the Lift Equation

Feature image: © Ivan Cholakov | Dreamstime.com

Like so many ‘first times’, my initial encounter with the lift equation was a fairly short and messy affair. My flying instructor wasn’t given to long preflight briefings in any case – and, to be fair, we’d agreed I would mostly do my own theory study.

A wave of panic

So imagine my surprise when I found myself facing a whiteboard that had been bisected by a line of code from, as a best guess, The Enigma Machine Of Dr. Moreau.

L = CL½⍴v2S

I still remember the wave of panic. The horror of High School maths and physics exams came back to haunt me. I’d never be able to apply that jibberish in a life-or-death situation on short finals would I? Was I about to wash out of my own flight training program??

“This is the lift equation,” the CFI counselled: “CL is a lift co-efficient for the airfoil, the little p thingy is actually rho, or air density; v is airspeed and S is wing area. As a pilot, the only bit you can really control is v, and because that’s squared, making a small change in your airspeed will generate a much bigger change in lift. Okay? Okay. Let’s go fly.”

The punctuation is almost entirely mine. And there may also have been an apology for having to bring such confronting algebra into our lives at all.

© Stephen Griffith | Dreamstime.com
© Stephen Griffith | Dreamstime.com

How I learned to stop worrying and love the equation

Happily, I committed the equation and its explanation to memory, and passed all my tests without having to worry about it again.

It was only much later that I came to appreciate the full elegance of the lift equation and it’s mathematical snapshot of a wing in flight.

So before you allow yourself to be put off by the mad assault of Arabic, Greek and Roman glyphs, let’s take a look under the equation’s hood. Because you don’t need a bunch of doctorates to appreciate it’s inner beauty – your interest in aviation is plenty.

Making it make sense

The way the equation gets written normally is convenient (and logical, believe it or not) for aerodynamicists – but it doesn’t help novices like me to explain what’s going on. So let’s begin by breaking it down into individual components, and then deal with them one by one.

equation

The elements that make up the lift equation are:

• CL
• times ½⍴v2,
• times S

You can put those three lumps into any order you choose.
I’ll explain why ½⍴v2 needs to stay together shortly. Meanwhile, the order I choose is this…

eSArea rules

I like to start with the simplest component, wing area S.

After all, it’s pretty self-explanatory: It doesn’t take a genius to appreciate that a bigger wing will generate more lift. Structural weight aside, if you could put A380 wings onto a Cessna Skyhawk, it would blow away in a chicken’s sneeze. Attach stubby F-104 blades and your Skyhawk would fly little better than a family sedan.

Small S, big v... Lockheed F-104A-10-LO, #56-0758. (U.S. Air Force photo)
Small S, big v… Lockheed F-104A-10-LO, #56-0758. (U.S. Air Force photo)

Surprisingly, you can change the area of your wing in flight. Yep, most high lift devices like flaps and slats add wing area, among other things, to give you extra lift when you need it.

The ‘structural weight aside’ note is an interesting point too. Weight, or gravity, has no place in calculating lift. If that seems crazy, just remember that weight and lift are opposing forces, not complementary. Lift goes up, Weight goes down. So Weight has no place in a Lift calculation.

On the other hand, if you load your Cessna down with A380 wings, or just too many golf clubs and cases of wine, the aircraft’s CL, ½⍴v2, and S numbers will need to deliver an equally higher result for L. (And then there’s all the extra drag to consider…)

eCFlying by numbers

Now let’s head back to the wing’s co-efficient of lift. Don’t let yourself get bamboozled by the science behind it – CL is simply a number, a convenient multiplier that tailors the rest of the equation to a specific wing setup…

Mind you, that’s only the boilerplate description, and my instructor telling me (or me telling you) that a pilot has no control over CL just plain wrong.

You see the lift coefficient changes – or can be changed – by just about anything. The obvious one is angle of attack. Every time you move the elevator, fly through an up- or down-draft, or just use the ailerons, you’ll get a different angle of attack and a new CL for the affected part of the wing.

Remember ailerons actually change the camber on their section of the wing too, which also contributes to a new CL. And for a more lasting effect, you can using high lift devices like flaps and slats, to change the camber, AoA, area and lift coefficient of a wing – giving you tons of control over lift while you work to stop the airspeed changing at all.

eSCL The wing is the thing

At this point, the lift equation has completely described the wing in all three dimensions. (See, I told you it was elegant.)

