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October 24, 2000, 01:02 
stiff numerical

#1 
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Hi, my name is Felipe and have the followin question:
In the Fluent manual, chapter 17, for the coupled solver say  "The NavierStokes equations become (numerically) very stiff at low Mach number due to the disparity between the fluid velocity v and the acoustic speed c (speed of sound)."  that it means that the equation becomes numerically stiff?. does this have some physical meaning? 

October 24, 2000, 01:57 
Re: stiff numerical

#2 
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(1). No, the physics does not change at low Mach number. (2).The physics does not change whether the air is stationary or the wind is blowing. (3). Only certain formulation and solution procedures will have problem in convergence at low Mach number.


October 24, 2000, 05:11 
Re: stiff numerical

#3 
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Hi Felipe,
the physical meaning you were looking for is already in your message: "disparity between the fluid velocity v and the acoustic speed c (speed of sound)." In the coupled solver Fluent solves a system of equations and not an equation at time as it does in the segregated solver (obvious to say this, but it is worth to repeat it). To clarify how this sentence answer your question let me make a step back in time. The term "stiff" has been introduced by Curtiss and Hirschfelder in 1952 (this two guys are "magister" for the Kinetic Theory of Gases) for a system of springs which had some springs much more "stiff" then others. Then this term has been used to indicate sistems of differential equations where the time constants have a variety of different values (they are spread over a wide range). From the physical point of view this means that the phenomena represented by this system of equations have different time scales. In the case you cited in your mail, you have that pressure waves move with the speed of sound c whereas the fluid has a velocity v which is much much lower than c (v<<c) , therefore the fluid motion has a time scale much lower than that of pressure perturbations. This can be hardly managed by "the coupled solver" unless you use an integration time step (related to the CFL) reallyreally small, which, provided you achieve stability, will make you waiting for years for a solution. The segregated solver makes of this "problem" an advantage introducing a series of semplifications of the equations (for example neglecting terms which depend on Ma^2 (being Ma <<1)) and to decouple the solutions procedure for the equations. I hope this helps. Ciao Maurizio Ref. Comincioli "Analisi Numerica" McGrawHill 

October 24, 2000, 05:21 
Re: stiff numerical (READ THIS ONE!!!))

#4 
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(the previous message I have posted was truncated. This is the original text. Sorry for the snag! Maurizio)
Hi Felipe, the physical meaning you were looking for is already in your message: "disparity between the fluid velocity v and the acoustic speed c (speed of sound)." In the coupled solver Fluent solves a system of equations and not an equation at time as it does in the segregated solver (obvious to say this, but it is worth to repeat it). To clarify how this sentence answer your question let me make a step back in time. The term "stiff" has been introduced by Curtiss and Hirschfelder in 1952 (this two guys are "magister" for the Kinetic Theory of Gases) for a system of springs which had some springs much more "stiff" then others. Then this term has been used to indicate sistems of differential equations where the time constants have a variety of different values (they are spread over a wide range). From the physical point of view this means that the phenomena represented by this system of equations have different time scales. In the case you cited in your mail, you have that pressure waves move with the speed of sound c whereas the fluid has a velocity v which is much much lower than c (v much smaller than c) , therefore the fluid motion has a time scale much lower than that of pressure perturbations. This can be hardly managed by "the coupled solver" unless you use an integration time step (related to the CFL) reallyreally small, which, provided you achieve stability, will make you waiting for years for a solution. The segregated solver makes of this "problem" an advantage introducing a series of semplifications of the equations (for example neglecting terms which depend on Ma^2 (being Ma much smaller then 1)) and to decouple the solutions procedure for the equations. I hope this helps. Ciao Maurizio Ref. Comincioli "Analisi Numerica" McGrawHill 

September 9, 2009, 18:42 
wot solver

#5 
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Nishant
Join Date: Mar 2009
Location: Glasgow, UK
Posts: 165
Rep Power: 9 
Hi Maurizio,
This is a very interest piece of information. Thanks for this. However I have a question for you here. I am working with a open source cfd code and the solver i am using is a basic PISO solver. I am trying to simulate a pulsating pressure field ( p=p0sin(w*t)) through a long pipe. The p0 value is very small.The solver is a compressible solver and it is solving the energypressure coupled equations in piso loop. I was wondering if this coupled energy and pressure equation is making my solver stiff?? Although time is not an issue in my case, but accuracy is definetely desirable. Should I use this coupled equation to solve or should I decouple the equations? regards, Nishant 

September 9, 2009, 20:02 
and

#6 
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Nishant
Join Date: Mar 2009
Location: Glasgow, UK
Posts: 165
Rep Power: 9 
In addition, my question is:
Although coupled solvers are stiff but are they more accurate (specially in case of sub sonic Piso based solvers)?? Nishant 

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