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

Hey,

Any particular reason to use either of the two methods when we look at viscous incompressible flow?

Telescoping of fluxes under a conservation form makes it easier to satisfy a divergence-free condition. At least that has been my experience.

What do you mean by easier? Will the solution of the poisson equation for pressure converge faster?

(Both methods should be divergence free upon convergence right?)

the discrete conservative form ensures a correct wave propagation

Are you talking about surface waves?

no, I am talking about convective waves... a good example is the Burgers equation:

- quasi-linear form: du/dt + u du/dx =0
- divergence form: du/dt + d/dx (u^2/2) = 0

in the continuous form such equations are mathematically equivalent but differences appear in the discretizations of the two forms, especially for high wavenumbers. That means for example a good or not description of turbulent waves.
In the book of Leveque you can find an example

If you use finite difference method you can use either of them.
For finite volume method conservative form (divergence form) is the must.

How does the incompressibility constraint affect such an equation (Burger's equation reduces to du/dt=0?)? The book by Leveque deals primarily with hyperbolic equations and shock waves not with incompressible systems. Am I missing something here?

Thank you for your reply, but could you please elaborate as to why in the case of finite difference?

in the case of incompressible flows, the momentum equation is parabolic but the continuity equation (div V=0) is hyperbolic.
In practice, the Burgers equation is a simple model to understand the formation of high gradients in the velocity field as those creating by the non-linear term in the momentum quantity.

Ok this is interesting. I have never seen this mentioned in any textbook. Usually we describe viscous flows as either elliptic (steady) or parabolic (unsteady). Inviscid flows can be elliptic or hyperbolic (steady) or hyperbolic (unsteady).

I have three questions

1. Is div U = 0 really hyperbolic? It seems elliptic to me.

2. If it is hyperbolic, does it mean that we are trying to solve a hyperbolic system although the pressure Poisson equation is elliptic? I don't understand how this works. Pressure disturbances are transmitted all across the domain at infinite speed in case of incompressible flow so there is no domain of dependence/domain of influence.

3. I have changed my original question so that it is clear that it is viscous flow I am interested in, i.e. a parabolic (or elliptic) system. How would your answer be in this case for my original question?


Thank you everyone for a nice discussion. :)

div(u) = 0 is a constraint and it does not have any dynamics in it. The momentum equation is a convection-diffusion equation. So the convection brings in some wave type behavior.

Thank you Praveen for you input, so how would you respond to my original question?

The original Burgers paper is dedicated to viscous flow, he treated the equation as a sample model for turbulence..
The continuity equation is intrinsically hyperbolic both for viscous and non viscous flows. The elliptic character "appears" under trasformation of the divergence-free constraint Div V= 0 in terms of the pressure equation Div(Grad phi) = q. The acustic waves are therefore "modelled" such as having infinite travelling velocity. But the convective waves have finite velocity and must be numerically well resolved. This is a typical issue in turbulence for example, owing to high gradients in the flow...
It is well known that discrete conservative formulations ensure a correct (convective) waves propagation...

Ok, but there seem to be some discrepancy in the subject. For instance:

http://www.flow3d.com/cfd-101/cfd-101-conservation.html

Particularly the part with unstructured grids.

I fully agree that conservation form is good when we have extremely sharp gradients, but it seems that there is more to it than just using conservation form all the time.

I totally disagree in what is stated in the post ... if you use a flux-balance for developing the conservative formulation, the numerical flux function is unique by construction and the method ensures conservation of the resolved variable on any type of grid!

I read it as:

On unstructured grids using conservation form the conservation is still ensured, however accuracy is not (if first order approximations are used).

Perhaps this is a no-issue since we generally do not want to use first order approximations anyway.

I don't know if their statement is correct or not, but I think it is worth discussing.

Ok to summarize this discussion so far:

agd advocates conservation form because it makes it easier to satisfy the divergence free condition.

lefix says conservation form is a must for FVM, but in the case of FDM either conservation or non-conservation form can be used.

FMDenaro advocates conservation form because it ensures correct wave propagation.

Flow3D (commercial software) use non-conservation form on unstructured grids.

Could anyone point me to a benchmark that can test the statements by agd and FMDenaro? Will standard test cases be enough (cavity flow, backward facing step, flow over cylinder)?

Even better, if someone can point me to a paper that illuminates these matters? :)

Have a nice weekend!

some years ago we performed spectral analysis about this issue:
http://onlinelibrary.wiley.com/doi/1...d.179/abstract

see also § 12.9 in the book of Leveque "Finite Volume Methods for Hypoerbolic Problems"

Thank you for the references. I am at home so I can not check the paper but I Will on Monday.

Regarding the book reference, yes I understand that in the case of a discontinuous solution we are better off using conservation form. And from your previous posts I understand that we should always expect discontinuous solutions in most incompressible flows and hence always use conservation form. Correct?

Cheers!

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.

[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.

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)

[QUOTE=leflix;417212]for the steady equation yes, but if you consider the unsteady momentum equation it is parabolic ;)

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. :confused:

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!

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.

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).

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.

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).