2.1. Experimental set-up

Experiments were carried out in two different closed-circuit wind tunnels, one at Harbin Institute of Technology, Shenzhen (HITsz), and the other at the Hong Kong Polytechnic University (HKPU), their working sections being 1.0 m × 0.8 m by 5.6 m long, and 0.6 m × 0.6 m by 2.4 m long, respectively. The flow non-uniformity was 0.1 % and the longitudinal turbulence intensity was no more than 0.4 % in the test section for both tunnels. The experimental set-ups were essentially the same, as shown schematically in figure 1(a). A flat plate with a clipper-built leading edge was installed horizontally above the floor of the test section, as a raised floor to control the boundary layer thickness. The dimensions of the plates were 2.6 m × 0.78 m × 0.015 m and 2.2 m × 0.59 m × 0.015 m, with clearances between the plate and the tunnel floor of 0.275 m and 0.088 m, in the HITsz and HKPU wind tunnels, respectively.

Figure 1.

(a) Schematic of the experimental arrangement. (b) Dimensions of a one-third-scaled Ahmed body (35°). (c) Pressure tap distribution on the rear end. The length unit is millimetre.

The vehicle model was a one-third-scaled Ahmed body (figure 1b) with φ = 35°, corresponding to a low-drag body. The full-scale model was described by Ahmed et al. (Reference Ahmed, Ramm and Faltin1984). The model had an overall length of L = 0.348 m, a width of W = 0.130 m and a height of H = 0.096 m, supported by four cylindrical struts of 16.67 mm in length and 10 mm in diameter. The front end of the model was 0.3 m downstream of the floor leading edge (figure 1a), where the boundary layer thickness was about 4 mm at ${U_\infty } = 8.33\;\textrm{m}\;{\textrm{s}^{ - 1}}$ or Re = 0.62 × 10⁵. The blockage ratios of the frontal surface of the model to the rectangular test section above the raised floor were approximately 2.2 % and 4.1 % in the HITsz and HKPU wind tunnels, respectively. Figure 1(b) shows the definition of the right-handed Cartesian coordinate system (x, y, z), with the origin O at the midpoint of the lower edge of the model vertical base. In this paper, the instantaneous velocity components in the x, y and z directions are defined as U, V and W, respectively, which can be decomposed into a time-averaged component and a fluctuating component, i.e. $U = \bar{U} + u$, $V = \bar{V} + v$ and $W = \bar{W} + w$, where overbar denotes time averaging and u, v and w are fluctuating velocities.

2.2. Measurement techniques

Single hot-wire probes fixed on a computer-controlled Dantec traverse system were used at HKPU to measure u for the estimate of the predominant frequencies in the flow (figure 1a). The sensing element was a tungsten wire of 5 μm diameter and about 1 mm in length. The wire was operated on a constant-temperature circuit (Dantec Streamline) at an overheat ratio of 1.8. The signal from the wire was offset, amplified, low-pass filtered at a cut-off frequency of 1.0 kHz, and digitized at a sampling frequency f_s of 3.0 kHz using a 16-bit analogue-to-digital converter (NI PCI-6143). The sampling duration was 3 min for each record. At least three records were collected for each measurement location. Flow is assumed to be symmetric with respect to the plane y* = 0, where superscript ‘*’ denotes normalization by $\sqrt A \textrm{ }( = 0.112\;\textrm{m})$ and/or U_∞. Therefore, the hot-wire measurements were performed only on the side of y* ≥ 0, covering the regions of x* = −1.27, y* ∈ [0, 0.43] and z* ∈ [0.91, 1.17] above the roof, x* = −1.27, y* ∈ [0.63, 0.69] and z* ∈ [−0.04, 0.90] over the side surface, x* ∈ [−0.54, 0], y* ∈ [0, 0.57] and z* ∈ [0.59, 0.91] over the rear window, and x* ∈ [0.09, 1.52], y* ∈ [0, 0.72] and z* ∈ [−0.04, 0.90] behind the vertical base. The total measurement points amount to more than 1500. The fast Fourier transform (FFT) algorithm was employed to calculate the power spectral density function E_u of u. The FFT window size N_w was 4096. The frequency resolution Δf in the spectral analysis depends on the sampling frequency and the FFT window size (Δf = f_s/N_w) and is estimated to be 0.73 Hz (Zhou et al. Reference Zhou, Du, Mi and Wang2012).

A Dantec high-speed two-dimensional PIV system (Litron LDY304-PIV, Nd:YLF) was used at the HITsz wind tunnel to measure the wake of the Ahmed body. The model surface and raised floor were painted black to minimize laser reflection noise. The flow was seeded by smoke generated from peanut oil using a TSI 9307-6 particle generator, with particles of about 1 μm in diameter. Flow illumination was provided by two pulsed laser sources of a 527 nm wavelength, each with a maximum energy output of 30 mJ pulse⁻¹. Particle images were taken using a charge-coupled device camera (PhantomV641, double frames, with a resolution of 2560 × 1600 pixels). Synchronization between image taking and flow illumination was provided by a Dantec timer box 80N77. PIV measurements were performed in four x–z planes of y* = 0, 0.14, 0.36 and 0.54, and eight y–z planes, i.e. x* = −1.27 at the roof and side surface of the body, x* = −0.36 and −0.09 above the rear window, and x* = 0.20, 0.43, 0.65, 0.86 and 1.29 behind the base. The PIV image covered an area of x* ∈ [−0.88, 2.80] and z* ∈ [−0.30, 1.99] for the x–z planes and y* ∈ [−0.60, 0.98] and z* ∈ [−0.35, 2.17] for the y–z planes. The image magnifications in both directions of each plane were identical, approximately 160 and 110 μm pixel⁻¹ in the x–z and y–z planes, respectively. The intervals between two successive pulses were 120 μs and 40 μs for measurements in the x–z and y–z planes, respectively. In processing PIV images, the adaptive PIV method (Dynamic Studio software) was used with a minimum interrogation area size of 32 × 32 pixels and a maximum size of 64 × 64 pixels. The number of particle densities per interrogation window was at least 25 or 20 in the x–z or y–z planes, respectively. The grid step size of 16 × 16 pixels produced 160 × 100 in-plane velocity vectors and the same number of vorticity data in the x–z and y–z planes.

Following Zhang et al. (Reference Zhang, Liu, Zhou, To and Tu2018), the uncertainty of PIV measurements was estimated based on image matching analysis. This method identifies particle image pairs in two successive exposures according to the measured displacement vectors and evaluates the residual distance or particle disparity vector between the particle image pairs, whose ensemble within the interrogation window is used to infer the instantaneous uncertainty of the measured displacement. For further details of this technique, please refer to Sciacchitano, Wieneke & Scarano (Reference Sciacchitano, Wieneke and Scarano2013). In the x–z planes, the estimated particle displacement uncertainties are below 0.15 pixels in both x and z directions, resulting in velocity uncertainties σ_U and σ_W, in U and W, less than 0.2 m s⁻¹ or 2.5 % of U_∞, respectively. Following Wen, Tang & Duan (Reference Wen, Tang and Duan2015), the uncertainty σ_ωy of the spanwise vorticity ω_y is given by

(2.1)\begin{equation}{\sigma _{\omega y}} = {\sigma _U}/\mathrm{\Delta }x,\end{equation}

where Δx is the grid size (16 pixels). Therefore, the σ_ωy is less than 7 % of the highest magnitude of ω_y (about 1100–1500 s⁻¹) in the x–z planes of y* ∈ [0, 0.54]. In the y–z planes of x* ∈ [−1.27, 1.29], the estimated particle displacement uncertainties are below 0.1 pixels in both y and z directions, accounting for velocity uncertainties of less than 3.3 % of U_∞ in both V and W. The uncertainty σ_ωx of the streamwise vorticity ω_x is less than 16 % of the highest magnitude of ω_x (about 900–1350 s⁻¹) in the y–z planes of x* ∈ [−1.27, 1.29].

PIV measurements were conducted twice with two different trigger rates in each plane. A lower trigger rate of 200 Hz was used for time averaging in order to obtain data with a longer time period. The other is set at 500 Hz to achieve a higher temporal resolution for instantaneous and proper orthogonal decomposition (POD) analyses. A total of 2000 PIV images were captured in the double frame mode for each test run. The percentage variations of all mean and root mean square velocities ${\bar{U}^\ast },{\bar{V}^\ast },{\bar{W}^\ast }$, $u_{rms}^\ast ,v_{rms}^\ast $ or $w_{rms}^\ast $ converge with increasing N to less than ±1 % once N > 1500, irrespective of the measurement plane or trigger rate. As such, N = 2000 is considered to be adequate for capturing the mean and fluctuating flow fields.

