Dynamic inflow model for a Floating Horizontal Axis Wind Turbine in surge motion

Floating Offshore Wind Turbines may experience large surge motions, which can cause blade-vortex interaction if they are similar to or faster than the local wind speed. Previous research hypothesized that this blade-vortex interaction phenomenon represented a turbulent wake state or even a vortex ring state, rendering the Actuator Disc Momentum Theory and the Blade Element Momentum Theory invalid. This hypothesis is challenged, and we show that the Actuator Disc Momentum Theory is valid and accurate in predicting the induction at the actuator in surge, even for large and fast motions. To accomplish 5 this, we develop a dynamic inflow model that simulates the vorticity-velocity system and the effect of motion. The model’s predictions are compared to other authors’ results, a semi-free wake vortex-ring model, other dynamic inflow models, and CFD simulations of an actuator disc in surge. The results show that surge motion and rotor-wake interaction do not result in a turbulent wake or vortex ring state, and that the application of Actuator Disc Momentum Theory and Blade Element Momentum Theory is valid and accurate when applied correctly in an inertial reference frame. In all cases, the results show 10 excellent agreement with the higher fidelity simulations. The proposed dynamic inflow model includes a modified Glauert’s correction for highly loaded streamtubes and is accurate and simple enough to be easily implemented in most Blade Element Momentum models. Copyright statement. TEXT

The complexity of the aerodynamics resulted in many interpretations of the phenomena. Several authors proposed that the flow could change from windmill to propeller state due to motion and changes in loading. Furthermore, several authors proposed that if the surge velocity is large enough, the combination of wind speed and surge velocity would be less than twice 25 the induction velocity, resulting in a turbulent wake state or even a vortex ring state (see Sørensen et al. (1998) for the definition of turbulent wake state and vortex ring state). Actuator Disc Momentum Theory, according to many authors, would no longer be valid under these conditions. Due to the fact that Blade Element Momentum Theory (BEM, see Glauert (1935)) is based on Actuator Disc Momentum Theory, the occurrence of turbulent wake state and vortex ring state would significantly limit the use of BEM for FOWTs. Given that BEM is the most commonly used tool for simulating the aerodynamics of horizontal axis 30 wind turbines (Madsen et al., 2020), this could have a significant impact on our design methods.
However, the prediction of turbulent wake state and vortex ring state for the actuator disc (wind turbine) in periodic surge motion appears to be in most cases the result of an invalid interpretation of the Actuator Disc Theory. As stated by Sørensen and Myken (1992), since the concept of the actuator disc was first formulated by Froude it has been closely related to the one-dimensional momentum theory and much confusion about its applicability in describing complex flow fields still exists.

