2.1. Large eddy simulations

The methodology used here for the LES is similar to that used for direct numerical simulations in Moise & Mathew (Reference Moise and Mathew2019) and Moise (Reference Moise2020a). The simulations were performed using incompact3d (Laizet & Li Reference Laizet and Li2011), an open-source flow solver which employs the projection method for incompressible flow simulations using a partially staggered Cartesian grid. The solver uses sixth-order, compact, finite difference schemes with spectral-like resolution (Lele Reference Lele1992) to compute derivatives and for interpolation, while the Poisson's equation for pressure is solved in spectral space (Laizet & Lamballais Reference Laizet and Lamballais2009; Lamballais, Fortuné & Laizet Reference Lamballais, Fortuné and Laizet2011). A low storage, third-order Runge–Kutta scheme was selected for time stepping.

The inflow conditions were chosen as the Maxworthy inflow profile (Ruith et al. Reference Ruith, Chen and Meiburg2004). In a cylindrical coordinate framework, the azimuthal, radial and axial velocity components are given by

(2.1)\begin{equation} \left.\begin{gathered} u_\theta = \frac{Sr}{2}\left(1-{erf}\left(\frac{r-1}{\delta}\right)\right) \\ u_r = 0 \\ u_x = 1-\frac{\alpha-1}{2\alpha} \left(1+{erf}\left(\frac{r-1}{\delta}\right)\right), \end{gathered}\right\} \end{equation}

where $r$ represents the radial distance from the swirling jet axis. This axisymmetric profile is dependent on three parameters: $S$, the swirl rate, $\alpha$, the core-to-coflow axial velocity ratio and $\delta$, representing the shear layer thickness of the swirling jet. The profile is in a dimensionless form, with the length and velocity scales chosen as the jet's radius and centreline axial velocity, respectively. The Reynolds number, $Re$, is defined based on these scales and all further references to lengths and velocities will be in the dimensionless form. Note that the swirl rate, $S$, represents the slope of the dimensionless azimuthal velocity at centreline (i.e. $\textrm {d} u_\theta /\textrm {d} r(r=0)$) and should not be confused with the more commonly used swirl number, which is usually denoted by the same symbol $S$ and is defined as $S = G_\theta /RG_x$ ($G_\theta$ and $G_x$ are the axial components of angular and linear momentum, respectively and $R$ is the pipe radius (e.g. Chigier & Chervinsky Reference Chigier and Chervinsky1967) or as the ratio of azimuthal to axial inflow velocity scales. These variables, both of which represent the inflow swirl strength, are directly proportional to each other for the Maxworthy profile, irrespective of the definition used. Thus, all further descriptions of inflow swirl strength are provided using the swirl rate alone. Henceforth, the symbol $S$ is also used to denote the same, following a convention similar to those used in previous computational studies (Ruith et al. Reference Ruith, Chen, Meiburg and Maxworthy2003, Reference Ruith, Chen and Meiburg2004).

In this study, only $S$ is varied, while the other parameters are fixed as $Re = 1000$, $\delta = 0.2$ and $\alpha = 100$. The inflow condition was inadvertently set as that of a clockwise swirling jet, implying that $S$ is negative in a right-hand coordinate framework, but the negative sign is ignored without loss of generality. The swirl rate, $S$, is a constant for a given simulation except when considering hysteresis effects, in which case, it varies linearly with time for a short duration till a target value is achieved, after which it remains constant (see Moise Reference Moise2020a). It is emphasized that in all simulations the inflow is modelled as a laminar swirling jet without adding any unsteady perturbations.

As noted above, the simulations were carried out in a Cartesian coordinate framework, with $x$ retained as the streamwise direction, while $y$ and $z$ represent the lateral directions. The lateral boundaries are assumed periodic for convenience, while the standard convective boundary condition was employed at the outflow. The domain is a cube of side 40 each. For this choice, the effect of lateral boundary conditions on VB remained negligible for most cases, except for the wide-open type of CVB (see § 3.3.2). A uniform grid spacing of $\varDelta = 1/12$ was used in all three directions, while the time step was chosen as 0.01. The Courant number based on these parameters is 0.12. Details on the requirement for LES and the effect of domain dimensions are provided in Pradeep (Reference Pradeep2019).

