Velocity fluctuations, variance and moment in turbulence

[ForTheGreater]

I'm studying CFD and I'm on turbulence. It states that the fluctuating part of velocity squared and time averaged is the variance but variance in statistical terms is the deviation from the mean squared and averaged. So what is variance in turbulent fluctuating velocity?

This variance is also called the second moment and I wonder if this has a physical representation or just a statistical tool? a moment as I understand it is a quantity distributed over a space. But what is the second moment representing?

The second moment especially becomes interesting when taking the second moment of velocities of different directions to model turbulence, so does this mean anything physically or as I said is it a statistical tool?

 

[MarcusAgrippa]

Turbulence is a statistical theory. The velocity field is a random variable. That is, at each point in space at a given time, the value of the velocity v(x,t) is a random variable.

Technically, one should imagine that one has set up in a laboratory ten trillion identical copies of the fluid system, and that at the sound of the psaltery, sackbut and harp, ten trillion experimenters bow down to take a reading of v(x,t) at position x at time t. Each experimenter, in general, measures a different value for v at x at time t. The collection of values obtained will have a mean, a variance, and so on. The square root of the variance is related to the standard deviation of v from its mean value. The measured value of v at x and t will have an associated statistical distribution that P(v) that tells you how often a given value occurs in the set of measurements (more accurately, it tells you the probability that the measured value of velocity will lie in the numerical range v to v+dv). The standard deviation measures (roughly) the width of the probability distribution P(v) around its mean value.

Because it is difficult to obtain ten trillion copies of your fluid system and ten trillion grad students to take the measurements, it is usual to assume two things: 1. that the flow is stationary, that is, that the statistics of the flow (or the expectation values if they are theoretically calculated rather than measured) do not change with time; and, 2. that we can replace the measurement of v over an ensemble (as described above) by measurements on a single stationary flow taken at ten trillion different times. One then calculates the statistics of the measurements of v, made at a fixed point x at different times. The assumption that time averages are the same as ensemble averages is called the ergodic hypothesis.

In short, the second moment BOTH means something physically, AND is a statistical tool. The answer to your final question is, therefore, yes.

 

[ForTheGreater]

Thank you for your reply. I understand the standard deviation, meaning the root of the second moment. But in the models the second moment is used directly.

For example the v³ is skewness and v⁴ is something else. Now I don't understand how v³ is skewness but I kind of understand what it is. I don't understand what v² should represent. I guess when it's in combination with density it becomes a measure of kinetic energy and therefore the fluctuating part of velocity multiplied with density becomes turbulent kinetic energy. And I guess when it's v'u' instead of v'² it's a measure of how strong the vorticity of the flow is, since turbulence is 3-D.

What if I put it like this, if everything was known and we didn't need statistical tools to describe the turbulent flow, what would v'² and v'u' describe?

 

[MarcusAgrippa]

The variance is the mean square deviation from the mean. The reason one does not use the standard deviation is because it involves a nasty square root. But they measure essentially the same thing: how much the distribution is spread out around the mean.

I am not sure what you are denoting by v and v'. Usually, v' is the random component of the velocity and v is either the total velocity or the mean velocity. Also, by v'^2 I assume you intend the average value of the random component squared. If I have guessed your notation correctly, v' = 0 when the flow is deterministic (non turbulent).

As for v^3 describing skewness, were the distribution completely symmetrical about the mean, there would be as many positive as there are negative values in the sum that calculates the mean value of v^3, so the sum would add to zero. The mean value of v^3 can only be non-zero if there are an excess of positive values, or of negative values, which means that the distribution is skewed.

The are, in the last analysis, qualitative statements and one needs to understand precisely what the moments of a distribution mean. For that, you would need to consult a book on mathematical statistics.

 

[ForTheGreater]

Yea I'm sorry I should be consistent with notation. I'm only talking about the fluctuating part. So second moment is really just measure of deviation from the mean and third moment is measure of skewness?

 

[MarcusAgrippa]

Yes, qualitatively. The mean value of the products u'v' is not related to vorticity (as far as I know) but measures the correlation of the components of the vector fluctuation. Look up a statistics book on the the meaning of correlation coefficients.

 

[MarcusAgrippa]

As I recall, a nice elementary text on turbulence is Tennekes and Lumley. They give some nice intuitive explanations of the meaning of the quantities of interest. The bible of turbulence is the Yaglom and Monin two volume work.