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Palinstrophy
Palinstrophy is the curl of the vorticity. It is defined as \frac \left( \nabla \times \omega \right), where \omega is the vorticity. Palinstrophy is mainly used in turbulence study, where there is a need to quantify how vorticity is transferred from one direction to the others. It is closely related to enstrophy In fluid dynamics, the enstrophy \mathcal can be interpreted as another type of potential density; or, more concretely, the quantity directly related to the kinetic energy in the flow model that corresponds to dissipation effects in the fluid. It i ..., the latter being more equivalent to the "power" of vorticity. References *Lesieur, Marcel. "Turbulence in fluids: stochastic and numerical modeling." NASA STI/Recon Technical Report A 91 (1990): 24106. *Pouquet, A., et al. "Evolution of high Reynolds number two-dimensional turbulence." Journal of Fluid Mechanics 72 (1975): 305-319. *{{cite book, title=Turbulence: An Introduction for Scientists and Engineers, author=D ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vorticity
In continuum mechanics, vorticity is a pseudovector (or axial vector) field that describes the local spinning motion of a continuum near some point (the tendency of something to rotate), as would be seen by an observer located at that point and traveling along with the flow. It is an important quantity in the dynamical theory of fluids and provides a convenient framework for understanding a variety of complex flow phenomena, such as the formation and motion of vortex rings. Mathematically, the vorticity \boldsymbol is the curl of the flow velocity \mathbf v: :\boldsymbol \equiv \nabla \times \mathbf v\,, where \nabla is the nabla operator. Conceptually, \boldsymbol could be determined by marking parts of a continuum in a small neighborhood of the point in question, and watching their ''relative'' displacements as they move along the flow. The vorticity \boldsymbol would be twice the mean angular velocity vector of those particles relative to their center of mass, orie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Curl (mathematics)
In vector calculus, the curl, also known as rotor, is a vector operator that describes the Differential (infinitesimal), infinitesimal Circulation (physics), circulation of a vector field in three-dimensional Euclidean space. The curl at a point in the field is represented by a vector (geometry), vector whose length and direction denote the Magnitude (mathematics), magnitude and axis of the maximum circulation. The curl of a field is formally defined as the circulation density at each point of the field. A vector field whose curl is zero is called irrotational. The curl is a form of derivative, differentiation for vector fields. The corresponding form of the fundamental theorem of calculus is Kelvin–Stokes theorem, Stokes' theorem, which relates the surface integral of the curl of a vector field to the line integral of the vector field around the boundary curve. The notation is more common in North America. In the rest of the world, particularly in 20th century scientific li ... [...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 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. The onset of turbulence can be predicted by the dimensionless Reynolds number, the ratio of kinetic energy to viscous damping ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Enstrophy
In fluid dynamics, the enstrophy \mathcal can be interpreted as another type of potential density; or, more concretely, the quantity directly related to the kinetic energy in the flow model that corresponds to dissipation effects in the fluid. It is particularly useful in the study of turbulent flows, and is often identified in the study of thrusters as well as in combustion theory and meteorology. Given a domain \Omega \subseteq \R^n and a once-weakly differentiable vector field u \in H^1(\R^n)^n which represents a fluid flow, such as a solution to the Navier-Stokes equations, its enstrophy is given by:where , \nabla \mathbf, ^2 = \sum_^n \left, \partial_i u^j \^2 . This quantity is the same as the squared seminorm , \mathbf, _^2of the solution in the Sobolev space H^1(\Omega)^n. Incompressible flow In the case that the flow is incompressible, or equivalently that \nabla \cdot \mathbf = 0 , the enstrophy can be described as the integral of the square of the vorticity \mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
