Near-field coherent structures in circular and fractal orifice jets

To investigate the influence of the orifice geometry on near-field coherent structures in a jet, Fourier proper orthogonal decomposition (Fourier-POD) is applied. Velocity and vorticity snapshots obtained from tomographic particle image velocimetry at the downstream distance of two equivalent orifice diameters are analyzed. Jets issuing from a circular orifice and from a fractal orifice are examined, where the fractal geometry is obtained from a repeating fractal pattern applied to a base square shape. While in the round jet energy is mostly contained at wave number $m=0$, associated to the characteristic Kelvin-Helmholtz vortex rings, in the fractal jet modal structures at the fundamental azimuthal wave number $m=4$ capture the largest amount of energy. In addition, energy is scattered across a wider range of wave numbers than in the round jet. The radial Fourier-POD profiles, however, are nearly insensitive to the orifice geometry, and collapse to a universal distribution when scaled with a characteristic radial length. A similar collapse was recently observed in POD analysis of turbulent structures in pipe flow. However, unlike in pipe flow, the azimuthal-to-radial aspect ratio of the Fourier-POD structures is not constant and varies greatly with the wave number. The second part of the paper focuses on the relationship between streamwise vorticity and streamwise velocity, to characterize the role of the orifice geometry on the lift-up mechanism recently found to be active in turbulent jets [P. Nogueira, A. Cavalieri, P. Jordan, and V. Jaunet, Large-scale streaky structures in turbulent jets, J. Fluid Mech. 873, 211 (2019)]. The averaging of the streamwise vorticity conditioned on intense positive fluctuations of streamwise velocity reveals a pair of vorticity structures of opposite sign flanking the conditioning point, inducing a radial flow towards the jet periphery. This pair of structures is observed in both jets, even if the azimuthal extent of this pattern is $30\%$ larger in the jet issuing from the circular orifice. The coupling between streamwise vorticity and velocity motions is also examined using Fourier-POD. The analysis reveals that in the jet with a circular orifice lower wave-number modes, corresponding to structures at larger scales, capture a larger fraction of the vorticity-velocity coupling. This evidences that the orifice geometry directly influences the interaction between velocity and vorticity.

Figure 1

The orifice geometries considered in this paper: (a) circular and (b) fractal. The red circle in panel (b) traces the outline of the circular orifice in panel (a).

Figure 2

Mean flow properties for both the round jet (red circles) and the fractal jet (cyan crosses): (a) spreading rate, (b) streamwise velocity decay, and (c) turbulence intensity.

Figure 3

In the first column, (a) mean streamwise velocity profiles at four different streamwise distances from the orifice, for the fractal (red continuous line) and round (cyan dashed line) orifices, and (b), (c) profiles of the streamwise and transversal rms velocity, respectively, for the two orifices at the same streamwise locations. In the second column, planar turbulent kinetic energy ${\overline{k}}^{*}\equiv \overline{(u_z^2+u_x^2)}/2$ normalized by the local streamwise velocity, for (d) the round jet and (e) the fractal jet.

Figure 4

(a, d) Maps of the mean streamwise velocity for (a) the round jet and (d) the fractal jet; (b, e) maps of the mean planar vorticity for (b) the round jet and (e) the fractal jet; (c) radial location of the maximum of ${\overline{\omega }}_{\perp }$ as a function of the azimuth $\theta$ for both jets; and (f) mean velocity components ${\overline{u}}_x$ and ${\overline{u}}_y$ on the $xy$ plane and map of the mean streamwise vorticity ${\overline{\omega }}_z$ for the fractal jet.

Figure 5

Maps of the root mean square (rms) of the (a, d) streamwise, (b, e) radial, and (c, f) azimuthal velocity components for (a–c) the round jet (top row) and (d–f) the fractal jet (bottom row).

Figure 6

Snapshots of the instantaneous vector fields containing the velocity components $u_x$ and $u_y$ on the $x\text{−}y$ plane and maps of the streamwise velocity fluctuations at $z_0/D_e=2$, for the (a–c) round jet and (d–f) fractal jet. Note that the velocity vector fields of the velocity components $u_x$ and $u_y$ overlap to the color maps of $u_z^{\prime }$, and they share the same Cartesian axes.

