3.1. Quadrant analysis and azimuthal vorticity distribution

Horseshoe vortices are closely related to the Reynolds shear stress in wall-bounded turbulence. In fact, one of the main criteria to detect horseshoe vortices involves determining the mean turbulent flow field about a point where the flow makes a strong contribution to the mean Reynolds shear stress (Adrian Reference Adrian2007). Therefore, it seems reasonable to inspect the Reynolds shear stress, $\overline {u_r u_z}$, in the turbulent jet. Here $u_r(=U_r-\overline {U_r})$ and $u_z(=U_z-\overline {U_z})$ denote the radial and streamwise fluctuating velocity components, respectively, and the overline indicates ensemble-averaged quantities. The contour map of Reynolds shear stress normalized by the centreline velocity, $\overline {u_r u_z}/U_{cl}^2$, against the normalized radial distance from the jet centreline, $r/d$, and the normalized axial distance from the jet nozzle, $z/d$, is displayed in figure 1. The locus of the maxima of the Reynolds shear stress is marked by the solid line, while the dashed lines correspond to the normalized shear stress level $\overline {u_r u_z}/U_{cl}^2=0.002$. The latter can be regarded as the indicator of the boundary between the shear and potential regions; the potential core is visible in the axial range $z/d=0-5$. Note that the criterion used to indicate the boundary between the shear and potential regions is not critical in our analysis, and any other indicator such as turbulence kinetic energy level or vorticity magnitude could be used instead. Nowhere in the jet field is the Reynolds shear stress negative, implying that, more often than not, $u_r$ and $u_z$ are of similar signs; that is, $u_r$ and $u_z$ are positively correlated in the turbulent jet. A quadrant analysis (Wallace, Eckelmann & Brodkey Reference Wallace, Eckelmann and Brodkey1972), which sorts the $u_r$ and $u_z$ fluctuations based on their sign, and presents their probability on a $u_r$–$u_z$ plane, can shed light on the contribution of various turbulent events to the Reynolds shear stress in the round jet.

Figure 1. The normalized Reynolds shear stress, $\overline {u_r u_z}/U_{cl}^2$, as a function of $z/d$ and $r/d$. The solid line marks the locus of maxima of $\overline {u_r u_z}/U_{cl}^2$ while the dotted lines indicate the contour level $\overline {u_r u_z}/U_{cl}^2=0.002$.

In the quadrant analysis, the $u_r$–$u_z$ plane is divided into 4 quadrants: $Q1$ where $u_r>0$ and $u_z>0$, $Q2$ where $u_r>0$ and $u_z<0$, $Q3$ where $u_r<0$ and $u_z<0$ and $Q4$ where ${u_r<0}$ and $u_z>0$. Here, a modified version of the quadrant analysis will be presented, which provides a better spatial understanding of these turbulent events (Kirchner et al. Reference Kirchner, Elliott and Dutton2020). To ensure that only the strong events associated with the large-scale horseshoe vortices are included in the analysis, first a three-dimensional (3-D) elliptic high-pass filter with the cutoff wavelengths $(\lambda _{r_c}, \lambda _{s_c}, \lambda _{z_c})=d\times (0.1,0.05,0.05)$ is applied to the flow fields. This filter attenuates fluid structures with wavelengths smaller than the cutoff wavelengths in the radial, azimuthal and axial directions. Then, a threshold, $H_Q=0.0004$, is chosen such that only the events that satisfy $| u_r u_z |/U_{cl}^2>H_Q$ are sorted into $Q1$–$Q4$ bins. This is the concept of the ‘hyperbolic hole’ first introduced by Willmarth & Lu (Reference Willmarth and Lu1972). The sensitivity of the quadrant analysis to the threshold was tested by comparing $Q1/Q3-1$ contours for $H_Q=0.0001$, 0.0004 and 0.0016 and no difference was observed. Given that $\overline {u_r u_z}>0$ everywhere, $Q1$ and $Q3$ events are dominant in the jet; therefore, our focus will be on $Q1$ and $Q3$ events. These are displayed in figure 2(a–d) where contribution of $Q1$ and $Q3$ events to the jet turbulence against $r/d$ and $z/d$ in the near and intermediate fields are plotted as contour maps. Here, the solid and dashed lines are the same as those in figure 1. The $Q1$ and $Q3$ events outside the shear region are not taken into account as they do not contribute to the horseshoe vortex structures. It is evident that each of $Q1$ and $Q3$ events constitute between $30\,\%$ and $45\,\%$ of the total turbulent events. Furthermore, $Q3$ events dominate the outer edge of the shear region. For a better comparison of $Q1$ and $Q3$ events, contour maps of $Q1/Q3-1$ are plotted in figure 2(e,f), which illustrate that $Q3$ events are considerably more frequent than $Q1$ events in the outer part of the shear region in the jet near field, while their dominance is less noticeable in the outer part of the shear region in the intermediate field. On the other hand, $Q1$ events exhibit a clear dominance near the potential core in the jet near field. Contributions of $Q1$ and $Q3$ events appear to be virtually equal below the locus of the maxima of the Reynolds shear stress (solid line in figure 2f) in the jet intermediate field. It is stressed that the dominance of either $Q1$ or $Q3$ events in the near field is more conspicuous than that of $Q3$ events in the intermediate field. As it will be shown, this difference between the near and intermediate fields affects the conditionally averaged horseshoe vortices in these domains.

Figure 2. Percentage contribution of (a,b) $Q1$ and (c,d) $Q3$ events to $\overline {u_r u_z}$. (e,f) Comparison of contributions of $Q1$ and $Q3$ events in the form of $Q1/Q3-1$. Panels (a), (c) and (e) are associated with the near field ($z/d=0-5$), while panels (b), (d) and (f) correspond to the intermediate field ($z/d=15-25$). The solid and dotted lines are same as those in figure 1. The open circle and square symbols in panel (e) indicate the location of conditional vortices displayed in figures 7 and 8, respectively. The open triangle and the filled star symbols in panel (f) indicate the location of conditional vortices illustrated in figures 9 and 12, respectively.

