Analytical models for the asymmetric wake of vertical

axis wind turbines

Pablo Ouroa,b,∗, Maxime Lazennecc
arXiv:2010.00301v1 [physics.flu-dyn] 1 Oct 2020

a
School of Mechanical, Aerospace and Civil Engineering, University of Manchester,
Manchester, M13 9PL, UK
b
Hydro-environmental Research Centre, School of Engineering, Cardiff University,
Cardiff, CF24 3AA, UK
c
École Polytechnique, 91120 Palaiseau, Paris, France

Abstract
Arrays of Vertical Axis Wind Turbines (VAWTs) can achieve larger power
generation per land area than horizontal axis turbines farms, due to the pos-
itive synergy between VATs in close proximity. Theoretical wake models
enable the reliable design of the array layout that maximises the energy out-
put, which need to depict the driving wake dynamics. VAWTs generate a
highly complex wake that evolves according to two governing length-scales,
namely the turbine rotor’s diameter and height which define a rectangular
shape of the wake cross-section, and feature distinct wake expansion rates.
This paper presents analytical VAWT wake models that account for an asym-
metric distribution of such wake expansion adopting a top-hat and Gaussian
velocity deficit distribution. Our proposed analytical Gaussian model leads
to an enhanced initial wake expansion prediction with the wake width (ε)
behind the rotor equal to (β/4π)1/2 with β being the ratio of initial wake
area to the VAWT’s frontal area, which addresses the limitations of previous
models that under-predicted the wake onset area. Velocity deficit predic-
tions are calculated in a series of numerical benchmarks consisting of a single
and an array of four in-line vertical axis wind turbines. In comparisons with
field data and large-eddy simulations, our models provide a good accuracy
to represent the mean wake distribution, maximum velocity deficit, and mo-

∗
Corresponding author: Dr Pablo Ouro (pablo.ouro@manchester.ac.uk)
Email addresses: pablo.ouro@manchester.ac.uk (Pablo Ouro),
maxime.lazennec@polytechnique.edu (Maxime Lazennec)

Preprint submitted to Elsevier October 2, 2020

mentum thickness, with the Gaussian model attaining the best predictions.
These models will aid to drive the design of VAT arrays and accelerate this
technology.
Keywords: Vertical Axis Wind Turbines, Wakes, Self-similarity,
Large-Eddy Simulation, Wind farm, VAWT

1. Introduction
In the global landscape of wind energy generation, all large-scale wind
farm projects comprise Horizontal Axis Wind Turbines (HAWTs) as a well-
established technology for both onshore and offshore environments, becom-
ing one of the most cost-effective resources to harness renewable energy [1].
Conversely, Vertical Axis Wind Turbines (VAWTs) are being developed at a
much slower pace with the remaining main challenge to prove their financial
viability, conditioned by the need for enhancing their power generation capa-
bilities. VAWTs offer a series of advantages over their HAWT counterparts
that can lead to innovative wind and hydro-kinetic energy projects, unfeasi-
ble if HAWTs were the chosen technology. For instance, they can effectively
harness kinetic energy from relatively low-to-medium flow velocity ranges as
those found in urban areas, rivers or tides, or be adopted in environmentally
sensitive regions as their slower rotational speeds can reduce fish collision
risk or acoustic contamination.
VAWT rotor blades rotate in a plane perpendicular to the approaching
flow direction, generating vortex structures over the upwind rotation half and
interacting with these over the downwind one. Despite such fluid-structure
complexity of VAT wakes, the research in characterising VAWT wakes is still
insufficient with limited knowledge about the governing flow mechanisms in
the far-wake dynamics. This led to extrapolate features of HAT wakes into
the dynamics of VAWT wakes which are fundamentally unequal, e.g. tip or
dynamic-stall vortices patterns [2].
Most experimental tests of VATs look at how to improve their perfor-
mance, leading to a limited number of studies analysing the near-wake dy-
namics and an almost absence of extensive tests that investigate the far-
wake. Few examples of the latter at small-scale laboratory scale are: Rolin
and Porté-Agel [3] measured with Particle Image Velocimetry (PIV) up to
10 diameters (D0 ) downstream of a VAT analysing the mean and turbulent
kinetic energy equations; Araya et al. [4] investigated the near- to far-wake

