Propeller Geometry

First some general geometrical de…nitions of the propeller are given. A sketch of a right handed propeller is given in …gure 4.17-a.

² The propeller blades are attached to the hub, which is attached to the end of the propeller shaft. The propeller rotates about the shaft center line. The direction of rotation is as viewed from behind, that is towards the shaft. In normal forward operation a right handed propeller rotates in clockwise direction when viewed from behind.

The propeller hub is, of course, rotational symmetric because it should not disturb the ‡ow. The attachment of the propeller blade to the hub is a gradual transition, which is done in the …llet area or the blade root. A streamlined cap is generally

…tted to the hub at the end of its shaft.

Figure 4.17: Sketch of a Propeller

² The front edge of the blade is called the leading edge or nose and the other end is the trailing edge or tail. The outermost position, where leading and trailing edges meet, is called the blade tip. The radius of the tip is called the propeller radius and the propeller diameter is, of course, twice its radius.

² The surface of the blade which is at the side from which the shaft comes is called the propeller back and the other side is the face of the propeller; when the ship moves forward, the propeller in‡ow is from its back. Because in forward speed the back side has a low average pressure and the face side has a high average pressure (this pressure di¤erence generates the thrust), the face is also called the pressure side and the back the suction side.

Consider now an arbitrary propeller as drawn in …gure 4.18-a.

Figure 4.18: Geometry of a Propeller

² The intersection of a cylinder with radius r and a propeller blade, the blade section, in …gure 4.18-a has the shape of an airfoil. Such a shape is also called just a foil or a pro…le, as given in …gure 4.18-b.

² The straight line between the leading and trailing edge of the pro…le is called the chord line of the pro…le and the distance between nose and tail is the chord length, c, of the foil.

² Generally, the origin of the local coordinate system of a pro…le is taken at the leading edge. The x-direction is towards the tail, the y-direction upwards, perpendicular to the chord. The angle between the nose-tail line and the undisturbed ‡ow (relative to the blade) is the angle of attack, ®.

² The distance between the suction side and the pressure side, measured perpendicular to the chord is the thickness, t(x), of the pro…le. The line of mid-thickness over the chord is the camber line.

² The blade sections of the propeller have a certain pitch. The chord line or nose-tail line of the blade section - a helix on the cylinder - becomes a straight pitch line, if the cylinder is developed on to a ‡at surface. The propeller pitch, P, is de…ned as the increase in axial direction of the pitch line over one full revolution 2¼r at each radius r; see …gure 4.19-a. The dimension of the pitch is a length. The ratio P=D is the pitch ratio. The pitch angle, µ = arctan (P=2¼r), is the angle between the pitch line and a plane perpendicular to the propeller shaft. The pitch distribution is given in a pitch diagram, which is simply a graph of the pitch at every radius.

² Figure 4.19-b shows the axial velocity V^e (speed of entrance) and the rotational velocity 2¼nr of the water particles at a radius r from the propeller axis. As a propeller is rotating in water, it can not advance P ¢ n and a certain di¤erence occurs.

The di¤erence between P ¢ n and Ve is called the slip of the propeller.

Figure 4.19: De…nition of Pitch and Slip

² A signi…cant radius, which is often used as representative for the propeller, is the radius at r=R = 0:7. If a pitch is given in the case of a variable pitch distribution it is usually the pitch at 0:7R. Note that half the area of the propeller disk is within a circle with radius 0:7R and that, consequently, the other half is outside this region;

so the pressure at this circular line is ”more or less” the average pressure over the full propeller disk.

Now, some important contours and areas will be de…ned:

² A plane perpendicular to the shaft and through the middle of the chord of the root section is called the propeller plane, see …gure 4.17-b.

² The projection of the blade contour on the propeller plane gives the projected blade contour and its area is called the projected blade area. The blade sections in this projection are segments of a circle.

² The blade sections in the cylinder of …gure 4.18-a are rotated into a plane parallel to the propeller plane. The angle of rotation at each radius is the pitch angle at that radius. This angle may vary over the radius; so it is a developed view. The ends of the developed blade sections form the developed blade contour. The blade sections in this projection remain circular.

² The circular blade sections in the developed contour can be expanded into a plane.

The contour thus derived is the developed and expanded contour, generally indicated only as the expanded contour. The chord line of the sections in this contour is now straight. Its area is called the expanded blade area.

Cavitation

Cavitation inception occurs when the local pressure is equal to the vapor pressure. The local pressure is expressed in non-dimensional terms as the pressure coe¢cient Cp. Similarly, the cavitation number is expressed non-dimensionally as:

¯¯ vapor pressure of water, ½ is the density of water and V is the undisturbed velocity in the

‡uid.