Another parameter that is closely associated with the horseshoe vortices in turbulent flows is the spanwise vorticity in wall-bounded flows, which is equivalent to the azimuthal vorticity ($\omega _\theta$) in the axisymmetric jet. Elsinga et al. (Reference Elsinga, Adrian, Van Oudheusden and Scarano2010) and Dennis & Nickels (Reference Dennis and Nickels2011) selected the signed spanwise swirl ($\lambda _{ci,span} \cdot \mathrm {sign}(\omega _{span})$) as their conditional averaging event in wall-bounded turbulence. The spanwise swirl ($\lambda _{ci,span}$) is the characteristic swirl of a horseshoe vortex head; while it has no sign on its own, it can be signed using the spanwise vorticity via $\mathrm {sign}(\omega _{span})$. Similarly, Kirchner et al. (Reference Kirchner, Elliott and Dutton2020) adopted the negative component of the signed azimuthal swirl as the signature event for conditional averaging of the hairpin structures in the supersonic wake flow. The signed azimuthal swirl is closely related to the azimuthal vorticity (Kirchner et al. Reference Kirchner, Elliott and Dutton2020); therefore, it is crucial to determine whether positive azimuthal vorticity ($\omega _{\theta,p}$) or negative azimuthal vorticity ($\omega _{\theta,n}$) events prevail in different regions of the jet flow. Since only the $\omega _\theta$ events that correspond to the large-scale horseshoe vortices are of interest, a threshold $H_{\omega _\theta }=0.05$ is adopted such that $\omega _{\theta,p}/\omega _{\theta _{max}}>H_{\omega _\theta }$ and $\omega _{\theta,n}/\omega _{\theta _{max}}<-H_{\omega _\theta }$ need to be satisfied. Here, $\omega _{\theta _{max}}$ is the maximum of ${\omega _\theta }$ in the flow field. Figure 3 presents contour maps of $(\omega _{\theta,p}-\omega _{\theta,n})/\omega _{\theta,pmax}$ in which $\omega _{\theta,pmax}$ is the maximum of $\omega _{\theta,p}$. Similar to figure 2, the dashed lines indicate the shear region borders. In the near field, $\omega _{\theta,p}$ events are more frequent in the shear region close to the borders with the potential core and potential outer layer, while $\omega _{\theta,n}$ events prevail in the core of the shear region. In the intermediate field, $\omega _{\theta,p}$ events are predominant closer to the edge of the shear region, while $\omega _{\theta,n}$ events predominate in the region closer to the jet centreline. It is noted that adopting other values for $H_{\omega _\theta }$ in the range 0.05–0.1 does not affect conclusions drawn.

Figure 3. Comparison of the positive azimuthal vorticity, $\omega _{\theta,p}$ and negative azimuthal vorticity, $\omega _{\theta,n}$, contribution to the total $\omega _\theta$ events presented as $(\omega _{\theta,p}-\omega _{\theta,n})/\omega _{\theta,pmax}$. (a) Near field. (b) Intermediate field. The lines and symbols are same as those in figure 2.

3.2. Horseshoe vortices

Horseshoe vortices and their association with ejection ($Q2$) and sweep ($Q4$) events have been well studied in wall-bounded flows. It was shown in § 3.1 that $Q1$ and $Q3$ events dominate the turbulent fluctuations in the round jet. Here, we present sample instantaneous flow visualizations to demonstrate the correspondence between vortical structures and $Q1$ and $Q3$ events, and also the rationale behind the conditional averaging procedure used for the statistical analysis of vortical structures later. The swirling strength, $\lambda _{ci}$, will be used to visualize the vortical structures in the present study. The swirling strength criterion, introduced by Zhou et al. (Reference Zhou, Adrian, Balachandar and Kendall1999), uses the imaginary part of the complex eigenvalue of the velocity gradient tensor to identify vortical structures in the flow. Examples of typical 3-D instantaneous vortical structures (grey isosurfaces) in the jet near field, visualized by isosurfaces of $\lambda _{ci}$, are displayed in figure 4(a) and 4(c). Overlaid on the $\lambda _{ci}$ isosurfaces are the $Q1$ and $Q3$ events, which are represented by red and cyan isosurfaces, respectively. These latter isosurfaces are defined by some constant values of $u_r u_z>0$, which define $Q1$ and $Q3$ events. Here, the mean velocities have been subtracted from all velocity components to reveal the fluctuating velocity fields.

Figure 4. Example near field instantaneous snapshots where the swirling strength isosurfaces with a surface level of $\lambda _{ci}=3.5U_{cl}/d$ are overlaid on the $Q1$ events (red isosurfaces with the surface level $\overline {u_z u_r}=0.02 U_{cl}^2$) and $Q3$ events (cyan isosurfaces with the surface level $\overline {u_z u_r}=0.01 U_{cl}^2$). (a) Outside view of the shear region, and (b) corresponding 2-D velocity vector field at $x/d=0$. (c) Inside view of the shear region, and (d) corresponding 2-D velocity vector field at $x/d=-0.18$.

