4.1 The $m=0$ modes

We first consider the axisymmetric ( $m=0$ ) perturbation modes. A similar linear stability problem was first studied analytically by Wang & Rusak (Reference Wang and Rusak1996a ). The present stability equations (2.11a–d ) result in a system of linear partial differential equations for the solution of the eigenvalue ${\it\sigma}$ . The problem has a continuum of solutions of perturbation modes that turn in sequence from being asymptotically stable to unstable as ${\it\omega}$ is increased above the first critical swirl ${\it\omega}_{0,1}$ , where ${\it\omega}_{0,1}=\sqrt{{\it\omega}_{B}^{2}+{\rm\pi}^{2}/(4L^{2})}$ (here ${\it\omega}_{B}=1.9159$ is Benjamin’s (Reference Benjamin1962) special swirl). The growth rates ${\it\sigma}_{r}$ (the real part of ${\it\sigma}$ ) as a function of the swirl ratio ${\it\omega}$ of the various $m=0$ perturbation modes on the solid-body rotation flow in a pipe with $L=6$ are shown in figure 2.

Figure 2.

The growth rates ${\it\sigma}_{r}$ (the real part of ${\it\sigma}$ ) as a function of the swirl ratio ${\it\omega}$ of the various inviscid axisymmetric ( $m=0$ ) perturbation modes on a solid-body rotation flow in a pipe with $L=6$ .

It is found that that all modes have real values of ${\it\sigma}$ that are negative and real shape functions when $0<{\it\omega}<{\it\omega}_{0,1}=1.9337$ , i.e. all the $m=0$ modes are asymptotically stable when the swirl ratio is below the first $m=0$ critical swirl. The ${\it\sigma}$ of the least stable mode is real and changes from negative to positive, and the flow becomes unstable as ${\it\omega}$ is increased from below the first critical swirl ${\it\omega}_{0,1}=1.9337$ to above it. The growth rate ${\it\sigma}_{r}$ of the $k=1$ mode increases in the range ${\it\omega}_{0,1}=1.9337<{\it\omega}<1.971$ , reaches a maximum value of $0.00475$ at ${\it\omega}=1.971$ and then decreases in the range $1.971<{\it\omega}<{\it\omega}_{0,2}$ , where ${\it\omega}_{0,2}=\sqrt{{\it\omega}_{B}^{2}+{\rm\pi}^{2}/L^{2}}=1.9862$ is the second critical swirl. This swirl level is also a fold point of ${\it\sigma}_{r}$ of the first least stable mode.

The growth rate ${\it\sigma}_{r}$ of the second ( $k=2$ ) mode changes from negative to zero as ${\it\omega}$ increases to the second critical swirl ${\it\omega}_{0,2}=1.9862$ . We note that the real growth rates of the two least stable modes merge to form the first fold point at ${\it\omega}_{0,2}=1.9862$ where for both modes ${\it\sigma}_{r}=0$ . All the other perturbation modes have real negative ${\it\sigma}$ and real shape functions and are asymptotically stable in the range ${\it\omega}_{0,1}<{\it\omega}<{\it\omega}_{0,2}$ . The two least stable modes combine together at ${\it\omega}_{0,2}=1.9862$ to form a pair of modes with complex-conjugate values of ${\it\sigma}$ and mode shape functions in the range ${\it\omega}>{\it\omega}_{0,2}$ . Only the real part ${\it\sigma}_{r}$ of these two modes is shown in figure 2; it increases in the range ${\it\omega}_{0,2}=1.9852<{\it\omega}<2.045$ , reaches a maximum of $0.0095$ at ${\it\omega}=2.045$ and then deceases in the range $2.045<{\it\omega}<{\it\omega}_{0,3}$ , where ${\it\omega}_{0,3}=\sqrt{{\it\omega}_{B}^{2}+9{\rm\pi}^{2}/(4L^{2})}=2.0706$ . These two complex-conjugate modes have an oscillatory behaviour of ${\it\sigma}_{r}$ as ${\it\omega}$ increases above ${\it\omega}_{0,3}=2.0706$ .

The ${\it\sigma}_{r}$ of the third ( $k=3$ ) least stable mode changes from negative to positive as ${\it\omega}$ increases from below the third critical swirl ${\it\omega}_{0,3}=2.0706$ to above it. The growth rate and shape functions of this mode are real and become dominant in the range ${\it\omega}_{0,3}<{\it\omega}<{\it\omega}_{0,4}$ , where ${\it\omega}_{0,4}=\sqrt{{\it\omega}_{B}^{2}+4{\rm\pi}^{2}/L^{2}}=2.1834$ . Additional $m=0$ perturbation modes with either real or complex values of the growth rates and shape functions appear in sequence as ${\it\omega}$ increases above an additional critical swirl ${\it\omega}_{0,n}=\sqrt{{\it\omega}_{B}^{2}+n^{2}{\rm\pi}^{2}/(4L^{2})}$ (here $n$ is a positive integer). By using the perturbation theory of the linear operators (see Wang Reference Wang2008), it can be shown that the critical swirl levels with even values of $n$ are also fold points of the ${\it\sigma}_{r}$ plots and complex-conjugate modes appear at these fold swirl levels. Also, the critical swirl levels with odd values of $n$ are points of exchange of stability of a mode with real ${\it\sigma}$ and shape functions. The dominant unstable $m=0$ perturbation at each swirl ratio is the perturbation mode with the greatest value of ${\it\sigma}_{r}$ at that swirl level.

Figure 3 shows a representative example of the solution of the most unstable $m=0$ (axisymmetric) mode shape functions at ${\it\omega}=2.13$ (between ${\it\omega}_{0,3}$ and ${\it\omega}_{0,4}$ ) in a pipe with $L=6$ for which ${\it\sigma}=0.025$ (on solution branch $k=3$ of $m=0$ case). Figure 3(a) presents the computed planar projected velocity vector, $(1+{\it\epsilon}\mathscr{W},{\it\epsilon}\mathscr{U})$ , of the mode in the flow domain; here $\mathscr{W}$ and $\mathscr{U}$ are real and ${\it\epsilon}$ is the perturbation’s amplitude such that there is a near stagnation point at the pipe outlet. Figure 3(b) describes a snapshot of the streaklines of the mode consisting of particles released at the pipe inlet ( $x=0$ ) on a circle with $r=0.15$ . It can be seen that the mode describes an axisymmetric axial growing wave with a wavelength of approximately $3.6$ and an amplitude that grows in $x$ . Similar figures can be given for all other computed $m=0$ modes.

Figure 3.

The shape functions of the dominant inviscid unstable $m=0$ mode at ${\it\omega}=2.13$ : (a) planar projected velocity vector field; (b) the streaklines of the mode consisting of particles released at the pipe inlet ( $x=0$ ) on a circle with $r=0.15$ .

