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Localized & non-localized quantity

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Old   September 25, 2013, 23:32
Default Localized & non-localized quantity
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Hi friends;

I read the book about turbulence in which it is mentioned that U (velocity) is a non-localized quantity, whereas vorticity is a localized quantity. Does anyone know that what localized quantity means? what characteristics it has? and also the differences between these two type quantities?

thanks in advance.
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Old   September 26, 2013, 23:19
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The concept in discussion is what is called a local or a non-local property of a field. That means that, if the property in discussion is local, its value can be defined in each point of a field and its value only depends on the values of other properties/variables in the point of study or observation. In other words, everything is locally determined by the values in each point.

A non-local property of a field is that where its value in each point in the field depends on the values of properties or variables all over the space where the field is defined. Due to the fact that for an incompressible fluid, the pressure field satisfies a Laplace equation with a source term (Poissons equation), it is possible to formally solve the pressure field as a function of the source term by means of the Biot-Savart law (really it is a Green's function). The result is an integral over all the definition space of the pressure field and the value of the pressure at each point depends on values of other properties/variables all over this definition space. Therefore, pressure is a non-local variable and so is velocity since it depends on the gradient of pressure, the gradient of this volume integral over the definition space. The vorticity equation has not a direct dependence on pressure and, therefore, vorticity is a local variable.
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Old   September 27, 2013, 03:46
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I suppose that the mathematical classification of a PDE can also be considered... hyperbolic and parabolic equations have real characteristic curves and the properties depends only upon specific portion of the domaina. Elliptic equations, such as the pressure equation, have no real characteristic curves so that the dependence is upon the whole integration domain.
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Old   September 28, 2013, 03:27
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Quote:
Originally Posted by FMDenaro View Post
I suppose that the mathematical classification of a PDE can also be considered... hyperbolic and parabolic equations have real characteristic curves and the properties depends only upon specific portion of the domaina. Elliptic equations, such as the pressure equation, have no real characteristic curves so that the dependence is upon the whole integration domain.
Dear FMDenaro;
thank you for your comment. I thought about your opinion, but I think it can not be governed for all equation. for example, vorticity equation (D omega/D t=(Omega.Grad) U+no (Laplacian Omega)) which is elliptic, but vorticity is local quantity. do you think so?
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Old   September 28, 2013, 03:31
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Quote:
Originally Posted by mb.pejvak View Post
Dear FMDenaro;
thank you for your comment. I thought about your opinion, but I think it can not be governed for all equation. for example, vorticity equation (D omega/D t=(U.Grad) Omega-no (Laplacian Omega)) which is elliptic, but vorticity is local quantity. do you think so?

Vorticity equation is not elliptic in the unsteady formulation
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Old   September 28, 2013, 22:15
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Quote:
Originally Posted by FMDenaro View Post
Vorticity equation is not elliptic in the unsteady formulation
thank you for the point. you are right, but how can explain U as a non-localized quantity based on its equation. NS can be parabolic and hyperbolic, but based on what I understand U is non-localized. is it only because of pressure gradient in NS equation?
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