#1
*2
+3
&4
~5
# Industrial Doctoral Centre for Offshore Renewable Energy, The King´s Buildings, EH9 3JL, Edinburgh, U.K.
1 alberto.perez.ortiz@ed.ac.uk
* Institute for Energy Systems, The University of Edinburgh, The King´s Buildings, EH9 3JL, Edinburgh, U.K.
2 alistair.borthwick@ed.ac.uk
+ College of Engineering, Mathematics and Physical Sciences, University of Exeter, Penryn Campus, TR10 9FE, Penryn, U.K.
3 h.c.m.smith@exeter.ac.uk
& Alstom Renewable Power - Ocean Energy, 8 th floor, Castlemead, BS1 3AG, Bristol, U.K.
4 paul.vigars@power.alstom.com
~ Department of Naval Architecture, Ocean and Marine Engineering, University of Strathclyde, G4 0LZ, Glasgow, U.K.
5 qing.xiao@strath.ac.uk
Abstract— Worldwide, many coastal sites with the potential for tidal power generation fall into the category of a strait between an island and a landmass. This paper characterizes the resource at an idealised strait and assesses the influence of the presence of the landmass on the extracted power from the strait. The results show that the island geometry and associated flow dynamics can alter estimates based on ideal flows. The presence of the island has a major influence on the tidal resource, with the landmass having a positive effect on the rate of maximum extracted power from the strait. The results confirm that the maximum power extracted is not directly related to the power naturally dissipated by bottom friction or the undisturbed kinetic flux in the strait.
Keywords
Tidal power, resource assessment, shallow water equations, island, landmass
I.
I NTRODUCTION
The tidal energy industry is presently transitioning from testing single megawatt scale tidal turbines towards developing the first pre-commercial turbine arrays. The success of such arrays will determine the development of commercial arrays that produce power in the order of hundreds of megawatts. At such levels of power extraction, commercial arrays could alter the large-scale flow dynamics and sediment transport processes. It is therefore important to have an accurate understanding of tidal sites and the hydrodynamic consequences of power extraction.
Two approaches have been taken to develop an understanding of power extraction from tidal sites. The first approach focuses on idealised coastal sites [1], with the relatively simple geometries allowing considerable flexibility when undertaking parameter studies. The lessons learned from analyses of ideal coastal sites can guide developers in their process of identification and classification of tidal sites.
Draper [2] identified four generic categories of tidal sites: a channel linking two infinite oceans; a channel linking an infinite ocean with an enclosed bay; a headland; and a strait between an island and a landmass. Garrett and Cummins [3] and Bryden and Couch [4] respectively explored the maximum extractable power and the associated effects on the free surface and velocity distribution along a channel linking two infinite oceans. Blanchfield et al . [5] estimated the extractable resource from a channel linking an ocean and an enclosed bay, and examined how the resource was affected by the geometry of the bay. Draper et al . [6] investigated the potential of a fence located in the vicinity of a headland and concluded that the maximum extractable power was linked neither to the undisturbed kinetic power nor the power naturally dissipated by bottom friction.
A second approach is to assess the tidal power availability of actual coastal sites using numerical models with grids fitted to the coastal geometry and extending beyond the continental shelf [7]. Adcock et al. [8] investigated the maximum power available from the Pentland Firth, located between the north mainland coast of Scotland and the Orkney Islands, which could be categorized as either a channel linking two infinite oceans or a strait between an island and a landmass. Blunden and Bahaj [9] evaluated the tidal stream resource at Portland
Bill, a headland located on the south coast of England.
This paper presents numerical predictions of the tidal resource of an idealised strait between an island and a landmass. Examples of this type of tidal site that have already been subject to numerical analysis are the Inner Sound [8], a strait between the island of Stroma and the north coast of
Scotland, and the Johnstone Strait between the island of
Vancouver and west coast of Canada [10].
When there is sufficient tidal head driving the flow, the presence of an island, island-landmass, or group of islands leads to high tidal current velocities that are suitable for power
generation. Fig. 1 shows a possible classification of such
island-tidal systems: a.
Island of similar length and width near a landmass.
Examples include the Inner Sound [8], Ramsey Sound [11] off the west coast of Wales, and the Kinmen Strait in the
China Sea [12]. b.
Island off the coast, e.g. Pentland Skerries at the east side of the Pentland Firth, Scotland.

c.
Island with a high length to width ratio near a landmass, e.g. Johnstone strait off the west coast of Canada [10]. d.
Island with a high width to length ratio near a landmass, e.g. strait of Kylerhea between the Isle of Skye and the north-west coast of Scotland. e.
Two islands off the coast, e.g. Cook Strait in New Zealand
[13]. f.