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This is particularly true for the case of an actuator in cyclic motion, as is the case of FOWTs.
The name of the theory is in itself misleading, because the Actuator Disc Momentum theory is in fact the theory of the mass and momentum balance of the streamtube that includes the actuator. The actuator disc is a physical model that enables a discontinuity of the pressure field into the governing flow equations as the reaction to an external force field. The added information that the pressure discontinuity occurs at the actuator allows us to estimate the velocity at the actuator by evaluating 40 stagnation pressure along the streamtube. Therefore, Actuator Disc Theory refers to the state of the streamtube defined in an inertial reference frame that contains the actuator which is static in the same inertial reference frame. Propeller state, windmill state, turbulent wake state, vortex ring state and propeller brake state do not refer to the state of the actuator but to the state of the streamtube (Sørensen et al., 1998). In an unsteady flow, an actuator might have an instantaneous loading as a propeller, while the streamtube remains in windmill state. Two examples of such inertial reference frames are the one attached to the 45 steady streamtube which includes the actuator disc associated with a stationary wind turbine (or propeller) in an incoming unperturbed wind speed U ∞ of any value, or the one attached to the steady streamtube that contains an actuator disc in a constant motion (not accelerated) in an incoming unperturbed wind speed U ∞ of any value.
When the actuator is moving in an inertial reference frame with a steady velocity, the streamtube and actuator are in the same inertial reference frame, and the reference unperturbed velocity of the wind used in the actuator disc model U ∞ ref is the sum 50 of the velocity of the wind in the inertial reference frame U ∞ with the moving velocity of the actuator in the inertial reference frame v act , as In the condition that v act is constant (time invariant)this condition, the actuator disc momentum theory applies, and the thrust coefficient C T is defined as where T is the thrust applied by the actuator, A is the area of the actuator and a is defined as the induction factor, such that the velocity perceived by the actuator U act (at the location of the actuator) is given by The model is derived using the following rationale. The surging actuator disc generates an unsteady flow, which violates the actuator disc model's assumption of steady flow. It is difficult to solve the unsteady momentum equation in an inertial or non-85 inertial reference frame using pressure-velocity solutions. However, whether the reference frame is accelerated or inertial, a lagrangian formulation of wake generation and convection and the resulting vorticity-velocity system solution of the induction field are invariant. A dynamic inflow model inspired by the lagrangian vorticity distribution should accurately predict the induction at the actuator, as demonstrated by Yu et al. (2019a) and Yu (2018). The wake and induction solutions are linear superpositions of a newly released wake (new wake) and a previously released wake (old wake), with respect to the reduced 90 time scale of the flow. The dynamic inflow models by Øye (1986), Larsen and Madsen (2013), Yu (2018) and Madsen et al. (2020) implicitly model this superposition and convection of the vorticity system, while explicitly defining the wake length and wake convection speed across time scales; these models should serve as a foundation for developing the proposed model.
The actuator's displacement dynamics can be interpreted as changing the vorticity system's relative convection speed, as it is invariant with respect to the reference frame. The quasi-steady solution for a fully developed wake with the strength of newly 95 shed wake elements can be determined using a modified 1D steady actuator disc model that simulates wake generation and convection caused by the force field. This 1D actuator disc model with dynamic inflow should be comparable to solutions from higher fidelity models, such as prescribed and (semi-) free-wake vortex-ring models, or CFD simulations.
In Section 1.3 we define the surge motion and thrust functions. Section 1.4 presents a summary of study cases found in literature, organised in distributions of the range of parameters that define the surge motion and thrust function. The simulations and analysis in this work use the following assumptions. The actuator surface is a circle of diameter D (radius R = D/2), and is always normal to the unperturbed free-stream U ∞ . The latter is uniform, steady, and aligned with the xdirection. The actuator moves in the x-direction according to Equation 4, where x act is the location of the actuator in the x-axis, A xact is the amplitude of the motion and ωΩ is the frequency of the motion, defined in relation to a reduced frequency 105 k as stated in Equation 5. The loading over the actuator is uniform and normal to the surface, and the thrust coefficient C T is defined by Equation 6 taking U ∞ as reference for the dynamic pressure, where C T0 is the average thrust coefficient, ∆C T is the amplitude of the variation of C T , ϕ is an additional phase difference between motion and loading, and t represents time. The sinusoidal loading approximates the load oscillations observed by other authors, as described in Section 1.4. The load change is a first-order result of the sinusoidal change in the non-entry boundary condition on the blades/actuator surface caused by the 110 sinusoidal motion (this is further expanded in Section 1.4). in Figure 1c . Orange symbols represent Eulerian Navier-Stokes simulations (commonly referred to as CFD), green symbols 120 represent Lagrangian vortex models, and blue symbols represent experiments (some also including simulations). Figure 1b is inspired by the work of Mancini et al. (2020). The survey shows that amplitudes of the motion are below 0.13D and reduced frequency k < 15. More importantly, the maximum surge velocity is v max < 1.15. The relation of ∆C T to C T0 shows that only in three cases the thrust reaches negative values. The almost linear relation of ∆C T to v max confirms the earlier observations by Mancini et al. (2020). An hypothesis is that the linear relation is explainable by the linear effect between the surge velocity 125 and the circulation on the blades, due to the change of the non-entry boundary condition on the blade surface. This hypothesis is expressed by the Equation 7, in which we consider the two-dimensional thrust coefficient at a given blade section. a ′ azimuthal induction is omitted. The aerodynamics of the blade section are approximated using a potential flow flat plate formulation.
The change in section thrust ∆C T blade section is then a function of the change in circulation ∆Γ and the rotor's local azimuthal velocity λ r U ∞ at radial position r (we disregard added mass effects). The change in circulation is a function of the chord c 130 of the section and the non-entry boundary condition, which is defined as the internal product of the section's normal − → n and the change in axial velocity ∆ − → v axial , the thrust variation equation is expressed as a function of the local variation of axial velocity, which is dominated by the surge motion.
In this work we will evaluate the proposed Actuator Disc Momentum theory with dynamic inflow correction in a motion and load space wider than (and encompassing) the one in Figure 1. The next section presents the Methods used in the research. It is followed by the Results and Discussion and finally the Conclusions.

