A BEM-RANS approach for the fast power output prediction of ducted vertical axis water turbines

A BEM-RANS approach for the fast power output

prediction of ducted vertical-axis water turbines

Favio DOMINGUEZ∗, Jean-Luc ACHARD∗, Jeronimo ZANETTE† and Christophe CORRE‡

∗LEGI UMR 5519, BP 53, 38041 Grenoble Cedex 9, FRANCE

E-mail: Favio.Dominguez/[email protected]†Hydroquest, Le Tarmac, 29 chemin du Vieux Chene, 38240 Meylan, FRANCE

E-mail: [email protected]‡LMFA UMR 5509, Ecole Centrale de Lyon, 36 av. Guy de Collongue, 69134 Ecully Cedex, FRANCE

E-mail: [email protected]

Abstract—A numerical modelling is proposed to efficientlycompute the power produced by a row of Vertical Axis WaterTurbines (VAWT) deployed in parallel within various types ofwater flows. As the computational cost of the unsteady ReynoldsAveraged Navier Stokes (URANS) approach is high, a coupledBlade Element Momentum (BEM) / (steady) Reynolds AveragedNavier-Stokes (RANS) approach is developed, restricted to a 2Dapproximation. More specifically, the HARVEST hydrokineticdevices considered in this study are made of twin contra-rotatingVAWTs of ducted H-Darrieus type rotors. Momentum sourceterms are derived for such rotors from URANS simulationstaking into account the presence of fairings. The source termsincluded in the BEM-RANS model are derived by also incor-porating the optimal tip speed ratio (TSR), using a procedurebased on the mass flow through each rotor and on local flowconditions upstream of the rotor path. When compared withreference URANS results, the BEM-RANS model yields anaccurate prediction for a cost reduced by orders of magnitude.This model is then applied to the targeted analysis of the powerproduced by a row of VAWTs through a river or a channel withvarious blockage ratios.

Index Terms—Vertical-axis ducted hydrokinetic turbines,Blade Element Momentum, Reynolds-Averaged Navier-Stokessimulations, power output prediction

I. INTRODUCTION

In order to capture significant energy, hydrokinetic turbines

need to be built in large arrays within regions such as rivers,

man-made channels or tidal straits where the local bathymetry

focuses the flow. The layout of these arrays of turbines can sig-

nificantly change the amount of energy captured from the flow

[1] [2]. The power output prediction expected from a specific

array is needed in order to minimize the risks taken by de-

velopers and stake holders. These predictions can be achieved

through experimental studies on small scale deployments in

laboratories. For instance simple farm models composed of

Darrieus-type turbines have been studied in a water channel

[3]. It is however expensive to perform experimental works to

study the device parameters in large scale turbine arrays and

in real environmental conditions. On the other hand, numerical

modelling lowers the risk and cost for power output prediction,

although there remains a need to validate the results against

measured data. The explicit modelling of turbine blades can

be achieved through the unsteady Reynolds Averaged Navier

Stokes (URANS) approach. As demonstrated by Zanette [4]

for the turbine geometry considered in the present study, the

URANS approach is useful for evaluating, in the immediate

vicinity of turbines, the complex flow conditions including

high levels of turbulence, shear and variable flow directions.

However, the computational cost of this approach is high and

remains within reasonable bounds for an isolated turbine only

or an array including a very small (up to 3 typically) number of

turbines. The simulation cost of an array of N water turbines is

indeed much larger than N times the cost of the simulation of

an isolated turbine, because the interactions between turbines

impact the optimal rotational speed of each turbine as will

be detailed below. Such a high fidelity URANS numerical

modelling appears in any case unrealistic to describe a farm

scale array.

For that reason, several fast calculation models, based on

simplified models of turbines, have been presented to predict

the performance of turbines within an array. A frequently

studied and validated model, especially for Horizontal Axis

Water Turbines (HAWTs), is the actuator disc model. Works

[5] [6] have used Computational Fluid Dynamics under the ac-

tuator disc approximation for wind turbine wake calculations.

Similarly, Nishino et al. [7] represented a tidal turbine through

an actuator disc model for the prediction of the hydrodynamic

limit of power extraction. Steady RANS coupled with a Blade

Element Momentum model (BEM) provides a hybrid model

in which the effect of the rotor on the flowfield is implicitly

introduced through source terms in the RANS momentum

equations applied in a disk volume swept by the spinning rotor.

