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**Wind Energy Science**
The interactive open-access journal of the European Academy of Wind Energy

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- Abstract
- Introduction
- The vortex ring model for the wake
- Evaluating the influence coefficient for an infinite array of vortex rings
- Straight segment approximation of vortex rings and its accuracy
- Using a finite array of rings to determine the influence coefficient
- Conclusions
- Code availability
- Competing interests
- Acknowledgements
- References

**Research article**
07 Jun 2018

**Research article** | 07 Jun 2018

The second curvature correction for the straight segment approximation of periodic vortex wakes

- Department of Mechanical and Manufacturing Engineering, University of Calgary, Calgary T2N 1N4, AB, Canada

- Department of Mechanical and Manufacturing Engineering, University of Calgary, Calgary T2N 1N4, AB, Canada

**Correspondence**: David H. Wood (dhwood@ucalgary.ca)

**Correspondence**: David H. Wood (dhwood@ucalgary.ca)

Abstract

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The periodic, helical vortex wakes of wind turbines, propellers, and helicopters are often approximated using straight vortex segments which cannot reproduce the binormal velocity associated with the local curvature. This leads to the need for the first curvature correction, which is well known and understood. It is less well known that under some circumstances, the binormal velocity determined from straight segments needs a second correction when the periodicity returns the vortex to the proximity of the point at which the velocity is required. This paper analyzes the second correction by modelling the helical far wake of a wind turbine as an infinite row of equispaced vortex rings of constant radius and circulation. The ring spacing is proportional to the helix pitch. The second correction is required at small vortex pitch, which is typical of the operating conditions of large modern turbines. Then the velocity induced by the periodic wake can greatly exceed the local curvature contribution. The second correction is quadratic in the inverse of the number of segments per ring and linear in the inverse spacing. An approximate expression is developed for the second correction and shown to reduce the errors by an order of magnitude.

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Wood, D. H.: The second curvature correction for the straight segment approximation of periodic vortex wakes, Wind Energ. Sci., 3, 345–352, https://doi.org/10.5194/wes-3-345-2018, 2018.

1 Introduction

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It is common for computational models of the wakes of helicopters, propellers, and wind turbines to use straight vortex segments whose position is iterated until they follow the local flow and the vortex is force free. Solving the Biot–Savart integral gives the induced velocity used in the iteration. Figure 1 shows a representation of a vortex trailing from a two-blade rotor with the straight segment approximation. The labels and symbols on the figure will be defined below. O'Brien et al. (2017) reviewed a range of computational models for wind turbines and Sarmast et al. (2016) describe a recent application of a free-wake vortex model using straight vortex segments.

A well-known difficulty of the straight segment approximation is that it does
not reproduce the binormal velocity due to the curvature of the vortex line
(e.g. Bhagwat and Leishman, 2014,
Govindarajan and Leishman, 2016, and Kim et al., 2016). This leads to the need for
the first curvature correction. To assess the errors of the straight segment
approximation and develop a correction,
Bhagwat and Leishman (2014) used a vortex ring whose binormal (axial) velocity, *U*,
is given by the well-known Kelvin equation

$$\begin{array}{}\text{(1)}& U={\displaystyle \frac{\mathrm{\Gamma}}{\mathrm{4}\mathit{\pi}}}\text{log}\left({\displaystyle \frac{\mathrm{8}}{a}}\right)-{\displaystyle \frac{\mathrm{1}}{\mathrm{4}}},\end{array}$$

where Γ is the circulation of one vortex, and *a* is the radius of the
vortex core (e.g. Saffman, 1992). Figure 2 shows the vortex ring
approximation to the helical wake in Fig. 1. Note that by Eq. (1) *U* increases as *a*
decreases. In this equation, and throughout this paper, all lengths are
normalized by the vortex radius (not the core radius *a*)
and all velocities by the wind speed. To reproduce Eq. (1), the
numerical evaluation of the Biot–Savart integral for the ring is “cut off”
by ignoring the contribution from distances smaller than *a* from the point
at which the velocity is required, the “control point”
shown in Figs. 1 and 2. It is
emphasized that the cut-off is a heuristic; Kelvin's equation (Eq. 1) is
usually derived from impulse considerations or other methods that do not use the
Biot–Savart law. Saffman (1992) documented many factors that
alter the vortex velocity from its Biot–Savart value: these include flow
along the vortex axis, differing distributions of swirl, etc. Nevertheless,
the Biot–Savart prescription is useful and computationally convenient.

Curvature in the wakes of rotors is often associated with vortex periodicity, the “return” of a vortex to the proximity of the control point, which can cause a significant contribution to the binormal velocity. Wood and Li (2002) and Wood (2004) used helical line vortices to analyze straight segment errors for this second effect of curvature, but their work has apparently not been considered in subsequent vortex modelling. Govindarajan and Leishman (2016) claimed that the second curvature correction is unnecessary and difficult to implement. The purpose of this paper is to document the importance of the second correction for wind turbine wakes under some operating conditions and to develop an effective and simple correction.

