This paper presents the computational models used by the authors at MARETEC/IST for hydrodynamic design and analysis of horizontal axis marine current turbines. The models combine a lifting line method for the optimization of the turbine blade geometry and an Integral Boundary Element Method (IBEM) for the hydrodynamic analysis. The classical lifting line optimization is used to determine the optimum blade circulation distribution for maximum power extraction. Blade geometry is determined with simplified cavitation requirements and limitations due to mechanical strength. The application of the design procedure is illustrated for a two-bladed 300 kW marine current turbine with a diameter of 11 meters. The effects of design tip-speed-ratio and the influence of blade section foils on power and cavitation inception are discussed. A more complete analysis may be carried out with an IBEM in steady and unsteady flow conditions. The IBEM has been extended to include wake alignment. The results are compared with experimental performance data available in the literature.

1. June 22 2011 OMAE 2011 Rotterdam 1
OMAE-2011
HYDRODYNAMIC DESIGN AND ANALYSIS OF
HORIZONTAL AXIS MARINE CURRENT TURBINES WITH
LIFTING LINE AND PANEL METHODS
João Baltazar, João Machado, José Falcão de Campos
Marine Environment and Technology Center (MARETEC)
Instituto Superior Técnico (IST)
Technical University of Lisbon
Portugal

2. June 24 2011 OMAE 2011 Rotterdam 2
Motivation
 Considerable interest in the use of Horizontal Axis Marine Current
Turbines (HAMCT) for tidal energy.
 Need for hydrodynamic design and analysis tools.
 General use of Blade Element Momentum (BEM) methods for design and
analysis.
 Lifting Line (LL) and Lifting Surface (SF) methods suitable for inverse
methods in blade design.
 Integral Boundary Element Methods (IBEM) suitable for complete analyis of
a marine current turbine with cavitation in a current velocity field.

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Objectives
 Application of Lifting Line to the blade design and analysis of horizontal
axis marine current turbines for:
 “Optimum” power extraction,
 High cavitation inception speeds;
 Preliminary estimation of mechanical stresses.
 Illustration of:
 Effect of design TSR.
 Effect of blade section camber and thickness.
 Application of Integral Boundary Element Method to the hydrodynamic
analysis of horizontal axis marine current turbines
 Steady analysis in uniform flow with wake alignment.
 Unsteady analysis in typical tidal velocity profile.

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Lifting Line Model
 Inviscid and incompressible flow
 Vortex model
 Induced velocities from Biot-Savart
 Axial force and Torque from Kutta-
Joukowski
 Viscous effects: section 2D drag/lift
ratio

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Lifting Line Model
Blade Forces
 Power and axial force coefficients
 Circulation distribution
 Blade section chord
 Blade pitch angle
 
1
2
( ) ( ) 1 tan
h
T t i
r
Z
C r v r dr  

   
 
1
2
(1 ) ( ) 1 cot
h
P a i
r
Z
C v r r dr

 

   
2
LC
Vc


i   
( )r
R
U



/D LC C 
K hrNumber of blades Hub radius
Tip speed ratio
Drag / Lift ratio
r
U
av
tv
V

i

dL
dD
/dQ r
dT

i 

6. June 24 2011 OMAE 2011 Rotterdam 6
Lifting Line Model
Optimization problem
 Vortex pitch aligns vortices with local
flow velocity al LL (propeller
moderately loaded theory).
 Induced velocities at the lifting line
calculated with the induction factor
method (Morgan and Wrench, 1966).
 Hub effect through vortex images at
infinitely long cylindrical hub.
 Classical optimization criterion in
uniform flow (Betz, 1919).
 Numerical solution: Vortex lattice
method
)()()(
rr i 

t
a
i
vr
v


 


1
tan 1
'
4
1
)(
1
dr
r-r'
i
rd
dΓ
π
ru
hr
a,t
a,t  

( )
, ( , , / )a ti Z r r
 



tan
tan i Lagrange multiplier

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HYDRODYNAMIC DESIGN PROCEDURE
1. Radial distribution of drag-to-lift ratio.
2. Optimize power at design TSR
3. Choice of design lift coefficient: compromise of maximum lift-to-drag to
cavitation inception margin. Results in the combination of maximum camber
and angle of attack. Chord and pitch are found.
4. Find maximum thickness for prescribed margin for mid-chord cavitation.
5. Estimate mechanical stresses. Go to step 3 if not satisfactory.
6. Evaluate Reynolds numbers for converged chords.
7. Return to step 1.

