Remark on the optimum propeller with a hub of finite length

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Remark on the optimum propeller with a hub

of finite length

by

Remark on the ontimum propeller with a hub of finite length.

Sunimary.

It is shown that with respect to optimization, the approximation of a hub by an infinitely long cylinder leads essentially to wrong results. It

is proved that in the case of a hub of finite length the circulation of the propeller blades at their roots has to be zero.

Introduction.

We will consider in this remark the influence of a hub of non zero dia-meter and finite length on the optimum spanwise load distribution of a screw propeller. Generally the hub is replaced by an infinitely long cylinder. Lerbs [] and Mc. Cormick [2] calculate on this base the circulation around the blades. Mc. Cormick comes to the conclusion that at the root of the blades the circulation has to be finite and does not tend to zero. Also the present author

[3]

In reply to [2] Tacbmindji [1f] argues by experimental evidence, that the circulation has to be zero at the root of the blades. He uses the same

representation for the hub as has been used in [2], viz, the infinitely long cylinder, but imposes an extra condition at the hub which has to ensure that the circulation of the blades tends to zero. This however is in contradiction with the nature of the mathematical problem. The elliptic partial differential

equation with the boundary conditions used by Mc. Cormick possesses a well defined unique solution and does not permit an extra boundary condition.

However we will show that for an optimum propeller with hub of finite length the physical feeling of Tachmandji is correct although the mathematical results of Mc. Cormick are correct with respect to the model he uses. The key to the solution lies in the fact that we have to take into account that the length of the hub is essentially finite. By this the fluid flow is converging at the end of the hub. Then the rotational velocities around the axis increase because roughly said the angular momentum around this axis of the fluid particles is constant. This means that theoretically there are heavy losses of thrust by the negative pressure at the converging part of the hub. This has been formulated in the last section of

_[5]

disc, from which the present remark arose. These losses are avoided when the circulation of the blades at the hub tends to zero.

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2. The optimum vorticity.

Consider a cylindrical coordinate system (x,r,e) embedded in an inviscid and incompressible fluid. The fluid has a velocity U in the positive x direction. We replace the hub by a rotationally symmetric body M. We assume that the flow

Fig. 1. The propeller with hub.

around the body (base flow) is known by numerical calculations and that the velocities induced by the blades of the propeller (disturbance velocities)

are sufficiently small. More explicitely when we introduce a small parameter e, then the total thrust of the system and the disturbance velocities are 0(c), while the velocities U and V in the x and r direction of the base flow are O(e°). The theory is valid for c,0.

Because of the linearity of the theory it is necessary that the blades B(L=1,...,n where n is the number of the blades) and the slopes _{of the blades} are in an C _{neighbourhood of reference surfaces H. which do not disturb}

tje

-'-the base fLow when -'-they rotate with -'-the angular velocity u of the screw. These urface are riot the "ordinary1' helicoidal surfaces.

O - = 0. (2.1)

By the inhomogeneity of the base flow the surfaces are distorted to some extent. For instance consider a not necessary straight line i. fixed to and hence

rotating with the body M. Then consider the surface H. built up by all the fluid particles of the base flow which have passed the line l. This surface contracts in the converging region of the base flow. It is clear that _infinitely far down stream H. consists of purely helicoidal lines although it is also

there not antordinarthelicoidal surface (2.1).

In the linear theory we assume that the blades B. of the propeller coincidé with the surfaces H. and that the free vorticity shed by the blades _remains on these H..

Now consider infinitely far down stream on the H., finite regions A.

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bound vorticity of such a strength that it sheds free vorticity which cancels exactly the free vorticity shed by the propeller blades B.. Then we consider the system of bound vorticity on B. and A. together. Because no free vorticity is behind this system it is clear that there is no resultant thrust on the combination of body, propeller and the regions A.. This means however that when we replace the body M arid the by the A. and reverse the sign of the bound vorticity on the latter, the thrust and the free vorticity shed by this system and hence the efficiency are the same as those of the original system

M B..

i

The original system has to be optimum, hence when we change the bound vorticity distribution by a function

Ar' = t,r(r,x). (2.2)

in such a way that the total thrust on M and B., together remains the same, the efficiency may change by a number of O(A2). The disturbed bound vorticity on the B. leaves behind a disturbed free vorticity, hence we have to change the equivalent bound vorticity on A. also of O(A) in order to annihilate again the free vorticity of the B.. Then the B. however the efficiency of the A. changes the same amount hence also a number of O(A2). This means that the system A. itself has to be optimum. Hence we can replace the optimum condition of the original system M,B. by the optimum condition for the A., for which however the classical theory is valid.

Now we have the result, the original system M,B is optimum when the free vorticity shed by it is such that it equals the vorticity on the far down stream H when these are "frozen" and placed in a homogeneous flow parallel

i

with the x axis, with a velocity which is proportional to the desired thrust. Then in agreement with

[6]

its neighbourhood which is in contradiction with the representation of free vorticity by a flow along the far down stream H.. This follows also from the

fact that such a concentrated vortex has around it per unit of length, an infinite amount of kinetic energy, which will reduce the efficiency to zero.

Having this free vorticity infinitely far down stram we can easily determine by means of the known base flow stream lines the strength of the free vorticity at the propeller blades B.. Then by integrating from the rootS, the circulation around the propeller can be determined. Because the free vorticity at x = + has a finite density in the neighbourhood of the x axi.s the bound circulation starts from zero at the roots of the IB. which is

i

different from the results belonging to the infinitely long hub.

of the induced velocity normal to the planes H. by which we find the local angle of incidence. Hence by an integration along the base flow stream lines we obtain the profiles of the blades B..

The hub can be taken into account in for instance two ways. First, it is possible to calculate the velociUes normal to the H. induced by the existence of the rotational symmetric hub with a fixed geometry, within the field of disturbance velocities of the vortices. This local angle of incidence has then to be added to the one of the previous paragraph. Second, we can calculate the normal component of the disturbance velocities at the hub and calculate the new geometry of the hub in order that the disturbed flow passes exactly along it. In the latter case we do not have to change the profiles of the previous paragraph however the hub is no longer rotationally symmetric. The hub becomes more or less part of the propeller of which the geometry has to he determined.

We can give also an interpretation of the result of Mc. Cormick. His solution belongs to the optimization of a propeller with blades B. adjusted to an "inner" shroud. Then as follows from the optimization theory we have to consider the surfaces on which free vorticity arises, hence the helicoidal

surfaces and the cylinder behind the inner shroud. These surfaces have to be placed (as two side infinite ones) in a parallel flow and we find in this way the optimum free vorticity on them. Again the bound vorticity on the blades can be calculated. However we have to calculate also the circu-lation around the inner shroud as has been done in [3] for the "outer" shroud which is in essence the same.

References.

Lerbs, H.W. Moderately Loaded Propeller Theory, Transaction of the Society of Naval Architects and Marine Engineers

(1952).

Mc. Cormick, B.W., "The Effect of a Finite Hub on the Optimum Propeller", JouÑial of Aeronautical Sciences, Vol. 22, No.

9, (1955).

Sparenberg, J.A., On Optimum Propellers with a Duct of Finite Length, Journal of Ship Research, June

1969.

24. Tacbmindji, A.J., The Potential Problem of the Optimum Propeller with Finite Hub. I.S.P. Vol. 3, no. 27,

(1956).

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