A few years ago, Serge Krauss mentioned that a more meaningful version of the Tail Volume Coefficient formula is:

Tail Moment measured in MACs * Ratio of Tail Area to Wing Area

The above is, of course, the exact algebraic equivalent to the more widely published version – which looks like gibberish (to me, anyway).
   One advantage of this version is you don’t need to memorize it or look it up. It is simply the familiar (to most every modeler) ratio of tail area to wing area with a simple tail moment leverage factor. We can immediately see, for example, that a ballpark TVC using average chord instead of MAC won’t be too far off the mark.
   We can see immediately see TVC is scalable, as the tail moment is in wing chords, not inches or centimeters. We can see it is a dimensionless number as both factors are ratios.
   The above format of the TVC formula gives us a pretty clear picture of what TVC does and does not describe about an aircraft. I think we should sh*tcan the other version.

Stability vs Damping

Do most people associate TVC with stability? However, isn’t it really damping? Or more like damping? The difference is not just nit-picky semantics, my opinion.
   Let’s do grade school level stability and damping descriptions (that is, describe them at my understanding level).
   Stable: A marble at the bottom of a bowl, if moved and released, will eventually come to rest again at the bottom of the bowl.
   Unstable: Turn the bowl upside down, and displace the marble. It will move even further away from the center of the bowl.
   Neutral Stability: Place the marble in the center of  a cookie sheet, and displace the marble. It does not have a tendency to move back towards the center or away from the center.
   Damped Motion: Pour syrup over each of the above

Suppose model Ding has a TVC of .35 and the CG is ½” in front of the wing 25% MAC point. We build a clone of the model, Dong, with 1” longer tail moment, but put the CG ½” behind the wing 25% MAC point. Dong also has a TVC of .35.
   Ding and Dong have equal TVCs. Do Ding and Dong have equal pitch stability?
   In the first Ding Dong example, we calculated tail moment (in wing chords) from the CG. Some prefer using the 25% MAC as the wing end of the tail moment measurement. If we build a new Ding and a new Dong with a lead ball mounted on an adjustable jackscrew, we can do the same mind experiment and reach the same conclusion – TVC is not a predictor of pitch stability, whichever TVC calculation you use.

At the magazine-article-level-of-aeronautical-knowledge level, TVC is a descriptor akin to syrup viscosity in the marble-in-a-bowl example. Using TVC as a stand alone descriptor (not as a coefficient), TVC is useful only as a comparative number. Knowing the TVC of one airplane doesn’t tell us much – we need TVCs of several airplanes, and how they fly, to make much practical use of TVC. Knowing the actual viscosity of our syrup is less important than knowing how much thinner or thicker it is than other syrups.
   I don’t see any clues in the raw TVC calculations as to what sorts of amplitudes or frequencies it might be better at damping. It probably dampens flutters better that phugoids, because syrup does.
   Does TVC tell us anything about potential maneuverability? Maybe, if we massage it a little. Whaddya think?

Include C_L in TVC?

Using the above formula, there is a mathematical assumption built in – that the relative lifting potential of both the wing and the tailplane are the same per square unit of area. Yet they have different C_Ls typically.
   In the “neutral point” formula this oversight is corrected with a second multiplier – the ratio of horizontal tailplane and wing lift curves. I’ve never seen a stunt ship where the tail was a miniaturized wing, so on the surface it seems a sensible correction. (There are also Reynolds number differences also – No, let’s not go there).
   We could easily modify TVC for our use by including the C_L in the numerator and denominator of the area ratios. We immediately run into a problem – we need to assume some AoA(s) to put in the “correct” C_L  . The whole calculation gets more and more speculative, and less and less real world.
   Let’s say we wanted to look at maneuver entries and exits using this approach. The calculation gets pretty wild at low AoAs, as the C_L of any airfoil at a very small AoA is very small. At one degree AoA, all airfoils are pretty much equal. Dividing one small number by another small number gives absurd ratios at times. (If we do that, it is no longer TVC, and we need a new name for the function.)
   For now, I’m treating this as a dead end approach. Has anyone done anything similar?
   