2
Figure 1: Wake evolution behind a VAWT of diameter D0 and height H0 over the (a)
horizontal (xy) plane at a mid-height elevation zh from the ground surface, and (b) vertical
(xz) plane through the rotor’s centre, with the thick solid line denoting the control volumes
of interest.

transition for turbine rotors with different number of blades and tip-speed
ratios using PIV to measure up to 11D0 downstream; and Ouro et al. [5]
measured the wake up to 14D0 downstream with acoustic Doppler velocime-
ter showing that remnants of the turbine-induced wake are still observed at
such far distances downstream.
The complex fluid dynamics in VAT wakes depend on the operational
regime accounted for by the Tip-Speed Ratio (TSR) which relates the blades’
angular speed to the free-stream velocity, and the turbine’s rotor geometric
solidity, which indicates the proportion of the swept perimeter occupied by
the blades [6, 7]. VAWTs designed with low solidity rotors operate at high
TSR which leads the blades to undergo light dynamic stall, i.e. flow separa-
tion occurs for effective angles of attacks larger than the static stall angle but
there is no full detachment of the generated leading-edge vortices preventing
a sudden drop in lift and torque. Alternatively, rotors with a higher solidity
operate at low TSR and blades experience deep dynamic stall, meaning the
attached leading-edge vortices enhance the lift-generation capabilities of the
blades but at a given angle that is normally when torque generation is maxi-
mum these energetic vortices are shed [8, 7]. Whilst the latter configurations
are common for Vertical Axis Tidal Turbines (VATTs), the design of VAWTs
normally adopt with a low solidity rotor and attain their optimal TSR, i.e.
relative rotational speed at which their power coefficient is maximum, in the
range of 4–5 [9], but these may suffer from self-starting issues. Hence, de-
signers are adopting more compact, higher solidity rotors that operate at
optimal TSR ranges of approx. 1.5–2.5, which is mostly adopted in VATTs
for hydro-kinetic applications at sea or in rivers [10, 8, 11].

3
 The horizontal-to-vertical asymmetric distribution of VAWT wakes is de-
picted in Fig. 1, which considers a control volume behind a turbine with
diameter D0 and height H0 over which mass and momentum needs to be
conserved [12]. Over the horizontal xy-plane, the main contribution to mo-
mentum entrainment over the lateral boundaries of the control volume results
from the blade-induced energetic vortices [13]. This differs from the wake
recovery dynamics over the xz-vertical plane (Fig. 1b), as shear layers gen-
erated in this direction result from the tip vortices [9]. These blade-induced
structures are not identical leading to potentially dissimilar wake mixing
rates over the horizontal (ky∗ ) and vertical (kz∗ ) planes, causing an asym-
metric wake recovery. Furthermore, VAWT rotors with height-to-diameter
aspect ratio different to unity leads to having two length-scales with which
the wake dynamics, e.g. velocity distributions, can be characterised, unlike
HAWT wakes that scale with its diameter [14]. As shown in Fig. 1, in the
horizontal plane D0 is the characteristic length scale while in the vertical
plane this is H0 .
More insights into the complex turbulence structures generated by verti-
cal axis wind or tidal rotor blades were gained from numerical simulations.
Research has evidenced the need for eddy-resolving closures, such as Large-
Eddy Simulation (LES), to resolve the flow within the VAT rotor and its
wake, as Reynolds-Averaged Navier Stokes models can fail to predict dy-
namic stall [15] or the replenishment of mean kinetic energy in the wake
which is driven by turbulence fluctuations [11, 5, 16]. Such flow complexity
requires fine numerical grids with three approaches mostly adopted to repre-
sent VAT rotors. Geometry-resolved simulations, e.g. adopting an immersed
boundary method, provide a high-resolution at the rotor capturing the vor-
tices induced by the blades [17, 8]. Actuator techniques enable to adopt
lower grid resolution while capturing the main wake dynamics but fail at
representing the dynamic stall vortices [18]. Actuator Line Method (ALM)
is the most widely adopted [19, 20, 21] whilst the Actuator Surface Method
(ASM) can provide further gains to improve the resolution of the flow at
rotor level [22].
There is still a need for developing analytical wake models tailored to
VATs that enable its prediction at almost no computational cost [23], which
is the aim of this article. Analytical models are widely adopted to design
array layouts by industry and researchers as they enable the evaluation of
multiple operational scenarios within a reasonable time frame [24], thus the
presented models will be invaluable for the future design of VAT arrays [25].

