Behavior of the square-back Ahmed body global modes at low ground clearance

The present work aims to study the evolution of the wake-flow dynamic, including the shear-layers interactions phenomena and the global modes behavior of a square-back Ahmed body with a width to height aspect ratio, $w/h$, of 1.346. Various ground-clearance configurations around the critical case associated with the onset of the lateral bistability and corresponding to a critical ground clearance, $g_c/w=0.1$, are investigated. For this purpose, vertical and transversal planar particle image velocimetry (PIV) and concurrent base pressure and force measurements are performed. Results show that the instantaneous wake-flow interactions between the side- or under-body flow and the wake shear layers depend mainly on the under-body flow rate and that a particular attention is required for the experimental under-body flow conditions. Pressure-velocity correlations and instantaneous filtered PIV snapshots analysis highlight the preferential shear-layer vortex shedding associated with the two selected stable and bistable wake-flow configurations. Present investigations underline the transition process from a stable wake configuration $(g/w<0.1)$ to a bistable state flow configuration $(g/w>0.1)$. The antisymmetric periodic global mode is strongly affected around the critical bistable case at the origin of a substantial drag reduction. The oscillation modes varies between stable and bistable states, and then the corresponding Strouhal numbers for the horizontal (respectively, vertical) evolve from 0.16 (respectively, 0.27) to 0.13 (respectively, 0.18). With regard to the present space-time evolution of the wake-flow features, a dynamical interpretation of the different interactions mechanisms is finally proposed for each case with a particular interest to the low-ground-clearance configurations poorly described in the literature.

Figure 1

Sketch of the experimental setup with PIV fields and base pressure sensors locations.

Figure 2

Static pressure measurements in the wake for ${\text{Re}}_h=2.86\times {10}^5$ and with $g/w=0.154.$ (a) Zero-pressure coefficient isosurface, (b) contours of static pressure in the plane y-z. Continuous black and white lines refers to positive and negative values, zero contour is denoted by dotted-dashed line and contour intervals are 0.1. Local minimum of mean pressure coefficient are located by the two white arrows.

Figure 3

Global drag ($\circ$ for $w/h\approx 1.350$ or $\square$ for $w/h\approx 1.178$) and base suction ($+$ for $w/h\approx 1.350$ or $\times$ for $w/h\approx 1.178$) coefficients against Reynolds numbers from Table 1 with the present measurements for $g/w=0.154$ (reported in Table 2). Subfigure shows the present global drag and base suction coefficients against $g/w$. Data from Bonnavion et al. [28] and Grandemange et al. [3] are also added. The black arrows highlights the minimum drag.

Figure 4

(a) Time histories and PDFs of the horizontal pressure gradient and the side-force coefficient for $g/w=0.154$ and ${\text{Re}}_h=2.86\times {10}^5$, (b) $\#N$ and (c) $\#P$ associated conditional averaging velocity fields. The white cross denotes the saddle point while the white circle denotes the vortex cores. The pink symbols denote the pressure probes location.

Figure 5

Root-mean-square velocity magnitude fields normalized by the free-stream velocity with time-average streamlines from PIV measurements for different Reynolds numbers and ground-clearance ratios. The blue line denotes the mean separatrix lines. Cyan circle symbol denotes the zero vertical velocity location (taken at $x/h=0.02$). The selected streamline, depicted in green, highlights the typical underflow behavior for each case. The right top figure gives the normalized under-body velocity profiles (extracted from PIV measurements) used to compute the normalized mean under-body flow velocity, $U_s^{*}$, reported in Table 3.

Figure 6

Sample of normalized instantaneous velocity magnitude frames in the middle plane $x\text{−}z$ showing evolution of flow structures. A POD filtering (retaining 300 first modes) is applied. To have an idea of the mean underflow stream behavior, the mean selected streamline, depicted in green in Fig. 5 is also reported in the present figures. $S_{\text{VC}}$ and $M_{\text{VC}}$ indicate the stationary and moving vortex core positions.

Figure 7

Spatiotemporal pressure/velocity correlations fields, $R_{\text{up}}$, computed between PIV velocity fields and the wall pressure sensors $s_{17}$ and $s_{19}$ in the vertical middle plane for (a) $g/w=0.063$, (b) $g/w=0.100,$ and (c) $g/w=0.154$ at ${\text{Re}}_h=2.86\times {10}^5$.

Figure 8

Profiles of $R_{\text{up}}$ and $R_{\text{vp}}$ following the dash-dot line plot in Fig. 7 for the three present $g/w$ cases and for the two sensors (a) $s_{17}$ and (b) $s_{19}$.

Figure 9

Sample of normalized instantaneous velocity magnitude frames in the middle plane x-y showing evolution of co-rotating shear-layer vortices corresponding to the $\#N$ state. The time between successive frames is, $t=14h/U_{\infty }$. A POD filtering (retaining 300 first modes) is applied.

Figure 10

Vortex center locations for the states $\#N$ and $\#P$ using the ${\mathrm{\Gamma }}_1$ criterion proposed by Graftieaux et al. [31]. The conditional time-average streamlines are also reported for clarifying the mean structures background for each state.

Figure 11

Spatiotemporal pressure/velocity correlations, $R_{\text{up}}$, computed between the PIV velocity fields (a) in the vertical middle plane and the pressure sensor, $s_{13}$, (b) in the transverse middle plane and the pressure sensor, $s_6$, respectively, for $g/w=0.063$ and 0.154 at ${\text{Re}}_h=2.86\times {10}^5$.

Figure 12

For the case, $g/w=0.154$, (a) magnitude-square coherence between the pressure sensors, (b) pressure cross-correlation, (c, d) spatiotemporal pressure/velocity correlations, (e) normalized wall pressure time-histories.

Figure 13

Autopower velocity spectra in the wake at the position 2 and 5 for $g/w=0.063$ and 0.154 at ${\text{Re}}_h=1.43\times {10}^5$ (black curve), ${\text{Re}}_h=2.86\times {10}^5$ (red curve), and ${\text{Re}}_h=4.29\times {10}^5$ (blue curve), obtained using hot-wire probe.

Figure 14

Autopower wall pressure spectra at ${\text{Re}}_h=2.86\times {10}^5$ for the three $g/w$ investigate.

Figure 15

Schematic representation of the proposed interpretation for the dynamical wake behavior in the case of lateral bistability (herein $\#P$ mode) and for the stable lateral case ($g/w<0.100$).