Is Turbulence at Point A Isotropic? Calculation Help

[Chestermiller]

((0.0625)(0.3125)+(0.4625)(0.1125)+(0.5624)(-0.2875)+...)/8

 

[nightingale]

I don't know how you got that average velocity plot, but it doesn't look right. How did you end up with the funny shape?
For the velocity gradient, I get: (7.8625 - 7.525)/0.8

Chet

I plot the average velocity at A (7.525), where I take A as the origin (0,0). Then I plot average velocity at B (7.5375) which is 0.4mm from A (0.4, 7..5375). Then I plot the average velocity at C (7.8625) at 0.8mm from A (0.8, 7.8625).
I then asked ms excel to generate the equation of the linear line and the gradient for me (0.4219).

Understood, (7.8625 - 7.525)/0.8= 0.421875 is a much simpler way to find the derivative,
Thank you Chet!

 

[nightingale]

((0.0625)(0.3125)+(0.4625)(0.1125)+(0.5624)(-0.2875)+...)/8

Ah, I see, thank you very much Sir! How clueless of me!

 

[nightingale]

This is not correct. You don't use the rms values of the fluctuations in the two directions. You use the average of the product of the individual fluctuations in the two directions.

You need to use the average velocities at the points A, B, and C to get the derivative of the average velocity.

No. Not for this.

The viscous shear stress is thus: τ = 1.8 * 10-5 * 0.4219.
I was wondering, for the turbulent shear stress, I consider both the fluctuations in the radial and axial direction.
But for the viscous shear stress I only consider the fluctuations at the axial direction.
Why and when do I have to only consider the viscous shear stress in the axial direction?

,

I also attempt to determine the eddy viscosity at B. I read some book which have rather different formulas for eddy viscosity:
McComb (2003) wrote that -ρ×u₁'×u₂' = ρ×V_T×(dU₁/dx₂) (this book only considers one direction, I don't know why), where V_T is the kinematic eddy viscosity.

Pope (2000) wrote that -ρ×u₁'×u₂' + 2/3 ×ρ×κ×δ₁₂ = ρ×V_T×(dU₁/dx₂)×(dU₂/dx₁) (this book considers two directions), where κ is the kinetic energy, δ is the Kronecker delta (I'm not sure what it is), V_T is the eddy viscosity (without kinematic).

And this lecture online (http://www.bakker.org/dartmouth06/engs150/10-rans.pdf) says that:

where the μ_t is the eddy viscosity and kinematic turbulent viscosity is actually V_T = μ_t/ρ.

I'm really confused on which equation I should chose and do I have to consider the kinetic energy and kronecker delta to determine the eddy viscosity?
Why does Pope (2000) states that V_T is the eddy viscosity (without kinematic) while McComb (2003) states that V_T is the kinematic turbulent viscosity?

Thank you very much for the guidance all this time.

 

[Chestermiller]

The viscous shear stress is thus: τ = 1.8 * 10-5 * 0.4219.
I was wondering, for the turbulent shear stress, I consider both the fluctuations in the radial and axial direction.
But for the viscous shear stress I only consider the fluctuations at the axial direction.
Why and when do I have to only consider the viscous shear stress in the axial direction?

Both the viscous stress and the turbulent stress you calculated are on a surface of constant r in the z direction, but also on a surface of constant z in the r direction.

View attachment 90985, View attachment 90986

I also attempt to determine the eddy viscosity at B. I read some book which have rather different formulas for eddy viscosity:
McComb (2003) wrote that -ρ×u₁'×u₂' = ρ×V_T×(dU₁/dx₂) (this book only considers one direction, I don't know why), where V_T is the kinematic eddy viscosity.

Pope (2000) wrote that -ρ×u₁'×u₂' + 2/3 ×ρ×κ×δ₁₂ = ρ×V_T×(dU₁/dx₂)×(dU₂/dx₁) (this book considers two directions), where κ is the kinetic energy, δ is the Kronecker delta (I'm not sure what it is), V_T is the eddy viscosity (without kinematic).

And this lecture online (http://www.bakker.org/dartmouth06/engs150/10-rans.pdf) says that:

View attachment 90987

where the μ_t is the eddy viscosity and kinematic turbulent viscosity is actually V_T = μ_t/ρ.

