2.1. Geometry and boundary conditions

Two notchback Ahmed bodies used for numerical simulations are shown in figure 1. The wake is expected to be asymmetric behind Model 1 and symmetric behind Model 2 following the experimental study by Sims-Williams et al. (Reference Sims-Williams, Marwood and Sprot2011). The same dimensions of the two models are expressed in the height of the model, $H = 0.096\;\textrm{m}$. The width of the model is $W = 1.35H$, the model length is $L = 3.82H$, the deck length is ${L_D} = 0.469H$ and the deck height is ${H_D} = 0.687H$. The clearance between the ground and the bottom of the model is ${H_C} = 0.21H$. The only difference between the two models is in the effective backlight angles, ${\beta _1} = 17.8\mathrm{^\circ }$ of Model 1 and ${\beta _2} = 21.0\mathrm{^\circ }$ of Model 2, giving the roof lengths ${L_{S1}} = 2.847H$ and ${L_{S2}} = 3.006H$. The origin of the coordinate is set at the middle body in the $x$ direction, the middle section of the model in the $y$ direction and the bottom body in the $z$ direction.

Figure 1. Geometric model: (a) aerial view of Model 1; (b) front view of the model; (c) side view of Model 1; (d) side view of Model 2.

The model is centred in a digital tunnel as illustrated in figure 2. The cross-sectional width of the tunnel domain is ${W_T} = 11.46H$ and the tunnel height is ${H_T} = 5.73H$. The inlet is set at a distance of ${D_{in}} = 8H$ upstream of the model and the outlet is located at ${D_{out}} = 19H$ downstream of the model. The Reynolds number, $Re = 5 \times {10^4}$, is based on the model height and a uniform inlet velocity profile ${U_{inf}} = 7.864\;\textrm{m}\;{\textrm{s}^{ - 1}}$. The pressure outlet with a constant pressure of $0\;\textrm{Pa}$ is used at the downstream boundary. The lateral and roof surfaces are set as symmetry planes. The no-slip wall boundary condition is applied on the surface of the models, and the ground of the domain is treated as a moving wall using the same velocity as ${U_{inf}}$.

Figure 2. The flow domain for simulations.

2.2. Numerical method and details

The governing LES equations solved with a commercial finite volume solver, Star CCM+ 2019.2, are the incompressible Navier–Stokes and the continuity equations filtered with the implicit spatial filter of characteristic width $\varDelta$. The filter width, $\varDelta$, is defined as $\varDelta = {({\varDelta _1}{\varDelta _2}{\varDelta _3})^{1/3}}$, where ${\varDelta _i}$ are the computational cell sizes in three coordinate directions. The wall-adapting local eddy-viscosity (WALE) model proposed by Nicoud & Ducros (Reference Nicoud and Ducros1999), which has been extensively tested and applied to predict flows around the hatchback (Aljure et al. Reference Aljure, Lehmkuhl, Rodríguez and Oliva2014), the squareback (Dalla Longa et al. Reference Dalla Longa, Evstafyeva and Morgans2019) and the notchback (He et al. Reference He, Minelli, Wang, Gao and Krajnović2021) Ahmed bodies, is used in this work. This subgrid-scale (SGS) model is based on the square of the velocity gradient tensor and accounts for the effects of the strain and rotation rates. The subgrid eddy viscosity in the WALE model is modelled as

(2.1)\begin{equation}{\mu _t} = \rho {({C_w}\varDelta )^2}\frac{{{{(S_{ij}^dS_{ij}^d)}^{3/2}}}}{{{{({{\tilde{S}}_{ij}}{{\tilde{S}}_{ij}})}^{5/2}} + {{(S_{ij}^dS_{ij}^d)}^{5/4}}}},\end{equation}

where ${C_w}$ is a constant, here ${C_w} = 0.544$. Tensor $\tilde{S}$ is the filtered rate of strain tensor and $S_{ij}^d$ is the traceless symmetric part of the square of the velocity gradient tensor ${g_{ij}}$, defined as

(2.2)\begin{equation}S_{ij}^d = {\textstyle{1 \over 2}}(\tilde{g}_{ij}^2 + \tilde{g}_{ji}^2) - {\textstyle{1 \over 3}}{\delta _{ij}}\tilde{g}_{kk}^2,\end{equation}

where ${g_{ij}}$ denotes $\partial {u_i}/\partial {x_j}$.

Convective fluxes are approximated by a blend of $98\,\%$ central difference scheme of second-order accuracy and 2 % first-order upwind scheme. The time integration is done using the second-order-accurate three-level time Euler scheme. The non-dimensional time step $\textrm{d}{t^\ast } = \varDelta {t_s}{U_{inf}}/H = 3.44 \times {10^{ - 3}}$, giving a Courant–Friedrichs–Lewy number lower than 1 in over 99 % of the cells during all time steps, including the simulation applying the fine mesh presented in Appendix A. Reynolds-averaged Navier–Stokes simulations are first conducted as initial conditions for all LES. Then, the flow is sampled after at least the initial ${t^\ast } = t{U_{inf}}/H = 62$. The sampling duration of the simulation for Model 1 is ${t^\ast } = 930$ corresponding to $30$ flow-through passages through the domain and that for Model 2 is ${t^\ast } = 124$.

2.3. Measurement set-up

The aerodynamic drag force ${F_d}$ is the component of the aerodynamic load in the $x$ direction associated with the coordinate system defined in figure 1. The aerodynamic coefficient of the drag, ${C_d}$, is defined as

(2.3)\begin{equation}{C_d} = \frac{{{F_d}}}{{{\textstyle{1 \over 2}}\rho AU_{inf}^2}},\end{equation}

where $\rho $ is the density of the fluid, A is the projected area of the frontal model surface in the x direction, as shown in figure 1(b), and ${U_{inf}}$ is the inlet velocity of the free stream.

