5.1 Solid-body rotation

Let the inlet flow describe a solid-body rotation with a constant unit axial velocity, constant unit temperature and constant inlet reactant mass fraction, i.e.

(5.1a−c ) $$\begin{eqnarray}\displaystyle v_{0}(y)=\sqrt{2y},\quad w_{0}(y)=T_{0}(y)=1,\quad Y_{0}(y)={\it\delta}\quad \text{for}~0\leqslant y\leqslant 1/2. & & \displaystyle\end{eqnarray}$$

Then, the base flow properties are

(5.2) $$\begin{eqnarray}\displaystyle & \displaystyle p_{0}(y)={\it\rho}_{0}(y)=\exp ({\it\gamma}Ma_{0}^{2}{\it\omega}^{2}y), & \displaystyle\end{eqnarray}$$

(5.3) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\Psi}(y)=\frac{1}{{\it\gamma}Ma_{0}^{2}{\it\omega}^{2}}[\exp ({\it\gamma}Ma_{0}^{2}{\it\omega}^{2}y)-1], & \displaystyle\end{eqnarray}$$

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

(5.5) $$\begin{eqnarray}\displaystyle & \displaystyle y({\it\Psi})=\frac{1}{{\it\gamma}Ma_{0}^{2}{\it\omega}^{2}}\ln ({\it\gamma}Ma_{0}^{2}{\it\omega}^{2}{\it\Psi}+1). & \displaystyle\end{eqnarray}$$

From (3.4)–(3.10), we find $K_{0}({\it\Psi})$ , $K_{0}^{\prime }({\it\Psi})$ , $H_{0}({\it\Psi})$ , $H_{0}^{\prime }({\it\Psi})$ , $s_{0}({\it\Psi})$ and $s_{0}^{\prime }({\it\Psi})$ ,

(5.6) $$\begin{eqnarray}\displaystyle & \displaystyle K_{0}({\it\Psi})=2{\it\omega}y({\it\Psi})=\frac{2}{{\it\gamma}Ma_{0}^{2}{\it\omega}}\ln ({\it\gamma}Ma_{0}^{2}{\it\omega}^{2}{\it\Psi}+1), & \displaystyle\end{eqnarray}$$

(5.7) $$\begin{eqnarray}\displaystyle & \displaystyle K_{0}^{\prime }({\it\Psi})=\frac{2{\it\omega}}{{\it\gamma}Ma_{0}^{2}{\it\omega}^{2}{\it\Psi}+1}, & \displaystyle\end{eqnarray}$$

(5.8) $$\begin{eqnarray}\displaystyle & \displaystyle H_{0}({\it\Psi})=1+\frac{({\it\gamma}-1)Ma_{0}^{2}}{2}\left(1+\frac{2}{{\it\gamma}Ma_{0}^{2}}\ln ({\it\gamma}Ma_{0}^{2}{\it\omega}^{2}{\it\Psi}+1)\right), & \displaystyle\end{eqnarray}$$

(5.9) $$\begin{eqnarray}\displaystyle & \displaystyle {H^{\prime }}_{0}({\it\Psi})=\frac{({\it\gamma}-1)Ma_{0}^{2}{\it\omega}^{2}}{({\it\gamma}Ma_{0}^{2}{\it\omega}^{2}{\it\Psi}+1)}, & \displaystyle\end{eqnarray}$$

(5.10) $$\begin{eqnarray}\displaystyle & \displaystyle s_{0}({\it\Psi})=-({\it\gamma}-1)\ln ({\it\gamma}Ma_{0}^{2}{\it\omega}^{2}{\it\Psi}+1), & \displaystyle\end{eqnarray}$$

(5.11) $$\begin{eqnarray}\displaystyle & \displaystyle {s^{\prime }}_{0}({\it\Psi})=-\frac{({\it\gamma}-1){\it\gamma}Ma_{0}^{2}{\it\omega}^{2}}{({\it\gamma}Ma_{0}^{2}{\it\omega}^{2}{\it\Psi}+1)}, & \displaystyle\end{eqnarray}$$

(5.12) $$\begin{eqnarray}\displaystyle & \displaystyle Y_{0}({\it\Psi})={\it\delta}. & \displaystyle\end{eqnarray}$$

To determine the outlet state, we use (2.28) and (2.29) to formulate the outlet functions $H({\it\Psi})$ and $s({\it\Psi})$ . Since in the present example the inlet reactant mass fraction is constant, we have $H^{\prime }({\it\Psi})={H^{\prime }}_{0}({\it\Psi})$ and $s^{\prime }({\it\Psi})={s^{\prime }}_{0}({\it\Psi})$ at the outlet. The effect of reaction appears only in computing $T$ and ${\it\rho}$ from $H({\it\Psi})$ and $s({\it\Psi})$ . Then, at the outlet, (3.14) and (3.15) become,

