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Conservative form of Navier-Stokes equation in cylindrical coordinates |
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#1 |
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Saravana
Join Date: Nov 2016
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All most all texts that I have seen have Navier-Stokes equation in non-conservative form cylindrical coordinates. Can anyone point me to a text that has Navier-Stokes equation in conservative form for cylindrical coordinates ?
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#2 |
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Nikhil
Join Date: Mar 2016
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Conservative (form of ) equation conserves (do not changes) their form with respect to co-ordinate system. that is why form is called conservative (Also these are expressed in terms of conservative variables - variables which conserves during a fluid flow such as mass, momentum, energy).
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#3 |
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Filippo Maria Denaro
Join Date: Jul 2010
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Consider the model equation:
d (rho*phi)/dt + Div f = 0 You can integrate over the Finite Volume [r:r+dr;theta:theta+dtheta;z:z+dz] and write the equation d [(rho*phi)]/dt + 1/|V| Int [S] n.f dS =0 where [(rho*phi)] is the volume averaged variable. Now you have to express the surface integral of the fluxes over the FV. Whatever the numerical flux reconstruction is used, the flux over each area is the same for adjacent volumes so that the conservation is guaranteed. |
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#4 |
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adam
Join Date: Apr 2013
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I also have the same problem.
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#5 |
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In cylindrical coordinates, the NS equations do not have conservative form. You can come close to it but with some source terms. See, e.g. eqn (8) here
https://arxiv.org/abs/1712.07765 You can remove the gravity source terms if you dont have gravity in your problem. |
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#6 | |
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Filippo Maria Denaro
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Quote:
I take the opportunity of this answer to give my opinion about such issue. There is no conservative form in general when the pointwise differential form is used. Often "conservative form" denote actually the divergence form. The key is that the conservative property requires a finite volume with an extension of non vanishing measure. That means using the integral formulation that states (conservative property) that the time variation of the volume-averaged estensive property depends only in the integral of the flux over the surface of the volume. A fact that is implied by the transport (Reynolds) theorem. When we use the differential form we can adopt the divergence form and discretize in such a specific way (depending on the accuracy order and adopted stencil) to write a numerically conservative scheme. The integral form of the equations allow us to write a conservative scheme in any type of geometry. |
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#7 |
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adam
Join Date: Apr 2013
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Hi saravana_ku, I think that the response to your question is in the following link. But it is written in vectorial form, and for more accuracy you have to multiply by the radius (r) and inter it inside different derivatives (take care when dowing that).
https://eprints.soton.ac.uk/49523/1/...r_Sandberg.pdf. |
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#8 |
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qutadah
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please see next comment.
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#9 | |
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qutadah
Join Date: Jun 2021
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Quote:
Thanks! Qutadah |
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Tags |
cylindrical coordinates, navier stokes equations |
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