3.1 The linear dynamics of a perturbed solid-body rotation flow

The axisymmetric flow simulations of Rusak & Wang (Reference Rusak and Wang2014) found that the initial evolution of a perturbed solid-body rotation is dominated by the growth (or decay) of the most unstable (or least stable) mode of perturbation that can be accurately predicted by the linear stability analysis of Wang & Rusak (Reference Wang and Rusak1996). In a recent study, Wang et al. (Reference Wang, Rusak, Gong and Liu2016) conducted an extended linear stability analysis of three-dimensional perturbation modes on the solid-body rotation flow in a finite-length pipe. It is expected that these stability results also provide a precise prediction of the linear stage of evolution of perturbations as computed by direct numerical simulations.

We first examine the initial flow evolution of a slightly perturbed solid-body rotation flow at two representative operational points: (i) $\unicode[STIX]{x1D714}=2.2$ and $Re=400$ and (ii) $\unicode[STIX]{x1D714}=2.55$ and $Re=214$ . Figure 4 describes the time history of the instantaneous growth rate $\unicode[STIX]{x1D70E}_{E}$ of the flow perturbation for the first case. It is found that, after a short transient stage of growth of the small initial perturbation during $0\leqslant t\leqslant 50$ , the simulated growth rate $\unicode[STIX]{x1D70E}_{E}$ approaches a near-constant value $\unicode[STIX]{x1D70E}_{E}=0.0297$ and stays at this value for a significant period of time from $t=50$ to $t=350$ . This time period is the linear growth stage of the perturbation. Figure 5 presents the snapshot of the streakline contour in the flow at $t=400$ (at the end of the linear growth stage), demonstrating that a dominant axisymmetric mode of perturbation has developed in the flow.

Figure 4. The time history of the instantaneous growth rate $\unicode[STIX]{x1D70E}_{E}$ for the case $\unicode[STIX]{x1D714}=2.2,Re=400$ . A near-constant growth rate of an axisymmetric perturbation mode with $\unicode[STIX]{x1D70E}_{E}=0.0297$ is found during the time $50<t<350$ .

Figure 5. Snapshot of streaklines consisting of massless particles released at the pipe inlet ( $x=0$ ) on a circle with $r=0.05$ for the case $\unicode[STIX]{x1D714}=2.2,Re=400$ at $t=400$ .

Table 1 shows a comparison of the simulated growth rate $\unicode[STIX]{x1D70E}_{E}$ of the dominant unstable mode with the theoretically predicted growth rates of the base flow at $\unicode[STIX]{x1D714}=2.2$ and $Re=400$ . In this case, the linear stability theory predicts that, indeed, the most unstable mode is axisymmetric ( $m=0$ ) with a growth rate $\unicode[STIX]{x1D70E}=0.028$ . This shows an agreement between the theoretical prediction and the computation according to the simulation. The second row in table 1 provides the theoretical prediction of the complex growth rate $\unicode[STIX]{x1D70E}=-0.049-1.68i$ of the second least stable mode that is a spiral ( $m=1$ ) mode. This mode is asymptotically stable and decays in time. Therefore, the dynamics of this mode cannot be revealed by computing $\unicode[STIX]{x1D70E}_{E}$ from the simulation.

Table 1. Theoretical linear stability perdition versus direct numerical simulation (DNS) result for the case where $\unicode[STIX]{x1D714}=2.2,Re=400$ , $L=2$ .

Figure 6 presents the time history of the instantaneous growth rate $\unicode[STIX]{x1D70E}_{E}$ of the flow perturbation for the second case, $\unicode[STIX]{x1D714}=2.55$ and $Re=214$ . As in the first case, it can be seen that after a short transient stage of growth of the small initial perturbation during $0\leqslant t\leqslant 9$ , the simulated growth rate $\unicode[STIX]{x1D70E}_{E}$ approaches a near-constant value $\unicode[STIX]{x1D70E}_{E}=0.036$ with a frequency of $1.611$ and stays at this value for the time between $t=9$ and $t=170$ . Again, this time period is the linear growth stage of the perturbation. Figure 7 presents snapshots of the rotating streaklines in the flow at two nearby times, $t=158.63$ (red) and $t=160.50$ (black), approximately $180$ degree in phase during the rotation cycle. This figure demonstrates that a dominant unstable three-dimensional, rotating spiral ( $m=1$ ) mode of perturbation has developed in the second flow case during the linear growth stage.

Table 2 shows a comparison of the simulated growth rate $\unicode[STIX]{x1D70E}_{E}$ of the dominant unstable spiral mode with the theoretically predicted growth rates of the base flow at $\unicode[STIX]{x1D714}=2.55$ and $Re=214$ . In this second case, the linear stability theory predicts that, indeed, the most unstable mode is a spiral ( $m=1$ ) mode with a growth rate $\unicode[STIX]{x1D70E}=0.034+1.609i$ . This shows again an agreement between the linear stability result and the simulated growth rate of the dominant perturbation, $\unicode[STIX]{x1D70E}=0.036+1.611i$ . The second row in table 2 provides the theoretical prediction of the growth rate $\unicode[STIX]{x1D70E}=-0.0290-0.129i$ of the second least stable mode that is an axisymmetric ( $m=0$ ) mode. This mode is asymptotically stable and decays in time. Therefore, the dynamics of this mode cannot be realized through the computation of $\unicode[STIX]{x1D70E}_{E}$ from the simulation.

Figure 6. The time history of the instantaneous growth rate $\unicode[STIX]{x1D70E}_{E}$ for the case $\unicode[STIX]{x1D714}=2.55$ , $Re=214$ . A near-constant growth rate of a spiral mode with $\unicode[STIX]{x1D70E}_{E}=0.036$ is found during the time $9<t<170$ .

Figure 7. Snapshots of streaklines at times $t=158.63$ (red) and $t=160.50$ (black) consisting of massless particles released at the pipe inlet ( $x=0$ ) on a circle with $r=0.05$ for the case $\unicode[STIX]{x1D714}=2.55,Re=214$ .

Table 2. Theoretical linear stability perdition versus simulation result for the case where $\unicode[STIX]{x1D714}=2.55$ , $Re=214$ , $L=2$ .