4.2 The $m=1$ modes

In parallel to the $m=0$ modes, the stability problem with $m=1$ also has a continuum of solutions of spiral perturbation modes that are studied in this paper for the first time. The growth rates ${\it\sigma}_{r}$ (the real part of ${\it\sigma}$ ) and frequency ${\it\sigma}_{i}$ (the imaginary part of ${\it\sigma}$ ) as a function of the swirl ratio ${\it\omega}$ of the various inviscid $m=1$ perturbation modes on the solid-body rotation flow in a pipe with $L=6$ are shown in figure 4. Each of the different modes is characterized by a continuous solution as a function of ${\it\omega}$ with a similar axial wavelength of the shape functions that increases with the mode number $k$ . The modes are numbered $k=1,k=2,$ and so on. Figure 4(a) describes ${\it\sigma}_{r}$ of the modes $k=1$ and $k=2$ in the range $1.5\leqslant {\it\omega}\leqslant 2.1$ and figure 4(b) describes the corresponding ${\it\sigma}_{i}$ of these modes. Figure 4(c) describes ${\it\sigma}_{r}$ of the modes $k=3$ and $k=4$ in the range $1.95\leqslant {\it\omega}\leqslant 2.35$ and figure 4(d) describes the corresponding ${\it\sigma}_{i}$ of these modes.

Figure 4.

The growth rates ${\it\sigma}$ as a function of the swirl ratio ${\it\omega}$ of the various $m=1$ spiral perturbation modes on a solid-body rotation flow in a pipe with $L=6$ ; (a) ${\it\sigma}_{r}$ of modes $k=1$ and $k=2$ , (b) ${\it\sigma}_{i}$ of modes $k=1$ and $k=2$ , (c) ${\it\sigma}_{r}$ of modes $k=3$ and $k=4$ , (d) ${\it\sigma}_{i}$ of modes $k=3$ and $k=4$ .

It is found that each one of these complex modes turns in sequence from being asymptotically stable to unstable as ${\it\omega}$ is increased. Mode $k=1$ changes its stability at the swirl level ${\it\omega}_{1,1}=1.613$ and reaches a local maximum positive value of ${\it\sigma}_{r}=0.000253$ at ${\it\omega}=1.629$ (figure 4 a). Then ${\it\sigma}_{r}$ of this mode continues with oscillations as ${\it\omega}$ is further increased and has maximum points of ${\it\sigma}_{r}=0.00335$ at ${\it\omega}=1.771$ , of ${\it\sigma}_{r}=0.00308$ at ${\it\omega}=1.954$ and so on. Similarly, mode $k=2$ changes its stability at the swirl level ${\it\omega}_{1,2}=1.766$ and reaches a maximum of ${\it\sigma}_{r}=0.00765$ at ${\it\omega}=1.826$ (figure 4 a). The ${\it\sigma}_{r}$ of this mode continues with oscillations as ${\it\omega}$ is further increased with a maximum point of ${\it\sigma}_{r}=0.0139$ at ${\it\omega}=1.978$ . The higher-order modes with $k=3$ and $k=4$ exhibit a similar behaviour (figure 4 c). The mode $k=3$ becomes unstable at swirl level ${\it\omega}_{1,3}=1.976$ and the mode $k=4$ becomes unstable at swirl level ${\it\omega}_{1,4}=2.199$ . We note that each mode reaches a maximum of ${\it\sigma}_{r}$ that increases with the mode number $k$ . This mode becomes the dominant unstable mode in the range of swirl around its maximum of ${\it\sigma}_{r}$ .

In contrast to the $m=0$ results, the growth rates ${\it\sigma}$ are all complex in the $m=1$ case since the equations contain $\text{i}{\it\omega}$ terms which reflect the rotation of the azimuthal wave. The imaginary part ${\it\sigma}_{i}$ of the modes describes the rate of rotation of the mode around the pipe centreline. For all modes ${\it\sigma}_{i}$ is negative, indicating a rotation of the mode in the direction of the base flow rotation. For this case with $m=1$ the values of ${\it\sigma}_{i}$ as a function of ${\it\omega}$ are close to $-{\it\omega}$ , i.e. the rate of rotation of the mode is close to the negative rate of rotation of the base flow (see reference dash-dot blue line ${\it\sigma}_{i}=-{\it\omega}$ in figures 4 b and 4 d). When $|{\it\sigma}_{i}|>{\it\omega}$ , the mode rotates around the centreline with an angular speed that is greater than that of the base flow and, when $|{\it\sigma}_{i}|<{\it\omega}$ , the mode rotates around the centreline with an angular speed that is less than that of the base flow. For example, mode $k=1$ has a faster rate of rotation than the base flow when ${\it\omega}<1.613$ and has a slower rate of rotation than the base flow when ${\it\omega}>1.613$ . Similarly, mode $k=2$ has a faster rate of rotation than the base flow when ${\it\omega}<1.766$ and has a slower rate of rotation than the base flow when ${\it\omega}>1.766$ . We find that for each of the modes $k=1$ through $k=4$ with $m=1$ , the value of ${\it\sigma}+\text{i}{\it\omega}=0$ at the swirl levels ${\it\omega}_{1,k}$ where ${\it\sigma}_{r}=0$ .

We note that ${\it\omega}_{1,1}$ is less than ${\it\omega}_{0,1}$ ; i.e. there exists a range of swirl ${\it\omega}_{1,1}\leqslant {\it\omega}\leqslant {\it\omega}_{0,1}$ where only $m=1$ unstable growing perturbations appear on the solid-body rotation flow and before the appearance of unstable axisymmetric $m=0$ modes. Additional comparison between the axisymmetric and the $m=1$ unstable modes is shown in the next section.

Figure 5 shows a representative example of the solution of the dominant unstable $m=1$ rotating spiral mode shape functions at ${\it\omega}=2$ in a pipe with $L=6$ for which ${\it\sigma}=0.0127-\text{i}1.840$ (on the solution branch of $k=2$ mode with $m=1$ ). Figure 5(a) presents the computed planar projected velocity vector, $(1+{\it\epsilon}\text{Re}(\mathscr{W}),{\it\epsilon}\text{Re}(\mathscr{U}))$ , of this mode in the flow domain; here ${\it\epsilon}$ is the perturbation’s amplitude such that there are two stagnation points inside the domain and near the pipe outlet. Figure 5(b) describes two snapshots of streaklines of the mode at two non-dimensional times that are ${\rm\Delta}t={\rm\pi}/(2|{\it\sigma}_{i}|)=0.854$ apart. Both streaklines consist of particles released at the pipe inlet ( $x=0$ ) on a circle with $r=0.15$ . It can be seen that the mode describes a rotating spiral perturbation in the direction of flow rotation. Figure 5(a) qualitatively resembles the velocity field that is shown in figures 1(a) and 1(b) in Spall & Gatski (Reference Spall and Gatski1991) at the initial stage of perturbation’s growth to a spiral type of breakdown. Similar figures can be given for all other computed $m=1$ modes.

Figure 5.

The shape functions of the dominant inviscid unstable $m=1$ mode at ${\it\omega}=2$ : (a) planar projected velocity vector field; (b) two streaklines of the spiral mode at times ${\rm\Delta}t={\rm\pi}/(2|{\it\sigma}_{i}|)=0.854$ apart that consist of particles released at the pipe inlet ( $x=0$ ) on a circle with $r=0.15$ .

