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Numerical analysis of rotational shape effect on skin friction coefficient in Couette-Taylor flows (with and without injection)

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Abstract

This research aims to conduct a numerical analysis of the effect of geometrical and injection parameters on the skin friction coefficient in Taylor-Couette flow. Two concentric cylinders and cones are chosen for this purpose, with the inner one rotating and the outer one fixed. The relevant equations are solved using FLUENT software for three-dimensional flows, and the flow is investigated in both with and without air bubbles injection. The skin friction coefficient is measured as a function of Reynolds number up to Re = 10,000. Finally, the effect of geometry and microbubble injection on the skin friction coefficient, skin friction ratio, and power gain are investigated. The results show that in conical and cylindrical systems, the skin friction coefficient dropped when the Reynolds number increased. The presence and the surge of bubble injection rate decreases the skin friction coefficient. The skin friction coefficient in the conical system in both injection and non-injection states is significantly lower than the cylindrical system. Besides, at high Reynolds numbers, the skin friction coefficient is equal in both cylindrical and conical flow.

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Availability of data and material

The study data are openly available.

Code availability

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Abbreviations

\(C_{f}\) :

Skin friction coefficient of the inner cylinder

\(C_{f0}\) :

The skin friction coefficient without bubbles

\(C_{fcy}\) :

Skin friction coefficient of cylinder

\(C_{fco}\) :

Skin friction coefficient of cone

\(f_{\beta }\) :

Coefficient in \(\omega\) equation

\(f_{{\beta^{ * } }}\) :

Coefficient in \(k\) equation

\(g\) :

Gravitational acceleration \(\left( {{\raise0.7ex\hbox{$m$} \!\mathord{\left/ {\vphantom {m {s^{2} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${s^{2} }$}}} \right)\)

\(k\,\) :

Turbulence kinetic energy \(\left( {{\raise0.7ex\hbox{${m^{2} }$} \!\mathord{\left/ {\vphantom {{m^{2} } {s^{2} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${s^{2} }$}}} \right)\)

\(\kappa_{cy}\) :

Power gain in cylindrical Taylor-Couette flow

\(\kappa_{co}\) :

Power gain in conical Taylor-Couette flow

\(P\) :

Pressure \(\left( {Pa} \right)\)

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

Reynolds number

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

Turbulence Reynolds number

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

Reynolds number in cylindrical Taylor-Couette flow

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

Reynolds number in conical Taylor-Couette flow

\(R_{avg}\) :

Average radius in conical Taylor-Couette flow \(\left( m \right)\)

\(S_{ij}\) :

Strain tensor

\(T\) :

Torque on inner cylinder in presence of bubbles \(\left( {N.m} \right)\)

\(T_{0}\) :

Torque on inner cylinder without bubbles \(\left( {N.m} \right)\)

\(t\) :

Time \((s)\)

\(U\) :

Linear velocity of inner cylinder \(\left( {{\raise0.7ex\hbox{$m$} \!\mathord{\left/ {\vphantom {m s}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$s$}}} \right)\)

\(U_{avg}\) :

Average linear velocity of inner cone \(\left( {{\raise0.7ex\hbox{$m$} \!\mathord{\left/ {\vphantom {m s}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$s$}}} \right)\)

\(u_{i}\) :

Mean velocity components

\(u_{i}^{\prime }\) :

Fluctuating velocity components

\(x_{i}\) :

Position vector \(\left( m \right)\)

\(\beta (t)\) :

Angular acceleration \(\left( {{\raise0.7ex\hbox{${rad}$} \!\mathord{\left/ {\vphantom {{rad} {s^{2} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${s^{2} }$}}} \right)\)

\(\eta\) :

Skin friction coefficient ratio

\(\mu \,\) :

Dynamic viscosity \(({\raise0.7ex\hbox{${N \cdot s}$} \!\mathord{\left/ {\vphantom {{N \cdot s} {m^{2} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${m^{2} }$}})\)

\(\mu_{t}\) :

Turbulence dynamic viscosity \(({\raise0.7ex\hbox{${N \cdot s}$} \!\mathord{\left/ {\vphantom {{N \cdot s} {m^{2} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${m^{2} }$}})\)

\(\upsilon\) :

Kinematic viscosity of main fluid \(({\raise0.7ex\hbox{${m^{2} }$} \!\mathord{\left/ {\vphantom {{m^{2} } s}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$s$}})\)

\(\tau_{w} \,\) :

Wall shear stress \(({\raise0.7ex\hbox{$N$} \!\mathord{\left/ {\vphantom {N {m^{2} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${m^{2} }$}})\)

\(\tau_{w - out}\) :

Wall shear stress on the bottom surface \(({\raise0.7ex\hbox{$N$} \!\mathord{\left/ {\vphantom {N {m^{2} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${m^{2} }$}})\)

\(\rho \,\) :

Density of fluid \(({\raise0.7ex\hbox{${kg}$} \!\mathord{\left/ {\vphantom {{kg} {m^{3} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${m^{3} }$}})\)

\(\delta_{ij}\) :

Kronecker delta

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Authors and Affiliations

  1. Department of Energy, Politecnico Di Milano, Via La Masa 34, 20156, Milan, Italy

    Mohammad Ahmadi

  2. Department of Mechanical Engineering, University of Sistan and Baluchestan, 98135-987, Zahedan, Iran

    Said Farahat

  3. Department of Mechanical Engineering, University of Gonabad, Gonabad, Iran

    Seyed Morteza Javadpour

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  1. Mohammad Ahmadi
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  2. Said Farahat
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  3. Seyed Morteza Javadpour
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Correspondence to Seyed Morteza Javadpour.

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Ahmadi, M., Farahat, S. & Javadpour, S.M. Numerical analysis of rotational shape effect on skin friction coefficient in Couette-Taylor flows (with and without injection). J Braz. Soc. Mech. Sci. Eng. 43, 573 (2021). https://doi.org/10.1007/s40430-021-03262-4

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