7 Theoretical
7.7 Pressure distribution calculations
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
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152 Marine propellers and propulsion
about an arbitrary wing form is divided into three stages as follows:
1. The establishment of relations between the flow in the plane of the wing section (ζ-plane) and that of the ‘near circle’ plane (z-plane).
2. The derivation of the relationship between the flow in z-plane and the flow in the true circle plane (z-plane).
3. The combining of the two previous stages into the final expression for the velocity distribution in the ζ-plane in terms of the ordinates of the wing section.
The derivation of the final equation for the velocity dis- tribution, equation (7.29), can be found in Abbott and van Doenhoff (Reference 11) for the reader who is inter- ested in the details of the derivation. For our purposes here, however, we merely state the results as
v= V[sin(α0+φ)+sin(α0+εT)][1+(dε/dθ)]eψ0 {( sinh2ψ+sin2θ)[1+(dψ/dθ)2]}
(7.29) wherevis the local velocity on any point on the surface of the wing section andV is the free streams velocity.
In order to make use of equation (7.29) to calculate the velocity at some point on the wing section it is neces- sary to define the coordinates of the wing section with respect to a line joining a point which is located mid- way between the nose of the section and its centre of curvature to the trailing edge. The coordinates of these leading and trailing points are taken to be (−2a, 0) and (2a, 0) respectively witha=1 for convenience. Next the values ofθandψare found from the coordinates (x, y) of the wing section as follows:
2 sin2θ=p+ p2+y a
2
(7.30) with
p=1−x 2a
2
−y 2a
2
and
y=2asinhψsinθ x=2asinhψcosθ
(7.31) The functionψ0=(1/2π)2π
0 ψdθhas then to be deter- mined from the relationship between ψ and θ. A first approximation to the parameter ε can be found by conjugating the curve of ψ against θ using the relationship
ε(φ)= 1 n
n k=1
(ψ−k−ψk) cot kπ
2n
(7.32) with
ψk=ψ
φ+kπ n
where the corrdinates in the z-plane are defined by z=ae(λ+iφ).
For most purposes a valuen=40 will give suffi- ciently accurate results.
Finally the values of (dε/dθ) and (dψ/dθ) are deter- mined from the curves ofεandψagainstθand hence equation (7.29) can be evaluated, usually in terms ofv/V.
For many purposes the first approximation to ε is sufficiently accurate; however, if this is not the case then a second approximation can be made by plotting ψagainstθ+εand re-working the calculation from the determination of the functionψ0.
This procedure is exact for computations in ideal fluids; however, the presence of viscosity to a real fluid leads to discrepancies between experiment and calcula- tion. The growth of the boundary layer over the section effectively changes the shape of the section, and one result of this is that the theoretical rate of changes of lift with angle of incidence is not realized. Pinkerton (Reference 12) found that fair agreement with exper- iment for the NACA 4412 aerofoil could be obtained by effectively distorting the shape of the section. The amount of the distortion is determined by calculating the incrementεTrequired to avoid infinite velocities at the trailing edge after the circulation has been adjusted to give the experimentally observed life coefficient. This gives rise to a modified function:
εα=ε+εT
2 (1−cosθ) (7.33)
whereεis the original inviscid function andεαis the modified value of the section.
Figure 7.17 shows the agreement obtained from the NACA 4412 pressure distribution using the Theodorsen and Theodorsen with Pinkerton correction methods.
The Theodorsen method is clearly not the only method of calculating the pressure distribution around an aerofoil section. It is one of a class of inviscid methods; other methods commonly used are those by Riegels and Wittich (Reference 13) and Weber
Figure 7.17 Comparison of theoretical and experimental pressure distributions around an aerofoil
Theoretical methods – basic concepts 153 (Reference 14). The Weber method was based origi-
nally on the earlier work of Riegels and Wittich, which in itself was closely related to the works of Goldstein, Thwaites and Watson, and provides a readily calculable procedure at either 8, 16 or 32 points around the aero- foil. The location of the calculation points is defined by a cosine function, so that a much greater distribution of calculation points is achieved at the leading and trail- ing edges of the section. Comparison of the methods, those based on Theodorsen, Riegels–Wittich and Weber, shows little variation for the range of aerofoils of interest to propeller designers so the choice of method reduces to one of personal preference for the user. The invis- cid approach was extended to the cascade problem by Wilkinson (Reference 15). In addition to the solutions to the aerofoil pressure distribution problem discussed here, the use of numerical methods based on vortex panel methods have been shown to give useful and reli- able results. These will be introduced later in the chapter.