4.3 The $m=2$ modes

The stability problem with $m=2$ has a continuum of solutions of double-spiral perturbation modes. For this case the growth rates ${\it\sigma}$ are all complex, ${\it\sigma}={\it\sigma}_{r}+\text{i}{\it\sigma}_{i}$ . The growth rate ${\it\sigma}_{r}$ and frequency ${\it\sigma}_{i}$ as a function of the swirl ratio ${\it\omega}$ of the various $m=2$ perturbation modes on the solid-body rotation flow in a pipe with $L=6$ are shown in figure 6. Again, each of the different modes is characterized by a continuous solution as a function of ${\it\omega}$ with a similar axial wavelength of the shape functions that increases with the mode number. Figure 6(a) describes ${\it\sigma}_{r}$ of the modes $k=1$ and $k=2$ in the range $2.34\leqslant {\it\omega}\leqslant 2.5$ and figure 6(b) describes the corresponding ${\it\sigma}_{i}$ of these modes. Figure 6(a) shows that mode $k=1$ changes stability at ${\it\omega}_{2,1}=2.377$ , reaches a maximum value of ${\it\sigma}_{r}=2.04\times 10^{-5}$ at ${\it\omega}=2.383$ and then ${\it\sigma}_{r}$ of this mode continues with oscillations as ${\it\omega}$ is further increased. Similarly, mode $k=2$ changes stability at the critical swirl ${\it\omega}_{2,2}=2.387$ , reaches a maximum of ${\it\sigma}_{r}=0.00147$ at ${\it\omega}=2.454$ and ${\it\sigma}_{r}$ of this mode also continues with oscillations as ${\it\omega}$ is further increased. Figure 6(b) shows that the frequency ${\it\sigma}_{i}$ of these modes is around $2{\it\omega}$ . Additional modes become unstable in sequence as ${\it\omega}$ is further increased.

Figure 6.

The growth rates ${\it\sigma}$ as a function of the swirl ratio ${\it\omega}$ of the various $m=2$ double-spiral perturbation modes on a solid-body rotation flow in a pipe with $L=6$ ; (a) ${\it\sigma}_{r}$ of modes $k=1$ and $k=2$ , (b) ${\it\sigma}_{i}$ of modes $k=1$ and $k=2$ .

Figure 7 shows a representative example of the solution of the dominant unstable $m=2$ double-spiral mode shape functions at ${\it\omega}=2.454$ in a pipe with $L=6$ for which ${\it\sigma}=0.00147-\text{i}4.86$ (on solution branch $k=2$ of $m=2$ case). Figure 7(a) presents the computed planar projected velocity vector, $(1+{\it\epsilon}\text{Re}(\mathscr{W}),{\it\epsilon}\text{Re}(\mathscr{U}))$ , of this mode in the flow domain; here ${\it\epsilon}$ is the perturbation’s amplitude such that there is a reversed flow at the middle of the pipe. Figure 7(b) describes a snapshot of streakline of the mode that consists of particles released at the pipe inlet ( $x=0$ ) on a circle with $r=0.15$ . It can be seen that the mode describes a rotating double-spiral perturbation in the direction of flow rotation with almost twice the base flow rate of rotation. Similar figures can be given for all other computed $m=2$ modes.

Figure 7.

The shape functions of the dominant inviscid unstable $m=2$ mode at ${\it\omega}=2.454$ : (a) planar projected velocity vector field; (b) one streakline of the double-spiral mode that consists of particles released at the pipe inlet ( $x=0$ ) on a circle with $r=0.15$ .

4.4 The comparison between the growth rates

The growth rates computed for the various perturbation modes at $m=0$ , $m=1$ and $m=2$ are compared in figure 8 in the range of swirl $1.6\leqslant {\it\omega}\leqslant 2.5$ . It is found that all perturbation modes are asymptotically stable when ${\it\omega}<{\it\omega}_{1,1}=1.613$ . The first four of the $m=1$ perturbation modes become unstable in sequence when ${\it\omega}_{1,1}=1.613\leqslant {\it\omega}\leqslant {\it\omega}_{0,1}=1.9337$ . Although axisymmetric perturbation modes appear in sequence as well when ${\it\omega}>{\it\omega}_{0,1}$ , their growth rates are less than those of the second and third $m=1$ spiral modes when ${\it\omega}_{0,1}=1.9337\leqslant {\it\omega}\leqslant 2.118$ . These $m=1$ spiral modes are the fastest growing perturbations in this range of swirl. An example of such a dominant unstable spiral mode at ${\it\omega}=2$ is shown in figure 5. The third axisymmetric $m=0$ mode has the greatest growth rate in the range of swirl $2.118<{\it\omega}<2.170$ and is the dominant unstable mode in those range. An example of such a dominant unstable axisymmetric mode at ${\it\omega}=2.13$ is shown in figure 3. Similarly, higher-order $m=1$ and $m=0$ perturbation modes become the dominant unstable modes in sequence and switch roles in certain ranges of swirl as the swirl ratio ${\it\omega}$ is further increased. We note that the $m=2$ double-spiral unstable modes appear only when ${\it\omega}_{2,1}=2.397<{\it\omega}<2.5$ but their growth rates ${\it\sigma}_{r}$ are much smaller than those of the axisymmetric and $m=1$ spiral modes and they are insignificant in terms of the dynamics of the flow in this range of swirl.

Figure 8.

A comparison between the growth rates of the various $m=0,1,2$ inviscid flow instability modes.

The comparison between the various computed perturbation modes shows that in the inviscid flow case the solid-body rotation flow undergoes a complicated dynamical process as the swirl ratio increases. When ${\it\omega}<1.613$ the flow is asymptotically stable and any axisymmetric or three-dimensional perturbation decays in time and the flow returns to a columnar state.

In the range of swirl ${\it\omega}_{1,1}=1.613<{\it\omega}<{\it\omega}_{0,1}=1.9337$ small perturbations grow as spiral waves. Moreover, Rusak & Wang (Reference Rusak and Wang2014) recently revealed that the swirl ratio ${\it\omega}_{0,1}$ has an important role in the global dynamics of perturbed inviscid and axisymmetric flows. Only when ${\it\omega}>{\it\omega}_{0,1}$ there exist two strong axisymmetric attractors of the flow dynamics; the first is a vortex breakdown state with a near-stagnation zone around the pipe centreline and the second is a wall-separation state with a quasi-stagnation zone attached to the pipe wall. Such flow attractors do not exist when $0\leqslant {\it\omega}<{\it\omega}_{0,1}$ . Therefore, these states are not global attractors of the flow dynamics as the specific unstable spiral $m=1$ modes grow in the range ${\it\omega}_{1,1}=1.613<{\it\omega}<{\it\omega}_{0,1}=1.9337$ . The evolution of the three-dimensional perturbations in this swirl range is yet unknown and may be studied by further analysis using numerical simulations.

In the range of swirl ${\it\omega}>{\it\omega}_{0,1}$ any initial sufficiently small perturbation evolves in the beginning according to the most unstable perturbation mode found in figure 8. However, the long-term dynamics of the perturbation growth, either spiral or axisymmetric, is not yet clear. It may either grow according to the linear theory for a long time and be attracted to one of the axisymmetric global attractors mentioned above (as shown theoretically and demonstrated numerically by Rusak & Wang (Reference Rusak and Wang2014)) or it may evolve into a complicated three-dimensional unsteady state that may include a spiral breakdown state. The current inviscid flow linear stability analysis reveals the initial evolution of perturbations and further analysis using numerical simulations is needed.

5.1 The $m=0$ modes

We first consider the axisymmetric ( $m=0$ ) perturbation modes. The growth rates ${\it\sigma}_{r}$ as a function of the swirl ratio ${\it\omega}$ of the various $m=0$ modes are shown in figure 9 for $Re=500$ (the red line) and for $Re=1000$ (the black line).