Methods and approach
The results presented and discussed in the Section Results and Discussion have five sources: the Navier-Stokes simulations of 140 an actuator disc in surge by de Vaal et al. (2014); simulations by a semi-free wake vortex-ring model of an actuator disc in (c) ∆CT vs. CT 0 . Orange symbols represent Eulerian Navier-Stokes simulations (commonly referred to as CFD), green symbols represent Lagrangian vortex models, and blue symbols represent experiments (some also including simulations).
surge motion developed in this work; dynamic inflow models derived by other authors; CFD simulations of an actuator disc with imposed thrust; and a 1D Actuator Disc Momentum model corrected for the unsteady surge motion and loading by using a dynamic inflow model derived in this work. The cases are defined by the surge motion and unsteady load on the actuator.
The results and discussion compare the estimated induction at the actuator disc. The higher fidelity results (Sections 3.1, 3.2 145 and 3.4) are used as benchmark for the results of the proposed dynamic inflow model. The impact of actuator motion is also demonstrated by comparing the proposed dynamic inflow model to other dynamic inflow models (Section 3.3).

Semi-free wake vortex ring model
The semi-free wake vortex ring model is a conventional model inspired by the approaches in the works of Yu et al. (2016), Yu (2018, van Kuik (2018) and van Kuik (2020). The "semi-free wake" description is due to the fact that the wake expands and 150 convects with self induction up to five diameters downstream of the actuator. After that location, the expansion is frozen and the wake convects with a velocity based on U ∞ and the velocity at the center of the wake.

CFD actuator disc model
OpenFOAM ( The loading is applied using the Equation 6 and is uniformly distributed over the actuator disc, whose position varies over time using the Equation 4. The disc average axial induction factors obtained with steady CFD simulations are compared to those predicted by momentum theory with Glauert correction for thrust coefficients ranging from CT = 0.2 to CT = 1.2 to validate 165 the model. The results are depicted in Figure 2. The results agree well with momentum theory at low thrust coefficients. At low thrust coefficients, the results agree well with momentum theory. The difference is 2.1% at C T = 0.8, and it grows larger as the thrust coefficient increases.

