i_, I_ NASA/CR-2002-211658 ICASE Report No. 2002-18 Lattice Boltzmann Equation on a 2D Rectangular Grid M'Hamed Bouzidi, Dominique D'Humieres, and Pierre Lallemand Universit_ Paris-Sud, Orsay Cedex, France Li-Shi Luo ICASE, Hampton, Virginia June 2002 https://ntrs.nasa.gov/search.jsp?R=20020063600 2020-06-29T20:04:16+00:00Z
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Lattice Boltzmann Equation on a 2D Rectangular Grid · LATTICE BOLTZMANN EQUATION ON A 2D RECTANGULAR GRID M'HAMEDBOUZIDI*,DOMINIQUED'HUMIERESt, PIERRE LALLEMAND_, AND LI-SH] LUO§
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i_, I_
NASA/CR-2002-211658
ICASE Report No. 2002-18
Lattice Boltzmann Equation on a 2D Rectangular Grid
M'Hamed Bouzidi, Dominique D'Humieres, and Pierre Lallemand
with a second order interpolation [16] are used for the boundary of the cylinder. The Reynolds number for
the flow is
UdRe = --.
u
We use the LBE model to simulate the flow at Re = 100 for which there is periodic vortex shedding behind
the cylinder.
q 22d *,
_5 (u, v) = (0, O)
16d
1 5dflow direction
d
(u, v) = (O, O)
FIG. 4. Configuration of a 2D flow past a cylinder asymmetrically placed in a channel.
The flow was computed on rectangular grids with several different values of the grid aspect ratio a, and
compared to the results with a square grid. The measured quantities are Strouhal number St, maximum drag
C_ a×, maximum lift coefficient C max minimum lift coefficient C rain and the pressure difference Ap. TheL ' L
results are summarized in Table 4.1. Table 4.1 also shows the lower and upper bounds of St, C_ a×, C_ _×, and
Ap, obtained by a number of conventional CFD methods presented in Ref. [17]. Overall the LBE simulation
results with square or rectangular grids agree well with each other, and with the CFD results in Ref. [17].
Figures 5 show the contours of the stream function tb(x, y) and the vorticity w(x, y) of the simulations on
a square grid of size N_ x N_ = 1401 x 308 and on a rectangular grid of size N_ x N_ = 1401 x 616. The
relative L2-norm difference of the two velocity fields is about 2.2 x 10 -4. Note that the aspect ratio for this
particular calculation is slightly different from that shown in figure 4, but this has negligible effect for the
present purpose of comparing results on the square and the rectangular grids.
The relative fluctuation of Strouhal number St is well under 1% and the values of St are well within the
bounds in Ref. [17]. The fluctuation of C_) ax is also under 1% but the values of C_ ax are all slightly lower
10
thantheresultsin Ref.[17].ThefluctuationofAP isabout1%andthevaluesofAp agree well with the
results in Ref. [17]. The values of lift coefficient obtained by the LBE simulations have a variation about
+670, which is much greater than the variations in other measured quantities.
I I E ' I I
0 200 400 600 800 1000 1200
FtG. 5. 2D flow past a cylinder asymmetrically placed in a channel at Re = 100. Top and bottom figure show contours of
the stream function W(x, y) and the vorticity _(x, y) of the flow, respectively. The dashed lines are the simulation results on
a square grid of size Nz × Nu = 1401 × 308, and the solid lines are that on a rectangular grid of size Nz × Nu = 1401 × 616.
A possible origin of the discrepancy in the lift coefficients is the following. The LBE method is intrin-
sically a compressible scheme and acoustic waves may be generated by, e.g., initial conditions that do not
include a proper pressure field or the flow itself that generates an oscillating pressure field as is the case
considered here. For a given value of the sound speed and a given choice of the boundary conditions at the
entrance and exit of the channel the frequency of some of the longitudinal acoustic modes can be close to
multiples of tile Strouhal frequency in the flow. This causes resonances between some of the acoustic waves
and the periodic shedding of vortices by the cylinder. The coupling between acoustic waves and vortex shed-
ding indeed affects the hydrodynamic fields, and in turn, various measured quantities. Among the measured
quantities, the lift coefficients are most sensitive to this effect. The mean drag coefficient is also affected but
to a much smaller extent. This problem is of broad interest. However it will be easier to study it with the
model of square grid for which the speed of sound and the bulk viscosity can be chosen in a broader range
than for the model of rectangular grid. A detailed study is beyond the scope of the present work and will
be addressed elsewhere.
