Modified Formulation of Laminar Kinetic Energy Transition Models by Means of Elastic-Net of a Big Experimental Database of Separated Flows

Abstract

New variants of the terms involved in the transport equations of Laminar Kinetic Energy (LKE) transition schemes are proposed. The model here proposed was tuned by means of a big experimental data set describing transitional separated flows. Data were acquired by means of time-resolved Particle Image Velocimetry (TR-PIV) measurements above a flat plate centered in a variable area opening channel. Tests were carried out for different free-stream turbulence intensities, Reynolds numbers and adverse pressure gradients. The large amount of test conditions provides the statistical response of the separated flow transition process to the parameter variation. For each condition, Proper Orthogonal Decomposition (POD) was applied to the ensemble of snapshots. Analysis of POD modes and spectra of related temporal coefficients allows us to define a scale separation criterion, and thus construct reduced order models of laminar and turbulent terms appearing in LKE schemes. Combinations of the POD modes of the two velocity components, their gradients and mixed terms involving velocity scales and mean flow strain rates are used to compute the targeting function vectors for the learning of the models. The role played by normal and shear stress-strain mechanisms in the different parts of the separated flow region was addressed for a fine tuning of both the laminar and the turbulent kinetic energy production terms. Additionally, a new definition of the energy transfer rate between the laminar and the turbulent scales is educated, using ingredients representing the work done by the finer scale on the larger one, inspired by the exact definition of the transfer rate appearing in the transport equations of coherent and incoherent fluctuations. An elastic-net technique was used to identify the best predictors, thus the model learning. The ability of the proposed model is verified by using it to predict results obtained in a different experimental database that did not participate to the education of the model.

Appendix

See Tables 2, 3 and 4.

Table 2 Coefficients of elastic net for $P_{K_L}$: $Tu=\frac{\sqrt{K_{tot}}}{U_{ext}}$, $Re=\frac{{\overline{U}}y}{\nu }{\frac{1}{10{,}000}}$ and $S=\left\| S\right\| \frac{y^{2}}{\nu }\frac{1}{10,000}$

Table 3 Coefficients of elastic net for $P_{K_T}$: $Tu=\frac{\sqrt{K_T}}{U_{ext}}$, $Re=\frac{\sqrt{K_{L}}y}{\nu }{\frac{1}{10{,}000}}$ and $S=\left\| S\right\| \frac{y^{2}}{\nu }\frac{1}{10,000}$

Table 4 Coefficients of elastic net for R: $Tu=\frac{\sqrt{K_{tot}}}{U_{ext}}$, $Re=\frac{{\overline{U}}y}{\nu }{\frac{1}{10{,}000}}$ and $S=\left\| S\right\| \frac{y^{2}}{\nu }\frac{1}{10,000}$