Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 9086246, 15 pages

http://dx.doi.org/10.1155/2016/9086246

## A Numerical Study of the Forces on Two Tandem Cylinders Exerted by Internal Solitary Waves

^{1}State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, Nanjing 210098, China^{2}College of Water Conservancy and Hydropower Engineering, Hohai University, Nanjing 210098, China^{3}College of Water Conservancy and Ecological Engineering, Nanchang Institute of Technology, Nanchang 330099, China^{4}College of Meteorology and Oceanography, PLA University of Science and Technology, Nanjing 211101, China

Received 20 January 2016; Revised 9 May 2016; Accepted 22 May 2016

Academic Editor: Maurizio Brocchini

Copyright © 2016 Yin Wang 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 three-dimensional numerical wave flume is employed to investigate the forces exerted by internal solitary waves (ISWs) on a pair of circular cylinders in tandem arrangement, using large-eddy simulation (LES) model. The effect of the centre-to-centre distance (*L*) ranging from 1.5 to 5 diameters (*D*) is studied for various ISWs amplitudes () in the two-layer fluid system. Vertical-averaged vorticity distribution and vertical-averaged pressure gradient distribution in each layer are presented to investigate the different hydrodynamic interference between cylinders and the ISWs forces on each cylinder at various* L*. Furthermore, the force behaviors of the two cylinders are also compared with that of an isolated cylinder in the same environment. The interaction between the two piles occurs in both layers, and it is found that, for , strong mutual interference appears between two cylinders; for , the two cylinders continue to influence each other in a weak-interference state; for , the interaction gradually decreases into a noninteracting state. This paper tries to provide some references to structural arrangement of double-cylinder structure and grouped-cylinder structure in stratified flow environment.

#### 1. Introduction

In oceans, estuaries, and lakes, the stable stratification of density happens while the fluid density changes along with the depth due to the variations of temperature, salinity, and other environmental factors. Internal solitary waves (ISWs) with different amplitudes may be produced by a tiny or weak perturbation in such stable stratified environment [1]. Based on the monitoring and measured data collected from the South China Sea, strong underwater currents caused by internal waves could be a serious threat to underwater structures, such as oil drilling platforms or supporting cylinders [2].

Thus more and more researchers got involved in the studies of IWs action on such structures. Laboratory experiment is an important approach to investigate the ISWs loading on cylinders. Ermanyuk and Gavrilov [3] experimentally studied the hydrodynamic loads exerted by ISW on a submerged circular cylinder in a two-layer system and discovered the locations of the maximum and minimum horizontal loads on the cylinder. Wei et al. [4] manufactured a new wave-maker to excite the ISWs and developed a measurement technique of hydrodynamic load to determine the interaction characteristics between IWs and an isolated cylindrical body in the tank.

Along with rapid development of computer and CFD technology, numerical methods have gotten more and more extensive application to investigate the loading induced by ISWs. The Morison empirical method [5], modal separation [6], and regression analyses [7] were applied to estimate the forces exerted by internal soliton on cylindrical piles. After that, a simplified estimation method of the force exerted by ISWs involving only the first mode internal wave was proposed [8]. The method can be used to estimate the force without observational current data and simplify the calculation procedure. For better solving the issues related to depression ISWs, an extended KdV model (EKdV) was employed to calculate the forces on a pile exerted by ISWs propagation from deep sea to shallow water [9]. Afterwards, Si et al. [10] used a generalized KdV model (GKdV) to obtain the vertical distribution of horizontal velocity of large-amplitude ISWs and discovered that the shear force was the largest at the turning point of the horizontal velocity. Based on water-wave theory, Linton and McIver [11] and Cadby and Linton [12] built a two-dimensional and three-dimensional numerical model, respectively, to study the interaction of waves with structures in two-layer fluids, and multipole expansions were used to solve the problems of wave radiation and scattering by a submerged structure in either the upper or lower layer. Then Sturova [13] conducted a study of radiation loads on interface piercing cylinder in a two-layer fluid of finite depth by a coupled element technique. By comparing with the ISWs force and the surface wave force, Du et al. [14] and Song et al. [15] found that the order magnitude of total force exerted by internal soliton was the same as that exerted by a surface wave, and the maximum total horizontal force caused by an internal wave amounted to 37.7% of that exerted by a surface wave [16]. Considering the above-mentioned, the ISWs force on underwater structures cannot be neglected.

Nevertheless, most researches before just focused on the cases of ISWs forces on an isolated cylindrical pile. In practical engineering, pile-supported structures are generally arranged in tandem or side by side with various centre-to-centre distance (*L*) to form the so-called multiple slender structures, such as the bundle of risers linking the seabed to the offshore platforms and the piles being used for supporting bridges [17]. In general, two cylinders of the structures submerged in water behave in a similar manner to a single cylinder when the two cylinders are sufficiently apart. In some situations the cylinders need to be placed at a close proximity [18], and the interference between the two bodies significantly changes the flow around them [19]. Different flow patterns can be characterized by the behavior of the wake region [20], and unexpected flow structures and forces can be generated as the spacing between two circular cylinders changes [21].

