Analysis of turbulent bending moments in tidal current boundary layersSpencer R. Alexander and Peter E. Hamlington Citation: Journal of Renewable and Sustainable Energy 7, 063118 (2015); doi: 10.1063/1.4936287 View online: http://dx.doi.org/10.1063/1.4936287 View Table of Contents: http://scitation.aip.org/content/aip/journal/jrse/7/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Wall pressure fluctuations beneath supersonic turbulent boundary layers Phys. Fluids 23, 085102 (2011); 10.1063/1.3622773 Effects of small-scale freestream turbulence on turbulent boundary layers with and without thermal convection Phys. Fluids 23, 065111 (2011); 10.1063/1.3596269 Inner/outer layer interactions in turbulent boundary layers: A refined measure for the large-scale amplitudemodulation mechanism Phys. Fluids 23, 061701 (2011); 10.1063/1.3589345 Boundary layers in rotating weakly turbulent Rayleigh–Bénard convection Phys. Fluids 22, 085103 (2010); 10.1063/1.3467900 A comparative study of near-wall turbulence in high and low Reynolds number boundary layers Phys. Fluids 13, 692 (2001); 10.1063/1.1344894

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Analysis of turbulent bending moments in tidal currentboundary layers

Spencer R. Alexander and Peter E. HamlingtonDepartment of Mechanical Engineering, University of Colorado, Boulder, Colorado 80309,USA

(Received 7 February 2015; accepted 9 November 2015; published online 1 December2015)

As ocean current turbines move from the design stage into production and

installation, a better understanding of localized loading is required in order to more

accurately predict turbine performance and durability. In this study, large eddy

simulations (LES) of tidal boundary layers without turbines are used to measure

the turbulent bending moments that would be experienced by an ocean current

turbine placed in a tidal channel. The LES model captures turbulence due to winds,

waves, thermal convection, and tides, thereby providing a high degree of physical

realism, and bending moments are calculated for an idealized infinitely thin

circular rotor disc. Probability density functions of bending moments are calculated

and detailed statistical measures of the turbulent environment are also examined,

including vertical profiles of Reynolds stresses, two-point velocity correlations, and

velocity structure functions. The simulations show that waves and tidal velocity

have the largest impacts on the strength of bending moments, while boundary layer

stability and wind speeds have only minimal impacts. It is shown that either

transverse velocity structure functions or two-point transverse velocity spatial

correlations can be used to predict and understand turbulent bending moments in

tidal channels. VC 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4936287]

I. INTRODUCTION

Deployments of renewable energy systems—including wind turbines, solar thermal collec-

tors, hydroelectric dams, and many other technologies—have grown at a considerable pace over

the past decade, and from 2005 to 2011 renewable energy capacity in the U.S. increased by

300%.1 In order to sustain this rapid increase, however, additional deployments and new sour-

ces of renewable energy are needed. To satisfy this need, energy from the ocean has shown

considerable promise as a predictable, abundant, and sustainable resource, and it is estimated

that ocean energy—comprising wave, tidal, current, and thermal energy—is capable of contrib-

uting an additional 400 TWh/yr, or 10% of the current U.S. energy needs, to the U.S. energy

portfolio.1

In order to increase the feasibility of large-scale ocean energy installations, further research

is required to understand the turbulent environment in which ocean energy devices operate.

Turbulent stresses and bending moments on ocean current turbines, in particular, are not well

understood and may contribute to large off-axis loads that can, in turn, lead to unexpectedly

short turbine lifetimes due to gearbox and other component failures. Such large off-axis loads

arise due to the intermittent nature of turbulence2 and are well known in the wind energy com-

munity to often exceed fatigue and extreme loads anticipated during the design process.3,4 Off-

axis load cases have not traditionally been included in the design process and such loads can

pose problems, in particular, for gears and bearings found in gearboxes.5

Although ocean current turbines may be placed in open water far from coasts, tidal chan-

nels have long been seen as the most viable locations for such turbines due to the strong and

highly predictable nature of tidal currents. The flow in tidal channels is, however, complex

and chaotic at a wide range of scales due to turbulence production by surface (i.e., wind) and

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JOURNAL OF RENEWABLE AND SUSTAINABLE ENERGY 7, 063118 (2015)

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bottom (i.e., no-slip) shears, waves, variable bathymetry (or bottom surface roughness), thermal

instabilities, and even turbines themselves (i.e., if a turbine is in the wake of another turbine).

Turbulence from all of these sources can create substantial small-scale temporal and spatial var-

iability of the tidal flow field, resulting in potentially large bending moments.

Understanding and predicting bending moments in such turbulent environments require si-

multaneous two-point spatial measurements across and downstream of the rotor disc.

Observational data for such two-point statistics is, however, not yet available in tidal channels

over the range of spatial scales (i.e., from meters to hundreds of meters) relevant to tidal tur-

bine arrays. By contrast, high fidelity computer simulations allow comparatively straightforward

measurements of two-point statistics over a broad range of scales. Using simulations, the turbu-

lent environment in which tidal current devices operate can be modeled, thereby enabling pre-

dictions of long-term turbine performance and turbulent fatigue loading.

Much of the current knowledge of tidal turbine loading is based on experience gained from

studies of wind turbines, where attempts have been made to understand turbulent atmospheric

environments and to assess the effects of bending moments on turbines. The interactions

between the atmospheric boundary layer and utility scale wind turbines have been studied and

classified using both experiments and numerical simulations. Using observational data, M€ucke

et al.,3 Morales et al.,6 and Milan et al.7 have connected extreme and fatigue turbine loads to

the intermittent nature of turbulent wind speed fluctuations, and Kelley et al.2 have further

examined the impacts of coherent turbulent structures on the dynamic response and bending

moments of wind turbines. Wind tunnel boundary layer experiments such as those performed

by Chamorro and Port�e-Agel8 have revealed additional properties of turbulent wakes formed

behind wind turbines, including the creation of non-axisymmetric wakes by inhomogeneous

boundary layer inlet flows.

Recently, numerical simulations have begun to provide substantial additional knowledge

regarding the connection between turbine loads, atmospheric boundary layer turbulence, and

turbine wakes. More specifically, a number of recent studies9–14 have used large eddy simula-

tions (LES) to model arrays of wind turbines in order to determine the extent to which turbine

wake interactions reduce power output and increase loading, and to examine how wake recov-

ery occurs in realistic atmospheric boundary layers (refer to Mehta et al.15 for a review of LES

methods applied to wind farm aerodynamics). Calaf et al.9 and Meyers and Meneveau,10,14 in

particular, have connected wake recovery to momentum entrainment from the atmospheric

boundary layer, with the kinetic energy transported vertically into the wake roughly equal to

the power extracted by turbines. The connection between properties of the turbulent flow field

and turbine loads are often examined using advanced turbine simulators, such as the Fatigue,

Aerodynamics, Structures, and Turbulence (FAST) code.16

Turbine loading and performance in the ocean have also been studied using numerical sim-

ulations, often with a focus on accurately modeling boundaries such as channel bathymetry and

turbine blade shape. Although there have been many studies of tidal turbines using Reynolds

averaged Navier-Stokes (RANS) approaches,17–23 LES has become increasingly common for

the study of tidal turbine performance and loading. Churchfield et al.24 and Gebreslassie et al.25

have both used LES to examine arrays of multiple tidal turbines in order to determine the

effects of array layout on wake interactions and power output. Other studies have used LES to

obtain high-resolution predictions for the near-field flow around single turbines, including the

studies of tidal turbine loading by Kang et al.26,27 and Afgan et al.,28 as well as the study of

tidal turbine noise generation by Lloyd et al.29 In addition to these computational studies, a

number of experimental studies have provided insights into loads, wake properties, power out-

put, and turbulence characteristics for both single tidal turbines30–39 and, more recently, two

turbines.40

By contrast to most prior computational studies of tidal turbine loading, the present paper

is specifically focused on understanding the turbulent bending moments that would be experi-

enced by a tidal current turbine placed in a realistic ocean environment, as well as how the

moments are connected with fundamental turbulence statistics. Such statistics are critical for

predicting moments and, ultimately, turbine performance and fatigue in advance of installation.

063118-2 S. R. Alexander and P. E. Hamlington J. Renewable Sustainable Energy 7, 063118 (2015)

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In order to resolve small-scale turbulent motions and increase the level of physical realism,

LES has been performed of an ocean tidal boundary layer for a range of different physical sce-

narios. Two of the scenarios loosely correspond to prospective tidal energy sites at Admiralty

Head and Nodule Point in Puget Sound. LES results are compared with observational measure-

ments of turbulence intensity (TI) profiles at these two locations from Thomson et al.41 The

observational measurements have been used previously to examine turbulence intermittency, co-

herence, and anisotropy42 and are used here to constrain the physical parameters used in the

simulations, as well as to gain confidence in the accuracy of the LES.

In the present study, a number of simulations for different physical scenarios have been

performed in order to understand the effects of winds, waves, tidal velocity, stability, and tidal

channel depth on the moments that would be experienced by a tidal turbine. Changes to the

wind and tidal velocities result in changes to the turbulence shear production, waves generate

Langmuir turbulence43–45 which increases vertical mixing near the surface, and boundary layer

stability strongly affects the creation and properties of convective turbulence. Previously, Li

et al.46 have shown using LES that Langmuir turbulence may play a relatively small role in the

near-surface turbulence dynamics, and the present study will examine the role of such wave-

driven turbulence on tidal turbine loading more specifically. The LES model used in the present

study has the capability to represent all of these effects and has been used previously for sev-

eral high-fidelity process studies of atmospheric and oceanic flows.44,45,47,48 Probability density

functions, two-point correlations, velocity structure functions, Reynolds stresses, and other sta-

tistics are used to understand the dependence of turbulent bending moments on each of these

physical effects, and an analytical connection is made between bending moments and two-point

turbulence statistics.