The paper is organized as follows. The next section introduces the vortex ring model of the wake. In the following section, the induced velocity for the periodic component of the wake over a range of vortex spacings is found in terms of its Biot–Savart integral. Section 4 describes the calculation of the induced velocity for the straight segment approximation, determines the second curvature correction, and tests its accuracy. The final section contains the conclusions.

2 The vortex ring model for the wake

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For a point with the same radius as a single vortex ring and distance *z*
from it, the Biot–Savart equation for *U* in the direction of the wind – the
binormal direction – is

$$\begin{array}{}\text{(2)}& U={\displaystyle \frac{\mathrm{\Gamma}}{\mathrm{4}\mathit{\pi}}}\underset{\mathrm{0}}{\overset{\mathrm{2}\mathit{\pi}}{\int}}{\displaystyle \frac{\mathrm{1}-\mathrm{cos}\mathit{\theta}}{(\mathrm{2}-\mathrm{2}\mathrm{cos}\mathit{\theta}+{z}^{\mathrm{2}}{)}^{\mathrm{3}/\mathrm{2}}}}\mathrm{d}\mathit{\theta},\end{array}$$

where *θ* is the vortex angle in cylindrical polar co-ordinates. If
*z*=0, the integral clearly has a logarithmic singularity as *θ*→0. The velocity, *U*_{1c}, requiring the first curvature
correction is

$$\begin{array}{}\text{(3)}& {U}_{\mathrm{1}\mathrm{c}}=U(z=\mathrm{0}),\end{array}$$

arising from the only ring containing the control point. The integral in
Eq. (2) and similar equations will be termed the “influence
coefficient” *I*, which has the same relative error characteristics as *U*.

The test case used here to investigate the second correction models the
far wake of a wind turbine as an infinite row of equispaced vortex rings of
constant spacing, *s*, radius, and Γ, extending to infinity on either
side of the control point at *z*=0. A row of rings is easier to analyze than
the helical vortices used by Wood and Li (2002) and
Wood (2004) but displays the same need at small separation for the second
correction. In addition, the discrete nature of the vortex rings helps to
localize the correction that is developed in Sect. 4.

The ring vortex wake is consistent with the “Joukowsky” model of the wake,
used by Sarmast et al. (2016); either the bound vorticity of the
blades is constant along their span or all the shed vorticity has rolled up
into tip and hub vortices before reaching the far wake. This is clearly a
simplification of wind turbine wakes in general, but the linearity of the
Biot–Savart law allows more complex wakes to be considered as an assembly of
elements such as rings. The velocity associated with the second correction,
*U*_{2c}, is induced by the vortices that do not contain the control point:

$$\begin{array}{}\text{(4)}& {U}_{\mathrm{2}\mathrm{c}}={\displaystyle \frac{\mathrm{\Gamma}}{\mathrm{4}\mathit{\pi}}}{I}_{\mathrm{2}\mathrm{c}}={\displaystyle \frac{\mathrm{\Gamma}}{\mathrm{4}\mathit{\pi}}}\underset{\mathrm{0}}{\overset{\mathrm{2}\mathit{\pi}}{\int}}\mathrm{2}\sum _{j=\mathrm{1}}^{\mathrm{\infty}}{\displaystyle \frac{\mathrm{1}-\mathrm{cos}\mathit{\theta}}{(\mathrm{2}-\mathrm{2}\mathrm{cos}\mathit{\theta}+(js{)}^{\mathrm{2}}{)}^{\mathrm{3}/\mathrm{2}}}}\mathrm{d}\mathit{\theta}.\end{array}$$

Equation (4) is not singular as *θ*→0, which is,
possibly, the reason why the need for the second correction has not been
appreciated. *s* can be identified with the pitch *p* of a helical vortex
wake and the number of blades, *N*_{b}, according to $s=\mathrm{2}\mathit{\pi}p/{N}_{\mathrm{b}}$. The
relationship between *p* and *s* can be seen by comparing Figs. 1
and 2.