The analysis of all other operational points described in this paper exhibits a similar agreement between the linear stability predictions and the numerical simulation results. The study of all of these cases clearly demonstrates the role of the dominant linear mode in the initial growth of a small perturbation of the base flow. Moreover, all of the computed examples demonstrate that the slightly perturbed solid-body rotation flow can take various complicated patterns of the initial evolution depending on the operational flow regime defined by the inlet swirl ratio $\unicode[STIX]{x1D714}$ and the flow Reynolds number $Re$ . Thus, a detailed examination of the simulated results at various values of $\unicode[STIX]{x1D714}$ and $Re$ in comparison to the theoretical prediction is needed. This would provide a convincing test for the new linear stability theory of Wang et al. (Reference Wang, Rusak, Gong and Liu2016) as well as a demonstration of the applicability of this stability theory to numerical simulations and to the physical mechanisms that govern the flow evolution.

Figure 8 describes in the operational range $2<\unicode[STIX]{x1D714}<2.6$ and $0<Re<800$ the predicted neutral line (which corresponds to $real(\unicode[STIX]{x1D70E})=0$ ) of $Re$ versus $\unicode[STIX]{x1D714}$ according to the linear stability theory of Wang et al. (Reference Wang, Rusak, Gong and Liu2016) for the case where the pipe non-dimensional length is $L=2$ . This border line is composed of two segments, the dash-dot line in the range $2.1<\unicode[STIX]{x1D714}<2.462$ , resulting from an analysis of the $m=0$ axisymmetric modes, and the solid line in the range $2.462<\unicode[STIX]{x1D714}<2.6$ , resulting from an analysis of the $m=1$ spiral modes. At operational points above these lines the solid-body rotation flow is unstable. Extensive successive simulations at various $\unicode[STIX]{x1D714}$ and $Re$ are also used to independently determine computed operational points along the border line of flow stability/instability. These points are also shown in figure 8. Here the circles (○) indicate operational points where an $m=0$ axisymmetric mode first becomes unstable as $Re$ is increased at a fixed $\unicode[STIX]{x1D714}$ and the triangles (▵) indicate operational points where an $m=1$ spiral mode first becomes unstable with the increase of $Re$ at a fixed $\unicode[STIX]{x1D714}$ . A good agreement between the theoretical linear stability predictions and the simulation results is exhibited.

Figure 8. The neutral linear stability line of $Re$ versus $\unicode[STIX]{x1D714}$ for flows in a pipe with $L=2$ (dot-dash line for $m=0$ axisymmetric mode and solid line for $m=1$ spiral mode). In addition, (○) indicates the neutral stability points with $m=0$ axisymmetric mode and ▵ indicates the neutral stability points with $m=1$ spiral mode, both obtained from the full three-dimensional time-dependent simulations (DNS).

The results of figure 8 demonstrate that the linear stability predictions capture, within an acceptable accuracy, the initial evolution of the simulated flows in the whole operational region. Above the neutral stability operating line the simulated flow perturbations initially grow with a linear pattern whose mode shapes and growth rates are in an agreement with the linear stability analysis. Moreover, the neutral stability line is formed by the two border line segments of the $m=0$ and $m=1$ modes, with a sharp intersection point between them at $\unicode[STIX]{x1D714}=2.462$ and $Re=263$ . The simulated flow neutral points nicely follow this trend. Furthermore, in the unstable operational range with $2.1<\unicode[STIX]{x1D714}<2.462$ , close to the neutral line, the simulated flow initially evolves with the most unstable axisymmetric mode while in the range with $2.462<\unicode[STIX]{x1D714}<2.6$ the flow evolves with the most unstable $m=1$ spiral mode. We conclude that the simulated initial evolution of the slightly perturbed flow is accurately predicted by the linear stability theory.

The underlying physical mechanism of the flow instability and the onset of the vortex breakdown process were revealed by Wang et al. (Reference Wang, Rusak, Gong and Liu2016), in a direct extension of the early work of Wang & Rusak (Reference Wang and Rusak1996) and the recent studies of Wang & Rusak (Reference Wang and Rusak2011), Rusak et al. (Reference Rusak, Wang, Xu and Taylor2012) and Rusak & Wang (Reference Rusak and Wang2014). The instability of both the axisymmetric and spiral perturbations above a certain value of the inlet swirl ratio is a result of the axial inhomogeneity of the flow caused by the different roles of the active inlet and the passive outlet non-periodic boundary conditions and the convective nature of the swirling flow. Therefore the classical stability theory that is based on periodic axial Fourier mode analysis (normal mode analysis) is not relevant to the initial evolution of perturbations as found in the direct numerical simulations. Instead, the evolution of the slightly perturbed solid-body rotation flow clearly develops an inlet–outlet non-axial periodic flow field as shown in figures 5 and 7. These figures can be alternatively obtained by using the most unstable eigenmode at the given operational conditions according to the linear stability analysis of Wang et al. (Reference Wang, Rusak, Gong and Liu2016), see their figures 3, 5, 7 and 11.

The linear stability analysis describes the initial growth of either axisymmetric or spiral instabilities of the base flow which initiate the breakdown process. However, direct numerical simulations are needed to complete the description of the nonlinear evolution of the flow as the perturbations grow from a small (infinitesimal) size to a finite size. The simulations described in the next section shed light on the various possible scenarios of the long-term flow dynamics.

3.2 The global dynamics of a perturbed solid-body rotation flow

The time history of the perturbations’ growth rates $\unicode[STIX]{x1D70E}_{E}$ presented in figures 4 and 6 shows that after the linear growth stage the growth rates decrease in time in both cases to a near-zero value. This indicates that in the long term, the simulated flows experience saturated nonlinear processes. We continue to examine the nonlinear evolution of the flow in the pipe in the operating regime of $Re$ versus $\unicode[STIX]{x1D714}$ . We find three different representative types of nonlinear evolution pattern. These are represented by the flow evolution at the operational points (i) $\unicode[STIX]{x1D714}=2.2,Re=300$ , (ii) $\unicode[STIX]{x1D714}=2.2,~Re=400$ and (iii) $\unicode[STIX]{x1D714}=2.55,~Re=214$ , respectively.