Figure 4(a) presents an instantaneous fluctuating flow field in the jet near field, viewed from the outer side of the shear region. Several arch-like vortical structures are visible in this snapshot. The heads of these arch-like structures are located farther away from the centreline, while their legs are closer to the jet centreline, i.e. ‘upright’ horseshoe vortices. Although some of these vortical structures are complete horseshoe-like vortices (with two visible legs), the majority of them are asymmetric and they usually appear to have one leg only. It can be seen that the majority of vortical structures surround $Q1$ isosurfaces, while the cyan $Q3$ isosurfaces are located on both sides of the vortical structures. Moreover, three horseshoe-like vortices labelled as V1, V2 and V3, form a group, inducing between their legs a long $Q1$ in the $z$ direction. To gain a better understanding of the link between the vortical structures and the $Q1$ streak, figure 4(b) presents the corresponding velocity vectors at $x/d=0$, after subtracting a convection velocity (the local mean velocity). The velocity vectors in the $z$–$y$ plane (figure 4b) display three swirling regions, marked by blue circles, which correspond to the heads of the horseshoe vortices V1, V2 and V3. An induced $Q1$ region is visible on the left-hand side of the horseshoe vortices’ heads in figure 4(b). Ejection events similar to those reported in wall-bounded flows appear to be induced by the horseshoe vortices in the jet flow. This mechanism in the jet involves ejections of high-speed fluid from the jet centreline to the low-speed regions away from the centreline. This is in contrast to the ejection in wall-bounded flows where low-speed fluid is ejected to the high-speed regions away from the wall (Robinson Reference Robinson1991). Recently, Nogueira et al. (Reference Nogueira, Cavalieri, Jordan and Jaunet2019) and Pickering et al. (Reference Pickering, Rigas, Nogueira, Cavalieri, Schmidt and Colonius2020) reported such events in the near field of a round jet and referred to them as lift-up motions, drawing an analogy between these events in the jet near field and the lift-up motion near the wall in wall-bounded turbulence (Kline et al. Reference Kline, Reynolds, Schraub and Runstadler1967). Given that these motions in the jet seem to be induced by a number of horseshoe-like vortices, which are angled in the streamwise direction, rather than elongated streamwise vortices associated with the lift-up motions in wall-bounded flows, sweep and ejection mechanisms, which are associated with horseshoe vortices, seem to better explain these events in the jet. The statistical evidence for the link between the horseshoe-like vortices and streaky structures in the jet near field will be provided in § 3.3.

Figure 4(c) presents an example 3-D instantaneous flow field in the jet near field, viewed from the inner side of the shear region (near the potential core); several horseshoe-like vortical structures are visible hugging $Q3$ events (cyan isosurfaces) in this snapshot. Contrast this with the voritcal structures in the outer edge of the shear region, where they hug $Q1$ events. Heads of the horseshoe vortices located near the potential core seem to be positioned close to the jet centreline, while their legs are farther away from the centreline; i.e. these are ‘inverted’ horseshoe vortices. Three horseshoe-like vortices are labelled as V1, V2 and V3 in figure 4(c); $Q3$ events (cyan isosurfaces) are visible between their legs. Moreover, $Q1$ events (red isosurfaces) are present near the heads of V2 and V3 on the side that faces the centreline. To further demonstrate the correlation between the $Q3$ events and the inverted horseshoe vortices, figure 4(d) presents the corresponding velocity vectors, after subtracting a convection velocity (local mean velocity), in the 2-D streamwise plane $x/d=-0.18$. The signatures of horseshoe vortices V1, V2 and V3 are marked in the 2-D velocity vector field. One can see that a $Q3$ region and a $Q1$ region are induced on the left-hand side and right-hand side of each of these horseshoe heads, respectively.

Vortical structures contribute to the $Q1$ and $Q3$ events in the jet intermediate field as well. An example 3-D instantaneous snapshot of the vortical structures (grey isosurfaces) together with the $Q1$ (red isosurfaces) and $Q3$ (cyan isosurfaces) events in the streamwise range $z/d=16-22$ is displayed in figure 5. Vortical structures in the jet intermediate field resemble upright horseshoe vortices, which were visible in the outer edge of the shear region in the jet near field displayed in figure 4(a) as well. They also appear to coincide with the $Q1$ regions. Moreover, very long $Q1$ and $Q3$ regions are present in the flow, stretching from $z/d=16$ to $z/d=22$. Samie et al. (Reference Samie, Lavoie and Pollard2021) referred to these structures as VLSMs and proposed that they are formed by the concatenation of large-scale horseshoe vortices; this is compatible with the instantaneous snapshot in which long high-speed regions are formed by the grouping of horseshoe vortices in the streamwise direction. They also showed that the VLSMs are statistically significant structures in the turbulent jet.

Figure 5. Example intermediate field instantaneous snapshot where the swirling strength isosurfaces (grey isosurfaces) with a surface level of $\lambda _{ci}=1.5U_{cl}/d$ are overlaid on the $Q1$ events (red isosurfaces with the surface level $\overline {u_z u_r}=0.03 U_{cl}^2$) and $Q3$ events (cyan isosurfaces with the surface level $\overline {u_z u_r}=0.02 U_{cl}^2$).

Statistical evidence for the presence of upright and inverted horseshoe vortices in the near and intermediate fields of the turbulent round jet is provided by conditionally averaged vortical structures. To illustrate the event used for conditional averaging, schematics of upright and inverted horseshoe vortices are presented in figures 6(a) and 6(b), respectively. These schematic vortices are inspired from the instantaneous vortical structures as well as the quadrant analysis and azimuthal vorticity distribution. The quadrant analysis revealed that $Q3$ events dominate the outer side of the shear region in both the near and intermediate fields, while $Q1$ events are dominant in the shear region close to the potential core in the near field. In all of these regions, the azimuthal vorticity, $\omega _{\theta }$, is overwhelmingly positive. Consistent with the azimuthal vorticity analysis, heads of both the upright and inverted schematic vortices are associated with positive azimuthal vorticity. The upright horseshoe vortex is accompanied by an induced $Q1$ event located between its legs and head, and an induced $Q3$ event above its head. In contrast, the region between the head and legs of the inverted horseshoe vortex is occupied by an induced $Q3$ event, while an induced $Q1$ event is located outside of the horseshoe vortex. It is noted that the appearance of $Q3$ ($Q1$) events outside of upright (inverted) horseshoe vortices in the instantaneous flow fields, such as those illustrated in figure 4, depends on the chosen level of isosurface. For example, in figure 4(c), if the level of isosurface is reduced, larger regions of $Q1$ events will appear. With the current isosurface levels one can still see $Q1$ regions (red isosurfaces) adjacent to the head of V2 and V3 close to the potential core in figure 4(c). Since the dominance of $Q1$ and $Q3$ events is observed near the edges of the shear region, and the azimuthal vorticity is overwhelmingly positive in these locations, $\omega _\theta >0$ is used as the condition event, and the shear region edges are considered for determining conditional vortical structures. Moreover, the head of the horseshoe vortex is sought, so the symmetry condition $u_\theta =0$, or more practically $\lvert u_\theta \rvert /u_{\theta,max}<\epsilon$, is used in addition to the positive azimuthal vorticity condition. Here, $u_\theta$ is the azimuthal fluctuating velocity, $u_{\theta,max}$ is its maximum in a snapshot and $\epsilon$ is a very small quantity; $\epsilon =0.01$ was used in the present study. Note that the conditional averaging criterion mentioned above is expected to yield a horseshoe-like vortex only in the regions where either $Q1$ or $Q3$ events are dominant. It will be shown that, in the shear regions where these events are in balance, more conditions are required to select between the upright or inverted horseshoe vortices.

