# Pitching Moment Test Results

Discussion in 'Hydrodynamics and Aerodynamics' started by PI Design, Oct 2, 2014.

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### PI DesignSenior Member

Hi,

If you measure the pitching moment coefficient of a foil in a wind tunnel do you get a different result to measuring the same foil at the same Reynolds number in water? I had previously assumed that the measured results would lead to the same coefficients, but the added mass is obviously greater in water than air and added mass appears to be the cause of the so-called Munk moment (M33-M11)VxVz.

I have not seen it written as such, but I believe that the measured pitching moment from experiment would be:

Pitching moment measured =Potential Munk moment - Viscous effects moment + camber induced moment + appendage induced moment

In which case, as the added mass, hence Munk moment, is less in air presumably the measured moment is too?

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### gonzoSenior Member

Water is not compressible, so there will be a different center of effort too. I assume that by added mass you mean displaced mass, or am I misunderstanding the question?

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### PI DesignSenior Member

Added mass as in the equivalent mass of fluid moved aside by an an accelerating body, leading to increased inertial forces and moments. Doesn't compressibility only matter at high speeds/Mach number? In my case, I am looking at in-water speeds less than 10kts, so about 70m/s in air for the same Re number - well in the subsonic region.

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### JoakimSenior Member

The only differences I can think of are speed of sound, cavitation, free surface effects and hydrostatic lift.

I have been working with fluid dynamics for 20 years and of course studied them as well. But I don't rememeber ever seeing a term called Munk moment before. According to this it is named after a Zeppelin designer and is caused by the pressure forces due to flow field: http://www.lemos.uni-rostock.de/fileadmin/MSF_Lemos/part_I_neu_.pdf

Certainly it is included in the measured coefficients, which only depend on Re as long as you are well in subsonic region, there is no cavitation and you are far from free surface.

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### PI DesignSenior Member

Hi Joakim,

Yes, the Munk moment is a destabilising moment that a body experiences when at an angle to flow (that is, it is a moment that will tend to make the body more misaligned rather than weathervane into the flow). It exists even in potential flow, unlike forces (D'Alambert paradox). Zeppelins have relatively high Added Mass (as the fluid to body density ratio is high) as do bodies submerged in water, so the Munk moment is significant. Which means, I think, that the measured moment coefficient in water will be different to that in air.

If you analyse a NACA 0012 section at 4 degrees angle of attack in CFD, both in air and water, with the same Re, subsonic, deeply submerged etc, do you get different pitching moment coefficients? It had not previously occurred to me that you would, and I can find no literature on it, but I think actually the moment coefficients in air and water would be different as a result of the differing Munk moments.

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### philSweetSenior Member

PI, can you please describe the planform you are working with, and also give the state vector on the wing. Your nomenclature is pretty confusing to me at this point. If you are just looking at a 2D foil using thin airfoil theory (largely developed by Monk, but we use Glauert's system of Fourier series to evaluate the problem now), then the position of zero moment about the span for a plane wing will have a location along the cord (25% for a flat plate) that won't change for incompressible flow regardless of the fluid. If you reference the lift force somewhere other than this location, then there will be a moment proportional to the lift and to the distance away from the lift reference.

Another point of confusion is that your question asks about the coefficients. The coefficients are normalized by dynamic pressure, so the fluid doesn't matter. When calculating actual forces, you have to multiply the coefficients by the dynamic pressure and any other nondimensionalizing quantities, so that is when the fluid density comes into play.

If you are talking about the change in moments do to an accelerating airfoil or to unsteady flow conditions (this is where added mass is considered), you need to provide details of the situation you are considering.

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### JoakimSenior Member

No. Why would we be using pitching moment coefficients, if they would depend on density as well?

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### PI DesignSenior Member

Well that's kind of my point of confusion!

Phil, I wrote this before Joakim's reply. Hopefully it explains things better.

