I know there is always a relationship between displacement vs. power in order to have a certain speed, but there seems that the faster you go the less difference it makes the final horsepower.
Bismarck had 160,000 hp and a speed of 30 knots.
Arizona had 35,000 and 20.7 knots.
The displacement relation was: Bismarck (50,000 ton)/Arizona (37,000 ton) = 1.35
So that Arizona will need 35% more horsepower to increase her final speed to be the same of Bismarck: 47,2967 hp. But we know that isn´t the case: 35% more speed means only 27 knots and I believe the Arizona with the extra power will not do it anyway. There is no direct proportion between displacement, power & speed.
Why?
The longer the ship the less the wave-making resistance. So a longer ship with the same displacement would need less power.
The waves forming in the bow coming in phase with the ones forming in the stern produce a step increase in the wave making resistance. This is what is called hull speed. Foeth can explain this more detailed than me, we should wait his answer.
You're right about wave power, but I can make a few additions. Your resistance consists mainly of frictional drag and wave drag (And a bit of form drag but as it scales with frictional drag, you could combine them). At low speeds your frictional drag is important, at higher speeds your wave resistance. (friction has a speed/power relation to the power 3, wave resistance +/- power 6, so really adds up fast).
As Marcelo already explained, interaction between waves of the hull determines the wave resistance. Two ships with an identical hull from but one 50% smaller than the other will have the same wave pattern at 70.7% (square root) of that speed. The longer ship will have an similar interference pattern at higher speeds, as you need longer waves to recreate the same pattern and longer waves are faster. Note that the power needed to overcome this drag is not the same. But, if the larger ship sails at the same speed, it's wave resistance is mroe likely to be low, especially if the smaller ship is already sailing at a point wiht horrendous wave interference, something warships tend to do at top speeds.
So, being longer really helps. The hull shape also matters significantly. Bismarck is a relatively slender ship compared to the Arizona.
Now, resistance is only one part, the other is propulsion. The difference between a 50% and 70% efficiency should be quite self-explanatory.
There is a direct relation between speed, power and hull form, but it's strongly non linear
Thanks to Marcelo and specially to Foeth.
So, two vessels, one longer than the other, let´s say by 25% of the lenght, at similar speeds would need less power to accomplished it. That´s it?
It´s interesting because, so in theory, a longer ship in need of less power to reach a certain speed is also more fuel efficient. In this case, displacement against displacement a CVN aircraft carrier is far more efficent than, let´s say: Iowa?
Thank
Yes, a larger ship is often more efficient at the same speed. Note that the resistance is rarely lower absolutely, only relatively. When sailing at the same speed, the advantage is with the longer ship. Note that at lower speeds the frictional resistance is (/ can be) worse for a larger ship.
Actually to double the speed of a ship you would need to cube the power output. so 2 cubed is 8 times the shp. to raise arizona from 20.7 to 30 knot or 1/3 more would be 1.5 cubed (3.375) so it would be on the order of 110,000 shp.
Also the Bismark hull is proprtioned close to what Arizona was
Arizona was 6.2:1 ratio 600 ft long 97 wide
BK was 6.45:1 ratio 820 ft long 127 wide
KGV was about 7:1
Washington was 7:1
South Dakota was 6.3:1 so closer to Bismark in ratio this is why she required more hp than Washington to make about the same speed. 130,000 for SD, 115,000 for Washington.
Iowa was close to 8:1 at 860 x 108
Hood was similar at 850 x 104
This is really simplified, because I don't have the time to get into the whole discussion.
There are 2 types of vessels out there. One type is called a target. If it isn't capable of silently doing 30+ knots at 2000 ft depth its always considered a target. The vessel that can silently go fast and deep is the one the targets are afraid of.
There are a lot of design features that could be used to get more speed or specifically "more efficiency" but they aren't present in the Bismarck design. Bismarck has a mild bulb in the bow, but not an efficient one like the Yamato, the transom stern is not present either. Having those you would see more efficiency so less power at high speed.
The statement longer means faster is not allways true. You have to consider the fineness of the hull when you make the statement. Longer is faster works if you say "for a given displacement the longer hull should be faster". That is pretty much a true statement, and it matches my comparison of the Washington and South Dakota classes, same displacement and width, but higher speed with less shp for the longer Washington class.
