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Keywords:

  • dynamic inflow;
  • actuator disc;
  • floating wind turbine;
  • surge motion;
  • aerodynamic damping

ABSTRACT

  1. Top of page
  2. ABSTRACT
  3. NOMENCLATURE
  4. INTRODUCTION
  5. INDUCED VELOCITY AND DYNAMIC INFLOW MODELLING
  6. ACTUATOR DISC MODEL
  7. VERIFICATION OF IMPLEMENTED MODELS
  8. DESCRIPTION OF TEST CASES
  9. COMPARISON AND EVALUATION
  10. CONCLUSIONS AND FURTHER WORK
  11. REFERENCES

Offshore wind turbines on floating platforms will experience larger motions than comparable bottom fixed wind turbines—for which the majority of industry standard design codes have been developed and validated. In this paper, the effect of a periodic surge motion on the integrated loads and induced velocity on a wind turbine rotor is investigated. Specifically, the performance of blade element momentum theory with a quasisteady wake as well as two widely used engineering dynamic inflow models is evaluated. A moving actuator disc model is used as reference, since the dynamics associated with the wake will be inherently included in the solution of the associated fluid dynamic problem. Through analysis of integrated rotor loads, induced velocities and aerodynamic damping, it is concluded that typical surge motions are sufficiently slow to not affect the wake dynamics predicted by engineering models significantly. Copyright © 2012 John Wiley & Sons, Ltd.


NOMENCLATURE

  1. Top of page
  2. ABSTRACT
  3. NOMENCLATURE
  4. INTRODUCTION
  5. INDUCED VELOCITY AND DYNAMIC INFLOW MODELLING
  6. ACTUATOR DISC MODEL
  7. VERIFICATION OF IMPLEMENTED MODELS
  8. DESCRIPTION OF TEST CASES
  9. COMPARISON AND EVALUATION
  10. CONCLUSIONS AND FURTHER WORK
  11. REFERENCES
Notations
a

axial induction factor (–)

A

rotor area (m2)

c

chord

CD

section drag coefficient (–)

CL

section lift coefficient (–)

CT

thrust coefficient (–)

D

drag per unit span (N m − 1)

f

force per unit area (N m − 2)

f ′ 

body force (N m − 3)

F

force per unit span (N m − 1)

Hs

significant wave height (m)

k

modelling constant (–)

L

lift per unit span (N m − 1)

ma

apparent additional mass (kg)

Mflat

blade flatwise bending moment (Nm)

Nb

number of blades (–)

r

radial coordinate (m)

R

rotor radius (m)

T

rotor thrust (N)

Tm

mean period (s)

u

surge velocity (m s − 1)

vave

disc-averaged induced velocity (m s − 1)

vi

induced velocity (m s − 1)

vint

intermediate induced velocity (m s − 1)

vqs

quasisteady induced velocity (m s − 1)

vr

fluid radial velocity (m s − 1)

vθ

fluid swirl velocity (m s − 1)

vz

fluid axial velocity (m s − 1)

V0

free-stream velocity (m s − 1)

Vz

axial velocity component at rotor disc (m s − 1)

Wax

aerodynamic work (axial) (J)

xA

displacement amplitude in surge (m)

z

axial coordinate (m)

z0

axial position of the actuator disc (m)

Greek letters
α

angle of attack (rad)

Δ

time step size (s)

ϵ

smearing parameter (–)

η

Gaussian smoothing function (–)

γ

blade twist angle (rad)

ωs

natural frequency in surge (rad s − 1)

Ω

rotor rotational velocity (rad s − 1)

ρ

air mass density (kg m − 3)

σs

surge standard deviation (m)

τ1

time constant (s)

τ2

time constant (s)

θ

blade pitch angle (rad)

θ0

collective blade pitch angle (rad)

INTRODUCTION

  1. Top of page
  2. ABSTRACT
  3. NOMENCLATURE
  4. INTRODUCTION
  5. INDUCED VELOCITY AND DYNAMIC INFLOW MODELLING
  6. ACTUATOR DISC MODEL
  7. VERIFICATION OF IMPLEMENTED MODELS
  8. DESCRIPTION OF TEST CASES
  9. COMPARISON AND EVALUATION
  10. CONCLUSIONS AND FURTHER WORK
  11. REFERENCES

Some of the latest offshore wind turbine concepts are placed on floating foundations that drastically reduce the stiffness of the tower and support structure so that significantly larger system motions can be expected than on an equivalent bottom fixed system. The environments (wind and wave) in which these wind turbines operate are by nature unsteady. The aerodynamic loads on the blades and structure are unsteady by nature.

