Drag reduction for swept flat plate flow

Drag reduction analyses for turbulent flow over a swept flat plate are conducted. To be more precise, large-eddy simulations of oblique traveling transversal surface waves in turbulent flat plate boundary layer flow at a momentum thickness based Reynolds number of ${\text{Re}}_{\theta }=1000$ are performed. For two parameter sets of the actuation parameters wavelength, period, and amplitude the angle between the wave propagation direction and the streamwise direction, i.e., the sweep angle $\phi$, is varied. The goal is to investigate the susceptibility of drag reduction, net power saving, and turbulence to different sweep angles of the traveling waves. The extension of the technique of spanwise traveling transversal surface waves allows partially up- or downstream traveling waves, i.e., opposite to or in the main flow direction. For downstream traveling waves, the periodic spanwise shear is decreased and the reduction of the wall-shear stress is diminished. Nevertheless, strong decreases of the turbulent stress tensor components persist even for the largest sweep angle. For upstream traveling waves, there is an increase of the lowered friction drag reduction. However, the turbulence level in the outer flow is also higher since the pressure drag is increased such that the total drag reduction is massively lowered. Compared to a purely spanwise traveling wave, i.e., $\phi =0^{\circ }$, the sweep angle $\phi \ne 0^{\circ }$ leads to a much stronger phase dependence of the skin friction since the partially up- and downstream traveling waves induce also periodic fluctuations in the streamwise direction. A shift of energy towards larger scale motions is observed which is also reflected by an increase of the integral length scale close to the wall.

Figure 1

Overview of the physical domain of the actuated turbulent boundary layer flow, where $L_x,L_y,$ and $L_z$ are the dimensions of the domain in the Cartesian directions, $\lambda$ is the spanwise wavelength of the traveling wave, $x_0$ marks the onset of the actuation, and $\phi$ is the angle of the wave propagation direction; the surface area for the integration of the wall-shear stress ${\tau }_w$ is shaded red between the streamwise positions $x_1$ and $x_2$.

Figure 2

Wall-normal distributions of (a) the mean streamwise velocity component and (b) the stochastic components of the Reynolds stress tensor scaled by the nonactuated reference friction velocity at $x=0$ for the LES in this paper (lines) and a DNS from [45] (symbols).

Figure 3

Temporal evolution of the instantaneous integrated drag for the nonactuated reference case $N_1$, and the actuated cases (a) $N_2$, (b) $N_3$, (c) $N_4$, and (d) $N_5$ and $N_{10}$ normalized by the averaged integrated drag of the reference case. $\phi$, sweep angle; do, downstream; up, upstream.

Figure 4

Variation of (a) the viscous, pressure, and total drag coefficient and (b) the net power saving.

Figure 5

Sinusoidal mesh deformation; spanwise averages are determined along $\zeta$, i.e., lines $\eta =\text{const}$.

Figure 6

Spanwise averaged wall-normal distribution of the mean spanwise velocity at $x=90$, scaled by (a) ${\overline{u}}^{+}$ the friction velocity of the nonactuated reference case $N_1$ with zooms of (b) the near-wall region and (c) the logarithmic region of the same data, and scaled by (d) ${\overline{u}}^{*}$ the friction velocity of each respective case. $\phi$, sweep angle; do, downstream; up, upstream.

Figure 7

Spanwise averaged wall-normal distribution of the spanwise velocity at $x=90$. $\phi$, sweep angle; do, downstream; up, upstream.

Figure 8

Ratio of the spanwise to the streamwise drag coefficient for (a) the total drag and (b) the viscous drag. $\phi$, sweep angle; do, downstream; up, upstream.

Figure 9

Bird's view of the three-dimensional contours of the random streamwise velocity fluctuations for $u^{\prime \prime +}=-3$ (blue) and $u^{\prime \prime +}=3$ (red) in the streamwise interval $-20\le x\le 145$ and in the near-wall region $0<y^{+}<20$ of (a) the nonactuated reference case $N_1$ and the actuated cases (b) $N_2$, (c) $N_5$, and (d) $N_{10}$. $\phi$, sweep angle; do, downstream; up, upstream.

Figure 10

Wall-normal distributions of the stochastic components of the Reynolds stress tensor scaled by the nonactuated reference friction velocity at $x=90$: (a) streamwise, (b) wall-normal, (c) spanwise, and (d) shear-stress component. $\phi$, sweep angle; do, downstream; up, upstream.

Figure 11

Wall-normal distributions of the stochastic turbulent kinetic energy $k_t=\frac{1}{2}(\overline{u^{″}{u}^{″}}+\overline{v^{″}{v}^{″}}+\overline{w^{″}{w}^{″}})$ at $x=90$ in (a) inner and (b) outer scaling. $\phi$, sweep angle; do, downstream; up, upstream.

Figure 12

Streamwise development of the spanwise averaged momentum thickness. $\phi$, sweep angle; do, downstream; up, upstream.

Figure 13

Wall-normal distributions of the spanwise averaged stochastic vorticity fluctuations at $x=90$: (a) streamwise $\overline{{\omega }_x^{″}{\omega }_x^{″}}$, (b) wall-normal $\overline{{\omega }_y^{″}{\omega }_y^{″}}$, and (c) spanwise component $\overline{{\omega }_z^{″}{\omega }_z^{″}}$. $\phi$, sweep angle; do, downstream; up, upstream.

Figure 14

Skin-friction coefficient at $x=90$: (a) spanwise distribution of the phase-averaged skin-friction coefficient and (b) spanwise averaged values of the skin-friction coefficient. $\phi$, sweep angle; do, downstream; up, upstream.

Figure 15

Normalized probability density function of the gradient of the streamwise velocity $\overline{u}+u^{″}$ at $x=90$. $\phi$, sweep angle; do, downstream; up, upstream.

Figure 16

Premultiplied energy spectra of the streamwise velocity fluctuations $u^{″}$ for the spatial distribution in the spanwise direction at $x=90$; all spectra are normalized by the total resolved energy based on the streamwise velocity component at the respective wall-normal height: (a) $y^{+}=10$ and (b) $y^{+}=200$. $\phi$, sweep angle; do, downstream; up, upstream.

Figure 17

Wall-normal distribution of the spanwise integral length scale ${\mathrm{\Lambda }}_z^{+}$ in inner scaling for the streamwise velocity component in the spanwise direction at $x=90$. $\phi$, sweep angle; do, downstream; up, upstream.

Figure 18

Distribution of the wall-normal gradient of the periodic velocity gradient $\partial \tilde{w}{/\partial y|}_{\mathrm{wall}}^{+}$ in the $x\text{−}z$ plane for the actuated cases (a) $N_2$, (b) $N_3$, (c) $N_4$, (d) $N_5$, and (e) $N_{10}$. (f) Distribution at $x=50$. $\phi$, sweep angle; do, downstream; up, upstream.