4
Our models improve the wake expansion both in terms of expansion rate and
wake onset area, and account for an asymmetric wake recovery, relevant for
VATs with height-to-diameter aspect ratios different to unity. The structure
of the paper is: §2 describes the underpinning basis our wake models are built
upon, which are derived in §3. Validation in four cases with comparisons
to large-eddy simulation results, field data and other available models are
presented in §4, with the main conclusions drawn in §5.

2. Novelty of wake models

The underpinning physics of our models for VAWT wakes lie in consid-
ering the wake to expand according to two length-scales, namely the wake
diameter (Dw ) and height (Hw ), featuring a rectangular shape analogously
to the projected area of the VAWT (A0 = D0 H0 ), as shown in Fig. 1. The
cross-sectional wake area Aw = Dw Hw at any downstream location (x) is
determined as the evolution of the wake from its onset location, xa , corre-
sponding to the location at which pressure equilibrium is reached, i.e. the
pressure at the wake centre is the same as the free-stream one. The latter
needs to be downstream of the swept perimeter region, i.e. x/D0 ≥ 0.5, and
for simplicity we set xa = 0.5D0 as the location of the wake onset in our
models, analogously to the hypothesis done in HAT wake models in which
such pressure equilibrium is attained almost immediately downstream of the
turbine rotor Frandsen et al. [26].
The wake expansion is determined from two non-dimensional parameters
corresponding to the horizontal (ky ) and vertical (kz ) directions, whose values
can be dissimilar as the shear layers generated by tip vortices over the vertical
direction are different to those from dynamic-stall vortices on the horizontal
one. Our wake expansion considerations allow to develop a Gaussian wake
model that accounts for a physically valid estimation of the initial wake area,
addressing limitations found in other VAT models partly build upon concepts
from HAWT wakes whose under-prediction of the wake area at xa leads to
non-physical values of velocity deficit in the near-wake, as discussed in §4.
Note that from hereafter we will use VAT to avoid distinction of wind or
tidal turbines as models can be applicable to either of these.

5
3. Derivation of the wake models
3.1. Momentum conservation
We build our momentum conserving wake models starting from the con-
servation form of the Reynolds Averaged Navier-Stokes equation for high
Reynolds numbers in the streamwise direction. After neglecting the pressure
and viscous terms, this equation reads

∂Uw (U0 − Uw ) ∂Vw (U0 − Uw ) ∂Ww (U0 − Uw ) ∂u0 u0 ∂u0 v 0 ∂u0 w0

+ + = + +
∂x ∂y ∂z ∂x ∂y ∂z
(1)
with (Uw , Vw , Ww ) being the vector of mean wake velocities in the stream-
wise, transverse and vertical directions respectively, and u0 u0 , u0 v 0 and u0 w0
denote time-averaged turbulent fluctuation correlations. We integrate 1 at
any streamwise location of a control volume that embeds the turbine and
expands over the y and z directions from −∞ to ∞, which together with the
assumption the shear stresses vanish when increasing the distance from the
wake centre provides the resulting RANS equation,
∞
Z
d
(Uw (U0 − Uw ) − u0 u0 ) dA ≈ 0 (2)
dx
−∞

The streamwise variation of u0 u0 is much reduced when compared to the

convective term [12], hence 2 can be simplified to obtain the momentum
integral [27] as
Z ∞
ρ (Uw (U0 − Uw )) dA = T (3)
−∞