It is found at $Re=500$ that all the modes have only real negative values of ${\it\sigma}$ when ${\it\omega}<1.985$ . At ${\it\omega}=1.985$ the modes $k=1$ and $k=2$ join together to form a fold point of their ${\it\sigma}_{r}$ curve. Then, these modes yield two branches of complex-conjugate growth rates with the same ${\it\sigma}_{r}<0$ that oscillates with ${\it\omega}$ when ${\it\omega}>1.985$ . The modes $k=3$ and $k=4$ exhibit a similar behaviour. They also have real negative growth rates when ${\it\omega}<2.18$ and form a fold point at ${\it\omega}=2.18$ from which two additional branches of complex-conjugate growth rates appear with same ${\it\sigma}_{r}<0$ that also oscillates with ${\it\omega}$ when ${\it\omega}>2.18$ . The mode $k=5$ also has real values of ${\it\sigma}_{r}$ when ${\it\omega}<2.475$ . This mode is the first mode that shows a change of the growth rate from negative to positive values at ${\it\omega}_{0,1}=2.3386$ . The swirl ratio ${\it\omega}_{0,1}=2.3386$ is the first critical swirl of the $m=0$ modes at $Re=500$ . All the $m=0$ modes are asymptotically stable for ${\it\omega}<2.3386$ at $Re=500$ . The growth rate of the $k=5$ mode is positive and this mode is unstable in the range of $2.3386<{\it\omega}<2.47$ . The swirl ratio ${\it\omega}_{0,2}=2.47$ is the second critical swirl of the $m=0$ modes at $Re=500$ at which this mode becomes asymptotically stable again. With further increase of ${\it\omega}$ , the modes $k=5$ and $k=6$ join together to form a fold point of their ${\it\sigma}_{r}$ curve at ${\it\omega}=2.475$ from which two additional branches of complex-conjugate growth rates appear. The ${\it\sigma}_{r}$ of these branches changes from negative to positive values at the third critical swirl ratio ${\it\omega}_{0,3}=2.495$ and then change back to negative values at the next critical swirl ratio ${\it\omega}_{0,4}=2.637$ . Additional higher-order instability modes appear in sequence as ${\it\omega}$ is further increased.

The case with $Re=1000$ exhibits a similar behaviour (see figure 9) but with reduced values of the critical swirl ratios where ${\it\sigma}_{r}$ changes its sign. In the $Re=1000$ case, the mode $k=3$ is the first to become unstable at ${\it\omega}_{0,1}=2.09$ and then becomes asymptotically stable at ${\it\omega}_{0,2}=2.177$ . The modes $k=3$ and $k=4$ join together at ${\it\omega}=2.18$ to form two complex-conjugate modes that become unstable at ${\it\omega}_{0,3}=2.205$ and then stabilize at ${\it\omega}_{0,4}=2.311$ . Again, additional higher-order instability modes appear in sequence as ${\it\omega}$ is further increased.

The comparison between the various ${\it\sigma}_{r}$ curves of the $m=0$ modes at $Re=500$ and $Re=1000$ shows that the values of ${\it\sigma}_{r}$ for each mode $k$ increase by a nearly constant value in the range of ${\it\omega}$ studied as $Re$ is increased. The effect of viscosity on the $m=0$ modes is passive. This is confirmed by the perturbation’s kinetic energy analysis in § 6.

Figure 9.

The growth rates ${\it\sigma}$ as a function of the swirl ratio ${\it\omega}$ of the various axisymmetric ( $m=0$ ) perturbation modes on a solid-body rotation flow in a pipe with $L=6$ ; the red line is for $Re=500$ and the black line is for $Re=1000$ .

The shapes of the various $m=0$ modes at both $Re=500$ and $Re=1000$ are similar to their counterparts in the inviscid flow case, shown for example in figure 3 for the $k=3$ mode at ${\it\omega}=2.13$ .

5.2 The $m=1$ modes

Similar to the inviscid flow case, the viscous flow stability problem with $m=1$ also has a continuum of solutions of spiral perturbation modes, all of them with a complex value of ${\it\sigma}={\it\sigma}_{r}+i{\it\sigma}_{i}$ . The growth rates ${\it\sigma}_{r}$ and frequency ${\it\sigma}_{i}$ as a function of the swirl ratio ${\it\omega}$ of the various $m=1$ perturbation modes on the solid-body rotation flow in a pipe of length $L=6$ are shown in figure 10. Each of the $m=1$ modes is characterized by a continuous solution as a function of ${\it\omega}$ with same axial wavelength of the shape functions that increases with the mode number $k$ where $k=1,2,3,4$ and so on.

Figure 10.

The growth rates ${\it\sigma}$ as a function of the swirl ratio ${\it\omega}$ of the various $m=1$ spiral perturbation modes on a solid-body rotation flow in a pipe with $L=6$ ; (a) ${\it\sigma}_{r}$ for $Re=500$ (red line) and $Re=1000$ (black line), (b) ${\it\sigma}_{i}$ for $Re=1000$ .

Figure 10(a) describes the curves of ${\it\sigma}_{r}$ as a function pf ${\it\omega}$ of the modes $k=1$ through $k=4$ in the range $1.6\leqslant {\it\omega}\leqslant 2.6$ for $Re=500$ (the red lines) and $Re=1000$ (the black lines). For the case $Re=500$ , we find that the $k=1$ mode has an oscillatory variation of ${\it\sigma}_{r}<0$ and is asymptotically stable in the entire range of swirl studied. Similarly, the $k=2$ mode also has an oscillatory change of ${\it\sigma}_{r}$ and is negative in the range of swirl studied. The $k=3$ mode is the first to become unstable at the swirl ratio ${\it\omega}_{1,1}=2.206$ ; i.e. all the $m=1$ perturbation modes are asymptotically stable when ${\it\omega}<{\it\omega}_{1,1}<2.206$ . The $k=3$ mode is unstable in the range $2.206<{\it\omega}<2.244$ and reaches a maximum value of ${\it\sigma}_{r}=0.0012$ at ${\it\omega}=2.225$ . The swirl ratio ${\it\omega}_{1,2}=2.244$ is the second critical swirl. The mode $k=3$ becomes asymptotically stable when ${\it\omega}>2.244$ with ${\it\sigma}_{r}<0$ that oscillates as ${\it\omega}$ is further increased. The $k=4$ mode changes its stability at the critical swirl ${\it\omega}_{1,3}=2.401$ , reaches a maximum values of ${\it\sigma}_{r}=0.0182$ at ${\it\omega}=2.483$ and ${\it\sigma}_{r}$ of this mode continues to stay positive and exhibits oscillations as ${\it\omega}$ is further increased within the specified swirl range. Higher-order critical swirl ratios and instability modes with $m=1$ appear in sequence and exhibit a similar behaviour. We note that the maximum of ${\it\sigma}_{r}$ of the modes with $m=1$ increases with the mode number $k$ .

The case with $Re=1000$ exhibits a similar behaviour (see figure 10 a) but with reduced values of the critical swirl ratios where ${\it\sigma}_{r}$ changes its sign. In the $Re=1000$ case, the mode $k=2$ is the first to become unstable at ${\it\omega}_{1,1}=1.960$ and then becomes asymptotically stable at ${\it\omega}_{1,2}=1.998$ . The $k=3$ mode becomes unstable at ${\it\omega}_{1,3}=2.022$ . Again, additional critical swirl ratios and instability modes appear in sequence as ${\it\omega}$ is further increased.

