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Kármán–Howarth Equation
In isotropic turbulence the Kármán–Howarth equation (after Theodore von Kármán and Leslie Howarth 1938), which is derived from the Navier–Stokes equations, is used to describe the evolution of non-dimensional longitudinal autocorrelation. Mathematical description Consider a two-point velocity correlation tensor for homogeneous turbulence : R_(\mathbf,t) = \overline. For isotropic turbulence, this correlation tensor can be expressed in terms of two scalar functions, using the invariant theory of full rotation group, first derived by Howard P. Robertson in 1940, :R_(\mathbf,t) = u'^2 \left\, \quad f(r,t) = \frac, \quad g(r,t) = \frac where u' is the root mean square turbulent velocity and u_1,\ u_2, \ u_3 are turbulent velocity in all three directions. Here, f(r) is the longitudinal correlation and g(r) is the lateral correlation of velocity at two different points. From continuity equation, we have :\frac=0 \quad \Rightarrow \quad g(r,t) = f(r,t) + \frac \fracf(r,t) Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isotropic
Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence '' anisotropy''. ''Anisotropy'' is also used to describe situations where properties vary systematically, dependent on direction. Isotropic radiation has the same intensity regardless of the direction of measurement, and an isotropic field exerts the same action regardless of how the test particle is oriented. Mathematics Within mathematics, ''isotropy'' has a few different meanings: ; Isotropic manifolds: A manifold is isotropic if the geometry on the manifold is the same regardless of direction. A similar concept is homogeneity. ; Isotropic quadratic form: A quadratic form ''q'' is said to be isotropic if there is a non-zero vector ''v'' such that ; such a ''v'' is an isotropic vector or null vector. In complex geometry, a line through the origin in the direction of an isotropic vecto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conservation Of Angular Momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed system remains constant. Angular momentum has both a direction and a magnitude, and both are conserved. Bicycles and motorcycles, frisbees, rifled bullets, and gyroscopes owe their useful properties to conservation of angular momentum. Conservation of angular momentum is also why hurricanes form spirals and neutron stars have high rotational rates. In general, conservation limits the possible motion of a system, but it does not uniquely determine it. The three-dimensional angular momentum for a point particle is classically represented as a pseudovector , the cross product of the particle's position vector (relative to some origin) and its momentum vector; the latter is in Newtonian mechanics. Unlike linear momentum, angular mom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equations Of Fluid Dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) and hydrodynamics (the study of liquids in motion). Fluid dynamics has a wide range of applications, including calculating forces and moments on aircraft, determining the mass flow rate of petroleum through pipelines, predicting weather patterns, understanding nebulae in interstellar space and modelling fission weapon detonation. Fluid dynamics offers a systematic structure—which underlies these practical disciplines—that embraces empirical and semi-empirical laws derived from flow measurement and used to solve practical problems. The solution to a fluid dynamics problem typically involves the calculation of various properties of the fluid, such as flow velocity, pressure, density, and temperature, as functions of space and time. Be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chandrasekhar Invariant
Chandrasekhar, Chandrashekhar or Chandra Shekhar is an Indian name and may refer to a number of individuals. The name comes from the name of an incarnation of the Hindu god Shiva. In this form he married the goddess Parvati. Etymologically, the name comes from the Sanskrit words "चन्द्र (''candra'')", meaning "moon", and "शेखर (''śekhara'')", meaning "crest" or "crown", which is an epithet of the Shiva. Notable people with this name In politics and activism: * Chandra Shekhar Azad (1906–1931), known as Azad ("The Free"), Indian revolutionary who organised the Hindustan Socialist Republican Army (HSRA) * K. M. Chandrasekhar (born 1948), Indian Cabinet Secretary * Sripati Chandrasekhar (1918–2001), an Indian demographer and politician * Chandra Shekhar Dubey (born 1946), member of the 14th Lok Sabha of India * Chandrashekhar Prasad (died 1997), assassinated Indian student activist * Kalvakuntla Chandrashekar Rao (born 1954), known as K.C.R., Indian politic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Batchelor–Chandrasekhar Equation
The Batchelor–Chandrasekhar equation is the evolution equation for the scalar functions, defining the two-point velocity correlation tensor of a homogeneous axisymmetric turbulence, named after George Batchelor and Subrahmanyan Chandrasekhar. They developed the theory of homogeneous axisymmetric turbulence based on Howard P. Robertson's work on isotropic turbulence using an invariant principle. This equation is an extension of Kármán–Howarth equation from isotropic to axisymmetric turbulence. Mathematical description The theory is based on the principle that the statistical properties are invariant for rotations about a particular direction \boldsymbol (say), and reflections in planes containing \boldsymbol and perpendicular to \boldsymbol. This type of axisymmetry is sometimes referred to as strong axisymmetry or axisymmetry in the strong sense, opposed to ''weak axisymmetry'', where reflections in planes perpendicular to \boldsymbol or planes containing \boldsymbol are no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Andrei Monin
