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Conservation v.s. nonconservation form in incompressible flows 

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March 19, 2013, 11:18 
Conservation v.s. nonconservation form in incompressible flows

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Any particular reason to use either of the two methods when we look at viscous incompressible flow? Last edited by Simbelmynė; March 20, 2013 at 02:17. Reason: Clarification that it is with regard to viscous flow. 

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March 19, 2013, 12:24 

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


March 19, 2013, 14:20 

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

March 19, 2013, 14:33 

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Filippo Maria Denaro
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the discrete conservative form ensures a correct wave propagation


March 19, 2013, 17:08 

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March 19, 2013, 17:25 

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

March 19, 2013, 17:30 

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March 19, 2013, 17:55 

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Quote:


March 19, 2013, 17:57 

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March 19, 2013, 18:16 

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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 nonlinear term in the momentum quantity. 

March 20, 2013, 02:28 

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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. 

March 20, 2013, 02:41 

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div(u) = 0 is a constraint and it does not have any dynamics in it. The momentum equation is a convectiondiffusion equation. So the convection brings in some wave type behavior.
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March 20, 2013, 03:49 

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March 20, 2013, 04:38 

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Filippo Maria Denaro
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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 divergencefree 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... 

March 20, 2013, 05:02 

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http://www.flow3d.com/cfd101/cfd101conservation.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. 

March 20, 2013, 05:27 

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I totally disagree in what is stated in the post ... if you use a fluxbalance 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! 

March 20, 2013, 06:03 

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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 noissue 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. 

March 22, 2013, 13:16 

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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 nonconservation form can be used. FMDenaro advocates conservation form because it ensures correct wave propagation. Flow3D (commercial software) use nonconservation 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! 

March 22, 2013, 13:43 

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Filippo Maria Denaro
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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" 

March 22, 2013, 20:18 

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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! 

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