Effects of approach flow conditions on the unsteady three-dimensional wake structure of a square-back Ahmed body

This paper investigates the effects of approach flow conditions on the unsteady three-dimensional wake characteristics of a square-back Ahmed body using improved delayed detached eddy simulations. Two approach flow conditions were investigated: a reference uniform flow (case A) and a thick turbulent boundary layer (TBL) (case B), where the Ahmed body is fully submerged in the TBL, $\delta >H$, where $\delta$ is the boundary layer thickness and $H$ is the total height of the body from the ground. Case A is studied extensively in the literature, whereas case B is less studied. The present results showed that the wake structure of case B is significantly different from that of case A. In particular, the wake of case A is dominated by a strong downwash flow from the top surface, but case B exhibits a much stronger upwash flow, which is induced by lower momentum of the fluid that emanates from the ground clearance and the pressure imbalance generated behind the body. For case A, conditional averaging of the flow field based on the sign of the drift force demonstrated the occurrence of the well-known wake switching (bimodality) event in the spanwise direction. However, the bimodality was completely suppressed in the wake of the Ahmed body submerged in the TBL. Both time-averaged turbulence statistics such as the Reynolds stresses and production terms, and time-resolved statistics including spectral analysis and temporal cross correlations are used to explore the differences between the wake structure of the Ahmed body in a uniform flow and thick TBL.

Figure 1

Schematic drawing of the Ahmed body showing the flow nomenclature and test conditions investigated. Case A: uniform approach flow or thin turbulent boundary layer (TBL), $\delta \ll H$, and case B: thick approach TBL, $\delta >H$.

Figure 2

Mesh configuration for both cases A and B.

Figure 3

Approach flow condition specified at the inlet of the simulation for case B compared with the experimental results of Essel et al. [18]. (a) Streamwise mean velocity profile normalized in outer coordinates ($U_{\infty }$ and $d$) and logarithmic form (inset), $U^{+}=1/\kappa \left(y^{+}\right)+B$, where the von Kármán constant $\kappa =0.41$, and the intercept is set at $B=5.0$, and (b) streamwise turbulence intensity profile in outer coordinate. The dashed horizontal lines in (a,b) represent the location of the bluff body in the outer coordinates, whereas the vertical solid lines in the inset of (a) represent the location of the bluff body in the inner coordinates.

Figure 4

Comparison between present numerical results and previous experiments for case A (Rodriguez et al. [13]) and case B (Essel et al. [18]). Case A: Profiles of (a) streamwise mean velocity $U/U_b$, and (b) Reynolds shear stress $-\overline{u^{\prime }{v}^{\prime }}/U_b^2$. Case B: Profiles of (c) streamwise mean velocity $U/U_b$, (d) streamwise Reynolds normal stress $\overline{{u^{\prime }}^2}/U_b^2$, (e) wall-normal Reynolds normal stress $\overline{{v^{\prime }}^2}/U_b^2$, and (f) Reynolds shear stress $-\overline{u^{\prime }{v}^{\prime }}/U_b^2$. The profiles of $U/U_b$ and $-\overline{u^{\prime }{v}^{\prime }}/U_b^2$ for case A are offset at an interval of 1.0 and 0.04, respectively, while the profiles of $U/U_b, \overline{{u^{\prime }}^2}/U_b^2, \overline{{v^{\prime }}^2}/U_b^2$, and $-\overline{u^{\prime }{v}^{\prime }}/U_b^2$ for case B are offset at an interval of 1.0, 0.1, 0.05, and 0.03 respectively. Profiles of Rodriquez et al. [13] were obtained from personal communication with the authors.

Figure 5

Planes of interest. $P_1$ is the wall-normal symmetry $(z/h=0)$ plane and $P_2$ is the spanwise symmetry $(y/h=0.5)$ plane of the Ahmed body. The cross-sectional $(y-z)$ planes are denoted as $P_3, P_4$, and $P_5$ with offset distances of $x/h=0.8$,1.4, and 2.0, respectively, from the rear end of the body.

Figure 6

Contours of the streamwise mean velocity at $P_1$ (a), (b) and $P_2$ (c), (d) for case A and case B, respectively. The red line represents the reverse flow region $(U<0)$ defined by the zero streamwise mean velocity contour line. The white dots denote the saddle points.

Figure 7

Contours of wall-normal mean velocity in $P_3$ (a), (d), $P_4$ (b), (e), and $P_5$ (c), (f) for case A and case B, respectively. Superimposed on the contours are the mean streamlines. The dashed black lines represent the rear surface of the Ahmed body.

Figure 8

Contours of spanwise mean velocity in $P_3$ (a), (d), $P_4$ (b), (e), and $P_5$ (c), (f) for case A and case B, respectively. Superimposed on the contours are the mean streamlines. The dashed black lines represent the rear surface of the Ahmed body.