3.2.1 The case study with $\unicode[STIX]{x1D714}=2.2,~Re=300$

Figure 9 describes the time history of the instantaneous growth rate $\unicode[STIX]{x1D70E}_{E}$ of the flow perturbation for the first case where $\unicode[STIX]{x1D714}=2.2$ , $Re=300$ . It is found that after a short transient stage of growth of the small initial perturbation, the simulated growth rate $\unicode[STIX]{x1D70E}_{E}$ approaches a near-constant value $\unicode[STIX]{x1D70E}_{E}=0.018$ and stays at this value for a significant period of time from $t=50$ to $t=500$ . This time period is the linear growth stage of the most unstable perturbation mode of the base flow.

Figure 9. The time history of the instantaneous growth rate $\unicode[STIX]{x1D70E}_{E}$ for the case $\unicode[STIX]{x1D714}=2.2,Re=300$ . A near-constant growth rate of an axisymmetric mode with $\unicode[STIX]{x1D70E}_{E}=0.018$ is found during the time $100<t<500$ .

Figure 10 presents snapshots of streaklines of the flow perturbation at various times from $t=400$ to $t=1200$ . The flow global evolution may be explained by the following scenario, also see figure 9.

Figure 10. The streaklines of flow perturbation consisting of massless particles released at the pipe inlet ( $x=0$ ) on a circle with $r=0.05$ for the case $\unicode[STIX]{x1D714}=2.2,Re=300$ at: (a) $t=400$ , (b) $t=650$ , (c) $t=700$ , (d) $t=1200$ .

During the time $1<t<50$ , the perturbation exhibits a transition period where all of the stable modes of the initial perturbation decay in time. At $t\sim 50$ , the perturbation takes the form of the most unstable mode at $\unicode[STIX]{x1D714}=2.2,~Re=300,~L=2$ that is an axisymmetric diverging flow mode around the centreline. During the linear growth rate period $50<t<500$ this axisymmetric perturbation mode grows in time with a nearly constant growth rate $\unicode[STIX]{x1D70E}_{E}=0.018$ (see figure 10(a) for the shape of the flow perturbation at $t=400$ ). As the perturbation continues to grow in time it forms an axisymmetric separation zone around the pipe centreline and the flow evolves during the time $500<t<650$ into an initial axisymmetric vortex breakdown state at time $t=650$ , see figure 10(b). With further increase of time, the axisymmetric breakdown zone grows in size (see figure 10(c) at $t=700$ ). It eventually nonlinearly saturates on a steady axisymmetric breakdown state for all $t>1000$ (see figure 10(d) at $t=1200$ ).

To further explore the global dynamics of the flow, we investigate a low-dimensional representation of the flow dynamics. We consider the phase portrait with time $t$ of the velocity components $(u_{r},u_{\unicode[STIX]{x1D703}},u_{x})$ at a certain point $r=0.5,~\unicode[STIX]{x1D703}=0,~x=1.5$ in the flow domain. It is found that phase portraits of these velocities at other points inside of the flow domain behave in a similar way. This indicates that the dynamics of the flow is sufficiently close to a low-dimensional manifold. Figure 11 shows the various phase portraits with time, each up to a certain time $t$ during the flow evolution from $t=0$ to $t=1200$ .

Figure 11. The phase portrait of $(u_{x},u_{y},u_{\unicode[STIX]{x1D703}})\Big|_{(1.5,0.5,0,t)}$ for $\unicode[STIX]{x1D714}=2.2$ and $Re=300$ , $t\in (1,1130)$ . (a) The initial transient stage (red); (b) the linear growth stage (green) and the nonlinear stage (blue) approaching the fixed point (circle) representing a nonlinear saturated state.

Figure 11(a) shows the velocity trajectory during the time from $t=0$ up to $t=120$ . Note that the scale of velocity changes during this time period is small, $O(10^{-3})$ . The initial flow dynamics consists of two stages of evolution. The first stage is the initial transient stage during $0\leqslant t\leqslant 50$ where the trajectory moves towards a linear spiral sink. This describes the decay of all of the stable modes, primarily the $m=1$ least stable mode, i.e. the velocity trajectory follows a spiral orbit towards a central point. The second stage is the development of the linear most unstable mode of the base flow starting at $t\sim 120$ . The velocity trajectory moves away from the centre of the spiral sink. It continues to evolve along a fixed straight line during the time elapsed from $t=120$ to $t=500$ , see figure 11(b) and the perturbation shape in figure 10(a) at $t=400$ . Note that at this large scale of velocity, the initial transient stage shrinks to almost a point. When $t>500$ , the trajectory continues to move along a near straight line (see the perturbation shapes in  10(b) and 10(c)). It asymptotically approaches a stable fixed point of the axisymmetric breakdown state shown in figure 10(d), and stays at this point for all $t>1200$ . At the given operational conditions $\unicode[STIX]{x1D714}=2.2,~Re=300$ , this stable fixed point of the axisymmetric breakdown state forms an attractor of the flow dynamics.

3.2.2 The case study with $\unicode[STIX]{x1D714}=2.2,~Re=400$

The growth rate $\unicode[STIX]{x1D70E}_{E}$ for the second case with $\unicode[STIX]{x1D714}=2.2,~Re=400$ is shown in figure 4. In addition, figure 12 presents snapshots of the streaklines of the flow perturbation at various times from $t=400$ to $t=1100$ . The flow global evolution in this case may be explained by the following scenario. During the initial time period $1<t<50$ , the perturbation exhibits a transition stage where again all of the stable modes of the initial perturbation decay in time. At $t$ around $50$ , the perturbation takes the form of the most unstable mode of the base flow at $\unicode[STIX]{x1D714}=2.2$ and $Re=400$ that is also an axisymmetric perturbation mode. During the linear growth stage $50<t<350$ this axisymmetric perturbation mode grows in time with a nearly constant growth rate $\unicode[STIX]{x1D70E}_{E}=0.0297$ , see figure 12(a). As the perturbation continues to grow in time it eventually forms an axisymmetric separation zone around the pipe centreline. The flow continues to evolve through a saturated nonlinear process during the time period $350<t<600$ into a axisymmetric vortex breakdown state at time $t=600$ , see the perturbation shape in figure 12(b). However, with further increase of time, an unstable spiral perturbation evolves in the axisymmetric vortex breakdown state, see figure 12(c) at $t=645$ . Through a second nonlinear evolution process during the time $645<t<1100$ , the axisymmetric breakdown state disappears and the perturbation evolves into a large spiral wave that rotates about the pipe centreline in the same direction of rotation as the base rotating flow, see figure 12(d) at two times $t=1099.04$ (red) and $t=1100.42$ (black), at approximately $180$ degree in phase during the rotation cycle. A three-dimensional, rotating spiral ( $m=1$ ) mode of perturbation dominates the flow for all $t>1000$ .

