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Numerical computation of channel flow past a square cylinder with an upstream control bi-partition

M.A. Moussaoui 1, M. Jami 1, A. Mezrhab 1,*, H. Naji 2

1 Faculté des sciences, Département de physique, Laboratoire de Mécanique & Energétique 60000 Oujda, Maroc 2 Université des Sciences et Technologies de Lille/Polytech’Lille/LML UMR 8107 F-59655 Villeneuve d’Ascq cedex, France

*Auteur correspondant : mezrhab@fso.ump.ma ; amezrhab@yahoo.fr Abstract:

The present paper deals with the application of the Multiple Relaxation Time-Lattice Boltzmann Equation (MRT-LBE) for simulation of channel flow with a bi-partition located upstream of a square cylinder in order to control the flow. Numerical investigations have been carried out for different heights and positions of the bi-partition at Reynolds number of 250. Key computational issues involved are the computation of fluid forces acting on the square cylinder, the vortex shedding frequency and the impact of such bluff-body on the flow pattern. A particular attention is paid to drag and lift coefficients on the square cylinder. The predicted results from MRT-LBE simulations show that in most cases, the interaction was beneficial insofar the drag of the square block was lower with the bi-partition than without it. Fluctuating side forces due to vortex shedding from the main body were also reduced for most bi-partition positions. 1. Introduction

The goal of the flow control is closely related to the reduction of the resistance and the magnitude of the fluctuating force acting on the body, as a consequence saving energy. There are several investigations in the literature aiming to alter or suppress the pattern of vortex shedding (e.g. Choi et al. [1]). Many methods categorized into two different strategies were proposed: these are passive and active controls. Passive control techniques, also called vortex suppression devices, which control the vortex shedding by modifying the shape of the bluff body or by including additional devices placed upstream or downstream. These devices disrupt or prevent the formation of organized two-dimensional structures of vortex shedding. As for active control techniques, which have been developed recently, they impart the external energy into the flow field in order to control the vortex shedding (see Sakamoto et al. [2]). Note that passive control methods are easier to implement than the active ones (e.g. Kwon and Choi, [3]).

The main objective of this study is to investigate numerically the reduction of fluid forces acting on a square cylinder placed downstream of a control bi-partition, using the Multiple-Relaxation-Time Lattice Boltzmann Equation (MRT-LBE) (see d’Humières et al. [4]). 2. Problem statement

In this paper, we target a channel flow equipped with a bi-partition, which is located symmetrically upstream of a square cylinder (see Fig. 1). The fluid flow (air) is assumed to be two dimensional, laminar and incompressible. The Reynolds number (Re), based on maximum incoming flow velocity (umax) and the cylinder width (d) is equal to 250. The center of the square cylinder and the tip of the bi-

partition are located on the center line at positions xb/d = 12 and xp, respectively. The two plates, of height h, are inclined respectively by 45° and -45° with respect to the horizontal axis. The distance between the nose of the bi-partition and the front face of the cylinder is w. The length and the height of the channel are set equal to L/d = 50 and H/d = 8 respectively.

A fully developed flow with a parabolic velocity profile with the maximum velocity umax is applied at the channel inlet. At the outlet, the velocity gradients are assumed to be zero. The downstream domain is chosen long enough in order to minimize the outflow effect on the flow upstream.

The viscous and pressure forces acting on the cylinder were used to calculate the drag and lift coefficients (Cd, Cl). These coefficients are defined as:

2max0.5

Dd

FC

u dρ= ,

2max0.5

Ll

FC

u dρ= (1)

where FD and FL are the drag and lift forces exerted by the fluid on the cylinder respectively. As the force on the cylinder caused by the viscous is too small, all forces mentioned herein refer to the force induced by the pressure distribution, which were obtained by integration. The root mean square (rms), also known as the quadratic mean, is a statistical measure of the magnitude of a varying quantity. The rms value is defined as

2

1

1( )

N

l l lirmsC C i C

N = = − ∑ (2)

where Cl and lC are the instantaneous and mean lift

coefficients, respectively, and N is the total number of discrete Cl values.