Multi-island system off the coast, such as the Alas strait in
Indonesia [14].
Fig. 1. Island-tidal system configurations suitable for power exploitation.
Island-mainland straits have attracted the attention of tidal developers interested in assessing power resource in terms of combinations of depths and flow regimes. The offshore side of the island is usually associated with deeper water but not necessarily slower flow regimes (e.g. the Outer Sound in the
Pentland Firth [8]). However, as tidal turbine technology advances into deeper waters, it is worth investigating the limits of extraction on the offshore side of the island-landmass system [15] [16]. In the future, this would become a two-path island-landmass system, whereby extraction levels at each side of the island may be interconnected [17].
Atwater and Lawrence [18] investigated the power potential of a split tidal channel and analysed the effects of power extraction from one side of the island on the bypass flow and the total flow through the channel. Polagye and
Malte [19] expanded this study to multiple channel networks.
However, both Atwater and Lawrence [18] and Polagye and
Malte [19] assumed that the levels of power extraction would not affect conditions at the channel boundaries, which may not be the case for limited size straits.
The paper is laid out as follows. Section II describes the numerical model used in the study. Section III presents the numerical model validation predictions against experimental results for flow past a submerged circular cylinder at
Reynolds numbers of 40 and 100, and for oscillatory laminar shallow water flow past a surface piercing circular cylinder.
Section IV discusses the set-up of the numerical model of island and landmass. Section V presents the results of the resource assessment of an island near a landmass. Section VI presents the results of the resource assessment of an island further off the coast. Section VII lists the conclusions.
II.
N UMERICAL M ODEL
The work presented in this paper was carried out using the finite element numerical code Fluidity [20] which solves the non-conservative form of the shallow water equations (SWE):
𝜕𝜂
𝜕𝑡
+ ∇ ∙ (ℎ𝑢̅) = 0
𝜕𝑢̅
𝜕𝑡
+ 𝑢̅ ∙ ∇𝑢̅ + 𝑔∇𝜂 + 𝐶 𝑓
|𝑢̅|𝑢̅ ℎ
= 0
(1)
(2) where η is the elevation of the free surface, 𝑢̅ is the horizontal velocity vector, t is time, ∇ is the horizontal gradient vector, h is the total water depth, g is the acceleration due to gravity, and C f
is the bottom drag coefficient.
The model setup follows guidelines given by the Fluidity developers [21]. For coastal and tidal power extraction modelling using Fluidity SWE, Martin-Short et al . [22] provide a detailed explanation of the application of Fluidity to assess the effects of power extraction on flow regime and sediment transport processes in the Inner Sound. The mixed finite element discretization scheme uses a piecewise linear discontinuous Galerkin approximation for velocity (P1DG) and a continuous Galerkin, piecewise quadratic formulation for the free surface (P2).
Power extraction is included in the model through the addition of an absorption term k f in the momentum equations applied at the area defined as the tidal farm. The absorption term is treated implicitly in the same way the friction term C f is computed in the momentum equations.
III.
V ERIFICATION AND V ALIDATION OF THE F LUIDITY
S HALLOW W ATER N UMERICAL M ODEL
Pérez-Ortiz et al . [23] have carried out several verification and validation tests which confirm that Fluidity satisfactorily models the flow field in the vicinity of a typical island coastal site, accounting for water depth changes and power extraction.
Further validation is presented herein. Numerical predictions of flow past a submerged circular cylinder at Reynolds numbers 40 and 100 are compared against data obtained experimentally by Countanceau and Bouard [24] and numerically by Collins and Dennis [25]. Predictions of an oscillating laminar shallow water flow past a surface-piercing circular cylinder are also compared against the experimental measurements undertaken by Lloyd et al . [26].
A.
Flow past a circular cylinder
Coutanceau and Bouard [24] characterized experimentally the unsteady wake of a circular cylinder in an impulsivelystarted flow at Reynolds numbers between 20 and 40. Collins and Dennis [25] investigated numerically the wake length of a circular cylinder immersed in an impulsively-started unidirectional flow at Reynolds numbers of 40, 100 and 500.
Fig. 2 depicts the rectangular model domain used in the present numerical model. A cylinder of diameter Ø c
is positioned on the flow centreline of the model, a distance
80 Ø c
downstream of the western inflow boundary. The domain length and water depth are 200 Ø c and 0.02
Ø c
, respectively. The ratio of the cylinder diameter to overall domain width is 0.07. The impulsively-started flow generates a reflection at the cylinder that, in the absence of an effective dissipation mechanism, advects throughout the domain. The

domain length is chosen to be sufficiently long to prevent the reflection generated at the cylinder from reaching the boundary, reflecting again, and returning back to the cylinder position over the time period analysed.
