4.1.1 Instability modes

The eigenvalue spectrum for the 2D varicose perturbations for the plume case $(Re,Fr,S)=(180,0.5,0.276)$ is shown in figure 3(a). The spectrum contains an isolated unstable eigenvalue at $\unicode[STIX]{x1D714}=5.70+0.92\text{i}$ , and an arc branch with eigenvalues along the real frequency that are unstable in the range $9<\unicode[STIX]{x1D714}_{r}<20$ . In the earlier studies on the global instability of circular buoyant jets and plumes (Coenen et al. Reference Coenen, Lesshafft, Garnaud and Sevilla2017; Chakravarthy et al. Reference Chakravarthy, Lesshafft and Huerre2018), similar features were observed. These studies had shown that while the isolated eigenvalue is responsible for global instability, the eigenvalues of the arc branch are artifacts that arise because of domain truncation in the streamwise direction and resulting spurious pressure feedback from the outflow boundary. In a recent study, Lesshafft (Reference Lesshafft2018) showed that the growth rates of these unphysical arc branch modes can be reduced by adding an absorption layer near the outflow boundary.

Figure 3.

(a) BiGlobal eigenvalue spectrum for the case $(Re,Fr,S)=(180,0.5,0.276)$ obtained for 2D ( $\unicode[STIX]{x1D6FD}=0$ ) varicose perturbations. (b) Eigenfunctions of the isolated BiGlobal mode.

In the present work, the absorption layer has been modelled by following an approach stated in Chakravarthy et al. (Reference Chakravarthy, Lesshafft and Huerre2018). The EVP has been solved by adding a damping term $-\unicode[STIX]{x1D706}(z)\hat{\boldsymbol{q}}$ on the left-hand side of (2.18). Following Chomaz (Reference Chomaz2003), the damping coefficient $\unicode[STIX]{x1D706}(z)$ is set to zero for $z<z_{ab}$ , and it increases smoothly from 0 to $\unicode[STIX]{x1D706}_{max}$ over a streamwise distance $z_{ab}<z<z_{max}$ as

(4.1) $$\begin{eqnarray}\unicode[STIX]{x1D706}(z)=\unicode[STIX]{x1D706}_{max}\left[1+\exp \left(\frac{z_{max}-z_{ab}}{z-z_{ab}}+\frac{z_{max}-z_{ab}}{z-z_{max}}\right)\right]^{-1},\end{eqnarray}$$

where $\unicode[STIX]{x1D706}_{max}$ is the maximum damping coefficient, $z_{ab}$ is the streamwise distance where the absorption layer starts and $z_{max}$ is the outflow boundary. The eigenvalue spectra obtained by adding the absorption layer with damping parameters $\unicode[STIX]{x1D706}_{max}=10$ and $z_{ab}=20$ are presented in figure 3(a). With the absorption layer, the growth rates of the arc branch modes decrease and these eigenvalues recede to the stable half-plane as seen from figure 3(a). It can also be observed that the addition of the absorption layer has a negligible effect on the isolated mode. In the parameter space investigated, only one isolated symmetric mode has been observed. This is in contrast to the case of circular plumes where two such isolated modes had been observed in a similar parametric space (Bharadwaj & Das Reference Bharadwaj and Das2017).

Further, we performed instability computations for larger domain lengths to investigate the dependence of isolated BiGlobal mode and the arc branch modes on domain length, and to check if the addition of the absorption layer produces any unintended effects. These computations are performed for the plume case presented in figure 3(a) and the corresponding results are included in appendix B. The analysis reveals that the isolated BiGlobal mode is converged with respect to domain length (figure 21 a) and the use of an absorption layer does not affect its frequency and growth rate.

Figure 3(b) shows the real parts of eigenvectors obtained for the unstable BiGlobal mode in figure 3(a). These eigenvectors show spatially periodic structures which is a characteristic of oscillatory instability. The imaginary parts of these eigenfunctions are similar to the real parts with a phase shift of about $90^{\circ }$ , i.e. the zeros of the imaginary parts are located at the extrema of the corresponding real parts and vice versa. This phase shift indicates that these structures convect downstream as they grow. Overall, their characteristics are found to be similar to those of the global unstable modes of circular plumes (Bharadwaj & Das Reference Bharadwaj and Das2017; Chakravarthy et al. Reference Chakravarthy, Lesshafft and Huerre2018).

