The helix wake and its properties: reduced order modeling through dynamic mode decomposition

Muscari, Claudia; Schito, Paolo; Viré, Axelle; Zasso, Alberto; van Wingerden, Jan-Willem

doi:https://doi.org/10.5194/wes-2024-149

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https://doi.org/10.5194/wes-2024-149

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02 Jan 2025

| 02 Jan 2025

Status: this preprint was under review for the journal WES. A revision for further review has not been submitted.

The helix wake and its properties: reduced order modeling through dynamic mode decomposition

Claudia Muscari, Paolo Schito, Axelle Viré, Alberto Zasso, and Jan-Willem van Wingerden

Abstract. Wind turbine wakes can be treated as a complex system of helical vortices. When this system destabilizes, the wake recovers its velocity deficit through mixing and entertainment of energy from the surrounding flow. How fast and effectively that happens depends on the inflow characteristics and can also be influenced by how the turbines are operated. Dynamic induction control techniques such as the helix affect the onset of instability and the transition from near to far wake, but the exact mechanisms are still unclear. Its potential in a wind farm context has been proved both numerically and experimentally, but a helix wake model does not exist yet. The goal of this study is to derive a data-driven model of the helix wake and characterize it. Dynamic mode decomposition of data generated with large eddy simulations is performed. We simulate the DTU 10 MW model turbine under a range of helix excitation frequencies and different inflows. We show that the helix modes are dominantly present both in laminar and turbulent flow. However, as turbulence intensity increases, they exhibit larger spatial decay and temporal amplitude. Additionally, we identify inflow modes related to the turbulence length scales of the inflow. We show that a very limited number of modes allows us to reconstruct the initial flow field accurately and that the optimum excitation frequency for the control technique depends on the turbulence intensity and on the position of the downstream turbines.

Received: 08 Nov 2024 – Discussion started: 02 Jan 2025

Competing interests: At least one of the (co-)authors is a member of the editorial board of Wind Energy Science.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this preprint. The responsibility to include appropriate place names lies with the authors.

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Claudia Muscari, Paolo Schito, Axelle Viré, Alberto Zasso, and Jan-Willem van Wingerden

Status: closed (peer review stopped)

RC1:
'Comment on wes-2024-149', Anonymous Referee #1, 06 Jan 2025
Review of the Paper: 'The Helix Wake and Its Properties: Reduced Order Modeling through Dynamic Mode Decomposition'

Summary of the Paper:
This study addresses a relevant and challenging topic in wind turbine wake dynamics by developing a data-driven model of the helix wake using Dynamic Mode Decomposition (DMD). The authors use large-eddy simulations (LES) to investigate how helix dynamics behave under varying inflow turbulence intensities. By applying DMD to the simulated data, the study aims to understand how inflow fluctuations affect helix-induced instabilities and identify a minimal set of modes that accurately reconstruct the wake field. The results provide insight into the role of turbulence in influencing helix dynamics and optimal excitation frequencies for downstream turbine control.
General Comments:
The paper tackles a significant problem in wind energy research, namely how dynamic control strategies such as helix excitation influence wake recovery in turbulent inflow conditions. The numerical setup is well-explained, and the description of data generation through LES appears to be sound and technically accurate. However, I have a few major concerns about the methodology and the clarity of the results presented.
Methodological Approach: While the application of DMD is appropriate for extracting coherent structures from wake flows, the use of this method in its current form seems overly complex for the primary objective of understanding helix-inflow interactions. A simpler and potentially more insightful approach would be to conduct a triple decomposition (Reynolds-Hussain) combined with discrete Fourier analysis. Given that the flow is periodically excited, Fourier modes would likely provide a clearer separation of helix-related dynamics from inflow turbulence. Moreover, Fourier modes and DMD modes are equivalent under steady-state conditions, making Fourier analysis a more straightforward tool in this context.

Mean Field in DMD: If the authors choose to continue using DMD, it is critical to subtract the mean flow field before performing the decomposition. Without this step, the leading DMD mode will always correspond to the mean field and dominate the energy content. This undermines the discussion of low-rank approximations and obscures the role of dynamic modes in capturing helix-related instabilities.

