Photo by Sean Trew, www.pacificfog.net
A recent post from the blog of the always interesting Charles Doane on a different formula to estimate hull speed that more accurately reflects the fact that lightweight boats are more capable of exceeding their nominal hull speed.
It’s been a while since we discussed all those mysterious numbers that both boat designers and journalists are always throwing around to confuse us when they talk about boats. You’ll recall last time I bloviated about displacement/length ratios. One big reason it’s a good idea to go to the trouble to calculate a boat’s D/L ratio is that you can use this number to refine your estimate of the boat’s maximum speed potential beyond the relatively rough estimate afforded by the classic hull-speed formula we discussed earlier.
This more accurate method of finding a boat’s potential top speed, devised by designer Dave Gerr and published in his tome, The Propeller Handbook, can also be used to estimate the hull speeds of multihulls (both catamarans and trimarans). Dave warns, however, that catamarans with very narrow hulls (for mysterious reasons that no one really understands) will often exceed the speeds predicted by his method.
To comprehend Dave’s formula, we first need to comprehend that the famous multiplier we used in the classic hull-speed formula (1.34) is in fact what is known as a speed/length ratio (S/L ratio). This ratio quantifies the relationship between a boat’s speed (BS), whatever it happens to be at any given point in time, and its waterline length, according to a formula that holds that S/L ratio equals a boat’s speed in knots divided by the square root of the boat’s load waterline length (S/L ratio = BS ÷ √LWL).
The brightest kids in class will instantly note that this is simply the classic hull-speed formula (HS = 1.34 x √LWL) run backwards to solve for the speed/length ratio instead of speed. What the classic hull-speed formula assumes is that 1.34 is the maximum S/L ratio that can ever be achieved (due to the characteristics of waves we discussed before) and thus always serves to limit a boat’s top speed potential.
What Dave’s formula does is estimate a boat’s maximum S/L ratio based on its D/L ratio so as to more accurately reflect the fact that lightweight boats are more capable of exceeding their nominal hull speed. Once we’ve arrived at a new and more accurate S/L ratio for a given boat, we can then plug it into the classic hull-speed formula to derive a new, more accurate estimate of that boat’s nominal hull speed.
Dave’s formula holds that a boat’s maximum S/L ratio equals 8.26 divided by its D/L ratio to the .311 power (Max S/L ratio = 8.26 ÷ D/L ratio↑.311). For a 12,000-pound boat with a 28-foot waterline and a D/L ratio of 244, we thus get the following results: 244 to the .311 power equals 5.53 (you’ll obviously need a scientific calculator to figure that out!), therefore 8.26 ÷ 5.53 equals a maximum S/L ratio of 1.49. Plug 1.49 into the hull-speed formula (1.49 x √LWL) and you get a new nominal hull speed of 7.9 knots (1.49 x 5.29 = 7.88), as compared to the boat’s old nominal hull speed of 7 knots (1.34 x 5.29 = 7.08).
This in itself is an appreciable difference, but it grows even larger as the boat grows lighter. Assume, for example, that our 12,000-pound boat has shed 3,000 pounds to become a 9,000-pound boat with the same load waterline length, and its D/L ratio drops to 183. Its old nominal hull speed, based solely on its LWL, remains exactly the same at 7 knots, but its new nominal hull speed, figured according to Dave’s method, now becomes 8.6 knots!
Things get even more exciting if you bear in mind that this revised hull-speed estimate still does not account for a boat’s potential to plane. That is, we’re still only talking about the top potential speed that may be achieved by a hull in displacement mode. Dave further warns that getting the extra hull speed his method predicts will require a lot of extra power, but he does maintain that his method is ultimately more accurate than the old one.
I, for one, happen to believe him. Running Dave’s formula on your own boat and finding out that its top speed is higher than you thought is an easy way to put a smile on your face. Unless you own a classic CCA-style cruiser with long overhangs. These boats have exaggerated D/L ratios, due to their short static waterlines (which get longer when the boat starts sailing and heels over) and when you run Dave’s formula on them you usually get a lower top speed. In cases like this, I always assume the classic hull-speed formula is more accurate.
How does he derive his formula? Why should speed-capability be rated by that formula?
From what I’ve read about what causes wavemaking-drag, I’d suggests the following:
SA * LWL / W
Where SA = sail-area, LWL = waterline length, & W = weight.
W/L is proportional to average underwater hull-cross-section. …which, in turn should be roughly proportional to maximum underwater hull-cross-section.
After all, the hull’s underwater cross-section is what’s pushing the waves up, and experiencing pressure from them.
And, regarding the explanation that, when approaching hull-speed, the stern is starting to fall into the bow-wave’s trough, making the boat sail uphill–Obviously the difficulty of dragging something uphill involves its weight. …and LWL is important as the quantity that postpones that problem.
…and of course thee hull-cross-section mentioned in the other paragraph is what makes that bow-wave in the first place, and plausibly is part of what influences its height.
The above-suggested sailboat speed-capability-rating formula agrees well with the order of various sailboat class’s official compensation-ratings for multi-class races.
So, my suggested formula is based on the two explanations that I’ve read, for wave-making-drag.
I should add that SA * L / W is intended as a rough measure of a boat’s ability to reach and exceed its hull-speed.
But if one wanted something that tries to roughly suggest a plausible speed for a displacement-boat, then SA * L / W could be multiplied by the square-root of L, giving:
SA * L^(3/2) / W
…where L^(3/2) means the 3/2 power of L.