Figure 4.

BiGlobal eigenvalue spectrum obtained for 2D ( $\unicode[STIX]{x1D6FD}=0$ ) sinuous perturbations for the case shown in figure 3(a).

In the parameter space investigated, the eigenvalue spectra obtained for sinuous perturbations do not contain any isolated modes. However, they contain an unstable arc branch which stabilizes on the addition of an absorption layer near the outflow boundary, as shown in figure 4. Thus, the planar plumes are BiGlobally unstable only for varicose perturbations and are stable for sinuous perturbations. This observation is consistent with the earlier experimental and numerical works on planar plumes. Cetegen et al. (Reference Cetegen, Dong and Soteriou1998) and Soteriou et al. (Reference Soteriou, Dong and Cetegen2002) revealed that the planar plumes undergo an instability in their near field which is characterized by self-sustained periodic oscillations that involve formation, growth and shedding of a symmetric vortex pair from the plume source at a unique frequency. Since it is observed that the planar plumes are BiGlobally stable for asymmetric perturbations, we confine our further discussion to symmetric perturbations only. The frequencies $\unicode[STIX]{x1D714}_{r}$ and growth rates $\unicode[STIX]{x1D714}_{i}$ of the unstable BiGlobal symmetric mode, evaluated at each discrete parameter set $(Re,Fr,S)$ over the entire parameter space, are used to obtain the BiGlobal frequency scaling and transition diagrams respectively.

4.1.2 Transition diagram and frequency scaling of the BiGlobal mode

The neutral curves obtained from the growth rates of the symmetric BiGlobal mode are presented in figure 5. These curves correspond to the contour lines of $\unicode[STIX]{x1D714}_{i}=0$ mapped in the $Re{-}Fr$ space, for each $S$ . The critical $Re$ decreases with a decrease in $Fr$ as well as with a decrease in $S$ . That is, plumes with larger buoyancy destabilize earlier at lower Reynolds numbers.

Figure 5.

Neutral stability curves of the BiGlobal mode in $Re{-}Fr$ space for various density ratios, obtained for spanwise wavenumber $\unicode[STIX]{x1D6FD}=0$ .

In earlier planar plume experiments, Cetegen et al. (Reference Cetegen, Dong and Soteriou1998) reported that the oscillatory plumes are not observed beyond a value of density ratio $S=0.5{-}0.6$ . However, similar to our observation in circular plumes, the present stability analysis shows that the planar plumes also exhibit the oscillatory instability beyond the threshold limit proposed by Cetegen et al. (Reference Cetegen, Dong and Soteriou1998). However, it is to be noted that with an increase in $S$ , the unstable region gets confined to the plumes of higher $Re$ and lower $Fr$ . In § 4.5 of Bharadwaj & Das (Reference Bharadwaj and Das2017), we estimated that the perturbation growth rates are governed by the relative strengths of buoyancy and viscous forces as buoyancy/viscous diffusion ${\sim}O(Re\unicode[STIX]{x1D707}^{\ast }/SFr^{2})$ . While the buoyancy tends to destabilize the plumes, the viscous diffusion tends to stabilize them. Therefore, plumes with higher $S$ exhibit instability only at higher $Re$ and lower $Fr$ where buoyancy dominates the viscous diffusion. As the density difference vanishes, plumes become globally stable as shown by Garnaud et al. (Reference Garnaud, Lesshafft, Schmid and Huerre2013).

A plausible reason for the absence of puffing for plumes of $S>0.6$ in the experiments of Cetegen et al. (Reference Cetegen, Dong and Soteriou1998) is that their experiments would have been confined to low $Re$ and high $Fr$ , at higher $S$ . While the $Fr$ range of Cetegen et al. (Reference Cetegen, Dong and Soteriou1998) is not readily available from their paper, their experiments were confined to $Re<150$ (one can infer this from figures 6 and 7 of Cetegen et al. Reference Cetegen, Dong and Soteriou1998). The neutral curves obtained from the present instability analysis (figure 5) show that the plumes of $Re<150$ largely remain stable for density ratios higher than 0.6. Since the experiments of Cetegen et al. (Reference Cetegen, Dong and Soteriou1998) are confined to $Re<150$ , they did not observe puffing for $S>0.6$ . Nevertheless, the present study shows that plumes exhibit puffing at any density ratio until $S\approx 1$ .

