Turbulence boundary conditions

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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	-0.91	3.30e-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.

Turbulence boundary conditions

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Introduction

Fully developed turbulent pipe-flow inlet

References

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