7 Theoretical
7.6 Thin aerofoil theory
Figure 7.8 showed the simulation of an aerofoil by a vor- tex sheet of variable strengthγ(s). If one imagines a thin aerofoil such that both surfaces come closer together, it becomes possible, without significant error, to con- sider the aerofoil to be represented by its camber line with a distribution of vorticity placed along its length.
When this is the case the resulting analysis is known as thin aerofoil theory, and is applicable to a wide class of aerofoils, many of which find application in propeller technology.
Consider Figure 7.13, which shows a distribution of vorticity along the camber line of an aerofoil. For the
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148 Marine propellers and propulsion
Figure 7.13 Thin aerofoil representation of an aerofoil
camber line to be a streamline in the flow field the com- ponent of velocity normal to the camber line must be zero along its entire length. This implies that
Vn+ωn(s)=0 (7.14)
whereVnis the component of free stream velocity nor- mal to the camber line, see inset in Figure 7.13; and ωn(s) is the normal velocity induced by the vortex sheet at some distances around the camber line from the leading edge.
If we now consider the components of equation (7.14) separately. From Figure 7.13 it is apparent, again from the inset, that for any pointQalong the camber line,
Vn=V sin α+tan−1
−dz dx
For small values of α and dz/dx, which are condi- tions of thin aerofoil theory and are almost always met in steady propeller theory, the general condition that sinθtanθθholds and, consequently, we may write for the above equation
Vn=V α− dz
dx
(7.15) whereα, the angle of incidence, is measured in radians.
Now consider the second term in equation (7.14), the normal velocity induced by the vortex sheet. We have previously stated that dz/dxis small for thin aerofoil theory, hence we can assume that the camber–chord ratio will also be small. This enables us to further
assume that normal velocity at the chord line will be approximately that at the corresponding point on the camber line and to consider the distribution of vorticity along the camber line to be represented by an identical distribution along the chord without incurring any sig- nificant error. Furthermore, implicit in this assumption is that the distancesaround the camber line approxi- mates the distancex along the section chord. Now to develop an expression forωn(s) consider Figure 7.14, which incorporates these assumptions.
From equation (7.6) we can write the following expression for the component of velocity dωn(x) normal to the chord line resulting from the vorticity element dξ whose strength isγ(ξ):
dωn(x)= − γ(ξ)dξ 2π(x−ξ)
Hence the total velocityωn(x) resulting from all the contributions of vorticity along the chord of the aerofoil is given by
ωn(x)= − c 0
γ(ξ)dξ 2π(x−ξ)
Consequently, by substituting this equation together with equation (7.15) back into equation (7.14), we derive the fundamental equation of thin aerofoil theory
1 2π
c
0
γ(ξ)dξ
(x−ξ)=V α− dz
dx
(7.16) This equation is an integral equation whose unknown is the distribution of vortex strengthγ(ξ) for a given
Theoretical methods – basic concepts 149
Figure 7.14 Calculation of induced velocity at the chord line
incidence angleαand camber profile. In this equation ξ, as in all of the previous discussion, is simply a dummy variable along theOxaxis or chord line.
In order to find a solution to the general problem of a cambered aerofoil, and the one of most practical importance to the propeller analyst, it is necessary to use the substitutions
ξ= c
2(1−cosθ)
which implies dξ=(c/2) sinθdθand x= c
2(1−cosθ0)
which then transforms equation (7.16) into 1
2π π
0
γ(θ) sinθdθ
cosθ−cosθ0)=V α− dz
dx
(7.17) In this equation the limits of integrationθ=πcorres- ponds toξ=candθ=0 toξ=0, as can be deduced from the above substitutions.