Figure 7

Relative energy distribution of the first three POD modes $i\in [1,3]$ at the first 11 azimuthal modes, $m\in [0,10]$, (a, b, c) for the round jet (left column) and (d, e, f) for the fractal jet (right column), at the downstream location of $z_0$. The POD analysis is made for the (a, b) streamwise (first row), (c, d) radial (second row), and (e, f) azimuthal (third row) components of the three-dimensional velocity vector field. The cumulative energy contents of the first and first five POD modes are represented by a cyan dashed line and a cyan continuous line, respectively. For the fractal orifice, the cumulative energy of the first $5(m+1)$ snapshot POD modes is reported as a thicker red line, to quantify suboptimality of the Fourier-POD analysis.

Figure 8

Relative energy contribution from the (a) streamwise, (b) radial, and (c) azimuthal velocity components to the energy contained within each of the first 30 wave numbers. Red circles represent the round jet and cyan crosses represent the fractal jet.

Figure 9

Streamwise component of the first 18 snapshot POD modes in the jet with fractal orifice, on an arbitrary color scale.

Figure 10

Radial profiles of the first three POD modes of the streamwise (first row), radial (second row), and azimuthal (third row) velocity components for azimuthal wave numbers $m=0,1,2,5,10$, for the round jet (left columns) and for the fractal jet (right column). The gray vertical lines indicate the radial location $r/D_e=0.5$.

Figure 11

Normalized radial peak location of the radial profiles for the first POD mode of the (a) streamwise, $R_z^p/D_e$, (b) radial, $R_r^p/D_e$, and (c) azimuthal, $R_{\theta }^p/D_e$, velocity components as a function of the azimuthal wave number $m$, for the round (red circles) and fractal (cyan crosses) orifices. The gray horizontal lines indicate the radial location $r/D_e=0.5$, i.e., the radial location of the edge of the round orifice.

Figure 12

Radial profiles of the streamwise (first row), radial (second row), and azimuthal (third row) velocity components of the first, $i=1$, second, $i=2$, and third, $i=3$, POD modes for the azimuthal wave numbers $m\in [2,40]$, for the round jet (red continuous line) and for the fractal jet (cyan dashed line).

Figure 13

Radial length scale of first Fourier-POD modes as a function of their azimuthal length scale for the (a) streamwise, (b) radial, and (c) azimuthal velocity components. Data points for wave numbers $m\in [1,40]$ are shown, moving from right to left for growing wave numbers, as shown by the arrows. The straight lines are used to estimate the aspect ratio of the eddies.

Figure 14

(a, b) Color maps of the normalized conditional average of the streamwise vorticity fluctuations, and labeled contours of the conditionally averaged streamwise velocity, for both (a) the round jet and (b) the fractal jet (see text for details on the conditioning procedure); (c, d) radial profiles at constant $r/D_e=0.5$ as a function of the azimuthal angle $\theta$ of (c) the streamwise vorticity and (d) the streamwise velocity, from the conditional averages presented in the maps in (a, b). The gray lines in panels (a) and (b) trace the nominal lip line location.

Figure 15

Relative energy distribution of the first three vector POD modes $i\in [1,3]$ at the first 21 azimuthal modes, $m\in [0,20]$, (a) for the round jet and (b) for the fractal jet. The Fourier-POD analysis is performed for the streamwise components of the vorticity and velocity fields. The cumulative energy contents of the first and first five POD modes are represented by a cyan dashed line and continuous line, respectively.

Figure 16

(a) Normalized radial peak location of the radial profiles of the streamwise vorticity component for the first vector POD mode; (b) normalized radial half size of the vorticity component of the first POD modes as a function of the wave number $m$; and (c) the quantity $\chi \left(m\right)$, defined in (6) as a function of $m$.

Figure 17

Radial profiles of the (a, b) streamwise vorticity and (c, d) streamwise velocity of the (a, c) first POD mode and (b, d) second POD mode, for the azimuthal wave numbers $m\in [5,40]$, for the round jet (red continuous line) and for the fractal jet (cyan dashed line). The gray vertical lines indicate the radial location $r/D_e=0.5$, i.e., the radial location of the edge of the round orifice.

Figure 18

Streamwise velocity of the three most energetic Fourier-POD modes, on an arbitrary color scale, and contours of the streamwise vorticity, with continuous lines identifying positive values and dashed lines identifying negative values, in the jet with (a, b, c) round orifice and with (d, e, f) fractal orifice.