Andrei Sergeevich Monin (russian: Андре́й Серге́евич Мо́нин; 2 July 1921 – 22 September 2007) was a Russian physicist, applied mathematician, and oceanographer. Monin was known for his contributions to statistical theory of turbulence and atmospheric physics. He served as the Director of the P.P. Shirshov Institute of Oceanology of the USSR Academy of Sciences. He was instrumental in developing the Shirshov Institute into one of the largest scientific centers for ocean and earth science studies. The Monin–Obukhov similarity theory and the Monin–Obukhov Length are named after Monin and Russian Academician Alexander Mikhailevich Obukhov. Life and work Monin was born on 2 July 1921 in Moscow to S. A. Monin, an Assistant Professor of the Moscow Pedagogical Institute. He joined the Mechanical and Mathematical Faculty of the Lomonosov Moscow State University in 1938 and received his bachelor's degree in 1942. Monin then enrolled for the post-graduate prog ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Philip Saffman
Philip Geoffrey Saffman FRS (19 March 1931 – 17 August 2008) was a mathematician and the Theodore von Kármán Professor of Applied Mathematics and Aeronautics at the California Institute of Technology.. Education and early life Saffman was born to a Jewish family in Leeds, England, and educated at Roundhay Grammar School and Trinity College, Cambridge which he entered aged 15. He received his Bachelor of Arts degree in 1953, studied for Part III of the Cambridge Mathematical Tripos in 1954 and was awarded his PhD in 1956 for research supervised by George Batchelor. Career and research Saffman started his academic career as a lecturer at the University of Cambridge, then joined King's College London as a Reader. Saffman joined the Caltech faculty in 1964 and was named the Theodore von Kármán Professor in 1995. According to Dan Meiron, Saffman "really was one of the leading figures in fluid mechanics," and he influenced almost every subfield of that discipline. He is kno ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Evgeny Lifshitz
Evgeny Mikhailovich Lifshitz (russian: Евге́ний Миха́йлович Ли́фшиц; February 21, 1915, Kharkiv, Russian Empire – October 29, 1985, Moscow, Russian SFSR) was a leading Soviet physicist and brother of the physicist Ilya Lifshitz. Work Born into a Ukrainian Jewish family in Kharkov, Kharkov Governorate, Russian Empire (now Kharkiv, Ukraine). Lifshitz is well known in the field of general relativity for coauthoring the BKL conjecture concerning the nature of a ''generic curvature singularity''. , this is widely regarded as one of the most important open problems in the subject of classical gravitation. With Lev Landau, Lifshitz co-authored ''Course of Theoretical Physics'', an ambitious series of physics textbooks, in which the two aimed to provide a graduate-level introduction to the entire field of physics. These books are still considered invaluable and continue to be widely used. Lifshitz was the second of only 43 people ever to pass Landau's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Turbulence
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between those layers. Turbulence is commonly observed in everyday phenomena such as surf, fast flowing rivers, billowing storm clouds, or smoke from a chimney, and most fluid flows occurring in nature or created in engineering applications are turbulent. Turbulence is caused by excessive kinetic energy in parts of a fluid flow, which overcomes the damping effect of the fluid's viscosity. For this reason turbulence is commonly realized in low viscosity fluids. In general terms, in turbulent flow, unsteady vortices appear of many sizes which interact with each other, consequently drag due to friction effects increases. This increases the energy needed to pump fluid through a pipe. The onset of turbulence can be predicted by the dimensionless Rey ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lev Landau
Lev Davidovich Landau (russian: Лев Дави́дович Ланда́у; 22 January 1908 – 1 April 1968) was a Soviet-Azerbaijani physicist of Jewish descent who made fundamental contributions to many areas of theoretical physics. His accomplishments include the independent co-discovery of the density matrix method in quantum mechanics (alongside John von Neumann), the quantum mechanical theory of diamagnetism, the theory of superfluidity, the theory of second-order phase transitions, the Ginzburg–Landau theory of superconductivity, the theory of Fermi liquids, the explanation of Landau damping in plasma physics, the Landau pole in quantum electrodynamics, the two-component theory of neutrinos, and Landau's equations for ''S'' matrix singularities. He received the 1962 Nobel Prize in Physics for his development of a mathematical theory of superfluidity that accounts for the properties of liquid helium II at a temperature below (). Life Early years Landa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
