Probability Density Functions in Fluid Mechanics

[pobatso]

Hi all,

For an exam I'm required to be able to plot the PDF of a fluctuating velocity function, say u(t)=sint(wt), using what they call the "graphical technique", but handily I can't find it anywhere in the lecture notes, and I'm struggling to find anything with a standard Google search.

Does anyone know an online resource or otherwise where I can learn this?

Cheers,
pobatso.

 

[pobatso]

And unfortunately I was foolish enough to leave this to 24hrs before the exam, please guys anything you can offer would be great!

 

[mikeph]

I'm trying to figure out what the question means, but how can there be a PDF when your velocity function is entirely deterministic?

 

[pobatso]

Apologies, I'll try to be more specific, see if that helps. Although the wording for the question applies also to random data, and the same question has been asked for random data, ie:

"Sketch a random signal u(t) as a function of time t and use the Graphical Technique to find the PDF"

Another part of the same q was:

"Repeat the above for a sine signal u(t)=sin(wt)"

Note that the u(t) function is reresenting the fluctuating value of the velocity signal around a mean value of a turbulent flow that doesn't change with time, say U', so that the absolute value U(t), is U(t)=U' + u(t).

Does this help at all?

 

[boneh3ad]

I'm trying to figure out what the question means, but how can there be a PDF when your velocity function is entirely deterministic?

Because unfortunately the Navier-Stokes equations are so highly nonlinear that accurate prediction of the quantities is near impossible in the turbulent regime given current technology. It is essentially a region of "spatiotemporal chaos" within which the quantities are often described statistically for modeling purposes.

 

[pobatso]

Think I'm a step closer - for the u=sin(wt) function, what I've done is basically made a histogram by taking small increments of du, say 10, and finding the range of t values that will occupy that specific segment. After normalising the areas so summed they all equal 1, you plot it as a histogram - voila, an estimation of the actual PDF.

However, how would you go about applying this to completely random data? How do you find the range of t values that would fit between it? Presumably the final graph will look something like a normal distribution, but how do you get those initial values for the histogram?