Journal of Wind Engineering and Industrial Aerodynamics, 27 (1988) 213-224

213

Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands

CALCULATING THE FLOWFIELD IN THE WAKE OF WIND TURBINES

J.F. AINSLIE
Central Electricity Generating Board
London EC4P 4EB, UK

S UMMARY
The paper briefly outlines the known features of wind turbine wakes.
A
numerical model is described which can be used to calculate the wake flowfield.
The effect of meteorological conditions on wake decay is examined in detail and
the way in which this is included in the model is described.
Comparison of
experimental data with model calculations is used to draw conclusions about the
wake flowfleld.
It is concluded that the numerical wake model can be used with confidence to
calculate the wake velocity field for wind turbines over a range of sizes and in
a variety of meteorological conditions.
It is considered that the model is
sufficiently simple and quick to use that the technique will be of considerable
use in the planning and design of windfarms and of turbines for wlndfarm
operation.

NOTATION
U,V
x,r

axial and radial velocities

axial distance coordinate
distance coordinate

(downwind from wind turbine) and radial

(from wake centreline)

uv

Reynolds stress cross-correlation

Uo

free stream windspeed

UN

free stream windspeed at hub height

Uc

wake centreline velocity

Von Karman constant

k1

dimensionless constant

(here taken to be 0.4)

Monin-Obukhov length

CT

wind turbine thrust coefficient

velocity deficit

Dm

centreline velocity deficit at 2 diameters downstream

do

centreline velocity deficit

wake width

KM

eddy diffuslvity of momentum

0167-6105/88/$03.50

1988 Elsevier Science Publishers B.V.

214
z

height above ground

zH

wind turbine hub height

wind turbine diameter

zo

roughness length

eddy viscosity

Ca

contribution to eddy viscosity due to ambient turbulence

INTRODUCTION
The grouping together of wind turbines into windfarms gives rise to
interactive effects due to the operation of the turbines in each others wakes.
The effects are both to reduce the power output of the wlndfarm and to increase
the fatigue loads experienced by the wind turbines.

These effects need to be

taken into account during the design of both the wind turbines themselves and
the overall windfarm.
The mathematical models of wind turbine wakes which have previously been
subject to most development have been based on empirical descriptions of a wake
(e.g. ref. i).

Although these models can be useful for estimating the effect of

wake decay characteristics

on windfarm efficiency,

they do not provide any

physical insight into the flow processes.

This paper describes the development of a wake model based on a numerical
solution of the differential equations governing the flow.

A relatively simple

eddy viscosity turbulence model is used, which results in an algorithm which is

both quick and easy to use.

The eddy viscosity has two contributions - one

describes the turbulent mixing due to turbulence generated within the shear
layer of the wake, the other describes the effect of ambient turbulence in the
atmosphere on wake mixing.

The ambient turbulence term is a dominant term in

many situations of practical interest and has considerable influence on the wake
velocity field.

The paper includes an examination of the influence of

meteorological conditions on the wake decay.

CHARACTERISTICS OF WIND TURBINE WAKES

Studies of turbine wakes, both in the field and in wind tunnel experiments,
indicate the existence of a complex near wake region which typically extends
over about 2-4 diameters downstream.

This region is dominated by the relaxation

of axial and radial pressure gradients caused by the extraction of energy from
the mean flow at the rotor disk.

This causes the centreline velocity to drop

and the wake width to increase as the air moves downstream from the rotor plane.
The minimum centreline velocity is reached between 1 and 2 diameters downstream,
beyond which the velocity begins to recover as fluid mixing takes over from
pressure gradient effects as the dominant process in the flow.

The tip vortices

215
from a turbine rotor decay within 2 to 3 diameters
turbulence

(ref. 2).

The bulk swirl introduced

due to the action of ambient

by the rotor is small, the swirl

angle being less than i0 degrees.

Large scale turbulence

is generated

in the annular shear layer of the wake

and this spreads into the core of the wake,

and 5 diameters

downstream.

reaching the centreline

Beyond 5 diameters downstream

between 3

the wake profile is

roughly Gaussian and the centreline deficit decays monotonically,

with the rate

of decay strongly dependent

Turbulence

on the ambient turbulence

intensity.

profiles across rotor wakes in the main decay region sometimes show off-axis
peaks where the velocity shear is a maximum
but intriguingly

(at Nasudden for instance,

this is not always the case (as at Aldborough,

ref. 3)

ref. 4).

