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Turbulence boundary conditions

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==Fully developed turbulent pipe-flow inlet==
==Fully developed turbulent pipe-flow inlet==
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For fully developed turbulent pipe flow the turbulence inlet properties can be estimated using the model presented by Basse in Table 1 of [10].
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For fully developed turbulent pipe flow the turbulence inlet properties can be estimated using the model presented by Basse in Table 1 of [1].
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:<math>l_{AA} = 0.14 \; \kappa_g \times \delta</math>
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Turbulence Intensity:
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:<math>I_{AA} = \sqrt{\left[ B_g + \frac{3}{2} A_g - \frac{8 C_g}{\sqrt[3]{Re_\tau}}\right] \times \frac{f}{8}}</math>
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:<math>I_{AA} = \sqrt{\left[ B_g + \frac{3}{2} A_g - \frac{8 C_g}{3 \sqrt{Re_\tau}}\right] \times \frac{f}{8}}</math>
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Turbulence Length-Scale:
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:<math>L_{AA} = 0.14 \; \kappa_g \times \delta</math>
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Turbulence Energy:
:<math>k_{AA} = U_m^2 \; I_{AA}^2</math>
:<math>k_{AA} = U_m^2 \; I_{AA}^2</math>
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:<math>\epsilon_{AA} = C_{\mu,AA}^\frac{3}{4} \times \frac{k_{AA}^\frac{3}{2}}{l_{AA}}</math>
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Turbulence Dissipation:
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:<math>\epsilon_{AA} = C_{\mu,AA}^\frac{3}{4} \times \frac{k_{AA}^\frac{3}{2}}{L_{AA}}</math>
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The model parameters <math>\kappa_g</math>, <math>A_g</math>, <math>B_g</math> and <math>C_g</math> can be computed using the following function:
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The subscript <math>AA</math> here denotes an area-averaged value. The model parameters <math>\kappa_g</math>, <math>A_g</math>, <math>B_g</math> and <math>C_g</math> can be computed using the following general function:
:<math>Q(Re_\tau) = a + b \cdot tanh(c \cdot [Re_\tau - d])</math>
:<math>Q(Re_\tau) = a + b \cdot tanh(c \cdot [Re_\tau - d])</math>
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Where the a, b, c and d constants have been fitted using Princeton Superpipe measurements [2]:
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Where the a, b, c and d constants have been fitted using Princeton Superpipe measurements [2] as described in equation S44 in [3] and table 1 in [4]:
{| class="wikitable" style="margin-left:20px; text-align: center;"
{| class="wikitable" style="margin-left:20px; text-align: center;"
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|}
|}
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:<math>\delta</math> is the boundary layer thickness, which in fully developed pipe flow is the radius, or half the [[hydraulic diameter]].
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:<math>\delta</math> is the boundary layer thickness, which in fully developed pipe flow is the radius, or half the [[hydraulic diameter]], <math>d_h</math>.
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:<math>f</math> is the friction factor.
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:<math>f</math> is the [https://en.wikipedia.org/wiki/Darcy%E2%80%93Weisbach_equation Darcy-Weisbach friction factor].
:<math>Re_\tau = \frac{u_\tau \cdot \delta}{\nu_k}</math> is the Reynolds number based on the friction velocity <math>u_\tau</math> and the kinematic viscosity <math>\nu_k</math>.
:<math>Re_\tau = \frac{u_\tau \cdot \delta}{\nu_k}</math> is the Reynolds number based on the friction velocity <math>u_\tau</math> and the kinematic viscosity <math>\nu_k</math>.
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A good estimate for the friction factor <math>f</math> in pipe flow is Cheng's correlation [13]:
A good estimate for the friction factor <math>f</math> in pipe flow is Cheng's correlation [13]:
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:<math>\frac{1}{f} = \left( \frac{Re_{d_h}}{64} \right) ^a \left( 1.8 \cdot log \frac{Re_{d_h}}{6.8} \right) ^ {2(1-a)b} \left( 2.0 \cdot log \frac{3.7 \; d_h}{k_s} \right) ^ {2(1-a)(1-b)}</math>
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:<math>a = \frac{1}{1 +  \left( \frac{Re_{d_h}}{2720} \right) ^ 9}</math>
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:<math>b = \frac{1}{1 +  \left( \frac{Re_{d_h}}{160 \; d_h / k_s} \right) ^ 2}</math>
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'''Information below is old and will be reformulated'''
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:<math>Re_{d_h} = \frac{U_m \cdot d_h}{\nu_k}</math>
