5.1. Solid-body rotation

In this example, the inlet flow is described by a solid-body rotation with a uniform axial velocity (axial plug flow) and a uniform temperature, i.e.

(5.1a,b ) $$\begin{eqnarray}v_{0}(y)=\sqrt{2y},\quad w_{0}(y)=T_{0}(y)=1\quad \text{for }0\leqslant y\leqslant 1/2.\end{eqnarray}$$

We find that

(5.2) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle p_{0}(y)={\it\rho}_{0}(y)=\exp ({\it\gamma}Ma_{0}^{2}{\it\omega}^{2}y),\\ \displaystyle {\it\Psi}(y)=\frac{1}{{\it\gamma}Ma_{0}^{2}{\it\omega}^{2}}[\exp ({\it\gamma}Ma_{0}^{2}{\it\omega}^{2}y)-1],\\ \displaystyle y({\it\Psi})=\frac{1}{{\it\gamma}Ma_{0}^{2}{\it\omega}^{2}}\ln ({\it\gamma}Ma_{0}^{2}{\it\omega}^{2}{\it\Psi}+1),\\ \displaystyle K({\it\Psi})=2{\it\omega}y({\it\Psi})=\frac{2}{{\it\gamma}Ma_{0}^{2}{\it\omega}}\ln ({\it\gamma}Ma_{0}^{2}{\it\omega}^{2}{\it\Psi}+1),\\ \displaystyle K^{\prime }({\it\Psi})=\frac{2{\it\omega}}{{\it\gamma}Ma_{0}^{2}{\it\omega}^{2}{\it\Psi}+1},\\ \displaystyle H({\it\Psi})=1+\frac{({\it\gamma}-1)Ma_{0}^{2}}{2}\left(1+\frac{2}{{\it\gamma}Ma_{0}^{2}}\ln ({\it\gamma}Ma_{0}^{2}{\it\omega}^{2}{\it\Psi}+1)\right),\\ \displaystyle H^{\prime }({\it\Psi})=\frac{({\it\gamma}-1)Ma_{0}^{2}{\it\omega}^{2}}{{\it\gamma}Ma_{0}^{2}{\it\omega}^{2}{\it\Psi}+1},\\ \displaystyle s({\it\Psi})=-({\it\gamma}-1)\ln ({\it\gamma}Ma_{0}^{2}{\it\omega}^{2}{\it\Psi}+1),\\ \displaystyle s^{\prime }({\it\Psi})=-\frac{{\it\gamma}({\it\gamma}-1)Ma_{0}^{2}{\it\omega}^{2}}{{\it\gamma}Ma_{0}^{2}{\it\omega}^{2}{\it\Psi}+1}.\end{array}\!\!\right\} & & \displaystyle\end{eqnarray}$$

Then, (3.17) becomes the system of ODEs

(5.3) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle w_{y}={\it\rho}\frac{{\it\omega}^{2}}{{\it\gamma}Ma_{0}^{2}{\it\omega}^{2}{\it\Psi}+1}\left(1+T-\frac{2}{{\it\gamma}Ma_{0}^{2}{\it\omega}^{2}y}\ln ({\it\gamma}Ma_{0}^{2}{\it\omega}^{2}{\it\Psi}+1)\right),\\ \displaystyle {\it\Psi}_{y}={\it\rho}w,\end{array}\!\!\right\} & & \displaystyle\end{eqnarray}$$

with boundary conditions

(5.4a,b ) $$\begin{eqnarray}{\it\Psi}(0)=0,\quad {\it\Psi}(1/2)={\rm\Psi}_{w}=\frac{1}{{\it\gamma}Ma_{0}^{2}{\it\omega}}\left[\exp \left(\frac{{\it\gamma}Ma_{0}^{2}{\it\omega}^{2}}{2}\right)-1\right],\end{eqnarray}$$

where

(5.5) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle T=1+\frac{({\it\gamma}-1)Ma_{0}^{2}}{2}\left(1+\frac{2}{{\it\gamma}Ma_{0}^{2}}\ln \left({\it\gamma}Ma_{0}^{2}{\it\omega}^{2}{\it\Psi}+1\right)-w^{2}\right.\\ \hspace{-24.0pt}\displaystyle \left.-\,\frac{2{\it\omega}^{2}}{y}\left(\frac{1}{{\it\gamma}Ma_{0}^{2}{\it\omega}^{2}}\ln ({\it\gamma}Ma_{0}^{2}{\it\omega}^{2}{\it\Psi}+1)\right)^{2}\right),\\ \displaystyle {\it\rho}=T^{1/({\it\gamma}-1)}({\it\gamma}Ma_{0}^{2}{\it\omega}^{2}{\it\Psi}+1).\end{array}\!\!\right\}\quad & & \displaystyle\end{eqnarray}$$