Testing corrections for the straight segment approximation requires an
accurate evaluation of the series in Eq. (4) and then an
integration in *θ*. This order is preferred because the integration in
*θ* of the summand results in incomplete elliptic integrals, which are
likely to be very difficult to sum. The innocuous looking series in
Eq. (4), however, does not appear to have a closed form sum. The standard
technique for summing infinite series of algebraic functions is via Laplace
transforms (e.g. Wheelon, 1954). This would be successful if the
exponent in the integrand was 1 instead of 3∕2, but for Eq. (4), the
method gave a principal value integral that could not be solved in closed
form. By the Cauchy integral test for series

$$\begin{array}{ll}{\displaystyle \frac{\mathrm{1}}{s}}& {\displaystyle}-{\displaystyle \frac{\mathrm{1}}{\sqrt{\mathrm{2}-\mathrm{2}\mathrm{cos}\mathit{\theta}+{s}^{\mathrm{2}}}}}\le {I}_{\mathrm{2}\mathrm{c}}\left(\mathit{\theta}\right)\le {\displaystyle \frac{\mathrm{1}}{s}}-{\displaystyle \frac{\mathrm{1}}{\sqrt{\mathrm{2}-\mathrm{2}\mathrm{cos}\mathit{\theta}+{s}^{\mathrm{2}}}}}\\ \text{(5)}& {\displaystyle}& {\displaystyle}+{\displaystyle \frac{\mathrm{2}(\mathrm{1}-\mathrm{cos}\mathit{\theta})}{(\mathrm{2}-\mathrm{2}\mathrm{cos}\mathit{\theta}+{s}^{\mathrm{2}}{)}^{\mathrm{3}/\mathrm{2}}}},\end{array}$$

where

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}{I}_{\mathrm{2}\mathrm{c}}=\underset{\mathrm{0}}{\overset{\mathrm{2}\mathit{\pi}}{\int}}{I}_{\mathrm{2}\mathrm{c}}\left(\mathit{\theta}\right)\mathrm{d}\mathit{\theta}\phantom{\rule{1em}{0ex}}\text{and}\\ \text{(6)}& {\displaystyle}& {\displaystyle}{I}_{\mathrm{2}\mathrm{c}}\left(\mathit{\theta}\right)=\sum _{j=\mathrm{1}}^{\mathrm{\infty}}{\displaystyle \frac{\mathrm{1}-\mathrm{cos}\mathit{\theta}}{(\mathrm{2}-\mathrm{2}\mathrm{cos}\mathit{\theta}+(js{)}^{\mathrm{2}}{)}^{\mathrm{3}/\mathrm{2}}}}.\end{array}$$

It is easy to show that the average velocity in the direction of the wind at
any radius *r*<1 within the Joukowsky wake is $\mathrm{1}-\mathrm{\Gamma}/s$ (e.g. Wood, 2011), and it is reasonable to assume that the total velocity of
the free-wake vortex rings is close to $\mathrm{1}-\mathrm{\Gamma}/\left(\mathrm{2}s\right)$ or the induced
velocity, $U\approx \mathrm{\Gamma}/\left(\mathrm{2}s\right)$. Further, the bounds in Eq. (5),
which both contain 1∕*s*, cause *U* to approach the average of
the wake and external velocities, provided the curvature singularity
does not contribute significantly to *U*. There is, unfortunately, only
limited experimental information on *a* and *U* for wind turbine wakes to
guide the assessment of the relative importance of the first and second
velocity fields and their corrections. Figure 3 shows the terms in
Eq. (5) for *s*=0.2. This typical value was obtained using the
following steps. For modern turbines, *N*_{b}=3, and *λ*≈7 for
most of the operating range. For optimal (Betz-Joukowsky) performance,
$U=\mathrm{1}/\mathrm{3}$, $p\approx \mathrm{2}/\left(\mathrm{3}\mathit{\lambda}\right)$, where *λ* is the tip speed ratio, and
${N}_{\mathrm{b}}\mathrm{\Gamma}\mathit{\lambda}/\mathit{\pi}=\mathrm{8}/\mathrm{9}$ (Wood, 2011). Thus Γ=0.133 and
*s*≈0.2. The sum in Eq. (4) is always zero when
*θ*=0, but, as *s* decreases, the bounds in Eq. (5) (and
hence the sum) tend to 2*π*∕*s* over an increasing range of *θ*.
Integrating over [0,2*π*] then leads to $U\approx {U}_{\mathrm{2}\mathrm{c}}\approx \mathrm{\Gamma}/\left(\mathrm{2}s\right)$, showing the potential importance of *U*_{2c}. The
integrand of *I*_{1c} for a small *θ* is also shown. Its integral and
*U*_{1c} depend on the cut-off, *a*; to match $U=\mathrm{2}/\mathrm{3}$ for the conditions in
Fig. 3, Eq. (1) requires $a\sim {\mathrm{10}}^{-\mathrm{23}}$, which does
not seem a reasonable value. Thus it is likely that *U*≈*U*_{2c} at the
small *p* and *s* typical of the operating conditions of modern wind
turbines.

