since air is sucked into the low pressure core of the vortex when it gets close to the surface. As appears from the figure, all the vortices created at the trailing edge tend to roll up into a single one left behind the yacht. Since this vortex contains rotational energy it gives rise to a resistance component, the induced resistance, discussed in the previous chapter.
At the tip the side force generated must go to zero, since no pressure jump between the two sides can exist in the flow at the tip. Near the root, on the other hand, the flow is uninfluenced by the tip and a large force may be generated, since the bottom acts as a wall, preventing the overflow. The variation between root and tip depends 011 the shape of the keel, and it may be shown that the best distribution of the force is an elliptical one. With this distribution the minimum amount of vertical energy is left behind, which means that the induced resistance is minimized. In Fig 6.3 (c) an elliptical distribution is shown. This may be imagined as one quarter of a full ellipse, as shown in (d). The simplest way to obtain an elliptical distribution of the side force is to make the keel planform elliptic. This has some disadvantages, however, and we will return shortly to the optimization of the planform.
An interesting phenomenon is indicated in Fig 6.3 (a) and (c). If the bottom of the hull may be considered as a flat plate of infinite extension, the flow around the keel would be the same as if the plate had been replaced by the mirror image of the keel in the plate. A flat wall parallel to the flow thus acts as a symmetry plane. Now. the bottom is neither flat nor infinite in reality, but for modern shallow hulls this is a reasonable approximation.
Definition of the keel The definition of the planform of a trapezoidal keel is given in Fig 6.4. planform First, it should be mentioned that the horizontal distance from nose to tail at all depths is called the chord. Two chords are specified in the
Fig 6.4 Definition of the planform
Aspect ratio: AR =
Taper ratio =
Sweep angle defined by 25% chord
Chord C, figure, namely the root and tip chords, C] and C2. These can be used to define a mean chord C = (C,+C2)/2. The most important parameter for the efficiency of the keel is the aspect ratio. AR, defined as AR = Tk/C, ie the keel depth divided by the mean chord. This is the geometric aspect ratio. As explained above, the effective aspect ratio AR., is twice as large, if the keel is attached to a large flat surface. The second parameter to be defined is the taper ratio, X, which is simply the ratio of the tip chord to the root chord, ie X - C2/CL.
Most keels are not exactly vertical, but sweep backwards to some extent. It is not obvious, however, how to define this sweep angle. The leading or trailing edges might be used for defining the angle, or perhaps the mid-line between the two, but the most appropriate choice turns out to be the line 25% of the chord length from the leading edge. As pointed out above, under certain ideal conditions, the centre of effort at every section lies along this line. Even though this is not exactly true in a real case, it is still a good approximation for fin keels and rudders of normal aspect ratios. We will return to the location of the centre of effort in Chapter 8, in connection with the balance of the yacht.
Classical wing theory One of the most well known and useful theories in aerodynamics is the so-
called lifting line theory for computing the lift and induced resistance (drag) of wings. Without going deeply into the mathematics, the basics of the theory may be explained with reference to Fig 6.5, which shows a wing with two free ends, symmetric about the centreline. It could also be inter-
Fig 6.5 Lifting line theory preted as a keel with its image reflected in the hull bottom. The wing is
dashed in the figure, since in the theory it is replaced by a set of vortices. There is thus one vortex along the span of the wing (from tip to tip). This is called the bound vortex, since it is fixed to the wing. However, as we have seen, vortices are shed backwards from the wing, particularly close to the tip. These are the free vortices, which align themselves with the local flow direction. There is a theorem stating that a vortex cannot have a free end in the flow. Thus, when a vortex filament bends backwards and leaves the bound part, the vortex strength of the latter is reduced by the strength of the filament. At the tip, all the vorticity has been shed backwards, and the bound vorticity is zero. Behind the wing, all the free vortex filaments roll up into one concentrated free vortex on each side. These two in turn are connected through the starting vortex (not shown in the figure), created when the wing started its motion.
The local force created by the vortex system is proportional to the component of the vortex at right angles to the local flow direction. Since the free vortices are parallel to the flow they do not create any force, but the bound vortices on the wing generate a force that is
L/ w proportional to the vortex strength. The theory also shows that the best distribution of vorticity, and hence force, on the wing is the elliptical one. In this case, the drag and lift coefficients, CD and CL of the wing, and the corresponding forces, can be obtained easily, as shown in Fig 6.5. CT2D ^e lift coefficient per degree in the two-dimensional case. For a symmetrical section in a frictionless fluid this coefficient may be obtained theoretically as 7T2/90 = 0.11. In a real flow it is slightly smaller due to viscosity, and 0.10 is a good approximation for all symmetrical sections.
If the force distribution on the wing is not elliptic, the effective aspect ratio should be used in the formula. This is always smaller than the elliptical one, but the difference is normally not very large, so the actual aspect ratio may be used for good estimates for non-elliptical loadings also. We may summarize the most important results as follows:
• the lift and induced drae coefficients can be estimated in most cases from the formulae of Fig 6.5
• the aspect ratio is the most important parameter for the lift and drag of a wing
• the elliptical force distribution is the best one.
The effect of the aspect ratio appears again in Fig 6.6, which is based on wind-tunnel experiments with wings of different aspect ratios. Lift and drag coefficients are given for varying angles of attack. In the lefthand diagram very different curves are obtained depending on ARe. For instance, at 5°, which is a typical leeway angle and hence angle of attack for a keel, the square wing with ARe = 1 produces less than one third of the lift coefficient of the two-dimensional wine, which has AR. = infinity. An effective ARC = 3 is relatively common for keels. It may be seen that this produces about twice as much lift as the square wing.
In the range of practical ARC the drag is relatively unchanged, but it should be kept in mind that this is for a given angle of attack, while in
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