The LESs were carried out using the explicit filtering approach proposed in Mathew et al. (Reference Mathew, Lechner, Foysi, Sesterhenn and Friedrich2003). This method is a reformulation of the approximate deconvolution model introduced in Stolz & Adams (Reference Stolz and Adams1999) into a simpler form. It is particularly suited for the present problem involving a transitional flow. Indeed, it has been demonstrated to accurately simulate various transitional flows (Visbal Reference Visbal2009; Rizvi & Mathew Reference Rizvi and Mathew2017). An approach closely related to this method has also been used to simulate transitional round jets (Bogey & Bailly Reference Bogey and Bailly2006, Reference Bogey and Bailly2009). The explicit filtering approach employs the same procedure as that used for a direct numerical simulation (performed using high-order differentiation schemes), but with an additional step of applying an appropriately chosen low-pass numerical filter to the computed velocity field in all directions at the end of each time step. Compact schemes for filtering applications (see Lele (Reference Lele1992), p. 40) were incorporated into the flow solver for this purpose. The filter is based on parameters, $n_f$ and $\alpha _f$. The former denotes the scheme's order, while the latter is inversely related to the damping effect of the filter and is chosen in the range $0 \leq \alpha _f \leq 0.5$. The implicit filter is defined by

(2.2)\begin{equation} \alpha_f u'_{i-1} + u'_i + \alpha_f u'_{i+1} = \sum_{n=0}^{n_f} a_n(u_{i+n}- u_{i-n}), \end{equation}

where $u_i'$ and $u_i$ represent the filtered and unfiltered velocity component at a grid point of index, $i$, (in a given direction), while $a_n$ are coefficients which depend on $\alpha _f$. For a periodic function on $[0,L]$ at $N = L/\varDelta$ grid points, a Fourier transform of the form $f = \varSigma _w \hat {f}_w \exp (2{\rm \pi} \textrm {i}wx/L)$ can be applied, where $\hat {f}_k$ is the Fourier coefficient, $w$ is the wavenumber (ranging from $-N/2$ to $N/2$) and $x$ is distance. This can be used to compute the transfer function in Fourier space, $H(w) = \hat {u}'_w/\hat {u}_w$. Using a scaled wavenumber of the form $k = 2{\rm \pi} w \varDelta /L$ (range $[0,{\rm \pi} ]$), the transfer function is then given by

(2.3)\begin{equation} H(k) = \frac{\varSigma_n a_n\cos(nk)}{1+2\alpha_f\cos(k)}, \end{equation}

which is shown in figure 1(a) for different $\alpha _f$ when $n_f = 10$. It is clear from the figure that the low wavenumber content remains mostly unaffected by such low-pass filters. In this study, $n_f = 10$ and $\alpha _f = 0.495$, unless otherwise mentioned.

Figure 1.

(a) Variation of the transfer function with wavenumber for tenth-order filter and different $\alpha _f$. (b) Schematic showing the extended ranges of existence of various VB flow states observed for $Re = 1000$. The dashed line at $S=1.2$ shows the upper limit of the range where $|m| = 4$ structures are observed. Arrows indicate the sense in which $S$ is changed in the hysteresis studies.

A ‘sponge layer’ was introduced at the outflow boundary that reduces the intensity of unsteady vortices associated with the turbulent flow locally while negligibly affecting the flow in other regions of the domain. This region had a streamwise length of 2 and spanned the entire lateral extent of the domain. In this region, the velocity field was strongly filtered at each time step using the same approach as that used for the LES but by employing a second-order filter ($n_f = 2$) with $\alpha _f = 0.4$.

All statistical quantities were computed by averaging flow fields at every time step after transients for a duration of ${\rm \Delta} T \approx 1700$. Thus, statistical convergence is expected even if coherent features of frequencies as low as 0.005 (time period 200, implying 8 cycles) exist. Further, azimuthal averaging was performed to increase statistical convergence as the mean flow is expected to be axisymmetric. Only data from the $y = 0$ and $z = 0$ planes was used for convenience.

