The log behaviour of the Reynolds shear stress in accelerating turbulent boundary layers

Direct numerical simulation of highly accelerated turbulent boundary layers (TBLs) reveals that the Reynolds shear stress, u′ v′⁺, monotonically decreases downstream and exhibits a logarithmic behaviour (e.g. -u′ v′⁺ = -(1/A_uv) ln y⁺ + B_uv) in the mesolayer region (e.g. 50 ≤ y⁺ ≤ 170). The thickness of the log layer of u′ v′⁺ increases with the streamwise distance and with the pressure gradient strength, extending over a large portion of the TBL thickness (up to 55 %). Simulations reveal that V⁺ ∂ U⁺/∂y⁺ ∼ 1/y⁺ ∼ ∂u′ v′⁺/∂y⁺, resulting in a logarithmic u′ v′⁺ profile. Also, V⁺ ∼ -y⁺ is no longer negligible as in zero-pressure-gradient (ZPG) flows. Other experimental/numerical data at similar favourable-pressure-gradient (FPG) strengths also show the presence of a log region in u′ v′⁺. This log region in u′ v′⁺ is larger in sink flows than in other spatially developing FPG flows. The latter flows exhibit the presence of a small power-law region in u′ v′⁺, which is non-existent in sink flows.