Laser-induced fluorescence (LIF) flow visualization was conducted at HKPU using a Dantec standard PIV system (PIV2100). Please refer to Zhang et al. (Reference Zhang, Zhou and To2015) for the details of the experimental set-up. Three rows of pinholes, each consisting of 64 circular orifices, 1 mm in diameter and equally separated by 1 mm, were made along lines parallel to and 7 mm upstream of the upper edge of the rear window, and the upper and lower edges of the vertical base, respectively. Smoke generated by a TSI 3079 particle generator using paraffin oil was pumped through one hollow cylindrical strut into a cavity in the rear part of the model and released into the flow from the pinholes. Flow images were taken in the x–z plane of y* = 0 and the x–y plane of z* = 0.72 over the rear window. The model surface, the raised floor and the tunnel working section walls hit by the laser sheets were all painted black to minimize reflection.

A PSI DTC Initium system was used to measure the time-averaged surface pressure on the model at HITsz. Twenty-eight pressure taps were made, whose locations are schematically shown in figure 1(c). The pressure taps were connected to an ESP scanner via plastic tubes of 1 mm inner diameter. The scanner was placed inside the test model to minimize the length of the tubes and hence the filtering effect of tubing in pressure measurements (Grandemange, Gohlke & Cadot Reference Grandemange, Gohlke and Cadot2013). The measurement accuracy was estimated to be approximately ± 2 Pa. The sampling frequency was 650 Hz, and a total of 40 000 data points were obtained for each measurement location. The time-averaged pressure coefficient is given by ${C_p} = (p - {p_0})/(0.5\rho U_\infty ^2)$, where p and p ₀ are the time-averaged local and free-stream static pressures, respectively, and ρ is the density of air.

The hot-wire and flow visualization measurements were carried out at ${U_\infty } \in [4,36]\;\textrm{m}\;{\textrm{s}^{ - 1}}$ and 3 m s⁻¹, corresponding to Re ∈ [0.3, 2.7] × 10⁵ and 0.22 × 10⁵, respectively. The PIV and pressure measurements were performed at ${U_\infty } = 8.3\;\textrm{m}\;{\textrm{s}^{ - 1}}$ (Re = 0.62 × 10⁵). The U_∞ value was measured using a Pitot-static tube connected to an electronic micro-manometer (Furness Control Ltd, FC510). Note that the Ahmed model has sharp edges at the rear end, and the flow structure at this relatively low Re range does not differ significantly from that at large Re of the order of 10⁶ (Vino et al. Reference Vino, Watkins, Mousley, Watmuff and Prasad2005).

3.1. Recirculation bubbles

Time-averaged streamlines in the x–z planes of y* = 0, 0.14, 0.36 and 0.54 (figure 2) show that the flow separates at the upper edge of the rear window and the lower edge of the vertical base, and rolls up behind the base, forming two recirculation bubbles, one above the other. The centre of the upper bubble is located at x* = 0.22 and z* = 0.52 near the symmetry plane (figure 2a,b), and deviates slightly from Tunay et al.'s (Reference Tunay, Sahin and Ozbolat2014) observation (x* = 0.26 and z* = 0.6), also from PIV measurements, in the symmetry plane (φ = 35°, Re = 0.17 × 10⁵). This deviation is not unexpected in view of differences in facilities, techniques, Re, etc., between the two experiments. In the x–z plane of y* = 0.36, the centre moves downstream to (x*, z*) = (0.35, 0.51) (figure 2c), suggesting that the upper bubble centre is tilted downstream from the symmetry plane to the off-symmetry plane. On the other hand, the centre of the lower bubble shifts upstream, from (x*, z*) = (0.47, 0.07) to (0.39, 0.09), from the plane of y* = 0.14 to that of y* = 0.36. In the x–z plane of y* = 0.54, both bubbles disappear (figure 2d), and an upwash flow occurs over the rear window and behind the base.

Figure 2.

PIV-measured time-averaged sectional streamlines in x–z planes of (a) y* = 0, (b) y* = 0.14, (c) y* = 0.36 and (d) y* = 0.54 at Re = 0.62 × 10⁵. The red ‘+’ denotes the focus of streamlines.

The vortex definition proposed by Zhou et al. (Reference Zhou, Adrian, Balachandar and Kendall1999) is presently adopted for the identification of vortices from the vorticity data and is briefly introduced below. A vortex core is a region where the velocity gradient tensor $\boldsymbol{\nabla }\boldsymbol{U}$ has complex eigenvalues (Chong, Perry & Cantwell Reference Chong, Perry and Cantwell1990). Then, $\boldsymbol{\nabla }\boldsymbol{U}$ may be written as

(3.1)\begin{equation}\boldsymbol{\nabla }\boldsymbol{U}\; = [\begin{array}{@{}ccc@{}} {{\boldsymbol{v}_r}}&{{\boldsymbol{v}_{cr}}}&{{\boldsymbol{v}_{ci}}} \end{array}]\left[ {\begin{array}{@{}ccc@{}} {{\lambda_r}}&{}&{}\\ {}&{{\lambda_{cr}}}&{{\lambda_{ci}}}\\ {}&{ - {\lambda_{ci}}}&{ - {\lambda_{cr}}} \end{array}} \right]{[\begin{array}{@{}ccc@{}} {{\boldsymbol{v}_r}}&{{\boldsymbol{v}_{cr}}}&{{\boldsymbol{v}_{ci}}} \end{array}]^{ - 1}},\end{equation}

where λ_r is the real eigenvalue with a corresponding eigenvector v_r, and λ_cr ± λ_cii are a conjugate pair of complex eigenvalues with complex eigenvectors v_cr ± v_cii. The local flow is either stretched or compressed along the axis v_r, while swirling in the plane spanned by vectors v_cr and v_ci. The strength of the local swirling motion is quantified by the imaginary part of the complex eigenvalue pair λ_ci, referred to as the local swirling strength of the vortex. This method is frame-independent and does not detect regions containing significant vorticity but no local swirling motion, such as shear layers. A vortex is identified if $\lambda _{ci}^{2\ast }$ is larger than a threshold, 0.4 in the x–z planes and 0.002 in the y–z planes, which are about 3 % and 1 % of the maximum $\lambda _{ci}^{2\ast }$, respectively.

As shown in the contours of instantaneous spanwise vorticity $\omega _y^\ast $ in each x–z plane from y* = 0 to 0.54 (figure 3a,c,e,g), one highly concentrated negative (or positive) vorticity layer separates from the upper edge of the rear window and another from the lower edge of the base. The former in the plane of y* = 0.54 (figure 3g) is deflected downwards towards the surface, as compared with its counterpart in the symmetry plane (figure 3a), spatially coinciding with the well-known C-pillar vortex. The wake is characterized by two regions, the upper of the negative $\omega _y^\ast $ concentrations and the lower of the positive, and the two regions appear contracting from y* = 0 to 0.54. This contraction is confirmed by the $\bar{\omega }_y^\ast $ contours (figure 3b,d,f,h) and is internally consistent with the streamlines (figure 2), where both bubbles are even invisible in the plane of y* = 0.54 (figure 2d).

Figure 3.

PIV-measured typical instantaneous (a,c,e,g) and time-averaged (b,d,f,h) spanwise vorticity contours in x–z planes of (a,b) y* = 0, (c,d) y* = 0.14, (e,f) y* = 0.36 and (g,h) y* = 0.54 at Re = 0.62 × 10⁵. Black thick lines are contours of the swirling strength $\bar{\lambda }_{ci}^{2\ast } = 0.4$ of the time-averaged flow. The magnitudes of $\bar{\omega }_{y,max}^\ast $ and $\bar{\omega }_{y,min}^\ast $ are marked in the figure.

Interestingly, a small corner vortex with positive vorticity occurs at the lower end of the rear window, which runs across almost the entire width of the window with $\bar{\omega }_{y,max}^\ast\approx 4.0$. In this paper, subscripts ‘max’ and ‘min’ denote extreme values in positively and negatively signed vorticity concentrations, respectively. Wang et al. (Reference Wang, Zhou, Pin and Chan2013) captured a similar structure based on the PIV measurements in the x–z plane of y* = 0 behind an Ahmed body (φ = 35°, Re = 0.62 × 10⁵), and suggested that this structure was induced by the negative circulation above the rear window. No further discussion was pursued. In the present work, a different view on the formation of the corner vortex, as well as its effect on the surface pressure, is provided based on the PIV, pressure and flow visualization data.