The velocity components in the $x$, $y$ and $z$ directions are ${U_x}$, ${U_y}$ and ${U_z}$, respectively. The associated streamwise, spanwise and vertical velocity components, u, v and w, are defined as

(2.4a–c)\begin{equation}u = \frac{{{U_x}}}{{{U_{inf}}}},\quad v = \frac{{{U_y}}}{{{U_{inf}}}},\quad w = \frac{{{U_z}}}{{{U_{inf}}}}.\end{equation}

The coefficient of pressure ${C_p}$ is defined as

(2.5)\begin{equation}{C_p} = \frac{{p - {p_\infty }}}{{{\textstyle{1 \over 2}}\rho U_{inf}^2}},\end{equation}

where p is the pressure and ${p_\infty }$ is the pressure in the upstream far field.

The base pressure ${C_{pBi}}$ on each surface of the rear body is defined as

(2.6)\begin{equation}{C_{pBi}} = \frac{1}{{{A_i}}}\int {{C_p}\cdot\textrm{d}{A_i}} ,\end{equation}

where $i \in \{ s,\; d,\; b\} $ stands for the slant, the deck and the back wall, respectively, and ${A_i}$ is the area of each surface.

Pressure probes are in the same way set at the two rear bodies, e.g. for Model 1 shown in figure 3. Two monitoring points, symmetrically distributed on the left and right sides, are arranged on each surface of the slant, the deck and the back wall. The distance between the central line and the corresponding side point is $dy = 0.417H$. The points on the slant are located at ${H_{P1}} = 0.844H$ above the bottom of the model (at half-height of the slant). The points on the deck are at ${D_{P2}} = 0.5{L_D}$ (half-length of the deck) from the trailing edge of the deck. The points on the back wall are at ${H_{P2}} = 0.5{H_D}$ (half-height of the deck) above the bottom of the model. The pressure is sampled at the last iteration of each time step during the simulation.

Figure 3. Pressure monitoring points on the rear surface of Model 1.

The extent of the asymmetry in the wake is quantified by the spanwise base pressure gradients, defined as

(2.7)\begin{equation}\frac{{\partial {C_{pi}}}}{{\partial y}} = \frac{{{C_p}({P_{ir}}) - {C_p}({P_{il}})}}{{2dy}},\end{equation}

where $i \in \{ s,\; d,\; b\} $ stands for the slant, the deck and the back wall, respectively, and ${C_p}(\; )$ represents the sampled pressure coefficient on each monitoring point. By the definition of the base pressure gradients, a higher absolute value of $\partial {C_{pi}}/\partial y$ indicates a higher degree of wake asymmetry. On the contrary, $\partial {C_{pi}}/\partial y$ fluctuating around $0$ suggests wake symmetry.

2.4. Description of modal analysis

Fast Fourier transform (FFT) and POD are performed for modal analysis of the wake flow. The POD approach provides an in-depth understanding of flow structures in terms of both energy contents and characteristic frequencies. As originally proposed by Lumley (Reference Lumley1970), and later introduced with the method of snapshots by Sirovich (Reference Sirovich1987), this method is based on the energy ranking of orthogonal structures predicted from a correlation matrix of the snapshots. A singular-value decomposition approach is used to conduct the POD analysis. The time step between snapshots is $\Delta t = 4.2 \times {10^{ - 4}}\;\textrm{s}$, 10 times $\mathrm{\Delta }{t_s}$ in simulations. Thus, the highest frequency considered in the modal analysis (the Nyquist frequency) is 1190 Hz. The lowest frequency captured is limited by the snapshot total sampling time to 13 Hz (the snapshots are collected more than ${t^\ast } = 62$ and at least ten periods should be captured for a reliable frequency evaluation). These two frequencies give a reliable real frequency range between 13 and 1190 Hz $(0.16 < S{t_H} < 14.52)$. The non-dimensional frequency $S{t_H}$ is the Strouhal number normalized with the free-stream velocity and the height of the model. The POD method has been successfully used in previously published works for flows of bluff bodies by Thacker et al. (Reference Thacker, Aubrun, Leroy and Devinant2013), Östh et al. (Reference Östh, Noack, Krajnović, Barros and Borée2014), Minelli et al. (Reference Minelli, Krajnović, Basara and Noack2016) and others. In the present work, the modal analysis used for the characterization of the wake dynamics is based on the energy content, identifying flow structures contributing to the asymmetry or symmetry of the wake. In particular, the POD algorithm is applied with a zonal approach for analysing the frequency content of the considered field portion.

3.1. Asymmetric and symmetric flows

The wake asymmetry behind Model 1 and the symmetry behind Model 2 are observed in the LES. Numerical accuracy based on the fine, the medium and the coarse meshes is established by a grid-independence examination presented in Appendix A. The time history of the base pressure gradients on the slant surface during the entire simulation for Model 1 is plotted in figure 4(a). Two stable asymmetric mirrored states observed during the simulation are expressed by two representative periods, ${S_a}$ during ${t^\ast } \in [0,\; 62]$ and ${S_b}$ during ${t^\ast } \in [558,\; 620]$, respectively shown in figures 4(b) and 4(c) by expanding the time scale. The two mirrored states establishing bi-stability of the wake are distinguished by the base pressure gradients that fluctuate with positive values during ${S_a}$ but with negative values during ${S_b}$. The pressure gradients during ${S_a}$ showing lower absolute values than ${S_b}$ are not linked to the transient period at the beginning of the computation, since the data are sampled after two flow-through passages through the domain ensuring a full flow development. Performing multiple simulations for capturing the switch of bi-stability, although the meshes are well resolved, the direction of the asymmetry remains sensitive to the mesh (see details in Appendix B). Therefore, the two states of bi-stability can be duplicated in the present LES, and the lower values of ${S_a}$ are attributed to the turbulent fluctuation in the flow evolution. In contrast to the asymmetry behind Model 1, the surrounding flow of Model 2 is significantly different in that the pressure gradients fluctuate around 0 indicating the symmetry. The time period ${t^\ast } \in [62,\; 124]$, denoted ${S_c}$, is selected from the flow around Model 2 as the representation of the stable symmetric state (see figure 4d).

Figure 4. Time history of the base pressure gradients on the slant: (a) the entire simulation for Model 1; (b) a stable asymmetric state for Model 1 during ${S_a}$; (c) an opposite stable asymmetric state for Model 1 during ${S_b}$; (d) a stable symmetric state for Model 2 during ${S_c}$.

