lambda_{2}}{\lambda_{1}}\right) We \ .$ Thus, in these simulations we used $We_{\bf effect} = 0.1, 0.3, 0.5$, respectively. Figure 6 shows the jet flowing inside the tube and then being extruded into the air. displays several snapshots taken from this run.