A new dynamic inflow model for vertical-axis wind turbines

This paper presents a new dynamic inflow model for vertical-axis wind turbines (VAWTs). The model uses the principle of Duhamel's integral. The indicial function of the inflow- and crossflow-induction required to apply Duhamel's integral is represented by an exponential function depending on the thrust coefficient and the azimuthal position. The parameters of this approximation are calibrated using a free wake vortex model. The model is compared with the results of a vortex model and higher fidelity computational fluid dynamic (CFD) simulations for the response of an actuator cylinder to a step input of the thrust and to a cyclic thrust. It is found that the discrepancies of the dynamic inflow model increase with increasing reduced frequency and baseline thrust. However, the deviations remain small. Analysing the application of a finite-bladed floating VAWT with non-uniform loading and validating it against actuator line CFD results that intrinsically include dynamic inflow shows that the new dynamic inflow model significantly outperforms the Larsen and Madsen model (which is the current standard in fully coupled VAWT models) and enhances the modelling of VAWTs.


INTRODUCTION
Offshore wind turbine technology has made significant and rapid progress since the first offshore wind farm was installed in 1991. 1 We advanced from fixed platforms to floating structures to be able to overcome deeper water depths. Onshore, horizontal-axis wind turbines (HAWTs) have reached a mature level of technology and dominate the market. Far offshore, the operational conditions are significantly different, raising the question whether other concepts such as vertical-axis wind turbines (VAWTs) could be more suitable and allow a reduction in the cost of energy.
The development of floating VAWTs is still at an early stage. A fundamental difference between onshore and offshore turbines is the additional complexity introduced by the motions of the floating platform. 1 Turbines are translating and rotating in three dimensions, as visualised in Figure 1, causing dynamic inflow conditions at the rotor.

Background
Dynamic inflow is the phenomena describing the unsteady relation between the induction at the rotor (or any other point of the flow) and the unsteady loading on the rotor and/or unsteady momentum of the flow. As an example, a variation of the loading can be caused by a change in the inflow wind speed and/or a change in for example pitch angle or rotor speed. The change in induction will lag the variation of the loading.
The gradual change of the induced velocity from one equilibrium to another is the essential characteristic of the dynamic inflow phenomenon.
In some modelling techniques such as computational fluid dynamics or vortex methods, phenomena like the dynamic inflow effect are represented inherently since the velocity field as a result of a variable force field and/or incoming flow is physically modelled in space and time. However, for fully coupled methods accounting for the aerodynamics, hydrodynamics, structural dynamics, and controller dynamics, these models are too time-consuming making them unsuitable for iterative processes. Simpler momentum-based models are often opted for; however, they need additional correction models to cope with unsteady effects such as dynamic inflow. 2 In previous work, 13 it has been identified that there is a clear difference in the dynamic response of the induction at the different azimuthal positions. This is not considered in dynamic inflow models developed for HAWTs. These models are only calibrated with respect to the rotor disk and thus do not necessarily consider the behaviour upwind and downwind.

Research objective
With this motivation, the objective of this paper is as follows: ''To develop a new engineering dynamic inflow model that enhances the modelling of VAWTs in dynamic inflow conditions.'' The paper consists of four parts. First, the main approach is highlighted, and the load cases and modelling techniques used in this work are briefly introduced. Second, a new dynamic inflow model is developed, and the derivation used to set up the model is explained step by step.
Third, the results are presented including a comparison of the new dynamic inflow model with respect to higher fidelity models. Fourth, the conclusions are summarised.

Approach
VAWTs are often represented by the actuator cylinder concept, in similarity to the actuator disc concept for HAWTs. The actuation surface, coinciding with the swept surface of the rotor, is loaded with the average normal and tangential blade forces occurring during one revolution of the rotor. The development of the new dynamic inflow model builds around this infinite-bladed representation of the VAWT. Because the normal load distribution of a VAWT can basically take any shape depending on variables such as the solidity, tip speed ratio, and blade pitch angle, it is decided to simplify the rotor loading. The average loads on the actuator are prescribed by a simple load distribution mimicking the overall shape of the average loading of a VAWT: a uniform load normal to the actuation surface pointing outwards for the upwind part of the The derivation and implementation of the new dynamic inflow model is verified by comparing the induction around an actuator cylinder obtained from the dynamic inflow model with the results obtained from the free wake vortex model. Since the results of the free wake vortex model are already used in the calibration, an external CFD model is used as an independent verification method to compute the velocity field around an unsteady actuator cylinder. The unsteady response of a step input in the thrust and a cyclic thrust is studied using the three methods.
Because the new dynamic inflow model is built using a simplified representation of the VAWT with infinite number of blades and a prescribed uniform load distribution, the second validation case study is a finite-bladed VAWT with non-uniform loading in dynamic conditions. The

Study load cases
For the calibration of the new dynamic inflow model as well as the verification and validation of it, different load cases are studied.
• Load cases for calibration: For the calibration of the dynamic inflow model, a database of step responses to the thrust of the simplified Actuator cylinder concept is generated using a 2D free wake vortex model. The database is set up for 17 baseline thrust coefficients ranging between 0.1 and 0.9. A step input in C T of 0.1 is introduced, and the induction response in x-and y-direction at 60 different azimuthal locations on the actuator is considered. Both a step increase and decrease are analysed. Note that the step input is applied only after a steady solution is found for the baseline thrust coefficient.

