*To those who fear flying, it is probably disconcerting that physicists and aeronautical engineers still passionately debate the fundamental issue underlying this endeavor: what keeps planes in the air?*(Kenneth Chang, New York Times, Dec 9, 2003)-
*Aeronautics was neither an industry nor a science. It was a miracle.*(Igor Sikorsky)

*Velocity at 10,14 and 17 degrees angle of attack: red high velocity, blue low velocity.*

*Pressure iso-surfaces for 5 degrees angle of attack: Low pressure on top and high pressure below leading edge, no high pressure on top at trailing edge = substantial lift.*

## The Mystery of Gliding Flight

The problem of explaining **why it is possible** to fly in the air using wings has haunted scientists since the birth of mathematical sciences. To fly, an upward force on the wing, referred to as **lift** **L**, has to be generated from the flow of air around the wing, while the air resistance to motion or **drag** **D**, is small.

The mystery is how a large ratio **L/D** can be created. In the gliding flight of birds and airplanes with fixed wings, **L/D **is typically between 10 and 20, which means that a good glider can glide up to 20 meters upon loosing 1 meter in altitude, or that a 400 ton jumbojet can cruise at an engine thrust of 20 tons, while about 400 tons is needed in take-off. By elementary Newtonian mechanics, upward lift must be accompanied by downwash with the wing redirecting air downwards.

The enigma of flight is how a wing generates substantial downwash; with downwash there is lift. Classical mathematics and mechanics could not give an answer: Newton computed the lift of a tilted flat plate bombarded by a horisontal stream of fluid particles from below and obtained a disappointingly small lift, proportional to the square of the tilting angle or **angle of attack.**

French mathematician d’Alembert followed up in 1752 with a computation based on **potential flow** (inviscid incompressible irrotational steady flow), showing that both the drag and lift of a wing is zero, referred to as d’Alembert’s paradox , since it contradicts observations and thus belongs to pure fiction. We recall

## The Secret Revealed in Four Basic Steps

- The flow is incompressible with small skin friction and thus can only separate at stagnation at the trailing edge (before stall).
- Main lift is created by low pressure (negative) on top of the leading edge of high speed flow in accordance with Bernoulli’s Principle.
- Main drag is created by high pressure (positive) on the leading edge by low speed flow in accordance with Bernoulli’s Principle.
- Lift and drag from the leading edge are preserved by a specific flow separation pattern at the trailing edge with alternating high and low pressure with zero mean.

Here 1. and 4. are the new elements of the New Theory, with 4. the most surprising and intriguing, which are combined with the classical elements 2. and 3. relating to Bernoulli’s Principle from 1738 bridging over 280 years.

# D’Alembert’s Paradox and Potential Flow

Potential flow around a wing profile is depicted in the following figure showing low pressure on top and high pressure below the leading edge creating lift:

Further inspection of the flow shows balancing high pressure on top and low pressure below the trailing edge, which cancels the lift created at the leading edge to give zero lift and also zero drag.

We also see that the direction of the outgoing flow is the same as the incoming flow, so that the the flow is not re-directed, or in other words there is no down-wash with the outgoing flow being directed downwards. The zero lift thus correspond to zero down-wash and thus zero upwards reaction force on the wing. Down-wash is necessary for the generation of lift.

D’Alembert could thus only give support to Newton’s prediction that flight is impossible. But of course the flight of birds which could be observed by everyone remained a mystery. To uncover the secret of flight D’Alembert’s Paradox had to be resolved, but nobody could figure out a solution.

It remained unsolved until 2008 as a a deep trench separating theoretical mathematical fluid mechanics from hydraulics as engineering fluid mechanics with

- theoretical fluid mechanics explaining phenomena which could not be observed (zero lift/drag)
- hydraulics observing phenomena which could not be explained (non-zero lift/drag)

as expressed by Nobel Laurate Cyril Hinslewood.

# Resolution of D’Alembert’s Paradox and Secret of Flight

In 2008 we presented a resolution of d’Alembert’s paradox which opened to an explanation of how lift is generated by a wing in incompressible flow.

Watching the movies of pressure and velocity of **turbulent computational solutions** of the **incompressible Navier-Stokes ****equations** with **slip/small friction boundary conditions**, around a three-dimensional **Naca0012** wing under increasing **angle of attack,** you can yourself uncover the secret of flight. What you see can be described in pictures as follows:

We see potential flow with high pressure at the separation on top of the wing before the trailing edge, being modified by

*low-pressure counter-rotating rolls of streamwise vorticity generated at separation**avoiding the build up of high/low pressure at the trailing edge of potential flow**thereby generating both lift and drag**with lift alternatively as reaction to downwash.*

The rolls of rotating flow generated at the trailing edge can be observed in the following picture from the Dryden Water Channel:

# Downwash

We understand that lift follows from downwash, and downwash from the low-pressure rotational slip separation which also gives drag. We understand that lift comes with drag.

Potential flow has zero drag but also zero lift.

There is no free lunch: To fly requires work to overcome drag.

Here is a movie showing the counter-rotating rolls of stream wise vorticity watching the trailing edge from behind.

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