# Talk:Lift coefficient

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## Scientific Usefulness

This article is not really helpful in defining a function for computing the lift coefficient other than to provide an equation to support measuring it through experiment, i.e., solving for the coefficient of lift using the equation for lift. It is like defining π = A/r2. This formulation would be helpful in experiments to compute π if methods of measuring the area of a circle were available (such as increasing the number of sides of a circumscribed polygon) but it does not provide an independent method to compute π such as the infinite series,

π = 4(1 - 1/3 + 1/5 - 1/7 ...) [Mathematics for Everyone, Laurie Buxton, 1984]

This is what I was looking for when I went to this article, an independent method. Further, working with lift necessarily includes working with angle of attack whereas this article specifically avoids angle of attack. It does, however, provide a nice graphic of a generic coefficient of lift with respect to the angle of attack of some undisclosed object.

I believe that this article could be improved upon significantly by providing typical lift coefficient computational methods for a series of nominal objects (flat plane, Clark-Y airfoil, a symmetric hypersonic wing, and my primary concern, a warhead reentry vehicle of the cylinder-cone-nose radius form) and include the affects of angle of attack. I'm working on the physics of lifting reentry vehicles and a simplified mathematical representation of this but I am not yet prepared to produce an entry about it - after all, at this time I came here to find those relationships, not explain them. If I do find those explanations, I will certainly recommend the improvement modifications I have outlined above.

(Please forgive my lack of knowledge of the Wiki markup language - I am just learning it).

--JEB 20:49, 27 May 2006 (UTC)

Exactly so. It is a framework used to find lift coefficient by experiment or to calculate lift when the coefficient is known. Unfortunately, unlike with finding π, there is no equation or even large collection of equations that can calculate lift independently. There are other simplifications of course for some conditions. In reality, lift is found by wind tunnel experiments or heavy computer work on small elements. Reality being what it is, the lift equation here has been a mainstay in aerodynamics for a very long time; no doubt about its usefulness. Keep in mind that this article is about this coefficient and corresponding equation, and not about lift in general. By the way, that graph for some unspecified object has a vertical scale that is unlikely. Meggar 23:23, 27 May 2006 (UTC)

Thank You, Meggar, for your insightful remarks. However, I tend to disagree about there not being some generic simplified equations with respect to the generation of lift coefficients. According to Van Nostrand's Encyclopedia Sixth Edition (Van Nostrand Reinhold Company Inc., 1983, Supersonic Aerodynamics, page 2731) "The lift coefficient for supersonic airfoils of infinite span at moderate angle of attack and Mach number is,

$C_L=\frac {4\alpha}{\sqrt{M^2 -1}}$

where α is the angle of attack in radians, and M is Mach Number." Using this equation, I have computed the supersonic CL for 6 angles of attack from 5 to 30 degrees and from Mach 1.2 to Mach 10 As presented in the following figure (This presumes, of course, no stall, which may or may not be the case at supersonic speeds. Note that I am concerned with atmospheric reentry and stall has not been an issue - the U.S. Space Shuttle, for instance, reenters the atmosphere with a positive angle of attack of 40 degrees and uses their control surfaces and the lift and drag forces as they are generated to produce their heat dissipating roll program.):

Possible Hypersonic Lift Coefficients

As can be seen, at Mach 1.2 (the first point in each curve), the CL values reach a maximum value of nearly 3.5. This prompts me to ask you why you feel that the "graph for some unspecified object has a vertical scale that is unlikely"? I'd like to know if you feel that it was too large or too small and under what conditions it would be unlikely.

To pursue this further with respect to supersonic drag (as I need both) and staying with Van Nostrand, on the same page the article says, "The drag coefficient of airfoils in supersonic flight is composed of several equally important parts. These are pressure drag, friction drag and drag due to lift."

Pressure Drag: Van Nostrand goes on to explain that pressure drag, also known as wave drag, "is similar to subsonic form drag" and is approximated by,

$C_D=\frac {5.33(t/c)^2}{\sqrt{M^2 -1}}$

which, since I don't like short cuts and unnecessary rounding, I prefer to write as,

$C_D=\frac {16(t/c)^2}{3\sqrt{M^2 -1}}$

where (t/c) is the thickness ratio or thickness to chord. I have a good friend who is also looking at these issues who says that a different representation of this same supersonic wave drag is presented in a book by Eshbach (I do not know the name or publisher) and does not included the squared factor, using (t/c) as a linear scalar. This is a difference issue that I need to resolve.

Friction Drag: Van Nostrand doesn't say much about supersonic friction drag other than to equate it with subsonic friction drag - so this is a question I have no answers for at this time and the theory looks a little messy - hopefully it is not a major issue at supersonic speeds through thin atmosphere - but it may be ... so I need to resolve this as well.