• Load cases for verification and validation:
Besides verifying the dynamic inflow model using the same step inputs to the thrust as used for the calibration of the model (see Section 4.1), the response to a cyclic thrust (see Section 4.2) is studied. Again, the infinite-bladed actuator cylinder with uniform load is considered. The cyclic thrust coefficient is defined by a baseline thrust (C T0 ), amplitude (ΔC T ), and reduced frequency (k). The cases that are considered in this research have a baseline thrust of 1∕9 and 7∕9, where the first one presents a low loaded case with a small induction and wake expansion and the second one presents a highly loaded case with a larger induction and considerable wake expansion. The thrust amplitude is fixed to 1∕9. The cyclic loading is applied with four different reduced frequencies, ie, 0.05, 0.2, 0.5, and 1. The frequency is non-dimensionalised using k = R∕V ∞ . The expression of the time-varying thrust coefficient is given by Equation (1). t is the time, R is the radius of the actuator circle, and V ∞ is the incoming wind speed.
The dynamic inflow model is developed for an infinite-bladed uniformly loaded VAWT. To validate that the dynamic inflow model also works in case of a finite-bladed non-uniformly loaded VAWT, simulations are performed for a floating VAWT (see Section 4.3). The analysis of the floating VAWT is adding significant complexity, and this will present the use of the new dynamic inflow model to an application or design case of the VAWT. In this work, only a surging motion (motion in the direction of the wind as shown in Figure 1) is considered. This will cause the velocity perceived by the turbine to be dynamic. The surging motion is prescribed by Equation (2), where s 0 is the baseline surging position, Δs is the surging amplitude, and k is the reduced frequency. The rotor loading is determined using the blade element theory. The baseline surging position is set to 0, and the amplitude to 1 m. The reduced frequencies studied in this work are 0.5, 1, and 2. The surging motions are randomly selected as a way of introducing dynamic inflow and do not necessarily comply with real conditions. (2)

Modelling techniques
The approach uses four modelling techniques: a 2D free wake vortex model, an actuator cylinder CFD model, the actuator cylinder model, and the actuator line CFD model. These models are used to analyse the study load cases and are introduced below.
• 2D free wake vortex model (VM): Because the 2D actuator cylinder in this paper is uniformly loaded upwind and downwind, vorticity will only be shed at the edges of the actuator or at the transition from upwind to downwind. In fact, the velocity field will be exactly the same as for a 2D uniformly loaded actuator disk, and thus, the surface on which the forces are applied is of non-importance. The time step is set to 0.01R∕V ∞ . This has shown converged solutions. Unsteady variations in C T are only applied after the steady solution was found for the baseline thrust coefficient. This model will be used to develop and tune the dynamic inflow model. The actuator cylinder model does not account for unsteady effects and as such should be extended using a dynamic inflow model. So far, the steady actuator cylindermodel has been combined (in, eg, SIMO-RIFLEX-AC 6 ) with the dynamic inflow model proposed by Larsen and Madsen. 18 In this model, dynamic inflow is modelled using a low pass filtering of the steady state induced velocities. The induced velocity filtered for the near and far wake is presented by Equations (3) to (5) in which a n−1 denotes the induced velocity of a previous time step. a s,n refers to the steady induced velocity of the current time step, and Δt is the size of the time step. nw and fw are the time constants for the near and far wake filter, respectively, and are non-dimensionalised with respect to the rotor radius and average wake velocity ( = ⋆ R∕V wake ).
The dimensional constants ⋆ nw and ⋆ fw are 0.5 and 2. 19 a nw = a n−1 exp a fw = a n−1 exp When using various models for verification and validation purposes of the dynamic inflow model, it is important to identify and quantify the differences and similarities of the results obtained by the models in absence of dynamic effects. The actuator cylinder model, the actuator cylinder CFD model and the free wake vortex model will be used to calculate the flow field around a simplified actuator cylinder with uniform normal loading upwind and uniform normal loading downwind. In Figure 3, a comparison of the x-induction at the midpoint of the rotor with respect to a steady thrust coefficient is presented. For the vortex model, a maximum difference of 5% is observed up to a C T of 0.7 compared with the momentum results of the actuator cylinder model. Above a C T of 0.7, the discrepancies increase. The CFD model agrees well with momentum theory for low thrust. The discrepancy of the induction at the centre of the actuator cylinder is slightly larger for the higher thrust values, with a maximum of 1.5%.