2.2. Spectral proper orthogonal decomposition

Coherent structures observed in the flow field were extracted using the spectral proper orthogonal decomposition (SPOD) approach (Lumley Reference Lumley1970; Picard & Delville Reference Picard and Delville2000; Towne, Schmidt & Colonius Reference Towne, Schmidt and Colonius2018). An SPOD mode is defined as the eigenfunction, $\boldsymbol \psi _i(\boldsymbol x,f)$, associated with the cross-spectral density tensor, ${\boldsymbol{\mathsf{S}}}(\boldsymbol x,\boldsymbol x',f)$. Here, $\boldsymbol x$ and $\boldsymbol x'$ denote position vectors, $t$ and $t'$, two time instants and $f = 1/|(t'-t)|$, the frequency based on the time interval between the instants. The eigenfunction satisfies

(2.4)\begin{equation} \langle {\boldsymbol{\mathsf{S}}}(\boldsymbol x,\boldsymbol x',f), \boldsymbol\psi_i(\boldsymbol x',f)\rangle_{\boldsymbol x'}^* = \lambda_i(f)\boldsymbol\psi_i(\boldsymbol x,f), \end{equation}

where $\langle ,\rangle _{\boldsymbol x'}$ represents an inner product with respect to $\boldsymbol x'$, ‘$^*$’ denotes conjugate transpose and $\lambda _i$ is the eigenvalue corresponding to $\boldsymbol \psi _i$, representing the relative energy associated with the SPOD mode (see Moise (Reference Moise2020a), pp. 7–8). The structures at a given frequency $f_0$ are given by $\boldsymbol \psi _i(\boldsymbol x,f_0)$ and can be identified as coherent when $\lambda _i(f_0)$ is non-negligible. To compute these, the MATLAB implementation of Towne et al. (Reference Towne, Schmidt and Colonius2018) was used. The code uses the Welch's method for better spectral estimation. A Hamming window was selected so as to reduce spectral leakage. In the LES, after the stationary flow state has been achieved, the velocity fields were stored at regular intervals in time. The cross-spectral density tensor was computed based on the velocity field, $\boldsymbol u(x,y,t)$, in the positive $xy$-plane alone, since only spiral modes of azimuthal wavenumber, $m= +1$ (counter-winding and co-rotating), were observed for the VB states examined in this study. The velocity field at each instant was arranged into a single column vector, referred to as a snapshot. The total number of snapshots collected for each case and the corresponding sampling frequency are denoted by $N$ and $f_S$, respectively. For all cases, $N \approx 1500$ and $f_S = 1$. These were divided into $N_B$ blocks with each containing $N_{SpB} = 256$ number of snapshots per block while a 50 % overlap between blocks (implying around 10 blocks in each case) was additionally used to increase the number of realizations.

3.1. Overview

The spatio-temporal features of various VB flow states that occur when the swirl rate, $S$, is varied for $Re = 1000$ are reported in this section, while aspects of bistability and SPOD are discussed in subsequent sections. The swirl rate range that was examined for this study is $0.8 \leq S \leq 2.5$. For all swirls, a transition to turbulence was observed at this Reynolds number.

For VB at $Re = 200$ studied in Moise & Mathew (Reference Moise and Mathew2019), an increase in $S$ beyond a critical value, $S_c$, leads to steady laminar VB, first with the development of a stagnation point and with a further increase, a steady bubble. By contrast, in the present study, intermittent flow reversal on the flow axis was seen at the lowest $S$ for which a bubble is present in the time-averaged velocity field. Thus, for convenience, $S_c$ is defined here as that below which no recirculation zones are observed in the mean flow. Similar intermittent behaviour about this critical swirl has been reported in experiments (e.g. Liang & Maxworthy (Reference Liang and Maxworthy2005) and, for fully turbulent swirling jet inflow conditions Oberleithner et al. Reference Oberleithner, Paschereit, Seele and Wygnanski2012). Here, it was observed that $S_c = 1.14$.