The very limited experimental information on the velocity of the vortices in
wind turbine wakes are in general agreement with this argument. Xiao et al. (2011)
measured the wake of a two-bladed turbine in a wind tunnel at
*λ*=4.91 using particle image velocimetry. They determined the vortex
velocity in the near wake as 10.8 m s^{−1} when the wind speed was 12 m s^{−1}. Thus
$U=(\mathrm{12}-\mathrm{10.8})/\mathrm{12}=\mathrm{0.1}$, which is lower than the value of 1∕3 that follows
from assuming optimal power output. Assuming *U*=*U*_{2c} and using the general
equation $p=(\mathrm{1}-U)/\mathit{\lambda}$ gives *s*=0.576 or 360 mm for the rotor of
radius 625 mm, which agrees very well with the value read from their Fig. 10.
This again implies that *U*_{2c}≫*U*_{1c}. Since most rotor wakes are
helices of some form, it is important to note that the equivalent inverse
pitch term dominates *U* for a helical vortex of sufficiently small *p*
(Kuibin and Okulov, 1998). There is a further reason to expect $U\approx {U}_{\mathrm{2}\mathrm{c}}\gg {U}_{\mathrm{1}\mathrm{c}}$ for many turbine wakes: *U*_{1c}, but not *U*_{2c},
is associated with the impulse necessary to form a vortex ring. If that
impulse and *U*_{1c} are significant it is unlikely that the wake can be
force free.

3 Evaluating the influence coefficient for an infinite array of vortex rings

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A closed form sum for *U*_{2c} in Eq. (4) could not be obtained
so the influence coefficients for a range of *s* values were determined as follows.
The Hermite–Hadamard inequality for monotonically decreasing functions that
tend to zero at large argument can be used simply to give a tighter bound on
*I*_{2c}(*θ*). It is

$$\begin{array}{}\text{(7)}& {I}_{\mathrm{2}\mathrm{c}}\left(\mathit{\theta}\right)={\displaystyle \frac{\mathrm{1}}{s}}-{\displaystyle \frac{\mathrm{1}}{\sqrt{\mathrm{2}-\mathrm{2}\mathrm{cos}\mathit{\theta}+{s}^{\mathrm{2}}}}}+{\displaystyle \frac{\mathrm{1}-\mathrm{cos}\mathit{\theta}}{(\mathrm{2}-\mathrm{2}\mathrm{cos}\mathit{\theta}+{s}^{\mathrm{2}}{)}^{\mathrm{3}/\mathrm{2}}}}+\mathit{\delta}\left(\mathit{\theta}\right),\end{array}$$

where the difference, *δ*(*θ*), is always positive but must be
determined numerically. The integral of the other terms on the right side of
Eq. (7) can be found exactly:

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}\underset{\mathrm{0}}{\overset{\mathrm{2}\mathit{\pi}}{\int}}\left({\displaystyle \frac{\mathrm{1}}{s}}-{\displaystyle \frac{\mathrm{1}}{\sqrt{\mathrm{2}-\mathrm{2}\mathrm{cos}\mathit{\theta}+{s}^{\mathrm{2}}}}}+{\displaystyle \frac{\mathrm{1}-\mathrm{cos}\mathit{\theta}}{(\mathrm{2}-\mathrm{2}\mathrm{cos}\mathit{\theta}+{s}^{\mathrm{2}}{)}^{\mathrm{3}/\mathrm{2}}}}\right)\mathrm{d}\mathit{\theta}\\ \text{(8)}& {\displaystyle}& {\displaystyle}={\displaystyle \frac{\mathrm{2}\mathit{\pi}}{s}}-{\displaystyle \frac{\mathrm{2}sE(-\mathrm{4}/{s}^{\mathrm{2}})}{{s}^{\mathrm{2}}+\mathrm{4}}}-{\displaystyle \frac{\mathrm{2}K(-\mathrm{4}/{s}^{\mathrm{2}})}{s}},\end{array}$$

where *E*(.) and *K*(.) are the complete elliptic integrals in standard notation.
The difference, *δ*, the integral of *δ*(*θ*) over [0,2*π*],
was evaluated using 2000 increments of *θ* and number of rings,
*N*_{r}=50 000. This value was chosen using the result obtained from Mathematica, that

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}\underset{\mathrm{0}}{\overset{\mathrm{2}\mathit{\pi}}{\int}}\mathrm{2}\sum _{j={N}_{\mathrm{r}}}^{\mathrm{\infty}}{\displaystyle \frac{\mathrm{1}-\mathrm{cos}\mathit{\theta}}{(\mathrm{2}-\mathrm{2}\mathrm{cos}\mathit{\theta}+(js{)}^{\mathrm{2}}{)}^{\mathrm{3}/\mathrm{2}}}}\mathrm{d}\mathit{\theta}\\ \text{(9)}& {\displaystyle}& {\displaystyle}\to R\left({N}_{\mathrm{r}}\right)={\displaystyle \frac{\mathrm{4}\mathit{\pi}\left(\mathit{\zeta}\right(\mathrm{3})-{H}_{{N}_{\mathrm{r}}}^{\left(\mathrm{3}\right)})}{{s}^{\mathrm{3}}}}\end{array}$$

