Assessment of an algebraic equilibrium wall-function for supercritical flows

In this work the application of an algebraic equilibrium wall-function to real-gas flows is presented and analyzed. The aim is to assess capabilities of existing algebraic wall-functions in supercritical conditions. In particular a systematic analysis on the coupled wall-function of Cabrit and Nicoud is carried out a-priori on a wall-resolved Large Eddy Simulations (WR-LES) database, featuring cryogenic para-hydrogen flow in a heated pipe at supercritical pressure. The model is shown to overestimate the wall-temperature for increasing values of the imposed heat flux and to slightly underestimate the skin friction velocity. The causes of the failure are investigated. In particular the focus is on the equilibrium boundary layer hypothesis, on the validity of the Van Driest transformation for supercritical, stratified flows and on the ideal-gas assumption employed in the original derivation of the model. For the latter, a consistent thermodynamic correction is proposed in order to extend the applicability of the mentioned wall-function to any arbitrary Equation of State (EOS). The proposed extension is tested a-priori and is shown to provide improved temperature and skin friction velocity predictions at wall, although still presenting relevant deviations from the reference database solution. The discrepancies seem to be addressed to the equilibrium assumption and to the Van Driest transformation. The former in particular accounts for 5-20% of the reference value for the skin friction velocity, among all cases and y^+ examined, and for 1-20% in terms of wall-temperature depending on the considered heat flux. The latter is shown to fail on the mentioned database for increasing stratifications, with errors between 1-40% depending on the considered case. An additional analysis of more recently proposed transformations for variable property flows reveals the same limitations.