APPENDICES

A1 ANALYSIS OF TORQUE REACTING FLOW THROUGH STARTING WINDMILL

The propeller boundary layer approximation of helical flow wrongly implies a centripetal pressure

gradient from the high optimal torque and reacting swirl of slow-moving inner ends of windmill blades.

Such a gradient would cause inward secondary flow in the boundary layer of the optimal blade at small

enough local speed ratio x. Then oddly the static 2D airfoil lift CL & drag CD coefficient data would not

even apply at the static tip speed ratio X=0. At larger x, an inwards pressure gradient reduces the outward

secondary flow and stall delay. Bigger actual stall delays then came as a performance surprise which

made the windspeed and stresses of “stall-regulation” higher than expected [1]

Upstream of a BEM (Blade Element Momentum Theory) windrotor of interference a, the non-

dimensional pressure rise is a(2-a) plus some decrease with radius with increasing expansion radial

velocity. The pressure drop across the rotor is

4a(1-a) [2]; so the pressure coefficient on the lee side of the rotor actuator disc actually decreases with

radius faster than a(3a-2) does, opposite to a centripetal gradient. Consistently the BEM does not actually

require the blade reaction flow to be cylindrical, and indeed is easily vector generalised to any coordinate

system and other configurations such as vertical axis windmills.[4] For Hawt’s the BEM predicts well at

all x and X.

Whereas Joukowski’s ‘refined’ general momentum propeller theory does take the reaction as

cylindrical swirl and becomes absurd at Hawt small x and X [5,6]. Unfortunately Fluent CFD also

assumes helical swirl with a centripetal pressure gradient. These motivated a direct visualisation Photo

1.4.1 of the flow through a x=X=0 “stator” with B=12 blades of 25 cm diameter in the DTU 50cm

square wind tunnel at flow of 10m/sec upstream. The mean chord lines of arc airfoils of camber about

15% chord were set at 45º. This 15º to the optimal BEM apparent wind of 60º at x=0 strived for the

BEM optimal CL=2 with solidity =1. The BEM through flow U would then be ¾ windspeed V and

exit lateral flow 2v would be √3V/2 (v =U/√ 3) at a net 49º to the wind, uniformly along the blade so

without radial gradients. The Reynolds number at midspan was 31,000..

Tufts on the rear of the blades confirmed attached chordwise flow with no secondary flow (or separation) observed. The smoke was smoothly bent by the stator through at least the BEM optimal 49º, emerging from each blade tangent and straight as in Photo 1.3.1, not spiralling about the rotor axis.

The flow in Photo 1.3.1 was as if in Figure A1 each material element was flung straight off the rotor

like an isolated particle under no force suggesting constant velocity. Conservation of angular momentum

definitely requires constant speed and then Bernoulli, constant pressure along each straight smokeline.

Figure 1 Sketch of SmokeLine through (BEM optimal) x=X=0 rotor

These (bicycle hub)‘tangent spoke’ paths are easily expressed in Lagrangian time ‘‘ since leaving the stator Y=0 plane tangential at ‘hub’ radius ‘ro ‘ and passing through angle ‘‘ to radius ‘r’

and downstream distance ‘Y’ Y=U, so sin = 2v /r=2vY/rU, cos ro /r The radial mass flux per unit azimuth is 2rvsin =4v2 constant at each Y, confirming an incompressible flow.. Resolving the polar azimuthal ‘Ua ‘and radial ‘Ur ‘flow velocity components, behind the rotor Ua=2v cos . Ur=2vsin =2vY/rU so as 1/r . sin is a simple Eulerian similarity variable between 0 and 1, or Y>0 and ro >0 or 0< < /2, or in the volume between the rotor plane and the cone downwind of the rotor center, here of half angle 49º. The cone generators ro =0, = /2 are (dividing) streamlines. Crocco’s relation can be checked that the vorticity ▼xU is parallel to the velocity to confirm the head H and so pressure are constant throughout this volume just like the speed. The vorticity is constant along a streamline/vortexline because the constant velocity doesn’t stretch it. This flung centrifuging flow and its vorticity naively intersect with the tip vorticity at a common ‘Rc

‘and ‘Yc ‘ because the tip r=R exit tangential velocity is v not 2v. Rc2=R2+v2Yc

2/U2=ro2+4v2Yc

2/U2 gives Yc

2=(R2-ro2)U2/ 3v2. So the strongest innermost ro=0 vorticity would meet the tip vortex at Yc =R and