The comparison between the various ${\it\sigma}_{r}$ curves of the $m=1$ modes at $Re=500$ and $Re=1000$ shows that the values of ${\it\sigma}_{r}$ for each mode $k$ increase by a nearly constant value in the range of ${\it\omega}$ studied as $Re$ is increased. The effect of viscosity on the $m=1$ modes is passive and this is also confirmed by the perturbation’s kinetic energy analysis in § 6.

The imaginary part ${\it\sigma}_{i}$ of the various modes with $m=1$ is shown in figure 10(b) for the case $Re=1000$ . Also shown in this figure is the reference line ${\it\sigma}_{i}=-{\it\omega}$ (the dash-dot blue line). The value of ${\it\sigma}_{i}$ describes the rate of rotation of the mode around the pipe centreline. We find that at the change of stability swirl ratio ${\it\omega}_{1,1}=1.960$ of the $k=2$ mode ${\it\sigma}_{i}$ is above the reference line, i.e. $-{\it\omega}_{1,1}<{\it\sigma}_{i}<0$ for ${\it\omega}>{\it\omega}_{1,1}$ . This means that the $k=2$ mode with $m=1$ rotates around the centreline at a clearly slower rate than the base flow. This is a direct result of the viscous effects.

Figure 11 shows a representative example of the solution of the dominant unstable $m=1$ rotating spiral mode shape functions at ${\it\omega}=2.48,Re=500$ in a pipe with $L=6$ for which ${\it\sigma}=0.0181-i2.438$ (on the solution branch of $k=4$ mode with $m=1$ ). Figure 11(a) presents the computed planar projected velocity vector, $(1+{\it\epsilon}\text{Re}(\mathscr{W}),{\it\epsilon}\text{Re}(\mathscr{U}))$ , of this mode in the flow domain. Figure 11(b) describes a snapshot of streaklines of the mode. The streakline consists of particles released at the pipe inlet ( $x=0$ ) on a circle with $r=0.15$ . It can be seen that the mode describes a rotating spiral perturbation that is similar to the inviscid mode shape shown in figure 5. Also, figure 11(a) qualitatively resembles the velocity field that is shown in figures 1(a) and 1(b) in Spall & Gatski (Reference Spall and Gatski1991) at the initial stage of perturbation’s growth to a spiral type of breakdown.

Figure 11.

The shape functions of the dominant unstable $m=1$ mode at ${\it\omega}=2.48$ : (a) planar projected velocity vector field; (b) the streaklines of the spiral mode that consist of particles released at the pipe inlet ( $x=0$ ) on a circle with $r=0.15$ .

5.3 The $m=2$ modes

The viscous flow stability problem with $m=2$ has a continuum of solutions of double-spiral perturbation modes with complex values of ${\it\sigma}={\it\sigma}_{r}+i{\it\sigma}_{i}$ and mode shapes. The curves of growth rate ${\it\sigma}_{r}$ and the frequency ${\it\sigma}_{i}$ as a function of the swirl ratio ${\it\omega}$ in the range $2.3\leqslant {\it\omega}\leqslant 3$ of the various $m=2$ perturbation modes on the solid-body rotation flow in a pipe with $L=6$ at $Re=1000$ are shown in figure 12. Again, each of the $k=1$ through $k=4$ modes is characterized by a continuous solution as a function of ${\it\omega}$ with the same axial wavelength of the shape function that decreases with the mode number $k$ .

Figure 12.

The growth rates ${\it\sigma}$ as a function of the swirl ratio ${\it\omega}$ of the various $m=2$ spiral perturbation modes on a solid-body rotation flow in a pipe with $Re=1000$ and $L=6$ ; (a) ${\it\sigma}_{r}$ as a function of ${\it\omega}$ , (b) ${\it\sigma}_{i}$ as a function of ${\it\omega}$ .

Figure 12(a) describes ${\it\sigma}_{r}$ of the modes $k=1$ through $k=4$ with $m=2$ . It is found that modes $k=1$ , $k=2$ and $k=3$ are asymptotically stable ( ${\it\sigma}_{r}<0$ ) in the entire range of swirl studied. The mode $k=4$ is the first to become unstable at the critical swirl ratio ${\it\omega}_{2,1}=2.9067$ . The $k=4$ mode reaches a maximum value of ${\it\sigma}_{r}=0.0066$ at ${\it\omega}=2.958$ and becomes asymptotically stable when ${\it\omega}$ increases above ${\it\omega}_{2,2}=2.999$ . The frequency ${\it\sigma}_{i}$ of the various modes with $m=2$ is shown in figure 12(b). Also shown in this figure is the reference line ${\it\sigma}_{i}=-2{\it\omega}$ (the blue dashed line). Within the studied range of swirl, the frequency of the modes $k=1$ through $k=4$ is close to or slightly less than twice the rate of rotation of the base flow. The shapes of the various $m=2$ modes at $Re=1000$ are similar to their counterparts in the inviscid flow case, shown for example in figure 7.

We also find that for the case $Re=500$ and $m=2$ the modes $k=1$ through $k=4$ appear in sequence and have similar ${\it\sigma}_{r}$ curves and mode shapes to those of the $Re=1000$ case but remain asymptotically stable ( ${\it\sigma}_{r}<0$ ) in the entire specified swirl range (not shown). The viscous effects stabilize these modes.

5.4 The neutral stability lines of the perturbation modes

We studied the stability of the solid-body rotation flow in a pipe with length $L=6$ for a wide range of $Re$ from $100$ up to $2000$ in the range of swirl ratios $0<{\it\omega}\leqslant 3$ . For each $Re$ we determined the swirl ratios of the $m=0$ , $m=1$ and $m=2$ modes at which ${\it\sigma}_{r}=0$ (a neutral mode) appears for the first time as ${\it\omega}$ is increased from zero. The neutral stability lines and the stability and instability domains are shown in the $Re$ versus ${\it\omega}$ plane in figure 13. The light solid lines represent the neutral lines for the $m=0$ modes, the heavy solid lines represent the neutral lines for the $m=1$ modes and the dotted lines represent the neutral lines for the $m=2$ modes. The shaded area is the unstable flow domain.

Figure 13.

The neutral stability lines dividing the stable domain (white area) and unstable domain (shaded area) for various perturbation modes with $L=6$ , where the light black lines are for the axisymmetric $m=0$ modes, the heavy black lines are for the single-spiral $m=1$ modes and the dashed lines are for the double-spiral $m=2$ modes.

We find from figure 13 that in the case with $L=6$ , the single-spiral $m=1$ instability modes are the first to appear as swirl ${\it\omega}$ is increased at a fixed $Re$ in the range $481<Re<2000$ . At lower $Re$ , $214<Re<481$ , the axisymmetric $m=0$ instability modes are the first to appear as swirl ratio is increased at a fixed $Re$ . The flow is stable for all ${\it\omega}<3$ when $Re<214$ . The $m=2$ instability modes do not affect the first change of flow stability within the studied range of $Re$ and ${\it\omega}$ . In addition, when ${\it\omega}$ is fixed and $Re$ changes, the $m=1$ modes may appear first as $Re$ is increased at certain ranges of ${\it\omega}$ , i.e. the ranges $0<{\it\omega}<2.122$ , $2.168<{\it\omega}<2.251$ , $2.301<{\it\omega}<2.366$ and so on, while at the in between ranges of ${\it\omega}$ the $m=0$ modes may appear first as $Re$ is increased.