This approach was originally proposed for vertical cross-axis

wind turbines by Rajagopalan and Fanucci [8] and later used

by Malki et al. [9] for the prediction of tidal stream turbine

performance in the ocean environment and by Antheaume

et al. [10] to determine the performance characteristics of

VAWTs interacting with others in a cluster of towers. This

model has also been used in [11] and [2] to study the

power output from different tidal turbine arrays and in [12]

to simulate the wake interaction of two wind turbines located

close to each other at the top of an ideal Gaussian hill. The

advantage of this type of model is that it provides viscous

simulations of the wake behaviour with fewer cells than a

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Proceedings of the 11th European Wave and Tidal Energy Conference 6-11th Sept 2015, Nantes, France

ISSN 2309-1983 Copyright © European Wave and Tidal Energy Conference 2015

high-fidelity URANS approach.

The HARVEST [13] hydrokinetic devices considered in

this study are made of twin contra-rotating VAWTs of H-

Darrieus type [14]. Each rotor consists of one or more vertical

straight blades that run along a cylindrical surface, having a

so-called airfoil wing profile which produces a lift force and

causes the blades to move proportionally faster than the speed

of the surrounding water. Calculations will be restricted in

this study to three-bladed rotors with a blade profile based

on a well-known NACA geometry. The resulting torque is

applied to a rotating vertical shaft, which is coupled to a

generator housed just above the water or on the sea or river

floor. Various extra arrangements characterize the HARVEST

technology. Firstly, these turbines are built into a supporting

structure allowing to stack several turbines in two counter-

rotating twin columns. In each column, rotors share the same

rotating vertical shaft. The possibility of combining several

identical turbines according to each site, enhances modularity

and optimises productivity. Secondly, the so-called supporting

structure is equipped with non-symmetrical lateral diffuser

type fairings that create overspeed events in the drive areas

of the turbines. Finally, the rotation speed of each column is

tuned so that each turbine systematically works at the optimum

level of its power curve. These three new features make even

more complex the implementation of simplified models for

each converter.

In the present work, a row of VAWTs will be described

in a 2D approximation, i.e. by a series of horizontal sections

for each full machine. This is consistent with one of the key

advantages of Cross Flow Water Turbines (CFWTs) which is

that, owing to their rectangular projected frontal area, they can

be stacked as part of fences capturing more of the cross sec-

tional area of the current flow than possible with the circular

projected frontal area of a row of HAWTs. Furthermore such

a simplified description is motivated by on-site experiments

with HARVEST turbines where 3D effects and free-surface

influence on the power output have been found very limited

[15]. The ducted design of the turbines prevents using the polar

plots available in the literature which usually provide pre-

determined force coefficients in terms of local incident velocity

and angle of attack, based on wind tunnel experiments, to

estimate the momentum source terms describing the rotor

effects. Consequently, momentum source terms will be derived

for the ducted vertical-axis cross flow HARVEST turbine

by building beforehand upon a set of high-fidelity URANS

simulations for an isolated turbine. Moreover spatial variations

of the incoming velocity are taken into account at the level of

the blade location itself and not only of the rotor. Thirdly, the

aforementioned optimal speed, also expressed as an optimal tip

speed ratio (TSR), the ratio between the rotational speed of the

tip of the blades and the free-stream horizontal velocity, is not

a priori known since it depends on the local flow conditions.

Therefore, correctly assessing the power of a water turbines

array requires a parametric study on the combination of TSR

values for all the turbines belonging to the array. To fix ideas

let us consider the elementary case of an array made of 3

turbines hence 6 rotors; a parametric study limited to 3 test

values of the TSR for each rotor would require 36 = 729calculations of the array to determine its power output in

true (optimal) operation conditions. The present work builds

a simplified model to describe the behavior of a rotor when

working at its optimal TSR by using a procedure both global

(based on the mass flow through each rotor) and local (based

on local flow conditions upstream of the rotor path) in order to

compute the source terms of the BEM model. Using this model

a turbine array is computed without requiring a parametric

study on the TSR values, thus making such a simulation

affordable.

Section II provides the main features of the HARVEST

CFWT and reviews the hydraulic operation of this turbine. In

particular, significant quantities of interest such as TSR and

power coefficient are introduced. Section III describes how the

performance of a CFWT can be accurately computed using a

high-fidelity (hence computationally expensive) URANS ap-

proach and also details how the need to assess the performance

of the turbine farm for an optimal set of TSR values leads

to an excessive computational cost, preventing the use of

the high-fidelity approach for practical assessment of a farm

power output. Section IV explains how a much cheaper turbine

modeling can be developed from a Blade Element Momentum

approach to account for the rotor description, inserted into

a RANS calculation of the stator part of the CFWT. The

proposed BEM-RANS approach is validated by comparison

with the reference URANS results previously obtained in

Section III. Eventually, the efficient BEM-RANS model is

applied in Section V to analyze the global performance of

a row of turbines located in a channel of rectangular section

with uniform inflow velocity.

