Mathematical Problems in Engineering

Volume 2018 (2018), Article ID 2394124, 14 pages

https://doi.org/10.1155/2018/2394124

## Higher-Order Spectral Analysis to Identify Quadratic Nonlinearities in Fluid-Structure Interaction

^{1}Department of Mechanical Engineering, College of Electrical & Mechanical Engineering, National University of Sciences & Technology (NUST), Islamabad 44000, Pakistan^{2}ANSYS, Inc., Houston, TX 7745, USA

Correspondence should be addressed to Imran Akhtar

Received 21 June 2017; Revised 8 October 2017; Accepted 1 November 2017; Published 9 January 2018

Academic Editor: Efstratios Tzirtzilakis

Copyright © 2018 Imran Akhtar and Mohammad Elyyan. 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

Hydrodynamic forces on a structure are the manifestation of fluid-structure interaction. Since this interaction is nonlinear, these forces consist of various frequencies: fundamental, harmonics, excitation, sum, and difference of these frequencies. To analyze this phenomenon, we perform numerical simulations of the flow past stationary and oscillating cylinders at low Reynolds numbers. We compute the pressure, integrate it over the surface, and obtain the lift and drag coefficients for the two configurations: stationary and transversely oscillating cylinders. Higher-order spectral analysis is performed to investigate the nonlinear interaction between the forces. We confirmed and investigated the quadratic coupling between the lift and drag coefficients and their phase relationship. We identify additional frequencies and their corresponding energy present in the flow field that appear as the manifestation of quadratic nonlinear interaction.

#### 1. Introduction

The fluid-structure interaction has its significance in flow physics and industrial applications. The flow behind a circular cylinder has become the canonical problem for studying such external separated flows [1–4]. When flow passes over a bluff body, an organized and periodic motion of a regular array of concentrated vorticity, known as the von Kármán vortex street, appears in the wake of the bluff body. This vortex shedding exerts oscillatory forces on the body, which are often decomposed into drag and lift components along the freestream and crossflow directions, respectively. If the body is capable of flexing or moving rigidly, these forces can cause it to oscillate and the classical vortex-induced vibration (VIV) problem takes place.

Many experimental and numerical studies have been performed to understand, model, and predict the phenomenon of VIV for fixed, excited, and elastic cylinders. The problem of a vibrating cylinder due to exerted forces goes back to the work of Strouhal [5] in the area of aeroacoustic and to the work of Rayleigh [6] on the oscillations of violin strings subject to the incoming wind. The first significant contribution to this problem is credited to Bishop and Hassan [1] who experimentally studied the flow past an externally oscillated cylinder over a Reynolds number range of 5,850 to 10,800 within the* lock-in* frequency range. The lock-in frequency range is the bandwidth where the lift frequency is entrained by the oscillating frequency of the cylinder. They reported that this bandwidth is a function of the Reynolds number and the amplitude of motion.

Williamson and Roshko [7] performed an experimental study of the flow over a cylinder undergoing simple harmonic motion in a low Reynolds number range, 300 to 1,000. They used aluminum particles on the fluid surface for visualization. They observed various vortex patterns and constructed a* map* of vortex synchronization regions using coordinates as a function of the nondimensional wavelength of cylinder motion (). They denoted certain vortex patterns by symbols, such as “P + S”, “S”, and “2P”. Here “S” denotes a single vortex, and “P” signifies a pair of vortices of opposite signs. Thus, the vortex pattern “P + S” is one in which a pair and a single vortex are shed during each oscillation cycle. Similarly, two vortex pairs are shed per cycle in a “2P” wake pattern, while two singlets are shed in a “2S” wake pattern.

Krishnamoorthy et al. [8] investigated the wake patterns of an oscillating cylinder at Reynolds number of 1,500. They varied the frequency of oscillation from one-third to three times the natural shedding frequency with a nondimensional amplitude of 0.22. They observed abrupt switching between some vortex patterns and a phase jump in the vortex shedding relative to the cylinder displacement.

Recent numerical studies on the flow around stationary and oscillating cylinders include the direct numerical simulations by Dong and Karniadakis [13], where they have simulated a cylinder excited at a nondimensional amplitude of 0.3 with at Re = 10,000. They used the spectral element method with polynomial orders ranging from 5 to 8 and 300 million degrees of freedom to resolve the flow at this high Reynolds number. The computational domain employed for the simulations extended 20 diameters upstream and 50 diameters downstream, with 40 diameters in the crossflow directions. The problem was solved in a coordinate system that is attached to the cylinder.

Al-Jamal and Dalton [14] performed a two-dimensional LES to compare the flow past a forced and an elastically vibrating cylinder within the lock-in region at Reynolds number of 8,000. They used a vorticity formulation instead of primitive-variable formulation with second-order central-difference scheme for all spatial derivatives. The problem was solved in an absolute coordinate system over an O-grid with a radius of five cylinder diameters. For the elastically mounted cylinder, spectral and complex demodulation analyses show that the phase angle between the lift and motion is strongly modulated, which is remarkably different from what is typically observed in an externally excited motion.

Kim and Williams [15] experimentally studied the nonlinear coupling of fluctuating drag and lift on a forced oscillating cylinder at = 15,200. They considered two cases of cylinder oscillations, crossflow and inline, at an oscillating frequency different from the von Kármán vortex shedding frequency. The nonlinear coupling between the von Kármán vortex shedding and the forcing frequency was investigated using higher-order spectral tools. Kim and Williams [15] reported quadratic nonlinear interaction between the vortex shedding modes and the oscillation frequency which produced sum and difference modes in the lift and drag coefficients spectra. However, their study is limited to small amplitudes of vibration.

