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Modelling of the flow field
surrounding tidal turbine
arrays for varying positions
in a channel
rsta.royalsocietypublishing.org
Research
Cite this article: Daly T, Myers LE, Bahaj AS.
2013 Modelling of the flow field surrounding
tidal turbine arrays for varying positions in a
channel. Phil Trans R Soc A 371: 20120246.
http://dx.doi.org/10.1098/rsta.2012.0246
One contribution of 14 to a Theme Issue ‘New
research in tidal current energy’.
Subject Areas:
energy, civil engineering
Keywords:
CFX, actuator fences, flow acceleration,
wake analysis, tidal arrays
Author for correspondence:
T. Daly
e-mail: td2e09@soton.ac.uk
T. Daly, L. E. Myers and A. S. Bahaj
Faculty of Engineering and the Environment, Sustainable Energy
Research Group, University of Southampton, Highfield,
Southampton SO17 1BJ, UK
The modelling of tidal turbines and the hydrodynamic
effects of tidal power extraction represents a relatively
new challenge in the field of computational fluid
dynamics. Many different methods of defining flow
and boundary conditions have been postulated and
examined to determine how accurately they replicate
the many parameters associated with tidal power
extraction. This paper outlines the results of numerical
modelling analysis carried out to investigate different
methods of defining the inflow velocity boundary
condition. This work is part of a wider research
programme investigating flow effects in tidal turbine
arrays. Results of this numerical analysis were
benchmarked against previous experimental work
conducted at the University of Southampton
Chilworth hydraulics laboratory. Results show
significant differences between certain methods of
defining inflow velocities. However, certain methods
do show good correlation with experimental results.
This correlation would appear to justify the use of
these velocity inflow definition methods in future
numerical modelling of the far-field flow effects of
tidal turbine arrays.
1. Introduction
The development of tidal turbine arrays is gradually
moving towards the development of full-scale
demonstration farms [1] and fully commercial arrays
[2]. Critical to the successful installation and operation
of these is the optimal positioning of turbines, both
within the farm or array itself and also in its respective
channel. This will ensure both cost effectiveness as
well as minimal environmental impact, which are likely
c 2013 The Author(s) Published by the Royal Society. All rights reserved.
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2
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to be critical to the success of any tidal farm or array project. Much research has examined flow
effects in marine current turbines, with the aim of contributing to the understanding of how
they affect the flow around them and perform under certain conditions. Garrett & Cummins [3]
examined issues such as the constraining of flow causing localized increase in velocity around
turbines. This work demonstrated that power output and overall efficiency were functions of flow
velocity in different regions, and hence were dependent on array size and proximity of arrays to
flow boundaries. This analysis was expanded further by Vennell [4], who looked at arranging tidal
turbine arrays such that the optimal flow velocities were achieved in these important regions. This
was done for different conditions including fixed numbers of rows of turbines and fixed crosssectional area blocked by turbines. The sensitivity of power output to flow velocities in different
regions demonstrated by these studies shows local flow magnitude increases and wake effects to
be of critical importance in tidal farm development.
Experimentation with tidal turbine rotors is the ideal method of investigating the many
parameters associated with their operation. However, the high costs involved make this method
unacceptable for smaller studies with relatively modest resources. Validated numerical modelling
therefore plays a crucial role in researching tidal turbine arrays. However, the many possible
variables associated with modern numerical modelling techniques means that a thorough investigation is required to ensure modelled results represent real-life effects as accurately as possible.
This paper outlines an investigation into the numerical modelling of actuator fences. These
have been used extensively in experimental work and numerical simulations to induce the
momentum changes in a flow which occur with tidal and wind turbines. Examples include an
investigation of tidal farm layouts (figure 1) in the study of Myers & Bahaj [5] and an analysis
of wind turbine wakes in the study of Sforza et al. [6]. Experiments were previously carried out
in the Chilworth hydraulics laboratory at the University of Southampton to examine wake and
local flow constraining effects associated with different positions of tidal turbine arrays in an
open channel [7]. Replicating experimental conditions using Ansys CFX software required the
construction of a mesh of the geometry of the experimental flume using the ICEM mesh generator.
Subsequently, a number of boundary conditions needed to be specified for different regions of
the mesh, including no-slip walls for the channel sidewalls of the flume and a free-slip wall to
represent the free surface of the flow. The inlet of the flume, where water is pumped into the
channel from the sumps, also needed to be specified, along with some mathematical expression
to represent the flow velocity magnitude throughout the depth of the flow at the inlet. A number
of different mathematical expressions were used to define this flow velocity magnitude. This was
to examine how the development of the flow changed with each expression and to examine which
method most closely replicated experimental conditions. Determining which method for inflow
velocity definition is most accurate can justify its continued use in future numerical modelling.
rsta.royalsocietypublishing.org Phil Trans R Soc A 371: 20120246
Figure 1. Artist’s impression of a tidal turbine array.