II. TURBINE DESCRIPTION AND OPERATION

The HARVEST turbine is equipped with two contra-rotating

three-bladed rotors mounted in a ducted diffuser, also referred

to as the turbine stator (see Fig. 1). The characteristic length

of the turbine is defined as the turbine width LT , distance

between the trailing edge of the stator blades; for the turbine

under study, LT = 4m. The duct length is equal to 0.47LT .

Each turbine rotor is made of 3 identical blades, with a chord

equal to 0.183m; the rotor diameter D and the hub diameter

d are respectively equal to 1m and 0.06m. These parameters

will be kept fixed throughout the present work.

A. Cross flow turbine operation

The HARVEST CFWT turbine blades rotate around the

vertical axis of the turbine with a characteristic rotation vector

ω (see Fig.2). The local flow velocity can be expressed either

in a fixed reference frame or in a moving reference frame,

attached to the rotating blade. If Vf denotes the absolute local

flow velocity upstream of the turbine blade, the corresponding

local velocity W of the fluid relative to the moving blade is

given by:

W = Vf − ω ∧R (1)

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Fig. 1. Schematic representation of a horizontal plane of a HARVEST turbine.

Fig. 2. Transversal cross section of a HARVEST rotor displaying a singleblade and the turbine hub. The flow is oriented from left to right.

where R denotes the position vector of the blade center

(mid-chord) in the 2D cylindrical coordinates system (0, θ, R)depicted in Fig.2. The angular position θ of the blade in

this reference frame is equal to 0 when the blade center

lies on the Y axis and increases when the blade is rotating

counterclockwise. The angle of attack α is defined as the angle

between the local relative velocity W upstream of the blade

and the blade chord line:

α = arctan

(

W · n

W · t

)

(2)

Three distinct reference frames can be distinguished in Fig.2:

i) the absolute reference frame, where cartesian coordinates

X , Y or cylindrical coordinates can be used; ii) the reference

frame relative to the flow, with axes respectively aligned with

the direction of the relative velocity and orthogonal to this

direction, in which the hydrodynamic force applied by the

fluid to the blade is decomposed into the drag D and lift Lcomponents; iii) the reference frame relative to the blade, with

axes respectively tangential and normal to the turbine rotation

center, in which the hydrodynamic force is decomposed into

a normal Fn and tangential Ft components. Using (Fn, Ft)facilitates the understanding of the cross flow turbine operation

since it is the tangential component Ft which creates the

driving torque Q of the rotor. The hydraulic operation of the

turbine can be eventually characterized by the power output

P = ωQ of each rotor, computed from the rotor torque Q and

the angular velocity ω = ‖ω‖. The instantaneous value of the

torque Q is computed for each angular position θ of the blade

from the instantaneous value of the tangential force component

Ft which yields an instantaneous value for the power output P .

It is customary to express the force components and also the

power output in non-dimensional form. Since the present study

is focused on 2D analysis of cross-flow turbine, assuming

effects in the vertical direction remain negligible, the drag and

lift coefficients and the normal and tangential force coefficients

of a turbine blade are defined as :

CD =D

12ρW

2c, CL =

L12ρW

2c

CT =Ft

12ρW

2c, CN =

Fn

12ρW

2c

(3)

where c denotes the blade chord and W is the magnitude

of the reference relative velocity W upstream of the blade.

These force coefficients depend on the angle of attack α and

the Reynolds number but the Reynolds effect can be neglected

in the present study because the typical value of the Reynolds

number is high enough (more than 2 millions based on inflow

velocity and rotor diameter) to ensure a fully turbulent flow.

The non-dimensional torque coefficient of the turbine rotor is

defined as:

CQ =Q

12ρV

2refDR

(4)

where the turbine diameter D and radius R are used as

characteristic lengths; the reference velocity Vref is an ab-

solute velocity which can be for instance the upstream farfield

velocity magnitude V0 in the case of a uniform upstream

farfield flow. The turbine power coefficient is defined as:

CP =P

12ρV

3refD

(5)

Since P = ωQ, the power coefficient can also be expressed

as:

CP = CQ × λ (6)

where λ denotes the TSR, defined as the ratio between the

angular velocity of the turbine blade and the reference velocity,

here computed as the incoming flow velocity magnitude V0:

λ =ωR

V0(7)

It must be emphasized the cross flow turbine operates in

practice at the optimal value of the TSR thanks to a regulation

system which adapts the rotational speed of the turbine until

the maximum power production is achieved.

III. URANS BASELINE SIMULATION

A. Model description

A straightforward approach to compute the power output P(or power coefficient CP ) of a turbine or an array of turbines

is to solve the URANS equations governing the unsteady

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turbulent flow around the turbine(s) :

∂Ui

∂xi

= 0

∂Ui

∂t+

∂Ui Uj

∂xj

= −1

ρ

∂P

∂xi

+∂

∂xj

[

ν

(

∂Ui

∂xj

+∂Uj

∂xi

)]

−∂u′

iu′

j

∂xj

(8)

where the Reynolds stress tensor components u′

iu′

j depend

on the fluctuations of the velocity components and must be

modeled, i.e. expressed in terms of averaged velocities Ui.

