4.1. Coherent structures

Coherent structures are believed to strongly affect turbulent mixing, hydrodynamic features, flow noise and various hydrodynamic and thermoacoustic instabilities (Fiedler Reference Fiedler1988; Zhang et al. Reference Zhang, Shahsavari, Rao, Yang and Wang2019, Reference Zhang, Shahsavari, Rao, Li, Yang and Wang2020a). Changing the coherent structure features can manipulate turbulent flow characteristics, which has great technical relevance. Enhancement and manipulation of coherent structures are possible by imposing periodic excitations on the jet (Ho & Huerre Reference Ho and Huerre1984). Moreover, it is of both practical and fundamental interest to study coherent structure features, when the jet experiences hydrodynamic instabilities. In the present paper, the supercritical jet is excited by various forcing waves to address both of the above objectives; manipulating coherent structures by periodic excitations and features of coherent structure in a hydrodynamically unstable jet.

This section discusses the effects of the imposed excitations on various aspects of the coherent structures in the investigated supercritical jets. To sum up the aims, the Q-criterion is used to educe the coherent structures, which are illustrated in figure 7 for the investigated cases. All coherent structures are shown at the value of 3 % of the maximum value of Q in each case for all the following analyses. In figure 7, the structures are coloured by the root mean square of density fluctuations to show the contribution of coherent motions in inducing density fluctuations and turbulent mixing. The results show that a multitude of coherent structure forms including helix, ring, line or rib, and hairpin appear in the supercritical jets. Similar to the most axisymmetric jets, coherent structures are single-helix-like near the injector for all the investigated cases. The single-helix structures are the first mode of the vortical structures in axisymmetric jets (Fiedler Reference Fiedler1988). However, the structures evolve into ring-like structures farther downstream of the injector due to the varicose perturbations in V1, VH1, VH2, VH3, VF1, VF2 and VF3 cases. Although V2 and V3 cases are also stimulated by the varicose perturbations, they are devoid of any ring-like structures. This is in line with the results presented in the previous sections, in which it is found that the varicose excitations have limited effect on the supercritical jet, when the forcing frequency matches the most amplified frequencies in the transient region. Figure 7 also shows that some sort of ring-like structures tilted toward the radial direction are also observed in the supercritical jet excited by flapping mode perturbations. It is interesting to note that the structures show similar features near the injector in the dual mode excited cases in comparison with the case V1. Therefore, the varicose part of the dual mode excitations has the dominant effect on the coherent structures in the potential core. The present results also show that line (or rib) and hairpin coherent structures appear between the ring-like structures in case V1 and dual mode excited cases. Such structures are generated during the formation of the ring-like elements from the helix structures. Farther downstream of the above-mentioned highly organized structures, a complex agglomeration of coherent structures observed in the flow field, which are produced from the cut-and-connect mechanism (Hussain Reference Hussain1986). This mechanism comprises of breakdowns, interactions and interconnections of the organized coherent structures. Such complex topology of the vortex filaments becomes irregular as the flow moves downstream. In all tested cases, the coherent structures induce negligible density perturbations in the potential core, while as the potential core breaks down, they induce huge density perturbations. Farther downstream, the density perturbations through the coherent structures are decreased significantly.

Figure 7.

Coherent structures shown by Q-criterion (@ 3 % of $Q_{Max}$) coloured by the root mean square density fluctuations.