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2. Literature review
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rsta.royalsocietypublishing.org Phil Trans R Soc A 371: 20120246
The use of momentum sink methods to simulate tidal turbines in numerical modelling is a
concept that has greatly simplified analysis of flow effects. Extensive work was carried out using
these techniques in the study of Harrison et al. [8]. The aim of the study was to compare the
characteristics of the wakes of actuator discs measured experimentally with those found using
computational fluid dynamics (CFD) analysis and the Reynolds-averaged Navier–Stokes (RANS)
equations. The experimental method involved measuring flow velocity magnitudes in wakes of
actuator discs in a circulating water channel. The numerical method involved using proposed
theory to calculate roughness coefficient and momentum sink for each actuator disc and applying
this to the regions of the flow in CFD in which the actuator disc was present. This momentum
sink method gave very similar far-wake properties in the numerical model to those experienced
in the flume during experiments. Here, the far wake is generally accepted to be the region five
actuator disc diameters or more downstream of the disc. For the actuator fence used in the
analysis presented in this paper, this diameter is represented by the actual height of the porous
rectangular fence.
The k–ω shear stress transport (SST) turbulence model was used in the simulation. The
justification given by the authors was that it performed better in applications where adverse
pressure gradients were present, and was suited to simulating a water flume due to the
development of boundary layers on the flume walls. The findings of this analysis showed that
velocity and turbulence parameters from CFD agreed well with those of experimental results.
Also as expected, the near-wake characteristics did not agree well between experiments and the
numerical model; however, in the far wake both methods correlated quite well. This is expected,
as momentum sink methods such as actuator discs and fences are not specifically designed for
the near-wake region. Device-generated turbulence and the initial velocity reduction in the near
wake of tidal turbines are not equivalent between actuator discs/fences and mechanical rotors.
However, a crucial factor mentioned in Harrison et al. [8] which justifies this method is that the
downstream spacing between tidal turbines in a farm is likely to be seven turbine diameters or
more, as with many wind turbine farms to date.
Momentum sinks were also used in Ansys CFX software in Lartiga & Crawford [9]. This
analysis aimed to examine methods of correcting for the wall blockage effects in wind and water
tunnels. These corrections are needed as blockage effects lead to inaccurate non-dimensional
turbine performance parameters, such as power coefficient (CP ) and tip speed ratio (λ). The
specific analytical correction methods that were analysed here were derived in Bahaj et al. [10].
The CFD analysis involved modelling both an unbounded domain with no blockage effects,
and a bounded water tunnel with various levels of blockage. Results showed that the analytical
expressions in the study of Bahaj et al. [10] agreed well with CFD for blockage of up to 10 per cent,
but that theory underestimated CP for higher blockage ratios.
The CFD software Ansys FLUENT was used in the study of Bai et al. [11] to investigate different
spacing of turbines in a tidal farm. A number of different turbines were added to a farm, and the
resulting effects on the central tidal turbine were examined. In the second instance, the number
of turbines in the farm was fixed at seven, but with different latitudinal and longitudinal spacing
used to examine the performance of the entire array. The computational domain used was almost
identical to the basic setup of a conventional water flume, with sidewalls and an inflow, outlet and
seabed. As with the work presented in Harrison et al. [8] and Lartiga & Crawford [9], horizontal
axis turbines were modelled using a momentum sink approximation. Once again the k–ω SST
turbulence model was used, with the authors justifying use of this method by stating that it
performs better in regions of adverse pressure gradients.
For the first investigated scenario of adding turbines to the farm, results showed that the
addition of turbines directly beside and downstream of the central turbine resulted in increases
in the central turbine power output of up to 7 per cent. However, adding turbines upstream
resulted in slightly decreased power produced. For the second scenario of varying the spacing
between a fixed number of turbines, results showed that spacing increased total array power
3
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(a) Previous experimental studies
Previous experimental work in the Chilworth hydraulics laboratory is outlined in detail in
Daly et al. [7]. A 21 m long and 1.37 m wide circulating water flume was used to carry
out the experimental work. Initially, an ambient flow map of the flume was taken using an
acoustic Doppler velocimeter (ADV). Flow velocity measurements were taken at several positions
downstream of the flume inlet and at lateral positions of 0.27, 0.54, 0.84 and 1.11 m from the
sidewall. Actuator fences were then used to represent tidal turbine arrays in a circulating water
flume. Wake velocity magnitude measurements were taken downstream of fences. Fences were
placed in different lateral positions 12 m downstream from the inflow to the flume to determine
how changing proximity to flow boundaries affects the structure of the wake of the array. The
12 m point was chosen, as examination of the ambient flow map of the flume showed the flow
had fully developed at this longitudinal location.
Results showed that the wake reduced in intensity and became more laterally extended as the
actuator fence was moved closer to the sidewall of the flume. A separate study was carried out to
examine the flow field between the actuator fence and sidewall owing to this change in proximity.
