In my research, I focused on the numerical simulation of the casting filling process, with a special emphasis on the calculation of metal flow confluences and the prediction of casting defects such as cold shuts. Casting is a complex thermal and mechanical forming process. During filling, liquid metal may split into several streams and later meet again. These meeting locations are often the weakest regions in the final product. A cold shut, for example, is a typical casting defect that forms when two molten metal fronts approach each other but do not fuse completely because their temperatures are too low. Although casting simulation software has been widely used to analyze filling, solidification, and shrinkage, the direct extraction and visualization of confluence positions have not been sufficiently addressed. Therefore, I proposed a systematic method to calculate the confluence positions during the filling process and applied it to cold shut casting defect prediction.
Modern cast components, including engine cylinder blocks and other complex structural parts, require careful control of the filling process. The interaction of multiple liquid metal streams can create local temperature drops, oxide film entrapment, gas porosity, and incomplete fusion. Thus, understanding where and when liquid fronts meet is essential for the effective prediction and prevention of casting defects. In this article, I summarize the governing equations used in casting filling simulation, explain my algorithm for extracting confluent positions, verify the method through post-processing visualization, and demonstrate the prediction of cold shut casting defects using an end cover casting case.
Mathematical Model of the Casting Filling Process
To analyze the flow behavior of liquid metal during casting filling, I established a mathematical model based on the conservation laws of mass, momentum, and energy. The real casting process involves many complex factors, including free surface flow, heat transfer, phase transformation, and mold interaction. For the purpose of numerical simulation, I introduced several simplifying assumptions. I assumed that the liquid metal is incompressible and that its density remains constant during filling. I also ignored the influence of gas pressure inside the cavity by assuming that the mold is sufficiently permeable. The flow was treated as single-phase, and the effect of solid particles or gas bubbles was neglected. I used a laminar flow model because the turbulent behavior of liquid metal is still difficult to model accurately and the filling stage often involves relatively low Reynolds numbers in thin-walled regions.
These assumptions allowed me to reduce the complexity of the problem while retaining the most important physical mechanisms that cause casting defects.
Continuity Equation
The continuity equation expresses the conservation of mass. For a three-dimensional control volume, the rate of mass increase must equal the net mass flux into the volume. The general continuity equation can be written as:
$$\frac{\partial \rho}{\partial t}+\frac{\partial(\rho u)}{\partial x}+\frac{\partial(\rho v)}{\partial y}+\frac{\partial(\rho w)}{\partial z}=0$$
Because I assumed that the density of the liquid metal is constant, the continuity equation is simplified to:
$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=0$$
where \(u\), \(v\), and \(w\) are the velocity components in the \(x\), \(y\), and \(z\) directions, respectively.
Momentum Conservation Equation
The momentum conservation equation for the liquid metal can be derived from Newton’s second law. The rate of change of momentum in a control volume is equal to the sum of all external forces acting on that volume, including body forces and surface forces. For a viscous fluid, the surface forces include both pressure and viscous stresses. The Navier-Stokes equations for an incompressible Newtonian fluid are:
$$\rho\left(\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}\right)=-\frac{\partial p}{\partial x}+\rho f_x+\mu\nabla^2 u$$
$$\rho\left(\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+w\frac{\partial v}{\partial z}\right)=-\frac{\partial p}{\partial y}+\rho f_y+\mu\nabla^2 v$$
$$\rho\left(\frac{\partial w}{\partial t}+u\frac{\partial w}{\partial x}+v\frac{\partial w}{\partial y}+w\frac{\partial w}{\partial z}\right)=-\frac{\partial p}{\partial z}+\rho f_z+\mu\nabla^2 w$$
In these equations, \(p\) is the pressure, \(\mu\) is the dynamic viscosity, \(\rho\) is the density, and \(f_x\), \(f_y\), and \(f_z\) are the components of the gravitational acceleration. The Laplacian operator \(\nabla^2\) represents the viscous diffusion term. This set of equations governs the velocity and pressure fields during the casting filling process.