Figure 6.

Variation of $St$ with $Ri$ for the unstable BiGlobal mode with spanwise wavenumber $\unicode[STIX]{x1D6FD}=0$ .

The non-dimensional frequency, or Strouhal number ( $St=\unicode[STIX]{x1D714}_{r}/(2\unicode[STIX]{x03C0})$ ), of the unstable BiGlobal mode, obtained for all the cases in the investigated parameter space, is plotted in figure 6 in terms of its variation with the plume source Richardson number, $Ri$ . The $St$ correlates well with $Ri$ , and the least-squares fit to the data yields a power law $St=0.54Ri^{0.44}$ ( $R$ -square $=$  0.998). Cetegen et al. (Reference Cetegen, Dong and Soteriou1998) measured oscillation frequencies in planar plumes of inlet $Ri$ range 1–100 and obtained a frequency correlation $St=0.55Ri^{\,0.45}$ from their experiments. The frequency correlation of the BiGlobal mode obtained from the present instability analysis is in excellent agreement with that obtained in the earlier experiments (Cetegen et al. Reference Cetegen, Dong and Soteriou1998).

4.1.3 Three-dimensional perturbations: effect of spanwise wavenumber $\unicode[STIX]{x1D6FD}$

The results presented above consider the perturbations to be 2D, i.e. $\unicode[STIX]{x1D6FD}=0$ . In this section, we briefly consider the effect of 3D perturbations, i.e. non-zero $\unicode[STIX]{x1D6FD}$ , on stability characteristics of planar plumes. To accomplish this, linear stability analysis has been performed by gradually increasing $\unicode[STIX]{x1D6FD}$ , for plumes of three Froude numbers $Fr=0.1$ , 0.5 and 1, and keeping $Re=180$ , $S=0.276$ . The frequency and growth rate of the most unstable BiGlobal mode obtained for various $\unicode[STIX]{x1D6FD}$ are plotted in figure 7. The analysis reveals that an increase in the spanwise wavenumber $\unicode[STIX]{x1D6FD}$ leads to a decrease in growth rates (figure 7 b) and eventually to stabilization of the plumes. However, its impact on the BiGlobal mode frequencies is negligible (figure 7 a). More importantly, this analysis shows that the 2D perturbations (i.e. $\unicode[STIX]{x1D6FD}=0$ ) always exhibit higher growth rates than the 3D perturbations (i.e. $\unicode[STIX]{x1D6FD}>0$ ). Therefore, the natural transitions in planar plumes that lead to BiGlobal instability will always be 2D.

Figure 7.

Variation of BiGlobal mode oscillation frequency and growth rate with spanwise wavenumber $\unicode[STIX]{x1D6FD}$ for plumes of $Re=180$ and $S=0.276$ .

4.2.1 Onset of oscillatory instability in rectangular plumes

The evolution of a rectangular plume with increase in inlet velocity $w_{0}$ , i.e. increase in inlet $Re$ and $Fr$ keeping $S$ constant, is presented to investigate the nature of instabilities that arise in these plumes. Velocity fields measured at 50 Hz using PIV are employed to characterize these instabilities, and the corresponding seed particle images are presented here as flow visualizations.

Figure 8.

Evolution of a planar plume from non-puffing to puffing state with an increase in inlet velocity: (a) $w_{0}=0.032~\text{m}~\text{s}^{-1}$ ( $Re=22$ , $Fr=0.05$ ), (b) $0.064~\text{m}~\text{s}^{-1}$ ( $Re=45$ , $Fr=0.1$ ), (c) $0.128~\text{m}~\text{s}^{-1}$ ( $Re=86$ , $Fr=0.2$ ) and (d) $0.194~\text{m}~\text{s}^{-1}$ ( $Re=126$ , $Fr=0.3$ ). The plume emanates from a rectangular orifice of width $B=0.042~\text{m}$ ( $\text{AR}=5$ ). Time gap between two successive snapshots is 0.04 s. The density ratio is kept constant at $S=0.276$ .