One well-established turbulence model is the k−ω SST model

which expresses the Reynolds constraints as a function of the

turbulent kinetic energy k and the specific dissipation ω. A

transport equation for both quantities must be solved on top

of the mass and momentum conservation equations (8) for the

mean flow. The resulting URANS equations are solved in this

work using the available commercial solver Fluent. System

(8) completed with the transport equations for the turbulent

quantities k and ω are solved using a second-order upwind

discretization.

B. Application to an isolated turbine in a channel

An HARVEST cross flow turbine is placed inside a rect-

angular channel and aligned with the incoming flow (hori-

zontal) direction. A view of an horizontal plane of the flow

configuration is provided in Fig.3. This plane corresponds to

the computational domain since a 2D analysis of the flow is

performed. The turbine center is located at (X,Y ) = (0, 0)in the absolute frame of reference. The ratio between the

turbine width LT and the channel width Wc = 3.5LT

defines the blockage ratio φ of the configuration. Assuming a

uniform horizontal inflow velocity magnitude V0, the physical

parameters of the computation are the Reynolds number Recbased on the blade chord and the velocity V0, the blockage

ratio φ and the TSR ω. It has already been emphasized that the

HARVEST turbine is working at an optimal TSR value thanks

to its regulation system. This means the power output predicted

for the turbine must be computed for this optimal TSR value,

which is not a priori known. Consequently, the prediction

of the power coefficient CP requires a series of numerical

simulations performed for a fixed value of the TSR parameter

λ in order to identify the power output corresponding to

the a priori unknown optimal value λ∗ of the TSR. Each

simulation is performed on the same grid, the topology of

which is illustrated in Fig.4. Note the simulations are actually

performed for only half of the channel (precisely for the

region y ≤ 0) with symmetry boundary conditions applied

along the x-axis and a slip boundary conditions applied at

the channel’s wall. This choice leaves a single rotor in the

computational domain, with a corresponding TSR λ. The inlet

of the computational domain is located 2.5LT upstream of

the turbine and the outlet is located 7.5LT downstream of the

turbine, ensuring no spurious effect of the far-field boundaries.

The computational mesh of the sole rotor region contains

110000 cells while the remainder of the domain, outside the

Fig. 3. Isolated turbine in a channel. Overview of the computational domain.

Fig. 4. View of the mesh resolution around (a) the turbine region and (b) theturbine rotors.

rotor region, contains about 135000 cells resulting in a total

number of cells approximately equal to 245000 for the case

under study (φ = 0.29). The computed quantity of interest CP

is plotted in Fig. 5 for V0 = 2.25m/s, φ = 0.29 and several

test values of the tip speed ratio λ for the turbine rotor: the

power coefficient of the isolated turbine is found to reach a

maximum value (CP )max = 1.059 for the optimal value of

TSR λ∗ = 2.3.

C. Application to a row of 3 turbines

As previously pointed out, several cross flow hydrokinetic

turbines are generally implemented on a given site in order

to exploit the stream associated with the site. Fig.6 displays

a schematic view of a single-row turbines array made of 3HARVEST turbines, hence a total of 6 turbine rotors. The

lateral spacing between the turbines is equal in that case to

the turbine length LT and the channel width WC is equal

to 24LT . The uniform freestream velocity V0 is set equal to

2.10m/s. Owing to the problem symmetry, the computational

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Fig. 5. Power coefficient of an isolated turbine in a channel (φ = 0.29 andV0 = 2.25m/s). Numerical prediction by the URANS model from a set ofcomputations performed for various (TSR) and by the BEM-RANS modelusing a single steady calculation at optimal TSR.

domain is reduced again to the lower half of the channel,

leaving a total of 3 rotors in the simulation. Each one of

these 3 rotors will be regulated in practice so that the rotor

power output is maximized. Because of the obvious interaction

between neighboring turbines, the optimal TSR value for one

rotor is likely to be different from the optimal value associated

with the neighboring rotor. In the simple case of 3 independent

rotors, a parametric study limited to 5 test values of the TSR

in the range [1.8, 3.0] for each rotor requires 125 calculations

of the array to determine its power output in true (optimal)

operation conditions. Even though physical considerations

based on the a priori analysis of the influence between turbines

help reducing the actual size of the parametric study, such a

numerical strategy cannot be applied to arrays that include

much more than 3 turbines. The variation of the row averaged

power coefficient CP with the TSR combination is plotted in

Fig.7; CP is computed as the sum of the power coefficients

associated with all the rotors divided by the number of rotors.

