On state instability of the bi-stable flow past a notchback bluff body

Abstract The wake of a notchback Ahmed body presenting a bi-stable nature is investigated by performing wind tunnel experiments and large-eddy simulations. Attention is confined to the Reynolds number ($Re$) influence on the wake state instability within $5\times 10^{4}\leq Re \leq 25\times 10^{4}$. Experimental observations suggest a wake bi-stability with low-frequency switches under low $Re$. The wake becomes ‘tri-stable’ with the increase of $Re$ with the introduction of a new symmetric state. The higher presence of the symmetric state can be considered as a symmetrization of the wake bi-stability with an increasing $Re$. The wake symmetry under high $Re$ attributed to the highly frequent switches of the wake is extremely sensitive to small yaw angles, showing the feature of bi-stable flows. The wake asymmetry is confirmed in numerical simulations with both low and high $Re$. The wake asymmetries are indicated by the wake separation, the reattachment and the wake dynamics identified by the proper orthogonal decomposition. However, the turbulence level is found to be significantly higher with a higher $Re$. This leads to a higher possibility to break the asymmetric state, resulting in highly frequent switches showing symmetry.

the realistic flow features, helping explore the drag reduction, stability, etc. Generally, aerodynamic studies of symmetric bodies assume the presence of symmetric flows. However, the surrounding flow of a notchback car observed by Cogotti (1986) suggests a symmetry breaking of the wake, showing switches between two asymmetric mirrored states. The bi-stability characterized by stochastic wake reversals behind notchback sedan models has been repeatedly observed also by Lawson, Garry & Faucompret (2007), Wieser et al. (2014) and Yan et al. (2019).
Not only sedans but also a car-like bluff body, the notchback Ahmed body, was found to produce wake asymmetry (Sims-Williams, Marwood & Sprot 2011). The wake bi-stability of the notchback geometry was confirmed, for the first time numerically, by He et al. (2021a) using large-eddy simulations (LES). A notchback Ahmed body maintains the frontal rounded surface of the original hatchback Ahmed body (Ahmed, Ramm & Faltin 1984) but with a trunk attached to the rear body. It is worth noting that the wake asymmetry has been observed for the squareback Ahmed body (Grandemange, Cadot & Gohlke 2012;Grandemange, Gohlke & Cadot 2013a,b). Particularly, as observed in the experiment (Grandemange et al. 2012) and later confirmed in LES (Evstafyeva, Morgans & Dalla Longa 2017), the presence of the symmetry breaking was found to depend on the Reynolds number in the laminar regime. The wake asymmetry persisting to the turbulent regime observed by Grandemange et al. (2013a,b) showed bi-stability characterized by stochastic switches of two mirrored asymmetric states.
So far, the wake bi-stability of the squareback body has been investigated extensively. However, the notchback configuration allowing the flow reattachment to the trunk is different from wake separations of a squareback body. Therefore, the wake bi-stability behind a notchback body deserves further investigation. The wake, particularly in the separation and reattachment region of the notchback car, has been found to be sensitive to the Reynolds number (Gilhome, Saunders & Sheridan 2001). Besides, bi-stable wake states with separation bubbles attached to the slanted rear were observed behind a slanted cylinder afterbody (Zigunov, Sellappan & Alvi 2020). In this work, the bi-stability was only found at a lower Reynolds number, Re = 2.5 × 10 4 . With higher Reynolds numbers, the separation bubble length was reduced, and the bi-stable phenomenon was not reported. Therefore, it can be deduced that the Reynolds number influences the wake state with slanted rears.
Although the wake bi-stability behind slanted rears has been reported in the literature, the underlying mechanism of the Reynolds number influence remains unclear. For this reason, the present work aims to investigate the instability of the bi-stable wake behind a notchback bluff body under the Reynolds number influence by performing both experiments and numerical simulations. The authors believe that the results of the present paper bring new insight into the natural wake bi-stability and help promote the understanding of the notchback body flow.
The manuscript is organized as follows. Section 2 presents the description of the numerical and experiment methods. In § 3, the results are analysed with the focus on the Reynolds number influence on the state instability of the bi-stable wake. Conclusions follow in § 4. The mesh resolution and validation are presented in the Appendix.