This equilibrium condition states that the momentum deficit flux in the
wake is proportional to the thrust force T exerted by the turbine. Note
that we consider the wakes to be in zero-pressure gradient flow and the
incoming velocity to be mostly uniformly distributed in the three directions
of space. The thrust force can be related to the thrust coefficient (CT ) from
the actuator disk theory as
1
T = CT ρA0 U02 (4)
2

6
3.2. Top-hat wake model
The value of the wake onset area Aa can be determined using the actuator
disk theory as Aa = βA0 , with β representing the relative initial wake expan-
sion at xa in terms of the turbine rotor’s cross-section A0 , independently of
whether such cross-section is circular (HATs) or rectangular (VATs). The ac-
tuator disk theory states the velocity over the plane at xa is U0 (1 − √
2a) whilst

1
at the rotor centre plane the velocity is U0 (1 − a), with a = 2 1 − 1 − CT
being the so-called induction factor. Hence, the value of β is determined
based on energy conservation as:
√
Aa 1−a 1 1 + 1 − CT
β= = = √ (5)
A0 1 − 2a 2 1 − CT
The wake area at any streamwise location is determined similarly to the
approach presented by Frandsen et al. [26] for HATs with Dw ∝ x1/2 . Hence,
the wake width is considered to expand asymmetrically over the horizontal
and vertical directions as:

 1/2  1/2
x − xa x − xa
Dw = D0 β + kwy , Hw = H0 β + kwz (6)
D0 H0
We now apply momentum balance to a control volume that embeds the
operating turbine expanding some distance upstream and downstream to ob-
tain the velocity deficit ∆U = U0 −Uw for the top-hat model with asymmetric
expansion over the horizontal and vertical directions:
 
v
∆U 1 u 2CT
= 1 − u 1−  (7)

1/2  1/2 
U0 2 t
x−xa x−xa
β + kwy D0 β + kwz H0
Values of kwy and kwz are considered equal to 2.0Iu , with Iu denoting
streamwise turbulence intensity. This value is half the one adopted in HAT
wakes [26] and in other VAT wake models [28]. We propose this value as
it provides better predictions of momentum thickness or maximum ∆U as
shown later in §4.

3.3. Gaussian wake model

A cornerstone in the derivation of a Gaussian wake model is to assume
the velocity deficit distribution to be self-similar, i.e. at any streamwise

7
distance ∆U can be directly determined from local scales of velocity and
length [27]. For axisymmetric wakes, as is the case of HAT wakes [29],
self-similarity is attained if the transverse distribution of ∆U consistently
follows a given function f (− 21 (r/σ)2 ), where r/σ is the distance from the
wake centre (r) normalised by the characteristic wake width (σ); and a given
velocity scale C(x), which is determined as the maximum normalised velocity
deficit (∆Umax /U0 ) at any streamwise distance. The self-similar normalised
∆U can be written as

1 r2
 
∆U U0 − Uw
= = C(x)f − 2 (8)
U0 U0 2σ
We assume ∆U to be self-similar following a Gaussian shape function,
with results in §4 showing this condition is deemed valid. Adopting a Gaus-
sian distribution allows a more physically-realistic description of the wake
velocity deficit compared to top-hat models that assume a uniform value
across the wake width. Whilst in most HAT wake cases the self-similar shape
function is defined only by the turbine diameter [30, 31], for VATs both its
diameter and height are characteristic length-scales that determine the dis-
tribution of ∆U/U0 over the horizontal and vertical directions of the wake.
The wake asymmetry is accounted in our model by adopting the superposi-
tion of two Gaussian distributions ∆U (y) and ∆U (z), whose characteristic
wake widths σy and σz scale depending on D0 and H0 , respectively. Following
Bastankhah and Porté-Agel [12], we propose these wake widths to expand
linearly in the downwind direction:

