NAVAL ARCHITECTURE

Chap-10.qxd 4~9~04 13:02 Page 183 PROPULSION 183 D FL V a Figure 10.6 Forces on blade section other in the direction of the flow is termed the drag, D. These two forces are expressed non-dimensionally as: CL ϭ L and CD ϭ D 1 rAV 2 1 rAV 2 2 2 Each of these coefficients will be a function of the angle of incidence and Reynolds’ number. For a given Reynolds’ number they depend on the angle of incidence only and a typical plot of lift and drag coeffi- cients against angle of incidence is presented in Figure 10.7. Lift and drag coefficients CL CD Angle of incidence, a No lift angle Figure 10.7 Lift and drag curves Initially the curve for the lift coefficient is practically a straight line starting from a small negative angle of incidence called the no lift angle. As the angle of incidence increases further the curve reduces in slope and then the coefficient begins to decrease. A steep drop occurs when the angle of incidence reaches the stall angle and the flow around the aerofoil breaks down. The drag coefficient has a minimum value near

Chap-10.qxd 4~9~04 13:02 Page 184 184 PROPULSION the zero angle of incidence, rises slowly at first and then more steeply as the angle of incidence increases. Lift generation Hydrodynamic theory shows the flow round an infinitely long circular cylinder in a non-viscous fluid as in Figure 10.8. BA Figure 10.8 Flow round circular cylinder At points A and B the velocity is zero and these are called stagnation points. The resultant force on the cylinder is zero. This flow can be transformed into the flow around an aerofoil as in Figure 10.9, the stag- nation points moving to AЈ and BЈ. The force on the aerofoil in these conditions is also zero. BЈ AЈ Figure 10.9 Flow round aerofoil without circulation In a viscous fluid the very high velocities at the trailing edge produce an unstable situation due to shear stresses. The potential flow pattern breaks down and a stable pattern develops with one of the stagnation points at the trailing edge, Figure 10.10. Direction of circulation Figure 10.10 Flow round aerofoil with circulation

Chap-10.qxd 4~9~04 13:02 Page 185 PROPULSION 185 The new pattern is the original pattern with a vortex superimposed upon it. The vortex is centred on the aerofoil and the strength of its cir- culation depends upon the shape of the section and its angle of inci- dence. Its strength is such as to move BЈ to the trailing edge. It can be shown that the lift on the aerofoil, for a given strength of circulation, ␶, is: Lift ϭ L ϭ ␳V␶ The fluid viscosity introduces a small drag force but has little influence on the lift generated. Three-dimensional flow The simple approach assumes an aerofoil of infinite span in which the flow would be two-dimensional. The lift force is generated by the differ- ence in pressures on the face and back of the foil. In practice an aero- foil will be finite in span and there will be a tendency for the pressures on the face and back to try to equalize at the tips by a flow around the ends of the span reducing the lift in these areas. Some lifting surfaces have plates fitted at the ends to prevent this ‘bleeding’ of the pressure. The effect is relatively greater the less the span in relation to the chord. This ratio of span to chord is termed the aspect ratio. As aspect ratio increases the lift characteristics approach more closely those of two- dimensional flow. Pressure distribution around an aerofoil The effect of the flow past, and circulation round, the aerofoil is to increase the velocity over the back and reduce it over the face. By Bernouilli’s principle there will be corresponding decreases in pres- sure over the back and increases over the face. Both pressure distribu- tions contribute to the total lift, the reduced pressure over the back making the greater contribution as shown in Figure 10.11. Reduced pressure on back Increased pressure on face Figure 10.11 Pressure distribution on aerofoil

Chap-10.qxd 4~9~04 13:02 Page 186 186 PROPULSION The maximum reduction in pressure occurs at a point between the mid-chord and the leading edge. If the reduction is too great in rela- tion to the ambient pressure in a fluid like water, bubbles form filled with air and water vapour. The bubbles are swept towards the trailing edge and they collapse as they enter an area of higher pressure. This is known as cavitation and is bad from the point of view of noise and effi- ciency. The large forces generated when the bubbles collapse can cause physical damage to the propeller. PROPELLER THRUST AND TORQUE Having discussed the basic action of an aerofoil in producing lift, the action of a screw propeller in generating thrust and torque can be con- sidered. The momentum theory has already been covered. The actu- ator disc used in that theory must now be replaced by a screw with a large number of blades. Blade element theory This theory considers the forces on a radial section of a propeller blade. It takes account of the axial and rotational velocities at the blade as deduced from the momentum theory. The flow conditions can be represented diagrammatically as in Figure 10.12. C b dL D B w1 aVa dD dL dT w2 Va(1 ϩ a ) NP b Va dM uw 0 2pNr (1 Ϫ aЈ) A dD 2pNr aЈ2pNr dr 2pNr b r Figure 10.12 Forces on blade element