Up to date, very few numerical investigations about the internal solitary waves (ISWs) action on a pair of circular cylinders in tandem arrangements can be found in the literatures. Thus, this paper aims to study numerically ISWs force behaviors of two tandem cylinders (placed parallel to wave direction). To begin with, large-eddy simulation model (LES) is employed to simulate the generation and propagation of depression ISWs in a three-dimensional numerical wave flume. With respect to two-phase stratified flow system, the horizontal current induced by ISWs reverses the flow direction between upper and lower layers. Therefore, the hydrodynamic interference occurring in each layer at diverse* L* ranging from 1.5*D* to 5*D* will be investigated, respectively, from the perspective of vorticity distribution and pressure gradient distribution. The changes in vorticity fields and pressure gradient fields induced by different pile-to-pile interactions will be employed to study and explain the ISWs force behaviors of the two tandem cylinders. Finally, the results are also compared with that of an isolated cylinder in the same environment.

#### 2. Theoretical Foundation

##### 2.1. Korteweg-de Vries (KdV) Theory

Many nonlinear equations, represented by KdV equation, are widely applied to describe the propagation of the internal solitary waves in horizontal direction. KdV equation can be written as where is the time; and represent the thicknesses of the upper layer and the lower layer; the linear velocity of linear IWs and the interfacial vertical displacement can be, respectively, expressed as where is the amplitude of incident wave; is the phase speed; is the characteristic wavelengths; is the density difference between two layers; is the gravity acceleration. and read

#### 3. Numerical Model

##### 3.1. Navier-Stokes Equations

The process of three-dimensional unsteady incompressible fluid motion governed by Navier-Stokes (N-S) equations can be described by where is the time; is the velocity component; is the Cartesian coordinate; is the pressure; is the kinematic viscosity; and is the body force that equals the gravity acceleration in the vertical direction.

##### 3.2. Scalar Transport Equation

The mass transfer between the two-layer water system has been taken into account. The scalar transport equation that governs the advection-diffusion effect is as follows:where is the volume concentration of brine water in the lower layer of the fluid system;* k* is the molecular diffusivity coefficient.

##### 3.3. LES Governing Equations

Applying a spatial filter to Navier-Stokes equations, the filtered momentum and mass equations can be written as where the overbar notation denotes the application of top-hat filter; the subgrid scale (SGS) stress tensor term in (9) is responsible for the momentum exchange between the subgrid scale and the resolved scale; is the subgrid scalar flux responsible for the scalar flux exchange between the subgrid scale and the resolved scale. Hence, subsequent modeling is required to determine the turbulent viscosity . As a function of the filter size and strain rate tensor, can be defined as where is the strain rate tensor; is the Smagorinsky constant; Δ is the filtered width. In general, the model coefficient varies in both time and space due to different flow conditions and even can be negative for a long time. Thus a dynamic procedure developed by Germano et al. [22] is employed to determine .

##### 3.4. Numerical Method and Boundary Condition

Large-eddy simulation model (LES) is employed to simulate the generation and propagation of ISWs of depression type. Velocity-pressure term is solved by SIMPLE algorithm to enforce mass conservation and to obtain the pressure field, while second-order centred differencing scheme is adopted for the spatial discretization, and the time step is discretized by second-order implicit scheme.

The left boundary, belonging to the wave generation area where gravity collapse happens, and the two sidewalls and bottom of the wave tank are specified as a rigid wall with no-slip condition. Sommerfeld radiation type to avoid wave reflection is adopted to specify the right boundary, and the “rigid lid” approximation is used here to filter the free surface mode to ignore the influence of the surface wave [23].

##### 3.5. Building of Numerical Wave Flume

A three-dimensional numerical wave flume established in current study is illustrated in Figure 1. The numerical wave tank in this paper has a dimension of 12 m × 0.5 m × 0.4 m in the stream-wise (), span-wise (), and vertical () direction, which represents the length, width, and height, respectively. This flume is divided into two parts, for one is the wave generation area which locates = 0.3 m from the left boundary in the direction and the remaining is the wave propagation area. The two piles are placed in tandem arrangement. The bottom centre of the upstream cylindrical pile (Pile_{1}) locates at from the coordinate origin, while* L* is the centre-to-centre distance between Pile_{1} and the downstream cylindrical pile (Pile_{2}).* D* is the diameter of two piles. The density is , and the thickness of each layer is* h*, of which the subscripts 1 and 2 represent the upper layer and the lower layer. The upper-layer fluid density is set to be 1000 kg/m^{3}, the lower-layer fluid density is set to be 1030 kg/m^{3}, and the volumetric concentration* C* of the brine water in the lower layer is around 3%. The step height Δ*h* we called here is the height difference of the pycnocline. The whole two-layer system keeps quiescent at the initial time.