There are 2 types of vessels out there. One type is called a target. If it isn't capable of silently doing 30+ knots at 2000 ft depth its always considered a target. The vessel that can silently go fast and deep is the one the targets are afraid of.
The frictional resistance is taken as a frictional coefficient (C_f) times the wetted surface (and half the density times velocity squared, called the dynamic pressure). This coefficient is weakly dependent on speed, so the total is more or less dependent of velocity squared. As resistance times velocity is the (effective) power, this power depends on the velocity cubed. This is the relation people are somewhat aware of. It’s basically the equivalent of the resistance of a flat plate being towed. Of course, a ship isn’t a flat plate, it has a form too.
The form resistance is a small fraction of the frictional resistance and can be taken quite constant (form factor k). Form and frictional resistance now equal (1+k)C_f. A standard formula can be taken for C_f, accepted internationally (called the ITTC-57 skin friction rule; taken from testing the resistance of flat plates having theoretically only frictional resistance. There are of course many variations according to the experience of modeling stations).
Now, the wave resistance coefficient (C_w) is not constant for various speeds and highly irregular and also starts off with a much higher power factor. But when you measure resistance on a model, you measure everything. This means you do not know k before hand and thus you do not know how much is wave resistance and how much is form drag in the total resistance. You *do* know the frictional resistance is 100% dominating when the ship is going very very slowy. Unfortunately, at very very low speeds you measure nearly nothing! And what you measure are the effects of a flow without turbulence so cannot be used to extrapolate to full scale. (Many amateurs do not know enough about the scaling of flows and claim to measure very high efficiencies for all kinds of hulls, propellers, wind mills, wave energy generations and so on. They cannot be convinced the efficiency of their inventions are actually waaaay worse when scaled up. There are exceptions, but we have to disappointed and disgruntled inventor from time to time. Wave energy is a hot topic but doomed to failure in all cases as inventors do not understand waves!)
Anyway, how do you know what your form factor is? Increase your speed to negate the scale effects somewhat and you get waves.
Fortunately, you can look at the ratio between various resistance parts and see what the ratio does. One guy called Prohaska thought up the devided this C_w by V^4, and you get a sort of linear relation with C_f. He said that the length of the wave coming from a ship depends on its velocity V. The energy in a wave depends on the waveheight h squared and h depends of V squared. This means already you have V to the power 4. Waves of different lengths depend on h squared, but longer waves contain more energy by their being longer. And as this depends on V as well, you now have a V to the power 5. Add to this that a wave has a width as well, and there is V to the power 6. So intially, wave resistance may go up to the power 6.
This means C_w scales with V^4 as C_w is wave resistance dv]ided by the dynamic pressure containing V squared. This method is of course quite dangerous if you have transom sterns or bulbous bows and the power 6 might then not work (resulting in an error in k and then you screw up your estimation of the resistance at full speed).
As you now know how wave resistance scaled at lower speeds, you can determine its relation for several points. Then you can determine this k, but it’s getting a bit detailed.
As your ship speed increases, waves coming from various points of the hull start to mix. This makes the wave resistance a sketchy and irregular pattern for most of its speed range. But at very high speeds, your dominant wave from bow and stern will interact most unfavourably. This is called hump speed. If you have either a very very slender hull such as a catamaran, or a planing hull as a speed boat, you can overcome this resistance. In fact, next time you are on board a large catamaran, try to stand near the rear. You can actually see the waves behind the ship to grow very large, and then reduce in size. The catamaran can overcome this hump in resistance and then sail at a lower resistance! Planing ships try to build up dynamic pressure by deflecting water downward and are a different typ eof hull (A water plane also has to overcome this wave hump, otherwise it will never get into the air).
Back to displacement ships. Unfortunately, battleships and battlecruisers are not very slender compared to say, a destroyer. Their wave resistance may not have a hump but continue to go up up up. The resistance may go up by powers of 5,6.. Or 10. This is the reason those ships have such an insane amount of power installed. The next knot may double your required engine power. This is also why of you loose an engine, your drop in speed is minor.
Stuart Slade has a few excellent hydro-articles at warships1.com. In on article he comments that Hipper’s ship without the center shaft only gains 1 knot. Slade concludes the center shaft is not very efficient. Although I really like his articles, this conclusion I strongly do not agree with. This is another discussion, but I think center shafts are good, efficiency wise.