The time-averaged pressure coefficient C_p on the rear window (figure 4a) drops gradually downstream from C_{p 1} = −0.32 at z* = 0.82 to C_{p 8} = −0.38 at z* = 0.50 in the symmetry plane of y* = 0. On the other hand, C_p on the vertical base increases almost linearly from C_{p 9} = −0.27 at z* = 0.39 to C_{p 14} = −0.18 at z* = 0.13. Similar observations have been made from C_{p 15} to C_{p 28} in the off-symmetry plane of y* = 0.36. The C_p distribution is in reasonable agreement with previous measurements on the rear end of an Ahmed body with φ = 35° by Lienhart & Becker (Reference Lienhart and Becker2003) and Meile et al. (Reference Meile, Ladinek, Brenn, Reppenhagen and Fuchs2016). There is an appreciable deviation in C_p, which is ascribed to the differences in experimental conditions such as Re and facilities among these studies. The pressure on the vertical base is appreciably higher than that on the rear window. The flow within the upper recirculation bubble near the vertical base moves upwards along the base under the pressure gradient, which is evident in figure 3. Such upward-moving fluid separates at the upper edge of the base and then rolls up before reattachment on the slanted surface under the pressure difference between the two surfaces, forming the corner vortex, which is very well illustrated by one typical photograph captured from flow visualization in the symmetry plane (figure 4b).

Figure 4.

(a) Time-averaged pressure coefficients C_p on the rear end in the symmetry plane of y* = 0 (red ‘×’) and an off-symmetry plane of y* = 0.36 (black ‘○’) at Re = 0.62 × 10⁵. The blue ‘$\Delta$’ and green ‘$\nabla$’ symbols denote C_p experimentally obtained in the plane of y* = 0 by Lienhart & Becker (Reference Lienhart and Becker2003) (Re = 8.93 × 10⁵) and Meile et al. (Reference Meile, Ladinek, Brenn, Reppenhagen and Fuchs2016) (Re = 6.70 × 10⁵), respectively, for an Ahmed body with φ = 35°. (b) A typical photograph of the flow structure in the symmetry plane at Re = 0.22 × 10⁵. Flow is from left to right.

Grandemange et al. (Reference Grandemange, Gohlke and Cadot2013) observed a bistability in the wake of a square-back Ahmed body, which is characterized by random spanwise switchings of the recirculation region between two preferred reflectional symmetry-breaking positions. The same observation was also made and investigated by Östh et al. (Reference Östh, Noack, Krajnović, Barros and Borée2014) and Barros et al. (Reference Barros, Borée, Cadot, Spohn and Noack2017). This phenomenon has, however, never been observed in the wake of the low-drag regime.

The measured time-averaged drag coefficient ${C_D}\textrm{ }({\equiv} {F_D}/(0.5\rho U_\infty ^2A)$, where F_D is aerodynamic drag) is 0.32, slightly higher than previous reports by Ahmed et al. (Reference Ahmed, Ramm and Faltin1984), Guilmineau (Reference Guilmineau2008), Meile et al. (Reference Meile, Ladinek, Brenn, Reppenhagen and Fuchs2016) and Rao et al. (Reference Rao, Minelli, Basara and Krajnović2018), which is probably due to the present lower Re. It is evident in table 2 that C_D decreases from 0.32 to 0.26 from Re = 1.67 × 10⁵ to 1.4 × 10⁶.

Table 2.

Drag coefficient C_D on an Ahmed body with φ = 35°.

3.2. Longitudinal structures

Figure 5 presents the PIV-measured instantaneous and time-averaged streamwise vorticities in the y–z planes. At x* = −0.36, a streamwise vortex with positive $\omega _x^\ast $ is captured near the upper corner of the rear window (figure 5a), accounting for a swirling motion as enclosed by the swirling strength $\bar{\lambda }_{ci}^{2\ast } = 0.002$ of the time-averaged flow with $\bar{\omega }_{x,max}^\ast{=} 1.8$ (figure 5b). This swirling region grows larger and the corresponding $\bar{\omega }_{x,max}^\ast $ increases to 2.1 downstream in the plane of x* = −0.09 (figure 5d). This longitudinal structure is apparently the C-pillar vortex, which is formed due to the rollup of the shear layer coming off the sidewall about the side edge of the rear window because of the pressure difference between the flows over the sidewall and the slanted surface (Ahmed et al. Reference Ahmed, Ramm and Faltin1984). The strength of the C-pillar vortex is substantially lower than that in the high-drag regime; its $\bar{\omega }_{x,max}^\ast $ in the y–z plane of x* = 0.20 was only about 10 % of the latter (Wang et al. Reference Wang, Zhou, Pin and Chan2013). It is noteworthy that the $\bar{\omega }_{x,max}^\ast $ location of the C-pillar vortex moves inwards and downwards from (y*, z*) = (0.59, 0.82) at x* = −0.36 to (0.53, 0.70) at x* = −0.09. Its angles are about 11° and 12° with respect to the slanted surface and the sidewall, respectively, larger than those (8° and 7°) at φ = 25° (Lehugeur et al. Reference Lehugeur, Gilliéron and Ta-Phuoc2005).

Figure 5.

PIV-measured instantaneous (a,c) and time-averaged (b,d) streamwise vorticity contours in the y–z planes of x* = −0.36 (a,b) and x* = −0.09 (c,d) at Re = 0.62 × 10⁵. The thick black curves are the contours of the swirling strength $\bar{\lambda }_{ci}^{2\ast } = 0.002$ of the time-averaged flow. The magnitudes of $\bar{\omega }_{x,max}^\ast $ and $\bar{\omega }_{x,min}^\ast $ are marked in the figure.

As shown in the $\bar{\omega }_x^\ast $ contours in the y–z plane of x* = −0.09 (figure 5d), a relatively strong $\bar{\omega }_x^\ast $ concentration is present in the vicinity of the C-pillar vortex, whose sign is opposite to that of the C-pillar vortex, with $\bar{\omega }_{x,min}^\ast $ reaching −0.8, though without swirling. This vorticity concentration is induced by the C-pillar vortex (Zhang et al. Reference Zhang, Zhou and To2015). A number of alternately signed $\omega _x^\ast $ concentrations are discernible at the same height as the roof in the y–z planes of x* = −0.36 and −0.09 (figure 5a,c), which are probably associated with the hairpin vortices wrapped with the longitudinal vortices above the roof. There is another streamwise swirling structure enclosed by the contour of $\bar{\lambda }_{ci}^{2\ast } = 0.002$ near the lower corner of the sidewall in both planes. This structure, referred to as the lower vortex, is formed along the lower edge of the sidewall (Wang et al. Reference Wang, Zhou, Pin and Chan2013).

It is insightful to examine the $\bar{\omega }_x^\ast $ contours, superimposed with sectional streamlines, in different y–z planes behind the vertical base (figure 6). At x* = 0.20 (figure 6a), the upper rightmost concentration marked by ‘C’ may be identified with the C-pillar vortex. Apparently, the C-pillar vortex starts to decay from $\bar{\omega }_{x,max}^\ast{=} 2.0$ to 1.3 from x* = 0.20 to 0.65 (figure 6a–c). A longitudinal swirling structure (marked by the contour of $\bar{\lambda }_{ci}^{2\ast } = 0.002$) with $\bar{\omega }_{x,min}^\ast{=}{-} 1.7$ occurs in x* = 0.20 on the inner side, close to the symmetry plane, of the C-pillar vortex, probably resulting from the interaction between the C-pillar vortex and the upper recirculation bubble. This induced structure starts to decay downstream with the C-pillar vortex. Behind the base, there is a large swirling structure enclosed by the contour of $\bar{\lambda }_{ci}^{2\ast } = 0.002$ (figure 6a), which coincides spatially with the upper bubble. Its vorticity concentration increases from $\bar{\omega }_{x,max}^\ast{=} 2.2$ at (y*, z*) = (0.45, 0.58) in the y–z plane of x* = 0.20 to $\bar{\omega }_{x,max}^\ast{=} 3.1$ at (y*, z*) = (0.38, 0.52) in the plane of x* = 0.43 (figure 6a,b). In both planes, the $\bar{\omega }_{x,max}^\ast $ value occurs at almost the same height (z* ≈ 0.52) as the upper bubble centre (figure 2). When moving downstream to x* = 0.65, $\bar{\omega }_{x,max}^\ast $ drops to 2.6 (figure 6c). Based on the streamlines in the x–z planes of y* ∈ [0, 0.36] (figure 2a–c), this longitudinal structure is linked to the upper bubble, which runs laterally outwards downstream. This longitudinally tilted large structure induces an upwash flow on its outer side (figure 6a–c), responsible for the upwash captured in the x–z plane of y* = 0.54 (figure 2d), and a downwash flow on the inner side close to the symmetry plane (figure 6b,c). Both the C-pillar vortex and the large longitudinal structure move downwards and towards the symmetry plane from x* = 0.43 to 0.65 (figure 6b,c) under a pressure difference between the outer flow and the wake. From x* = 0.86, the C-pillar vortex appears to merge into the longitudinal large structure (figure 6d,e), that is, the far wake is dominated by one single pair of longitudinal trailing vortices. The lower vortex (figure 5) decays gradually in the wake from $\bar{\omega }_{x,max}^\ast{=} 1.5$ at x* = 0.20 to $\bar{\omega }_{x,max}^\ast{=} 0.8$ at x* = 1.29 (figure 6).