This rich combination of change of stability of the $m=0$ and $m=1$ modes in viscous flows shows that the solid-body rotation flow in a straight circular rotating pipe may have various possible instability routes to form finite-amplitude spiral waves as well as vortex-breakdown or wall-separation states. We note that the recent experiments of Dennis et al. (Reference Dennis, Seraudie and Poole2014) studied a flow in a straight circular rotating pipe and found the appearance of either an axisymmetric wave or a spiral mode that grows in time and eventually evolves into an axisymmetric breakdown state. The present results provide a base case study for additional detailed numerical simulations and experimental studies.

6.1 The difference between classical stability theory and the present analysis

The results of § 4 show that the solid-body rotation flow in a finite-length pipe is generically changing its stability as the base flow swirl ratio increases above certain levels ${\it\omega}_{0,1}$ , ${\it\omega}_{1,1}$ , ${\it\omega}_{2,1}$ and so on, and unstable modes with $m=0,1,2,\ldots ,$ respectively, appear in sequence as swirl is further increased for both the inviscid and the viscous flow cases. At any given swirl ratio, the perturbation’s dynamics is determined by the mode with the greatest growth rate ${\it\sigma}_{r}$ . This result is strikingly different from the predictions according to the classical normal mode linear stability theory which proved the well-known result that the solid-body rotation flow in an infinitely long pipe is stable to any type of infinitesimal perturbations (Kelvin Reference Kelvin1880; Rayleigh Reference Rayleigh1916; Synge Reference Synge1933). Thus, it is necessary to further understand the difference between the classical linear stability theory and the present stability analysis of the solid-body rotation flow in a finite-length pipe.

In the classical vortex stability theory, the analysis is based on the method of normal modes; a technique, that following Fourier, can be effectively used to analyse the flow stability problem for an axially independent base flow. Namely, the perturbation’s fields of velocity vector and pressure are assumed to follow the solution form $\boldsymbol{u}^{\prime }(x,r,{\it\theta},t)=\tilde{\boldsymbol{u}}(r)\text{e}^{\text{i}m{\it\theta}+\text{i}{\it\alpha}x+{\it\sigma}t}$ and $p^{\prime }(x,r,{\it\theta},t)=\tilde{p}(r)\text{e}^{\text{i}m{\it\theta}+\text{i}{\it\alpha}x+{\it\sigma}t}$ , respectively. The axial wavenumber ${\it\alpha}$ must be real to preserve the boundness of the perturbation in the pipe as $x\rightarrow \pm \infty$ and $m$ must be an integer to preserve the perturbation’s azimuthal periodicity in the circular pipe. We note that the classical normal mode form of solution of the linear stability problem may be specific in its shape. Substituting this solution form of the perturbation into the linearized flow problem (2.11a–d ) and using the axial homogeneity of the problem, results in an eigenvalue problem for solving ${\it\sigma}$ as a function of $m$ and ${\it\alpha}$ that involves ODEs for the mode shape functions that are functions of the radial distance $r$ only. The classical normal mode solution approach significantly simplifies the analysis of the vortex stability problem (2.11a–d ) but this simplification may restrict the scope of the applicability of the theory. We emphasize that the classical stability analysis is but a linear problem with special boundary conditions, the far-field boundness conditions as $x\rightarrow \pm \infty$ .

The classical normal mode analysis provides the linear stability of a base columnar vortex flow in an infinitely long pipe. Di Prima & Habetler (Reference Di Prima and Habetler1969) proved a completeness theorem for the non-self-adjoint eigenvalue problem derived by the method of normal modes, and determined the class of initial disturbances that can be investigated by the method of normal modes. Therefore, all types of perturbations in a columnar vortex flow in an infinitely long pipe can be described by the normal mode shapes or a linear combination of these modes. Basically, the classical stability theory is applicable to a perturbation with a finite kinetic energy in a columnar vortex in an infinitely long pipe. However, this creates a delicate situation since such entities of infinite extent do not exist in practice. When inlet and outlet conditions are specified and interact with the flow in the bulk, they may generate an interference between the perturbation and the inlet and outlet states. Note that the normal modes are not solutions of the present stability problem (2.11a–d ) with the boundary conditions (2.13a–e ). The eigenfunction and eigenvalue are determined by the whole problem, including the characteristic equations and the assumed boundary conditions.

In a real flow, when the evolution of disturbances is only locally affected by the upstream and downstream states, the classical theory may well predict the flow linear stability characteristics in the bulk, such as in the cases of Taylor’s experiments of Taylor–Couette flow, the Crow instability of two trailing counter-rotating vortices and a rotating flow in an annular device. However, in the case where the perturbation interacts with the inlet (or upstream) and outlet (or downstream) states, there is an active transfer of kinetic energy between the perturbation and the base flow at the pipe outlet and inlet. We refer the reader to the detailed analyses in Gallaire & Chomaz (Reference Gallaire and Chomaz2004) and in Wang & Rusak (Reference Wang and Rusak2011) for the transfer mechanism of the perturbation’s kinetic energy in the axisymmetric case. Such an energy transfer mechanism is associated with the necessary loss of the perturbation’s periodicity due to the flow convective nature from the inlet to the outlet. The perturbation becomes naturally active in absorbing kinetic energy through the boundaries to sustain the possible perturbation’s growth in the bulk as the inlet swirl ratio is increased. Such a situation cannot be described by the classical vortex stability normal modes where the modes are naturally passive in the transfer of kinetic energy to the bulk. In the following we demonstrate a similar mechanism for the three-dimensional perturbations.

In the present physical model, the swirling flow is generated by a fixed in place and fixed in time vortex generator ahead of the pipe inlet and is convected into a finite-length pipe. The flow interacts with the outlet discharge device and moves out of the domain. Within the inviscid flow approach for a swirling flow in the pipe, three conditions must be specified at the pipe inlet to describe the incoming flow properties and only one condition is set at the outlet to specify how the flow convects out of the pipe through the discharge device. In the viscous flow case, two additional boundary conditions must be also included at the pipe outlet but these are relatively passive at high $Re$ situations and have only a minor effect on the perturbation’s behaviour in the bulk. This generates a fundamentally different physical set-up than the infinitely long pipe case; the inlet and outlet states have essentially a different influence on the flow in the bulk and, therefore, periodic inlet–outlet conditions are not realistic and are inconsistent with the present physical set-up. This leads to a natural interaction of the perturbation with the boundaries, specifically at high swirl levels, and the onset of non-axial Fourier instability modes as the swirl ratio is increased. This is the root cause for the difference between the present stability approach and the classical vortex stability theory.

