# On the CFD Modelling of Slamming of the Metal Melt in High-Pressure Die Casting Involving Lost Cores

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## Abstract

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## 1. Introduction

## 2. Model Equations and Simulation Methodology

#### 2.1. Model Equations

**U**=

**0**), at ambient temperature (T = ${T}_{amb}$) and at atmospheric pressure (p = ${p}_{amb}$). The initial conditions for the turbulence model are set via the length scale of the largest eddies and the turbulence intensity. Those quantities were set to 2 mm and 5 %, respectively.

#### 2.2. Simulation Methodology

## 3. Results and Discussion

#### 3.1. The Concept of the Dimensionless Slamming Factor

#### 3.2. Determining the Slamming Factor with Respect to Mesh Resolution

#### 3.3. Effect of Turbulence

#### 3.4. Effect of Courant Number

#### 3.5. Response of the Core to the Spike-Like Force Impact

## 4. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. The Menter SST k-ω Model (Menter 1994)

## References

- Kaufmann, H.; Uggowitzer, P. Metallurgy and Processing of High-Integrity Light Metal Pressure Castings; Schiele & Schön: Berlin, Germany, 2007. [Google Scholar]
- Nogowizin, B. Theorie und Praxis des Druckgusses, 1st ed.; Schiele & Schoen: Berlin, Germany, 2010. [Google Scholar]
- Brunnhuber, E. Praxis der Druckgussfertigung; Schiele & Schoen: Berlin, Germany, 1991. [Google Scholar]
- Campbell, J. Complete Casting Handbook: Metal Casting Processes, Metallurgy, Techniques and Design; Elsevier Science: Amsterdam, The Netherlands, 2015. [Google Scholar]
- Jelínek, P.; Adámková, E. Lost cores for high-pressure die casting. Arch. Foundry Eng.
**2014**, 14, 101–104. [Google Scholar] [CrossRef] - Kohlstädt, S.; Rabus, U.; Goeke, S.; Kuckenburg, S. Verfahren zur Herstellung Eines Metallischen Druckgussbauteils Unter Verwendung Eines Salzkerns Mit Integrierter Stützstruktur und Hiermit Hergestelltes Druckgussbauteil. DE Patent DE10,201,410,221,359A1, 21 April 2016. [Google Scholar]
- Schneider, T.; Kohlstädt, S.; Rabus, U. Gehäuse mit Druckgussbauteil zur Anordnung Eines Elektrischen Fahrmotors in Einem Kraftfahrzeug und Verfahren zur Herstellung eines Druckgussbauteils. DE Patent DE10,201,410,221,358A1, 21 April 2016. [Google Scholar]
- Graf, E.; Soell, G. Vorrichtung zur Herstellung Eines Druckgussbauteils mit Einem Kern und Einem Einlegeteil. DE Patent DE10,145,876A1, 10 April 2003. [Google Scholar]
- Graf, E.; Söll, G. Verfahren zur Herstellung Eines Zylinderkurbelgehäuses. WO Patent WO2,008,138,508A1, 20 November 2008. [Google Scholar]
- Fuchs, B.; Körner, C. Mesh resolution consideration for the viability prediction of lost salt cores in the high pressure die casting process. Prog. Comput. Fluid Dyn.
**2014**, 14, 24–30. [Google Scholar] [CrossRef] - Kohlstädt, S.; Vynnycky, M.; Gebauer-Teichmann, A. Experimental and numerical CHT-investigations of cooling structures formed by lost cores in cast housings for optimal heat transfer. Heat Mass Trans.
**2018**, 54, 3445–3459. [Google Scholar] [CrossRef] - Kohlstädt, S.; Vynnycky, M.; Neubauer, A.; Gebauer-Teichmann, A. Comparative RANS turbulence modelling of lost salt core viability in high pressure die casting. Prog. Comput. Fluid Dyn.
**2019**, 19, 316–327. [Google Scholar] [CrossRef] - Fuchs, B.; Eibisch, H.; Körner, C. Core viability simulation for salt core technology in high-pressure die casting. Int. J. Met.
**2013**, 7, 39–45. [Google Scholar] [CrossRef] - Fuchs, V. Numerische Modellierung von Fluid-Struktur-Wechselwirkungen an Wellenbeaufschlagten Strukturen; Kassel University Press GmbH: Kassel, Germany, 2014. [Google Scholar]
- Campbell, T.; Weynberg, P. Measurement of Parameters Affecting Slamming—Final Report; Technical Report, Wolfson Marine Craft Unit Report No. 440; University of Southampton: Southampton, UK, 1980. [Google Scholar]
- Abrate, S. Hull slamming. Appl. Mech. Rev.