Figure 6.

Downstream evolution of PIV-measured time-averaged streamwise vorticity contours and sectional streamlines in y–z planes (Re = 0.62 × 10⁵). The red ‘+’ and the thick black curves are as in figures 2 and 5, respectively. The blue broken lines and red dashed-dotted lines are indicative for the upper and lower borders of the swirling regions corresponding to the C-pillar vortex and the leg of the upper recirculation bubble, respectively. The magnitudes of $\bar{\omega }_{x,max}^\ast $ and $\bar{\omega }_{x,min}^\ast $ are marked in the figure.

3.3. Unsteady wake vortices and scaling

Unsteady flow structures behind the base are investigated via E_u measured at different heights in the symmetry plane for a number of downstream stations (figure 7). There is no prominent peak in E_u measured at x* = 0.2 over the range of z* ∈ [0, 0.72]. A rather pronounced peak occurs at f* = St_w = 0.30 at B₁ (z* = 0) and B₃ (z* = 0.25) at x* = 0.43, C₁ (z* = 0), C₃ (z* = 0.25) and C₄ (z* = 0.32) at x* = 0.65, and D₁ (z* = 0) and D₃ (z* = 0.25) at x* = 0.86, but not at B₂, C₂ and D₂ (z* = 0.13), where a minor peak is discernible only at the second harmonic of St_w = 0.30, i.e. f* = 0.60. The peak at St_w = 0.30 becomes more pronounced with increasing x* for lower positions B₁ through D₁ and more visible at higher z* (B₃, z* = 0.25; C₄, z* = 0.32) from x* = 0.43 to 0.65. The observations suggest that the unsteady structures associated with St_w = 0.30 are related to the two recirculation bubbles behind the base. The peak at St_w = 0.30 is no longer visible for x* ≥ 1.29, suggesting the disappearance of these unsteady structures.

Figure 7.

Power spectral density function E_u of the hot-wire signal measured along A₁–A₆, B₁–B₆, C₁–C₆, D₁–D₆ and E₁–E₆, respectively, at y* = 0 in the wake, at Re = 0.62 × 10⁵.

Zhang et al. (Reference Zhang, Zhou and To2015) captured a predominant frequency of St_w = 0.44, larger than the present St_w, behind the vertical base of an Ahmed model with φ = 25° (Re = 0.62 × 10⁵). Other investigators (e.g. Minguez et al. Reference Minguez, Pasquetti and Serre2008; Joseph, Amandolese & Aider Reference Joseph, Amandolese and Aider2012; Kohri et al. Reference Kohri, Yamanashi, Nasu, Hashizume and Katoh2014) also reported this predominant frequency, varying rather wildly, in the wake of an Ahmed body with various φ (figure 8a). One may wonder whether there is any scaling law to govern this predominant frequency. In the high-drag regime, the flow separates from both the upper and lower edges of the base, forming two recirculation bubbles behind the base. The St_w is attributed to the alternate emanation of unsteady structures from the two bubbles (Zhang et al. Reference Zhang, Zhou and To2015). We may surmise that the height h of the vertical base is a characteristic length in this case. On the other hand, H might be the characteristic length in the low-drag regime where the two recirculation bubbles result from flow separation from the upper edge of the rear window and the lower edge of the base. Note that the Ahmed body with the critical angle φ = 30° can be in either high- or low-drag regimes (Sims-Williams Reference Sims-Williams2001; Conan, Anthoine & Planquart Reference Conan, Anthoine and Planquart2011; Kohri et al. Reference Kohri, Yamanashi, Nasu, Hashizume and Katoh2014). Furthermore, it was observed even in the high-drag regime that the separated flow over the rear window did not fully reattach on the slanted surface and, instead, merged with the upper bubble behind the base (Vino et al. Reference Vino, Watkins, Mousley, Watmuff and Prasad2005; Conan et al. Reference Conan, Anthoine and Planquart2011). That is, H should be considered as the characteristic length for φ = 30° in both regimes. As such, we may define

(3.2a,b)\begin{equation}St_w^{\#} = fS/{U_\infty },\quad S = \left\{ {\begin{array}{@{}ll} {h,}&{\varphi \; \lt \; 30^\circ ,}\\ {H,}&{\varphi \; \ge \; 30^\circ ,} \end{array}} \right.\end{equation}

where S is the vertical distance between the separation points of the two bubbles. The square-back Ahmed body may be described by φ = 0° or 90°, corresponding to S = h or H, respectively, where h equals H.

Figure 8.

Dependences of (a) St_w, (b) $St_w^{\#} $ and (c) $St_w^ + $ on φ. Symbols: ○, present measurement (Re = 0.62 × 10⁵); ×, Kohri et al. (Reference Kohri, Yamanashi, Nasu, Hashizume and Katoh2014, Re = 0.73 × 10⁵); +, Zhang et al. (Reference Zhang, Zhou and To2015, Re = 0.89 × 10⁵); Δ, Sims-Williams (Reference Sims-Williams2001, Re ∈ [3.10, 6.40] × 10⁵); ∇, Vino et al. (Reference Vino, Watkins, Mousley, Watmuff and Prasad2005, Re = 4.40 × 10⁵); ${\triangleleft} $, Grandemange et al. (Reference Grandemange, Gohlke and Cadot2013, Re = 1.07 × 10⁵); ${\triangleright} $, Barros et al. (Reference Barros, Borée, Noack and Spohn2016a, Re = (2.33–6.98) × 10⁵); □, Joseph et al. (Reference Joseph, Amandolese and Aider2012, Re = 4.49 × 10⁵); ◊, Minguez et al. (Reference Minguez, Pasquetti and Serre2008, Re = 8.93 × 10⁵); ⋆, Volpe, Devinant & Kourta (Reference Volpe, Devinant and Kourta2015, Re = 5.42 × 10⁵); , Lahaye, Leroy & Kourta (Reference Lahaye, Leroy and Kourta2014, Re = 6.41 × 10⁵). The uncertainty bars for the present data are determined from the standard deviation of six repeated measurements.