Figure 12. Snapshots of the streaklines consisting of massless particles released at the pipe inlet ( $x=0$ ) on a circle with $r=0.05$ for the case $\unicode[STIX]{x1D714}=2.2,Re=400$ at: (a) $t=400$ , (b) $t=600$ , (c) $t=645$ , (d) $t=1099.04$ (red) and $t=1100.42$ (black).

We study the phase portrait of the velocity components $(u_{r},u_{\unicode[STIX]{x1D703}},u_{x})$ at the point $r=0.5,\unicode[STIX]{x1D703}=0,~x=1.5$ with time $t$ , see figure 13. Figure 13(a) shows the velocity trajectory during the time from $t=1$ up to $t=130$ . The flow dynamics consists of two stages of evolution. The first stage is again the transient stage during $1\leqslant t<60$ where the velocity trajectory moves towards a linear spiral sink, which again describes the decay of all stable modes, primarily the $m=1$ least stable perturbation mode of the base flow, i.e. the velocity trajectory follows a spiral orbit towards a centre point. The second stage is the development of the linear most unstable mode starting at $t\sim 60$ . The velocity trajectory moves away from the centre of the spiral sink along a fixed straight line during the time elapsed from $t=60$ to $t=130$ . Notice however that the velocity trajectory changes in small scales, of the order of $10^{-3}$ . The simulated flow is able to accurately capture the dynamics of this small-scale perturbation.

Figure 13. The phase portrait of $(u_{x},u_{y},u_{\unicode[STIX]{x1D703}})|_{(1.5,0.5,0,t)}$ for $\unicode[STIX]{x1D714}=2.2$ , $Re=400$ . (a) $t\in (10,130)$ ; the initial transient stage. (b) $t\in (1,680)$ ; the black straight line is the linear evolution stage. The green near straight line is the saturated nonlinear stage and the trajectory approaches another fixed point. The red line is a spiral source orbit with linear growth rate estimated as $\unicode[STIX]{x1D70E}=0.078+i2.0583$ . (c) $t\in (1,1100)$ ; the green line shows the nonlinear transient stage. (d) $t\in (800,1100)$ ; the blue line is the orbit approaching to a limit cycle, shown by a closed yellow line.

Figure 13(b) shows the velocity trajectory during the time from $t=0$ up to $t=680$ on a larger scale. The initial transient stage is too small to be seen at this scale and it shrinks to almost a point. The velocity trajectory moves first along a straight line (the black coloured line) from $t=50$ to $t=350$ . This represents the exponential linear growth of the velocity components of the axisymmetric perturbation during the linear growth stage, $50<t<350$ . This stage is followed by a near-straight-line trajectory till $t=600$ . This part corresponds to the nonlinear evolution stage with a decreasing growth of the perturbation. The velocity trajectory approaches a fixed point at $t=600$ . This fixed point corresponds to the establishment of the axisymmetric vortex breakdown flow state found at $t\sim 600$ , see figure 12(b). However, this fixed point is unstable to a three-dimensional mode of perturbation. The velocity trajectory moves away from this fixed point, following an outward spiral orbit (the red coloured line), which is a typical linear spiral source, see the perturbation shape in figure 12(c). The growth rate of the linear spiral source is estimated from the simulation to be $\unicode[STIX]{x1D70E}\sim 0.078+2.0583i$ .

Figure 13(c) shows the complete trajectory for the time from $t=0$ to $t=1100$ . The green coloured trajectory shows the continuation of the linear spiral source which forms a nonlinear transient stage. The trajectory is eventually attracted to a closed orbit, forming a near-limit-cycle orbit of the rotating spiral breakdown described in figure 12(d). The blue coloured trajectory shows the dense orbit surrounding the limit-cycle orbit (the yellow coloured line). Figure 13(d) shows a closer view of the limit-cycle orbit and the surrounding orbit during the time from $t=800$ to $t=1100$ . The rotation of the spiral breakdown is estimated to have a frequency of approximately $2$ and is in the direction of the base flow rotation. At the given operational conditions $\unicode[STIX]{x1D714}=2.2,~Re=400$ and $L=2$ , this limit-cycle orbit of the rotating spiral wave breakdown forms an attractor of the flow dynamics.

3.2.3 The case study with $\unicode[STIX]{x1D714}=2.55,~Re=214$

For the case $\unicode[STIX]{x1D714}=2.55,~Re=214$ , figure 14 presents snapshots of the streaklines of the flow. The flow global evolution may be explained by the following scenario, also see the time history of the instantaneous growth rate for this case in figure 6. During the time $0<t<9$ , the perturbation exhibits a transition period where all the stable modes of the initial perturbation decay in time. At $t$ around $9$ the perturbation takes the form of the most unstable mode of the base flow at $\unicode[STIX]{x1D714}=2.55,~Re=214,~L=2$ that is a spiral mode. During the linear growth period $9<t<170$ this spiral perturbation mode grows in time with a nearly constant growth rate $\unicode[STIX]{x1D70E}_{E}=0.036$ . Figure 14(a,b) presents snapshots of streaklines of the growing rotating spiral perturbation at two times, $t=139.38$ (red) and $t=141.13$ (black), and at two other times, $t=179.99$ (red) and $t=179.77$ (black), respectively (both at approximately $180$ degree in phase during the rotation cycle). As the perturbation continues to grow in time it is stabilized at a saturated nonlinear limit-cycle process when $t>290$ and evolves into a large spiral wave with a constant amplitude that rotates about the pipe centreline in the same direction of rotation as the base rotating flow, see figure 14(c) for two snapshots of streaklines of the rotating spiral breakdown at times $t=599.69$ (red) and $t=601.87$ (black), again, at approximately $180$ degree in phase during the rotation cycle. This process of a direct evolution from a spiral instability mode to a spiral breakdown state, without first establishing an axisymmetric breakdown state, was not described before.