Figure 9.

The time traces of streamwise velocity along the centreline ( $x=0$ ) at three different heights $z=1$ , 2 and 3, and corresponding FFTs for the four planar plume cases shown in figure 8, i.e. (a) $w_{0}=0.032~\text{m}~\text{s}^{-1}$ , (b) $0.066\text{m}~\text{s}^{-1}$ , (c) $0.128\text{m}~\text{s}^{-1}$ and (d) $0.194\text{m}~\text{s}^{-1}$ .

Figure 8 shows the evolution of a plume emanating from a rectangular orifice of $B=0.042~\text{m}$ ( $\text{AR}=5$ ), with an increase in inlet velocity $w_{0}$ from 0.032 to $0.194~\text{m}~\text{s}^{-1}$ , resulting in a simultaneous variation of $Re$ and $Fr$ in the ranges $22\leqslant Re\leqslant 126$ and $0.05\leqslant Fr\leqslant 0.3$ . At each inlet velocity, the behaviour of the plume with time is also presented to show its unsteady nature. At lower inlet velocities, the plume develops a long laminar column which is relatively steady near the inlet. It undergoes a whiplash-like instability away from the source which involves sinuous meandering (figure 8 a,b). With an increase in inlet velocity, beyond a critical value, the plume undergoes a periodic instability in the near field which involves shedding of a pair of symmetric vortices with opposite circulation, as observed in figure 8(c,d). Additionally, away from the plume source, the central stem exhibits asymmetric undulations similar to that observed at lower inlet velocities. Overall, we observe two types of instabilities in planar plumes: (1) asymmetric instability away from plume source which involves sinuous meandering and (2) symmetric instability near the plume source which involves periodic formation and shedding of vortex pairs of opposite circulation.

To characterize these two behaviours, we follow the criterion proposed by Sreenivasan, Raghu & Kyle (Reference Sreenivasan, Raghu and Kyle1989) who investigated the instabilities in variable-density jets. Sreenivasan et al. (Reference Sreenivasan, Raghu and Kyle1989) differentiated the occurrence of absolute instability and convective instability in momentum-dominated variable-density jets by investigating their frequency spectra. They observed that the frequency spectrum of the absolute instability is characterized by very sharp peaks at a characteristic frequency and its harmonics. Whereas, the convective instability is characterized by a broadened spectrum, with much less prominent and distributed spectral peaks. Note that the absolute instability is a local analogue to the globally unstable, oscillatory behaviour (Schmid & Henningson Reference Schmid and Henningson2012). The presence of a finite region of absolute instability is a necessary condition for a flow to be globally unstable (Huerre & Monkewitz Reference Huerre and Monkewitz1990), i.e. flows that are absolutely unstable tend to become globally unstable. Therefore, in the present study, we calculated the frequency spectra from the velocity fields obtained from PIV to identify the global instability. Figure 9 shows the time traces of velocity at three streamwise locations $z=1$ , 2 and 3 along the plume centreline ( $x=0$ ) and their corresponding fast Fourier transforms (FFTs), for the four plume cases shown in figure 8.

At lower inlet velocities (figures 8 a,b and 9 a,b), the FFTs show a broadband frequency spectrum with the peaks distributed over a set of frequencies, which is a characteristic of convective instability. Therefore, the sinuous meandering or the asymmetric instability observed in plumes corresponds to the convective instability. This observation is consistent with the earlier works of Pera & Gebhart (Reference Pera and Gebhart1971), Yang (Reference Yang1992) and Ravier et al. (Reference Ravier, Abid, Amielh and Anselmet2006) concerning planar plumes. From local-spatial and local-temporal linear stability analysis in planar plumes, they obtained that these plumes are convectively unstable for both varicose and sinuous perturbations, with the sinuous perturbations always exhibiting higher growth rates than the varicose perturbations. The occurrence of sinuous instability in the far field of the planar plumes was also observed in earlier experiments (Pera & Gebhart Reference Pera and Gebhart1971) and computations (Ravier et al. Reference Ravier, Abid, Amielh and Anselmet2006). The spectral behaviour changes beyond the critical inlet velocity, i.e. at higher inlet velocities (figures 8 c,d and 9 c,d) the FFTs show sharper peaks at a characteristic frequency and its harmonics, with higher amplitudes, which implies that the symmetric periodic instability observed in the near field corresponds to the global instability. The periodic nature of this instability is evident from the near-field ( $x=0$ and $z=1$ ) velocity traces presented in figure 9(c,d). The flow visualizations (figure 8 c,d) show that this global oscillatory instability is symmetric in nature consistent with our observation from the present BiGlobal linear stability analysis. This oscillatory instability in plumes is commonly termed ‘puffing’. The dominant frequency of pulsation in the near field is termed the ‘puffing frequency’ of the plume. Further away from the plume source, the flow becomes more complex and additional frequencies which correspond to the sub/super harmonics of the puffing frequency dominate the flow ( $z=3$ in figure 9 c,d).

