*Anatoly I. Ruban*

- Published in print:
- 2015
- Published Online:
- October 2015
- ISBN:
- 9780199681747
- eISBN:
- 9780191761614
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199681747.003.0003
- Subject:
- Physics, Soft Matter / Biological Physics

Chapter 2 is devoted to the thin aerofoil theory for subsonic flows. The chapter first assumes that the free-stream Mach number M∞ is close to zero, and use the Euler equations for an incompressible ...
More

Chapter 2 is devoted to the thin aerofoil theory for subsonic flows. The chapter first assumes that the free-stream Mach number M∞ is close to zero, and use the Euler equations for an incompressible fluid to describe the flow past an aerofoil. If the aerofoil is thin, then explicit integral formulae can be deduced for the lift force and the pressure distribution on the surface of an arbitrary aerofoil. The chapter then shows that the theory can be easily generalized to subsonic flow with arbitrary M∞ < 1. This is achieved with the help of the so-called Prandtl–Glauert transformation. Chapter 2 also includes a discussion of unsteady flows past thin aerofoils, and the flows with separation. The Chapter concludes with Prandtl’s theory of the flow past a large aspect ratio wing.Less

Chapter 2 is devoted to the thin aerofoil theory for subsonic flows. The chapter first assumes that the free-stream Mach number *M*_{∞} is close to zero, and use the Euler equations for an incompressible fluid to describe the flow past an aerofoil. If the aerofoil is thin, then explicit integral formulae can be deduced for the lift force and the pressure distribution on the surface of an arbitrary aerofoil. The chapter then shows that the theory can be easily generalized to subsonic flow with arbitrary *M*_{∞} < 1. This is achieved with the help of the so-called Prandtl–Glauert transformation. Chapter 2 also includes a discussion of unsteady flows past thin aerofoils, and the flows with separation. The Chapter concludes with Prandtl’s theory of the flow past a large aspect ratio wing.

*S. G. Rajeev*

- Published in print:
- 2018
- Published Online:
- October 2018
- ISBN:
- 9780198805021
- eISBN:
- 9780191843136
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198805021.003.0004
- Subject:
- Physics, Soft Matter / Biological Physics, Condensed Matter Physics / Materials

Some solutions of Euler’s equations are found here. The simplest are the steady flows: water flowing out of a tank at a constant rate, the Venturi and Pitot tubes. Another is the static solution of a ...
More

Some solutions of Euler’s equations are found here. The simplest are the steady flows: water flowing out of a tank at a constant rate, the Venturi and Pitot tubes. Another is the static solution of a self-gravitating fluid of variable density (e.g., a star). If the total mass is too large, such a star can collapse (Chandrasekhar limit). If the flow is both irrotational and incompressible, it must satisfy Laplace’s equation. Complex analysismethods can be used to solve for the flow past a cylinder or inside a disk with a stirrer. Joukowski used conformal transformations on the cylinder to find the lift of the wing of an airplane, in the limit of zero viscosity. Waves on the surface of a fluid are studied as another example. The speed of these waves is derived as a function of their wavelength and the depth of the fluid.Less

Some solutions of Euler’s equations are found here. The simplest are the steady flows: water flowing out of a tank at a constant rate, the Venturi and Pitot tubes. Another is the static solution of a self-gravitating fluid of variable density (e.g., a star). If the total mass is too large, such a star can collapse (Chandrasekhar limit). If the flow is both irrotational and incompressible, it must satisfy Laplace’s equation. Complex analysismethods can be used to solve for the flow past a cylinder or inside a disk with a stirrer. Joukowski used conformal transformations on the cylinder to find the lift of the wing of an airplane, in the limit of zero viscosity. Waves on the surface of a fluid are studied as another example. The speed of these waves is derived as a function of their wavelength and the depth of the fluid.

*S. G. Rajeev*

- Published in print:
- 2018
- Published Online:
- October 2018
- ISBN:
- 9780198805021
- eISBN:
- 9780191843136
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198805021.003.0006
- Subject:
- Physics, Soft Matter / Biological Physics, Condensed Matter Physics / Materials

When the speed of a fluid exceeds that of sound, discontinuities in density occur, called shocks.The opposite limit from incompressibility (constant density) is constant pressure. In this limit, we ...
More

When the speed of a fluid exceeds that of sound, discontinuities in density occur, called shocks.The opposite limit from incompressibility (constant density) is constant pressure. In this limit, we get Burgers equation. It can be solved exactly in one dimension using the Cole–Hopf transformation. The limit of small viscosity is found not to be the same as zero viscosity: there is a residual drag no matter how small it is. The Maxwell construction of thermodynamics was adapted by Lax and Oleneik to derive rules for shocks in this limit. The Riemann problem of time evolution with a discontinuous initial density is solved in one dimension. These simple solutions provide the basic intuition for more complicated shocks.Less

When the speed of a fluid exceeds that of sound, discontinuities in density occur, called shocks.The opposite limit from incompressibility (constant density) is constant pressure. In this limit, we get Burgers equation. It can be solved exactly in one dimension using the Cole–Hopf transformation. The limit of small viscosity is found not to be the same as zero viscosity: there is a residual drag no matter how small it is. The Maxwell construction of thermodynamics was adapted by Lax and Oleneik to derive rules for shocks in this limit. The Riemann problem of time evolution with a discontinuous initial density is solved in one dimension. These simple solutions provide the basic intuition for more complicated shocks.