Piet Lammertse on propeller theory

TIP LOSS

17 Tip loss

17.1 The nature of the tip loss

Tip loss differs essentially from the losses we have encountered so far.

Consider the ( ring ) thrust equation (7.5), repeated here :

d T = d m^{•} . w

(17.1)

or its non-dimensional version ( 7.12 ) :

d T ′ (x) = d m ′ . b (x)

(17.2)

Thrust is the product of mass flow and induction.

All of the efficiency analyses and optimizations we applied so far acted exclusively on the induction term b (x). Loss of ( momentum ) efficiency occurs only when b (x) is uniformly inflated back to restore the design thrust, while maintaining the optimal shape of the distribution.

Certainly in the case of light loading, the ring induction does not influence the mass flow at all, so mass flow and efficiency remain independent even if the induction changes.

The b (x) schedulings ( 10.8 ) and ( 16.12 ) optimize rotational and viscous efficiency. These efficiencies are functions of the local pitch angle, and by geometrical relations like (2.4) of the non-dimensional radius x. Changing the mass flow distribution does not change the optimal distribution of the induction with blade radius.

Tip loss does not change the shape of the induction distribution at all. But it does affect the mass which receives that induction. Some of the air in the original momentum tube escapes around the blade tips, and is not accelerated. By (17.1), for a given induction and therefore for a given efficiency, there will be less thrust. To compensate, the induction over the active part of the prop disk will have to increase. By this indirect route, the tip loss reduces the efficiency for a given thrust after all.

TODO - MENTION THAT IN THE END BOTH ARE REDUCTIONS ON THE THRUST AT A GIVEN EFFICIENCY, AND THE LOSS OCCURS WHEN UNIFORMLY INFLATING IT BACK AGAIN.

17.2 Thrust loss in the spaces between the blades

The previous chapters all gave velocity diagrams at the blade element. This was enough to deduce local efficiencies, and to schedule an optimal shape for the distribution of the local induction.

In the presence of tip loss, the axial induction velocities in the spaces between the blades will be less than the induced velocity at the blades, but this does not affect the reasoning at the blade elements themselves.

However, it would seem proper to determine the average induction in a full ring of the flow tube, and find a reduced thrust from this in equation (17.1).

But it is not a simple matter to find the induction in the spaces between the blades, and we will find that in some way it will not be needed.

The other option is to find an average, "effective" value for the ring mass flow dm^●. Conceptually, the blades still give the local air the full induction b(x), but in Newton's law of (17.1) they do not "feel" the resistance of the full mass flow through the ring pushing back on them.

By this concept, (17.1) with the full induction b(x) applies for the whole ring, provided we can find the reduced ring mass flow dm^●.

Tip loss theory is built around this concept.

17.3 A local mass flow ratio

We made the local thrust (7.4) non-dimensional in (7.5), by introducing a non-dimensional mass flow dm ′.

We now define a function K ( x ) for the reduction of the effective local mass flow, due to "tip loss". This changes the light-loading momentum flow from (7.11) into :

d m ′ \equiv K (x) . \frac{2 x . d x}{X^{2}}

(17.3)

This equation defines the mass reduction factor K ( x ).

Like before, dm ′ is just the effective non-dimensional frontal area of the ring at radius x. Without K ( x ) all the rings add up to a non-dimensional frontal area of 1, or ( 1 + ½ b ) if we retain the heavy loading term in (7.10).

With K ( x ), the effective frontal area of a ring is reduced by that factor. The local thrust (17.2) becomes :

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

(17.4)

17.4 Combining K ( x) with the optimal rotation

Equation ( 10.7 ) gave us the optimal axial induction distribution b(x) for a lightly loaded, rotationally optimal propeller of constant marginal loss C, and constant overall loss ratio ½ C :

b (x) = C . {cos}^{2} φ s

(17.5)

In (17.4) this yields :

d T ′ = C . K (x) . {cos}^{2} φ . \frac{2 x . d x}{X^{2}}

(17.6)

The ring thrust is reduced by a factor of cos² φ on the induction by (17.5), and by the factor K ( x ) on the mass flow by (17.3). Although the two reductions are very different in nature, both reductinis are needed if we wish to maintain the basic actuator disk efficiency.

We define a notation for the combined equal-efficiency thrust reduction factor :

G (x) \equiv K (x) . {cos}^{2} φ

(17.7)

The symbol G for this factor is a common convention in the literature. It is probably named after Sidney Goldstein, who made the first rigorous analysis of it in 1929.

We will find that unfortunately, K ( x ) is not completely independent of the distribution of b ( x ). But clearly, the case for the optimal cos² φ distribution is of special interest.

17.5 The overall thrust reduction

With the tip reduction K ( x ), the thrust integral for the rotationally optimal propeller changes from ( 11.2 ) to :

T ′ = C . \int_{0}^{X} K (x) . {cos}^{2} φ . \frac{2 x . d x}{X^{2}}

(17.8)

By (17.6), we can also write this as :

T ′ = C . \int_{0}^{X} G (x) . \frac{2 x . d x}{X^{2}}

(17.9)

In (11.3), we defined the symbol κ_φ for the integral of the local thrust reduction factor cos²φ over the prop disk. We define a similar factor κ_G for the combined reduction G ( x ) :

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

(17.10)

17.6 Thrust and induction

By direct comparison with chapter 11.2, we can now write the same analysis with κ_G in the place of κ_φ .
From (17.9) and (17.10) we have :

T ′ = C . κ_{G}

(17.11)

C = \frac{T ′}{κ_{G}}

(17.12)

From (17.5) we have :

b (x) = T ′ . \frac{{cos}^{2} φ}{κ_{G}}

(17.13)

For the same thrust, the induction gets larger relative to 1 / κ_φ, to compensate for the loss in mass flow.

From (17.6) we have :

d T ′ = T ′ . K (x) . \frac{{cos}^{2} φ}{κ_{G}} . \frac{2 x . d x}{X^{2}}

= T ′ . \frac{G (x)}{κ_{G}} . \frac{2 x . d x}{X^{2}}

(17.14)

These are the direct equivalents of ( 11.11 ) and ( 11.12 ) before tip loss.

17.7 The effect on the efficiency

From ( 9.3 ), the local loss ratio at any radius follows only from the local induction, regardless of the mass flow.

Substituting the induction (17.13) into the efficiencies ( 9.6 ) and ( 9.7 ), we have :

\frac{d P_{m} ′}{d T ′} = ½ . \frac{T ′}{κ_{G}}

(17.15)

η_{m} = \frac{1}{1 + ½ . \frac{T ′}{κ_{G}}}

(17.16)

The combined tip and rotational loss of (17.11) increases the momentum loss of the propeller by a factor of 1 / κ_G relative to the ideal actuator disk efficiency (5.12). This loss is larger than that of (11.9) for rotation alone.

The question now remains : how are we going to find the shape of K ( x ) c.q. G ( x ), and how are we going to integrate it into a κ_G.