Proper orthogonal decomposition analysis and modelling of the wake deviation behind a squareback Ahmed body

We investigate numerically the three-dimensional (3D) flow around a squareback Ahmed body at Reynolds number $\text{Re}={10}^4$. Proper orthogonal decomposition (POD) is applied to a symmetry-augmented database in order to describe and model the flow dynamics. Comparison with experiments at a higher Reynolds number in a plane section of the near wake at midheight shows that the simulation captures several features of the experimental flow, in particular the antisymmetric quasisteady deviation mode. 3D POD analysis allows us to classify the different physical processes in terms of mode contribution to the kinetic energy over the entire domain. It is found that the dominant fluctuating mode on the entire domain corresponds to the 3D quasisteady wake deviation, and that its amplitude is well estimated from 2D near-wake data. The next most energetic flow fluctuations consist of vortex shedding and bubble pumping mechanisms. It is found that the amplitude of the deviation is negatively correlated with the intensity of the vortex shedding in the spanwise direction and the suction drag coefficient. Finally, we find that despite the slow convergence of the decomposition, a POD-based low-dimensional model reproduces the dynamics of the wake deviation observed experimentally, as well as the main characteristics of the global modes identified in the simulation.

Figure 1

Numerical configuration. The streamwise, spanwise, and wall-normal positions are referred to as $x$, $y$, $z$. The length, width, and height of the body are respectively $L$, $W,$ and $H$. The origin of the axes (indicated in red) is taken at the top and foremost position of the Ahmed body in the vertical symmetry plane.

Figure 2

Streamwise velocity contours of the mean flow; left) horizontal plane at body midheight $z=-0.5$; right) vertical midplane at $y=0$.

Figure 3

Near-wake 2D POD spectrum in the simulation and experiment.

Figure 4

2D wake POD modes 1 to 4; top row: simulation; bottom row: experiment.

Figure 5

Amplitudes of 2D POD modes 1 to 4 (from top to bottom); (left) simulation; (right) experiment.

Figure 6

Histogram comparison of 2D POD modes 1 to 4 (from left to right); top row: simulation; bottom row: experiment.

Figure 7

2D wake POD modes 5 to 8; top row: simulation; bottom row: experiment.

Figure 8

2D POD amplitude power spectral density of the amplitudes in the numerical simulation $|{\widehat{a}}_n{|}^2$ for modes 3–8; the red and black dashed lines, respectively, correspond to the frequencies 0.08 and 0.2.

Figure 9

3D POD eigenvalue spectrum.

Figure 10

Streamwise velocity contours on body midheight plane $z=-0.5$; top: projection of an instantaneous field $\underline{u}$ onto the first ten modes $u_{\mathrm{recons}}={\sum }_{n=1}^{10}{a}_n\left(t\right)\underline{\varphi }\left(\underline{x}\right)$ where $a_n=\int \underline{u}(\underline{x},t).\underline{\varphi }\left(\underline{x}\right)d\underline{x}.$; bottom: difference field $|u-u_{\mathrm{recons}}|$.

Figure 11

Streamwise velocity contours of 3D POD modes 1 (top row) and 2 (bottom row); (left) horizontal section on body mid-height plane $z=-0.5$; (right) vertical section at $y=-0.4$.

Figure 12

(Left) amplitudes of the first two POD modes; (right) power spectral density of the quasisteady deviation mode $|{\widehat{a}}_2{|}^2$; the red and black lines, respectively, correspond to the two frequencies 0.08 and 0.2.

Figure 13

Streamlines of the field viewed from downstream (left) mode 1 $\sqrt{{\lambda }_1}{\underline{\varphi }}_1$; (right) combination of mode 1 and mode 2 $\sqrt{{\lambda }_1}{\underline{\varphi }}_1+\sqrt{{\lambda }_2}{\underline{\varphi }}_2$.

Figure 14

Streamwise velocity contours of 3D POD spatial modes 3–10 (from top to bottom); (left) horizontal section on body midheight plane $y=-0.5$; (right) vertical section at $z=-0.4$.

Figure 15

(Left) amplitudes of the modes $a_n$, $n=3,5,7,9$; (right) power spectral density of the 3D POD mode amplitudes in the simulation $|{\widehat{a}}_n{|}^2$; the red and black lines, respectively, correspond to the two frequencies 0.08 and 0.2.

Figure 16

(Left) energy of POD modes corresponding to quasisteady deviation $a_2^2$, spanwise ($r_{Kh}^2=a_3^2+a_4^2$) and vertical ($r_{Kv}^2=a_5^2+a_6^2$) vortex shedding intensities; vertical lines correspond to minima of $a_2^2$; (right) correlation coefficient between vortex shedding energy and quasisteady deviation energy. The largest value of the correlation coefficient is $-0.6$ at zero time delay and is related with spanwise vortex shedding.

Figure 17

(Left) base suction coefficient $C_B=-C_p$ obtained by integrating the pressure over the rear of the body; (right) correlation between the base suction coefficient and the POD amplitudes corresponding to the steady deviation ($a_2$, top), wake pumping mode ($a_7$, middle), and vortex shedding modes ($r_{Kh,v}^2$, bottom). The time delays corresponding to vertical lines in the figure correspond to a value of four time units for $a_2$ and $-1.5$ for $a_7$.

Figure 18

Power spectral density of the 3D POD mode amplitudes 3–10 $|{\widehat{a}}_n^{3D,M_0}{|}^2$ in the simplified model; the red and black lines, respectively, correspond to the two frequencies 0.08 and 0.2.

Figure 19

(Left) POD amplitude $a_2$; top: experiment (2D); bottom: model M (3D); (right) histogram of $a_2$.

Figure 20

(Left) amplitudes of 3D POD amplitudes of modes 3, 5, 7, and 9 in the model; (right) power spectral density of POD amplitudes in the model $|{\widehat{a}}_n^{3d,M}{|}^2$; the red and black lines, respectively, correspond to the two frequencies 0.08 and 0.2.