1. The Buhl High-Induction Correction for Blade Element Momentum Theory Applied to Tidal Stream Turbines Dr. Ian Masters (Swansea University) Dr. Michael Togneri* (Swansea University) Marine Energy Research Group, Swansea University Singleton Park, Swansea, SA2 8PP, United Kingdom

2. What is BEMT? Synthesis of two simple turbine models: – Stream tube & enclosed actuator disc – Hydrodynamic forces on 2D foils Rotor disc enclosed in streamtube, with velocity and pressure variation. Image from Hansen, M “Aerodynamics of Wind Turbines”, Earthscan Flow velocities for blade segment at radius r. Image from Burton, T et al, “Wind Energy Handbook”, John Wiley & Sons

3. Characteristics of BEMT Simpler problem than full CFD – Turbine effects on fluid ignored – Requires less computational power – Can obtain results much faster – Allows rapid investigation of wide range of cases Simplifying assumptions: – Inflow/wake can be regarded as an enclosed streamtube – No wake mixing – Momentum change described by two parameters: Axial induction factor (AIF, a), tangential induction factor (TIF, b)

4. High induction state AIF values in excess of 0.5 non-physical in classical BEMT U wake = (1 – 2a)U ∞ Semi-empirical correction necessary Must be validated against experiment

5. High induction correction schemes Graphs show high-induction corrections with and without tip/hub loss correction Current model uses Buhl-derived formulation

6. High induction correction schemes Mathematical formulation straightforward Momentum flux through annular element equated with hydrodynamic forces on corresponding portion of rotor blade: – f 1 : axial momentum flux; f 2 : axial blade forces; g 1 : tangential momentum flux; g 2 : tangential blade forces Each term a function of AIF and TIF Minimise (f 1 – f 2 ) 2 + (g 1 – g 2 ) 2 across (a,b)-space to determine solution High induction correction simply modifies f 1 for high values of AIF (e.g., a > 0.4)

7. High induction correction schemes Classical Buhl formulation of axial force for a > a c : Assumes perfect reversal of flow (i.e., C Fa = 2) for a = 1 Other values are plausible - e.g., 3D drag coefficient for a flat plate gives C Fa (a = 1) = 1.3 In general, denoting C Fa (a = 1) by C Fa1 :

8. Validation against experiment Experimental data from work by Tedds et al., Mason-Jones et al.

9. Effects of HI correction on thrust Uncorrected solution has higher thrust More pronounced nearer the tip

10. Effects of HI correction on thrust Uncorrected solution has near-tip region of relatively high annular thrust Coincides with the region where uncorrected AIF reaches physically meaningful limit

11. HI correction for an existing rotor 5 o increase in rotor pitch moves rotor into HI regime

12. HI correction for an existing rotor 10 o increase in pitch has more pronounced effect Difficulties finding solution without HI correction

13. Combining HI correction with tip/hub losses HI correction has greater effect in conjunction with tip/hub losses Losses lead to greater AIF values

14. Summary Classical BEMT does not deal with high induction, semi-empirical correction needed Modified Buhl correction validated against experiment – Good agreement for power, less good for thrust Correction works in conjunction with tip/hub losses BEMT results for a high-induction rotor without HI correction not physically meaningful