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diaw September 27, 2005 08:06

Name for 'events per metre'?
An odd question, but here goes...

In the temporal frame we have 'events/second' = Hertz

I was wondering if there is a name for 'events/metre' in the spatial frame.

I can't think of any at the moment, but thought I'd ask the world-at-large.



Jonas Larsson September 27, 2005 08:33

Re: Name for 'events per metre'?
Well, perhaps you could argue that the wave number is events/meter. For a defitinion see:

diaw September 27, 2005 09:10

Re: Name for 'events per metre'?
Hi Jonas,

Thanks so much for that link. That is very interesting.

For instance, lets say we are tring to understand turbulence & would like to refer to the 'scale' - say 50 eddies per meter.

Interesting to see the 'wave number' coming into the picture. Could this refelect something about the physics?


Mark Carlson September 28, 2005 02:16

Re: Name for 'events per metre'?
samples per length seems like a Resolution to me. SPI = Samples Per Inch. DPI=Dots per inch PPI=pixels per inch.

diaw September 28, 2005 02:33

Re: Name for 'events per metre'?
Thanks Mark,

This would go along with the common relationship in time eg. rad/s - number of circular measures per time.

So we have a 'resolution' in time scale of (1/s)=Hz - in length scale of (1/m)=geoz [Gz] (as in 'geometric' - new word).

In some of my research I have found a parallel phenomenon in the spatial frame to what we are so comfortable with in the time frame eg. frequency domain, Laplace domain etc. I was trying to find a suitable available word to use. If none exists then I'll 'invent' one until someone shouts loud-enough... :)


------------------ Mark Carlson wrote: samples per length seems like a Resolution to me. SPI = Samples Per Inch. DPI=Dots per inch PPI=pixels per inch.

Harry Fulmer September 28, 2005 07:32

Re: Name for 'events per metre'?
"Gertz" [Gz]

I like it!!

diaw September 28, 2005 08:25

Re: Name for 'events per metre'?
Harry, you win the prize... :)

'Gertz' [Gz] it is then...

----------- Let us then extend the 'Gertz' domain into a parallel form of the 'Sampling Theorem' which states that in order to accurately sample a continuous signal, the sampling frequency needs to be at least (preferably greater than) 2 times the maximum frequency within the system being sampled.

The 'Gertz' equivalent than states that in order to accurately sample an 'event' in the 'per metre' (1/m) domain, one needs to sample at a Gertz of at least 2 (preferably greater).

(1/m)sample >= 2 * max in (1/m)domain

=> (m)sample <= (1/2) * min in (m) domain...

--------- Does this ring a bell in terms of critical cell Peclet & critical cell Reynolds?

Could it be that we 'don't see' many phenomena in simulation because we use the incorrect scales?

More interesting things happen when there are 2 dominant 'frequencies', or Gertz.


---------- Harry Fulmer wrote: "Gertz" [Gz]

I like it!!

taw September 29, 2005 01:02

Re: Name for 'events per metre'?
Hi diaw, I was reading throgh the index, interesting. when I read the sampling theorem, and when reading of the "two" actually what rang to me was not pecelt number. I have been dealing on scales level as i have told you last time, I think what it is refering or relating here is the differencing scheme of your discretization, that it is when you traverse or integrate in temporal scales , your biggest frequency , which for me is the same as your "smallest time", or at least a basis, step should encompass or consider it ..... but what follows is if your temporal discetization or molecule is not made up of only two temporally consecutive nodes, it should then no longer be two, so it should or may be three, five etc, actually most temporal differencing are two then it holds, does that sound sensible. i have been trying to make frequency related concepts by the time marching concept, if i am wrong correct me before i mislead other thanks

diaw September 29, 2005 06:31

Re: Name for 'events per metre'?
Hi Taw,

If I understand what you are saying, you are refering to the commonly-held notion that the 2 comes from the so-called discretisation procedure most commonly originating in the Finite Difference Method. It seems that this '2' or 1/(1/2) is used to explain the reason behind the critical Peclet #, or critical Reynolds number as applied to a computational element.

My findings have shown that this '2' is actually a result of a singularity in the N-S equations themselves - well before going into spatial transformation (commonly called discretisation). In the 'momentum eqn' this evidences itself nicely due to the 2 in stress term.

For the Energy equation it is not so clear, because, I believe that the Energy Equation commonly derived for incompressible flow in most text-books, is in error by a factor of precisely 2 for the convection term. This anomaly can be seen if a full development of the energy-equation is performed (in say 2D, I haven't extended to 3d yet) without making the compressible assumption & then later applying the incompressible assumption to the full energy form. Try it for yourself, but be forewarned - the derivations get long...

The reason for the apparent anomaly is that the incompressible assumption reduces the continuity equaton to a simple form - which is then plugged in later as the energy eqn is expanded. If the density is allowed to vary, then final form contains a few extra terms - leading to the '2'.

For the record, my research is pointing towards the first stage of so-called discretisation as being a 'continuous transformation' ie. a squeezing of the flow field to nodal points. The fact that it is a 'continuous process' is evidenced that you can reverse the process at any point & arrive back at the original governing equations.

I perceive the true discretisation stage to be when we move into the non-linear loop - in 'discrete steps' - after working with say Newton-Raphson, one cannot simply reverse back to the original situation... it is now discrete & lossy...

Anyway, these are rather deep philosophical viewpoints which may not sit well with many folks, but they are based on my 'big-picture' experience with the Control Engineering field.

So, there you have it...


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