## Abstract

We present a new mathematical theory explaining the fluid mechanics of subsonic flight, which is fundamentally different from the existing boundary layer-circulation theory by Prandtl–Kutta–Zhukovsky formed 100 year ago. The new theory is based on our new resolution of d’Alembert’s paradox showing that slightly viscous bluff body flow can be viewed as zero-drag/lift potential flow modified by 3d rotational slip separation arising from a specific separation instability of potential flow, into turbulent flow with nonzero drag/lift. For a wing this separation mechanism maintains the large lift of potential flow generated at the leading edge at the price of small drag, resulting in a lift to drag quotient of size 15–20 for a small propeller plane at cruising speed with Reynolds number \({Re\approx 10^{7}}\) and a jumbojet at take-off and landing with \({Re\approx 10^{8}}\) , which allows flight at affordable power. The new mathematical theory is supported by computed turbulent solutions of the Navier–Stokes equations with a slip boundary condition as a model of observed small skin friction of a turbulent boundary layer always arising for \({Re > 10^{6}}\) , in close accordance with experimental observations over the entire range of angle of attacks including stall using a few millions of mesh points for a full wing-body configuration.

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## References

- 1
Hoffman J., Johnson C.: Computational Turbulent Incompressible Flow. Springer, New York (2007)

- 2
Hoffman J., Johnson C.: Mathematical Secret of Flight. Normat

**57**, 145–169 (2009) - 3
Hoffman J., Johnson C.: Resolution of d’Alembert’s paradox. J. Math. Fluid Mech.

**12**(3), 321–334 (2010) - 4
von Karman T., Sears W.R.: Airfoil theory for non-uniform motion. AIAA J.

**41**(7A), 5–16 (2003) - 5
Peters, D.A., Karunamoorthy, S., Cao, W.-M.: Finite state induced models part 1: two-dimensional thin airfoil. AIAA J. Aircraft.

**32**(2), 313–322 (1995) - 6
Prandtl, L., Deans, W.M.: Essentials of fluid dynamics (1953)

- 7
Talay, T.A.: Introduction to the Aerodynamics of Flight. In: Scientific and Technical Information Office, vol. 367, National Aeronautics and Space Administration (1975)

- 8
Schlichting, H., Truckenbrodt, E.: Aerodynamics of the Airpoplane. Mac Graw Hill, New York (1979)

- 9
Anderson, J.D.: Fundamentals of Aerodynamics (2010)

- 10
McCormick, B.W.: Aerodynamics, aeronautics, and flight mechanics, vol. 2, Wiley, New York (1995)

- 11
Krasnov.: Aerodynamics (1978)

- 12
Von K\'arm\'an, T.: Aerodynamics. Cornell University press, Mc Graw-Hill company (1963)

- 13
von Mises, R.: Theory of Flight. Courier Corporation (1959)

- 14
Birkhoff, G.: Hydrodynamics: a study in fact logic and similitude (1950)

- 15
Birkhoff, G.: Hydrodynamics: a study in fact logic and similitude (1950) I think that to attribute dAlemberts paradox to the neglect of viscosity is an unwarranted oversimplification. The root lies deeper, in lack of precisely that deductive rigour whose importance is so commonly minimised by physicists and engineers (1950)

- 16
On the motion of a fluid with very small viscosity. In: Proceedings of 3rd International Mathematics Congress, pp. 484–491 (1904)

- 17
Stoker, J.: On the other hand, the uninitiated would be very likely to get wrong ideas about some of the important and useful achievements in hydrodynamics from reading this chapter.In the case of air foil theory, for example, the author treats only the negative aspects of the theory. It has always seemed to the reviewer that the Kutta-Joukowsky theory of airfoils is one of the most beauti- ful and striking accomplishments in applied mathematics. The fact that the introduction of a sharp trailing edge makes possible a physi- cal argument, based on consideration of the effect of viscosity, that leads to a purely mathematical assumption regarding the behavior of an analytic function which in its turn makes the solution to the flow problem unique and also at the same time furnishes a value for the lift force, represents a real triumph of mathematical ingenuity. Rev. [14] Bull. Am. Math. Soc. 57(6), 497–499

