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Turbulence intensity

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==Definition==
==Definition==
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The turbulence intensity, also often referd to as turbulence level, is defined as:
+
The turbulence intensity, also often refered to as turbulence level, is defined as:
-
:<math>I \equiv \frac{u'}{U}</math>
+
:<math>I \equiv \frac{u'}{U}</math>,
-
Where <math>u'</math> is the root-mean-square of the turbulent velocity fluctuations and <math>U</math> is the mean velocity ([[Reynolds averaging|Reynolds averaged]]).
+
where <math>u'</math> is the root-mean-square of the turbulent velocity fluctuations and <math>U</math> is the mean velocity ([[Reynolds averaging|Reynolds averaged]]).
If the turbulent energy, <math>k</math>, is known <math>u'</math> can be computed as:
If the turbulent energy, <math>k</math>, is known <math>u'</math> can be computed as:
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===Fully developed pipe flow===
===Fully developed pipe flow===
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For fully developed pipe flow the turbulence intensity at the core can be estimated as:
+
For fully developed duct flow the turbulence intensity at the core can be estimated as [1]:
-
:<math>I = 0.16 \; Re_{d_h}^{-\frac{1}{8}}</math>
+
:<math>I = 0.16 \; Re_{d_h}^{-\frac{1}{8}}</math>,
-
Where <math>Re_{d_h}</math> is the [[Reynolds number]] based on the pipe [[hydraulic diameter]] <math>d_h</math>.
+
where <math>Re_{d_h}</math> is the [[Reynolds number]] based on the pipe [[hydraulic diameter]] <math>d_h</math>. Additional details on the derivation can be found in [2].
 +
 
 +
Russo and Basse published a paper [3] where they derive turbulence intensity scaling laws based on CFD simulations and Princeton Superpipe measurements. The turbulence intensity over the pipe area is defined as an arithmetic mean (AM). The measurement-based scaling laws are:
 +
 
 +
:<math>I_{\rm Smooth~pipe~axis} = 0.0550 \; Re^{-0.0407}</math>
 +
:<math>I_{\rm Smooth~pipe~area,~AM} = 0.227 \; Re^{-0.100}</math>
 +
 
 +
Scaling using other turbulence intensity definitions is investigated in [4,5]. Here, it is also found that turbulence intensity scales with the friction factor, both for smooth- and rough-wall pipe flow. Code for an example in [5] can be found in [6]. A high Reynolds number transition in the scaling has been characterized in [7,8]. Turbulence intensity scaling extrapolated to extreme Reynolds numbers is studied in [9].
 +
 
 +
== References ==
 +
 
 +
{{reference-paper|author=[1] ANSYS, Inc.|year=2022|title=ANSYS Fluent User's Guide, Release R1|rest=Equation (7.71)}}
 +
 
 +
{{reference-paper|author=[2] Basse, N.T.|year=2022|title=Mind the Gap: Boundary Conditions for Turbulence Modelling|rest=https://www.researchgate.net/publication/359218404_Mind_the_Gap_Boundary_Conditions_for_Turbulence_Modelling}}
 +
 
 +
{{reference-paper|author=[3] Russo, F. and Basse, N.T.|year=2016|title=Scaling of turbulence intensity for low-speed flow in smooth pipes|rest=Flow Meas. Instrum., vol. 52, pp. 101–114}}
 +
 
 +
{{reference-paper|author=[4] Basse, N.T.|year=2017|title=Turbulence intensity and the friction factor for smooth- and rough-wall pipe flow|rest=Fluids, vol. 2, 30}}
 +
 
 +
{{reference-paper|author=[5] Basse, N.T.|year=2019|title=Turbulence intensity scaling: A fugue|rest=Fluids, vol. 4, 180}}
 +
 
 +
{{reference-paper|author=[6] Basse, N.T.|year=2019|title=Python code to calculate turbulence intensity based on Reynolds number and surface roughness.|rest=https://www.researchgate.net/publication/336374461_Python_code_to_calculate_turbulence_intensity_based_on_Reynolds_number_and_surface_roughness}}
 +
 
 +
{{reference-paper|author=[7] Basse, N.T.|year=2021|title=Scaling of global properties of fluctuating and mean streamwise velocities in pipe flow: Characterization of a high Reynolds number transition region|rest=Physics of Fluids, vol. 33, 065127}}
 +
 
 +
{{reference-paper|author=[8] Basse, N.T.|year=2021|title=Scaling of global properties of fluctuating streamwise velocities in pipe flow: Impact of the viscous term|rest=Physics of Fluids, vol. 33, 125109}}
 +
 
 +
{{reference-paper|author=[9] Basse, N.T.|year=2022|title=Extrapolation of turbulence intensity scaling
 +
to Re_tau>>10^5|rest=Physics of Fluids, vol. 34, 075128}}

Latest revision as of 16:24, 23 July 2022

Contents

Definition

The turbulence intensity, also often refered to as turbulence level, is defined as:

I \equiv \frac{u'}{U},

where u' is the root-mean-square of the turbulent velocity fluctuations and U is the mean velocity (Reynolds averaged).

