Near-Wall Determination of the Turbulent Prandtl Number Based on Experiments, Numerical Simulation and Analytical Models

Abstract

The Reynolds-averaged computation of turbulent flow with heat transfer most commonly models the turbulent heat flux as directly related to the turbulent flux of momentum through the turbulent Prandtl number. Its significant deviation from a uniform bulk flow value for high molecular Prandtl numbers needs to be adequately described to predict accurately the heat transfer. The present study derives a model for the near-wall variation of this important parameter, used as input into an analytical solution of heated turbulent pipe flow. The basic functional form of the profile of the turbulent Prandtl number is determined from direct numerical simulations (DNS), and experimental data are used for model calibration. The analytically predicted Nusselt numbers agree very well with experimental measurements, proving the reliability of the proposed model for the turbulent Prandtl number also for Reynolds numbers well beyond the scope of DNS. The validation against experiments further highlights the significant effect of the temperature-dependent material properties of the considered high Prandtl number liquids. Numerical simulations often discard this aspect to reduce the computational effort. The present combination of DNS, analytical solution, and experiments appears as a convenient approach for modeling turbulent key quantities such as the turbulent Prandtl number, which is well applicable to other convective flow conditions and Prandtl number regimes, as well.

Acknowledgements

The financial support from the Austrian Research Promotion Agency (FFG), the Virtual Vehicle competence center Project-number B2T3, and AVL List GmbH are gratefully acknowledged. The simulations were conducted using the Vienna Scientific Cluster (VSC).

Additional information

Notes on contributors

Helfried Steiner

Helfried Steiner is an Associate Professor at the Institute of Fluid Mechanics and Heat Transfer at Graz University of Technology, Austria. His main research activities are in the field of numerical simulation and modeling of complex turbulent flow, using the methods direct numerical simulation (DNS), Large-Eddy Simulation (LES), as well as Reynolds-averaged Navier–Stokes (RANS). He made various contributions to the modeling of turbulent combustion, flow boiling, as well as multiphase flow. His research is currently focused on the analysis and modeling of the near wall turbulent convective heat transfer to describe accurately thermal boundary layers when applying LES or RANS methods.

Christoph Irrenfried

Christoph Irrenfried is a Ph.D. student in the Faculty of Mechanical Engineering at Graz University of Technology under the supervision of Prof. H. Steiner. The focus of his doctoral research is on the detailed investigation of heat transfer in wall-bounded flow, using direct numerical simulations. He received his master's degree in 2013, and his bachelor's degree in 2011, both from the Graz University of Technology.

Günter Brenn

Günter Brenn studied aerospace engineering at the University of Stuttgart in Germany. He received his Ph.D. from the same university for research on drop shape oscillations. After a two years post-doc stay in Japan, in 1992 he joined the Chair of Fluid Mechanics (LSTM) at the University of Erlangen (Germany), where he did his habilitation in fluid mechanics in 1999. In 2002, he took his present position of a full professor and head of the Institute of Fluid Mechanics and Heat Transfer at the University of Technology in Graz, Austria. His research interests are spray flows, the rheology, and rheometry of complex liquids, heat and mass transfer, the stability of free-surface flows, and optical flow measuring techniques.