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Conservation- v.s. non-conservation form in incompressible flows

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Old   March 23, 2013, 05:47
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Quote:
Originally Posted by Simbelmynė View Post
Any particular reason to use either of the two methods when we look at viscous incompressible flow?
It depends on the type of flow you are trying to solve and the methods you are using. Local or global numerical conservation is a property one normally wants to build in but obviously it cannot be done for all quantities. For example, in highly swirling flows one often wants to conserve angular momentum in order for the simulation to be reasonable but the swirl-related velocity component being solved may well be something else. Rearranging terms to conserve angular momentum will mean the equation is in non-conservation form in terms of what is being solved.

Whether a conservative or non-conservative 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 non-conserved. 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 non-conservative 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.
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Old   March 29, 2013, 12:33
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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.
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Old   March 29, 2013, 12:46
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Originally Posted by Simbelmynė View Post
Thank you for your reply, but could you please elaborate as to why in the case of finite difference?

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)
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Old   March 29, 2013, 14:11
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[QUOTE=leflix;417212]
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Originally Posted by FMDenaro View Post
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.
for the steady equation yes, but if you consider the unsteady momentum equation it is parabolic
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Old   March 29, 2013, 19:33
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Originally Posted by leflix View Post
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)
Thank you for your reply leflix.

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 non-conservation 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 non-conservation 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!
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Old   March 30, 2013, 05:19
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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 quasi-linear 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.
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Old   April 1, 2013, 04:19
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Originally Posted by FMDenaro View Post
Ok, first do not think about convection or diffusion, consider instead the total flux of some quantities as f.
Yes I understand this. If you check my previous post again you will see that I put the word "usually" when I said that conservation/non-conservation form usually comes to the treatment of the convective term. This is because the diffusive term in FDM is in conservation form (unless you are not doing something strange when discretizing that term you end up with a numerically conservative expression).

Quote:
Originally Posted by FMDenaro View Post
But often in FDM one discretizes the quasi-linear 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.
Yes. I agree (see above). However I think it is strange that Leveque is using the terminology "quasi linear form", it sounds like some ad-hoc method to be used in stead of the "real" method. If we are deriving the momentum equations we can use Newton's second law as a starting point OR we can derive the momentum equation by setting up a conservative momentum flux expression assuming momentum is a conserved property (and yes the methods both satisfy Newton's second law). In case of the first approach we will end up with a non-conservation expression where the Substantial derivative describes the acceleration terms. Going from non-conservation form to conservation form is done with the aid of the continuity equation.

Quote:
Originally Posted by FMDenaro View Post
Said that, my advice is to use always the conservative form.
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 non-conservation form is preferred for incompressible flows (and why).
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