the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Force Partitioning Analysis of VortexInduced Vibrations of Wind Turbine Tower Sections
Abstract. VortexInduced Vibrations (VIV) of wind turbine towers during installation is an aerostructural problem of significant practical relevance. Vibrations may happen in the tower structure, especially when the rotornacelle assembly is not yet attached to the tower or if the rotor blades are not yet connected to the towernacelle assembly. The complexity of aeroelastic phenomena involved in VIV makes the modeling and analysis challenging. Therefore, the aim of the current research is to investigate the fundamental mechanisms causing the onset and sustenance of vortexinduced vibrations. To gain more understanding of the nature of vibrations, a methodology is established that distinguishes different components of the forces at play. This approach allows identifying how various force components impact the oscillation of a rigid body. The method is executed using the OpenFOAM opensource software. Numerical simulations are conducted on a twodimensional smooth cylinder at both subcritical and supercritical Reynolds numbers to establish a correlation between wind turbine tower vibrations and the force mechanism. The analysis involves performing Unsteady ReynoldsAveraged Navier Stokes (URANS) simulations using the modified pimpleFoam solver with the kω SST turbulence model. Both fixed and freevibrating cases are studied for smooth cylinders. For the high Reynolds number cases, a setup matching the tower top segment of the IEA 15MW reference wind turbine was chosen. Studying the flow around a cylinder at a subcritical Reynolds number reveals that the primary force involved is the vorticity force. The combined force due to viscosity, added mass, and vorticity contributes most to the overall force. For a freely vibrating cylinder with a single degree of freedom in the crossflow direction, the analysis indicates that the force component associated with the cylinder's motion is crucial and significantly affects the total force. Moreover, analysing the energy transfer between the fluid and the structure, a positive energy contribution by the vorticityinduced force is observed on or before the dominant Strouhal velocity. This confirms observations at low Reynolds numbers in the literature that the vortex shedding predominantly contributes to the initiation of oscillations during VIV. The kinematic force contributes to the energy transfer of the system, but the mean energy transfer per cycle is negligible.
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RC1: 'Comment on wes202410', Anonymous Referee #1, 22 Feb 2024
The authors present the study of a canonical vortexinduced vibrations (VIV) problem, namely the circular cylinder in transverse oscillations, by means of 2D simulations. The intention is to approximate the effects of such aeroelastic phenomenon for wind turbine towers, using a simplified setup. The study focuses on Reynolds numbers between 1e6 and 1e7, and additional simulations in the laminar regime were performed for validation purposes. Force partitioning is used to analyze the aeroelastic phenomenon.
The manuscript is clearly structured and wellwritten. It contains almost all the information required to reproduce the work done (some suggestions provided below). However, the main criticism of this reviewer revolves around the novelty and impact of the performed work. It does not seem to be clear what the specific contribution of this work is. All the numerical methods, including the force partitioning approach, seem to have already been employed in previous work. Indeed, the FSI strategy employed seems to correspond to the one used in Vire et al. (2020), published in this journal, so that most of the manuscript seems to be somehow a duplicate. Along the same lines, the discussion derived from the application of the force partitioning did not seem to bring new information to the aeroelastic phenomena characterization, as it is uncertain to what extent the observations made are novel (knowing the amount of literature available through classical force decompositions). The authors are therefore encouraged to improve the writing of the manuscript, contextualizing how their findings are placed with respect to the state of the art, and how the new findings do impact what we already know about VIV.
A series of specific comments are included below, to be considered by the authors in order to improve the manuscript.
Major comments
 The rationale behind the 3D>2D casting should be further illustrated in the manuscript. The authors have selected a section corresponding to the tower top. Is it the most critical section with regards to a potential lockin?
 This reviewer found problems seeing how the values of Table 2 (m*,c*) were computed, and what their relation is to the mechanical properties and dimensions of the full tower.
 Regarding the FSI implementation, did the authors employ a weak or a strong coupling approach? How was the mesh deformed?
 Consider including a validation for the turbulent setup, or citing Viré et al. (2020) to that end.
 Equation 10: The amplitude factor y_0 was never introduced. Even if not relevant for this work or for the derivation, the generalized form of the equation may also take an initial phase. I found the comment "The mode shape at any time represents the ratio of the displacement at the point z to the reference displacement y(t)." rather confusing. The mode shape is timeindependent, and the reference displacement has not been introduced. The choice of axes notation is also confusing, as e.g. the axial direction of Figure 1 corresponds to "x".