- 18
Bloor, D.: The Enigma of the Aerofoil: Rival Theories in Aerodynamics, 1909–1930. The University of Chicago Press (2011)

- 19
Hoffren, J.: Quest for an improved explanation of lift. In: AIAA Aerospace Sciences Meeting and Exhibit, 39th Reno, NV (2001)

- 20
Hoffren, J.: Quest for an improved explanation of lift. In: AIAA Aerospace Sciences Meeting and Exhibit, 39th Reno, NV (2001) Quest for an Improved Explanation of Lift. AIAA J The classical explanations of lift involving potential flow, circulation and Kutta conditions are criticized as abstract, non-physical and difficult to comprehend. The basic physical principles tend to be buried and replaced by mystical jargon. Classical explanations for the generation of lift do not make the essence of the subject clear, relying heavily on cryptical terminology and theorems from mathematics. Many classical texts even appear to have a fundamental error in their underlying assumptions. Although the subject of lift is old, it is felt that a satisfactory general but easily understandable explanation for the phenomenon (of lift), is still lacking, and consequently there is a genuine need for one (2001)

- 21
Moin, P., Kim, J.: Tackling turbulence with supercomputers. Scientific American

**276**(1), 46–52 (1997) - 22
Hoffman, J., Jansson, J., Jansson, N., Vilela De Abreu, R.: Time-resolved adaptive FEM simulation of the DLR-F11 aircraft model at high Reynolds number. In: Proceeding of 52nd Aerospace Sciences Meeting, AIAA (2014)

- 23
Chang, K.: Staying Aloft: What Does Keep Them Up There. New York Times (2003)

- 24
Stack, J., Lindsay, W.F.: Tests of N-85, N-86 and N-87 airfoil sections in the 11-inch high speed wind tunnel NAC A Technical Report 665 (1938)

- 25
Herrig, L.J., Emery, J.C., Erwin, J.R.: Effect of section thickness and trailing-edge radius on the performance of NACA 65-series compressor blades in cascade at low speeds (1951)

- 26
Duchon J., Robert R.: Inertial energy dissipation for weak solutions of incompressible Euler and Navier–Stokes equations. Nonlinearity

**13**, 249–255 (2000) - 27
Hoffman J., Johnson C.: Blowup of Euler solutions. BIT Numer. Math.

**48**(2), 285–307 (2008) - 28
Hoffman J., Jansson J., Vilela De Abreu R.: Adaptive modeling of turbulent flow with residual based turbulent kinetic energy dissipation. Comput. Meth. Appl. Mech. Eng.

**200**(37–40), 2758–2767 (2011) - 29
Bangerth W., Rannacher R.: Adaptive finite element methods for differential equations. Springer, New York (2003)

- 30
Hoffman J., Johnson C.: A new approach to computational turbulence modeling. Comput. Method Appl. Mech. Eng.

**195**, 2865–2880 (2006) - 31
Hoffman J.: Adaptive simulation of the turbulent flow past a sphere. J. Fluid Mech.

**568**, 77–88 (2006) - 32
Hoffman J.: Computation of mean drag for bluff body problems using adaptive dns/les. SIAM J. Sci. Comput.

**27**(1), 184–207 (2005) - 33
Hoffman J.: Efficient computation of mean drag for the subcritical flow past a circular cylinder using general galerkin g2. Int. J. Numer. Meth. Fluid

**59**(11), 1241–1258 (2009) - 34
Jansson, J., Hoffman, J., Jansson, N.: Simulation of 3D flow past a NACA 0012 wing section, CTL technical report kth-ctl-4023

- 35
Leray J.: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math.

**63**, 193–248 (1934) - 36
Hoffman J., Jansson J., Vilela de Abreu R., Degirmenci C., Jansson N., Müller K., Nazarov M., Spühler J. Hiromi: Unicorn: parallel adaptive finite element simulation of turbulent flow and fluid-structure interaction for deforming domains and complex geometry. Comput. Fluids

**80**, 310–319 (2013) - 37
De Abreu, R.V., Jansson, N., Hoffman, J.: Adaptive computation of aeroacoustic sources for rudimentary landing gear. In: Proceedings for Workshop on Benchmark problems for Airframe Noise Computations I (2010)