The

axial turbulent velocity on the centreline decays towards the ambient level with
an exponent around -2/3 in most cases.
introduced

The lengthscale

of the turbulence

by the wake is of the order of the rotor diameter.

The total momentum deficit in the wake is determined

coefficient
conditions

by the thrust

of the wind turbine - and is thus a function of the operating

(tip-speed-ratio,

pitch angle etc.).

The rate of recovery of the

wake downstream of the near wake region is governed by both the turbulence
generated

in the wake shear layer and the level of ambient turbulence

atmosphere.
surface

The atmospheric

roughness

turbulence

is a function of the site - primarily

- the hub height of the turbine

near the ground) and atmospheric

in the

stability.

(turbulence

levels are higher

Although strong winds are normally

associated with neutral stability turbines operating near the cut-in wind speed
(which occurs for a relatively large proportion of the turbine operating time)
could be operating in more stable conditions.
to reduced turbulence
also correspond
Comparison

intensities

These conditions will give rise

and hence a more persistent wake, and will

to high thrust coefficient

values.

of wind tunnel wake measurements

with available field data

indicate that wake deficits measured in the field are often smaller than those
from wind tunnel experiments.
wind - particularly

This is thought to be because variability

wind direction - gives rise to wake meandering,

of the

which means

that a measured deficit averaged over a few minutes will in fact be averaged
over a portion of the wake profile.

DEVELOPMENT

OF THE EQUATIONS

The wake will be considered

flow with zero circumferential

to be an axlsymmetric,
velocities

fully turbulent,

flowfield will be assumed to be stationary with time.

Pressure gradients

co-flowlng fluid outside the wake will be assumed to be negligible.

first few diameters

wake

(since swirl is small), and the

downstream the gradients

of mean quantities

in the

Beyond the

in the radial

216
direction will be very much greater than the gradients in the axial direction.
The Navier-Stokes equation can then be replaced with its equivalent thin shear
layer approximation and the viscous terms dropped, giving:
U ~ U/~x + V~U/~r = -(I/r)

~ (r~'~)/~r

(i)

The turbulent viscosity concept is used to describe the shear stresses with
an eddy viscosity defined by:
-u--v = E ~U/~r,

= iw (x) Uw (x) + C a

(2)

where iw and U w are suitable length and velocity scales describing the wake
shear layer, and E a is the ambient turbulence contribution to the eddy
viscosity.

The length and velocity scales are taken to be proportional to the

wake width b and the velocity difference Uo-U c across the wake shear layer; these
scales are therefore characteristic of the downstream distance x and independent
of r.
The contribution G a to the eddy viscosity is given by the same parameter
used by boundary layer modellers to describe momentum transfer in the atmosphere
- the eddy diffusivity of momentum k M (ref. 5) which has the same dimensions as
and is defined in an analogous way.
A modification is required to the equation for the eddy viscosity in the
near wake - up to about 5 diameters downstream of the rotor.

This is because

there is a lack of equilibrium between the mean velocity field and the
turbulence field in the near wake region, and the relation (2) does not apply
(ref. 6).
More understanding

is required of the non-equilibrium nature of the flow in

this region before it can be successfully modelled.

A clue, however,

to the

treatment of the near wake region is given by turbulence data which indicates
the build-up of turbulence in the shear layer of turbine wakes.

The effect of

this build-up of turbulence appears consistent with a filter function F of the

form:
F = 0.65 + [(x - 4.5)/23.32]

1/3

x <

5.5

5.5

(3)
=

which gives the equation for the eddy viscosity as:

= F[k I b (U o - U c) + K M]

(4)

Numerical solution of the equations

The momentum equation (I) is solved, along with the equation for radial
velocity, using equation (4) to calculate the eddy viscosity terms.

The

solution scheme adopted by the author is an implicit numerical finite difference

217
scheme, with a simple forward difference for the axial advection term.

The

solution routine only requires the inversion of a tri-diagonal matrix and it can
be performed quickly on a small desk-top computer.

INPUT PARAMETERS
Fixin~ the value of k 1
The model as described contains one constant, kl, which is expected to be
a property of the shear layer and largely independent of ambient turbulence and
the details of the wake producing object.