Energy Conservation Equation
The temperature distribution during filling has a strong influence on casting defect formation. I also solved the energy conservation equation, which balances the convective and conductive heat transfer with the internal energy change. The energy equation can be written as:
$$\rho c_p\left(\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}+w\frac{\partial T}{\partial z}\right)=\lambda\left(\frac{\partial^2 T}{\partial x^2}+\frac{\partial^2 T}{\partial y^2}+\frac{\partial^2 T}{\partial z^2}\right)$$
where \(T\) is the temperature, \(c_p\) is the specific heat capacity, and \(\lambda\) is the thermal conductivity. This equation allowed me to obtain the temperature history of the metal at every location in the mold cavity. The temperature at the moment when different liquid fronts meet is critical for the prediction of cold shut casting defects.
Numerical Discretization and Solution Method
The governing equations are nonlinear partial differential equations, and they cannot be solved analytically for arbitrary casting geometries. I used the finite difference method to discretize the equations on a structured Cartesian grid. The spatial domain was divided into cuboid control volumes. The transport equations were then converted into algebraic equations that can be solved iteratively.
I compared several numerical algorithms commonly used for casting filling simulation. The SIMPLE algorithm solves the pressure-velocity coupling using a pressure correction procedure. The MAC algorithm uses marker particles to track free surfaces, but it requires a large amount of memory and is relatively slow. The SOLA-VOF algorithm combines the SOLA pressure correction method with the volume-of-fluid free surface tracking method. The VOF method uses a scalar function to represent the volume fraction of liquid in each cell. This greatly reduces the computational cost and is particularly suitable for complex three-dimensional casting filling problems. Therefore, I selected the SOLA-VOF algorithm for my simulations.
| Algorithm | Main feature | Advantage | Disadvantage |
|---|---|---|---|
| SIMPLE | Pressure correction | Widely used for incompressible flow | Slow convergence for free-surface problems |
| MAC | Marker particles | Accurate free-surface tracking | High memory and computational cost |
| SOLA-VOF | Volume fraction and pressure correction | Fast, stable, low storage | Requires careful treatment of free surface |
Confluence Calculation Method
The main objective of my research was to calculate the confluence positions in the casting filling process. A confluence is formed when two or more liquid metal fronts meet inside the mold cavity. These positions are closely related to casting defects. If the fronts are too cold, the resulting casting defect is a cold shut. If the fronts contain oxide films or gas bubbles, the defect may appear as an inclusion or porosity. Therefore, the accurate calculation of confluence positions is extremely valuable for casting defect analysis.
Flow Front Extraction Based on the VOF Method
In the VOF method, the free surface is represented by the volume fraction function:
$$F(x,y,z,t)=\frac{V_l(x,y,z,t)}{V(x,y,z)}$$
where \(V_l\) is the volume of liquid metal in the control volume and \(V\) is the total volume of that control volume. A value of \(F=1\) means that the cell is completely filled with liquid metal. A value of \(F=0\) means that the cell is empty. When \(0<f<1\), \(f\)="" a="" at="" cell="" cell.
Construction of the Time Field
To extract the confluence positions, I needed to know the time at which each spatial location was first reached by the liquid front. I defined a time field \(T(x,y,z)\) such that:
$$T(x,y,z)=t \quad \text{if } F(x,y,z,t)>0 \text{ and } F(x,y,z,t-\Delta t)=0$$
In practice, I obtained the flow front positions at different time steps from the VOF field. When the liquid front passed through a cell for the first time, I assigned the current time value to that cell. I stored the time values as integers to reduce memory consumption. This time field provided a compact representation of the filling sequence. The three-dimensional, time-varying flow field was compressed into a stationary scalar field that can be easily processed and visualized.
For example, in a simple one-dimensional case, if a liquid stream flows along the positive \(x\)-direction, the value of \(T(x)\) increases along the flow direction. If two liquid streams flow toward each other, the time field reaches a maximum at the location where the two streams meet. This property is exactly what I used to identify confluence locations.