(5.13) $$\begin{eqnarray}\displaystyle & \displaystyle w_{y}={\it\rho}\frac{{\it\omega}^{2}}{({\it\gamma}Ma_{0}^{2}{\it\omega}^{2}{\it\Psi}+1)}\left(1+T-\frac{2}{{\it\gamma}Ma_{0}^{2}{\it\omega}^{2}y}\ln ({\it\gamma}Ma_{0}^{2}{\it\omega}^{2}{\it\Psi}+1)\right), & \displaystyle\end{eqnarray}$$

(5.14) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\Psi}_{y}={\it\rho}w, & \displaystyle\end{eqnarray}$$

with

(5.15) $$\begin{eqnarray}\displaystyle T & = & \displaystyle 1+{\it\beta}{\it\delta}+\frac{({\it\gamma}-1)Ma_{0}^{2}}{2}\left\{\vphantom{\displaystyle \left(\frac{1}{{\it\gamma}Ma_{0}^{2}}\right)^{2}}1+\frac{2}{{\it\gamma}Ma_{0}^{2}}\ln ({\it\gamma}Ma_{0}^{2}{\it\omega}^{2}{\it\Psi}+1)\right.\nonumber\\ \displaystyle & & \displaystyle -\left.w^{2}-\frac{2{\it\omega}^{2}}{y}\left(\frac{1}{{\it\gamma}Ma_{0}^{2}{\it\omega}^{2}}\ln ({\it\gamma}Ma_{0}^{2}{\it\omega}^{2}{\it\Psi}+1)\right)^{2}\right\},\end{eqnarray}$$

(5.16) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\rho}=T^{1/({\it\gamma}-1)}\frac{({\it\gamma}Ma_{0}^{2}{\it\omega}^{2}{\it\Psi}+1)}{(1+{\it\beta}{\it\delta})^{{\it\gamma}/({\it\gamma}-1)}}, & \displaystyle\end{eqnarray}$$

and with boundary conditions,

(5.17a,b ) $$\begin{eqnarray}{\it\Psi}(0)=0,\quad {\it\Psi}(1/2)={\it\Psi}_{w}=\frac{1}{{\it\gamma}Ma_{0}^{2}{\it\omega}^{2}}\left[\exp \left(\frac{{\it\gamma}Ma_{0}^{2}{\it\omega}^{2}}{2}\right)-1\right].\end{eqnarray}$$

We determine type 0, 1 and 2 solutions of system (5.13)–(5.17) in the range of $0\leqslant {\it\omega}\leqslant 4$ for various $0\leqslant {\it\beta}{\it\delta}\leqslant 2$ . Examples of these types of solutions are shown in figures 2–4.

Figure 5.

Contour plots of temperature ( $T$ ) and circulation ( $K$ ) for ${\it\beta}{\it\delta}=0.05$ with various ${\it\omega}$ : (a) ${\it\omega}=0.4$ , (b) ${\it\omega}=0.7$ , (c) ${\it\omega}=0.8$ , (d) ${\it\omega}=0.82$ . There are 15 equispaced lines of $T$ (solid lines) in the respective range [1, 1.05] and 15 equispaced lines of $K$ (dashed lines) in the respective range: (a) [0, 0.4], (b) [0, 0.69], (c) [0, 0.79], (d) [0, 0.81]. Here, the inlet flow is the Lamb–Oseen vortex with $b=4$ with unit axial velocity and temperature. Taken from Choi et al. (Reference Choi, Rusak and Kapila2007) with permission from Taylor & Francis Ltd., http://www.tandfonline.com.

Type 0 solution of the ODEs system (5.13)–(5.17) is the outlet state of a type 0 solution of the PDEs system (2.1)–(2.6) for zero swirl level. The type 0 solution of the PDE system describes a parallel flow in the pipe with zero swirl and a (classical) straight thin reaction zone across the pipe. The temperature increases due to the heat release across the reaction zone from an inlet value, $T(0,y)=1$ , to an outlet temperature, $T(x_{0},y)=1+{\it\beta}{\it\delta}$ . Thus, density decreases along the pipe. The position of reaction zone in the pipe is dictated by the reaction variables, activation energy ${\it\theta}$ and reaction rate $\mathscr{A}$ . Accordingly, from the continuity equation, the flow accelerates as a parallel flow across the reaction zone from an inlet axial velocity, $w(0,y)=1$ , to an outlet axial velocity, $w(x_{0},y)=1+{\it\beta}{\it\delta}$ . The numerical simulations in figure 5(a), taken from Choi et al. (Reference Choi, Rusak and Kapila2007), for ${\it\omega}\leqslant 0.4$ demonstrate an example of these flow and temperature patterns for a low swirl condition.