To compare the asymmetric and symmetric wakes, mean flows around Model 1 (during ${S_b}$) and Model 2 (during ${S_c}$) are presented. Figure 5 shows the distribution of the mean surface pressure (left-hand column) and the probability density function (PDF) of the base pressure gradients against the base pressure. For Model 1, the right-hand side of the slant shows negative pressure. On the right-hand side of the deck, a positive pressure region can be found (see figure 5a). In figure 5(b), however, the symmetry of the surface pressure is indicated by showing equivalent distributions on the two sides. The flow states can be quantitatively assessed by the base pressure gradients. As shown in figure 5(c), the PDF indicates that the values of the base pressure gradients are concentrated in the negative or positive region as a result of the asymmetric flow patterns in the wake. On the other hand, figure 5(d) demonstrates the symmetry of the wake behind Model 2, showing the concentration of the pressure gradients around $0$. Moreover, focusing on the base pressure, particularly ${C_{pBs}}$ and ${C_{pBd}}$ on the slant and deck, the values are lower in the case of Model 2. The base pressure contributing to the difference in the aerodynamic drag between the two models is discussed later in this section.

Figure 5. The mean surface pressure distributed on the rear body: (a) Model 1, during ${S_b}$; (b) Model 2, during ${S_c}$. The PDFs of the base pressure gradients against the base pressure on the slant, the deck and the back wall: (c) Model 1, during ${S_b}$; (d) Model 2, during ${S_c}$.

The distribution of the surface pressure on the rear body is associated with the flow structures in the wake. Visualizing the iso-surface of the mean pressure, the averaged flow structure ${V_c}$ behind the slant of the two models is presented in figure 6(a,b). During ${S_b}$ and ${S_c}$, the placement of ${V_c}$ is found to be different between the two models. While its extension for Model 1 is limited to the region near the upper-right corner, Model 2 produces a structure stretching between the two upper corners (see figure 6b). The shape of ${V_c}$ in the two cases is in agreement with the negative pressure on the slant surface observed in figure 5(a,b).

Figure 6. Flow topology visualized by iso-surfaces of the mean pressure at $\overline {{C_p}} ={-} 0.8$, coloured by the mean streamwise velocity, $\bar{u}$: (a) Model 1, during ${S_b}$; (b) Model 2, during ${S_c}$. Averaged near-wall flow traces on the rear surface: (c) Model 1, during ${S_b}$; (d) Model 2, during ${S_c}$. (e) Profiles of the streamwise vorticity, $\overline {{\varOmega _x}} $. The horizontal probe line is located at the near-wall region, $0.01H$ away behind the roof's trailing edge.

Furthermore, the negative pressure inside ${V_c}$ is attributed to the separation from the upper slant, as illustrated by the near-wall traces in figure 6(c,d). The flow separating from the upper slant constitutes two counter-rotating vortices, ${V_{cl}}$ on the left-hand side and ${V_{cr}}$ on the right-hand side. During ${S_b}$, the region of ${V_{cr}}$ is larger, forming a recirculation bubble at the upper-right of the slant resulting in the asymmetric formation of ${V_c}$. Meanwhile, as indicated by the red arrows in figure 6(c), the flow separating from the upper part of the slant reattaches to the right-hand side of the deck, resulting in the reattachment ${R_{dr}}$. This explains the positive pressure on the right-hand side of the deck as shown in figure 5(a). Another reattachment, ${R_{dl}}$, with a shorter reattaching length is found on the left-hand side of the deck. However, ${R_{dl}}$ is driven by the flow separating from the left C-pillar, as indicated by the yellow arrow in figure 6(c). Shown in figure 6(d), the recirculation bubbles shaped by the two shedding vortices can be observed at the two sides during ${S_c}$, forming the symmetry of ${V_c}$ on the slant. As illustrated by the white arrows, the flow separating from the upper slant reattaches to both sides of the deck. The two reattachments, ${R_{dl}}$ and ${R_{dr}}$, are in agreement with two symmetric regions of positive pressure on the deck presented in figure 5(b).

The extent of the asymmetry characterized by the separation on the slant is quantitated by the horizontal profile illustrating the streamwise vorticity, $\overline {{\varOmega _x}} $, behind the near-wall region of the roof's trailing edge (see figure 6e). The vorticity is normalized by the free-stream velocity and the height of the model. Vorticity $\overline {{\varOmega _x}} $ for Model 1 is found to be positive for most of the lateral extension, indicating the asymmetry of the separation from the slanted surface. In the case of Model 2, $\overline {{\varOmega _x}} $ is found to be negative on the left-hand side but positive on the right-hand side, as a result of the symmetric separation on the slant.

The identification of the positions of flow separations and reattachments visualized by the spanwise vorticity, $\overline {{\varOmega _y}} $, and the streamlines projected on the planes in the lateral direction are presented in figure 7. For quantitative analysis, the separation angle, $\theta $, is defined as the angle between the streamwise direction ($x$ coordinate) and the line linking the separation point and the position on the contour at $\overline {{\varOmega _y}} = 3.7$ with the maximum $x$ coordinate. The reattachment length, ${L_R}$, is defined by the distance between the positive–negative transition point of near wall $\overline {{\varOmega _y}} $ and the trailing edge of the deck. As indicated by the streamlines and distribution of $\overline {{\varOmega _y}} $, the flow first separates from the shear layer near the roof's trailing edge. Therefore, a recirculation bubble associated with ${V_c}$ forms above the slant. A second separation from the near-wall region of the deck produces a recirculation bubble, ${V_b}$, behind the back wall. Near the ground, the vortex shedding from the shear layer of the underbody generates another recirculation region ${V_g}$.

Figure 7. Mean streamlines and the spanwise vorticity, $\overline {{\varOmega _y}} $, on the left ${Y_1}$ plane (a,c) and the right ${Y_2}$ plane (b,d). The distance between ${Y_1}$ and ${Y_2}$ is ${D_{P3}} = 0.5W$ (half-width of the model). Model 1, during ${S_b}$: (a) on the ${Y_1}$ plane; (b) on the ${Y_2}$ plane. Model 2, during ${S_c}$: (c) on the ${Y_1}$ plane; (d) on the ${Y_2}$ plane.

