12 PROPELLERS AND PROPULSION 12.1 Introduction We discuss in this section the nature of steady and unsteady propulsion. In many marine vessels and vehicles, an engine (diesel or gas turbine, say) or an electric motor drives the (Continued on next page)12.2 Steady Propulsion of Vessels 51 propeller through a linkage of shafts, reducers, and bearings, and the effects of each part are important in the response of the net system. Large, commercial surface vessels spend the vast majority of their time operating in open-water and at constant speed. In this case, steady propulsion conditions are generally optimized for fuel efficiency. An approximation of the transient behavior of a system can be made using the quasi-static assumption. In the second section, we list several low-order models of thrusters, which have recently been used to model and simulate truly unsteady conditions. 12.2 Steady Propulsion of Vessels The notation we will use is as given in Table 1, and there are two different flow conditions to consider. Self-propelled conditions refer to the propeller being installed and its propelling the vessel; there are no additional forces or moments on the vessel, such as would be caused by a towing bar or hawser. Furthermore, the flow around the hull interacts with the flow through the propeller. We use an sp subscript to indicate specifically self-propulsion conditions. Conversely, when the propeller is run in open water, i.e., not behind a hull, we use an o subscript; when the hull is towed with no propeller we use a t subscript. When subscripts are not used, generalization to either condition is implied. Finally, because of similitude (using diameter D in place of L when the propeller is involved), we do not distinguish between the magnitude of forces in model and full-scale vessels. Rsp N Rt N T N ne Hz nm Hz ne np Hz η Qe Nm Qp Nm �g Pe W Pp W D m U m/s Up m/s Qm Nm f kg/s fm kg/s f hull resistance under self-propulsion towed hull resistance (no propeller attached) thrust of the propeller rotational speed of the engine maximum value of rotational speed of the propeller gear ratio engine torque propeller torque gearbox efficiency engine power propeller shaft power propeller diameter vessel speed water speed seen at the propeller maximum engine torque fuel rate (or energy rate in electric motor) maximum value of Table 1: Nomenclature52 12 PROPELLERS AND PROPULSION 12.2.1 Basic Characteristics In the steady state, force balance in self-propulsion requires that Rsp = Tsp. (150) The gear ratio η is usually large, indicating that the propeller turns much more slowly than the driving engine or motor. The following relations define the gearbox: Qne = ηnp (151) p = �g ηQe, Uand power follows as Pp = �g Pe, for any flow condition. We call J = Up/npD the advance ratio of the prop when it is exposed to a water speed Up; note that in the wake of the vessel, p may not be the same as the speed of the vessel U . A propeller operating in open water can be characterized by two nondimensional parameters which are both functions of J : ToKT = (thrust coefficient) (152)πn2D4 pKQ = Qpo (torque coefficient). (153)πn2D5 pThe open-water propeller efficiency can be written then as ToU J (U )KT�o = = . (154)2βnpQpo 2βKQ This efficiency divides the useful thrust power by the shaft power. Thrust and torque co-efficients are typically nearly linear over a range of J , and therefore fit the approximate form: KT (J ) = α1 − α2J (155) KQ(J ) = ρ1 − ρ2J. As written, the four coefficients [α1, α2, ρ1, ρ2] are usually positive, as shown in the figure. We next introduce three factors useful for scaling and parameterizing our mathematical models: • Up = U (1 − w); w is referred to as the wake fraction. A typical wake fraction of 0.1, for example, indicates that the incoming velocity seen by the propeller is only 90% of the vessel’s speed. The propeller is operating in a wake. In practical terms, the wake fraction comes about this way: Suppose the open water thrust of a propeller is known at a given U and np. Behind a vessel moving at speed U , and with the propeller spinning at the same np, the prop creates some extra thrust.12.2 Steady Propulsion of Vessels 53 1.0 0.10η KT KQ00.8 0.08 K T 0.6 0.06 K Qη 0 0.4 0.04 0.2 0.02 0.0 0.00 0.0 0.2 0.4 0.6 0.8 1.0 J Figure 4: Typical thrust and torque coefficients. w scales U at the prop and thus J; w is then chosen so that the open water thrust coefficient matches what is observed. The wake fraction can also be estimated by making direct velocity measurements behind the hull, with no propeller. • Rt = Rsp(1 − t). Often, a propeller will increase the resistance of the vessel by creating low-pressure on its intake side (near the hull), which makes Rsp > Rt. In this case, t is a small positive number, with 0.2 as a typical value. t is called the thrust deduction even though it is used to model resistance of the hull; it is obviously specific to both the hull and the propeller(s), and how they interact. The thrust deduction is particularly useful, and can be estimated from published values, if only the towed resistance of a hull is known. • Qpo = . The rotative efficiency �R, which may be greater than one, translates �RQpsp self-propelled torque to open water torque, for the same incident velocity Up, thrust T , and rotation rate np. �R is meant to account for spatial variations in the wake of the vessel which are not captured by the wake fraction, as well as the turbulence induced by the hull. Note that in comparison with the wake fraction, rotative efficiency equalizes torque instead of thrust. A common measure of efficiency, the quasi-propulsive efficiency, is based on the towed resis-tance, and the self-propelled torque. RtU �QP = (156)2βnpQpsp To(1 − t)Up�R = 2βnp(1 − w)Qpo = �o�R (1 − t) . (1 − w)54 12 PROPELLERS AND PROPULSION To and Qpo are values for the inflow speed Up, and thus that �o is the open-water propeller efficiency at this speed. It follows that To(Up) = Tsp, which was used to complete the above equation. The quasi-propulsive efficiency can be greater than one, since it relies on the towed resistance and in general Rt > Rsp. The ratio (1−t)/(1−w) is often called the