Energy thickness in turbulent boundary layer flows

Kurzfassung

In this study, we investigate the properties of energy thickness δ3 in turbulent boundary layer (TBL) flows, a parameter derived solely from the mean streamwise velocity (U) profile. Through an analysis of the energy integral equation for zero pressure gradient TBLs, we establish a close relationship between turbulent kinetic energy (TKE) production and δ3, offering a practical method to estimate TKE production, which is particularly useful in physical experiments where direct measurements are challenging. The significance of δ3 becomes even more pronounced in TBLs under pressure gradient. Through extensive analysis of numerical and experimental data, we show that the ratio between δ3 and the momentum thickness δ2 is a promising criterion for predicting flow separation. Moreover, we derive a new energy integral equation for TBLs under arbitrary pressure gradients, and provide approximations for TKE productions terms by Ruv ∂U/∂y and Ruu ∂U/∂x, and dissipation term by the mean shear. Here, x, y represent the streamwise and wall-normal directions, respectively, and Ruu and Ruv are the Reynolds normal and shear stresses. The accuracy and robustness of the new energy integral equation and the approximation equations are validated using direct numerical simulations data. Our results show that the TKE production by Ruv ∂U/∂y and the overall productions consistently remain positive, reflecting a continuous conversion of mean kinetic energy into TKE across all TBLs. However, under strong favourable pressure gradients, TKE production by Ruu ∂U/∂x becomes negative, indicating a reverse energy transfer from TKE to mean kinetic energy.