Hi,
Imagine a simple, symmetric NACA 0012 foil with an aspect ratio of 1 (so square in planform). Set it to have an angle of attack of 0 degrees and move it through a stationary fluid at a steady 3m/s. The body will experience drag and nothing else. Now set the angle of attack to 4 degrees. What happens? If you had an ideal fluid (potential flow, no viscosity, no circulation) there would be no lift and no drag but the foil would experience a moment that would rotate it to an angle of attack of 90 degrees (in fact it would oscillate around 90 degrees due to the lack of drag). The fact that it has rotated from 4 degrees to 90 degrees, rather than going back to zero degrees shows that the moment is a destabilising moment. This moment is called the Munk moment, after the aerospace guru Max Munk who found it to be important in the world of Zeppelins (because without tail fins it means the zep won’t stay pointing forward). This moment exists even in inviscid fluid because even though there can be no lift or drag in such a fluid (see D’Alembert paradox) there is still a moment due to the offset in the upper and lower stagnation point locations (which are equal and opposite, hence no force). So, unlike lift and drag forces, the Munk moment is not a viscous effect but a wholly geometric one.

It can be shown that the (pitch) Munk moment is equal to (A33 – A11)Vsin(alpha)cos(alpha)

where A33 and A11 are the Added Mass of the body in the pure heave and surge directions (generally Added Mass is Aij indicating force in i direction and acceleration in j direction)
V is velocity
alpha is the angle of attack.

Even though Added Mass is usually associated with accelerations (to produce a force), as discussed above you can still have a moment without a force and hence even at a constant speed the Added Mass is creating the Munk moment.

Now, in a real fluid there is viscosity and the affect of that is to reduce the Munk moment because the stagnation points move closer together. The viscosity also is the cause of lift and drag. For a symmetric foil at low angles of attack the lift and drag act at the quarter chord point, so there is no additional moment. For a cambered foil if the lift and drag are applied at aerodynamic centre (as per convention) there is a constant non-zero moment. Tail fins on the back will also create a restoring moment to counter the destabilising Munk moment.

If a water tunnel experiment was conducted and the moment measured, the total measured moment would therefore, I think, be:

Pitching moment measured =Potential Munk moment - Viscous effects moment - camber induced moment - appendage induced moment

And this should be les than zero if the foil is to be stable.

Now, the Munk moment is an Added mass term and Added mass is essentially a fraction of the bodies volume multiplied by the fluid density. So, it is greater in water than air. Consequently if the same foil was tested in air and water and the resulting moment non-dimensionalised by dividing by qSc (dynamic pressure, area and chord) the coefficient of pitching moment would be more positive (less stable) in water than air.

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### PI DesignSenior Member

They depend on viscosity, so why not density too?

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### JoakimSenior Member

No! They depend only on Re.

Your equation for Munk moment is wrong in post #8. It has V^2 not just V. Added mass is equal to added volume * rho, thus Munk moment is proportional to dynamic pressure just like everything else determing Cm.

Measuring Cm in different densities at the same Re will give the same results.

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### philSweetSenior Member

I'm still lost as far as using added mass in this way.

What happens with a foil is that the Kutta condition is applied to the trailing edge and you do have circulation. This directly affects the calculation of the pressures on the foil and the associated moments. If you want to talk about potential flow without the Kutta condition, it would be better to consider slender blunt bodies such as ellipsoids.

But the point seems to have been made - the coefficient does not depend on the density of the ideal fluid since aerodynamic coefficients are nondimentionalized in a way that eliminates density as one of the independent variables. This is done through the dynamic pressure, 1/2*rho*v^2.

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### PI DesignSenior Member

Ah yes I see now, thanks. Of course the Munk component is a function of dynamic pressure and so coefficient wise it is constant. Thanks.

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### jehardimanSenior Member

Huuummmmm, sorta kinda. Reynolds number is based on the kinematic viscosity which is the dynamic viscosity divided by the density. So for boat sized items (say 6 feet) at boat speed Rn's (say 2.5E06) then the Va for the water is ~ 1/10 the Va for air.

Since the secondary pressure effects are only a function of del Va and fluid mass, the same pertubation absolute magnitude has a very different effect between water and air; i.e. the same cross velocity that gives a 4 degree angle of attack in air gives a 34 degree angle of attack in water. This is important with respect to body stability deritive differences between aero and hydro.

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### gonzoSenior Member

I thought the question was about foils acting like active stabilizers do.

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