Proper orthogonal decomposition analysis of the large-scale dynamics of a round turbulent jet in counterflow

Although the mean flow features of the turbulent jet in counterflow have been studied in the past, the large-scale dynamics of this flow configuration remain unexplored. The present work presents the modal analysis, through proper orthogonal decomposition (POD), of large eddy simulations (LESs) of counterflowing jets at different jet-to-counterflow velocity ratios $(\alpha =2.2,3.4,5.1)$ reported in detail by Rovira, Engvall, and Duwig [Phys. Fluids 32, 045102 (2000)]. In the present study, a qualitative investigation of the three-dimensional (3D) turbulent structure of this jet configuration is performed by vortex identification. Additionally, a simplified description of the origin and development of these coherent structures is presented. Planar two-dimensional (2D) POD results for the case with $\alpha =3.4$ are directly compared and found to be in close agreement with the results available literature. In this case, over an equal time interval, temporal and spatial resolutions are observed to have a minor effect on mode energy content. All three cases are also analyzed with 3D POD and the evolution of the peak mode frequency with $\alpha$ is studied. Additionally, by employing a wavelet transform, the intermittent behavior of the fundamental mode dynamics is evidenced for the first time. Finally, the jet in counterflow case with $\alpha =5.1$ is analyzed with a 3D spectral POD. Varying jet penetration, precession, and a stretching and contracting motion are found to be the most dominant modes.

Figure 1

Cross-sectional view of the instantaneous axial velocity for the jet in counterflow at $\alpha =5.1$ along with boundary conditions.

Figure 2

Visualization of the instantaneous jet vortices visualized by isocontours of Q-criterion ($Q\sim 1.5\times {10}^4$ for the large-scale structures and $Q\sim 1.8\times {10}^5$ for small-scale structures) and colored by nondimensional axial velocity. The jet in counterflow depicted has $\alpha =5.1$. The time-averaged stagnation surface (isocontour of axial velocity zero) is represented by a gray transparent isocontour. Both the stagnation surface and the large-scale structures are clipped by the $y=0$ plane for visibility. Additionally, large-scale structures are clipped at $U_x/U_j<0.3$ which would otherwise overlap with the small-scale structures. The direction of the counterflow is also shown for reference.

Figure 3

Mean velocity (i.e., mode zero) and the first three spatial POD modes for the LES case at $\alpha =3.4$. The 2D POD is performed with two velocity components. It should be noted that the spatial domain shown here is smaller than the computational domain employed for the present 2D POD study. These coordinate limits and the accompanying visual representation was employed to replicate that employed by Bernero and Fiedler [6] to assist the reader in the comparison. In their work, Bernero and Fiedler [6] only present the time-averaged velocity and the first two modes.

Figure 4

Effect of the spatial sampling size and the dimensions for the POD study (2D or 3D) on the evolution of the normalised energy corresponding to each POD mode at $\alpha =3.4$. All LES results are studied with either 2D or 3D POD and have a temporal resolution of $f\approx 31$ Hz and $T=29$ s. They are compared with the 2D experimental data by Bernero and Fiedler [6] with $f\approx 25$ Hz and $T=16$ s. Here, $f$ is the sampling frequency, $\mathrm{\Delta }x/d$ is the spatial sampling size normalized with the nozzle diameter (note that $\mathrm{\Delta }x=\mathrm{\Delta }y$) and $T$ is the total sampling time.

Figure 5

Effect of the sampling frequency (left) and the sampling time (right) on the evolution of the normalised energy corresponding to each POD mode at $\alpha =3.4$. All LES results are studied with 2D POD and have a spatial resolution of $\mathrm{\Delta }x/d\approx 0.1$. They are compared with the 2D experimental data by Bernero and Fiedler [6] with $\mathrm{\Delta }x/d\approx 0.2$. Here, $f$ is the sampling frequency, $\mathrm{\Delta }x/d$ is the spatial sampling size normalized with the nozzle diameter (note that $\mathrm{\Delta }x=\mathrm{\Delta }y$) and $T$ is the total sampling time.

Figure 6

Wavelet transform for the first three POD modes with $f\approx 62$ Hz and $\mathrm{\Delta }x/d\approx 0.1$ for (a) 2D and (b) 3D POD at $\alpha =3.4$. The solid white lines represent the peak frequency for each mode reported by Bernero and Fiedler [6] while the white dashed lines represent the range of frequencies of the main peaks in the power spectra of the jet centroid for counterflowing jets over the range $2.9\le \alpha \le 5.1$ reported by Tsunoda and Takei [7]. Roman numerals highlight mode events that are discussed in the main text.

Figure 7

Evolution of the peak value of the Strouhal number for the first six POD modes for several cases of the jet in counterflow. Results for 2D POD by Bernero and Fiedler [6] and for 3D POD by Duwig and Revsted [8] are also included for comparison.

Figure 8

Wavelet transform of the first six SPOD modes with filter size $N_f=366$.

Figure 9

(Left) Reconstruction of the temporal evolution of the first SPOD mode visualized by the stagnation surface viewed on the $x\text{−}z$ plane. (Middle) Reconstruction of the temporal evolution of the second and third SPOD modes visualized by the stagnation surface viewed on the $y\text{−}z$ plane. (Right) Reconstruction of the temporal evolution of the fourth and fifth SPOD modes visualized by the stagnation surface viewed on the $y\text{−}z$ plane. The black arrows indicate the direction forwards in time and the black cross represents the jet centerline.

Figure 10

(Left) Phase portrait showcasing the relationship between the temporal coefficient of the second and third SPOD modes. The solid gray line represents the complete time period from $t_0=0$ s to $t_{\text{end}}=35.6$ s while the solid black line, which partially covers the gray line, depicts the period between $t_1=8.3\mathrm{s}$ and $t_2=13.9\mathrm{s}$. (Middle and right) Wavelet transform of the second and third SPOD modes. The vertical gray dashed lines highlight the time period when the most intense event (i.e., brightest spot) occurs which matches the period for which the black line in the phase diagram is drawn.