Figure 6. Schematic of horseshoe vortices (grey isosurface) together with the induced $Q1$ (red isosurface) and $Q3$ events (cyan isosurface). (a) Upright horseshoe vortex and (b) inverted horseshoe vortex. The vertical dashed line represents the jet centreline.

The conditions $\omega _\theta >0$ and $\lvert u_\theta \rvert <0.01u_{\theta,max}$ were applied at $(r_{ref}/d,z_{ref}/d)=(0.33,2.7)$ (marked with an open circle symbol in figures 2e and 3a), to obtain a velocity field around the reference point $(r_{ref}, z_{ref})$ satisfying the condition event as

(3.1)\begin{align} &\tilde{u}_i ({\rm \Delta} r, {\rm \Delta} \theta, {\rm \Delta} z) \nonumber\\ &\quad=\langle u_i(r_{ref}+{\rm \Delta} r, {\rm \Delta} \theta, z_{ref}+{\rm \Delta} z) \,|\, \omega_\theta (r_{ref}, z_{ref})>0\ \&\ \lvert u_\theta (r_{ref}, z_{ref}) \rvert <0.01u_{\theta,max} \rangle, \end{align}

where angled brackets indicate spatial and temporal averaging, tilde signifies conditional features and $i \in \{r,\theta,z\}$. The swirling strength was then computed for the conditional field. The isosurface $\lambda _{ci}=0.22\lambda _{ci,max}$ is used to visualize the vortical structure as displayed in figure 7. Further, a high-speed (positive $u_z$) and a low-speed (negative $u_z$) region are highlighted in the figure, using the isosurface levels $\tilde {u}_z=0.5 \tilde {u}_{z,max}$ and $\tilde {u}_z=0.7 \tilde {u}_{z,min}$, respectively. It is reiterated that the event seeks the points in the flow that correspond to a positive azimuthal vorticity and a negligible azimuthal velocity, i.e. part of a straight vortex tube that is parallel with the $x\unicode{x2013}y$ cross-stream planes. The fact that the resulting averaged vortex tube is part of a horseshoe-like vortex implies that horseshoe vortex structures are statistically significant features at the location of conditional averaging. This is in addition to the presence of such structures in the instantaneous flow field illustrated earlier. As expected from the instantaneous flow field observations, the conditional vortical structure at $(r_{ref}/d,z_{ref}/d)=(0.33,2.7)$ (which is a location in the near field shear region near the potential core) is an inverted horseshoe vortex. The leg of the conditional horseshoe vortex is at a $25^{\circ }$ angle with respect to the jet centreline, and the distance between the legs, based on the in-plane vector field illustrated in figure 7(c), is approximately $0.14d$. This is estimated by measuring the distance between the centres of two counter-rotating streamwise vortices. The in-plane velocity vectors in figure 7(b) reveal a region of zero radial velocity below the horseshoe vortex head in the domain ${\rm \Delta} z/d=2.5\unicode{x2013}2.7$ and ${\rm \Delta} y/d=0.18\unicode{x2013}0.22$, while above the horseshoe vortex head in the region ${\rm \Delta} z/d>2.7$, a positive radial mean velocity is observed. This is compatible with the positive mean radial velocity near the jet centreline in round jets, and demonstrates how inverted horseshoe structures contribute to the mixing of the potential core with the turbulent shear region in the jet near field by inducing large-scale transport of fluid mass from the potential core to the turbulent shear region. The presence of such inverted horseshoe-like vortices near the outer edge of the shear region results in detrainment, i.e. transport of fluid from the turbulent shear region to the non-turbulent region.

Figure 7. Conditional horseshoe vortex with its head located at $(r_{ref}/d,z_{ref}/d)=(0.33,2.7)$. The red and cyan isosurfaces correspond to the conditional high-speed and low-speed regions, respectively. (a) Isometric view. (b) Side view with the vector field located at ${\rm \Delta} x/d=0$. (c) Top view with the vector field located at ${\rm \Delta} z/d=2.67$. Grey isosurface: $\lambda _{ci}=0.22 \lambda _{ci,max}$. Red isosurfaces: $\tilde {u}_z=0.5 \tilde {u}_{z,max}$. Cyan isosurface: $\tilde {u}_z=0.7 \tilde {u}_{z,min}$. Events used for conditional averaging are $\omega _\theta >0$ and $\lvert u_\theta \rvert <0.01u_{\theta,max}$.

The same events used earlier (i.e. $\omega _\theta >0$ and $\lvert u_\theta \rvert <0.01u_{\theta,max}$) were used to conditionally average the 3-D velocity field at $(r_{ref}/d,z_{ref}/d)=(0.82,2.7)$. This location is marked with open square symbols in figures 2(e) and 3(a), where $Q3$ events dominate $Q1$ events, and positive azimuthal vorticity events are more frequent than negative counterparts. The conditional horseshoe vortex associated with this event is displayed in figure 8 visualized by the swirling strength isosurface $\lambda _{ci}=0.22 \lambda _{ci,max}$. Consistent with the instantaneous vortical structures observed in the outer edge of the jet near field shear region, the conditional vortical structure is an upright horseshoe vortex. The leg of this horseshoe vortex is at a $50^{\circ }$ angle with respect to the jet centreline, and the distance between the legs, based on the in-plane vector field illustrated in figure 8(c), is approximately $0.27d$. Aside from the high-speed region between the head and legs of the conditional horseshoe vortex and the low-speed region above its head, there are two nearly symmetrical low-speed regions on both sides of the horseshoe vortex legs (shown in figure 8c only). These low-speed events are consistent with those on both sides of the vortical structures in the instantaneous flow field presented in figure 4(a). A stagnation line is visible in the in-plane velocity vector field shown in figure 8(b) above the horseshoe vortex head in the streamwise range ${\rm \Delta} z/d>2.7$. However, in the streamwise range ${\rm \Delta} z/d<2.7$, the mean radial velocity is negative. From this velocity vector field one can conclude that a net flow of fluid from outside of the shear region is induced by the horseshoe vortex into the shear region. This scenario is consistent with the negative mean radial velocity at this location and the positive entrainment of irrotational fluid from outside of the shear region into the turbulent jet. This implies that the large-scale vortical structures are tied to the entrainment in the turbulent jet as suggested by Philip & Marusic (Reference Philip and Marusic2012).