**2011**, 64, 060803. [Google Scholar] [CrossRef] - Kohlstädt, S.; Vynnycky, M.; Jäckel, J. Towards the modelling of fluid-structure interactive lost core deformation in high-pressure die casting. Appl. Math. Mod.
**2020**, 20, 319–333. [Google Scholar] [CrossRef] - Hirt, C.; Nichols, B. Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys.
**1981**, 39, 201–225. [Google Scholar] [CrossRef] - Dahle, A.; Arnberg, L. The rheological properties of solidifying aluminum foundry alloys. JOM
**1996**, 48, 34–37. [Google Scholar] [CrossRef] - Ferrer, P.; Causon, D.; Qian, L.; Mingham, C.; Ma, Z. A multi-region coupling scheme for compressible and incompressible flow solvers for two-phase flow in a numerical wave tank. Comput. Fluids
**2016**, 125, 116–129. [Google Scholar] [CrossRef] [Green Version] - Mayon, R.; Sabeur, Z.; Tan, M.Y.; Djidjeli, K. Free surface flow and wave impact at complex solid structures. In Proceedings of the 12th International Conference on Hydrodynamics, Egmond aan Zee, NL, USA, 18–23 September 2016. [Google Scholar]
- Brackbill, J.; Kothe, D.; Zemach, C. A continuum method for modeling surface tension. J. Comput. Phys.
**1992**, 100, 335–354. [Google Scholar] [CrossRef] - Versteeg, H.; Malalasekera, W. An Introduction to Computational Fluid Dynamics: The Finite Volume Method; Pearson Education Limited: London, UK, 2007. [Google Scholar]
- Menter, F. 2-equation eddy-viscosity turbulence models for engineering applications. AIAA J.
**1994**, 32, 1598–1605. [Google Scholar] [CrossRef] [Green Version] - Koch, M.; Lechner, C.; Reuter, F.; Köhler, K.; Mettin, R.; Lauterborn, W. Numerical modeling of laser generated cavitation bubbles with the finite volume and volume of fluid method, using OpenFOAM. Comput. Fluids
**2016**, 126, 71–90. [Google Scholar] [CrossRef] - White, F. Fluid Mechanics, 7th ed.; McGraw-Hill: New York, NY, USA, 2011. [Google Scholar]
- Assael, M.; Kakosimos, K.; Banish, R.; Brillo, J.; Egry, I.; Brooks, R.; Quested, P.; Mills, K.; Nagashima, A.; Sato, Y.; et al. Reference data for the density and viscosity of liquid aluminum and liquid iron. J. Phys. Chem. Ref. Data
**2006**, 35, 285–300. [Google Scholar] [CrossRef] [Green Version] - Jasak, H.; Jemcov, A.; Tukovic, Z. OpenFOAM: A C++ Library for Complex Physics Simulations. In Proceedings of the International Workshop on Coupled Methods in Numerical Dynamics IUC, Dubrovnik, Croatia, 19–21 September 2007. [Google Scholar]
- Jasak, H. OpenFOAM: Open source CFD in research and industry. Int. J. Naval Archit. Ocean Eng.
**2009**, 1, 89–94. [Google Scholar] - Weller, H.; Tabor, G.; Jasak, H.; Fureby, C. A tensorial approach to computational continuum mechanics using object orientated techniques. Comput. Phys.
**1998**, 12, 620–631. [Google Scholar] [CrossRef] - Maric, T.; Höpken, J.; Mooney, K. Openfoam Technology Primer; Stan Mott: Hannover, Germany, 2014. [Google Scholar]
- Chen, G.; Xiong, Q.; Morris, P.; Paterson, E.; Sergeev, A.; Wang, Y. OpenFOAM for computational fluid dynamics. Not. AMS
**2014**, 61, 354–363. [Google Scholar] [CrossRef] - Jasak, H. Error Analysis and Estimation for the Finite Volume Method with Applications to Fluid Flows. Ph.D. Thesis, Imperial College London (University of London), London, UK, 1996. [Google Scholar]
- Droniou, J. Finite volume schemes for diffusion equations: Introduction to and review of modern methods. Math. Model. Methods Appl. Sci.
**2014**, 24, 1575–1619. [Google Scholar] [CrossRef] - Robertson, E.; Choudhury, V.; Bhushan, S.; Walters, D. Validation of OpenFOAM numerical methods and turbulence models for incompressible bluff body flows. Comput. Fluids
**2015**, 123, 122–145. [Google Scholar] [CrossRef] - Higuera, P.; Lara, J.; Losada, I. Simulating coastal engineering processes with OpenFOAM
^{®}. Coast. Eng.**2013**, 71, 119–134. [Google Scholar] [CrossRef] - Issa, R. Solution of the implicitly discretised fluid flow equations by operator-splitting. J. Comput. Phys.
