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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.

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 [

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 [

Williamson and Roshko [

Krishnamoorthy et al. [

Recent numerical studies on the flow around stationary and oscillating cylinders include the direct numerical simulations by Dong and Karniadakis [

Al-Jamal and Dalton [

Kim and Williams [

Marzouk and Nayfeh [

Nayfeh et al. [

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 [

The manuscript is organized as follows. Section

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.

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

The governing equations are solved on a nonstaggered grid topology [

In this study, an “O”-type grid is employed to simulate the flow over a circular cylinder as shown in Figure

A 2D layout of an “O”-type grid in the (

The computational domain is decomposed using two-dimensional decomposition, which allows us to take advantage of scalable parallel computers (e.g., distributed-memory clusters, message-passing programming model). In the domain decomposition technique, each processor gets a “slice” of the grid, as shown in Figure

(a) A 2D layout of “O” grid distributed among 8 processors in the

The fluid force on the cylinder is the manifestation of the pressure and shear stresses acting on the surface of the cylinder, which can be decomposed into two components, namely, lift and drag forces. Lift and drag coefficients are expressed in terms of the pressure and shear stresses as follows:

To validate our numerical results, we perform two- and three-dimensional numerical simulations of the flow past a stationary circular cylinder at Reynolds numbers of 525 and 1,000. For the flow past a circular cylinder, the wake starts to get turbulent after Re = 500 and becomes fully turbulent at Re = 1,000. We compute the lift and drag coefficients using (

We perform two- and three-dimensional direct simulations of the flow over a stationary circular cylinder at

Flow parameters at

Data from | | St |
---|---|---|

Experiment [ | 1.15–1.2 | − |

Experiment [ | − | |

3D DNS [ | | |

3D DNS (present) | | |

2D DNS [ | | |

2D DNS (present) | | |

The instantaneous vorticity isosurfaces (level = 0.5, 1.0, and 1.5) at

Similar to the

Spanwise vorticity contours of one vortex shedding cycle over a stationary cylinder at

For the three-dimensional calculation, we performed a DNS on a grid size of

Geometry of the cylinder with arrows showing the flow direction and checkerboard pattern indicating 8 processors in the spanwise direction (a) and isosurfaces of

Table

Flow parameters at

Data from | | St |
---|---|---|

Experiment [ | | |

2D [ | | |

2D (present) | | |

3D DNS [ | | |

3D DNS (present) | | |

Figure

1D energy spectrum at

In order to ensure the sufficiency of the grid resolution and domain size, we conduct grid and domain independence studies of our domain. Grid and domain independence studies are performed for two- and three-dimensional grids for flow past a stationary cylinder at

For the two-dimensional geometry arrangement, we considered two grid resolutions at domain size of

Grid dependence study

Dim | Grid size ( | Domain size | CPU ( | | St |
---|---|---|---|---|---|

2 | | | | | |

2 | | | | | |

| |||||

3 | | | | | |

3 | | | | | |

In order to establish domain independence, we increased the domain size to

Domain dependence study

Dim | Grid size ( | Domain size | CPU ( | | St |
---|---|---|---|---|---|

2 | | | | | |

2 | | | | | |

| |||||

3 | | | | | |

3 | | | | | |

In a fluid-structure interaction, a structure may be elastic or excited externally. The behavior of a structure undergoing vibrations depends mainly on the natural frequencies of the structure and vortex shedding. If the frequency of vortex shedding is close to the natural frequency of the structure, it might oscillate with high amplitudes, eventually leading to its damage or complete structural failure.

In order to simulate the motion of the structure, three basic methods can be employed for the numerical discretization of the flow problem, namely, the Immersed Boundary (IB) method, the Arbitrary Lagrangian-Eulerian (ALE) method, and the Accelerating Reference Frame (ARF) method. In the Immersed Boundary (IB) method, the flow is simulated with immersed boundaries on a grid that does not conform to the shape of these boundaries. The advantage of IB method appears in the grid generation, which does not require coordinate transformation at every increment of body motion. However, applying appropriate boundary conditions is not straightforward and resolving the boundary layer at high Reynolds numbers increases the grid-size requirement faster than a corresponding body-conformal grid. For detailed description and comparison of this method, readers are referred to Mittal and Iaccarino [

In Arbitrary Lagrangian-Eulerian (ALE) method, the mesh local to the structure is distorted continuously in time as the structure moves, and the boundary conditions on the body and in the far field are usually fixed in time. In terms of computational cost, the ALE method is more computationally expensive than the IB method due to the continuous remeshing and the temporal changes of the mesh interpolation functions. Unlike ALE, the mesh in the Accelerating Reference Frame (ARF) method is fixed to the structure and the momentum equations and corresponding boundary conditions are modified to allow for the motion. In this way, the computational overhead associated with the coordinate transformation at every time step can be avoided; however, the ARF method is limited in its application.

