Piet Lammertse on propeller theory

AXIAL MOMENTUM

7 Concentric tubes

7.1 The independence of the rings

Most of propeller theory rests on the idea that the momentum tube of figure 4.1 can be subdivided into set of thin-walled concentric tubes, sliding ( and rotating ) inside each other like a pirate's telescope. These tubes will cut the propeller disk into nested, concentric rings. Figure 2.2 already showed such a tube.

The thin-walled tubes are supposed not to influence each other very much. But such independence is by no means obvious. The justification can be found in vortex theory, which we will discuss much later.

In practice, the independence of the thin-walled tubes holds up much better than might reasonably be expected.

If the rings and tubes of a propeller do not interact, then each ring is like a separate propeller. It has its own local induction ratio, its own pitch angle, and its own blade elements. The propeller as a whole is just an assembly of independent, nested ring propellers flying in close formation. We can derive the conditions for every single ring, and then integrate the ring contributions into an overall value for the whole propeller.

7.2 Integrating over the rings

We will often wish to know the total, or average of some value g ( r ) over the prop disk. This could be the overall thrust, or an average loss factor. Such values can be found by integrating the individual ring contributions. We will use the prefix "d" for variables belonging to a ring of "infinitesimal" width dr.

The frontal area of a ring is 2 π r . dr, and the contribution of a local value g ( r ) to the total will be g ( r ) . 2 πr . dr. Summing over "infinitely narrow" rings of width dr represents an integral. The average of g ( r ) over the disk is the integral divided by the disk area. We will call this average G :

G = \frac{1}{π R^{2}} . \int_{0}^{R} g (r) . 2 π r . d r

(7.1)

or, by dividing out the factor of π :

G = \frac{1}{R^{2}} . \int_{0}^{R} g (r) . 2 r . d r

(7.2)

By (2.12), we can replace r ⁄ R by x ⁄ X, and dr ⁄ R by dx ⁄ X, yielding :

G = \frac{1}{X^{2}} . \int_{0}^{X} g (x) . 2 x . d x

(7.3)

We can write either g ( r ), g ( x ) or g ( φ ), whichever is the most natural expression for the function at that moment. They are easily converted into each other by (2.5), when the time comes to evaluate the integral. In many cases, we will simply write g without an argument.

We can interpret 2 π r . dr as d ( π r² ), the increase of the disk area with r, or 2 x . dx as the increase of d ( x² ). This is rarely useful from an analytical point of view, but it offers the opportunity to plot g ( x ) against x² instead of against x. Such a plot can be useful to get a graphical feeling for the contribution of a ring. The function value is unchanged on the vertical axis, but the horizontal "strips" d ( x ² ) get progressively narrower towards the hub. This gives the proper visual impression of their relative area contribution to the integral.

7.3 Numerical integration

Somwetimes the analytical integral (7.3) of f ( x ) over the prop disk is complicated. In such cases, numerical integration can come to the rescue.

The typical functions in propeller theory have a teardrop shape, starting from zero at the hub and going back to zero at the outer end with a more or less elliptical tip. Such functions are easily integrated numerically. The integral over the infinitesimal width dx becomes a sum over finite widths.

Replacing dx in (7.3) by Δ x = X ⁄ N and bringing it outside the summation yields :

G = \frac{1}{X . N} . \sum_{0 . Δ x}^{N . Δ x} g (x) . 2 x

(7.4)

This expression sums over N + 1 points ranging from x = 0 to x = X , and divides by N rings of width Δ x. With the end values of f (x) both zero, this is equivalent to the trapezoidal rule.

A typical value for N used to be N = 10 in the days of the slide rule. The result was often improved by Simpson's weighting rule, with end correction for the rounded tip. Today, N = 10,000 is more common and the simple numerical integration of (7.4) does not need such refinement.

7.4 The thrust per ring

The thrust of a uniform actuator disk follows from (5.1). We define the equivalent for a ring :

d T = d m^{•} . w = d m^{•} . b . V

(7.5)

The non-dimensional version follows just like in (5.8) :

d T ′ = \frac{d m^{•}}{ρ V^{2} . π R^{2}} . b . V

(7.6)

The difference is that now dm^● is the air flowing through a ring, instead of the air flowing through the whole disk, and that the induction ratio b may vary among the rings.

By analogy with (5.2) and (5.3), the mass flow through a ring is :

d m^{•} = ρ . V . 2 π R . d r . (1 + ½ b)

(7.7)

≅ ρ . V . 2 π R . d r

We can make dm^● non-dimensional relative to the whole disk area in the same way as m ′ in (5.4) :

d m ′ \equiv \frac{d m^{•}}{ρ V . π R^{2}}

(7.8)

Applying (7.7) to (7.6), the ρ .V terms cancel, and we have :

d m ′ = \frac{2 π r . d r}{π R^{2}} . (1 + ½ b)

(7.9)

Replacing r ⁄ R by x ⁄ X and dr ⁄ R by dx ⁄ X like we did in (7.3) yields :

d m ′ = \frac{2 x . d x .}{X^{2}} . (1 + ½ b)

(7.10)

For light loading, this reduces to :

d m ′ = \frac{2 x . d x}{X^{2}}

(7.11)

Not surprisingly, the non-dimensional mass flow is simply the ( effective ) frontal area of the ring.

Using this in (7.6) we have :

d T ′ = d m ′ . b

(7.12)

For a "heavily" loaded ring this yields :

d T ′ = \frac{2 x . d x .}{X^{2}} . (1 + ½ b) . b

(7.13)

and for light loading :

d T ′ = \frac{2 x . d x}{X^{2}} . b

(7.14)

7.5 The momentum loss per ring

The momentum loss for a uniform actuator disk follows from (4.15). The equivalent for a ring is :

d P_{m} = ½ d m^{•} . w^{2} = ½ d m^{•} . b^{2} . V^{2}

(7.15)

We can make this non-dimensional in the same way as we did for the thrust. The result is :

d P_{m} ′ = ½ d m ′ . b^{2}

(7.16)

For a "heavily" loaded ring this yields :

d P_{m} ′ = \frac{2 x . d x .}{X^{2}} . (1 + ½ b) . ½ b^{2}

(7.17)

and for light loading :

d P_{m} ′ = \frac{2 x . d x .}{X^{2}} . ½ b^{2}

(7.18)