Lift Induced Drag: Van Nostrand points out the following, "Drag due to lift is the component of the resultant pressure force which acts in the drag direction. It too has an exact counterpart in subsonic flow. Drag due to lift is,

$C_D=\frac {4\alpha^2}{\sqrt{M^2 -1}}$

The term induced drag used for subsonic flow is not applied to supersonic drags. Induced drag [in] the subsonic case is due to tip losses. Tip losses occur in supersonic flow, but not in the same proportions as in subsonic flow."

Well, the above lift induced drag discussion has me a little baffled and I'm not sure what to do with it. It implies that it is not significant for supersonic flight - so, if this is true, I should probably not be attempting to model it.

BOTTOM LINE: The bottom line here is that I do not feel that I have captured the essence of modeling lifting reentries with the above information. These are my issues:

1. I believe that the lift coefficient modeling is sufficient for my needs. I can assume symmetry of the object and zero lift when the angle of attack is zero. But is the α term really radians, the sine of the angle of attack, or degrees?
2. The pressure drag equation looks reasonable but is the (t/c) term squared or not?
3. When you go to the actual lift and drag equations which are essentially identical with only the choice of the coefficient, lift or drag, there's a question regarding the definition of the aerodynamic reference area. In drag discussions, it is pretty apparent that the drag reference area is the projection of the object on a plane perpendicular to the medium [air] relative velocity or direction of flight through the medium. With respect to lift, the rule appears to be the area of the wing - this is why I believe that the α variable really should be sin(α) because sin(α) times the wing area represents the projection of the object on that same plane normal to the direction of flight. When the angle of attack is small, this projected area is small and hence the lift is small. When the angle is large, the projection and the lift are large. This also suggests to me that the lift induced drag should be a major player in the supersonic physics as well.

Enough said ... I have written this because I am a little frustrated in resolving a simplified lift/drag model which will give me first order answers and possibly second order. I believe that it can be done - tell me one way or the other. --JEB 03:52, 28 May 2006 (UTC)

I'm not a physicist, etc, but here's my take which I've mentioned on other related pages. Anytime you come across a 'coefficient of...' in a physical equation, basically what you are looking at is a cop-out. What it means is that the observed phenomenon is apparently linear, but too complicated to describe in "real" terms. All the odd unknowns are simply bundled into a number which is arrived at to make the equation fit the observations. That doesn't mean it's useless, because it's relatively straightforward to either test your aerofoil in a wind tunnel and measure the coefficient (i.e. fudge it so the numbers come out to fit the equation) or else consult published tables of data for already tested sections. Then you can go on to design a practical aeroplane or whatever. Doing the reverse - deriving the coefficient from first principles - is possibly of academic interest but probably not of interest to the practical engineer. It's interesting that the equations you state above are for foils operating in certain very narrowly (and somewhat exotic) conditions. I'd suggest that a more general solution is too complex to be found, the Navier-Stokes equations being the intractable beasts that they are. Graham 15:58, 30 May 2006 (UTC)
About that aerodynamic reference area: For wings the plan area is used for both lift and drag coefficients. Lift/drag ratio can then be found simply be dividing on by the other. Also we wouldn't want the reference area going to zero for a thin section at zero angle of attack. I wonder Jebsys, if you have discovered the link to the report server over in the NACA article. There should be some helpfull information there, for example - [1] Meggar 04:27, 1 June 2006 (UTC)
Would it be appropiate to include specific soulutions to known geometric bodies on this page? i.e. the Cl value for a sphere is well known and can be derived, also there are general solutions for cylenders, of inifinite length, etc.--203.45.15.164 15:41, 1 June 2006 (UTC)

## Proposed move

This page should be moved to Lift coefficient to be consistent with Pressure coefficient, and Drag coefficient, among others. -- Kaszeta 15:55, 2 August 2005 (UTC)

Yes. Meggar 16:32, 2005 August 2 (UTC)

• Moved. Uncontroversial, no admin assistance was required. Dragons flight 19:53, August 5, 2005 (UTC)

## Intro wording

Some may see this point as trivial, but I think it should be addressed. The first sentence of the article reads as follows:

The lift coefficient (CL or CZ) is a number associated with a particular shape of an airfoil,

Given that airfoils are 2-D, infinite bodies, they have a section lift coefficient (Cl) as stated later in the article. As such, there is an apparent contradiction in the beginning of the article. An easy resolution would be to reword the sentence as follows:

The lift coefficient (CL or CZ) is a number associated with a particular shape of an wing or lifting surface,

Thoughts? Jeff220 05:18, 3 February 2007 (UTC)

