Lattice Boltzmann model for incompressible axisymmetric flows
Sheng Chen,
*
Jonas Tölke,
†
Sebastian Geller, and Manfred Krafczyk
‡
Institute for Computational Modeling in Civil Engineering, Technical University, Braunschweig 38106, Germany
Received 15 May 2008; published 8 October 2008
A lattice Boltzmann model for incompressible axisymmetric flow is proposed in this paper. Unlike previous
axisymmetric lattice Boltzmann models, which were based on “primitive-variables” Navier-Stokes equations,
the target macroscopic equations of the present model are vorticity-stream-function formulations. Due to the
intrinsic features of vorticity-stream-function formulations, the present model is more efficient, more stable,
and much simpler than the existing models. The advantages of the present model are validated by numerical
experiments.
DOI: 10.1103/PhysRevE.78.046703 PACS numbers: 47.11.-j
I. INTRODUCTION
In the last 2 decades, the lattice Boltzmann model has
matured as an efficient alternative and promising numerical
scheme for simulating and modeling complicated physical,
chemical, and social systems 1–18. However, most existing
lattice Boltzmann models predicting the flow of incompress-
ible fluids are based on the Cartesian coordinate system, al-
though numerous practical flow problems exist for which
axial symmetry holds 19,20. If the azimuthal velocity
equals zero, a three-dimensional axisymmetric flow can be
reduced to a quasi-two-dimensional problem for conven-
tional Navier-Stokes NS solvers in the cylindrical coordi-
nate system 21. If standard lattice Boltzmann models are
employed, one has to use the Cartesian coordinate system to
solve such kind of cylindrical flow problems, which means
that we have to use a three-dimensional lattice model to
solve a quasi-two-dimensional problem, which obviously de-
creases the efficiency of the simulation. Therefore an axi-
symmetric lattice Boltzmann model which will depend on
only two coordinates is highly desirable 19,20,22,23.
The available current literature on axisymmetric lattice
Boltzmann models is growing but still quite sparse. Halliday
et al. are the pioneers in this field 22. They recovered the
axisymmetric NS equations within a two-dimensional Carte-
sian framework by adding a “geometrical forcing” term into
the evolving equation of particle distribution functions. The
first- h
i
1
and second-order h
i
2
terms in an expansion of
the “geometrical forcing” term are carefully chosen so that
the terms in the lattice continuity and momentum equations,
respectively, arising from the cylindrical polar coordinate
system are recovered. Furthermore, the authors proved that
the forcing strategy of their axisymmetric lattice Boltzmann
model is consistent with that developed by Luo 24,25.
Then Premnath and Abraham designed a lattice Boltzmann
model for axisymmetric multiphase flows following the line
of Halliday 26. The additional temporally and spatially de-
pendent forcing terms, which account for the axisymmetric
contributions of the order parameter of the fluid phases and
inertial, viscous, and surface tension forces, are included in
their model. Peng and his copartners extended Halliday’s
idea to axisymmetric thermal systems 23. Besides the ra-
dial and axial velocity components solved by the lattice Bolt-
zmann formulation, the azimuthal velocity component and
the temperature are obtained using a finite difference
scheme. Recently Huang et al. proposed a revised version of
Peng’s model for axisymmetric thermal flows 27. In Hua-
ng’s model, the incompressible D2Q9 model proposed by He
and Luo 28 is used instead of the standard D2Q9 model 1
to improve the numerical stability and to reduce the com-
pressibility effect.
Because the general philosophy of Halliday’s model is to
treat the additional geometrical forcing terms resulting from
the axisymmetric contributions similar to the forcing terms
in the standard lattice Boltzmann equations 19,22,27, Reis
and Phillips 19 rederived h
i
1
and h
i
2
following the line of
Guo et al. 29. In Ref. 29, it has been proven that only
Guo’s forcing strategy allows one to obtain the correct NS
equations with forcing terms while others may produce non-
physical terms. Reis and Phillips argued that the expressions
of Halliday’s model are incorrect and gave simpler ones.
Lately, they validated their model through several bench-
marks 20.
All axisymmetric lattice Boltzmann models referred to
above are identical, as they described targeting at the axi-
symmetric NS equations, thus we refer to them as
“primitive-variables-based” axisymmetric lattice Boltzmann
models in this paper 30. The intrinsic disadvantages of the
primitive-variables-based axisymmetric lattice Boltzmann
models are obvious. The first, in contrast to the general strat-
egy of the lattice Boltzmann equations with additional forc-
ing given by Guo et al., in which the forcing terms need to
be expanded just to first-order to recover the additional term
in the lattice momentum equation 29, in order to get the
correct target macroscopic equations, the geometrical forcing
term of these models has to be expanded up to second-order
to recover the additional terms in the lattice continuity equa-
tion besides the lattice momentum equation i.e., Eqs. 8 and
9 in Ref. 22, respectively. So it is inevitable that the
derivation process for the detailed forms of h
i
1
, h
i
2
and the
constraints on h
i
2
, to which careful attention must be paid,
are complex 19,20,22,23,26,27. The second and most im-
portant one is that there are too many complex differential
*
chen@irmb.tu-bs.de
†
toelke@irmb.tu-bs.de
‡
kraft@irmb.tu-bs.de
PHYSICAL REVIEW E 78, 046703 2008
1539-3755/2008/784/0467038 ©2008 The American Physical Society 046703-1