 Statistics shown in Figure 8 may suffer from the short total times simulated. Could be interesting to include a comment in the manuscript, and to state if that could be a limitation for the analysis made.
 In Figure 12, for the moving average, was the considered window a multiple of the period of motion? In the opinion of this reviewer, that could facilitate future comparisons.
 "As the oscillations are almost sinusoidal, the net energy per cycle becomes almost zero." This reviewer had problems understanding this statement. Even with pure sinusoids, a phase difference should lead to a nonzero energy transfer (when considering what happens at the frequency of motion). Note that, at the LCO of e.g. Figure 9.a, one should be "pumping" energy into the system to counteract the structural damping and maintain the same amplitude of vibration.
 While the distribution of the OpenFOAM implementation performed by the authors is highly appreciated, and it is well described in the manuscript, it could be interesting to improve the writing so that nonOpenFOAM practitioners do not get lost. For instance, one could include more "fundamental" descriptions of what the "0" folder contains, or how blockMesh generates the grids.
Minor comments
 Could be interesting to comment on how the method should be extended to situations where the structure is characterized by more than one mode.
 Is "Cartesian grid" the best way to describe the mesh? I believe near the cylinder your grid lines become normal to the surface.
 "Nevertheless, the more recent study by Viré et al. (2020) gives confidence in performing URANS simulations": while this is true, could passing to another turbulence model eventually have an impact on the conclusions of this work, for instance regarding the relative contribution of the forces?
 Would the provided OpenFOAM modifications work for 3D cases in its present form?
 The link to Zenodo of the manuscript did not seem to be correct (but a working one was indicated by the authors in the submission).
 Equation 9: the symbol used here for the natural frequency is later on used in the manuscript for displacements.
 Equation 11: for consistency with the rest of the derivation, maybe it is a good idea to use '' for the second derivative.
 Equation 16: there is a jump in the derivation here, from continuous to discrete. Was it intended?
 "frequency of the tower is calculated to be 0.48" > 0.48 Hz
 Figure 3: misses the coordinates for the center of the cylinder.
 Some words on the computational cost of the simulations would be appreciated.
Citation: https://doi.org/10.5194/wes202410RC1 
RC2: 'Comment on wes202410', Anonymous Referee #2, 06 Mar 2024
# General Comments
The primary objective of the present study is to investigate the different force components during at different stages of vortexinduced vibrations (VIV). VIV can appear on different engineering structures but they are particularly of interest for wind turbines during transportation and construction. To this end, the work studies the force components during VIV on a circular cylinder section model by means of 2D URANS simulations with the kω SST turbulence model in the laminar and transcritical Reynolds number regime. To investigate the force components, the forcepartitioning method has been implemented in OpenFOAM. Analysis is conducted for fixed sections and for a SDOF oscillator. For the transcritical regime, the dynamic properties resemble a segment of the IEA15MW reference wind turbine tower.
The analysis shows the relevant force components and their energy contributions to the oscillation. At high Re, the vorticity and kinematic forces are most dominant during oscillation. The contributions of these two components vary depending on the inflow velocity. At low U/Ust ratio, the vorticity force is dominant, whereas kinematic forces become more relevant as oscillation amplitude increase. The energy transfer from the fluid to the structure is mostly due to the vorticity force.
The paper is well written.
# Specific Comments
 The "subcritical" Reynolds number regime is generally understood to be different than the laminar Reynolds number regime (see for example the cited work by Belloli (2015)). In the present work, both terms are used interchangeably.
 Chapter 3.1: The reviewer is missing boundary conditions at the inlet for $k$ and $\omega$. There is information about $k$ and $\omega$ in lines 256ff but they are, at least for $k$, introduced as initial values.
 Chapter 3.1: The reviewer is interested in the time steps used and the resultant Courant number. Has the used time step been verified?
 Chapter 3.1: The reviewer is interested in the coupling approach of fluid and body.
 Figure 3/numerical domain: Information about the size of the upstream and downstream part of the domain is missing.
 Figure 3/numerical domain: The domain in the acrosswind direction seems rather small with 15 d and a blockage ratio that is above 7 %. The cited work by Viré (2020) uses 100 d in the acrosswind and 100 d in the alongwind direction. Verification of the domain size can show the appropriateness of the chosen domain size.
 Chapter 3.2: The grid convergence study is conducted for the surface pressures on a fixed cylinder in transcritical Reynolds number regime. The main aim of the paper, however, is to investigate the situations for a SDOF oscillator. In the reviewer's view, the grid convergence study should investigate a critical situation during free vibration. Additionally, the increase of number of cells by a factor of 2 seems rather small even for a 2D simulation.
