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!