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Prandtl–Glauert Transformation
The Prandtl–Glauert transformation is a mathematical technique which allows solving certain compressible flow problems by incompressible-flow calculation methods. It also allows applying incompressible-flow data to compressible-flow cases. Mathematical formulation Inviscid compressible flow over slender bodies is governed by linearized compressible small-disturbance potential equation: :\phi_ + \phi_ + \phi_ = M_\infty^2 \phi_ \quad \mbox together with the small-disturbance flow-tangency boundary condition. :V_\infty n_x + \phi_y n_y + \phi_z n_z = 0 \quad \mbox M_\infty is the freestream Mach number, and n_x, n_y, n_z are the surface-normal vector components. The unknown variable is the perturbation potential \phi(x,y,z), and the total velocity is given by its gradient plus the freestream velocity V_\infty which is assumed here to be along x. :\vec = \nabla \phi + V_\infty \hat = (V_\infty + \phi_x) \hat + \phi_y \hat + \phi_z \hat The above formulation is valid only ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compressible Flow
Compressible flow (or gas dynamics) is the branch of fluid mechanics that deals with flows having significant changes in fluid density. While all flows are compressible, flows are usually treated as being incompressible when the Mach number (the ratio of the speed of the flow to the speed of sound) is smaller than 0.3 (since the density change due to velocity is about 5% in that case).Anderson, J.D., ''Fundamentals of Aerodynamics'', 4th Ed., McGraw–Hill, 2007. The study of compressible flow is relevant to high-speed aircraft, jet engines, rocket motors, high-speed entry into a planetary atmosphere, gas pipelines, commercial applications such as abrasive blasting, and many other fields. History The study of gas dynamics is often associated with the flight of modern high-speed aircraft and atmospheric reentry of space-exploration vehicles; however, its origins lie with simpler machines. At the beginning of the 19th century, investigation into the behaviour of fired bullets led to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Incompressible Flow
In fluid mechanics or more generally continuum mechanics, incompressible flow ( isochoric flow) refers to a flow in which the material density is constant within a fluid parcel—an infinitesimal volume that moves with the flow velocity. An equivalent statement that implies incompressibility is that the divergence of the flow velocity is zero (see the derivation below, which illustrates why these conditions are equivalent). Incompressible flow does not imply that the fluid itself is incompressible. It is shown in the derivation below that (under the right conditions) even compressible fluids can – to a good approximation – be modelled as an incompressible flow. Incompressible flow implies that the density remains constant within a parcel of fluid that moves with the flow velocity. Derivation The fundamental requirement for incompressible flow is that the density, \rho , is constant within a small element volume, ''dV'', which moves at the flow velocity u. Mathem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lifting-line Theory
The Prandtl lifting-line theory is a mathematical model in aerodynamics that predicts lift distribution over a three-dimensional wing based on its geometry. It is also known as the Lanchester–Prandtl wing theory. The theory was expressed independently by Frederick W. Lanchester in 1907, and by Ludwig Prandtl in 1918–1919 after working with Albert Betz and Max Munk. In this model, the bound vortex loses strength along the whole wingspan because it is shed as a vortex-sheet from the trailing edge, rather than just as a single vortex from the wing-tips. Introduction It is difficult to predict analytically the overall amount of lift that a wing of given geometry will generate. When analyzing a three-dimensional finite wing, the first approximation to understanding is to consider slicing the wing into cross-sections and analyzing each cross-section independently as a wing in a two-dimensional world. Each of these slices is called an airfoil, and it is easier to understand an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Airfoil
An airfoil (American English) or aerofoil (British English) is the cross-sectional shape of an object whose motion through a gas is capable of generating significant lift, such as a wing, a sail, or the blades of propeller, rotor, or turbine. A solid body moving through a fluid produces an aerodynamic force. The component of this force perpendicular to the relative freestream velocity is called lift. The component parallel to the relative freestream velocity is called drag. An airfoil is a streamlined shape that is capable of generating significantly more lift than drag. Airfoils can be designed for use at different speeds by modifying their geometry: those for subsonic flight generally have a rounded leading edge, while those designed for supersonic flight tend to be slimmer with a sharp leading edge. All have a sharp trailing edge. Foils of similar function designed with water as the working fluid are called hydrofoils. The lift on an airfoil is primarily the r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ludwig Prandtl
