Local relaminarization mechanism induced by a dynamic free-slip boundary

Applying a dynamic free-slip boundary in a turbulent boundary layer has been shown recently to shift outward the near-wall transverse vorticity away from the wall and reduces the wall skin friction by more than 40%. Herein we present a local relaminarization mechanism induced by the dynamic free-slip boundary, from the perspective of energy exchange and transportation. The spatial evolution of the energy components associated with the mean motion, turbulent motion, and a shear-free oscillatory motion is presented. An analysis of the average energy exchange process in the near-wall region suggests that the energy of turbulence is transferred to the mean motion, against the canonical downward turbulent energy cascade. A considerable amount of energy is supplied to the shear-free motions, which displaces the highly turbulent and shear motions away from the wall. The relaminarization mechanism is associated with outward-shifted transverse vorticity and the depletion of the shear motions near the wall. As an effective method to manipulate the critical region for wall shear stress generation, the dynamic free-slip boundary produces a much stronger effect than the conventional relaminarization process, which can be employed for efficient drag reduction and boundary layer control.

Figure 1

Analytical expressions representing the energy exchange process with (a) Reynolds decomposition and (b) triple decomposition. The $\langle ·\rangle$ represents the phase average. The arrows indicate the direction of energy transfer.

Figure 2

Snapshots of the (a) expanding and (b) contracting dynamic free-slip boundaries. The air-water interface is highlighted by red dotted lines and the movement direction of the interface is qualitatively shown by the arrows. The normalized $x$ coordinate ($\frac{x{u}_{\tau 0}}{\nu }$) is shown along the bottom edge, which will be referred to for presenting the results.

Figure 3

Profiles of (a) $\overline{U}$ and (b) and (c) $\tilde{U}$ in the (b) upstream and (c) downstream regions of the dynamic boundary array. In (a) diamonds denote upstream, crosses denote downstream, and the solid line shows the baseline case. In comparison with the upstream region, $\overline{U}$ decreases but $\tilde{U}$ increases in the downstream region.

Figure 4

Profiles of (a) turbulent kinetic energy $\overline{k}=1/2(\overline{u^{\prime 2}}+\overline{v^{\prime 2}})$ and (b) transverse vorticity $\overline{{\omega }_z}$ exhibit similar evolution trends in the $x$ direction. Both $\overline{k}$ and $\overline{{\omega }_z}$ decrease near the wall and develop an outer peak. The streamwise locations ($\frac{x{u}_{\tau 0}}{\nu }$) are given along the top edge of (a). The profiles of the baseline case are shown by the black solid line.

Figure 5

Profiles of the individual components of ${\mathrm{TKP}}_{\mathrm{MT}}$ at two representative $x$ locations (a) $\frac{x{u}_{\tau 0}}{\nu }=270$ and (b) $\frac{x{u}_{\tau 0}}{\nu }=660$, where negative and positive values of ${\mathrm{TKP}}_{\mathrm{MT}}$ are observed. The baseline $\mathrm{TKP}$ is shown for reference. Note that the scale of the $y$ axis is different in the two panels.

Figure 6

Contours for (a) ${\mathrm{TKP}}_{\mathrm{MT}}$, (b) ${\mathrm{TKP}}_{\mathrm{MO}}$, and (c) ${\mathrm{TKP}}_{\mathrm{OT}}$ in the near-wall region. Air films are represented as green half ovals.

Figure 7

Average value of the energy transfer process in the near-wall region between $\frac{y{u}_{\tau 0}}{\nu }=30$ and $\frac{y{u}_{\tau 0}}{\nu }=65$ and between $\frac{x{u}_{\tau 0}}{\nu }=10$ and $\frac{x{u}_{\tau 0}}{\nu }=750$. The direction of energy transfer is shown by the arrows. As a reference, the average value of the baseline TKP in the same region is 0.03.

Figure 8

Contours of the normalized velocity gradient (a) $\frac{\partial \overline{U}}{\partial y}\frac{\nu }{u_{\tau 0}^2}$ and (b) $\frac{\partial \overline{U}}{\partial x}\frac{\nu }{u_{\tau 0}^2}$. As the major component of $\overline{{\omega }_z}$, the distribution of $\frac{\partial \overline{U}}{\partial y}$ implies the existence of stationary vortices above the air film array. Beneath these vortices, strong positive $\frac{\partial \overline{U}}{\partial x}$ is generated.