Alternative Techniques: Another potentially fruitful approach could be spectral Proper Orthogonal Decomposition (POD) or Biorthogonal Spectral Decomposition (BOD), which are increasingly recognized in the literature as robust methods for analyzing periodically excited turbulent flows. These methods might offer a clearer distinction between the dominant flow structures and inflow turbulence.

Specific Comments:
Line 270: I find it difficult to visually identify the hub vortex instability and its destabilization from a single snapshot. A more detailed explanation or visual aid highlighting these features would be helpful.

Line 299: The reasoning behind the choice of velocity components for DMD is unclear. Could the authors provide further justification or reference existing literature that supports their selection?

Line 369: Referring to the model as 'simple' may be misleading, given that it includes the mean field, which contains significant information. It would also help if the authors clarified the intended future application of the model—whether it is for flow state estimation, control, or something else.

Discussion of Reconstruction Errors: The discussion linking inflow turbulence timescales and helix excitation frequency is difficult to follow based solely on reconstruction errors. These errors provide an indirect measure of mode energy but do not directly illustrate interactions between inflow turbulence and helix dynamics. A more quantitative approach, such as interaction maps generated via bispectral mode decomposition, might be more appropriate.

Recommendations:
Consider adopting a triple decomposition approach with discrete Fourier analysis for a clearer separation of mean flow, periodic structures, and turbulence.

2. If DMD is retained, subtract the mean field before decomposition to ensure meaningful low-rank approximations.

3. Explore alternative decomposition techniques such as spectral POD or BOD to gain deeper insights into mode interactions.

4. Clarify key aspects of the methodology, including the choice of velocity components and the purpose of the model.

5. Improve the discussion of inflow turbulence interactions by incorporating more quantitative metrics, such as interaction maps or bispectral analyses.

Conclusion:
This paper presents a promising approach to modeling helix wake dynamics, a topic with significant implications for wind farm optimization. While the numerical setup and data generation are robust, the current analysis could benefit from methodological simplification and additional quantitative insights. Addressing these concerns would significantly enhance the clarity and impact of the study.
Citation: https://doi.org/10.5194/wes-2024-149-RC1
RC2: 'Comment on wes-2024-149', Anonymous Referee #2, 21 Jan 2025