The present study is differentiated from prior research by the extent of ocean physics

included in the simulations and by the focus on bending moments that would be experienced

by a tidal current turbine. Prior studies of tidal current turbines have often neglected the effects

of wind, waves, or tidal motions in order to decrease computational cost or to focus on impacts

from other physical processes. Conversely, prior fundamental studies of tidal flow and coastal

boundary layers have often not connected physical processes with the loads and turbulent flow

fields experienced by tidal turbines. For example, in one of the most comprehensive prior stud-

ies of tidal flows, Li et al.49 studied temporally varying tidal currents in an estuarine boundary

layer, but did not consider waves, different stability conditions, or varying tidal channel depth,

and were not specifically focused on tidal turbine loading. Similarly, the LES studies by Taylor

et al.50 and Gayen et al.51 have provided considerable insights into boundary layer structure

and turbulence properties for stratified boundary layers in open channel flows, but once again

were not focused on bending moments generated in well-mixed tidal boundary layers, such as

those examined here.

Finally, it should be noted that the present study does not include turbines in the simula-

tions. Rather, turbines are represented as idealized infinitely thin circular discs during the calcu-

lation of bending moments and an attempt is made to understand how the bending moments

that would be experienced by a tidal turbine are affected by different physical characteristics of

the flow field, independent of the particular choice of turbine design.

II. TURBULENT BENDING MOMENTS

Quantification of the expected bending moments on turbines in realistic ocean environ-

ments is an important objective of the present study. Prior research has been performed on

wind turbine loading for different atmospheric conditions,2,16,52,53 including the specific loading

associated with the blade structure. Similar computational and experimental studies have been

performed on loads experienced by ocean current turbines,23,24,26–28,32,54–57 but the dependence

of such loads on characteristics of the oceanic boundary layer such as wind and wave shear,

boundary layer depth, and stability conditions have yet to be examined.

In the present paper, the bending moments (which are sometimes referred to as off-axisloads) that would be experienced by a tidal turbine are considered without restricting the

063118-3 S. R. Alexander and P. E. Hamlington J. Renewable Sustainable Energy 7, 063118 (2015)

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analysis to a specific choice of turbine design. The moments at location x and time t, denoted

as M(x, t), are measured for a rigid, infinitely thin circular disc of radius D and with a hub-

height above the ocean bottom d, as shown in the schematic in Fig. 1. The direction normal to

the disc is the x direction and the resulting moments are calculated from an integral over the

area of the disc as

Mðx; tÞ ¼ð

A

q½r� uðxþ r; tÞ�uxðxþ r; tÞdrydrz; (1)

where r¼ [rx, ry, rz] is the distance from the central point to a location on the disc, u¼ [ux, uy,

uz] is the velocity vector, ux is the velocity normal to the disc, and A is the cross-sectional area

of the disc. Since the disc is assumed to be infinitely thin, rx¼ 0 and Eq. (1) can be expanded

as

Mxðx; tÞ ¼ð

A

q½ryuzðxþ r; tÞ � rzuyðxþ r; tÞ�uxðxþ r; tÞdA; (2)

Myðx; tÞ ¼ð

A

qrzu2xðxþ r; tÞdA; (3)

Mzðx; tÞ ¼ �ð

A

qryu2xðxþ r; tÞdA : (4)

The component Mx is about the axis normal to the turbine disc and provides a torque that either

accelerates or decelerates the turbine. The components My and Mz are bending moments (or

off-axis loads) about the horizontal and vertical bisects of the turbine disc, respectively (see

Fig. 1). The expressions in Eqs. (3) and (4) are used to calculate My and Mz in Section IV.

Bending moments are of particular concern in the design of turbine gearboxes, since such off-

axis loads can place unexpectedly large stresses on gears and bearings that are not typically

addressed during the design process.5

Bending moments are related to the characteristics of the turbulent flow field, and in particular

to transverse velocity differences over different spatial separations across the rotor disc. This can

be seen by assuming the rotor disc to be a circle with diameter D that is symmetric about the y

FIG. 1. Schematic of an idealized circular rotor disc of diameter D and hub-height d, showing the direction of bending

moments Mx, My, and Mz defined in Eqs. (2)–(4). The hub-height d is measured from the bottom boundary of the tidal

channel.

063118-4 S. R. Alexander and P. E. Hamlington J. Renewable Sustainable Energy 7, 063118 (2015)

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and z axes and by decomposing the velocity into mean and fluctuating parts as ux ¼ �ux þ u0x,

where ð�Þ indicates an average over time and horizontal directions. The resulting bending moments

from Eqs. (3) and (4) are then written as (the x coordinate is suppressed for brevity)

Myðx; tÞ ¼ð

ry

ðrz�0

qa1rz½Drzu0xðyþ ry; z; tÞ� þ qa2rz½Drz

�uxðzÞ�drzdry; (5)

Mzðx; tÞ ¼ �ð

rz

ðry�0

qa3ry½Dryu0xðy; zþ rz; tÞ�drydrz; (6)

where the ai variables have been introduced to simplify the notation and are given by

a1 � u0xðyþ ry; zþ rz; tÞ þ u0xðyþ ry; z� rz; tÞ þ 2�uxðzþ rzÞ ;a2 � �uxðzþ rzÞ þ �uxðz� rzÞ � 2u0xðyþ ry; z� rz; tÞ ;a3 � u0xðy� ry; zþ rz; tÞ þ u0xðyþ ry; zþ rz; tÞ þ �uxðzþ rzÞ :

(7)

The velocity difference terms introduced in Eqs. (5) and (6) are defined as

Drzu0xðyþ ry; z; tÞ ¼ u0xðyþ ry; zþ rz; tÞ � u0xðyþ ry; z� rz; tÞ ;

Dryu0xðy; zþ rz; tÞ ¼ u0xðyþ ry; zþ rz; tÞ � u0xðy� ry; zþ rz; tÞ ;

Drz�uxðzÞ ¼ �uxðzþ rzÞ � �uxðz� rzÞ ; (8)

where the last expression involving �ux is connected to the mean vertical shear d�ux=dz. Note

that since the tidal channel is assumed here to be horizontally homogeneous, mean flow statis-

tics depend only on z and other shear components such as d�ux=dy are zero. Moments of the

fluctuating velocity differences in Eq. (8) are, in turn, connected to turbulent structure functions.

Properties of structure functions in tidal channels and their connection to bending moments will

be discussed in more detail in Section IV.

From Eq. (5), My is seen to depend on two primary effects: a contribution due to variations

in turbulent fluctuations over the disc and a contribution due to variations in the mean velocity

over the disc. The latter effect is otherwise known as the mean shear, and thus My is termed

the mean shear bending moment. By contrast, Eq. (6) indicates that Mz depends only on the dif-

ferences in turbulent fluctuations over the disc; this moment is thus termed the turbulent eddybending moment. In Section IV, Eqs. (5) and (6) will be used to understand bending moments

in tidal channels for a range of physical scenarios.

III. DETAILS OF THE NUMERICAL SIMULATIONS

The National Center for Atmospheric Research (NCAR) LES model is used to perform the

simulations.44,47 In order to represent the effects of winds, waves, and tides, the simulations

solve the forced Wave-Averaged Boussinesq (WAB) equations given by58–62

@u

@tþ uL � ru ¼ �rpþ b� uL � rusð Þ> þ fc þ sgs; (9)

@h@tþ uL � rh ¼ sgs; (10)

r � u ¼ 0; (11)

where u¼ [ux, uy, uz] is the three-dimensional Eulerian flow velocity, x � r� u is the Eulerian

vorticity, uL¼ uþus is the Lagrangian velocity, us is the wave-induced Stokes drift velocity, pis the pressure normalized by the background density q0, b is the buoyancy, fc is a driving term

used to create the tidal current, h is the potential temperature, and sgs denotes subgrid-scale

063118-5 S. R. Alexander and P. E. Hamlington J. Renewable Sustainable Energy 7, 063118 (2015)

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(SGS) terms introduced by the LES modeling. The WAB equations account for the effects of

wave forcing through advection by the Lagrangian velocity uL and through the forcing term

uL�(rus)> on the right-hand side of Eq. (9), both of which include the wave-induced Stokes-

drift velocity us. Note that in the present study Coriolis effects are not considered; a similar

approach has been used in prior studies of tidal boundary layers49 and rotational effects are not

expected to make a significant contribution to the dynamics at the range of scales examined

here. An equation of state is used for the density q, namely,

q ¼ q0½1þ bTðh0 � hÞ�; (12)

where bT¼ 2� 10�4 K�1 is the thermal expansion coefficient and salinity effects have been

neglected. The buoyancy is b ¼ bz where b¼�gq/q0. Parameter values used to obtain the

buoyancy are q0¼ 1000 kg/m3, h0¼ 290.16 K, and g¼ 9.81 m/s2.

The WAB equations in Eqs. (9)–(12) have been used previously in a number of computa-

tional studies of wave-driven Langmuir turbulence.44,45,48 Langmuir turbulence consists of

disordered collections of counter-rotating vortical cells (typically called “Langmuir” cells).