Will the Real TVC Please Stand Up?

Another TVC question – should the tail moment be measured from the CG or the wing 25% MAC? My answer is “Yes ”. Here’s why:
   I like to think of the CG TVC as the “trimmed” TVC. In flight, any aircraft rotates around the honest-to-gosh CG, not some theoretical point on a drawing. If we add nose weight, we always increase TVC (by lengthening the tail moment). All of us have read Ted Fancher’s explanation(s) of how and why models with larger TVCs and more rearward CGs do better in gusty conditions.
   The more theoretical TVC, calculated from the wing 25% MAC is useful in a different way. Let’s call that MAC TVC. (We can never have too many acronyms). We can compare TVC values among various designs without the “corruption” of trimming preferences the designer has incorporated into the CG location. MAC TVC “levels the playing field” in making design comparisons. MAC TVC gives a truer picture of the differences among Classic era ships, and a truer picture of how the Classics differ from more modern designs.
   Having both TVCs in our tool kit makes it easy to understand some things – such as Ted Fancher’s explanation of why modern ships fly easier in gusty winds mentioned above. A model with CG TVC = MAC TVC is a windy weather airplane.
   Rather than debate which is the “real” TVC, I vote for using both CG TVC and MAC TVC – but knowing the difference. I prefer MAC TVC in forum posts since it is the more unbiased comparator of the two.
   
A Rose, by Any Other Name, . . .

The Volume of what? When someone says “tail volume”, I picture dunking the tail in water then measuring (preferably calculating) what volume of water is displaced. In one version of the formula, we multiply wing area by average wing chord, which hints of three dimensions, or at least a power of three cousin of sorts. But, as we (or any high school freshman) can see, that is only because of a clumsy way of writing the formula. The version at the start of this post doesn’t look cubic or anything other than planar. It is a planar leverage dimension times an area, pure and simple.
   Suppose we read a comic strip and measure a panel, getting the area of the panel. We measure the distance of our eye to the comic strip and multiply that times the area. We now have the Comic Strip Volume Coefficient.
   Truth is, we are doing a leverage calculation. We don’t know the forces we are levering, so we use tailplane and wing areas as stand-ins for the forces. The areas are only there because they are roughly proportional to the forces.
   TVC can be calculated from a silhouette of a model – two dimensional to the core. An Impact with a 12% airfoil and ¼” sheet stab and elevator would have the same TVC as the more popular versions flying today. A silhouette is a two dimensional representation of a three dimensional object. We cannot “add back in” a third dimension using only planar measurements taken on the same plane as the others.
   Ah well, engineers aren’t judged by their mastery of naming things, but by their skill at doing things. However, we shouldn’t be misled by the name into thinking we are calculating something we are not.

       Larry Fulwider

This was all pretty much figured out 70 years ago.  It's in textbooks.  There is a range of rigor that people use to look at this stuff.  Tail volume coefficient is on the crude side.  . . .

. . .

One of the reasons I thought this was worth posting was to show that you couldn't do much more with TVC than what has already been done. At least as far as I could see. I think it is a handy comparator, but limited. I don't know that I would call TVC crude, just limited pretty much to what I said in the opener.
        The other reason was to see if I was missing something. That TVC had some analytical value that I could not see.