Traditionally, unsteady aerodynamic effects can be divided in mainly two parts, namely unsteady profile aerodynamics and dynamic inflow effects. [1] The first part accounts for the dependence of sectional aerodynamic coefficients on the time-varying angle of attack. Time scales associated with these dynamic effects are proportional to the ratio of blade chord length to the relative effective velocity seen by the blade section, approximately c ∕ Ωr, typically fractions of a second. [1]

Dynamic inflow effects account for the influence of time-varying shed vorticity on the induced velocity at the rotor plane. In this case, characteristic time scales are of the order of the rotor diameter to free-stream velocity, 2R ∕ V0, typically 5 to 20 s for modern multimegawatt wind turbines. [1]

Considering a floating wind turbine, the natural periods for platform motions can—depending on the nature of the platform and the particular motion—be in the order of approximately 5 to 100 s. [2, 3] Of the two unsteady aerodynamic effects considered, it is only the dynamic inflow effect that has time constants of comparable order of magnitude as the platform motions.

Analysis codes used to verify the stability and the ability of a floating wind turbine structure to withstand experienced loads (e.g. Bladed, FAST [4] or HAWC2 [5]) all base aerodynamic calculations on the blade element momentum (BEM) method. The classical BEM method determines induced velocities at the rotor plane by assuming equilibrium between applied aerodynamic loads and the induced flow field. The dynamics of this process (inertia between applied loads and induced velocities) is only modelled by means of engineering type dynamic inflow models. [6, 1]

In this work, the back and forth surge motion of a floating wind turbine is investigated. Because of the surge motion, the wind turbine rotor experiences an apparent wind effect and, in addition, will be moving into and out of its own wake. Both these effects could be important to consider in a dynamic inflow model. Results for different dynamic inflow models are compared with an actuator disc model in FLUENT, where wake dynamics are inherently accounted for as part of the flow solution.

INDUCED VELOCITY AND DYNAMIC INFLOW MODELLING

  1. Top of page
  2. ABSTRACT
  3. NOMENCLATURE
  4. INTRODUCTION
  5. INDUCED VELOCITY AND DYNAMIC INFLOW MODELLING
  6. ACTUATOR DISC MODEL
  7. VERIFICATION OF IMPLEMENTED MODELS
  8. DESCRIPTION OF TEST CASES
  9. COMPARISON AND EVALUATION
  10. CONCLUSIONS AND FURTHER WORK
  11. REFERENCES

A wind turbine rotor extracts kinetic energy from the air flow passing through it, reducing the velocity of the air flowing through the rotor. Thus, the rotor induces a velocity deficit onto the free-stream, with this deficit referred to as the induced velocity. For a wind turbine, the induced velocity, vi, is defined as a positive quantity if it slows down the free-stream, V0. The velocity at the rotor disc is therefore

  • display math(1)

where a = vi ∕ V0 is referred to as the axial induction factor.

Basic principles of dynamic inflow modelling

According to standard BEM theory, flow properties (e.g. velocity and pressure) at the rotor disc are determined on the basis of steady axial momentum theory with assumed flow properties infinitely far upstream and downstream of the rotor disc itself. To model unsteady aerodynamic loads on a rotor, the dynamics associated with the development of induced velocity when changing the thrust have to be taken into consideration as well. This necessitates an alternative approach to the simple steady axial momentum balance. Early steps in dynamic inflow modelling were taken by Carpenter and Fridovich, [7] who described induced velocity in the form of an ordinary differential equation to represent the time lag associated with the build-up of induced velocity, on the basis of studies of the unsteady development of induced velocity on a rotor disc in response to changes in blade pitch and input thrust. This original work has since seen significant further development, especially in application to helicopter dynamics, [8] but also more specifically for wind turbine applications by, e.g., Suzuki. [9]

The principle of the dynamic inflow approach can be illustrated in its simplest form by considering the uniform axial induced velocity over an actuator disc. With axial momentum theory used, the rotor thrust and inflow can be related by

  • display math(2)

where the term dvi ∕ dt represents the additional force on the rotor disc due to accelerating inflow. [10] The value of ma represents the apparent mass associated with a solid circular disc oscillating in a stagnant fluid. A value of ma = (8 ∕ 3π)ρAR has been suggested by Carpenter and Fridovich. [7]

Pit–Peters-type dynamic inflow model

Equation (2) represents the most basic form of a dynamic inflow equation. In this form, it is limited to a uniform time-varying induced velocity distribution over the entire rotor disc. By dividing the rotor into annular sections and applying the dynamic inflow equation independently to each section, one obtain a model that allows for radial variation of induced velocity—a much more realistic situation. Dynamic inflow models in the aerodynamic codes of GH Bladed and AeroDyn [11, 9] are, e.g., based on the assumption that the dynamic inflow equation can be applied on independent annular rings. Accordingly, a dynamic inflow equation for each independent annular ring section is

  • display math(3)

with the rotor divided into k = 1 … Nk annular rings, as shown in Figure 1.

image

Figure 1. A wind turbine rotor discretized into annular ring elements, indicating the notation used in this paper.