If the turbulent energy, k, is known u' can be computed as:

u' \equiv \sqrt{\frac{1}{3} \, ( u_x'^2 + u_y'^2 + u_z'^2 )} = \sqrt{\frac{2}{3}\, k}

U can be computed from the three mean velocity components U_x, U_y and U_z as:

U \equiv \sqrt{U_x^2 + U_y^2 + U_z^2}

Estimating the turbulence intensity

When setting boundary conditions for a CFD simulation it is often necessary to estimate the turbulence intensity on the inlets. To do this accurately it is good to have some form of measurements or previous experince to base the estimate on. Here are a few examples of common estimations of the incoming turbulence intensity:

  1. High-turbulence case: High-speed flow inside complex geometries like heat-exchangers and flow inside rotating machinery (turbines and compressors). Typically the turbulence intensity is between 5% and 20%
  2. Medium-turbulence case: Flow in not-so-complex devices like large pipes, ventilation flows etc. or low speed flows (low Reynolds number). Typically the turbulence intensity is between 1% and 5%
  3. Low-turbulence case: Flow originating from a fluid that stands still, like external flow across cars, submarines and aircrafts. Very high-quality wind-tunnels can also reach really low turbulence levels. Typically the turbulence intensity is very low, well below 1%.

Fully developed pipe flow

For fully developed duct flow the turbulence intensity at the core can be estimated as [1]:

I = 0.16 \; Re_{d_h}^{-\frac{1}{8}},

where Re_{d_h} is the Reynolds number based on the pipe hydraulic diameter d_h. Additional details on the derivation can be found in [2].

Russo and Basse published a paper [3] where they derive turbulence intensity scaling laws based on CFD simulations and Princeton Superpipe measurements. The turbulence intensity over the pipe area is defined as an arithmetic mean (AM). The measurement-based scaling laws are:

I_{\rm Smooth~pipe~axis} = 0.0550 \; Re^{-0.0407}
I_{\rm Smooth~pipe~area,~AM} = 0.227 \; Re^{-0.100}

Scaling using other turbulence intensity definitions is investigated in [4,5]. Here, it is also found that turbulence intensity scales with the friction factor, both for smooth- and rough-wall pipe flow. Code for an example in [5] can be found in [6]. A high Reynolds number transition in the scaling has been characterized in [7,8]. Turbulence intensity scaling extrapolated to extreme Reynolds numbers is studied in [9].

References

[1] ANSYS, Inc. (2022), "ANSYS Fluent User's Guide, Release R1", Equation (7.71).

[2] Basse, N.T. (2022), "Mind the Gap: Boundary Conditions for Turbulence Modelling", https://www.researchgate.net/publication/359218404_Mind_the_Gap_Boundary_Conditions_for_Turbulence_Modelling.

[3] Russo, F. and Basse, N.T. (2016), "Scaling of turbulence intensity for low-speed flow in smooth pipes", Flow Meas. Instrum., vol. 52, pp. 101–114.

[4] Basse, N.T. (2017), "Turbulence intensity and the friction factor for smooth- and rough-wall pipe flow", Fluids, vol. 2, 30.

[5] Basse, N.T. (2019), "Turbulence intensity scaling: A fugue", Fluids, vol. 4, 180.

[6] Basse, N.T. (2019), "Python code to calculate turbulence intensity based on Reynolds number and surface roughness.", https://www.researchgate.net/publication/336374461_Python_code_to_calculate_turbulence_intensity_based_on_Reynolds_number_and_surface_roughness.

[7] Basse, N.T. (2021), "Scaling of global properties of fluctuating and mean streamwise velocities in pipe flow: Characterization of a high Reynolds number transition region", Physics of Fluids, vol. 33, 065127.

[8] Basse, N.T. (2021), "Scaling of global properties of fluctuating streamwise velocities in pipe flow: Impact of the viscous term", Physics of Fluids, vol. 33, 125109.

[9] Basse, N.T. (2022), "Extrapolation of turbulence intensity scaling to Re_tau>>10^5", Physics of Fluids, vol. 34, 075128.

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