These cells are typically 10 m deep in the vertical and can be up to 1 km long in the horizontal

direction (although typically they are much shorter, as in the present simulations). They create

surface convergence zones where foam, plankton, and other debris collect, resulting in charac-

teristic “windrows.”43 The primary effect of Langmuir turbulence on the flow field is to create

more intense vertical mixing, which may substantially alter the loads experienced by a turbine

in a tidal channel.

The NCAR LES model solves Eqs. (9)–(12) on a structured, rectilinear grid. A spectral

method is used in horizontal (x, y) directions and a second-order finite difference method is

used in the vertical (z) direction for the velocities. A third-order finite difference method is

used in the vertical direction for h. A three-step explicit Runge-Kutta method is used to

advance the solution in time, and the Poisson pressure equation associated with Eq. (11) is

solved iteratively at each time step. A variable time step is used with a constant Courant-

Friedrichs-Lewy (CFL) number of 0.5. The SGS model used in the simulations is described by

Sullivan et al.,63 and takes into account not only the subgrid-scale influences proportional to

the resolved-scale strain rate but also the influences proportional to the horizontal average strain

rate. At the bottom boundary, the SGS model is calibrated to match Monin-Obukhov similarity

theory. This SGS model has been used in previous studies of Langmuir turbulence.44,45,48

The forcing term, fc ¼ fcx, in Eq. (9) is used to create a tidal current in the x direction and

is constant in both space and time. Following Li et al.,49 future work may include temporal

variations in fc in order to consider situations in which the tidal stream changes magnitude and

direction. Here, however, a time-independent fc is used in the simulations to create uniform

tidal velocities of 1.0 m/s, 2.0 m/s, and 3.0 m/s.

The effects of wave forcing are parameterized in Eqs. (9) and (10) via the Stokes drift

velocity us appearing in the Lagrangian velocity uL. The Stokes drift velocity is modeled as

us ¼ usðzÞ½cosð#sÞx þ sinð#sÞy�; (13)

where us(z) is the Stokes drift vertical profile and #s is the wave direction. Using the Donelan

empirical spectrum64,65 for the ocean wave field, the resulting vertical profile us(z) decreases

rapidly with depth, as shown in Fig. 2. In the simulations, the Stokes drift velocity is assumed

to be the same at all horizontal (x, y) locations with an angle of #s ¼ 0�. That is, the wave field

is assumed to be perfectly aligned with the tidal direction. The magnitude of the Stokes drift

and the corresponding profile us(z) are determined by setting the 10 m wind speed, denoted U10,

in the Donelan spectrum. Profiles of us(z) used in the present simulations are shown in Fig. 2.

These profiles correspond to U10¼ 5.0 m/s and 10.0 m/s, which give spectral significant wave

heights of 0.7 m and 2.7 m, respectively. These wave heights have been chosen based on the

typical (i.e., less than 1 m) and maximum (i.e., 2.3 m) wave heights observed at Admiralty Inlet

in Puget Sound.66

063118-6 S. R. Alexander and P. E. Hamlington J. Renewable Sustainable Energy 7, 063118 (2015)

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All of the simulations performed in the present study are fully periodic in the horizontal

(x, y) directions for all simulation variables. With the tidal forcing fc ¼ fcx, this results in an

inflow along the x direction that is directly tied to the outflow. On the top boundary, the hori-

zontal velocities (i.e., ux and uy) are driven by a wind shear stress of magnitude s in the #w

direction. In the simulations, #w is always chosen to be in the positive x direction, resulting in

a wind shear friction velocity us in the x direction. The vertical velocity uz is constrained to be

zero at the top boundary in all simulations. Depending on the intended thermal stability of the

simulations, either a cooling, warming, or adiabatic temperature flux, Q0, is applied at the

top boundary. At the bottom boundary, a no-slip condition is used for velocities in all three

directions and the temperature flux is zero. The surface roughness for the bottom no-slip bound-

ary is z0¼ 0.001 m, and the SGS model used in the LES matches the log-law for the mean

velocity at the lower boundary (as described in Sullivan et al.63). Note that the bottom bound-

ary is assumed in this study to be flat and absent of any bathymetric features.

The physical setup and conditions used in the numerical simulations have been loosely

guided by the wind, wave, and tidal conditions found at the Admiralty Head and Nodule Point

locations in Puget Sound. These sites have been characterized in Thomson et al.41 and recently

collected data allows some limited comparisons to be made between the simulation and experi-

mental results. Since the present simulations are, however, more broadly focused on the effects

of winds, waves, boundary layer depth, instabilities, and tidal velocity, a series of 13 simula-

tions corresponding to 11 different physical scenarios have been performed, as summarized in

Table I. The baseline simulation (denoted “B” in Table I) has horizontal lengths Lx¼ 600 m

and Ly¼ 300 m with depth H¼ 39 m, a tidal velocity of Ut¼ 2 m/s, upper wind forcing resulting

in us¼ 6.3 mm/s, waves created by a 10 m wind of U10¼ 5.0 m/s, and no heat flux at the

surface (corresponding to a neutral boundary layer). Figure 2 shows the Stokes drift profile

used in the baseline case. The horizontal dimensions of the computational grid in the baseline

case are Nx¼ 512 and Ny¼ 256, resulting in a horizontal grid resolution of Dx,y¼ 1.2 m. The

vertical grid size is Nz¼ 128 giving a vertical resolution of Dz¼ 0.30 m. The turbulent

Langmuir number in the baseline simulation is Lat ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffius=usð0Þ

p¼ 0:25, which is a standard

value observed in realistic ocean environments,44 although it may be larger than the values

found in typical tidal channels.

A pair of simulations with identical physical parameters to the baseline simulation but with

doubled (Bþ) and halved (B�) resolution have been performed in order to ascertain grid conver-

gence. All other simulations have identical values of Lx, Ly, Nx, and Ny to the baseline case

FIG. 2. Profiles of the Stokes drift magnitude us(z) for the B (solid black lines), B� (red dashed lines), Bþ (blue dashed-

dotted lines), and Waþ (heavy black lines) simulations. The profiles for the B, B�, and Bþ simulations correspond to

U10¼ 5.0 m/s, and the profile for the Waþ simulation corresponds to U10¼ 10.0 m/s, as summarized in Table I. The inset

shows the same profiles with a logarithmic axis for us(z).

063118-7 S. R. Alexander and P. E. Hamlington J. Renewable Sustainable Energy 7, 063118 (2015)

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(i.e., simulation B), but h is varied to 22 m and 56 m in two of the simulations (denoted H� and

Hþ in Table I). These two values of h correspond to the locations at Nodule Point and

Admiralty Head, respectively, allowing qualitative comparisons of the simulations with observa-

tions from Thomson et al.41 Note, however, that the comparisons with observations are limited

since the present LES only matches the tidal channel depths at the Nodule Point and Admiralty

Head locations. More rigorous validation of the simulations requires additional information

from the observational locations about realistic wind and wave forcing, bathymetry, and inlet

flow properties. As such, the focus of the present study is on understanding the effects of differ-

ent physical scenarios on current turbine bending moments, as opposed to rigorously modeling

specific real-world tidal boundary layer locations.

As summarized in Table I, the tidal velocity (T� and Tþ), wind strength (Wi� and Wiþ),

wave strength (Wa� and Waþ), and surface heat flux (In� and Inþ) have each been varied in

the simulations to be successively smaller and larger than the baseline values. Section IV will

show that relatively small changes in the tidal velocity Ut result in large changes in flow prop-

erties, and so the T� and Tþ simulations correspond to values of Ut that are only 1 m/s smaller

and larger, respectively, than the baseline value of Ut¼ 2.0 m/s. In order to determine wind and

wave effects, the surface wind was turned off for the Wi– simulation and the Stokes drift forc-

ing was turned off in the Wa� simulation. The Wiþ and Waþ simulations correspond to surface

wind and wave forcings, respectively, which are twice as large as the baseline values. Stronger

forcings produce even greater changes in flow properties, but become increasingly less realistic

in comparison to the standard values used for the baseline case. Profiles of the Stokes drift us(z)

used for the Bþ, B�, and Waþ simulations are shown in Fig. 2. Finally, ocean surface cooling

of Q0¼�5.0 W/m2 has been found previously45 to create an unstable mixed layer in the pres-

ence of Stokes drift forcing, and the same cooling is used here for the In– simulation. In order

to perform a more meaningful examination of stability effects, heating with the same magnitude

but opposite sign is used for the Inþ simulation.

TABLE I. Summary of the physical setup used in the simulations. Horizontal and vertical domain sizes are denoted as Lx,

Ly, and h, computational domain sizes are given by Nx, Ny, and Nz, Ut is the tidal velocity, us is the top surface friction ve-

locity, s is the wind stress, #w is the wind direction with respect to the x-axis, U10 is the wind speed 10 m above the ocean

surface, us(0) is the Stokes drift velocity at the surface, Lat ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffius=usð0Þ

pis the turbulent Langmuir number, #s is the

Stokes drift direction with respect to the x-axis, Q0 is the surface heat flux, and z0 is the roughness height of the bottom

boundary. Parameters varied in each of the simulations are shown in bold.