Dynamic inflow models by other authors
In this paper, we compare the results of induction using the proposed dynamic inflow model and five previously published 170 dynamic inflow models. The five models are Pitt and Peters (1981) as described by Yu (2018), by Øye (1986) as described by Yu (2018), the model by Larsen and Madsen (2013), Yu (2018) (also described by Yu et al. (2019b)) and Madsen et al. The new dynamic inflow model presented in this work is labelled as Ferreira. The reader is also directed to the ECN model (see Schepers (2012)), which expands on the model developed by Pitt and Peters (1981). Madsen (2018). The Madsen dynamic inflow model is conceptualized as a curve fit of the solution of an unsteady actuator disc in a step function that uses two time scales to better approximate the radial dependency of the unsteady induction, implicitly as a near wake and far wake time scales. This is also the interpretation proposed in the work of De Tavernier and Ferreira (2020) when reviewing the implementation for Vertical Axis Wind Turbines (see also Larsen and Madsen (2013)), discussing the time scales as near wake and far wake. The model presented by Pirrung and Madsen (2018) predicts several corrections for loading 190 and radial effects and is calibrated against higher fidelity simulations. The two time constant filter approach was previously proposed by Øye (1986), and represents a departure from the approach by Pitt and Peters (1981) of the solution of the pressure-velocity towards the solution of vorticity-velocity problem. This solution of the vorticity-velocity problem was discussed by Øye (1986), Larsen and Madsen (2013) and Madsen et al. (2020) as a dynamic filter of near and far wake solutions.
In this work we take inspiration of the two time scales approach for representing the contribution of the wake generated 195 previously and the newly shed wake, and to distinguish between the induction at streamtube scale from the induction at the actuator. The solution of the vorticity-velocity system does not require the time integration of the flow acceleration, but it is calculated directly from the vorticity system at each time step. The wake solution and the induction solution are the linear superposition of a newly released wake (new wake) and a previously released wake (old wake), in relation to the reduced time scale of the flow. The convection of the two wake systems must be determined. We therefore define two reference values of 200 induction, namely the streamtube induction velocity u str and the induction velocity at the location of the actuator u act . We use these velocities to determine the convection of the vorticity system in the streamtube and in relation to the actuator.
The first variable of the dynamic inflow model is thestep of the algorithm is to define an unperturbed reference velocity on of the inertial reference frame that contains the streamtube and the actuator. In the case of the actuator in an oscillating surge, the reference velocity can be defined as in Equation 8 205 The second variable of the model is thestep of the algorithm is to define a streamtube wake-convection reference velocity, as defined in Equation 9. U str is determined by averaging the two induction terms u str and u act ; the equal weighing of the two induction terms reflects the balance between the proximity of the short newly shed wake to the region where the velocity is evaluated (actuator) and the distance to the longer previously shed vorticity system. Although different averaging weights 210 can lead to more fine tuned solutions, this relation appears to be sufficiently accurate., where we average the two induction velocities.
We can calculate an equivalent quasi-steady solution of the induction velocity of a vorticity system generated by a thrust C T and wake convected in streamtube with reference velocity U str (Equation 10) to be later used as a forcing function of a 215 steady solution of the newly shed wake. It is important to note that this forcing function differs from the one commonly used in dynamic inflow models (usually the steady induction for a given thrust coefficient as defined in Equation 2). Equation 10 approaches the 1D steady actuator disc thrust equation, taking U str as the mass flow rate that experiences a momentum change of u qs (per unit fluid density). If the system converges to a steady flow, Equation 10 converges to Equation 2.
We can choose to apply a form of Glauert's correction for the case of heavily loaded streamtubes and instantaneous C T > 0, inspired in the formulation presented by Burton et al. (2011). The heavily loaded streamtube criterion is defined as with C T1 = 1.816.

If the criterion in Equation 11
applies, the value of u qs can be determined by Equation 12, curve fitted from Glauert's 225 correction as described by Burton et al. (2011).
Due to the fact that wake convection varies along the streamtube, we now define length scales for actuator/near wake L act and streamtube/far wake scale L str in Equations 13 and 14. The choice of one and five diameters are suitable for near and far wake scales ; at one diameter the wake has achieved over 90% of its expansion and increase in induction, and the vorticity We now define time scales of convection of the wake for actuator/near wake scale and streamtube/far wake scale. For the streamtube scale we define one time scale τ str given by Equation 15, used for the convection of the old vorticity system and the convection of the generation of the new vorticity system.
For the actuator/near wake scale we need to define two time scales: one for the convection of the old vorticity system Following the approach by Larsen and Madsen (2013), we can now calculate the new solutions of the streamtube induction velocity u str and the induction velocity at the location of the actuator u act by the implicit integration in time of the effect of the filtered forcing function u qs . The approach is similar to that of Øye (1986) which, however, has an explicit integration in time of the filtered forcing function.
can also be written as Equation 20.
Equation 20 shows the effect of the actuator motion (v act is defined in the same reference frame as U ∞ ). As the actuator 255 moves away fromform the previously shed wake, the effective induction decreases. As the actuator moves into the wake, the effective induction increases.
The modelalgorithm can be generalised to the case of actuator motions that have a non-zero average displacement, e.g. an actuator travelling in forward motion with periodic oscillations. In this case, the most suitable inertial reference frame needs to be updated and so does U ∞ ref . The varying reference wind speed can be determined by A third filtering can be applied in the 260 form of Equation 21 An example of the implementation of the model as an algorithm in Python is shown in Appendix A: Implementation of the model as an algorithm in Python.
In the Results section, the induction at the actuator is represented by its non-dimensioned form a, defined by Equation 22.  a and C T are, as in the remaining of this work, defined in relation to the unperturbed wind speed U ∞ref = U ∞ .

