# Probability Density Functions in Fluid Mechanics

• pobatso
In summary, the conversation is about finding resources to learn how to plot the PDF of a fluctuating velocity function using the "graphical technique". The function in question is u(t) = sin(wt) and the conversation also touches on the challenges of predicting quantities in the turbulent regime. The method for finding the PDF involves creating a histogram and normalizing the areas to equal 1. The final graph is expected to resemble a normal distribution, but there is uncertainty about how to obtain the initial values for the histogram.
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.

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

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

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?

Last edited:
MikeyW said:
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.

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?

## 1. What is a probability density function (PDF) in fluid mechanics?

A probability density function in fluid mechanics is a mathematical function that describes the probability of a fluid particle having a certain velocity or other physical property at a given point in space. It is often used to model the behavior of a large number of particles in a fluid system.

## 2. How is a PDF different from a velocity distribution function?

A velocity distribution function (VDF) describes the distribution of velocities of particles in a fluid system, while a PDF describes the probability of a particle having a certain velocity. The VDF is a physical quantity, while the PDF is a mathematical concept used for modeling.

## 3. What are the applications of PDFs in fluid mechanics?

PDFs are commonly used in fluid mechanics to model the behavior of turbulent flows, which are characterized by chaotic and unpredictable fluctuations in velocity. They are also used in the study of particle transport and mixing in fluids, as well as in the analysis of boundary layers and other flow phenomena.

## 4. How are PDFs calculated or measured in experiments?

In experiments, PDFs are typically measured using techniques such as particle image velocimetry (PIV) or hot-wire anemometry. These methods involve tracking the motion of particles or measuring the velocity of fluid flow at multiple points in space and then using statistical analysis to calculate the PDF.

## 5. Can PDFs be used to predict the behavior of a specific fluid system?

PDFs are not typically used for predicting the behavior of a specific fluid system, as they are based on statistical analysis and cannot account for all the complex dynamics of a real fluid system. However, they can provide valuable insights and help researchers understand and model the behavior of fluid systems in general.

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