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Stig Øye dynamic inflow model

Another commonly used dynamic inflow model is the one proposed by S. Øye (see Hansen [12] and Snel and Schepers [1]), where dynamic effects are simulated by passing quasisteady values of induced velocity through two first-order filters

  • display math(4)

and

  • display math(5)

where

  • display math(6)

The quasisteady induced velocities (vqs), determined from the standard BEM method, are adjusted in series first to intermediate values (vint) and then to dynamically adjusted values (vi) to take the time delays associated with a change in the wake state into account. The time constants in this model have been calibrated using a discrete vortex ring model and ensure that the tip reacts faster to a change in loads.

Possible pitfalls of dynamic inflow modelling

Both the dynamic inflow models introduced are implemented in a BEM code. In case of a wind turbine rotor undergoing a periodic surge motion, the instantaneous axial velocity at the rotor disc becomes

  • display math(7)

where u is the (harmonic) velocity of structural motion, in this paper expressed as

  • display math(8)

The appropriate treatment of this additional velocity term is not immediately apparent. It is evident that the instantaneous mass flow through the rotor disc will be affected by this term; what is less evident is how this term will affect the global mass balance on which BEM theory is based. Integrated over an oscillation period, the mean mass flow through the rotor disc should be unaffected. As alluded to in the introduction, the time constants related to dynamic inflow effects are typically smaller than the natural surge period of a floating wind turbine, meaning the mass flow and steady assumptions of BEM theory could be adequate.

However, this is not the only source of uncertainty encountered in dynamic inflow modelling; further modifications are introduced to both models. These include Prandtl's tip loss correction to account for non-uniform loading on the rotor disc (or annular ring).

An empirical Glauert correction for heavily loaded discs (or annular rings) is implemented to improve the accuracy of the BEM method in the turbulent wake region when 1 ∕ 2 ≤ a ≤ 1. This introduces another adjustment factor

  • display math(9)

and the axial induction factor a needed in the BEM method is now calculated, using (7), as

  • display math(10)

This adjustment factor is used to modify the axial velocity/mass flow through the rotor disc

  • display math(11)

The addition of these engineering models, which are all influenced by and dependent on modelling assumptions, to the standard BEM formulation may potentially lead to a situation where modelling uncertainties become significant.

Furthermore, both engineering models were derived and calibrated on the basis of experiment where loads were rapidly changed through pitch regulation on the blades. The source of dynamic loading on a rotor oscillating in surge is thus somewhat different than what the dynamic wake models were originally intended for.

The true validity and applicability of all modelling assumptions involved can only be judged by comparing the models to the results from a more comprehensive and accurate simulation tool that includes more physics. One such model is a non-uniform axisymmetric actuator disc model based on the full incompressible Navier–Stokes (NS) equations. This is used as reference when validating the engineering type models for a wind turbine oscillating in surge.

ACTUATOR DISC MODEL

  1. Top of page
  2. ABSTRACT
  3. NOMENCLATURE
  4. INTRODUCTION
  5. INDUCED VELOCITY AND DYNAMIC INFLOW MODELLING
  6. ACTUATOR DISC MODEL
  7. VERIFICATION OF IMPLEMENTED MODELS
  8. DESCRIPTION OF TEST CASES
  9. COMPARISON AND EVALUATION
  10. CONCLUSIONS AND FURTHER WORK
  11. REFERENCES

The actuator disc model [13-15] provides a means of efficiently introducing the forces from a real rotor blade into a computational flow field. This is accomplished by distributing the forces over a permeable disc with an area proportional to the rotor swept area.

Basic principles

In this particular implementation, the flow field is resolved in the commercial software FLUENT using a finite volume discretization of the incompressible NS equations, according to which the momentum equation is expressed as

  • display math(12)

The velocity vector in cylindrical coordinates is given by inline image , and the body force vector inline image is used to model the effect of a rotor by introducing source terms in the momentum equation.

The body force vector represents a force per unit volume; the force from the rotor applied on an annular area of the differential size is therefore

  • display math(13)

where the load inline image is a vector of aerodynamic force per spanwise length on the rotor.

The aerodynamic forces acting on the rotor are determined from local velocities and by utilizing two-dimensional aerofoil characteristics. For every aerofoil element, the relative velocity, Vrel, and flow angle, ϕ, with respect to the rotor plane are determined. Note that in case the actuator disc is moving within the flow field at a given velocity, e.g. u, this velocity component is subtracted from the measured fluid velocity, vz, in a cell to find the relative velocity experienced by a blade element. The lift and drag forces per spanwise length are calculated once the local angle of attack, α, is found taking local pitch, γ, into account, and lift and drag coefficients, CL and CD , have been determined by table lookup.

  • display math(14)
  • display math(15)
  • display math(16)
  • display math(17)

Finally, a force vector is found by projecting the lift and drag forces along the axial and tangential directions.

