Higher Cp at higher speeds equates to less resistance?

Discussion in 'Hydrodynamics and Aerodynamics' started by dustman, Dec 1, 2020.

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dustmanSenior Member

I keep reading this all over but it doesn't compute. I would think the more gradual the transition is for the water going around the hull the less form and wave drag there would be, which seems important at higher speeds. Is it because a hull with fuller ends has less wetted surface area? Doesn't form and wave drag increase with speed at a greater rate than frictional resistance?

Can someone enlighten me?

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Mr EfficiencySenior Member

Skin friction is the speed killer in faster boats, increasing with the square of the speed, roughly, that can't be underestimated, obviously low block co-efficient tends to high surface area per unit of displacement.

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TANSLSenior Member

@dustman, I don't know if you are referring to Cp, prismatic coefficient or Cb, block coefficient. As you have specifically mentioned the Cp, I am going to talk about him. Analyzing the formulas of the Holtrop-Mennen method we can see that Cp is involved in frictional resistance but also in wave-making and wave-breaking resistance. It intervenes by adding, subtracting or dividing other factors. Therefore, it is not easy to know how its value affects the total resistance. Intuitively, you would think that a lower value of Cp would give rise to a finer hull, with lower resistance, but I would not ensure anything

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tlouth7Senior Member

I assume you are talking about high displacement speeds, approaching hull speed (yes I know it isn't real). In this regime wavemaking drag will tend to dominate. Consider a boat with fine ends (low Cp), sitting in the classic trough between bow and stern wave. It will have most of its submerged volume in the middle, right in the trough. In contrast a boat with high Cp will have more of its volume in the bow and stern, where the peaks of the waves are. Thus a boat with higher Cp will float higher (sort of) in this situation, resulting in lower drag. Actually I think it is better to think of those full ends pushing the waves further apart, which has the effect of increasing the effective hull speed of the boat. Thus the boat is operating at a lower effective speed/length ratio, which can result in lower drag.

Of course this comes with a penalty of less fine entry and exit, resulting in higher drag at lower speeds. Hence there is a balance to be struck, and the exact position of that balance depends of the proportion of hull speed that the designer is optimising for.

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DCockeySenior Member

From Principles of Naval Architecture Volume II 1939/1947, pp 93-94:

3. Major Effect of Prismatic Coefficient on Residual Resistance. Fig. 31, in which the curves of area of some of the forms in Fig. 29 are drawn to scale, emphasizes the rapid blunting of the ends which accompanies increase of prismatic coefficient, and which explains the very marked effect of this variable on the residual resistance. In general, the blunting increases the separate wave-making tendencies of the ends, particularly that of the fore body. At the same time, however it forces the waves nearer to the extremities. Thus, while corresponding humps are increased in magnitude, the natural wave-making lengths of the ship are increased also. ....

.....

It will be noted that an increase in prismatic coefficient sometimes causes a reduction of residual resistance. A relatively high speed-length ratios, where the resistance curves shoot upward very steeply, the natural wave-making lengths are much more important in fixing the relative portions of the curves than the magnitudes of the natural wave-making resistance of the ends.

Note that the discussion above relates to wave-making resistance which is only one component of resistance. Viscous skin friction is another important component which is affected by the overall shape of the hull.

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dustmanSenior Member

Thanks guys, that makes sense.

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fastsailingSenior Member

But ignores one thing.
Take a look at:
1) how resistance formulas are made for cars, submarines, etc, for equipment operating in a single fluid, not boundary of two of them like boats.
2) how prismatic coefficient is defined, it has nothing to do how full the ends are. It is strictly defined by the volume, length and cross sectional area at the largest part.

From 1, do you realize a car can have less air resistance at same speed than other car, even if it has larger Cd, if it also has less cross sectional area?
From 2, what happens to maximum cross sectional area when length and displacement volume is kept constant, but Cp is increased?
Combine the results, and see the reasoning you won't find in the text book, although it is obvious.

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DCockeySenior Member

Cars, submarines and other vehicles operating in a single fluid, not at the boundary of two fluid, do not have any wave making resistance. Wave making resistance can be signficicant and even dominant for boats depending on the speed.

Correct that the definition of prismatic coefficient does not include anything about how much displacement is in the ends. However prismatic coefficient (Cp) is the ratio of the average cross sectional area to the maximum cross sectional area. (Derivation below) A boat with a higher Cp will tend to have displacement distributed more evenly along the entire length of the boat while a lower Cp will have displacement more concentrated close to the area of maximum cross sectional area. (This assumes a normal distribution of cross sectional area with the cross sectional area staying constant or decreasing with distance from the area of maximum cross sectional area.)

Yes, it is possible for a car or submarine with larger maximum cross sectional area to have lower resistance.

It is also possible and entirely consistant with the Cp guidance in text books for a boat with larger maximum cross sectional area to have lower resistance at certain speeds than a boat with with smaller cross sectional area. Assuming two boats have the same displacement and length and similar hull shapes, the boat with lower Cp will have a larger maximum cross sectional area than a boat with a higher Cp. If the Cp of the second boat is higher than optimum for the speed being considered te boat with the larger maximum cross sectional area and lower Cp may have lower resistance.

Derivation of Prismatic Coefficient equals Average cross sectional area / maximum cross sectional area

Cp - Prismatic coefficient
Vol - Displacement as volume
LWL - Waterline length
Amax - Maximum cross sectional area
A(x) - Cross sectional area at station x
Aavg - Average cross sectional area

Cp = Vol / (LWL x Amax)

Vol = Integral A(x) over waterline length

Aavg = (Integral A(x) over waterline length) / LWL = Vol/LWL

Aavg / Amax = (Vol/LWL) / Amax = Vol / (LWL x Amax) = Cp

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DogCavalryI aim to misbehave.

Now I read DCockey explain fundamental concepts of hydrodynamics and I'm much happier!

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DCockeySenior Member

Apparently it doesn't take much to make you happy.

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Or how grim the news is.☹

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fastsailingSenior Member

You are moving the goalposts.
The implied question was: why does higher Cp equate to less resistance?
It was not if there are cases when that does not hold true. Therefore the discussion is about the case when that does hold true.
The answer has everything to do with cross sectional area. And is closely related to how much water needs to be pumped out of the way in a given time interwall at a given boatspeed. Wavemaking resistance is dependent on that, but not only on that.

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DogCavalryI aim to misbehave.

And even though it's irrelevant, the prismatic coeficient of a car is bigger than anything other than a barge. A submarine is also very very high.

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DCockeySenior Member

From two earlier posts:

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DogCavalryI aim to misbehave.

See? I'm already more relaxed. Of course I'm drilling holes in a foundation for rebar with a huge hammerdrill, before I pour the slab, so I'm already pretty relaxed.

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