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Journal of Visualization

, Volume 21, Issue 4, pp 525–532 | Cite as

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

  • Massimiliano Breda
  • Oliver R. H. Buxton
Regular Paper

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

Previous studies have investigated the role of the coherent structures (the so-called Kelvin–Helmholtz vortex rings) in an axisymmetric round jet. As shown by Yule (1978) and Dimotakis et al. (1983), vortex rings are emitted at the exit of a round jet and these large-scale structures are important for both entrainment and mixing. Since the 1980s, various authors have tried to modify the state of these coherent structures to obtain entrainment and mixing benefits. It was found that both large wetted perimeter (Shakouchi and Iriyama 2014) and a combination of corners and flat sides (Gutmark et al. 1989) would allow the augmentation of both. Hence, a repeating fractal pattern (fractal dimension \(D_\mathrm{f}=1.5\) and 3 iterations) is proposed, which maximises the number of corners and wetted perimeter for a given open area. The same geometry has been studied for axisymmetric wake generators (Nedić et al. 2013), where a breakup of the coherent structures and a reduction of the shedding energy were observed. We therefore examine round and fractal orifices to investigate the effects on the near and far field due to the different embedded coherence with the aid of planar and tomographic particle image velocimetry (PIV). The repetitive fractal pattern is applied to a base square geometry, leading to a different aspect ratio \(A\!\!\!R\), defined as
$$\begin{aligned} A\!\!\!R= \frac{\text {Major axis}}{\text {Minor axis}} . \end{aligned}$$
(1)

2 Methodology

The experiments are conducted in a jet facility at Imperial College London. The air flow is conditioned with a honeycomb and two screens of decreasing mesh size before it enters a \(5\mathrm{th}\) order polynomial contraction leading to a turbulence intensity of \(0.5\%\) at the centreline. Two orifices of identical open area (\(D_\mathrm{e}^2\)), as shown in Fig. 1, were attached to the jet exit. The flow was evaluated using a cylindrical coordinate system (x, r, \(\theta\)), as shown in Fig. 2a.
Fig. 1

Exit geometries studied, major axis and minor axis are shown

Initially, a low-resolution two-dimensional two-component (2D–2C) planar particle image velocimetry (LRPIV) was conducted across the \(x-r\) plane between 0 and 23 \(D_\mathrm{e}\), placing the laser sheet across the jet’s centreline, where \(D_\mathrm{e}\) is the equivalent diameter of the orifices and \(U_\mathrm{e}\) the equivalent velocity
$$\begin{aligned} D_\mathrm{e} = \sqrt{\text {Exit area}}= 15.8 \, \mathrm{mm} \quad U_\mathrm{e} = \frac{\text {Volumetric flow rate}}{\text {Exit area}} =9.93\,\mathrm{m}/\mathrm{s} . \end{aligned}$$
(2)
For the fractal jet, tests were conducted across both major and minor geometrical axes, as shown in Fig. 1. Henceforth, “major” and “minor” axes will specifically refer to the planes aligned with their respective axes at the jet exit. An Imager SX 4M camera was used for the acquisition. In total, 1500 images were acquired per jet configuration. Subsequently, a high resolution planar PIV (HRPIV) experiment was conducted between 23 and 26 \(x/D_\mathrm{e}\), to study the effect of the exit geometries on the turbulent lengthscales of the jet in the far field. This time, 4500 images were acquired using the same camera. Following the findings from these two experiments, 3D–3C tomographic PIV (TPIV) (Elsinga and Scarano 2006) was conducted in a slice of the jet of thickness \(5\,\mathrm{mm}\), perpendicular to the \(r-\theta\) plane, with 1500 images acquired (\(U_x\) is the out-of-plane component), imaging the flow in forward scatter using four cameras Imager SX 4M. All experiments were conducted at a global Reynolds number of \(Re_\mathrm{G}=10{,}000\), where
$$\begin{aligned} Re_\mathrm{G} = \frac{U_\mathrm{e} D_\mathrm{e}}{\nu }. \end{aligned}$$
(3)
The laser sheet (or volume for the TPIV case) was produced in both experiments by a low-speed double cavity Nd:YAG laser at \(15\,\mathrm{Hz}\), whilst the image processing was done in the commercial software DaVis 8.4. A summary of the processing steps of the experiments is detailed in Table 1.
Fig. 2

a Jet’s schematics and position of TPIV volume. b Processing time of a single TPIV time step in DaVis. The computer has two processors, Intel(R) Xeon(R) CPU ES-2690 v2 at 3 GHz and 128 GB of RAM