  • display math(18)

A real rotor has a finite number of blades that will produce a system of tip vortices in the wake. This wake will not be the same as that behind a constantly loaded idealized actuator disc. To account for the finite number of blades, a Prandtl tip-correction factor, Fc, is implemented to correct the aerodynamic force components in (18), which then becomes

  • display math(19)

The actuator disc appears as a one-dimensional (1D) line in an axisymmetric plane. To prevent spatial oscillations due to discontinuous variations over the actuator disc, forces are smeared axially and symmetrically away from the line using a Gaussian smoothing kernel, [15] inline image, where

  • display math(20)

The axial position of the actuator disc at the beginning of a given time step is indicated by z0. In case the axial position of the actuator disc is moving within the flow field, the value of z0 will be updated at the end of a time step. The parameter ϵ controls the smearing and is set equal to ϵ = ϵiΔz with ϵi of the order 1 ≲ ϵi 2 so that the final distributed force as a spatial function is

  • display math(21)

Moving actuator disc

Modifications to the actuator disc model have to be made to simulate a wind turbine rotor oscillating in surge motion. The approach followed in this implementation is to displace the virtual position of the actuator disc within a stationary omputational mesh. This is to ensure that realistic interaction between the actuator disc and the wake structure is modelled. [15]

Starting at an initial position, located at the origin z = z0 = 0, the position of the actuator disc z0 is updated at the end of every time step, as z0(t) = xA sin(ωst). The updated position is then used to determine the column index of the grid cells in which the actuator disc is located. This is indicated schematically in Figure 2, where the actuator disc is represented by the red line and the column of grid cells in which it is located is shaded in grey. In the part of the computational grid where the moving rotor can be located, indicated by dashed lines in Figure 2, equidistant cell spacing is used in both the radial and axial directions. Outside of this region, grid cells are gradually stretched in either the axial or the radial direction. Fluid velocities at the actuator disc's location need to be sampled to evaluate equations (14)-(17), which determine the volume forces that will be applied to the flow field in the next time step. Quadratic interpolation is used to determine these velocities; the velocities in the current grid cells in addition to the axial neighbouring cells directly upstream and downstream are required for this calculation. Once the loading on the actuator disc itself has been updated, the Gaussian spreading function, (21), is used to determine the corresponding volume force that should be applied to each cell in the computational grid.

image

Figure 2. Notation around the moving actuator disc.

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The just described actuator disc model is the most physically complete and realistic model considered in this work. No additional assumptions regarding the wake geometry, definition of the axial induction factor in case of a moving rotor, its value when the rotor is highly loaded or unsteady modelling of induced velocity need to be made—all of which is necessary for the BEM models. These parameters are naturally included in the actuator disc model and result as part of the flow field solution when solving the NS equations numerically. Differences between the two engineering dynamic inflow approaches of an oscillating wind turbine rotor and the more advanced computational fluid dynamics implementation will be investigated in the next sections, where it will hopefully shed light on the validity of the engineering models for taking own motion into account.

VERIFICATION OF IMPLEMENTED MODELS

  1. Top of page
  2. ABSTRACT
  3. NOMENCLATURE
  4. INTRODUCTION
  5. INDUCED VELOCITY AND DYNAMIC INFLOW MODELLING
  6. ACTUATOR DISC MODEL
  7. VERIFICATION OF IMPLEMENTED MODELS
  8. DESCRIPTION OF TEST CASES
  9. COMPARISON AND EVALUATION
  10. CONCLUSIONS AND FURTHER WORK
  11. REFERENCES

For a low and uniform thrust loading (i.e. actual rotor geometry is neglected at this point) on the actuator disc, steady-state predictions of the numerical actuator disc model can be verified against simple momentum theory relations between thrust and induced velocity (axial induction factor). At CT = 0.4, the integrated axial induction on the actuator disc matches the axial momentum theory results extremely well, as is shown in Figure 3. The actuator disc and momentum theory results for CT = 0.8 and CT = 0.89 also agree closely. For higher thrust loadings CT = 1.0 and especially CT = 1.15, deviations from the momentum theory results are more pronounced, whereas it appears to be in good agreement with the empirical Glauert relation that is used in the BEM method.

image

Figure 3. CT(a) and CP(a) according to momentum theory and as determined using the actuator disc model, for CT = 0.4, 0.8, 0.89, 1.0 and 1.15.