• S defines the plan view, span and chord, giving the wing area.

• CL takes a vertical slice through the wing, describing the airfoil’s shape and angle of attack simply by rating its efficiency.

Neat huh? There’s an infographic a bit further down. But first, let’s go flying.

© Steirus | Dreamstime.com
© Steirus | Dreamstime.com

e1:2 It’s all about energy

Despite being voted the Bit Most Likely To Do Your Head In, I love this component of the lift equation.

A wing sitting still on the ground (well, anywhere) won’t fly. What’s missing? Energy. And that’s where ½⍴v2 comes in.

Normally, the kinetic energy of anything is worked out as ½mv2, or half of mass times the velocity squared. (The ½ is another convenient multiplier, that keeps the mv2 part in step with the rest of physics.)

Of course the wing, being attached to a moving airplane, is what has the kinetic energy. But for lift, what really matters is the kinetic energy of the relative airflow – the wind passing around the wing. Happily, the magic word ‘relative’ means we can just take the same energy equation and replace the meaningless mass of the wing with the much more meaningful mass of the relative airflow.

It’s a beautiful solution: Instead of considering a wing tearing through still air, we think about the air tearing around a still wing. Either way, the amount of kinetic energy is the same.

Graphic

Rho what?’ you ask

Of course, the accurate way to measure air mass is by volume, or how many molecules there are in each three dimensional ‘block’. And that’s density, in a nutshell.

To bring it full circle, the total number of air molecules acting around a wing will determine how much energy is imparted to the wing in flight. And two things determine how many air molecules that’s likely to be:

  1. The number of molecules in each passing parcel of air (the density, or rho); and
  2. The number of those parcels that you can fly through each nanosecond, or hour, if you prefer more conventional measures of velocity like knots and mph.

That’s mass and volume right there. Just run them through Physics 101 and you have the Kinetic Energy of your wing/wind system:

EK = ½mv2, or ½⍴v2, depending on whether you want to study the wing or the air flowing around it.

(Incidentally, EK = ½mv2 doesn’t hold true as v gets closer to light speed, but I figured that wouldn’t come up in day-to-day flying.)

© airscape photo
© airscape photo

All together now

And there we have it.

Lift comes from the three dimensional shape of a wing (CL ﹡ S), multiplied by the kinetic energy of the air flowing around it it (½⍴v2).

Want to see that again? Okay. L = (CLS)(½⍴v2), which is just another way of arranging L = CL½⍴v2S

equation

And far from only being able to control lift with velocity, we have three ‘levers’ we can pull. Three from three, in fact. Short of searching around for more or less dense air, you can control every element of the lift equation.

Having that range of options (plus their various combinations) and knowing when to choose the right one one, will help to keep your flying safe and sustainable.

So don’t be afraid of the Lift Equation. Learn to love it.

Lift = ... well, total weight. Smoke and yarn tufts highlight the varied airflow acting on an F/A-18 airframe in flight. (NASA photo).
Lift = … well, total weight. Smoke and yarn tufts highlight the varied airflow acting on an F/A-18 airframe in flight. (NASA photo).

Addendum: Need a Lift?

You may have noticed that there’s one part of the equation I haven’t mentioned yet… ‘L = ‘

I’ve left it to last because, frankly, it’s kind of useless. Normally, the sum would be the whole point of an equation, but not this one. And it gets even more counter-intuitive because, for pilots, the lift equation is no way to calculate lift.

For one thing, lift in flight equals weight. Yep, L = W, and never mind all the CL½⍴v2S add-ons. If you know what the aircraft weights, you’ll know how much lift it needs to fly.

Total lift may differ from all-up weight in a climb or descent (with power or gravity making up the difference) but knowing whether you’re flying up, down or level is far more valuable than knowing a numerical value for L anyway.

Plus, even though I’ve talked about wings all the way through, various other bits of the airframe will be producing lift, or even downforce, during flight. So you’d need to know CL for the whole airframe in all it’s configurations and angles of attack to even get started.

The true value of the equation is in understanding the factors that contribute to lift, and how they weigh in to the total. Knowing that airspeed contributes exponentially, for example.

Because lift may be energy, but knowledge is power.

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