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

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

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

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

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.

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

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.

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

cheers mates!!!

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.

The simplest answer is to mod both sides of your courant equation, then you will not need a time-machine...

Basically, the courant relationship can be also be seen as a relationship between time-scales & 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...

So what is the sign of nothing?

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.

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

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

That is because you do not know How

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)

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

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.

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