[saravana_ku]

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 ?

[nkalkote]

Originally Posted by saravana_ku

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 ?

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).

[FMDenaro]

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.

[dahmen]

I also have the same problem.

[praveen]

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.

[FMDenaro]

Originally Posted by praveen

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.

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.

[dahmen]

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.

[qutadah.r]

please see next comment.

[qutadah.r]

Originally Posted by praveen

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.

I dont understand you point, the momentum, mass, or energy is also conserved across an integral volume, (using whatever domain and boundaries I use for my finite volume), which could be cylindrical or spherical coordinates.... How is that not possible with cylindrical coordinates as you say?


Thanks!


Qutadah