Chap-10.qxd 4~9~04 13:02 Page 187 PROPULSION 187 Consider a radial section at r from the axis. If the revolutions are N per unit time the rotational velocity is 2␲Nr. If the blade was a screw rotating in a solid it would advance axially at a speed NP, where P is the pitch of the blade. As water is not solid the screw actually advances at a lesser speed, Va. The ratio Va/ND is termed the advance coefficient, and is denoted by J. Alternatively the propeller can be considered as having ‘slipped’ by an amount NP Ϫ Va. The slip or slip ratio is: Slip ϭ (NP Ϫ Va)/NP ϭ 1 Ϫ J/p where p is the pitch ratio ϭ P/D In Figure 10.12 the line OB represents the direction of motion of the blade relative to still water. Allowing for the axial and rotational inflow velocities, the flow is along OD. The lift and drag forces on the blade element, area dA, shown will be: dL ϭ 1 rV 12CL dA ϭ 1 rCL[V 2a(1 ϩ a)2 ϩ 4p2r 2(1 Ϫ aЈ)2]bdr 2 2 where: V 2 ϭ V 2a(1 ϩ a)2 ϩ 4p2r 2(1 Ϫ aЈ)2 1 dD ϭ 1 rV 21CD dA ϭ 1 rCD[V 2a(1 ϩ a)2 ϩ 4p2r 2(1 Ϫ aЈ)2]bdr 2 2 The contributions of these elemental forces to the thrust, T, on the blade follows as: dT ϭ dL cos w Ϫ dD sin w ϭ dL  cos w Ϫ dD sin w dL ϭ 1 rV 12CL(cos w Ϫ tan b sin w)bdr 2 where: tan ␤ ϭ dD/dL ϭ CD/CL. ϭ 1 rV 12CL cos(w ϩ b) bdr 2 cos b Since V1 ϭ Va(1 ϩ a)/sin ␸, dT ϭ 1 rC L V 2a(1 ϩ a)2 cos(w ϩ b) bdr 2 sin2 w cos b

Chap-10.qxd 4~9~04 13:02 Page 188 188 PROPULSION The total thrust acting is obtained by integrating this expression from the hub to the tip of the blade. In a similar way, the transverse force acting on the blade element is given by: dM ϭ dL sin w ϩ dD cos w ϭ dL  sin w ϩ dD cos w dL ϭ 1 rV 12CL sin(w ϩ b) bdr 2 cos w Continuing as before, substituting for V1 and multiplying by r to give torque: dQ ϭ rdM ϭ 1 rC L V 2a(1 ϩ a)2 sin(w ϩ b) brdr 2 sin2 w cos b The total torque is obtained by integration from the hub to the tip of the blade. The thrust power of the propeller will be proportional to TVa and the shaft power to 2␲NQ. So the propeller efficiency will be TVa/2␲NQ. Correspondingly there is an efficiency associated with the blade elem- ent in the ratio of the thrust to torque on the element. This is: blade element efficiency ϭ Va ϫ 1 2pNr tan(w ϩ b) But from Figure 10.12, Va ϭ Va(1 ϩ a) ϫ 1 Ϫ aЈ ϭ 1 Ϫ aЈ tan w 2pNr 2pNr(1 ϩ aЈ) 1 ϩ a 1 ϩ a This gives a blade element efficiency: 1 Ϫ aЈ ϫ tan w 1 ϩ a tan(w ϩ b) This shows that the efficiency of the blade element is governed by the ‘momentum factor’ and the blade section characteristics in the form of the angles ␸ and ␤, the latter representing the ratio of the drag to lift coefficients. If ␤ were zero the blade efficiency reduces to the ideal effi- ciency deduced from the momentum theory. Thus the drag on the blade leads to an additional loss of efficiency.