Criterion for Confluence Identification
If the time field has a local maximum in any spatial direction, then the fluid is moving away from that location in opposite directions. Therefore, that location must be the result of two liquid fronts meeting. Mathematically, a necessary condition for a local maximum is that the second derivative is negative. I extended this idea to three dimensions. A cell is considered to be a confluence position if any of the three second derivatives is negative:
$$\frac{\partial^2 T}{\partial x^2}<0 \quad \text{or} \quad \frac{\partial^2 T}{\partial y^2}<0 \quad \text{or} \quad \frac{\partial^2 T}{\partial z^2}<0$$
Using the finite difference approximation, the second derivative in the \(x\)-direction can be written as:
$$\frac{\partial^2 T}{\partial x^2}\approx\frac{T(x-\Delta x,y,z)-2T(x,y,z)+T(x+\Delta x,y,z)}{(\Delta x)^2}$$
Similar expressions are used for the \(y\)- and \(z\)-directions. If one of the three discrete second derivatives is negative, I marked the corresponding cell as part of a confluence region.
Optimization for Large Time Steps
In an ideal numerical simulation, the time step should be small enough that the liquid front travels no more than one cell per time step. In that ideal case, the time values of adjacent cells are different along the flow path, and the second derivative criterion works perfectly. However, in real engineering simulations, the time step is often large because of computational efficiency. The liquid front may travel through several cells in one time step. In this situation, several consecutive cells may have the same time value. The second derivative at the confluence position may become zero, and the confluence would not be detected.
To solve this problem, I moderately relaxed the confluence extraction criterion. When the second derivative is zero, I gradually increased the detection distance \(\Delta x\). I allowed \(\Delta x\) to increase from one cell length to five cell lengths. If the second derivative became negative after increasing \(\Delta x\), the cell was recognized as a confluence position. This optimization made the algorithm robust even when the time step was relatively large.
| Detection distance | Second derivative sign | Conclusion |
|---|---|---|
| \(\Delta x = 1\) cell | Zero | Cannot determine |
| \(\Delta x = 2\) cells | Negative | Confluence detected |
| \(\Delta x = 3\) to 5 cells | Negative | Confluence detected |
| \(\Delta x > 5\) cells | Zero or positive | Not a confluence |
Confluence Scale and Filtering
In real casting filling processes, the metal front is not always smooth. Local instabilities, surface waves, and velocity variations can cause the front to split into small streams. These small streams may meet again after traveling only a few cells. These small-scale confluence events are numerous and not important for casting defect analysis. They are not the major areas where cold shuts or other casting defects are likely to form.
I introduced a quantity called the confluence scale to distinguish major confluences from minor ones. The confluence scale \(L\) is defined as the smaller distance along the two reverse flow directions over which the time field continuously decreases from the confluence position. If the time value begins to increase again, the counting stops. A larger confluence scale means that the two liquid fronts traveled a long distance before meeting. This usually corresponds to a more important confluence area.
I then applied a threshold to filter out small-scale confluences. Only confluence regions whose scale was greater than the threshold were retained. The threshold was usually set to about one percent of the maximum casting length. This filtering procedure removed unstable local front interactions and preserved the main confluence regions that have a strong relationship with casting defects.
Post-Processing Implementation and Verification
To verify my confluence calculation method, I developed a post-processing visualization program using Qt and VTK. Qt is a cross-platform C++ graphical user interface framework. VTK is a powerful visualization library that can process large scientific data sets. The combination of Qt and VTK allowed me to create an interactive three-dimensional visualization environment for casting simulation results. I used the QVTKWidget class to embed the VTK rendering window inside the Qt interface. The program can display mold geometry, temperature fields, velocity fields, pressure fields, and the calculated confluence positions.
The VTK data pipeline was organized into a visualization model and a graphical model. I read the finite difference simulation results into structured grid data sets. Then I used filters to extract the mold region, the cavity region, and the confluence region. A mapper converted the filtered data into graphical primitives. Finally, the renderer displayed the result in the rendering window. I also implemented functions for rotation, zooming, clipping, animation, and screenshot export.
This post-processing tool was essential for comparing the calculated confluence positions with the observed flow behavior in the filling simulation. If the calculated confluence positions match the locations where liquid fronts are observed to meet, then the algorithm is validated.
Verification on a Simple Casting
I first tested my algorithm on a casting with a relatively simple shape. The casting was designed with a gating system that allowed liquid metal to enter from two sides. During filling, the two liquid fronts moved toward each other and collided near the center of the cavity. After the collision, the liquid rebounded toward the side walls and caused additional confluence events. I observed these events in the velocity field display and then compared them with the confluence results computed by my algorithm.