for large *N*_{r}, where *ζ*(.) is the zeta function, and *H* is
the harmonic number in standard notation. In later use of this result, *R*(*j*) will
be called the “remainder”. Using 2*π*∕*s* as an estimate for the integral in Eq. (4),
and using Mathematica to evaluate *H*_{3}(*N*_{r}), gave the relative
error in truncating the sum at *N*_{r}=50 000 as $\mathrm{2}/{s}^{\mathrm{2}}\times {\mathrm{10}}^{-\mathrm{10}}=\mathrm{2}\times {\mathrm{10}}^{-\mathrm{8}}$
for the smallest value of *s* considered here, *s*=0.1. To the number of decimal places
used in Table 1, truncation does not alter the integral of the terms in Eq. (7)
over [0,2*π*]. For every calculation up to Sect. 5, *N*_{r}=50 000.
The *θ*−integral of the sum in Eq. (7) was found using the MATLAB
quadrature routine “integral” with an absolute tolerance of 10^{−8}.
All integrands are symmetric about *θ*=*π* and so were obtained over [0,*π*].
Table 1 also shows the approximate *λ* for a Betz–Joukowsky optimal rotor.
*δ* is small: combining Eqs. (6) and (7) leads to

$$\begin{array}{}\text{(10)}& \mathrm{0}\le \mathit{\delta}\le {\displaystyle \frac{\mathrm{2}K(-\mathrm{4}/{s}^{\mathrm{2}})}{s}}-{\displaystyle \frac{\mathrm{2}sE(-\mathrm{4}/{s}^{\mathrm{2}})}{{s}^{\mathrm{2}}+\mathrm{4}}},\end{array}$$

which is satisfied by all *δ* values in Table 1. *δ* is less than 4 % of
*I*_{c} for the worst case of *s*=0.8; Eq. (8) is an increasingly
good approximation to the sum and the influence coefficient approaches
2*π*∕*s* as *s* decreases and *λ* increases.
The values of *I*_{2c} in Table 1 will be compared to the values from the straight segment approximation.

4 Straight segment approximation of vortex rings and its accuracy

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Each of the rings not containing the control point was approximated by an
even number, *N*_{s}, of straight segments. The *i*th segment of each ring
started at $\mathit{\theta}=\mathrm{2}\mathit{\pi}i/{N}_{\mathrm{s}}+{\mathit{\theta}}_{\mathrm{0}}$ when measured from the control
point and finished at $\mathrm{2}\mathit{\pi}(i+\mathrm{1})/{N}_{\mathrm{s}}+{\mathit{\theta}}_{\mathrm{0}}$. *θ*_{0} is the angular
displacement between the control point and the start of the first (*i*=0)
segment. A straightforward application of Eq. (10.115) of Katz and Plotkin (2001) gives *I*_{2c}(*i*,*j*), the contribution of the *i*th
segment on the two *j*th vortex rings to *U*_{2c}, as

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}{U}_{\mathrm{2}\mathrm{c}}(i,j)={\displaystyle \frac{\mathrm{\Gamma}}{\mathrm{4}\mathit{\pi}}}\sum _{j=\mathrm{1}}^{{N}_{\mathrm{r}}}\sum _{i=\mathrm{1}}^{{N}_{\mathrm{s}}}{I}_{\mathrm{2}\mathrm{c}}(i,j),\text{where}\\ \text{(11)}& {\displaystyle}& {\displaystyle}{I}_{\mathrm{2}\mathrm{c}}(i,j)={\displaystyle \frac{{A}_{i}{B}_{i,j}}{{C}_{i,j}}},\end{array}$$

and

$$\begin{array}{ll}\text{(12)}& {\displaystyle}{A}_{i}& {\displaystyle}=\mathrm{8}\mathrm{sin}({\displaystyle \frac{\mathit{\pi}i}{{N}_{\mathrm{s}}}}+{\displaystyle \frac{{\mathit{\theta}}_{\mathrm{0}}}{\mathrm{2}}})\mathrm{sin}({\displaystyle \frac{\mathit{\pi}(i+\mathrm{1})}{{N}_{\mathrm{s}}}}+{\displaystyle \frac{{\mathit{\theta}}_{\mathrm{0}}}{\mathrm{2}}}),{\displaystyle}{B}_{i,j}& {\displaystyle}=\mathrm{sin}{\displaystyle \frac{\mathit{\pi}}{{N}_{\mathrm{s}}}}({\displaystyle \frac{\mathrm{1}}{{b}_{i+\mathrm{1},j}}}+{\displaystyle \frac{\mathrm{1}}{{b}_{i,j}}})\\ \text{(13)}& {\displaystyle}& {\displaystyle}+\mathrm{sin}({\displaystyle \frac{(\mathrm{2}i+\mathrm{1})\mathit{\pi}}{{N}_{\mathrm{s}}}}+{\mathit{\theta}}_{\mathrm{0}})({\displaystyle \frac{\mathrm{1}}{{b}_{i+\mathrm{1},j}}}-{\displaystyle \frac{\mathrm{1}}{{b}_{i,j}}}),{\displaystyle}{C}_{i,j}& {\displaystyle}=\mathrm{cos}{\displaystyle \frac{\mathrm{2}\mathit{\pi}}{{N}_{\mathrm{s}}}}+{b}_{i,j}^{\mathrm{2}}+{b}_{i+\mathrm{1},j}^{\mathrm{2}}\\ \text{(14)}& {\displaystyle}& {\displaystyle}+\mathrm{cos}({\displaystyle \frac{\mathrm{2}\mathit{\pi}(\mathrm{2}i+\mathrm{1})}{{N}_{\mathrm{s}}}}+\mathrm{2}{\mathit{\theta}}_{\mathrm{0}})-\mathrm{2},\end{array}$$