THE DERIVATION OF A NEW DYNAMIC INFLOW MODEL
The development of the new engineering dynamic inflow model for VAWTs is based on the methodology developed in Yu et al. 11 It uses a similar approach as Wagner's model for 2D unsteady airfoil aerodynamics. Duhamel's integral, as given in Equation (6), is used to represent the total induction response of the system to an arbitrary thrust excitation by superimposing the response to step inputs. a st represents the steady induction; Φ is referred to as the indicial response function.

Database of step responses
The database of step responses is generated using a 2D free wake vortex model. Because vortex models describe the wake in space and time, it captures the unsteady responses intrinsically. The database is set up for various baseline thrust coefficients ranging between 0.1 and 0.9, and the induction response in x-and y-direction at a range of azimuthal locations on the actuator is considered. Both a step increase and decrease in the thrust are analysed on the simplified infinite-bladed actuator cylinder concept. In Figure 5,

Indicial function
The indicial step responses of the induction, calculated using the 2D free wake VM, are represented using an exponential approximation using two exponential terms. Similar as found by other researchers such as Pirrung and Madsen 10 and Yu et al 11 two exponential terms seem to be the optimal option since one term is under-fitting and three terms are over-fitting the data. The indicial functions, denoted by Φ, are described by  Equation (7). The parameter t ⋆ is the time non-dimensioned with U 0 ∕R.
For every thrust coefficient and azimuthal position with a step increase and decrease in the thrust, the parameters , 1 , and 2 are determined using a least-square method. The time series is discretised logarithmically to concentrate more time steps right after the thrust coefficient jump.
In Figure 6

Coefficients of indicial function
With the coefficients , 1 , and 2 calibrated for the database, a relation can be identified between the coefficients of the exponential approximation on one hand and the thrust coefficient and azimuthal position on the other hand. Note that the parameters for the step increase  (8) and (9).
In between the upwind and downwind region, a linear fit is added. The polynomial coefficients are provided in Table 1. In the appendix, Table A1 presents the polynomial coefficients for the y-induction.
As an example Figures 7 and 8 are provided. In Figure 7, the trend between the exponential coefficients and the azimuthal position is shown for three different thrust coefficients. Also, the polynomial fit is added. The polynomial fit is symmetric explaining why all uneven polynomial terms are cancelled. The fit represents the data well; however, larger deviations are present near the edges of the actuator cylinder. In Figure 8, the relation between the polynomial coefficients and the thrust coefficient is presented for the upwind and downwind region separately. This second fit is only presented for the P 1 and P 2 as a representative example for all parameters.

Duhamel's integral
Because now the indicial function is known for every combination of baseline thrust coefficient and azimuthal position, Duhamel's integral can be solved. The application of Duhamel's integral for dynamic inflow assumes that the induction response of an actuator cylinder can be built up as a superposition of responses to a series of step changes. However, because the induction is non-linear with respect to the thrust (C T = 4a(1 − a) according to the momentum theory), this assumption is challenged at high thrust. Duhamel's integral can be solved numerically, as given by Equation (10). At every time step t, a new step change to the thrust is introduced. The response of the induction at time step t is the sum of all responses to the previous step changes introduced at = [0, t] and evaluated at time step t. d is the size from one time step to the other.

Step in thrust
To verify the development and implementation of the new dynamic inflow model, the response of the induction to a step input in the thrust is considered. Because this is the basis of the development of the dynamic inflow model, it is expected that the results match rather well. In