Three fences of varying porosity (ratio of open to total area of fence) were gradually moved from
the centre of the flume towards the sidewall. At each of these positions of the array, the flow
velocity magnitude at a fixed point 0.5 fence diameters out from the sidewall of the flume was
measured, and compared with the velocity at the same position when no fence was present in the
flume. Results of this analysis showed an increase in flow velocity of up to 15 per cent occurred
by decreasing the proximity of the array to the channel sidewall compared with that existing
upstream. The aim of the numerical modelling outlined here was to validate these experimental
results with a view to expanding on them in future research.
In order to compare experimental results to those of a numerical RANS solver, ICEM CFD
software was used to recreate the geometry of the Chilworth flume. ICEM CFD is geometry
creation software compatible with a number of numerical solvers. In this case, the Ansys CFX
solver was used to replicate experimental conditions.
(b) Geometry creation and meshing
After creating the basic geometry of the Chilworth flume in ICEM CFD, a region of the flow
domain, given the name ‘fence’, was also specified as its own part. This region represented the
......................................................
3. Experimental work and numerical modelling
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output by up to 1.3 per cent, but could also lead to decreases of up to 8 per cent. However,
as the numerical work was not validated against experimental work, it is difficult to draw any
quantitative conclusions.
These studies clearly showed many similarities in certain aspects of their numerical modelling
approach. All three used the momentum sink approach to simulate the changes to the flow
environment caused by an energy-extracting turbine. Both Harrison et al. [8] and Bai et al. [11]
used the same k–ω SST turbulence model, citing its ability to cope better with near-wall results
than other turbulence models. Lartiga & Crawford [9] also used a k–ω model, but it is not clear
whether this was the same model as used by the authors of the other two studies. However, all
three contrasted differed in the method in which they defined their inflow velocity. Both Harrison
et al. [8] and Bai et al. [11] used different boundary laws, whereby the velocity at any point from
the seabed or flume bed up to the water surface is some function of the height above the bed.
Lartiga & Crawford [9] simply gave a single value for inflow velocity of 2 m s−1 . None of the
authors gave any discussion of what other possible methods could be used, or any reason for
using their particular method over others available. For any study that looks at the hydrodynamic
effects of tidal turbines, determining the appropriate inflow velocity definition method could be
crucial. It would ensure that the use of these methods in future work to expand on knowledge
already gained would be strongly justified.
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5
region where the porous actuator fence used in experiments was present. The effect of the actuator
fence on the flow was replicated by applying an additional momentum source term to this region,
the value of which is directly related to the porosity of the actuator fence [8]. This was included
in the RANS equations by Ansys CFX. Isolating this momentum sink region was done using
the block splitting function of ICEM. A hexahedral mesh was applied to the geometry, with a
finer mesh around more critical areas such as the close to the walls of the flume and around the
actuator fence. The momentum source term S to be applied to this region is given by the following
expression
K ρ
Ud |Ud |.
(3.1)
S=
xi 2
Here Ud represents the flow velocity at the face of the actuator fence and K is the drag coefficient.
An approximation for K proposed by Taylor [12] and used in Harrison et al. [8] is given by the
expression
1
θ2 =
.
(3.2)
1+K
The resistance loss coefficient of the fence is then given by K/xi where xi is the thickness of
the fence, which in this case is 4 mm. The experimental conditions for a 0.38 porosity fence were
represented in CFX. Using equations (3.1) and (3.2), this corresponds to a value for resistance loss
coefficient of 1481. An example of the ICEM geometry and mesh is displayed in figures 2 and 3.
(c) Inflow velocity boundary condition
CFX-PRE was the software used for specifying the boundary conditions of the problem. In
specifying the inflow boundary conditions to the computational domain, it can be useful to
......................................................
Figure 3. Shaded area showing section of mesh representing actuator fence and hence momentum sink region.
rsta.royalsocietypublishing.org Phil Trans R Soc A 371: 20120246
Figure 2. Example of ICEM mesh.
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6
0.15
0.10
0.05
0.22
0.24
0.26
0.28
0.30
0.32
velocity magnitude
0.34
0.36
0.38
(m s–1)
Figure 4. Curves showing fit of expression of inlet velocity methods to experimental data. Crosses, measurement; long-dashed
line, exponential fit; short-dashed line, Dyer law; dotted line, 1/8th power law.
height above flume bed (m)
0.25
0.20
0.15
0.10
0.05
0.22
0.24
0.26 0.28 0.30 0.32 0.34
velocity magnitude (m s–1)
0.36
0.38
Figure 5. Velocity profiles from ambient flow map of Chilworth flume from experiments and inlet velocity methods. Crosses,
measurement; short-dashed line, Dyer law; long-dashed line, exponential fit; dotted line, 1/8th power law; solid line,
recirculation.
have some actual experimental data that can be used to define flow velocity and turbulence
parameters. The plane along which actuator fences were placed during the experiments in Daly
et al. [7] was 12 m from the inflow of the flume. A high-resolution flow field map was also taken
to determine the ambient flow conditions in the flume in the absence of any obstructions. The
closest downstream position to this at which data were taken from was 9 m downstream. This 9 m
position was therefore taken in CFX analysis as the inflow point to the computational domain.
