Aerofoils: Lift, Drag, and Circulation - Small Wind Turbines: Analysis, Design, and Application - page 68

The first leg in Fig. 4.8 is upstream of the blades. As it is also a component of a
circular contour centred on the turbine axis, the first leg cannot contribute to C
because the wind is assumed steady, one-dimensional and spatially uniform, and
hence inviscid, and the circulation can never be changed in an inviscid fluid. (This
simple and elegant argument was devised by the famous British aerodynamicist
G.I. Taylor in 1915, see [ 16 ].)
Now the second leg is traversed from upstream to downstream whereas the
fourth leg is traversed in the opposite direction. If the width of the contour is r -1 ,
then the contribution to C from legs 2 and 4 will cancel. Thus C is determined
entirely from the third leg and so

NC ¼ 2prW 2

ð 4 : 9 Þ

where W 2 is the average circumferential velocity in the wake. Note that this
equation is identical to Eq. 2.8 . Combining with ( 3.5 ) and ( 3.11 ) for zero drag,
gives

L ¼ qU T C ð 4 : 10 Þ

If the blades are stationary, then U T ? U 0 as r ? 0, and
L ¼ qU 0 C

ð 4 : 11 Þ

which is the famous Kutta-Joukowski equation for aerofoils. Circulation around
aerofoils manifests itself as the so-called upwash upstream and the downwash
downstream of the aerofoil.
It is interesting that the derivation of ( 4.11 ) for aerofoils is much more difficult
than this ''proof'' for a cascade; see, for example, Sect. 6.4 of [ 17 ]. There is,
however, one major difference between aerofoils and blade elements of finite
solidity: whereas no circumferential velocity can be ''induced'' upstream of
blades, there is no such constraint on aerofoils or indeed, on a cascade of aerofoils.
In other words, there can be no net upwash for wind turbine blades. The
assumption that half the far-wake circumferential velocity defines the value of a 0
for the velocity triangle in Fig. 3.2 , must, therefore, be viewed as an attempt to
redistribute W so as to produce an upwash for the aerofoils comprising the blade
elements. Aerofoil behaviour can, therefore, only be an approximation for blade
elements at sufficiently small values of W in the wake; typically W increases as r
increases.
In addition to its fundamental connection with angular momentum, two
important features of C are that it measures vortex strength and it is a conserved
quantity in an otherwise inviscid flow; if the blades have a ''bound'' vorticity of
strength C, then the strength of the vortices trailing from the blades is also C. Note
that it is the circulation that is conserved, not the vorticity. The significance of
conservation of circulation is that the simplest wake structure for a wind turbine,
with a uniform U ? and leading to the Betz-Joukowsky limit, is when the N hub
vortices lie along the turbine axis and the N tip vortices are constant diameter
helices in the far-wake. The trailing helical vortices in this so-called Joukowsky