The flow states that occur for $S < S_c$ are categorized as pre-VB states. For these, a non-rotating, large-scale, spiral structure of azimuthal wavenumber, $|m| =4$, was observed. This structure, which is similar to that seen in laminar pre-VB states in Moise & Mathew (Reference Moise and Mathew2019), seems to be spurious and probably arises due to the choice of the Cartesian coordinate framework used. The BVB occurs for $S > S_c$, but this structure is also present in the swirl range of $1.14 \leq S \leq 1.2$. Hence, features at VB onset are not scrutinized in detail and this study focusses mainly on VB flow states that occur for $S \geq 1.3$, which do not contain this $|m| = 4$ structure (see figure 1(b) and table 1).

Table 1.

Approximate ranges of existence of turbulent flow states observed for streamwise-invariant initial conditions based on the Maxworthy inflow profile for simulations in range $0.8 \leq S \leq 2.5$. Additionally, the range where the possibly spurious $|m| = 4$ structures occur is provided. Here, $S^l$ and $S^u$ represent the lower and upper limits of $S$, respectively.

While the highest swirl examined is $S = 2.5$, it is cautioned that with increasing $S$, VB occurs closer to the inflow plane. This is a cause for concern at high swirls ($S \geq 1.66$) due to the steady nature of the imposed inflow conditions, but the results are still expected to be relevant, as elaborated in § 6.3. The ranges of $S$ in which different flow states are observed in the simulations when the initial conditions used are streamwise-invariant (i.e. $\partial \boldsymbol u/\partial x = 0$ at all $x$ and based only on the inflow profile) are provided in table 1. It is possible to sustain the VB states for a larger range by exploiting hysteresis behaviour. A schematic based on these extended and overlapping swirl ranges is shown in figure 1(b). The boundaries of each range are only accurate up to a difference in $S$ of $\delta S = 0.05$. As seen from the figure, BVB is observed for relatively lower swirl rates, while the wide-open type of CVB occurs only for very high values of $S$. The same trends have also been observed for laminar VB at $Re = 200$ (Moise & Mathew Reference Moise and Mathew2019). Note that the swirl range associated with streamwise-invariant initial conditions (table 1) of each VB state is indicative of that part of the extended range (figure 1b) in which that state is relatively more stable.

3.2. Bubble form of vortex breakdown

As noted above, the $|m|=4$ structures are absent for $S \geq 1.3$. Thus, this study focusses on VB states that occur for $S \geq 1.3$, but for completeness, features about the onset of VB ($S_c = 1.14$) are documented briefly in appendix A. It is emphasized that these results are inconclusive due to the presence of the $|m| = 4$ structures. Nevertheless, one interesting feature seen is that the flow exhibits coherent periodic oscillations for $S = 1.13$ and 1.14 (figure 21b), which is also reported in experiments on transitional annular swirling jets close to the onset of VB (Vanierschot & Ogus Reference Vanierschot and Ogus2019).

An overview of the features of BVB observed in the range of $1.3 \leq S \leq 1.5$ is provided first. Instantaneous flow features for $S = 1.3$ are shown in figure 2. The flow remains laminar for $x \leq 9$ at this swirl, as indicated by the pressure field shown (see also figures 3 and 5b). Two isolated recirculation zones are present over approximately, $2 \leq x \leq 6$ and $8 \leq x \leq 10$. The upstream one is referred to here as the ‘bubble’ and has a two-celled structure. In the entire range of $1.3 \leq S \leq 1.5$, the BVB has a two-celled structure, but at higher $S$, the flow becomes turbulent within the bubble region and thus, the two-celled structure can be inferred only from the mean flow and not instantaneous features. The mean radius of the bubble increases with increasing $S$, while the mean streamwise position of the bubble decreases. The bubble has no stagnation point at its nose and is toroidal (see streamline pattern in figure 2a) in the swirl range of $1.3 \leq S \leq 1.45$ (and also for $1.2 \leq S \leq 1.3$, for which $|m| = 4$ structures are present). By contrast, this stagnation point is present for $S = 1.5$. Similar trends were also reported for laminar BVB (Moise Reference Moise2020a). The two-celled BVB in swirling jets can be inferred from experiments (see Billant et al. (Reference Billant, Chomaz and Huerre1998), figure 7(a), p. 196, where the contours show presence of four stagnation points in the bubble region) and is also commonly observed in other swirling flows (Faler & Leibovich Reference Faler and Leibovich1978). However, the toroidal structure seems not to be so commonly reported, but might occur in some swirling flows (Lucca-Negro & O'Doherty (Reference Lucca-Negro and O'Doherty2001); also see § 6.1.1). The time-averaged streamwise position of the bubble moves upstream with increasing $S$, but stabilizes for $S \geq 1.4$ while the bubble's size continues to increase, a trend similar to those observed in experiments (Escudier & Keller Reference Escudier and Keller1985; Oberleithner et al. Reference Oberleithner, Paschereit, Seele and Wygnanski2012; Manoharan et al. Reference Manoharan, Frederick, Clees, O'Connor and Hemchandra2020). These features are further elucidated below by comparing two typical cases of $S = 1.3$ and $1.4$.