The calculation of the viscous pressure distribution around an aerofoil is a particularly complex procedure, and rigorous methods such as those by Firmin (Refer- ence 16) need to be employed. Indeed the complexity of these methods has generally precluded them from propeller analysis and many design programmes at the present time in favour of more approximate methods, as will be seen later.
If the section thickness distribution and camber line are of standard forms for which velocity distributions are known, such as the NACA forms, then the resulting velocity distribution can be readily approximated. The basis of the approximation is that the load distribution over a thin section may be considered to comprise two components:
1. A basic load distribution at the ideal angle of attack.
2. An additional distribution of load which is propor- tional to the angle of attack as measured from the ideal angle of attack.
The basic load distribution is a function only of the shape of the thin aerofoil section, and if the section is considered only to be the mean line, then it is a function only of the mean line geometry. Hence, if the parent camber line is modified by multiplying all of the ordin- ates by a constant factor, then the ideal design of attack αiand the design lift coefficientcliof the modified cam- ber line are similarly derived by multiplying the parent values by the same factor.
The second distribution cited above results from the angle of attack of the section and is termed the additional load distribution; theoretically this does not contribute to any additional moment about the quarter chord point of the aerofoil. In practice there is a small effect since the aerodynamic centre in viscous flow is usually just astern of the quarter chord point. This additional load distribution is dependent to an extent on aerofoil shape and is also non-linear with incidence angle but can be calculated for a given aerofoil shape using the methods cited earlier in this chapter. The non-linearity with inci- dence angle, however, is small and for most marine
engineering purposes can be assumed linear. As a consequence, additional load distributions are normally calculated only for a series of profile forms at a repre- sentative incidence angle and assumed to be linear for other values.
In addition to these two components of load, the actual thickness form at zero incidence has a velocity distribution over the surface associated with it, but this does not contribute to the external load produced by the aerofoil. Accordingly, the resultant velocity dis- tribution over the aerofoil surface can be considered to comprise three separate and, to a first approxima- tion, independent components, which can be added to give the resultant velocity distribution at a particular incidence angle. These components are as follows:
1. A velocity distribution over the basic thickness form at zero incidence.
2. A velocity distribution over the mean line corres- ponding to the load distribution at its ideal angle of incidence.
3. A velocity distribution corresponding to the add- itional load distribution associated with the angle of incidence of the aerofoil.
