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

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

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)

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

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

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)

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 :

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

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

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

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

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

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

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