Relationship between the base pressure and the velocity in the near-wake of an Ahmed body

We investigate the near-wake flow of an Ahmed body which is characterized by switches between two asymmetric states that are mirrors of each other in the spanwise direction. The work focuses on the relationship between the base pressure distribution and the near-wake velocity field. Using direct numerical simulation obtained at a Reynolds number of ${10}^4$ based on incoming velocity and body height as well as Bonnavion and Cadot's experiment [G. Bonnavion and O. Cadot, J. Fluid Mech. 854, 196 (2018)], we perform proper orthogonal decomposition (POD) of the base pressure field. The signature of the switches is given by the amplitude of the most energetic, antisymmetric POD mode. However, switches are also characterized by a global base suction decrease, as well as deformations in both vertical and lateral directions, which all correspond to large-scale symmetric modes. Most of the base suction reduction is due to the two most energetic symmetric modes. Using the linear stochastic estimation technique of Podvin et al. [Exp. Fluids 59, 58 (2018)], we show that the large scales of the near-wake velocity field can be recovered to some extent from the base pressure modes. Conversely, it is found that the dominant pressure modes and the base suction fluctuation can be well estimated from the POD velocity modes of the near-wake.

Figure 1

POD pressure spectrum in the simulation and in the experiment [27].

Figure 2

POD pressure modes. Left: experiment and right: simulation.

Figure 3

Evolution of the normalized pressure POD amplitudes in the simulation (left) and the experiment (right): (top row) modes 1 and 2; (bottom row) modes 3 and 4. A moving average of ten convective time units was used.

Figure 4

Comparison of the joint histograms of the POD amplitudes $a_1^{pn}$ and $a_k^{pn}$ for $k=2,3,4$. The color contours correspond to the numerical data. The dashed line contours correspond to the same levels for the experiment. The contour levels are the following: (a) $k=2$: 0.003, 0.005, 0.01, 0.02, 0.03; (b) $k=3$: 0.0025, 0.005, 0.01, 0.024; (c) $k=4$: 0.002, 0.005, 0.01, 0.015, 0.02, 0.025. A moving average of 10 convective time units was used.

Figure 5

Reconstruction of the mean pressure coefficient distribution at the base $c_p(y,z)$ using the symmetrized mean and the first four POD modes based on the characteristic amplitudes $a_n^{p,eq}$ and $a_n^{p,s}$: (left) quasistationary state $C_p+\sqrt{{\lambda }_1^p}{\varphi }_1^p+{\sum }_{n=2}^4\sqrt{{\lambda }_n^p}{a}_n^{p,eq}{\varphi }_n^p$; (center) switch state $C_p+{\sum }_{n=2}^4\sqrt{{\lambda }_n^p}{a}_n^{p,s}{\varphi }_n^p$; (right) symmetric quasistationary state $C_p-\sqrt{{\lambda }_1^p}{\varphi }_1^p+{\sum }_{n=2}^4\sqrt{{\lambda }_n^p}{a}_n^{p,eq}{\varphi }_n^p$.

Figure 6

(Left) Relative contribution of the POD modes to the variations of the base pressure coefficient (see text). (Right) Contribution of the modes corresponding to the characteristic states $a_n^{p,eq}$ and $a_n^{p,s}$.

Figure 7

Top left: mean flow in the midvertical plane. The vertical dashed line represents the cross section on which the streamwise component of the POD modes is represented in the right two columns. The black contour corresponds to a zero mean velocity.

Figure 8

Velocity modes computed over the near-wake volume (delimited with the thick black line) and extended to the full domain, viewed in a horizontal plane at midheight.

Figure 9

(a), (b) Velocity mode contribution to first POD mode $M_{1n}^{vp}$ (a) and second POD mode $M_{2n}^{vp}$ (b). (c), (d) Velocity POD amplitudes for the first (c) and second mode (d). The top row corresponds to the exact coefficient $a_n^v$. The bottom row corresponds to the estimated coefficient $a_n^{v,e}$. The dotted line represents the instantaneous signal, and the red thick line to the signal filtered with a moving average of five time units. The vertical dashed line corresponds to the limit of the POD snapshot acquisition. The three red dotted lines correspond to the times selected for reconstruction in Fig. 11.

Figure 10

(a), (b) Influence coefficients $M_{9n}^{vp}$ (a) and $M_{10n}^{vp}$ (b). (c), (d) Velocity POD amplitudes for the ninth (c) and tenth mode (d). Legend as in Fig. 9.

Figure 11

Streamwise velocity field at three instants $t=300,385,495$ materialized by the vertical dotted lines in Figs. 9 and 10: (top row) DNS field; (middle row) field projected on the first ten POD velocity modes; (bottom row) field estimated from the first ten POD pressure modes.

Figure 12

Base pressure POD coefficients estimated from the near-wake velocity field. The top row of each subfigure corresponds to the exact coefficient $a_n^p$. The bottom row corresponds to the estimated coefficient $a_n^{p,e}$. The dotted line represents the instantaneous coefficient and the red thick line to it being filtered with a moving average of five time units. The dashed vertical line corresponds to the limit of the POD snapshot acquisition. (a) $n=1,$ (b) $n=2,$ (c) $n=3,$ and (d) $n=4$.

Figure 13

Pressure contribution to second velocity mode $M_{2n}^{pv}$ (a) and third velocity mode $M_{3n}^{pv}$ (b).

Figure 14

Base drag coefficient: (top) DNS, (middle) reconstruction based on pressure modes 2 and 3, and (bottom) estimation based on POD velocity modes. The dashed vertical line corresponds to the limit of the POD snapshot acquisition.