### Blade Element Momentum Theory for Tidal Turbine Simulation with

```Blade Element Momentum Theory for Tidal Turbine
Simulation with Wave Effects: A Validation Study
* H. C. Buckland, I. Masters and J. A. C. Orme
*[email protected]
Introduction
Fast and robust turbine computer simulation:
Performance, periodic stall
Survivability, extreme wave climate
Fatigue
Fluid flow conditions
Outline
Turbine Performance simulation BEMT
Tidal flow boundary layer
Stream function wave theory
Wave acceleration
Tidal flow + Wave disturbance
Validation study
Inflow profile
•Waves
•Tidal stream
dFa1(a,b)
dT1(a,b)
Momentum theory
dFa2(a,b)
dT2(a,b)
Numerical aim:
dFa1(a,b) = dFa2(a,b)
dT1(a,b) = dT2(a,b)
Minimise g:
g=[ dFa1(a,b) - dFa2(a,b) ] 2 + [ dT1(a,b) - dT2(a,b) ] 2
Momentum Theory
U (1  2a)
U (1  a)
U
 0
Cavitation
Closed System:
Unknowns: a, b, T Fa
Two pairs of equations:
dT_{1}, dFa_{1}, dT_{2}, dFa_{2}
Optimiser ‘fmincon’ for a closed
BEMT system
Inflow profile
•Waves
•Tidal stream
dFa1(a,b)
dT1(a,b)
Momentum theory
dFa2(a,b)
dT2(a,b)
Numerical aim:
dFa1(a,b) = dFa2(a,b)
dT1(a,b) = dT2(a,b)
Minimise g:
g=[ dFa1(a,b) - dFa2(a,b) ] 2 + [ dT1(a,b) - dT2(a,b) ] 2
Tidal boundary layer
Bed friction -> boundary layer
Permeates the whole water column
Power law approximation for boundary layers
Assume a constant mean free surface height
h
uB  u(h / H s )1/10
x
Chaplin’s stream function wave theory
Finite depth, 2D irrotational wave of permanent form
Frame of reference moves with the wave
Finite depth wave theory:
Incompressible flow
Boundary condition
C
v
u
2  0
d
0
dy
Kinematic free surface condition:

v

x u  C
Bernoulli equation on the free surface:
N
L
2n(d  y)
2nx
 ( x, y)   x   an sinh
cos
T
L
L
n 1
Mean stream flow
Wave Disturbance
v 2  (u  C ) 2
Q   ( x) 
2g
Tidal flow +wave forces
Problems:
Depth dependent tide velocity
U w  uw  ut cos
Coupling:
Doppler effect
Alter moving frame of reference
C  ut cos
Accelerative forces:
The Morison equation
FME  Fdr  Fin
dFin  Cm Ax
dU
dl
dt
Fain
FT in
Cm  1  C A  1 
MA
Ax dl
M A  W 2dl
Axial oscillatory inflow:
Tangential oscillatory inflow:
M A  (c sin  )2 dr
M A  (c cos ) 2 dr
dFa  dFa  dFain
dFT  dFT  dFT in
c
The Barltrop Experiments
Tidal turbine in a wave tank
2 seperate investigations
Mx 
r R
 dF r
r  Rhub
Barltrop, N. Et al. (2006) Wave-Current Interactions in
Marine Current Turbines.
a
350mm turbine diameter
200 rpm
0.3m/s 1m/s
Wave height 150mm
Long waves 0.5Hz
Steep waves 1Hz
Bending Moments Mx My
Towed to simulate tidal flow!
My 
r R
 dF r
r  Rhub
T
Self Weight bending moment
Mx My results: 1m/s current
The Barltrop Experiments
Tidal turbine in a wave tank
2 seperate investigations
N
Fa  
rR
 dF
n 1 r  Rhub
Barltrop, N. Et al. (2006) Wave-Current Interactions in
Marine Current Turbines.
a
350mm turbine diameter
200 rpm
0.3m/s 1m/s
Wave height 150mm
Long waves 0.5Hz
Steep waves 1Hz
Bending Moments Mx My
400mm turbine diameter
90rpm
0.7m/s
0.833Hz
Varying wave heights
00mm 35mm 84mm 126mm
Torque T
Axial force Fa
Towed to simulate tidal flow!
Barltrop, N. Et al. (2007) Investigation into WaveCurrent Interactions in Marine Current Turbines.
N
T 
rR
 dF r
n 1 r  Rhub
T
Axial force and torque
TSR vs Ct, Cp and Cfa
Conclusion
Validation of wave theory
Compatibility of dynamic inflow with BEMT
Validation of self weight torque
Wave effect on performance is dependent on TSR curve
profiles
Further work
Wave superposition
Sea spectra, random phase sampling
Storm event simulation
Two way wave and current coupling
```