Type 1 solution of the ODEs system (5.13)–(5.17) is found in the range $0<{\it\omega}<{\it\omega}_{0{\it\delta}}({\it\beta}{\it\delta})$ and describes the outlet state of a type 1 solution of the PDEs system (2.1)–(2.6). Here, ${\it\omega}_{0{\it\delta}}$ is a critical swirl for the first appearance of breakdown states. In this solution, for a fixed heat release, ${\it\beta}{\it\delta}$ , the centreline outlet axial velocity $w(x_{0},0)<1+{\it\beta}{\it\delta}$ and decreases with increase of the inlet swirl, ${\it\omega}$ . This is the result of the increasing nonlinear interaction between convection of azimuthal vorticity and vorticity stretching related to axial gradient of the circulation which decreases azimuthal vorticity along the centreline, and thereby flow deceleration is increased, as also found in an inert flow. This mechanism is explained in details in Rusak et al. (Reference Rusak, Kapila and Choi2002) and also dominates reacting flows with swirl. By continuity, the wall outlet axial velocity $w(x_{0},1/2)>1+{\it\beta}{\it\delta}$ and increases with the increase of the inlet swirl, ${\it\omega}$ . As a result, we infer the structure of the flow and temperature in the pipe. The centreline axial velocity accelerates across the reaction zone from the inlet value $w(0,0)=1$ to $w=1+{\it\beta}{\it\delta}$ and then decelerates downstream of the reaction zone to the outlet axial velocity, $w(x_{0},0)<1+{\it\beta}{\it\delta}$ . Along the pipe wall, the axial velocity increases from the inlet value $w(0,1/2)=1$ to $1+{\it\beta}{\it\delta}$ at the reaction zone and to the outlet wall axial velocity of $w(x_{0},1/2)>1+{\it\beta}{\it\delta}$ . It is clear that the effect of incoming swirl is to decelerate the flow near the centreline and accelerate the flow along the wall. Therefore, the reaction zone is curved in the radial direction toward downstream, i.e. the reaction zone is pushed toward the inlet around the centreline and pushed downstream near the wall with respect to type 0 solution with no swirl. The temperature increases across the curved reaction zone from $T=1$ to an outlet temperature, $T=1+{\it\beta}{\it\delta}$ . The position of the reaction zone in the pipe is again dictated by the reaction variables, ${\it\theta}$ and $\mathscr{A}$ . The numerical simulations in figure 5(b,c), taken from Choi et al. (Reference Choi, Rusak and Kapila2007), for ${\it\omega}=0.7$ and 0.8 demonstrate these flow and temperature patterns for type 1 solutions.

With heat release from the reaction, flow is accelerated across the reaction zone. Then, with increase of heat release at a fixed inlet swirl level, type 1 solutions are also affected by the nonlinear interaction between convection of azimuthal vorticity and increased baroclinic effects resulting from the coupling between steepening temperature gradients and velocity gradients. This interaction makes a smaller decrease of azimuthal vorticity due to swirl near the centreline, and thereby the centreline outlet axial velocity is less decreased. These effects are also discussed in Rusak et al. (Reference Rusak, Kapila and Choi2002). The numerical simulations in figures 4(b) and 6(b) for ${\it\omega}=0.6$ in Choi et al. (Reference Choi, Rusak and Kapila2007) demonstrate these flow and temperature patterns.