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To support the interpretation of the results in Figure 3, Table 1 presents for each sub-case (labelled by the reduced frequency k) the average thrust coefficient C T , the amplitude of the variation of thrust coefficient ∆C T , the time average of the area-weighted and Prandtl-tip-corrected average inductionā deV aal , the time average of the area-weighted average induction obtained with the semi-free wake vortex ring modelā sf wm , the time average of the induction at the center of the actuator predicted by the dynamic inflow modelā dynamic inf low , the time average of the induction calculated using steady Actuator 290 Disc Theory a (C T ) steady and the steady induction of the time average of thrust coefficient a (CT ) steady (the last two predicted using steady 1D actuator disc theory).
The results in Figure 3 and Table 1 show that: Prandtl's tip correction), it results thatā deV aal −ā sf wm < 0.02.

2.
Although the dynamic inflow model is one-dimensional, the difference to the semi-free wake vortex ring model prediction is, in all cases, less than ∆a < 0.01 for the region r/R ⩽ 0.8.  In the next section we will compare the predictions of the dynamic inflow model with the results of the semi-free wake vortex ring model for a more diverse and more challenging set of cases.   occurs in the case of lowest frequency, with the difference at some points of the cycle being ∆a = 0.02. In this low frequency, the streamtube is significantly accelerated due to the slowly changing load, and the dynamic inflow model must capture this acceleration.

Comparison
Figure 5 allows us to distinguish the effect of motion from the effect of varying thrust. Figures 5a and 5b allow to compare the effect of increasing the reduced frequency of the motion while the thrust remains constant. Due to motion, the induction is higher when the actuator is in the downwind region (the actuator moves faster than the wake and immerses in its own wake), 330 and lowers as the actuator moves upwind (lower density of vorticity in the near wake). The increasing frequency of motion increases the amplitude of the induction and shifts its phase. Although it shifts towards the phase of the position of the motion, it is actually shifting towards a π/2 shift in relation to the velocity of the motion. Figures 5c and 5d show the cases of a static actuator where the load is phase shifted by π between the two figures. The inductions are naturally also phase shifted by π. Although trivial, these two cases are important to understand Figures 5e and 5f. Figure 5e corresponds to the typical 335 case experienced by a surging wind turbine, where the loading is highest when the actuator moves upwind and lowest when the actuator moves downwind. The effects of motion on the near wake density and the effects of thrust are out of phase and mostly cancel each other. Figure 5f shows a case that is mostly infeasible in a floating wind turbine (and probably undesirable as it could be unstable), where the thrust and motion are in phase and accumulate. This theoretical case allows us to push the dynamic flow model to one of the more challenging cases as it results in a larger amplitude of induction. However, even in this 340 case, the dynamic inflow model is in good agreement with the results of the semi free wake model. . The increased frequency leads to higher changes of loading, but the variation is so fast that the streamtube does not change the velocity significantly.

Comparison of results of the dynamic inflow model with those of other dynamic inflow models
In this section, we compare the results of induction using the semi-free wake vortex ring model to those of the proposed dynamic Glauert's correction for heavily loaded actuator in their quasi-steady forcing function term.