Moreover, in this case,

(5.6) $$\begin{eqnarray}\displaystyle E(0) & = & \displaystyle (1+{\it\gamma}Ma_{0}^{2})\int _{0}^{1/2}\exp ({\it\gamma}Ma_{0}^{2}{\it\omega}^{2}y)\,\text{d}y=\frac{1+{\it\gamma}Ma_{0}^{2}}{{\it\gamma}Ma_{0}^{2}{\it\omega}^{2}}\left[\exp \left(\frac{{\it\gamma}Ma_{0}^{2}{\it\omega}^{2}}{2}\right)-1\right]\nonumber\\ \displaystyle & = & \displaystyle (1+{\it\gamma}Ma_{0}^{2}){\it\Psi}_{w}.\end{eqnarray}$$

The pressure $p(y)$ is found from $p(y)={\it\rho}(y)T(y)$ .

As mentioned before, representative examples of four types of solutions of the columnar problem of the ODE system (5.3) with (5.4) at $Ma_{0}=0.5$ and various values of ${\it\omega}$ are shown in the various panels of figure 2; see the discussion in § 3. The resulting bifurcation diagrams of ${\it\Psi}_{y}(0)$ for each type of solution as a function of the swirl ratio ${\it\omega}$ at various Mach numbers, $Ma_{0}=0$ , $0.1$ , $0.3$ and $0.5$ , are shown in figure 3. The type 0 branch of solutions is characterized by ${\it\Psi}_{y}(0)=1$ for all ${\it\omega}$ and $Ma_{0}$ . At $Ma_{0}=0$ , the solutions exhibit a jump (vertical) bifurcation of type 1 solutions (with $0<{\it\Psi}_{y}(0)<1$ ) from the type 0 branch of solutions at the special swirl ratio ${\it\omega}={\it\omega}_{B}=1.91586$ . Moreover, there is a jump bifurcation of type 2 solutions (with $1<{\it\Psi}_{y}(0)<3.485$ ) at ${\it\omega}_{B}$ . Type 3 solutions with ${\it\Psi}_{y}(0)=0$ start at ${\it\omega}_{\ast }={\it\omega}_{B}$ and develop as ${\it\omega}$ increases above ${\it\omega}_{\ast }$ with an increasing size of $y_{0}$ ; see figure 4.

Figure 3.

The bifurcation diagram of the centreline axial velocity ${\it\Psi}_{y}(0)$ versus the swirl ratio ${\it\omega}$ for solutions of the columnar flow (ODE) problem for a base solid-body rotation flow with a uniform axial velocity and a constant temperature at Mach numbers of $Ma_{0}=0,0.1,0.3$ and 0.5.

Figure 4.

The size of the stagnation zone $y_{0}$ versus the swirl ratio ${\it\omega}$ for the type 3 breakdown solutions of the columnar flow (ODE) problem for a base solid-body rotation flow with a uniform axial velocity and a constant temperature at various Mach numbers.

As the axial flow Mach number, $Ma_{0}$ , is increased from zero, we find a transcritical bifurcation of type 1 and 2 solutions from the base type 0 branch of solutions at ${\it\omega}={\it\omega}_{B}(Ma_{0})$ . The swirl level ${\it\omega}_{B}$ increases monotonically with the increase of $Ma_{0}$ . For each $0<Ma_{0}<1$ , we also find that the type 1 solutions describe a decelerated flow near the centreline where ${\it\Psi}_{y}(0)$ decreases from $1$ to 0 as ${\it\omega}$ decreases from ${\it\omega}_{B}(Ma_{0})$ . At ${\it\omega}={\it\omega}_{\ast }(Ma_{0})$ , the type 1 solutions reach a state with ${\it\Psi}_{y}(0)=0$ . It should be noted that ${\it\omega}_{\ast }$ increases monotonically with the increase of $Ma_{0}$ . A branch of type 3 solutions (with ${\it\Psi}_{y}(0)=0$ ) starts at ${\it\omega}={\it\omega}_{\ast }$ and develops for ${\it\omega}>{\it\omega}_{\ast }$ . The size, $y_{0}$ , of the stagnation zone of the type 3 solutions as a function of ${\it\omega}$ is shown in figure 4 for $Ma_{0}=0,0.1,0.3$ and $0.5$ . For these Mach numbers, $y_{0}$ increases with ${\it\omega}$ . Moreover, for a fixed ${\it\omega}$ , $y_{0}$ decreases with the inlet axial flow Mach number. In addition, type 2 solutions (with ${\it\Psi}_{y}(0)>1$ ) exist for ${\it\omega}>{\it\omega}_{B}(Ma_{0})$ .

