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

Abstract

Direct numerical simulation of highly accelerated turbulent boundary layers (TBLs) reveals that the Reynolds shear stress,$\overline{u^{\prime }v^{\prime }}^{+}$, monotonically decreases downstream and exhibits a logarithmic behaviour (e.g. $-\overline{u^{\prime }v^{\prime }}^{+}=-(1/A_{uv})\ln y^{+}+B_{uv}$) in the mesolayer region (e.g. $50\leqslant y^{+}\leqslant 170$). The thickness of the log layer of$\overline{u^{\prime }v^{\prime }}^{+}$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$V^{+}\,\partial U^{+}/\partial y^{+}\sim 1/y^{+}\sim \partial \overline{u^{\prime }v^{\prime }}^{+}/\partial y^{+}$, resulting in a logarithmic$\overline{u^{\prime }v^{\prime }}^{+}$profile. Also,$V^{+}\sim -y^{+}$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$\overline{u^{\prime }v^{\prime }}^{+}$. This log region in$\overline{u^{\prime }v^{\prime }}^{+}$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$\overline{u^{\prime }v^{\prime }}^{+}$, which is non-existent in sink flows.