Chap-10.qxd 4~9~04 13:02 Page 189 PROPULSION 189 The simple analysis ignores many factors which have to be taken into account in more comprehensive theories. These include: (1) The finite number of blades and the variation in the axial and rotational inflow factors. (2) Interference effects between blades. (3) The flow around the tip from face to back of the blade which produces a tip vortex modifying the lift and drag for that region of the blade. It is not possible to cover adequately the more advanced propeller theories in a book of this nature. For those the reader should refer to a more specialist treatise (Carlton, 1994). Theory has developed greatly in recent years, much of the development being possible because of the increasing power of modern computers. So that the reader is famil- iar with the terminology mention can be made of: (1) Lifting line models. In these the aerofoil blade element is replaced with a single bound vortex at the radius concerned. The strength of the vortices varies with radius and the line in the radial direction about which they act is called the lifting line. (2) Lifting surface models. In these the aerofoil is represented by an infinitely thin bound vortex sheet. The vortices in the sheet are adjusted to give the lifting characteristics of the blade. That is they are such as to generate the required circulation at each radial section. In some models the thickness of the sections is represented by source-sink distributions to provide the pressure distribution across the section. Pressures are needed for studying cavitation. (3) Surface vorticity models. In this case rather than being arranged on a sheet the vortices are arranged around the section. Thus they can represent the section thickness as well as the lift char- acteristics. (4) Vortex lattice models. In such models the surface of the blade and its properties are represented by a system of vortex panels. PRESENTATION OF PROPELLER DATA Dimensional analysis was used in the last chapter to deduce meaning- ful non-dimensional parameters for studying and presenting resist- ance. The same process can be used for propulsion.

Chap-10.qxd 4~9~04 13:02 Page 190 190 PROPULSION Thrust and torque It is reasonable to expect the thrust, T, and the torque, Q, developed by a propeller to depend upon: (1) its size as represented by its diameter, D; (2) its rate of revolutions, N; (3) its speed of advance, Va; (4) the viscosity and density of the fluid it is operating in; (5) gravity. The performance generally also depends upon the static pressure in the fluid but this affects cavitation and will be discussed later. As with resistance, the thrust and torque can be expressed in terms of the above variables and the fundamental dimensions of time, length and mass substituted in each. Equating the indices of the fundamental dimensions leads to a relationship: T ϭ rV 2D2  f1  ND  ,  n f3  gD     Va  f2  VaD  ,     V 2  a As required this gives thrust in the units of force and the various expressions in brackets are non-dimensional. f1 is a function of advance coefficient and is likely to be important. f2 is a function of Reynolds’ number. Whilst relevant to the drag on the propeller blades due to vis- cous effects its influence is likely to be small in comparison with the other dynamic forces acting. It is therefore neglected at this stage. f3 is a function of Froude number and is concerned with gravity effects. Unless the propeller is acting close to a free surface where waves may be created, or is being tested behind a hull, it too can be ignored. Hence for deeply immersed propellers in the non-cavitating condi- tion, the expression for thrust reduces to: T ϭ rV 2 D 2 ϫ f T  ND  a  Va  For two geometrically similar propellers, operating at the same advance coefficient the expression in the brackets will be the same for both. Hence using subscripts 1 and 2 to denote the two propellers: T1 ϭ r1 ϫ V 2 ϫ D 2 T2 r2 V a1 1 2 a2 D 2 2

Chap-10.qxd 4~9~04 13:02 Page 191 PROPULSION 191 If it is necessary to take Froude number into account: gD1 ϭ gD2 V 2 V 2 a1 a2 To satisfy both Froude number and advance coefficient: T1 ϭ r1 ϫ D 3 ϭ r1 l3 T2 r2 1 r2 D 3 2 where ␭ is the ratio of the linear dimensions. Since ND/Va is constant: N1 ϭ Va1 ϫ D2 ϭ 1 N 2 Va2 D1 l0.5 Thus for dynamic similarity the model propeller must rotate faster than the corresponding ship propeller in the inverse ratio of the square root of the linear dimensions. The thrust power is the product of thrust and velocity and for the same Froude number: PT1 ϭ r1 l3.5 PT2 r 2 Correspondingly for torque it can be shown that: Q ϭ rV 2 D 3 ϫ fQ  ND  a  Va  The ratio of torques for geometrically similar propellers at the same advance coefficient and Froude number will be as the fourth power of the linear dimensions. That is: Q 1 ϭ r1 l4 Q 2 r2 Coefficients for presenting data It has been shown that: T ϭ rV 2 D 2[ f T( J )] and Q ϭ rV 2 D 3[ f Q ( J )] a a