The maximum value of CP is equal to 1.18 and reached for

(λ∗

1, λ∗

2, λ∗

3) = (2.3, 2.2, 2.2).

IV. DEVELOPMENT OF THE BEM-RANS MODEL

A. General design principles

The BEM-RANS approach solves the steady RANS equa-

tions in a computational domain where the rotor blades

have been suppressed and replaced with computational cells

discretizing the surface swept by the rotor blades over a

period of revolution. Fig.8 displays a schematic view of this

discretized swept surface used for the rotor description and

Fig.9 provides a general and a more detailed view of an actual

computational grid for BEM-RANS calculations. The RANS

equations solved by the BEM-RANS approach are similar

to the URANS system (8); the continuity equation remains

Fig. 6. Single row three turbines schematic representation .

Fig. 7. Averaged power coefficient of a 3-turbine row in a channel (φ = 0.13and V0 = 2.10m/s). Numerical prediction by the URANS model from a setof computations performed for various combinations of TSRi (i = 1, 3)and by the BEM-RANS model using a single steady calculation at optimal(TSR1, TSR2, TSR3).

unchanged while the momentum equation reads now:

∂Ui

∂t+

∂Ui Uj

∂xj

=

−1

ρ

∂P

∂xi

+∂

∂xj

[

ν

(

∂Ui

∂xj

+∂Uj

∂xi

)]

−∂u′

iu′

j

∂xj

+ Si

(9)

where the source term S is locally active in each cell of the

virtual rotor surface previously defined. It must be defined in

such a way it accounts for the time-averaged (over a period

of rotation for the rotor) hydrodynamic effects of the rotating

blades. The instantaneous force applied on the flow by the

blade element is expressed in the fixed system coordinate

(0, X, Y ) by projecting (−FL,−FD) or (−FN ,−FT ) onto

the X and Y axes. These instantaneous values must be time-

averaged over one period of revolution before being inserted in

(9). The X and Y components of the momentum source term

S(k) introduced in the kth cell of the virtual rotor, identified

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by the angular position θ(k), is computed as:

S(k)X,Y = −

N × F(k)X,Y

2πRwr

(10)

where N is the number of rotor blades (N = 3 here) and

wr is the width of the virtual rotor, corresponding to the

thickness of the blade. The components F(k)X,Y of the force

applied in the kth cell of the virtual rotor can be deduced

from the lift and drag computed in this same cell or from the

normal and tangential force components using the geometric

transformations:(

F(k)X

F(k)Y

)

=

(

− cos(θ(k)) − sin(θ(k))− sin(θ(k)) cos(θ(k))

)(

FT

FN

)

(11)(

F(k)T

F(k)N

)

=

(

− cos(α(k)) sin(α(k))− sin(α(k)) − cos(α(k))

)

(

F(k)D

F(k)L

)

(12)

where the angle of attack α(k) is associated with the kth cell of

the virtual rotor. The lift and drag can be eventually computed

using the non-dimensional force coefficients introduced in (3):

F(k)L,D = CL,D(α(k))

1

2ρ[W (k)]2c (13)

Since α(k) can be computed from W(k) using (2), the source

term (10) is entirely defined once the local reference velocity

W(k) is itself defined and the non-dimensional lift and drag

coefficients CL,D for the rotor blade are available. A custom-

ary practice when applying the BEM-RANS approach is to

use polar plots for CL,D which are already available for the

blade profile used in the turbine design. Such a straighforward

approach cannot be applied in the present case because of the

ducted design of the turbine which introduces confinement

effect not properly taken into account in polar plots obtained

from free-flow experiments performed on the blade profile.

This observation means preliminary URANS simulations are

needed to build relevant tables of values for CL,D. Once these

tables obtained and stored, the solution procedure of the BEM-

RANS model reads:

• calculation of the flow through the HARVEST ducted

turbines using fixed grids only, with the moving rotor

grid of the baseline URANS simulations replaced with a

fixed grid of the surface swept by the rotor

• at each iteration of the solution process applied to (9), S

is computed using values W(k) extracted from the current

velocity field

• the new velocity field obtained by advancing (9) in time

yields new values of W(k), leading to new values of S

• the iterative process is continued until a steady state is

reached, which provides steady (averaged) force compo-

nents hence the expected converged value for the power

coefficient.

The cost reduction expected from the BEM-RANS model with

respect to the baseline URANS approach is thus obtained both

from a reduced number of cells in the computational domain,

since the highly refined mesh around the rotor blades is

Fig. 8. Schematic view of the discretized swept surface used in the BEM-RANS model. A value of the BEM source term S is computed in each gridcell.

