Vol.4 : Computational Turbulent Incompressible Flow

Secret of Flying

Since the dawn of history man has asked upon observation of flying birds, why it is than some heavier-than-air objects can fly, but others cannot? The critical question is how a lift force from the pressure distribution around the object can be created balancing the gravitational force.

d'Alembert showed in 1752 that in potential flow both the lift and drag is zero, supporting Newton´s prediction using a particle model of air flow of a very small lift, and da Vinci´s unsuccessful attempts of human powered flying machines. So from both mathematical and practical point of view the old dream of flying like the birds seemed out of reach for humans, when the two brothers Orwille and Wilbur Wright in 1903 showed that powered heavier-than-air flight, indeed was possible.

Quickly thereafter, the mathematicians Kutta and Jukowski changed the zero lift potential solution of a wing section, by adding a flow circulating around the wing creating an unsymmetric pressure distribution with substantial lift, with the circulation and resulting lift so determined that the flow separated at the trailing edge of the wing, referred to as the Kutta-Jukowski condition.

Further, in 1904 Prandtl presented his resolution of d'Alembert's paradox of zero lift based on laminar boundary layer effects of vanishing viscosity, and mathematical order seemed to be resurrected: Indeed powered flight was possible as predicted by mathematics! But only after flying was demonstrated to be possible in practice!

Vol 4 shows that Prandtl's resolution is incorrect and that there is no large scale circulation around a wing and asks from where the Kutta-Jukowski condition comes? Does really commonly accepted fluid dynamics show that flying is possible?

Vol 4 shows that the answer is NO, because the true turbulent nature of the flow has to be taken into account, and it is not in classical aero dynamics of laminar boundary layer theory and Kutta-Jukowski potential flow.

Vol 4 shows that without turbulence, there would be no lift, and with turbulence the lift comes along with drag, so flying consumes energy, which the migrating birds know very well, as well as the airline companies.

Vol 4 shows more precisely that the turbulent flow around a wing generates a lift increasing with the angle of attack up to around 20 degrees, whereafter it quickly drops and the drag increases drastically corresponding to stall. The lift is generated because the wing redirects the flow of air to a downward direction after the wing (referred to as downwash), and the key question to answer is why a wing generates downwash. If there is downwash, then there is lift.

Vol 4 shows that downwash occurs because of the generation of turbulent streamwise vorticity with low pressure at the trailing edge, which depletes the high pressure of the potential solution with zero lift, which is not realized because of exponential instability at the trailing edge.

Vol 4 shows that in a fictional ideal perfect mathematical World of infinite precision, the potential solution with zero lift could occur and thus prevent flying. However, in the real imperfect World in which we live, the potential solution could never occur because of exponential instability, and from the instability a low-pressure turbulent wake always develops effectively generating stable lift. Thus stable flying in a real imperfect World is possible, but not in a fictional prefect World.

Vol 4 shows that the turbulent streamwise vorticity generated at the trailing edge, is similar to the vortex generated in a bathtub drain, also referred to as vortex stretching.

Vol 4 opens the possibility that the lift and drag of the turbulent flow around an entire aircraft can be accurately simulated by G2 for the Euler equations (Euler/G2) without resolving the boundary layers, thereby reducing the number of mesh points from the commonly predicted impossible 10 millions times 10 millions to the entirely possible around 10 millions mesh points with a reduction factor of 10 millions.

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