Figure 7 also reveals that VH1, VH2, VH3, VF1, VF2 and VF3 cases experience bifurcations at downstream regions of the injector, through which structures are shed toward different branches, which are shown by solid lines. This is consistent with the bifurcations reported in subcritical jets (Danaila & Boersma Reference Danaila and Boersma2000; Gohil et al. Reference Gohil, Saha and Muralidhar2015; Tyliszczak Reference Tyliszczak2015). Previous investigations show that the bifurcation in dual varicose–helical mode excited subcritical jets is the result of self-induced velocity normal to a vortex ring induced by the tilting of the vortex due to the uneven distribution of vorticity and convective velocity inside the vortex ring (Gohil et al. Reference Gohil, Saha and Muralidhar2015). Such uneven distributions are due to the helical perturbations (Gohil et al. Reference Gohil, Saha and Muralidhar2015). Details of the bifurcation phenomenon in the present supercritical jet are illustrated in figure 8, which plots the spatial distribution of the instantaneous tangential velocity alongside with the isosurface of Q criterion and the isoline of density. Here, the isosurface of the density at 99 % of the maximum value of density is used to show the jet potential core. It can be seen in the zoomed-in graph in figure 8 that the self-induced velocity normal to the vortex ring (tangential velocity) is already present upstream of the potential core breakdown and the bifurcation location. Therefore, although the self-induced velocity is the necessary criterion to induce bifurcation in the dual mode excited cases, it is not sufficient. Results show that the self-induced velocity results in bifurcation downstream of the potential core, where the stratified shear layer stabilization mechanisms decay significantly. Therefore, the bifurcation takes place, if the self-induced velocity overthrows the stabilization mechanisms, which necessarily takes place downstream of the potential core.

Figure 8.

Spatial distribution of the instantaneous azimuthal velocity alongside with isosurface of $0.03Q_{Max}$ criterion and isoline of $0.99\rho _{Max}$ in case VF1.

As it is shown in figure 3, the supercritical jet experiences the highest amplitude perturbations at two distinct azimuthal locations in VH1, VF1–VF3 cases, while in VH2 and VH3 cases, five and nine distinct locations in the azimuthal direction experience the highest amplitude perturbations, respectively. Although the twofold coherent structure shedding is clearly observed in figure 7 for VH1, VF1–VF3 cases, the fivefold and ninefold patterns are rarely observed in VH2 and VH3 cases. In order to further shed light onto the bifurcations in the dual mode excited cases, figure 9 shows isolines of the time-averaged axial velocity at the values of 0.5 and $2\ \textrm {m}\ \textrm {s}^{-1}$ along with isosurfaces of the axial velocity at the value of $2 \ \textrm {m}\ \textrm {s}^{-1}$ coloured by the time-averaged density. The results show that the circular jet cross-section at $Z/D<5$ evolves into a more elliptic shape as the supercritical jets in VH1, VF1–VF3 cases move into the downstream regions ($Z/D>5$). Among these numerically tested cases, just VH1 and VF1 cases experience twofold patterns at $Z/D>10$. The results show that the jet cross-section is almost circular at different axial locations in VH2 and VH3 cases. Therefore, the high modes of the bifurcation (fivefold and ninefold cases, VH2 and VH3) are hardly trackable in the averaged results under the supercritical condition. Reynolds et al. showed that the existence of the multiarm patterns in subcritical fluid flows is highly dependent on the flow Reynolds number (Reynolds et al. Reference Reynolds, Parekh, Juvet and Lee2003). The multiarm patterns are well produced at low Reynolds numbers (e.g. 2000–3000) (Gohil et al. Reference Gohil, Saha and Muralidhar2015; Tyliszczak Reference Tyliszczak2015). However, the present study shows that the averaged flow field is devoid of the multiarm patterns at high Reynolds number supercritical fluid flows, due to the high level of turbulent mixing under the high-Reynolds-number condition. The density distribution on the isosurfaces of the axial velocity shows that the density field does not follow exactly the same multifold patterns as the velocity field. For instance, in the cases of VH1 and VF1, the fluid density is at the maximum value at the central part of the jet at $Z/D>5$.

Figure 9.

Isolines of the time-averaged axial velocity (at 0.5 and $2\ \textrm {m}\ \textrm {s}^{-1}$) along with the isosurface of the axial velocity at $2\ \textrm {m}\ \textrm {s}^{-1}$ coloured by the time-averaged density.