For each type of solution of the columnar ODE system, ${\it\Psi}(y)$ , we also computed the value of the columnar flow force, $E({\it\Psi})$ , and the values of ${\rm\Delta}E=E(0)-E({\it\Psi})$ . The resulting bifurcation diagrams of ${\rm\Delta}E$ as a function of ${\it\omega}$ for $Ma_{0}=0.3$ and $0.5$ are shown in figure 5. For the type 0 solutions, ${\rm\Delta}E=0$ by definition. It can be seen that, for each Mach number, ${\rm\Delta}E$ of the type 1 solutions equals ${\rm\Delta}E$ of the type 3 solutions at ${\it\omega}={\it\omega}_{\ast }(Ma_{0})$ . The ${\rm\Delta}E$ values of type 1 solutions are negative for the range of swirl ${\it\omega}_{\ast }(Ma_{0})<{\it\omega}<{\it\omega}_{B}(Ma_{0})$ and become positive for the type 2 solutions when ${\it\omega}>{\it\omega}_{B}(Ma_{0})$ . However, the ${\rm\Delta}E$ values of the type 3 solutions increase from negative to positive and vanish at ${\it\omega}={\it\omega}_{0}(Ma_{0})$ , which is less than ${\it\omega}_{B}(Ma_{0})$ . The ${\rm\Delta}E$ values of the type 3 solutions are positive when ${\it\omega}>{\it\omega}_{0}(Ma_{0})$ . It should be noted that ${\it\omega}_{0}$ increases with the increase of Mach number, $Ma_{0}$ .

Figure 5.

The bifurcation diagrams of the difference of columnar flow force ${\rm\Delta}E$ versus the swirl ratio ${\it\omega}$ for solutions of the columnar flow (ODE) problem for a base solid-body rotation with a uniform axial velocity and a constant temperature at (a) $Ma_{0}=0.3$ and (b)  $Ma_{0}=0.5$ .

The results of solutions of the ODE problem shed light on the steady-state flows in the finite-length pipe. They show that the columnar solutions of type 0 are the outlet states of the parallel columnar flow solutions (2.18) in the pipe for all ${\it\omega}$ (the type 0 solutions of the PDE (2.13) with (2.14) and (2.15)). On the other hand, the columnar type 1 solutions are not outlet states of solutions of the problem in the range ${\it\omega}_{\ast }(Ma_{0})<{\it\omega}<{\it\omega}_{B}(Ma_{0})$ , while the columnar type 3 solutions are the outlet states of solutions of the problem that describe breakdown states in the pipe when ${\it\omega}\geqslant {\it\omega}_{0}(Ma_{0})$ (denoted by type 3 solutions of the PDE (2.13) with (2.14) and (2.15)).