Figure 2.

Instantaneous flow features of bubble form of vortex breakdown (BVB) at $S = 1.3$ shown on the $z = 0$ plane using (a) projected streamlines overlaid on axial velocity contours and (b) pressure contours.

Figure 3.

The spatial structure of the bubble form of vortex breakdown (BVB) is shown using $\lambda _2$-criterion. Isosurfaces with $\lambda _2 = -0.4$ for $S = 1.3$ (a) and $S = 1.4$ (b), both at an arbitrary time instant, are plotted and coloured based on $u_x$.

3.2.1. Comparison of instantaneous features

The instantaneous vortical structures in the flow were identified using isosurfaces of $\lambda _2$ (Jeong & Hussain Reference Jeong and Hussain1995). These are shown for the two cases of $S = 1.3$ and 1.4 in figure 3. Note that with the formation of the RZ, the swirling jet develops an ‘inner’ shear layer in addition to the ‘outer’ shear layer with the coflow (Liang & Maxworthy Reference Liang and Maxworthy2005). In the figures, this swirling region (sandwiched between the two shear layers) appears as the spheroidal structure in $3\leq x\leq 6$ and represents the bubble's envelope. For $S = 1.3$, the transition to turbulence is positioned well downstream of the bubble ($x \approx 10$), while for $S = 1.4$, it occurs at the rear end of the bubble ($x \approx 6$). A spiral coherent structure can be discerned downstream of the bubble for the former case. This structure has an azimuthal wave number $m = +1$ (counter-winding and co-rotating) and does not affect the bubble (see also § 5) implying that this is not the PVC. The position of transition to turbulence moves upstream with increasing $S$ which can also be inferred from the figure. Nevertheless, in the entire swirl range associated with the BVB, most of the bubble region remained approximately axisymmetric and laminar.

The dynamics of the flow for the two cases are best understood by examining the animations provided as supplementary movies 1, 2, 3 and 4 available at https://doi.org/10.1017/jfm.2021.118 which show temporal variation of axial velocity and vorticity magnitude on the $z= 0$ plane for $S = 1.3$ and 1.4. The streamwise oscillations of the bubble can be clearly seen from these animations. Further, the bubble changes in size due to these oscillations. There are interesting similarities between this flow state and the pulsating type of BVB reported in Moise & Mathew (Reference Moise and Mathew2019). The oscillations can also be inferred from figure 4 where the centreline variation of axial velocity is plotted at different time instants. For both swirls, the flow is unsteady, including in the laminar upstream region. The mild unsteadiness observed close to the inflow plane might be of concern due to the imposition of steady inflow conditions and it is possible that the characteristics of the streamwise oscillations might be different if these artificial constraints are removed. However, streamwise oscillations for BVB are also reported in experiments (Liang & Maxworthy Reference Liang and Maxworthy2005; Oberleithner et al. Reference Oberleithner, Paschereit, Seele and Wygnanski2012) and there are strong reasons to expect the results to be valid at such low values of inflow swirls, as elaborated in § 6.3.

Figure 4.

Axial velocity variation along the swirling jet axis for the bubble form of vortex breakdown (BVB) at different time instants and for (a) $S = 1.3$ and (b) $S = 1.4$.