and

$$\begin{array}{}\text{(15)}& {b}_{i,j}=\sqrt{\mathrm{2}-\mathrm{2}\mathrm{cos}({\displaystyle \frac{\mathrm{2}\mathit{\pi}i}{{N}_{\mathrm{s}}}}+{\mathit{\theta}}_{\mathrm{0}})+(js{)}^{\mathrm{2}}}.\end{array}$$

For the first calculations, the junction of the first and *N*_{s}th segment was
aligned with the control point so *θ*_{0}=0. The influence coefficients
calculated from Eqs. (11)–(14) are compared to the results
from Table 1 in Fig. 4 in terms of the relative error using the
integral over [0,*π*] as the denominator since the integrand must be
symmetric about *θ*=*π*. The error is defined as the exact integral
minus the straight segment approximation so the latter is always an
underestimate. Two separate ranges of *θ* are considered: for $\mathrm{0}\le \left|\mathit{\theta}\right|\le \mathit{\pi}/\mathrm{2}$ the error is 1 to 2 orders of magnitude
higher than for $\mathit{\pi}/\mathrm{2}\le \left|\mathit{\theta}\right|\le \mathit{\pi}$ and the errors in the
first range increase with decreasing *s* whereas in the second range they
decrease. Figure 5 shows the reason. For this test case, no aligned
segment contributes to *U*_{2c} as the Biot–Savart velocity must lie in the
plane containing the segment and the control point. Otherwise, both errors
scale as $\mathrm{1}/{N}_{\mathrm{s}}^{\mathrm{2}}$, as was found in the helix simulations of
Wood (2004). As *s* decreases, however, the aligned segment error
increases proportionally to 1∕*s*, but the error for the remaining range of
*θ* decreases at constant *N*_{s}.

Figure 5 shows the angular contribution to the influence
coefficients for *s*=0.2 and *N*_{s}=40. The value of *θ* used for
plotting is the midpoint of each segment. The solid line shows the exact
integral from *θ*_{i} to *θ*_{i+1}. As was found by
Wood and Li (2002) and Wood (2004), the errors are localized near
*θ*=0. A correction for the error for the aligned segments can be
developed from the small-*θ* expansion of the series in
Eq. (4):

$$\begin{array}{ll}{\displaystyle}{I}_{\mathrm{2}\mathrm{c}}\left(\mathit{\theta}\right)& {\displaystyle}=\mathrm{2}\sum _{j=\mathrm{1}}^{\mathrm{\infty}}{I}_{\mathrm{2}\mathrm{c}}(\mathit{\theta},j)=\sum _{j=\mathrm{1}}^{\mathrm{\infty}}{\displaystyle \frac{\mathrm{2}-\mathrm{2}\mathrm{cos}\mathit{\theta}}{(\mathrm{2}-\mathrm{2}\mathrm{cos}\mathit{\theta}+(js{)}^{\mathrm{2}}{)}^{\mathrm{3}/\mathrm{2}}}}\\ \text{(16)}& {\displaystyle}& {\displaystyle}\to {\displaystyle \frac{\mathit{\zeta}\left(\mathrm{3}\right){\mathit{\theta}}^{\mathrm{2}}}{{s}^{\mathrm{3}}}}\text{as}\mathit{\theta}/s\to \mathrm{0}\end{array}$$

and *ζ*(3)=1.2026. Equation (16) has two important implications.
First, the best possible error for periodic straight segments
scales as (*N*_{s}*s*)^{−3} but it is likely that an unrealistically high value
of *N*_{s} would be required to achieve this. Second, 1∕*ζ*(3) or over
80 *%* of the correction to *U*_{2c} is due to the two rings (*j*=1) on
either side of the control point. This is the justification for pointing out
the aligned segments in Figs. 1 and 2. A general form of the correction,
therefore, can be based on the returned vortex on either side of the control
point. Since the distance from the control point to the vortex segments must
be calculated in a free-wake simulation, it should not be difficult to
determine the proximity in terms of *θ*∕*z* and apply a correction. A more
general correction is Δ(*θ*_{s}), where ${\mathit{\theta}}_{\mathrm{s}}=\mathrm{2}\mathit{\pi}/{N}_{\mathrm{s}}$ is
obtained by integrating in *θ* only for *j*=1 and then using *ζ*(3)
to correct approximately for the remaining rings. The result
is