Figure 7(a,b) showing the two section planes for Model 1 indicates that the separation angle ${\theta _1}$ on the left-hand side $(5.8\mathrm{^\circ })$ is smaller than that on the right-hand side $(17.6\mathrm{^\circ })$. The separations at two different angles influence the downstream flow. The flow coming from the upper part of the slant, on the left-hand side of the body, results in a large slant angle ${\theta _2}$ $(27.5\mathrm{^\circ })$ of the separation from the deck. As a result, the recirculation bubble ${V_b}$ is attached to the vertical base wall. Although the flow on the left-hand side moves over the deck shown by the streamlines, ${L_R}$ with a shorter reattaching length is observed on the deck. As previously discussed, this reattachment is formed by the flow separating from the left C-pillar (see figure 6c). However, on the right-hand side, the flow coming from the upper part of the slant reattaches to the deck, resulting in a smaller ${\theta _2}$ $(21.3\mathrm{^\circ })$. Thus, ${V_b}$ extends further on the left-hand side. Comparing the core position of ${V_b}$ on the two sides identifies the downstream asymmetry behind the back wall. As above, the wake asymmetry behind Model 1 is characterized by asymmetric wake separations and reattachments.

On the other hand, during ${S_c}$, symmetric separations are found behind the slant of Model 2, as shown in figure 7(c,d). The flow separating for the angle, ${\theta _1} = 12.2\mathrm{^\circ }$ on the left and ${\theta _1} = 12.5\mathrm{^\circ }$ on the right, shapes the identical length of ${L_R}$ on both sides of the deck. Besides, the second separation from the deck, with ${\theta _2} = 24.2\mathrm{^\circ }$ on the left and ${\theta _2} = 23.8\mathrm{^\circ }$ on the right, results in comparable recirculations behind the back wall on the two sides. Therefore, the placement of the separations and the reattachments confirms the symmetry of the wake behind Model 2.

For the explanation of the flow separating from the shear layer of the upper slant, schematic representations of streamlines behind the two models are presented in figure 8. Figure 8(a) represents the case of wake asymmetry behind Model 1, illustrating that, on one side, the flow separating for a lager angle from the upper slant reattaches to the deck (shown in red). Thereby, the downstream flow separates for a small angle from the deck, forming a recirculation bubble away from the back wall. On the opposite side, the flow separating for a small angle from the upper slant rejects reattaching to the deck, moving downstream over the deck (shown in blue). For Model 2, the wake symmetry is illustrated in figure 8(b), indicating consistent separations from the upper slant on both sides, followed by symmetric downstream reattachment and the second separation.

Figure 8. Sketch of the streamlines describing the flow separating from the shear layer of the upper slant: (a) Model 1, during ${S_b}$; (b) Model 2, during ${S_c}$.

The recirculation bubble formed by the second separation from the deck influences the downstream flow structures behind the back wall. In figure 9(a,b), two reverse recirculations, ${V_{bsl}}$ and ${V_{bsr}}$, are formed by the vortices shedding from two side edges of the back wall. Another two counter-rotating recirculation bubbles, ${V_{bl}}$ and ${V_{br}}$, are found further away from the lateral edges of the body. Here, ${V_{bl}}$ and ${V_{br}}$ are the left- and right-hand parts of ${V_b}$ generated by the separation from the deck. During ${S_b}$, for Model 1, these vortices are asymmetric as ${V_{bl}}$ and ${V_{br}}$ deflect to the left. Thus ${V_{bsr}}$ on the right-hand side develops into a larger structure. The corresponding vortices for Model 2 (during ${S_c}$) are found to be symmetric around the symmetry line of the body (figure 9b). The pressure distribution behind the back wall is dependent on the recirculation bubbles. In figure 9(c–f), visualizing the iso-surface of negative mean pressure shapes a toroidal vortex behind the rear model. The toroidal vortex is known as an important structure in the wake of bluff bodies and has already been repeatedly observed by Krajnović & Davidson (Reference Krajnović and Davidson2003, Reference Krajnović and Davidson2005) and Roumeas, Gillieron & Kourta (Reference Roumeas, Gillieron and Kourta2009). The form of the toroidal vortex follows the recirculation bubbles behind the back wall. Once ${V_b}$ extends further downstream on the right-hand side (see figure 7b), the right-hand part of toroidal vortex tilts backwards, as shown in figure 9(c). On the other hand, when the recirculations behind the back wall are symmetric (see figure 7c,d), the toroidal vortex tilts on both sides, as shown in figure 9(d).

Figure 9. Mean streamlines and vertical vorticity, $\overline {{\varOmega _z}} $, on the ${Z_2}$ plane located at ${H_{P2}} = 0.5{H_D}$ (half-height of the deck) above the bottom model: (a) Model 1, during ${S_b}$; (b) Model 2, during ${S_c}$. Flow structures visualized by iso-surfaces of mean pressure at $\overline {{C_p}} ={-} 0.4$. Top view: (c) Model 1, during ${S_b}$; (d) Model 2, during ${S_c}$. Perspective view: (e) Model 1, during ${S_b}$; (f) Model 2, during ${S_c}$.

By visualization of the mean iso-surface of the $Q$-criterion, the statistically averaged flow structures behind the slant are presented in figure 10(a–d). The non-dimensional $Q^{\ast}$ is normalized by $Q^{\ast}=Q(H/U_{inf})^{2}$. Here, the left C-pillar vortex ${V_l}$ consists of two subvortices, ${V_{l1}}$ and ${V_{l2}}$, separating from the left-hand edges of the slant and the deck. On the right-hand side, ${V_r}$ consists of ${V_{r1}}$ and ${V_{r2}}$ separating from the right-hand edges of the slant and the deck. Meanwhile, ${V_c}$ formed by separations from the shear layer of the roof's trailing edge is found on the slant. Moreover, fragmented structures ${V_{fl}}$ and ${V_{fr}}$ are formed on the deck, showing opposite spanwise velocity.