Fig. 2. Rectangular model geometry and cylinder with five boundaries and two mesh regions used to simulate numerically the flow past a circular cylinder (not scaled in the x direction).
Fig. 3.
Early stage evolution of wake length at Re = 40: experimental results by Coutanceau and Bouard [24] (solid line); Fluidity predictions for Mesh 1
(diamond), 2 (circle), 3 (square), 4 (plus sign), 5 (cross) and 6 (point).
The model domain has five boundaries: two open boundaries Γ
1
and Γ
4
at the west and east sides of the domain; closed boundaries Γ
2
and Γ
3
at the north and south sides defined by a free-slip condition; and the circular cylinder Γ
5 defined by a no-slip condition. The model is set as an inflowoutflow channel with prescribed stream-wise velocity U
∞
at Γ
1 and a zero pressure condition at Γ
4
. The value of kinematic viscosity is used to yield the desired Reynolds number.
The domain is spatially discretised using the mesh generator software GMSH [27]. A different mesh edge length is defined for each of the three regions into which the domain is split: upstream of the cylinder; downstream of cylinder; and at the cylinder itself. Table 1 shows the mesh edge length used at each region of the domain and total number of mesh elements for each of the six spatial discretization cases analysed in the mesh convergence process.
TABLE I
S PATIAL D ISCRETIZATION C ASES A SSESSED IN THE M ESH C ONVERGENCE
A NALYSIS .
E LEMENT E DGE L ENGTH U SED IN THE T HREE M ESH R EGIONS AND
T OTAL N UMBER OF M ESH E LEMENTS .
Element edge length (m)
Mesh Upstream Downstream Cylinder
1
2
3
4
5
6
Cylinder
Ø c
Ø c
/2
Ø c
/4
Ø c
/4
Ø c
/4
Ø c
/4
Cylinder
3
Ø c
/4
3 Ø c
/8
3
Ø c
/16
3 Ø c
/16
3
Ø c
/16
3
Ø c
/16
π
π
π
Ø
π
π
π
Ø
Ø
Ø
Ø
Ø c c c c c c
/12
/28
/36
/76
/156
/316
Mesh
Elements
11,854
12,826
184,072
190,076
197,654
207,322
Figs. 3 and 4 plot the numerical predictions of early stage temporal variation of the wake length for Re = 40 and 100, where t * is the non-dimensional time, defined as t·U
∞ / Ø c
. The plots also depict the results from Coutanceau and Bouard [24] and Collins and Dennis [25]. It can be seen that the numerical predictions converge close towards the experimental and alternative numerical solutions as the mesh resolution becomes finer.
Fig. 4.
Early stage evolution of wake length at Re = 100: numerical results by
Collins and Dennis [25] (solid line): Fluidity predictions for Mesh 1
(diamond), 2 (circle), 3 (square), 4 (plus sign), 5 (cross) and 6 (point).
B.
Flow past a surface piercing circular cylinder in oscillatory laminar shallow water flow
The second set of validation tests concerns wake formation in oscillatory laminar shallow flows past a surface-piercing circular cylinder. Lloyd et al . [26] investigated the type of wake formation for different island geometries and flow conditions, in a laboratory basin. The rectangular domain is defined according to the dimensions of the shallow water flume tank (length 11 m, width 3.3 m, depth 0.06 m). A circular cylinder of 0.6 m diameter is located at the centre of the domain ( x = 0, y = 0). Bed and lateral wall friction is defined using a non-dimensional friction coefficient C f
=
10.592 x 10 -3 . Kinematic viscosity of water is ν = 10 -6 m 2 s -1 .
The spatial discretization of the domain is based on the specifications used for Mesh 4 in Table 1. However, due to the oscillatory nature of the flow, the element edge length upstream and downstream of the cylinder is set to 3
Ø c
/16. The total number of mesh elements in the model is 11,140.
A free-slip condition is applied at the north and south boundaries of the domain. A no-slip boundary condition is applied to the surface of the cylinder. To replicate the flow conditions in the laboratory basin used by Lloyd et al . [26], a half-wave sine inlet velocity boundary condition of period T =
140 s and amplitude U o
= 0.04 m/s is set at the west and east
ends of the rectangular domain (Fig. 5). For zero inlet
velocities at the boundaries, zero pressure and free-stress conditions are also applied.

Fig. 5.
Half-wave sine inlet velocity boundary conditions set at west (solid line) and east (dashed line) ends of the rectangular domain.