The dependence of $St_w^{\#} $ on φ is presented in figure 8(b). The thus defined $St_w^{\#} $ displays a good collapse, increasing from 0.23 to about 0.30 as φ varies from 12.5° to 30°, and dropping once φ > 30°. It seems plausible that St_w indeed scales with S. There is an appreciable variation in $St_w^{\#} $, especially for φ < 30°. Zhang et al. (Reference Zhang, Zhou and To2015) discovered at φ = 25° that the unsteady structures of St_w resulted from the nearly periodic growth-and-burst cycle of the two bubbles behind the base. Naturally, St_w may depend on the bubble size, and a big bubble is associated with a long growth period before bursting and thus a low St_w. On another note, the bubble size is affected by the effective separation angle θ, with respect to the streamwise direction, of the separated shear layer. A large θ may act to cut short the bubble size, thus increasing $St_w^{\#} $. For the high-drag regime, θ is obviously the same as φ; for the low-drag regime, flow separates from the upper edge of the rear window and θ should be given by the angle between the streamwise direction and the local tangential velocity at x* = 0 of the streamline through the separation point (figure 8c). The choice of the point at x* = 0 on the streamline ensures that the effective separation angles are taken at the same streamwise location. For the square-back Ahmed body, θ is 0°. Then, we may define a new non-dimensional frequency by

(3.3a,b)\begin{equation}St_w^ + = \alpha St_w^{\#} = f\alpha S/{U_\infty },\quad \alpha = \frac{{90\mathrm{^\circ } - \theta }}{{90\mathrm{^\circ }}},\end{equation}

where α is a deflection coefficient with a value of α ∈ [0, 1], depending on θ. This $St_w^ + $ collapses well to a constant of about 0.20, regardless of φ (figure 8c). For the present data, θ is estimated to be 13° from the PIV-measured time-averaged streamlines in figure 2(a). We use θ = φ = 30° when rescaling Sims-Williams’ (Reference Sims-Williams2001) and Vino et al.'s (Reference Vino, Watkins, Mousley, Watmuff and Prasad2005) data in the high-drag regime. Kohri et al.'s (Reference Kohri, Yamanashi, Nasu, Hashizume and Katoh2014) data at φ = 30°, but corresponding to the low-drag regime, are not included in figure 8(c), as the velocity field is not reported and hence θ could not be estimated. The slight scatter in the $St_w^ + $ data is probably due to a difference in facilities, Re and measurement uncertainties. The product of αS in (3.3a,b) may be interpreted as a corrected characteristic length of the separation bubbles, which depends on both S and θ.

It is of interest to understand the physics behind the unsteady structures of St_w = 0.30. Figure 9 shows PIV-measured instantaneous flow structures in one typical cycle with a period $T_w^\ast \textrm{ }({\approx} 3.3)$ of the unsteady structure generation in the symmetry plane of the wake. Following Zhang et al. (Reference Zhang, Zhou and To2015), the length l* of each bubble is defined by the maximum longitudinal length of the region where U ≤ 0. At the beginning (t* = 0) of the cycle, one organized structure characterized by the concentration of quite strong positive $\omega _y^\ast $ (figure 9a), as highlighted by a red broken curve, appears pinched off from the lower recirculation bubble at x* ≈ 0.4, shooting slightly upwards. At t* = $0.25T_w^\ast $, the upper recirculation bubble undergoes a decrease in size from l* = 0.76 at t* = 0 to l* = 0.60, accompanied by the emanation of a structure of negative $\omega _y^\ast $ from it (figure 9b). Then, both bubbles gradually grow in size, with l* increasing to 0.81 at t* = $0.75T_w^\ast $ (figure 9c,d). During the whole cycle, the upper and lower bubbles undergo a nearly periodic expansion and contraction, l* varying over [0.60, 0.81] and [0.33, 0.81], respectively. Kohri et al. (Reference Kohri, Yamanashi, Nasu, Hashizume and Katoh2014) detected a predominant frequency St ≈ 0.27 behind the Ahmed body of φ = 35° at Re = 0.73 × 10⁵, which was ascribed to the fluctuation of an upwash flow from the lower edge of the base. They advocated that the fluctuation was induced by periodic vortex shedding from the bottom end. However, the present hot-wire and PIV data suggest a different scenario. The unsteady structures of St_w = 0.30 may result from the nearly periodic expansion and contraction of both bubbles when the organized structures are either pinched off from the lower bubble or emanated from the upper bubble, with their onset at x* ≈ 0.4. More elaboration on these unsteady structures and nearly periodic variation of the bubbles will be given below.

Figure 9.

PIV-measured instantaneous sectional streamlines and isocontours of spanwise vorticity $\omega _y^\ast $ in one typical cycle in the symmetry plane of y* = 0 at Re = 0.62 × 10⁵. The red solid curve indicates the bubble size, determined from the sectional streamline that encloses the region of recirculation bubbles. The red broken line highlights the structure emanated from each bubble.

The spectral coherence $Co{h_{{u_1}{u_2}}}\textrm{ }( = (Co_{{u_1}{u_2}}^2 + Q_{{u_1}{u_2}}^2)/{E_{{u_1}}}{E_{{u_2}}})$ between two fluctuations ${u_1}$ and ${u_2}$ provides a measure for the correlation between the spectral components of ${u_1}$ and ${u_2}$ (Antonia, Zhou & Matsumara Reference Antonia, Zhou and Matsumura1993), where $C{o_{{u_1}{u_2}}}$ and ${Q_{{u_1}{u_2}}}$ are the co-spectrum and quadrature spectrum of ${u_1}$ and ${u_2}$, respectively. The spectral coherence $Co{h_{u{B_1}u{B_3}}}$ (figure 10a) between two simultaneously measured hot-wire signals at B₁ and B₃ (x* = 0.43) is characterized by one predominant peak around St_w = 0.30, reaching nearly 0.35, and the corresponding spectral phase shift ${\varPhi _{u{B_1}u{B_3}}}\textrm{ }( = {\tan ^{ - 1}}({Q_{u{B_1}u{B_3}}}/C{o_{u{B_1}u{B_3}}}))$ is about 0.5${\rm \pi}$ (figure 10b). B₁ and B₃ (and also C₁ and C₃ or D₁ and D₃ that follow) are as in figure 7. Similar observations are also made at x* = 0.65 (C₁ and C₃) and x* = 0.86 (D₁ and D₃), corresponding to a phase shift of 0.63${\rm \pi}$ and 0.70${\rm \pi}$ (not shown), respectively. Note that there is a difference in the convecting velocity between the flows at z* = 0 and 0.25, that is, the velocity of the flow from the gap between the body and the ground is higher than that in the wake around z* = 0.25 (not shown). This difference varies downstream in the near wake, accounting for the increase in $\varPhi$ from x* = 0.43 to 0.86. Clearly, the unsteady structures detected at B₁ and B₃, C₁ and C₃ or D₁ and D₃ are highly correlated. This phase shift is distinct from that of near ${\rm \pi}$ associated with the unsteady structures emanated from the two recirculation bubbles behind the base in the high-drag regime (Zhang et al. Reference Zhang, Zhou and To2015).

Figure 10.

(a) Spectral coherence $Co{h_{u{B_1}u{B_3}}}$ and (b) phase ${\varPhi _{u{B_1}u{B_3}}}$ between simultaneously measured hot-wire signals $u{B_1}$ and $u{B_3}$, at Re = 0.62 × 10⁵. Please refer to figure 7 for the locations of B ₁ and B ₃.

To gain an in-depth understanding of the unsteady structures of St_w = 0.30 and why ${\varPhi _{u{B_1}u{B_3}}} \approx 0.5{\rm \pi} $, we performed a POD analysis of the time-resolved PIV data obtained over x* ∈ [−0.61, 1.93] and z* ∈ [−0.07, 1.34] in the symmetry plane. The POD is introduced in detail by Sirovich (Reference Sirovich1987), for example. Briefly, the spatiotemporal fluctuating velocity field u(x, t) is decomposed into an optimal set of spatial base functions ${\boldsymbol{\varPsi }_{\boldsymbol{n}}}(\boldsymbol{x})$ and temporal mode coefficients a_n(t), viz.

(3.4)\begin{equation}\boldsymbol{u}(\boldsymbol{x},\; t) = \boldsymbol{U}(\boldsymbol{x},\; t) - \bar{\boldsymbol{U}}(\boldsymbol{x}) = \sum\limits_n {{a_n}(t){\boldsymbol{\varPsi }_{\boldsymbol{n}}}(\boldsymbol{x})} ,\end{equation}

where U is the instantaneous velocity vector at the position x and n is the POD mode number. The fluctuating velocity fields from M snapshots can be arranged in a matrix ${\boldsymbol{u}_M}$ as

(3.5)\begin{equation}{\boldsymbol{u}_M} = [\,\boldsymbol{u}({t_1})\quad \boldsymbol{u}({t_2})\quad \ldots \quad \boldsymbol{u}({t_M})\,].\end{equation}

Then, the problem can be solved by

(3.6)\begin{equation}\boldsymbol{u}_M^\textrm{T}{\boldsymbol{u}_M}{\boldsymbol{\varPhi }_n} = {\lambda _n}{\boldsymbol{\varPhi }_n},\end{equation}

where the superscript ‘T’ denotes the transpose operator, ${\boldsymbol{\varPhi }_n}$ is the eigenvector and λ_n is the eigenvalue or mode energy of the nth mode, representing the contribution from the corresponding POD mode to the total turbulent kinetic energy. Solutions are sorted in order of decreasing λ_n so that the most energetic modes are the first few modes. Since the mean velocity field is subtracted, the Mth eigenvalue will always be zero (neglecting round-off errors). Then the normalized and spatially orthogonal base function ${\boldsymbol{\varPsi }_n}(\boldsymbol{x})$ can be given by