The extraction procedure produced several intermediate results. First, I displayed the time field, which showed the arrival time of the liquid front at every position. The time values increased smoothly along the flow paths. Second, I displayed the raw confluence results before filtering. Many small scattered confluence cells appeared on the surface. These cells were caused by front instability and local flow fluctuations. Third, I displayed the confluence scale field. It was clear that the largest confluence scale appeared near the center of the casting. Finally, after applying the threshold, only the main confluence regions remained. These regions agreed well with the locations where two liquid fronts visibly met during the filling process. This agreement confirmed the correctness of my method.
Application to a Steering Wheel Casting
I also applied the method to a more complex production component: a magnesium alloy steering wheel produced by high-pressure die casting. The casting process design included four gates and eight overflow slots. I analyzed the filling simulation and identified six major confluence locations by carefully observing the velocity field. The liquid metal from the two central gates met at the middle region first. Then the metal from the long side gates produced another set of confluence areas. Finally, two streams met near the overflow slots at the end of filling.
Using my confluence calculation algorithm, I obtained a set of confluence regions on the casting surface. The calculated regions were numbered by their confluence scale. The six main regions identified by the algorithm matched the six regions observed in the flow field. This result showed that the method is not limited to simple laboratory-shaped castings but can also work reliably for complex industrial components.
Cold Shut Casting Defect Prediction
Cold shut is one of the most important casting defects. It occurs when liquid metal streams meet after their temperatures have dropped near the liquidus or solidus temperature. If the flowing fronts are partially solidified or covered by oxide films, they cannot fuse completely. The resulting discontinuity creates a crack-like defect in the casting. Cold shuts reduce the mechanical strength of the casting and increase the risk of failure during service. Therefore, predicting cold shuts from numerical simulation results is highly valuable.
Traditionally, engineers try to predict cold shut casting defects by visually inspecting the velocity field and temperature field. However, when the filling process is complex, this is very difficult. In three dimensions, the view may be obstructed, and many liquid streams may move simultaneously. It is almost impossible to track every confluence event and determine its corresponding temperature. My confluence calculation method solves this problem by automatically identifying the main confluence locations. After that, I only need to extract the temperature history at these selected locations and judge whether the temperature is too low for proper fusion.
End Cover Casting Process Design
To demonstrate the cold shut prediction procedure, I selected an end cover casting made of ductile iron QT400-18. The casting had a complex back surface with several hollow regions, ribs, and bosses. The overall dimensions were 522 mm by 522 mm by 148 mm. The maximum wall thickness was about 30 mm, while the minimum wall thickness was only about 10 mm. This large variation in wall thickness made the casting prone to flow-related casting defects.
I designed a sand casting process with two cavities in one mold. The mold dimensions were 1305 mm by 790 mm by 435 mm. The gating system consisted of a sprue, three runners, and eight ingates. Each cavity was connected to four ingates. In addition, five risers were placed above each cavity to feed shrinkage during solidification.
| Property | Value | Unit |
|---|---|---|
| Solid density | 7255 | kg/m³ |
| Thermal conductivity | 54.48 | W/(m·K) |
| Young’s modulus | 111.32 | GPa |
| Shear modulus | 43.57 | GPa |
| Liquid dynamic viscosity | 4.54×10⁻³ | Pa·s |
| Liquid specific heat | 794.8 | J/kg/K |
| Liquid surface tension coefficient | 1.59 | N/m |
The finite difference mesh was generated with a uniform cell size of 4 mm. The number of cells in the \(x\), \(y\), and \(z\) directions was 339, 207, and 115, respectively. The total number of cells was about eight million. This mesh resolution was sufficient to capture the important flow features in the thin-walled regions of the casting.
| Parameter | Value | Unit |
|---|---|---|
| Liquid metal density | 6702 | kg/m³ |
| Air density | 1 | kg/m³ |
| Gravity acceleration | 9.8 | m/s² |
| Inlet velocity | 1 | m/s |
| Pouring temperature | 1350 | °C |
| Initial mold temperature | 20 | °C |
| Initial air temperature | 20 | °C |
| Heat transfer coefficient between casting and air | 10 | W/(m²·K) |
| Heat transfer coefficient between casting and mold | 1000 | W/(m²·K) |
Simulation Results and Flow Complexity
I performed the numerical simulation and obtained the velocity field and temperature field at different filling times. The liquid metal first entered the mold through the sprue, passed through the runners, and then entered the cavities through eight ingates. Because the bottom surface of the cavity was complex, the metal immediately split into several small streams. These streams met again in the lower part of the casting. This process happened many times at different locations. The flow pattern was so complex that it was practically impossible to track all of the confluence events manually.