6.2 Production mechanisms of the perturbation’s kinetic energy

We study the various productions of the perturbation’s kinetic energy inside the bulk and at the boundaries based on the Reynolds–Orr equation for the perturbation’s specific integrated kinetic energy $E$ ,

For an axisymmetric base flow with velocity components, $\boldsymbol{u}_{0}=(u_{0x},u_{0r},u_{0{\it\theta}})$ , the Reynolds–Orr equation (see, for example, Wu, Ma & Zhou (Reference Wu, Ma and Zhou2006)) relates the rate of change in time of $E$ to various source terms. In appendix A we derive the Reynolds–Orr equation in the cylindrical coordinates, i.e. for all $t>0$

Here $S_{1}$ and $S_{2}$ are the pipe inlet and outlet sections, respectively; the boundary $\partial V_{C}$ is the domain’s control surface (the inlet, outlet and wall); $\partial /\partial n$ is the directional derivative in the outer normal direction of $\partial V_{C}$ . Also, $\unicode[STIX]{x1D63D}$ is the symmetric strain-rate tensor of the base flow, given in cylindrical coordinates by

The source terms on the right-hand side of (6.2) include, in order, the production of $E$ inside the flow domain, the production of $E$ by the work of the pressure perturbations at the pipe inlet and the outlet, the transfer of $E$ resulting from the convection of $E$ by the base flow axial velocity $u_{0x}$ at the inlet and the outlet and the production of $E$ by the viscous dissipation. The Reynolds–Orr equation applies to any type (axisymmetric or three-dimensional) and to any size of perturbation and is therefore demonstrating the nonlinear kinetic energy exchange between the base flow and the perturbation to determine the fate of the perturbation.

For a base columnar solid-body rotation flow with a uniform axial velocity, $u_{0x}=1$ , $u_{0r}=0$ and $u_{0{\it\theta}}={\it\omega}r$ , and the symmetric strain-rate tensor $\unicode[STIX]{x1D63D}$ vanish in the entire flow domain. Therefore, the first integral on the right-hand side of (6.2) vanishes. In addition, the two integrals over the inlet surface $S_{1}$ also vanish due to the no velocity perturbation conditions at the inlet. As a result, the rate of change of the perturbation’s kinetic energy $E$ in time is given for all $t>0$ by,

The first term on the right-hand side of (6.4) represents the work done by the pressure perturbation due to the axial velocity perturbation at the outlet. This term cannot be computed without a solution of the flow perturbation. The solution reflects the flow capability to absorb kinetic energy through the interaction with the outlet at various swirl ratios ${\it\omega}$ . Yet, once a solution of the perturbation structure and evolution is found, we can compute this term, compare it with the other terms, and determine from (6.4) the base flow stability or instability mechanism. The second term on the right-hand side of (6.4) represents the perturbation’s kinetic energy that convects out of the domain, and is thus always negative. The last two terms in (6.4) represent the viscous effect throughout the domain as well as at the boundaries.

We note that the eigenfunctions are in general complex functions and the velocities and pressure perturbations are real components of the eigenfunctions and are given by

Here $\bar{\mathscr{U}},\bar{\mathscr{V}},\bar{\mathscr{W}}$ and $\bar{\mathscr{P}}$ are the complex-conjugate functions of $\mathscr{U},~\mathscr{V},~\mathscr{W}$ and $\mathscr{P}$ , respectively. Inserting (6.5) into  (6.4) we obtain for all $t>0$ ,

Here,

The term $E_{vis1}$ represents the viscous effects in the bulk on $\text{d}E(t)/\text{d}t$ ,

The term $E_{vis2}$ represents the viscous effects from the boundaries (wall and outlet) on $\text{d}E(t)/\text{d}t$ ,

where

The derivation of these formulas is straightforward and follows a similar pattern. Note that the integrals of the terms $\text{e}^{2\text{i}m{\it\theta}+{\it\sigma}t}$ and its complex conjugate $\overline{\text{e}^{2\text{i}m{\it\theta}+{\it\sigma}t}}$ with respect to ${\it\theta}$ vanish because of the periodicity in the interval $0\leqslant {\it\theta}<2{\rm\pi}$ .

Similarly, we also find from (6.1) that

where

From the formulation (6.11) it follows that

Then, from (6.4) and (6.5), we can use the Reynolds–Orr equation to replace $\text{d}E/\text{d}t$ in (6.13). Then, together with (6.11), we find that time-dependent terms cancel each other in the formula for ${\it\sigma}_{r}$ and

where

The term ${\it\sigma}_{r1}$ represents the growth rate of the perturbation as a result of the integrated perturbation’s pressure work at the outlet. The term ${\it\sigma}_{r2}$ represents the growth rate of the perturbation as a result of the integrated convective loss of perturbation’s kinetic energy at the outlet, and the term ${\it\sigma}_{r3}={\it\sigma}_{internal}+{\it\sigma}_{boundary}$ , where ${\it\sigma}_{internal}=E_{vis1}/(2\mathscr{E}_{0}Re)$ and ${\it\sigma}_{boundary}=E_{vis2}/(2\mathscr{E}_{0}Re)$ , represents the growth rate of the perturbation as a result of the viscous dissipation of the perturbation’s kinetic energy. The sum of these three terms gives the perturbation’s growth rate in the bulk.

6.3 The inviscid flow case

We now use (6.4)–(6.15) to explore the mechanism of the perturbation’s loss of stability in the inviscid flow case where

We consider a representative example of the solution of the flow perturbation from § 4 for the $m=1$ case, the $k=2$ mode, at ${\it\omega}=2$ . This mode is the most unstable mode at ${\it\omega}=2$ . Figure 14(a) describes for this mode the radial distributions at the outlet $(x=L)$ of $-E_{1}(r)$ (the dashed line) and $-E_{2}(r)$ (the dash-dot line). Also shown in figure 14(a) is the radial distribution of the sum of these two terms (the solid line). We can see that these two terms are maximized or minimized around $r=0.5$ and decrease in size near the centreline $(r=0)$ and near the wall $(r=1)$ . Also, for this swirl level and this mode, the pressure production term is greater, in its absolute size, than the convective loss of the perturbation’s kinetic energy around $r=0.5$ , such that the sum of the terms is positive and maximized around $r=0.5$ and only slightly negative near the centreline and the wall. The integral of the sum of these terms across the outlet section is positive. It indicates that, at this swirl level and this mode shape, the net perturbation’s pressure work all across the outlet is greater than the net convective loss of perturbation’s kinetic energy all across the outlet to make $\text{d}E/\text{d}t>0$ and form the instability of the base flow with this mode of perturbation. Figure 14(b) presents similar results for the benchmark axisymmetric ( $m=0$ ) third ( $k=3$ ) instability mode at ${\it\omega}=2.15$ . On the other hand, for all $m$ studied here, when ${\it\omega}<{\it\omega}_{m,1}$ , the situation reverses and the net perturbation’s pressure work all across the outlet is less than the net convective loss of perturbation’s kinetic energy all across the outlet to make $\text{d}E/\text{d}t<0$ and form the asymptotic stability of the base flow with the various modes of perturbation.

Figure 14.

The radial distribution at the outlet $(x=L)$ of the perturbation’s energy production terms for (a) the $m=1,~k=2$ perturbation mode at ${\it\omega}=2$ ; (b) the $m=0,k=3$ perturbation mode at ${\it\omega}=2.15$ .

Figure 15.

The growth rates ${\it\sigma}_{r1}$ , ${\it\sigma}_{r2}$ , ${\it\sigma}_{r1}+{\it\sigma}_{r2}$ and ${\it\sigma}_{r}$ as a function of ${\it\omega}$ : (a) the $m=1,~k=4$ perturbation mode; (b) the $m=0,~k=3$ perturbation mode.