Description of numerical simulations
The investigated geometry is a notchback Ahmed body. The wake of this body is expected to be bi-stable as previously observed by Sims-Williams et al. (2011). The model dimensions expressed in the body height, H = 0.096 m, are presented in figure 1. The body width is W = 1.35H, and the length is L = 3.82H. The afterbody is characterized as a notchback configuration, presenting a slant and a deck. The deck length is L D = 0.469H, and the deck height is H D = 0.687H. The effective backlight angle is β = 17.8 • , allowing the roof length L S = 2.847H. The body is suspended off the ground with a ground clearance of H C = 0.21H. The model is placed in the central width of a computation tunnel, under zero yaw angle, with the tunnel width W T = 11.46H and the height H T = 5.73H. The inlet is set as a uniform inlet velocity profile, located at 8H upstream of the model. The pressure outlet with a constant 0 Pa is set at 19H downstream of the model. The lateral surfaces and the roof are set as symmetry planes. The no-slip wall is applied to the surface of the model and the ground. The governing LES equations are solved with the commercial finite volume solver, Star CCM+ 2019.2. The subgrid eddies are modelled by the wall-adapting local eddy-viscosity (WALE) model (Nicoud & Ducros 1999). Convective fluxes are approximated by a blend of 98 % central difference scheme of second-order accuracy and 2 % upwind scheme. The time integration is done using the second-order accurate three-level time Euler scheme. The Reynolds-averaged Navier-Stokes (RANS) approach is first employed for the initial condition. An initial time t * = tU inf /H = 62 (t is the simulation time, U inf is the free stream velocity) corresponding to two flow-through passages through the domain is considered for the physical establishment of the flow for LES. Then, the data are sampled at the last of 12 iterations in each time step with a sampling duration of t * = 124. The time step set-up, mesh resolution and validation are presented in the Appendix. The numerical method used in the present work has been applied to the studied flow by the same authors in He et al. (2021a,b,c).

Experimental set-up
Experiments are carried out in the closed-circuit wind tunnel at Chalmers University of Technology. The flow turbulence level is within 0.15 % at a frequency range of 1 to 10 000 Hz. The tested model has a height H * = 2H = 0.192 m, giving the same shape as the model used in the simulation but twice as large. The model is mounted in a test section of 6.5H * × 9.4H * × 15.6H * (height × width × length) with a ground clearance of 0.21H * . The body is supported by four vertical cylinders with a diameter of 0.1H * .
The model is equipped with pressure taps on the rear body. The pressure is obtained using a Scanivalve system (NetScanner TM model 9116). This system has an accuracy of ±0.2 Pa for the used pressure range (±300 Pa) with a sampling frequency of 62.5 Hz. The hot-wire is mounted on a computer-controlled three-dimensional traversing mechanism to characterize the flow frequency. The hot-wire is connected to a constant temperature circuit (Dantec 56C01 CTA) with an over-heat ratio of 1.7. The velocity signal is filtered at p.d.f. a cutoff frequency of 20 000 Hz and digitized at a sampling frequency of 10 000 Hz, giving a reliable real frequency range between 5 and 5000 Hz.