In order to construct the bifurcation diagram of steady states in the pipe according to the PDE (2.13) with (2.14) and (2.15), we also need to account for the effect of pipe length on the various solutions bifurcating at ${\it\omega}_{1}(Ma_{0})$ , as found in Wang & Rusak (Reference Wang and Rusak1997a ) and Rusak et al. (Reference Rusak, Choi and Lee2007). Rusak et al. (Reference Rusak, Choi and Lee2007) showed a transcritical bifurcation of solutions of this problem from the branch of type 0 states in the pipe at ${\it\omega}={\it\omega}_{1}(Ma_{0})$ , which is greater than ${\it\omega}_{B}(Ma_{0})$ . The difference between ${\it\omega}_{1}(Ma_{0})$ and ${\it\omega}_{B}(Ma_{0})$ is inversely proportional to the square of the non-dimensional pipe length, $1/x_{0}^{2}$ . As the pipe length increases, ${\it\omega}_{1}(Ma_{0})$ approaches ${\it\omega}_{B}(Ma_{0})$ . For the type 0 states in the pipe the outlet centreline axial velocity is ${\it\Psi}_{y}(0,x_{0})=1$ . The bifurcating solutions at ${\it\omega}_{1}(Ma_{0})$ were also described analytically in Rusak et al. (Reference Rusak, Choi and Lee2007). They form type 1 and 2 states of the PDE (2.13) with (2.14) and (2.15) as ${\it\omega}$ changes around ${\it\omega}_{1}(Ma_{0})$ . These states describe type 1 flows in the pipe with deceleration and temperature increase near the centreline, and with a decreasing value of ${\it\Psi}_{y}(0,x_{0})$ as ${\it\omega}$ is decreased from ${\it\omega}_{1}(Ma_{0})$ . For ${\it\omega}>{\it\omega}_{1}(Ma_{0})$ , they describe type 2 flows in the pipe with acceleration and temperature decrease along the centreline. The outlet centreline axial velocity ${\it\Psi}_{y}(0,x_{0})$ of these states increases from 1 with the increase of ${\it\omega}$ above ${\it\omega}_{1}(Ma_{0})$ . As ${\it\omega}$ is decreased from ${\it\omega}_{1}(Ma_{0})$ , the type 1 states exhibit a zero value of ${\it\Psi}_{y}(0,x_{0})$ at ${\it\omega}={\it\omega}_{+}(Ma_{0})$ . Following the global analysis of incompressible swirling flows of Wang & Rusak (Reference Wang and Rusak1997a ), it is expected that the structure of the bifurcation is continuous when the Mach number is increased from zero to a higher value. Therefore, the type 1 states are connected at ${\it\omega}_{+}(Ma_{0})$ to type 3 states in the pipe for which ${\it\Psi}_{y}(0,x_{0})=0$ . As ${\it\omega}$ is further decreased from ${\it\omega}_{+}(Ma_{0})$ to ${\it\omega}_{0}(Ma_{0})+{\it\epsilon}(x_{0})$ , the branch of type 3 states appears. Here, ${\it\epsilon}(x_{0})>0$ is the effect of pipe length on ${\it\omega}_{0}$ and it decreases to zero as $x_{0}$ is increased. Our numerical simulations show that the correction is small when $x_{0}>6$ , but ${\it\epsilon}(x_{0})$ is not small when $x_{0}<6$ . The type 3 branch of states folds at ${\it\omega}={\it\omega}_{0}(Ma_{0})+{\it\epsilon}(x_{0})$ , and as ${\it\omega}$ is increased above ${\it\omega}_{0}(Ma_{0})+{\it\epsilon}(x_{0})$ , it forms the branch of breakdown states in the pipe. For the type 3 breakdown states in the pipe, ${\it\Psi}_{y}(0,x_{0})=0$ and the breakdown zone size at the outlet is given by $y_{0}$ as a function of $Ma_{0}$ (see figure 4).

The bifurcation diagram of steady-state flows in a finite-length straight circular pipe according to the PDE (2.13) with (2.14) and (2.15) is shown in figure 6 for a representative case where $Ma_{0}=0.5$ and $x_{0}=3$ . The horizontal axis is the swirl ratio, ${\it\omega}$ , of the inlet flow. The vertical axis represents the outlet centreline axial velocity at the pipe outlet centreline, ${\it\Psi}_{y}(0,x_{0})$ . This is a representative parameter of the various types of steady swirling flows in the pipe. For the type 0 columnar state in the pipe, ${\it\Psi}_{y}(0,x_{0})=1$ for all ${\it\omega}$ . The type 1 and 2 states bifurcate at ${\it\omega}={\it\omega}_{1}(Ma_{0})$ . The type 1 states are represented by the dashed line in figure 6 and the type 2 states are represented by the solid line. Both types are described by the solutions in Rusak et al. (Reference Rusak, Choi and Lee2007). The type 3 states that connect ${\it\omega}_{+}(Ma_{0})$ and ${\it\omega}_{0}(Ma_{0})+{\it\epsilon}(x_{0})$ are also represented by the dashed line with ${\it\Psi}_{y}(0,x_{0})=0$ . The type 3 states in the range ${\it\omega}>{\it\omega}_{0}(Ma_{0})+{\it\epsilon}(x_{0})$ are described by the solid line with ${\it\Psi}_{y}(0,x_{0})=0$ . Also shown for reference are solutions of the columnar ODE problem (5.3) with (5.4). The swirling flow states in the finite-length pipe are related to the solutions of columnar ODEs only for the type 0 columnar and type 3 breakdown states. For each $Ma_{0}$ , we find that the columnar state is a unique steady state of the PDE problem when ${\it\omega}<{\it\omega}_{0}(Ma_{0})+{\it\epsilon}(x_{0})$ . There exist three steady states, type 0, type 1 and type 3, when ${\it\omega}_{0}(Ma_{0})+{\it\epsilon}(x_{0})\leqslant {\it\omega}\leqslant {\it\omega}_{1}(Ma_{0})$ and three steady states, type 0, type 2 and type 3, when ${\it\omega}>{\it\omega}_{1}(Ma_{0})$ .