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In all figures, tables and discussions, the following consistent naming convention will be used when referring to the different models considered:

acd

Moving actuator disc model

bem-qs

Quasi steady BEM model, no dynamic inflow model

bem-pp

BEM model including a Pit–Peters-type dynamic inflow model

bem-oye

BEM model including a Stig Øye dynamic inflow model

Steady-state loads

Although the actuator disc model performs well in case a uniform axial load is applied to it, its performance under loading more representative of a real rotor—including axial and tangential loads, as well as the effects of a finite number of blades—should also be verified. Figure 4 shows a comparison between the axial (Figure 4(a)) and tangential (Figure 4(b)) load distributions along a NREL 5 MW [16] rotor blade operating at rated conditions (V0 = 11.2 m s − 1, Ω = 1.2671 rad s − 1, θ = 0°), as calculated with the actuator disc model, the implemented BEM method and the industry standard analysis tool HAWC2 as reference.

image

Figure 4. Force per unit span (a) normal and (b) tangential to an NREL 5 MW rotor blade at rated operating conditions, as calculated with the actuator disc model, blade element momentum model, and analysis code HAWC2 as reference.

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For this steady-state reference case, there is excellent agreement of the spanwise loads (both normal and tangential to the rotor plane) predicted by all the analysis models. This gives confidence that the actuator disc model is also well capable of representing the loads on a real wind turbine rotor.

Transient loads

The effect of including dynamic inflow models, as opposed to an equilibrium wake assumption, in a BEM code can, e.g., be illustrated by simulating pitching transients. The particular case considered here is based on measurements taken on the Tjæreborg test turbine (see Snel and Schepers [1] for details), where after an initial period with θ0 = 0.07°, the blade pitch is rapidly increased to θ0 = 3.716°, maintained at this level for 30 n and then rapidly decreased to its original value. The pitch change takes place while the wind turbine is operating in a steady wind of V0 = 8.7 m s − 1 with rotational speed of Ω = 2.304 rad s − 1. This test case is also simulated with the numerical actuator disc model. Measured data for the blade flatwise moment at r = 4.21 m from the rotor axis are available for comparison. The simulated blade flatwise moments and disc-averaged induced velocities for all the models are plotted as non-dimensional values in Figure 5(a),(b), respectively. The quantities were non-dimensionalized according to

  • display math(22)

where q = actual time series value (Mflat or vave), qs1 = stationary value at initial conditions and qs2 = stationary value after step in θ0.

image

Figure 5. Non-dimensional (a) flapwise moment and (b) disc-averaged induced velocity predicted for the Tjæreborg wind turbine during a pitch step change from θ0 = 0.07° to θ0 = 3.716° and back, while operating with V0 = 8.7 m s − 1 and Ω = 2.304 rad s − 1.

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Figure 5(b) shows how the induced velocity gradually approaches new steady-state values associated with each pitch setting for all the dynamic models, whereas Figure 5(a) indicates an initial overshoot of the flatwise moment directly after a fast pitch change that gradually decays to steady-state values. The behaviour of the different models is of interest. The quasisteady BEM model immediately assumes the new steady-state values, as expected, whereas both the dynamic inflow models show similar overshoot and decay as the measurements. The Pit–Peters-type model, however, decays faster than the Øye model. The actuator disc model also predicts similar results, although somewhat slower decay than both dynamic inflow models, especially as the loading on the rotor is increasing.

DESCRIPTION OF TEST CASES

  1. Top of page
  2. ABSTRACT
  3. NOMENCLATURE
  4. INTRODUCTION
  5. INDUCED VELOCITY AND DYNAMIC INFLOW MODELLING
  6. ACTUATOR DISC MODEL
  7. VERIFICATION OF IMPLEMENTED MODELS
  8. DESCRIPTION OF TEST CASES
  9. COMPARISON AND EVALUATION
  10. CONCLUSIONS AND FURTHER WORK
  11. REFERENCES

The test cases that are to be used as basis for comparison between the different models will be introduced as briefly described in this section.

Parameter selection for surge motion

The surge motion experienced at the rotor of a floating offshore wind turbine is greatly dependent on substructure topology and the prevailing sea state. The focus of this investigation is not on the mechanisms that generate the surge motion but rather on when given a particular motion, how it should be taken into account in aerodynamic calculations. However, to ensure that a realistic range of values are considered, values for surge motion frequency and amplitude are obtained from a conceptual study [3] performed for an NREL 5 MW wind turbine mounted on substructures developed by NREL. The standard deviations of the surge displacement and the surge natural frequency for six different designs in the base case considered (200 m water depth, V0 = 11.2 m s − 1, Hs = 10 m and Tm = 13.6 s) are listed in Table 1. The surge motion amplitude values are approximated by assuming the surge oscillation at the natural frequency to be a narrow-banded process.

Table 1. Standard deviations and natural frequency of system surge motion. [3]
 ωs (rad s − 1)σs (m)inline image (m)
Shallow drafted barge0.77022.5983.674
Tension leg platform (TLP) surface0.1272.9124.118
TLP submerged0.1462.7983.957
TLP reserve buoyancy = 20.1854.1965.934
TLP reserve buoyancy = 60.24612.36717.490

The values in Table 1 are interpreted as good reference values and not absolute correct values. To investigate the effect of surge motion more thoroughly, simulations will be run with surge motion determined by a selection of frequencies between 0.127 and 1.0 rad s − 1, and the amplitude varied between 2 and 16 m at each frequency value.