$$\begin{array}{ll}{\displaystyle}\mathrm{\Delta}\left({\mathit{\theta}}_{\mathrm{s}}\right)& {\displaystyle}\approx \mathrm{2}\mathit{\zeta}\left(\mathrm{3}\right)[{\displaystyle \frac{F\left({\mathit{\theta}}_{\mathrm{s}}/\mathrm{2},-\mathrm{4}/{s}^{\mathrm{2}}\right)}{s}}-{\displaystyle \frac{sE\left({\mathit{\theta}}_{\mathrm{s}}/\mathrm{2},-\mathrm{4}/{s}^{\mathrm{2}}\right)}{{s}^{\mathrm{2}}+\mathrm{4}}}\\ \text{(17)}& {\displaystyle}& {\displaystyle}-{\displaystyle \frac{\mathrm{2}\mathrm{sin}{\mathit{\theta}}_{\mathrm{s}}}{\left({s}^{\mathrm{2}}+\mathrm{4}\right)\sqrt{\mathrm{2}-\mathrm{2}\mathrm{cos}{\mathit{\theta}}_{\mathrm{s}}+{s}^{\mathrm{2}}}}}],\end{array}$$

where *E*(.) and *F*(.) are the incomplete elliptic integrals. Equation (17)
is shown in Fig. 3 to give a better estimate for the aligned
segments. An alternative, simpler correction than Eq. (17) can be found
by using $\mathrm{2}-\mathrm{2}\mathrm{cos}\mathit{\theta}\sim {\mathit{\theta}}^{\mathrm{2}}$ for small *θ* values to give

$$\begin{array}{}\text{(18)}& \mathrm{\Delta}\left({\mathit{\theta}}_{\mathrm{s}}\right)\approx \mathrm{2}\mathit{\zeta}\left(\mathrm{3}\right)\left[\mathrm{log}\left({\displaystyle \frac{{\mathit{\theta}}_{\mathrm{s}}+\sqrt{{\mathit{\theta}}_{\mathrm{s}}^{\mathrm{2}}+{s}^{\mathrm{2}}}}{s}}\right)-{\displaystyle \frac{{\mathit{\theta}}_{\mathrm{s}}}{\sqrt{{\mathit{\theta}}_{\mathrm{s}}^{\mathrm{2}}+{s}^{\mathrm{2}}}}}\right],\end{array}$$

which gives almost the same correction for the aligned segments; Fig. 5.
These results for the application of Eqs. (17) and
(18) to the aligned segments are similar at the other values of *s*
as well, but are not shown in the interests of brevity. It is noted that the
correction developed here is simple in the sense that the vortex curvature is
known a priori. As pointed out by
Govindarajan and Leishman (2016), however, and shown by the analysis of
Kim et al. (2016), the modelling of three-dimensional wakes of varying geometry
can be considerably more complex.

One of these complexities is that the control point may not align with the
junction of segments on (in this case) adjacent rings. The effect of this can
be investigated by using non-zero *θ*_{0} in Eqs. (12)–(15). The results are shown in Fig. 6 for $\mathrm{20}\le {N}_{\mathrm{s}}\le \mathrm{160}$ and $\mathrm{0}\le {\mathit{\theta}}_{\mathrm{0}}\le \mathit{\pi}/{N}_{\mathrm{s}}$ and *s*=0.1. As was found
for other values of *s*, there is remarkably little variation in the error
with *θ*_{0} except for the lowest *N*_{s}, suggesting that the correction
derived above for the aligned case (*θ*_{0}=0) is also applicable to other
values. This is not an immediately obvious result from Eqs. (12)–(15). For ${\mathit{\theta}}_{\mathrm{0}}=\mathit{\pi}/{N}_{\mathrm{s}}$ and $\mathit{\theta}=\mathrm{2}\mathit{\pi}/{N}_{\mathrm{s}}$:

$$\begin{array}{ll}{\displaystyle}{I}_{\mathrm{2}\mathrm{c}}(\mathit{\theta},\mathrm{1})& {\displaystyle}={\displaystyle \frac{\mathrm{8}{\mathrm{sin}}^{\mathrm{2}}(\mathit{\theta}/\mathrm{2})\mathrm{sin}\mathit{\theta}}{\phantom{\rule{0.33em}{0ex}}\left(\mathrm{3}-\mathrm{4}\mathrm{cos}\mathit{\theta}+\mathrm{cos}\mathrm{2}\mathit{\theta}+\mathrm{2}{s}^{\mathrm{2}}\right)\sqrt{\mathrm{2}-\mathrm{2}\mathrm{cos}\mathit{\theta}+{s}^{\mathrm{2}}}}}\\ \text{(19)}& {\displaystyle}& {\displaystyle}\to ({\displaystyle \frac{\mathit{\theta}}{s}}{)}^{\mathrm{3}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\text{as}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\mathit{\theta}/s\to \mathrm{0},\end{array}$$

which suggests a difference from the case when *θ*_{0}=0.

