Thanks to their high reliability, efficiency and precision, process simulation software is increasingly used on a daily basis: many of the defects commonly found in foundry parts are already successfully grasped by existing numerical models and can therefore be prevented. These technologies are rapidly evolving and, thanks to the increasing availability of computing power, you are now able to simulate very complex problems by dividing the domain with millions of cells. Despite this, some important physical aspects have dimensions that remain much smaller than the size of the grid cell: in the modeling of these phenomena are introduced the main approximations, not being able to simulate them directly.
Among the numerical challenges that propose the “sub-grid” models – those that involve precisely physical phenomena with a characteristic length smaller than the cell size – one of the largest is to accurately and effectively simulate the smallest air bubbles that remain embedded within the metal.
THE SOFTWARE FLOW-3D® CAST
The aim is therefore to show the current solutions available for the simulation and analysis of air incorporation and to propose an innovative solution able to overcome the limits of traditional methods.
This was done using FLOW-3D® CAST software, one of the most accurate software for modelling a wide range of foundry processes. Its peculiarities are the ability to interpret with great accuracy the geometries of the piece through the FAVOR algorithm, despite the use of a structured grid, and the absolute precision in modeling the movement of the fluid alloy during the filling phase, using the TruVOF algorithm. In addition, the software has numerous numerical models able to simulate all the physics and particularities characterizing the foundry processes, from the thermal cycle of preheating to the final extraction of the piece from the mould. Thanks to this it is possible to carry out simulations with a high precision in determining the position of the defects related to the filling phase.
Furthermore, FLOW-3D® CAST has a partially open-source code that can be easily customised: allowing the creation of new numerical models or the improvement of existing ones, the software adapts perfectly to the purpose of this work.
NUMERICAL MODELS TO SIMULATE AIR ENTRAPMENT
Within FLOW-3D® CAST there are different approaches to simulate air incorporation, of increasing complexity. The simplest approximation is to completely ignore the influence of air on the metal, except for a condition of constant and uniform pressure imposed on the exposed surface of the liquid alloy. With this approximation, the metal behaves as if a perfect vacuum had been created inside the mold, or similarly as if the air could be evacuated instantly from anywhere in the mold. The most complete modeling, on the other hand, consists in simulating also the dynamics of the air, calculating its speed at each point.
Although it is possible to consider the complete and coupled fluid dynamics of metal and air simultaneously, in most cases this is not necessary. Due to the relatively small influence of air on the dense and viscous surface of the metal, it is possible to significantly reduce the calculation time by concentrating most of the resources on resolving the motion of the alloy. On the other hand, completely neglecting the influence of air does not allow to obtain realistic solutions to the problem, both in terms of filling dynamics and with regard to the identification of defects related to the presence of air.
To obtain the best compromise between simulation speed and precision of the result, considering all the physical phenomena that have a significant influence during the filling phase of a die-casting process, within FLOW-3D® CAST it was decided not to calculate the complete dynamics of the air but to approximate its main contributions with two additional numerical models: In fact, air can be incorporated in the fluid because it is completely surrounded by it, in the form of compressible bubbles (bubble model), or trapped as a quantity dispersed by the effect of turbulence (air entrainment model).
Air Entrainment model
The air entrainment model  was developed in the 1990s to simulate the effect of turbulence on the surface of a moving liquid. During filling, in fact, the high speeds at which the fluid is found are sufficient to disturb the surface to the point of incorporating air in the form of microscopic bubbles. Every single air particle is much smaller than the calculation cell: for this reason the air is represented as a diffused quantity in the metal.
Despite the fact that the individual bubbles, being microscopic, are not able by themselves to create a significant defect in the piece, their presence, at different concentrations, can affect the movement of the metal itself. The presence of air in the alloy, in fact, causes the density and local viscosity of the fluid to vary significantly. This, combined with the effects of buoyancy and the variation in resistance to motion, allows to obtain a much more accurate filling dynamic, without, however, excessively weighing down the calculation by not resolving the motion of the gas directly and at every point.
Moreover, at high concentrations, even small bubbles can give rise to important defects. For this reason it is important to trace during the simulation the volume of air incorporated, analyze the regions and times in which it is most captured within the alloy, and study the evacuation through appropriate wells. Finally, since very high pressures are reached in the die-casting process, the compressibility of the air also plays an important role. The mass of gas incorporated in the metal does not vary with the pressure, but the volume is strongly dependent on it: for this reason the analysis of the two separate outputs, mass and volume of air, allows to obtain a clear vision of the phenomenon both in the phase of incorporation and in terms of quantity of porosity distributed by gas (Fig. 1).