Analysis and discussion
3.1. The flow state of bi-stability The flow state under the Reynolds number influence observed in the wind tunnel experiment is discussed in this section. For the notchback Ahmed body, the asymmetric flow reattachment on the deck leads to the pressure difference on the two sides (He et al. 2021a). To obtain the pressure signals, two monitoring points, P dl and P dr , are set on the deck (figure 2a). The distance between the central line and each point is dy = 0.417H * . The points are at D P2 = 0.5L * D from the deck's trailing edge, where L * D = 0.469H * is the deck length. Therefore, the asymmetry degree can be quantified by the deck pressure gradient, defined as where C p ( ) represents the sampled pressure coefficient on each monitoring point. By the definition, a higher absolute value of ∂C pd /∂y indicates a higher degree of wake asymmetry.  At the low Re region with Re ≤ 10 × 10 4 , ∂C pd /∂y concentrates at both negative and positive values. This means that the statistic wake undergoes two asymmetric states. In figure 2(c,d), for Re = 5 × 10 4 and Re = 10 × 10 4 , the wake bi-stability is characterized as ∂C pd /∂y stochastically switches between the two states, S A and S B . The two states are considered to be mirrored since the absolute values of ∂C pd /∂y during S A and S B are similar. Under the low Re, the higher amplitude of the ∂C pd /∂y fluctuation indicates a lower accuracy because the ratio of the measuring deviation to pressure signals is higher.
For 12.5 × 10 4 ≤ Re ≤ 17.5 × 10 4 , ∂C pd /∂y not only concentrates with negative and positive values but also in the centre region approaching zero. Thus, the wake seems to be 'tri-stable'. For example, in figure 2(e), under Re = 15 × 10 4 , ∂C pd /∂y undergoes S A , S B and the symmetric state, S C . As the Reynolds number increases further, the proportion of S C gradually increases. For Re ≥ 20 × 10 4 , the wake seems symmetrized as ∂C pd /∂y remains in the centre. Shown in figure 2( f ) is that S C becomes dominant among the three states. Looking at figure 2(a), for 5 × 10 4 ≤ Re ≤ 15 × 10 4 , the degree of the asymmetry gradually increases with the higher Reynolds number since the absolute values of the positive and negative peaks become higher. Furthermore, comparing figure 2(c-f ), it can be observed that the wake switches more frequently with a higher Re, increasing the proportion of S C .
Although ∂C pd /∂y presented in figure 2 can be analysed regarding the bi-stability, its fluctuation, particularly at the higher Re, interferes with the distinction between the wake states. When the signal of ∂C pd /∂y is filtered with an average filter over windows of 0.5 s, like examples shown in figure 3, the interference of the fluctuation reduces. Therefore, the wake states can be quantitatively depicted. When the wake is in the asymmetric state, ∂C pd /∂y corresponding to ±0.62 and ±0.8 are respectively observed for Re = 5 × 10 4 and Re = 20 × 10 4 . To distinguish the wake states, the dash lines at ∂C pd /∂y = ±0.31 (figure 3a) and ∂C pd /∂y = ±0.4 (figure 3b) are employed as the boundary lines.
Inspired by Grandemange et al. (2013a), the statistics of the bi-stable states are analysed by the probability distribution. For example, the conditional probability obtained under Re = 5 × 10 4 and Re = 20 × 10 4 is listed in table 1. At the low Re, S A and S B states are dominant, and the wake tends to maintain the current asymmetric state. However, at the high Re, the proportion of S C becomes higher, leading the S A and S B states to be more unstable. Table 1. Probabilities of the current wake state, S t , depending on the previous states, S t−1 . P(E 1 = E 2 ) is the conditional probability of the event E 1 , given by the event E 2 . The events are considered at 0.5 Hz. To study the instability of the wake states, the probability distribution of the wake switch is investigated. Assuming P switch = P(S t / = S t−1 ), S t−1 ∈ (S A , S B ) to be the switching rate independent of the instant t. Therefore, P switch = 0.0147 for Re = 5 × 10 4 and P switch = 0.0741 for Re = 20 × 10 4 can be given by the results listed in table 1 with (3.2) Note that the switching event considered in (3.2) considers the possibility of the asymmetry breaking, describing the wake shifting from an asymmetric state to S C or directly to the opposite asymmetric state. In most instances, the wake undergoes the S C state during the switching process. However, the wake shifting from the asymmetric state to S C does not ensure a successful switch to the opposite asymmetric state. Sometimes the wake returns from S C to the initial asymmetric state, presenting an attempt to switch. This phenomenon has been discussed by He et al. (2021a). Therefore, (3.2) counts both the successful and unsuccessful switches since the attempt to switch also breaks the asymmetric state. Taking into account the events considered at 0.5 Hz, the dimension of P switch can be considered as 'per 0.5 s'. Thus, the expected value of the duration (in seconds) for the wake state can be represented by the reciprocal of 2P switch . Assuming that when not switching, the wake is possibly in one of the asymmetric states or in the S C state. Thus, the expected value of the duration time for the asymmetric state needs to eliminate the interference of S C , given by Applying (3.3) for all tested Re, T m is a function of Re following an exponential law illustrated in figure 4. The T m for each Reynolds number is obtained from the pressure signals monitored over 1.8 × 10 3 s, using an averaging filter over windows of 0.5 s. The standard deviations of T m are calculated with window lengths of 3 × 10 2 s. It can be seen that T m decreases with higher Re. Moreover, under the low Re, the standard deviations are higher since the asymmetric state can be maintained possibly from a few seconds to tens of seconds. The duration time of the asymmetric state reduces with increasing Re symmetrizing the wake. Therefore, exploring the instability of the symmetric state requires an examination under small yaws. For the wake bi-stability behind the squareback body, the discontinuous Similarly, for the notchback case, the wake is sensitive to small yaw angles, showing in the experiment a phase jump around zero yaws. As shown in figure 5, under Re = 25 × 10 4 , the model placed at zero yaws presents the fluctuation of ∂C pd /∂y around 0, indicating wake symmetry. However, under positive or negative small yaw angles, ∂C pd /∂y stays respectively in positive or negative regions, suggesting asymmetry. Therefore, for the notchback body, the symmetry of the wake at high Re remains unstable. This means that the wake still has the bi-stable nature, being different from general symmetric flows of symmetric bluff bodies.

