(80c) Convergence Control and Convergence Improvement in Lagrangian Predictions of Particulate Two-Phase Flows

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.