Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 6381256, 18 pages

https://doi.org/10.1155/2017/6381256

## Numerical Study on Flow around Four Square-Arranged Cylinders at Low Reynolds Numbers

^{1}Ocean College, Zhejiang University, Zhoushan 316021, China^{2}China Communications Construction Company, Beijing 100088, China

Correspondence should be addressed to Xi-zeng Zhao; nc.ude.ujz@oahzgnezix

Received 16 October 2016; Accepted 4 January 2017; Published 27 March 2017

Academic Editor: Nicolas Gourdain

Copyright © 2017 Yang-yang Gao 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

In this paper, numerical simulations of flow past four square-arranged cylinders are carried out at different spacing ratios (; is the center to center distance; is the cylinder diameter) and Reynolds numbers . The effects of spacing ratio and Reynolds number on the wake flow characteristics are investigated, such as the instantaneous vorticity contours, force coefficients, and vortex shedding frequencies. The results show that the flow characteristics behind the four-cylinder cases are significantly affected by the spacing ratios and Reynolds numbers. At the same spacing ratio, the transformation of flow pattern is advanced quickly with increasing of Reynolds numbers, the values of force coefficients are correspondingly fluctuated with large amplitude, and the vortex shedding frequency is increased significantly with .

#### 1. Introduction

Investigation of unsteady viscous flow past a single or array of circular cylinders is of great interest for many ocean and offshore engineering applications. Compared to an isolated single cylinder, it is more difficult to predict flow around multicylinder configurations, depending on the geometric parameters such as spacing ratio between the cylinders ( is the center-center distance between cylinders, and is the cylinder diameter) and the incidence angle relative to the free-stream flow. The flow behaviors behind multicylinders are much more complex due to the complicated dynamic interaction between the shear layers, shed vortices, and Karman vortex streets in the wake flow. Most of the researches are focused on flow around an isolated circular cylinder or two cylinders, as pointed in the reviews by Zdravkovich [1], Sumner [2], and Bearman [3]. However, few investigations are performed to reveal the flow dynamics of a four-square-arranged cylinder configuration, which is essential for various marine structures as a fundamental arrangement. Four square-arranged cylinders could be found in many offshore engineering applications, such the pile foundation system of offshore platforms and offshore wind turbine, risers, and pipelines. When the four cylinders are in close proximity configuration, flow-induced vibration may arise, which would increase the risk of fatigue failure in engineering. Therefore, the numerical simulation was carried out to investigate the flow characteristics of flow past four square-arranged cylinders.

The earliest experimental investigations on flow past four square-arranged cylinders were reported by Sayers [4, 5]. The force coefficients and vortex shedding frequency for each cylinder were investigated at and (Reynolds number is defined as , is the diameter of cylinder, is the free-stream velocity, and is the kinematic viscosity) in an open-jet wind tunnel. The nondimensional vortex shedding frequency St (defined ; is the vortex shedding frequency) was found to be varied largely across wake at and equal to that for a single isolated cylinder when . Soon after that, Lam and Lo [6] categorized the flow characteristics behind four square-arranged cylinders into three flow patterns varied with ranging from 1.28 to 5.96 at : (i) the shear layers of upstream cylinder shielded the downstream cylinder; (ii) the generated free shear layers from upstream cylinder reattached onto the downstream cylinder; (iii) the vortices shed from upstream cylinder and impinged the downstream cylinder. Lam and Fang [7] experimentally studied force coefficients and the effects of flow interference between four cylinders on the mean pressure distribution while ranges from 1.26 to 5.80 with and . They confirmed that is a critical spacing ratio and the forces characteristics vary largely with incidence angles (0° ≤ * α* ≤ 45°) and . Similarly, Lam et al. [8] experimentally measured the mean and fluctuating forces and Strouhal numbers on each cylinder at spacing ratio in the range 1.69 to 3.83, different angles of incidence

*ranging from 0° to 180° at a 15° interval, and . As a result, three types of flow pattern were observed depending on : (i) shear layers around cylinder oscillate freely and vortex forms at large , (ii) shear layers reattach on or shield the downstream cylinder, and (iii) narrow gap flows form at small . Lam and Zou [9, 10] summarized several distinct flow patterns at subcritical Reynolds numbers and concluded that the turbulence flow patterns are affected by the spacing ratio and Re by using LDA (laser Doppler anemometry) and PIV (particle imaging velocimetry) to measure the turbulent flows around four cylinders in an in-line square configuration with different spacing ratios of 1.5, 2.5, 3.5, and 5.0. Lam et al. [11] carried an experimental investigation on flow past four cylinders in a square configuration with a spacing ratio of 4.0 and at by using PIV and LIF (laser induced fluorescence) flow visualization technique. Concerning*

*α**, they observed three basic flow patterns including the formation of a jet flow which could lead to strong flow-induced vibration. Recently, Wang et al. [12] further investigated the vortex shedding characteristics using PIV at various from 2.0 to 5.0, , and inclination angle ranging from 0° to 45°. The fluctuating drag and lift coefficients on four cylinders were measured and the relation between the flow patterns and forces on the cylinders was further studied. By using LIF and PIV technology, Zou and Lin [13] captured the biased flow pattern and classical bistable phenomenon for flow past four square-arranged cylinders at , , and (*

*α**H*is the length of cylinder immersed in water), which is seldom investigated by researchers.

Apart from the above experimental studies, numerical simulations on four-cylinder array have also been investigated at small Reynolds numbers, except for some LES study [9, 10, 14] at and the FEM study by Zhao and Cheng [15] at . Farrant et al. [16] employed the cell boundary element method (CBEM) to investigate the two-dimensional incompressible flow around four-cylinder array at and inclination angle * α* = 0° and 45°. The in-phase and anti-phase vortex shedding behavior were observed in their study. Lam et al. [17] carried out a two-dimensional simulation on cross-flow past four square-arranged cylinders at and 200. The results of their study showed that, depending on the spacing ratio , three distinct flow patterns were observed, and the alternation of forces and pressure distribution on cylinders when different flow patterns transform were also discussed. Han et al. [18] used spectral element method (SEM) to simulate a two-dimensional laminar flow with incidence angle = 0° and 45°. Three distinct flow patterns exist with variation of in each incidence angle, and the force coefficients and Strouhal numbers changed significantly when one kind of flow patterns switched to another. Abbasi et al. [19] further performed 2D numerical investigation of flow past four square cylinders in an in-line square configuration using LBM (lattice Boltzmann method). Four distinct flow patterns were observed at different Reynolds numbers () when spacing ratio was set at 2.0, 4.0, and 7.0, respectively. They also noted that the variation of was contributed much to the transformation of flow pattern and the characteristics of wake flow when spacing ratio is small (). Tong et al. [20] carried out a three-dimensional simulation on flow past four square-arranged cylinders at and . Four wake flow regimes were observed at different , and they found that the root-mean-square (RMS) values of dynamical parameters, length of wake flow, and phase angle of lift coefficient on downstream cylinders were all changed significantly when the flow transited from one regime to another. Generally, the published works discussed above are summarized in Table 1.