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Conservation v.s. nonconservation form in incompressible flows 

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March 23, 2013, 05:47 

#21  
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andy
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Whether a conservative or nonconservative form is better behaved can vary from term to term within a transport equation depending on what they physically represent and what else is conserved or nonconserved. For example, in curvilinear grid orientated coordinates one can be faced with a choice of something like (area_ef*f_ef  area_wf*f_wf) or (vol_p*dfdi_p). For a constant value of f the former will not be constant if the face areas vary whereas the nonconservative form will be. Not that people use curvilinear grid orientated coordinates much these days. If the coefficient matrix is singular like the pressure related variable in many pressure correction schemes then a conservation form has to be used for the rhs in order to get a solution. Starting with a conservation form and moving to something else if it misbehaves in testing is probably not an unwise way to go about things. 

March 29, 2013, 12:33 

#22 
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[QUOTE=FMDenaro;415111]in the case of incompressible flows, the momentum equation is parabolic but the continuity equation (div V=0) is hyperbolic.
Filipo I guess momentum equation for incompressible flows is rather elliptic. 

March 29, 2013, 12:46 

#23  
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It is due to the way finite volume method is built and how the Gauss theorem is used for integration. As example if you consider the laplacian operator integral[ lap(FI)]dv = integral[ Div(grad(FI))]dv = integral [grad(FI).n]ds It is the way you should proceed with finite volume method. In finite difference you would write: lap(FI) = d²FI/dx² + d²FI/dy² and you discretize the second derivative operator directlywith any scheme you like. Or you could use also d/dx (grad(FI)_x ) + d/dy(grad(FI)_y) which is the divergence operator applied to the vector grad(FI). And then you use any scheme for first derivative operator d/dx and d/dy. grad(FI)_x ) and (grad(FI)_y are respectively the x and y componnent of vector grad(FI) 

March 29, 2013, 14:11 

#24 
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Filippo Maria Denaro
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[QUOTE=leflix;417212]for the steady equation yes, but if you consider the unsteady momentum equation it is parabolic


March 29, 2013, 19:33 

#25  
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Unfortunately I don't understand it at all. I'll try to explain why and hopefully you can help me understand. 1. Since you replied to my previous post (which was a reply to your post before that) I assume that this post is to explain "why we can use either conservation or nonconservation form when using FDM". In this context I also assume that you are replying to my original question. 2. You explain how conservation is assured in case of FVM. Is this so that I should understand something about point 1 above? 3. Then you explain that we could use any FDM scheme when approximating the first and/or second derivatives. Your example is not from the convective terms and since it is in the treatment of the convective terms we (usually) have the difference between conservation and nonconservation form I just don't follow this reasoning. 4. What makes me even more confused is that FMDenaro likes the post which means that he agrees (I assume) although when looking at his posts in this thread he seems to be of the opinion that conservation form should always be used. I hope that I am not sounding rude in this post, this is not my intention. I really am confused about some of the posts though. Well perhaps I can blame it on the late hour Good Night! 

March 30, 2013, 05:19 

#26 
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Filippo Maria Denaro
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Ok, first do not think about convection or diffusion, consider instead the total flux of some quantities as f.
Now, in FVM it is automatic to ensure the discrete conservation since we discretize the integral: Int [S] n.f dS => sum [k=1..Nsurfaces] (fn A)_k and the numerical flux function fn is unique by construction. Some controversial appears for the FDM. Again, I think is useful to consider the discretization of the continuos term. If you discretize the divergence form, you have on cartesian grids Div.f => sum df_i/dx_i for example, in the momentun quantity, the convective and diffusive terms are written as Div.(vv) and Div.(2mu S) and the discretization are conservative if the numerical flux functions are unique. But often in FDM one discretizes the quasilinear form, for example in the (incompressible flow) momentun quantity the convective and diffusive terms are written as v.Grad v and mu Lap v and no discretizations can ensure conservation. Said that, my advice is to use always the conservative form. 

April 1, 2013, 04:19 

#27  
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Yes and this is in reply to my original question without the "why" part. I guess you have already mentioned the "why" in your previous posts. To summarize: I am not confused about how the FDM and FVM methods discretize the equations. I simply wish to know if conservation or nonconservation form is preferred for incompressible flows (and why). 

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