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February 22, 2006, 06:38 
COURANT NUMBER

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Hi Dear Friends,
Nowadays, I am simulating high Reynolds number flows by using the Bounded High Order QUICKEST scheme. In this scheme, there is the Courant number as a parameter. My question is this: This Courant number may be less that zero? Thanks, Ferreira 

February 22, 2006, 11:22 
Re: COURANT NUMBER

#2 
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not unless your fluid is travelling back in time......


February 22, 2006, 18:16 
Re: COURANT NUMBER

#3 
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Write down the definition of the Courant number, then tell us how could we get negative values


February 23, 2006, 02:11 
Re: COURANT NUMBER

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a basic Courant number definition would be
C = a(dt/dx) in case of flow reversal the courant can be negative is this not the case of a negative Courant number but as in the case of negative Courant numbers Upwind methods and flow dependent methods would be used also the case of flux splitting methods is very much suitable for these cases This is what i think any thing contradicting please reply bye Ravi Kiran 

February 23, 2006, 03:27 
Re: COURANT NUMBER

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Commercial flow solvers can be a blessing because they enable to do flow simulations without the need to know the numerical details.
They can, however, be disastrous for exactly the same reason! My advice would be: Learn a bit about numerics before you use it!! I have seen quite a few results presented by people in industry that defied any common sense  still, they were used to design products. If only the people had known what they were doing, they could have used the commercial software efficiently. Most commercial solvers are actually really good pieces of software. 

February 23, 2006, 06:17 
Re: COURANT NUMBER

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Hi Dear Frinds,
Sorry! But it is a confused discussion. Anyway, as posted by Ravi, the Courant number is defined by adt/dx, where a is convected velocity. In the fluid flow problem that I have in hand, my convected velocity is, for example, u(x+0.5h) on the previous time level. And this convected velocity may be greater or smaller than zero. So, the Courant number is negative or positive one. VG Ferreira 

February 23, 2006, 07:26 
Re: COURANT NUMBER

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The Courant number is a NonDimensional number. Can we carry your confused logic to other non dimensional numbers? Why not start by telling us the meaning of Negative Reynolds, Nusselt, etc.. There is definitly a need to understand the basics of Science before applying it. Numerical Computations produce results of all kinds, it is only the expert eye that distinguishes between what is physically possible and what is absolute rubbish. Commercial programmes will produce rubbish if fed rubbish, a well known fact, a good example of this is the dissipation of energy, under certain stability conditions, negative dissipation has been produced and reported, in plain English, Negative Entropy, then the question is if we Can accept these results and forget about the Second Law of Thermodynamics Or start looking for the reasons of the failure in the computational Scheme. It is up to your Logic.


February 23, 2006, 07:54 
Re: COURANT NUMBER

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Hi CFD guys
Courant number is only a stability criterion for a method only CFL condition would be more appropriately used where we used the absolute velocity hence value of CFL cannot be negative even if the convective velocity is negative upwind methods can me safely used 

February 23, 2006, 08:03 
Re: COURANT NUMBER

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cheers mates!!!


February 23, 2006, 09:04 
Re: COURANT NUMBER

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Ferreira, the Courant equation and the stability criterion I'm sure you already know, thus, let me try to contribute with my own naive interpretation about the subject:
Besides being a stability criterion, the Courant number is a measure of solution mobility, so, if you have a CFL number equal to 1 it's equivalent to say that your solution is passing through at least 1 cell per time step. Regards Renato. 

February 23, 2006, 10:11 
Re: COURANT NUMBER

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The simplest answer is to mod both sides of your courant equation, then you will not need a timemachine...
Basically, the courant relationship can be also be seen as a relationship between timescales & convection scales. In this sense, there is no understanding of positive, or negative velocity. Thus courant number does not go negative. It is not a vector relationship. diaw... 

February 23, 2006, 14:24 
Re: COURANT NUMBER

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So what is the sign of nothing?


February 23, 2006, 17:55 
Re: COURANT NUMBER

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Yes. But the stability condition is actually:
0 < c < 1 for a > 0 (FTBS) 0 > c > 1 for a < 0 (FTFS) Assuming your condition came from a FTBS or FTFS discretization of the advection equation. 

February 25, 2006, 07:40 
Re: COURANT NUMBER

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Sonin,A.A. The Physical Basis Of Dimensional Analysis


February 25, 2006, 11:03 
Re: COURANT NUMBER

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The Courant number is not obtained using Dimensional Analysis. It is obtained from analyis of the stability of the numerical scheme.


February 25, 2006, 12:15 
Re: COURANT NUMBER

#16 
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That is because you do not know How


February 25, 2006, 13:02 
Re: COURANT NUMBER

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You are perfectly correct. I don't know how to use dimensional analysis to prove the stability of a numerical scheme. I do however know how to use the standard mathematical tools (Von Neumann stability analysis, method of equivalent PDE, and matrix based approach).
Perhaps you could show me how dimensional analysis works by applying it to the following simple scalar advection equation (standard textbook example): dudt + a dudx = 0 where a < 0 and the equations are to be discretized with first order finite difference. Use forward in time for dudt and forward in space for dudx (note using Von Neumann stability analysis you can show this is stable subject to 1 =< C =< 0). Where C is defined as: C = a dt/dx (dt=time step, dx=grid size) 

February 25, 2006, 13:12 
Re: COURANT NUMBER

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Read the reference I mentioned in my previous post (There are a couple of good references there too)


February 25, 2006, 13:32 
Re: COURANT NUMBER

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But will the method provide the correct answer (ie that the Courant number MUST be negative) ?
You have said in a previous post that negative Courant numbers have no meaning. 

February 25, 2006, 13:45 
Re: COURANT NUMBER

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My friend, there is nothing like negative nondimensional numbers, not just the Courant Number, in simple language, it has not been defined yet and I doubt that it will ever be as long as we are living on this planet. In our physical world, Nondimensional numbers are all positive and I do not know about their behaviour in a metaphysical world. Cheers


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