Cyclic thrust
Because dynamic inflow can be realised by dynamic thrust, a cyclic thrust coefficient is applied on the infinite-bladed AC concept with various baseline thrusts (C T0 ), amplitudes (ΔC T ), and reduced frequencies (k), as described earlier in the methodology section. In Figure 10, the x-induction is presented at three locations (upwind at ≈45 • , centre at ≈10 • , and downwind at ≈ −5 • ) for a baseline thrust of 1/9, variation of 1/9 and a reduced frequency of 0.5. The results are again presented for the newly developed dynamic inflow model, the VM, and the CFD model as external reference code. The dynamic inflow model is again applied using the results of the VM as steady input values. In general, a good fit can be recognised for all three locations, with a slightly larger deviation at the downwind location. This is true for most cases.
To evaluate this further, other baseline thrust coefficients and reduced frequencies are considered. A range of reduced frequencies is studied going from quasi-steady (k ≤ 0.05) to moderate unsteady (0.05 < k ≤ 0.2) and highly unsteady conditions (k > 0.2). In order to quantify the hysterical response, the amplitude and phase delay of the induction with respect to the thrust coefficient are calculated using Lissajous' graphical  method. 22 Saying that A is the induction amplitude and C is the zero-crossing height, the phase difference Φ between the induction and thrust is given by Equation (11).
The results are presented in Figures 11 and 12 for a baseline thrust of 1/9 and 7/9, respectively. The figures reveal that in general, there is a good match between all three models. For the lowest baseline thrust, the results from the CFD model and vortex model match almost exactly.
The error of the newly developed dynamic inflow model with respect to the other models seems to slightly increase with reduced frequency.
Also, for the largest baseline thrust, the deviations between the dynamic inflow model and the other models increase with increasing reduced frequency. For the larger baseline thrust, the difference between the vortex model and CFD model is slightly larger. This might be expected since in steady case, it was already observed that the deviations between both models increased for increasing thrust coefficient (see Figure 3). The  linear. This is one of the assumptions made when applying Duhamel's equation. At large reduced frequencies, the unsteadiness increases, and this is emphasising every error of the model further.

Floating VAWT
To validate that the dynamic inflow model, developed for an infinite-bladed VAWT, also works in case of a finite-bladed non-uniformly loaded The normal and tangential loadings of all models are presented in Figure 13A and B for a reduced frequency of 1.
The variation of the TRAC value for the actuator cylinder model without dynamic inflow model, with the old Larsen and Madsen 18 dynamic inflow model, and with the new dynamic inflow model for different reduced frequencies is presented in Figure 14. Figure 14A presents the value for the normal loading time series, while Figure 14B considers the tangential loading time series. It is clear that both dynamic inflow models outperform the predictions of the actuator cylinder model without dynamic inflow model. Also, the TRAC value decreases with increasing reduced frequency, confirming that the dynamic inflow effects are larger at larger frequencies.
When comparing the TRAC value of the old and the new dynamic inflow model, one can conclude that the new model outperforms the old one. Only at the reduced frequency of 0.5, the old model is slightly better than the new model if the criteria is applied on the normal loading.
However, the values are very close to each other, and the differences in the time series are small. The TRAC value in dynamic conditions is still lower than in steady conditions and decreasing further for more unsteadiness (ie, larger reduced frequency).
It should be remarked that although the dynamic inflow model is also derived for the y-induction, it is of significantly less importance than the x-induction. When not including the dynamic inflow model for the y-induction, there is no visible difference for the normal and tangential loading. The model is mainly derived for completeness.

CONCLUSION
Because wind turbines are often operating in dynamic inflow conditions, it is of great importance to have an accurate engineering dynamic inflow model available. In this paper, a new dynamic inflow model is developed specifically for VAWTs.
The development of the model is based on the methodology presented by Yu et al. 11 and is using the principle of Duhamel's integral. The indicial response functions of the x-and y-induction, required to be able to apply Duhamel's equation, are determined using a three-step process.
First, a database is built using a free wake vortex model to define the response to a step input in the thrust coefficient. The indicial responses are consequently approximated by a 2nd order exponential function of which the coefficients are calibrated based on the database. Thirdly, a relationship is identified between the coefficients of the exponential approximation on one hand and the thrust coefficient and azimuthal position at the rotor on the other hand.
The newly developed dynamic inflow model is verified by evaluating the predictions to a step in thrust on the infinite-bladed uniformly loaded actuator cylinder and comparing with the results of the free wake vortex model and a CFD model as independent model. Furthermore, the induction of a 2D actuator cylinder subjected to a cyclic thrust has been compared for various baseline thrust values and reduced frequencies using the same models. It is found that the discrepancies of the dynamic inflow model increase with reduced frequency and baseline thrust; however, the deviations remain small. Also, larger deviations are observed for the downwind location. The deviations can be mainly attributed to the limitations of Duhamel's integral.
Finally, the dynamic inflow model is implemented in the actuator cylinder model and compared against the actuator line CFD model as a way to validate that the model also works in case of a finite-bladed non-uniformly loaded VAWT and enhances the modelling of VAWTs in dynamic inflow conditions. A floating motion is introduced as extra complexity to approach design conditions. Simulations are done for a VAWT in unsteady operation because of a surging motion. The normal and tangential loadings are calculated, and the TRAC values of the loading time series indicate a clear improvement in the predictions in both normal and tangential loadings compared with the predictions performed in absence of a dynamic inflow model and the Larsen and Madsen 18 dynamic inflow model that was originally developed for horizontal-axis wind turbines.