# Effects of multiscale geometry on the large-scale coherent structures of an axisymmetric turbulent jet

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### Abstract

In this study, the effect of multiscale geometry on the near-field structure of an axisymmetric turbulent jet is examined at a global Reynolds number of \(Re_\mathrm{G}=10{,}000\). With the aid of tomographic particle image velocimetry, the suppression of the coherent structures due to this fractal geometry is analysed and the changes to the near-field vorticity are evaluated. This particular geometry leads to the breakup of the azimuthal vortex rings present for round jets and to the formation of radial and streamwise opposite-signed patches of vorticity. The latter are found to be responsible for the axis switching of the jet, a phenomenon observed for some noncircular jets where the major axis shrinks and the minor one expands in the near field, effectively switching the two axes of the jet. This was the first time, to the knowledge of the authors, that axis switching has been observed for a jet where the coherent structures have been suppressed. Following the significant differences found in the near field, the far field is examined. There, the integral lengthscale of the large scale eddies \(\mathscr {L}_{ur}\) and the size of the jet evaluated in terms of the jet half-width \(r_{1/2}\) are found to evolve in a similar fashion, whilst the ratio \(\mathscr {L}_{ur}/r_{1/2}\) is found to be higher for the fractal jet than for the round jet, for which the near-field structures have not been suppressed.

### Graphical abstract

## Keywords

Axisymmetric jet Coherent structures Tomographic PIV Multiscale geometry Noncircular jet## 1 Introduction

## 2 Methodology

*x*,

*r*, \(\theta\)), as shown in Fig. 2a.

^{1}followed by two iterations of motion tracking enhancement (MTE) (Novara et al. 2010) to reduce the number of ghost particles and a final recursive direct correlation.

^{2}A summary of the processing steps together with the computational time is provided in Fig. 2b. Finally, to correct the non-zero divergence, the data were corrected using the divergence correction scheme (DCS) as in De Silva et al. (2013). This scheme is based on a non-linear optimisation-based constraint which minimally alters the acquired velocity field, whilst restricting the magnitude of the divergence to a maximum tolerance. The objective function to be minimised is the ensemble average of the “turbulent kinetic energy” added to the measured velocity field, i.e. \(\tilde{k} = \sum _{i=1}^{3} \langle (U_i - \overline{U}_i)\rangle\), where the velocity is split following the Reynolds decomposition as:

^{3}after a convergence study, which led to an average \(\tilde{k}/\overline{U}_\mathrm{cl}< 5\%\) (compensated by the jet streamwise centreline velocity).

Processing details for planar and tomographic PIV

LRPIV | HRPIV | TPIV | |
---|---|---|---|

Initial interrogation window | \(64\times 64\) pixels | \(64\times 64\) pixels | \(160\times 160\times 160\) voxels |

Final interrogation window | \(16\times 16\) pixels | \(12\times 12\) pixels | \(48\times 48\times 48\) voxels |

Overlap (%) | 50 | 50 | 75 |

Spatial resolution (worst case) | \(22 \eta\) | \(5.5\eta\) (far field) | \(11 \eta\) |

Digital Resolution (px/mm) | 16.7 | 28.0 | 45.7 |

## 3 Results

*a connected turbulent fluid mass with instantaneously phase-correlated vorticity over its spatial extent*. This motivated the acquisition of the full three-dimensional vorticity vector to study and evaluate the state of such structures. Despite the sole availability of 2D PIV data, Breda and Buxton (2017) suggested that the fractal geometry would lead to a breakup of the coherent structures for an axisymmetric jet, evidenced by the suppression of the negative anti-correlation in the two-point correlation between the radial velocity component

*v*in the streamwise direction

*x*, usually associated with the Kelvin–Helmholtz (K–H) vortex rings. In this study, we further investigate the state of the coherent structures by analysing the vorticity, calculated from the tomographic PIV data, in polar coordinates \(\omega = [\omega _r, \omega _\theta ,\omega _x]\). An initial evaluation can be done by looking at the time averaged mean vorticity in the near field across all time steps acquired:

This phenomenon has been found for various noncircular jets and it was explained through the contribution of \(\omega _\theta\) (Hussain and Husain 1989) and \(\omega _x\) (Zaman 1996). The former was mainly linked to the different induced velocities between the major and the minor axes across an elliptical vortex ring, leading to the faster segments curling up and protruding from the jet core. For the fractal jet at issue, however, this appears not to be the leading cause of axis switching due to the breakup of the azimuthal vorticity.

*u*in the radial direction

*r*. Due to the limited size of the field of view, the integration was performed between \(r=0\) and the first zero-crossing \(r^\prime\), which has been confirmed to be sufficient for an integral scale evaluation by Nicolaides et al. (2004).

## 4 Conclusions

In this study, the effects of suppressing the coherent structures with the aid of multiscale-fractal geometry have been investigated. It is found that the fractal geometry breaks up the coherent vortex rings as shown by the azimuthal vorticity and by the mean vorticity; however, it also re-orients the vorticity vector, increasing the magnitude of \(\omega _r\) and \(\omega _x\). The latter is found to play a key role for the axis switching of the fractal jet due to the opposite-signed patches of vorticity injecting fluid towards the jet centre. Despite the significant differences in the near field, it is observed that the eddies and the jet half-width evolve in a similar fashion in the far field. However, the suppression of coherence is still visible in the ratio between \(\mathscr {L}_{ur}\) and \(r_{1/2}\) in the far field, which is observed to be higher for the fractal jet.

## Footnotes

- 1.
The first recursive direct correlation was done in steps of specific correlation volume voxel size (passes): \(160\times 160\times 160\) (10) \(\rightarrow 96\times 96\times 96\) (2) \(\rightarrow 64\times 64 \times 64\) (2). The volume overlap was 75%.

- 2.
The second recursive direct correlation was: \(160\times 160\times 160\) (10) \(\rightarrow 96\times 96\times 96\) (2) \(\rightarrow 64\times 64 \times 64\) (2) \(\rightarrow 48 \times 48 \times 48\) (2). The volume overlap was 75%.

- 3.
The notation \(\tilde{\cdot }\) is used to denote the corrected velocity field for tomographic PIV after the DCS is applied. Later, it is dropped since the analysis on the 3D–3C data is based solely on the corrected fields.

## Notes

### Acknowledgements

The authors gratefully acknowledge EPSRC for funding through EPSRC Grant no. EP/L023520/1.

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