The Waterskier's Paradox

Rudimentary drag theory suggests that high-speed waterskiing should not be possible. But it is, of course, and this leads to some interesting thoughts on getting the most out of a planing hull.

Uffa Fox gave the yachting world some crucial inventions. During his time in International Fourteens, between the wars, he introduced the kicking strap (the vang), trapeze and the planing hull to dinghy racing. The kicking strap was accepted with little argument. The benefits of the trapeze were so clear that it was immediately banned! But a dinghy which could plane downwind was perhaps the most dramatic invention of all.

The Problem

However, the planing hull is also, perhaps, the yachting world's biggest "can of worms". What the planing mode actually means - in terms of its effect on hull drag, and on the way the craft must be handled - has been the source of enough contention to fill a century of post-race arguments.

When I taught a course called "The Science of Sailing", I spent a fair slice of time sorting out the issue for my students. Later, when my attention was drawn to Daniel Savitsky's "Hydrodynamic Design of Planing Hulls" (Marine Technology, Vol. 1, No. 1, Oct. 1964, pages 71-95) the mathematical theory was better sorted out - although the basic physics of the process remained hidden.

But as always seems to be the case, the answer came from simplifying the problem, not from complicated mathematics. What was the simplest planing craft? Easy - a water ski. Next to no displacement, with almost all its load carried by dynamic pressure.

But when I threw a few simple bits of physics into the analysis of the ski, I was hit by a puzzle. It goes as follows

a skier, starting from rest, quickly comes to a near upright position - with, perhaps, a backward lean of about ten degrees at a speed of 10 knots or so.

The lean is determined by the ratio of the the drag force to the upward force acting on the ski - the skier's weight.

As speed increases, drag normally increases, usually with the square of speed.

So at 20 knots, the lean should increase by a factor of (20/10) squared - that is, 40 degrees or so

... and, as drag increased further, true high-speed skiers would lean almost on their backs! But, as we know, this is not what happens. Thus the "waterskier's paradox".

Figure 1. If we ignore skin friction, the drag and lift forces add to generate a force at right angles to the ski.

The Solution

As with most apparent paradoxes, the cause is a weakness in an assumption. It is true that forces like skin friction and form drag do increase with the square of flow speed - but what about the drag due to the lift itself? And what about other drag considerations, such as the wetted surface?

Here we do need some physics and mathematics - so please, if you can, be patient. The destination is worth the effort.

If you must, you can jump forward the the next heading ("Was All This Worth It?") - but don't complain if the result you see shocks you! It will only do so because you have not taken time to follow it through.

Skin friction is the easy one to handle. It is simply the product of some drag coefficient, the flow speed squared, and the area of contact. We don't even need to know the drag coefficient, for now.

The drag due to the lift is tied totally to the inclination of the ski. A look at figure 1 shows this force is just the slope of the ski on the water (i.e., its pitch) , multiplied by the total weight of the skier. But without knowing this pitch, we are still none the wiser.

This pitch is the source of the lift - just as the angle of attack of an aeroplane's wing is the source of its lift. So, as for an aeroplane wing, we can throw in a lift coefficient - again, value not a problem for now. The lift force depends on the pitch of the ski ("angle of attack"), and also on the area of ski on the water.

Lift force, like most drag forces, also varies with the square of flow speed.

But now we pause, and remind ourselves that lift force does not change - it stays equal to the skier's weight. Mathematically,

WEIGHT = LIFT COEFFICIENT x AREA x PITCH x SPEED ^ 2

So at any speed, the product of area and pitch is constant. The higher the angle of the ski on the water, the less the amount of ski that stays on it.

If a skier places the ski flat on the water, the drag due to lift will be reduced - but the large wetted area will increase skin friction. At the other end of the scale, a large pitch will minimise skin friction, but maximise drag due to lift.

Figure 2. As we increase the pitch of the ski, skin friction reduces, but the drag due to lift increases.