(3.7)\begin{equation}{\boldsymbol{\varPsi }_n}(\boldsymbol{x}) = \frac{{{\boldsymbol{u}_M}{\boldsymbol{\varPhi }_n}}}{{||{\boldsymbol{u}_M}{\boldsymbol{\varPhi }_n}||}},\quad n = 1,2, \ldots ,M - 1,\end{equation}

where ||…|| denotes the discrete 2-norm. The mode coefficients are determined by projecting u(x, t) onto the POD modes:

(3.8)\begin{equation}{a_n}(t) = \boldsymbol{\varPsi }_n^\textrm{T}(\boldsymbol{x})\boldsymbol{u}(\boldsymbol{x},\; t).\end{equation}

A total of M = 1000 snapshots were analysed, in which the mean velocity field, i.e. mode 0, was subtracted in data processing. Figure 11(a) shows the sensitivity of the POD relative energy contributions ${\lambda _n}/\sum {{\lambda _n}}$ to the number of snapshots. The ${\lambda _n}/\sum {{\lambda _n}}$ of the first four modes (mode 0 is not included) change little when M exceeds 700. The result implies that M = 1000 is adequate if these lower POD modes are of major interest. The ${\lambda _n}/\sum {{\lambda _n}}$ for the first 20 POD modes (figure 11b) indicates clearly that the first four modes are the most energetic, accounting for 7.6 %, 7.0 %, 4.9 % and 3.6 % of the total turbulent kinetic energy, respectively. Therefore, our attention will be focused on those four modes in the following discussion.

Figure 11.

(a) POD eigenvalue convergence as a function of PIV snapshot number M. (b) Relative contribution of individual POD modes for mode n ∈ [1, 20], where M = 1000. The POD analysis is conducted over x* ∈ [−0.61, 1.93] and z* ∈ [−0.07, 1.34] in the symmetry plane. Here Re = 0.62 × 10⁵.

In order to extract the periodic structures from random fluctuations, a triple decomposition is applied to the PIV data, following Hussain and Reynolds (Reference Hussain and Reynolds1970), viz.

(3.9)\begin{equation}\boldsymbol{U}(\boldsymbol{x},t) = \bar{\boldsymbol{U}}(\boldsymbol{x}) + \boldsymbol{u}(\boldsymbol{x},t) = \bar{\boldsymbol{U}}(\boldsymbol{x}) + \tilde{\boldsymbol{u}}(\boldsymbol{x},\; \phi (t)) + \boldsymbol{u^{\prime\prime}}(\boldsymbol{x},t),\end{equation}

where u is further decomposed into a nearly periodic part $\tilde{\boldsymbol{u}}$ and a residual $\boldsymbol{u^{\prime\prime}}$, and ϕ is the phase angle of $\tilde{\boldsymbol{u}}$. The nearly periodic or coherent motion may be given in terms of $\bar{\boldsymbol{U}}(\boldsymbol{x}) + \tilde{\boldsymbol{u}}(\boldsymbol{x},\; \phi (t))$, obtained by a POD reconstruction. The power spectral density functions of the POD coefficients display a pronounced peak at f* ≈ 0.30 for modes 1 and 2 (figure 12a,b) but not for modes 3 and 4 (figure 12c,d) or higher modes (not shown). The observation suggests that only modes 1 and 2 are associated with the nearly periodic structures of St_w = 0.30.

Figure 12.

Power spectra of the first four POD coefficients: (a) mode 1; (b) 2; (c) 3; and (d) 4. The POD analysis is conducted over x* ∈ [−0.61, 1.93] and z* ∈ [−0.07, 1.34] in the symmetry plane and M = 1000. Here Re = 0.62 × 10⁵.

A phase-resolved flow field may be reconstructed approximately with a low-order flow model including the mean flow (mode 0) and modes 1 and 2 (Ben Chiekh et al. Reference Ben Chiekh, Michard, Grosjean and Bera2004), viz.

(3.10)\begin{equation}{\boldsymbol{U}_{LOM}}(\boldsymbol{x},\phi ) = \bar{\boldsymbol{U}}(\boldsymbol{x}) + {a_1}(\phi ){\boldsymbol{\varPsi }_1}(\boldsymbol{x}) + {a_2}(\phi ){\boldsymbol{\varPsi }_2}(\boldsymbol{x}),\end{equation}

where ${a_1} = \sqrt {2{\lambda _1}} \sin (\phi )$ and ${a_2} = \sqrt {2{\lambda _2}} \cos (\phi )$ (van Oudheusden et al. Reference van Oudheusden, Scarano, Van Hinsberg and Watt2005), viz.

(3.11a,b)\begin{equation}\frac{{{a_1}(i)}}{{\sqrt {2{\lambda _1}} }} = {r_i}\sin ({\phi _i}),\quad \frac{{{a_2}(i)}}{{\sqrt {2{\lambda _2}} }} = {r_i}\cos ({\phi _i}).\end{equation}

The r_i in (3.11) is the ‘radius’ of the normalized ellipse, viz.

(3.12)\begin{equation}r_i^2 = \frac{{{a_1}{{(i)}^2}}}{{2{\lambda _1}}} + \frac{{{a_2}{{(i)}^2}}}{{2{\lambda _2}}}.\end{equation}

Figure 13 presents the reconstructed phase-resolved sectional streamlines viewed in a reference frame fixed on the model, superimposed with the isocontours of spanwise vorticity. The upper and lower recirculation bubbles are highlighted by red-coloured thick broken lines. The variation of the flow structure with the phase provides an insight into the formation of the nearly periodic structures of St_w = 0.30. At ϕ = 0°, one structure, marked by the contour of $\lambda _{ci}^{2\ast } = 0.4$ with $\omega _{y,max}^\ast{=} 10.0$, near the hot-wire measurement point B₁ (x* = 0.43, z* = 0), appears to pinch off from the lower recirculation bubble (figure 13a). The observation conforms to the emanated structure from the lower bubble, as captured in figure 9(a). As a result, the length of the lower bubble retreats markedly from its maximum l* = 0.82 at ϕ = 315° (figure 13h) to l* = 0.30 at ϕ = 0° (figure 13a). Then, the lower recirculation bubble grows gradually in size from ϕ = 0° to 315°, because of the continuous entrainment of fluid from the lower edge, until part of the bubble is pinched off again in the next cycle. Obviously, the lower bubble undergoes a nearly periodic variation in size over l* ∈ [0.30, 0.82]. This is internally consistent with the observation from the instantaneous flow structure images in figure 9. Meanwhile, the separated structure is advected downstream and its maximum vorticity decays from $\omega _{y,max}^\ast{=} 10.0$ at x* = 0.66 (ϕ = 0°) to $\omega _{y,max}^\ast{=} 1.9$ at x* = 1.16 (ϕ = 315°) (figure 13a–h). This structure is invisible beyond x* = 1.5 during the entire cycle, implying that this structure vanishes rather rapidly.

Figure 13.

Phase-resolved reconstructed sectional streamlines (white) and spanwise vorticity within one typical cycle in the symmetry plane y* = 0. The white symbol ‘×’ denotes the saddle point. The red line indicates the bubble size, determined from the sectional streamline that encloses the region of recirculation bubbles. Black thick lines are contours of $\lambda _{ci}^{2\ast } = 0.4$. The magnitudes of $\omega _{y,max}^\ast $ and $\omega _{y,min}^\ast $ are marked in the figure. Here Re = 0.62 × 10⁵.

The upper recirculation bubble also varies in size. At ϕ = 45°, this bubble grows to its maximum with l* = 0.90 (figure 13b). With continuously entrained fluid from the upper edge of the slanted surface, the bubble cannot maintain itself any more and bursts at ϕ = 90°, causing its size to collapse from l* = 0.90 at ϕ = 45° to l* = 0.69 at ϕ = 90° (figure 13b,c). Then, the bubble grows again, reaching l* = 0.81 at ϕ = 315° (figure 13c–h). This growth-and-burst process reflects the upper bubble variation captured in figure 9. The burst is accompanied by the separation of one swirling structure, marked by the contour of $\lambda _{ci}^{2\ast } = 0.4$ with $\omega _{y,min}^\ast{=}{-} 2.3$, from the upper recirculation bubble. This structure is then advected downstream (figure 13c–f). The result is consistent with the structure emanation captured in figure 9(b). This structure decays quickly downstream, from $\omega _{y,min}^\ast{=}{-} 2.3$ at ϕ = 90° to $\omega _{y,min}^\ast{=}{-} 1.6$ at ϕ = 225°, and vanishes before x* = 1.0 at ϕ = 270° (figure 13c–g). Note that the emanation of structures from the lower and upper bubbles occurs at ϕ = 0° and 90°, respectively, which agrees approximately with ${\varPhi _{u{B_1}u{B_3}}} \approx 0.5{\rm \pi} $ at f* = 0.30 (figure 10).