Fig. 9. Typical fixed mesh for BEM-RANS calculations. Top : overview ofthe stator and the surface swept by the rotor blades (in green). Bottom : close-up on rotor region showing the local grid simplification when the real bladesare replaced with the BEM-RANS swept surface.

replaced with a much coarser 1D mesh of the surface swept by

the blades, and from the replacement of an unsteady simulation

with a steady simulation. However, since the quantity W(k)

is a local velocity relative to the blade, it must be computed

from a local absolute velocity V(k)f relative to cell k of the

virtual rotor and from the rotational speed of the turbine ωusing (1). This observation implies the RANS-BEM model

should be applied for several values of ω or the TSR λ until

the optimal value of λ is reached. The cost of the RANS-BEM

model for predicting the power output of a turbine array would

thus remain rather high because of the persistent need for a

parametric study leading to the identification of the optimal

set of TSR values for all the turbine rotor. To overcome this

difficulty, a methodology to derive a RANS-BEM model for

turbines working at their optimal TSR is now described.

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B. Methodology for model derivation

The force components FX,Y used in (10) to compute the

momentum source terms are function of the angular position

θ along the path of the turbine rotor, the rotational speed ωof the turbine and the upstream farfield conditions ξ. In the

simple case of a uniform upstream farfield flow in a channel of

variable width Wc, ξ reduces to the flow velocity magnitude V0

and the channel width Wc or the blockage ratio φ; in the case

of non-uniform upstream flow, ξ gathers all the parameters

needed to describe this farfield condition. When computed

from (13), (11), (12) for an optimal value ω∗ of the turbine

rotational speed, the force components should be such that:

FX,Y (θ, ξ, ω∗) = [A(θ, α)CD(α) +B(θ, α)CL(α)]

1

2ρW 2c

(14)

where α = α(θ, ξ, ω∗) and W = W (θ, ξ, ω∗). Deriving a

BEM-RANS model thus requires:

• to define a process for computing the local incidence αand the reference relative velocity W for each angular

position θ of the blade,

• to build tables for the lift and drag coefficient CD, CL.

The methodology followed to derive a BEM-RANS model

relies on a series of URANS computations which has been per-

formed for various inflow conditions ξ and various rotational

speeds ω. For each set ξ of inflow conditions, the optimal value

ω∗ yielding the maximum power output has been identified.

Different ways to compute the reference relative velocity Wand the angle of attack α for each angular position θ of the

blade have then been assessed and universal relationships for

CL,D, which do not depend or only weakly depend on ξ and

ω, have been identified by using (14) with CL,D as unknowns,

all other quantities (α, W , FX,Y ) being available.

One of the drawback of such a local approach is its

computational cost. Even though the variations for the lift and

drag coefficients are tabulated once for all, the calculation of

the local reference velocity W and the local incidence α must

be performed for each cell of the BEM rotor disk at each

iteration of the BEM-RANS calculation - this condition can

be slightly relaxed since W and α can be frozen for a few it-

erations without significantly altering the convergence process

to a steady-state. This motivated the search for an alternative

calculation process where FX,Y are directly expressed in the

form:

FX,Y (θ, ξ, ω∗) =

1

2ρV 2

d (ξ, ω∗) cCX,Y (θ) (15)

where Vd is an absolute reference velocity, to be determined.

The methodology to derive a BEM-RANS model of the

form (15) is similar to the one previously described for the

local model (14). It relies on the same series of URANS

computations performed for various inflow conditions ξ and

optimal rotational speeds ω∗, which is post-processed to

assess whether universal variations for the force coefficients

CX,Y can be identified when dividing the computed force

components FX,Y by a dynamic pressure based on a well-

chosen global reference velocity Vd.

C. Hybrid local / global model

The computing process for α and W must be developed in

such a way it can be applied to the BEM-RANS configuration

where the blade geometry is no longer represented in the flow

simulation. The process applied to the post-analysis of the

URANS flowfield, averaged over one revolution of the turbine,

is summarized as follows:

• i) for each angular position θ of the blade, the averaged

relative velocity field W is deduced from the averaged

absolute velocity field V using (1).

• ii) the streamline (computed from the local relative ve-

locity field) passing through the center of the blade is

described in the upstream direction until the variation

of the flow direction along this streamline is below a

prescribed threshold.

• iii) the relative velocity W is computed at the monitoring

point reached at the end of step ii) and its components

are used to define the angle of attack using (2) while its

magnitude defines the reference velocity used in (14).

• iv) the lift and drag coefficient CL,D can then be deduced

from the known values of FX,Y and the computed α and

W using (14).

Applying the local BEM-RANS model derived from this post-

analysis of URANS calculations is a two-step process:

• for each angular position θ discretizing the virtual rotor

(see Fig.9), the reference velocity vector W is computed

by following the above steps i)-iii).