In order to further analyse the coherent structure features under the imposed excitations, figure 10 presents the pitch distance between the coherent structures in the axial direction averaged over the snapshots of the simulations ($\langle L_{P}\rangle$). The pitch distance (as shown in the embedded graph in figure 10) is an axial distance between the crest of structures at Z–r planes. Here, just helix and ring structures are considered to measure the pitch distance. Both the axial coordinate and the pitch distance are normalized by the injector diameter. In the dual mode excited cases, only coherent structures before the bifurcations are considered here. Figure 10 shows that unexcited, V2, V3 and helical cases have a similar pitch distance pattern in the axial direction. In such cases, the pitch distance increases linearly with the axial direction. However, in the other numerically tested cases (V1 and dual modes), the pitch distance grows in the axial direction in a logistic pattern (increases exponentially at low Z/D, while it levels off at the downstream regions). Here, the distance between the injector outlet and the axial location, where the last coherent helix or ring structures are observed, is defined as the coherent structure penetration depth ($\langle L_{CS}\rangle$). The normalized values of the penetration depth are given in the legend of figure 10. Obtained results show that coherent structures in case F have the least penetration depth, while they have the maximum penetration depth in case V1 among the investigated cases. The results show that the coherent structures have more penetration depth in the dual varicose/helical excited cases as compared with the dual varicose/flapping excited cases.

Figure 10.

Non-dimensionalized coherent structure pitch distance along the non-dimensionalized axial direction.

Coherent structures are not necessarily highly energetic in turbulent flows. The level of the energy through the coherent structures is dependent upon the flow regime. In fully developed flows, the turbulent energy through coherent structures is comparable with incoherent turbulent motions, while in transitional flows, they contain the majority of the turbulent energy (Hussain Reference Hussain1986). The question posed here concerns the key effects of the excitations on the turbulent energy level though the coherent structures in the present supercritical jet. This energy can be monitored by evaluating the fraction of the total turbulent kinetic energy attributed to the coherent structures ($k_{CS}(\%)=100\times E_{CS}/E$, where E and $E_{CS}$ are the total kinetic energy and the corresponding energy through the coherent structures, respectively). The previous investigations show that $k_{CS}$ is approximately 20 % in plane mixing layers, 25 % in accelerated mixing layers, 50 % in near jets, 10 % in axisymmetric far-jets, 25 % in near wakes and 20 % in plane far-wakes (Fiedler Reference Fiedler1988). In order to calculate $k_{CS}$ in the present supercritical jets, a cylindrical subdomain of the computational domain near the injector with the length and diameter of 10D and 5D, respectively, is chosen to calculate both the total turbulent kinetic energy and the corresponding value through the coherent structures. Here, the coherent structures are discerned via the Q-criterion at the value of 3 % of the maximum value of Q in each case. Then, the kinetic energy through the coherent structures ($E_{CS}$) is integrated in the marked volume using the Q-criterion. Moreover, the total value of the kinetic energy (E) is calculated by integrating the kinetic energy through the subdomain near the injector. Then, the averaged value of $k_{CS}$ is measured using the snapshots of the numerical simulations. The present obtained results show that this ratio is 15.6 % in the unexcited jet. In order to compare $\langle k_{CS}\rangle$ in different cases appropriately, $\langle k_{CS}\rangle$ in the excited cases are normalized by the corresponding value in the unexcited jet. Figure 11 compares the normalized value of the fraction of the kinetic energy attributed with the coherent structures in the investigated cases. It is interesting to note that $\langle k_{CS}\rangle$ drops considerably in the excited cases as compared with the unexcited case. More specifically, the fraction of the turbulent kinetic energy through the coherent structures in V1 and dual mode excited cases is less than 80 % of $\langle k_{CS}\rangle$ in the excited case. This is consistent with the previously observed turbulence suppression near the exit of an axisymmetric jet under external excitations (Zaman & Hussain Reference Zaman and Hussain1981). Figure 7 shows that the coherent structures are more coherent and larger in V1 and dual mode excited cases as compared with the corresponding structures in the other cases. It can be concluded that the coherent structures carry less turbulent energy, if the imposed excitations create large scale coherent structures. As it is observed from the cases of V2, V3, H and F, figure 7, structures are smaller than the structures observed in V1 and the dual mode excited cases. Therefore, $\langle k_{CS}\rangle$ is higher in V2, V3, H and F cases as compared with V1 and the dual mode excited cases.