Now the solution of equation (7.17), which obeys the Kutta condition at the trailing edge, that isγ(π)=0, and make the camber line a streamline to the flow, is found to be
γ(θ)=2V
A0
1+cosθ sinθ
+∞
n=1
An sin (nθ)
(7.18) in which the Fourier coefficients A0 and An can be shown, as stated below, to relate to the shape of the camber line and the angle of the incidence flow by the substitution of equation (7.18) into (7.17) followed by some algebraic manipulation:
A0=α− 1 π
π
0
dz dx
dθ0 (7.18a)
An=2 π
π
0
dz dx
cos(nθ0)dθ0
For the details of this manipulation the reader is referred to any standard textbook on aerodynamics.
In summary, therefore, equations (7.18) and (7.18a) define the strength of the vortex sheet distributed over
a camber line of a given shape and at a particular incidence angle so as to obey the Kutta condition at the trailing edge. The restrictions to this theoretical treatment are that:
1. the aerofoils are two-dimensional and operating as isolated aerofoils,
2. the thickness and camber chord ratios are small, 3. the incidence angle is also small.
Conditions (2) and (3) are normally met in propeller technology, certainly in the outer blade sections. How- ever, because the aspect ratio of a propeller blade is small and all propeller blades operate in a cascade, Con- dition (1) is never satisfied and corrections have to be introduced for this type of analysis, as will be seen later.
With these reservations in mind, equation (7.18) can be developed further, so as to obtain relationships for the normal aerodynamic properties of an aerofoil.
From equation (7.7) the circulation around the cam- ber line is given by
= c
0
γ(ξ)dξ
which, by using the earlier substitution of ξ=(c/2) (1−cosθ), takes the form
=c 2
c
0 γ(θ) sinθdθ (7.19)
from which equation (7.18) can be written as =cV A0
π
0
(1+cosθ)dθ +
∞ n=1
An π
0
sinθsin(nθ)dθ
which, by reference to any table of standard integrals, reduces to
=cV πA0+π
2A1
(7.20) Now by combining equations (7.1) and (7.3), one can derive an equation for the lift coefficient per unit span as
c1= 2 Vc
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150 Marine propellers and propulsion from which we derive from equation (7.20)
c1=π[2A0+A1] (7.21)
Consequently, by substituting equations (7.18a) into (7.21) we derive the general thin aerofoil relation for the lift coefficient per unit span as
c1=2π α+ 1 π
π
0
dz dx
(cosθ0−1)dθ0
(7.22) Equation (7.22) can be seen as a linear equation between candαfor a given camber geometry by splitting the terms in the following way:
c1=2πα+2 π
0
dz dx
(cosθ0−1)dθ0
... ... ... ...
Lift Lift at zero incidence slope
in which the theoretical life slope dc1
dα =2π/rad (7.23)
Figure 7.15 shows the thin aerofoil characteristics schematically plotted against experimental single and cascaded aerofoil results. From the figure it is seen that the actual life slope curve is generally less than 2π.
The theoretical zero lift angleα0is the angle for which equation (7.22) yields a value ofc1=0. As such it is seen that
α0= −1 π
π
0
dz dx
(cosθ0−1)dθ0 (7.24)
Figure 7.15 Thin aerofoil and experimental aerofoil characteristics
Figure 7.16 Calculation of moments about the leading edge
Again from Figure 7.15 it is seen that the experimental results for zero lift angle for single and cascaded aero- foils are less than these predicted by thin aerofoil theory.