Stansby [28] observed that numerical models based on the hydrostatic shallow water equations cannot resolve properly the complex flow dynamics past surface-piercing conical islands. With this in mind, the Fluidity model is also run solving the unsteady 3D Navier Stokes equations. The spatial discretization in the x y plane used is the same as that in the
SWE version. The water column is spatially discretised using
15 sigma layers of uniform vertical length. The total number of mesh elements is approximately 167,100. No turbulence model is used to resolve the power dissipated due to the viscous forces being of smaller scale than the grid size.
Fig. 6 compares vorticity contour plots obtained
experimentally at a Keulegan-Carpenter number KC = 9.3 by
Lloyd et al . [26] with those computed using Fluidity SWE and
3D. The plots are extracted from the third tidal cycle. In general, the vorticity distributions from Fluidity SWE and 3D compare reasonably well with those from Lloyd et al . [26].
The flow direction around the circular cylinder and local variability in flow rotationality are reasonably well captured by both SWE and 3D. This provides confidence in the capability of Fluidity SWE to capture the flow dynamics around a surface-piercing circular cylinder in an oscillatory laminar shallow flow.
Figs 7 and 8 show the time histories of stream-wise velocities at x = 0 m, y = 0.45 m and x = 0 m, y = -0.45 m. The range, profile, and phase of the velocity time series are satisfactorily reproduced by Fluidity SWE and 3D. From the plots, it is not clear that Fluidity 3D provides a better approximation to the experimental results than Fluidity SWE.
Certain discrepancies in range and shape between the experimental and numerical solutions may have been influenced by differences between the numerical model and experimental set-up.
Fig. 6 Vorticity contour plots for oscillatory flow past a surface-piercing cylinder at KC = 9.3 during the third tidal cycle: experimental measurements by Lloyd et al . [26] (left column); numerical predictions by Fluidity SWE
(centre column); and numerical predictions by Fluidity 3D with 15 sigma layers (right column). The dark/light colour vorticity contour plots taken from
Lloyd et al . [26] correspond to blue/red colour vorticity contour plots in
Fluidity and indicate negative/positive vorticity. (a) t = T / 4, (b) t = T / 2, (c) t
= 5 T / 8, (d) t = 3 T / 4.
Fig. 7.
Time histories of non-dimensional stream-wise velocity component during 3 tidal periods at x = 0 m and y = 0.45 m at KC = 9.3: experimental data, Lloyd et al . [26] (solid line); numerical predictions by Fluidity SWE
(dashed line); and numerical predictions by Fluidity 3D with 15 sigma layers
(dotted line).

remainder of the domain. Here, the increase in water depth near the open boundaries is implemented to imitate the effect of a continental shelf and to prevent reflections from the island reaching the open boundaries [29] [30].
Fig. 8.
Time histories of non-dimensional stream-wise velocity component during 3 tidal periods at x = 0 m and y = -0.45 m at KC = 9.3: experimental data, Lloyd et al . [26] (solid line); numerical predictions by Fluidity SWE
(dashed line); and numerical predictions by Fluidity 3D with 15 sigma layers
(dotted line).
IV.
N UMERICAL M ODEL OF I SLAND AND L ANDMASS S ET UP
The definition sketch in Fig. 9 depicts the model geometry, tidal parameters, and location of the tidal farm used in the numerical assessment of tidal power extractable from a strait located between an island and a semi-infinite landmass. The rectangular domain comprises five boundaries: open west and east boundaries Γ
1
and Γ
4
; solid north and south boundaries Γ
2 and
Γ
3
, the latter representing the landmass; and a solid boundary Γ
5
which represents the island. The strait is effectively defined by boundaries Γ
3
and Γ
5
.
Fig. 10. Water depth contours in the island landmass domain.
The cross-sectional mean velocity at the narrowest section of the strait is defined as 𝑈 mean sea level at the Γ
1
and is set to an M ℎ 𝑤
= 𝑎 𝑜
𝐴 sin(𝜔𝑡) 𝑎 𝑜
= 0.5 (1 − cos ( 𝜔𝑡
4
))
Γ
4
are defined as h w zero surface elevation condition is set at Γ
4
. The surface elevation at
Γ
1 2 𝑜
. Sea surface elevations above
tidal constituent:
(3)
(4)
and h e
. A where A and ω is the amplitude and frequency of the M
2
tidal wave (3 m and 1.41 x 10 -4 rad/s respectively). The parameter a o
smooths the tidal signal during the first two tidal cycles:
In all cases, a free slip condition is applied at the north solid boundary Γ
2
. Free slip and no slip conditions are applied
(separately) at both landmass
Γ
3 and island
Γ
5 boundaries, and hereby referred as free-slip and no-slip scenarios.
Seabed friction is characterized by the dimensionless coefficient C f
= 0.0025. Turbulence is included using a depth averaged parabolic eddy viscosity 𝜀̅ [31]: 𝜀̅ = 𝑘
6
[𝐶 𝑓
(𝑢 2 +𝑣 2 )]
1/2 ℎ (5)
Fig. 9.