• the relationships for CL,D previously derived (once for

all) in step iv) are applied for the computed angle of

attack α and the momentum source terms (10) inserted

in the BEM-RANS system (9) are computed using (14).

In order to build the relevant CL,D curves, the URANS

simulation of an isolated turbine in a channel (see Fig.3) has

been performed for a wide range of upstream conditions ξ and

systematically optimal values ω∗ of the angular velocity. The

similarity between the computed polar plots was sufficient to

extract a unique average polar plot for the lift coefficient and

the drag coefficient, at least when the angle of attack varies

between about 10◦ and 30◦, which corresponds to an angular

position θ between 30◦ and 150◦. These curve for CL, CD are

displayed in Fig.10. Deriving a universal curve for CL,D when

θ is outside this interval seemed dubious as no clear trend was

observed when superimposing plots corresponding to various

upstream conditions ξ. Thus, for angular position θ between 0◦

and 30◦ and between 150◦ and 360◦, the previous model based

on local flow parameters is replaced with a model of type

(15). The careful analysis of the URANS simulations for the

isolated turbine, performed for various flow conditions ξ and

corresponding optimal TSR ω∗, yields a universal CX,Y (θ)curve to describe the force components in this range of angular

position when Vd is computed from the flow rate through each

rotor (see Fig.10).

A BEM-RANS simulation is a steady simulation where no

actual blade motion is taking place. For the proposed BEM-

RANS model, the source terms FX,Y are computed so as to

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Fig. 10. Hydrodynamics coefficients curves used to feed the BEM-RANSmodel. Top : model based on the global velocity computed from the flowratethrough the turbine. Bottom : model based on local flow parameters.

reproduce the effect of the actual rotor when it is rotating at

the optimal TSR λ∗. To apply the local model, an estimate

of λ∗ is needed to compute the (virtual) relative velocity field

W from the absolute velocity field V provided by the BEM-

RANS calculation. It is obtained from a correlation between

the global velocity Vd and the optimal TSR λ∗ which was

also established from the thorough analysis of the URANS

calculations for an isolated turbine in a channel.

D. BEM-RANS model validation

The BEM-RANS model is applied to the test-problem of the

isolated turbine in a channel, described in section III-B. Note

that the flow configuration (φ = 0.29, V0 = 2.25m/s) does

not belong to the set of URANS calculations used to derive

the BEM-RANS model which makes the test problem relevant

for validation purpose. The predicted value for CP using

BEM-RANS is displayed in Fig.5 and compares very well

with the URANS prediction : CPBEM−RANS = 1.065 (with

λ∗ = 2.29) which exceeds the reference URANS value by

less than 0.6%. This agreement is a direct consequence of the

accurate prediction of the averaged (over a period of rotation)

force components FX , FY using the BEM-RANS model;

this agreement can be observed in Fig.11 where the URANS

prediction is plotted along the BEM-RANS prediction. The

accuracy of the BEM-RANS calculation is also illustrated in

Fig.12 where velocity profiles computed using BEM-RANS

and URANS are superimposed. The computational cost of

the BEM-RANS reduces to a single steady simulation on a

grid of 148000 cells only since the rotor blades are no longer

discretized; the total number of cells corresponds to the same

135000 cells for the stator part also used for the URANS

Fig. 11. Comparisons of the angular evolution of the forces (FX , FY ) actingon the rotor.

Fig. 12. Computed velocity profiles using the URANS model (in black) andthe BEM-RANS model (in red) on top of the BEM-RANS velocity contours.

simulation but the number of cells added by the discretization

of the rotor path amounts only to 13000 cells. Outside the

rotor region, the BEM-RANS computational grid is the same

as the grid used for the previous URANS calculations. The

BEM-RANS cost of 1 hour must be compared with the

cost of several URANS simulations, for various values of λ,

performed over several periods of rotation until the periodic

unsteady flow is established on the grid of 245000 cells

(including the required fine discretization of 110000 cells over

the rotor blades), which eventually amounts to 144 hours of

calculation.

The next validation is performed for the test-problem of

the 3-turbine row described in section III-C. The predicted

value for CP using BEM-RANS is displayed in Fig.7 and

compares again very well with the URANS reference result.

The predicted BEM-RANS value is equal to CP = 1.21(obtained for (λ∗

1, λ∗

2, λ∗

3) = (2.42, 2.36, 2.33)), a difference of

+2.5% only with respect to the reference URANS calculation.

The computational cost of the numerical prediction is divided

by a factor larger than 300 when using the BEM-RANS model

instead of the high-fidelity URANS approach.

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Fig. 13. Schematic representation of some of the turbines layouts used toanalyse the effect of the lateral distance and the channel blockage ratio onthe power output.

