I've seen one book and I've uploaded picture of wondering part.
(Statistical Fluid Mechanics: Mechanics of Turbulence, Volume 1, P.278)
Actually I can't understand whole part eventhough I've read it.
But I can talk main wondering point of this part.
At Eq(5. 27), we can see that RMS value is function of distance from the wall and is proportional to friction velocity which represent mean wall shear stress.
I want to know why.
Is it proved mathematically? Or just assumption by observation and intuition for turbulence?
Why RMS fluctuation is proportional to mean wall shear stress and should be determined by distance from the wall.
I want to know physical reason.
What I've guessed is below.
First thing is relation between RMS fluctuation and mean wall shear stress.
If wall shear stress is high, it means that velocity gradient is high because wall shear stress is multiplication of viscosity and velocity gradient.
And if velocity gradient is high, it means that production of turbulence is active. We can see it from both k-epsilon model and LES model.
I think this can be reason why RMS fluctuation is proportional to friction velocity which represents wall shear stress.
Second one is function of distance from the wall.
As leave away from the wall, turbulence is activated because vicinity of the wall, turbulence is suppresed by dominant molecule viscosity.
We can see it from viscous sublayer.
So as leave away from the wall, because we leave away from the viscous sublayer, production of turbulence can be more activated.
Here are my physical guess but I'm not sure. Please let me know right thing.
Thanks :)
As the distance from the wall increases, there is more space for the fluctuations to occupy. As wall shear increases, the tendency to create and sustain fluctuations in the stream increases because this is what drives the fluctuations. Here's an old paper on the subject http://dudleybenton.altervista.org/publications/A%20Turbulent%20Burst%20Model%20for%20Boundary%20Layer%20Flows%20with%20Pressure%20Gradient.pdf
In other words, as you go farther from the wall, it is expected the entrainment of primary and secondary flow gets enhanced. This may cause rms fluctuations to vary.
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