[granzer]

I have been trying to understand the conservative and non-conservative forms of the conservation equations in CFD. I do understand what they are from the various posts I have read, but all of them only ever talk about how the conservative form is useful for compressible flow and nothing is told about the non-conservative form of the equation. What is the use of the non-conservative form and where can it be used over conservative form?

[FMDenaro]

Originally Posted by granzer

I have been trying to understand the conservative and non-conservative forms of the conservation equations in CFD. I do understand what they are from the various posts I have read, but all of them only ever talk about how the conservative form is useful for compressible flow and nothing is told about the non-conservative form of the equation. What is the use of the non-conservative form and where can it be used over conservative form?

They are mathematically equivalent but not numerically. Mathematically, the quasi-linear form can have some advatage but numerically I strongly reccomend the use of the conservative form.
Note that you find in literature the term "conservative" both for the differential form (divergence form) and integral form. The letter being formally more correct as it defines the conservation by means of the unicity of the integral of the flux function.

[Ford Prefect]

They are only mathematically equivalent when density can be considered constant.

Also, the conservation form seems much more "logical" to use on a fixed mesh since it is derived on a volume fixed in space. The non-conservation form is derived for a volume moving with the flow.

[FMDenaro]

Originally Posted by Ford Prefect

They are only mathematically equivalent when density can be considered constant.

Also, the conservation form seems much more "logical" to use on a fixed mesh since it is derived on a volume fixed in space. The non-conservation form is derived for a volume moving with the flow.

No, this is not correct, you simply use the density equation to get the quasi linear form, no constraint on the density being necessary.
Just consider as example the derivation of the quasi-linear form of the 1D Euler equations for compressible flows.

[Ford Prefect]

Yes, of course you are correct, my bad.

[granzer]

Originally Posted by FMDenaro

They are mathematically equivalent but not numerically. Mathematically, the quasi-linear form can have some advatage but numerically I strongly reccomend the use of the conservative form.
Note that you find in literature the term "conservative" both for the differential form (divergence form) and integral form. The letter being formally more correct as it defines the conservation by means of the unicity of the integral of the flux function.

Ok is it that conservative form is always more advantageous? and if so what is the use of studying about non-conservative form.

[FMDenaro]

Originally Posted by granzer

Ok is it that conservative form is always more advantageous? and if so what is the use of studying about non-conservative form.

I personally suggest using always the conservative form when the problem must be solved numerically. I don't see any gain in using the quasi-linear form in the numerical discretization. Conversely, some problems are typical of such a discretization.

[Martin Hegedus]

In general, as has been stated above, the conservative form is used.

However, for higher order finite difference methods, getting the discretization "correct" (for lack of a better term) near a boundary is challenging. One method is Summation by Parts (SBP). As far as I can tell, (I don't claim to be all that knowledgeable about it) part of the premise is to minimize the error/difference between conservative and non-conservative forms. I think the motivation has something to do with energy methods involving d(u^2)/dx and an attempt is made to get d(u^2)/dx = u*du/dx numerically equal in addition to having the integral numerically equal to u^2.

[naffrancois]

Originally Posted by granzer

Ok is it that conservative form is always more advantageous? and if so what is the use of studying about non-conservative form.

Studies about non-conservative form of Euler equations is, among other applications, a step towards building numerical schemes for hyperboolic PDE's which cannot be written under conservative form. Remind that hyperbolic conservative PDE's numerical schemes can be derived "easilly" through Rankine-Hugoniot jump relations. Some fluid flow models, such as multiphase flow models of Baer-Nunziato type for example typically involve a conservative part and non-conservative products which yield a non-conservative hyperbolic set of equations for which you cannot directly apply Rankine-Hugoniot relations and build a numerical scheme in a straighforward way.

[FMDenaro]

As far as the quasi-linear form is concerned, the hystory of CFD is written by the first simulations at Los Alamos, just consider the well known MAC method wherein the quasi-linear form was well suited to develop a lagrangian update. This approach has been one of the reasons for other hystorical approaches. For example the Arakawa method for the vorticity-stream function formulation. When the computational resources were enough to solve small turbulent flow, researchers tryied to make the method energy-conserving to gain stability in the simulation.
However, modern researches showed the superiority of the conservative methods. The book of Leveque has some nice comparison between the formulations.

[randolph]

Dr. Denaro,

In the early implementation of K-Epsilon model of OpenFOAM, the following equation is used.

fvm::ddt(epsilon_)
+ fvm::div(phi_, epsilon_)
- fvm::Sp(fvc::div(phi_), epsilon_)

recall U & grad epsilon = div(U epsilon) - epsilon div(U).
It looks like they are using the quasi-linear form of the Epsilon.

Dr. Jasak indicates "boundedness of k when the model is used with simpleFoam (and alike) and partial convergence of the pressure equation" in this post "https://www.cfd-online.com/Forums/openfoam/64602-origin-fvm-sp-fvc-div-phi_-epsilon_-kepsilon-eqn.html#post217245"

I am very confused. It looks like they trying to use the quasi-linear form of Epsilon to improve the boundness???

Does the quasi-linear form have any numerical advantage over the conservative form in terms of boundness?

Thanks,
Rdf

[FMDenaro]

Originally Posted by randolph

Dr. Denaro,

In the early implementation of K-Epsilon model of OpenFOAM, the following equation is used.

fvm::ddt(epsilon_)
+ fvm::div(phi_, epsilon_)
- fvm::Sp(fvc::div(phi_), epsilon_)

recall U & grad epsilon = div(U epsilon) - epsilon div(U).
It looks like they are using the quasi-linear form of the Epsilon.

Dr. Jasak indicates "boundedness of k when the model is used with simpleFoam (and alike) and partial convergence of the pressure equation" in this post "https://www.cfd-online.com/Forums/openfoam/64602-origin-fvm-sp-fvc-div-phi_-epsilon_-kepsilon-eqn.html#post217245"

I am very confused. It looks like they trying to use the quasi-linear form of Epsilon to improve the boundness???

Does the quasi-linear form have any numerical advantage over the conservative form in terms of boundness?

Thanks,
Rdf

Kinetic energy is not subject to a conservation equation, production terms being present coupling the dissipation of kinetic energy to internal energy. Only the total energy does.

[randolph]

Dr. Denaro,

Here is another case regards to scalar transport. ("https://www.cfd-online.com/Forums/openfoam-solving/67964-scalartransportfoam-rtd-calculations.html#post280210")

In the post of chegdan, he states that :

         solve
            (
                fvc::ddt(C)
              + fvc::div(phi, C)
	      - fvc::laplacian(D, C)
              - fvc::laplacian(Dturbulent, C)
            );

was unbounded even with the bounded schemes. However, when he added another term to the equation:

            solve
            (
                fvm::ddt(C)
              + fvm::div(phi, C)
	      + fvm::SuSp(-fvc::div(phi), C)
	      - fvm::laplacian(D, C)
              - fvm::laplacian(Dturbulent, C)
            );

Is this case relevant to our discussion?

Thanks,
Rdf

[FMDenaro]

The use of the laplacian operator can be not correct in a general conservation framework ... For example, the turbulent viscosity is a pointwise function, therefore Div (2*mu_t*S), that is the conservative form, is not equivalent to mu* Lap v. The same happens for the heat flux.



However, conservation is not related to boundness in the solution