Case Baseline Wind Wave Depth Stability Tide

Simulation B B� Bþ Wi� Wiþ Wa� Waþ H� Hþ In� Inþ T� Tþ

Lx (m) 600 600 600 600 600 600 600 600 600 600 600 600 600

Ly (m) 300 300 300 300 300 300 300 300 300 300 300 300 300

h (m) 39 39 39 39 39 39 39 22 56 39 39 39 39

Nx 512 256 1024 512 512 512 512 512 512 512 512 512 512

Ny 256 128 512 256 256 256 256 256 256 256 256 256 256

Nz 128 64 256 128 128 128 128 72 184 128 128 128 128

Ut (m/s) 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 1.0 3.0

us (mm/s) 6.3 6.3 6.3 0.0 12.6 6.3 6.3 6.3 6.3 6.3 6.3 6.3 6.3

s (N/m2) 0.04 0.04 0.04 0.0 0.08 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04

#w (deg) 0 0 0 0 0 0 0 0 0 0 0 0 0

U10 (m/s) 5.0 5.0 5.0 5.0 5.0 0.0 10.0 5.0 5.0 5.0 5.0 5.0 5.0

us(0) (m/s) 0.10 0.10 0.10 0.10 0.10 0 0.20 0.10 0.10 0.10 0.10 0.10 0.10

Lat 0.25 0.25 0.25 0.0 0.35 ‘ 0.18 0.25 0.25 0.25 0.25 0.25 0.25

#s (deg) 0 0 0 0 0 0 0 0 0 0 0 0 0

Q0 (W/m2) 0 0 0 0 0 0 0 0 0 �5.0 5.0 0 0

z0 (m) 10�3 10�3 10�3 10�3 10�3 10�3 10�3 10�3 10�3 10�3 10�3 10�3 10�3

063118-8 S. R. Alexander and P. E. Hamlington J. Renewable Sustainable Energy 7, 063118 (2015)

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IV. RESULTS

After an initial spin-up period during which the boundary layer turbulence is allowed to

develop, each of the simulations summarized in Table I were run for a total of 6 h of virtual

time, with an average timestep of 0.15 s for simulations with the baseline resolution (i.e., the

resolution of case B). The higher resolution simulation Bþ required a smaller average timestep

and thus more overall timesteps were necessary to reach 6 h of virtual time. The total simulated

time in each case was sufficient to allow turbulence to fully develop, and the ux velocity and

Reynolds stress profiles were found to be stable and unchanging with additional simulation

time.

The basic flow state in each of the simulations at the end of 6 h is represented by the snap-

shot of the ux velocity field for case B shown in Fig. 3. As expected for a tidal boundary layer,

the velocity is generally smallest at the bottom boundary and greatest near the surface, and

there is substantial spatial variability resulting from the turbulent nature of the flow field. The

domain shown in Figure 3 is also sufficiently large in the horizontal (x, y) directions to contain

many statistically independent turbulent structures; this will be confirmed quantitatively in

Section IV C where it will be shown that the turbulent integral scales are roughly 50 m in the xdirection and 10 m in the y direction, both of which are small relative to the domain size

[Lx, Ly]¼ [600m, 300 m].

In the following, a validation of the simulation results is outlined, including a study of grid

convergence. Single point statistics of the velocity and temperature for the different simulations

are then presented, followed by an analysis of two point turbulence statistics. Finally, off-axis

bending moments for each of the different physical scenarios summarized in Table I are

outlined.

A. Validation and grid convergence

Validation of the physical parameters and numerical methods used in the simulations is

critical for grounding the simulations in reality and for identifying any potential numerical

issues. Validation of the simulations is performed here through comparison with theory, prior

studies, and observational data from Thomson et al.41 Prior work by Sullivan and Patton in the

atmospheric convective boundary layer67 indicates that mesh sizes of 2563 in a domain of size

(Lx, Ly, Lz)¼ (5120, 5120, 2048)m are sufficient for achieving scale separation between the

large scale energy containing eddies and the filter cutoff scale, and the present simulations have

a similar dynamic range in a smaller domain with finer grid resolution. This computational grid

size has been used as a baseline for the present simulations (roughly corresponding to the size

of case B), and in the following validation results are presented for three different grid resolu-

tions in order to determine grid convergence.

FIG. 3. Three-dimensional snapshot of the horizontal velocity field ux after 6 h of virtual time for the B simulation.

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Figure 4 shows vertical profiles of �ux along with diagonal and off-diagonal Reynolds

stresses for the B, B�, and Bþ simulations. Here, ð�Þ denotes an average taken over horizontal

(x, y) directions as well as time t. The profile of �ux shows good agreement with the theoreti-

cally predicted log-layer profile and the qualitative behavior of the diagonal and off-diagonal

Reynolds stresses is consistent with prior studies of wall bounded shear flows. That is, u02x >u02y > u02z and u0xu0z < 0, with the greatest magnitudes of all stresses occurring close to the bot-

tom boundary. The Reynolds stress profiles also show good agreement with previous studies of

the atmosphere47,67 and tidal channels.68

From a more quantitative perspective, deviations of the simulation results from the theoreti-

cal log-layer profile are characterized by the quantity /m, which is defined as63,68

/m zð Þ � jz

u

@U

@z; (14)

where U � ð�ux þ �uyÞ1=2; j ¼ 0:41, and u* is the friction velocity at the bottom no-slip bound-

ary. While the /m profile for case B, shown in Fig. 5, is not precisely unity, as would be

expected in the case of a perfect log-layer profile, the small deviations are consistent with those

found in previous studies63,68 of atmospheric and oceanic boundary layers. Figure 5 shows that

the deviations are largest near the channel bottom, approaching a high value of 1.2 and a low

value of 0.7. In the simulations, grid anisotropy (i.e., the ratio of Dx,y to Dz) was found to have

a large effect on the /m profile, with higher values of the ratio (greater than 10) resulting in

more substantial deviations from unity. A grid anisotropy ratio of approximately four, as used

in the present simulations, results in /m profiles that are in line with previous studies.63,68

Figures 4 and 5 also show the effects of varying grid resolution on the velocity and

Reynolds stress profiles for low (B�), base (B), and high (Bþ) resolutions. The vertical profiles

of �ux in Fig. 4(a) indicate that there is good agreement between the base and high resolution

runs, with slight deviations near the top of the boundary layer for the low resolution run. The

diagonal Reynolds stresses in Fig. 4(b) show moderate variations for the three resolutions along

the entire depth of the domain. The u0xu0z Reynolds stress in Fig. 4(c) shows good agreement for

z/h between 0.2 and 1, with more substantial deviations in the lower resolution run. The /m

profiles in Fig. 5, show similar deviations from unity for all three resolutions, but the deviations

occur at heights proportional to the grid size. In particular, the /m profile for the highest resolu-

tion run does not show substantially less deviation from unity, just a deviation at a height closer

to z/h¼ 0. Overall, the simulations with different grid resolutions indicate that the resolution of

the base run (B) captures most statistics with acceptable accuracy relative to a simulation with

grid resolution doubled in each direction. In the interest of exploring as large a parameter space

FIG. 4. Vertical profiles of (a) �ux, (b) diagonal Reynolds stresses, u02x (solid), u02y (dashed), and u02z (dashed–dotted), and (c)

off-diagonal Reynolds stresses, u0xu0z (solid), u0xu0y (dashed), and u0yu0z (dashed–dotted). Each panel shows values for three

different grid resolutions, corresponding to simulations B (black lines), B– (blue lines), and Bþ (red lines), as a function of

z/h. Additionally, a theoretical log layer velocity profile, u ¼ ðu=jÞlnðz=z0Þ, is shown in grey in (a) (this profile is obscured

by the overlapping lines for the B, B�, and Bþ simulations).

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as possible, the resolution of the base simulations thus appears to be appropriate for developing

insights into the variation of off-axis loads for varying winds, waves, stability, and other physi-

cal parameters.

Finally, comparisons with the observational data of Thomson et al.41 are made using the

TI, which is defined as

TI � u02x1=2

�ux: (15)

The resulting vertical profiles of TI are shown in Fig. 6 for the simulations B (corresponding to a

39 m depth), H� (corresponding to a 22 m depth at the Nodule Point location), and Hþ (corre-

sponding to a 56 m depth at the Admiralty Head location).41 Figure 6 shows that the simulations

correctly capture the basic shape of the observational TI profiles, but slightly underestimate the

magnitude of TI (note that observational data were only available up to roughly 18 m above the

ocean bottom). Increasing the grid resolution (i.e., simulation Bþ) resulted in a slight (roughly 5%)

increase in the magnitude of the TI profile, but the LES results for all values of h do display the

same qualitative behavior seen in the observations; namely, a large increase in TI near the bottom

of the ocean boundary layer as u02x increases and �ux decreases (note that immediately at the bottom

boundary TI¼ 0 since u02x ¼ 0). While the roughness length z0 used in the simulations (see Table

I) was chosen to give a close match to the observed TI profiles from Thomson et al.,41 more com-

plete agreement with the observations most likely requires more realistic modeling of bathymetry

and inlet flow conditions, both of which are beyond the scope and intent of the present study.

Such improved agreement would also be necessary to obtain more accurate predictions of wake

expansion rate in simulations that include models for tidal turbines. Nevertheless, the good qualita-

tive agreement and approximate quantitative agreement between the LES results and observations

shown in Fig. 6 provides confidence in the validity of the simulations.

B. Single point statistics

Single point statistics are commonly used for understanding and characterizing tidal current

environments and provide valuable information to developers, designers, and modelers regard-

ing mean flow magnitudes, turbulence intensities, and transient effects. However, the ability of

single point statistics to predict bending moments and off-axis loads is limited. In this section,

FIG. 5. Vertical profiles of /m, defined in Eq. (14), for three different grid resolutions, corresponding to the simulations B

(solid black lines), B� (blue dashed lines), and Bþ (red dashed-dotted lines), as a function of z/h.