Actually CZ usually refers to a normal force (perpendicular to the body), whilst CL to lift force, which is perpendicular to the velocity vector. Lift is generated by bodies and wings, and in, for example guided weapons, the body force can be dominant. The reference area is not always the wing planform; rockets, shells and missiles use the body cross section as the reference area, it is quite arbitrary. In principle, the square of the mean aerodynamic chord could be used, but nobody does. What needs to be addressed is why we use force and moment coefficients at all. Fundamentally, by using appropriate scale lengths, speeds, areas and densities the governing equations may be written in terms of force and moment coefficients, Reynolds number and Mach number. Solutions obtained from wind tunnels or computational fluid mechanics may be applied to a wide range of dynamically similar cases. The starting point is consideration of dynamic similarity and dimensional analysis. Gordon Vigurs 20:02, 17 June 2007 (UTC)

## I am trying to find the answer to a question

How do changes in camberline/chordline affect stall speed and critical AOA? Anybody can help? --Natasha2006 17:59, 3 April 2007 (UTC)

Since more camber, more lift and more drag; less camber, less lift and less drag. Can I deduct that camber changes proportionally to lift coefficient? If so, stall speed decreases and critical AOA increases with increase of camber. Please correct me if I am wrong! --Natasha2006 20:02, 3 April 2007 (UTC)

It depends on how the camberline shape changes. There is no simple formula. Generally, adding camber will increase the maximum lift coefficient. Since the lift curve slope is essentially fixed, this means stall will occur at a higher angle of attack. Botag 21:35, 4 April 2007 (UTC)

## Defining the Lift Coefficient

I propose more effort be applied into defining what the lift coefficient is, why it is used, and how it is used.

Currently, the article is backward engineering the lift coefficient, defining it in terms of the lift equation [i.e. as the ratio of lift vs (dynamic pressure x planform area)]. The article does not explain where the lift coefficient came from. Engineers didn't just magically come up with the lift equation, everything has significance and was chosen for a reason. I hope we can alter the article to reflect this line of thinking. I say we because my knowledge of the concept is not perfect, I am currently a struggling aerodynamics student. For everyones sake, I hope terms and concepts can be presented consistently in layman's terms. If Joe Schmoe can't understand the concept, the article has failed.

The lift coefficient is a simplification engineers use to save time and money when trying to understand the lift an aerodynamic body produces. Determining an object's lift is not simple, due to the many variables lift depends on. While there are only two aerodynamic forces acting on a body, pressure and shear stress, the composition of these forces are dependent on:

1. Free stream velocity V
2. Free stream density ρ
3. Aerodynamic surface size S
4. Angle of attack α
5. Airfoil Shape
6. Viscosity coefficient μ
7. Compressibility of the airflow a

Since an object's lift is dependent on each variable, an engineer attempting to understand an aerodynamic body's lift in real world conditions must test each value independently. That is, a body will have its lift measured while the free stream velocity is altered and variables 2-7 are fixed. Later, the same body will have its lift measured while the free stream density is altered and variables 1 & 3-7 are fixed. This process continues until all the variables have been independently altered, which amounts to a lot of time and money spent in the wind tunnel.

To help alleviate the complexity, dimensional analysis was performed on the preliminary lift function:

L = f(V,S,μ,a)

Rather than show the lengthy analysis, the final result and its defining features will be presented. A major result is the familiar lift equation:

L = q*S*Cl

In turn, the Cl can be defined through the lift equation as:

Cl = L/(q*S)

The biggest feature of this new equation is not its simplicity, but that it is purely a function of the angle of attack, Reynolds number, and the Mach number. During the dimensional analysis, the presence of terms that met the definition of the Reynolds and Mach number were condensed. Groups of terms that were originally singularly defined have been grouped together under symbols that describe complex relationships. In essence, the preliminary Lift function has been redefined into the much more managable Lift Coefficient function:

Cl = f(α, M, Re)

Now, an engineer who wants to determine the lift of a body at a predetermined angle of attack will only have to perform the tedious task of singleing out variables twice. By only altering the free stream Mach number and the Reynolds number, the engineer only has to accomplish 2/7 of the original workload.

More importantly though, the M and Re number have a special property that has dubbed them as similarity parameters. An aerodynamicist can test two proportional models under the same M and Re number and come up with the same exact Cl. The power lies in the fact that theoretically a low cost, scale model can now stand in for a multimillion dollar aircraft during testing - as long as the M and Re numbers match.

Alright, so far I've describe what the Cl is and why it is used. Later when I feel more motivated I'll explain how its used. I can guarantee you this probably isn't 100% right, so feel free to suggest corrections. LostCause 17:43, 23 August 2007 (UTC)

## air, water and other fluids

it shouldnt say air, because this principles can apply to any other fluid that have a similar behaviour than air. it must be more generic for other fluids as well. --Cokepe (talk) 14:09, 17 January 2008 (UTC)

"$\vec v\,$ is true airspeed, (speed of the body relative to a static point on the earth's surface)"