Definition sketch for a strait between island and landmass, a) model geometry and tidal parameters; and b) geometry and location of the tidal farm.
The dimensions of the coastal domain are given by the length L , width B and water depth h = 40m. The island is defined as an ellipse of length L i
and width B i
. Herein, the island is circular, such that L i
= B i
= Ø i
= 2000m.
The domain length and width are set to L = 70 Ø i
and B = 20 Ø i
. The width of the domain is chosen so that it enables the free stream velocity U
∞
to be approached at the north boundary, far from the island. The circular island is located midway along the domain in the stream-wise direction. The parameter s defines the width of the strait; in this case, s =
Ø i
.
Fig. 10 shows the water depth contours in the domain. The
water depth is set to a fixed value of h in the stream-wise direction for distances 25 Ø i
upstream and downstream from the island centre. The water depth is linearly increased from h to 75 h at distances 25 to 30 Ø i
upstream and downstream of the island centre. The water depth is set to 75 h throughout the where k = 0.41 is the von Kármán constant, and u and v are the stream-wise and transverse velocity components.
The tidal farm (or area where power extraction occurs) is located at the narrowest section of the strait and its geometry is defined by its plan-wise length L f
and width B f
.
A separate mesh convergence analysis is performed for the free-slip and no-slip scenarios with steady-state boundary conditions. The convergence analysis is based on the mesh edge length cases used in the flow past a cylinder validation test, and presented in Table 1. For the free-slip scenario, the solution converges on Mesh 4. For the no-slip scenario, although the solution converges at the narrowest section of the strait for Mesh 4, a further increase in mesh resolution is required to resolve the wake behind the island. Although the wake behind the island is not accurately captured, the results from the validation test against the measurements by Lloyd et al . [26] show that Mesh 4 captures reasonably well the primary flow features in the vicinity of the island.

Fig. 11 shows the unstructured mesh used to discretise the
island-landmass domain. The mesh contains 8,027 vertices and 16,054 elements. A regular grid of 80 isosceles triangles is situated in the area where power extraction is implemented.
The width B f
of the farm is equal to s . The length L f
of the area of power extraction is equal to h and it is of same order of size as the element mesh edge length used to discretise the island and strait regions.
Fig. 11. Unstructured spatial discretization of the island landmass domain, with a regular grid used to delineate the tidal farm at the strait.
The time step is chosen to comply with the Courant-
Friedrichs-Lewy (CFL) stability criterion. The simulation is carried out for seven tidal periods, with the first two tidal periods corresponding to ramp-up of the system. The following two tidal periods correspond to spin-up of the system. The final three tidal periods are used for assessment of the tidal resource at the strait. tidal period-averaged results: undisturbed kinetic power 𝑃̅ 𝑘𝑜
, defined as the kinetic power in the strait with no power extraction; natural power dissipated on the seabed in the strait in the absence of power extraction 𝑃̅ 𝑠
; kinetic power in the strait 𝑃̅ 𝑘
with the tidal farm present; and power extracted from the flow by the tidal farm 𝑃̅ 𝑒
. 𝑃̅ 𝑒
represents an upper limit to power extraction. Power available for tidal turbines will be lower than 𝑃̅ 𝑒
because the methodology employed for power extraction does not account for losses due to support structure drag and mixing around tidal turbines [32]. It appears that the free-slip and no-slip cases represent lower and upper bounds on power extracted depending on the boundary condition selected. Flow separation occurs solely in the no-slip scenario, leading to the disparity in the results between the two scenarios. However, the results obtained for the non-uniform seabed scenario show that for a more realistic case, the value of maximum power extracted lies between that obtained for the free-slip and no-slip scenarios.
The maximum extracted power does not exhibit a relationship with the natural power dissipated in the strait in any of the three cases. The results from the no-slip case may indicate that the extracted power is linked to the undisturbed kinetic power. However, this does not occur in either the freeslip or the non-uniform seabed scenarios. The slope of decrease in the kinetic power along the strait is higher for the free-slip and non-uniform seabed scenarios than for the noslip scenario when k f
< 0.5, but is approximately the same for k f
> 0.5.
V.
I SLAND IN THE V ICINITIES OF THE L ANDMASS
This section presents the tidal power resource assessments obtained for the strait between an island and a landmass.
Three scenarios are considered: a free-slip condition at the solid boundaries of the island and landmass; a no-slip condition at the island and landmass boundaries; and a domain
with a water profile set-up at the strait region (Fig. 12) hereby
referred as the non-uniform seabed scenario.