Figure 14. The snapshots of the streaklines of the spiral mode perturbation consisting of massless particles released at the pipe inlet ( $x=0$ ) on a circle with $r=0.05$ for the case $\unicode[STIX]{x1D714}=2.55,~Re=214$ at: (a) $t=139.38$ (red) and $t=141.13$ (black), (b) $t=177.99$ (red) and $t=179.77$ (black), (c) $t=599.69$ (red) and $t=601.87$ (black).

For a low-dimensional representation of the flow dynamics, we again consider the phase portrait of the velocity components $(u_{r},~u_{\unicode[STIX]{x1D703}},~u_{x})$ at the point $r=0.5,~\unicode[STIX]{x1D703}=0,~x=1.5$ with time $t$ . Figure 15 shows various velocity phase portraits up to certain times $t$ during the flow evolution from $t=1$ to $t=600$ .

Figure 15. The phase portrait of $(u_{x},u_{y},u_{\unicode[STIX]{x1D703}})|_{(1.5,0.5,0,t)}$ for $\unicode[STIX]{x1D714}=2.55$ and $Re=214$ . (a) $t\in (1,20)$ ; the initial transient stage. (b) $t\in (1,170)$ . (c) $t\in (1,600)$ ; linear (red) and nonlinear spiral source (yellow). The trajectory is approaching a limit cycle, shown by a closed black cycle.

Figure 15(a) shows the velocity trajectory at the point $r=0.5,\unicode[STIX]{x1D703}=0,~x=1.5$ during the initial stage from $t=1$ up to $t=20$ . It again consists of two stages of evolution. The first is the transient stage during $0\leqslant t\leqslant 9$ where the trajectory moves in a complicated way which describes the decay of all stable modes. The second stage is the development of the linear most unstable mode of the base flow starting at $t\sim 9$ . Now the velocity trajectory moves away from the centre of a spiral source along a spiral out orbit (the red coloured line) during time $t=9$ to $t=20$ . Notice again that the velocity changes occur again at small scales, of the order of $10^{-3}$ . The simulated flow correctly captures the dynamics of this fine small-scale perturbation.

Figure 15(b) shows the velocity trajectory from $t=1$ to $t=170$ on a larger scale. The initial transient stage is too small to be seen in this scale and shrinks to almost a point. The velocity trajectory moves along an outward spiral orbit during the linear growth stage, $9<t<170$ , see the perturbation shape in  14(a) and 14(b).

Figure 15(c) shows the complete velocity trajectory during the time period from $t=1$ to $t=600$ . The trajectory moves for all time along an outward spiral orbit. After the linear stage of the orbit (the red coloured line), the second part of the orbit presents a continued nonlinear saturated process with a decreasing growth of the perturbation size (the yellow coloured line). The velocity trajectory is eventually attracted to a closed limit-cycle orbit (the black coloured line). The rotating spiral breakdown shape for this state is shown in figure 14(d). At the given operational conditions $\unicode[STIX]{x1D714}=2.55,~Re=214$ , this limit-cycle orbit of the rotating spiral wave breakdown forms an attractor of the flow dynamics.

6.1 Local stability versus global stability

It is important to understand further the nature of the instability leading to the vortex breakdown states as arises from the three-dimensional numerical simulations. There has been a considerable effort in the literature to relate the instability of vortex flows to the global stability of a weakly non-parallel developing flow along its main axis in an infinitely long axial domain. This approach relates to the classical absolute/convective instability analysis. In a slightly non-parallel flow governed by a small parameter $\unicode[STIX]{x1D716}$ , $0<\unicode[STIX]{x1D716}\ll 1$ , a rescaling of the axial length $x$ by $X=\unicode[STIX]{x1D716}x$ reveals the role of flow stability at various axial cross-sections, and is called local stability analysis. At the dominant order of the problem is the classical stability problem of an axially parallel base flow. The local linear stability properties of the flow at an axial cross-section $x$ is determined by the nature of growth of normal mode perturbations on the local flow as a base flow in an infinitely long homogeneous domain. A flow can be classified as locally stable when all perturbations decay in time or absolutely/convectively unstable based on the base flow wave guiding nature, i.e.

1. (i) Convective instability (CI) occurs when the flow is locally unstable, but, in a fixed laboratory frame, the perturbation diminishes over time.

2. (ii) Absolute instability (AI) occurs when a flow is locally unstable, but, in a fixed laboratory frame, the perturbation grows in time.

The significance of CI–AI for a spatially weakly developing flow is that such a flow can stay globally linearly stable although it may be characterized locally with a CI. On the other hand, a flow with a sufficient finite region of AI may become a self-sustained wave maker and trigger a global linear instability by a transition from convective to absolute local instability, see for example the pioneering papers by Huerre & Monkewitz (Reference Huerre and Monkewitz1985), Chomaz, Huerre & Lg (Reference Chomaz, Huerre and Redekopp1988) and Huerre & Monkewitz (Reference Huerre and Monkewitz1990).

The CI–AI concept captures the competitive effects of the perturbation’s growth against its convection by the base flow. The underlying physical concept of CI–AI explains that perturbations in a CI region can be amplified but are deemed to convect out of the instability region and that perturbations in an AI region can sustain growth to generate a global mode of instability. We emphasize that the local/global stability (instability) is a closely connected pair of concepts in the stability theory of a weakly non-parallel flow. On the other hand, the various instability processes of the solid-body rotation flow found in the present simulations is of a completely different physical mechanism where the local/global stability (instability) pair does not even exist; the solid-body rotation flow is linearly stable to all normal mode perturbations and is a wave guide (Kelvin Reference Kelvin1880). The instabilities found in this paper are induced by the finite-domain effect, and no local normal mode instability is necessary for the onset of instabilities of the rotating flow in a finite-length pipe.