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the impacts of various physical parameters on single point statistics are analyzed, and in

Sections IV D and V the connections between both single and two point statistics and bending

moments are examined in more detail.

Figures 7(a) and 7(c) show that profiles of the mean velocity �ux do not change substantially

as either the wind or the stability conditions change. Changes in the wave strength do, however,

affect �ux near the surface, where �ux for Waþ decreases below the corresponding values for B

and Wa�, as shown in Fig. 7(b). The increased wave strength for the Waþ simulation, modeled

as a stronger Stokes drift velocity (see Table I) results in an anti-Stokes effect, as previously

observed by Hamlington et al.45 and explained by McWilliams and Fox-Kemper.69 Profiles of

�ux also change for T� and Tþ as compared to the base case due to the large changes in mean

tidal velocity. Overall, the general uniformity of the mean velocity profile with respect to

changes in the wind, wave, and stability conditions is significant for the understanding of

turbine loads, since a drastic change in vertical velocity profile for different physical parameters

would result in added complexity towards addressing loads associated with the mean vertical

shear.

FIG. 6. Turbulence intensity (TI) as a function of z/h for h¼ 22 m, 39 m, and 56 m, corresponding to simulations H�, B,

and Hþ, respectively. Observational results from Thomson et al.41 are also shown for the 22 m Nodule Point location (red

dashed line) and the 56 m Admiralty Head location (red dashed-dotted line).

FIG. 7. Vertical profiles of �ux for (a) Wi6, (b) Wa6, (c) In6, and (d) T6. In each panel, solid black lines are baseline results

for case B, dashed blue lines show Wi�, Wa�, In�, and T�, and red dashed-dotted lines show Wiþ, Waþ, Inþ, and Tþ.

063118-12 S. R. Alexander and P. E. Hamlington J. Renewable Sustainable Energy 7, 063118 (2015)

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Vertical profiles of Reynolds stress are shown in Fig. 8 and provide insights into the level

of turbulent activity within the flow. Both the Waþ and Inþ simulations show increases in u02xcompared to the base case, although little change is seen in either u02z or u0xu0z (note also that the

increase in u02x is relatively small for Inþ). An increase in u02x represents an increase in the larg-

est component of turbulence kinetic energy, and will impact not only the bending moments dis-

cussed in this study but also transient torque loads. Large deviations from the base Reynolds

stress profiles are present in the T� and Tþ profiles, and the magnitudes of the changes are

worth noting: a 50% increase of mean velocity approximately doubles the u02x Reynolds stress,

while halving the mean velocity results in roughly a factor of four decrease in u02x .

The mean temperature profiles remain close to the background value h0 for all but the

unstable (In�) and stable (Inþ) cases. Figure 9 shows that there is a sharp increase in the mean

potential temperature h� h0 for Inþ due to heating at the surface while there is a sharp

decrease for In� due to surface cooling. These large changes in the temperature profiles are

accompanied by corresponding changes in the temperature fluxes, as shown in Fig. 10. For the

unstable case (In�), u0xh0 < 0, corresponding to an anti-correlation between u0x and h0; while for

the stable case (Inþ), u0xh0 > 0. In Fig. 10(c), u0zh

0 > 0 for the unstable case, corresponding to a

downward flux of low temperature fluid as the surface is cooled; while u0zh0 < 0 for the stable

case, corresponding to a downward flux of high temperature fluid as the surface is heated. The

horizontal fluxes in Fig. 10(b) for the stable and unstable cases remain close to zero. It should

FIG. 8. Vertical profiles of Reynolds stresses u02x (solid), u0xu0z (dashed), and u02z (dashed-dotted) for (a) Wi6, (b) Wa6, (c)

In6, and (d) T6. In each panel, black lines are baseline results for case B, blue lines show Wi�, Wa�, In�, and T�, and red

lines show Wiþ, Waþ, Inþ, and Tþ. Note that the horizontal axis of panel (d) is roughly twice as large as the horizontal

axes for panels (a)–(c), all of which are equal.

FIG. 9. Vertical profiles of h� h0 for unstable In� (blue dashed line), baseline B (black solid line), and stable Inþ (red

dashed-dotted line) simulations.

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be noted that there are large variations in u0xh0 and u0yh

0 for the base case (B). These variations

are due primarily to large h02 in the base simulation resulting from no imposed constraint on

the correlation between the horizontal turbulence velocities and h0, other than the condition that

the vertical flux of h0 be zero at the surface.

C. Two point statistics

Two-point velocity correlation functions and integral length scales have been shown in

prior studies to be accurate predictors of turbine loading.2,3,6 In particular, these studies have

shown that when the integral length scale of the turbulence is similar to the diameter of the tur-

bine rotor, the loads on the turbine are greatest; turbulence with this characteristic length scale

results in large velocity gradients across the rotor swept area and correspondingly large bending

moments. In Section II, it was explicitly shown that two-point statistics such as turbulence

structure functions across the rotor disc are closely connected to bending moments.

Figure 11 shows the longitudinal velocity correlation, f11, and transverse velocity correla-

tion, f12, as functions of height, where the correlation functions are given by

f11 z;Dxð Þ ¼u0x x; y; zð Þu0x xþ Dx; y; zð Þ

u02x; f12 z;Dyð Þ ¼

u0x x; y; zð Þu0x x; yþ Dy; zð Þu02x

; (16)

FIG. 10. Vertical profiles of (a) u0xh0 , (b) u0yh

0 , and (c) u0zh0 for In� (blue dashed lines), B (black solid lines), and Inþ (red

dashed-dotted lines) simulations.

FIG. 11. Vertical profiles of the (a) longitudinal, f11(z, Dx), and (b) transverse, f12(z, Dy), correlation functions for the base-

line simulation (B). Colors show values of f11 and f12 at each z and Dx or Dy, and solid black lines show the integral scales

K11 and K12 defined in Eq. (17).

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and the corresponding integral scales K11 and K12 are defined here as

K11ðzÞ ¼ðLx=2

0

f11ðz; x0Þdx0; K12ðzÞ ¼ðLy=2

0

f12ðz; y0Þdy0 : (17)

Note that the upper bounds of the integrals in Eq. (17) are taken to be half the full horizontal

domain in the x and y directions in order to avoid spuriously high integral scales due to the pe-

riodicity of the simulations in horizontal directions.

For the base case simulation B, Fig. 11 shows that over the full tidal channel depth the lon-

gitudinal integral scale K11 is between roughly 20 and 50 m and the transverse integral scale

K12 is between 0 and 15 m. Figure 11(a) shows that, generally, the turbulence remains longitu-

dinally correlated for larger Dx as z/h increases, although the correlation approaches zero at the

upper surface; these variations are reflected in the increase of K11(z) with increasing z/h. For

the transverse correlation f12 in Fig. 11(b), K12(z) increases monotonically with increasing z/h,

reflecting increased transverse correlations for larger Dy from the bottom to the top of the tidal

channel.

As shown in Fig. 12, the transverse correlation f12, which was shown in Section II to be

closely connected to the bending moment Mz, is qualitatively similar for all of the simulations

performed in the present study. As in Figs. 7 and 8 for the mean velocities and Reynolds

stresses, respectively, some of the largest changes in f12 occur as the wave forcing varies, with

approximately a factor of 2 increase in K12 at all z/h for the Waþ simulation. This indicates

that the greatest changes in bending moments may occur as the wave forcing is varied. The

transverse length scale also varies for the other simulations, but magnitude changes are more

modest, on the order of 10%.

In addition to understanding the scale of the turbulent fluxes present in the ocean boundary,

it is also important to understand the magnitude and distribution of velocity variations—these

two properties feed directly into the distribution of bending moments experienced by the tur-

bine, as discussed in Section II. To understand these spatial velocity differences, Fig. 13 shows

structure functions S1j defined as

S1j z; rjð Þ ¼ju0x xþ rjð Þ � u0x x� rjð Þj

u02x1=2

; (18)

FIG. 12. Vertical profiles of the transverse correlation f12(z, Dy) for (a) Wi6, (b) Wa6, (c) In6, and (d) T6. The top row

shows Wi�, Wa�, In�, and T�, and the bottom row shows Wiþ, Waþ, Inþ, and Tþ. Colors show values of f11 and f12 at

each z and Dy, solid black lines are K12 for each simulation, and dashed black lines are K12 for the base case (B).

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where x is a position in space and rj is a vector of length r in the j (i.e., x, y, or z) direction.

Each of the structure functions shown in Fig. 13 follows an approximate r1/3 scaling, in accord-

ance with classical Kolmogorov theory,70 although the departures from this scaling are greatest

for S11 and S12 near the bottom boundary. In both cases, the structure functions have reduced

slope, indicating greater prominence of small scale motions relative to large scales.

Finally, Fig. 14 shows probability density functions (pdfs) of the transverse velocity incre-

ments Dryux for different values of Dry

, where Dryux is defined in Eq. (8). These increments

play a central role in determining the magnitude of Mz, as shown by the integral in Eq. (6). In

general, the pdfs become increasingly non-Gaussian as Drydecreases, and the pdfs for Dry

at

typical rotor radii (5 m) show indications of non-Gaussianity and intermittency. The implica-

tions of this are significant for models that assume a normal distribution of velocity differences,

since Fig. 14 indicates that extreme events, leading to greater bending moments, will occur

more frequently than expected from a normal distribution.