Figure 11.

Normalized kinetic energy and helicity through the coherent structures.

Helicity is one of the key features showing connections between coherent structures and the supercritical turbulent flow. This variable measures the orthogonality of the coherent structures or vortices with the local velocity (Moffatt Reference Moffatt1969). If the coherent structures are orthogonal to the local velocity, the helicity is zero. This means that the structures have two-dimensional motions in nature. On the other hand, the non-zero helicity indicates that the structures have three-dimensional motions. Here, helicity is integrated over the coherent structures in each case using a similar methodology as the one used to measure $\langle k_{CS}\rangle$. Next, the helicity in each excited case is normalized by the corresponding value of the unexcited jet. Figure 11 shows that the helicity drops in all excited cases as compared with the unexcited case.

The results clearly reveal that the degree of three-dimensionality of the coherent structures reduces when the supercritical jet stimulated by any kind of harmonic perturbations. Such effects are at the maximum value in V1 and the dual mode excited cases. Among the numerically tested cases, the dual varicose/flapping mode excitations with the frequency ratio of 2 have the maximum effect on the helicity. However, the varicose mode excitations with the frequency matching the most amplified frequency in the transient region (V2 and V3 cases) as well as the helical and the flapping mode excitations have the least effect on the local helicity.

4.2. Jet development

Characterizing the effect of the excitations on the flow development region is achieved by comparing the potential core length in the unexcited and excited cases. This is shown in figure 12. The potential core length is measured based upon both density and velocity profiles, $L_{\langle \rho \rangle }$ and $L_{\langle Uz\rangle }$, respectively, which are normalized by the injector diameter (D). The end of the potential core is measured by tracking the location, where the centreline density (or velocity) profile drops to 99 % of the corresponding variables at the inlet.

Figure 12.

Non-dimensionalized potential core length based upon density and velocity profiles.

Figure 12 reveals that, unlike the unexcited and excited cases under the varicose excitations (V1–V3 cases), in cases under the helical (case H), the flapping (case F) and the dual mode (VH1–VH3 and VF1–VF3 cases) excitations, the potential core length calculated based on the density profile is considerably shorter than the potential core length measured based on the axial velocity profiles. The results show that the length of the flow development region drops drastically under the flapping (case F) and the dual mode excitations (VH1–VH3 and VF1–VF3 cases), while the varicose excitations with the frequency matching the most amplified frequencies in the transient region (V2 and V3 cases) have negligible effect on the potential core length.

The effects of the imposed excitations on the supercritical jet are further studied by evaluating the pseudo-similarity behaviours of the supercritical jet under both unexcited and excited conditions. The aim is to search for a similarity between the unexcited and excited supercritical jets as well as subcritical constant and variable density jets. To such an aim, non-dimensional density and axial velocity are plotted in figure 13 along the jet axis in search for pseudo-similarity. Here, the density and the axial velocity are normalized as

(4.1)\begin{gather} {u}^{*}=\frac{\langle U_{z}^{inj}\rangle }{\langle {{U}_{z}}\rangle}, \end{gather}