V. EXPLOITATION OF THE BEM-RANS MODEL

The validated BEM-RANS model is applied to the analysis

of the power output produced by a single row of n CFTWs

aligned across a channel of fixed width WC = 37.5LT =150m. The spacing LS between each turbine is expressed as

L∗

S ×LT with L∗

S ≥ 1. The row is supposed to be symmetric

with respect to the channel center and the distance between

the first and last turbine of the row and the closest channel

bank is equal to L∗

B×LT with L∗

B ≥ 1. It is straightforward to

establish the relationship : 2L∗

B+(n−1)L∗

S+n = n/φ where

the blockage ratio φ is such that n/φ = WC/LT = 37.5.

Consequently, the power produced by the row of turbines

depends on two parameters (assuming a constant inflow

velocity can be maintained) : the lateral spacing between

turbines characterized by L∗

S and the blockage ratio φ (or

equivalently the number of turbines). A given blockage ratio,

for instance φ = 0.08 for n = 3 (see Fig.13), can be

achieved for various values of L∗

S ; reversely, a fixed spacing,

for instance L∗

S = 1 (see Fig.13), can be maintained for

various values of φ. In practice, The BEM-RANS is applied

to assess the variation of the averaged power coefficient of the

row (CP = (∑n

i=1(CP )i) /n) with L∗

S for a fixed blockage

ratio φ = 0.08 and with φ for a fixed lateral spacing L∗

S = 1.

A. Influence of lateral spacing

Eleven single-row configurations corresponding to LS rang-

ing from 1 to 17 with n = 3 or φ = 0.08 are efficiently

computed using the BEM-RANS model since a single steady

flow computation is needed for each value of L∗

S , even though

Fig. 14. Velocity contours of three turbines in two configurations with (a)L∗

S= 1 and (b) L∗

S= 9.

Fig. 15. Variation of the average power coefficient (BEM-RANS prediction)for a single-row array with n = 3 (φ = 0.08) as a function of the lateralspacing L∗

S.

the optimal combination of TSR values varies with the change

of lateral spacing. Typical velocity contours, computed for

L∗

S = 1 and L∗

S = 9 are displayed in Fig.14 and qualitatively

illustrate the variation of the flow around and through each tur-

bine with the evolution of the lateral spacing. The quantitative

information is summarized in Fig.15 where CP is plotted as a

function of L∗

S . The power coefficient is maximum when L∗

S is

equal to its minimum value (L∗

S = 1 in the present analysis),

because of the beneficial flow acceleration produced by the

ducted turbines when close to one another. Next, the value of

CP decreases when L∗

S increases because the flow acceleration

is reduced with the turbines increasingly apart. Eventually,

the power coefficient increases again because of the flow

acceleration produced by the outer ducted turbines getting

close to the side-boundaries of the computational domain,

corresponding to the channel banks.

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Fig. 16. Velocity contours of fifteen turbines arrayed in a single-rowconfiguration corresponding to a channel blockage ratio of 0.39 with L∗

S= 1

(Case 7).

Fig. 17. Variation of the average power coefficient (BEM-RANS prediction)for a single-row array with L∗

S= 1 as a function of the blockage ratio φ.

B. Influence of blockage ratio

The velocity contours computed using the BEM-RANS

model for φ = 0.39 (n = 15) and L∗

S = 1 are displayed

in Fig.16 and illustrate again the beneficial flow acceleration

produced by the ducted turbines when in close proximity. The

value of CP plotted in Fig. 17 continuously increases with

the blockage ratio but the rate of increase tends to level off

between φ = 0.30 and φ = 0.40 before it strongly increases

again because of the flow acceleration produced near the side

boundaries corresponding to the channel banks.

VI. CONCLUSION

The BEM-RANS model developed in this work yields a

fast and accurate prediction of the power output produced by

a turbine row which can include up to 19 CFWTs, while a

similar analysis is definitely out of reach for a high-fidelity

URANS strategy. The tremendous cost reduction achieved

using the BEM-RANS model instead of the URANS model

results from two main effects : i) the replacement of an

unsteady calculation on a grid including the fine discretization

of each rotor blade with a steady computation on a stator grid

with each rotor coarsely described by a 1D grid; ii) the direct

calculation of each rotor behaviour for an optimal value of

its own TSR, without the need for a parametric study over

the TSR combination. The BEM-RANS model is currently

being improved in order to be applied to multi-row turbine

arrays where wake effects will be significant. It is also used

as a baseline strategy to derive lower fidelity model, yielding

an estimate of a rotor power output from purely geometric

information on the rotor location with respect to other rotors

included in the array.

ACKNOWLEDGMENT

The first author gratefully acknowledges the funding of his

PhD thesis by the Carnot institute ”Energies du Futur” in the

framework of the ORPHEE project.

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A BEM-RANS approach for the fast power output prediction of ducted vertical axis water turbines

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