Resistance Although resistance and propulsion are dealt with separately in this book this is merely a convention. In reality the two are closely inter- dependent although in practice the split is a convenient one. The res- istance determines the thrust required of the propulsion device. Then propulsion deals with prmfding that thrust and the interaction between the propulsor and the flow around the hull. When a body moves through a fluid it experiences forces opposing the motion. As a ship moves through water and air it experiences both water and air forces. The water and air masses may themselves be mov- ing, the water due to currents and the air as a result of winds. These will, in general, be of different magnitudes and directions. The resist- ance is studied initially in still water with no wind. Separate allowances are made for wind and the resulting distance travelled corrected for water movements. Unless the winds are strong the water resistance will be the dominant factor in determining the speed achieved. FLUID FLOW Classical hydrodynamics leads to a flow pattern past a body of the type shown in Figure 9.1. Figure 9.1 Streamlines round elliptic body As the fluid moves past the body the spacing of the streamlines changes, and the velocity of flow changes, because the mass flow within streamlines is constant. Bernouillis theorem applies and there are 143 144 RESISTANCE corresponding changes in pressure. For a given streamline, if p, p, v and h are the pressure, density, velocity and height above a selected datum level, then: _ v 2 P + - + gh = constant p 2 Simple hydrodynamic theow deals with fluids without iscosity. In a non-viscous fluid a deeply submerged body experiences no resistance. Although the fluid is disturbed by the passage of the body, it returns to its original state of rest once the body has passed. There will be local forces acting on the body but these will cancel each other out when integrated over the whole body. These local forces are due to the pres- sure changes occasioned by the changing velocities in the fluid flow. In studying fluid dynamics it is useful to develop a number of non- dimensional parameters with which to characterize the flow and the forces. These are based on the fluid properties. The physical properties of interest in resistance studies are the density, p, x4scosity,/* and the static pressure in the fluid, p. Taking R as the resistance, Vas velocity and L as a typical length, dimensional analysis leads to an expression for resistance: R = fLaVbp)xagep f The quantities involved in this expression can all be expressed in terms of the fundamental dimensions of time, 7, mass, M and length L. For instance resistance is a force and therefore has dimensions ML/T 2, p has dimensions M/L 3 and so on. Substituting these fundamental dimensions in the relationship above: Equating the indices of the timdamental dimensions on the two sides of the equation the number of unknown indices can be reduced to three and the expression for resistance can be written as: RESISTANCE 145 The expression for resistance can then be written as: Thus the analysis indicates the following non-dimensional combina- tions as likely to be significant: R VL p V p pV2t 2 p (gL) 0.5 pV 2 The first three ratios are termed, respectively, the resistance coefficient, Reynolds number, and Froude number. The fourth is related to cavitation and is discussed later. In a wider analysis the speed of sound in water, o and the surface tension, o-, can be introduced. These lead to non- dimensional quantities V/o, and o/g pL 2 which are termed the Mach number and Weber number. These last two are not important in the con- text of this present book and are not considered further. The ratio tx/p is called the kinematic viscosity, and is denoted by u. At this stage it is assumed that these non-dimensional quantities are independent of each other. The expression for the resistance can then be written as: Consider first J) which is concerned with wave-making resistance. Take two geometrically similar ships or a ship and a geometrically similar model, denoted by subscripts 1 and 2. (g/t and Rwl = iOlVlIf2 t V 2 ) Hence: 146 RESISTANCE The form of J is unknown, but, whatever its form, provided gL1/V 2 = gL2/V 2 the values ofj will be the same. It follows that: Since L1/V 2 = L2/V, this leads to: Rw2 _ P2L Rw2 A, or R,q PiLl Rwl A 1 For this relationship to hold Vl/(gL1) 5 = V,2/(gL2) 5 assuming p is constant. Putting this into words, the wave-making resistances of geometrically similar forms will be in the ratio of their displacements when their speeds are in the ratio of the square roots of their lengths. This has become known as Froudes law of comparison and the quantity V/(gL) 5 is called the Froude number. In this form it is non-dimensional. Ifgis omit- ted from the Froude number, as it is in the presentation of some data, then