Outline of simulation approach

All simulations start off with a stationary rotor located at the origin, operating in a steady wind, with fixed rotor speed and blade pitch settings as indicated in Table 2. For the first 80 s of the simulation, this situation is maintained. Thereafter, the surge oscillations are started, and depending on oscillation frequency, at least seven full cycles are completed before the end of the simulation run to ensure that all start-up transients have been eliminated. Figure 6 shows an example of the prescribed surge displacement. As a representative cycle, data from the last complete surge oscillation are extracted for comparison between all models (see the green line in Figure 6). Furthermore, comparisons at fixed time intervals during a representative cycle (namely 0T, 0.25T, 0.5T and 0.75T) are also made (indicated by red markers in Figure 6).

Table 2. Operating conditions for the NREL 5 MW wind turbine during all simulations.
ParameterValue
Wind speed, V011.2 m s − 1
Rotor speed, Ω1.2671 rad s − 1
Blade pitch, θ00.0°
image

Figure 6. Prescribed surge displacement of the rotor, for ωs = 0.127 rad s  − 1 and xA = 8 m. The last complete cycle is indicated in green, with red markers indicating specific time instants considered during an oscillation.

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All calculations with the moving actuator disc model are performed on a computational grid that extends 10R upstream and 20R downstream in the axial direction from the neutral position of the actuator disc, and 10R in the radial direction from the rotor axis. The axial inflow velocity is prescribed on the upstream boundary, and a pressure outlet is defined on the downstream boundary, with a symmetry boundary condition on the lateral boundary. Following observations in Mikkelsen, [15] the flow Reynolds number is set to Re = 5000, and the Gaussian smearing parameter to ϵi = 1.5. The equidistant part of the grid where the actuator disc is located has 41 × 40 grid cells in the axial and radial directions, respectively, giving a total cell count of 201 × 140 for the entire grid, whereas a time step of Δt = 0.1 s is used throughout.

COMPARISON AND EVALUATION

  1. Top of page
  2. ABSTRACT
  3. NOMENCLATURE
  4. INTRODUCTION
  5. INDUCED VELOCITY AND DYNAMIC INFLOW MODELLING
  6. ACTUATOR DISC MODEL
  7. VERIFICATION OF IMPLEMENTED MODELS
  8. DESCRIPTION OF TEST CASES
  9. COMPARISON AND EVALUATION
  10. CONCLUSIONS AND FURTHER WORK
  11. REFERENCES

In this section, the models introduced in the paper will be applied to a number of test problems, and comparisons will be made between the different results. The induced velocity, integrated rotor loads and the work done by aerodynamic forces will all be investigated to draw final conclusions.

Axial development of induced velocity

Axial momentum theory dictates that if the axial induced velocity at the rotor disc is vi, it will be zero far upstream of the rotor disc, and 2vi far downstream in the wake. Figure 7 shows the axial induced velocity, extracted at r = 0.7R, when the NREL 5 MW reference turbine is operating with the parameter settings indicated in Table 2, for which CT = 0.78.

image

Figure 7. Variation of the axial induced velocity for the NREL 5 MW reference turbine operating at rated speed.

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If the rotor disc experiences significant axial motion, Figure 7 appears to indicate that the rotor might interact with an induced velocity gradient during the axial motion, which could possibly be of significance. If the oscillation is slow enough for the resulting apparent wind effect to be, comparatively, very small, it is expected that the wake will be able to adjust to the change in loading brought on by the apparent wind, in which case axial variation of the induced velocity gradient will not be an appreciable influence.

Simulations are run for a number of oscillation frequencies (ωs) and amplitudes (xA) so that the axial variation of induced velocity can be compared between the different cases. In Figures 8-10, the induced velocity upstream and downstream of the rotor disc at various stages of an oscillation cycle are compared for ωs = 0.127 rad s − 1, ωs = 0.246 rad s − 1 and ωs = 0.50 rad s − 1, respectively. For each oscillation frequency, three different oscillation amplitudes are also considered, namely xA = 4.0 m, xA = 8.0 m and xA = 16.0 m.

image

Figure 8. Axial variation of induced velocity for a low surge oscillation frequency.

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image

Figure 9. Axial variation of induced velocity for an intermediate surge oscillation frequency.

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image

Figure 10. Axial variation of induced velocity for a high surge oscillation frequency.

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Integrated rotor loads and induced velocity

Both the actuator disc and the BEM models discretize the rotor disc into annular rings. To investigate the overall differences between the models, integrated values for the entire rotor will be considered. The total thrust force on the rotor disc is obtained by summing the contributions from each of the annular rings and is expressed in terms of a thrust coefficient, CT. The area-averaged induced velocity, vave, is used a representative value for the rotor disc.