The following sections outline the different methods for defining inflow velocity magnitudes
that were used. The curves of the expressions for the following three methods are shown in
figure 4, with the experimental readings from 9 m downstream of the inlet of the flume and 0.54 m
from the sidewall of the flume shown for reference. A comparison of the ambient flow predicted
by both the experiments and all the methods is shown in figure 5, with the location being 12 m
from the inlet of the flume and 0.54 m from the flume sidewall.
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0.20
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height above flume bed (m)
0.25
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Table 1. Results of exponential expression curve-fitting exercise.
7
b
c
d
SSE
R2
0.27
0.324
0.331
−0.323
−78.79
0.0001
0.99
0.54
0.316
0.401
−0.315
−85.31
0.0002
0.99
0.84
0.323
0.241
−0.323
−69.55
0.0001
0.99
1.1
0.311
0.22
−0.311
−70.38
0.0001
0.99
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(i) Power law
A commonly used method of defining inflow velocities, also used in the study of Bai et al. [11],
is to use a power-law profile. This is originally derived from boundary layer theory. This power
law assumes the no-slip condition, which states that there is some region of flow towards the
bottom of the channel where flow velocity magnitude equates to zero. The law further assumes
that throughout the remainder of the hydraulic depth Dh , the velocity U at any point is a function
of the distance from the bottom of the channel (Z). The expression for this power law is given by
U
Z 1/n
=
.
(3.3)
Umax
Dh
Here, Umax is the maximum flow magnitude throughout the depth and n is any arbitrary constant.
Although it is possible to use any value for n to attempt to replicate flow conditions, a comparison
of several values for n was considered to be beyond the scope of this study. Therefore, one of
the most commonly used values, 8, was inputted to the CFX simulation. Based on experimental
results, 0.34 m s−1 was chosen as the value for Umax .
(ii) Exponential velocity profile expression
The next method tested was to set the inflow velocity conditions to the CFX simulation as close
as possible to the boundary profile that was observed at the inflow point of the flume. The 9 m
downstream from the flume inflow point was chosen as the inflow point to the CFX simulation.
Data for flow velocity magnitude against vertical height from the bed were entered into MATLAB
software, and a curve was fitted to these data in an attempt to determine an expression which
would give the closest value possible to the observed flow velocity magnitude for any input
value of height above the bed. Several possible curve fittings were attempted, and it was found
that fitting an exponential curve gave the most accurate results. The generic expression given
by this curve-fitting exercise for any function f (x) whose value is dependent on the value of an
independent variable x is
(3.4)
f (x) = aebx + cedx ,
where a, b, c and d are arbitrary constants. In this case, f (x) will be the flow velocity, and the
independent variable x is the vertical height above the flume bed. At the 9 m point from the inflow
during experimentation, the change in flow velocity with height was recorded at four lateral
positions along the width of the Chilworth flume. These were at distances from the sidewall
of the flume of 0.27, 0.54, 0.84 and 1.11 m. The data recorded were entered into the curve-fitting
exercise to determine values for a, b, c and d, as well as the value for certain statistical measures
of accuracy. Table 1 gives the values returned for each data entry. Here, SSE represents the sum
of squared errors, while R2 represents the coefficient of determination. This has been defined by
Ryan et al. [13] as the fraction of the variation in the dependent variable that can be explained by
the fitted equation. It can therefore be considered a measure of the accuracy of the equation in
predicting observed real-life values, with a maximum of 1.
As table 1 clearly demonstrates, the expressions outputted by MATLAB very accurately
replicate the experimental conditions at the 9 m downstream position, and the differences
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a
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lateral
position (m)
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Table 2. Results of Dyer boundary law curve-fitting exercise.
8
A
SSE
R2
0.27
0.009481
0.1663
0.0001633
0.9226
0.54
0.009513
0.1661
0.0001123
0.9458
0.84
0.009838
0.1523
0.000278
0.8829
1.1
0.009076
0.1548
0.0002792
0.8647
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between each are very small. Given these small differences and the time required, it was decided
that using all four expressions for simulation was unnecessary. Given the small differences
between them, any one of the lateral positions could have been chosen. In the end, the expression
from a lateral position of 0.54 m was chosen.
(iii) Dyer boundary law
The third method to be tested was the Dyer boundary law. The theory behind this method is
outlined in detail in Dyer [14], and it was also used in the analysis in Harrison et al. [8]. The
Dyer boundary law states that velocity at any depth y (U(y)) is given as a function of the depth
according to the following relationship:
∗
yU
+ A,
(3.5)
U(y) = 2.5U∗ ln
ν
where U∗ represents the friction velocity, A is a constant and ν represents the kinematic viscosity
of water. The values for U∗ and A were found by curve-fitting experimental data to the above
equation. This was done by curve-fitting data from the 9 m downstream point of the Chilworth
flume, as was done with the curve fitting for the exponential velocity profile expression. This is
done using the curve-fitting tool of MATLAB, and using the custom equation creator to specify the
above equation as that to which the data are fitted. This was done with all the lateral data found
from the Chilworth flume. The values for U∗ and A, as well as the sum of squared errors and
R2 values found from the curve-fitting exercise, are displayed in table 2. To ensure consistency
between all inflow velocity methods, the data for the 0.54 m lateral position fit were used in
Ansys CFX.