The paper treats a very interesting subject using advanced methodologies and provides a sufficiently new contribution to the problem of wind farm optimization. However, I have serious doubts on the numerics used. Concerning the LES, the computational mesh seems very rough for an LES, the domain is rather limited, and the boundary conditions are not clear. Concerning the DMD, evidence of convergence of the spectrum is not provided, and the choice of some important parameters such as the Delta t between two snapshots are not sufficiently justified with respect to the physics of the problem. Finally, the literature discussion is rather limited, and should be expanded. All in all, I consider the paper to be not ready for publication since the robustness and physical relevance of the results is unsure.
Detailed comments about the above general points are given in the following:
- Introduction: many relevant contributions about DMD analyses of turbine's wakes have been completed skipped. Some of them also discuss the effect of inflow turbulence of the DMD modes, which is very relevant for this work. Please discuss, for instance, Iungo et al. "Data-driven Reduced Order Model for prediction of wind turbine wakes", J. Phys. 2015, Debnath et al "Towards reduced order modelling for predicting the dynamics of coherent vorticity structures within wind turbine wakes", Proc. Roy Soc. 2017, De Cillis et al. "The influence of incoming turbulence on the dynamic modes of an NREL-5MW wind turbine wake", Ren. En 2022, De Cillis et al. "Dynamic-mode-decomposition of the wake of the NREL-5MW wind turbine impinged by a laminar inflow", 2022, Manganelli et al. "The effect of Coriolis force on the coherent structures in the wake of a 5MW wind turbine" En. Conv Man. 2025 , among many others
- Line 208: "The precursor mesh does not need a fine resolution"--> the resolution should be sufficiently high to describe the inertial range of turbulence. Please provide a spectrum of the fluctuations which are injected in the successor domain showing that the inflow is indeed turbulent.
- Table 1: the resolution of the successor seems totally insufficient (only 48 points in z means a few points on each blade). Please provide evidence of the convergence of this results on the chosen mesh.
- Line 218: "snapshot is taken once every 2 s." --> The Delta t between two successive snapshots should describe well the rotation of the blades and also the pitch variation. How many degrees of rotation of the blades correspond to these 2 seconds? Is this sampling time sufficient for describing accurately (at least, say, 10-15 snapshots per period) the pitch angle variation for all the Strouhal number considered? These points should be described in detail.
- Line 229: "500 m downstream of the inlet" --> the inflow is placed only 2.8 diameters from the turbine, which is usually not sufficient to let the turbulent inflow adapt. This, together with the rough discretization used for the precursor simulation, let one doubt about the features of the turbulent flow that impinges the turbine. Please show some turbulent spectra also in the successor domain, upstream and downstream of the turbine (see, for instance, figures 7 and 8 of Korb et al. JFM 2023 )
- Line 230: the boundary conditions are not clear. Why in the successor simulation, west and east directions have two different conditions (time varying mapped fixed value and inletOutlet)?
- Section 4.2.1-4.2.2: Please provide the DMD spectrum and assess is convergence with respect to 1) the chosen Delta t (time between two snapshots), 2) the POD rank, 3) the number of snapshots (800 is probably not sufficient). Also, is the DMD performed on the whole domain?
- Eq. 14: Please provide a measure of the error also using the cumulative percentage distribution of turbulent kinetic energy, used in many of the references cited above.
- Figure 10: Please provide a plot of the amplitude vs Strouhal number of all DMD modes, not only of the first 7. Also, it is not clear how these modes are chosen, are the ones with the largest amplitude? Is any algorithm for selecting the amplitudes and/or reconstructing the flow used, such as for instance the sparsity promoting algorithm? Or the largest Strouhal numbers are simply cut off? Please provide details on this important points.
- line 316: "the amplitude can be related to the total energy content". This is not true. DMD does not evaluate the modes with respect to their energy but to their dynamical relevance. Please refer to relevant literature about this point.
- Figure 12: the fact that only two modes provide a visually good reconstruction is remarkable but surprising. How about the 3D flow field and/or the kinetic energy reconstructed?
- Section 4.2.2: please provide the DMD spectrum and the amplitude with respect to the Strouhal number for all the DMD modes, and assess the convergence of the spectrum with respect to 1) the chosen Delta t (time between two snapshots), 2) the POD rank, 3) the number of snapshots. I really doubt that 300 snapshots for a turbulent case are sufficient for describing the turbulent flow. Also, I am not sure with less snapshots are used for the turbulent case with respect to the uniform one.

Citation: https://doi.org/10.5194/wes-2024-149-RC2

Status: closed (peer review stopped)

RC1:
'Comment on wes-2024-149', Anonymous Referee #1, 06 Jan 2025
Review of the Paper: 'The Helix Wake and Its Properties: Reduced Order Modeling through Dynamic Mode Decomposition'

Summary of the Paper:
This study addresses a relevant and challenging topic in wind turbine wake dynamics by developing a data-driven model of the helix wake using Dynamic Mode Decomposition (DMD). The authors use large-eddy simulations (LES) to investigate how helix dynamics behave under varying inflow turbulence intensities. By applying DMD to the simulated data, the study aims to understand how inflow fluctuations affect helix-induced instabilities and identify a minimal set of modes that accurately reconstruct the wake field. The results provide insight into the role of turbulence in influencing helix dynamics and optimal excitation frequencies for downstream turbine control.
General Comments:
The paper tackles a significant problem in wind energy research, namely how dynamic control strategies such as helix excitation influence wake recovery in turbulent inflow conditions. The numerical setup is well-explained, and the description of data generation through LES appears to be sound and technically accurate. However, I have a few major concerns about the methodology and the clarity of the results presented.
Methodological Approach: While the application of DMD is appropriate for extracting coherent structures from wake flows, the use of this method in its current form seems overly complex for the primary objective of understanding helix-inflow interactions. A simpler and potentially more insightful approach would be to conduct a triple decomposition (Reynolds-Hussain) combined with discrete Fourier analysis. Given that the flow is periodically excited, Fourier modes would likely provide a clearer separation of helix-related dynamics from inflow turbulence. Moreover, Fourier modes and DMD modes are equivalent under steady-state conditions, making Fourier analysis a more straightforward tool in this context.