When comparing these values over one oscillation period, some interesting observations can be made. Figure 11(a),(b) shows that for a low oscillation frequency, the difference between the two dynamic wake models and a quasisteady BEM model is very small. However, as the oscillation frequency is increased, larger differences between the respective models begin to emerge. The differences are observed most clearly in the calculated induced velocities shown in Figure 11(f),(h). For these two cases, the surge oscillation periods are of similar magnitude as the characteristic time scales associated with dynamic wake effects. Subsequently, the choice of dynamic inflow model has a strong influence on the predicted results. It was previously shown (see Figure 5(b)) how induced velocity approaches the quasisteady values after rapid changes in loading on the rotor. Similar behaviour is observed in the results shown here; only now the changes in rotor loading are not due to rapid pitch changes but because of the apparent wind caused by the rotor surge motion. The quasisteady BEM model responds instantaneously to the change in rotor loading. The Pit–Peters model, which was shown to have the fastest decay to steady state values, follows the quasisteady value the closest, although with some delay and subsequently also a lower amplitude. For the Stig Øye dynamic inflow model, which showed slower decay than the Pit–Peters model in the pitching transients, there is an even more marked difference compared with the quasisteady result. However, it compares favourably with the induced velocity predicted by the numerical actuator disc model.

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Figure 11. Thrust coefficient (left) and average axial induced velocity (right) predicted by various methods for xA = 8 m and the oscillation frequency increasing from ωs = 0.127 to ωs = 0.770 rad s  − 1.

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For oscillation frequencies ωs = 0.500 rad s − 1 and ωs = 0.770 rad s − 1, Figure 11(e),(g) shows that the rotor thrust coefficient exceeds unity during the surge motion. This indicates that the rotor is operating in the turbulent wake state, where reliance is made on the empirical Glauert correction to relate the applied force to an induced velocity. The difference on the thrust is, however, very small when comparing the different BEM implementations and the actuator disc model.

Vorticity distribution

Contours of wake vorticity give a visual representation of the wake behind the wind turbine rotor. In Figures 12-14, the instantaneous vorticity distribution at the start of an oscillation cycle (when the rotor is about to start moving into its own wake) is shown on the left, whereas a distribution 0.125T later is shown on the right. The figures represent surge oscillations at ωs = 0.127, 0.246 and 0.500 rad s − 1, respectively, whereas the displacement amplitude is fixed at xA = 16 m in each case. In Figure 12, for the lowest oscillation frequency, the root and tip vortices closely resemble those for a stationary rotor, with slight deviations in the wake only visible approximately three rotor radii downstream. For the intermediate oscillation frequency, there is a slight increase in wake expansion visible near the rotor from Figure 13(a),(b). The wake structure downstream of the rotor is also more visibly affected by the time varying load on the rotor due to the surge oscillations. At the highest oscillation frequency considered, the wake vortex structures in Figure 14(a),(b) differ significantly from those observed for the lower frequency surge oscillation frequencies (as was already seen in Figure 10(c)).

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Figure 12. Wake vorticity distribution for a rotor oscillating with ωs = 0.127 rad s  − 1, xA = 16.0 m at the beginning of an oscillation cycle (left) and 0.125T later (right).

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image

Figure 13. Wake vorticity distribution for a rotor oscillating with ωs = 0.246 rad s  − 1, xA = 16.0 m at the beginning of an oscillation cycle (left) and 0.125T later (right).

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image

Figure 14. Wake vorticity distribution for a rotor oscillating with ωs = 0.500 rad s  − 1, xA = 16.0 m at the beginning of an oscillation cycle (left) and 0.125T later (right).

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Aerodynamic work

For the purpose of global dynamic analysis of a floating wind turbine, the calculation of aerodynamic damping introduced by the rotor plays an important role in the overall system dynamics. Since the aerodynamic forces are directly coupled to induced velocity, which is in turn dependent on the wake modelling, it is necessary to understand the effect/influence of different dynamic wake models on calculated aerodynamic damping.

The aerodynamic work carried out by the axial component of the aerodynamic forces on the rotor blades during one surge oscillation cycle can be derived as [17]

  • display math(23)

where T is the instantaneous aerodynamic rotor thrust force, being integrated over one of the harmonic surge displacement cycles made by the rotor. Figure 6 showed an example of such a cycle.

The aerodynamic work for different displacement amplitudes over a range of oscillation frequencies was calculated with the moving actuator disc model and the three BEM models. To assess the relative aerodynamic work done by the rotor thrust force according to the actuator disc and BEM approaches, the aerodynamic work is normalized with the moving actuator disc result as reference.

