# Hull Asymmetry and Minimum Wave Drag

Discussion in 'Hydrodynamics and Aerodynamics' started by DCockey, May 28, 2011.

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### Remmlingerengineer

This is exactly right. Years ago I taught a course in the Uni-lab about supersonic flight. We used a circulating water channel where a water-film of a few millimeters depth ran down a glass plate 3x2 meters. Below the plate there was a spot-light that projected the wave pattern on to a screen above the channel. The resulting pictures of the aircraft models on the glass plate were very similar to the Schlieren-pictures form the supersonic wind tunnel. Mathematically the differential equations in both cases are actually the same.
Uli

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### DCockeySenior Member

Thanks for the clarification.

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### philSweetSenior Member

Well, its a pretty fuzzy idea at the moment. This is what I'm muddling with. Water is modelled as an incompressible fluid and a gravitational free surface. Air is compressible, that's why we need the extra nondimensional Mach number to handle the extra variable. Lets look a 2D horizontal region in the water near a free surface. In 2D, the region is compressible because the water can escape upwards displacing all the water above it up to the surface. We can extend the idea to 3d by considering vertical pipes of water that include the free surface. The closer you get to the surface, the easier it is to compress the region. Thus I think an additional nondimensional, but oriented variable could be constructed that works similarly to Mach. It characterizes the response of a horizontal region of water as a function of the ratio of static pressure at depth D to stagnation pressure at depth D. The idea is to replace the free surface with this extra item and solve a symmetrical case of vessel traveling though a medium which has a gradient with respect to this peculiar compressibility. I'm playing fast an loose with some boundary conditions and I'm tossing out some momentum terms, but so do other methods.

As far as relating pressure waves to gravity waves in this manner, I don't know. Would a breaking gravity wave be analogous to a shockwave? It is related to steepness, thus a vertical gradient issue. That might help to describe a mock mach 1 condition.

Ha! once in a great while my intuition is right, or at least agrees with wiki.-

from here-http://en.wikipedia.org/wiki/Wake

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### philSweetSenior Member

So now I think I can characterize this asymmetry a bit. What is missing from thinship theory that will produce asymmetry is the idea of entropy increasing along the axis of the hull. The rate of entropy increase depends on the geometry, but also on the amount of free energy remaining to draw on. So there is a logrithmic decay to radiated energy and the change in entropy and stagnation pressure along the length of the hull even if the geometry suggests a constant energy loss. And I think the details are strongly dependant on depth.

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### Leo LazauskasSenior Member

If you are interested in minimising wave resistance, then you should look at Ward's
Optimum Symmetric Ships. (I have posted photos in several threads). They are a lot
lumpier than the Sears-Haack body.

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### DCockeySenior Member

Your use of compressible in your last sentence is incorrect based on the usage of the term and its implications in any standard version of fluid mechanics. "Compressible" means that the density of the fluid changes sufficiently due to pressure varations to have a signficant affect on the flow. The variation in density of water due to pressure variations in any flow around a boat or ship is negligably small.

The second sentence if fundamentally incorrect given the standard use of "compress", etc.
What equations are you using? What definition of compressibile are you using?
A hydraulic jump from supercritical, shallow water flow to subcritical flow is analogous to a shock wave, a breaking gravity wave is not (in general) analogus to a shock wave except in a very loose manner.

A good example of why Wikipedia is not a authoritative source. As discussed in an earlier post free surface waves in general are only in a very loose sense analogous to shock waves. Also the wikipedia entry is mixing and confusing "waves" and "wakes".

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Agreed.

Wiki is NOT an authoritative source for anything, other than just "general" and "basic" stuff. Not always correct in that regard too.

If anyone wishes to cite an article from Wiki as their premise for 'something'....doesn't bode well.