σy x − xa σz x − xa
= ky∗ + εy , = kz∗ + εz (9)
D0 D0 H0 H0
Wake expansion rates ky∗ and kz∗ are estimated as 0.35Iu as proposed by
Bastankhah and Porté-Agel [12], which are proven adequate given the wake
predictions presented in §4 considering εy and εz that represent the wake
onset width are well determined. The wake velocity 8 can be re-written as

y2 z2
  
Uw = U0 1 − C(x)exp − 2 − 2 (10)
2σy 2σz
This is an algebraic equation with one unknown, the velocity scale C(x),
which is solved equating the momentum integral 3 to the turbine thrust force
4 to obtain the definition of the wake velocity 10 as

8
 ∞
Z
y2 z2 y2 z2
   
U02 C(x)exp − 2− 2 1 − C(x)exp − 2 − 2 dA
2σy 2σz 2σy 2σz
−∞
1
= CT A0 U02 (11)
2
R ∞ Considering
2
the rectangular
2
√ cross-section
R∞ of the wakes √
2 2
Aw =Dw Hw and
−∞
exp(−y /(2σy )) dy= 2πσy and −∞ exp(−y /σy )dy= πσy , 11 can be
integrated to determine the normalised maximum velocity deficit, C(x), as
1
σy σz πC(x)2 − 2πσy σz C(x) + CT A0 = 0 (12)
2
From the two possible solutions to this quadratic equation, the value that
provides a physical solution for the characteristic velocity scale is then
s
∆Umax CT A0
C(x) = =1− 1− (13)
U0 2πσy σz
Our proposed Gaussian model for VAT wakes is obtained from 10 and 13
as
s !
y2 z2
 
∆U CT D0 H0
= 1− 1− exp − 2 − 2 (14)
U0 2πσy σz 2σy 2σz
The values of εy and εz are determined from the mass flow deficit rate
immediately behind the turbine’s rotor at xa by equating that predicted by
the top-hat model 7 and to the expression from the Gaussian model 14,
providing the following relation:

s ! s !
Da Ha 2CT CT
1− 1− = 2πD0 εy H0 εz 1− 1− (15)
2 β 2πεy εz

Hence, εy and εz expressions for VAT wakes are determined as

β
εy εz = (16)
4π
Note that the initial wake expansion rates are just a function of β, i.e. only
depends on the thrust coefficient CT . The lack of an extensive experimental

9
campaign focused on VAT far-wakes prevents from individually accounting
for each of these wake expansion rates. Hence, for simplicity, we assume
the normalised wake onset width is identical in the horizontal and vertical
directions, i.e. εy = εz , with their value becoming
1 p
εy = εz = √ β (17)
4π
It is noteworthy that our√formulation of the wake onset width (17) differs
from the definition ε = 0.25 β proposed by Abkar and Dabiri [19] and Abkar
[28], which corresponds to the value for HATs. Our formulation overcomes
limitations from these models in relation to non-physical estimations of ∆U
at short distances behind the turbines, which
√ resulted from
√ an incorrect
definition of ε in 14. Comparing ε ≈ 0.282 β (17) and 0.25 β, the latter is
approx. 12% smaller, thus leading to and underestimation of the wake onset
width at x = xa .

4. Prediction of the wake models

We present the validation of the wake models in four cases with compar-
isons to laboratory experiments, field data, and large-eddy simulations. Re-
sults from the Gaussian wake model by Abkar [28], whose underlying physics
were partially based on HATs, are also included to compare the accuracy and
reliability of both model predictions. From hereafter, the origin of coordi-
nates are at the turbine centre of rotation and mid-height and for simplicity
the turbine diameter and height are represented by D and H instead of D0
and H0 .

4.1. Single VAWT operating in a turbulent boundary layer flow

Cases 1a to 1c correspond to a single VAWT operating in a turbulent
boundary layer flow representing three scenarios in which the device attains
different Tip-Speed Ratios (TSR), thrust coefficient (CT ), and aspect ratio
[28]. In cases 1a and 1b, the turbine has a diameter D = 26m with a height
H = 24 m whilst in case 1c its height is equal to 48 m, which yields diameter-
to-height aspect ratios close to unity for former cases and almost of two for
case 1c. We are interested in comparing cases 1a and 1b as the VAWT
operates at TSR of 3.8 and 2.5 respectively, attaining thrust coefficients of
0.65 and 0.34 that lead to different wake dynamics. In case 1c, the turbine
has a CT equal to 0.64 and operates at TSR = 3.8, as in case 1a, but its

