Similarity analysis for turbulent boundary layer with pressure gradient: The outer flow

The 'equilibrium'-type similarity analysis of George and Castillo¹¹ for the outer part of zero pressure gradient boundary layers has been extended to include boundary layers with pressure gradient. The outer velocity profiles (for fixed upstream conditions) are found to scale with U_∞, the Reynolds shear stress profiles with U²_∞dδ/dx, and the condition for 'equilibrium' similarity is Λ ≡ [δ/(dδ/dx) [(1/ρU²_∞)dP_∞/dx] = - [δ/(dδ/dx)] [(dU_∞/dx)/U_∞] = constant. For Λ ≠ 0 it follows that δ ~ U-^1/Λ_∞Three major results are obtained: First, most pressure gradient boundary experiments are found to be equilibrium flows (by the new definition), and non-equilibrium flows appear to be the exception. Second, the Reynolds number dependence observed in some profiles is primary due to the upstream conditions, but this effect can be removed by using U_∞δ*/δ to scale the velocity deficit. Third, there appear to be only three values of Λ and three corresponding outer boundary layer velocity profiles: one for all adverse pressure gradients (APG, Λ ≈ 0.22), one for all favorable pressure gradients (FPG, Λ ≈ -1.92), and one for all zero pressure gradients (ZPG, Λ = 0).