Note that although the plumes at low inlet velocities are convectively unstable to asymmetric perturbations, they are globally stable since these perturbations eventually convect downstream away from the plume source, thereby stabilizing the plumes in the near field in the long-time limit. Since the present study is only confined to global instability, we consider these plumes to be globally stable, and refer to them as ‘non-puffing plumes’ in our further discussion.

4.2.2 Effect of AR

The effect of AR on puffing has been examined by comparing the plume states and puffing frequencies of rectangular plumes with those of the results obtained from the BiGlobal instability analysis with $\unicode[STIX]{x1D6FD}=0$ . The case of $\text{AR}=1$ , i.e. square plumes, will be discussed separately in § 4.2.4.

The plume states are classified as non-puffing and puffing as described in § 4.2.1, and are mapped in the $Re{-}Fr$ plane at each density ratio $S$ . Figure 10 shows the plume states for rectangular plumes of two ARs, 3.4 and 5, and four density ratios, $S=0.138$ , 0.276, 0.414 and 0.552. The filled markers in figure 10 represent the puffing plumes, whereas the non-puffing plumes are denoted by open markers. It is evident from figure 10 that in spite of their low ARs, the rectangular plumes closely follow the transition criterion obtained from BiGlobal instability analysis with $\unicode[STIX]{x1D6FD}=0$ . This shows that the neutral curves obtained from instability analysis could effectively delineate the puffing and non-puffing regions for rectangular plumes of low AR. However, the square plumes (discussed in § 4.2.4) deviate from the predictions of the instability analysis, which indicates that there is an AR limit for the validity of these BiGlobal instability predictions.

Figure 10.

Mapping of plume behaviour on $Re{-}Fr$ plane for various $S$ , for both rectangular and square plumes. The open and filled markers correspond to non-puffing and puffing plumes respectively. The dashed lines are the neutral curves obtained from BiGlobal linear stability analysis with $\unicode[STIX]{x1D6FD}=0$ (shown in figure 5). The continuous lines are the transition curves obtained by Soteriou et al. (Reference Soteriou, Dong and Cetegen2002) from their simulations.

Further, we compare our transition curves with those obtained earlier by Soteriou et al. (Reference Soteriou, Dong and Cetegen2002). By performing Lagrangian simulations in planar plumes, Soteriou et al. (Reference Soteriou, Dong and Cetegen2002) obtained transition $Re$ at which the plumes become puffing based on a transition criterion. The stability boundaries obtained by them in $Fr$ range 0.1–0.3 (extracted from figure 14 of Soteriou et al. Reference Soteriou, Dong and Cetegen2002) are also presented in figure 10 for comparison. A comparison of their transition curves with that of the present instability analysis and experiments shows that their transition criterion yielded a more conservative estimate of critical $Re$ at which the planar plumes turn puffing, while the present work reasonably delineates these regions. The reason for this difference between the earlier work of Soteriou et al. (Reference Soteriou, Dong and Cetegen2002) and the present work lies in the transition criterion chosen by them for demarcating non-puffing and puffing regions, which is briefly described here.