Marzouk and Nayfeh [16] performed two-dimensional unsteady RANS simulations of the flow around stationary and crossflow oscillating cylinders with different frequencies and amplitudes in the* lock-in* regime at Re = 500. The problem was solved on a curvilinear coordinate system on an O-grid mesh with a radius of 25 cylinder diameters. They employed the artificial compressibility method to solve the governing equations with second-order accurate schemes in both time and space: central differences for the diffusion term and an upwind scheme for the convective terms. The one-equation Spalart-Almaras model is applied to represent the unresolved scales in the flow. They observed discontinuities between the frequency response and phase angle of the lift relative to the displacement and identified two distinctive modes, namely, a low-lift mode and a high-lift mode. In the low-lift mode, the lift and displacement are out-of-phase while, in the high-lift mode, they are in-phase. Moreover, they reported that the motion amplitude must exceed a certain threshold for these discontinuities to take place. For a comprehensive review of the VIV phenomenon, the reader is referred to Williamson and Govardhan [17].

Nayfeh et al. [18] investigated two wake-oscillator models to represent the lift, namely, the van der Pol and Rayleigh oscillators. Using a higher-order spectral moments analysis, they found that the phase angle between the lift components at and is around . Based on this finding, they concluded that the van der Pol oscillator is more suitable for modeling the* steady-state* lift coefficient; that is,The angular frequency in (1) is related (but not equal) to the* angular shedding frequency * and the parameters and represent the linear and nonlinear damping coefficients, respectively. The values of the parameters in (1) depend on the Reynolds number. Nayfeh et al. [19] modified the computational fluid dynamics (CFD) based analytical model [18] to include the transient flow over the cylinder. Marzouk et al. [20] added a Duffing-type cubic term to the van der Pol oscillator model of the lift to match the phase between different harmonics. Later, Akhtar et al. [21, 22] extended the model to a more general shape of elliptic cylinders with different eccentricities. Unlike the lift oscillator model, the drag coefficient is modeled as a function of the or depending upon their phases, such thatEquation (2) represents the coupling between the lift and drag coefficients.

Modeling of the lift and drag coefficients is complex and requires a thorough understanding of flow physics even in the stationary case. This fluid-structure interaction becomes more complex in the nonstationary case where the cylinder is either forced to oscillate or vibrates under the influence of hydrodynamic forces. Due to inherent nonlinearity in this phenomenon, one would expect interaction of multiple frequencies in the dynamical system. For three-dimensional flows, which is often the case in real applications, the hydrodynamic forces are affected by the turbulent structures formed in the wake and can not be modeled without including the effects of these structures.

The objective of the current study is to quantify the nonlinear coupling between the hydrodynamic forces on stationary and oscillating circulars [23]. In the current study, we perform numerical simulations of the flow past two- and three-dimensional flows past stationary and oscillating cylinders at and . Higher-order spectral techniques, such as auto-power spectrum, cross-bispectrum, and cross-bicoherence, are employed to identify the quadratic coupling between the lift and drag coefficients acting on the cylinder. In this study, modes refer to various frequencies that appear as the manifestation of quadratic coupling.

The manuscript is organized as follows. Section 2 presents the numerical methodology to solve the incompressible Navier-Stokes equations. The numerical results are then validated and verified. In Section 3, we discuss the Accelerating Reference Frame (ARF) methodology employed to simulate the moving structure, that is, oscillating cylinder. Again, the numerical results are compared with the experimental results for an oscillating cylinder with different frequencies and amplitudes. We define cross-bispectrum and cross-bicoherence in Section 4. In Section 5, we present the numerical results and analyze the nonlinear interaction for different flow configurations. We identify quadratic coupling between the lift and drag forces. This coupling provides an insight of the flow physics and is useful in developing reduced-order models for the hydrodynamic forces on the structure and computing the phase relationship between the forces.

#### 2. Methodology and Numerical Scheme

The following section outlines the computational technique employed in the current study. Validation of the solver is performed by comparing the results with existing experimental and numerical data.

##### 2.1. Numerical Methodology

The flow in the current problem is governed by the incompressible continuity and momentum (Navier-Stokes) equations, which can be represented as follows.

*Continuity Equation*

*Momentum Equation*where ; represents the Cartesian velocity components ; is the pressure; is the fluid density; and is the fluid kinematic viscosity. In order to broaden the spectrum of the application, we employ body fitted curvilinear coordinates for the governing equations. Thus, the irregular physical domain is transformed into a regular computational space. Therefore, the time-dependent incompressible continuity and Navier-Stokes equations are solved in a generalized coordinate system . Furthermore, we nondimensionalize the governing equations with the cylinder diameter (*D*) as the characteristic length and freestream velocity () as the characteristic velocity. The flow Reynolds number is defined as , where is the kinematic viscosity. The nondimensional governing equations in the generalized coordinates are written in a strong-conservative form aswhere the flux () is defined asIn (6), is the inverse of the Jacobian or the volume of the cell; is the volume flux normal to the surface of constant ; and is the “mesh skewness tensor.”

The governing equations are solved on a nonstaggered grid topology [24]. A second-order central-difference scheme is used for all spatial derivatives except for convective terms where a QUICK scheme is employed [25]. Temporal advancement is performed using semi-implicit predictor-corrector scheme, where an intermediate velocity is calculated at the predictor step and updated at the corrector step by satisfying the pressure-Poisson equation.

In this study, an “O”-type grid is employed to simulate the flow over a circular cylinder as shown in Figure 1. Inflow and outflow boundary conditions at the domain boundaries are simulated using Dirichlet and Neumann boundary conditions, and no-slip no-penetration boundary conditions are applied at the cylinder surface.