fliud flow mechanics?

Hi

My fluid fow study has been in aero, not marine. In aero, we have the lift equation:

L = 1/2 x p x V^2 x S x Cl

I seem to recall at the time of study that the term 1/2 was due to the compressability of air. So my question is that in relation to foils, c-boards and rudders, is this term still 1/2 or does it become one due to the incompressability of water?


Regards,
Ken

 

Ken,

You're right about the formula adjusting for incompressible fluids, but I don't recall exactly what the factor is. You might check with the International Hydrofoil Society (http://www.foils.org). That's where the REAL foils geeks hang out...

 

Gas compression only becomes significant as you approach sonic speeds. Even in a sonic or supersonic formula, wouldn't compressability be more complicated than a simple fraction?

 

Thanks Deering, I'll check that out.

Skippy the 1/2 term is standard in subsonic formula. Low Re as for model gliders and sailing is the area I am interested in so I don't know what's used in supersonic applications. It may be noted however that if one is consistant with this term, using the same term to derive coefficients from tunnel tests, then whatever term is used simply gets reflected in the derived coefficient values.

The term 1/2 gives a drag coefficient of 1 for an open end tube and this may well be the original basis of this term being adopted as the standard.


Regards,
Ken

 

The 1/2 originates from the formula for kinetic energy Wk= 1/2 * m*V^2. The formula is derived from some basic assumtions like incompressibility of the fluid.
(The unit calculation gives a hint 1/2*p*V^2 / 2 gives Energy/Volume. Multiply with Area and you get Energy/length and that equals force.)

Anders M

 

Lift and drag will both change due to compressibilty, but this is taken care of in Cl and Cd. With CFD codes, it is common either to output pressures directly, or to output Cls/Cds on the surface for each element.

The difficult questions are what happens with cavitation, and what happens when things aren't rigid.

Tim B.

 

Thanks for that,

Then the 1/2 term is valid for hydro dynamics?

Regards,
Ken

 

Should the coefficients be taken care of by just using the right values for the appropriate reynolds number or do we have 2 sets of coefficients depending on compressability?

CFD codes?

Ken

 

It depends on how you're doing your sums. At low speeds both water and air are as near to incompressible as makes no difference, but the cl and cd values will change as you increase mach number (for given angle of attack and reynolds number). I have just run an example in X-foil (the results are a little unrealistic but it proves the point): Setup as follows-

NACA 0009 - 189 pts
Re 1.5 million
alpha=1

Obviously to do a realistic example, I should have changed Re as well. you are highly unlikely to get to mach 0.8 with an Re of 1.5 million. remember that V=mach no. * 330 (m/s) in air.

Results are attached in a PDF file

Once we have the Cl and Cd, we just use 1/2*rho*V^2*Area as before. The effect of mach number (ie. compressibility) is taken into account in the Cl and Cd.

Tim B.

 

Yes.

You have to realize that the (1/2 rho V^2) bit is really a convention. The coefficient is defined as what's left over after you divide the actual force by the planform area and (1/2 rho V^2). The coefficients for hydrodynamics, subsonic aerodynamics, and supersonic aerodynamics all use the same convention. So the fact that "dynamic pressure" = (1/2 rho V^2) has nothing to do with compressibility.

The convention hasn't always been universally adopted. In the early days of aviation, there was a lot of disagreement as to whether the 1/2 should be there or if it should effectively be included in the coefficient. IIRC, the European engineers tended not to show the 1/2 and the American engineers did.

I suspect the latter convention won out because it's measurable. At low speeds, the difference between the total pressure and static atmospheric pressure is approximately (1/2 rho V^2). So you could measure it directly with a differential pressure guage hooked up to a pitot probe and a static port. That made it a convenient quantity to use in non-dimensionalizing test data to coefficient form.

However, here's where compressibility comes in. The pressure difference between total pressure and staic pressure - what we now call "impact pressure", qc - isn't really (1/2 rho V^2). The subsonic formula for impact pressure is:

qc = Pa * {[1 + 0.2 * (V/a)^2]^3.5 - 1}
qc = impact pressure
Pa = static pressure
a = speed of sound
V = fluid velocity

(V/a) is also known as Mach number. For water, the speed of sound is very high and the difference between dynamic pressure and impact pressure is negligible.

Would it be better to use impact pressure to define the coefficients instead of dynamic pressure? Maybe. But the convention was established before airplanes started to fly fast enough for it to matter. (Impact pressure is commonly used to determine position error coefficients when calibrating aircraft pitot-static systems precisely because it is what you measure - see the USAF Test Pilot School text)

BTW, the use of planform area is also a convention. It could be anything that has units of length^2. And things like including the wing area inside the fusealge in the reference area instead of just the exposed area is also conventinal practice.

So whenever you see a non-dimensional coefficient, be aware that it's just part of a force-moment bookkeeping system, and you need to know what the rules are of that bookkeeping system are. A good example in boats is the residuary resistance.

Residuary resistance is the hull drag left over after the skin friction drag is subtracted out. When you subract out the wave drag (and the lift-induced drag if any) from the residuary resistance, you have the form drag. And by convention, the skin friction drag is figured using the wetted area and the ITTC friction line. The ITTC is the International Towing Tank Comittee, a group of folk that get together periodically to exchange lessons learned and to settle on conventions like just how to calculate residuary resistance so the data from one tank can be compared to the data from another tank. It doesn't mean the ITTC friction line is the most accurate for a given model - it's pretty good for a lot of different ones, but you might be able to estimate the skin friction more accurately.

It's all whay they call fluid dynamics, "The science of the non-constant constant."