D. Bending moments

The final component of the analysis regards bending moments, which are outlined analyti-

cally in Section II. Bending moments are primary drivers of turbine design, and understanding

FIG. 13. Structure functions (a) S11, (b) S12, and (c) S13 for the baseline simulation (B), where the structure functions are

defined in Eq. (18). Results are shown for eight different heights, for z/h between the bottom boundary at z/h¼ 0 (blue) and

the top surface at z/h¼ 1 (red). Black dashed lines show r1/3 scaling relations, corresponding to the prediction from

Kolmogorov.70

FIG. 14. Probability density functions (pdfs) PðDryuxÞ for (a) Dry

¼ 1; 5, and 25 m at z¼ 15 m and (b) Dry¼ 5 m at z¼ 5,

15, and 25 m. Each of the pdfs is shifted by two orders of magnitude for clarity and solid black lines show Gaussian

distributions.

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the magnitude and physical properties of such moments will aid in the design and performance

prediction of ocean current turbines. Two primary types of bending moments are considered

here: shear moments My and eddy moments Mz. The eddy moments are so named since they

have zero means for the x–y homogeneous tidal channels examined in the present study. These

moments are calculated using Eqs. (3) and (4) for a circular infinitely thin rotor disc of diame-

ter D¼ 10 m. This diameter was chosen to be approximately representative of Oð10 mÞreal-world turbines, such as the 11 m diameter Seaflow turbine,71 the 16 m diameter OpenHydro

turbine,72 and the 20 m diameter Seagen turbine.73 The moments are calculated at a range of

different hub heights d (see Fig. 1) between 5 m and 34 m.

Figure 15(a) shows that My is strongly dominated by mean shear in the flow and increases

substantially as z/h approaches zero where the mean shear is largest. The magnitude of the me-

dian bending moment at the channel bottom is three times larger than the median bending

moment at the surface, with half of the change occurring in the bottom 10% of the boundary

layer. The 90th percentile bending moment is approximately double the median bending

moment, and the two vertical profiles are qualitatively very similar.

Changes to the My bending moments for the different simulation cases, as shown in

Fig. 16, are subtle but important. In Fig. 16(b), increasing the strength of the wave forcing

results in an increase in My near the surface and a small decrease in the middle half of the

boundary layer. This change is attributable, in part, to a similar change in the mean velocity

profile, shown in Fig. 7, where the increased wave forcing case was the only case to substan-

tially alter the mean velocity profile. Changes in the surface wind forcing and boundary layer

stability have little impact on My, as shown in Figs. 16(a) and 16(c), respectively. The bending

moments are very sensitive to changes in mean tidal velocity, as shown in Fig. 16(d), with sig-

nificantly weaker moments in the T– case and larger moments in the Tþ case.

The eddy bending moment Mz, shown for the baseline case in Fig. 15(b), is driven by hori-

zontal imbalances in ux velocities across the rotor disc. The median moments at the channel

bottom are nearly double the surface value, and the increase between the surface and the bot-

tom is nearly linear. At mid-depths, the magnitude of Mz is similar to the magnitude of My.

Much like My, the magnitude of the 90th percentile bending moments is approximately double

the mean. For Mz, waves again have the most noticeable impact on the magnitude and vertical

profile of the bending moment. In particular, increasing the wave strength (bottom panel of

Fig. 17(b)) decreases the bending moments by 10%–15% through the entire domain. Changes

in wind shear (Fig. 17(a)) and stability (Fig. 17(c)) result in minimal changes to the bending

moment, while changes to the mean velocity (Fig. 17(d)) again result in large changes to the

magnitude of Mz.

FIG. 15. Probability density functions (pdfs) of bending moments (a) My(z) and (b) Mz(z) as functions of height z/h for the

baseline simulation (B). The magnitude of the pdf is represented by the color, the median bending moments are shown as

solid black lines, and 10th and 90th percentiles are shown as black dashed-dotted lines. The logarithm of pdf magnitudes is

shown for clarity.

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V. DISCUSSION

In Sections II–IV, methodologies for simulating flows in ocean tidal channels have been

outlined and results pertaining to loading of ocean current turbines in these environments have

been presented. These methods and results have implications for turbine designers, observatio-

nal campaign managers, and modelers; implications of the present results for each of these

three groups are outlined in more detail below.

A. Implications for turbine designers

Based upon the present results, the most important consideration for a turbine designer is

the choice of hub height. Turbines mounted near the ocean floor not only see much larger bend-

ing moments (Fig. 15) but also see much lower velocities (Fig. 7(a)), while turbines higher in

FIG. 16. Probability density functions (pdfs) of shear bending moments My for (a) Wi6, (b) Wa6, (c) In6, and (d) T6. The top

row shows Wi�, Wa�, In�, and T� and the bottom row shows Wiþ, Waþ, Inþ, and Tþ. The median bending moments for each

simulation case are shown as solid black lines, and 10th and 90th percentiles are shown as black dashed-dotted lines. Baseline

results for the median moments are shown as black dashed lines. The logarithm of pdf magnitudes is shown for clarity.

FIG. 17. Probability density functions (pdfs) of eddy bending moments Mz for (a) Wi6, (b) Wa6, (c) In6, and (d) T6. The top

row shows Wi�, Wa�, In�, and T� and the bottom row shows Wiþ, Waþ, Inþ, and Tþ. The median bending moments for each

simulation case are shown as solid black lines, and 10th and 90th percentiles are shown as black dashed-dotted lines. Baseline

results for the median moments are shown as black dashed lines. The logarithm of pdf magnitudes is shown for clarity.

063118-18 S. R. Alexander and P. E. Hamlington J. Renewable Sustainable Energy 7, 063118 (2015)

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the ocean boundary layer harness power from much higher velocity flows with lower magnitude

bending moments. One of the main drivers of turbulence near the ocean surface—strong

waves—have been shown to actually slightly decrease the bending moment at moderate depths

(Section IV D); it should be noted, however, that this analysis only considers bending loads and

not transient torque loads, which may depend differently on waves and other physical

characteristics.

Effects of length scale and structure function magnitude have also been shown to combine

in interesting ways that warrant additional investigation by turbine designers. The inclusion of

stronger waves in the simulations results in a large increase in the transverse length scale,

which is a likely contributor to overall decreased bending moments at many depths for the

strong wave simulation. Design flexibility in rotor diameter could allow for an opportunity to

decrease overall loads by ensuring rotor diameter is of a different order of magnitude than the

characteristic scale of turbulent eddies.

Finally, turbine designers should note that the underlying drivers of bending moments have

been shown to deviate from a normal distribution (Fig. 14). Extreme loads will occur more fre-

quently than a Gaussian distribution would imply. Designers should account for these higher

probabilities in assessing turbine loads.

B. Implications for observational campaigns

Observational campaigns are critical for understanding the specific environment in which a

proposed turbine will be installed. These campaigns allow for very accurate understanding of

mean velocities and the primary component of turbulence kinetic energy. Of all physical prop-

erties investigated in this paper, the mean tidal velocity has shown the largest impact on overall

bending moment magnitude.

Through the analytical derivation (Section II), loads have been shown to tie closely to two

point correlations and structure functions. For these properties to be understood in a specific

installation environment, two point measurements are needed—a single point measurement de-

vice cannot capture transverse length scales and structure functions. While capturing a variety

of measurement widths and physical conditions will help in understanding bending moments,

the present study indicates that priority should be given to capturing information on transverse

lengths of the same scale as the turbine diameter.

Finally, as waves have been shown to have a significant impact on bending moments

(Section IV D), campaign managers should consider capturing wave specific data, such as am-

plitude and period, to accompany data on mean velocities and turbulence kinetic energy. These

wave properties will help designers and modelers better understand the turbulent ocean environ-

ment. Stability was found to be less important in understanding loads for this idealized analysis,

but could be of substantial importance when understanding wake recovery in arrays of real-

world turbines.

C. Implications for tidal channel simulations

Tidal channel simulations have the potential to supplement and enhance measurement

campaigns in the design process. However, for maximum effect, simulations should include the

physics most relevant to ocean current boundary conditions.

In the present analysis of bending moments, it was found that the most important variables

to consider in the simulation process were mean tidal velocity and waves. Simulations should

assess a variety of tidal velocities that are anticipated at the installation site, not just the mean

velocity. Waves were found to be important, particularly in the top 70% of the boundary layer,

and should be included in simulation physics.

One parameter that was found to have little effect on the resulting bending moments was

stability. The present set of simulations use the Boussinesq hypothesis (as described in Section

III), but for estimating mean and 90th percentile loads, stability effects were negligible.

However, this study has only investigated moderate levels of surface heating and cooling; for

063118-19 S. R. Alexander and P. E. Hamlington J. Renewable Sustainable Energy 7, 063118 (2015)

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areas with more substantial heating and cooling effects, additional analysis is required. Stability

could also be of primary importance when considering wake recovery in turbine arrays.

Comparisons of the simulation results to the measurement campaign by Thomson et al.41

show that the simulations slightly underestimate the turbulence intensity profile in the tidal

channel. Possible explanations for this include the influences of bathymetry and coastal features

on the inlet and bottom boundaries. In the case of a complex surrounding environment, free

stream turbulence in the simulation may need to be supplemented to account for these non-

local effects.

Finally, it was found in the simulations that high levels of grid anisotropy (the ratio of Dx,y

to Dz) resulted in substantial deviations from a log-layer profile. While the ocean channel envi-

ronment lends itself well to anisotropic grids (since the horizontal domain length is typically

much larger than the vertical depth), efforts should be made to either keep the grid anisotropy

below four or to adapt the SGS model to account for the grid anisotropy.