Figure 8. Conditional horseshoe vortex with its head located at $(r_{ref}/d,z_{ref}/d)=(0.82,2.7)$. The red and cyan isosurfaces show the conditional high-speed and low-speed regions, respectively. (a) Isometric view. (b) Side view with the vector field at ${\rm \Delta} x/d=0$. (c) Top view with the vector field at ${\rm \Delta} z/d=2.8$. Grey isosurface: $\lambda _{ci}=0.22 \lambda _{ci,max}$. Red isosurface: $\tilde {u}_z=0.5 \tilde {u}_{z,max}$. Cyan isosurfaces: $\tilde {u}_z=0.7 \tilde {u}_{z,min}$ in panels (a) and (b) and $\tilde {u}_z=0.3 \tilde {u}_{z,min}$ in panel (c). Events used for conditional averaging are $\omega _\theta >0$ and $\lvert u_\theta \rvert <0.01u_{\theta,max}$.

The same events used earlier (i.e. $\omega _\theta >0$ and $\lvert u_\theta \rvert <0.01u_{\theta,max}$) were used to conditionally average the 3-D velocity field at $(r_{ref}/d,z_{ref}/d)=(2.1,17.4)$ in the jet intermediate field. This location is marked with an open triangle symbol in figures 2(f) and 3(b). The conditional vortical structure associated with this event is displayed in figure 9 using the swirling strength isosurface $\lambda _{ci}=0.12 \lambda _{ci,max}$. This is an upright horseshoe vortex, the leg of which is at an approximately $40^{\circ }$ angle with respect to the jet centreline. Similar to the conditional horseshoe vortex at the outer edge of the shear region in the jet near field, the conditional horseshoe vortex in the intermediate field induces a negative mean radial velocity, contributing to the entrainment of the irrotational fluid into the turbulent shear region. This is because of the stagnation line in the streamwise range ${\rm \Delta} z/d>17.3$ (above the hairpin vortex head), and a negative mean radial velocity in the streamwise range ${\rm \Delta} z/d<17.3$ (below the horseshoe vortex head), which are visible in figure 9(b). Based on the velocity vectors in the $z\unicode{x2013}y$ plane, the induced negative radial velocity due to the horseshoe vortex in the intermediate field is noticeably smaller than that corresponding to the horseshoe vortex in the jet near field.

Figure 9. Conditional horseshoe vortex with its head located at $(r_{ref}/d,z_{ref}/d)=(2.1,17.4)$. The red and cyan isosurfaces correspond to the conditional high-speed and low-speed regions, respectively. (a) Isometric view. (b) Side view with the vector field at ${\rm \Delta} x/d=0$. (c) Top view with the vector field at ${\rm \Delta} z/d=17.5$. Grey isosurface: $\lambda _{ci}=0.12 \lambda _{ci,max}$. Red isosurfaces: $\tilde {u}_z=0.6 \tilde {u}_{z,max}$. Cyan isosurfaces: $\tilde {u}_z=0.8 \tilde {u}_{z,min}$ (in panels a and b) and $\tilde {u}_z=0.2 \tilde {u}_{z,min}$ in panel (c). Events used for conditional averaging are $\omega _\theta >0$ and $\lvert u_\theta \rvert <0.01u_{\theta,max}$.

The conditional horseshoe vortices at various streamwise points near the edge of the boundary between the shear region and the potential core in the jet near field were calculated, and their evolution is presented in figure 10. The dashed line represents the jet centreline, while the dotted line denotes the boundary between the shear region and the potential core. It is noted that these conditional vortical structures have been calculated separately, and presented in a single plot for comparison. The conditional vortical structures in this region are inverted horseshoe vortices, and their legs are at a $30^{\circ}\unicode{x2013}45^{\circ }$ angle with respect to the jet centreline. Moreover, their size appears to remain virtually unchanged with $z/d$. This trend is consistent with the nearly constant jet half-radius $r_{0.5}$, implying that these large-scale horseshoe vortices probably scale on $r_{0.5}$. The jet half-radius is determined as the radial location at which the mean axial velocity is equal to half of the mean axial velocity at the centreline.

Figure 10. Evolution of the conditionally averaged horseshoe vortex close to the potential core in the near field. (a) Isometric view. (b) Side view. (c) Top view. Isosurfaces correspond to $\lambda _{ci}=0.22 \lambda _{ci,max}$. The dashed line represents the jet centreline, while the dotted line denotes the boundary between the shear region and potential core.

The evolution of horseshoe vortices near the outer edge of the shear region in the jet intermediate field is illustrated in figure 11. Similar to those presented in figure 10, the conditional horseshoe vortices here have been calculated separately, and presented in a single plot for comparison. The $x$ and $y$ axes in figures 11(a) and 11(b) are normalized with the jet diameter $d$, while the jet half-radius $r_{0.5}$ is used to normalize the $x$ and $y$ axes in figure 11(c). Evidently, the conditional vortical structures in the jet intermediate field are upright horseshoe vortices and, moving along the streamwise direction, their azimuthal extent increases proportional to $r_{0.5}$. Legs of these horseshoe vortices are at an angle in the range $35$–$45^{\circ }$ with respect to the jet centreline. Comparing the conditional horseshoe vortices in the intermediate field with those in the near field, it is noted that they distinctly differ in shape. Specifically, while the near field conditional vortical structures are horseshoe-shaped structures, those in the intermediate field look like an ‘H’ with shorter upper caps compared with their lower caps. This is highlighted in figure 11(c) by the blue dashed lines. It is probably due to the fact that the conditional vortical structures in the jet intermediate field are formed by both the upright and inverted horseshoe-like vortices. This is because of the weak dominance of $Q3$ over $Q1$ events near the outer edge of the shear region in the jet intermediate field, which stems from the weak dominance of upright vortical structures over inverted ones in that region. This is one possible explanation for this observation. Alternative valid explanations may exist for this phenomenon.