Mean Field in DMD: If the authors choose to continue using DMD, it is critical to subtract the mean flow field before performing the decomposition. Without this step, the leading DMD mode will always correspond to the mean field and dominate the energy content. This undermines the discussion of low-rank approximations and obscures the role of dynamic modes in capturing helix-related instabilities.

Alternative Techniques: Another potentially fruitful approach could be spectral Proper Orthogonal Decomposition (POD) or Biorthogonal Spectral Decomposition (BOD), which are increasingly recognized in the literature as robust methods for analyzing periodically excited turbulent flows. These methods might offer a clearer distinction between the dominant flow structures and inflow turbulence.

Specific Comments:
Line 270: I find it difficult to visually identify the hub vortex instability and its destabilization from a single snapshot. A more detailed explanation or visual aid highlighting these features would be helpful.

Line 299: The reasoning behind the choice of velocity components for DMD is unclear. Could the authors provide further justification or reference existing literature that supports their selection?

Line 369: Referring to the model as 'simple' may be misleading, given that it includes the mean field, which contains significant information. It would also help if the authors clarified the intended future application of the model—whether it is for flow state estimation, control, or something else.

Discussion of Reconstruction Errors: The discussion linking inflow turbulence timescales and helix excitation frequency is difficult to follow based solely on reconstruction errors. These errors provide an indirect measure of mode energy but do not directly illustrate interactions between inflow turbulence and helix dynamics. A more quantitative approach, such as interaction maps generated via bispectral mode decomposition, might be more appropriate.

Recommendations:
Consider adopting a triple decomposition approach with discrete Fourier analysis for a clearer separation of mean flow, periodic structures, and turbulence.

2. If DMD is retained, subtract the mean field before decomposition to ensure meaningful low-rank approximations.

3. Explore alternative decomposition techniques such as spectral POD or BOD to gain deeper insights into mode interactions.

4. Clarify key aspects of the methodology, including the choice of velocity components and the purpose of the model.

5. Improve the discussion of inflow turbulence interactions by incorporating more quantitative metrics, such as interaction maps or bispectral analyses.

Conclusion:
This paper presents a promising approach to modeling helix wake dynamics, a topic with significant implications for wind farm optimization. While the numerical setup and data generation are robust, the current analysis could benefit from methodological simplification and additional quantitative insights. Addressing these concerns would significantly enhance the clarity and impact of the study.
Citation: https://doi.org/10.5194/wes-2024-149-RC1
RC2: 'Comment on wes-2024-149', Anonymous Referee #2, 21 Jan 2025