(iv) Flow recirculation
The final method used differs quite substantially from the other methods used and those used in
the studies of Harrison et al. [8], Lartiga & Crawford [9] and Bai et al. [11]. It involved using the
interface capabilities of Ansys CFX to constantly recirculate flow from the outlet of the Chilworth
flume back to the inflow, in order to allow the flow to continually develop. With the other methods
which have been previously discussed, it was not possible to tell whether or not the flow had
actually reached a point where it had fully developed by the time it had reached the outlet
of the domain. It is possible that if a longer computational domain were constructed, the flow
may have continued to develop within the numerically modelled domain. By allowing flow to
recirculate from the outlet back to the inflow continually, it was possible to allow the flow to
continually develop until it had reached a steady-state condition immediately upstream of the
numerical domain.
Applying recirculation like this in Ansys CFX required some different boundary conditions
from the other methods. Rather than a domain inflow and outlet, these areas were instead defined
as two sides of an interface. Then a constant mass flow rate was defined across the interface. The
flow then automatically developed throughout the depth and width of the flume according to
other boundary conditions. Also unlike other methods, turbulence boundary conditions did not
need to be specified, as the development of turbulence was automatic according to the roughness
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U∗ (m s−1 )
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lateral
position (m)
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(d) Other boundary conditions
Following application of the correct conditions for the inflow velocity, the remaining boundary
conditions were specified. The sidewalls of the flume and the bed were set as no-slip walls, while
the flow surface, as it was not a physically constrained boundary, was set as a free-slip wall.
As all real walls are not hydraulically smooth, any information on how rough they might
be is needed for CFD simulations so that turbulence can develop accurately. This is particularly
important in the case of the recirculation method, as turbulence parameters cannot be specified
here and so the development of turbulence is entirely driven by wall roughness. CFX requires an
equivalent sand grain roughness to be specified for each wall, rather than the actual roughness
itself. The relationship between sand grain roughness ks and Manning’s coefficient n was
originally developed by Nikuradse [15] and was expanded on subsequently by various authors.
A final expression derived by Webber [16] and also presented in Marriott & Jayaratne [17] is
n 6
.
(3.7)
ks =
0.0389
Manning’s coefficient for glass of 0.010 was entered into the above equation, and the equivalent
sand grain roughness subsequently calculated entered into CFX.
As the method of inflow velocity definition, it was also possible to specify turbulence
conditions for the inflow boundary condition. This was necessary for all the methods except the
recirculation method for reasons already mentioned. Expressions were entered for the turbulent
kinetic energy k and eddy dissipation ε. These quantities are functions of the flow velocity at any
point and are given by the following expressions:
3
2
k = I2 Uavg
2
(3.8)
k3/2
,
0.3Dh
(3.9)
and
ε=
where I represents the turbulence intensity, Uavg represents the average freestream flow velocity
and Dh represents the water depth, which in this case is 0.3 m. The turbulence intensity entered
was given by the following expression:
−1/8
I = 0.16Rel
,
(3.10)
where Rel is the Reynolds number associated with the characteristic length. If the hydraulic
diameter (which in this case is the water depth) of the flow is L, then the characteristic length
is given by
l = 0.07L.
(3.11)
The flow velocity used to compute the Reynolds number was 0.34 m s−1 , which, as with the 1/8th
power law in numerical modelling, was chosen on the basis of observed experimental results.
......................................................
where ρ is the fluid density and A the cross-sectional area of the flume. The mass flow rate was
then applied and the flow was allowed to fully develop. Once it had developed fully, the results
for one plane of the flume were exported and used as the inflow condition for all meshes where
an actuator fence was present.
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of the flume bed and the behaviour of the RANS equations. Once the flow had constantly been
recirculated and found to have fully developed, a plane section of flow conditions was taken and
then used as the inflow condition for the case where an actuator fence is present in the flume.
The mass flow rate through the flume was found from interpolation of the experimental
data. The depth of flow was divided into the eight sections of the depth from which flow
velocity measurements were taken. The mass flow rate for each section was then found using
the expression for mass flow rate ṁ:
ṁ = ρ ŪA,
(3.6)
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Table 3. Results of mesh independence study.