Figure 9

Isosurfaces of mean pressure coefficient $C_p\approx -0.22$ colored by streamwise mean velocity in the wake of (a) case A and (b) case B.

Figure 10

Distributions of (a) local maximum mean velocity $U_m/U_{\infty }$, (b) velocity difference $\mathrm{\Delta }U/U_{\infty }$, and vorticity thickness (c) ${\delta }_{\omega ,z}/h$, and (d) ${\delta }_{\omega ,y}/h$.

Figure 11

Contours of Reynolds stresses in $P_1$. Streamwise Reynolds normal stress $\overline{{u^{\prime }}^2}/U_b^2$ (a), (b), wall-normal Reynolds normal stress, $\overline{{v^{\prime }}^2}/U_b^2$ (c), (d), spanwise Reynolds normal stress $\overline{{w^{\prime }}^2}/U_b^2$ (e), (f), and Reynolds shear stress $-\overline{u^{\prime }{v}^{\prime }}/U_b^2$ (e), (f), for case A and case B, respectively. The red line represents the reverse flow region $(U<0)$ defined by the zero streamwise mean velocity contour line.

Figure 12

Contours of Reynolds normal stresses in $P_4$. Streamwise Reynolds normal stress $\overline{{u^{\prime }}^2}/U_b^2$ (a), (d), wall-normal Reynolds normal stress $\overline{{v^{\prime }}^2}/U_b^2$ (b), (e), and spanwise Reynolds normal stress $\overline{{w^{\prime }}^2}/U_b^2$ (c), (f), for case A and case B, respectively. The dashed black lines represent the rear surface of the Ahmed body.

Figure 13

Contours of Reynolds shear stresses $-\overline{u^{\prime }{v}^{\prime }}/U_b^2$ (a), (c), and $-\overline{u^{\prime }{v}^{\prime }}/U_b^2$ (b), (d), in $P_4$ for case A and case B, respectively. The dashed black lines represent the rear surface of the Ahmed body.

Figure 14

Distributions of the maximum (a) streamwise, (b) wall-normal, and (c) spanwise Reynolds normal stresses and the corresponding Reynolds stress anisotropy, (d) $b_{11}$, (d) $b_{22}$, and (d) $b_{33}$, in the near wake of case A and case B. The vertical dashed line denotes the $L_r/h$.

Figure 15

Distributions of maximum Reynolds shear stresses, (a) $|\overline{u^{\prime }{v}^{\prime }}|/U_b^2$ and (b) $|\overline{u^{\prime }{w}^{\prime }}|/U_b^2$, and the Reynolds stress anisotropy, (c) $b_{12}$ and (d) $b_{13}$, in the near wake of case A and case B. The vertical dashed line denotes the $L_r/h$.

Figure 16

Production terms of the transport equations for the Reynolds normal stresses and turbulent kinetic energy. $P_{uu}h/U_b^3$ (a), (c), $P_{vv}h/U_b^3$ (b), (d), $P_{ww}h/U_b^3$ (e), (h), and $P_{k}h/U_b^3$ (f), (h), for case A and case B, respectively. The red line represents the reverse flow region defined by the zero streamwise mean velocity contour line.

Figure 17

Isosurfaces of ${\lambda }_2$ colored by instantaneous streamwise velocity: (a), (b) 3D vortical structures based on ${\lambda }_2{h}^2/U_b^2=-5$, and (c)–(h) planar views of the vortical structures based on ${\lambda }_2{h}^2/U_b^2=-50$.

Figure 18

Temporal histories of drift (a), (b) and lift (c), (d) coefficients along with their corresponding normalized probability density function (PDF) and joint-PDF (e), (f). The time interval of wake switching event is outlined in (a) as $t_{\mathrm{switch}}$.

Figure 19

Contours of $C_p$ on rear surface (a)–(c) and spanwise mean velocity $W/U_b$ in $P_2$ with streamlines superimposed (d)–(f) for case A. Conditional averaging based on the P state (a), (d) and the N state (b), (e). Total ensemble averaging based on both P and N states (c), (f). The red line represents the reverse flow region defined by the zero streamwise mean velocity contour line.

Figure 20

Instantaneous snapshots during the interval of bimodal wake switching event indicated in Fig. 18 as $t_{\mathrm{switch}}^{}$. (a)–(e) Snapshots of instantaneous pressure coefficient contour in $P_2$, and (f)–(j) isosurfaces of instantaneous pressure coefficient $C_p=-0.3$.

Figure 21

Premultiplied frequency spectra profiles of the fluctuations of the (a) lift coefficient and (b) drift coefficient, and (c) temporal cross correlation between the lift and drift coefficients.