I'm really confused on which equation I should chose and do I have to consider the kinetic energy and kronecker delta to determine the eddy viscosity?
Why does Pope (2000) states that V_T is the eddy viscosity (without kinematic) while McComb (2003) states that V_T is the kinematic turbulent viscosity?

Thank you very much for the guidance all this time.

Your questions are indicating that your fundamental understanding of turbulence and turbulent stresses needs beefing up. You need to understand the fundamentals before you start trying to apply it to problems. I'm going to recommend another book that I hold in high regard, and hope you will consider using it: Transport Phenomena by Bird, Stewart, and Lightfoot, Chapter 5. Physics Forums is just not structured to present a complete primer on Turbulent Flow.

Chet

 

[nightingale]

Both the viscous stress and the turbulent stress you calculated are on a surface of constant r in the z direction, but also on a surface of constant z in the r direction.

Your questions are indicating that your fundamental understanding of turbulence and turbulent stresses needs beefing up. You need to understand the fundamentals before you start trying to apply it to problems. I'm going to recommend another book that I hold in high regard, and hope you will consider using it: Transport Phenomena by Bird, Stewart, and Lightfoot, Chapter 5. Physics Forums is just not structured to present a complete primer on Turbulent Flow.

Chet

Thank you Sir.

I have borrowed the book you recommend and read chapter 5 and chapter 1 which talk about kronecker delta. I really helps me to understand turbulence better.
I'm still intrigued on why Pope (2000) simply said V_T as 'eddy viscosity' when it is, I think, 'kinematic eddy viscosity' but I decided I'll ask my teacher next time I met her.

I now know that the formula I should be using is:

-ρu'v' = μ_T * dU/dy

Where the μ_T here is the eddy viscosity.

Earlier the turbulent shear stress is:
τ = -ρu'v'
τ = - 1.2 * ((0.0625)(0.3125)+(0.4625)(0.1125)+(0.5624)(-0.2875)+...)/8
τ = 0.11343

and the velocity gradient is 0.4219

Thus the eddy viscosity is:
0.11343 = μ * 0.4219
μ = 0.27
Did I do right? Thank you very much.

 

[Chestermiller]

Thank you Sir.

I have borrowed the book you recommend and read chapter 5 and chapter 1 which talk about kronecker delta. I really helps me to understand turbulence better.
I'm still intrigued on why Pope (2000) simply said V_T as 'eddy viscosity' when it is, I think, 'kinematic eddy viscosity' but I decided I'll ask my teacher next time I met her.

I now know that the formula I should be using is:

-ρu'v' = μ_T * dU/dy

Where the μ_T here is the eddy viscosity.

Earlier the turbulent shear stress is:
τ = -ρu'v'
τ = - 1.2 * ((0.0625)(0.3125)+(0.4625)(0.1125)+(0.5624)(-0.2875)+...)/8
τ = 0.11343

and the velocity gradient is 0.4219

Thus the eddy viscosity is:
0.11343 = μ * 0.4219
μ = 0.27
Did I do right? Thank you very much.

Yes. But please make sure the you show the units.

 

[nightingale]

Yes. But please make sure the you show the units.

Thank you, Sir. I take the unit of the eddy viscosity as Pa.s.

I attempt to do the last question, which asked me to determine the production of the turbulence energy at B.
I found that the corresponding formula is:

Production = -2 u'v'*(dU/dy)
and thus the production is:
Production = -2 *((0.0625)(0.3125)+(0.4625)(0.1125)+(0.5624)(-0.2875)+...)/8 * (0.4219)
Production = -2 * -0.094525 * 0.4219
Production = 0.0798

Did I do it correctly?
Thank you very much for the guidance.

 

[Chestermiller]

I can't help you here. I'm not familiar with the equation you wrote for the rate of production of turbulent energy. But, it doesn't seem to have the right units.

Chet

 

[nightingale]

I can't help you here. I'm not familiar with the equation you wrote for the rate of production of turbulent energy. But, it doesn't seem to have the right units.

Chet

I understand, I will continue to read about the topic. Thank you very much for the patience and the guidance all this time Chet! You are a wonderful teacher!