Figure 11. Evolution of the conditionally averaged horseshoe vortex in the jet intermediate field. (a) Isometric view. (b) Side view. (c) Top view. Isosurfaces correspond to $\lambda _{ci}=0.11 \lambda _{ci,max}$. The dashed line represents the jet centreline, while the dotted line denotes the boundary between the shear region and outer non-turbulent region. Note that the $x$ and $y$ axes in panels (a) and (b) are normalized with the jet diameter $d$, while the jet half-radius $r_{0.5}$ is used to normalize the $x$ and $y$ axes in panel(c).

The conditional vortical structures presented so far were computed near the boundaries of the shear region with the potential core or the non-turbulent surrounding fluid. According to figure 2, the regions near the boundaries are dominated by either $Q1$ or $Q3$ events. It is proposed that the prevalence of $Q1$ or $Q3$ events corresponds to one of the two types of horseshoe-like vortex, i.e. an upright or inverted horseshoe vortex. To further illustrate this, the same events used earlier (i.e. $\omega _\theta >0$ and $\lvert u_\theta \rvert <0.01u_{\theta,max}$) were used to compute the conditional vortical structure at $(r_{ref}/d, z_{ref}/d)=(1.3,20.7)$ (marked with a black star in figure 2f), where $Q1$ and $Q3$ events are in balance. The result, which is displayed in 12(a), is an azimuthally aligned vortex tube (grey isosurface) with no sign of preference for either an upright or inverted horseshoe structure. An induced high-speed region (red isosurface) and an induced low-speed region (cyan isosurface) are also illustrated in figure 12(a). If the constraints $u_r>0$ and $u_z>0$, i.e. a $Q1$ event, are added to the previous conditions at the same location in the flow, an upright conditional horseshoe vortex will be obtained as presented in figure 12(b) in which the red isosurface is a high-speed region corresponding to the conditional streamwise velocity $\tilde {u}_z=0.5 \tilde {u}_{z,max}$. On the other hand, if a $Q3$ event is applied as the additional constraint, an inverted conditional horseshoe-like vortex is resolved as plotted in figure 12(c). These results clearly demonstrate that, where $Q1$ and $Q3$ events are in balance, both upright and inverted horseshoe vortices coexist with similar probabilities. They also support the notion that horseshoe vortices correspond closely to $Q1$ and $Q3$ events and therefore to the entrainment and turbulence production.

Figure 12. Conditional horseshoe vortex with its head located at $(r_{ref}/d,z_{ref}/d)=(1.3,20.7)$. The red and cyan isosurfaces show the conditional high-speed and low-speed regions, respectively. The condition events are (a) $\omega _\theta >0$ and $\lvert u_\theta \rvert <0.01u_{\theta,max}$, (b) $\omega _\theta >0$, $\lvert u_\theta \rvert <0.01u_{\theta,max}$, $u_r>0$ and $u_z>0$ and (c) $\omega _\theta >0$, $\lvert u_\theta \rvert <0.01u_{\theta,max}$, $u_r<0$ and $u_z<0$. Grey isosurface: $\lambda _{ci}=0.18 \lambda _{ci,max}$. Red isosurface: $\tilde {u}_z=0.5 \tilde {u}_{z,max}$. Cyan isosurface: $\tilde {u}_z=0.7 \tilde {u}_{z,min}$.

3.3. Long structures

In the instantaneous flow fields presented in figures 4 and 5, besides horseshoe vortices visualized by the swirling strength, long regions of $Q1$ and $Q3$ are visible. It appears that the high-speed/$Q1$ regions, extended in the $z$-direction, are induced by the streamwise organization of large-scale horseshoe vortices. Such long structures have been extensively studied in wall-bounded turbulent flows and are referred to as large-scale motions and VLSMs depending on their extension in the streamwise direction (e.g. Hutchins & Marusic Reference Hutchins and Marusic2007; Monty et al. Reference Monty, Stewart, Williams and Chong2007; Lee et al. Reference Lee, Sung and Adrian2019). These structures in wall-bounded flows are low-speed regions, flanked by high-speed regions on either side, and it is believed that they are formed by organized concatenation of large-scale eddy packets (Kevin, Monty & Hutchins Reference Kevin, Monty and Hutchins2019; Lee et al. Reference Lee, Sung and Adrian2019).

Isosurfaces of $u_z$-fluctuations can be used to visualize these long structures, as shown in figure 13(a), where a sample instantaneous flow field from the jet near field is displayed. To obtain the flow field, the mean streamwise velocity is subtracted to obtain the $u_z$-fluctuations, and isosurfaces of $u_z=0.1U_j$ (red isosurfaces) and $u_z=-0.1U_j$ (cyan isosurfaces) represent the high-speed and low-speed regions, respectively. Isosurfaces with the surface level $u_z=0.1U_j$ envelope all the isosurfaces with $u_z>0.1U_j$, and similarly isosurfaces with the surface level $u_z=-0.1U_j$ encompass all the isosurfaces with $u_z<-0.1U_j$. It is worth noting that the overall shapes of the long structures are not overly sensitive to the choice of the isosurface level. Figures 13(b) and 13(c) present isocontours of $u_z$-fluctuations on the cross-stream plane $z/d=2.8$ and the unwrapped streamwise–azimuthal plane at $r/d=0.6$, respectively. Several long streaky high-speed regions flanked by low-speed regions on either side are seen in the 3-D flow field as well as the unwrapped streamwise–azimuthal plane. The length and azimuthal width of these structures increase with $z$. The cross-stream $u_z$-contours displayed in figure 13(b) exhibit a clear periodic behaviour between the high-speed and low-speed structures in the azimuthal direction; each of these structures spans almost the entire shear region in the radial direction ($r/d=0.3\unicode{x2013}1$). In this particular instance, six high-speed and low-speed events are visible in the cross-stream plane at $z/d=2.8$. In the majority of cross-stream instances at this streamwise location, four to six high-speed and low-speed events dominate the flow. This is consistent with the findings of Jung et al. (Reference Jung, Gamard and George2004), whose results revealed that, for a round jet, the first POD mode energy distribution peaks at the azimuthal modes $m=4\unicode{x2013}6$ in the streamwise range $z/d=2\unicode{x2013}3$.