The TPIV images were processed using the MART algorithm (6 iterations), a multiplicative algebraic reconstruction (Herman and Lent 1976). To improve the quality of the reconstruction, the dark background previously acquired was subtracted. Sliding minimum subtraction, Gaussian smoothing using a \(3 \times 3\) pixels kernel (to reduce the background noise) and sharpening (to reduce the number of ghost particles) were also performed on the raw images, allowing the particle diameter to be of the order of 2 pixels and the ppp to be within 0.04 and 0.05. Volume self-calibration (Wieneke 2008) was then applied to the reconstructed particle-image volumes to improve the data accuracy. To reduce the postprocessing time, sparse cross-correlation (Discetti and Astarita 2012) was implemented. The vectors were reconstructed in three steps: an initial recursive direct correlation,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:
$$\begin{aligned} U_i = \overline{U}_i + u_i. \end{aligned}$$
(4)
The divergence error was set to be \(|\partial \tilde{u}_i /\partial x_i \le 10 s^{-1}|\)3 after a convergence study, which led to an average \(\tilde{k}/\overline{U}_\mathrm{cl}< 5\%\) (compensated by the jet streamwise centreline velocity).
Finally, the experiments’ spatial resolution was evaluated by estimating the dissipation rate \(\varepsilon\) using all nine velocity gradients (George and Hussein 1991) from the TPIV data and deriving the Kolmogorov lengthscale \(\eta = (\nu ^3/\varepsilon )^{1/4}\). Additionally, an HRPIV test was conducted in the near field, \(2 \le x/D_\mathrm{e} \le 6\), to estimate the dissipation rate since a spatial resolution, in terms of cross-correlation window, that is twice as fine as that of the TPIV achieved. The resolution of this HRPIV data was estimated to be between \(6-9\eta\) (worst case). The dissipation rate estimated from the TPIV data set and the HRPIV data set, using the assumption of local axisymmetry (George and Hussein 1991), was found to differ by up to 50% due to the coarse spatial resolution of the TPIV. In fact, both estimates of dissipation are likely to be underestimates since a resolution of \(\sim 3\eta\) is considered necessary to resolve the dissipative scales of a turbulent flow (Worth et al. 2010; Buxton et al. 2011). Nevertheless, in terms of the longitudinal Taylor lengthscale \(\lambda _\mathrm{f}\), the spatial resolution of the TPIV was always less than \(0.6 \lambda _\mathrm{f}\), which is deemed more than sufficient to resolve the large-scale vortical structures, which are the focus of this study.
Table 1

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

Since the work of Brown and Roshko (1974), coherent structures have attracted numerous studies; however, the terminology “coherent structures” itself did not receive a formal definition until the work of Hussain (1986): 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:
$$\begin{aligned} |\overline{\omega }| = \overline{\sqrt{\omega _r^2 + \omega _\theta ^2 + \omega _x^2}}, \end{aligned}$$
(5)
compensated by the jet streamwise centreline velocity \(\overline{U}_\mathrm{cl}\) and jet half-width \(r_{1/2}\) defined as:
$$\begin{aligned} \overline{U}_x (x,r_{1/2}) = 0.5 \overline{U}_\mathrm{cl}(x). \end{aligned}$$
(6)
For an axisymmetric round jet, it is expected that the coherent structures will appear as a coherent region of vorticity surrounding the jet core, due to the development of the K–H vortex rings. This is presented in Fig. 3. For the fractal jet, the multiscale geometry generates isolated areas of high absolute vorticity, even though it appears there is still a largely coherent ring of vorticity around its core, as shown in Fig. 3b. However, to complete the analysis on the state of the coherent structures, the single components of vorticity need to be analysed. In fact, as discussed by Violato and Scarano (2011), modifying the state of coherence using a Chevron nozzle led to a reduction of the azimuthal vorticity \(\omega _\theta\) and to an increase of the importance of the radial \(\omega _r\) and streamwise \(\omega _x\) components. Hence, the orientation of the vorticity vector is now evaluated to verify that the large-scale vorticity remains coherent across at least one component.
Fig. 3

Mean vorticity \(|\overline{\omega }|\) at \(x/D_\mathrm{e} = 2\); the exit geometry is shown in grey (TPIV data). The major axis (black dashed line) and the minor axis (white dashed line) are shown for the fractal jet

The first component to be investigated is \(\omega _\theta\), which is the dominant one of the K–H vortex rings for a round jet. As shown by the contour of mean \(\omega _\theta\) in Fig. 4a, the coherence of the round jet is shown by a well-defined ring of vorticity. Further, it is confirmed that most of the contribution to the overall mean vorticity is due to the azimuthal vorticity, since the radial and streamwise components are negligible in comparison, as shown in Figs. 5a and 6a
$$\begin{aligned} |\omega _\theta | \gg |\omega _r| \approx |\omega _x| \approx 0. \end{aligned}$$
(7)
Having shown a clear example of coherent vorticity, the fractal jet is now evaluated. Differently from the round jet, this time the azimuthal vorticity is found to be split into patches of strong vorticity as shown in Fig. 4b, which are divided by the fractal geometry.
Fig. 4

Mean azimuthal vorticity \(\omega _\theta\) at \(x/D_\mathrm{e} = 2\) (TPIV data)