10
−1
9 m from
inlet
12.7 m
15 m
18 m
21 m
acceleration point
(figure 6)
1 626 570
0.3119
0.1097
0.2562
0.2883
0.3046
0.3317
3 107 982
0.3119
0.1081
0.2562
0.2886
0.3052
0.3322
6 339 168
0.3118
0.1064
0.2561
0.2888
0.3054
0.3327
extrapolated value
0.3119
0.1013
0.2565
0.2900
0.3079
0.3339
% fine error
0.028
1.654
0.07
0.044
0.046
0.172
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Finally, the k–ω SST turbulence model was selected. It was decided its good performance with
adverse pressure gradients and separated flow meant it was ideal for the current flow problem.
(e) Mesh independence study
Because of approximations and averaging used in the RANS equations, the number of cells in any
given CFD mesh can directly affect the results after processing. A greater number of cells results in
a more accurate solution to a problem for a given set of boundary conditions and solver settings.
However, increasing mesh density also requires greater processing time. For any given problem,
the optimal mesh gives sufficiently accurate results for certain important flow parameters in the
shortest possible processing time. In this particular study, those important parameters are the flow
velocity magnitudes throughout the computational domain. To verify the robustness of the mesh
used, the flow velocity magnitude in the direction of flow at six different points was gathered for
the scenario where an actuator fence was placed in the centre of the flume. The points chosen were
at the mid-height and mid-lateral position 9 m from the inflow to the Chilworth flume (the inflow
point to the CFX domain for reasons explained previously), 12.7, 15, 18, 21 m and the point 50 mm
from the sidewall of the flume at which increases in flow velocity magnitude were observed and
measured during experimentation. The 12.7 m position was examined as this lies in the immediate
far wake of the actuator fence when it was in the flume centre, which was placed at 12 m from the
inflow to the Chilworth flume. As replicating far-wake effects is what the actuator fence method
is specifically designed for, ensuring robustness in this region of flow is of critical importance.
The method used is a standard method recommended by the American Society of Mechanical
Engineers (ASME), outlined in Celik et al. [18] and also used in Harrison et al. [8]. The authors
describe a grid convergence method, which is based on the Richardson extrapolation method and
has been used and tested in hundreds of CFD studies. The interested reader is referred to Celik
et al. [18] for a more in-depth description of this method. The results are displayed in table 3.
4. Results and discussion
(a) Mesh independence study results
The results of the mesh independence study are given in table 3. This was carried out for the
actuator fence positioned in the centre of the flume. The inflow velocity definition method used
for this was the 1/8th power law. The parameters across the top row are the X-direction flow
velocity magnitudes in the centre of the flume at various distances downstream of the inflow.
The last column records flow velocity magnitude at the point where flow localized acceleration
was recorded in experiments. The parameters down the first column are some of the calculated
values which [18] instructs to report. The coarse, medium and fine meshes had element numbers
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number of
mesh elements
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velocity at various longitudinal positions downstream of inlet (m s )
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positions of actuator fence centre
position of ADV
Figure 6. Graphical display of positions of actuator fence and ADV in Chilworth flume for flow acceleration analysis.
X direction velocity (m s–1)
0.40
0.38
0.36
0.34
0.32
0.30
0.28
0
0.1
0.5
0.2
0.3
0.4
distance between sidewall of flume and
edge of actuator fence (m)
0.6
Figure 7. Flow velocity magnitudes at centre depth and at distance of 0.5 fence heights from flume sidewall caused by
constraining of flow by actuator fence. Cross symbol, measurement; open square, exponential fit; filled square, Dyer law; filled
circle, 1/8th power law; open circle, recirculation method.
of 1 626 570, 3 107 982 and 6 339 168, respectively. In most of the parameters the errors are almost
negligibly small. The largest errors occur at the point of the flume where velocity measurements
were taken, and also at the 12.7 m point, which is in the far wake of the fence. The most probable
reason for this point having higher deviation between meshes is that it is in an area where there
are greater fluctuations in velocity magnitude between cells in all directions. Even these errors
are not higher than 2 per cent. Overall, the results appear to support the view that the mesh with
3 107 982 cells gives sufficient accuracy to allow it to be used more extensively.
(b) Constrained flow analysis
Figure 6 gives a plan view description of the experimental domain. An ADV was placed at a
fixed point 50 mm from the sidewall of the flume and three actuator fence heights downstream
of the fence. The fence was initially placed in the centre of the flume, and then gradually
moved closer to the channel sidewall. For each of the positions, the flow velocity magnitude
is measured by the ADV. This procedure was replicated in Ansys CFX, with the inflow velocity
being defined using all the methods previously outlined. The results of each of these methods
compared with experimental results are shown in figures 7 and 8. Figure 7 gives the flow velocity
magnitudes recorded by the ADV at the point specified in figure 6. Figure 8 gives these ADV
readings normalized against the natural flow velocity present at the same point with no actuator
fence present.
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actuator fence
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1.22
12
1.16
1.14
1.12
1.10
1.08
1.06
1.04
0
0.1
0.2
0.3
0.4
0.5
0.6
distance between sidewall of flume and
edge of actuator fence (m)
Figure 8. Non-dimensional increase in flow velocity at centre flow height and at distance of 0.5 fence heights from flume
sidewall caused by constraining of flow by actuator fence. Cross symbol, measurement; open square, exponential fit; filled
square, Dyer law; filled circle, 1/8th power law; open circle, recirculation method.
