Vortex Theory

Other blade element theories differ from the momentum-blade element theory principally in the way in which the induced velocities are calculated.

Numerically based methods use vortex lattice models, either with a pres­cribed geometry or using vortices that are free to align themselves with the resultant flow. These numerical approaches generally require computer run times that are extremely long.

Goldstein’s classical vortex theory for predicting propeller performance is not much more difficult to apply than the momentum-blade element ap­proach. The vortex theory is described in detail in Reference 3.3 and is briefly outlined here.

Figure 6.10 shows the induced velocity at the propeller plane in more detail. The resultant velocity w is normal to Ve and is composed of a tangential component, w„ and an axial component, wa. w0 is a fictitious velocity to be discussed later. From the geometry, w, and wa are related by

V + Wg _ W, wr — Wt wa

This can be solved for wa as a function of wt.

= |[-V + W2 + 4 wt((or-wt)]

It is convenient to express all velocities in terms of Vr, so that the preceding equation becomes

t4b+V^tFt)] <«5>

Goldstein’s vortex theory relates w, to the bound circulation, Г, around any blade station by

.ВГ = 47jtkw, (6.36)

к is known as Goldstein’s kappa factor. This factor is not expressible in a closed form, but it is available in graphical form in the literature. An approximation of к is Prandtl’s tip loss factor, F, which becomes more exact as J becomes smaller or as the number of blades increases. Thus

ВГ = 4tttFw, (6.37)

where

F = fees" exp (6.38)

фт is the helix angle of the propeller’s helical trailing vortex system at the tip. For a lightly lpaded propeller,

фт = tan-1 A (6.39)

However, the lift must vanish at the tip of a propeller blade, which generally means that the local angle of attack at the tip must be zero. Thus, another expression for фт can be obtained by reference to Figure 6.6.

фт = /3 t (6.40)

For most cases, Equation 6.40 is preferred to Equation 6.39.

From the Kutta-Joukowski theorem,

L = pVT

Г = |cC(Ve (6.41)

Substituting Equation 6.41 into Equation 6.37, the result can be expressed as

o-C, £ = 8xF (6.42)

V j V t

Q can be calculated from

С/ = a 3 – tan-1 (6.43)

and VJVT from

Equations 6.35 to 6.44 can be solved iteratively for w, IVT■ C( and the other quantities can be found to evaluate Equation 6.32a and 6.32b.

If a, is assumed to be small, Reference 3.3 shows that a, can be solved directly by assuming that

The result is

«і * i(- X + /X2 + 4Y) (6.46)

where

X = tan4>+5-~—t ^ 8xF cos ф

Y _ <ra(g – ф)

8xF cos ф

As the flow passes through a propeller, the axial component of velocity increases gradually. An estimate of this variation of wa with axial distance, s, can be obtained from

w-‘w-(l+v? m) <6-47>

Woq is the value of wa at the plane of the propeller. Note that far ahead of the propeller (s = — °°), wa vanishes, while far behind the propeller (s = +°°), wa equals twice its value at the propeller plane.

The tangential component, w„ increases from zero just ahead of the propeller to 2w, just behind the propeller. This rapid change in w, through the propeller results in a curved flow field that effectively reduces the camber of the blade sections. Expressed as a reduction in the section angle of attack, Да, Reference 3.3 derives the following expression for Да. Assuming Да to be small,

As just given, Да is in radians.

wa and w, can be obtained approximately from

w, = Vra, sin (ф + a,)

wa = V&i cos (ф + a,)

As an example in the use of the vortex theory, consider the three-bladed propeller having the geometry shown in Figure 6.11. Wind tunnel testing, of this particular propeller, designated 5868-R6, Clark-Y section, three blades, is reported in Reference 6.3. These measurements are presented in Figures 6.12,

6.ІЗ, and 6.14.

This particular propeller has nearly a constant pitch from the 35% radius station out to the tip corresponding to a 15° blade angle at the 75% station. From the definition of the pitch,

p = 2irr tan /3

Thus, p/D. = xv tan /3. For an x of 0.75 and a j3 of 15°, plD = 0.631. Thus, for
a blade angle of 15° at x = 0.75, /3 at any other station will equal

0.631°

TOC

This propeller is a variable pitch propeller and the curves shown in Figures 6.12, 6.13, and 6.14 are for different values of /3 at the 75% station. If /З0.75 denotes this angle, then /3 will generally be given by

The preceding /3 is measured relative to the chord line of the Clark-Y airfoil. The angle between the zero lift line and the chord line for this airfoil is equal approximately to

a(o = 46 t/c°

where tic is the section thickness-to-chord ratio. Thus, to obtain /3 of the zero lift line, с*!,, is added to the geometric pitch angle given by Equation 6.47. Consider a value for /З0.75 and 35° and an x of 0.6. From Figure 6.7,

^ = 0.073 ’

«

– = 0.103

c

Using these values, a(o, /3, and a are calculated to be

«,„ = 4.7°

0- = 0.139 /3 = 38.5°

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 V nD Figure 6.13 Power coefficient curves for propeller 5868-9, Clark-Y section, three blades.

For this example, assume the propeller to be operating at an advance ratio of

1.4. J and A are related by

J = 7tA

Thus, for this example,

A = 0.446

For the Clark-Y airfoil,

a — 6.0 C(/rad

Cd = 0.006 + 0.010 (C, – 0.15)2

These are for an 8% thick airfoil. cd, of course, will vary slightly with thickness. With these values for a and Cd, the following quantities can be calculated in the order listed.

Since both induced and rotational losses are included ih the vortex theory, it is more accurate than the momentum-blade element theory. The vortex theory also models the physical propeller more accurately, since the loading vanishes at the blade tip.

The vortex theory was programmed and numerically integrated using a programmable calculator. These calculated points are included in Figures 6.12, 6.13, and 6.14 for a blade angle of 35°. The theory 4s seen to predict the shapes of the thrust and power curves rather well and results in values of CT and Cp that correspond to within Г of the blade angle. In fact, if a blade angle
of 34° is used in the calculations, CV and Cp values are obtained that lie almost exactly on the 35° experimental line over a range of J values from 1.1 to 1.8. Below an advance ratio of 1.1, the section angles of attack become large, so that the blades begin to stall. At this point the prediction of the propeller characteristics becomes very questionable.