In this scenario, water depth is increased linearly from h = 5 m at the island and landmass boundaries to h = 40 m at a distance 0.1
Ø i
off the boundaries.
Fig. 12.
Top (left) and lateral (right) sketch of water depth profile defined in the island-landmass non-uniform seabed scenario. h = 5m (solid line), h =
20m (dashed line) and h = 40m (dotted line).
The free-slip scenario aims to reduce the complexity of the analysis. However, the results from the validation test against
Lloyd et al . [26] show that a no-slip condition on the circular cylinder is necessary to recreate realistic flow conditions and wake generation.
The three island-landmass scenarios are run for a range of extraction levels k f
between 0 and 4.5. Fig. 13 shows three
Fig. 13.
Power profiles as functions of k f
for a strait between an island and landmass: free-slip (black), no-slip (red) and non-uniform seabed (green) scenarios. Extracted power for tidal farm located in the strait (solid line), kinetic power for the strait with the tidal farm present (dash-dot line), kinetic power for undisturbed conditions in the strait (dotted line) and natural power dissipated on the seabed at the strait (dashed line).
It is interesting to investigate the ratio of the power extracted 𝑃̅ 𝑒
to the undisturbed kinetic power 𝑃̅ 𝑘𝑜
, 𝑃̂ , as an estimate of the latter may be the only information available to developers during the preliminary stages of a project. For maximum power extraction 𝑃̅ 𝑒
the power ratios are 𝑃̂ = 0.49,
0.96, and 0.74
for the free-slip, no-slip, and non-uniform seabed scenarios respectively. The value of 𝑃̂ for the nonuniform seabed scenario is close to the mean of the free-slip and no-slip values, which perhaps provide lower and upper bounds on the estimates of power extracted.

At maximum 𝑃̂ , the consequent flow deficit will make tidal projects commercially unviable and lead to unacceptably large flow changes from an environmental point of view.
In order to better understand what the levels of power extraction k f
represent, it is possible to convert k f
into an approximate equivalent number of turbines N
T
MW power rated P
R
. Consider a 1 turbine of 20 m diameter Ø
T
with cut-in speed U
C of 1m/s and rated speed U
R
of 2.5 m/s. The power output of the turbine is defined by the power coefficient C
P function:
𝐶
𝑃
= 0 , 𝑈 < 𝑈
𝐶
𝐶
𝑃
= 0.4, 𝑈
𝐶
< 𝑈 < 𝑈
𝑅
𝐶
𝑃
=
2𝑃
𝑅 𝜌𝐴
𝑇
𝑈 3
, 𝑈 > 𝑈
𝑅
(6)
(7)
(8)
The power generated and capacity factor figures presented in Table 2, when combined with an economic model, will identify the optimal economic number of turbines at the tidal farm. However, the reduction in flow velocities at the optimal economic power extraction level may cause unacceptable environmental effects at the site.
Fig. 14 plots the ratio of the three tidal period-averaged
scalar flow to the scalar flow in the absence of power extraction across the section at the middle of the strait and across a section of equal length on the offshore side of the island for the various island-landmass scenarios and levels of power extraction considered. The diminishing rates in kinetic
power across the strait shown in Fig. 14 are in agreement with the trends of the scalar flow across the strait. Fig. 14 also
reveals that the reduction in scalar flow through the strait is not converted into an equivalent increase of flow rate through the offshore side of the island. This means that there is a fraction of the power lost from the system which is not recovered.
The above power curve is valid for an isolated turbine or for a small array of turbines. However, turbines in large arrays will experience differences in C
P
that will depend on the dynamics of the site and tidal array arrangement [33]. Due to the dependency between C
P
, power extracted by turbines and flow reduction in the site, maximizing power output from turbines will require a certain degree of design tuning [34].
Let the projected area of the turbine support structure A
S
be equivalent to 10% of the turbine rotor swept area A
T
. Both thrust C
T
and supporting structure drag C
D
coefficients are assumed constant and equal to 0.8 and 0.9 respectively. Thus, the equivalent number of turbines is computed as:
𝑁
𝑇
=
𝐴
𝑆
𝐶
2𝑘
𝐷 𝑓
𝐴 𝑓
+ 𝐴
𝑇
𝐶
𝑇 where A f
(9)
is the area through which power is extracted from the flow, in this case it is equal to the area of the tidal farm.
Table 2 presents the equivalent number of turbines N
T
for extraction levels k f
between 0 and 1.12. Table 2 also includes three tidal period-averaged generated power 𝑃̅
𝑇 by the tidal array based on N
T and C
P
curve. The capacity factor CF of the tidal farm is computed based on 𝑃̅
𝑇
, N
T
and P
R
. Finally, Table
2 shows the percentage decrease in the mean strait velocity 𝑈 ∗ 𝑜 and mean kinetic power in the strait 𝑃̅ ∗ 𝑘 over three tidal periods against the case with no power extraction in the strait.