The temperature field also showed some important features. The temperature of the liquid metal decreased as it flowed through the runner system. Inside the cavity, the temperature difference was relatively small, so the color map did not clearly reveal the dangerous regions. The temperature at different locations was affected by the local geometry and the flow history. Without a quantitative extraction method, I could not easily determine which locations were likely to produce a cold shut casting defect.
Confluence Calculation for the End Cover
I used my confluence extraction algorithm to process the filling simulation results. The raw confluence result contained many small and scattered regions. These small regions were caused by the unstable flow of the liquid front and the complex geometry of the casting. I calculated the confluence scale for each region and applied a threshold to remove insignificant regions. The remaining main confluence regions were located in the areas farthest from the ingates. These are exactly the areas where the liquid metal has traveled the longest distance and lost the most heat.
I selected several points in the main confluence regions and extracted their temperature histories from the simulation results. The temperature data showed that when the liquid fronts arrived at these confluence positions, the temperature was close to the liquidus or solidus range. Some positions even showed a temperature drop below the point where the metal could still flow freely. This means that the metal fronts were partially solidified when they met. Under such conditions, complete fusion is impossible. Therefore, I predicted that cold shut casting defects would appear at these main confluence positions.
The advantage of this method is clear. Instead of visually searching the entire filling process, I only needed to analyze the automatically extracted confluence regions. This greatly reduced the workload and improved the reliability of casting defect prediction. The method also provides a quantitative basis for evaluating whether a confluence region is dangerous.
Experimental Validation
To verify my prediction, I produced the end cover casting using the designed casting process. After the casting had cooled and the risers were removed, I examined the casting surface. The results showed obvious cold shut defects on the side wall in the middle part of the casting. The observed defect locations matched the predicted confluence regions very well. This confirmed that the confluence calculation method can be used as a reliable tool for cold shut casting defect prediction.
In addition, the experimental casting showed that the defects were not randomly distributed. They appeared exactly where the time field had a local maximum and the local temperature was too low. This indicates that the combination of time-field-based confluence extraction and temperature analysis is an effective strategy for casting defect prediction.
Conclusion
In this research, I developed a method for calculating the confluence positions of liquid metal during the casting filling process and applied it to the prediction of cold shut casting defects. I derived the governing equations for casting filling simulation, including the continuity equation, the momentum equation, and the energy equation. I used the VOF method to extract the flow front and constructed a time field that records the arrival time of the liquid front at every location. From the time field, I identified confluence positions by checking the sign of the second derivative. I also optimized the algorithm for large time steps and introduced the confluence scale to filter out insignificant small-scale confluence events.
The post-processing program I built with Qt and VTK allowed me to visualize the time field, the raw confluence field, the confluence scale, and the final main confluence regions. Through verification on a simple casting and a complex steering wheel casting, I confirmed that the calculated main confluence positions agreed with the flow field observations. The method is robust and applicable to complex industrial casting geometries.
For the end cover casting, I demonstrated a complete workflow from process design to numerical simulation to cold shut casting defect prediction. The calculated main confluence regions indicated the most dangerous locations. The temperature histories at these locations showed that the metal fronts were too cold to fuse completely. The subsequent casting experiment confirmed the appearance of cold shuts at the predicted positions. This method significantly improves the efficiency of casting process analysis and provides a scientific basis for the quantitative prediction of cold shut casting defects.
In the future, I believe that this confluence calculation approach can be extended to other filling-related casting defects, such as gas entrapment, oxide inclusions, and misruns. Further research can also be carried out to establish a direct criterion that combines the confluence scale, cooling rate, and critical solid fraction for more accurate casting defect prediction. With the continuous development of digital casting simulation, this method will contribute to the intelligent design and optimization of casting processes.
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