       Larry Fulwider

I have a 3/16" foam board "model" of a proposed model, where wings and tail surfaces are not particularly distinguishable. I know from reading about neutral point computation that just what has been mentioned is necessary: lift curve slopes, down-wash angles, etc. So I concluded much as Howard that I was getting into the "crude" zone, especially at these RN's. So I did a crude version of the "cruder" stuff and just assumed a certain percent efficiency for aft surfaces and weighted MAC 1/4 points by area and this approximate efficiency. Bill Netzeband is playing with this shape in his own way, but we have started with the same shape. My predicted a.c. locations came out about the same using two different methods, but now I am ready to find the neutral point (Edit: by experiment) and try to predict a usable c.g. placement. I've set my initial prediction based on the approximations mentioned above (mostly relying on Simons figures) and want to see how close they come with different "tail" positions. I think Bill will try to get an approximate c.g. for a glide, but I'd really like to find the actual N.P. first.

Well, it's so approximate now that I hesitate to even mention what I'll try, if I can sneek into a basketball court this winter. I know that any up elevator will change everything (speed, c.g.) for a good glide, but that actually finding the NP without giving "up" elevator implies finding the c.g. position for the least disasterous non-glide (neutrally stable). However, if the aft surfaces are less efficient, then there should already be some de-stabilizing effect calling for a forward movement of the c.g., which might allow a glide. That c.g. should be awfully close to the NP. Then we'll see whether the static margins predicted by "Ted's Rule" ( tail area ratio equalling the c.g. placement along the MAC) in on-line calculators really works. Anyway, I'm looking for that point between pitch-up and dive with coplanar surfaces. The thing has 2 degrees of dihedral.

I don't know whether you followed that - or even want to - but that's my "plan." It's based on the idea that the actual fancier N.P. computations are as much disturbed by the inaccuracies of prediction as my little experiment is.

'think I can find that wierd little balance point this way?

SK

Thanks for the comments, Larry-

I'm experimenting to see whether I can break up the quote; 'hope this doesn't make a mess...

Gradually moving the CG back and reducing the incidence until only, what, 1 test glide out of 10 stays on track? That would be one data point on Ted’s curve? My opinion, the key is to set a low standard for success for a successful glide at any given setting. As you said, you do want to be right on the edge. I’d imagine you will also be on the edge of testing your patience  ...Your plan is to then change the tailplane area and find another data point on Ted’s curve?...You are not using Simon’s .6 horizontal “tailplane efficiency”, but calculated your own? Or did I misread that?

This is pretty much what I have in mind, except that any coincidence with "Ted's Curve" will be unlikely with this very different plan form. So I'm not really looking to verify or compare. I just cited that previous experiment on-line with the c.g./NP calculator as a source for the apparent need for 15-20+% static margins in CL planes. I thought that I'd like to see whether that turns out to hold for this model too. Hence the need to know the neutral point position. I do want to vary the stabilizer's area and position (if it has a discernable stab) to  see how that affects the N.P. and c.g. I just chose an rfficiency factor that I thought suited the best position, much as Martin Simons suggests, but I went higher than .6.

Another little nicety, that you’ve probably already thought of, is that arctan of the glide angle is ~= to the AoA. You get the tan of the glide angle by simply measuring the distance of the glide and knowing the launch height. In a gliding experiment I helped my son with (Science Fair, many years ago), we were “lucky” in that one constant was that even with all the variables, the lifting surfaces stayed nearly parallel to the floor on successful glides. However, a side photo at any point in the glide would show the angle of the lifting surfaces to the surroundings, and in any case you add or subtract that from the glide angle and find a pretty darned accurate AoA.

This is a nice method that I had not considered. I'll think about that for future reference. There are probably some interesting things that this idea might help in investigating.

Edit: I know you are not particularly interested in AoA here, I was thinking ahead -- critics might say your experiment is at such a low Reynolds you are mimicking a piece of down floating to the floor, and it is not an aerodynamic experiment at all. I'd think an AoA of even 10 degrees would squelch that criticism.Larry Fulwider

Well, I think my "wing" loading will take it out of the feather class. At 3.2 oz for 234 in^2 before at least an ounce of ballast, we're looking at 2.0 - 2.6 oz/ft^2. It will have some speed.