For $S = 1.3$, in the bubble region (approximately $2 \leq x \leq 6$, see figure 3a), flow reversal on the axis is absent at all times. By contrast, intermittent flow reversal on the axis was observed in the same region for $S = 1.4$, as shown in figure 4(b). This implies that for the case of $S = 1.3$, the RZ remains toroidal at all times and a core region of flow along the streamwise direction is present in the bubble region, while for higher $S$, this occurs only intermittently.

3.2.2. Comparison of time-averaged features

The time-averaged features for the two cases are shown in figure 5(a) using projected mean streamlines on the meridional plane, along with contours of mean axial velocity. For $S = 1.3$, two recirculation zones can be seen in the regions $1 \leq x \leq 6$ and $8 \leq x \leq 10$. The upstream one is relatively larger in size and is the two-celled bubble. There is no stagnation point at the nose of the bubble, which is expected, since the instantaneous flow fields show that the stagnation point is absent at all times (figure 4a). A projected streamline passes through the bubble, highlighting the toroidal nature of this upstream RZ. The downstream RZ is one-celled and associated with the spiral mode shown in figure 3(a) (see also § 5). For $S = 1.4$, although there is intermittent stagnation point formation at the bubble's nose, the mean flow still shows none implying that core flow reversal is not dominant. More importantly, the downstream RZ is absent for this case. The implications of these results are further considered in § 6.1.1. Contours of the root-mean-square (rms) of the fluctuating axial velocity component are shown in figure 5(b). It is clearly seen for $S = 1.3$ that the fluctuations are of relatively high intensity in the core region for $6 \leq x \leq 8$, which is associated with the streamwise oscillation of the spiral coherent structure. This feature is absent for $S = 1.4$.

Figure 5.

Time-averaged features based on azimuthally-averaged fields with (a) projected streamlines on meridional plane and axial mean velocity contours and (b) contours of rms of the fluctuating axial velocity field shown for cases $S = 1.3$ (a,b) and $S = 1.4$ (c,d).

3.3. Conical form of vortex breakdown

The CVB was observed for $S \geq 1.4$. Both the regular and wide-open types appear in overlapping swirl ranges (see figure 1b), features of which are discussed below.

3.3.1. Regular type

For both BVB and CVB there exist inner and outer shear layers downstream of VB associated with the core RZ and the coflow regions, respectively, with the jet flow confined within these. For the CVB, this flow takes the shape of a ‘conical sheet’ in which the swirl strength becomes negligible due to strong radial expansion. This sheet is highlighted for the regular type of CVB for a typical case of $S = 1.8$ in figure 6(a) using isosurfaces of vorticity magnitude. The characteristic conical shape associated with this form of VB in the upstream regions is clearly visible in the figure (cf. figure 3). The transition to turbulence occurs well downstream of the cone's vertex and the presence of a coherent structure in this region can also inferred. This vortical structure was found to be a spiral of azimuthal wavenumber, $m = +1$ (counter-winding and co-rotating, see § 2) and is highlighted in figure 6(b) using $\lambda _2$ isosurfaces. This feature is further examined in § 5 using SPOD. Incidentally, it is also noted that unlike the bubble form, no pronounced streamwise oscillation was seen for this VB form at any swirl rate at which it exists. Additionally, the stagnation point at the cone's vertex was observed to be present at all times for all cases of $S$ where the CVB was observed.

Figure 6.

Three-dimensional spatial structure of regular conical form of vortex breakdown (CVB) for $S = 1.8$, visualized using isosurfaces of (a) vorticity magnitude, $|\boldsymbol \omega | = 1.5$ and (b) $\lambda _2 = -2$. Inflow swirl is clockwise (see § 2) implying a counter-winding spiral vortical structure.

An important feature of turbulent CVB that can be observed in figure 6 is the relatively lower radial spread of the flow downstream of VB when compared to the laminar CVB (cf. Moise & Mathew (Reference Moise and Mathew2019), figure 9, p. 336). Indeed, the maximum radius of the RZ for the present turbulent case is approximately half that achieved by the latter, which suggests the possibility that the transition to turbulence might be the cause for the limited spread of the flow.

