# The Physics Of Fast Sailing

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Sailing boats exploit the discontinuity in fluid flow that exists at the air/water interface in order to propel themselves. We may consider the water to be at rest and describe the velocity of the wind by a vector VT. The magnitude of this vector is equal to the wind speed with respect to the water and the direction of the vector coincides with the wind direction. Under the influence of the wind, the boat moves at a speed VH in a direction given by the vector \B. Motion of any object through still air with a velocity \B gives rise to an induced wind velocity - V0 in the opposite direction, hence the total or apparent wind velocity felt by the boat is the vector sum of the true and induced winds, that is

This vector equation can be represented schematically by the triangle shown in Fig. 1-1. In this figure each of the sides of the triangle is given by a line whose length is proportional to the magnitude of the

I IK I I I he sailing triangle showing (he apparent win'1 <•( ihe true wind and induccd wind.

I IK I I I he sailing triangle showing (he apparent win'1 <•( ihe true wind and induccd wind.

,cctor sum igh speed sailing appropriate vector; the directions of the lines are those of the corresponding wind directions. These directions are described in the figure by the angles y and /?. The interior angle opposite the side VA is (180° — y) and since the sum of the interior angles of any plane triangle is 180°, the angle between the vectors Vr and V^ must be

This simple wind vector triangle can be used to derive a very useful relation. The law of sines in trigonometry states that the ratio of any side of a triangle to the sine of the opposite angle is a constant. Hence

In terms of VT and VB, this gives

where cos /?/sin fi = ctn (\$ (the cotangent of /?) varies with as shown in Fig. 1-2.

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Note that Eq. (1-3) contains no approximation; it is simply an alternate statement of Eq. (1-1) and applies equally to any sort of body moving through air. Since VT is known from a static measurement and the course angle to the true wind y is known, we see that VB is a function only of the angle (3. In order to see what /? is, we must now begin to describe sailing boats.

The essential elements for momentum transfer from the wind are a vertical aerofoil (sail) and a vertical hydrofoil (keel, centreboard, etc.) as shown in Fig. 1-3. In practice, the vertical downward force of

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I ig 1-3. The ideal yacht: essential elements for momentum transfer.

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I ig 1-3. The ideal yacht: essential elements for momentum transfer.

gravity requires the buoyant support of a hull or hulls. These hulls do not play any role in the momentum transfer in the limit where the hydrofoil function is wholly fulfilled by a keel or board. In the horizontal plane, the foils shown from above in Fig. l-4(a) give rise to the forces indicated in l ig. I-4(b). The water How — VH over the hydrofoil and the airflow VA over the aerofoil produce forces Fu and FA respectively. In order that the boat be in equilibrium (unaccclcratcd)

Fig. 1-4. Illustrating the course theorem.

The force ¥H generated by the hydrofoil can be decomposed into a component of drag in the — Vg direction <2>H and a component of lift at right angles to the course, , as shown in Fig. l-4(c). The angle ôH = arcctn(i?H/^H) is the hydrodynamic drag angle. The aerodynamic vector F^ can be decomposed into components parallel and perpendicular to \B: Fx, the driving force in the direction of the motion and Fy, the sideforce. It can also be decomposed parallel and perpendicular to V^ as the aerodynamic drag and aerodynamic lift a for which the aerodynamic drag angle ôA = arcctn(J?A/@A) can be defined. These two decompositions are shown in Fig. l-4(d). Since Fx = and Fy = ï£H, thus the angle between F^ and ¥y is SH. The lift is perpendicular to V^ and ¥y is perpendicular to VB ; thus the angle between A and Fy must be equal to the angle between V4 and Vfl, that is

This relation, known as the course theorem, and the definitions

of the drag angles constitute a mathematical description of a sailing boat.

We see from Eq. (1-3) that for values of y not too near 180°, ctn /? the physics should be as large as possible in order to provide a high value of VB/VT. of fast Figure 1-2 shows us that this corresponds to small values of /?. sailing Equations (1-5) and (1-6) imply that the use of foils having large lift-to-drag ratios will result in low values of /?.

Clearly, the knowledge of dH and SA for all values of y and VT amounts to a complete solution of the problem. The accurate calculation of the lifts and drags for all VT and y is a complicated problem and the results of any mathematical model are subject to question. For the purpose of establishing a firm foundation to our basic design criteria, we shall look at some measured drag angles for a broad spectrum of sailing craft and try to draw some conclusions.

In Fig. 1-5 we have plotted the aero drag angle <5^ for an International

Fig. 1-5. The aerodynamic drag angle as a function of course angle to the true wind.

as functions of the course angle y. The data points taken from the table on p. 313 of Ref. 1, are admittedly few and the choice of curves used to fit them reflects my bias for a number 59 ship curve which I have always found to fit physical data quite well.

The dinghy with its relatively low aspect ratio sail plan and high parasitic wind resistance has the largest drag angle. The 12-metre boat and the Tornado, both with high aspect ratio rigs and aerodynamically clean' decks have nearly identical values of SA for all y. The ice boat, using a fully battened unarig is somewhat more efficient.