TABLE II
E XTRACTION L EVELS AND E QUIVALENT NUMBER OF TURBINES IN THE STRAIT .
T HREE T IDAL P ERIOD -A VERAGED T IDAL F ARM P OWER G ENERATED , T IDAL
F ARM C APACITY F ACTOR , P ERCENTAGE D ECREASE IN M EAN S TRAIT
VELOCITY AND P ERCENTAGE D ECREASE IN M EAN K INETIC P OWER .
Fig. 14. Changes in the scalar flow to flow in undisturbed conditions ratio for the free slip (black), no slip (red) and bathymetry (green) scenarios for different levels of power extraction. Scalar flow across the tidal farm (solid line) and through a cross-section of identical length at the offshore side of the island (dashed line).
To understand better the relationship between levels of power extraction and bypass flow, the free slip scenario is run at steady-state conditions with h w
- h e
= A.
Fig. 15 and 16
show the velocity streamlines without power extraction and with high power extraction where k f
= 2.24. The large resistance in the strait affects the flow path such that the streamlines divert towards the offshore side of the island where the resistance is lower. k f
0.02 0.03 0.07 0.14 0.28 0.56 1.12
N
T
10 20 40 80 160 320 640
𝑃̅
𝑇
[MW] 3.7
CF [%] 37.2
7.2
35.9
13.4
33.5
23.2
29.0
35.5
22.2
47.2
14.8
54.2
8.5
̅
𝑃̅
∗ 𝑜
∗ 𝑘
[%]
[%]
1.6
5.4
3.1
10.2
5.9
18.8
10.7
32.1
17.7
48.6
26.9
65.3
37.6
79.1
Fig. 15.
Steady-state velocity streamlines for the free-slip island-landmass scenario in the absence of power extraction at the strait.

Fig. 16.
Steady-state velocity streamlines for the free-slip island-landmass scenario with power extraction in the strait corresponding to k f
= 2.24.
Draper [2] found that the power at a strait between an island with a high width to length ratio and a landmass is different to that predicted for a channel connecting two infinite ocean basins [3] because of the change in the driving head caused by the presence of power extraction. Draper [2] concluded that when this difference was accounted, the
Garrett and Cummins channel model [3] provided a satisfactory representation of the basic physics in the strait given that there was negligible bypass flow at the island
geometrical scale assessed. However, Fig. 14 shows that the
bypass flow is not negligible in this case. Fig. 17 plots the
head driving the flow in the strait h wi
– h ei
(see Fig. 9) using
the free-slip scenario for zero, low k f
= 0.14
and high k f
= 2.24
power extraction levels. Increasing power extraction raises the amplitude of the head driving the flow. However, the calculation of power that can be extracted from a channel linking two infinite ocean basins assumes that the head driving the flow is constant and unaffected by the levels of power extraction. system is considered as an isolated offshore island. Power extraction takes place at the south side of the island over a rectangular area of dimensions L f x B f
, which is equal to that used in the island-landmass system. The model is spatially discretised, following Mesh 4 (Table 1).
As in Section V, both free-slip and no-slip scenarios are
considered. From the resource assessments in Fig. 18, it can
be seen that the free-slip and no-slip scenarios appear to set lower and upper bounds for the power extracted (in the context of the wall boundary conditions) for the isolated offshore island, as for the island-landmass system. There is no relationship between the maximum extracted power and the power naturally dissipated at the seabed. As would be expected, the undisturbed kinetic power, measured at a cross section (of width equal to B f
) south of the island does not provide a useful measure by which to estimate the maximum power extracted. As for the island-landmass system, the rate of decrease of kinetic power at low extraction levels ( k f
< 0.14) is considerably higher with free-slip than no-slip.
Fig. 17.
Flow driving head between entrance and exit of the strait for the freeslip scenario: no power extraction (solid line); low extraction k f
= 0.14
(dashed line); and very high extraction k f
= 2.24 (dotted line).
In the free-slip scenario, based on the amplitude of the head
driving the flow shown in Fig. 17, the Garrett and Cummins
[3] channel model predicts a maximum extracted power in the order of 45MW. Consequently, the Garrett and Cummins channel model may not apply to this case, where the island geometry allows bypass flow effects and there is an increase in driving amplitude due to increasing power extraction rates.
VI.
I SOLATED O FFSHORE I SLAND
A comparison is now made between the resource assessments for the previous island-landmass system and for an island located sufficiently far from the coast for the landmass to have negligible effect. In the latter case, the distance s between island and landmass is set to 9.5
Ø i
, and the
Fig. 18.