Rc2=4R2/3 before the final expansion to 3R2/2. Thus the blade trailing vorticity is very quickly flung into

the tip vorticity where their equal and opposite axial components can cancel in the violent eddies seen, leaving the following wake effectively free of net axial vorticity. So simple is the physical mechanism by which the kinetic energy of lateral velocity is lost, as has long been baldly said (if anything) to justify the BEM. There is likely a free-standing vortex ring with its outermost circulating streamline forming the swirl-free inner side of the cone of influence. The sharp bend in the smoke would mark the ring axial location and the bypassing flow’s expansion inwards around the back of the dividing streamline is highly prone to eddying even for solid body flow and without the axial vorticity mixing upstream. In the DTU wind tunnel the smoke literally exploded in all directions as it bent quite sharply inwards and the wake downwind was very turbulent. Ultimately the wake is effectively the 2D parallel flow Uo(r) to the axis and wind with no radial velocity or significant swirl or difference from ambient pressure. (If there is an infinitesimal amount of net inner vorticity left inside the rotortube it is sufficient to induce the original angular momentum flux in an infinitely/ weak potential vortex outside.) For X>0 the straight paths still might be realised intermittentently in turbulent flow, so naively

calculating them for the optimal rotor and ignoring the assisting outwards BEM pressure gradient shows

all intersect with the tip vortices within a diameter downstream for X at least 2.

The following Appendix 2 considers boundary layer equations consistent with the experiment’s

observations of no secondary flow at small x and no radial pressure gradient behind the rotor despite the

heavy torque.

A2 Stall Delaying Effect of Hawt Rotation A2.1 BOUNDARY LAYER TREATMENT OF ROTATIONAL EFFECTS

So at least at =0 the reaction v drives a straight line flow in the lab frame, whereas the rotation

obviously causes a circular ‘self’ wind r in the blade frame. The (non-linear) boundary layer

equations are naturally in the blade frame and conventionally [3] r +v was taken as circular there too.

So [3] to the standard planar boundary layer equations 5.7.11 in [7] were added Coriolis -2xu and

centrifugal terms (3.2 in [7]) due to the blade rotation as well as cylindrical curvilinear terms (App 2

in [7]). Viewing upwind in Figure 2 the blade path curves to the right, so then the Coriolis apparent force

is to bend velocities to the left. Consider the chordwise velocity ‘w’ over the leeward side of a flat blade

aligned to the outer apparent wind ‘W’ at the angle‘‘ to the azimuth. Then the azimuthal component

ua=w cos forced a spanwise flow in the boundary layer by curvilinear ua2/r , Coriolis -2wcos , and

centrifugal 2r terms. For no radial outer flow these combined [3] into

dp/dr=(Wcos-r)2/r=v2/r (1)

or the curvilinear term in the implied cylindrical lab frame with inwards dp/dx singular at x=0 for the

optimum BEM. Instead the X=0 lab frame flow was straight in the tunnel and the actual pressure behind

the rotor confirmed1 uniform at X=0, and slightly decreases with radius for optimal and practicable rotors

in the BEM.

Via dp/ dr, the net effect of these terms in forcing secondary spanwise flow in boundary layer was the

reduction of the skin-friction-retarded w values by their W ones. The centrifugal term had no and the

curvilinear – Coriolis factored as

wcos[wcos-2r]/r (2)

had deficit of (W-w)cos { 2r-

(W+w)cos}/r (3)

Because Wcos =r+v, the forcing simplified to (W-w)cos

{r-w cosv}/r (4)

Then taking the average value of w as ½W, the bracket { }≈½ r(1-3a’) which predicted inwards

secondary flow for a’>⅓ or optimal x<.7. The forcing could have been expanded as

(W-w)cos {1-(wcosv)/ (Wcos -v)} (5)

So to prevent the anomaly and the inward implicit dp/dr, since the experiment at x=0 saw neither

secondary flow or dp/dr, ignore the explicit a’ or v terms, at all x not just large, and take the forcing as

≈ (W-w)cos {1-w/W}= W{1-w/W}2 (6)

where =cosis the component of rotation normal to the blade. This simple high x function of only

the apparent wind W is also correctly zero at =0, has the same sign at all x>0, and allows the solution

to be pursued further analytically with more insight than numerically[3] . At mid boundary layer w =½W,

the forcing of u will be about ¼ the Coriolis alone. The shear layer of any directly outwards radial flow

such as due to the actual BEM outwards dp/dr will be superimposed, once known.

The azimuthal equation (4) had curvilinear -urua/r + 2ur Coriolis. Adding cos times this equation to

sin times ordinary axial uY equation to get the chordwise equation made its “rotation” term

urcos[2r -wcos]/r

again the same difference of Coriolis and curvilinear as before in (2), but here without dp/dr and outer ur

to reverse the effect.