Figure 1-6 shows a plot of the hydro drag angles SH for the same boats. We see that the dinghy and the 12-metre boat have nearly identical values of ¿H. This curve is very nearly linear and is a common feature of all monohull sailing craft. The Tornado catamaran, typical of fast multihulls, has a much lower value of SH and the ice boat with its runners operating on 'rails* as it were, has an almost perfect curve with ^ 0 until a value of y * 180° is approached.

1 80

Fig. 1-6. The hydrodynamie drag angle as a function of course angle to the true wind.

1 80

Fig. 1-6. The hydrodynamie drag angle as a function of course angle to the true wind.

Fig. 1-7. p versus y for the International 12' dinghy, a 12-metre yacht, a Tornado catamaran, and an ice boat.

1 40

1 20

J3 100

INT 1 2

1 2-IVl

In order to give these data for drag angles some physical reality, we shall use their sum, SA + dH = shown plotted as a function of y in Fig. 1-7, and Eq. (1-3) to compare VB/VT for our sample sailing craft. This comparison is shown in the form of polar plots of VB/VT versus y with y increasing ccw from the horizontal. In Fig. 1-8 the dinghy, the 12-metre, and the catamaran are compared, and in Fig. 1-9, the catamaran and the iceboat are shown. As you can see, iceboating at 80 knots plus is another world altogether!

1 60

1 80

1 60

Fig. 1-8. Polar curves for the Tornado, 12-Metre, and International 12' dinghy for vt ^ 10 knots.

Fig. 1-8. Polar curves for the Tornado, 12-Metre, and International 12' dinghy for vt ^ 10 knots.

I ig. 1-9. Polar curves for the Tornado and the ice boat for vt - 10 knots.

I ig. 1-9. Polar curves for the Tornado and the ice boat for vt - 10 knots.

The main factor determining the aero drag angle is the ratio of sail area to parasitic (non lift-producing) area. In the case of the hydro drag angle, it is the leeway resistor area versus the area of the vertical 7

jgh speed projection of the non lift-producing hull. In many cases these functions sailing cannot be clearly separated, that is, the keel and hull are not distinct.

This fact is a major complicating factor in attempting an analytical treatment.

For a given boat, these drag angles are a function not only of y, the course angle, but also of VT, the wind speed. The data given in Figs. 1-5 and 1-6 was taken for wind speeds in the 5-10 knot range. At higher wind speeds, heeling and the need to reduce sail area causes dA to increase and the effect of heeling and wind and motion induced waves causes SH to increase as well. Thus VB/VT decreases with increasing VT as seen in Fig. 1-10 in which the polar plots of VB/VT for a 12-metre boat are compared for VT = 10 and 20 knots.

Fig. 1-10. Showing the decrease of vb/vt with increasing vt for a 12-Metre yacht.

Fig. 1-10. Showing the decrease of vb/vt with increasing vt for a 12-Metre yacht.

If we view the ideal yacht of Fig. 1-3 from dead ahead, (see Fig. 1-11) we see that the sideways components of the sail force Fy acting through the centre of effort of the sail and the lift exerted by the keel = Fy lie in antiparallel directions along lines separated by a distance h. This forms a couple hFy that tends to heel the boat to leeward. As the boat heels the couple formed by the shift of the centre of buoyancy to leeward with respect to the centre of gravity increases until it cancels the heeling torque:

where b is the horizontal distance between the centres of gravity and buoyancy. Since the heeling force Fy is proportional to the sail area, thus the maximum sail area that can be carried for a given apparent wind speed is proportional to the product of W and b/h. In the case of a heeled monohull as shown in Fig. 1-12, the distance b is small, hence the weight W must be large. In a boat like the 12-metre, this weight is carried in the form of lead at the bottom of the keel. In the case of the dinghy, crew shift to windward provides the righting moment.

Multihulls such as the catamaran shown in Fig. 1-13 arc unballasted and depend upon their large lateral spread and consequent large value 8 of b for their righting moment. The maximum righting moment is

Fig. 1-13. The righting moment for a heeled catamaran.

achieved in a catamaran at a heel angle of 10° or less as compared to 30° or so for a monohull. Hence the effect on SA of the slightly greater relative parasitic drag area of the Tornado is compensated by the greater effective sail area associated with the lower angle of heel For speed under sail, it is clear that multihulls are the way to go.

Multihull sailing craft may employ any of three possible hull configurations: a double outrigger or trimaran, a reversible single outrigger or proa, and a double canoe or catamaran. The trimaran consists of a central load-carrying hull stabilized by outriggers, one on either side. The proa has two dissimilar hulls. In order always to keep the heavy hull to windward, the proa must be able to sail in either direction. The catamaran features two similar hulls separated by a distance of about half the boat length.

In the following chapters we shall discuss in some detail the principles involved in the design of high speed sailing craft. In this discussion we shall constantly bear in mind the basic criterion for fast sailing craft as developed above, namely that the aerodynamic and hydro-dynamic lift-to-drag ratios should be as large as possible for various sized craft sailed in sheltered and offshore waters.