Mathematical Problems in Engineering

Volume 2019, Article ID 1752803, 14 pages

https://doi.org/10.1155/2019/1752803

## Proximity Effects on Characteristics of Flow around Three Inline Square Cylinders

^{1}Mathematics Department, Air University, Islamabad, 44000, Pakistan^{2}Mathematics Department, COMSATS Institute of Information Technology Islamabad, 44000, Pakistan^{3}Mathematics and Statistics Department, Bacha Khan University, Charsadda, 44000, Pakistan

Correspondence should be addressed to Waqas Sarwar Abbasi; moc.liamtoh@555-saqaw

Received 20 August 2018; Accepted 18 December 2018; Published 10 January 2019

Academic Editor: Sergey A. Suslov

Copyright © 2019 Waqas Sarwar Abbasi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This work presents the numerical investigations performed to study the proximity effects on fluid flow characteristics around three inline square cylinders using the lattice Boltzmann method. For this purpose the gap spacing () is systematically varied in the range 0.5 to 16 diameters of cylinder by keeping Reynolds number fixed at 200. Five different flow patterns are observed at different values of spacing: bluff body flow, gap trapped flow, irregular flow, alternate shedding, and modulated shedding. These patterns have a significant effect on flow induced forces and vortex shedding frequency. The spacing value = 2 is found to be critical due to sudden changes in fluid flow characteristics. The flow parameters of first cylinder are found to be closer to single cylinder values but for middle and third cylinder the differences confirm the wake interference effect even at large values of spacing.

#### 1. Introduction

Numerical analysis of the problems concerning interaction of fluid with solid structures has received increasing attention recently because of its wider applications in real life, industrial and engineering problems. Examples can be found in many areas such as aircraft, heat exchanger tubes, cooling systems for nuclear power plants, transmission cables, bridges, high rise buildings, and electronic equipment. Similarly at low Reynolds numbers these applications can be found in microelectromechanical systems (MEMS) and cooling of fibers. Square cylinder serves as a basic component in the design of such structures. These structures often interact with fluids like air or water and experience flow induced forces which can lead to their failure under certain unfavorable circumstances. Therefore it is important to fully understand these interactions and their resulting effects on the structures in order to improve the structural designs and also to avoid industrial loss.

When one cylinder is immersed in the wake of another, the fluid flow characteristics strongly depend on the gap spacing () between cylinders, geometric parameters related to cylinders arrangement, Reynolds number (*Re*), and shape of the cylinders [1–4]. Despite numerous investigations of the flow patterns and force coefficients of the flow past circular cylinders, very little attention has been directed toward the flow interference effects around an inline array of square cylinders [5–10]. The main difference between circular and square cylinder geometry is the point of separation of flow. In case of former one it is not fixed while for the latter one it is fixed [11]. This alters the flow interference effects considerably. Also the characteristics of flow are different for different shapes of cylinders like circular, square, and rectangular [12–14]. Further, the flow interference effects also change with different arrangements of cylinders even at the same values of spacing and* Re* [15].

Very few experimental and numerical studies have been carried out for the flow around inline arranged square cylinders. Some representative experimental studies are those of Kim et al. [16], Liu and Chen [17], Sakamoto et al. [18], Sayers [19], and Lam and Lo [20]. Kim et al. [16] experimentally measured the flow fields around two square cylinders in a tandem (inline) arrangement. Their results showed that the flow patterns at gap spacing ≤ 2 were drastically different from those at ≥ 2.5. Liu and Chen [17] observed two different kinds of flow patterns by progressively increasing and decreasing the gap spacing, from 1.5 to 9, between two tandem square cylinders. Sakamoto et al. [18] experimentally observed significant changes in time-averaged forces of square cylinders for gap spacing above and below = 4 at* Re* = 27,600. Sayers [19] measured the lift coefficient (*CL*) and drag coefficient (*CD*) of a single cylinder in a group of four equally spaced cylinders placed in an open-jet wind tunnel. Lam and Lo [20] reported three different kinds of behaviors of shear layers while investigating the flow around four inline cylinders. While searching the literature, it can be observed that most of the experimental results are limited to high values of* Re* and less information is available about the phenomena of low* Re* regarding the experimental studies. On the other hand, numerical techniques are capable of capturing the flow characteristics at both high and low values of* Re* efficiently. Inoue et al. [21] numerically examined the sound generation in a uniform flow at low Mach number (*Ma*) and* Re* for flow past two inline square cylinders. Etminan [22] numerically investigated the fluid flow and heat transfer for flow past square cylinders at 1 ≤* Re* ≤ 200 and Prandtl number* Pr* = 0.71. Their results showed that the flow was steady for* Re* ≤ 35 and unsteady-periodic for* Re* ≥ 40. Bao et al. [23] investigated the flow past an inline array of six square cylinders at* Re *= 100 and ranging from 1.5 to 15. They observed six different kinds of flow patterns: steady wake, non-fully developed vortex street in single-row and double-row, fully developed vortex street in double-row, fully developed vortex street in partially recovered single-row, and fully developed multiple vortex streets. They also observed that the first and second cylinder behave similar to the two inline cylinders configuration in terms of aerodynamic force coefficients while the other four cylinders experience periodic variation of forces with the increase in gap spacing. For more details about fluid flow from an inline array of cylinders, interested readers are referred to [24–26] and references therein.

In the present study, the case of flow interacting with an inline array of three square cylinders is considered, because inline array of cylinders is a fundamental element in any tube array, offshore structures, and microelectromechanical system (MEMS) devices. In this work special attention is paid to the investigation of the characteristics of vortex shedding and flow induced forces caused by the flow interference of multiple cylinders. We will mainly focus on the existence of different flow patterns under the effect of by fixing* Re *at a specific value, dependence of force coefficients on , and existence of critical spacing values (if any) where flow characteristics change abruptly.

#### 2. Lattice Boltzmann Method

Lattice Boltzmann method, evolved from lattice gas cellular automata (LGCA) [27], was firstly introduced by McNamara and Zanetti [28]. The lattice Boltzmann evaluation equation with Bhatnagar-Gross-Krook collision model [29] is defined aswhere is particle distribution function along the particle speed direction at position and time . denotes the equilibrium distribution function and is the single relaxation time parameter.

The macroscopic velocity and density are calculated by the following relations.The equilibrium distribution function depending on density and velocity is given as [30] where is the lattice speed while and are the lattice width and time step, respectively. In the present work which means that . The weighting factors for the two-dimensional and nine-velocity particle (D_{2}Q_{9}) model [30], considered in the present study, are , , and . Furthermore, the speed of sound is and the kinematic viscosity is taken to be . Further details about LBM can be found in [31].

#### 3. Geometry of Problem

The schematic flow diagram for the problem under consideration is given in Figure 1. Three square cylinders having the same size () are placed, inline to each other, inside a rectangular domain of length* L* and height* H*. The first cylinder (*c*_{1}) is placed at a distance* L*_{u} = 6 from the entrance of computational domain. The second cylinder (*c*_{2}) is placed at a gap distance from* c*_{1} while the third cylinder (*c*_{3}) is also placed at a gap distance from* c*_{2}. Similarly the distance from* c*_{3} to the exit position of computational domain is* L*_{d} = 25. All the lengths are nondimensionalized using the cylinder size “”. Uniform inflow condition (*u* = and* v* = 0) is incorporated at entrance of computational domain. While at exit position, convective boundary condition () is applied in terms of distribution functions [32]. At solid surfaces, including the surface of cylinders and walls of channel, the bounce back rule is applied which resembles to the no-slip boundary condition [33].