[rere]

I've been wracking my brain over this question, can anyone shed some light:

Convert 3 dimensional unsteady compressible NS equations to axisymmetric 2 dimensional incompressible laminar form for a cylindrical pipe.ignore gravity

I figured I need to use the cylinderical form of NS equations, which is r, theta and z component. But then I'm lost...

[FMDenaro]

if I am right, a typical set of equations is described in:

J. Fluid Mech., 179, 179-299, 1987

[rere]

Originally Posted by FMDenaro

if I am right, a typical set of equations is described in:

J. Fluid Mech., 179, 179-299, 1987

Thanks FMDenaro,

I can't seem to access the journal, can you suggest another source?

[FMDenaro]

many good textbook illustrate the set of equations you want ...
Maybe, you can find them aslo in the wiki page

[FMDenaro]

you can start from here http://ingforum.haninge.kth.se/armin..._STOKES_EQ.pdf

and setting the symmetry constraint

[mprinkey]

Originally Posted by rere

I've been wracking my brain over this question, can anyone shed some light:

Convert 3 dimensional unsteady compressible NS equations to axisymmetric 2 dimensional incompressible laminar form for a cylindrical pipe.ignore gravity

I figured I need to use the cylinderical form of NS equations, which is r, theta and z component. But then I'm lost...

Since you are to start from unsteady compressible NS equations, you need to do more than just coordinate transformation and reduction. Start with the compressible equation set in cylindrical coordinates. Then you need to zero out all of the transient terms, take the limit as the density goes to a constant and div U goes to zero. Axisymmetry implies that all theta derivative terms go to zero as well. And eliminate the body force term.

The mass equation will collapse to div U = 0. The energy equation will decouple because there is no density dependence on temperature. The momentum equation will lose the second coefficient of viscosity term because div U = 0. Note that U in the theta direction may be non-zero (swirl velocity), so only d/dtheta terms vanish, not all U_theta terms.

[FMDenaro]

YEs, I agree ... It is also quite common to start from the set of equations in general vectorial expression, setting all hypotheses (constant density, constant temperature, steady flow, etc.) and derive the vector form of the incompressible equation.

Then the vectors and nabla operators can be expressed in the reference system you want to use

[rere]

Originally Posted by mprinkey

Since you are to start from unsteady compressible NS equations, you need to do more than just coordinate transformation and reduction. Start with the compressible equation set in cylindrical coordinates. Then you need to zero out all of the transient terms, take the limit as the density goes to a constant and div U goes to zero. Axisymmetry implies that all theta derivative terms go to zero as well. And eliminate the body force term.

The mass equation will collapse to div U = 0. The energy equation will decouple because there is no density dependence on temperature. The momentum equation will lose the second coefficient of viscosity term because div U = 0. Note that U in the theta direction may be non-zero (swirl velocity), so only d/dtheta terms vanish, not all U_theta terms.

mprinkey, this is the question i mentioned in the PM. Thank you.