5 Using a finite array of rings to determine the influence coefficient

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The second curvature error was shown in the last section to be caused largely
by aligned segments on the rings either side of the control point. For
increasing *θ*, *I*_{2c}(*θ*) becomes dominated by rings at a larger
distance from the control point. This is shown in Fig. 7, which
implies that *N*_{r} either must be large to ensure an accurate determination
of *I*_{2c} or a suitable remainder term be used. This allows an
approximate determination of the influence coefficient for the case in which
*θ*_{0}=0 according to

$$\begin{array}{}\text{(20)}& {I}_{\mathrm{2}\mathrm{c}}\approx \mathrm{2}\mathrm{\Delta}\left({\mathit{\theta}}_{\mathrm{s}}\right)+\sum _{j=\mathrm{1}}^{{N}_{\mathrm{r}}}\sum _{i=\mathrm{1}}^{{N}_{\mathrm{s}}}{I}_{\mathrm{2}\mathrm{c}}(i,j)+R\left({N}_{\mathrm{r}}\right),\end{array}$$

where one possibility for the remainder *R*(*N*_{r}) is given by Eq. (9).

The terms in Eq. (20) are listed in Table 2 for *s*=0.2. A
significant number of vortex rings, *N*_{r}, or equivalently a large
stream-wise distance is needed to make the remainder, *R*(*N*_{r}), accurate.
Typically, *N*_{r}≥20 for this *s*, and then *R*(*N*_{r}) is comparable to
Δ(*θ*_{s}). For *N*_{s}=20, for example, after applying an accurate
remainder, the second curvature correction changes the relative error from
3.4 % to less than 0.2 %, which is a reduction by 2 orders of magnitude.

6 Conclusions

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The widely used straight segment approximation for approximating the curved and periodic vortex wakes of wind turbines, propellers, and helicopters can have two errors associated with the wake curvature. The first is the well-known error in reproducing the locally induced binormal velocity. This is usually accommodated by a cut-off in the Biot–Savart determination of the vortex velocity using Eqs. (2) and (3) at a distance comparable with the radius of the vortex core. The second, less well-known error is the subject of this paper. It arises from the alignment of the segments of the periodic vortex returning to the proximity of the point at which the velocity is being determined.

By modelling the far wake of a wind turbine as an infinite row of equispaced vortex rings, two important results were obtained. First, it was shown that the velocity associated with the second error dominates at the small spacings typical of modern wind turbine operation. The available experimental evidence on wake structure is consistent with this finding. Then it is shown that the second error is quadratic in the number of segments per revolution and inversely proportional to the spacing of the rings, which is proportional to the pitch of a more realistic, but more difficult, helical wake. The model to investigate the second correction is artificial in that a single, infinite row of vortex rings of constant spacing, radius, and circulation is not applicable to the near wake. Nevertheless the model demonstrated the general importance of the rings adjacent to the control point at which the velocity is being calculated. These adjacent rings contribute over 80 % of the correction that is needed because the straight segment approximation does not correctly determine the contribution to the induced velocity from the closest parts of the adjacent rings, called the aligned segments.

It was also shown that the best behaviour possible for the second error is cubic in the product of the number of segments per revolution and the vortex spacing. It is likely, however, that larger numbers of vortex segments would be needed to achieve this error than are used in practice. This result was generalized to develop a second correction that improves the computed induced velocity by nearly 1 order of magnitude.

Code availability

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Code availability.

The MATLAB codes used in this study are available from the author.

Competing interests

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Competing interests.

The author declares that he has no conflict of interest.

Acknowledgements

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Acknowledgements.

This work is part of a research project on wind turbine aerodynamics funded by the NSERC Discovery Grants Program.

Edited by: Alessandro Bianchini

Reviewed by: Joseph Saverin and Wang Xiaodong

References

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Short summary

The vortices in the wakes of wind turbines are often approximated by short, straight vortex segments, which cannot reproduce the curvature singularity in the induced velocity. They can also have a second error due to the periodicity: the vortices return to close proximity of the point at which the velocity is calculated. The second error is assessed by representing the far wake of a turbine as a row of vortex rings. The error is quantified and a simple correction is developed.

The vortices in the wakes of wind turbines are often approximated by short, straight vortex...

Wind Energy Science

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