10
Figure 2: Normalised velocity deficit profiles for case 1a with CT = 0.65, TSR = 3.8, D
= 26 m and H = 24 m. Comparison of our proposed top-hat (dashed line) and Gaussian
(solid black line) analytical wake models, with the Gaussian model proposed by Abkar
[28] (solid red line) and LES-ALM results from Abkar and Dabiri [19] (circles).

aspect ratio of two promotes a larger wake asymmetry when comparing its
recovery over the horizontal and vertical directions. Turbines are equipped
with three NACA 0018 blades with a chord length c = 0.75 m, leading to a
solidity value Nb c/πD0 ≈ 3%. The free-stream velocity at hub-height (U0 ) is
7.0 ms−1 with a turbulent intensity (Iu ) of 9.1%.
Validation of normalised velocity deficit predictions obtained with the
proposed analytical wake models are presented in Fig. 2 with horizontal
(y) and vertical (z) profiles across the turbine wake centre at downstream
distances of x/D = 3, 6, 9 and 12. For completeness, we include the LES-
ALM results from Abkar and Dabiri [19]. Our Gaussian model provides a
similar accuracy to that of Abkar [28] at most profiles, with ours with an
slight underprediction of ∆U/U0 for x/D = 3 observed in both vertical and
horizontal profiles. Note that the model from Abkar [28] slightly overpredicts
the LES-ALM velocity deficit values at the wake centre at distances of x/D =

11
Figure 3: Normalised velocity deficit profiles for case 1b with CT = 0.34, TSR = 2.5, D
= 26 m and H = 24 m. Same legend as Fig. 2.

9 and 12, whereas ours attains a closer match to the LES data.
Figure 3 presents the results obtained for case 1b in which the VAWT
operates at a lower TSR which decreases its CT in comparison to case 1a
(and case 1c). We observe our Gaussian wake model provides again a good
estimate of the maximum velocity deficit at the wake centre over the vertical
and horizontal directions, and the wake width is also in agreement with the
LES and wake model from Abkar [28]. As VAWT wakes asymmetry depends
on its aspect ratio [14], in cases 1a and 1b this is approx. unity leading
the wake to feature a Gaussian distribution over the horizontal and vertical
directions at x/D ≥ 3, which supports the self-similarity assumption. Some
degree of asymmetry in ∆U/U0 is appreciated at x/D = 3 with its maximum
value over the horizontal plane being slightly larger than in the vertical one.
The turbine in case 1c has an aspect ratio close to two that promotes an
uneven recovery over the vertical and horizontal directions which is well ob-
served in Fig. 4 with the distribution of ∆U/U0 over the y-direction profiles
attaining a Gaussian profile whilst, over the vertical direction, the profiles

12
Figure 4: Normalised velocity deficit profiles for case 1c with CT = 0.63, TSR = 3.8, D
= 26 m and H = 48 m. Same legend as in Fig. 2.

nearer to the VAWT exhibit an almost top-hat distribution. After x/D = 9

which corresponds to x/H = 4.5, the LES data indicate the velocity deficit
in the VAWT wake over the vertical direction recovers the Gaussian distri-
bution. Our Gaussian wake model provides a satisfactory representation of
∆U/U0 as at x/D = 3 its maximum value is closer to the LES data than the
one predicted by the model from Abkar [28] which overpredicts this value.
Far downstream of the turbine, at x/D = 9 and 12, our wake model achieves
a good agreement with the LES results for the maximum velocity deficit.
Further analysis of the wake predictions is presented in Fig. 5 with the
evolution of maximum velocity deficit (∆Umax /U0 ) over the streamwise di-
rection for cases 1a to 1c, which represents the velocity scale adopted in the
wake model adopted in the Gaussian and top-hat velocity distributions. In
comparison to the LES results, our Gaussian model estimates a faster ve-
locity recovery, i.e. lower maximum wake velocity deficit, than that from
Abkar [28], which is a result of the considered rectangular wake onset area
at xa represented by εy and εz . For cases 1a and 1b, all analytical models