With u=ur henceforth, again ignoring explicit v urcos[2 -wcosWcos -

v)] ≈ u [2 –w/W] (7)

eliminates an unlikely reversal at small x, has a single universal ‘shape’ and reduces the number of

perturbation terms. Again taking average w as ½W then the new average at a’=0 is about ¾ of the

Coriolis part alone. Thus in the blade frame the net secondary flow will itself bend by about ¾ of the

Coriolis acceleration towards the trailing edge. This second Coriolis effect will be shown to dominate

other side effects of the net secondary flow to increase the chordwise wall shear, delaying the onset of

stall and separation with zero wall shear in the adverse chordwise pressure gradient on the rear of the

airfoil due to its angle of attack. That angle of attack (and foil shape) effect can be considered for

simplicity as a separate competing perturbation to the basic flat plate boundary layer of Figure 2.

In the lab frame fluid dragged by rotating surface points on the inner blade is flung outwards in the

secondary flow where it lags fluid dragged by the faster outer blade, so it further thins the already thinner

outer boundary layer.

Figure 2 Twisted Flat Blade & Lee Boundary Layer Viewed from Downwind Far from Axis & Rotating

with Blade

A2.3. PERTURBATION OF FLAT PLATE BOUNDARY LAYER EQUATIONS FOR ROTATION TERMS

With z=0 the leading edge and z=c the trailing edge and the kinematic viscosity, the flat plate boundary

layer has depth

= (z/W)½ which scales its normal coordinate y as =y/

d/dz=½/z d/dW=-½/W d/dz=-½/z d/dW=½/W (8)

Now denoting d/dz as dz, , etc the dependent velocity v normal to the plate is the sum of vw =-dz from

dyvw =- dzw and

vu=-dr from dyvu =-dru. Expanding the chordwise stream function in z

W{f(+F2(z2 +..} so w= dy=W(f ’+F2’z2 +..) & vd2w/dy2=W2(f ”’/z +F2”’z +..) (9)

vw ½Wf/z + ½Wf’/z -5WF2 z/2 + ½WF2’z +… (10)

So via (7) the modified chordwise boundary layer equation (wdzw)+vwdyw -vd2w/dy2=u(2-w/W) -udrw –

vudyw (11)

has leading -W2/z LHS terms ½ f f”+f”’=0 with the cancellation of ½ f’f” terms due to the z

dependence of in

dydz -dzdy (which will likewise cancel for F2 and upcoming g)

f=f’=0 and f”=.332 achieves the outer stream f’=1 at =∞ the basic Blasius solution in Fig3

Via (6) the modified spanwise boundary layer equation, wdzu+vwdyu-vd2u/dy2=W{1-w/W}2-

udru–vudyu (12)

here requires gz +.. u=dy=zg’ for leading terms W times f’g’-½f g”-g”’=(1-f’)2

This centrifuging of the deficit from the outer flow is the reduction of the Coriolis 2(1-f’) by the high x

curvilinear 1-f’2.

At =0 g=g’=0, g”=.663 achieves g’=0 at =∞ . g’ has a peak of about .275 at small =1 due to the

concentration at the surface of squaring the chordwise velocity deficit. This square would stay larger

further away from the wall for a profile closer to separation such as the Falkner-Skan of negative

exponent, whose secondary flow would be more extensive and of slightly higher shear. The spanwise to

chordwise wall shear ratio is eg”/f ” with the perturbation factor of r,

e=z/W=cos(xV/W)z/r (13)

For the BEM optimal rotor designed at minimum CLd with minimum CD / CL with B blades

c=8r(1-cosd)/ BCLd (14)

So then at midchord e= 4cos(1-cosd)xV/ WBCLd (15)

For B=3 blades at CLd=1 , then the midchord spanwise to Blasius wall shear is plotted in green in Fig 3

for x=xd =d and peaks at .8 at x=⅔ At large x, there is little change in using the design values of x and

W instead of the operating values. The variation is as 2 or x -2 from the rapid reduction in solidity with

large x and constant CLd. At small speed ratio x with W near √3V/2 the main variation in e is the x term,

which should be the actual operating value, generally lower at stall than the design value, which lowers

the perturbation. Due to practical limits on the blade chord, B=3 design is sub optimal at small x with

c<in (14) and higher CLd , further reducing the perturbation factor from eqn (15). Waterpumping

windmills approach optimal low x≈1 design but their B of 18 or 24 (presumably to avoid low blade

aspect ratio and high end losses) makes the secondary flow insignificant.

