Cylindrical vortex wake model: right cylinder

Branlard, Emmanuel; Gaunaa, Mac

doi:10.1002/we.1800

Cited by 50 publications

(62 citation statements)

References 16 publications

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“…The newly shed wake is generated at the edge of the disc and convected downstream by the local velocity. By applying the equation of velocity induced by a finite vortex cylinder derived in Branlard and Gaunaa, the axial velocity at a position ( z p , r p ) induced by the wake vorticity system can be calculated by the following integral:

normalΦ false(z_{p}, r_{p}, t false) = true \int_{0}^{\infty} \frac{- γ_{t} false(z, t false)}{4 π \sqrt{r_{p} R false(z, t false)}} {([], false(z - z_{p} false) m ((), K false(m^{2} false) + \frac{R false(z, t false) - r_{p}}{R false(z, t false) + r_{p}} normalΠ false(m_{0}^{2}, m^{2} false)))}_{z}^{z + d z},

where < z p , r p > are the coordinates of point P where the velocity is to be calculated, γ t ( z , t ) is the tangential vorticity of the surface at location z at time t . K ( m 2 ), and

normalΠ false(m_{0}^{2}, m^{2} false)

are the complete elliptic integrals of the first and third kind.…”

Section: Methodsmentioning

confidence: 99%

New dynamic‐inflow engineering models based on linear and nonlinear actuator disc vortex models

Tavernier

Ferreira

et al. 2019

Wind Energy

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Two new engineering models are presented for the aerodynamic induction of a wind turbine under dynamic thrust. The models are developed using the differential form of Duhamel integrals of indicial responses of actuator disc type vortex models. The time constants of the indicial functions are obtained by the indicial responses of a linear and a nonlinear actuator disc model. The new dynamic‐inflow engineering models are verified against the results of a Computational Fluid Dynamics (CFD) model and compared against the dynamic‐inflow engineering models of Pitt‐Peters, Øye, and Energy Research Center of the Netherlands (ECN), for several load cases. Comparisons of all models show that two time constants are necessary to predict the dynamic induction. The amplitude and phase delay of the velocity distribution shows a strong radial dependency. Verifying the models against results from the CFD model shows that the model based on the linear actuator disc vortex model predicts a similar performance as the Øye model. The model based on the nonlinear actuator disc vortex model predicts the dynamic induction better than the other models concerning both phase delay and amplitude, especially at high load.

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normalΦ false(z_{p}, r_{p}, t false) = true \int_{0}^{\infty} \frac{- γ_{t} false(z, t false)}{4 π \sqrt{r_{p} R false(z, t false)}} {([], false(z - z_{p} false) m ((), K false(m^{2} false) + \frac{R false(z, t false) - r_{p}}{R false(z, t false) + r_{p}} normalΠ false(m_{0}^{2}, m^{2} false)))}_{z}^{z + d z},

where < z p , r p > are the coordinates of point P where the velocity is to be calculated, γ t ( z , t ) is the tangential vorticity of the surface at location z at time t . K ( m 2 ), and

normalΠ false(m_{0}^{2}, m^{2} false)

are the complete elliptic integrals of the first and third kind.…”

Section: Methodsmentioning

confidence: 99%

New dynamic‐inflow engineering models based on linear and nonlinear actuator disc vortex models

Tavernier

Ferreira

et al. 2019

Wind Energy

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“…Some examples are Okulov and Sørensen and Branlard et al . . The effect of the number of blades is studied by Okulov and Sørensen using vortex theory, and by van Kuik describing the limit transitions from rotor to disc.…”

Section: Introductionmentioning

confidence: 99%

“…Some examples are Okulov and Sørensen 7,8 and Branlard et al. 9,10 The effect of the number of blades is studied by Okulov and Sørensen 8 using vortex theory, and by van Kuik 11 describing the limit transitions from rotor to disc. Textbooks like Schaffarczyk 12 and reviews like Wald 13 and Sørensen 14 present an overview of rotor aerodynamics and the role of the actuator disc herein.…”

Section: Introductionmentioning

confidence: 99%

Potential flow solutions for energy extracting actuator disc flows

Kuik

Lignarolo

2015

Wind Energy

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The actuator disc is the oldest representation of a rotor, screw or propeller. Performance prediction is possible by applying momentum theory, giving integrated values for power and velocity. Computational fluid dynamics has provided much more flow details, but a full potential flow solution zooming in on these flow details was still absent. With the wake boundary discretized by vortex rings, flow states for energy extracting discs have been obtained for thrust coefficients up to 0.998. Boundary conditions are met with an accuracy of a few . Results from momentum theory are confirmed.Most rotor design codes use momentum theory in annulus or differential form, assuming that the axial velocity v x at the disc is uniform. However, the absolute velocity jvj is found to be uniform, and arguments for this are presented. The non-uniformity of v x is an inherent part of the flow solution caused by, in terms of momentum theory, the pressure acting at the annuli. This makes the annuli not independent from each other as assumed in current design codes. Although this was already known, it is now confirmed up to the highest thrust coefficients.Optimizing a rotor design should be carried out for the non-uniform distribution of v x . To enable this, an equation for the non-uniformity as function of thrust and radial position is presented, being a surface-fit to the calculated data. Qualitatively, the non-uniform distribution does the same as the Prandtl-Glauert-Shen tip correction applied to a uniform distribution.

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“…The vortex cylinder model applied to the actuator disk concept yields a simple expression characterising the induction (Branlard and Gaunaa, 2015;Medici et al, 2011) that can be integrated into a WFR model. This simple induction model is one-dimensional.…”

Section: Combined Wind-induction Modelmentioning

confidence: 99%

Wind field reconstruction from nacelle-mounted lidar short-range measurements

et al. 2017

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Abstract. Profiling nacelle lidars probe the wind at several heights and several distances upstream of the rotor. The development of such lidar systems is relatively recent, and it is still unclear how to condense the lidar raw measurements into useful wind field characteristics such as speed, direction, vertical and longitudinal gradients (wind shear). In this paper, we demonstrate an innovative method to estimate wind field characteristics using nacelle lidar measurements taken within the induction zone. Model-fitting wind field reconstruction techniques are applied to nacelle lidar measurements taken at multiple distances close to the rotor, where a wind model is combined with a simple induction model. The method allows robust determination of free-stream wind characteristics. The method was applied to experimental data obtained with two different types of nacelle lidar (five-beam Demonstrator and ZephIR Dual Mode). The reconstructed wind speed was within 0.5 % of the wind speed measured with a mast-top-mounted cup anemometer at 2.5 rotor diameters upstream of the turbine. The technique described in this paper overcomes measurement range limitations of the currently available nacelle lidar technology.

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Cylindrical vortex wake model: right cylinder

Cited by 50 publications

References 16 publications

New dynamic‐inflow engineering models based on linear and nonlinear actuator disc vortex models

New dynamic‐inflow engineering models based on linear and nonlinear actuator disc vortex models

Potential flow solutions for energy extracting actuator disc flows

Wind field reconstruction from nacelle-mounted lidar short-range measurements

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