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Numerical simulation of turbulent forced convection of a power law fluid flow in an axially rotating pipe

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Abstract

Large eddy simulation with an extended Smagorinsky model has been carried out to study a fully developed turbulent forced convection of thermally independent shear-thinning (n = 0.75) fluid through a heated axially rotating pipe. Uniform constant heat flux has been imposed at the wall as a thermal boundary condition. The Reynolds and Prandtl numbers of the simulation have been assumed to be Res = 4000 and Prs = 1, respectively, over a rotation rate range of 0 ≤ N ≤ 3. The computations procedure is based on a finite difference scheme, second-order accurate in space and time; the numeric resolution is 653 grid points in axial, radial and circumferential directions, respectively, with a computational domain length of 20R. The aim of the present study is to investigate the effects of the rotating pipe wall on the turbulent and thermal statistics. The emerged findings suggest that the centrifugal force induced by the rotating pipe wall results in a marked enhancement of the mean axial velocity and an attenuation of the temperature profiles along the radial coordinates; this trend is more pronounced as the rotation rate increases. The increased rotation rate also induces a significant reduction in the temperature fluctuations intensity and consequently, in the axial turbulent heat flux. The predictions also show that the friction factor and Nusselt number are reduced when the rotation rate varies from 0 to 0.5, while they are enhanced with increasing N for N ≥ 1.

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Abbreviations

\( U_{b} \) :

Average velocity (m s−1)

\( U_{\tau } \) :

Friction velocity \( U_{\tau } = \left( {{{\tau_{\text{w}} } \mathord{\left/ {\vphantom {{\tau_{\text{w}} } \rho }} \right. \kern-0pt} \rho }} \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} \) (m s−1)

\( U_{\text{CL}} \) :

Centreline axial velocity for analytical fully developed laminar profile \( U_{\text{CL}} = {{\left( {3n + 1} \right)U_{b} } \mathord{\left/ {\vphantom {{\left( {3n + 1} \right)U_{b} } {\left( {n + 1} \right)}}} \right. \kern-0pt} {\left( {n + 1} \right)}} \) (m s−1)

\( R \) :

Pipe radius (m)

\( n \) :

Flow index

\( K \) :

Consistency index (pa sn)

\( q \) :

Heat flux

\( k \) :

Turbulent kinetic energy

\( T_{\text{ref}} \) :

Reference temperature \( T_{\text{ref}} = {{q_{\text{w}} } \mathord{\left/ {\vphantom {{q_{\text{w}} } {\rho C_{p} U_{\text{CL}} }}} \right. \kern-0pt} {\rho C_{p} U_{\text{CL}} }} \) (K)

\( Y^{ + } \) :

Wall distance \( Y^{ + } = {{\rho U_{\tau } Y} \mathord{\left/ {\vphantom {{\rho U_{\tau } Y} {\eta_{\text{w}} }}} \right. \kern-0pt} {\eta_{\text{w}} }} \)

\( T_{\text{w}} \) :

Wall temperature (K)

\( \varOmega_{k} \) :

Rotational velocity of the pipe wall

\( N_{k} \) :

Rotation rate \( N_{k} = {{2\varOmega_{k} R} \mathord{\left/ {\vphantom {{2\varOmega_{k} R} {U_{\text{CL}} }}} \right. \kern-0pt} {U_{\text{CL}} }} \)

\( f \) :

Friction factor \( f = {{2\tau_{W} } \mathord{\left/ {\vphantom {{2\tau_{W} } {\left( {\rho U_{b}^{2} } \right)}}} \right. \kern-0pt} {\left( {\rho U_{b}^{2} } \right)}} \)

\( {\text{Nu}} \) :

Nusselt number \( {\text{Nu}} = {{hD} \mathord{\left/ {\vphantom {{hD} k}} \right. \kern-0pt} k} \)

\( \bar{S}_{ij} \) :

Strain rate tensor \( \bar{S}_{ij} = {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}\left( {{{\partial \bar{u}_{i} } \mathord{\left/ {\vphantom {{\partial \bar{u}_{i} } {\partial x_{j} }}} \right. \kern-0pt} {\partial x_{j} }} + {{\partial \bar{u}_{j} } \mathord{\left/ {\vphantom {{\partial \bar{u}_{j} } {\partial x_{i} }}} \right. \kern-0pt} {\partial x_{i} }}} \right) \)

\( T_{\varTheta j} \) :

Subgrid heat flux tensor \( T_{\varTheta j} = {{ - \alpha_{t} \partial \bar{T}} \mathord{\left/ {\vphantom {{ - \alpha_{t} \partial \bar{T}} {\partial x_{j} }}} \right. \kern-0pt} {\partial x_{j} }} \)

\( \text{Re}_{\text{s}} \) :

Reynolds number of the simulations \( \text{Re}_{\text{s}} {{ = \rho U_{\text{CL}}^{2 - n} R^{n} } \mathord{\left/ {\vphantom {{ = \rho U_{\text{CL}}^{2 - n} R^{n} } K}} \right. \kern-0pt} K} \)

\( \text{Re}_{\text{MR}} \) :