5. Conclusion and discussion. In this paper we have successfully proposed a two-dimensional nine-
velocity generalized lattice Boltzmann model with multiple relaxations on a rectangular grid with arbitrary
aspect ratio a = _u/_x. We have numerically validated the model by using the model to simulate several
benchmark problems, and have obtained satisfactory results. In contrast to the previous two-dimensional,
nine-velocity, multi-relaxation model on a square grid [11], the model on a rectangular grid is more prone
to instability, and the admissible maximum value of local velocity magnitude is much less than that in
the model on a square grid. It should also be stressed that, although this work is in part motivated by a
previous work [2], it is realized that the nine-velocity lattice BGK equation cannot possibly work properly
11
ona rectangulargrid. Specifically,the latticeBGKequationdoesnot havesufficientdegreesof freedomto satisfytheconstraintsimposedby isotropyandGalileaninvariance.With ninediscretevelocitiesintwo-dimensions,it isnecessaryto usethemulti-relaxationsto construct an LBE model on a rectangular
grid.
This work is our first attempt to construct a lattice Boltzmann model on an arbitrary unstructured grid.
As discussed in Ref. [12], one difficulty encountered in the LBE model on an unstructured grid is due to
tile fact that Ve_f # e_Vf because the discrete velocity set {e_ } has spatial dependence. In this work, we
found that there are additional issues in the LBE model on an unstructured grid needed to be addressed.
First, we found that tile local grid structure severely affects the local sound speed. If the sound speed
varies spatially depending on local grid structure, then the model is unphysical. Correct acoustic propagation
is an essential part of the lattice Boltzmann method. Secondly, the constraints of isotropy and Galilean
invariance are difficult to satisfy by using the lattice BGK model, as proposed in Ref. [12], unless the discrete
velocity set includes a large number of velocities. Thirdly, the numerical stability is severely affected by the
local grid structure even for uniform structured grid, as we have demonstrated in this work. Stability is of
key importance to an effective lattice Boltzmann algorithm. However, we have not yet developed a method
to systematically improve the stability of the lattice Boltzmann method. We believe that the aforementioned
issues must be resolved before we can construct a lattice Boltzmann model on an arbitrary unstructured
grid.
Acknowledgments. D.d'H. and P.L. would like to acknowledge the support from ICASE for their
visit to ICASE in 1999 2000, during which part of this work was performed. L.S.L. would like to acknowl-
edge partial support from NASA Langley Research Center under the program of Innovative Algorithms for
Aerospace Engineering Analysis and Optimization. The authors would like to thank Dr. M. D. Salas, the
director of ICASE, for his support and encouragement of this work.
[1] D.
[2] J.
[3] u.
[4] x.
[5] --
[6] --
[7] T.
[8] x.
REFERENCES
D'HuMI_RES, Generalized lattice-Boltzmann equation, in Rarefied Gas Dynamics: Theory and Simu-
lations, Progress in Astronautics and Aeronautics _vbl. 159, edited by B. D. Shizgal and D. P. Weaver
(AIAA, Washington, DC, 1992), pp. 450 458.
M. V. A. KOELMAN, A simple lattice Boltzmann scheme for Navier-Stokes fluid flow, Europhys.
Lett., 15 (1991), pp. 603-607.
FRISCH, B. HASSLACHER AND Y. POMEAU, Lattice-gas automata for the Navier-Stokes equation,
Phys. Rev. Lett., 56 (1986), 1505 1508.
HE AND L.-S. Luo, A priori derivation o] the lattice Boltzmann equation, Phys. Rev. E, 55 (1997),
pp. R6333 R6336.