A different numerical model is necessary to consider the effect of the air macro-regions that are trapped in the metal. These bubbles are created because a liquid metal front completely surrounds a region of air, until it is isolated and incorporated into the liquid alloy. When they form, they are generally much larger than the mesh cells and can therefore be simulated directly by them.
It is therefore possible to carry out a direct modeling of the problem, implemented in FLOW-3D® CAST through the numerical model of the adiabatic bubbles, or “bubble model” . Using this approach each contiguous region of cells without liquid metal within them defines a “vacuum region”, or bubble. Each vacuum region is characterized by pressure, temperature, volume, and uniform inertia; while the local motion of the air, and thus the friction at the interface with the fluid, is neglected. These hypotheses are generally valid if the density of the gas is much lower than that of the fluid and the speed of the gas is comparable to that of the fluid, a typical situation of the process of filling a mould with liquid metal.
If a flow contains several bubbles, pressure and volume in each of them vary independently, exerting different efforts in different regions of the liquid. In fact, if the thrust of the liquid is such as to compress a bubble, it increases its internal pressure and consequently returns a greater force to the fluid itself (Fig.2). Finally, a bubble can only be evacuated from the mould by means of special vents that regulate the outgoing air flow through suitable pressure drops, which take into account the size of the outgoing section and the compressibility effects that characterize high speed air flows.
The adiabatic bubble model, therefore, is a powerful model that gives the possibility to monitor the gas regions during the filling process, influencing the flow dynamics and giving the user the possibility to estimate the position of some typical defects related to the entrapment of gas bubbles (Fig. 3).
In order to correctly estimate the size of the defect, it is not possible to use scalar models, such as the air entrapment model, which mediate the information of the single bubble inside the calculation cells and which tend to disperse the quantity inside the fluid. With a direct approach this does not happen: the quantity of gas is localized, it can be divided into several bubbles but it is not generally diffused in the metal.
CAPTURE THE SMALLEST BUBBLES: A NEW MODEL OF FLOW-3D® CAST
Although powerful, the model of the adiabatic bubbles has an intrinsic limitation: to be able to represent a region of air it is necessary that it is larger than a cell of the calculation grid. If a bubble becomes smaller than the cell it can no longer be modelled directly: the air region collapses and disappears from the simulation, losing the relative information and the effect on the motion of the fluid.
In order to be able to estimate the position of the defects related to them, traditionally we visualize the regions in which the last bubbles, those with the highest pressure, disappear from the simulation. This approach can certainly give an idea of the area in which it is likely to have defects but does not allow to know if the bubble, and therefore the defect, has moved after being removed from the simulation. In addition, it does not provide any information about the size of the defect itself, because it does not take into account the evolution of the volume of the bubble and its pressure.
In order to overcome this limitation, thus capturing even the bubbles much smaller than the size of the cell while maintaining the typical characteristics of the model of the adiabatic bubbles (each bubble treated individually, not dispersed, containing precise information on size and pressure), a new numerical model has been developed. Thanks to this model, each bubble, when it collapses inside a cell, is replaced by one or more properly defined point particles that contain all the information about the disappearance of the gas region.
These new particles can move following the path of the metal, deviating from it for its resistance to motion and for the effects of buoyancy. The size of the particles can also change depending on the local pressure of the metal, following the same physical law that characterizes the bubbles. Finally, gas particles can rejoin existing bubbles, adding their contribution in terms of air mass to the gas region. At the end of filling, the intensification pressure compresses each particle, giving the user the final position and the exact size of the defect.
Numerical results and validations
Many tests have been carried out to validate the new model, from simple test cases to complete simulation of real pieces, comparing numerical and experimental results. Below are presented the results obtained on a geometry provided by Form Srl, compared with the experimental analysis carried out on the molded part.
In the real piece many and repeatable defects have been found that can be clearly traced back to porosity from gas that cannot be adequately represented through the classical numerical analyses (Fig.4).
Thanks to the new model, each bubble collapsed into a cell generates some gas particles that are moved and compressed throughout the filling process. Looking at the result at the end of the simulation (Fig. 5) you can see that the highest concentration of gas particles is in the same region where the defects were recorded in the real part.
A new numerical model has been developed to accurately monitor the smallest bubbles trapped in the metal during a die-casting filling, and has been successfully validated against experimental evidence. The model extends the capabilities of FLOW-3D® CAST by improving the possibilities of understanding and visualizing filling defects.
 C. W. HIRT, Modeling Turbulent Entrainment of Air at a Free Surface, Flow Science Report 01-12, (2012)
 C. W. HIRT, Void Regions and Bubble Models in FLOW-3D, Flow Science Report 01-13, (2013)
 W. G. WALKINGTON, Die Casting Defects Troubleshooting Guide, NADCA, Illinois (2003)