For a fixed $Fr$ and $S$ , Soteriou et al. (Reference Soteriou, Dong and Cetegen2002) performed simulations by incrementing $Re$ , starting from the lowest possible value (see figure 13 in Soteriou et al. Reference Soteriou, Dong and Cetegen2002). At each $Re$ , they tested for unsteadiness at various streamwise locations along the plume column, and considered the plume to be non-puffing as long as the plume column remained steady. As the plume column turned unsteady, they considered the plume to be puffing. However, in figures 8 and 9, we have seen that the unsteadiness in these plumes can arise as a result of asymmetric instability as well. This is also supported by the fact that the initial instability observed in their work is also asymmetric (second and third columns in figure 13 of Soteriou et al. Reference Soteriou, Dong and Cetegen2002), which arises because of convective instability. Also, in the present experiments, we observed the appearance of asymmetric instability at low flow rates (figure 8 a,b) prior to the occurrence of symmetric puffing mode at high flow rates (figure 8 c,d). Therefore, the transition criterion of Soteriou et al. (Reference Soteriou, Dong and Cetegen2002) yielded a more conservative estimate of critical $Re$ . The FFT-based criterion of Sreenivasan et al. (Reference Sreenivasan, Raghu and Kyle1989) followed here seems to be a robust transition criterion to separate the puffing plumes from unsteady non-puffing plumes.

Figure 11.

Strouhal number variation with Richardson number for rectangular and square plumes.

Figure 12.

Streamwise visualizations. Evolution of a rectangular plume of $Re=480$ , $Fr=0.3$ and $S=0.553$ , emanating from an orifice of $B=62~\text{mm}$ ( $\text{AR}=3.4$ ), at various instants during a single puffing cycle. Puffing frequency for this case is 4.3 Hz. The dashed lines correspond to the $z$ locations at which the spanwise visualizations (figure 13) are performed.

Figure 13.

Spanwise visualizations performed at various $z$ locations for the case shown in figure 12. Panels (a–h) correspond to the same time instants as in figure 12.

The puffing frequencies measured for various rectangular plumes are represented as their Strouhal number variation with Richardson number in figure 11. The $St{-}Ri$ correlation obtained from BiGlobal linear stability analysis, i.e. $St=0.54Ri^{0.44}$ , is also plotted for comparison. Overall, the rectangular plumes of various ARs closely follow the $St{-}Ri$ correlation obtained from the BiGlobal stability analysis. However, at a fixed $Ri$ , it can be observed that the Strouhal numbers exhibit a marginal scatter with respect to the plume ARs. For instance, in figure 11, the $St$ values of $\text{AR}=3.4$ largely lie above the $St{-}Ri$ correlation of the unstable BiGlobal mode, whereas the $St$ values of $\text{AR}=5$ lie below it. Also, this scatter can be prominently observed at low Richardson numbers. Thereby, rectangular plumes exhibit a weaker dependence on AR.

Overall, the global dynamics of rectangular plumes is in reasonable agreement with that of the BiGlobal instability predictions, which are obtained assuming these plumes to be 2D in nature. This shows that the puffing dynamics in rectangular plumes largely remains 2D even at such low ARs. To further verify this inference, we performed both streamwise and spanwise flow visualizations (figure 2) in puffing rectangular plumes to examine their three-dimensionality. These measurements are performed in a phase-resolved manner to resolve various time instants during a puffing cycle, as explained in § 3. The streamwise visualizations assist in identifying the flow features observed in spanwise visualizations (e.g. puffing vortex (PV), central stem (CS), etc.).

4.2.3 Three-dimensionality: phase-locked streamwise and spanwise visualizations

Figure 12(a–h) shows the streamwise evolution of a typical puffing rectangular plume in the $x{-}z$ plane at various time instants during a single puffing cycle. Figure 13(a–h) shows the corresponding spanwise visualizations in the $x{-}y$ plane, at four different heights (i.e. $z=0.08$ , 1, 2 and 3). The position of PV in figure 13 has been identified by correlating it with figure 12. The spanwise visualizations show that the flow remains uniform near the plume inlet at all time instants of a puffing cycle, as observed at $z=0.08$ in figure 13. As the plume fluid convects downstream, two phenomena occur. First, the global instability leads to the periodic formation of spanwise vortex pair as discussed earlier and leads to puffing (marked as PV in figures 12 and 13). Second, the distortion of shear layer at the corners (marked as D in figure 13) leads to the formation of four streamwise vortices, aligned with the corners (shown as corner vortices (CVs) in figure 13). These CVs are also observed in non-puffing steady laminar plumes, demonstrated here in figure 14 for a laminar rectangular plume of $\text{AR}=10$ . This implies that these vortices are not related to the global instability in plumes.