Figure 13. Example instantaneous normalized $u_z$-fluctuations in the near field. (a) Red isosurface: $u_z=0.1U_j$, cyan isosurface: $u_z=-0.1U_j$. (b) Contours of normalized $u_z$-fluctuations at $z/d=2.8$ denoted by a horizontal plane in panel(a). (c) Contours of normalized $u_z$-fluctuations on a azimuthal–axial plane at $r/d=0.6$. The radial location of this plane is indicated by a circle in panel (b).

Moving on to the intermediate field, an example 3-D instantaneous flow field visualized by isosurfaces of $u_z$-fluctuations is displayed in figure 14(a). In this plot, the isosurface levels $u_z=\pm 0.2U_{cl}$ (where $U_{cl}$ is the jet centreline velocity) are chosen for presenting the high-speed and low-speed events. Corresponding 2-D contour maps are plotted in figures 14(b) and 14(c) at the cross-stream plane $z/d=21$ and the unwrapped streamwise–azimuthal plane at $r/d=1.5$, respectively. A long spiral high-speed region is visible in figure 14(a) extending from $z/d=16$ to 27. This is equivalent to an axial length that is $\sim$6.7 times the jet half-radius at the centre of this spiral structure. The signature of a 3-D spiral VLSM in an unwrapped streamwise–azimuthal plane is a long structure with an oblique angle relative to the main flow direction. Three such structures are marked with solid lines in figure 14(c). Note that the lines in this figure are for presentation purposes only to highlight the obliqueness of the structures, and are not used in the statistical analysis presented afterwards. An objective detection scheme is presented later (see figure 15). Axisymmetry requires that the probability of occurrences of right-handed and left-handed spirals are equal. In figure 14(c), signatures of both right-handed and left-handed spirals are visible; that is, VLSMs with both negative and positive oblique angles with the main flow direction are present in the flow. Also, seen in this figure is a straight long high-speed event stretching from $z/d=15$ to $z/d=20$. The long coherent structures appear to extend more than $10d$ in the streamwise direction, as seen in figure 14(c), and span between the centreline and the outer edge of the shear region, as it is evident in figure 14(b). Moreover, the cross-stream velocity field corresponds to an azimuthal mode $m=2$ in this snapshot, i.e. two high-speed and two low-speed regions. Recently, Samie et al. (Reference Samie, Lavoie and Pollard2021) used two-point velocity measurements, and proposed that an organized concatenation of large-scale structures along the $z$-direction results in the formation of spiral VLSMs in the round jet intermediate field. They reached this conclusion based on the configuration of the VLSM component of (two-dimensional) two-point correlation maps. According to Samie et al. (Reference Samie, Lavoie and Pollard2021), these spiral structures consist of high-speed regions flanked on either side by low-speed regions, and are formed by the concatenation of horseshoe vortices. Samie et al. (Reference Samie, Lavoie and Pollard2021) showed that, depending on the radial location, VLSMs constitute between $30\,\%$ and $80\,\%$ of the total streamwise turbulence intensity in the jet flow. Therefore, VLSMs notably contribute to the dynamics of the jet flow.

Figure 14. Example instantaneous normalized $u_z$-fluctuations in the fully turbulent region. (a) Red isosurface: $u_z=0.2U_{cl}$, cyan isosurface: $u_z=-0.2U_{cl}$. (b) Normalized $u_z$-fluctuations at $z/d=21$ denoted by a horizontal plane on panel (a). (c) Normalized $u_z$-fluctuations on an unwrapped azimuthal–axial plane at $r/d=1.5$. The radial location of this plane is indicated by the circle in panel (b).

Figure 15. Illustration of the conditional averaging scheme for long coherent structures on an axial–azimuthal plane. (a,b) Two modes of schematic oblique structures. (c) A schematic axially aligned structure.

Instantaneous flow fields such as those presented above reveal long streaky structures in the jet near field and axially aligned and spiral VLSMs in the jet intermediate field. The statistical significance of these coherent features is investigated in greater detail using volumetric conditional averages. The long coherent structures are detected on 2-D streamwise–azimuthal planes using the conditions illustrated in figure 15. Two modes of spiral VLSMs as well as axially aligned VLSMs are detected by this scheme, which relies on a streamwise length ($l_z$) and an azimuthal length ($l_s$) as the input.

The detection scheme was used to conditionally average axially aligned streaky coherent structures in the jet near field ($z/d=1\unicode{x2013}5$). That is because only these structures were observed in the instantaneous flow field in this region. A 3-D view of the conditional $u_z$ flow field around the reference point $(z_{ref},r_{ref})=(2.8d,0.6d)$, denoted by $\tilde {u}_z$, is displayed in figure 16(a), while corresponding 2-D contour maps on a cross-stream plane at $z/d=2.8$ and on an unwrapped streamwise–azimuthal plane at $r/d=0.6$ are presented in figures 16(b) and 16(c), respectively. Following the dimensions of the instantaneous streaky structures, the detection input values $l_z=d$ and $l_s=0.03d$ were used here. Different values of $l_z$ and $l_s$ were tested to explore the sensitivity of the conditional structure dimensions to the choice of $l_z$ and $l_s$ (see figure 18 in the Appendix for details). One can see that, while the final conditional velocity field is not sensitive to the value of the azimuthal span $l_s$, the length of the main structure increases with increasing $l_z$. Nonetheless, this does not affect the qualitative features of the conditional streaky structures. The azimuthal periodicity of the coherent structures, which were observed in the instantaneous flow field, are confirmed by the conditional structure. The $\tilde {u}_z$ flow field contains four positive and four negative $\tilde {u}_z$ regions corresponding to the azimuthal mode $m=4$, implying that the azimuthal mode $m=4$ statistically dominates other azimuthal wavenumbers. However, given the superposition of the flow fields associated with all azimuthal wavenumbers in the conditional averaging, structures other than the central structure (located at $({\rm \Delta} x/d, {\rm \Delta} y/d)=(0.5, 0)$ in figure 16b) are weak. In addition to the azimuthal periodicity, a streamwise periodicity is implied from figure 16. That is, a negative $\tilde {u}_z$ region follows the main positive $\tilde {u}_z$ region in the streamwise direction, which is in turn flanked on either side by two positive $\tilde {u}_z$ regions. In-plane conditional velocity vectors obtained for a conditional horseshoe vortex in the near field (presented in figure 8) is overlaid on the 2-D contour plots in figure 16(b,c). It is noted that the vector fields are not computed along with the streaky structures using the detection scheme illustrated in figure 15; instead, they correspond to the conditional horseshoe vortex computed in § 3.2 at the same streamwise location. The comparison of the velocity vector field and the $\tilde {u}_z$ field reveals that the size of the conditional horseshoe vortex matches that of the conditional streaky structure. This observation lends support to the notion that the streaky structures are induced by 3-D horseshoe vortices that are inclined to the axial direction, rather than purely streamwise vortices.