Thin aerofoil theory also predicts the pitching moment of the aerofoil. Consider Figure 7.16 which shows a more detailed view of the element of the vortex sheet shown in Figure 7.14. From Figure 7.16 we see that the moment per unit span of the aerofoil is given by
MLE = − c
0 ξ(dL)= −V c
0 ξγ(ξ)dξ
which by substituting in the distribution of vorticity given by equation (7.18) and again using the transform- ationξ=(c/2)(1−cosθ) gives
MLE = −V2c2 2
π 0
A0(1−cos2θ)dθ +
π
0
∞ n=1
Ansinθsin(nθ)dθ
− π
0
∞ n=1
Ansinθcosθsin(nθ)dθ
which, by solving in an analogous way to that forc1
and using the definition of the moment coefficient given in equation (7.3), gives an expression for the pitch- ing moment coefficient about the leading edge of the aerofoil as
cmLE = −π
2 A0+A1−A2
2
or by appeal to equation (7.21) cmLE = −c1
4 +π
4(A1−A2)
(7.25) and since from equation (7.4)
cmLE = −c1
4 +cmc/4
we may deduce that cmc/4= π
4[A2−A1] (7.26)
Equation (7.26) demonstrates that, according to thin aerofoil theory, the aerodynamic centre is at the quarter chord point, since the pitching moment at this point
Theoretical methods – basic concepts 151 is dependent only on the camber profile (see equation
(7.18a) for the basis of the coefficientsA1andA2) and independent of the lift coefficient.
Equations (7.23) to (7.26) are significant results in thin aerofoil theory and also in many branches of pro- peller analysis. It is therefore important to be able to calculate these parameters readily for an arbitrary aerofoil. The theoretical lift slope curve presents no problem, since it is a general result independent of aero- foil geometry. However, this is not the case with the other equations, since the integrals behave badly in the region of the leading and trailing edges. To overcome these problems various numerical procedures have been developed over the years. In the case of the aero lift angle, Burrill (Reference 6) and Hawdonet al. (Ref- erence 7) developed a tabular method based on the relationship
α0= 1 c
19 n=1
fn(x)yn(x) degrees (7.27) where the chordal spacing is given by
xn=cn
20 (n=1, 2, 3, 4,. . ., 20)
The multipliersfn(x) are given in Table 7.2 for both sets of references. The Burrill data is sufficient for most con- ventional aerofoil shapes; however, it does lead to inac- curacies when dealing with ‘S’ shaped sections, such as might be encountered when analysing controllable pitch propellers in off-design pitch settings. This is due to its being based on a trapezoidal rule formulation. The Haw- don relationship was designed to overcome this problem
Table 7.2 Zero lift angle multiplies for use with equation (7.27)
n xc fn(x)Burrill fn(x)Hawdonet al.
LE
1 0.15 5.04 5.04
2 0.10 3.38 3.38
3 0.15 3.01 3.00
4 0.20 2.87 2.85
5 0.25 2.81 2.81
6 0.30 2.84 2.84
7 0.35 2.92 2.94
8 0.40 3.09 3.10
9 0.45 3.32 3.33
10 0.50 3.64 3.65
11 0.55 4.07 4.07
12 0.60 4.64 4.65
13 0.65 5.44 5.46
14 0.70 6.65 6.63
15 0.75 8.59 8.43
16 0.80 11.40 11.40
17 0.85 17.05 17.02
18 0.90 35.40 −22.82
19 0.95 186.20 310.72
TE
by using a second-order relationship and systematic tests with camber lines ranging from a parabolic form to a symmetrical ‘S’ shape showed this latter relationship to agree to within 0.5 per cent of the thin aerofoil results.
With regard to the pitching moment coefficient a simi- lar approximation method was developed by Pankhurst (Reference 8). In this procedure the pitching moment coefficient is given by the relationship
cmc/4= 1 c
14 n=1
Bn(yb(xn)+yf(xn)) (7.28) where yb and yf are the back and face ordinates of the aerofoil at each of the xn chordal spacings. The coefficientsBnare given in Table 7.3.
Table 7.3 Pitching moment coefficient multipliers for equation (7.28) (taken from Reference 11).
n xn Bn
1 0 (LE) −0.119
2 0.025 −0.156
3 0.05 −0.104
4 0.10 −0.124
5 0.20 −0.074
6 0.30 −0.009
7 0.40 0.045
8 0.50 0.101
9 0.60 0.170
10 0.70 0.273
11 0.80 0.477
12 0.90 0.786
13 0.95 3.026
14 1.00 (TE) −4.289