VI. CONCLUSIONS

A series of large eddy simulations have been performed in order to better understand the

tidal channel environment and turbulent bending moments experienced by ocean current turbine

rotors, with a focus on the physical parameters that impact these moments. The simulations

were validated against theoretical predictions as well as compared to observational data from

the Admiralty Head and Nodule Point inlets off the coast of Washington. In addition to simula-

tion results and statistics related to the understanding of bending moments, an analytical deriva-

tion was presented that explicitly links two-point turbulence statistics to bending moments.

The simulations were analyzed to provide information on mean velocity, Reynolds stresses,

longitudinal and transverse correlations, length scales, structure functions, and bending moment

distributions. Bending moments were calculated for an idealized infinitely thin circular rotor

disc and vertical and horizontal bending moments were found to be of the same order of magni-

tude at moderate depths, although horizontal bending moments increased substantially near the

ocean floor. The mean tidal velocity profile and wave strength were found to be the most im-

portant physical factors in determining the bending moment magnitude and distribution.

In future work, extending the analysis to include statistics and distributions on transient tor-

que loads is an important task. The analysis could also be expanded to run the resulting simula-

tion velocity snapshots through an advanced turbine simulator to provide statistics on all forms

of turbine loads. As with off-axis loads, torque loads can have negative effects on gearbox and

drivetrain components,5 and also reflect intermittent variations in the speed of the incoming

flow, which can affect blade, tower, and other structural loads.3 Consistent with prior stud-

ies2,3,6,7 for wind turbines, the present study indicates that intermittency can also play an impor-

tant role in determining bending moments in tidal boundary layers. As opposed to the idealized

analysis of turbine loading presented here, using velocity snapshots from the current study as

input to an advanced turbine simulator will therefore provide a more complete picture of the

blade, drivetrain, and tower loads experienced by realistic ocean current turbines. Additionally,

comparisons against other observational campaigns could further validate the simulation param-

eters while also providing guidance on improved modeling of inflow turbulence conditions for

different types of environment. Finally, future work will include realistic models for tidal tur-

bines in order to examine wake effects in turbine arrays. The present study has specifically

focused on the effects of different physical sources of turbulence on bending moments, for

which it was convenient to remove turbine-turbine interactions, but waking of one turbine by

another is a primary real-world contribution to bending moments that deserves further study.

ACKNOWLEDGMENTS

Helpful discussions with Dr. Peter Sullivan, Dr. Baylor Fox-Kemper, Dr. Jim Thomson, and

Dr. Katherine McCaffrey are gratefully acknowledged. The work of Katherine Smith in performing

final test simulations is also greatly appreciated. The authors were partially supported by NSF

1258995. This research utilized the Janus supercomputer, which was supported by NSF (Award No.

063118-20 S. R. Alexander and P. E. Hamlington J. Renewable Sustainable Energy 7, 063118 (2015)

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CNS-0821794) and the University of Colorado at Boulder. The Janus supercomputer is a joint effort

of the University of Colorado at Boulder, the University of Colorado Denver, and the National

Center for Atmospheric Research.

1K. Burman and A. Walker, “Ocean energy technology overview,” Technical Report No. DOE/GO-102009-2823, U.S.Department of Energy, Federal Energy Management Program, 2009.

2N. D. Kelley, B. J. Jonkman, G. N. Scott, J. T. Bialasiewicz, and L. S. Redmond, “The impact of coherent turbulence onwind turbine aeroelastic response and its simulation,” in Windpower 2005 Conference Proceedings (2005).

3T. M€ucke, D. Kleinhaus, and J. Peinke, “Atmospheric turbulence and its influence on the alternating loads on windturbines,” Wind Energy 14, 301–316 (2011).

4A. Sathe, J. Mann, T. Barlas, W. A. A. M. Bierbooms, and G. J. W. van Bussel, “Influence of atmospheric stability onwind turbine loads,” Wind Energy 16, 1013 (2013).

5F. Oyague, “Gearbox modeling and load simulation of a baseline 750-kW wind turbine using state-of-the-art simulationcodes,” NREL Technical Report No. NREL/TP-500-41160, 2009.

6A. Morales, M. W€achter, and J. Peinke, “Characterization of wind turbulence by higher-order statistics,” Wind Energy15, 391–406 (2012).

7P. Milan, M. W€achter, and J. Peinke, “Turbulent character of wind energy,” Phys. Rev. Lett. 110, 138701 (2013).8L. P. Chamorro and F. Port�e-Agel, “A wind-tunnel investigation of wind-turbine wakes: Boundary-layer turbulenceeffects,” Boundary-Layer Meteorol. 132(1), 129–149 (2009).

9M. Calaf, C. Meneveau, and J. Meyers, “Large eddy simulation study of fully developed wind-turbine array boundarylayers,” Phys. Fluids 22, 015110 (2010).

10J. Meyers and C. Meneveau, “Large eddy simulations of large wind-turbine arrays in the atmospheric boundary layer,”AIAA Paper 2010-827, 2010.

11F. Port�e-Agel, Y.-T. Wu, H. Lu, and R. J. Conzemius, “Large-eddy simulation of atmospheric boundary layer flowthrough wind turbines and wind farms,” J. Wind Eng. Ind. Aerodyn. 99, 154–168 (2011).

12M. J. Churchfield, S. Lee, P. J. Moriarty, L. A. Martinez, S. Leonardi, G. Vijayakumar, and J. G. Brasseur, “A large-eddysimulation of wind-plant aerodynamics,” AIAA Paper 2012-537, 2012.

13A. W. Lavely, G. Vijayakumar, J. G. Brasseur, E. G. Paterson, and M. P. Kinzel, “Comparing unsteady loadings on windturbines using TurbSim and LES flow fields,” AIAA Paper 2012-0818, 2012.

14J. Meyers and C. Meneveau, “Flow visualization using momentum and energy transport tubes and applications to turbu-lent flow in wind farms,” J. Fluid Mech. 715, 335–358 (2013).

15D. Mehta, A. H. van Zuijlen, B. Koren, J. G. Holierhoek, and H. Bijl, “Large eddy simulation of wind farm aerodynam-ics: A review,” J. Wind Eng. Ind. Aerodyn. 133, 1–17 (2014).

16J. M. Jonkman and M. L. Buhl, Jr., “FAST user’s guide,” Technical Report No. NREL/EL-500-38230, NationalRenewable Energy Laboratory, 2005.

17S. Gant and T. Stallard, “Modelling a tidal turbine in unsteady flow,” in Proceedings of the Eighteenth (2008)International Offshore and Polar Engineering Conference (2008), pp. 473–479.

18W. M. J. Batten, M. E. Harrison, and A. S. Bahaj, “Accuracy of the actuator disc-RANS approach for predicting the per-formance and wake of tidal turbines,” Philos. Trans. R. Soc. A 371, 20120293 (2013).

19M. E. Harrison, W. M. J. Batten, L. E. Myers, and A. S. Bahaj, “Comparison between CFD simulations and experimentsfor predicting the far wake of horizontal axis tidal turbines,” IET Renewable Power Gener. 4, 613–627 (2010).

20J. H. Lee, S. Park, D. H. Kim, S. H. Rhee, and M.-C. Kim, “Computational methods for performance analysis of horizon-tal axis tidal stream turbines,” Appl. Energy 98, 512–523 (2012).

21X. Sun, J. P. Chick, and I. G. Bryden, “Laboratory-scale simulation of energy extraction from tidal currents,” RenewableEnergy 33, 1267–1274 (2008).

22S. R. Turnock, A. B. Phillips, J. Banks, and R. Nicholls-Lee, “Modelling tidal current turbine wakes using a coupledRANS-BEMT approach as a tool for analysing power capture of arrays of turbines,” Ocean Eng. 38, 1300–1307 (2011).

23R. Malki, A. J. Williams, T. N. Croft, M. Togneri, and I. Masters, “A coupled blade element momentum-computationalfluid dynamics model for evaluating tidal stream turbine performance,” Appl. Math. Modell. 37, 3006–3020 (2013).

24M. J. Churchfield, Y. Li, and P. J. Moriarty, “Large-eddy simulation study of wake propagation and power production inan array of tidal-current turbines,” Philos. Trans. R. Soc. A 371, 20120421 (2013).

25M. G. Gebreslassie, G. R. Tabor, and M. R. Belmont, “Investigation of the performance of a staggered configuration oftidal turbines using CFD,” Renewable Energy 80, 690–698 (2015).

26S. Kang, I. Borazjani, J. A. Colby, and F. Sotiropoulos, “Numerical simulation of 3D flow past a real-life marine hydroki-netic turbine,” Adv. Water Resour. 39, 33–43 (2012).

27S. Kang, A. Lightbody, C. Hill, and F. Sotiropoulos, “High-resolution numerical simulation of turbulence in naturalwaterways,” Adv. Water Resour. 34(1), 98–113 (2011).

28I. Afgan, J. McNaughton, S. Rolfo, D. D. Apsley, T. Stallard, and P. Stansby, “Turbulent flow and loading on a tidalstream turbine by LES and RANS,” Int. J. Heat Fluid Flow 43, 96–108 (2013).

29T. P. Lloyd, S. R. Turnock, and V. F. Humphrey, “Assessing the influence of inflow turbulence on noise and performanceof a tidal turbine using large eddy simulations,” Renewable Energy 71, 742–754 (2014).

30A. S. Bahaj, W. M. J. Batten, and G. McCann, “Experimental verifications of numerical predictions for the hydrodynamicperformance of horizontal axis marine current turbines,” Renewable Energy 32, 2479–2490 (2007).

