SCIENCE CHINA Physics, Mechanics & Astronomy • Article •
October 2012 Vol.55 No.10: 1873–1885 doi: 10.1007/s11433-012-4876-5
Lattice Boltzmann simulation of flow around two, three and four circular cylinders in close proximity KANG XiuYing* & SU YanPing Department of Physics, Beijing Normal University, Beijing 100875, China Received July 22, 2011; accepted February 29, 2012; published online August 23, 2012
Cross-flows around two, three and four circular cylinders in tandem, side-by-side, isosceles triangle and square arrangements are simulated using the incompressible lattice Boltzmann method with a second-order accurate curved boundary condition at Reynolds number 200 and the cylinder center-to-center transverse or/and longitudinal spacing 1.5D, where D is the identical circular cylinder diameter. The wake patterns, pressure and force distributions on the cylinders and mechanism of flow dynamics are investigated and compared among the four cases. The results also show that flows around the three or four cylinders significantly differ from those of the two cylinders in the tandem and side-by-side arrangements although there are some common features among the four cases due to their similarity of structures, which are interesting, complex and useful for practical applications. This study provides a useful database to validate the simplicity, accuracy and robustness of the Lattice Boltzmann method. Lattice Boltzmann method, curved boundary condition, cross-flow around cylinders PACS number(s): 47.11.Qr, 47.32.cb, 47.85.LCitation:
Kang X Y, Su Y P. Lattice Boltzmann simulation of flow around two, three and four circular cylinders in close proximity. Sci China-Phys Mech Astron, 2012, 55: 18731885, doi: 10.1007/s11433-012-4876-5
1 Introduction Cross-flow of fluid around a group of cylinders has practical importance in engineering applications, such as offshore oil and gas pipelines, cooling towers and antennas. The flow experiences considerable resistance, and typically involves separation and reattachment of the boundary layers from cylinders. Also the formation of a significant interference wake downstream depends on the configuration parameters of these bodies relative to the flow. Investigation of the flow around cylinders can provide a better understanding of fluid dynamics including vortex dynamics, pressure distribution and body forces which is critical for engineering design practice. In general, circular cylinders are always the classical examples of bluff-bodies because they are commonly *Corresponding author (email:
[email protected]) © Science China Press and Springer-Verlag Berlin Heidelberg 2012
observed structures in nature, and employed in engineering and other applications. This paper presents a detailed numerical study of cross-flow around two, three and four circular cylinders close arranged in various configurations at Reynolds number 200. Numerous studies have been performed on the flow around two cylinders over a wide Reynolds number range. Zdravkovich [1,2] has reviewed the problem of flow interference when two cylinders are placed in tandem, side-byside and staggered arrangements in a steady current. A variety of flow patterns characterized by the behavior of the wake region were identified according to the spacing between two cylinders related with Reynolds number. Also, distributions of pressure and force on cylinder surfaces have been investigated experimentally and numerically by various authors [3–8]. Furthermore considerable progress has been made towards understanding the mechanisms of flow phys.scichina.com
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dynamics phenomena observed in such problems. Investigations on three- and four-cylinder arrays have been relatively scarce. Gu and Sun [9] investigated three parallel circular cylinders in equilateral-triangular configurations in cross uniform flow by wind tunnel. They studied the classification of flow pattern with different spacing ratios at a higher Reynolds number, and measured the surface pressure for several cylinder arrangements. Bao et al. [10] numerically investigated the effects of flow interference among three cylinders equidistantly arranged and force distributions on their surfaces at a low Reynolds in detail. Others experimentally and numerically examined four cylinders in an inline square configuration with high and low Reynolds numbers [11–14]. Flow pattern, pressure and force investigations were performed. Comparing the observed features around four cylinders with those of two cylinders, it has now been realized that the principle of superposition cannot be applied to the highly nonlinear situation of cross flow around cylinders. The wake around three- or fourcylinders is far more complex than that in one or two cylinders. Contrary to experimental studies, numerical simulation has