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Blade Element Momentum Theory for Tidal Turbine Simulation with Wave Effects: A Validation Study * H. C. Buckland, I. Masters and J. A. C. Orme *513924@swansea.ac.uk

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Introduction Fast and robust turbine computer simulation: Performance, periodic stall Survivability, extreme wave climate Fatigue Fluid flow conditions

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Outline Turbine Performance simulation BEMT Tidal flow boundary layer Stream function wave theory Wave acceleration Tidal flow + Wave disturbance Validation study

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Blade element theory dF a1 (a,b) dT 1 (a,b) Inflow profile Waves Tidal stream Numerical aim: dF a1 (a,b) = dF a2 (a,b) dT 1 (a,b) = dT 2 (a,b) Minimise g: g=[ dF a1 (a,b) - dF a2 (a,b) ] 2 + [ dT 1 (a,b) - dT 2 (a,b) ] 2 Momentum theory dF a2 (a,b) dT 2 (a,b)

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Blade Element Momentum Theory BEMT Momentum Theory

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Closed System: Unknowns: a, b, T Fa Two pairs of equations: dT_{1}, dFa_{1}, dT_{2}, dFa_{2} Cavitation Blade Element Theory Blade Element Momentum Theory BEMT

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Optimiser ‘fmincon’ for a closed BEMT system b

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BEMT steady state example

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Blade element theory dF a1 (a,b) dT 1 (a,b) Inflow profile Waves Tidal stream Numerical aim: dF a1 (a,b) = dF a2 (a,b) dT 1 (a,b) = dT 2 (a,b) Minimise g: g=[ dF a1 (a,b) - dF a2 (a,b) ] 2 + [ dT 1 (a,b) - dT 2 (a,b) ] 2 Momentum theory dF a2 (a,b) dT 2 (a,b)

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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 x

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Chaplin’s stream function wave theory C u v Finite depth, 2D irrotational wave of permanent form Frame of reference moves with the wave Finite depth wave theory: Incompressible flow Boundary condition Kinematic free surface condition: Bernoulli equation on the free surface: Mean stream flow Wave Disturbance

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Tidal flow +wave forces Problems: Depth dependent tide velocity Steady state BEMT Coupling: Doppler effect Alter moving frame of reference

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Accelerative forces: The Morison equation c Axial oscillatory inflow: Tangential oscillatory inflow:

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The Barltrop Experiments 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! Barltrop, N. Et al. (2006) Wave-Current Interactions in Marine Current Turbines. Tidal turbine in a wave tank 2 seperate investigations

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Self Weight bending moment

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Mx My results: 1m/s current

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The Barltrop Experiments Barltrop, N. Et al. (2007) Investigation into Wave- Current Interactions in Marine Current Turbines. 350mm turbine diameter 200 rpm 0.3m/s 1m/s Wave height 150mm Long waves 0.5Hz Steep waves 1Hz Bending Moments Mx My Barltrop, N. Et al. (2006) Wave-Current Interactions in Marine Current Turbines. 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! Tidal turbine in a wave tank 2 seperate investigations

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Axial force and torque

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TSR vs Ct, Cp and Cfa

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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

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Further work Wave superposition Sea spectra, random phase sampling Storm event simulation Two way wave and current coupling

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Cascade theory The theory in this lecture comes from: Fluid Mechanics of Turbomachinery by George F. Wislicenus Dover Publications, INC. 1965.

Cascade theory The theory in this lecture comes from: Fluid Mechanics of Turbomachinery by George F. Wislicenus Dover Publications, INC. 1965.

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