Refer to "computational Fluid Mechanics & HeatTransfer", 2 ed., Tannehill J.C., Anderson D.A. & Pletcher R.H., Taylor & Francis.

Specifically refer to pages 88 & 89.

abs(c*dt/dx)<=1

where: c*dt/dx = Courant number.

Hopefully this reference should convince you of the reasoning behind the Courant Number. There will also be many others.

diaw...

Thanks for the reference DIAW (some sanity at last).

You do have to be careful using that definition with my previous examples though as you require the additional conditions c > 0 or c < 0 (I used a instead of c in my example) depending on whether the scheme is a FTBS or FTFS discretization (I'm sure you know this, I'm sure its in the text you reference but the comment may be useful for those who are just starting out). If the Courant number can;t be negative then concepts like upwinding and the Courant, Friedrichs, Levy condition would have no or very limited usefulness.

A little aside for our fiend A:

On the subject of negative non-dimensional numbers:

Pressure coefficient Lift Coefficient Pitching moment coefficient

are all examples of non-dimensional numbers that can have negative values without breaking the laws of Physics - and there are many, many, many more.

I believe you may be confusing the concept of non-dimensional numbers and similarity parameters. As I have said previously the Courant number is *not* a similarity parameter.

This will be my last word on the subject as there is very little left to say - the maths doesn't lie.

Courant number can be negetive if one can buy a negetive apple to quench ones negetive hunger.

compressible NS equations along with energy and continuity in cylindrical coordinates in conservative form, where to get those?

All what you have mentioned are, in your own words, Coefficients, in simple language, a group of definitins agreed upon by professionals of the science. NonDimensional variables on the other hand, arise as a result of nondimensionalising an equation describing a phenomenun, be it a physical, human science, biological, etc... Do you get the difference? You can do the same as Diaw, get a reference book, read and understand it ..... Good Luck