Total mechanical energy transport lines and attractors in separating turbulent boundary layers

In order to provide new insights into the energetics of turbulent boundary layer flows with separation, we compute and characterize energy transport lines, which are defined as the lines tangent to the total mechanical energy transport vector field. Separation is induced in a ${\text{Re}}_{\theta }=490$ turbulent boundary layer by imposing two types of transpiration velocity profiles at the top boundary of the computational domain: one with suction and blowing, and the other with suction only. For both separation bubbles, we find that energy transport lines can exhibit attractors, which are a manifestation of local dissipation in the flow. We identify several attracting sets along the bottom wall and attracting spiral nodes inside the separation bubbles and their corresponding basins of attraction. The size and positions of the energy basins of attraction help us understand where the available total mechanical energy is transported to due to the mean flow, and turbulent Reynolds and viscous stresses, before it is fully dissipated. The suction-and-blowing separation bubble leads to slightly smaller losses of total mechanical energy than those for a reference nonseparated boundary layer, while the suction-only case shows larger losses than the nonseparated boundary layer case.

Figure 1

Contour plots of streamwise velocity in snapshots of the simulations, also showing aspects of the DNS configuration. (a) TSB-SB case; (b) TSB-SO case. Dashed vertical line: recycling plane; dotted vertical line: reference plane; thick solid line near the top boundary corresponds to a profile of $50V_{\text{top}}/U_o+100$; the thin solid line shows the separating streamline of the mean flow.

Figure 2

Contour plots of the mean total mechanical energy density $E(x,y)$. (a) TSB-SB; (b) TSB-SO. The dashed line shows the separating streamline.

Figure 3

Selected mean-flow streamlines (dashed lines) and energy transport lines (solid lines). (a) TSB-SB; (b) TSB-SO. The streamlines and energy lines are plotted starting from $x=0$, $y/{\theta }_o=2,4,...,10$. The source/sink of energy [i.e., the right-hand side (RHS) of Eq. (1)] is shown with filled contours. Energy lines near the spiral node attractor inside the separation bubble are not plotted for clarity; a closer view is shown in the upper-left inset. The upper-right inset in (a) shows the source of energy [i.e., positive RHS of Eq. (1)] near the crest of the TSB-SB bubble. The dotted line denotes the locus of $U=0$. The dash-dotted line is the separating streamline. The center of the attractor inside the bubble is marked by $+$ ($[x,y]/{\theta }_o=[317.3,17.6]$ for TSB-SB; $[313.6,16.2]$ for TSB-SO). The $y$ axis is stretched by a factor of 5 for clarity.

Figure 4

Basins of the energy attractors in TSB-SB (a) and TSB-SO (b). The classification for the various regions I–VI is described in the text. The various solid lines mark energy transport lines that act as boundaries between the basins of attraction. The $y$ axis is stretched by a factor of 5 for clarity.

Figure 5

Basins of the energy attractors at $x=0$ with reference to the mean streamwise velocity in wall units. (a) TSB-SB; (b) TSB-SO. The classification for the five regions refer to the text.

Figure 6

(a) Comparison of energy transport lines separating basins I+II+III+IV from region V (lines), and between region V and a location outside all dissipative regions (inflow below $y/{\theta }_o=20$) (lines with circle), for the three flows considered: TSB-SB (dashed lines), TSB-SO (long dashes and dots), and ZPG TBL [19] (solid lines). Panel (b) shows the downstream evolution of the total energy loss between the lines shown in (a): lines without symbol measure between the lines in (a) and the wall, and lines with triangles measure between the lines and lines with symbols in (a); lines with open square show the total energy loss in the flow domain.