I like your analysis of what I'm doing, and I may well end up using some elevator deflection. Right now, though, it's just the bare minimum (not even streamlined sections), with some accessories.

SK

Lift/drag is proportional to the arctangent of glide angle.  Lift is approximately proportional to angle of attack in the region of angle of attack below stall. 

Implying Serge should Or just an additional analytical tip?

      Larry Fulwider
       

I'm not sure of your point.  Tail volume coefficient is a simple way of looking at the moment capability of a tail or the rate of change of moment with angle of attack.  Moment is a force acting through an arm.  Tail area is a surrogate for force or for rate of change of force with angle of attack.  For a given tail lift coefficient, air density, and velocity, force on the tail is approximately proportional to tail area.  When you compare different tails at the same conditions, the velocity and density cancels out, so what's left has units of area.  Multiplying that area by the moment arm gives units of volume.  We typically nondimensionalize this by dividing by wing area and mean aerodynamic chord.  These things are arbitrary, but wing area and MAC are pretty useful for comparisons, so we've pretty much standardized on them.  The practice of nondimensionalizing for comparing wings and stuff goes back at least to the Wright Brothers' wind tunnel tests.  

I have a 3/16" foam board "model" of a proposed model, where wings and tail surfaces are not particularly distinguishable. I know from reading about neutral point computation that just what has been mentioned is necessary: lift curve slopes, down-wash angles, etc. So I concluded much as Howard that I was getting into the "crude" zone, especially at these RN's. So I did a crude version of the "cruder" stuff and just assumed a certain percent efficiency for aft surfaces and weighted MAC 1/4 points by area and this approximate efficiency. Bill Netzeband is playing with this shape in his own way, but we have started with the same shape. My predicted a.c. locations came out about the same using two different methods, but now I am ready to find the neutral point (Edit: by experiment) and try to predict a usable c.g. placement. I've set my initial prediction based on the approximations mentioned above (mostly relying on Simons figures) and want to see how close they come with different "tail" positions. I think Bill will try to get an approximate c.g. for a glide, but I'd really like to find the actual N.P. first.

Well, it's so approximate now that I hesitate to even mention what I'll try, if I can sneek into a basketball court this winter. I know that any up elevator will change everything (speed, c.g.) for a good glide, but that actually finding the NP without giving "up" elevator implies finding the c.g. position for the least disasterous non-glide (neutrally stable). However, if the aft surfaces are less efficient, then there should already be some de-stabilizing effect calling for a forward movement of the c.g., which might allow a glide. That c.g. should be awfully close to the NP. Then we'll see whether the static margins predicted by "Ted's Rule" ( tail area ratio equalling the c.g. placement along the MAC) in on-line calculators really works. Anyway, I'm looking for that point between pitch-up and dive with coplanar surfaces. The thing has 2 degrees of dihedral.

I don't know whether you followed that - or even want to - but that's my "plan." It's based on the idea that the actual fancier N.P. computations are as much disturbed by the inaccuracies of prediction as my little experiment is.

'think I can find that wierd little balance point this way?

SK

I think so, but it might help to have a rudimentary airfoil on the wing so the wing stalling won't befuddle the data.

Implying that arctangent of glide angle is not proportional to angle of attack.  Of course, it depends what you hold constant.  If you control the airplane to hold pitch angle zero, -glide angle = angle of attack.

Serge said, “ . . . Anyway, I'm looking for that point between pitch-up and dive . . .” which is a constant pitch angle.

The pitch angle doesn’t have to be zero, just a constant for any given glide. Since a word is only worth 10^-3 pictures, I added some pictures below.

So, for our data, it is true that:

AoA = glide angle + C (where C is the angle of the wing to the floor)

Your first sentence applies only to glides that are too poor to be entered into the data. Your third sentence is the special case of C = 0.

. . . Tail volume coefficient is a simple way of looking at the moment capability of a tail . . .

Exactly

. . . or the rate of change of moment with angle of attack.  . . .

