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March 10, 2000, 06:40 
Channel Flow

#1 
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1.) After an answer that I gave in this forum some time ago, I have a disagreement with a collegue: Considering a laminar flow in a channel with low Reynolds number (Re=300700) and a temperature arise of 500700 K, which form of the Navier Stokes Equations should used? Additionaly the Fluid is air and the pressure drop is negligible. Bird Lightfoot(transport phenomena) gives two Forms of the Navier Stokes equations. (Page 85 Eq. A B C) use the general Formulation with stress tensors, which can be simplified under the assumption of konstant density and give Eq. D E F. Can we use Eq. D E F to solve the flow problem pressented above? of course the density variation has to be considered or we should use the compressible form (Eq. A B C)
2.) Regardless the answer of the first question: Using the compressible Form of Navier Stokes Eq. (A B C) in cylindrical Koordintes how do we treat the stress tension terms using Finite Volumes and a collocated grid? Should we simplify them until we get many small simple terms like 4/3 m/r (dv/dr)+ ... + d/dz(m du/dr) or there is an other smarter way to treat them? how do we handle such terms containing single or mixed derivatives? 3.) Anyone here with a Literature cite or experience with compressible NS in cylindrical coordinates? Thanks in advance I.Dotsikas 

March 10, 2000, 13:06 
Re: Channel Flow

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(1). If the pressure is roughly constant, then instead of using rho=p/(R*T) to compute the density, you can use rho=p_ref/(R*T), where T comes from the solution of the energy equation. In this case, you don't have to assume that the density is constant. (2). If you are taking the Finite Volume approach, you need to use the control volume form of governing equation, then carry out the surface and volume integrals. You can find this part from the CFD text books.(3). For the finitedifference approach, you start with the differential form of the governing equations. On the other hand, you need to start with the integral form of the governing equations when using the finitevolume approach.


March 11, 2000, 23:35 
Re: Channel Flow

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Neither of the above written things give an answer to what had been asked. There would be a thousand and one postings from you now in reply to this. But sorry  I won't be able to reply, would be away for some time you see.
For your question Dostikas  I haven't come across any compressible DNS case in cylindrical coordinates. But there are a number of cases for incompressible flow, you might be already familiar: (1) R. Verzicco & P. Orlandi. J. Comp. Phys. 123, 402414, 1996. (2) J.G.M. Eggels et al. J. Fluid Mech. 268, 175, 1994. (3) J.G.M. Eggels. Ph.D. thesis. Delft Univeristy of Tech., 1994. (4) P. ORLANDI & M. FATICA. J. Fluid Mech. 343, 4372, 1997. 

March 16, 2000, 17:34 
Re: Channel Flow

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Concerning your point (3), I do work with compressible NS equations in 2D cylindrical coordinates.
Concerning point (1), do you mean the Boussinesq approximation? PG 

March 16, 2000, 20:36 
Re: Channel Flow

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For axisymmetric compressible NS, I found the following to be helpful:
Broglia, Manna, Deconick, Degrez. Development and Validation of an Axisymmetric Navier Stokes Solver for Hypersonic Flows. Von Karman Institute for Fluid Dynamics. 1995. It is a technical report. Bob 

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