**1986**, 62, 40–65. [Google Scholar] [CrossRef] - Patankar, S.; Spalding, D. A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows. In Numerical Prediction of Flow, Heat Transfer, Turbulence and Combustion; Elsevier: Berlin/Heidelberg, Germany, 1983; pp. 54–73. [Google Scholar]
- von Karman, T.H. The Impact on Seaplane Floats during Landing; Technical Report; National Advisory Committee on Aeronautics: Washington, DC, USA, 1929; p. 8. [Google Scholar]
- Wagner, H. Über Stoß- und Gleitvorgänge an der Oberfläche von Flüssigkeiten. Zamm J. Appl. Math. Mech. Angew. Math. Und Mech.
**1932**, 12, 193–215. [Google Scholar] [CrossRef] - Wienke, J. Druckschlagbelastung auf schlanke zylindrische Bauwerke durch brechende Wellen: Theoretische und großmaßstäbliche Laboruntersuchungen. Ph.D. Thesis, TU Braunschweig, Braunschweig, Germany, 2001. [Google Scholar]
- Campbell, T.; Weynberg, P. An Investigation into Wave Slamming Loads on Cylinders; Technical Report, Wolfson Marine Craft Unit Report No. 317; University of Southampton: Southampton, UK, 1977. [Google Scholar]
- Bodily, K.; Carlson, S.; Truscott, T. The water entry of slender axisymmetric bodies. Phys. Fluids
**2014**, 26, 072108. [Google Scholar] [CrossRef] - Wienke, J.; Sparboom, U.; Oumeraci, H. Breaking wave impact on a slender cylinder. In Coastal Engineering 2000; ASCE: Reston, VA, USA, 2001; pp. 1787–1798. [Google Scholar]
- Ghadimi, P.; Dashtimanesh, A.; Djeddi, S.R. Study of water entry of circular cylinder by using analytical and numerical solutions. J. Braz. Soc. Mech. Sci. Eng.
**2012**, 34, 225–232. [Google Scholar] [CrossRef] [Green Version] - Cointe, R.; Armand, J.L. Hydrodynamic impact analysis of a cylinder. J. Offshore Mech. Arct. Eng.
**1987**, 109, 237–243. [Google Scholar] [CrossRef] - Logvinovich, G. Hydrodynamics of Free-Boundary Flows; Israel Program for Scientific Translation: Jerusalem, Israel, 1972. [Google Scholar]
- Duff, G.; Naylor, D. Differential Equations of Applied Mathematics; Wiley: New York, NY, USA, 1966. [Google Scholar]
- Rusche, H. Computational Fluid Dynamics of Dispersed Two-Phase Flows at High Phase Fractions. Ph.D. Thesis, Imperial College London (University of London), London, UK, 2003. [Google Scholar]
- Bo, W.; Grove, J.W. A volume of fluid method based ghost fluid method for compressible multi-fluid flows. Comput. Fluids
**2014**, 90, 113–122. [Google Scholar] [CrossRef] - Fedkiw, R.; Aslam, T.; Merriman, B.; Osher, S. A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method). J. Comput. Phys.
**1999**, 152, 457–492. [Google Scholar] [CrossRef] - Roenby, J.; Bredmose, H.; Jasak, H. A computational method for sharp interface advection. R. Soc. Open Sci.
**2016**, 3, 160405. [Google Scholar] [CrossRef] [Green Version] - Vukčević, V. Numerical Modelling of Coupled Potential and Viscous Flow for Marine Applications. Ph.D. Thesis, Fakultet strojarstva i brodogradnje, Sveučilište u Zagrebu, Zagreb, Croatia, 2016. [Google Scholar]
- Menter, F.; Kuntz, M.; Langtry, R. Ten years of industrial experience with the SST turbulence model. Turbul. Heat Mass Transf.
**2003**, 4, 625–632. [Google Scholar] - Yakhot, V.; Orszag, S.A.; Thangam, S.; Gatski, T.B.; Speziale, C.G. Development of turbulence models for shear flows by a double expansion technique. Phys. Fluids A Fluid Dyn.
**1992**, 4, 1510–1520. [Google Scholar] [CrossRef] [Green Version] - Courant, R.; Friedrichs, K.; Lewy, H. On the partial difference equations of mathematical physics. IBM J. Res. Dev.
**1967**, 11(2), 215–234. [Google Scholar] [CrossRef] - Beitz, W.; Shields, M.; Dubbel, H.; Davies, B.; Küttner, K. DUBBEL—Handbook of Mechanical Engineering; Springer: London, UK, 2013. [Google Scholar]
- Salencon, J.; Lyle, S. Handbook of Continuum Mechanics: General Concepts—Thermoelasticity; Physics and Astronomy Online Library; Springer: Berlin, Germany, 2001. [Google Scholar]
- Groezinger, D. Water-Soluble Salt Cores. U.S. Patent US8,403,028B2, 26 March 2013. [Google Scholar]

