Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 4126123, 8 pages

http://dx.doi.org/10.1155/2016/4126123

## Large-Scale Computations of Flow around Two Cylinders by a Domain Decomposition Method

^{1}School of Engineering, Sun Yat-sen University, Guangzhou 510275, China^{2}Department of Mathematical and Physical Sciences, Japan Women’s University, Tokyo 112-8681, Japan

Received 25 September 2015; Revised 31 December 2015; Accepted 28 February 2016

Academic Editor: Uchechukwu E. Vincent

Copyright © 2016 Hongkun Zhu 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

A parallel computation is applied to study the flow past a pair of cylinders in tandem at Reynolds numbers of 1000 by Domain Decomposition Method. The computations were carried out for different sets of arrangements at large scale. The modeling by domain decomposition was validated by comparing available well-recognized results. Two cylinders with different diameters were further investigated; for different diameter ratios, the wake width ratio and some properties of the critical space ratio that dominates the flow regime were discovered. This result has important implications on future industrial application efforts as well as codes and standards related to the two-cylinder structure.

#### 1. Introduction

Cylinder is one of the most common structures in engineering, such as piers and chimney, the struts of offshore platform, the pipe of condenser, and more. When fluid flows through the cylinders, the shedding vortex may cause interference effect. This effect may lead to the vibration of the cylinder structures or fatigue of the materials and as a result, the structure may be destroyed. Due to its importance in engineering applications, flow past two cylinders has been studied by experimental and numerical investigations for several decades [1]. As early as 1977, Zdravkovich classified the characteristic of flow past two cylinders in tandem into three regimes at low Reynolds number (Re). For more complicated applications, Wu et al. [2] studied two cylinders in tandem with wind tunnel and water tunnel at the Re of 1000. The flow visualization in the water tunnel showed the existence of streamwise vortices in spanwise direction. To make out the relationship between different arrangements of two cylinders and Re, Mittal et al. [3] studied incompressible flows past two cylinders in tandem and staggered arrangements (Re = 100 and Re = 1000) by a stabilized finite element method (FEM). Jester and Kallinderis [4] investigated the incompressible flow about fixed cylinder pairs numerically and cylinder arrangements include tandem, side-by-side, and staggered at Reynolds numbers of 80 and 1000. Hysteresis effects and bistable biased gap flow in tandem arrangements were reproduced. Different combinations of Re and arrangements of cylinders have been reported by many [5–7], as is reported in [8–17] and summarized by Sumner [1]; most of the early researches on this problem contribute to the flow structures induced by different spacing ratios (), Re and variant cylinder shapes.

Recently, there has been a lot of interest on the diameter ratio () of two cylinders in tandem. Zhao et al. studied turbulent flow past two cylinders with different diameters numerically. The hydrodynamic force and vortex shedding characteristics were proved to depend on the relative position of small cylinders around the main cylinder [18]. Mahbub Alam and Zhou investigated the Strouhal number, hydrodynamic forces and flow structures, and vortex shedding frequency of flow past two cylinders in tandem with different diameters in wind tunnel [19]. The diameter of upstream cylinder varied from 0.24 to 1 of the downstream cylinder diameter and the distance between two cylinders remains 5.5 times of the diameter of the upstream cylinder. Ding-Yong et al. conducted the simulation of different spacing ratios and different diameter ratios at Re = 200 using Fluent [20]. Besides two cylinders, Zhang et al. determined the influence of the diameter ratio on the flow past three cylinders in two dimensions by applied finite element method [21].

As far as we know, the flow structure of two cylinders in tandem affected by the changing of diameter ratio and spacing ratio is still uncertain, and the computation scale of numerical experiments published research is very limited. To investigate the flow regime more concretely and more subtly, large-scale simulations by Domain Decomposition Method (DDM), which is considered to have better accuracy and less time cost when comparing with conventional methods, are implemented [22, 23]. The large-scale modeling was validated by comparing with others’ reports for two cylinders of the same diameter with different spacing ratios. Moreover, to study its influence to the flow past two cylinders in tandem more comprehensively, the flow structures at specified diameter ratios ( = 0.5 and = 1) with different spacing ratios (: 3–6) are investigated. The critical spacing ratio that affects the flow regime is expected to be determined for different diameter ratios.

This paper is organized as follows: Section 2 introduces the governing equations to be solved as well as the domain decomposition method (DDM). In Section 3, the models and the boundary conditions are described in detail. Numerical results for different spacing ratio and different diameter ratios ( and ) are present and compared with others’ work in Section 4. At last, Section 5 draws the conclusions of this work.

#### 2. Formulations

##### 2.1. Governing Equations

Let be the boundary of a three-dimensional polyhedral domain . is the first order Sobolev space and is the space of 2nd power summable functions on . Under the assumption that the flow field is incompressible, viscous, and laminar, the solving of the model can be summarized as finding such that for any , the following set of equations hold [24]:where is the velocity []; is the pressure [Pa]; is the density (const.) [kg/m^{3}]; is the body force [N/m^{3}]; is the stress tensor [N/m^{2}] defined bywith the Kronecker delta and the viscosity [kg/(m·s)].

An initial velocity is applied in at Dirichlet boundary conditionsand Neumann boundary conditionsare also applied, where and is the outward normal direction to

##### 2.2. Characteristic Finite Element Scheme

Using the definition a characteristic finite element scheme approximates the material derivative in (1) at as follows: [23]where is a position function; see Figure 1.