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 C-H Kuo November 11, 1998 10:20

What is Density-Based method?

This is a great place for post-school learning. I appreciate the large and diversified knowledge from this website.

I have been working on CFD several years, and once a while I hear people talking about pressure-based and density-based solution methods. I really don't know what it is, could not find from literature, and felt shame to ask. I asked several times and did not get it. I think it is an ambiguous term or the person I asked is same as me--don't really know it.

Could someone give a detailed description on this method, and the advantage and disadvantage of it. I will appreciate this, and I belive many out there will also appreciate it.

 Hongjun Li November 11, 1998 13:09

Re: What is Density-Based method?

There is realy no explict difinition of 'density-based' or 'pressure based' (or as far as I know). But the two solution methods are quite different. The main difference is how to link the continuity equation to the other equations (momentum and energy) so that the conservations of mass, momentum and energy are satisfied simutanously during the course of solution.

The 'density-based' (DB in short) methods are widely used in compressible N-S equations. No matter for steady-state or unsteady state solutions, the algorithm must be developed from the unsteady compressible N-S equations. The 'links' between the continuty Eq. and the momentum and energy eqs. relies on the density change which is obtained from the continuity eq. Put it in a simle way (in reality, it is not that simple, because everythin is strongly coupled) if the mass is not conservative, there will be a density residual from the continue eq. This density residual will lead to velocity and temperature residuals through the momentum and energy equations. That will updata all flow quantities until conservation arrived. Most(but not all) compressible codes are density-based (MacCormack, Beam-Warming, Jameson, Van Leer, as well as all high order Godunov-type schemes like MUSCL, ENO, TVD etc.) Some advantages of DB schemes are (among others) (1) all equations can be in strongly conservation form, the solution methods are the same for each eq. (2) strong shock waves can be captured. The disadvantage is its poor capability to solve low Mach flows (slow and hard to converge). Most DB code will fail to converge to the RIGHT solution for Mach number below 0.1.

The 'pressure-based' (PB) method used the pressure to link the continuity equation and the others. In this method, the continuity equation must be solved separatly. There are several approaches in this area (please refer to numerical methods for incompressible N-S equations in CFD text books). The PB method was originally for incompressible flows (density term does not appear in continuity equation, even for unsteady flow). A good example of PB code is SIMPLE. But later, this method is extended to solve compressible flows as well, with less ability to handle strong shock waves.

In short statement but not absolute (I have to say that because there are a few people love to make nonsense argument or even personal attack). If you need to due with compressible flow (M=0.3 to very high), you may want to use DB approach. If you need to solve incompressible flows, you may choose PB method. If your problem is more complicated including both low and high speeds, you may try the newly developed 'all-speed method' which is still not very popular yet.

Hongjun

 jay November 11, 1998 18:06

Re: What is Density-Based method?

In the density based method, the algorithm uses a constitutive relation such as the ideal-gas law to iterate for the pressure. In the pressure based method, no constitutive relation is used; pressure is never even iterated for, only the pressure difference is calculated iteratively (in SIMPLE)

That is the gist of the matter. For an explanation as to why: it has to do w/ the structure of the equations that describe compressible and incompressible flows. One is elliptic, another is parabolic (in basic form; I bet there are people out there who'll jump on me for saying this!)

W.F Ames's book on Numerical Analysis is a good place to start to understand these math details. Someone below has suggested Fletcher for a beginners book. That also is a very good reference to understand this concept.

 Aldrin Wong November 18, 1998 01:09

Re: What is Density-Based method?

Where can I find more information regarding the "all-speed" method ? Is there a proper name for it ?

 Hongjun Li November 18, 1998 17:11

Re: What is Density-Based method?

I did some work in this area about 4-5 years ago. I need to dig out some papers from my files and come back to you later.

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