31A. S. Bahaj, A. F. Molland, J. R. Chaplin, and W. M. J. Batten, “Power and thrust measurements of marine current tur-bines under various hydrodynamic flow conditions in a cavitation tunnel and a towing tank,” Renewable Energy 32,407–426 (2007).

32L. Myers and A. S. Bahaj, “Near wake properties of horizontal axis marine current turbines,” in Proceedings of the 8thEuropean Wave and Tidal Energy Conference (2009), pp. 558–565.

33B. Gaurier, P. Davies, A. Deuff, and G. Germain, “Flume tank characterization of marine current turbine blade behaviourunder current and wave loading,” Renewable Energy 59, 1–12 (2013).

063118-21 S. R. Alexander and P. E. Hamlington J. Renewable Sustainable Energy 7, 063118 (2015)

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

67.190.30.33 On: Thu, 03 Dec 2015 02:33:45

34L. P. Chamorro, C. Hill, S. Morton, C. Ellis, R. E. A. Arndt, and F. Sotiropoulos, “On the interaction between a turbu-lence open channel flow and an axial-flow turbine,” J. Fluid Mech. 716, 658–670 (2013).

35L. P. Chamorro, D. R. Troolin, S.-J. Lee, R. E. A. Arndt, and F. Sotiropoulos, “Three-dimensional flow visualization inthe wake of a miniature axial-flow hydrokinetic turbine,” Exp. Fluids 54, 1459 (2013).

36E. Fernandez-Rodriguez, T. J. Stallard, and P. K. Stansby, “Experimental study of extreme thrust on a tidal stream rotordue to turbulent flow and with opposing waves,” J. Fluids Struct. 51, 354–361 (2014).

37P. Mycek, B. Gaurier, G. Germain, G. Pinon, and E. Rivoalen, “Experimental study of the turbulence intensity effects onmarine current turbines behaviour. Part I: One single turbine,” Renewable Energy 66, 729–746 (2014).

38I. A. Milne, A. H. Day, R. N. Sharma, and R. G. J. Flay, “Blade loading on tidal turbines for uniform unsteady flow,”Renewable Energy 77, 338–350 (2015).

39T. Stallard, T. Feng, and P. K. Stansby, “Experimental study of the mean wake of a tidal stream rotor in a shallow turbu-lent flow,” J. Fluids Struct. 54, 235–246 (2015).

40P. Mycek, B. Gaurier, G. Germain, G. Pinon, and E. Rivoalen, “Experimental study of the turbulence intensity effects onmarine current turbines behaviour. Part II: Two interacting turbines,” Renewable Energy 68, 876–892 (2014).

41J. Thomson, B. Polagye, V. Durgesh, and M. C. Richmond, “Measurements of turbulence at two tidal energy sites inPuget Sound, WA,” IEEE J. Oceanic Eng. 37(3), 363–374 (2012).

42K. McCaffrey, B. Fox-Kemper, P. E. Hamlington, and J. Thomson, “Characterization of turbulence anisotropy, coher-ence, and intermittency at a prospective tidal energy site: Observational data analysis,” Renewable Energy 76, 441–453(2015).

43I. Langmuir, “Surface motion of water induced by wind,” Science 87, 119–123 (1938).44J. C. McWilliams, P. P. Sullivan, and C. H. Moeng, “Langmuir turbulence in the ocean,” J. Fluid Mech. 334(1), 1–30

(1997).45P. E. Hamlington, L. P. Van Roekel, B. Fox-Kemper, K. Julien, and G. P. Chini, “Langmuir-submesoscale interactions:

Descriptive analysis of multiscale frontal spin-down simulations,” J. Phys. Oceanogr. 44, 2249–2272 (2014).46S. Li, M. Li, G. P. Gerbi, and J.-B. Song, “Roles of breaking waves and Langmuir circulation in the surface boundary

layer of a coastal ocean,” J. Geophys. Res.: Oceans 118, 5173–5187 (2013).47C.-H. Moeng, “A large-eddy-simulation model for the study of planetary boundary-layer turbulence,” J. Atmos. Sci.

41(13), 2052–2062 (1984).48L. P. Van Roekel, B. Fox-Kemper, P. P. Sullivan, P. E. Hamlington, and S. R. Haney, “The form and orientation of

Langmuir cells for misaligned winds and waves,” J. Geophys. Res.-Oceans 117, C05001 (2012).49M. Li, S. Radhakrishnan, U. Piomelli, and W. R. Geyer, “Large eddy simulation of the tidal cycle variations of an estua-

rine boundary layer,” J. Geophys. Res. 115, C08003, doi:10.1029/2009JC005702 (2010).50J. R. Taylor, S. Sarkar, and V. Armenio, “Large eddy simulation of stably stratified open channel flow,” Phys. Fluids 17,

116602 (2005).51B. Gayen, S. Sarkar, and J. R. Taylor, “Large eddy simulation of a stratified boundary layer under an oscillatory current,”

J. Fluid Mech. 643, 233–266 (2010).52L. Y. Pao and K. E. Johnson, “A tutorial on the dynamics and control of wind turbines and wind farms,” in American

Control Conference, 2009. ACC’09 (2009), pp. 2076–2089.53J. H. Laks, L. Y. Pao, and A. D. Wright, “Control of wind turbines: Past, present, and future,” in American Control

Conference, 2009. ACC’09 (2009), pp. 2096–2103.54A. S. Bahaj, W. M. J. Batten, A. F. Molland, and J. R. Chaplin, “Experimental investigation into the hydrodynamic per-

formance of marine current turbines,” Sustainable Energy Series Report No. 3, 2005.55L. Myers and A. S. Bahaj, “Simulated electrical power potential harnessed by marine current turbine arrays in the

Alderney Race,” Renewable Energy 30(11), 1713–1731 (2005).56A. S. Bahaj and L. Myers, “Analytical estimates of the energy yield potential from the Alderney Race (channel islands)

using marine current energy converters,” Renewable Energy 29(12), 1931–1945 (2004).57R. McSherry, J. Grimwade, I. Jones, S. Mathias, A. Wells, and A. Mateus, “3D CFD modelling of tidal turbine perform-

ance with validation against laboratory experiments,” in Proceedings of the 9th European Wave and Tidal EnergyConference, University of Southampton, UK (2011).

58A. D. D. Craik and S. Leibovich, “A rational model for Langmuir circulations,” J. Fluid Mech. 73, 401–426(1976).

59I. Gjaja and D. D. Holm, “Self-consistent Hamiltonian dynamics of wave mean-flow interaction for a rotating stratifiedincompressible fluid,” Physica D 98, 343–378 (1996).

60D. D. Holm, “The ideal Craik-Leibovich equations,” Physica D 98(2–4), 415–441 (1996).61J. C. McWilliams, J. M. Restrepo, and E. M. Lane, “An asymptotic theory for the interaction of waves and currents in

coastal waters,” J. Fluid Mech. 511, 135–178 (2004).62N. Suzuki and B. Fox-Kemper, private communication (2015).63P. P. Sullivan, J. C. McWilliams, and C.-H. Moeng, “A subgrid-scale model for large-eddy simulation of planetary

boundary-layer flows,” Boundary-Layer Meteorol. 71(3), 247–276 (1994).64M. A. Donelan, J. Hamilton, and W. H. Hui, “Directional spectra of wind-generated waves,” Philos. Trans. R. Soc.

London Ser. A 315(1534), 509–562 (1985).65A. Webb and B. Fox-Kemper, “Wave spectral moments and Stokes drift estimation,” Ocean Modell. 40(3–4), 273–288

(2011).66B. Polagye and J. Thomson, “Tidal energy resource characterization: Methodology and field study in Admiralty Inlet,

Puget Sound, WA (USA),” Proc. IMechE Part A: J. Power Energy 227, 352 (2013).67P. P. Sullivan and E. G. Patton, “The effect of mesh resolution on convective boundary layer statistics and structures gen-

erated by large-eddy simulation,” J. Atmos. Sci. 68(10), 2395 (2011).68M. J. Churchfield, Y. Li, and P. J. Moriarty, “A large-eddy simulation study of wake propagation and power production

in an array of tidal current turbines,” NREL, Technical Report No. CP-5000-51765, 2011.69J. C. McWilliams and B. Fox-Kemper, “Oceanic wave-balanced surface fronts and filaments,” J. Fluid Mech. 730,

464–490 (2013).

063118-22 S. R. Alexander and P. E. Hamlington J. Renewable Sustainable Energy 7, 063118 (2015)

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

67.190.30.33 On: Thu, 03 Dec 2015 02:33:45

70A. N. Kolmogorov, “The local structure of turbulence in incompressible fluid for very large Reynolds number,” Dokl.Akad. Nauk SSSR 30, 299–303 (1941).

71European Commission, Directorate General for Research and European Commission, Directorate General for Research,Energy, New and Renewable Energy Sources, and IT Power (Organization), “SEAFLOW: World’s first pilot project forthe exploitation of marine currents at a commercial scale: Contact JOR3-CT98-0202: Final Publishable Report,” Officefor Official Publications of the European Communities, 2005.

72OpenHydro, http://fundyforce.ca/technology/openhydro-nova-scotia-power/ for Fundy Ocean Research Center forEnergy, 2015.

73SeaGen-S 2 MW, MCT Product Brochure, Marine Current Turbines, 2013.

063118-23 S. R. Alexander and P. E. Hamlington J. Renewable Sustainable Energy 7, 063118 (2015)

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

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