Fig. 1. Schematic representation of the configuration and nomenclature

3. Lattice Boltzmann Method

In this study, we use the well known D2Q9 model (see Figure 2) on a square lattice with lattice spacing δx=δy=1 (where D refers to space dimensions and Q to the number of particles at a computational node). The MRT has been initially devised by d’Humieres [4]. This method attempts to relax different modes with different relaxation times so

xb xp

h

w

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that the bulk and shear viscosities can be determined and adjusted independently. The principle of this method can be described as follows: at each node of the lattice, one calculates nine moments coming from the nine functions of distribution. These are related by the following linear transformation (see Lallemand [5] for details):

m Mf= (3) where the vectors m and et f are arranged in the

following order:

( , , , , , , , , )Tx x y y xx xym e j q j q p pρ ε= and

0 1 2 3 4 5 6 7 8( , , , , , , , , )Tf f f f f f f f f f= , the superscript T

being the transpose operator.

Fig. 2. D2Q9 square lattice model

Following the kinetic theory of gases and making use of the symmetry feature of the discrete velocity set, the transformation matrix M of the incompressible lattice Boltzmann model can be written as:

1 1 1 1 1 1 1 1 1

4 1 1 1 1 2 2 2 2

4 2 2 2 2 1 1 1 1

0 1 0 1 0 1 1 1 1

0 2 0 2 0 1 1 1 1

0 0 1 0 1 1 1 1 1

0 0 2 0 2 1 1 1 1

0 1 1 1 1 0 0 0 0

0 0 0 0 0 1 1 1 1

M

− − − − − − − − − − − − = − − − − − −

− − − − − − −

(4)

The nine elements of the moment vector m are: the fluid density (ρ), the kinetic energy (e), the velocity (ε) which is related to the square of e, the x and y components of the

momentum, mass flux, (jx, jy), the energy fluxes (qx, qy) and the diagonal and off-diagonal components of the viscous stress tensor (pxx, pxy).

The collision modifies the moments according to the following mechanism: some of the moments are conserved (density and quantity of movement), whereas the other moments are calculated from the relaxation equations, which can be expressed as:

( )= + −c eqm m S m m (5)

with S = diag(0,s1,s2,0,s4,0,s6,s7,s8, where si is the relaxation rate. The quantity mc is the moment after collision and meq is the moment at the equilibrium:

(0, , ,0, ,0, , , )eq eq eq eq eq eq eq Tx y xx xym e q q p pε= (6)

The new functions of distribution fc are calculated from the new moments:

-1 c cf M m= (7)

M-1can be easily computed using the fact that M is an orthogonal matrix and thus M.MT is a diagonal matrix. The fluid density and the components of the momentum (mass flux) are obtained by:

( , )

ii

x y i ii

f

J j j u f e

ρ

ρ

=

= =∑

∑� (8)

The boundary conditions are quite easy to implement starting from the simple “bounce-back” of particles on boundaries. When the solid boundaries are not located at the half grid spacing beyond the last fluid nod, a combination of the “bounce-back” scheme and a spatial interpolation of first or second order have been adopted here (see Bouzidi et al. [6] for more details). 4. Results and discussions

4.1. Effect of the bi-partition height

Fig. 3 shows the streamline contours of the flow pattern in the case where the bi-partition is located at w/d = 5 and for various plate heights.

Looking at this figure, one can note that an alternating Kàrmàn vortex street is produced in the wake region downstream of the square cylinder for all cases. For low values of h* ( )/( 2)h d= , the vortex shedding behind the bi-

partition is very weak and the Kàrmàn vortex street is very narrower behind the square cylinder. When the bi-partition height increases, the alternating vortex shedding appears in the wake of the cylinder. The vortex shedding becomes more intense and affects strongly the flow topology. In the case of h*

= 0.5 (see Fig. 3(c)), the streamlines pattern indicates that the separated shear layer from the bi-partition reattaches to the cylinder and rolls up in a quasi-steady manner, meaning that the separated shear layers of the control bi-partition cylinder form a pair of standing vortices. As the bi-partition height is further increased h*

= 0.9 as shown in Figs. 3(d), the flow behind the bi-partition changes considerably and becomes less unstable. Two asymmetrically huge vortices are formed in the gap between the two bodies. This can be explained by the fact that the limited space between the two bodies prevents the formation of the Kàrmàn vortex street even though the Reynolds number is larger than its critical value for such bi-partition heights.

