Faculty of Foundry Engineering

FACULTY OF FOUNDRY ENGINEERING

INTERNATIONAL WORKSHOP

connected with

XXXVI SCIENTIFIC CONFERENCE FOUNDRYMAN' DAY 2012

Krakow, 22 – 23 Nov. 2012

INCREASING NUMERICAL STABILITY OF THE LATTICE BOLTZMANN MODEL FOR MOULD FILLING PROCESS BY THE FRACTIONAL STEP ALGORITHM

M. Szucki1, J. S. Suchy2, J. Sobczyk3

1, 2 AGH University of Science and Technology. Faculty of Foundry Engineering.

Reymonta 23, 30-059 Krakow, Poland

3 The Strata Mechanics Research Institute of the Polish Academy of Sciences.

Reymonta 27, 30-059 Krakow, Poland

; ;

Keywords: Lattice Boltzmann Method, LBM, Fractional Step, mould filling

Abstract

The aim of this work is to develop an algorithm which allows the use of the Fractional Step method to stabilize the numerical model of the mould filling process presented in [1, 2].

Nowadays, a computer simulation software has become an indispensable tool for supporting engineers in designing the optimum casting technology [3]. Among factors that have asignificant impact on the final quality of the cast, the mould filling process is one of the most important and widely analysed numerically. Under real conditions a metal flow is accompanied by anumber of additional processes, such as a heat transfer and crystallization, which can significantly affect the behaviour of the liquid alloy in the casting mould. This makes the problem of modelling mould filling a very complex one. Thus, simulation of aliquid metal movement usually requires complicated and time-consuming calculations. Therefore, scientist and engineers are searching for new efficient solutions in this domain. One of the newest approach in the Computational Fluid Dynamics (CFD) is called the Lattice Boltzmann Method (LBM). The available literature, including works published by the authors [1, 4], indicates, that this method can be successfully used to simulate mould filling processes. Among many advantages, such as simplicity of the calculation algorithm and the efficient use of modern multi-core CPUs, the LBM has a major drawback, which is a low numerical stability in the case of high Reynolds number flows.

Typically, the stability of free surface and multiphase flow models (widely used in the foundry industry) is increased with the Smagorinsky Subgrid Scale model [5]. This approach, however, raises some doubts because it involves interference in the process physical basis. In the present study the authors proposed the use of the Fractional Step method (FS) [6] to stabilize the Lattice Boltzmann model of the mould filling process. In this case, the numerical stability is improved by dividing the calculation procedure into two stages, called the predictor and corrector, respectively. In the predictor step calculations are done based on pre-established kinematic viscosity, which guarantee the stability of the numerical process. While in the corrector step, by solving the appropriate diffusion equation, a velocity field is corrected corresponding to the actual viscosity of the fluid:

(1)

where:

u - fluid velocity, t - time.

Parameter b is defined as follows:

, (2)

where:

- kinematic viscosity of fluid, - fictitious viscosity which guarantee a numerical stability.

The study included the analysis of the stability of free surface and two-phase flow (liquid metal - air) models. The simulation results for the models without stabilization as well as stabilized with the Smagorinsky algorithm and Fractional Step method were compared. It was also demonstrated, that by using the FS the proposed free surface Lattice Boltzmann model can be applied to solve real problems in the foundry industry. For this purpose simulation results were compared with the data obtained experimentally.

The carried out analysis indicates that without the proper stabilization method, numerical modelling of the mould filling process may not be possible. In addition to the viscosity, a factor that significantly affects the stability of the LBM is the actual size of the analysed domain (this parameter is also directly reflected in the value of the Reynolds number). With the use of the Fractional Step method it is possible to simulate low-viscosity flows, two-phase flow (in which the viscosity of the two fluids is significantly different) or flows in large areas. Additionally, through the application of FS algorithm, local variation of the viscosity can be included in the model as shown in [7].

Acknowledgements

The authors acknowledge the financial support from The Polish Ministry of Science and Higher Education through the Dean’s Grant, AGH No. 15.11.170.452.

References

1.  Szucki M., Suchy J. S., Gurgul D., Odwzorowanie powierzchni swobodnej przy modelowaniu procesu wypełniania formy odlewniczej metodą siatkową Boltzmanna. XXXIII Konferencja Naukowa z okazji Święta Odlewnika 2009, Kraków, Poland, 11-12 Dec., 2009

2.  Szucki M., Suchy J. S., Żak P., Lelito J., Gracz B., Extended free surface flow model based on the Lattice Boltzmann approach. Metallurgy and Foundry Engineering 36 (2010), No. 2, pp. 113-121

3.  Piwowarski G., Krajewski W. K., Lelito J., Optimization of casting technology of the pressure die cast AZ91D Mg-based alloy. Metallurgy and Foundry Engineering 36 (2012), No. 2, pp. 105-111

4.  Szucki M., Żak P., Lelito J., Suchy J S., Thermal Lattice Boltzmann method accuracy analysis for AZ91 alloy and AZ91/SiC composite. 7TH International PhD Conference, Brno, Czech Republic, 24 June 2010

5.  Hou S., Sterling J., Chen S., Doolen G. D., A Lattice Boltzmann Subgrid Model for High Reynolds Number Flows. Amer. Mathematical Society 6 (1994), pp. 151-166

6.  Shu C., Niu X. D., Chew Y. T., Cai Q. D., A fractional step lattice Boltzmann method for simulating high Reynolds number flows. Mathematics and Computers in Simulation 72 (2006), pp. 201-205

7.  Szucki M., Suchy J. S., A method for taking into account local viscosity changes in single relaxation time the lattice Boltzmann model. Metallurgy and Foundry Engineering 38 (2012), No. 1, pp. 33-42