In our problem of the flow past an oscillating cylinder, the ARF method is a suitable candidate for simulating moving boundaries. The momentum equations can be directly coupled with the cylinder motion by adding a frame acceleration term and modifying the boundary conditions to accommodate the moving reference frame [

At the domain boundary, the velocity boundary condition is modified to include the effect of moving body, such that

In the current study, the equation governing the nondimensional displacement (forced-oscillation) of the cylinder is modeled by a simple harmonic motion in the inline and crossflow directions as follows:

A schematic of the possible stationary and moving boundary cases.

To validate the moving boundary feature in the parallel CFD solver, we simulate different flow configurations for a stationary and crossflow oscillating cylinders as shown in Table

Crossflow oscillations.

Case | | | | Comments |
---|---|---|---|---|

A | 200 | − | − | Stationary |

B | 200 | | | Crossflow |

C | 200 | | | Crossflow |

D | 200 | | | Crossflow |

E | 200 | | | Crossflow |

F | 200 | | | Crossflow |

G | 200 | | | Crossflow |

H | 200 | | | Crossflow |

Frequency-amplitude plot:

The fluid-structure interaction is a highly nonlinear phenomenon. One of the important properties of the nonlinear systems is that different frequency components in the system can interact with each other and generate new frequencies. These new components typically appear as the sum and the difference of combining frequencies. The phase of the frequency provides the information if it is a result of a nonlinear interaction of other frequencies. Analysis of the time series of any fluctuating physical quantity, for example, lift and drag coefficients, starts by determining the frequency distribution of power of the fluctuation, which can be found by calculating the auto-power spectrum of the time series. The power spectrum helps in identifying the dominant frequencies of the domain and their harmonics, but it does not provide information about whether a nonlinear wave-wave interaction exists. For any wave-wave interaction to be considered nonlinear, it must satisfy the following selection rules or resonance conditions:

In the current study, we investigate the nonlinear (quadratic) coupling between the time histories of the lift,

From the symmetry properties of the cross-bispectrum and cross-bicoherence, the calculation of these quantities is limited to two regions: Region S, which defines the sum interaction region and is calculated on

Symmetry properties of bispectrum reduce the region to sum (S) and difference (D).

Simulations are performed for six different cases of stationary and crossflow oscillating cylinders at

Crossflow oscillations at

Case | | | Comments |
---|---|---|---|

J | 525 | − | 2D stationary |

K | 525 | | 2D crossflow |

L | 525 | | 2D crossflow |

M | 1000 | − | 2D stationary |

N | 1000 | | 2D crossflow |

P | 1000 | − | 3D stationary |

Time histories of the lift and drag coefficients of the cylinder are divided into periodic segments, which are used to calculate the different statistical variables, that is, power spectrum, cross-bispectrum, and cross-bicoherence, and Ensemble averaging is used to reduce statistical variance of the results. The power spectra presented here are calculated using three time segments with 4096 samples each. For higher-order spectral moments, the number of realizations is increased 12 with 1024 data points for each realization.

First, we compute the auto-power spectrum of the lift coefficient for each case as shown in Figure

Power spectra of lift coefficients

Case J

Case K

Case L

Case M

Case N

Case P

Figure

Power spectra of drag coefficients

Case J

Case K

Case L

Case M

Case N

Case P

In order to identify the quadratic coupling between lift and drag coefficients, we perform the cross-bispectrum/bicoherence analysis of the lift and drag signals. Figure

Cross-bispectrum

Case J

Case K

Case L

Case M

Case N

Case P

which is consistent with the findings of Kim and Williams [

High values of

We numerically simulated the flow past stationary and oscillating cylinders and compute the hydrodynamic forces on it. We performed higher-order spectral analysis of the lift and drag coefficients and identified different frequencies generated due to nonlinear interaction. Bispectrum/bicoherence analyses show the energy content of the quadratic coupling between the modes present in the lift and drag. In the future, we would perform 3D simulations of the flow past oscillating cylinders and analyze the effect of third-dimensionality on the nonlinear interaction between different frequencies and how it differs from the two-dimensional flows.

All the authors declare that there are no conflicts of interest regarding the publication of this article.