Power profiles as functions of k f
for a tidal farm located south of an isolated offshore island: free-slip (black); and no-slip (red) solid boundaries.
Power extracted at farm located south of the island (solid line); kinetic power measured across the tidal farm (dash-dot line); kinetic power measured across the tidal farm in undisturbed conditions (dotted line); and natural power dissipated on the seabed south of the island (dashed line).
The ratios of extracted power to the undisturbed kinetic power for the free-slip and no-slip scenarios are 0.47 and 0.79 respectively. Compared to the island-landmass system, there is an 18% decrease for the no-slip scenario while the free-slip values are similar. Under the same tidal conditions, the landmass acts to constrain the flow, and therefore increases the kinetic power available, compared to an isolated island.
However, this effect is diminished when considering the ratio of power extracted to the undisturbed kinetic power, and the rates of decay of kinetic power for low levels of power extraction.
Fig. 19 presents the non-dimensional scalar flows through
cross sections of width B f
at the south and north sides of the
island. Comparison between Fig. 19 and Fig. 14 reveals that
the rates of decrease of flow due to power extraction and consequent increase of bypass flow in the island system are very similar to those computed for the island-landmass system.

linking two infinite oceans. The main causes for the underestimate are the change in the head driving the flow through the strait and the bypass flow around the island.
Assessment was also made of the tidal resource on one side of an isolated island located far off the coast. In this case, neither the power dissipated by the seabed nor the kinetic power in natural conditions approximated the maximum power extracted. Comparison against results from the islandlandmass system showed that ratios of power extracted to undisturbed kinetic power are very similar for the free-slip scenario, but for the no-slip scenario the ratio was 18% lower for the offshore island. The rate of decay and increase of the scalar flow through the area of power extraction and through the opposite site of the island were found to be very similar to that obtained for the island-landmass system.
Fig. 19.
Changes in the scalar flow to flow in undisturbed conditions ratio for the free-slip (black) and no-slip (red) scenarios at different levels of power extraction: scalar flow across the tidal farm (solid line); and scale flow through a cross-section of identical length at the north side of the island
(dashed line).
VII.
C ONCLUSIONS
As a preliminary to tidal power resource assessment, two sets of validation tests were used to assess the adequacy of
Fluidity as modelling tool for the shallow flow hydrodynamics of island tidal systems. Comparison between the model predictions, alternative numerical predictions [25], and high quality experimental visualisations [24] of impulsively-started flow past a cylinder at Reynolds numbers of 40 and 100 demonstrated that Fluidity captures correctly the mechanics of early stage wake evolution. Further comparison of model predictions against experimental data for flow past a surfacepiercing circular cylinder in an oscillatory laminar shallow flow [26] confirmed that Fluidity SWE is able to reproduce reasonably well the flow dynamics around an island (noting that the experiments were at laboratory scale). The results obtained with Fluidity SWE were in good agreement with those obtained with Fluidity 3D.
A parameter study was undertaken to investigate the tidal resource at a strait between an island and a landmass. The estimates of power extraction were sensitive to the choice between free-slip and no-slip wall conditions, especially at large power extraction rates. Flow separation in the no-slip scenario was found to be the main driver behind the discrepancies in the two sets of results. Analysis of a strait with realistic bathymetry yielded power extraction estimates between the free-slip and no-slip values. This points towards the free-slip and no-slip estimates representing lower and upper bounds (in terms of the effect of choice of wall condition) for the power extracted from the strait. The peak in power extracted represents an upper limit to power extraction, and the actual power available for generation will be lower due to mixing and support structure losses not accounted in the methodology for power extraction employed. In all cases, the extracted power was found not to be a function of the naturally dissipated power by the seabed at the strait. The undisturbed kinetic power at the strait gave a slightly better measure of the power extracted for the no-slip scenario than the free-slip and non-uniform seabed scenarios. It was also found that the extracted power from a strait between an island and mainland is underestimated by a model of a channel
VIII.
F URTHER W ORK
The authors aim to extend this study to the numerical
resource assessment of cases c, d, e and f indicated in Fig. 1.
The effects of basin depth and frictional environment on the tidal resource will be also assessed for a coastal island near a landmass. Future work will include the implementation of a turbine model that accounts for sub-grid scale mixing losses and blockage [35].
A CKNOWLEDGMENT
The authors would like to thank Alstom Ocean Energy and in particular to James McNaughton for their support. The authors would like to thank the Applied Modelling and
Computation Group at Imperial College of London for free access to the software Fluidity and their support. The
Industrial Doctoral Centre for Offshore Renewable Energy is funded by the Energy Technologies Institute and the RCUK
Energy Programme, grant number (EP/J500847/1).
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