Figure 10. Flow structures visualized by iso-surfaces of mean $Q$-criterion. For $Q^{\ast}=14.9$: (a) Model 1, during ${S_b}$; (b) Model 2, during ${S_c}$. For $Q^{\ast}=4.5$: (c) Model 1, during ${S_b}$; (d) Model 2, during ${S_c}$. Distributions of the mean Reynolds shear stress component, $\overline {u^{\prime}v^{\prime}} $, on the ${Z_1}$ plane behind the slant, located at ${H_{P1}} = 0.844H$ above the bottom of the model (half-height of the slant): (e) Model 1, during ${S_b}$; (f) Model 2, during ${S_c}$.

During ${S_c}$, figure 10(a) suggests that ${V_c}$ primarily forms on the right-hand side of the upper slant, confirming the flow structure visualized by pressure iso-surfaces in figure 6(a). Furthermore, as shown in figure 10(b), ${V_c}$ deflects to the left-hand side interacting with ${V_{l1}}$. On the right-hand side, ${V_{r1}}$ and ${V_{r2}}$ merging to form ${V_r}$ moving downstream are not affected by ${V_c}$. However, during ${S_c}$, as shown in figure 10(c,d), ${V_c}$ separates from both sides of the upper slant. In this case, ${V_c}$ is in the centre. Thus, ${V_{l1}}$ or ${V_{r1}}$ on either side is not disturbed by ${V_c}$.

The status of ${V_c}$ is found to be relevant to the fragmented structures on the deck. Once ${V_c}$ deflects to the left-hand side, ${V_{fr}}$ on the right-hand side of the deck is more dominant than ${V_{fl}}$. Likewise, during the symmetric state, ${V_{fl}}$ and ${V_{fr}}$ are equivalent to maintain ${V_c}$ to the centre. The reverse ${V_{fl}}$ and ${V_{fr}}$ located on the deck are considered related to the ‘decklid’ vortices observed by Jenkins (Reference Jenkins2000), who suggested that these two vortices induce reverse flow over the decklid dominating the separation of a notchback car.

The distribution of $\overline {u^{\prime}v^{\prime}} $ behind the slant given in figure 10(e,f) suggests that, during ${S_b}$, both ${V_{r1}}$ and ${V_{r2}}$ can be observed on the right-hand side. However, on the left-hand side, only ${V_{l2}}$ appears due to the interaction of ${V_{l1}}$ and ${V_c}$. During ${S_c}$, on the other hand, the C-pillar vortex containing two subvortices can be clearly found on each side. Comparing $\overline {u^{\prime}v^{\prime}} $ inside the dashed frames, one can observe the deflection of ${V_c}$ during ${S_b}$ but the symmetry of ${V_c}$ during ${V_c}$. Moreover, the absolute value of $\overline {u^{\prime}v^{\prime}} $ is found to be higher with the symmetric wake, indicating a more turbulent flow behind Model 2. For this reason, the wake instability involving the unsteadiness of separations requires a deeper analysis.

The FFT algorithm is employed to assess the power spectral density (PSD) function of pressure on the ${Z_1}$ plane behind the slant (see figure 11). The frequency spectrum shows approximated peaks for the two models, giving $S{t_H} = 0.53$ for Model 1 and $S{t_H} = 0.6$ for Model 2. Therefore, it is interesting to inquire how the flow structures behind the two models show similar primary frequencies but behave inconsistently.

Figure 11. The PSD of pressure signals sampled on the ${Z_1}$ plane behind the slant. The considered sampling period for Model 1 is ${S_b}$ and for Model 2 is ${S_c}$.

For an in-depth exploration of the unsteady structures in the wake, POD is applied on the pressure field on the ${Z_1}$ plane. For example, figure 12(a) shows the energy content of the first six pressure modes concerning the considered period, the asymmetric state during ${S_b}$. Here, Mode 1 corresponds to the averaged pressure field containing most of the energy. The rest of the modes are distributed in pairs. Figure 12(b) shows the temporary coefficients obtained in the left- and right-hand parts of Mode 2. The two curves shown in red and black are similar but shifted in time, providing the steps to reconstruct the motion of the considered flow structures.

Figure 12. Proper orthogonal decomposition analysis. (a) Energy content of the first six modes. (b) Temporary coefficients: ${b_2}$ (black) and ${b_3}$ (red).

The wake structures recognized by pressure snapshots and the POD distributions with associated modes are presented in figure 13, showing the wake dynamics during ${S_b}$ and ${S_c}$. The flow structures ${V_c}$ associated with the vortex shedding from the shear layer of the slant, and the two C-pillar vortices, ${V_l}$ and ${V_r}$, are distinguished by the instantaneous pressure in figure 13(a,b). During ${S_b}$, ${V_c}$ deflects to the left-hand side to interact with ${V_l}$. Meanwhile, ${V_r}$ extending further downstream on the right-hand side is not affected by ${V_c}$. During ${S_c}$, however, given that ${V_c}$ symmetrically separating from the upper slant is located in the central region, the two C-pillar vortices are equivalent on both sides.

Figure 13. Proper orthogonal decomposition analysis for the pressure projected on the ${Z_1}$ plane. The snapshot of instantaneous pressure: (a) Model 1, during ${S_b}$; (b) Model 2, during ${S_c}$. Spatial distribution and the associated PSD of the temporary coefficient of Mode 2: (c) Model 1, during ${S_b}$; (d) Model 2, during ${S_c}$. Mode 3: (e) Model 1, during ${S_b}$; (f) Model 2, during ${S_c}$. The dotted line is in the central symmetry plane at $y = 0$.

The main moving direction of ${V_c}$ is identified in Mode 2 by showing the wave packets, suggesting that ${V_c}$ moves downstream on the left-hand side during ${S_b}$ (figure 13c), but on the centre during ${S_c}$ (figure 13d). The associated main frequencies of Mode 2 are similar to the results of FFT (figure 11). Therefore, the main frequency containing most of the energy describes the marked separation shedding from the shear layer of the upper slant. Furthermore, expressed by Mode 3, secondary directions of ${V_c}$ are indicated, showing left-hand deflection described by the flow patterns in figure 13(e). On the other hand, in figure 13(f), the secondary direction describes ${V_c}$ moving downstream in the centre, following the main direction. The main frequency of Mode 3 is slightly lower for Model 1 than for Model 2. As a result, the wake dynamics shown in figure 13 confirms the wake asymmetry for Model 1 and the symmetry for Model 2.