The physical process is schematically described, based on the above observations, in figure 14 for the nearly periodic variation of two recirculation bubbles and the ensuing separation of the coherent structures, responsible for St_w = 0.30. The flow separated from the lower edge of the base is entrained into the lower recirculation bubble continuously, leading to an increase in pressure within the bubble. After growing to its maximum with a certain pressure level, this bubble cannot sustain itself any more and bursts at ϕ = 0°, meanwhile ejecting one structure. The bubble contracts greatly at ϕ = 0°. This sudden contraction may correspond to a significant pressure drop in the bubble. On the other hand, the upper recirculation bubble is much bigger in size but also weaker in vorticity strength than the lower one (figure 13). This difference is markedly bigger than that in the high-drag regime (Zhang et al. Reference Zhang, Zhou and To2015). The sudden contraction and pressure drop of the lower bubble may naturally disturb the upper bubble and induce its bursting at ϕ = 90°, accompanied by the emanation of a coherent structure. This may explain the phase shift of about 0.5${\rm \pi}$ at f* = 0.30 (figure 10b). This phase shift may imply a relatively strong interaction between the structures emanated from the two bubbles and hence a short lifespan, vanishing by x* = 1.29 (figure 7). In contrast, their counterparts in the high-drag regime are in antiphase and persist till x* = 1.40 (Zhang et al. Reference Zhang, Zhou and To2015). After bursting, the bubble starts its growth in size and pressure of the next cycle, and the separated structures are advected downstream, though decaying rapidly.

Figure 14.

Schematic of the physical process for the nearly periodic motion of two recirculation bubbles and separated flow structures that result.

The second moments of the fluctuating velocities in the wake may have an important impact on the surface pressure on the rear part of a bluff body (e.g. Balachandar, Mittal & Najjar Reference Balachandar, Mittal and Najjar1997; Barros et al. Reference Barros, Borée, Noack, Spohn and Ruiz2016b). Following Grandemange et al. (Reference Grandemange, Gohlke and Cadot2013), we may write the streamwise mean momentum balance of the recirculation region, where the viscous force is neglected, behind a low-drag Ahmed body as

(3.13)\begin{align}\langle {C_p}\rangle {H^\ast }{W^\ast } & = \int_{\partial B} {{C_{pw}}(\boldsymbol{n}\boldsymbol{\cdot }{\boldsymbol{i}_x})\,\textrm{d}{B^\ast }} + 2\int_{\partial B} {\overline {{u^\ast }{u^\ast }} (\boldsymbol{n}\boldsymbol{\cdot }{\boldsymbol{i}_x})\,\textrm{d}{B^\ast }} + 2\int_{\partial B} {\overline {{u^\ast }{v^\ast }} (\boldsymbol{n}\boldsymbol{\cdot }{\boldsymbol{i}_y})\,\textrm{d}{B^\ast }} \nonumber\\ & \quad + 2\int_{\partial B} {\overline {{u^\ast }{w^\ast }} (\boldsymbol{n}\boldsymbol{\cdot }{\boldsymbol{i}_z})\,\textrm{d}{B^\ast }},\end{align}

where i_x, i_y and i_z are the unit vectors of the coordinate system, n is the local unit vector normal to the differential element dB* of the three-dimensional recirculation boundary ∂B, $\langle {C_p}\rangle $ is the spatially averaged C_p on the rear end, and C_pw is the local time-averaged pressure coefficient in the wake. Equation (3.13) shows a close connection between the Reynolds stresses along the recirculation region boundary and the pressure drag. Given the symmetry conditions in the plane of y* = 0, $\boldsymbol{n}\boldsymbol{\cdot }{\boldsymbol{i}_y}$ is zero so that the term $\overline {{u^\ast }{v^\ast }} (\boldsymbol{n}\boldsymbol{\cdot }{\boldsymbol{i}_y})$ vanishes in the mean recirculation region, as Barros et al. (Reference Barros, Borée, Noack, Spohn and Ruiz2016b) did for a square-back Ahmed body. Then, the sum of the integrations on the right-hand side of (3.13) may be reduced to a two-dimensional form (Balachandar et al. Reference Balachandar, Mittal and Najjar1997), namely,

(3.14)\begin{equation}C = \int_{\partial b} {{C_{pw}}(\boldsymbol{n}\boldsymbol{\cdot }{\boldsymbol{i}_x})\,\textrm{d}{b^\ast }} + 2\int_{\partial b} {\overline {{u^\ast }{u^\ast }} (\boldsymbol{n}\boldsymbol{\cdot }{\boldsymbol{i}_x})\,\textrm{d}{b^\ast }} + 2\int_{\partial b} {\overline {{u^\ast }{w^\ast }} (\boldsymbol{n}\boldsymbol{\cdot }{\boldsymbol{i}_z})\,\textrm{d}{b^\ast }} ,\end{equation}

where db* is the differential element of the in-plane recirculation boundary ∂b in the plane of y* = 0.

Figure 15 presents the time-averaged normal stresses $\overline {{u^\ast }{u^\ast }} $ and $\overline {{w^\ast }{w^\ast }} $, and shear stress $\overline {{u^\ast }{w^\ast }} $ in two x–z planes of y* = 0 and 0.36, where the recirculation boundary enclosing the upper and lower recirculation bubbles is highlighted by a red-coloured broken curve as determined from the time-averaged sectional streamlines (figure 2). Several observations may be made. Firstly, the $\overline {{u^\ast }{u^\ast }} $ contours display two concentrations (figure 15a,b). One occurs behind the rear window, which is apparently associated with the shear layer separated from the upper edge of the slanted surface. The other takes place downstream of the lower edge of the base. In the symmetry plane, this concentration spatially coincides with the structure of St_w = 0.30 detected behind the base (figure 7) and is connected to the nearly periodic motions of the upper and lower bubbles and hence the structures emanated from the bubbles. Secondly, the $\overline {{w^\ast }{w^\ast }} $ contours display one highly concentrated region behind the vertical base (figure 15c,d). The rollup of the gap flow and the downwash flow from the rear window appear to clash with each other near (x*, z*) ≈ (0.78, 0.08) in the symmetry plane of y* = 0 and (0.75, 0.42) in the off-symmetry plane of y* = 0.36 based on the PIV-measured streamlines (figure 2a,c), which may account for this concentration. Thirdly, two opposite-signed concentrations are evident in the $\overline {{u^\ast }{w^\ast }} $ contours in both x–z planes (figure 15e,f). The upper is negative while the lower is positive, which are linked to the upper and lower recirculation bubbles, respectively.

Figure 15.

Isocontours of PIV-measured time-averaged in-plane Reynolds stresses in the x–z planes of y* = 0 and y* = 0.36. Thick broken red curves indicate the recirculation boundary determined from sectional streamlines in figure 2. The black symbol ‘×’ denotes the dividing point in the boundary where $\overline {{u^\ast }{w^\ast }} = 0$. Here Re = 0.62 × 10⁵.