, Theory o/the lattice Boltzmann method: l_rom the Boltzmann equation to the lattice Boltzmann
equation, Phys. Rev. E, 56 (1997), 6811-6817.
, Lattice Boltzmann model/or the incompressible Navier-Stokes equation, J. Stat. Phys., 88 (1997),
pp. 927-944.
ABE, Derivation of the lattice Boltzmann method by means o/the discrete ordinate method for the
Boltzmann equation, J. Computat. Phys., 131 (1997), pp. 241-246.
HE AND G. D. DOOLEN, Lattice Boltzmann method on a curvilinear coordinate system: Vortex
shedding behind a circular cylinder, Phys. Rev. E, 56 (1997), pp. 434-440.
12
[9]--, Lattice Boltzmann method on curvilinear coordinates system: Flow around a circular cylinder, J.
Computat. Phys., 134 (1997), pp. 306-315.
[10] O. FILIPPOVA AND D. HANEL, Grid refinement/or lattice-BGK models, J. Computat. Phys., 147
(1998), pp. 219-228.
[11] P. LALLEMAND AND L.-S. LUO, Theory of the lattice Boltzmann method: Dispersion, dissipation,
isotropy, Galilean invariance, and stability, Phys. Rev. E, 61 (2000), pp. 6546 6562.
[12] I. V. KARLIN, S. SUCCI, AND S. ORSZAG, Lattice Boltzmann method for irregular grids, Phys. Rev.
Lett., 82 (1999), pp. 5245 5248.
[13] Y.H. QIAN, D. D'HuMIRES, AND P. LALLEMAND, Lattice BGK models/or Navier-Stokes equation,
Europhys. Lett., 17 (1992), pp. 479 484.
[14] H. CHEN, S. CttEN, AND W.H. N'IATTHAEUS, Recovery of the Navier-Stokes equations using a lattice-
gas Boltzmann method, Phys. Rev. A, 45 (1992), pp. R5339-R5342.
[15] We noted that although the model in Ref. [2] proposed to use face-centered rectangular grid, only the
square grid was used in the numerical tests in Ref. [2].
[16] M. BOUZIDI, M. FIRDAOUSS AND P. LALLEMAND, Momentum transfer of a lattice-Boltzmann fluid
with boundaries, Phys. Fluids, 13 (2001), pp. 3452-3459.
[17] M. SCHXFER AND S. TUREK, Benchmark computations of laminar flow around a cylinder, in Notes in
Form ApprovedREPORT DOCUMENTATION PAGE OMBNo. 0704-0188
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4. TITLE AND SUBTITLE
Lattice Boltzmann equation oi1 a 2D rectangular grid
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6. AUTHOR(S)
M'Hamed Bouzidi, Dominique D'Humieres Pierre Lallemand, and Li-Shi Luo
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)
ICASE
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NASA Langley Research Center
Hampton, VA 23681-2199
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Hampton, VA 23681-2199
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8. PERFORMING ORGANIZATION
REPORT NUMBER
ICASE Report No. 2002-18
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NASA/CR-2002-211658
ICASE Report No. 2002-18
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Final Report
Journal of Computational Physics, Vol. 172, pp. 704-717, 2001.
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13. ABSTRACT (Maximum 200 words)
We construct a nmlti-relaxation lattice Boltzmann model on a two-dimensional rectangular grid. The model is partly
inspired by a previous work of Koelman to construct a lattice BGK model on a two-dimensional rectangular grid.
The linearized dispersion equation is analyzed to obtain the constraints on the isotropy of the transport coefficients
and Galilean invariance for various wave propagations in the model. The linear stability of the model is also studied.
The model is numerically tested for three cases: (a) a vortex moving with a constant velocity on a mesh periodic
boundary conditions; (b) Poiseuille flow with an arbitrary inclined angle with respect to the lattice orientation; and
(c) a cylinder asymmetrically placed in a channel. The numerical results of these tests are compared with either
analytic solutions or the results obtained by other methods. Satisfactory results are obtained for the numerical