(4.2)\begin{gather}{{\rho }^{*}}=\frac{\langle {{\rho }^{inj}} \rangle -\langle {{\rho }^{\infty }} \rangle }{\langle \rho \rangle -\langle {{\rho }^{\infty }} \rangle }, \end{gather}

where, $\langle {\rho }^{\infty } \rangle$ is the time-averaged far field density, $\langle U_{z}^{inj} \rangle$ and $\langle {\rho }^{inj} \rangle$ are the time-averaged axial velocity and density at the inlet, respectively. Variable-density jets show no similarity when the coordinate is normalized by the injector diameter (Schmitt et al. Reference Schmitt, Selle, Ruiz and Cuenot2010). In this light, the coordinate is usually normalized by an effective diameter ($d_{\alpha }^{eff}$ ), which is defined as

(4.3)\begin{equation} d_{\alpha }^{eff}=D{{\left( \frac{\langle {{\rho }^{\infty }} \rangle }{\langle {{\rho }^{inj}} \rangle } \right)}^{\alpha }}, \end{equation}

where, $\alpha$ is a constant. Here, $\alpha$ is set to be $-1/2$ and 1/2 to plot the normalized velocity and density profiles, respectively. Moreover, the potential core length is subtracted from the axial coordinate to ignore the negligible changes of the axial velocity and the density in the potential core. Some of the previous analyses carried out on the low-pressure constant density (Chassaing_1) and variable density with ${\rho }^{inj}$/${\rho }^{\infty }=1.43$ (Chassaing_1.43) jets as well as a high-pressure jet with ${\rho }^{inj}$/${\rho }^{\infty }=7.9$ (Zong_7.9) are also presented in figure 13 for reference (Chassaing Reference Chassaing1979; Zong Reference Zong2005). It is worth noting that ${\rho }^{inj}$/${\rho }^{\infty }$ is 3.6 for the present supercritical jets, where ${\rho }^{inj}$ is the fluid density at the injector outlet and ${\rho }^{\infty }$ is the far-field fluid flow density.

Figure 13.

Normalized (a) velocity and (b) density distributions along the normalized axial direction. The potential core lengths based upon the velocity and density profiles ($L_{\langle Uz\rangle }$ and $L_{\langle \rho \rangle }$) are subtracted from the axial direction in panels (a) and (b), respectively.

It can be seen from figure 13 that almost all the numerically investigated cases show linear decay in the normalized velocity and density profiles, except the non-dimensional velocity in VH1 and VF1 cases in which it is impossible to fit a linear curve through the non-dimensional velocity profile. The slopes of the velocity and density curves, denoted by $K_{u}$ and $K_{\rho }$ are presented in the legends. The present investigations show that the normalized velocity and density curves in the unexcited V2 and V3 cases have almost identical slopes as compared with the low-pressure jets (Chassaing_1 and Chassaing_1.43). Therefore, unexcited, V2, and V3 jets favourably compare with low-pressure jets. The results also reveal similarity between V1 and F jets. Furthermore, figure 13 shows that the normalized density distributions collapse into a single curve in the dual varicose/flapping excited jets. The normalized velocity distribution in the high-density ratio high-pressure case (Zong_7.9) favourably compares with VH1. It is not surprising that exciting the supercritical jet with the dual mode perturbations leads to significant changes in pseudo-similarity of the jets, since such excitations alter the jet topology (figure 9). However, excited jets exhibit similarity under specific excitation modes (e.g. similarity between V1 and F or similarity between V2 and V3). The results show that $K_{u}$ and $K_{\rho }$ are increased significantly under the dual mode excitations as well as the varicose mode excitations (when the forcing frequency matches the most amplified frequency in the jet potential core). This clearly shows that the supercritical jets decay more rapidly in V1, VH1–VH3 and VF1–VF3 cases as compared with the other unexcited and excited cases. Besides the turbulent mixing induced by the excitations, the jet bifurcation in the dual mode cases plays a role in the supercritical jet decay. As the jet bifurcates, the axial velocity drops considerably on the jet axis, since the high velocity jet deviates from the jet axis toward the radial direction in the multiarm cases.