_a

_b

_c

_d

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Fig. 3. Streamlines for different bi-partition heights at w/d =5 and Re=250; (a) without control, (b) h*= 0.1, (c) h* = 0.5, (d) h* = 0.9

4.2. Reduction of the Drag coefficient

Fig. 4 depicts the time-averaged drag coefficient Cd versus the spacing w/d for different heights of the bi-partition. It should be noted that the Cd value obtained in the case of the square cylinder without the bi-partition is about 1.53.

The presence of the bi-partition causes a noticeable change in the inflow to the square cylinder; consequently, the drag acting on the cylinder is obviously minimized as can be seen in Fig. 4. As the height of the bi-partition increases, the alternating vortex shedding that occurs between the two bodies becomes largest (see Fig.3). This indicates that the pressure in front of the cylinder becomes lower and lower as the bi-partition height increases, showing the important role played by the bi-partition height. As the control bi-partition height increases, the drag of the square cylinder decreases, except for w/d = 1 and when h* exceeds 0.4. When the bi-partition height is large enough, the drag on the square cylinder is negative. This shows that the horizontal force acting on the cylinder has modified from a drag to a propulsive force. The square cylinder is inside the near wake behind the bi-partition and therefore is in a low pressure region. In these cases, the drag of the square cylinder is usually negative. Note that the spacing at which the square cylinder drag changes sign is called drag inversion spacing or critical spacing, and it depends on the Re. When the bi-partition height reaches a certain value (h* = 0.4), the total drag begins to increase.

An interesting feature found in results show that the drag increases significantly for 3 ≤ w/d ≤ 5 and 0.4 ≤ h* ≤ 0.6. This is due to the occurrence of the Kàrmàn vortex street behind the bi-partition. The position of the bi-partition has also an important contribution in the reduction of the drag acting on the square cylinder for any height of the bi-partition.

Fig. 4. Drag coefficient

4.3. rms lift coefficient

The rms values are deduced from temporal lift coefficient fluctuations. In Fig.5, we present this parameter versus the bi-partition location for various heights. Note that the rmsCl obtained without control is about 0.78. From these results, we can conclude that the fluctuating lift coefficient is very sensitive to the spacing w/d. The amplitude of the fluctuating lift acting on the square cylinder is reduced as the spacing w/d increases, until this spacing exceeds a

certain value, for which obvious alternate vortex shedding occurs behind the bi-partition and becomes more important. In this situation, the amplitude of the fluctuating lift increases significantly, this explains the increase of rms Cl from its value corresponding to w/d = 3.

Fig. 5. rms lift coefficient

5. Conclusion

This paper has reported the numerical results of multiple relaxation time lattice Boltzmann method of incompressible flow in a horizontal channel with a square cylinder placed downstream of a control bi-partition. The simulation was conducted at a Reynolds number of 250, based on the square width, and the maximum incoming flow velocity. In the light of the obtained results, the following conclusions can be drawn: - In general, the drag on the square cylinder decreases as the control bi-partition height increases; - The maximum reduction of the drag acting on the square cylinder and the optimum position of the control system are obtained for a particular control bi-partition height; - Negative drag on the square cylinder is achieved for large heights of the bi-partition; - The amplitude of the fluctuating lift on the square cylinder is successfully suppressed using the control bi-partition; - The fluctuating lift can be completely suppressed by choosing carefully the height and the position of the control bi-partition. 6. Références

[1] Choi H., Jeon W-P., Kim J., 2008, Control of Flow Over a Bluff Body, Annu. Rev. Fluid Mech., Vol. 40 pp.113-139. [2] Sakamoto, H, Haniu, V, 1994. Optimal suppression of fluid forces acting on a circular cylinder. Journal of Fluids Engineering 116, pp. 221–227. [3] Kwon, K., Choi, H., 1996. Control of laminar vortex shedding behind a circular cylinder using splitter plates. Physics of Fluids, Vol. 8, No. 2, pp. 479–486. [4] d’Humières, D., 1992. Generalized lattice Boltzmann equations. In Rarefied gas dynamics: theory and simulations (ed. B. D. Shizgal & Weaver). Prog. Aeronaut. Astronaut. 159, pp. 450-458. [5] Lallemand, P., Luo, L.S. 2000. Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance, and stability. Phys Rev E, Vol. 61, pp. 6546-6562. [6] Bouzidi M., Firdaouss M., Lallemand P., 2001, Momentum transfer of a Boltzmann-lattice fluid with boundaries, Phys. of Fluids, vol. 13, No. 11, pp. 3452-3459.