Focusing on the C-pillar vortices, the dynamics of ${V_l}$ and ${V_r}$ can be identified by POD modes employed on the divided regions shown in figure 13a,b. The modality of the C-pillar vortex is presented in figure 14. During ${S_b}$, since ${V_c}$ deflects to the left-hand side, ${V_r}$ moving downstream on the right-hand side can be explicitly captured by Mode 2 with a highly concentrated main frequency of $S{t_H} = 2.22$ (see figure 14a). On the left-hand side, in figure 14(b), ${V_l}$ is disturbed by the deflection of ${V_c}$. Therefore, the direction of ${V_l}$, moving downstream with a deviation to the centre, is vague until Mode 4. In the spectrum of the associated frequency, in addition to the main frequency of ${V_l}$ at $S{t_H} = 2.55$, peak clusters with lower frequencies are detected due to the interaction with ${V_c}$. However, in agreement with ${V_c}$ moving downstream without deflections, dynamics of ${V_r}$ and ${V_l}$ can be both described by Mode 2 in the symmetric case during ${S_c}$ (see figure 14c,d). The main frequencies of ${V_r}$ and ${V_l}$ are identified by evident peak values, confirming that the two C-pillar vortices are not affected by ${V_c}$. It can be noted that the frequency of the C-pillar vortex is lower for Model 2 than Model 1, which is attributed to the discrepancy between the two effective backlight angles.

Figure 14. Dynamics of the C-pillar vortex and the associated PSD of the temporary coefficient: (a) ${V_r}$ for Model 1 during ${S_b}$; (b) ${V_l}$ for Model 1 during ${S_b}$; (c) ${V_r}$ for Model 2 during ${S_c}$; (d) ${V_l}$ for Model 2 during ${S_c}$.

The different behaviour of the wake structures between the asymmetric and symmetric flows influences the aerodynamic drag. In order to discuss the asymmetry of the force signals, the two models are divided into left and right parts. Based on the projected area of the entire frontal model surface, the coefficients of the aerodynamic drag are displayed in figure 15. During ${S_b}$, as shown in figure 15(a), the time history curves and the probability density illustrate that the drag ${C_d}$ is markedly lower on the left part of Model 1, demonstrating an explicit flow asymmetry. In figure 15(b), however, ${C_d}$ shows high degrees of consistency between the two parts. Notably, during ${S_c}$, ${C_d}$ on the two sides is characterized by frequent switches between two highly consistent curves, showing an instability of the symmetric wake.

Figure 15. Time history curves of the aerodynamic drag coefficient ${C_d}$: (a) Model 1, during ${S_b}$; Model 2, during ${S_c}$. The right-hand side of each graph shows PDF of ${C_d}$.

Considered as a whole, the averaged drag of the entire Model 1 is $\overline {{C_d}} = 0.324$, being lower than $\overline {{C_d}} = 0.345$ of Model 2, which is related to the base pressure suggested by Roshko (Reference Roshko1993). The averaged base pressure $\overline {{C_{pBi}}} $ on the rear of Model 1 and Model 2 is presented in figure 16. The difference of $\overline {{C_{pBi}}} $ between the two models can be found on the slant and the deck, showing lower absolute values obtained for Model 1. By the base pressure definition given in § 2.3, the lower absolute value of $\overline {{C_{pBi}}} $ suggests a lower value of negative pressure on the rear surface, leading to a lower drag of the model. Therefore, $\overline {{C_{pBi}}} $ shown in figure 16 follows the lower $\overline {{C_d}} $ observed for Model 1. Moreover, it can be seen from figure 6 that, compared to Model 1 (figure 6a), the flow structure ${V_c}$ has a larger contact area attached to the slant and the deck of Model 2 because ${V_c}$ forms on both sides (figure 6b). For this reason, the negative pressure inside ${V_c}$ contributes to the higher absolute value of the negative base pressure for Model 2.

Figure 16. Averaged base pressure $\overline {{C_{pBi}}} $ on the rear model; $i \in \{ s,\; d,\; b\} $ stands for the slant, the deck and the back wall.

3.2. Switching process of the wake bi-stability

For the wake bi-stability of squareback Ahmed bodies, the asymmetric wake has been observed to randomly switch to the opposite mirrored state (Grandemange et al. Reference Grandemange, Gohlke and Cadot2013a,b; Östh et al. Reference Östh, Noack, Krajnović, Barros and Borée2014; Volpe et al. Reference Volpe, Devinant and Kourta2015; Dalla Longa et al. Reference Dalla Longa, Evstafyeva and Morgans2019; Haffner et al. Reference Haffner, Borée, Spohn and Castelain2020). In terms of the notchback configuration, the switching process of wake bi-stability was experimentally observed by Cogotti (Reference Cogotti1986) and Lawson et al. (Reference Lawson, Garry and Faucompret2007). In this work, the first numerical result showing a switching period for the bi-stability behind the notchback body (Model 1) is presented, denoted ${S_d}$ during ${t^\ast } \in [63,\; 125]$, shown figure 17(a). The switch of the wake is identified by $\partial {C_{ps}}/\partial y$ dropping from positive to negative values during ${S_d}$, showing bi-stability. The expectation for the second wake switch has not been achieved during the simulation lasting for 30 flow-through passages through the domain on account of the large time scale of the bi-modality. However, the possibility of triggering the switching process is signified by the illustration of $\partial {C_{ps}}/\partial y$ repeatedly approaching $0$. Therefore, the time period, ${S_e}$ during ${t^\ast } \in [252,\; 314]$, showing the attempt to switch, is selected to compare with ${S_d}$ to analysis the switching process.

