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I got Aircraft Design: A conceptual approach for Christmas, and I'm having a hard time with lift coefficients because I honestly have no idea what "lift force per unit span" means, so can someone please explain this to me?

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    $\begingroup$ If the total lift of a wing is L, and the span of the wing is X, then isn't the lift per unit span just L/X? $\endgroup$ – Riccati Jan 6 '16 at 23:36
  • $\begingroup$ It's not a rectangular wing, and it also has an uneven taper. $\endgroup$ – ptgflyer Jan 6 '16 at 23:41
  • $\begingroup$ So the question now becomes, how to find the mean aerodynamic chord of a wing that does not have a straight leading and/or trailing edge. $\endgroup$ – ptgflyer Jan 6 '16 at 23:43
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    $\begingroup$ Maybe the book is just talking about a thin slice of the wing then. Lift per span might mean (delta L)/(delta X). In answer to your latest comment: No, chord and span are different measurements. $\endgroup$ – Riccati Jan 6 '16 at 23:45
  • $\begingroup$ Yes, yes they are. I get things mixed up sometimes. Thanks. $\endgroup$ – ptgflyer Jan 6 '16 at 23:48
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The concept of lift force per unit span comes from potential flow theory. It will need some background information to explain what it means, so bear with me.

In the early years of flight, electricity was new and exciting, and it just happened that the equations which could calculate the strength of an electromagnetic field worked equally well when calculating the local flow change effected by a wing. What is the electrical current in a wire became the vorticity in a vortex, and the strength and orientation of the induced magnetic field were equivalent to the induced flow changes. So the vocabulary of electricity was copied over to aerodynamics, just like brain research used vocabulary from computer science when that was a hot topic.

Now we are left with abstract concepts like induced drag or lift per unit span. It would be so much more descriptive to use proper names, but the authors of technical books learned it that way and are much too lazy to explain aerodynamics any better.

In potential flow theory, you have sources, sinks and vortices. Sources and sinks are used to generate the displacement effect of a physical body moving through air, and vortices are used to explain why wings bend the flow and create lift. In order to calculate the lift force $L$ of a single vortex in two-dimensional flow, the circulation strength $\Gamma$ of the vortex is multiplied by the airspeed $u_{\infty}$ and air density $\rho$. You will find an equation like $L = -\Gamma\cdot u_{\infty}\cdot\rho$ in many treatises about numerical aerodynamics.

To expand that into the third dimension (and, consequently, into reality), you need to add something measured in spanwise direction - but you have already lift, and adding the third dimension would give a moment (lift times distance) where only lift would make sense. Therefore, this two-dimensional lift is now called "lift per unit of span" so there is still space for a third dimension where two-dimensional flow did already produce lift (counter to any sound intuition).

And no, this is never constant over span. In all cases the vorticity is gradually reduced towards the tips, or explained in a better way, the suction force acting on the wing is gradually reduced when you approach the tips because when the wing ends, nothing can prevent the air from flowing from the high-pressure region below to the low-pressure region on the upper surface of the wing.

While the potential flow mentioned above is the mathematical way of looking at aircraft, lift coefficients are the engineer's way of expressing things. From tests it was soon clear that the lift force of a wing scales with the dynamic pressure $q$ of the flow, that is the product of air density and the square of airspeed: $q = \frac{\rho}{2}\cdot v^2$.

The next observation of engineers was that lift also scales with wing area $S$. To make the lifting force independent of wing size and dynamic pressure, they stripped both from the lift (physical unit of Kilopond, Newton or pound-force) so they arrived at a dimensionless figure which they called lift coefficient $c_L$. Doing so made it much easier to compare measurements or scale up known designs for the next, better design. The lift equation now becomes $L = c_L\cdot S\cdot\frac{\rho}{2}\cdot v^2$

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    $\begingroup$ Another excellent answer, helpful even for us knuckle-dragging operators who never got too far past the "houses bigger/houses smaller" explanation of flight controls! In your 4th paragraph, you talk about the "strength of the vortex" -- which I visualize as the swirl coming off the wingtips -- a 3D phenomena. In the 2D context, what does "vortex" refer to here? Thanks! $\endgroup$ – Ralph J Jan 7 '16 at 15:40
  • $\begingroup$ @RalphJ: In the 2D context this is just something which swirls the air, like the drain of a bathtub viewed from above (if you allow for the column of water to have zero height, ideally). Also, this swirl is not coming off the wingtip, but in a sheet of infinite swirls along the whole span. Another detail the lazy authors mostly get wrong. $\endgroup$ – Peter Kämpf Jan 7 '16 at 15:54
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Imagine that the wing is a carrot, and chop it up as you would chop a carrot into discs. The lift (force) produced by a slice of thickness 1 is the lift (force) per unit span of that slice. ("Thickness 1" could be in whatever units you choose, so another way to look at that is to divide the lift by the thickness of the slice.)

For a uniform (straight, not tapered, swept, or twisted) wing, every slice produces the same amount of lift, so as Riccati points out, the lift per unit span is just the total lift divided by the wing span. However, on a wing whose shape varies from the fuselage to the tip, each slice is slightly different. A tapered wing might look a bit like a very conical carrot, and the lift per unit span decreases smoothly from the root to the tip, just like the diameter of each disc decreases as you get towards the tip of the carrot. (I'm not saying the shape of the carrot is related at all: it's just a way of thinking about considering each slice separately.)

While you can use total lift to compare different wings, you can use lift per unit span to compare wings in a way that's independent of their span. A wing twice as long will produce twice the lift (ignoring real-world effects like flex and prop wash), but it will have the same lift per unit span, because it has the same thickness and shape as the shorter wing. More usefully, you can use it to look at different parts of the same wing: to compare the root and the tip. Later in your book, you'll see charts showing how the lift per unit span varies along the length for different shapes/designs of wing.

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Consider a finite (three dimensional) wing producing lift. It would be difficult for us to calculate the total lift and exact lift distribution of the wing unless it is quite simple.

One way to deal with it is to 'slice' the wing into a number of segments for which the lift force can be found and the take into account the effects of variation of various wing parameters like:

  • Chord

  • Geometric twist

  • Aerodynamic twist (airfoil shape).

The lift per unit span of the wing can be found from the lift coefficient of the airfoil- basically, we are assuming that the flow over a finite wing can be treated as locally two-dimensional and finding the forces on the wing using this.

As an example, take a three dimensional wing and then slice it into small pieces so that the lift is essentially constant within each (i.e. the airfoil section and angle of attack is constant). For each of the slices, it is possible to find the lift (from airfoil and flow characteristics). This gives the lift per unit span (unit span here means the size satisfying above conditions).

Now, the total lift can be found by simply adding the lifts from various sections. Another thing is that the spanwise variation of the lift per unit span gives the lift distribution of the wing- helping us to compare various wing planforms- like elliptical vs rectangular etc.

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