Figure 6.

The bifurcation diagram of the outlet centreline axial velocity ${\it\Psi}_{y}(0,x_{0})$ versus the swirl ratio ${\it\omega}$ for solutions of the axisymmetric flow problem (2.13) with (2.14) and (2.15) for a swirling flow in a finite-length straight circular pipe of non-dimensional length $x_{0}=3$ with an incoming solid-body rotation flow with a uniform axial velocity and a constant temperature at $Ma_{0}=0.5$ . The solid line represents stable solutions while the dashed line represents unstable solutions. The thin line describes for reference the ODE bifurcation diagram at $Ma_{0}=0.5$ .

Figure 7 summarizes the variation of the important swirl levels, ${\it\omega}_{\ast }$ , ${\it\omega}_{0}$ and ${\it\omega}_{B}$ , with the Mach number, $Ma_{0}$ . These swirl ratios increase monotonically with increase of the Mach number. It can be seen that ${\it\omega}_{B}$ increases faster than ${\it\omega}_{0}$ (and ${\it\omega}_{\ast }$ ) as the Mach number increases. Figure 7 together with figure 4 show that the appearance of vortex breakdown in a pipe is delayed with increase of Mach number. For a fixed ${\it\omega}$ , the breakdown zone is smaller when $Ma_{0}$ increases.

Figure 7.

The swirl ratios ${\it\omega}_{\ast }$ , ${\it\omega}_{0}$ and ${\it\omega}_{1}$ as a function of the Mach number for a swirling flow in a finite-length straight circular pipe of non-dimensional length $x_{0}=3$ with an incoming solid-body rotation flow with a uniform axial velocity and a constant temperature.

5.2. Lamb–Oseen vortex model

Let the inlet flow describe a Lamb–Oseen vortex with a uniform axial speed (axial plug flow) and a uniform temperature, i.e.

(5.7a,b ) $$\begin{eqnarray}v_{0}(y)=\frac{1}{\sqrt{2y}}(1-\exp (-2by)),\quad w_{0}(y)=T_{0}(y)=1\quad \text{for }0\leqslant y\leqslant 1/2.\end{eqnarray}$$

Here, $b$ is the vortex core size parameter, where the vortex core radius is $r_{c}=1.12/\sqrt{b}$ . Then, the base pressure profile is given by

(5.8) $$\begin{eqnarray}p_{0y}={\it\gamma}Ma_{0}^{2}{\it\omega}^{2}p_{0}\frac{v_{0}^{2}(y)}{2yT_{0}(y)}={\it\gamma}Ma_{0}^{2}{\it\omega}^{2}p_{0}\frac{(1-\exp (-2by))^{2}}{4y^{2}},\end{eqnarray}$$

with a centreline condition, $p_{0}(0)=1$ . The function $p_{0}(y)$ is computed by a fourth-order accurate Runge–Kutta integration scheme. Since $T_{0}(y)=1$ , ${\it\rho}_{0}(y)=p_{0}(y)$ , and we have ${\it\Psi}_{y}=p_{0}(y)$ . Then, ${\it\Psi}(y)=\int _{0}^{y}p_{0}(\bar{y})\,\text{d}\bar{y}$ , which again can be computed by the fourth-order accurate Runge–Kutta integration scheme. To invert ${\it\Psi}(y)$ to $y({\it\Psi})$ , we approximate