Figure 17. Time history curves of base pressure gradients on the slant surface: (a) the entire simulation for Model 1; (b) the switching period during ${S_d}$, in which ${S_{d1}}$ is ${t^\ast } \in [84,92]$, ${S_{d2}}$ is ${t^\ast } \in [92,96]$, ${S_{d3}}$ is ${t^\ast } \in [96,104]$; (c) the attempt to switch during ${S_e}$, in which ${S_{e1}}$ is ${t^\ast } \in [273,281]$, ${S_{e2}}$ is ${t^\ast } \in [281,285]$, ${S_{d3}}$ is ${t^\ast } \in [285,293]$.

Probability distributions of base pressure gradients on the rear of Model 1, during ${S_d}$ and ${S_e}$, are presented in figure 18. In figure 18(a), concentrations with both positive and negative values are observed on the rear body, particularly on the slant and the deck. Therefore, the identification of the wake bi-stability behind Model 1 is indicated, suggesting that the wake during ${S_d}$ experiences two asymmetric mirrored states. However, in figure 18(b), it is evident that the pressure gradients on each rear surface are concentrated on one side, showing the absence of the opposite state.

Figure 18. From left to right, PDFs of the base pressure gradients against the base pressure on the slant, the deck and the back wall: (a) during ${S_d}$, the successful switching period; (b) during ${S_e}$, the failed switching period.

To focus on the switching periods, the central parts of ${S_d}$ and ${S_e}$ are extracted, denoted ${S_{d1 - 3}}$ and ${S_{e1 - 3}}$, respectively, shown in figure 17(b,c). During these periods, the wake starts to switch (${S_{d1}}$ and ${S_{e1}}$), is switching (${S_{d2}}$ and ${S_{e2}}$), and reverses to the opposite state (${S_{d3}}$) or returns to the original state (${S_{d3}}$).

The mean streamlines and the spanwise vorticity distribution during ${S_{d1 - 3}}$ and ${S_{e1 - 3}}$ are presented in figure 19. For the switching period ${S_{d1 - 3}}$, as indicated in figure 19(a), the flow separating from the upper part of the slant reattaches to the deck on both sides. Before the switch, ${S_{d1}}$, the reattachment length ${L_R}$ on the left-hand side of the deck is found to be longer than that on the right-hand side due to the larger separation angle ${\theta _1}$. As a consequence, the downstream separation angle ${\theta _2}$ has a smaller value on the left-hand side. During ${S_{d2}}$, once the wake is switching, a temporary symmetric state with comparable separations and reattachments on the two sides can be observed. After a successful switch, the wake turns asymmetric on the opposite side during ${S_{d3}}.$ Angle ${\theta _1}$ on the left-hand side is found to be larger, resulting in a longer ${L_R}$ and a smaller ${\theta _2}$.

Figure 19. Mean streamlines and spanwise vorticity, $\overline {{\varOmega _y}} $, on the left ${Y_1}$ plane (left-hand column) and the right ${Y_2}$ plane (right-hand column), the distance between ${Y_1}$ and ${Y_2}$ planes being ${D_{P3}} = 0.5W$: (a) during ${S_{d1 - 3}}$; (b) during ${S_{e1 - 3}}$.

Figure 19(b) shows the separations and the reattachments during the period ${S_{e1 - 3}}$. The wake asymmetry during ${S_{e1}}$ is characterized by a smaller ${\theta _1}$, shorter ${L_R}$ and larger ${\theta _2}$ on the left-hand side. During ${S_{e2}}$, although the mean flow fails to show a perfect symmetry, the asymmetry is found to become weaker than for ${S_{e1}}$. During ${S_{e3}}$, the smaller ${\theta _1}$, shorter ${L_R}$ and larger ${\theta _2}$ are still found on the left-hand side, showing that the wake returns to the original asymmetric state as a result of the failed switch.

Flow mechanisms of the wake switch are analysed from the perspective of the flow structures behind the slant. The instantaneous distribution of the streamwise vorticity showing separations behind the roof's trailing edge is presented in figure 20. It can be seen that the wake structure ${V_c}$ consists of two substructures, ${V_{cl}}$ and ${V_{cr}}$, forming on the left- and right-hand sides of the upper slant, respectively. On the left-hand side, ${V_{cl}}$ moves downstream, deflecting to the right-hand side. On the right-hand side, ${V_{cr}}$ deviates towards the left-hand side. Using the POD method, modal analysis is applied for the two switching periods ${S_{d1 - 3}}$ and ${S_{e1 - 3}}$, with the aim of exploring the dynamics of the wake structures, as shown in figure 21.

Figure 20. Instantaneous distribution of the streamwise vorticity, ${\Omega _x}$, on the ${Z_0}$ plane behind the roof's trailing edge: (a) during ${S_{d1 - 3}}$; (b) during ${S_{e1 - 3}}$. The dotted line is the central symmetry plane at $y = 0$.

Figure 21. Proper orthogonal decomposition analysis for the pressure distribution projected on the ${Z_1}$ plane behind the slant during the switching process of the wake bi-stability. Each panel shows a snapshot of instantaneous pressure, Mode 2 and Mode 3. During the temporary states: (a) ${S_{d1}}$; (b) ${S_{e1}}$; (c) ${S_{d2}}$; (d) ${S_{e2}}$; (e) ${S_{d3}}$; (f) ${S_{e3}}$. The dotted line is in the central symmetry plane at $y = 0$.

Snapshots of the wake structures behind the upper part of the slant during the period ${S_{d1 - 3}}$ in which a successful switch of the bi-stability occurs are presented in figure 20(a). At ${t^\ast } = 84$, ${V_{cl}}$ is dominant, showing a larger separation region attached to the upper slant. Meanwhile, ${V_c}$ combining the downstream parts of ${V_{cl}}$ and ${V_{cr}}$ is located at the right-hand side. In figure 21(a), the main direction of ${V_c}$, moving downstream on the right-hand side, is described by Mode 2 of POD, showing the wake dynamics before the switch. Moving to ${t^\ast } = 89$, the region of ${V_{cr}}$ gradually becomes larger. This can be explained by an energetic movement of ${V_{cr}}$ in the secondary direction, deflecting to the left, as captured by Mode 3 in figure 21(a). However, during ${S_{d1}}$ the movement of ${V_{cl}}$ is invisible in the first three POD modes. At ${t^\ast } = 94$, the structures ${V_{cl}}$ and ${V_{cr}}$ are found to be of similar extension, resulting in a temporary symmetric state. In figure 21(b), the direction of the flow structures showing the movement in the downstream direction is presented with Mode 2 and Mode 3. After the successful switch, the extension of ${V_{cr}}$ continues to increase, as shown in the snapshots at ${t^\ast } = 99$ and ${t^\ast } = 104$. Therefore, ${V_c}$ starts to move downstream at the left-hand side, as described by the distribution of the streamwise vorticity and by Mode 2 given in figure 21(c). This means that the asymmetric wake reverses to the opposite state after the switch. Moreover, it is noted that Mode 3 in figure 21(c) indicates a secondary direction showing the movement of ${V_{cr}}$ with a deflection to the left. This means that ${V_{cr}}$ remains more energetic than ${V_{cl}}$, showing the stability of the asymmetric wake after the switch.