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For fully developed duct flow the turbulence intensity at the core can be estimated as [1]:
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The [[hydraulic diameter]] <math>d_h</math> is the diameter of a circular pipe. For a rectangular pipe with width <math>a</math> and height <math>b</math> the hydraulic diameter can be computed from <math>d_h = 2 \; \frac{a b}{a + b}</math>.
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:<math>I = 0.16 \; Re_{d_h}^{-\frac{1}{8}}</math>,
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The equivalent sand-grain-roughness <math>k_s</math> is dependent on the pipe surface properties.  
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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].
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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:
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:<math>I_{\rm Smooth~pipe~axis} = 0.0550 \; Re^{-0.0407}</math>
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:<math>I_{\rm Smooth~pipe~area,~AM} = 0.227 \; Re^{-0.100}</math>
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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].
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== References ==
== References ==
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{{reference-paper|author=[1] ANSYS, Inc.|year=2022|title=ANSYS Fluent User's Guide, Release R1|rest=Equation (7.71)}}
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{{reference-paper|author=[1] Basse, N.T.|year=2023|title=An Algebraic Non-Equilibrium Turbulence Model of the High Reynolds Number Transition Region|rest=Water 2023, 15, 3234. https://doi.org/10.3390/w15183234}}
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{{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}}
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{{reference-paper|author=[2] Hultmark M, Vallikivi M, Bailey SCC and Smits AJ.|year=2013|title=Logarithmic scaling
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of turbulence in smooth- and rough-wall pipe flow|rest=J. Fluid Mech. 728, 376-395}}
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{{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}}
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{{reference-paper|author=[3] Basse, N.T.|year=2023|title=Supplementary Information: An algebraic non-equilibrium turbulence model of the high Reynolds number transition region|rest=https://www.researchgate.net/publication/373108195_Supplementary_Information_An_algebraic_non-equilibrium_turbulence_model_of_the_high_Reynolds_number_transition_region}}
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{{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}}
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{{reference-paper|author=[4] 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, https://arxiv.org/abs/2109.11626}}
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{{reference-paper|author=[5] Basse, N.T.|year=2019|title=Turbulence intensity scaling: A fugue|rest=Fluids, vol. 4, 180}}
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{{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}}
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{{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}}
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{{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}}
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{{reference-paper|author=[9] Basse, N.T.|year=2022|title=Extrapolation of turbulence intensity scaling
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to Re_tau>>10^5|rest=Physics of Fluids, vol. 34, 075128}}
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{{reference-paper|author=[10] Basse, N.T.|year=2023|title=An Algebraic Non-Equilibrium Turbulence Model of the High Reynolds Number Transition Region|rest=Water 2023, 15, 3234. https://doi.org/10.3390/w15183234}}
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{{reference-paper|author=[11] Basse, N.T.|year=2023|title=Supplementary Information: An algebraic non-equilibrium turbulence model of the high Reynolds number transition region|rest=https://www.researchgate.net/publication/373108195_Supplementary_Information_An_algebraic_non-equilibrium_turbulence_model_of_the_high_Reynolds_number_transition_region}}
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{{reference-paper|author=[12] Hultmark M, Vallikivi M, Bailey SCC and Smits AJ.|year=2013|title=Logarithmic scaling
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of turbulence in smooth- and rough-wall pipe flow|rest=J. Fluid Mech. 728, 376-395}}
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Latest revision as of 22:07, 8 December 2024

This section is under construction, please do not trust the information available here yet

Introduction

Fully developed turbulent pipe-flow inlet

For fully developed turbulent pipe flow the turbulence inlet properties can be estimated using the model presented by Basse in Table 1 of [1].