The condition that cavitation occurs when the local pressure is equal to the vapor pressure, means that a pro…le or a propeller will start to cavitate when the lowest pressure is near the vapor pressure. This is expressed in non-dimensional terms as:

¾ = ¡C^p(min) (4.62)

The cavitation number or cavitation index, ¾, is non-dimensional. This means that it is the parameter which has to be maintained when propeller model tests are carried out. It determines the pressure in the test section of the cavitation tunnel.

Preservation of the cavitation number in a physical ”ship+propeller” model is di¢cult. It is largely for this reason that MARIN built a towing tank in a vacuum chamber, about two decades ago. This allowed them to reduce the atmospheric pressure and thus to keep the ¾-value in their model more like that in the prototype.

Blade Area Ratio

An important parameter of the propeller is the blade area ratio, given as the ratio between the area of all blades and the area of the propeller plane, A0 = 0:25¼D²:

- the projected blade area ratio, A_P=A₀ and - the expanded blade area ratio, AE=A0

in which AP is the projected blade area and AE is the expanded blade area.

The latter, the expanded blade area ratio, AE=A0, is physically most signi…cant; when no further indication is given, this blade area ratio is generally meant.

From the frictional resistance point of view, the blade area should be as low as possible.

But a low blade area will result in low pressures (to obtain the required thrust) with high risks on cavitation. The blade area has to be chosen such that cavitation is avoided as much as possible.

Figure 4.20 shows the pressure distribution on a cavitating foil. The pressure at the nose on the back of the propeller drops down to the vapor pressure; a local cavity is the result.

A lower pitch ratio (resulting in a less low pressure at the back) and a higher blade area ratio (so a larger foil area) will deliver the same thrust (with a bit lower e¢ciency) without cavitation.

Figure 4.20: Cavitating Foil

An old and very simple formula for the minimum projected blade area ratio is that by Taylor:

AP

A₀ = 1:067 ¡ 0:229 ¢ P

D (4.63)

A handsome indication for the minimum expanded blade area ratio gives the formula of Auf ’m Keller:

AE

A₀ = (1:3 + 0:3 ¢ Z) ¢ T

(p ¡ pv) ¢ D² + k (4.64)

in which the constant k varies from 0.00 for fast naval vessels to 0.20 for highly powered full ships.

Number of Propeller Blades

The number of propeller blades, Z, is chosen in relation to possible vibrations. An 8 cylinder engine and a 4 bladed propeller may su¤er from resonance frequencies because the blade passage frequency (the frequency at which a blade passes close to the rudder for example) and the engine frequencies have common harmonics. In that case vibrations can become extensive, resulting in damage.

The structure of the wake is also important for the choice of the number of blades. When the wake has strong harmonics equal to the number of blades, thrust variations may become large; these, too, can lead to vibrations.

Wageningen Propeller Series

Systematic series of propeller models have been tested by MARIN to form a basis for propeller design. The starting point of a series is its parent form. The extent and applica-bility of the series depends on the parameters which are varied and on the range of their variations. Several series are described in the literature.

The most extensive and widely used …xed pitch open propeller series is the Wageningen B-series. The basic form of the B-series is simple and it has a good performance. The extent of the series is large; some 210 propellers have been tested.

The following parameters of the B-series propellers have been varied:

- the expanded blade area ratio, A_E=A₀, from 0.3 to 1.2 - the number of blades, Z, from 2 to 7

- the pitch ratio, P =D, from 0.5 to 1.4

The propellers are indicated by their number of blades and expanded blade area ratio. For instance, propeller B-4.85 has four blades and an expanded blade area ratio of 0.85. The test results are given in so-called open water diagrams per series of one blade number, one area ratio and a range of pitch ratios. The open water tests of the B-series were done at various rotation rates, n, so at various Reynolds numbers. These results were corrected to a standard Reynolds number of Rn = 2 ¢ 10⁶ along the lines of the ITTC method, as given by equation 4.42. However, these corrections are very small.

In addition to open …xed pitch propellers, MARIN has also developed series of …xed pitch propellers in an accelerating nozzle (Ka-series) and series of controllable pitch propellers in a nozzle. All available information on MARIN systematic propeller series has been summa-rized by [Kuiper, 1992]. The propulsive characteristics have been described by polynomials to make them usable in computer programs. Some other research institutes have also made their own propeller series, but the Wageningen Propeller Series are the most well known.