Reynolds number of Metzner and Reed \( \text{Re}_{\text{MR}} = {{8\rho D^{n} U_{b}^{2 - n} } \mathord{\left/ {\vphantom {{8\rho D^{n} U_{b}^{2 - n} } {K\left( {6 + {2 \mathord{\left/ {\vphantom {2 n}} \right. \kern-0pt} n}} \right)}}} \right. \kern-0pt} {K\left( {6 + {2 \mathord{\left/ {\vphantom {2 n}} \right. \kern-0pt} n}} \right)}}^{n} \)

\( \text{Re}_{\text{g}} \) :

Generalised Reynolds number \( \text{Re}_{\text{g}} = {{\rho U_{b} D} \mathord{\left/ {\vphantom {{\rho U_{b} D} {\eta_{\text{w}} }}} \right. \kern-0pt} {\eta_{\text{w}} }} \)

\( \text{Re}_{\text{cr}} \) :

Critical Reynolds number \( \text{Re}_{\text{cr}} = 2100\left( {{{\left( {4n + 2} \right)\left( {5n + 3} \right)} \mathord{\left/ {\vphantom {{\left( {4n + 2} \right)\left( {5n + 3} \right)} {3\left( {3n + 1} \right)^{2} }}} \right. \kern-0pt} {3\left( {3n + 1} \right)^{2} }}} \right) \)

\( \Pr_{\text{s}} \) :

Prandtl number of the simulations \( \Pr_{\text{s}} = {K \mathord{\left/ {\vphantom {K {\rho \alpha R^{n - 1} U_{\text{CL}}^{1 - n} }}} \right. \kern-0pt} {\rho \alpha R^{n - 1} U_{\text{CL}}^{1 - n} }} \)

\( F\left( {U_{i}^{'} } \right) \) :

Flatness factor \( F\left( {U_{i}^{'} } \right) = {{\left\langle {U_{i}^{'4} } \right\rangle } \mathord{\left/ {\vphantom {{\left\langle {U_{i}^{'4} } \right\rangle } {\left\langle {U_{i}^{'2} } \right\rangle }}} \right. \kern-0pt} {\left\langle {U_{i}^{'2} } \right\rangle }}^{2} \)

\( S\left( {U_{i}^{'} } \right) \) :

Skewness factor \( S\left( {U_{i}^{'} } \right) = {{\left\langle {U_{i}^{'3} } \right\rangle } \mathord{\left/ {\vphantom {{\left\langle {U_{i}^{'3} } \right\rangle } {\left\langle {U_{i}^{'2} } \right\rangle }}} \right. \kern-0pt} {\left\langle {U_{i}^{'2} } \right\rangle }}^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0pt} 2}}} \)

\( \alpha_{\text{t}} \) :

Thermal diffusivity \( \alpha_{\text{t}} = {{\nu_{\text{t}} } \mathord{\left/ {\vphantom {{\nu_{\text{t}} } {\Pr_{\text{t}} }}} \right. \kern-0pt} {\Pr_{\text{t}} }} \) (m2 s−1)

\( \dot{\gamma } \) :

Shear rate \( \dot{\gamma } = \sqrt {S_{ij} S_{ij} } \)

\( \Delta \) :

Computational grid

\( \eta \) :

Apparent viscosity \( \eta = K\dot{\gamma }^{n - 1} \)

\( \varTheta \) :

Dimensionless temperature

ρ :

Density

\( \bar{\tau }_{ij} \) :

Subgrid stress tensor \( \bar{\tau }_{ij} = - 2\nu_{\text{t}} \bar{S}_{ij} \)

d:

Dimensionless

\( z,r,\theta \) :

Axial, radial, tangential velocity

b :

Average

C:

Centreline

L:

Laminar

t:

Turbulent

s:

Simulation

w:

Wall

g:

Generalised

〈〉:

Statistically averaged

\( \left( {} \right)^{ + } \) :

Normalised by \( U_{\tau } \), or \( T_{\tau } \)

\( \left( {} \right)^{\prime } \) :

Fluctuation component

\( \left( {\overline{{}} } \right) \) :

Filtered variable

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Acknowledgements

The authors would like to acknowledge the Laboratory of Industrial Technologies (Ibn-Khaldoun University Tiaret, Algeria). The authors would like to thank Pr. Paolo Orlandi for his useful discussion and comments.

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Authors and Affiliations

  1. Laboratoire de Mécanique Appliquée, Université USTO Mohamed Boudiaf, BP 1505, EL M’Naouar, Oran, Algeria

    Mohamed Abdi & Abdelkader Noureddine

  2. Laboratoire de Modélisation et Simulation Multi Echelle, MSME, Université Paris-Est, UMR 8208, CNRS, 5 bd Descartes, Marne-la-Vallée, 77454, Paris, France

    Meryem Ould-Rouiss

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  1. Mohamed Abdi
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Correspondence to Mohamed Abdi.

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Abdi, M., Noureddine, A. & Ould-Rouiss, M. Numerical simulation of turbulent forced convection of a power law fluid flow in an axially rotating pipe. J Braz. Soc. Mech. Sci. Eng. 42, 17 (2020). https://doi.org/10.1007/s40430-019-2099-7

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