Snapshots of the wake structures during the period of the attempt to switch, ${S_{e1 - 3}}$, for the observation of the process of the failed switch, are presented in figure 20(b). At ${t^\ast } = 273$, the separation region of ${V_{cr}}$ is larger than that of ${V_{cl}}$. At the same time, ${V_c}$ is mainly on the left-hand side, moving downstream as described by Mode 2 in figure 21(d). The asymmetric wake state before the attempt to switch is unstable, due to deflection to the right, as seen in Mode 3 in figure 21(d). This proves that the movement of ${V_{cl}}$ is more energetic, despite a smaller extension. Thereby, the region of ${V_{cl}}$ becomes larger at ${t^\ast } = 278$ and matches ${V_{cr}}$ at ${t^\ast } = 283$, showing the transition of the wake to the temporary symmetric state. The movement of ${V_c}$ to the downstream wake in the centre is described by Mode 2 and Mode 3 in figure 21(e). Although the temporary symmetric state at ${t^\ast } = 283$ is found to be similar to that at ${t^\ast } = 94$, the wake fails to switch as the region of ${V_{cr}}$ remains larger and ${V_c}$ stays on the left-hand side (see snapshots at ${t^\ast } = 288$ and ${t^\ast } = 293$). This means that the wake returns to the original asymmetric state after the failed switch. In figure 21(f), the direction of ${V_c}$ moving downstream on the left-hand side is indicated in Mode 2. Furthermore, Mode 3 in figure 21(f) indicates the deflection to the left, showing that ${V_{cr}}$ with a larger region tends to be more energetic. This produces a stable wake asymmetry after the failed switch.

In order to present the three-dimensional flow structures behind the slant during ${S_{d1 - 3}}$ and ${S_{e1 - 3}}$, ${V_c}$ identified by evaluating iso-surfaces of instantaneous pressure at ${C_p} ={-} 0.8$ is displayed in figure 22. Particularly, figure 22(a) shows the process of the successful switch. In the beginning, at ${t^\ast } = 84$, ${V_c}$ forms at the upper-left of the slant with a deflection to the right-hand side. Moving to ${t^\ast } = 89$, with the deflection to the right, ${V_c}$ starts to form at the upper-right corner of the slant. Afterwards, ${V_c}$ moves to the right-hand side, showing the appearance of ${V_c}$ on the two upper corners of the slant, as described in the snapshot at ${t^\ast } = 94$. As indicated in snapshots at ${t^\ast } = 99$ and ${t^\ast } = 104$, ${V_c}$ disappears on the upper-left corner and eventually switches to the right-hand side, completing the switching process. As for the process of the failed switch, figure 22(b) shows that, at ${t^\ast } = 273$, ${V_c}$ forms at the upper-right corner of the slant with a deflection to the left. However, in the following procedure, although showing an attempt to reach the switch, the left shift of ${V_c}$ is insufficient to force the structure to form at the left-hand side of the slant, culminating in the failed switch.

Figure 22. Instantaneous iso-surfaces of pressure at ${C_p} ={-} 0.8$, coloured by the instantaneous streamwise velocity, $u$: (a) during ${S_{d1 - 3}}$; (b) during ${S_{e1 - 3}}$.

Observations of figures 20–22 allow the deduction that, as described by the secondary direction, the deflection of the near-wall structures behind the slant to the opposite side triggers the process of the switch. The decisive factor activating a successful switch is the consequence of the deflection that can drive the wake structures to the opposite side after a temporary symmetric state. Once the flow structure, ${V_c}$, starts to form at the upper slant on the opposite side, a switch of the bi-stability is finalized. On the contrary, when the deflection of ${V_c}$ is inadequate to sustain its formation on the opposite side, the wake fails to switch and returns to the initial state. As a result, the wake develops into a stable asymmetric state after the successful or failed switching process.

Furthermore, the downstream toroidal vortex is presented in figure 23. The toroidal vortex behind the squareback Ahmed body observed by Dalla Longa et al. (Reference Dalla Longa, Evstafyeva and Morgans2019) was sensitive to the switching process of the wake bi-stability. For the notchback Ahmed body, it is found that the status of the toroidal vortex is dependent on the wake state. In figure 23(a), during ${S_{d1}}$ before the switch, the left part of the toroidal vortex tilts backwards. Then, the toroidal vortex tilts on both sides when the wake is switching during ${S_{d2}}$, in which a temporary symmetry is observed. After the successful switch, following the wake turning to be asymmetric on the opposite side, the right-hand part of the toroidal vortex tilts backwards. For the period of the attempt to switch shown in figure 23(b), the right-hand part of the toroidal vortex tilts backwards during ${S_{e1}}$ before the switch. The temporary symmetry of the toroidal vortex is thereby formed as both sides tilt during ${S_{e2}}$. However, the difference is that the right-hand part of the toroidal vortex tilts again after the failed switch, as observed during ${S_{e3}}$. From figure 23, the underlying mechanism of the downstream flow structure is found sensitive to the upstream wake state in the notchback region.

Figure 23. Iso-surfaces of mean pressure at $\overline {{C_p}} ={-} 0.4$, coloured by the mean streamwise velocity, $\bar{u}$: (a) during ${S_{d1 - 3}}$; (b) during ${S_{e1 - 3}}$.