Both figures show the same general trend. The flow velocity magnitude and increase compared
with ambient conditions at the fixed point is highest when the actuator fence is closest to the
sidewall of the flume, with a gradual reduction as the fence is moved further towards the
centreline of the channel. This is an expected result and can simply be explained by the continuity
equation. If the flow has less area through which to flow, the flow velocity needs to increase so
as to ensure continuity of mass, with the effect becoming less pronounced as the area expands.
However, both figures show quite contrasting performance of each numerical method in terms
of both how they compare with measurements and the percentage increase in velocity due to
constraining.
If figure 7 is examined independently, it can be seen that the recirculation method and 1/8th
power law have results which replicate those of experiments most closely. Significant deviations
from experimental results can be seen with both the Dyer boundary law and exponential curve
fit. One reason postulated for the poor accuracy of these particular methods is that they may
not replicate the natural boundary layer development off the bed of the Chilworth flume.
Experimental work has shown that there is a gradual growth in the boundary layer from the inlet
of the flume until the vertical velocity profile stabilizes at approximately 9–12 m downstream. This
is analogous to the development of flow over a flat plate, as explained by Hamill [19]. In general,
the boundary layer thickness settles to a constant value at some point downstream where the
energy loss disruption to flow is replaced by momentum being transferred from faster moving
upper layers of fluid to the slower lower layers. With the Dyer boundary law, Ryan et al. [13]
claim that flow follows this profile in a region which begins just outside the viscous sublayer
and finishes at 10–20% of the overall hydraulic depth. But Ryan et al. [13] also suggest that
power-law profiles have been shown to be more appropriate for describing flow development
in this outer region. Also crucial is that the 1/8th power law has also been extensively used in
pipe flow, where the development of boundary layers is similar to that which occurs with flat
plates. When considering the performance of the recirculation method, the no-slip condition is
an important consideration. As this method does not strictly define flow velocity throughout the
depth, the application of the no-slip boundary condition in CFX is one which will dictate flow
development, along with wall roughness and turbulence parameters. This is therefore also more
likely to replicate the boundary layer development than the exponential fit and Dyer boundary
layer profile.
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1.18
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X direction velocity (m s–1)
1.20
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actuator fence
Figure 9. Graphical display of positions of fence centre in Chilworth flume for wake analysis.
Figure 8 shows some interesting results for the prediction of increase in flow velocity compared
with ambient conditions. When comparing results all numerical methods predict the measured
localized increases in velocity owing to the constrained nature of the flow to within 5 per cent.
In fact, the exponential curve fit, 1/8th power law and Dyer curve fit all have almost identical
results to each other. This is an interesting result, as it shows that the diversion of flow around
the actuator fence and the subsequent increase in local flow velocities is a phenomenon which is
replicated equally by all methods. This diversion of flow appears to be handled quite differently
by the recirculation method. Although in some instances it predicts magnitude of velocity increase
close to the values of experiments, it does give much higher values at certain points. The fact that
it does not have identical results to the other inflow methods is another observation which can
most probably be attributed to the lack of strict rules on flow velocity through the depth.
The results of the above analysis would appear to suggest that the 1/8th power law is overall
the method whose results correlate closest to those of experiments, and appear to suggest that it
may be the most suitable for future numerical modelling of constrained flow. The next section will
outline how all the inlet velocity definition methods compare when analysing the wake effects
associated with tidal turbine arrays.
(c) Wake modelling results
Figure 9 shows a plan view of the setup for the wake modelling of the actuator fence. The
fence was initially placed in the centre of the flume, and the ADV was used to take flow
velocity measurements at several points downstream. This procedure was then repeated for cases
when the fence was moved closer to the sidewall. These cases were 200 mm, or two actuator
fence heights from the centre, and 400 mm or four actuator fence diameters from the centre.
Results of this wake mapping outlined in Daly et al. [7] showed a smaller reduction of velocity
as the fence was moved closer to flow boundaries, and also the wake became more laterally
stretched. Figures 10–12 show the velocity profiles downstream of the fence along a vertical plane
intersecting the centre of the fence. Both experimental and numerical results (using CFX) are
presented.