Model predictions were compared with

wind tunnel data at low ambient turbulence intensity to fix the value of k I.
The experiment was performed with an axisymmetric simulator suspended on fine
wires in the centre of the wind tunnel.

Comparisons of wake decay were made at

thrust coefficients of 0.79, 0.62 and 0.31, and good agreement was found in all
3 cases when a value of 0.015 was used for k I.
thrust coefficient in Fig.

Comparison is shown for one

I, and a comparison is also made with rotor data

collected under similar conditions; the agreement illustrates that the constant
is equally valid for rotors.
Initial wake parameters
It is envisaged that the solution will be started at a downstream distance
of about 2 diameters,
dominate the flow.

i.e. at such a distance that pressure gradients no longer

The wake decay rate is comparatively insensitive to the

initial velocity profile and a Gaussian profile I-U/U o = D M exp (-3.56(r/b) 2)

will normally be used.

~ecessary input parameters are the initial velocity

deficit D M and wake width b.

Available experimental data on centreline velocity

deficits at 2 diameters indicate that simulators may behave slightly different

to rotors in this respect, as shown in Fig. 2; the field data from wind turbines
also show some departures from the wind tunnel rotor data, perhaps due to
averaging effects.

An equation based on the wind tunnel studies of rotors is:

D M = C T - 0.05 - (16 C T - 0.5) A/IO00

(5)

where A is the ambient turbulence intensity (%).

The wake width b can then be calculated by conservation of momentum:
b = [3.56 CTI(8 D M (i - 0.5 DM))]

(6)

Methods of estimatin~ K M
KM can be described in terms of the normal boundary layer parameters.

In

the surface layer up to I00 m or so in height,

K M = ku,

zl#m(z/L)

(7)

218

REF 11 ]
REF 10 .~

WIND TUNNEL ROTOR DATA

Z~ CERL
-f- REF 12
REF 14
+ REF 4
'O REF 3

SIMULATOR
ORKNEY
NIBE
ALDBOROUGH
NASUDDEN (HOURLY AVERAGES)

FIGURES REFER TO THRUST COEFFICIENTS

09

08-

0.7" 0
DM

~ .

0,6"

0.5-

t 0'78

0.6

0,7

O.8
+0.7
04-

~o~.~_.~.

~0.55

--

55

- - 0.31

0.3-

' ":Z

,~ ' '

(~

"'

'

'

"

1~

1~2

14

'

i ~

16

AMBIENT TURBULENCE INTENSITY/%

Fig. I.
thrust.

Initial velocity deficit as a function of ambient turbulence and roto~

219
where u w

is the friction velocity and

~m (=I for neutral conditions)

the influence of stability on the mixing processes.

taken as representative

reflects

If a logarithmic profile is

of neutral conditions:

u = (u, /k) in (z/z o)

(8)

this equation links the value of ~ a in a simple way to the turbine hub height
and the surface roughness.
eddy viscosity

E / ( U H D) = Fk I (b/D)
Empirical
conditions

In neutral conditions

the complete equation for the

is:
(i - Uc/UH) + Fk2/in (ZH/Zo)

relationships

(9)

must be used to determine K M in the case of stable

(see, for example,

ref. 5).

THE TREATMENT OF WAKE MEANDERING

The wake model as described above predicts
the externally

imposed windfield

never be perfectly

the wake velocity field assuming

is stationary with time.

In fact this will

true for a wind turbine operating in the atmospheric

layer, although the approximation

neutral and unstable atmospheric

becomes reasonable
conditions

boundary

in stable conditions.

In

the wake will meander relative to an

observer fixed on the ground due to fluctuations

in wind direction.

This will

result in the centreline velocity deficit measured by the fixed observer, ~ ,

being less than the stationary value because the measurement

point sweeps across

a region of the wake profile during the averaging period of the measurement.
An expression has been developed by Ainslie
meandering;

J%
d = d o [i + 7.12 ( ~ e

x /b)2] -

where d o is the uncorrected

centreline

(I0)
deficit and

deviation of wind direction fluctuations,

as the wake measurements
A particular
meteorological
purposes
6-O =

(ref. 7) to correct for wake

this correction gives the deficit measured by an observer as:

is the standard

measured over the same averaging time

(but not including very short timescale fluctuations).

field experiment

data to enable ~

may include sufficient

to be estimated.

background

However,

a simple estimate is given by the approximate

for predictive

relation:

~v/U

(11)

Since estimates

of

~'v are available for neutral conditions

surface roughness and height

values of 6"8

(ref. 8, for instance)

can be drawn up (Table I).

model predictions

in the next section.

as a function of

a table of approximate

These are the values used for the

220
MODEL PREDICTIONS
The predictions of the wake model are shown in figs. 3 and 4.
calculations are for neutral stability conditions,

The

a wind turbine hub height of

around 50 m and surface roughness of 0.05 m (i.e. gentle grassy farmland).