Moreover, the CI–AI approach is limited in its scope and accuracy of calculations to cases where the base flow is only weakly developing along the axial direction. Moreover, since this approach is based on the local stability properties of the flow profile at various axial cross-sections, it does not allow the full interactions in the flow, i.e. no interaction between the sections properties and the inlet and outlet conditions is described. This may apply to the nature of thin boundary layer or wake flows and of certain regions of vortex flows but not to flows where there may be a strong global interaction between the flow in the bulk and the domain boundaries, including the pipe wall, inlet and outlet. The latter is the case of the stability of a vortex flow in a finite-length domain. In such a case, only a linear stability analysis of the base flow which also includes all the boundary conditions used in the simulations accurately represents the flow dynamics as computed by direct numerical simulations. In the following subsections we further explain this view.

6.2 The study of local/global stability of swirling flows

Ruith et al. (Reference Ruith, Chen, Meiburg and Maxworthy2003) studied the local CI–AI properties of a numerically simulated base axisymmetric and steady vortex breakdown state of a Grabowski–Berger vortex (Grabowski & Berger Reference Grabowski and Berger1976) injected into a semi-infinite domain at $Re=200$ and swirl level $S=1$ , and that contains a centreline bubble separation zone. They assumed that at any axial cross-section the base flow may be approximated by a Batchelor vortex profile (Batchelor Reference Batchelor1964) with specific parameters of the vortex core size and strength. They used the results of CI–AI analyses of the base Batchelor vortex by Delbende, Chomaz & Huerre (Reference Delbende, Chomaz and Huerre1998) and Olendraru, Sellier & Huerre (Reference Olendraru, Sellier and Huerre1999) to determine the nature of the flow local stability at various axial cross-sections. They found a finite local region of AI that is attached to the rear part of the axisymmetric breakdown bubble and qualitatively attributed this region to the loss of stability of the axisymmetric breakdown state to a certain three-dimensional perturbation and its evolution to a spiral breakdown state. However, they realized that the base flow state has a significant radial expansion around the breakdown bubble and is not a slightly parallel flow state. Therefore, their CI–AI analysis may not lead to any quantitatively accurate and conclusive link between the existence of an absolute instability (AI) region in the axisymmetric breakdown state and the spiral mode of instability of the breakdown bubble.

Realizing the limitations of the CI–AI analysis, and in an attempt to shed additional light on the flow dynamics, Ruith et al. (Reference Ruith, Chen, Meiburg and Maxworthy2003) also developed another strategy to estimate the global instability of the breakdown bubble. They were the first to use three-dimensional direct numerical simulation to describe the shape and growth rate of the rotating spiral mode that destabilizes the bubble and evolves to a spiral breakdown state.

In a follow-up study, Gallaire et al. (Reference Gallaire, Ruith, Meiburg, Chomaz and Huerre2006) further explored the relationship of CI–AI to the global nonlinear mode of instability of the breakdown bubble. They used the base axisymmetric vortex breakdown state computed by Ruith et al. (Reference Ruith, Chen, Meiburg and Maxworthy2003) and assumed that it is a weakly non-parallel base flow. They conducted a thorough linear local stability analysis at all the axial cross-sections of this simulated base flow. According to their CI–AI analysis, they inferred that a nonlinear global mode develops in the axisymmetric breakdown state, triggered by the appearance of a locally absolutely unstable region behind the breakdown bubble. However, the predicted frequency for the dominant $m=1$ mode for the case $Re=200$ and swirl parameter $S=1$ is $\unicode[STIX]{x1D70E}_{i}\sim 1.3$ , which is higher than the frequency of the dominant mode found in the simulation of Ruith et al. (Reference Ruith, Chen, Meiburg and Maxworthy2003) where $\unicode[STIX]{x1D70E}_{i}=1.18$ . More seriously, their study showed that the transition from CI to AI for the $m=1$ mode first occurs at the axial location near the nose of the breakdown bubble where the base flow exhibits a significant radial expansion, as shown by their non-parallelism parameter in figure 2 of Gallaire et al. (Reference Gallaire, Ruith, Meiburg, Chomaz and Huerre2006). Thus, the weakly non-parallel flow assumption of the base flow is not justified. Moreover, they also do not show any evidence that the predicted shape of the absolutely unstable mode indeed becomes a wave maker and is similar to the shape of the linear growth mode of global instability simulated by Ruith et al. (Reference Ruith, Chen, Meiburg and Maxworthy2003). This invalidation can be clearly observed by the reported absolutely unstable wavelength of $\unicode[STIX]{x1D706}_{0}=13.1$ , which is far greater than any length scale of the AI region and cannot be considered as a slightly non-parallel flow. The authors claimed, however, that this length scale found from the CI–AI analysis is consistent with the Ruith et al. (Reference Ruith, Chen, Meiburg and Maxworthy2003) simulation result for the spiral mode, but this is an ad hoc conclusion drawn with no direct correlation between the CI–AI results and the simulation results. In conclusion, they did not establish a sufficient convincing evidence that CI–AI analysis can indeed induce the global spiral instability of the breakdown bubble.

6.3 A vortex stability approach based on real physical boundary conditions

We refer again to the study of Ruith et al. (Reference Ruith, Chen, Meiburg and Maxworthy2003) who deduced the linear stability characteristics of the destabilizing spiral mode of the axisymmetric breakdown state directly from results of direct numerical simulation of flow evolution. This led them for the first time to the discovery of a clean linearly unstable spiral mode that is responsible for a spiral type of vortex breakdown. The specific growth rates and frequencies of the unstable spiral mode quantitatively identified by their study became the benchmark for further study. For the case $Re=200$ and $S=1$ , the $m=1$ mode is the most unstable perturbation and is characterized by a complex growth rate, $\unicode[STIX]{x1D70E}=0.0359+1.18i$ . The mode shape of the linear eigenmode has a typical wavenumber of approximately $k=1/5$ , evidently not supporting the weakly non-parallel assumption in the local/global stability relationship. Moreover, the unstable mode (presented in figure 30 of Ruith et al. (Reference Ruith, Chen, Meiburg and Maxworthy2003) shows a spatial development in the axial direction and the corresponding mode shape is spiral (not helical). This mode is formed by the strong interaction of the vortex flow in the bulk and the imposed physical constraints at the inlet (upstream state) where the velocity profiles are fixed, at the radial far field and at the downstream. The eigenmode obtained does not show a correlation to the CI–AI transition location described by the analysis of Gallaire et al. (Reference Gallaire, Ruith, Meiburg, Chomaz and Huerre2006).

