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Calculations of drag coefficients via hydrodynamic cellular automata
G. Kohring
To cite this version:
G. Kohring. Calculations of drag coefficients via hydrodynamic cellular automata. Journal de Physique
II, EDP Sciences, 1992, 2 (3), pp.265-269. �10.1051/jp2:1992130�. �jpa-00247629�
J. Pbys. II France 2 (1992) 265-269 MARDI 1992, PAGE 2G5
Classification Physics Abstracts
05.50 07.55M
Short Communication
Calculations of drag coefficients via hydrodynamic
cellular automata
GA- Kohring
Institut fir Theoretische Phj'sik, Uni,~ersitit zu I16111, Zfilpicherst.rafle 77, D-5000 I161n 41, Germany
(RecHved 17 December 1991, accepted 13 January 1992)
Abstract The drag coefficient, CD, of an arbitrary shaped object ca;; be calculated by two- dimensional hydrodynamic cellular automata. Results spanning nearly four orders of magnitude
in the Reynolds number (0.I < Re < 10~) are presented for a simple, hexagonal object, and
good quantitative agreement with previous experiments on cylinders is obtained.
Hydrodynamical cellular automata (IIDCA), also called "lattice gases", are believed to re-
produce the physics described by the Navier-Stokes equations [1-6]. Many different researchers
have shown during the past few years how conventional hydrodynamic phenomena can be sim- ulated using HDCA [2-4] and they have identified important applications not easily studied by
conventional methods, such as the simulation of low Reynolds number flows in porous media [3, 7, 8]. More recently, Duarte and Brosa have suggested that HDCA may also be fruitfully applied to the problem of calculating drag coefficients for an obstacle in a viscous fluid [4].
Their initial simulations covered Reynolds numbers in the range 8 < Re < 80 and demon- strated good agreement with the experimental results of Wieselsberger [5]. In this paper, their application of the HDCA is put to a more stringent test, by simulating flows for which the
Reynolds number ranges over nearly four orders of magnitude. Indeed, to our knowledge, this is the largest range of Reynolds numbers ever investigated by IIDCA for a single application.
Such a large range of Reynolds numbers is not easily obtained within a single IIDCA model,
because each model has an intrinsic number, ll~(p) [6], which sets the scale for the usable
Reynolds number. With the following definition for ll~:
lL(P) + ~((~
,(l)
where cs is the speed of sound in the HDCA, p is the particle density per site, g(p) is the lat-
tice anisotropy factor and v(p) + q/p is the kinematic viscosity, the Reynolds number is defined
2fif' JOURNAL DE PIIYSIQUE11 N03
by:
Re +
'~°~°~*~~~
(2)
cs
where uo is the characteristic speed of the flow, and Lo is the characteristic length scale of the system. At fixed lL, the Reynolds number can only be changed by changing uo or to,
this however presents several problems. First, the length scale, to, is bounded from below by
the mean free path of the HDCA particles ii, 8]. Experience has shown that to should be
no smaller than about seven mean-free paths in order to get reasonable agreement with the
asymptotic (I.e., I/£o
~0) results [8]. Second, the characteristic speed, uo, cannot be too large
since the maximum flow velocity of uo
=I (in units where the lattice spacing and the time step are equal to unity) corresponds to pure streaming without any particle interactions. On the other hand, if uo is too small, then the velocity fluctuations become too large [9]. Again, experience has shown that a value of uo in the range [0.1, 0.3] provides a good compromise to
thesi problems [8, 9].
The Reynolds number can also be adjusted by varying R+(p). In fact, ll~(p) can be made arbitrarily small by simply taking p
~0 [6], however, such small densities require excessively large lattices in order to obtain meaningful space averages. Theoretically, it is also possible to
get R+
=0 by taking p near 3 where g(p)
=0 [6], however, our simulations here have shown that this method does not yield results compatible with previous experiments. This problem
is most likely due to a breakdown of the Boltzmann approximation for large densities [6]. We therefore use different variants of HDCA to efficiently vary R+(p).