13
Figure 5: Normalised maximum velocity deficit for cases 1a to 1c. Same legend as in Fig.
2.

predict similar values of ∆Umax /U0 for distances x/D0 ≥ 8. Slightly larger
differences are observed for case 1c in which the wake is more asymmetric,
with our Gaussian analytical wake model providing a closer prediction to
those from the LES. We also note that the latter is able to provide wake
velocity values at any streamwise distance even immediately downstream of
the VAWT, whilst Abkar [28] model fails at providing physical values for
x/D ≤ 2.0 in case 1c due to determining the value of β partly based on HAT
wake characteristics.
In turbine array modelling, any analytical wake model needs to provide
an accurate downstream evolution of the wake deficit so wake-to-wake inter-
actions are reliably accounted for, which is key to estimate the performance
of the individual devices as power scales according to P ∝ U03i . Fig. 6
presents the streamwise evolution of the normalised total momentum deficit
at the centre planes of the wake, i.e. computed over the horizontal (My ) and
vertical (Mz ) directions at z = 0 and y = 0 respectively, which result from
combining 3 and 4, as
R
2 Uw (U0 − Uw ) dxi
Mi = (18)
A0 U02
An overall good agreement of the total momentum deficit is observed with
the Gaussian models providing a closer match with the LES data, except
for My in case 1a with our top-hat model attains a better match. In line
with the maximum velocity deficit results, our Gaussian model predicts lower
total momentum deficit which appear closer to the LES data compared to
the estimates from the other theoretical models. LES results from case 1c

14
Figure 6: Predictions of momentum thickness for cases 1a, 1b and 1c. Same legend as in
Fig. 2.

depict the asymmetric wake distribution with larger values for Mz than My ,
with the latter decaying at a slower rate than the vertical momentum deficit,
which is well predicted by all wake models. The top-hat model appears to
consistently underestimate the value of momentum deficits in most cases.

4.2. Array of four aligned VAWTs

The largest experimental facility with full-scale VAWTs, FLOWE (US),
has 18 devices rated at 1.2MW and research undertaken at this site en-
abled the world-first quantification of a VAWT array performance and flow
mechanisms driving the wake recovery [32]. To quantify the accuracy of our
analytical models, we consider a configuration with four VAWTs fully aligned
with the onset flow, i.e. the second to fourth turbines operate in complete
waked conditions, with a separation of 11D between them. The turbines
have a diameter and height of 1.2 m and 6.1 m respectively, with a separa-
tion of 3 m from the bottom tip to the ground level, i.e. a total height of 9.1
m, and operate at TSR = 2.3. The free-stream velocity is 8.45 ms−1 with
a turbulence intensity of 11% considered for all turbines, and flow statistics
in the wakes are obtained at seven equally-spaced bins that span over the

15
Figure 7: Normalised maximum velocity deficit profiles for the case with four full-scale
VAWT. Comparison of our proposed top-hat (dashed line) and Gaussian (solid black line)
analytical wake models, with the Gaussian model proposed by Abkar [28] (solid red line)
and field data from Kinzel et al. [32] (circles).

whole turbine height, whose measures are then integrated into a single verti-
cally averaged value. We estimate a thrust coefficient of 0.652 based on the
field power coefficient of 0.134 calculated in Kinzel et al. [32] based on the
free-stream velocity, using the induction factor of the actuator disk theory.
Comparison of the maximum velocity deficit over the centre-line of the
turbines is presented in Fig. 7, including the field data which measured ve-
locities at 2D and 8D downstream of each device, and our two proposed
wake models with that from Abkar [28]. Our Gaussian wake model provides
a good accuracy when compared to the field data for those points further
downstream of each turbine whilst slightly over predicts the maximum ve-
locity deficit at a distance of 2D downstream of the first and fourth turbine.
The latter could be attributed to the wake velocity distribution not featuring
full self-similarity in the wake profiles at such short distance given the aspect
ratio of these VAWTs is almost of five. These results evidence the relevance
of the initial wake expansion correctly accounted for in our Gaussian model
compared to the one proposed in Abkar [28] which overestimates the wake
velocity deficit throughout the array. Our top-hat model notably underes-
timates the maximum velocity deficit, in line with the results presented in
§4.1, but these proved useful in the computation of the momentum thickness
in the far-wake, thus somewhat valuable to predict the performance of the
turbines in arrays.
A key feature of analytical wake models is that they need to provide