**Figure 1.**The geometry for investigating the slamming on a salt core in channel; all dimensions are in mm.

**Figure 2.**An example of a computational grid created with the utilities blockMesh and mirrorMesh for a mesh spacing of 2 mm.

**Figure 3.**Pressure Implicit with Splitting of Operator (PISO) algorithm before and after the adjustments.

**Figure 5.**Phase distribution of the melt at impact and immediately afterwards: (

**a**) $\frac{t\xb7U}{R}$ = 0.2; (

**b**) $\frac{t\xb7U}{R}$ = 0.205; (

**c**) $\frac{t\xb7U}{R}$ = 0.21. Here, $U={U}_{in}$ with ${U}_{in}$ = 20 ms${}^{-1}$.

**Figure 6.**The pressure field at the times of impact and immediately afterwards: (

**a**) $\frac{t\xb7U}{R}$ = 0.2; (

**b**) $\frac{t\xb7U}{R}$ = 0.205; (

**c**) $\frac{t\xb7U}{R}$ = 0.21. Here, $U={U}_{in}$ with ${U}_{in}$ = 20 ms${}^{-1}$.

**Figure 7.**The velocity magnitude field at the times of impact and immediately afterwards: (

**a**) $\frac{t\xb7U}{R}$ = 0.2; (

**b**) $\frac{t\xb7U}{R}$ = 0.205; (

**c**) $\frac{t\xb7U}{R}$ = 0.21. Here, $U={U}_{in}$ with ${U}_{in}$ = 20 ms${}^{-1}$.

**Figure 9.**Comparison of the computed result with reference studies in previously published articles; mesh cell spacing 0.025 mm.