Figure 7.18 demonstrates the procedure and the velocity distributions for standard NACA aerofoil forms
Figure 7.18 Synthesis of pressure distribution (Reproduced with permission from Reference 11)
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154 Marine propellers and propulsion
Table 7.4Typical NACA data for propeller type sections
x y (v/V)2 v/V va/V
(%c) (%c)
0 0 0 0 5.471
1.25 0.646 1.050 1.029 1.376
2.5 0.903 1.085 1.042 0.980
5.0 1.255 1.097 1.047 0.689
7.5 1.516 1.105 1.051 0.557
10 1.729 1.108 1.053 0.476
15 2.067 1.112 1.055 0.379
20 2.332 1.116 1.057 0.319
30 2.709 1.123 1.060 0.244
40 2.927 1.132 1.064 0.196
50 3.000 1.137 1.066 0.160
60 2.917 1.141 1.068 0.130
70 2.635 1.132 1.064 0.104
80 2.099 1.104 1.051 0.077
90 1.259 1.035 1.017 0.049
95 0.707 0.962 0.981 0.032
100 0.060 0 0 0
LE radius: 0.176 %c NACA 16-006 basic thickness form
c1i=1.0 αi=1.40◦ cmc/4= −0.219
x y dyc/dx PR v/V=PR/4
(%c) (%c)
0 0
0.5 0.281 0.47539 0.75 0.396 0.44004
⎫⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎬
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎭ 1.25 0.603 0.39531 2.5 1.055 0.33404
1.092 0.273 5.0 1.803 0.27149
7.5 2.432 0.23378
10 2.981 0.20618
15 3.903 0.16546
20 4.651 0.13452
⎫⎪
⎪⎬
⎪⎪
⎭1.096 0.274
25 5.257 0.10873
30 5.742 0.08595
35 6.120 0.06498
40 6.394 0.04507
1.100 0.275
45 6.571 0.02559
50 6.651 0.00607
55 6.631 −0.01404 1.104 0.276 60 6.508 −0.03537
65 6.274 −0.05887 1.108 0.277 70 5.913 −0.08610 1.108 0.277 75 5.401 −0.12058 1.112 0.278 80 4.673 −0.18034 1.112 0.278 85 3.607 −0.23430 0.840 0.210 90 2.452 −0.24521 0.588 0.147 95 1.226 −0.24521 0.368 0.092 100 0 −0.24521 0 0
Data for NACA mean linea=0.8 (modified)
Theoretical methods – basic concepts 155 can be obtained from Reference 11. By way of example
of this data, Table 7.4 shows the relevant data for a NACA 16-006 basic thickness form and a NACA a=0.8 modified mean line. It will be seen that this data can be used principally in two ways: first, given a section form at incidence, to determine the resulting pressure distribution and secondly, given the section form and lift coefficient, to determine the appropriate design incidence and associated pressure distribution.
In the first case for a given maximum camber of the subject aerofoil the value ofc1i,αi, cmc/4 and the v/V distribution are scaled by the ratio of the max- imum camber–chord ratio, taking into account any flow curvature effects from that shown in Table 7.4.
In the case of thea=0.8 (modified) mean line:
camber scale factor (Sc)y/cof actual aerofoil 0.06651
(7.34) The values ofv/V relating to the basic section thick- ness velocity distribution at zero incidence can be used directly from the appropriate table relating to the thickness form. However, the additional load velocity distribution requires modification since that given in Table 7.4 relates to a specific lift coefficientc1: in many cases this lift coefficient has a value of unity, but this needs to be checked (Reference 11) for each particu- lar application in order to avoid serious error. Since the data given in Reference 11 relates to potential flow, the associated angle of incidence for the distribution can be calculated as
α∗ =CL∗
2π (7.35)
Hence theva/V distribution has to be scaled by a factor of
additional load scale factor (SA)= α−αi
α∗
(7.36) The resultant velocity distribution over the surface of the aerofoil is then given by
(u/V)U= v V +Sc
v V
+SA
va
V
(u/V)L= v V −Sc
v V
−SA
va
V
(7.37)
where the suffices U and L relate to the upper and lower aerofoil surfaces respectively.
In the second case, cited above, of a given section form and desired lift coefficient an analogous procedure is adopted in which the camber scale factor, equation (7.34), is applied to thev/Vdistribution. However, in this case equation (7.36) is modified to take the form
SA=
CL−CLi CL∗
(7.36a)
The resultant surface velocity distribution is then cal- culated using equations (7.37).
The pressure distribution around the aerofoil is related to the velocity distribution by Bernoulli’s equation:
p∞+1
2V2=pL+1
2u2 (7.38)
wherep∞andpLare the static pressures remote from the aerofoil and at a point on the surface where the local velocity isurespectively. Then by rearranging equation (7.38), we obtain
pL−p∞= 1
2(V2−u2)
and dividing by the free stream dynamic pressure12V2, whereVis the free stream velocity far from the aerofoil, we have
pL−p∞
1
2V2 = 1−u V
2
(7.39) The term [(pL−p∞)/12V2] is termed the pressure coefficient (CP) for a point on the surface of the aero- foil; hence in terms of this coefficient equation (7.39) becomes
CP= 1−u V
2
(7.40)