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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] |
Flow Velocity
In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the flow velocity vector is scalar, the ''flow speed''. It is also called velocity field; when evaluated along a line, it is called a velocity profile (as in, e.g., law of the wall). Definition The flow velocity ''u'' of a fluid is a vector field : \mathbf=\mathbf(\mathbf,t), which gives the velocity of an '' element of fluid'' at a position \mathbf\, and time t.\, The flow speed ''q'' is the length of the flow velocity vector :q = \, \mathbf \, and is a scalar field. Uses The flow velocity of a fluid effectively describes everything about the motion of a fluid. Many physical properties of a fluid can be expressed mathematically in terms of the flow velocity. Some common examples follow: Steady flow The flow of a fluid is sai ... [...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] |
Fluid Dynamics
In physics, physical chemistry and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids – liquids and gases. It has several subdisciplines, including (the study of air and other gases in motion) and (the study of water and other liquids in motion). Fluid dynamics has a wide range of applications, including calculating forces and moment (physics), moments on aircraft, determining the mass flow rate of petroleum through pipeline transport, pipelines, weather forecasting, predicting weather patterns, understanding nebulae in interstellar space, understanding large scale Geophysical fluid dynamics, geophysical flows involving oceans/atmosphere and Nuclear weapon design, 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 fl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sobolev Space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, i.e. a Banach space. Intuitively, a Sobolev space is a space of functions possessing sufficiently many derivatives for some application domain, such as partial differential equations, and equipped with a norm that measures both the size and regularity of a function. Sobolev spaces are named after the Russian mathematician Sergei Sobolev. Their importance comes from the fact that weak solutions of some important partial differential equations exist in appropriate Sobolev spaces, even when there are no strong solutions in spaces of continuous functions with the derivatives understood in the classical sense. Motivation In this section and throughout the article \Omega is an open subset of \R^n. There are man ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuum Mechanics
Continuum mechanics is a branch of mechanics that deals with the deformation of and transmission of forces through materials modeled as a ''continuous medium'' (also called a ''continuum'') rather than as discrete particles. Continuum mechanics deals with ''deformable bodies'', as opposed to rigid bodies. A continuum model assumes that the substance of the object completely fills the space it occupies. While ignoring the fact that matter is made of atoms, this provides a sufficiently accurate description of matter on length scales much greater than that of inter-atomic distances. The concept of a continuous medium allows for intuitive analysis of bulk matter by using differential equations that describe the behavior of such matter according to physical laws, such as mass conservation, momentum conservation, and energy conservation. Information about the specific material is expressed in constitutive relationships. Continuum mechanics treats the physical properties of ... [...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] |
Atmospheric Circulation
Atmospheric circulation is the large-scale movement of Atmosphere of Earth, air and together with ocean circulation is the means by which thermal energy is redistributed on the surface of the Earth. The Earth's atmospheric circulation varies from year to year, but the large-scale structure of its circulation remains fairly constant. The smaller-scale weather systems – middle latitudes, mid-latitude low-pressure area, depressions, or tropical convective cells – occur chaotically, and long-range weather predictions of those cannot be made beyond ten days in practice, or a month in theory (see chaos theory and the butterfly effect). The Earth's weather is a consequence of its illumination by the Sun and the laws of thermodynamics. The atmospheric circulation can be viewed as a heat engine driven by the Sun's energy and whose heat sink, energy sink, ultimately, is the blackness of space. The work produced by that engine causes the motion of the masses of air, and in that process ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Viscosity
Viscosity is a measure of a fluid's rate-dependent drag (physics), resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for example, syrup has a higher viscosity than water. Viscosity is defined scientifically as a force multiplied by a time divided by an area. Thus its SI units are newton-seconds per metre squared, or pascal-seconds. Viscosity quantifies the internal friction, frictional force between adjacent layers of fluid that are in relative motion. For instance, when a viscous fluid is forced through a tube, it flows more quickly near the tube's center line than near its walls. Experiments show that some stress (physics), stress (such as a pressure difference between the two ends of the tube) is needed to sustain the flow. This is because a force is required to overcome the friction between the layers of the fluid which are in relative motion. For a tube ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Incompressible Flow
In fluid mechanics, or more generally continuum mechanics, incompressible flow is a flow in which the material density does not vary over time. Equivalently, the divergence of an incompressible flow velocity is zero. Under certain conditions, the flow of compressible fluids can be modelled as incompressible flow to a good approximation. 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. Mathematically, this constraint implies that the material derivative (discussed below) of the density must vanish to ensure incompressible flow. Before introducing this constraint, we must apply the conservation of mass to generate the necessary relations. The mass is calculated by a volume integral of the density, \rho : : = . The conservation of mass requires that the time derivative of the mass inside a control volume be equal to the mass flux, J, acro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Seminorm
In mathematics, particularly in functional analysis, a seminorm is like a Norm (mathematics), norm but need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some Absorbing set, absorbing Absolutely convex set, disk and, conversely, the Minkowski functional of any such set is a seminorm. A topological vector space is locally convex if and only if its topology is induced by a family of seminorms. Definition Let X be a vector space over either the real numbers \R or the Complex number, complex numbers \Complex. A real-valued function p : X \to \R is called a if it satisfies the following two conditions: # Subadditive function, Subadditivity/Triangle inequality: p(x + y) \leq p(x) + p(y) for all x, y \in X. # Homogeneous function, Absolute homogeneity: p(s x) =, s, p(x) for all x \in X and all scalars s. These two conditions imply that p(0) = 0If z \in X denotes the zero vector in X while 0 denote the zer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Potential Density
The potential density of a fluid parcel at pressure P is the density that the parcel would acquire if adiabatically brought to a reference pressure P_, often 1 bar (100 kPa). Whereas density changes with changing pressure, potential density of a fluid parcel is conserved as the pressure experienced by the parcel changes (provided no mixing with other parcels or net heat flux occurs). The concept is used in oceanography and (to a lesser extent) atmospheric science. Potential density is a dynamically important property: for static stability potential density must decrease upward. If it doesn't, a fluid parcel displaced upward finds itself lighter than its neighbors, and continues to move upward; similarly, a fluid parcel displaced downward would be heavier than its neighbors. This is true even if the density of the fluid decreases upward. In stable conditions (potential density decreasing upward) motion along surfaces of constant potential density (isopycnals) is energetically fav ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Navier–Stokes Equations
The Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances. They were named after French engineer and physicist Claude-Louis Navier and the Irish physicist and mathematician George Gabriel Stokes. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842–1850 (Stokes). The Navier–Stokes equations mathematically express momentum balance for Newtonian fluids and make use of conservation of mass. They are sometimes accompanied by an equation of state relating pressure, temperature and density. They arise from applying Isaac Newton's second law to fluid motion, together with the assumption that the stress in the fluid is the sum of a diffusing viscous term (proportional to the gradient of velocity) and a pressure term—hence describing ''viscous flow''. The difference between them and the closely related Euler equations is that Navier–Stokes equat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