In order to evaluate the effect of the $\overline {{u^\ast }{u^\ast }} $ and $\overline {{u^\ast }{w^\ast }} $ concentrations of each bubble on $\langle {C_p}\rangle $, ∂b in the symmetry plane is divided into two parts (∂b_u and ∂b_l) by the dividing point where $\overline {{u^\ast }{w^\ast }} $ equals zero, as marked by the black symbol ‘×’ at (x*, z*) ≈ (0.78, 0.2) (figure 15a,g). The subscripts u and l denote the upper and lower bubbles, respectively. Then, the sum of the integrations of $\overline {{u^\ast }{u^\ast }} $ and $\overline {{u^\ast }{w^\ast }} $ along ∂b_u and ∂b_l can be written as

(3.15)\begin{equation}C_i^R = \int_{\partial {b_i}} {\overline {{u^\ast }{u^\ast }} (\boldsymbol{n}\boldsymbol{\cdot }{\boldsymbol{i}_x})\,\textrm{d}{b^\ast }} + \int_{\partial {b_i}} {\overline {{u^\ast }{w^\ast }} (\boldsymbol{n}\boldsymbol{\cdot }{\boldsymbol{i}_z})\,\textrm{d}{b^\ast }} ,\quad i \in \{ u\;\textrm{or}\;l\} .\end{equation}

The $C_u^R$ and $C_l^R$ are calculated to be approximately −0.005 and −0.007 in the symmetry plane, which contribute negatively to $\langle {C_p}\rangle $. As will be shown in § 4, the structures of St_w = 0.30 show up in the region of x* ∈ [0.43, 0.86], y* ∈ [−0.29, 0.29] and z* ∈ [−0.04, 0.32]. The Reynolds stress distribution in an off-symmetry plane of y* = 0.14 also displays the concentrations of ${\overline {{u^\ast }{u^\ast }} _{max}} = 0.052$ at (x*, z*) = (0.69, 0.02) and ${\overline {{u^\ast }{w^\ast }} _{max}} = 0.034$ at (x*, z*) = (0.81, 0.05) (not shown), due to the presence of this unsteady structure. It may be inferred from the relationship between the Reynolds stresses and $\langle {C_p}\rangle $ that this unsteady structure may have a considerable impact on the base and rear window pressures in the off-symmetry planes, as in the symmetry plane. The structures emanated alternately from the upper and lower bubbles may produce a substantial turbulent kinetic energy k* $=0.5(\overline {{u^\ast }{u^\ast }} + \overline {{w^\ast }{w^\ast }} )$, as supported by the $\overline {{u^\ast }{u^\ast }} $ and $\overline {{w^\ast }{w^\ast }} $ concentrations associated with these structures (figure 15). If we consider a vehicle cruising in ambient still fluid, the energy loss due to the turbulent kinetic energy consumes part of the work done on the fluid, and hence increases the vehicle pressure drag. That is, the emanated structures contribute to a rise in the drag.

3.4. Roof, sidewall and gap vortices

The E_u measured above the roof at x* = −1.27 and z* = 0.94 displays two major peaks (figure 16a) at f* = St_r = 0.20 for F₁ (y* = 0) and from F₃ to F₅ (y* ∈ [0.29, 0.43]) and at f* = St_a = 0.14 for F₂ (y* = 0.14). A streamwise vortex with negative vorticity is captured instantaneously above the roof around y* = 0.14 (figure 16c), which is responsible for the time-averaged swirling structure (marked by $\bar{\lambda }_{ci}^{2\ast } = 0.002$) with $\bar{\omega }_{x,min}^\ast{=}{-} 0.8$ at y* = 0.14 and z* = 0.98 (figure 16d). This vortex is spatially coincident with the detected St_a = 0.14. The St_r = 0.20 is associated with the hairpin vortices emanated from the separation bubble near the leading edge of the roof, and St_a = 0.14 is linked to the oscillation of one pair of longitudinal counter-rotating vortices behind the bubble, whose origins are the two foci at each end of the flow separation line (Zhang et al. Reference Zhang, Zhou and To2015). When viewed in the y–z plane, the vortex at y* = 0.14 is clockwise and the other at y* = −0.14 must be anticlockwise. Therefore, the tangential velocity at the inner side of each longitudinal vortex about the symmetry plane is upwards. Thus, at y* ∈ [−0.14, 0.14], the longitudinal vortices may act to push up the hairpin vortices to the outer flow with a higher convection velocity to get diffused, which may be responsible for the relatively weak peak of St_r = 0.20 in E_u at F₁ (y* = 0) (figure 16a).

Figure 16.

Power spectral density function E_u of the hot-wire signal measured along (a) F₁–F₅ (x* = −1.27, z* = 0.94) above the roof, and (b) G₁–G₇ (x* = −1.27, y* = 0.63) near the sidewall, respectively. PIV-measured (c) typical instantaneous and (d) time-averaged streamwise vorticity contours in the y–z plane of x* = −1.27. Thick black closed contours correspond to the swirling strength $\bar{\lambda }_{ci}^{2\ast } = 0.002$ of the time-averaged flow. The magnitudes of $\bar{\omega }_{x,max}^\ast $ and $\bar{\omega }_{x,min}^\ast $ are marked in the figure. Here Re = 0.62 × 10⁵.

The E_u measured over the sidewall at x* = −1.27 and y* = 0.63 (figure 16b) shows one pronounced peak at f* = St_s = 0.27 over z* ∈ [0.14, 0.72] (G₂–G₆). As above the roof, St_s is associated with the side vortices emanated from the separation bubble near the leading edge of the sidewall. At G₄ (z* = 0.43) near the mid-height of the sidewall, this peak is less pronounced, similarly to the case at F₁ above the roof in the symmetry plane (figure 16a), with the same flow physics involved, as is evident from one pair of counter-rotating longitudinal vortices over the sidewall about the mid-height of the body (figure 16c,d). Another predominant frequency of f* = St_l = 2.12 is detected at G₁–G₂ near the lower edge of the sidewall (figure 16b), which is ascribed to the vortices generated by the upstream struts located at x* = −2.52 and y* = ±0.49 (Zhang et al. Reference Zhang, Zhou and To2015).

In view of the strong three-dimensionality of the flow, the hot-wire measurements were performed at various spanwise locations over the slanted surface, as shown in figure 17. A peak at St_r = 0.20 is discernible at H₃–H₅ (x* = −0.36, y* ∈ [0.29, 0.43], z* = 0.91) near the upper edge of the rear window and at I₃–I₅ (x* = −0.09, y* ∈ [0.29, 0.43], z* = 0.91) further downstream (figure 17a,b). However, this peak becomes less pronounced at H₂ and I₂ (y* = 0.14) near the symmetry plane, and even vanishes at H₁ and I₁ (y* = 0), which may be linked to the interaction between these structures with the longitudinal vortices above the roof near the symmetry plane (figure 17a,c,d). In addition, St_r is not detectable in E_u measured at J₁–J₄ and K₁–K₅ at the height of the roof (z* = 0.86) (figure 17c,d) and also at lower positions (not shown), indicating that these vortices travel above the upper recirculation bubble.

Figure 17.

Power spectral density function E_u of the hot-wire signal measured along (a) H₁–H₆ (x* = −0.36, z* = 0.91), (b) I₁–I₆ (x* = −0.09, z* = 0.91), (c) J₁–J₆ (x* = −0.36, z* = 0.86) and (d) K₁–K₆ (x* = −0.09, z* = 0.86), respectively, over the slant. Here Re = 0.62 × 10⁵.

A peak occurs at St_s = 0.27 in E_u measured at H₆, I₆, J₆ and K₆ near the side edge of the rear window (y* = 0.57) (figure 17), where the C-pillar vortex takes place (figure 5). A typical photograph captured in the x–y plane of z* = 0.72 over the rear window (figure 18) exhibits nearly periodic structures right above the side edges of the rear window. Obviously, this image should cut through the C-pillar vortices. Then, these nearly periodic structures reconfirm Zhang et al.'s (Reference Zhang, Liu, Zhou, To and Tu2018) assertion that the side vortices may wrap around and get tangled with the C-pillar vortex.

Figure 18.

Typical photograph of the flow structure in the x–y plane of z* = 0.72 at Re = 0.22 × 10⁵. Flow is from left to right.

In order to detect possible nearly periodic structures formed in the gap between the ground and the underside of the Ahmed body, hot-wire measurements were conducted along L₁–L₇ (x* = 0.09, z* = 0), downstream of the lower edge of the base (figure 19). Two prominent frequencies occur at L₃–L₇ (y* ∈ [0.36, 0.79]) in E_u. One is f* = St_g = 1.77 at L₃–L₅ over y* ∈ [0.36, 0.65], spatially corresponding to the gap vortex captured by Wang et al. (Reference Wang, Zhou, Pin and Chan2013). Another is f* = St_l = 2.12 at L₆–L₇ over y* ∈ [0.72, 0.79], outside the lower corner of the base, spatially coinciding with the lower vortex, which originates from the vortices detected at G₁–G₂ at the lower edge of the sidewall (figure 16b).

Figure 19.

Power spectral density function E_u of the hot-wire signal measured along J₁–J₇ (x* = 0.09, z* = 0) in the wake, at Re = 0.62 × 10⁵.