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Conservative versus Non-conservative forms

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Old   April 30, 2010, 13:47
Default Conservative versus Non-conservative forms
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Miguel Caro
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Hi,
i readed in the Leveque's book and the Toro's book about the difference between the conervative and non conservative formulation. But it is not clear how they explained the difference between these two forms.

Does anybody has a clear explanation of this?

Thanks.
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Old   April 30, 2010, 16:13
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I have been trying to understand this myself.

Both forms are conservation laws but the basic idea is that the mathematical character of the equation may be suited better towards different physics or have other advantages/disadvantages.

My comments won't be too helpful but here's some random information that I've gathered.

See discussion at
http://www.cfd-online.com/Forums/main/10994-conservative-non-conservative-form.html

There may also be information at
http://books.google.com/books?id=J5y...page&q&f=false
http://ocw.mit.edu/NR/rdonlyres/Eart...A870/0/ch1.pdf

For energy conservation, it is better to solve the Total Energy Equation than the Thermal Energy Equation if there are shocks. A seemingly common misconception is that only one of these forms of the equations includes friction. Both are fundamental conservation laws which are related to each other and which include friction. The thermal energy equation, for example, is the total energy equation minus the dot product of the velocity vector and conservation of momentum.

I was also told by an ANSYS engineer that
- Method is inherently conservative
- Guarantees conservation of fluxes through the control volume
- Overall solution will be conservative in nature but may not be the actual solution
- Formulation sensitive to distorted elements which can prevent convergence if such elements are in critical flow regions
Unfortunately, I do not know which form of the equation to which these apply.

Maybe this will give a starting point for issues to look out for. I'm sorry that I do not have any clear information.

Good luck.
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Old   April 30, 2010, 19:51
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The conservative form derives from expressing the conservation equations in a divergence form, i.e. the equations have the form of local time rate of change + divergence of flux = source terms. The non-conservative form is a form of the equations that is not expressible in this form. One of the principle advantages to the the conservative form is that once the equations are discretized, the flux terms "telescope", that is if you sum the fluxes into and out of a row of cells, the intercell fluxes cancel and the net flux is just the flux out of one end of the row - flux in the other end.
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Old   May 1, 2010, 08:32
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Conservative form has an advantage: its integral form allows discontinuous solutions (leading to Rankine-Hugoniot relation and correct shock speed). Non-conservative form does not have such a character (it allows only smooth differentiable solutions).

Discrete conservation is important in computing shocks. If conservation is violated, a shock may travel at a wrong speed. Successful non-conservative schemes may turn out to satisfy some form of discrete conservation.
Rather than conservation itself, it is more important to use a right conservation form (need to pick right quantities that should be conserved) since a wrong conservation form can lead to a wrong shock speed.
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Old   May 1, 2010, 17:32
Default Thank you all, it is now a little more clear
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Quote:
Originally Posted by gory View Post
Conservative form has an advantage: its integral form allows discontinuous solutions (leading to Rankine-Hugoniot relation and correct shock speed). Non-conservative form does not have such a character (it allows only smooth differentiable solutions).

Discrete conservation is important in computing shocks. If conservation is violated, a shock may travel at a wrong speed. Successful non-conservative schemes may turn out to satisfy some form of discrete conservation.
Rather than conservation itself, it is more important to use a right conservation form (need to pick right quantities that should be conserved) since a wrong conservation form can lead to a wrong shock speed.
Thank you all!!
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Old   May 3, 2010, 09:10
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Does CFX allow you to choose which form of the equations are solved?

I see that CFX-Pre allows one to choose "Thermal Energy" or "Total Energy." Which one is the conservative form?

I do not see where there is a choice for the momentum equations.

Thanks so much for any information!
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Old   May 5, 2010, 09:12
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Nicholas F Camus
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Jade the manual explains the difference between the "Thermal Energy" or "Total Energy" options. If I remember correctly the Total Energy option is just applied as a power [kg m^2 s^-3] to be applied to a domain, subdomain or boundary - the solver then calculates the heat flux accross each cell according to volume/surface ratios of the cells to the domain selected. Using the Thermal Energy option, you are required to supply the power per unit volume.

Both of these options supply the solver with an enery source term (a addtional term in the energy equation), thus the solver is working with the conservative variables. The CFX solver always works with conservatives, although transformations to primatives are bound to occur during the soultion process in order to calculate various quantities such as fluxes etc.

There is options to apply momentum source terms; again this is clearly described in the manuals.

All the best.
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Old   May 5, 2010, 09:34
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Nicholas F Camus
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I have also just read your confused posts on variable conservation. OK.

The books by R. LeVeque and F. Toro are primaraly (almost completely) concerned with hyperbolic systems. These are systems which can be cast into purely conservative form (i.e. in the form of a system of conservation laws (mass, momentum and energy)). There are man many system that can be written in this form: ideal gas dynamics, relativistic hydrodynamics, magnetohydrodynamics, and relativistic hydrodynamics (including multi-fluid systems) to name the major ones.

The physics of such hyperbolic systems is different to those parabolic systems of viscous unsteady flow that codes such as CFX are designed to deal with. The wave structure of the hyperbolic flows is crucial, as shock formation, and strong rarefraction wave can seriously effect flow dynamics (typically only very fast flows). CFX and the like are not concerned with such wave probagation as at slow speed the wave strengths are very weak indeed - hence their used of a very feable (but robust) upwind scheme.

Where the variable conversion comes into such hyperbolic code is in the calculation of the intercell fluxes, the calculation of which often needs sound speed and primative variables - density, pressure ect. The solution for these intercell fluxes for the hyperbolic systems is not streight forward, and it is not possible to use the physical fluxes F = Variable*velocity. This is what the Riemann problem is based around, and this is well documented in the above books.

The type of variable used has no bairing on the shock capturing ability of the scheme in question. The ability to resolve strong shocks in such system is wholly dependent on the Riemann solver used. For Riemann solvers there are two main types Exact solutions (only for hydrodynamics and relativistic HD), and approxiamte solutions (the cleaver stuff). The second of these can be split into linear (Roe-type (not quite right)) Riemann solvers using eigenvalue formulations, which can use primative varibles as well as conservatives, and basic approxiamte solvers which use conservatives and are based around conservative formulations (see HLL Riemann solver for a popular example).

Please feel free to ask any more questions. I think this should be enough for you to read up on for now!

All the best, and good luck.
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Old   May 6, 2010, 08:36
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Thanks Killercam. Actually, I am simply asking which equations, "Thermal Energy" or "Total Energy" is considered the conservative form. DO you know?

Also I do not see where there is a choice for the momentum equations in the software.

Thanks for any information.
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Old   May 6, 2010, 09:01
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Nicholas F Camus
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In CFX these are just terms used for specifing energy sources. If you are asking about equations concerned with purely thermal energy, and total energy...

They are both energy equations and energy is a conservative quantity. As for their form, both when dervived correctly should be fully conservative, for any finite volume method.

In CFX - if I can remember, here goes. For any boundary (Inlet, outlet etc.) you can specify boundary source terms for continuity, energy and turbulence. You can also specify momentum sources by introducing a sub-domain into you fluid domain. Sources - Boundary Source - Sources - and I think then use sources for continuity. Good luck.
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Old   May 6, 2010, 09:33
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In that case, all the conservation equations are conservative? What is meant by the conservative and non-conservative forms then?
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Old   July 20, 2010, 11:13
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hey anyone please explain about the non conservative form also ..
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