Consider the effect of this secondary flow on the primary w boundary layer. Firstly it induces a normal

velocity

vu=-dr= -gzdr+½z(g-g’)drW/ W (16)

due to the twist of and increase in the apparent wind with r. Physically the twist increase in and so u

with r requires an inflow towards the wall. Whereas the increase in apparent wind with r means the

boundary layer thickness towards the tip is thinner which makes vu outflow. Forming the net forcing

terms of power z on the RHS of (11)

zg’(2-f’) + zWdr f”g - z drW {g’f’ -½gf ”}

Where the standard cancellation occurs in the last -udrw –vudyw origin term. Terms of order z with

W2F2=fcc + Wdrfc - drW fca , subscripts denoting Coriolis on Coriolis, twist on Coriolis, and

apparent wind on Coriolis effects respectively, give the equations

L(fcc)=2f’f ’cc -½f f’’cc -5f’’fcc/2 -f’’’cc=2-f’ (17) L(fc)= g f” (18) L(fca)= g’f’ -½gf ” (19)

Shooting from =0 fcc= f c =fca=f’cc= f ’c = f’ca =0 with fcc”=.465 f “c =.0822 fca”=.134 achieves f’cc=

f’c = f’ca =0 at =∞ as in Fig. 3

The ratio of second to first factors Wdr = -tan sec (W/V)dx Estimating using the BEM optimal

rotor, the ratio declines rapidly with x from 1 at x=.3 to .3 at x=1 and f “c /fcc” is weak. But the ratio of

third to first

drW / =dx(W/V) /cos reaches 1 at large x and is ½ at x=.2, so the apparent wind variation effect

reduces the double Coriolis feedback effect more significantly. The biggest net reduction of both

corrections to the first wall shear is 28% at large x, so the Coriolis veer is dominant at worst .72x.75=.54

its naïve strength in the perturbation to chordwise wall shear,

e2 [fcc”-fc” Wdr-fca” drW /]/f” (20)

The plot in Fig 4 for B=3 CLd =1 at midchord shows a relative perturbation to the Blasius wall shear

peaking at .2 at x=.6. Competing is the negative perturbation from (wdzw) to wall shear proportional to

angle of attack at midchord. The of zero wall shear at midchord ,as the approximate separation

point, will be proportional to 1+(20) or very roughly the fractional correction of stall s and stall CLs will

be proportional to the plotted factor (20).

Due to the stronger secondary flow for profiles of nearly reversing /separating flow, and for the actual

outwards dp/dr, such preliminary analysis is an underestimate.

A2.3 Conclusions In the rotating blade frame the Coriolis forcing of secondary flow is reduced significantly by

curvilinear terms but reversal at low x is rejected by experiment at x=0 showing no secondary flow and

that the torque reaction flow in the lab frame is straight and not curved by any inward pressure gradient.

As a better (and simpler) approximation the net difference of Coriolis minus curvilinear simply at high x

is simply used at all x. The Blasius flat plate boundary layer is perturbed by expansion in z/W. Then at

midchord the peak secondary flow shear is .8 of the chordwise flat plate shear. The thinning of the

chordwise boundary layer by the tailwise veering of this effective Coriolis secondary flow is moderated

by its own transport from thicker slower inner boundary layer to faster thinner outer. The relative wall

shear increase and stall delay is small enough at high x and in practice at low x to justify the perturbation

approach.

References [1].Hansen, A.C. & Butterfield, C.P. Windmill Aerodynamics, Annual review of Fluid

Mechanics 25 p115-149(1993)

[2] Glauert, H., (1935) Chap X1 Windmills and Fans in Division L Airplane Propellors Aerodynamic Theory 3 (ed.W.F. Durand), J. Springer, Berlin, reprinted by Dover Publications New York 1963.

[3] Martínez GG Sorenson JN & Shen W Z 3D boundary layer study on a rotating wind turbine blade

Journal of Physics: Conference Series 75 (2007) 012032 doi:10.1088/1742-6596/75/1/012032

[4] Farthing, S. P. (2007) “Optimal Robust and Benign Horizontal and Vertical Axis Wind Turbines,” Journal of Power and Energy Vol 221 No. 7, p971-979.

[5]. Farthing, S. P. (2010) “Robustly Optimal Wind Turbine for Swirl Expansion and Decay”Journal of Power and Energy Vol 224 No.8.

[6]. Sorenson,J.N. & van Kuik, G. General momentum theory for wind turbines at low tip speed ratios Wind Energy , 2010

[7] Batchelor, G.K. (1967) An Introduction to Fluid Mechanics Cambridge University Press