Figure 15(a) presents ${\it\sigma}_{r1}$ , ${\it\sigma}_{r2}$ and ${\it\sigma}_{r1}+{\it\sigma}_{r2}$ as a function of ${\it\omega}$ for the $m=1$ case, the $k=4$ mode, in the range of swirl ratio $2.15<{\it\omega}<2.4$ . In this range of the swirl ratio, the $m=1$ and $k=4$ mode changes its stability from asymptotically stable to unstable at ${\it\omega}_{1,4}=2.199$ . Also shown in figure 15(a) is the growth rate ${\it\sigma}_{r}$ of this mode of perturbation as a function of ${\it\omega}$ , taken from figure 4(c). It can be seen that the line of ${\it\sigma}_{r1}+{\it\sigma}_{r2}$ nicely matches with the computed line of ${\it\sigma}_{r}$ . We emphasize that ${\it\sigma}_{r1}$ and ${\it\sigma}_{r2}$ are computed according to (6.15a ) and (6.15b ) based on the mode shape eigenfunctions at the outlet and in the bulk (through $\mathscr{E}_{0}$ ) as they change with ${\it\omega}$ and do not involve the values of ${\it\sigma}_{r}$ or ${\it\sigma}_{i}$ of the mode. Moreover, both ${\it\sigma}_{r1}$ and ${\it\sigma}_{r2}$ exhibit a non-monotonic dependence on the swirl ratio ${\it\omega}$ and reflect the change of the perturbation’s stability as ${\it\omega}$ increases around ${\it\omega}_{1,4}=2.199$ . We see that when ${\it\omega}<2.199$ , ${\it\sigma}_{r1}<-{\it\sigma}_{r2}$ , the net convective loss of perturbation’s kinetic energy all across the outlet dominates the perturbation’s pressure work at the outlet, and the mode is asymptotically stable. On the other hand, when $2.199<{\it\omega}<2.4$ , ${\it\sigma}_{r1}>-{\it\sigma}_{r2}$ , the perturbation’s pressure work at the outlet dominates the net convective loss of perturbation’s kinetic energy all across the outlet, and the mode is unstable. These two terms depend on the mode eigenfunctions at the outlet and in the bulk and they change with ${\it\omega}$ . This change with ${\it\omega}$ reflects the capability of the perturbation in the bulk to absorb energy through the outlet and thereby affect the base flow stability. We note however, that the outlet serves here only as an agent to transfer the perturbation’s kinetic energy to the bulk. Similar results for the change of stability of the benchmark axisymmetric ( $m=0$ ) third mode are presented in figure 15(b) in the range of swirl ratio $2.06<{\it\omega}<2.18$ and are also found for all the other $m=0,1,2$ perturbation modes (not shown).

6.4 The viscous flow case

Figure 16 presents the computed values of ${\it\sigma}_{r1}$ (the dashed line), ${\it\sigma}_{r2}$ (the dash-dot line), ${\it\sigma}_{internal}$ (the red line), ${\it\sigma}_{boundary}$ (the dash-dot blue line) and ${\it\sigma}_{r1}+{\it\sigma}_{r2}+{\it\sigma}_{r3}$ (the open circles) as a function of ${\it\omega}$ for the $m=1$ and $k=4$ mode at $Re=1000$ and $L=6$ in the range of swirl ratio $2.15<{\it\omega}<2.4$ (the same mode and range of swirl studied in the inviscid flow case in figure 15 a). In this case, this mode changes its stability from asymptotically stable to unstable at ${\it\omega}_{1,8}=2.261$ . Also shown in figure 16 is the computed growth rate ${\it\sigma}_{r}$ of this mode of perturbation, taken from figure 10(a) (the solid line). It can be seen that the line of ${\it\sigma}_{r1}+{\it\sigma}_{r2}+{\it\sigma}_{r3}$ nicely matches with the computed line of ${\it\sigma}_{r}$ . The lines of ${\it\sigma}_{r1}$ and ${\it\sigma}_{r2}$ are similar to the inviscid flow case. Moreover, we find that ${\it\sigma}_{boundary}$ is close to zero and that the viscous effect through ${\it\sigma}_{r3}$ is dominated by ${\it\sigma}_{internal}$ that is negative and is nearly constant in the studied range of ${\it\omega}$ . This indicates that the viscous effect is stabilizing the perturbation’s mode and is only slightly dependent on ${\it\omega}$ . It means that viscosity plays only a passive role in decreasing ${\it\sigma}_{r}$ and thereby increasing the swirl level of change of stability of the mode from ${\it\omega}=2.199$ in the inviscid flow case to ${\it\omega}=2.261$ at $Re=1000$ . Also, it demonstrates that the perturbation mode shape is dominated by the inviscid flow solution at the same ${\it\omega}$ and $L$ .

Figure 16.

The growth rates ${\it\sigma}_{r1}$ , ${\it\sigma}_{r2}$ , ${\it\sigma}_{internal}$ , ${\it\sigma}_{boundary}$ , ${\it\sigma}_{r1}+{\it\sigma}_{r2}+{\it\sigma}_{r3}$ and ${\it\sigma}_{r}$ as a function of ${\it\omega}$ for the $m=1$ and $k=4$ perturbation mode at $Re=1000$ with $L=6$ .

To further substantiate this idea, we compute the growth rate ${\it\sigma}_{r}$ for the $m=1$ and $k=4$ mode as a function of $Re$ at a fixed ${\it\omega}=2.3$ and $L=6$ according to the formulas (6.14) and (15a–c ), where we replace in these formulas the exact viscous flow mode shapes with the inviscid flow mode shapes at ${\it\omega}=2.3$ and $L=6$ . From this computation we find that ${\it\sigma}_{r1}+{\it\sigma}_{r2}=0.0274$ is the inviscid flow growth rate and $(E_{vis1}+E_{vis2})/(2\mathscr{E}_{0})=-17.275$ . Thus, ${\it\sigma}_{r}$ for the $m=1$ and $k=4$ mode at ${\it\omega}=2.3$ and $L=6$ may be approximated at high $Re$ flows by

These approximate growth rates are shown in figure 17 (the solid line with $+$ symbol) in the range of $0<Re<500$ (figure 17 a) and in the range of $0<Re<30\,000$ (figure 17 b). Also shown in figure 17(a,b) are the computed results from the solution of the viscous flow problem at each $Re$ with ${\it\omega}=2.3$ (the open circles). In addition, figure 17(b) shows the inviscid flow result for this case, ${\it\sigma}_{r}=0.0274$ (the dashed line). We can see that the computation of ${\it\sigma}_{r}$ based on the inviscid flow mode shapes is close to the exact results of the viscous flow cases for all $Re>300$ . The agreement between the two computations becomes better as $Re$ is increased. The exact viscous flow results for ${\it\sigma}_{r}$ approach the inviscid flow result as $Re$ is increased, indicating that the inviscid flow result is the inviscid-limit result of the viscous flow problem when $Re$ is above $20\,000$ and tends to infinity, as described by (6.14) and (15a–c ). This shows again that the perturbation mode shape at any given $Re>1000$ is dominated by the inviscid flow solution at the same ${\it\omega}$ and $L$ . We emphasize that similar results are found for all other perturbation modes at various ${\it\omega}$ and $L$ . Essentially, when $Re>1000$ , the viscous flow three-dimensional instability modes found in this paper are close to the inviscid flow solution with a small correction of their ${\it\sigma}_{r}$ due to the viscous effects.

Figure 17.

The growth rates ${\it\sigma}_{r}$ as a function of $Re$ at ${\it\omega}=2.3$ and $L=6$ : (a) $0<Re\leqslant 500$ and (b) $0<Re\leqslant 30\,000$ .