The flow structures
In order to identify the flow structures and explore the underlying mechanism of the switch depending on Re, numerical simulations using LES are performed at Re = 5 × 10 4 , Re = 15 × 10 4 and Re = 20 × 10 4 . The wake asymmetry is predicted in all three cases, but the wake switch is not observed due to the limitation of the simulations to simulate sufficiently long physical time. For example, for Re = 20 × 10 4 , the present experiment suggests that the expected period of the asymmetric state estimated by (3.3) is T m = 2.189 s. Although the numerical simulation considering the flow averaged for t * = 124 corresponding to four flow-through passages through the domain, the physical time t = t * H/U inf = 0.38 s is much shorter than T m . However, the wake bi-stability is observed since the case of Re = 5 × 10 4 is in the S B state, but the other two cases show the S A state due to the random appearance of asymmetric states. For clarity, the numerical results are presented in the same normalized state by mirroring S A to S B .

R6-8
On state instability of the bi-stable flow Re = 5 × 10 4 Re = 15 × 10 4 Re = 20 × 10 4 The distribution of the mean streamlines and the spanwise vorticity, Ω y , projected on the section planes, Y 1 and Y 2 , are presented in figure 6. The vorticity is normalized with U inf and H. The wake separation can be identified by Ω y , showing the vortex V c separating from the near-wall region of the roof, V b from the deck's trailing edge and V g from the bottom body. The flow reattaching to the deck is indicated in the streamlines. For quantitative analysis, L VC indicating the separation length of V c is defined as the distance between the roof and the local maximum x-coordinate on the contour line of Ω y = 3.7. The reattachment length L R is defined as the distance between the positive-negative transition point of near-wall Ω y and the deck's trailing edge. The asymmetry of the mean flow is predicted in the three cases. The readers are referred to He et al. (2021a) for details of the asymmetric flow structures behind the notchback Ahmed body.
Focusing on the notchback region, the flow structures projected on the horizontal Z 1 plane behind the rear slant are presented in figure 7. The distribution of the mean streamwise velocity,ū, normalized by U inf , is shown in figure 7(a-c). For the three cases, the flow structure V c deflecting to the left-hand side indicates asymmetry. Therefore, the C-pillar vortex on the left-hand side is disturbed by V c . On the other hand, the right C-pillar vortex, V r , extends further downstream. The similarity of theū field can be quantitated by the profile on the probe line, L p , illustrated in figure 7(j).
The wake dynamics is identified by the modal analysis using the proper orthogonal decomposition (POD). As originally proposed by Lumley (1970), and later introduced with the method of snapshots by Sirovich (1987), the POD method is based on the energy ranking of orthogonal structures predicted from a correlation matrix of the snapshots. A singular value decomposition (SVD) approach is used for the POD analysis. The POD method applied to the pressure snapshots in the Z 1 plane recognizes the modes of the flow structures based on the energy content. This approach has been successfully used for the studied flow in previous published works by the same author (He et al. 2021a,b). The mean  However, large differences are found in the distribution of turbulence kinetic energy (TKE) presented in figure 7(g-i). Under Re = 5 × 10 4 , TKE in the wake is lower. For the two cases with the higher Re, TKE increases sharply. The distinction of TKE is quantitated by the profiles shown in figure 7(k). Compared with the two high Re cases, TKE is much lower with the low Re. Therefore, the wake with a higher Re is more turbulent. For the wake bi-stability behind the squareback Ahmed body, literature has shown the sensitivity to the underbody (Barros et al. 2017) or the background (Burton et al. 2017) turbulence. Therefore, it can be deduced that the high turbulent wake for the notchback case leads to unsteadiness, resulting in the higher possibility to break the asymmetric state. For this reason, the wake switching frequency increases with the higher Re, culminating in the higher proportion of S C observed in the experiment.