16
Figure 8: Evolution of the power ratio between the first two turbines at FLOWE for differ-
ent levels of turbulence intensity. Comparison between our proposed Gaussian analytical
wake model (black line) and that of Abkar [28] (red line).

reliable power estimates of turbines in arrays, being this dependent on the

velocity field such as P ∝ U 3 . In Fig. 8 we provide a quantitative comparison
of the ratio of power generated by the first turbine T1 , assuming UT1 = U0 ,
and the turbine T2 immediately behind, with UT2 = Uw (x). The power ratio
P1 /P2 is estimated for various distances between these turbines, xT1 − xT2 ,
considering 11D as in FLOWE, 8D and 5D, and over a range of turbulence
intensities 0.01 < Iu < 0.15 to consider scenarios representative of offshore
(Iu = 6–8%) or onshore (Iu = 10–12%) locations [33]. Results show power
production from the downstream turbine obtained with our Gaussian model
is consistently larger than with Abkar’s model, in line with the evolution of
velocity deficit seen in Fig. 7. Considering the well-known HAWT array of
Horns Rev which can feature turbulence intensities of approx. 7–8%, with
turbine spacing of 11D, 8D and 5D the differences in predictions of P1 /P2 are
6%, 9% and 14% respectively, with Abkar’s model underestimating the values
as it accounts for larger velocity deficit as shown in Fig. 7. At high turbu-
lence intensity values, variation in power estimates between models narrows
down as the wake expansion increases with Iu leading to faster momentum
recovery, in turn diminishing the relevance of the onset wake width prediction
(Eq. 17). Overall, this power-prediction sensitivity shows that the presented
improvements in our Gaussian model can avoid an underestimation of the
array efficiency, especially for environments with low turbulence levels.

17
5. Conclusions
This paper presents a new set of top-hat and Gaussian wake models for
Vertical Axis Turbines (VATs) built upon considering a rectangular wake
cross-section that evolves according to two length-scales and including an
asymmetric expansion over the horizontal and vertical planes. In comparison
to previous wake models, we included three main features: the location at
which the wake pressure is balanced is attained at xa = D/2, with D being
the turbine diameter; the wake at this location has a rectangular cross-section
as the VAT rotor projected area that scales with a factor ε = (β/4π)1/2 ;
and, the asymmetric wake recovery dynamics is represented by uneven wake
expansion rates.
We validated the accuracy of our proposed models in comparison to large-
eddy simulation results and field data in four cases involving a standalone
and an array of vertical axis wind turbines. We proved the suitability of our
models to estimate the wake velocities for all cases at any downstream loca-
tion, providing improvements compared to other existing VAT wake models.
Our analytical models provide a good estimation of the maximum velocity
deficit and momentum thickness for the three cases with a single turbine
that operated at two tip-speed ratios and had two diameter-to-height aspect
ratios. In application to an array of four full-scale turbines, the models com-
pared well with field data with results showing that our Gaussian wake model
attains a good estimation of the maximum velocity deficit, which verifies its
reliability to predict the interaction between turbines in arrays. In relation to
power generation, we performed a sensitivity analysis over a range of turbu-
lence intensity values and turbine spacing showing there is a larger deviation
between models for low turbulence values, with our Gaussian model provid-
ing larger power generation values due to its better prediction of the velocity
deficit field.
Overall, the results confirmed an enhanced accuracy of our wake models
compared to other previously proposed models, which will lead to improved
designs of VAT arrays increasing the development pace of this promising
technology.

Acknowledgements
This research was partially supported by the UK’s Engineering and Physi-
cal Sciences Research Council (EPSRC) through the project EP/R51150X/1.

18
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