 Table 1: The reviewer is interested which y+ value is shown. Is it the mean value?
 line 209f: How is the separation point calculated and where is it located?
 Figure 5, 6, and 8: It is not clear how the statistical values are obtained. Is the whole time history used, or is an initial time of the simulation discarded?
 Figure 5, 6, and 8: The reviewer is missing information about the verification of the convergence of the results. It is not clear how the chosen simulation time has been deemed appropriate to calculate the statistical values.
 One of the main concerns of the reviewer are the already mentioned diverging forces in Figures 9 c) and e). To the best of the reviewer's knowledge, the lift force does not diverge during VIV. See for instance the wind tunnel results by the coauthor Belloli (2012) and Belloli (2015). In the numerical simulations of the coauthor Viré (2020) it is shown that the lift force decreases significantly after reaching a maximum in Figure 11 $\omega* = 0.95$. Is this change of lift coefficient in time a particular situation that only appears in URANS simulations or has it been already observed in wind tunnel or fullscale?
 The reviewer is interest to know if the oscillation amplitudes also diverge. VIV are selflimited and the oscillation amplitudes should reach a maximum even during lockin, contrary to aeroelastic phenomena such as galloping or flutter.
 Figures 9 c) and e): The reviewer is missing information about the criterion used to stop the simulations after ~450 s and ~525 s, respectively. What happens after the shown time histories?# Technical Corrections
 line 22f: "challenge" instead of challenges, "as taller" instead of "as a taller"
 Figure 4: The definition of the normalized xcoordinate x* is missing.
 The direction of lift and drag and the respective formulas for the coefficients could be defined for clarification.
 Figure 5 caption: remove "citep".
 line 236 "ydirection,corresponding" add space after comma.
 Table 2: Damping ratio has been introduced as $\zeta$ on p.8 instead of $c^*$ here. Is there a difference between these two damping ratios?
 line 264: $y_{wall}$ is not defined.
 The dataset is not available at https://10.0.20.161/zenodo.10529197 .Citation: https://doi.org/10.5194/wes202410RC2 
RC3: 'Comment on wes202410', Anonymous Referee #3, 12 Mar 2024
The authors propose a numerical study on the vortex induced vibrations (VIV) in a 2D cylinder. The latter is simulated using URANS simulations and a kw SST model in OpenFoam. The authors implement a forcepartitioning method in this code, while determining the nature of the forces related to the VIV in the system. The overall aim is to emulate the dynamics of a wind turbine tower, that is simulated as a twodimensional cylinder attached to a springmassdamper system. The role of the force terms in triggering the oscillations and on the dynamics of the VIV are studied. The analysis also includes the energy transfer between the cylinder and the fluid.
I find the paper well written and the simulations carefully planned and performed. The system studied and the results and properly justified and of interest for the community. I have, nonetheless, the following remarks:
Within the text, the force related to shedding is called, alternatively, vorticity force and vortexinduced force. The authors should uniformise how they refer to it.
The added mass force, first term in the RHS of equation 4, is never properly defined. Moreover, while maybe obvious, the difference between the volume V and the surface B in integrals is not given in the text.
Figures 1 and 3 are inconsistent. Moreover, if the cylinder is 2D, what is the height H in table 1? It would help to understand this parameter if it is added to figure 3.
It is not clear which criteria were used to choose the set of parameters (c, EI, ks) from equations 918 (or the mass ratio in table 2).
Why no comparison with the literature has been made for the cylinder in the turbulent regime? This could be done, at least, for a nonoscillating system. This would be still relevant as the turbulent wake may be conditioned by the turbulence model, separation point,etc.
Overall, the labels in figures are too small and hard to read even in electronic form.
Citation: https://doi.org/10.5194/wes202410RC3 
AC1: 'Comment on wes202410', Shyam VimalKumar, 13 May 2024
Dear reviewers,
The authors would like to thank the reviewers for their time in providing feedback on the manuscript. The changes are incorporated into the manuscript, and the responses to all the comments from the reviewers are available in the attached file.
Data sets
Dataset for Force Partitioning Analysis of VortexInduced Vibrations of Wind Turbine Tower Sections Shyam VimalKumar https://doi.org/10.5281/zenodo.10529197
Model code and software
Force Partitioning Method  OpenFOAM Shyam VimalKumar https://gitlab.tudelft.nl/svimalkumar/fpfoam
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