Drag reduction mechanisms of a car model at moderate yaw by bi-frequency forcing

A bi-frequency open-loop control strategy aiming to combine both high- and low-frequency forcing effects is used to experimentally reduce the drag of a simplified car model at a slight yaw angle of $5^{\circ }$. The unforced mean wake features a lateral asymmetry which induces a low base pressure footprint close to the leeward side and increases drag compared to the aligned model. Forcing is performed with pulsed jets along the windward trailing edge. High-frequency forcing acts as a time-invariant flap. The fluidic flap effect deviates the windward shear layer towards the leeward side and reduces the wake bluffness, but the lateral asymmetry of the near wake is still observed. The drag reduction related to this high-frequency forcing is about 6% with a high actuation efficiency. A modulation of the high-frequency forcing with a low-frequency component is then introduced in order to modify the mass and momentum exchange in the separating shear layer at the windward trailing edge. We find that the modulated forcing provides the ability to manipulate the mean wake orientation while maintaining the fluidic flap effect. Among all wake orientations, those reducing drag are the ones having a mean symmetric wake. The bi-frequency control strategy leads to a maximum drag reduction of 7% for the best choice of frequencies. Importantly, the bi-frequency control is more efficient than the single high-frequency forcing, the actuator requiring only half the actuation energy and presenting an actuation efficiency multiplied by 3. Finally, the physical mechanisms related to drag reduction are carefully analyzed. In particular, we show that the wake symmetrization reduces the global production of turbulent kinetic energy in the shear layers. These results open up opportunities for closed-loop control of wake asymmetries.

Figure 1

Sketch of the yawed model setup. (a) Wind tunnel and model geometries. (b) Top view of the yawed model. (c) Perspective view. (d) Locations of the pressure sensors over the base surface.

Figure 2

(a) Time series of the jet velocity at $f=50$ and $500\mathrm{Hz}$. (b) The evolution of the effective jet velocity as a function of the frequency.

Figure 3

Unforced flow. Mean wake in the mid-height plane $z=0.67$ and the mean base pressure. From (a) to (c): distribution of the mean streamwise velocity $\overline{u}$, the mean spanwise velocity $\overline{v}$, and the mean base pressure. Black lines in (a) are iso-contour lines at $\overline{u}=\pm 0.25$.

Figure 4

Unforced flow. Distribution of Reynolds stresses in the mid-height plane $z=0.67$. From (a) to (c): $\overline{u^{\prime }{u}^{\prime }},\overline{v^{\prime }{v}^{\prime }}$, and $\overline{u^{\prime }{v}^{\prime }}$.

Figure 5

Effect of the windward high-frequency forcing ${\text{St}}_H=6$ on the wake. (a) Streamlines of the mean velocity field overlapped with the contour maps of the velocity magnitude $‖\overline{\mathit{u}}‖$. (b) Iso-contour lines of time-averaged streamwise velocity $\overline{u}\in \{-0.25,0.25,0.6\}$. Black line: unforced flow; red dashed line: high-frequency forced flow. (c) Streamwise evolution of the velocity angle $\theta =arctan(\overline{v}/\overline{u})$ of the streamline issuing from the leeward $(x,y)=(0,0.6)$ and windward separation point $(x,y)=(0,-0.6)$. (d) $\overline{C_p}$ on the mid-height line.

Figure 6

Bi-frequency actuation command generated by the multiplication of two periodic square waves at low and high frequency respectively. $T_{LF}$ corresponds to the period of the low-frequency component.

Figure 7

(a) Phase-averaged jet velocity $[[V_{\text{Jet}}]]/U_{\infty }$ for ${\text{St}}_H=0.24$ (left) and ${\text{St}}_{H_{\text{bf}}}=0.24\otimes 6$ (right). $T_{LF}$ is the period of the low-frequency forcing $\left({\text{St}}_H=0.24\right)$. (b) Power spectral density (PSD) of the jet velocity signal at ${\text{St}}_{H_{\text{bf}}}=0.24\otimes 6$.

Figure 8

Bi-frequency forcing results. (a) Variation of ${\gamma }_D\left(▵\right)$ and $\overline{\partial C_p/\partial y}(*)$ as a function of ${\text{St}}_{H_{\text{low}}}$. For comparison, the value ${\gamma }_D=0.94$ for ${\text{St}}_H=6$ is shown on the left $y$ axis and the value $\overline{\partial C_p/\partial y}=-0.11$ for both the unforced flow and ${\text{St}}_H=6$ is shown on the right $y$ axis. (b) $\overline{C_p}$ values on the mid-height line. (c) The evolution of PDF of $\partial C_p/\partial y$ with increasing ${\text{St}}_{H_{\text{low}}}$. The PDFs are normalized by their maximum value. The letter “u” in the abscissa stands for the unforced flow. (d) The joint PDF of $\langle C_p\rangle$ versus $\partial C_p/\partial y$ for some highlighted configurations. The letters “A” and “S” indicate the asymmetric and symmetric state, respectively.

Figure 9

Power spectral density (PSD) of the lateral pressure gradient $\partial C_p/\partial y$ in the frequency range ${\text{St}}_H=[0.01,1]$.

Figure 10

Effects of bi-frequency actuation on the near wake. From left to right: unforced flow, windward forced flows at ${\text{St}}_{H_{\text{bf}}}=0.24\otimes 6$ and ${\text{St}}_{H_{\text{bf}}}=0.48\otimes 6$. (a) Distributions of the time-averaged spanwise velocity $\overline{v}$ on the mid-height plane $z=0.67$. (b) Streamlines of the mean velocity field overlapped with the contour maps of the velocity magnitude $‖\overline{\mathit{u}}‖=\sqrt{{\overline{u}}^2+{\overline{v}}^2}$.

Figure 11

Streamwise evolution of the velocity angle $\theta =arctan(\overline{v}/\overline{u})$ of the streamline issuing from the leeward $(x,y)=(0,0.6)$ and windward separation point $(x,y)=(0,-0.6)$.

Figure 12

Effects of bi-frequency actuation on the shear layers. From left to right: unforced flow, ${\text{St}}_{H_{\text{bf}}}=0.24\otimes 6$, and ${\text{St}}_{H_{\text{bf}}}=0.48\otimes 6$. (a) Distribution of $\overline{u^{\prime }{v}^{\prime }}$ along the windward shear layer in the streamwise range $x\in [0,0.5]$. (b) Distribution of the spatial correlation coefficient ${\rho }_{v^{\prime }{v}^{\prime }}$ along the windward shear layer. The reference point $(x_0,y_0)$ for each case is marked by “+” in the figure. At the reference point, ${\rho }_{v^{\prime }{v}^{\prime }}=1$. The dashed line and arrow indicate the zone specified in (a).

Figure 13

Streamwise evolution of the maximum $\overline{u^{\prime }{v}^{\prime }}$ along the windward shear layer.

Figure 14

Distribution of the production rate $\mathrm{\Pi }$ in the 2D plane $(x,y)\in [0,2]\times [-0.8,0.8]$.

Figure 15

Synthesis of drag reduction performances and mechanisms. The inserted sketches illustrate qualitatively the recirculating flow in the wake. HF and LF represent high frequency and low frequency respectively.