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The findings corroborate previous discussions. For non-moving actuators (Figures 7a and 7c), the various dynamic flow models agree reasonably well, with the more advanced/complex models (Yu, Madsen, and Ferreira) agreeing better with the semi-free wake vortex ring model results. The agreement between models decreases as the average C T and reduced frequency

Comparison of results of the dynamic inflow model with CFD simulations
In this section, we compare the induction results obtained using the suggested dynamic inflow model (labeled Ferreira), the semi-free wake vortex ring model, and the actuator disc simulations in OpenFOAM (labeled CFD). The results indicate a high degree of agreement. For average C T = 0.5 (Figures 8a and 8b), the CFD and semi-free wake vortex models produce induction differences of less than 0.01 at various radial places. Even for the case with motion ( Figure   370 8b), the findings demonstrate a minor radial variation in induction. The dynamic inflow model agrees well with the higher-   is significantly faster than the unperturbed wind speed. Previous pronouncements of this effect were based on an inaccurate interpretation of the actuator's accelerated reference frame.
Additionally, the results confirmed that, while increasing frequency of motion can result in increased loading and velocity amplitudes, the streamtube's inertia results in essentially constant induction. The effect of motion tends to cancel out the variation in thrust (assuming a DeltaCT proportional to the surge velocity), and the variance in induction at the actuator 395 decreases with greater frequency.
The model formulates wake generation and convection in lagrangian terms, and the resulting vorticity-velocity system solution of the induction field is frame-invariant. This allows the accelerating actuator's induction to be predicted. The model is based on the well-established techniques developed by Øye (1986), Larsen and Madsen (2013), Madsen et al. (2020), and Yu (2018) and Yu (2018).

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The straightforward approach is simply implementable in BEM models. The existing implementation already addresses the scenario of heavily loaded streamtubes; yet, even for static actuators, this region remains challenging. For future work, the model's simplicity and analytical formulation make it well-suited for optimizing and controlling FOWTs. The prediction of induction at the tip region is postponed till further research is completed.
In this work we proposed the test and verification of several hypotheses (described in the introduction) that would allow actuator was twenty times that of the unperturbed wind speed and the amplitude of thrust variation was more that twice the average loading. The results also allowed us to confirm that although the increased frequency of motion can lead to higher amplitude of loading and velocity, the inertia of the streamtube results in an almost constant induction. The effect of motion tends to counter the variation of thrust (assuming a ∆C T proportional to the surge velocity), and the induction at the actuator has a smaller variation with increased frequency. As hypothesized, there is no occurrence of vortex ring state or even turbulent 415 wake condition. Although this had been raised by previous works, it was actually the result of an incorrect application of actuator disc momentum theory to the accelerated reference frame of the actuator (inspired by the conventional application when the actuator is an inertial reference frame e.g. a flying propeller). The proposed dynamic inflow model is inspired by the work of Madsen, and in its current formulation, is simpler than the formulation presented by Madsen et al. (2020). The simple algorithm can easily be implemented in BEM models. The proposed dynamic inflow model showed a very good prediction up to 420 r/R = 0.8. However, it can be further developed to account for the outer 20% of the radius, including blade vortex interaction.
The current implementation already addresses the case of heavier loaded streamtubes; this region remains challenging, even for static actuators. For future work, the simplicity of the model and its analytical formulation makes it suitable for optimisation and control of FOWTs. The prediction of the induction at the tip region still needs to be improved. Reply to referees 580 The authors would like to thank the referees' valuable comments. As a result to answering the reviewers comments, significant changes were done, namely: 1. A more detailed explanation of the equations of the model was added.

2.
A new section was added (Section 3.3), where the proposed dynamic inflow model is compared with several other dynamic inflow models, namely the one by Pitt and Peters (1981) as described by Yu (2018), by Øye (1986) as described 585 by Yu (2018), the model by Larsen and Madsen (2013), the model by Yu (2018)  are also compared with those of the semi-free wake vortex ring model.

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Several editorial changes were also done. The answers to each specific comment by the referees are found in the next pages, including a list of changes. The changed text is often printed in blue, except for figures and tables. Many editorial changes are not identified by marking the text blue, as not to overload the text.
We believe all comments have been addressed and the new additional content further proves the relevance of the model and of the work. We hope the referees agree.