Figure 3 shows the wake decay for various thrust coefficient values without any
meandering correction;

the results thus correspond to a near-instantaneous

profile, and can be compared with data from wind tunnel experiments undertaken
using a simulated atmospheric boundary layer (which does not include the longtimescale meandering effects).

The agreement is very encouraging.

Figure 4 shows the wake decay for various thrust coefficient values, but
including the effects of wake meandering on the averaged wake deficits.

These

calculations

therefore correspond to the wake deficits measured in field

experiments,

and available experimental data is included in Fig. 4.

Again, the

overall agreement is very good, and it supports the use of a wake meandering
correction.

Detailed comparisons on a case-by-case basis requires good quality

information on the meteorological

conditions which is not always available.

indication of the comparison in Fig. 4, however,

formulated,

The

is that the wake model as

and making use of the input parameters suggested in this paper,

provides a very good estimate of the wake velocity deficits which will be
encountered in practice.

FUTURE DEVELOPMENTS
The main uncertainties
region.

requiring further work relate to the near wake

The development of the shear stresses

the near wake is still not well understood,

had to be used in this model.

(and hence the eddy viscosity)

and the empirical equation

in

(3) has

The initial wake velocity deficits used at

2 diameters come from a parameterisation of available experimental data; this

again leads to uncertainty in the predicted deficits in the first 3 or 4
diameters downstream,

although the uncertainty will reduce as more field data

becomes available.
The model as presented here is essentially a 2-dimensional model, taking
advantage of axisymmetric symmetry.
in a 3-dimensional formulation,

It would be interesting to pursue the model

which could then include vertical profiles of

windspeed and turbulence.

The wake model describes the flowfield in the wake of a single turbine.

For

small windfarms the interactions will be between pairs of turbines, and the
relevant effects can be calculated directly.

However,

for groups of more than

about 5 turbines the interactions will involve more than 2 turbines, and effects
such as the growth of one wake within the flowfield of an upstream wake become
important.

The method of dealing with this is usually to take single wake

221

CENTRELINE VELOCITY
DEFICIT
1.00.8-

0 ROTOR, 0.5% TURBULENCE

(REF 10)
Z~ CERL SIMULATOR, 2% TURBULENCE
0

0.5"

"-- 0.74

0.30.2-

0.31

0.1
4

(~

81'0

1'5

DOWNSTREAM DISTANCE/DIAMETERS

Fig. 2. C o m p a r i s o n of model calculations w i t h w i n d tunnel data at low ambient

turbulence.

WIND TUNNEL DATA: [] CERL ROTOR

'~ REF 11
1.0"
V REF 10
0.8"

I-.L)
m
M.
LM

0.60.5-

A=10%
8%
12%

CT ~- 0.9
0.7

0.40'5V

c,

CT ---- 0.78
0.84,
0.54,

0.3-

0.20
,--I

LU
C=

Iz

ILl

0.1

[]

0.08

DOWNSTREAM DISTANCE/DIAMETERS
Fig. 3. W a k e decay as a f u n c t i o n of rotor thrust for a typical neutral b o u n d a r y
layer (no m e a n d e r i n g correction).
Calculations for z H = 50 m, z o ffi 0.05 m,
K H /(UHD) = 0.023, A = 14%.

222

FI ELD DATA:
V
I
O
O

0.8"
i0.6-

REF13
REF13
REF12
REF 3
REF14

CT ---- 0.7
0.64
0.6
0.8
0.67

CT =

~0.4I--

0.5

D
U14.1

13
A

~ 0.2"

,--I
ILl

>

LLI

Z
LU

ntI--

Z
"'

!0.1'

I0.05

10
DO~/VNSTREAM

DISTANCE/DIAMETERS

Fig. 4. Wake decay as a function of rotor thrust for a typical neutral boundary
layer (including meandering correction).
Parameters as for Flg 3, wlth
~=0.1.