Figure 16. Conditionally averaged axially aligned streaky structures in the near field based on the conditional scheme illustrated in figure 15(c) with the reference point at $(z_{ref}, r_{ref})=(2.8d, 0.6d)$. (a) Red isosurfaces: $\tilde {u}_z=0.55\tilde {u}_{z,max}$ and $0.12\tilde {u}_{z,max}$ , cyan isosurface: $\tilde {u}_z=0.12\tilde {u}_{z,min}$. (b) Contour map of $\tilde {u}_z/\tilde {u}_{z,max}$ at $z/d=2.8$ denoted by a horizontal plane in panel(a). (c) Contour map of $\tilde {u}_z/\tilde {u}_{z,max}$ on a streamwise–azimuthal plane at $r/d=0.6$. The radial location of this plane is indicated by the circle in panel (b).

The evidence for the statistical significance of spiral VLSMs is provided here by conditionally averaging these structures using the detection scheme presented in figure 15, after applying a filter to the DNS flow fields that removes small-scale motions. The conditional inputs used for the spiral VLSMs are $l_z=7d$ and $l_s=3d$, and the reference location of the detection is $(z_{ref},r_{ref})=(19.7d,1.48d)$. Different combinations of $l_z$ and $l_s$ were applied to obtain the right-handed spiral (see figure 19 in the Appendix for details). The salient features of the resulting conditional structures remain unchanged for the values of $l_z$ and $l_s$ considered here, which were chosen upon inspection of several instantaneous flow fields. Three-dimensional views of the conditional VLSMs (two spiral VLSM modes and one axially aligned VLSM) are presented in figure 17(a–c), while corresponding 2-D contour maps on the unwrapped streamwise–azimuthal plane corresponding to the radial location $r=1.48d=r_{0.5}$ are displayed in figure 17(d–f). The 3-D structures are visualized by $\tilde {u}_z$-isosurfaces, in which conditional high-speed regions correspond to $\tilde {u}_z/\tilde {u}_{z,max}=0.1$ and 0.6 (red isosurfaces), and conditional low-speed regions are associated with $\tilde {u}_z/\tilde {u}_{z,max}=-0.15$ (cyan isosurfaces). In total, 500 independent flow fields were used for the conditional averaging; $34\,\%$ of the total detected VLSMs were right-handed spirals, $38\,\%$ were left-handed spirals and $28\,\%$ were axially aligned VLSMs. While the percentage contribution from the right-handed and left-handed spirals to the VLSMs are close, they are expected to be equal due to axisymmetry. The observed difference between the percentage for the spiral cases is not statistically significant given the finite number of snapshots available. These percentage contribution values reveal that the spiral VLSMs play a key role in the dynamics of turbulent round jets. The conditional spiral VLSMs are comprised of two high-speed and two low-speed spiral structures, where one high-speed structure is more pronounced than the other one. This is consistent with the azimuthal mode $m=2$ in the instantaneous flow fields (figure 14) as well as dominance of the POD eigenspectra (integrated over frequency) at azimuthal modes $m=2$ and $m=1$ in the jet intermediate and far fields reported in Gamard, Jung & George (Reference Gamard, Jung and George2004). The combination of modes 1 and 2 results in the conditional VLSM displayed in figure 17 in which the second high-speed structure is weak. It can be seen that the long VLSMs constitute three large-scale high-speed features that are aligned diagonally (in spiral VLSMs) or axially (in the axially aligned VLSM) along the $z$-direction. The in-plane velocity vector field corresponding to the conditional horseshoe vortex obtained in § 3.2 (see figure 9) is overlaid on the $\tilde {u}_z$ contour map depicted in figure 17(d). It is stressed that the velocity vector field has been computed through a separate conditional scheme than the one introduced for the conditional VLSM, and is presented here to compare the conditional horseshoe vortex width with that of the VLSM. The widths of the horseshoe vortex and the spiral VLSM appear to be comparable, promoting the notion that VLSMs are formed by the diagonal concatenation of horseshoe vortices. These 3-D conditional structures are very similar to the conceptual model of spiral VLSMs proposed by Samie et al. (Reference Samie, Lavoie and Pollard2021).

Figure 17. Conditionally averaged VLSMs in the axial range $z/d=14.5-26.8$ based on the conditional scheme illustrated in figure 15 with the reference point at $(z_{ref}, r_{ref})=(19.7d, 1.48d)$. (a–c) Isometric view of VLSMs. The red isosurfaces correspond to $\tilde {u}_z/\tilde {u}_{z,max}=0.1$ and 0.6, while the cyan isosurfaces are associated with $\tilde {u}_z/\tilde {u}_{z,max}=-0.15$. (d–f) Contour map of $\tilde {u}_z/\tilde {u}_{z,max}$ on a streamwise–azimuthal plane at $r=r_{ref}$ associated with the (d) right-handed spiral, (e) left-handed spiral and (f) axially aligned VLSM.