In constant-density jets ejecting from rectangular orifices and ducts, the formation of CVs had been widely reported in earlier works (Quinn Reference Quinn1992; Zaman Reference Zaman1996; Gutmark & Grinstein Reference Gutmark and Grinstein1999; Sau Reference Sau1999, Reference Sau2002). They concluded that these streamwise vortices represent the Prandtl secondary flow of first kind that occurs due to unequal flow acceleration and resulting shear layer distortion nearer to the corners. In orifice-generated homogeneous jets, this secondary flow occurs as a result of flow acceleration because of vena-contracta (Quinn Reference Quinn1992). A secondary flow phenomenon, similar to that observed in orifice-generated constant-density jets, occurs in the present case as a result of buoyancy, which leads to the formation of these CVs. This phenomenon will be briefly described below.

As the plume fluid ejects out into the atmosphere, it undergoes a lateral contraction due to buoyant acceleration as observed at all time instants in figure 12. However, the buoyant acceleration, and thereby the lateral contraction, is not uniform over the entire span as observed from spanwise visualizations at $z=0.08$ in figure 13. The plume fluid near the shorter edges accelerates more slowly than the plume fluid at the centre. This is evident from spanwise visualizations at $z=0.08$ in figure 13 in which the plume fluid converges laterally all along the span, while it remains intact near the shorter edges, on either extremes of the span. This unequal acceleration of plume fluid results in the distortion/skewing (D) of the shear layer near the corners as highlighted at $z=0.08$ in figure 13(d), which results in the generation of streamwise vorticity at the corners. Further downstream, this skewed shear layer at four corners rolls up into four streamwise vortices aligned with plume corners, similar to that observed by Zaman (Reference Zaman1996) and Sau (Reference Sau1999) in the case of free rectangular jets. Thus, it can be inferred that the CVs correspond to the secondary flow superposed over the primary plume flow because of the presence of corners. Zaman (Reference Zaman1996) had shown that the orientation of these CVs influences the downstream development of the jet by promoting or resisting the axis-switching phenomenon, observed in non-circular jets (Gutmark & Grinstein Reference Gutmark and Grinstein1999). In the present buoyant plumes, the CVs formed are of ‘outflow’ type as shown at $z=1$ in figure 13(d), exactly similar to that observed by Zaman (Reference Zaman1996) in the case of free rectangular homogeneous jets. In this ‘outflow’ configuration, the two CV pairs located at either side of the span tend to elongate the plume cross-section along the span as shown at $z=1$ in figure 13(d), thus resisting the axis-switching of plume cross-section in buoyant plumes. Therefore, no axis-switching phenomenon has been observed in the present buoyant plumes.

Figure 14.

Spanwise visualization of a non-puffing steady plume of $\text{AR}=10$ at two arbitrary time instants (a,b). The visualizations show the presence of CVs even in the absence of puffing.

Now, we revert back to our inference drawn previously in § 4.2.2 regarding puffing dynamics in rectangular plumes. In spite of smaller ARs, we had observed in the previous section that the puffing frequencies and plume transition in rectangular plumes are in good agreement with those obtained from BiGlobal instability analysis with $\unicode[STIX]{x1D6FD}=0$ . The spanwise visualizations of puffing plumes in the near field, i.e. $z\leqslant 1$ in figure 13, show that, barring secondary flow, the plume CS and the PVs indeed remain largely 2D without any spanwise variations. This 2D nature CS and PV is more pronounced for higher AR plumes. This shows that puffing is 2D in a rectangular plume, even at such low AR. However, further downstream, the spanwise undulations develop along the plume stem as observed at $z=2$ in figure 13, which leads to a rapid breakdown of plume fluid and transition to turbulence.

Up to now, we have investigated the puffing behaviour in rectangular plumes of $\text{AR}\geqslant 3$ . These plumes exhibited a weaker dependence on the ARs. Next, we consider the effect of a further decrease in AR by investigating the puffing dynamics in square plumes, i.e. $\text{AR}=1$ .