223

characteristics
fields.

as shown in figs.

This will inevitably

3 and 4 and to superpose

give rise to inaccuracies

the wake velocity

(ref. 9) -for instance

the higher turbulence within the windfarm due to upstream wakes is not often
taken into account.

What limited data is available

overlapping wakes indicates

in error.

that the superposltion

on the behaviour of
assumption

could be seriously

Clearly this is an area where much further work is needed.

CONCLUSIONS
A single wake model has been presented which aims to calculate the wake
velocity field behind a wind turbine taking account of all relevant
meteorological

influences.

input parameters.

Recommendations

are made as to the appropriate

Comparison with available experimental

data shows good

agreement with both wind tunnel studies and field data, provided the effects of
wake meandering are modelled in the latter case.
The model is comparatively
on a desk-top computer.

simple and the equations

It is, therefore,

to provide reliable estimates

suggested

of wake deficits

can be solved speedily

that the model can be used

for use in the planning and

designing of windfarms.

ACKNOWLEDGEMENT
This paper is published

by permission of the Central Electricity

Generating

Board.
REFERENCES
1
P.B.S. Lissaman and E.R. Bate, Energy effectiveness of arrays of wind energy
conversion systems, Aerovironment Inc. Report AV FR 7058 (1977), Pasadena,
USA
J.R. Connell and R.L. George, The wake of the MOD-OA wind turbine two rotor
diameters downwind on 3rd December 1981. Report no. PNL-4210, Pacific
Northwest Laboratory, Battelle, USA, 1982
U. Hogstrom, D.N. Asimakopoulos, H. Kambezidis, C.G. Helmis and A. Smedman,
A field study of the wake behind a 2 MW wlnd turbine.
Meteorology
Department report, Uppsala University, 1987.
G.J. Taylor, Wake and performance measurements on the Lawson-Tancred
horizontal axis windmill, Proc lEE (A), 130, No. 9, 604-612, 1983
E. Plate (ed.), Engineering Meteorology,

Elsevier,

17 m

1982

J.F. Ainslie, Development of an eddy viscosity model for wind turbine wakes,
Proc. 7th BWEA Wind Energy Conf., Oxford, MEP, 1985
J.F. Ainslie, Wake modelling and the prediction o f turbulence properties,
Proc. 8th BWEA Wind Energy Conf., Cambridge, MEP, 1986

224
8

Engineering Sciences Data Unit, Characteristics

near the ground, Part II, ESDU Datasheet 74031

J.F. Ainslie, A wake interaction model for calculating

Proc. 6th BWEA Wind Energy Conf., Reading, CUP, 1984

I0

A.M. Talmon, The wake of a horizontal axis wind turbine model; measurements
in uniform approach flow and in a simulated atmospheric boundary layer,
Report MT-TNO 85-010121, 1985, TNO Apledoorn, The Netherlands

II

P.H. Alfredsson, J.A. Dahlberg and P.E.J. Vermeulen, A comparison between

predicted and measured data from wind turbine wakes, Jnl Wind Engineering,
6 no. 3 (1982)

12

R.S. Haines, D.J. Milborrow, D.I. Page, A.S. Scott, W.G. Stevenson and G.J.
Taylor, Wake interactions between the Howden HWP-300 and the WEG MS-1 wind
turbine generators on Orkney, U.K., Proc. 1986 EWEC Conf., Rome

13

R.W. Baker and S.N. Walker, Measured wind turbine generator wake
characteristics, Proc. AWEA Wind Power Conf., San Francisco, 1985

14

G.J. Taylor, D.J. Milborrow, D.N. Mclntosh and D.T. Swift-Hook,

Wake
measurements on the Nibe windmills, Proc. 7th BWEA Wind Energy Conf.,
Oxford, MEP, 1985

TABLE

of atmospheric

turbulence

cluster efficiencies,

Wind direction fluctuation

values~

neutral stabili~.x

%
Zo = 0. i
Rough farmland

Zo = 0.05
Smooth farmland

Zo = 0.01
Level grass plane

10 m

0.16

0.14

0.II

20 m

0.14

0.12

0.i0

40 m

0.12

0.11

0.09

60 m

0.ii

0.i0

0.08

I00 m

0.10

0.09

0.07

Height