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### Leo LazauskasSenior Member

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### philSweetSenior Member

Okay, I've been able to spend a little time collecting my thoughts. The mock Mach number I was searching for turns out to be the depth Froude number. Depth Froude number is analogous to Mach in the following way- Both identify flow regimes which have discontinuities with respect to entropy. In liquid flow, the discontinuity involves vorticity and takes up a lot more space than the nonreversible action in a compressible gas shock wave. Take a look at what Remmlinger said. How did they adjust the flow of water to mimic a specific mach number? is there an rule for that?
So in the case of thinship theory applied to surface ships, it works, and Michell's integral works, for waterlines deeper than the critical depth of Fr=1. For shallower waterlines than this, this situation is that of an aircraft that is supersonic, and the aero folks have already done the heavy lifting. If my memory serves, Michell's integral involves creating a projection of the ship onto the centerplane. For shallow waterlines, the hull needs to be projected along a forward angled line as in the aero case.

<< deleted a sentence>>

The two numbers happen to be set up as inverses of one another. Criticality occurs above M=1 and below Fr=1, but that hardly matters.

In defense of wiki, kinda, I find that topics that have had stable classical solutions for decades prior to the web, which includes vector field math and the the equasions governing fliud flow, waves, etc. are done done quite well. At any rate, it was offered as support, nothing more. And I think the discussion since has done quite a lot to support the statement I highlighted and nothing to call it into question.

I still find it strange that this thread of DCockey's has had such wide interest for so long and yet no one had mentioned that there was a well known case where thinship theory produced a minimum drag body with fore/aft asymmetry.

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### DCockeySenior Member

Phil, it sound like you are trying to treat the 3D flow as a series of 2D slices. That approach is occasionaly useful provided the interaction between the layers is properly accounted for and all the layers are solved for together. Classic lifting line wing theory is an example where such an approach can be useful.

The equations Remmlinger was referring to are for flow in open channels. This is a topic of considerable interest for civil engineers designing dams, flood control structures, and and similar. (At the other end of the size spectrum I used it for analysis of windshield wiper performance though in that case I had to also include the effects of surface tension.) The transition from subcritical to supercritical channel flow occurs at a Froude number based on depth of 1. Fd less than 1 and the flow is subcritical, Fd greater than 1 and the flow is supercritical. Fd, depth Froude Number is based on the total depth of the water from the free surface to the bottom, not the depth of a particular "slice" beneath the surface. And the bottom needs to be solid without flow through it.

It is not valid to consider the flow for an arbitrary shallow layer near the surface of water of considerably deeper depth to be equivalent to the flow over a shallow, solid bottom. The fundamental difference is the no flow goes though the shallow, solid bottom but flow definately goes though the imaginary "bottom" of an arbitrary shallow layer near the surface of water of considerably deeper depth. This difference in boundary conditions fundamentally changes the physics.

Thinship theory is fundamentally different than the slender body theory used for the Whitcombe area rule, Sears-Haack body, etc. Superficially they appear similar but there are fundamental differences in the physics and mathematics, assuming depth Froude number is significantly less than 1 for the thin ship theory.

For depth Froude number greater than 1 there are similarities.

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### Leo LazauskasSenior Member

If you want more fun with analogies, then X. Chen showed that the problem
of a slender body moving in an ideal wave medium has the same dynamic
properties as Einstein's theory of Special Relativity. See:
Chen, X-N.,
"Hydrodynamics of Wave-Making in Shallow Water",
PhD Thesis, Uni. of Stuttgart, 1999.

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### philSweetSenior Member

Yep, this is a serious concern alright. Or perhaps a better way to say this is that this is the problem with the current solution that we need to repair. It is handleable. As you optimize for minimum wake, the discontinuity becomes smaller and its behavior smoother. There is a singularity at the surface that has to be patched. In practice, the grid construction would handle it. I think the end result of an unrestrained optimization would have the same numerical wake drag result as what Michlet produces now , just a differently shaped upper hull. There are some minor changes. The length of a waterline above the critical depth would trend longer. There would be some stretch down into the lower part of the hull.

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### philSweetSenior Member

Thanks Leo. I'll have that for dessert.

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### Leo LazauskasSenior Member

I'm not convinced of that.
Ward's Optimum Symmetric Ship is the result of a mathematical minimization
of Michell's Integral with a constraint on fixed volume. The only reason the
public release of Michlet does not find the similar ships to Ward is that I do
not allow as much freedom for the offsets. When I allowed more freedom in
the offsets I got similar bumpy shapes to Ward: at low Froude numbers the
minimum wave drag hulls had large bulbs at the ends; at higher Froude
numbers there are large bumps around midships.

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