Figure 14 shows the axial distribution of the radial location of the half-maximum value of the jet mean density ($r_{\rho 1/2}$) for the investigated cases. Both axes are normalized by the injector diameter. Linear curves are fitted to the numerical data to obtain the jet virtual origin. The intercept and the slope, (a,b), of the linear curves are presented in the legend of figure 14. The virtual origin ($Z_0=-a/b$) is a location 1, where $r_{\rho 1/2}$ is zero (Mi et al. Reference Mi, Nobes and Nathan2001). Here, $r_{\rho 1/2}$ is spatially averaged in the azimuthal direction in each jet cross-section. It can be seen that the width of the excited jets increases faster in the axial direction than the unexcited jet width. Among the excited cases, VF1, VF2 and VF3 cases have the fastest jet development, while V2 and V3 have the least development rates. Furthermore, the development rate of the jet width is almost identical in V1, H and F cases. The results reveal that exciting the jet with the varicose mode perturbations at the forcing frequency matching the most amplified hydrodynamic frequency in the potential core (preferred mode) (case V1) has more effect on the jet development as compared with the excited jets with the forcing frequencies matching the most amplified frequencies in the unexcited jet transient region (V2 and V3 cases).

Figure 14.

Non-dimensionalized radial location of the half-maximum value of jet mean density profile along the non-dimensionalized axial direction.

It is known that the cross-section of circular gas jets changes considerably due to mixing and hydrodynamic instabilities, as the jet emerges from the injector. In order to quantitatively compare the jet cross-section development in the axial direction, the maximum and mean radial locations of the half-maximum value of the jet mean density are subtracted from each other at each flow cross-section, to show the uniformity of the jet cross-section. Figure 15 shows the axial distribution of the jet radius evolution. In order to compare different cases appropriately, the virtual origin location is subtracted from the axial direction in each case. Results show that the jet becomes quickly non-circular in the dual varicose/flapping mode excited cases (VF1, VF2 and VF3 cases). The helical excitations (case H) do not induce considerable changes in the jet cross-section as compared with the unexcited case, while the varicose (case V1) and flapping (case F) modes have drastic effects on the cross-section shape. Moreover, unlike VF2 and VF3 cases, in VH2 and VH3 cases, the jet cross-section development rates are almost identical to the unexcited jet.

Figure 15.

Non-dimensionalized jet cross-section diameter along the non-dimensionalized axial direction. The virtual origin location is subtracted from the axial direction.

Round non-swirling jets are usually devoid of any significant circulation or vorticity production along the jet direction. However, vorticity source terms such as vortex tilting and vortex stretching induce three-dimensional flow motions, which result in a weak circulation. The local circulation is a good measure to extract such weak motions in the azimuthal direction. This parameter is the derivative of the circulation in the azimuthal direction ($\textrm {d}\varGamma /\textrm {d}\theta =r\langle U_{\theta }\rangle$), which takes into account the effects of the non-homogeneous velocity distribution in the azimuthal direction (Ianiro et al. Reference Ianiro, Lynch, Violato, Cardone and Scarano2018). Figure 16 plots the local circulation normalized by the injector diameter multiplied by the injection velocity of the supercritical jet ($U_z^{inj}$) along the jet axis for the unexcited and excited cases. Here, the local circulation at each axial location is integrated over the whole chamber cross-section. Moreover, as it is shown in figure 16, the potential core length ($L_{\langle \rho \rangle }$) is subtracted from the axial coordinate to properly compare the local circulation in different cases.

Figure 16.

Non-dimensionalized local circulation along the non-dimensionalized axial direction. The potential core length based upon the density profile is subtracted from the axial direction.