I don’t see any rates of change or AoAs implied in the calculation, but that’s not an issue

. . .  Moment is a force acting through an arm.  Tail area is a surrogate for force . . . .  . For a given tail lift coefficient, air density, and velocity, force on the tail is approximately proportional to tail area.  When you compare different tails at the same conditions, the velocity and density cancels out, so what's left has units of area. . . .

Exactly

. . .  Multiplying that area by the moment arm gives units of volume.  . . .

Whoa down! Where did that come from? You just said above, “Moment is a force acting through an arm”, but as soon you get the arm, you suddenly decide it stops being a moment and instantly becomes a volume? What kind of logic is that? What happened to the “moment” you were trying to calculate? Did it evaporate?

Are you saying that any time you multiply three dimensions together, you get a volume? Always? Even if the numbers are not dimensions of the object?

Read my examples, and tell me which you believe are volumes:
     Comic Strip Volume Coefficient
     Box hanging by a string, using the length of the string
     Torque Wrench
     Witch Caricature Index

      Larry Fulwider

"I don’t see any rates of change or AoAs implied in the calculation, but that’s not an issue"

I agree with the first, but not the second.  The rate of change of moment with angle of attack is the main point of the exercise.



"Whoa down! Where did that come from? You just said above, “Moment is a force acting through an arm”, but as soon you get the arm, you suddenly decide it stops being a moment and instantly becomes a volume? What kind of logic is that? What happened to the “moment” you were trying to calculate? Did it evaporate?

Are you saying that any time you multiply three dimensions together, you get a volume? Always? Even if the numbers are not dimensions of the object?"

Although I am accustomed to being both paid and respected more when I do flight mechanics calculations, I'll take another cut at it:

First, a definition.  By "moment", I mean the conventional physics and engineering definition, not the stunt definition.

"Tail volume coefficient" is a number you can calculate for one airplane or tail configuration to compare it to another.  It can be used as a measure of:  1) the amount of moment available from the tail to rotate the airplane, and 2) the stabilizing moment from the tail when the airplane is disturbed in pitch.  You guys have been discussing the second of these. Aerodynamic moment due to the tail increases sorta linearly as angle of attack of the tail changes from the equilibrium point.  For number 2) the tail volume coefficient is a simplified way of comparing the rate at which moment changes as angle of attack (alpha) changes among different airplanes or tail configurations.  Consider the following comparison between two airplanes.  

(dM/d(alpha))₁ vs. (dM/d(alpha))₂
Units are distance * force / angle, eg. ft. lb./radian.

If you do the comparison at the same increment of angle of attack away from equilibrium angle of attack, the comparison becomes
M₁ vs. M₂
If (dM/d(alpha))₁ = 3 * (dM/d(alpha))₂, then M₁ =3 * M₂
Units are moment = distance * force.

Now to expand the moments.  Assuming tail lift acts perpendicular to tail moment arm, M = l * L, where l is tail moment arm and L is tail lift.  l is a distance and L is a force.  The comparison becomes
l₁ * L₁ vs. l₂ * L₂
If M₁ =3 * M₂, then l₁ * L₁  = 3 * l₂ * L₂
Units are distance * force.

Now to expand the force.  The tail volume coefficient simplification assumes the same lift curve slope for any shape tail, hence lift on the tail is proportional to velocity squared, air density, tail area, and angle of attack.  Angle of attack was assumed to be the same for both tails, so the comparison becomes
l₁ * V₁² * rho₁ * St₁ vs. l₂ * V₂² * rho₂ * St₂,
where l tail moment arm, a distance,
V is velocity with units distance / time,
rho is air density with units mass / volume = mass / distance cubed,
St is tail area with units distance squared.
If  l₁ * L₁  =3 * l₂ * L₂, then l₁ * V₁² * rho₁ * St₁ = 3 * l₂ * V₂² * rho₂ * St₂
Units are distance squared * mass / time squared.