These results show that each numerical method displays the same general trend, with a quite
dramatic decrease in velocity in the region directly behind the actuator fence. The low velocity
magnitudes towards the mid-height of the wake are simply explained by the dramatic and
sudden drop in momentum owing to the presence of the fence, while the higher values both below
and above the fence are simply because of a gradual recovery towards freestream conditions
owing to flow being diverted around the edges of the fence. When comparing the correlation
of each velocity inflow method with experimental results, a very noticeable difference between
each method is in the lower 10 per cent of the depth. The 1/8th power law fits quite well in
this region, while other methods deviate quite substantially from experimental results. This once
again supports the postulation made in the previous section that power laws perform better in
regions well outside the viscous sublayer, further than the initial 10 per cent of the depth. It also
supports the postulation that the Dyer law and exponential curve fit are incapable of replicating
the natural boundary layer development in the Chilworth flume. As for the recirculation method,
the deviation of this method from experimental results could be attributed to the relative lack
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positions of actuator fence centre
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13.7 fence heights
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0.20
0.15
0.10
0.05
0
0.10
0.15
0.20
0.25
X velocity (m s–1)
0.30
0.35
Figure 10. Plot showing velocity profiles behind actuator fence from experiments and CFX when fence is positioned in centre of
flume. Cross symbol, measurement; long-dashed line, exponential fit; short-dashed line, Dyer law; dotted line, 1/8th log profile;
solid line, recirculation method.
height above flume bed (m)
0.30
0.25
0.20
0.15
0.10
0.05
0
0.10
0.15
0.20
0.25
X velocity (m s–1)
0.30
0.35
Figure 11. Plot showing velocity profiles behind actuator fence from experiments and CFX when fence is positioned 200 mm
from centre of flume. Cross symbol, measurement; long-dashed line, exponential fit; short-dashed line, Dyer law; dotted line,
1/8th log profile; solid line, recirculation method.
of control of flow development associated with it. While the other methods quite strictly control
the flow velocity magnitude throughout the depth, the recirculation method relies solely on the
inputted mass flow rate, wall roughness and turbulence parameters. It is not clear how sensitive
the results of the analysis are to each of these parameters, but getting values for each which are
exactly the same as in the Chilworth flume may be quite difficult. As an example, the mass flow
rate was calculated using interpolation from experimental results taken at a lateral distance of
0.54 m from the sidewall of the flume. It is most probable that the mass flow rate would have
been different if experimental results were taken from a different position, leading to different
wake results.
There are also quite considerable deviations in other regions of the depth, in particular, at
depths furthest away from the centreline where simulations under-predict wake velocity deficits.
One possible reason for this particular disagreement between modelled and experimental results
may be a limitation of the use of momentum loss coefficients in CFX. The porosity of the actuator
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0.25
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height above flume bed (m)
0.30
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0.20
0.15
0.10
0.05
0
0.10
0.15
0.20
0.25
X velocity (m s–1)
0.30
0.35
Figure 12. Plot showing velocity profiles behind actuator fence from experiments and CFX when fence is positioned 400 mm
from centre of flume. Cross symbol, measurement; long-dashed line, exponential fit; short-dashed line, Dyer law; dotted line,
1/8th log profile; solid line, recirculation method.
fence used was 0.38; however, there is no definite rule for how the hole pattern of the fence
should be defined in order to achieve this. In operation, the highest induction factor and hence
momentum drop are close to the blade tips for a conventional horizontal axis turbine. Modelling
and simulating a membrane that has varying porosity might go some way towards achieving
better replication of full-scale conditions and closer agreement between experimental models and
numerical simulations.
However, it is not possible to account for this when applying the momentum loss coefficient
calculated directly from the porosity in CFX using equations (3.1) and (3.2). Instead the
momentum loss has to be applied either uniformly across the fence region, or by specifying
individual losses in the X-, Y- and Z-directions. The uniform loss was chosen in this instance, but
neither of these would be able to account for different fence hole patterns. Manually introducing
hole patterns would undoubtedly lead to significantly greater computational time but could be a
potential route to reducing the disparity between observed and simulated results.
The results of both the wake and constrained flow analysis appear to definitively rule out the
continued use of both the Dyer boundary law and exponential curve fits. While they may be
applicable to some particular instances, the above results strongly suggest that they are incapable
of replicating natural flow development of the flume at the Chilworth hydraulics laboratory, and
also do not perform as well as other suggested methods in replicating the wake of the actuator
fence. The only method which appears to perform consistently well throughout the flow depth
and as the fence is moved closer to flow boundaries is the 1/8th power law. The initial curve fit in
figure 4 suggested that it may not be the most accurate method. However, its strong performance
in the bottom 10 per cent of the flow depth is very significant, as it strongly suggests that it is
able to replicate the natural development of the flow in the Chilworth flume, as already outlined
by Hamill [19]. This could possibly be attributed to the fact that this method was originally
developed from pipe flow, where the gradual transition of the flow velocity magnitude from no
slip at the wall to freestream conditions is identical to that of a flat plate. The correlation of the
results of this method with experimental results in analysing both wake and flow acceleration
effects strongly supports its continued use in future numerical modelling of actuator fences.
5. Conclusions and future work
A study to investigate varying methods of defining the vertical velocity profiles at the inlet of a
numerical model simulating a channel containing porous surfaces has been presented. Numerical
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0.25
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height above flume bed (m)
0.30
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providing the funding for a PhD studentship.
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modelling was carried out using Ansys CFX software and for each inlet definition the structure
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size and flow acceleration magnitude in the surrounding hydrodynamic environment, as well as
the possible implications for the power output of other turbine arrays in the channel.
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