Conclusions
The bi-stable flow past a notchback Ahmed body is investigated by wind tunnel experiments and LES. The Reynolds number influence on the wake instability is analysed. Experimental results suggest that for Re ≤ 10 × 10 4 , the wake is bi-stable with low-frequency switches. For 12.5 × 10 4 ≤ Re ≤ 17.5 × 10 4 , the wake becomes 'tri-stable' due to the presence of a symmetric state. With Re increasing further, the proportion of the symmetric state increases, symmetrizing the bi-stable flow.
The wake switching frequency is assessed by calculating the conditional probability of the wake states observed in experiments. A higher frequency of the switch between two mirrored states is found with increasing Re, leading the duration time of the asymmetric state to decrease following an exponential law. Therefore, the increasing proportion of the symmetric state under higher Re is attributed to highly frequent wake switches. The symmetric state at high Re still has the bi-stable feature since the unstable wake remains sensitive to small yaw angles approaching zero.
The wake asymmetry is confirmed in the LES at Re = 5 × 10 4 , Re = 15 × 10 4 and Re = 20 × 10 4 . The consistency of the asymmetric wake is indicated by the wake separation, the reattachment and the wake dynamics identified by POD. However, the turbulence level is found significantly higher in the two higher Re cases. Therefore, a higher turbulent wake is considered to trigger a higher possibility to break the asymmetric state, increasing the wake switching frequency, which in turn produces a higher proportion of the symmetric state.

Appendix. Mesh resolution and validation
The multi-block hexahedral conforming computational grids are built using the Pointwise grids generator. The mesh resolutions are presented in table 2. The applicability of these mesh resolutions for the studied flow, established by a grid independence examination, has been discussed in the previous work (He et al. 2021b). For the higher Re case, to keep the wall-normal resolution n + < 1 and fulfil the requirement of the streamwise resolution + s and the spanwise resolution + l , smaller computational cells are used, leading to a higher number of cells. The non-dimensional time step is dt * = t s U inf /H = 3.44 × 10 −3 for Re = 5 × 10 4 , dt * = 1.15 × 10 −3 for Re = 15 × 10 4 and dt * = 8.6 × 10 −4 for Re = 20 × 10 4 , allowing the Courant-Friedrichs-Lewy number for all simulations being lower than one in over 99 % of the cells during all time steps.
The present LES is validated by comparing it to the experimental results. For the experiment, the wake frequency is obtained from velocity signals sampled by a hot-wire. The measuring point is placed at 0.1H * behind the half-height of the C-pillar (the one which is not disturbed by the deflection of V c during the asymmetric state). The asymmetric wake state for sampling is identified by ∂C pd /∂y. The velocity and pressure signals are sampled throughout 2 s during the asymmetric state. For the LES, the frequency is obtained from the velocity magnitude signals monitored at the position following the hot-wire measurement.
The comparison of ∂C pd /∂y is presented in table 2. It can be seen that the pressure and the degree of the wake asymmetry obtained in LES are in good agreement with those observed in the experiment. Moreover, the comparison of the shedding frequency of the Number of cells Wall-normal Streamwise (fore and rear parts) Streamwise (middle body) Spanwise Re = 5 × 10 4 3.3 × 10 7 n + < 1 3 < + s < 28 3 < + s < 55 3 < + l < 25 Re = 15 × 10 4 5.8 × 10 7 n + < 1 3 < + s < 28 3 < + s < 55 3 < + l < 25 Re = 20 × 10 4 8.1 × 10 7 n + < 1 3 < + s < 28 3 < + s < 55 3 < + l < 25 Table 2. Spatial resolutions of the grids.  C-pillar vortex is presented in figure 8. The frequency spectrum is obtained by the fast Fourier transform. The non-dimensional frequency F + is the Strouhal number (St H * for experiments and St H for LES) normalized with U inf and the model height. The peak values of the normalized power spectral density (PSD) illustrate that the main frequency observed in the experiment is in accordance with that obtained in LES. Therefore, the accuracy of the LES is established by the experimental validation.