Ludwig Prandtl (4 February 1875 – 15 August 1953) was a German fluid dynamicist, physicist and aerospace scientist. He was a pioneer in the development of rigorous systematic mathematical analyses which he used for underlying the science of aerodynamics, which have come to form the basis of the applied science of aeronautical engineering. In the 1920s he developed the mathematical basis for the fundamental principles of subsonic aerodynamics in particular; and in general up to and including transonic velocities. His studies identified the boundary layer, thin-airfoils, and lifting-line theories. The Prandtl number was named after him. Early years Prandtl was born in Freising, near Munich, in 1875. His mother suffered from a lengthy illness and, as a result, Ludwig spent more time with his father, a professor of engineering. His father also encouraged him to observe nature and think about his observations. He entered the Technische Hochschule Munich in 1894 and graduated wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermann Glauert
Hermann Glauert, FRS (4 October 1892 – 6 August 1934) was a British aerodynamicist and Principal Scientific Officer of the Royal Aircraft Establishment, Farnborough until his death in 1934. Early life and education Glauert was born in Sheffield, Yorkshire; his father Louis Glauert was a cutlery manufacturer. He attended King Edward VII School, Sheffield and Trinity College, Cambridge. Career Glauert wrote numerous reports and memoranda dealing with aerofoil and propeller theory. His book, ''The Elements of Aerofoil and Airscrew Theory'' was the single most important instrument for spreading airfoil and wing theory around the English speaking world. Glauert independently developed Prandtl-Glauert method from the then-existing aerodynamic theory and published his results in '' The Proceedings of the Royal Society'' in 1928. In the 1930s, he was the academic supervisor of aerodynamicist and educationalist Gwen Alston. Death Glauert died aged 41 in an accident in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jakob Ackeret
Jakob Ackeret, FRAeS (17 March 1898 – 27 March 1981) was a Swiss aeronautical engineer. He is widely viewed as one of the foremost aeronautics experts of the 20th century. Birth and education Jakob Ackeret was born in 1898 in Switzerland. He received his diploma degree in mechanical engineering from ETH Zurich in 1920 under the supervision of Aurel Stodola. From 1921 to 1927 he worked with Ludwig Prandtl at the "Aerodynamische Versuchsanstalt" in Göttingen, witnessing a legendary period in the development of modern fluid dynamics. He received his PhD from ETH Zurich in 1927. Academic career After completing his PhD, Ackeret worked at Escher Wyss AG in Zurich as chief engineer of hydraulics, where he applied, with great success, modern aerodynamics to the design of turbines. He became a professor of Aerodynamics at ETH Zurich in 1931, where Wernher von Braun was one of his students. Research Ackeret was an expert on gas turbines and was known for his research on propel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Singularity
In mathematics, a singularity is a point at which a given mathematical object is not defined, or a point where the mathematical object ceases to be well-behaved in some particular way, such as by lacking differentiability or analyticity. For example, the real function : f(x) = \frac has a singularity at x = 0, where the numerical value of the function approaches \pm\infty so the function is not defined. The absolute value function g(x) = , x, also has a singularity at x = 0, since it is not differentiable there. The algebraic curve defined by \left\ in the (x, y) coordinate system has a singularity (called a cusp) at (0, 0). For singularities in algebraic geometry, see singular point of an algebraic variety. For singularities in differential geometry, see singularity theory. Real analysis In real analysis, singularities are either discontinuities, or discontinuities of the derivative (sometimes also discontinuities of higher order derivatives). There are four kin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prandtl–Glauert Singularity
The Prandtl–Glauert singularity is a theoretical construct in flow physics, often incorrectly used to explain vapor cones in transonic flows. It is the prediction by the Prandtl–Glauert transformation that infinite pressures would be experienced by an aircraft as it approaches the speed of sound. Because it is invalid to apply the transformation at these speeds, the predicted singularity does not emerge. The incorrect association is related to the early-20th-century misconception of the impenetrability of the sound barrier. Reasons of invalidity around Mach 1 The Prandtl–Glauert transformation assumes linearity (i.e. a small change will have a small effect that is proportional to its size). This assumption becomes inaccurate toward Mach 1 and is entirely invalid in places where the flow reaches supersonic speeds, since sonic shock waves are instantaneous (and thus manifestly non-linear) changes in the flow. Indeed, one assumption in the Prandtl–Glauert transformation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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