The time-averaged flow features for $S = 1.8$ are shown in figure 7. It can be seen that the RZ has a one-celled structure. While the projected streamlines are seen to expand approximately conically downstream to the stagnation point, they strongly curve as the maximum radius of the RZ is attained. Indeed, the downstream parts of the RZ strongly resembles in shape the corresponding regions of the bubble. This similarity will be further discussed while examining bistability features in § 4.2.

Figure 7.

Time-averaged flow structure of regular conical form of vortex breakdown (CVB) at $S = 1.8$ on meridional plane: (a) projected streamlines overlaid on contours of mean axial velocity and (b) rms of the fluctuating axial velocity component.

3.3.2. Wide-open type

Features of the wide-open type of CVB are discussed using a typical case of $S = 2.15$. Instantaneous features for this case are shown at an arbitrary instant in figure 8. It can be inferred from these that the flow, downstream of VB, has a large radial expansion and moves approximately parallel to the inflow plane (with $u_x \approx 0$), as is typical of this type of the conical form. Note that this large radial expansion leads to reduced velocity magnitudes, since the flow is now spread over a larger area, while the flow rate remains constant. It was observed that this strong radial spread leads to the flow interacting with the lateral boundaries, which can cause artificial confinement effects. However, further extending the boundary incurs a large numerical expense and hence, only the case of this swirl rate ($S = 2.15$) was studied for in a laterally extended domain. This is discussed in appendix B, where it is shown that most qualitative features of this type, including hysteresis behaviour, remain the same irrespective of the domain chosen. The only differences observed were that, in the extended domain, the RZ's radius is larger while its streamwise extent is reduced. Note that the characteristic features of this type of CVB are observed in the vicinity of the inflow plane, while downstream, there is negligible flow. Since the former is captured well in both domains, the smaller domain was employed at all other swirl rates examined here, so as to reduce numerical expense.

Figure 8.

Instantaneous features for wide-open conical form of vortex breakdown (CVB) at $S = 2.15$: axial velocity contours on (a) $z = 0$ and (b) $x = 2$ planes and (c) isosurfaces of vorticity magnitude ($|\boldsymbol {\omega }| = 0.3$). The axial velocity range shown is chosen for clarity and is smaller than the actual range.

The time-averaged features of the wide-open CVB are shown in figure 9. It is seen from the figure that the approximately conical sheet downstream of VB sharply reverses direction and moves towards the inflow plane (see region of $r \approx 5$ and $1 \leq x \leq 3$) as it continues to expand radially. Indeed, the strongest flow reversal ($u_x < 0$) in the mean flow field is observed in this region and not in the RZ, while the fluctuating component has maximum intensity here. Note that the flow interacts with the lateral boundary, at which the projected streamlines become almost parallel to the boundary ($7 \leq x \leq 20$ and $r \approx 20$). This effect was found to be absent when the domain is extended (see appendix B). Additionally, the RZ extends almost to the streamwise boundary for the present domain which is not the case when the lateral boundaries are extended.

Figure 9.

Time-averaged flow features for wide-open conical form of vortex breakdown (CVB) at $S = 2.15$: (a) projected streamlines and contours of $u_x$ and (b) rms of fluctuating axial velocity on meridional plane.

4.1. Regular and wide-open types of the conical form

Projected streamlines on the meridional plane based on the time-averaged velocity fields of the regular and wide-open types of CVB at $S = 2.1$ are compared in figure 10(a). The stark difference in the size of the RZ between the two types is evident, with the ‘eye’ of this zone positioned at $x \approx 5$ and $r \approx 4$ for the regular type, while $x \approx 12.5$ and $r \approx 14$ for the wide-open type. However, it was also observed that the flow structure upstream of VB ($x \approx 1$) is approximately the same for both cases. This is highlighted in figure 10(b), which shows the centreline mean axial velocity for the two types. It can be clearly seen that the differences are negligible upstream of the position of VB (i.e. the first point along the axis where $u_x = 0$, see inset). The differences manifest downstream, with the reverse flow relatively weaker for the wide-open type. The existence of differences only downstream of VB and the hysteresis behaviour observed suggest that the wide-open type might be sustained due to the Coanda effect. This was confirmed by examining the pressure fields and is discussed in § 6.2.3, but it is apparent from the figure that there are large differences between the two types, clearly establishing that it is useful to classify the wide-open type as distinct from the regular type of CVB, albeit closely related.