The finding of the linearly unstable spiral modes of Ruith et al. (Reference Ruith, Chen, Meiburg and Maxworthy2003) is an important contribution to the vortex stability theory. It clarifies the nature of the linear mode. This study raised the issue of the importance of quantitative evidence in supporting a theory. This is particularly important as highly accurate computation is available and thus quantitative evidence must be established for testing of a theory. Along this line, in a recent study, Meliga & Gallaire (Reference Meliga and Gallaire2011) performed a linear stability analysis of the base axisymmetric vortex breakdown state obtained by Ruith et al. (Reference Ruith, Chen, Meiburg and Maxworthy2003) where they used the same boundary conditions of Ruith et al. (Reference Ruith, Chen, Meiburg and Maxworthy2003). They confirmed with high accuracy the benchmark linear stability results obtained originally by Ruith et al. (Reference Ruith, Chen, Meiburg and Maxworthy2003). For example, for the case $Re=200$ and $S=1$ they found that the $m=1$ mode is the most unstable mode with growth rate $\unicode[STIX]{x1D70E}=0.0387+1.16i$ , which is close to the benchmark result, especially the frequency has three digits accuracy. Moreover, the corresponding mode shape presented in their study is sufficiently close to the one found in Ruith et al. (Reference Ruith, Chen, Meiburg and Maxworthy2003). This shows that only when the same boundary conditions are used in the linear stability analysis as in the simulation, can an accurate prediction of the spiral mode of instability of the breakdown bubble be found. It should be pointed out that the study of Meliga & Gallaire (Reference Meliga and Gallaire2011) is essentially in the same spirit as the analysis of Wang & Rusak (Reference Wang and Rusak1996), where realistic constraints on the vortex flow were used. The instability is, therefore, induced by the strong influence of the imposed physical conditions, which effectively eliminate the weakly non-parallel nature of the base flow. Such an accurate result cannot be obtained by CI–AI analysis since the crucial boundary conditions are ignored in this approach.

6.4 The strong non-parallel flow effect on vortex stability

The significance of the current study is that the selected nominal base flow, solid-body rotation flow with a uniform axial velocity, is the only strain-free flow. Therefore, it exhibits very special stability characteristics, i.e. lack of any instability mechanism including transient growth within the framework of classical normal mode analysis (Kelvin Reference Kelvin1880). This is equally true for any type of perturbation, axisymmetric ( $m=0$ ) or helical ( $m\neq 0$ ). Thus, CI–AI is completely unrelated to the instability mechanism of the solid-body rotation flow in a finite-length pipe found in the present simulations. This is the reason we conducted in § 5 the analysis of energy transfer mechanism based on the Reynolds–Orr equation. This analysis clearly demonstrates that the instability mechanism, discovered originally by Wang & Rusak (Reference Wang and Rusak1996) and followed by Wang et al. (Reference Wang, Rusak, Gong and Liu2016) and by the present study, is essentially a new flow instability mechanism, beyond the scope of classical normal mode analysis, i.e. it is a finite-length domain instability. As shown in the present study, this new ingredient of flow stability is the only mechanism that leads to the onset of the various types of vortex breakdown in a solid-body rotation flow in a finite-length pipe.

The current study also consolidates the view that the instability arising from the unstable spiral mode of Ruith et al. (Reference Ruith, Chen, Meiburg and Maxworthy2003) is also a finite-length domain instability and is beyond the scope of the classical flow instability based on the normal mode or CI–AI analyses. To further illustrate this idea, we conducted an energy analysis based on the Reynolds–Orr equation (A 1) where in this case the base flow is the axisymmetric breakdown state established by a special axisymmetric simulation at $\unicode[STIX]{x1D714}=2.2$ and $Re=400$ with the same boundary conditions stated in § 2. We computed the linear spiral mode of instability perturbation that evolves in this base flow state and that stabilizes over time on the rotating spiral vortex breakdown state as described in § 3.2.2. We computed for this linear mode of instability the various production (loss) terms of the perturbation’s kinetic energy in the right-hand side of (A 1). We first note from the inlet conditions that the second and fourth terms on the right-hand side of (A 1) vanish. Figure 18(a) shows (dash line) the axial distribution of the integrated energy production at cross-sections $x$ in the bulk between the inlet and the outlet as found from the inviscid effect (the axial distribution of the radially integrated integrand in the first term in (A 1)). Also shown is the axial distribution of the viscous damping effect in the bulk (dash-dot line, from the sixth term in (A 1)). The sum of these two bulk effects is also shown (solid line). We note that the seventh term of the viscous effects is very small. It is clear that the overall perturbation’s kinetic production in the bulk (the integrated combined effects due the sum of the two terms) is negative and contributes an integrated negative growth rate of the perturbations kinetic energy in the amount of $\unicode[STIX]{x1D70E}_{E,internal}\sim -0.097$ . If the base flow was weakly non-parallel and the AI mechanism was the major source for the onset of the linear spiral mode of instability in the axisymmetric breakdown state, then this integrated combination of effects in the bulk should have been positive and not negative. This rules out the relationship between AI and the mode instability of the axisymmetric breakdown state.

