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Normal and cross-flow Reynolds stresses: differences between confined and semi-confined flows

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Abstract

Understanding turbulent wall-bounded flows remains an elusive goal. Most turbulent phenomena are non-linear, complex and have broad range of scales that are difficult to completely resolve. Progress is made only in minute steps and enlightening models are rare. Herein, we undertake the effort to bundle several experimental and numerical databases to overcome some of these difficulties and to learn more about the kinematics of turbulent wall-bounded flows. The general scope of the present work is to quantify the characteristics of wall-normal and spanwise Reynolds stresses, which might be different for confined (e.g., pipe) and semi-confined (e.g., boundary layer) flows. In particular, the peak position of wall-normal stress and a shoulder in spanwise stress never described in detail before are investigated using select experimental and direct numerical simulation databases available in the open literature. It is found that the positions of the \( \left\langle {v'{^2} } \right\rangle^{ + } \)-peak in confined and semi-confined flow differ significantly above δ + ≈ 600. A similar behavior is found for the position of the \( \left\langle {u'v'} \right\rangle^{ + } \)-peak. The upper end of the logarithmic region seems to be closely related to the position of the \( \left\langle {v'{^2} } \right\rangle^{ + } \)-peak. The \( \left\langle {w'{^2} } \right\rangle^{ + } \)-shoulder is found to be twice as far from the wall than the \( \left\langle {v'{^2} } \right\rangle^{ + } \)-peak. It covers a significantly large portion of the typical zero-pressure-gradient turbulent boundary layer.

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Notes

  1. Strict self-similarity is here understood as the state where all moments of the velocity fluctuations taken at different streamwise positions and properly normalized collapse in single curves for the entire wall flow. The equilibrium adverse-pressure-gradient turbulent boundary layer investigated by Skåre and Krogstad (1994) is an appropriate example for that state.

  2. Comte-Bellot’s (1965) data (red symbols in Fig. 1) will be discussed in one of the following paragraphs.

  3. Note that Österlund’s (1999) log law has parameters κ = 0.38 and B = 4.1, which differ from the classical κ = 0.41 and B = 5 and may affect the determination of the mean-flow logarithmic region.

  4. Of course there is still a third, the acherontic option, that due to possible super-grid problems, the influence of the large scales on the v-fluctuations is not properly captured by recent DNS and therefore all of them are discredited. However, because there are no cogent arguments for this option, we will not follow this track.

Abbreviations

u′ v′ w′ :

Velocity fluctuations in streamwise, wall-normal, and spanwise direction

\( u_{e} \) :

Velocity at the edge of the boundary layer or the channel’s centerline

\( u_{\tau } \) :

Wall skin-friction velocity

\( \left\langle {u'v'} \right\rangle^{ + } \) :

Reynolds shear stress, normalized with inner variables

\( \left\langle {v'{^2} } \right\rangle^{ + } \) :

Wall-normal stress, normalized with inner variables

\( \left\langle {w'{^2} } \right\rangle^{ + } \) :

Spanwise stress, normalized with inner variables

y :

Wall-normal coordinate

\( y_{puv}^{ + } = {{y_{puv} u_{\tau } } \mathord{\left/ {\vphantom {{y_{puv} u_{\tau } } \nu }} \right. \kern-\nulldelimiterspace} \nu } \) :

Peak position of Reynolds shear stress, normalized with inner variables

\( y_{pv}^{ + } = {{y_{pv} u_{\tau } } \mathord{\left/ {\vphantom {{y_{pv} u_{\tau } } \nu }} \right. \kern-\nulldelimiterspace} \nu } \) :

Peak position of wall-normal stress, normalized with inner variables

\( y_{sw}^{ + } = {{y_{sw} u_{\tau } } \mathord{\left/ {\vphantom {{y_{sw} u_{\tau } } \nu }} \right. \kern-\nulldelimiterspace} \nu } \) :

Shoulder position of spanwise stress, normalized with inner variables

\( \delta \) :

Boundary-layer thickness, half channel height or pipe radius

\( \delta^{ + } = {{\delta u_{\tau } } \mathord{\left/ {\vphantom {{\delta u_{\tau } } \nu }} \right. \kern-\nulldelimiterspace} \nu } \) :

Kármán number

\( \Upgamma_{1} ,\Upgamma_{2} \) :

Diagnosis functions

\( \eta_{pv} = {{y_{pv} } \mathord{\left/ {\vphantom {{y_{pv} } \delta }} \right. \kern-\nulldelimiterspace} \delta } \) :

Peak position of wall-normal stress, normalized with outer variables

\( \nu \) :

Kinematic viscosity

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Acknowledgments

This paper is dedicated to the extensive and distinguished career of our friend and longtime editor of Experiments of Fluids, Professor Donald O. Rockwell. We thank all scientists who supported us with their data. Special thanks go to Professor Peter Bradshaw who provided us with his translation of Genevieve Comte-Bellot’s doctoral thesis. Finally, the authors would like to thank an anonymous reviewer for encouraging us to extend the paper significantly. A shorter conference version of this article was presented at the Sixth International Symposium on Turbulence, Heat and Mass Transfer, September 14–18, 2009, Rome, Italy.

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  1. Institut für Luft- und Kältetechnik Dresden, Bertolt-Brecht-Allee 20, 01309, Dresden, Germany

    Matthias H. Buschmann

  2. Department of Mechanical Engineering, Virginia Commonwealth University, Richmond, VA, 23284-3015, USA

    Mohamed Gad-el-Hak

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Correspondence to Mohamed Gad-el-Hak.

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Buschmann, M.H., Gad-el-Hak, M. Normal and cross-flow Reynolds stresses: differences between confined and semi-confined flows. Exp Fluids 49, 213–223 (2010). https://doi.org/10.1007/s00348-010-0834-z

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