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Turbulence boundary conditions

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#redirect [[Turbulence free-stream boundary conditions]]
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<!-- #redirect [[Turbulence free-stream boundary conditions]] -->
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'''This section is under construction, please do not trust the information available here yet'''
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==Introduction==
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==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 [1].
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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}{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:
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:<math>k_{AA} = U_m^2 \; I_{AA}^2</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 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:
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:<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] as described in equation S44 in [3] and table 1 in [4]:
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{| class="wikitable" style="margin-left:20px; text-align: center;"
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|-
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! style="width:80px;"|Parameter !! style="width:60px;"|a !! style="width:60px;"|b !! style="width:60px;"|c !! style="width:60px;"|d
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|-
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| <math>\kappa_g</math> || style="text-align:right;"|−1.18 || style="text-align:right;"|1.52 || style="text-align:right;"|2.15e-4 || style="text-align:right;"|-8786
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|-
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| <math>A_g</math> || style="text-align:right;"|2.21 || style="text-align:right;"|-0.60 || style="text-align:right;"|3.97e-5 || style="text-align:right;"|11186
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|-
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| <math>B_g</math> || style="text-align:right;"|1.28 || style="text-align:right;"|-0.32 || style="text-align:right;"|5.85e-5 || style="text-align:right;"|4609
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|-
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| <math>C_g / \sqrt{Re_\tau}</math> || style="text-align:right;"|1.03 || style="text-align:right;"|-0.91||  style="text-align:right;"|3.30e-5 || style="text-align:right;"|-11755
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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]], <math>d_h</math>.
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:<math>f</math> is the [https://en.wikipedia.org/wiki/Darcy%E2%80%93Weisbach_equation Darcy-Weisbach friction factor].
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:<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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The friction velocity <math>u_\tau</math> can be computed using the friction factor <math>f</math> and the mean pipe flow velocity <math>U_m</math> using the formula:
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:<math>u_\tau = \sqrt{\frac{f}{8}} \cdot U_m</math>
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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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:<math>Re_{d_h} = \frac{U_m \cdot d_h}{\nu_k}</math>
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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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The equivalent sand-grain-roughness <math>k_s</math> is dependent on the pipe surface properties.
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== References ==
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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] 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] 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=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}}

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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