It can be seen from figure 16 that the local circulation increases toward the axial direction in the potential core with approximately identical slopes in the axial direction in different unexcited and excited cases. Comparison of figures 6 and 16 reveals that the intensive growth of the local circulation is accompanied by the vorticity productions, specifically by vortex stretching and vortex titling source terms. It should be noted that in the present confined round jet, flow disturbances grow on the shear layer around the jet in the potential core are of a helical type (figure 7). Therefore, it is expected that even in the unexcited jets and the jets stimulated via the varicose perturbations, the jets experience the local circulation. Figure 16 shows that the local circulation drops downstream of the potential core, since the produced vorticity (or circulation) is utilized by the jet in the potential core breakdown process to produce turbulent structures. The local circulation drops faster in F, H and dual mode excited cases as compared with the varicose mode excited cases. Farther downstream, the local circulation increases again, most probably due to chaotic motions induced by different hydrodynamic instability modes of the present confined supercritical jets. The results indicate that the increases in the local circulation are more intense in the F, H and dual mode excited cases as compared with other cases. This can be a symptom of a more intensely triggered hydrodynamic instability modes in the downstream regions in H, F and dual mode excited cases as compared with the other investigated cases.

4.3. Turbulent kinetic energy and mixing

In order to characterize the role of the excitations on the turbulent flow characteristics, figure 17 plots the turbulent energy spectra along the jet axis for the investigated cases. The energy spectra are obtained by performing a fast Fourier transformation on the axial velocity fluctuations at 120 spatial locations on the jet centreline. The length of the potential core is subtracted from the axial coordinate to clearly show the turbulent characteristics in the potential core. The excitation frequencies are marked on the $y$-axes in figure 17. Besides the effects of the imposed coherent excitations on the potential core, the overall shape of the spectrum is almost identical in all the investigated cases. The jets have a relatively low turbulent energy in the potential core. However, as the potential core breaks down, the energy increases in the turbulent structures with the characteristic frequency matching the jet preferred mode. As the jet moves downstream, the energy is transferred to the large-scale oscillations. Comparisons between the unexcited, H and F cases show that the imposed helical and flapping perturbations have negligible effects on the turbulent energy in the potential core, while they increase the large-scale oscillations in the downstream regions. The flapping mode excitations increase the large-scale oscillations more as compared with the helical mode excitations.

Figure 17.

Energy spectra of the axial velocity fluctuations along the non-dimensionalized axial direction. The potential core length based upon the density profile is subtracted from the axial direction.

Figure 17 reveals that the varicose and the dual mode excitations increase the turbulent energy in the potential core in a narrow band frequency range, which corresponds to the frequency of the imposed excitations. Then, this energy is transferred from the coherent oscillations to the large-scale structures in the downstream regions. It is found that the distribution of the turbulent energy from the imposed varicose excitations to the large-scale structures is slower in V2 and V3 cases as compared with case V1, since the energy of the large-scale structures starts growing in a shorter axial location from the jet breakdown location in V1 as compared with V2 and V3. This reveals why the varicose excitations with the forcing frequency matching the most amplified frequencies in the jet transient region have less effect on the supercritical jet characteristics as compared with the varicose mode excitations with the forcing frequency matching the jet preferred mode. Among the investigated cases, the dual varicose/flapping mode excitations have the fastest and the most intense energy distribution from the imposed excitations to the large-scale oscillations, since the energy of the large-scale structures starts growing considerably at a relatively shorter axial distance from the jet breakdown location in the dual varicose/flapping mode excited cases (marked by stars in figure 17) as compared with the other cases. Although the energy spectra presented in figure 17 can be used as a primary index for the turbulent mixing, this parameter is not solely enough to make a conclusion about the effect of the imposed excitations on the turbulent mixing. It is known that the decay of the velocity and/or the jet spreading are useful indices to compare the turbulent mixing in different cases (Charonko & Prestridge Reference Charonko and Prestridge2017). In addition to these quantities, the jet cross-section development presented in figure 15 is another index to track the turbulent mixing. Considering the above-mentioned parameters presented in figures 13–15, it can be concluded that the dual varicose/flapping mode excitations have the highest impact on the turbulent mixing. This conclusion is consistent with the fast and considerable high-energy distribution from the coherent excitations to the large-scale turbulent structures in the dual varicose/flapping mode excited cases in figure 17.