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Design Example
 General design data.
 2 blades.
 Diameter: 11 m
 Nominal current speed: 2.5 m/s
 Nondimensional hub radius: 0.15
 Design TSRs: 3.5 and 5.
 Blade sections:
 NACA 63812 (f/c=0.08)
 NACA 63815 (f/c=0.08)
 NACA 63818 (f/c=0.08)
 NACA 65415 (f/c=0.04)
Lift and Drag data for
NACA 63xxx sections

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Blade Section Data
Minimum pressure envelopes
of NACA 63xxx foils
Lift-to-Drag ratio for
NACA 63xxx foils

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Blade Design – Radial distribution of circulation

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Blade Design – Radial Distributions of Chord and Pitch
PitchChord

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Power Coefficient

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Cavitation Inception

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Turbine Analysis with Integral Boundary Element Method
(IBEM) – Potential Flow Problem
 Inviscid and incompressible
 Velocity Field
 Perturbation Potential
 Undisturbed velocity
 Laplace equation
 Boundary Conditions
 Infinity
 Boundaries
 Wakes
 Kutta at sharp t.e.
U
( , , , ) ( , , , ) ( , , , )V x y z t V x y z t x y z t  
( , , )eV U x r t x   
2
0 
( , , , )x r t 
0 if andx x    
on B Hn U S S
n



   

, = on Wn U p p S
n n
  
  
 
   
 
  

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Integral Equation IBEM
 Integral equation
 where
2 ( , ) ( , ) ( , ) ( , ) ,
B H Wq q qS S S
B H
G G
p t G p q q t dS q t dS
n n n
p S S

  

   
    
    
 
 
1
( , )
G
R p q
 

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Vortex Wake Models
 Aligned wake model for steady flow.
 Empirical expansion.
 Vortex pitch alignment.
 Aligned wake model for unsteady flow.
 Expansion neglected.
 Vortex pitch alignment for for time-averaged axisymmetric inflow.
 Unsteady vortex shedding tangential convection velocity is the blade rotational
velocity.

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Viscous Effects
 Corrections to axial force and torque due to:
 Blade section drag.
 Viscous effect on the lift force.
 Corrections are applied sectionwise along the radius by:
 Determining the inflow velocity to the blade sections from Kutta-Joukowski law in
quasi-steady flow and inviscid hydrodynamic pitch angle.
 Correcting elemental axial force and torque using section viscous lift and drag.
Viscous lift is determined from potential lift by using 2D blade section data.
 Integrating along the radius.

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Numerical Method
(Code PROPAN)
 Surface Discretization:
 Structured surface grid with quadrilateral hyperboloidal elements
 Panel Method
 Integral equation solved by collocation method
 Constant source and dipole distributions
 Influence coefficients from formulation of Morino and Kuo (1974)
 Iterative pressure Kutta condition (IPK)

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Turbine Grid with Aligned Wake

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IBEM Steady Analysis in Uniform Flow
Axial Force Coefficient Power Coefficient

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IBEM Unsteady Analysis in Tidal Velocity Profile
Axial Force and Power Coefficient Fluctuations

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Concluding Remarks
 Lifting line is useful as inverse optimization method to design the blades of a horizontal
axis marine current turbine, including:
 Optimization of power extraction.
 Cavitation inception constraints.
 Preliminary estimates of mechanical strength constraints.
 The panel method is useful to analyse the hydrodynamic performance in:
 Steady and unsteady flow.
 Check cavitation inception margins in wetted flow at design and off-design conditions.
 Further work:
 Prediction of blade cavitation in steady and unsteady flow conditions.
 Further validation studies.