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 UniversitySingleton 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

Uwake = (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

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

a

CF

a

BEMT CFa

-a curve

Spera-corrected CFa

-a curve

Glauert-corrected CFa

-a curve

Buhl-corrected CFa

-a curve

High induction correction schemes

• Mathematical formulation straightforward• Momentum flux through annular element equated with

hydrodynamic forces on corresponding portion of rotor blade:– f1: axial momentum flux; f2: axial blade forces; g1: tangential momentum flux;

g2: tangential blade forces

• Each term a function of AIF and TIF• Minimise (f1 – f2)2 + (g1 – g2)2 across (a,b)-space to determine

solution• High induction correction simply modifies f1 for high values of

AIF (e.g., a > 0.4)

High induction correction schemes

• Classical Buhl formulation of axial force for a > ac:

• Assumes perfect reversal of flow (i.e., CFa = 2) for a = 1• Other values are plausible - e.g., 3D drag coefficient for a

flat plate gives CFa(a = 1) = 1.3

• In general, denoting CFa(a = 1) by CFa1 :

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

• 5o increase in rotor pitch moves rotor into HI regime

HI correction for an existing rotor

• 10o 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

0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

TSR

CF

a

Uncorrected curveHI correction onlyTip/hub loss correction onlyBoth corrections

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