The HDCA models used for these simulations have all been discussed before II, 10]. The
I-bit, FIIP-II model was used to achieve very large Reynolds number flows, while the FIIP-I model was used for intermediate Reynolds numbers near unity. Very small Reynolds numbers
were achieved using a recently introduced momentum non-conserving model [10]. The simula- tions were performed on a 2-D channel, with a hexagonal obstacle placed in the center of the
channel. (Circles would have rather rough surfaces on a lattice and are known from earlier simulations [4] to have approximately the same drag coefficients.) In most of the simulations, the span of the object, to, was fixed at 20 il of the channel height, H, and kept larger than the 71 limit indicated above. This value of H also seemed to be large enough to avoid problems
associated with the objects wake reflecting off the channel walls and small enough to avoid excessive computational effort. The channel length, L, was varied between one and four times the channel height, I-e-, H < L < 4 x L, but the data showed no systematic variation as a
function of L. The average channel velocity used in our simulations varied between 0.I and
0.2, and since the system was initialized with a Poiseuille flow profile, this means that the maximum channel velocity Was never greater than about 0.3.
The drag coefficient, CD is defined by:
CD + ~
,
(3)
~2 ~
2~ °
where, F is the force acting on the object and A is the object's cross-sectional area perpen- dicular to the direction of flow [5]. For 2-D flow, the cross-sectional area is replaced by the
cross-sectional length, £o. The force acting on the object can be easily calculated within the
cellular automata approach as the sum of the momentum change over all particles which hit
N°3 CALCULATIONS OF DRAG COEFFICIENTS VIA HDCA 267
the object at some time t [4]. Since we use no slip boundary conditions, such particles undergo
a momentum change which is simply equal to twice the momentum with which they hit the object.
Both~ the definitions of the drag coefficient and the Reynolds numbers are ambiguous with respect to the characteristic length and characteristic velocity. Following Wieselsberger [5]~
we define £o as the total span of the hexagon~ although some authors prefer to use the radius [4]. Furthermore~ we take uo as the average channel velocity~ whereas other authors would use the maximum channel velocity [4]. In a real experiment~ one would try to use a homogeneous
flow profile, however, this is not efficient in our simulations because the channel walls cannot be made arbitrarily wide without excessive computational effort. Since the average channel
velocity is 2/3 of the maximum velocity for a Poiseuille flow, the different definitions here would give at most a multiplicative adjustment to the results, but the qualitative results would be
unchanged.
A statement about the simulation time is also needed. For simple flows, such as Couette
flow, it can be shown that the relaxation time from a random initial condition is on the order of (see for example Lim in [2]) :
~l/2
"~~(~)"
(~)
In our simulations, we do not use a random initial condition, rather we initialize the channel to
a Poiseuille profile. For Reynolds numbers outside the turbulent regime, the final flow should
only be a perturbation upon the Poiseuille flow and therefore one could expect a faster relax- ation to equilibrium than that suggested by equation (4). Indeed, we find that the relaxation to equilibrium from our starting state occurs approximately 100 times faster than estimated by
equation (4). Hence, we allow for equilibration a number of iterations equal to approximately
HL/(100~~v). After the system has come into equilibrium, a number of iterations equal to four times the relaxation time was typically used for making measurements, with measure-
ments taken every 100 time steps. However, for the FHP-I model at small Reynolds number,
many more measurement iterations were needed in order to compensate for the inefficiency of the algorithm. At our smallest Reynolds number for the FHP-I model 10~ iterations were
performed.
Finally, it should be mentioned that the programs used here have been discussed before
[iii, and, for the FHP-I model, it is within 2 percent of the optimal speed according to the criteria of Bagnoli [12] for scalar and vector machines. The only reason for not using Bagnoli's algorithm is that it is not well suited to the Connection Machine, CM-2, where the previous algorithms are faster and consume less memory.
The data is presented in figure I. The solid line in the figure represents the experimental
work of Wieselsberger for a cylinder in three dimensions. As can be seen, the three different models taken together yield results which are in good agreement with the experimental data
over four orders of magnitude. Now, of course one could have used the FHP-II model over the entire range of Re, but as explained above, this would have been extremely inefficient
computationally. The problem of attaining larger Reynolds numbers would seem to be only a problem of using a computer with larger memory. Although, we would like to note that much
higher values for the Reynolds number can be obtained if the obstacle is omitted and one works with flow through a free channel, in which case £o equals the channel width H. For this
situation we obtained Reynolds numbers up to Re m 6000, however, no significant deviations
from laminar Poiseuille flow were found.
268 JOURNAL DE PHYSIQUE II N°3
10~
~
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