In an attempt to characterize the effects of the imposed excitations on the small-scale turbulent mixing, figure 18 plots the axial distribution of the time-averaged subgrid scale turbulent kinetic energy integrated over the jet cross-section at each axial location. Here, the length of the potential core ($L_{\langle \rho \rangle }$) is subtracted from the axial direction to appropriately compare the turbulent kinetic energy in the small-scale structures in different cases. In this study, the subgrid scale kinetic energy is calculated by solving the following transport equation:

(4.4)\begin{align} &\frac{\partial {\rho }{{{{k}}}_{sgs}}}{\partial t}+\frac{\partial {\rho }{{\bar{U}_{j}}{{{k}}}_{sgs}}}{\partial {x}_{j}}-\frac{\partial {}}{\partial {x}_{j}}\left[{\rho }(\nu+\nu_{sgs})\frac{\partial{{{{k}}}_{sgs}}}{\partial {x}_{j}}\right] \nonumber\\ &\quad ={\frac{2}{3}}{\rho {k}_{sgs}}{\frac{\partial{\overline{{U}_{k}}}}{\partial {{x}_{k}}}}+ {\rho}{\nu_{sgs}}{\frac{\partial{\overline{{U}_{i}}}}{\partial {{x}_{j}}}}({2}{\bar{D}_{ij}}-{\frac{1}{3}}{\textrm{tr}({{2}{\bar{D}_{ij}}})}{\delta_{ij}})-{\frac{\rho {{C}_{\epsilon}}{{k}_{sgs}^{3/2}}}{\varDelta}} \end{align}

where, $k_{sgs}$ is the subgrid scale kinetic energy, $\nu$ is the kinematic viscosity, $\nu _{sgs}$ is the subgrid scale kinematic viscosity, ${D_{ij}}$ is the resolved-scale strain rate tensor, $\delta _{ij}$ is the Kronecker delta, $\varDelta$ is the LES filter size and $C_\epsilon =1.048$ is the model constant.

Figure 18.

Time-averaged small-scale turbulent kinetic energy integrated over the jet cross-section along the non-dimensionalized axial direction. The potential core length based upon the density profile is subtracted from the axial direction.

Figure 18 shows that the small-scale kinetic energy decays in the potential core in all cases due to the natural tendency of the density stratifications and the vorticity sinks to damp turbulent fluctuations. The decay rates of the small-scale turbulent kinetic energy in the potential core are almost identical in different unexcited and excited cases. However, downstream of the potential core ($Z>L_{\langle \rho \rangle }$), the small-scale turbulent kinetic energy increases in all cases due to the three-dimensional motions generated during the potential core breakdown. It is found that the dual mode excited cases have very high small-scale mixing effects on the jet in the transient region.

The question posed here concerns the effects of the above-mentioned turbulent kinetic energy on the supercritical jet turbulent diffusion. To such an aim, cross-correlations between the imposed modulations at the inlet and the density fluctuations on the jet centreline are calculated for the investigated cases. Figure 19 presents the results along the jet axis. Here, the length of the potential core is subtracted from the axial direction to appropriately compare the data. The results show that the excitations induce negligible density perturbations and hence diffusion effects in the potential core. However, the correlations grow drastically at the potential core breakdown location ($Z=L_{\langle \rho \rangle }$), where the turbulent structures induce a considerable amount of the turbulent diffusion. As the supercritical jet moves downstream, such highly correlated relations between the excitations and the density fluctuations disappear slowly, most probably due to the entrainment.

Figure 19.

Cross-correlation between the density fluctuations with the imposed excitations along the non-dimensionalized axial direction. The potential core length based upon the density profile is subtracted from the axial direction.