Negative production of turbulent kinetic energy in a turbulent separation bubble

DNS data are used to examine the behavior of turbulence in the boundary layer separating from a flat plate, and reattaching. Particular attention is given to a region of negative production of turbulent kinetic energy. The inlet Reynolds number $R_\theta$ based on momentum thickness is equal to 300, 600 and 900. In all cases, the production $P_k$ is weak across the bubble and goes negative with a smaller magnitude than the dissipation at the top, where the streamline curvature is convex. An indicator of streamwise curvature $U_{2,1}$, which comes from a rapid pressure-driven change of the mean strain rate, is indeed associated with negative $P_k$. That is, the budget term arising from $U_{2,1}$ yields negative Reynolds shear stress ($-\overline{uv}<0$), and then the product of $-\overline{uv}$ and $U_{1,2}$ contributes to negative $P_k$. There is no one-to-one correspondence in a region between negative $-\overline{uv}$ and negative $P_k$. The correspondence is however excellent when the Reynolds shear stress is defined in the streamline orthogonal coordinate system, i.e., $\overline{ab}\equiv\left((\overline{vv}-\overline{uu})U_1 U_2+\overline{uv}(U_1^2-U_2^2))\right/(U_1^2+U_2^2)$, which underlines that the streamline curvature is an important ingredient for negative $P_k$.