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Cordially, Prof. Carlos Ferreira Anonymous Referee # 1 Referee review of "Dynamic inflow model for a Floating Horizontal Axis Wind Turbine in surge motion" by Ferreira et al.
Referee's comment The paper deals with a timely and interesting set of questions, related to the state of actuator disk/momentum theory for the case of oscilatory disk motions. Clearly this area is of interest for wind turbines placed on off-shore 600 platforms that will oscillate back and forth and will change the inflow velocity being seen by the system. The overall conclusions, which this referee finds reasonable and interesting, is that if properly formulated, standard actuator disk approach still works, as long as the correctly chosen U-infinity(t) is used. The introduction and motivation are well described and the survey of prior work (in particular Fig 1) is very good. The introduction also gives the impression that a more fundamentals oriented rational method will be proposed to deal with non-inertial to inertial reference frames etc etc. that has caused confusions in 605 the past. So that all seemed very promising.
Answer: Thank you for this positive starting comment. The objective of the work is to derive a model based on correct physics.
Referee's comment However, once the "meat" of the contribution starts being described, the material is suddently presented as an "algorithm" to be implemented in python, etc. and there seems to be no connection whatsoever with any actual physics 610 or principles being invoked. That is to say, where did Eqs. 20 and associated Eqs. for u a ct, u s tr etc, Eqs. 17 & 18 come from?
There seems to be no connection with any actual physics or principles being invoked.
Answer: Thank you for this helpful comment. The expression "algorithm" has been replaced by model. More impor- tantly, text has been added/modified to explain the derivation of the equations.
Referee's comment More specifically, in line 195 authors claim to be computing "new solutons for the streamwise induction 615 velocity at actuator". What equation is being solved exactly and how is the solution obtained? Up to this point in the paper there is not a single dynamical evolution equation being presented. One would expect some equation of the form du/dt = ... and then the solution is Eqs 17,18 etc. Instead, what the authors seem to be doing is simply a-priori assuming that a time filtering will have benefits of some sort to be used as inflow for the model implementation to come later, but it does not look like Eqs. 17 and 18 are "solutions" to anything in particular. Only in point 6 of the introductory sentences there is a reference 620 to a time-filtering method (Larsen-Madsen model). In that paper the time-filtering was motivated simply by saying something along the lines of "engineering model for response functions" including inertia of structures etc. How is that approach really justified in light of the very fundemantal sounding comments made in the introduction of the paper? This paper should provide a clear discussion of these aspects.
Answer: Several points are mentioned in this comment. We will aim to address all. The model aims to present an 625 equivalent solution of the vorticity-velocity problem, in the perspective vorticity is shed at each time step and previously shed vorticity is convected away from the actuator at each time step. This can be approximated by a convolution of the current solution and a new steady state solution. This approach is on the basis of dynamic inflow models such as the one by Øye (1986), Larsen and Madsen (2013), Yu (2018) and Madsen et al. (2020). These models often referred to a filtering approach of the near and far wake, which is a reasonable description; we opted for the same description, but the language is not totally correct. The text has been modified to avoid the word "filter" and instead present the evolution from one vorticity system to a new vorticity system.
Regarding the point of the need of an explicity du/dt = ... formulation, here one politely disagrees with the referee. The dynamic inflow model by Pitt and Peters (1981) (and also the ECN model) has an explicit time integration of du/dt because it models a linearized form of the unsteady momentum equation. The model of Øye (1986), although it presents a du/dt = ...

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formulation, is in fact solving the same convolution problem as the models by Larsen and Madsen (2013), Yu (2018) and Madsen et al. (2020), just with a different numerical integration procedure. The formulation of solutions that decay with time through an exponential of time (as these last cited models and the model proposed in this work) provides an implicit form of time integration and, a clearer interpretation of the phenomena. However, the equation presented in this work can be converted to a du/dt = ... formulation as Øye (1986). We have added/modified the text to make this clearer.