Generally, you'd want to compare tail configurations or airplanes at the same air density and velocity: people talk about tail volume coefficient, not tail-volume-times-dynamic-pressure coefficient.  So you can assume that V₁² * rho₁ = V₂² * rho₂ .  The comparison becomes
l₁ * St₁ vs. l₂ * St₂
If l₁ * V₁² * rho₁ * St₁ = 3 * l₂ * V₂² * rho₂ * St₂, then
l₁ * St₁ = 3 * l₂ * St₂
Units are distance cubed = volume, hence the name tail volume.  It is a valid, although inadequate, concept, and the units are indeed of volume.  If you are just complaining that calling distance cubed "volume" is a misnomer when it doesn't refer to a space, I suggest that an easier aero nomenclatural target would be "phugoid".

Tail volume is a simplified way of comparing the tail effectiveness of different airplanes the same size or of different tail configurations on the same airplane.  Aeros like to have rules of thumb that apply to airplanes of different sizes, so they nondimensionalize tail volume by dividing by Sw * MAC, where Sw is wing area, with units distance squared, and MAC is mean aerodynamic chord, a distance.  You can think of it as dividing tail area by wing area and tail moment arm by MAC, hence tail area is expressed as a fraction of wing area and tail moment arm is expressed as a fraction (probably > 1) of MAC.  The comparison then becomes
(l₁ * St₁) / (MAC₁ * Sw₁) vs. (l₂ * St₂ ) / (MAC₂ * Sw₂)
Ain't no units; it's nondimensional.  This is the tail volume coefficient.  


"Read my examples, and tell me which you believe are volumes:
     Comic Strip Volume Coefficient
     Box hanging by a string, using the length of the string
     Torque Wrench
     Witch Caricature Index"

No, thanks, but you can explain to me why 50-ohm antenna cable is 50 ohms, regardless of length.  

Howard,

THANKS! IMHO, you nailed it with the the comment, buried in your equation-heavy post (#10?), that any result of the multiplication of three terms in units of length will be a term with units of length, cubed.

Cube, to most of us, without any other reference, suggests a rectilinear solid 3-D (solid geometric) figure with all sides of equal length. A "cube" has 'volume' associated with the length of any side. Tail Volume, the product of three length-unit terms, a length-cubed term, got confusing. ...An Emily Litella moment: "Never mind," (or alternately: "Phugoid about it...")

You got the relationship right among pitch angle (C, you called it), glide angle, and angle of attack.  I think what you meant was that for still air, nothing happening sideways, and steady conditions, pitch angle is small, which is a coincidence.  Saying that glide angle is ~= to the angle of attack implies that tangent of angle of attack ~= drag / lift.  

Howard –

I don’t know if it implies that or not. If you say it does, it does.

You and I don’t see eye-to-eye on two issues, neither of which are disagreements on the aerodynamics. Your knowledge of aerodynamics is far superior to mine, and most everyone else’s on this forum. Your ability to apply that knowledge to practical problems is well known.

My statement was a practical application of Euclid’s interior angles theorem as a handy data collection tool in the experiment Serge described. You (originally) said my statement was an error.

There is no “coincidence” involved, as the experiment produces data only when we get straight glides in still air. That is in the description of the experiment. What I meant to say was exactly what I said.

The application of the theorem is illustrated below.

     Larry Fulwider

I like xkcd, too.  That's a particularly good one.

. . .
3. I had to look above to see what the "simplified formula" is.  I take it that you mean ratio of tail area to wing area.  Tossing in tail length / MAC does tell you more, but it's still not enough, I think, to be of a lot of use in designing or tweaking a stunt plane.
 . . .

No, this one:

Tail Moment measured in MACs * Ratio of Tail Area to Wing Area

(the tail moment is already tossed in)

Ed,

Have you been feeding your kitty too many sugar mice?      

Jim Pollock,