Similarity analysis of favorable pressure gradient turbulent boundary layers with eventual quasilaminarization

Using similarity analysis, the scales and similarity constraints for a favorable pressure gradient (FPG) turbulent boundary layer with eventual quasilaminarization are obtained. In order to achieve equilibrium in the boundary layer, the pressure parameter $Λ$ must be a constant; thus, a power relation between the boundary layer thickness $δ$ and the free-stream velocity $U∞$ exists. Consequently, the power is given by the pressure parameter $Λ$ as $δ∼U∞−1/Λ$⁠. Through an analysis using the pressure parameter, two quadrants are found: quadrant I describes FPG turbulent flows and quadrant II corresponds to quasilaminar flows. Moreover, a horizontal line exists for zero pressure gradient flows. Different values of the pressure parameter are found for equilibrium FPG flows, contrary to the findings of Castillo and George [“Similarity analysis for turbulent boundary layer with pressure gradient: Outer flow,” AIAA J. 39, 41 (2001)]. In the case of strong FPG flows with quasilaminarization, the pressure parameter reaches a maximum value of 0.47. At this point, a sudden reduction in the skin friction of about 57% is observed and a redistribution of the Reynolds stresses throughout the boundary layer is achieved. The mean velocity deficit profiles are also found to be attenuated when scaled using the free-stream velocity $U∞$ or $U∞δ∗/δ$⁠. For flows in quadrant II, a reduction in the outer flow of the $⟨u2⟩$ component of the Reynolds stress is observed, whereas the $⟨v2⟩$ and $−⟨uv⟩$ components nearly vanish impending quasilaminarization. Due to the presence of the $⟨u2⟩$ component in the boundary layer, the flow never reaches a full laminar state and a more uniform redistribution of the component is observed as the skin friction decreases due to the imposed FPG. Furthermore, the shape of the profile remains the same until a quasilaminar state is reached, where the profile no longer shows high values of the stress on the proximity to the wall. In addition, the production term $−⟨uv⟩dU/dy$ is nearly zero for flows in quadrant II. Also, the boundary layer parameters such as the shape factor $H$ and the ratio of the displacement thickness to the boundary layer thickness, $δ∗/δ$⁠, increase as the flow achieves a quasilaminar state.