**Figure 10.**Influence of the selected turbulence model on the computed result for the slamming factor.

**Figure 11.**Results of the time step size $\Delta t$ study: (

**a**) Results for ${F}_{stat}$; (

**b**) Results for ${F}_{max}$.

**Table 1.**Boundary conditions for the presented model. $\mathbf{n}$ denotes the outward unit normal vector.

Boundary | $\mathit{\gamma}$ | U/ms${}^{-1}$ | p/Pa | T/K | ||||
---|---|---|---|---|---|---|---|---|

inlet | $\gamma $ | = 1 | $\mathbf{U}$ | = ${U}_{in}{\mathbf{e}}_{\mathbf{x}}$ | $\mathbf{n}\xb7\nabla p$ | = 0 | T | = ${T}_{melt}$ |

outlet | $\mathbf{n}\xb7\nabla \gamma $ | = 0 | $(\mathbf{n}\xb7\nabla )\mathbf{U}$ | = 0 | p | = ${p}_{amb}$ | $\mathbf{n}\xb7\nabla T$ | = 0 |

salt core | $\mathbf{n}\xb7\nabla \gamma $ | = 0 | $\mathbf{U}$ | = 0 | $\mathbf{n}\xb7\nabla p$ | = $\rho (\mathbf{g}\xb7\mathbf{n})$ | $\mathbf{n}\xb7\nabla T$ | = 0 |

wall | $\mathbf{n}\xb7\nabla \gamma $ | = 0 | $\mathbf{U}$ | = 0 | $\mathbf{n}\xb7\nabla p$ | = $\rho (\mathbf{g}\xb7\mathbf{n})$ | $\mathbf{n}\xb7\nabla T$ | = 0 |

${\mathit{c}}_{{\mathit{v}}_{\mathit{g}}}$ | 720 J kg${}^{-1}$ K${}^{-1}$ |
---|---|

${c}_{{v}_{l}}$ | 1000 J kg${}^{-1}$ K${}^{-1}$ |

${k}_{g}$ | 0.026 W m${}^{-1}$K${}^{-1}$ |

${k}_{l}$ | 70 W m${}^{-1}$K${}^{-1}$ |

M | 0.028 kg mol${}^{-1}$ |

${T}_{amb}$ | 293 K |

${p}_{amb}$ | ${10}^{5}$ Pa |

${T}_{melt}$ | 823 K |

${U}_{in}$ | 10, 20 m s${}^{-1}$ |

${\mu}_{g}$ | 1.8$\times {10}^{-5}$ Pa s |

${\mu}_{l}$ | 1.62$\times {10}^{-3}$ Pa s |

${\rho}_{l}$ | 2520 kg m${}^{-3}$ |

$\sigma $ | 0.629 N m${}^{-1}$ |

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**MDPI and ACS Style**

Kohlstädt, S.; Vynnycky, M.; Goeke, S.
On the CFD Modelling of Slamming of the Metal Melt in High-Pressure Die Casting Involving Lost Cores. *Metals* **2021**, *11*, 78.
https://doi.org/10.3390/met11010078

**AMA Style**

Kohlstädt S, Vynnycky M, Goeke S.
On the CFD Modelling of Slamming of the Metal Melt in High-Pressure Die Casting Involving Lost Cores. *Metals*. 2021; 11(1):78.
https://doi.org/10.3390/met11010078

**Chicago/Turabian Style**

Kohlstädt, Sebastian, Michael Vynnycky, and Stephan Goeke.
2021. "On the CFD Modelling of Slamming of the Metal Melt in High-Pressure Die Casting Involving Lost Cores" *Metals* 11, no. 1: 78.
https://doi.org/10.3390/met11010078