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Referee's comment Presentation of results  show one cycle of resulting induction factor for various conditions and good results compared with the semi-free wake vortex ring model are shown. Was the inflow velocity time-filtering approach simply proposed by noting empirically from such plots that time-filtering the input would yield desired results? And parameters obtained by fitting the observed behaviors? That may be a fine approach for very applied settings, but unless better justified by analysis of governing equations, it it does not seem to rise to the level of a scientific contribution since it does not seem 645 convincing that it can be generalized in any way to other conditions.
Answer: Choice of the formulation of the model was not based on what works. Once it was defined that the model needs to evaluate the solution in the inertial reference frame and accounts for the motion of the actuator, it was necessary to have a formulation that was invariant with the reference frame, and that is the vorticity-velocity formulation. The model needs to account for the change of the vorticity system, as new wake is shed and old wake is convected, and the relative position of 650 the actuator in relation of the vorticity system. The text was modified to better explain this.
Referee's comment In view of the above comments, it is recommended that the authors aim to justify and derive the "timefiltering" approach somehow, if that is possible. If not possible, publication in WES is perhaps not fully justified and also, then the characterization of prior work (references to past "confusions") should be reworded to avoid raising the readers' hopes that the present paper will clarify these things.

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Answer: The changes to the text should clarify the physics behind the derivation of the model.

Some additional comments for minor revisions, if useful:
Referee's comment Abstract, first sentence: the statement " ..surge motions .. when faster than the local wind speed, cause rotor-wake interaction." Do the authors mean to imply that only if surge motion is larger than, say, 8 m/s (local wind speed), there will be rotor-wake interactions? One would expect "interactions" even at much lower surge motion speeds.. Needs more 660 precise wording. It seems when authors say "interactions" they have something very specific in mind but at this stage of the paper readers will have more general interpretations of "interactions" in mind.
Answer: The text is modified to specify blade-vortex interaction. The abstract is also revised.
Referee's comment Line 24: do the authors mean to say "a turbulent wake with the wake in front of the turbine?" since the normal state of turbine wakes is a turbulent wake state in the first place.

Emmanuel Branlard (Referee)
Referee's comment In this paper the authors present a dynamic inflow model suitable for FOWT, and verify the results against 680 high and mid fidelity simulations. This is a nicely written paper, with interesting methods and conclusions. I have some general comments that I hope can improve the revision of the paper.
Answer: Thank you for the kind comment. You comments have been very useful towards improving the work. Thank you.
Referee's comment 685 -I believe the paper would benefit from adding more justifications for each of the important equations of the model. You'll find several specific comments in the pdf regarding this.My general comments are the following: -I believe the paper would benefit from adding more justifications for each of the important equations of the model. You'll find several specific comments in the pdf regarding this.
Answer: The text ws modified to address this, including the comments in the pdf, which are listed below.

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Referee's comment -Some results for various radial positions would probably be needed to support the conclusion that the model compare well with the ring model for up to r/R=0.8.
Answer: Section 3.4 was added, where the model is compared with CFD simualtions and semi-free wake vortex model simulations, includign results at different radial positions. These results are used to support the discussion and conclusion, which are modified.

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beyond the region of validity of the model and the vortex-ring-based models, so it will have to be treated with care -I do not expect the vortex-ring based model to accurately capture the vortex-ring state which will be highly turbulent and diffusive.).
The question that could be answered and would be really interesting would be whether the vortex ring state model occurs 715 "sooner" (for some low frequencies maybe) than one would expect from the steady conditions (zero frequency), or "later", or simply "at the same time". I think such an investigation will really add to the paper (again, keeping the limitations of both models in mind). At least a small moderation on the fact that the vortex ring state was not really "tested" would be great (I understand that the study still makes a point that it was not reached for "moderately loaded" rotors).
Answer: Previous authors claimed that high thrust coefficients occurred because the perceived velocity in the ref-720 erence frame of turbine becomes very low or negative, and that this represented a vortex ring state . That interpretation is incorrect. However, regardless of the motion, the streamtube can enter vortex ring state if a large loading is applied for a long enough time. So, the work does not mean that vortex ring state cannot occur, only that the interpretation of the velocity perceived in the reference frame of the wind turbine does not represent vortex ring state. The text is modified to further clarify this.