Since the surge acceleration of the rotor is proportional to inline image, the frequency range for each of the displacement amplitude values is limited so that the highest acceleration experienced by the rotor is approximately the same.

The results are shown in Figures 15-17 for xA = 4 m, xA = 8 m and xA = 16 m, respectively. It is observed that the aerodynamic work done by the BEM models with dynamic inflow models is slightly larger than the case where the induced velocity is calculated in a quasisteady manner. Furthermore, this figure suggests that aerodynamic damping calculated with the BEM dynamic wake models is larger than that calculated with the actuator disc model by a very small margin. The accuracy of the moving actuator disc results is dependent on the grid discretization and time step size. As an example, repeating the case where ωs = 0.5 rad s − 1 and xA = 8 m on a computational grid with double as many grid cells in both the axial and radial directions (i.e. 401 × 280) and using a time step nearly three times smaller (Δt = 0.035 s), the aerodynamic work calculated is only 0.22% smaller than that for the current setup (thus, the differences with respect to the BEM models will be marginally larger than reported).

image

Figure 15. Relative aerodynamic work done by the rotor thrust force over an oscillation cycle according to the various methods, normalized with respect to the moving actuator disc result; in all cases, xA = 4 m.

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image

Figure 16. Relative aerodynamic work done by the rotor thrust force over an oscillation cycle according to the various methods, normalized with respect to the moving actuator disc result; in all cases, xA = 8 m.

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image

Figure 17. Relative aerodynamic work done by the rotor thrust force over an oscillation cycle according to the various methods, normalized with respect to the moving actuator disc result; in all cases, xA = 16 m.

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CONCLUSIONS AND FURTHER WORK

  1. Top of page
  2. ABSTRACT
  3. NOMENCLATURE
  4. INTRODUCTION
  5. INDUCED VELOCITY AND DYNAMIC INFLOW MODELLING
  6. ACTUATOR DISC MODEL
  7. VERIFICATION OF IMPLEMENTED MODELS
  8. DESCRIPTION OF TEST CASES
  9. COMPARISON AND EVALUATION
  10. CONCLUSIONS AND FURTHER WORK
  11. REFERENCES

In this work, two industry standard dynamic inflow models, originally derived and calibrated for rapidly changing loads through pitch regulation, were compared with an actuator disc model for simulation of a floating wind turbine rotor that is oscillating in surge. From the simulations results, it can be concluded that even though there are some differences in the local induced velocity for an oscillating rotor compared with a stationary one, and the resulting wake geometry does not exactly resemble the idealized momentum theory streamtube model, these differences are not significant enough for the BEM theory-based models to fail. Despite the additional dynamic degree of freedom, the BEM method with appropriate engineering models is still a reasonable tool to use for analysing floating wind turbines where large rotor surge motions are present.

The integrated rotor thrust loads as determined by all of the methods considered agreed quite closely. Comparisons of the average induced velocity over a rotor disc showed larger differences between the models. Compared with the results from the actuator disc model, the time constant of the Pit–Peters model appears to be slightly too low, whereas that of the Stig Øye model matches the actuator disc results more closely; however, there are still some differences.

With the current results, it appears that aerodynamic damping is marginally over-predicted by the BEM methods including dynamic wake models when compared with the actuator disc model, which is the more physically correct model and thus assumed to be the most accurate reference. The differences are smallest at low oscillation frequencies and increase slightly as the surge oscillation frequency increases but never exceeding 4% of the actuator disc value for all the test conditions considered.

In case surge oscillation frequencies are high enough so that the natural periods of oscillation are of similar magnitude as the time constants associated with dynamic inflow effects, calculated induced velocities are noticeably influenced by the choice of dynamic inflow model in a BEM code. However, the effects of these variations in induced velocity on integrated rotor loads, such as the rotor thrust, and also aerodynamic work associated with the surge oscillations, are limited. Therefore, it is concluded that current engineering models for wake dynamics seem to be sufficiently capable of dealing with the additional unsteady surge motion of a wind turbine rotor in a global force analysis.

Surge motion is only one of six possible degrees of freedom in which a floating wind turbine could possibly experience large motions—compared with a land-based system. It would be useful to also study the effect of other motions, such as yaw and pitch in particular, especially since this requires further engineering inflow distribution models to be included in the standard BEM modelling approach. This would, however, be a non-axisymmetric problem implying that the current implementation of the moving actuator disc model is no longer applicable.

REFERENCES

  1. Top of page
  2. ABSTRACT
  3. NOMENCLATURE
  4. INTRODUCTION
  5. INDUCED VELOCITY AND DYNAMIC INFLOW MODELLING
  6. ACTUATOR DISC MODEL
  7. VERIFICATION OF IMPLEMENTED MODELS
  8. DESCRIPTION OF TEST CASES
  9. COMPARISON AND EVALUATION
  10. CONCLUSIONS AND FURTHER WORK
  11. REFERENCES
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