Convergence Control and convergence improvement in Lagrangian predictions of particulate two-phase flows

One popular method for the simulation particulate two-phase flows is the Euler-Lagrange-Method. In this case, the continuous phase is simulated by an Eulerian approach commonly using a two-equation turbulence model to solve the Reynolds-Averaged-Navier-Stokes equations. The particle motion is modelled by the Lagrangian approach by simulating discrete particle trajectories based on Newton's equation of motion. One distinguishes between one way coupling; i.e. only the particle motion is influenced by the flow field, and two-way-coupling, i.e. also the influence of particles on the flow field by momentum transfer and turbulence modification is addressed as well. The latter approach has to be chosen in cases of higher particle concentrations [2]. In either case turbulent fluid fluctuations have to be simulated for the particle tracking. Those fluctuations have a stochastic characteristic and are modelled using random variables [1]. Consequently, the overall process is of a stochastic nature as well. The paper elaborates the disadvantages of the commonly used models to simulate two-way coupling processes. After that, a new model to overcome those disadvantages will be introduced. Finally, the results obtained by the new model are discussed.

Talking about one- or two-way coupling the direction of momentum transfer is addressed: For one-way coupling only the momentum transfer from the continuous phase to the particles is modelled by the drag force. In the case of two-way coupling a momentum source on the governing equation of the flow field has to be considered additionally. This is done by implementing source terms within the momentum equation of the continuous phase [3]. These source terms have a stochastic characteristic caused by the random nature of the Lagrangian trajectories as mentioned before. In general, the two-way coupling is implemented by an iterative solution of the flow field and a pre-specified number of particle trajectories. Since, the solution of the continuous phase depends now on stochastic values of the local source terms after each iteration step calculating new particle trajectories the flow field will change just due to the randomness of these source terms. This will finally affect the convergence behaviour of the flow field solution in a negative way.

Considering the local momentum source as a random variable originating from particle trajectories crossing the given volume element we have to ensure that the variance between different particle trajectory iterations is small, i.e. statistical reliability of the calculated mean value of the momentum source is high. This can be accomplished by considering a large number of particle trajectories within the volume element which can be achieved in two different ways: Either increasing the total number of calculated particle trajectories [4] – which in turn causes higher computational costs – or averaging over a larger volume – which in turn compromises spatial resolution. The latter might be acceptable in regions with a small momentum transfer but is not in regions with high momentum transfer and / or strong gradients of the momentum source term.

The approach of our new model is as follows. The continuous phase is calculated by a conventional solver based on a two-equation turbulence model. The disperse phase is calculated by a code developed in our group based on the Lagrangian approach using a continuum random walk model [1]. The source terms are calculated by a new model to overcome the disadvantages addressed above using two dedicated parameters allowing for a self-adaptive control of the total number of calculated particle trajectories and the applied local averaging of momentum source terms. Those source terms are finally used as input parameter for the continuous phase simulation.

The first step of the new model is the same as in conventional models [5]. The source term per element of the numerical mesh is calculated for a certain number of particle tracks. Subsequently, two criteria are checked within the new model: The first check quantifies the relative change in momentum and simultaneously the second check quantifies the statistical reliability of the momentum source. As a result three different cases have to be considered. If the proof of reliability has failed and the momentum source is high then the overall number of particle tracks has to be increased. Second, in case of failed reliability and a small momentum source a new process called spatial averaging is conducted. That means, the source terms of neighbouring mesh elements will be averaged. This is important for regions with low particle concentrations. In the third case, i.e. the requested statistical reliability is given for the elements, nothing has to be done.

The presentation will compare the results obtained by a conventional approach to the results of the new simulation approach. The results will show a significant increase in convergence performance of the overall process. The examples presented will give an overview from simple pipe flows to practical, complex applications.

In summary, a new model is presented to overcome deficiencies and numerical disadvantages in convergence behaviour of the conventional approaches. It comprises a self-adapting proof of statistical reliability and the improvement of efficiency.

[1] S. Elghobashi, On Predicting Particle-Laden Turbulent Flows, Applied Scientific Research, 52, 309-329, 1994.

[2] H.-J. Schmid, L. Vogel, On the modelling of the particle dynamics in electro-hydrodynamic flow-fields: I. Comparison of Eulerian and Lagrangian modelling approach, Powder Technology, 135-136,118-135, 2003.

[3] F. Durst, D. Milojevic, B. Schönung, Eulerian and Lagrangian predictions of particulate two-phase flows: a numerical study, Appl. Math. Modelling, 8, 101-115, 1984.

[4]  G. Kohnen, M. Rüger, M. Sommerfeld, Convergence behaviour for numerical calculations by the Euler/Lagrange method for strongly coupled phases, Numerical Methods for Multiphase Flow 1994, (Eds. C. T. Crowe, R. Johnson, A. Prosperetti, M. Sommerfeld, Y. Tsuji) ASME Fluids Engineering Division Summer Meeting, Lake Tahoe, U.S.A., June 1994, ASME FED-Vol. 185, 191-202, 1994.  

[5] C. T. Crowe, The Particle-Source-In Cell (PSI-CELL) Model for Gas-Droplet Flows, Journal of Fluids Engineering, 99, 325-332, 1977.