The suppression of coherence leads to the formation of opposite-signed radial patches of vorticity as shown in Fig. 5b, with magnitude significantly higher compared to the round jet. These patches are found to form uniformly around the core of the fractal jet. Patches of opposite-signed streamwise vorticity, of lower intensity compared to the radial component, are also observed in Fig. 6b
$$\begin{aligned} |\omega _\theta | \simeq |\omega _r| \gtrsim |\omega _x|. \end{aligned}$$
(8)
As shown, none of the vorticity vector components show a coherent behaviour, since the azimuthal vorticity is split into patches, whilst the streamwise and radial ones are divided into areas of opposite sign. Hence, following the definition of Hussain (1986), no large-scale coherent structures can be found in the mean, suggesting the multiscale geometry is responsible for suppressing them. Of particular interest is the distribution of \(\omega _x\) and \(\omega _\theta\), which have been found before to promote “axis switching”. This phenomenon, where the major axis of the noncircular jet shrinks as the minor one expands, has been previously reported for square jets (Quinn 1992; Zaman 1996), but it has not been observed for jets where the coherence was suppressed due to, for example, the fractal geometry. A first observation can be drawn from Fig. 3b, where it is observed that the regions of high vorticity across the major axis are moving towards the jet centre compared to the exit geometry, while the ones across the minor axis are expanding away from the core. In Fig. 8a, axis switching is identified too at \(x/D_\mathrm{e}=2.3\), at which the jet half-width imaged across the major axis for the fractal jet becomes smaller than the one at the minor axis.
Fig. 5

Mean radial vorticity \(\omega _r\) at \(x/D_\mathrm{e} = 2\) (TPIV data)

Fig. 6

Mean streamwise vorticity \(\omega _x\) at \(x/D_\mathrm{e} = 2\). The opposite-signed vortex regions leading to the axis switching are shown by the arrows (TPIV data)

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.

The study of Zaman (1996), who tested rectangular jets fitted with tabs alternatively on the major or on the minor axes, suggested that axis switching would take place only if the noncircular geometry emitted regions of opposite-signed streamwise vorticity with the signs as in Fig. 7b, where flow would be injected towards the jet centre, thereby shrinking the major axis. However, if the signs of the streamwise vorticity were as in Fig. 7a, the flow would be ejected towards the ambient and therefore axis switching would not occur. As shown in Fig. 6b, the fractal geometry generates regions of opposite-signed streamwise vorticity, whose direction makes them inject fluid towards the jet centre, indicating that for this jet the \(\omega _x\) dynamics is likely responsible for the axis switching. \(\omega _x\) does not exclude a relevant contribution of \(\omega _\theta\) too, since its asymmetrical distribution could also be responsible for this phenomenon. However, the streamwise vorticity certainly plays the key role of shrinking the major axis, allowing this phenomenon to take place.
Fig. 7

Schematics of the jets fitted with tabs from Zaman (1996)

Despite the significant differences in the near field, it remains to be clarified if the coherent structures have any residual effect in the far field, where the jet is fully developed. Since the work of Townsend (1976), it has been believed that flows would achieve self-similarity, becoming independent of the initial conditions (i.e. independent from the state of the near-field coherent structures), due to the entrainment of background fluid into the turbulent stream. George (1989) suggested even that the initial conditions could affect the spreading rate of the jet, but not the self-similar profiles. To further investigate the influence of the coherent structures in the far field, the region between 23 and 26 \(x/D_\mathrm{e}\) is examined, where the mean velocity and Reynolds stresses are found to be self-similar. We do so by evaluating the integral lengthscale \(\mathscr {L}_{ur}\), calculated at the jet centreline from the jet two-point correlation \(R_{ur}\) of the streamwise velocity fluctuations 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).
$$\begin{aligned} R_{ur} (x,r, \psi )= & {} \frac{ \langle u(x,r) u(x, r+ \psi ) \rangle }{u^{\prime } (x,r) u^{\prime } (x,r+ \psi ) },\end{aligned}$$
(9)
$$\begin{aligned} \mathscr {L}_{ur} (x)= & {} \int _{0}^{r^\prime } R_{ur}(x,\psi ) \mathrm {d} \psi . \end{aligned}$$
(10)
The integral lengthscale \(\mathscr {L}_{ur}\), which represents the size of the large-scale turbulent eddies, is compensated by the jet half-width \(r_{1/2}\), which can be interpreted as the macroscopic lengthscale of the jet as a whole.
As shown in Fig. 8b, the ratio \(\mathscr {L}_{ur}/r_{1/2}\) is a constant for both jets [verifying an assumption made in George (1989)], indicating that the large-scale eddies and the whole jet are evolving in a similar fashion to one another. However, this constant is larger for the fractal jet. This suggests that traces of the suppression of the coherent structures can be found in the far field, even though the jet is evolving independently of the initial conditions.
Fig. 8

a Axis switching in the near field (LRPIV data) and b \(\mathscr {L}_{ur}/r_{1/2}\) in the far field (HRPIV data). The location of the axes switch is shown by the green dashed line. A marker is plotted every ten points to avoid clutter

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. 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. 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. 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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Copyright information

© The Visualization Society of Japan 2018

Authors and Affiliations

  1. 1.Department of AeronauticsImperial College LondonLondonUK

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