- 38
Jansson N., Hoffman J., Nazarov M.: Adaptive simulation of turbulent flow past a full car model. In: Proceedings of the 2011 ACM/IEEE International Conference for High Performance Computing, Networking, Storage and Analysis, SC ’11 (2011)

- 39
Vilela De Abreu R., Jansson N., Hoffman J.: Computation of aeroacoustic sources for a complex landing gear geometry using adaptive FEM. In: Proceedings for the Second Workshop on Benchmark Problems for Airframe Noise Computations (BANC-II), Colorado Springs (2012)

- 40
Vilela de Abreu R., Jansson N., Hoffman J.: Adaptive computation of aeroacoustic sources for rudimentary landing gear. Int. J. Numer. Meth. Fluids

**74**(6), 406–421 (2014) - 41
Hoffman J.: Simulation of turbulent flow past bluff bodies on coarse meshes using General Galerkin methods: drag crisis and turbulent Euler solutions. Comput. Mech.

**38**, 390–402 (2006) - 42
Hoffman, J., Jansson, N.: A computational study of turbulent flow separation for a circular cylinder using skin friction boundary conditions. In: Proceedings for Quality and Reliability of Large–Eddy Simulations II, Pisa, Italy (2009)

- 43
Fefferman, C.L.: Existence and smoothness of the Navier-Stokes equation. The Millennium Prize Problems, pp. 57–67 (2000)

- 44
Hadamard, J.: Sur les problémes aux dérivées partielles et leur signification physique. Princeton University Bulletin, vol. 13, pp. 49–52 (1902)

- 45
Ladson, C.: Effects of independent variation of Mach and Reynolds numbers on the low-speed aerodynamic characteristics of the NACA 0012 airfoil section (1988)

- 46
Gregory, N., O’Reilly, C.L.: Low speed aerodynamic characteristics of airfoil profiles including effects of upper surface roughness simulating hoar frost (1970)

- 47
Rumsey, C.: 2nd AIAA CFD High Lift Prediction Workshop HiLiftPW-2, (http://hiliftpw.larc.nasa.gov/) (2013)

- 48
Sagaut P.: Large Eddy Simulation for Incompressible Flows (3rd edn.). Springer, Berlin (2005)

- 49
Zdravkovich, M.M.: Flow around circular cylinders: a comprehensive guide through flow phenomena, experiments, applications, mathematical models, and simulations. Vol. 1 [Fundamentals], Oxford Univ. Press, Oxford (1997)

- 50
Korotkin, A.I.: The three dimensionality of the flow transverse to a circular cylinder. Fluid Mech. Soviet Res.

**5**, 96–103 (1976) - 51
Humphreys J.S.: On a circular cylinder in a steady wind at transition Reynolds numbers. J. Fluid Mech.

**9**(4), 603–612 (1960) - 52
Schewe, G.: Reynolds-number effects in flow around more-or-less bluff bodies. J. Wind Eng. Ind. Aerodyn.

**89**(14), 1267–1289 (2001) - 53
Gölling, B.: Experimentelle Untersuchungen des laminar-turbulenten \"Uberganges der Zylindergrenzschichtstr\"omung, Doctoral dissertation, DLR (2001)

- 54
UIUC Airfoil Data Site, http://www.ae.illinois.edu/m-selig/ads.html

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## Additional information

*Editorial Foreword* The special character of this article requires some comments by the editors on the purpose of its publication. Though, its mathematical content does not meet the degree of mathematical rigor usually expected by articles in this journal, the implications of the argument and the accompanying novel numerical computations are of such far reaching importance for technical fluid dynamics, particularly for the computation of certain features in turbulent flow, that
it deserves serious consideration. The main purpose of this publication is therefore to stimulate critical discussion among
the experts in this area about the relevance and justification of the view taken in this article and its possible consequences
for modeling and computation of turbulent flow.

Communicated by R. Rannacher

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### Cite this article

Hoffman, J., Jansson, J. & Johnson, C. New Theory of Flight.
*J. Math. Fluid Mech.* **18, **219–241 (2016). https://doi.org/10.1007/s00021-015-0220-y

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### Mathematics Subject Classification

- 76G25
- 76E09

### Keywords

- Flight
- lift theory
- d’Alembert’s paradox