The paper treats a very interesting subject using advanced methodologies and provides a sufficiently new contribution to the problem of wind farm optimization. However, I have serious doubts on the numerics used. Concerning the LES, the computational mesh seems very rough for an LES, the domain is rather limited, and the boundary conditions are not clear. Concerning the DMD, evidence of convergence of the spectrum is not provided, and the choice of some important parameters such as the Delta t between two snapshots are not sufficiently justified with respect to the physics of the problem. Finally, the literature discussion is rather limited, and should be expanded. All in all, I consider the paper to be not ready for publication since the robustness and physical relevance of the results is unsure.
Detailed comments about the above general points are given in the following:
- Introduction: many relevant contributions about DMD analyses of turbine's wakes have been completed skipped. Some of them also discuss the effect of inflow turbulence of the DMD modes, which is very relevant for this work. Please discuss, for instance, Iungo et al. "Data-driven Reduced Order Model for prediction of wind turbine wakes", J. Phys. 2015, Debnath et al "Towards reduced order modelling for predicting the dynamics of coherent vorticity structures within wind turbine wakes", Proc. Roy Soc. 2017, De Cillis et al. "The influence of incoming turbulence on the dynamic modes of an NREL-5MW wind turbine wake", Ren. En 2022, De Cillis et al. "Dynamic-mode-decomposition of the wake of the NREL-5MW wind turbine impinged by a laminar inflow", 2022, Manganelli et al. "The effect of Coriolis force on the coherent structures in the wake of a 5MW wind turbine" En. Conv Man. 2025 , among many others
- Line 208: "The precursor mesh does not need a fine resolution"--> the resolution should be sufficiently high to describe the inertial range of turbulence. Please provide a spectrum of the fluctuations which are injected in the successor domain showing that the inflow is indeed turbulent.
- Table 1: the resolution of the successor seems totally insufficient (only 48 points in z means a few points on each blade). Please provide evidence of the convergence of this results on the chosen mesh.
- Line 218: "snapshot is taken once every 2 s." --> The Delta t between two successive snapshots should describe well the rotation of the blades and also the pitch variation. How many degrees of rotation of the blades correspond to these 2 seconds? Is this sampling time sufficient for describing accurately (at least, say, 10-15 snapshots per period) the pitch angle variation for all the Strouhal number considered? These points should be described in detail.
- Line 229: "500 m downstream of the inlet" --> the inflow is placed only 2.8 diameters from the turbine, which is usually not sufficient to let the turbulent inflow adapt. This, together with the rough discretization used for the precursor simulation, let one doubt about the features of the turbulent flow that impinges the turbine. Please show some turbulent spectra also in the successor domain, upstream and downstream of the turbine (see, for instance, figures 7 and 8 of Korb et al. JFM 2023 )
- Line 230: the boundary conditions are not clear. Why in the successor simulation, west and east directions have two different conditions (time varying mapped fixed value and inletOutlet)?
- Section 4.2.1-4.2.2: Please provide the DMD spectrum and assess is convergence with respect to 1) the chosen Delta t (time between two snapshots), 2) the POD rank, 3) the number of snapshots (800 is probably not sufficient). Also, is the DMD performed on the whole domain?
- Eq. 14: Please provide a measure of the error also using the cumulative percentage distribution of turbulent kinetic energy, used in many of the references cited above.
- Figure 10: Please provide a plot of the amplitude vs Strouhal number of all DMD modes, not only of the first 7. Also, it is not clear how these modes are chosen, are the ones with the largest amplitude? Is any algorithm for selecting the amplitudes and/or reconstructing the flow used, such as for instance the sparsity promoting algorithm? Or the largest Strouhal numbers are simply cut off? Please provide details on this important points.
- line 316: "the amplitude can be related to the total energy content". This is not true. DMD does not evaluate the modes with respect to their energy but to their dynamical relevance. Please refer to relevant literature about this point.
- Figure 12: the fact that only two modes provide a visually good reconstruction is remarkable but surprising. How about the 3D flow field and/or the kinetic energy reconstructed?
- Section 4.2.2: please provide the DMD spectrum and the amplitude with respect to the Strouhal number for all the DMD modes, and assess the convergence of the spectrum with respect to 1) the chosen Delta t (time between two snapshots), 2) the POD rank, 3) the number of snapshots. I really doubt that 300 snapshots for a turbulent case are sufficient for describing the turbulent flow. Also, I am not sure with less snapshots are used for the turbulent case with respect to the uniform one.

Citation: https://doi.org/10.5194/wes-2024-149-RC2

Claudia Muscari, Paolo Schito, Axelle Viré, Alberto Zasso, and Jan-Willem van Wingerden

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Short summary

This paper presents the findings of a study aimed at describing the flow system downstream of a wind turbine operated with a novel control technology. Results from heavy high-fidelity simulations are used to obtain a low-fidelity model that is quick enough to be used for the optimization of such technologies. Additionally, we were able to retrieve an improved understanding of the physics of such systems under different inflow conditions.


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