Moreover, we find that the main energy production from the third and fifth terms of right-hand side of (A 1) comes from the outlet state as shown by figure 18(b). It can be seen that the radial distribution of the work performed at the outlet by the pressure perturbation on the axial velocity perturbation (the dash-dot line) overcomes for all $r$ the radial distribution of the convective perturbation’s kinetic energy loss at the outlet (dash line). The combined radial distribution of effects at the outlet (the solid line) is positive all across the outlet section from the centreline to the wall. This combined effect contributes an integrated positive growth rate of perturbation kinetic energy production at the outlet where $\unicode[STIX]{x1D70E}_{E,outlet}=+0.174$ . The net effect from the bulk and the outlet produces a growth rate of the linear spiral mode of perturbation that is $\unicode[STIX]{x1D70E}_{E}=\unicode[STIX]{x1D70E}_{E,outlet}+\unicode[STIX]{x1D70E}_{E,internal}\sim 0.077$ . The use of the Reynolds–Orr equation gives a growth rate of the spiral instability mode in the axisymmetric breakdown state that is similar to that found in the simulation, where $\unicode[STIX]{x1D70E}_{E}\sim 0.078$ . It shows that this instability is caused by the axial flow inhomogeneity in the strong non-parallel base flow and is driven by the imbalance between the base flow in the bulk, the inlet state and the outlet state.

Figure 18. The various production (loss) terms of the perturbation’s kinetic energy: (a) distribution along the pipe axis inside of the bulk, and (b) at the outlet.

We also present figure 19, which shows the spatial distribution of the production of an inviscid perturbation’s kinetic energy in the bulk ( $0\leqslant x\leqslant 2$ and $0\leqslant r\leqslant 1$ ) as computed from the integrand of the first term on the right-hand side of (A 1). We can see that there is a significant loss of energy along the pipe, specifically around $x=1.4$ and $r=0.25$ , and production of energy near the outlet at $r\sim 0.55$ . The energy pattern has a nearly one wavelength distribution and this cannot be described by a weakly non-parallel assumption about the flow used in CI–AI analyses. The figure actually demonstrates the strong non-parallel flow effect on the perturbation mode that is induced by the boundary conditions which form the axial inhomogeneity in the flow. This is further evidence that the instability mode cannot be driven by an CI–AI mechanism.

Figure 19. The inviscid perturbation’s kinetic energy productions’ density inside of the domain.

In the case of a general vortex flow in a finite-length domain, the Reynolds–Orr equation shows that the instability is caused essentially by the non-axial homogeneity of the base flow and the imbalance of the exchange of the perturbation’s kinetic energy between the base flow in the bulk (except the solid-body rotation flow) and the boundary settings at the domain inlet and outlet. Note that the non-axial homogeneity is also an element in a weakly non-parallel flow approach but it has a completely different role in the instability onset.

In the local/global stability pair, the instability is caused by a sufficiently strong AI region to sustain the perturbation’s growth. Actually, a weak non-parallel flow inhomogeneity by itself modulates the wave but is not the cause of instability. The instability in such an approach depends on the local sectional instability nature as found from normal mode analysis. In a sharp contrast to that, a strong non-parallel flow inhomogeneity induces instability by itself as found in the present study for the solid-body rotation flow in a finite-length pipe. This is made even more striking since in the current study the finite domain is a pipe with an axial length scale that is the same as the diameter of pipe. In such a case the weakly non-parallel flow assumption has no value. Extending the present analysis to other base flows such as the Lamb–Oseen vortex (Gong Reference Gong2017) or the a Grabowski–Berger vortex (Ruith et al. Reference Ruith, Chen, Meiburg and Maxworthy2003) shows a similar dynamical behaviour where the underlying instability mechanism remains the same. We therefore conjecture that the instability found in the current simulations is a general vortex stability behaviour for a confined vortex flow with sufficient swirl. The root cause of this instability was first revealed by the Wang & Rusak (Reference Wang and Rusak1996) analysis and is clearly explained by the use of the Reynolds–Orr equation as a diagnostic tool of the simulation results. An alternative approach is to use the direct numerical simulation to identify the instability process, as was first done by Ruith et al. (Reference Ruith, Chen, Meiburg and Maxworthy2003), in tandem with analysis based on the Reynolds–Orr equation.

6.5 The role of instability in the onset of vortex breakdown

This study provides strong quantitative evidence to support the stability results and the onset mechanism of various types of vortex breakdown and answers the crucial question as to how the vortex breakdown process starts and evolves into various types of breakdown states. The main results shown in figures 8, 16 and 17 demonstrate the consistency of the direct numerical simulation results with the linear stability theories of Wang & Rusak (Reference Wang and Rusak1996) and of Wang et al. (Reference Wang, Rusak, Gong and Liu2016). These include:

1. (i) The simulated flows strictly obey the predicted swirling flow dynamics found in the linear stability analysis when the perturbation is sufficiently small and thereby clarify the physical conditions $Re$ versus $\unicode[STIX]{x1D714}$ for the onset of instabilities and the vortex breakdown process.

2. (ii) The simulated nonlinear dynamics of the flow can be understood by instantaneous flow snapshots and the related phase portraits of the velocity trajectory at a point in the domain. This provides a clear explanation of the various flow evolution paths, the forms of vortex breakdown states and the possible transitions among them.

3. (iii) The analysis of the energy transfer mechanism reveals an essential ingredient of the physical mechanism that drives the linear as well as the nonlinear flow dynamics, i.e. the loss of flow axial homogeneity between the inlet and outlet by the flow convective nature, and the associated competition between energy production or loss inside the bulk and at the boundaries.

These findings provide an integrated dynamical picture for the onset of the vortex breakdown process and evolution to various breakdown states on the solid-body rotation flow in a finite-length pipe. They also provide a consistent and unified theoretical explanation of the vortex breakdown phenomenon, axisymmetric or spiral, in a solid-body rotation flow in a finite-length pipe. In particular, we point out that the onset of the process of vortex breakdown in a solid-body rotation flow is not related to Benjamin’s vortex breakdown theory, where the axisymmetric vortex breakdown is interpreted as a result of a supercritical–subcritical jump transition analogous to the hydraulic jump phenomenon. The flow transition in vortex breakdown is essentially smooth and neither a super–sub critical transition nor a hydraulic jump phenomenon are found in the simulation.

We shall emphasize that the solid-body rotation flow is closely related to other flow models. The result can be used for large core size of Lamb–Ossen vortices (a family of Batchelor vortex) and can also be readily extended by computational efforts to other vortex flow models including the Batchelor vortex. More importantly the fundamental mechanism revealed by this study evidently has a general nature in regard to the understanding of the vortex breakdown phenomenon in various flow configurations.