Naturally, there is a "happy medium" between the two extremes. Introducing terms "A" for wetted area, "S" for skin friction coefficient, "L" for lift coefficient and "V" for the velocity (speed) of the ski, the total drag is

Skin friction + Drag Due to Lift

= S x A x V ^ 2 + WEIGHT x PITCH

We can get rid of the pitch term using the previous equation, to give total drag

S x A x V2 + WEIGHT2/(L A V ^ 2)

The first part increases with wetted surface A, the second decreases (as in figure 2). The best point occurs where the two parts are equal - i.e, where

A V ^ 2 = WEIGHT / square root(SL)

and the drag here is

S (WEIGHT / square root(*SL)) + WEIGHT ^ 2/(L WEIGHT / square root(SL))

= 2 x WEIGHT / square root(L/S)

Was All This Worth It?

Look carefully at the above expression. It has lost something. In fact, it has lost a lot.

There is no surface area, nor is there a pitch term. But ski pitch is the ratio of drag to weight - so equals 2 / square root(L/S), a constant. And wetted surface decreases with the square of the speed.

But where has the speed term, V, gone? Could drag be truly independent of speed?

The answer is, surprisingly, yes. This is the resolution of the skier's paradox. With some presumptions, a skier experiences no more drag - so must lean no further - at 30 knots than at 10 knots.

However, when we look at the wetted surface we need, to keep to the optimum point of the drag curve, we find a catch.

In going from 10 knots to 30 knots, we have to reduce wetted surface by a factor of 9. On a one-metre ski, only the last 10 cm or so must touch the water. This means the weight of the skier must also move to the back of the ski. But such a change is not a too big a problem, when all of the weight of the skier is mobile.

Application to Yachts and Power Boats

While it is easy to move the full weight of a skier to the back, moving the centre of gravity of a hull to within a metre of its transom is rarely possible.

In extreme racing hulls, the solution is a step in the planing surface, close to the centre of gravity. As the hull lifts, the transom, effectively, moves forward. Unfortunately, issues like seakeeping and light-wind performance tend to make such solutions of little general use.

There are, however, several points which can be usefully kept in mind. As speed increases, wetted surface MUST decrease. Use the idea of constancy of pitch with care, since it will depend of the shape of the hull's bottom.

In particular, keep in mind that the correct location for the centre of gravity of the hull will depend on the intended operating speed. To see this, we need to pass on to a more detailed analysis.

A Look at the Full Story

The professional versions of my hull CAD program, Hullform, include several drag analysis schemes, including Savitsky's. It is worthwhile to look at a hull, both to see how well the constant-drag rule performs, and to see what else can be learnt.

In figure 3, I have taken a large planing hull as an example. Hullform allows drag curves to be plotted for a range of centre of gravity positions, and I have located it at positions from 12 to 20 metres from the stem (The design location was just over 12 metres) The Savitsky scheme is only applicable in the "full planing" mode, which commences just over 20 knots for this hull - this is why the lower part of the speed range has been deleted.

Figure 3. Savitsky full-planing drag analysis for a 23 metre, 46 tonne hull.

It is obvious, firstly, that the design location of the centre of gravity is about the best for speeds up to 30 knots. It is unlikely that the craft would operate any faster than this - a rather fortunate coincidence, because the centre of gravity position becomes an increasing problem at higher speeds.

At about 35 knots, the best position is about 16 metres from the stem. Beyond 40 knots, it moves back to 20 metres. While a position here would not be hard to generate, such a design would have big problems at lower speeds - the drag at 20 knots is nearly three times that for a centre of gravity at 12 metres.

But then, naval architecture has never been all science - the art of compromise is also a big part of the profession.

You can also see the constancy of drag coming in to effect, beyond 30 knots. The least drag for all curves stays close to 6 tonne, right up to 60 knots.

Finally, in case any mathematically-oriented reader gets to wondering, the 46 tonne weight and 6 tonne drag give a ratio of lift coefficient to skin friction coefficient of about 250. The skin friction coefficient is about 0.002, so hull lift coefficient works out to be 0.5. A perfect flat surface with perfect flow has a lift coefficient of 1, and allowing for imperfections in flow, and the hull's deadrise, the smaller value is quite reasonable.

Its comforting to find that simple science can end so close to the mark!