obvious advantages in low-Re flow investigations. It can quickly provide flow information, such as the instantaneous full-field information of velocity, vorticity and pressure fields, which are very difficult to obtain experimentally. In addition, aided by the rapid development of computer technology and advance in numerical computational methods, more multi-body problems are dealt numerically. Since the solution procedure is explicit, and easy to implement and parallelize, the lattice Boltzmann method [15,16] has increasingly become an attractive alternative computational method for solving fluid dynamics problems in various systems [17–20], especially those with complex boundaries and structures [21,22]. On the other hand, at low Reynolds number two-dimensional simulation can be used to give some insight about the details of the vortex dynamics in the flow wake and force distribution on the bodies as an approximation due to its high efficiency, although vortex shedding is a three-dimensional phenomenon. In this paper, the incompressible lattice Boltzmann model is used to simulate the two-dimensional wake interference and pressure and force effects at Reynolds number 200 around two, three, and four equal circular cylinders in several configurations with the centre-to-centre transverse or/and longitudinal pitch ratio L/D 1.5, where D is the circular cylinder diameter while L is the center-to-center spacing between two cylinders. It is hoped that the studies could provide important information and insight of the physics of fluid-structure interaction of multi-cylinders in a cross-flow as well as a useful database to validate the simplicity, accuracy and robustness of the Lattice Boltzmann method. This paper is organized as follows: the incompressible lattice Boltzmann model, second-order accuracy curved boundary condition and momentum-exchange force evalua-
October (2012) Vol. 55 No. 10
tion method are briefly described in the first section. The numerical results and analysis for the various simulated cases are presented in the final section.
2 Numerical model Computationally, the Lattice Boltzmann method (LBM) [16] belongs to a class of the pseudo-compressible solvers of the Navier Stokes equations and can be classified as a Lagrangian, local equilibrium (finite-hyperbolicity) approximation. We consider a model in which the collision assumptions are simplified to a single-time relaxation form and it is referred to as the Bhatnagar-Gross-Krook (BGK) approximation [15]. The Boltzmann equation with the BGK collision model is discretized in velocity space by introduction of a finite set of velocities, e, and associated density distribution functions, f (x, e, t), yielding
1 t f e f ( f feq ).
(1)
On the left hand side of eq. (1) the first term represents the effect of the propagating fluid motion and the second term describes the convection. The factor 1/ is the inverse of the relaxation time and the equilibrium distribution function feq represents the invariant function under collision dependent on the microscopic velocity vector of the molecule e and the macroscopic velocity u. He and Luo [23] proposed a new incompressible lattice Boltzmann model. In this paper, we use this model to simulate the bluff-body flow. p (x, t) is a non-negative real number describing the distribution function of the fluid pressure at node x and time t moving in direction e instead of the density distribution function f (x, e, t) in eq. (1). The distribution function evolves with the single relaxation time Bhatnagar-Gross-Krook operator according to the Boltzmann equation: p ( x e t , t t ) p ( x , t )
peq p
,
(2)
where peq is the equilibrium distribution function in direction related to pressure p and velocity u. A common choice is 3e u 9 (ea u) 2 3 u 2 peq p p0 2 2 c4 2 c2 c
,
(3)
where c =x/t, x and t are the space and time step, p0= cs2ρ, and ρ is the flow density. cs is the sound speed and cs2=c2/3. For D2Q9 LBM model, the discrete velocity set {e=0,1,…,8} is e0=(0, 0), e1,3=(1, 0)c, e2,4=(0, 1)c, and e5,6,7,8=(1, 1)c. The corresponding weight function ωα is 0=4/9, 1,2,3,4=1/9, and 5,6,7,8=1/36.
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The lattice Boltzmann model presented here yields the correct hydrodynamic behavior for an incompressible fluid with the limit of low Mach and Knudsen numbers. The macroscopic pressure p, velocity u and kinetic viscosity are calculated by
u
p p ,
(4a)
1 p e , cs2
(4b)
1
1
c 2 t . 3 2
(4c)
To achieve second-order accuracy on no-slip cylinder walls numerically, Filippova and Hänel curved boundary method [24,25] illustrated in Figure 1 is used in our work. The lattice nodes on the solid and fluid side are denoted as b and f or ff, respectively, the boundary location is denoted as w, and the fraction of an intersected link in the fluid region =(xfxw)/(xfxb), 0