(5.9) $$\begin{eqnarray}y({\it\Psi})=\mathop{\sum }_{n=1}^{n=N}a_{n}{\it\Psi}^{n},\end{eqnarray}$$

where typically $N=25$ , and the coefficients $a_{n}$ are found from a curve-fitting procedure to the numerical solution of ${\it\Psi}$ versus $y$ . It should be noted that the approximation (5.9) must satisfy

(5.10a,b ) $$\begin{eqnarray}y(0)=0,\quad y\left({\it\Psi}_{0}=\int _{0}^{1/2}{\it\rho}_{0}(\bar{y})\right)=1/2.\end{eqnarray}$$

Using (5.9), we can compute

(5.11) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle K({\it\Psi})={\it\omega}(1-\exp (-2by({\it\Psi}))),\\ \displaystyle K^{\prime }({\it\Psi})=\frac{2b{\it\omega}\exp (-2by({\it\Psi}))}{p_{0}(y({\it\Psi}))},\\ \displaystyle H^{\prime }({\it\Psi})=\frac{({\it\gamma}-1)Ma_{0}^{2}{\it\omega}^{2}v_{0}(y({\it\Psi}))v_{0y}(y({\it\Psi}))}{p_{0}(y({\it\Psi}))},\\ \displaystyle s^{\prime }({\it\Psi})=-\frac{{\it\gamma}({\it\gamma}-1)Ma_{0}^{2}{\it\omega}^{2}(1-\exp (-2by({\it\Psi})))^{2}}{4y^{2}p_{0}(y({\it\Psi}))}.\end{array}\!\!\right\} & & \displaystyle\end{eqnarray}$$

These are used to solve the system of ODEs (3.17) with $T$ and ${\it\rho}$ according to (3.15) and (3.16) and with boundary conditions (3.18).

In all computed examples for the Lamb–Oseen vortex, we focus on a representative case where $b=4$ ( $r_{c}=0.56$ ). The results are similar to those found for the solid-body rotation flow in § 5.1. The bifurcation diagrams of ${\it\Psi}_{y}(0)$ for the solutions of the columnar ODE problem as a function of the swirl ratio, ${\it\omega}$ , at various Mach numbers, $Ma_{0}=0$ , $0.1$ , $0.3$ and $0.5$ , are shown in figure 8. Unlike the solid-body rotation case, at $Ma_{0}=0$ , the solutions exhibit a transcritical bifurcation of type 1 and type 2 solutions from the type 0 branch of solutions at the special swirl ratio ${\it\omega}={\it\omega}_{B}=0.8825$ . Moreover, unlike the solid-body rotation case, type 3 solutions with ${\it\Psi}_{y}(0)=0$ start at ${\it\omega}_{+}=0.7382$ and develop as ${\it\omega}$ decreases down to ${\it\omega}_{\ast }=0.709$ . At ${\it\omega}_{\ast }$ , the branch of type 3 solutions folds and the type 3 solutions develop for ${\it\omega}>{\it\omega}_{\ast }$ . As the axial flow Mach number, $Ma_{0}$ , is increased from zero, we find a similar behaviour of the bifurcation diagram, but with higher values of ${\it\omega}_{B}$ , ${\it\omega}_{+}$ and ${\it\omega}_{\ast }$ that increase with $Ma_{0}$ . The corresponding size, $y_{0}$ , of the stagnation zone of the type 3 solutions as a function of ${\it\omega}$ is shown in figure 9 for various values of $Ma_{0}$ . Unlike the solid-body rotation case, for these Mach numbers, $y_{0}$ first increases with ${\it\omega}$ decreasing from ${\it\omega}_{+}$ to ${\it\omega}_{\ast }$ and then folds as ${\it\omega}$ is increased from ${\it\omega}_{\ast }$ . There exist two type 3 solutions in the range of ${\it\omega}_{\ast }\leqslant {\it\omega}\leqslant {\it\omega}_{+}$ , one with a small stagnation zone and the other with a larger stagnation zone. Moreover, for a fixed ${\it\omega}$ greater than ${\it\omega}_{+}$ , $y_{0}$ decreases with increase of the inlet axial flow Mach number.

Figure 8.

The bifurcation diagram of the centreline axial velocity ${\it\Psi}_{y}(0)$ versus the swirl ratio ${\it\omega}$ of solutions of the columnar flow (ODE) problem for a base Lamb–Oseen vortex with a uniform axial velocity and a constant temperature at Mach numbers $Ma_{0}=0,0.1,0.3$ and 0.5.