Figure 10.

Comparison of time-averaged flow features for $S = 2.1$ using (a) projected streamlines on meridional plane and (b) centreline axial velocity variation (inset showing variation in the vicinity of the stagnation point), for conical form of vortex breakdown (CVB) types: ---------, black (thick), regular; ---------, red (thin), wide-open.

4.2. Bubble and conical forms

A comparison of the bistable bubble and conical forms of VB for $S = 1.5$ is made in figure 11. The instantaneous axial velocity contours on the $z = 0$ plane clearly highlight the spheroidal and conical recirculation zones of the BVB and CVB respectively. The remaining plots show the time-averaged features. The presence of a stagnation point at the RZ's nose in the time-averaged flow field for both forms can be inferred from the streamwise variation of centreline velocity plotted in figure 11(c) (see inset). The two-celled structure of the BVB and the relatively larger one-celled RZ of the CVB are compared in figure 11(d). It is instructive to compare the present turbulent case ($Re = 1000$) with that at the same $S$ in the laminar regime (cf. Moise (Reference Moise2020a), figure 3, p. 11, for $Re = 200$). For the latter, the recirculation zones for the CVB and BVB have a maximum radius of around 15 and 1, respectively, while in the present case, it is 6 and 2, respectively. Thus, it is seen that the differences between the two forms is considerably reduced with an increase in $Re$. The implications of these are further discussed in § 6.1.2. This difference between the two forms was observed to be further reduced at a lower swirl of $S = 1.4$, as shown in figure 12(a) (see also, movies 5 and 6 and the hysteresis diagram, figure 12b).

Figure 11.

A comparison of the bistable turbulent bubble and conical forms of vortex breakdown for $S = 1.5$. Instantaneous axial velocity contours of (a) bubble form of vortex breakdown (BVB) and (b) conical form of vortex breakdown (CVB) shown along with (c) centreline velocity (inset showing variation in the vicinity of the stagnation point) and (d) projected streamlines based on time-averaged velocity fields comparing the two forms: ---------, blue (thin), BVB; ---------, black (thick), CVB.

Figure 12.

(a) Comparison of time-averaged flow features using projected streamlines on meridional plane for bistable forms at $S = 1.4$: ---------, blue (thin), bubble form of vortex breakdown (BVB); ---------, black (thick), conical form of vortex breakdown (CVB). (b) A hysteresis curve based on maximum radial extent of a streamline starting from $x = 0$ and $r = 0.2$, both obtained as swirl rate, $S$, is varied for $Re = 1000$ (regions of overlap shaded grey): ($\triangle$, blue), pre-VB; ($\circ$, green), oscillating BVB; ($\times$, black), regular CVB; ($\square$, red), wide-open CVB.

4.3. Hysteresis diagram

A hysteresis diagram is shown in figure 12(b) based on all the stationary, sustained VB states that occur in overlapping $S$ ranges for $Re = 1000$ and the inflow condition, the Maxworthy profile with $\alpha = 100$ and $\delta = 0.2$. This is based on the maximum radius, $\sigma _{max}$, achieved by a projected streamline starting at $x = 0$ and $r = 0.2$, computed using the mean velocity field. Note that the streamline chosen is arbitrary but represents the qualitative features of all the VB states well and $\sigma _{max}$ is commensurate to the RZ size. As $S$ is increased beyond $S_c = 1.14$, $\sigma _{max}$ associated with the BVB is seen to gradually increase (green symbols), representative of the increase in the RZ's size with swirl. By contrast, for regular CVB, the parameter remains approximately a constant for most $S$, except close to the threshold of $S = 1.4$. This indicates that the transition to turbulence limits the radial spread of the conical sheet for the regular CVB. The wide-open CVB is seen to increase in size with increasing $S$, although confinement effects prevent further scrutiny.