4.2.4 Square plumes

Experiments are performed using a square orifice of dimensions $B=L=62~\text{mm}$ , at four density ratios $S=0.138$ , 0.276, 0.414 and 0.552, by systematically increasing the plume inlet velocity at each density ratio, similar to that performed in the case of rectangular plumes. These square plumes are categorized into non-puffing and puffing states based on temporal flow visualizations and FFT approach as explained in § 4.2.1, and the resulting plume states are presented in figure 10 along with rectangular plumes. Also, the Strouhal numbers of the puffing square plumes are presented in figure 11.

Overall, it is observed that the puffing dynamics in square plumes differs from that in 2D or rectangular plumes. Figure 10 shows that square plumes are more stable than 2D/rectangular plumes, i.e. puffing tends to initiate in square plumes at higher critical $Re$ when compared to 2D/rectangular plumes. Also, for a fixed $Ri$ , the puffing Strouhal number in a square plume is higher than that in a 2D/rectangular plume. The puffing frequencies in square plumes correlate as $St=0.65Ri^{0.44}$ as shown in figure 11. It is interesting to note that the power-law exponent for square plumes is same as that for the rectangular plumes, i.e. $St\propto Ri^{0.44}$ . However, the proportionality constant is higher for square plumes. We further highlight the differences observed between rectangular and square plumes in both non-puffing and puffing regimes.

Figure 15.

Flow field in non-puffing (a) rectangular and (b) square plumes of $Re=60$ , $Fr=0.05$ and $S=0.414$ .

Figure 15 compares the flow field of a non-puffing rectangular ( $\text{AR}=3.4$ ) and square plume with similar inlet conditions, i.e. same ( $Re$ , $Fr$ , $S$ ). From the time evolution of these plumes, it can be observed that the rectangular plumes undergo asymmetric instability as observed in figure 15(a) and as discussed earlier in § 4.2.1. Whereas, the square plumes remain stable even in the far field as observed from their time evolution in figure 15(b). The far field asymmetric convective instability observed in rectangular plumes disappears as $\text{AR}\rightarrow 1$ .

Figure 16.

Streamwise visualizations. Evolution of a square plume of $Re=235$ , $Fr=0.3$ and $S=0.28$ , emanating from an orifice of width $B=62~\text{mm}$ , at various instants during a single puffing cycle. Puffing frequency for this case is 6.2 Hz. The dashed lines correspond to $z$ locations at which the spanwise visualizations (figure 17) are performed.

Figure 17.

Spanwise visualizations (in $x{-}y$ plane) at various $z$ locations for the case shown in figure 16. Panels (a–h) correspond to the same time instants as in figure 16 (CVP, corner vortex pairs).

Further, we compare the PV of a square plume with that of a rectangular plume and attempt to explain the reason for higher Strouhal numbers observed in square plumes. Figure 16 shows the streamwise visualizations of a puffing square plume at various time instants during a single puffing cycle. The corresponding spanwise visualizations performed at various heights are shown in figure 17. It is evident from the spanwise visualizations that the structure of the PV (shown in figures 16 and 17) is highly 3D, whereas a PV in rectangular plumes remains largely 2D as seen earlier in figure 13. Cetegen & Ahmed (Reference Cetegen and Ahmed1993) investigated the puffing mechanism in circular plumes by introducing various mechanisms that prevent the formed PV from interacting with the plume fluid nearer to the source. From their experiments, they proposed that the puffing frequencies are strongly coupled with the downstream development of the PV. They observed that the perturbations introduced by the downstream convecting PV act as source of instability to the plume fluid nearer to the source, which result in the formation of the next PV. That is, the induction and convection of the formed PV provide the coupling to sustain these periodic oscillations. Therefore, we attribute the difference observed in puffing frequency correlations of square and rectangular plumes to their PV dynamics. We conjecture that the 3D structure and dynamics of PVs in square plumes might have resulted in an increase in their puffing frequencies compared to those of the rectangular plumes.

Additionally, the flow field in square plumes is dominated by four corner vortex pairs (as shown in figure 17) similar to that observed in momentum-dominated homogeneous square jets (Quinn Reference Quinn1992; Gutmark & Grinstein Reference Gutmark and Grinstein1999).