Turbulence Intensity:

I_{AA} = \sqrt{\left[ B_g + \frac{3}{2} A_g - \frac{8 C_g}{3 \sqrt{Re_\tau}}\right] \times \frac{f}{8}}

Turbulence Length-Scale:

L_{AA} = 0.14 \; \kappa_g \times \delta

Turbulence Energy:

k_{AA} = U_m^2 \; I_{AA}^2

Turbulence Dissipation:

\epsilon_{AA} = C_{\mu,AA}^\frac{3}{4} \times \frac{k_{AA}^\frac{3}{2}}{L_{AA}}

The subscript AA here denotes an area-averaged value. The model parameters \kappa_g, A_g, B_g and C_g can be computed using the following general function:

Q(Re_\tau) = a + b \cdot tanh(c \cdot [Re_\tau - d])

Where the a, b, c and d constants have been fitted using Princeton Superpipe measurements [2] as described in equation S44 in [3] and table 1 in [4]:

Parameter a b c d
\kappa_g −1.18 1.52 2.15e-4 -8786
A_g 2.21 -0.60 3.97e-5 11186
B_g 1.28 -0.32 5.85e-5 4609
C_g / \sqrt{Re_\tau} 1.03 -0.913.30e-5 -11755
\delta is the boundary layer thickness, which in fully developed pipe flow is the radius, or half the hydraulic diameter, d_h.
f is the Darcy-Weisbach friction factor.
Re_\tau = \frac{u_\tau \cdot \delta}{\nu_k} is the Reynolds number based on the friction velocity u_\tau and the kinematic viscosity \nu_k.

The friction velocity u_\tau can be computed using the friction factor f and the mean pipe flow velocity U_m using the formula:

u_\tau = \sqrt{\frac{f}{8}} \cdot U_m

A good estimate for the friction factor f in pipe flow is Cheng's correlation [13]:

\frac{1}{f} = \left( \frac{Re_{d_h}}{64} \right) ^a \left( 1.8 \cdot log \frac{Re_{d_h}}{6.8} \right) ^ {2(1-a)b} \left( 2.0 \cdot log \frac{3.7 \; d_h}{k_s} \right) ^ {2(1-a)(1-b)}
a = \frac{1}{1 + \left( \frac{Re_{d_h}}{2720} \right) ^ 9}
b = \frac{1}{1 + \left( \frac{Re_{d_h}}{160 \; d_h / k_s} \right) ^ 2}
Re_{d_h} = \frac{U_m \cdot d_h}{\nu_k}

The hydraulic diameter d_h is the diameter of a circular pipe. For a rectangular pipe with width a and height b the hydraulic diameter can be computed from d_h = 2 \; \frac{a b}{a + b}.

The equivalent sand-grain-roughness k_s is dependent on the pipe surface properties.

References

[1] Basse, N.T. (2023), "An Algebraic Non-Equilibrium Turbulence Model of the High Reynolds Number Transition Region", Water 2023, 15, 3234. https://doi.org/10.3390/w15183234.

[2] Hultmark M, Vallikivi M, Bailey SCC and Smits AJ. (2013), "Logarithmic scaling of turbulence in smooth- and rough-wall pipe flow", J. Fluid Mech. 728, 376-395.

[3] Basse, N.T. (2023), "Supplementary Information: An algebraic non-equilibrium turbulence model of the high Reynolds number transition region", https://www.researchgate.net/publication/373108195_Supplementary_Information_An_algebraic_non-equilibrium_turbulence_model_of_the_high_Reynolds_number_transition_region.

[4] 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, https://arxiv.org/abs/2109.11626.

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