Figure 9.

The size of the stagnation zone $y_{0}$ versus the swirl ratio ${\it\omega}$ for the type 3 breakdown solutions of the columnar flow (ODE) problem for a base Lamb–Oseen vortex with a uniform axial velocity and a constant temperature at various Mach numbers.

In addition, as in the solid-body rotation case, we find that the values of ${\rm\Delta}E$ of the type 1 solutions are negative in the range of ${\it\omega}_{\ast }\leqslant {\it\omega}\leqslant {\it\omega}_{B}$ , while the values of ${\rm\Delta}E$ of the type 2 solutions are positive in the range of ${\it\omega}>{\it\omega}_{B}$ . We also find from this analysis the values of ${\it\omega}_{0}$ as a function of $Ma_{0}$ for which breakdown states of the PDE problem appear for the first time as swirl is increased. In all cases, ${\it\omega}_{\ast }<{\it\omega}_{0}<{\it\omega}_{+}$ along the branch of type 3 solutions that fold back.

The results of solutions of the ODE problem for the Lamb–Oseen vortex show that for all ${\it\omega}$ the columnar solutions of type 0 are the outlet states of the type 0 parallel flow solutions in the pipe. The type 1 and connected type 3 solutions are not outlet states of solutions of the problem in the range ${\it\omega}_{\ast }(Ma_{0})<{\it\omega}<{\it\omega}_{B}(Ma_{0})$ . The columnar type 3 solutions are the outlet states of type 3 breakdown states in the pipe when ${\it\omega}\geqslant {\it\omega}_{0}(Ma_{0})$ .

The bifurcation diagram of solutions of steady-state flows in a finite-length straight circular pipe according to the PDE problem is shown in figure 10 for a representative case where $b=4$ , $Ma_{0}=0.5$ and $x_{0}=3$ . The type 0 columnar states in the pipe are represented by ${\it\Psi}_{y}(0,x_{0})=1$ for all ${\it\omega}$ . The type 1 states are represented by the dashed line in figure 10 and the type 2 states are represented by the solid line. Both are described by the solutions of Rusak et al. (Reference Rusak, Choi and Lee2007). The type 3 breakdown solutions of the PDE problem are characterized by ${\it\Psi}_{y}(0,x_{0})=0$ for ${\it\omega}>{\it\omega}_{0}(Ma_{0})+{\it\epsilon}(x_{0})$ . Also shown for reference are solutions of the columnar ODE problem. The solutions of the swirling flow problem in the finite-length pipe are related with the solutions of columnar ODEs only for the type 0, 2 and 3 breakdown states.

Figure 10.

The bifurcation diagram of the outlet centreline axial velocity ${\it\Psi}_{y}(0,x_{0})$ versus the swirl ratio ${\it\omega}$ for solutions of the axisymmetric flow problem (2.13) with (2.14) and (2.15) of a swirling flow in a finite-length straight circular pipe of non-dimensional length $x_{0}=3$ with an incoming Lamb–Oseen vortex with a uniform axial velocity and a constant temperature at $Ma_{0}=0.5$ . The solid line represents stable solutions while the dashed line represents unstable solutions. The thin line describes for reference the ODE bifurcation diagram at $Ma_{0}=0.5$ .

The monotonic increase of the swirl levels, ${\it\omega}_{\ast }$ , ${\it\omega}_{0}$ and ${\it\omega}_{B}$ , as a function of the Mach number, $Ma_{0}$ , for the Lamb–Oseen vortex with $b=4$ is presented in figure 11. We find that ${\it\omega}_{0}$ (and ${\it\omega}_{\ast }$ ) slowly increases with the Mach number while ${\it\omega}_{B}$ increases faster. Figures 9 and 11 show for the Lamb–Oseen vortex that the appearance of vortex breakdown in a finite-length pipe is delayed by the compressibility effect.

Figure 11.

The swirl ratios ${\it\omega}_{\ast }$ , ${\it\omega}_{0}$ and ${\it\omega}_{1}$ as a function of the Mach number for a swirling flow in a finite-length straight circular pipe of non-dimensional length $x_{0}=3$ with an incoming Lamb–Oseen vortex with a uniform axial velocity and a constant temperature.