Type 2 solution of the ODEs system (5.13)–(5.17) is found in the range ${\it\omega}>{\it\omega}_{0{\it\delta}}({\it\beta}{\it\delta})$ and forms the outlet state of a type 2 solution of the PDEs system (2.1)–(2.6), which describes a reacting swirling flow in the pipe with a finite size breakdown zone around the centreline. The width of the breakdown zone at the outlet is $y_{s}({\it\omega},{\it\beta}{\it\delta})$ . In such a flow state, for a fixed heat release, the nonlinear interactions between the convection of azimuthal vorticity and stretching of vorticity by axial gradient of circulation are stronger than in type 1 solutions and form the breakdown zone. The centreline axial velocity decelerates from the inlet value, $w(0,0)=1$ , to stagnation, $w=0$ , at a certain point along the centreline and remains zero from that point to the outlet. By continuity, along the pipe wall, the axial velocity increases from $w(0,1/2)=1$ to the outlet wall axial velocity of $w(x_{0},1/2)\gg 1+{\it\beta}{\it\delta}$ . Comparing figures 3 and 4, both at the same heat release ${\it\beta}{\it\delta}=2$ , it is clear that the effect of a high incoming swirl is to further decelerate the flow near the centreline and establish a centreline breakdown zone and to further accelerate the flow along the wall. Therefore, at higher incoming swirl levels, the reaction zone is much more curved in the radial direction toward downstream, i.e. the reaction zone is pushed further toward the inlet around the centreline and pushed further downstream near the wall. The temperature increases across the curved reaction zone from $T=1$ to an outlet temperature, $T=1+{\it\beta}{\it\delta}$ . The position of reaction zone in the pipe is again dictated by the reaction variables, ${\it\theta}$ and $\mathscr{A}$ . With increase of heat release at a fixed inlet swirl level, type 2 solutions also experience baroclinic effects, similar to those discussed above in type 1 solution. These effects delay the appearance of breakdown in the swirling reacting flow to higher swirl levels. The numerical simulations in figure 5(d) for ${\it\omega}=0.82$ demonstrate the flow and temperature patterns for type 2 solutions. In addition, figures 3–6 in Choi et al. (Reference Choi, Rusak and Kapila2007) for ${\it\omega}=0.86$ , 0.88 demonstrate the effect of increasing heat release on flow and temperature patterns of type 2 solutions. In all cases computed, the reaction zone is upstream of the leading stagnation point and wraps around the vortex breakdown separation zone. This is similar to flow and reaction zone patterns observed in the experimental study in figures 4 and 5 in Umeh et al. (Reference Umeh, Rusak and Gutmark2012).

We note that, in realistic situations, the velocity inside the breakdown zone is slightly negative due to viscous diffusion of momentum from the rotating bulk into the breakdown zone. The simulations of high Reynolds number inert flows show that the negative axial velocity inside the breakdown zone becomes smaller with increase of Reynolds number (Beran Reference Beran1994; Lopez Reference Lopez1994). In the inviscid limit case, it is expected that the zone becomes stagnant. This is supported by inviscid flow simulations which show natural evolution of perturbations in the flow to form a time-asymptotic near-stagnant breakdown zone (Rusak et al. Reference Rusak, Wang, Xu and Taylor2012).

Figure 6.

Bifurcation diagram of the outlet centreline axial velocity $w(x_{0},0)$ as a function of ${\it\omega}$ at various values of ${\it\beta}{\it\delta}$ for an inlet solid-body rotation flow with $Ma_{0}=0.001$ .

The ODEs system (5.13)–(5.17) is integrated numerically for a variety of values of ${\it\omega}$ and ${\it\beta}{\it\delta}$ with ${\it\gamma}=1.4$ and $Ma_{0}=0.001$ . The results of the computations are summarized in the bifurcation diagrams of solutions shown in figure 6. For a fixed ${\it\beta}{\it\delta}$ , the outlet centreline axial velocity $w(x_{0},0)=1+{\it\beta}{\it\delta}$ when ${\it\omega}=0$ , it decreases with increase of inlet swirl ${\it\omega}$ and becomes zero at the critical swirl level, ${\it\omega}_{0{\it\delta}}$ . For higher swirl, ${\it\omega}>{\it\omega}_{0{\it\delta}}$ , the outlet centreline axial velocity is zero. As the heat release, ${\it\beta}{\it\delta}$ , increases from zero, the outlet centreline axial velocity $w(x_{0},0)$ increases for a fixed inlet swirl in the range $0\leqslant {\it\omega}<{\it\omega}_{0{\it\delta}}$ . The first appearance of a vortex breakdown state is delayed with the increase of ${\it\beta}{\it\delta}$ . Figure 7 shows the change of ${\it\omega}_{0{\it\delta}}$ with ${\it\beta}{\it\delta}$ at $Ma_{0}=0.001$ . It shows that the critical swirl for the first appearance of breakdown in the reacting flow increases monotonically with ${\it\beta}{\it\delta}$ . The size, $y_{s}$ , of the breakdown zone at the outlet as a function of ${\it\omega}>{\it\omega}_{0{\it\delta}}$ is shown in figure 8 for various ${\it\beta}{\it\delta}$ . The size $y_{s}$ increases with ${\it\omega}$ at a fixed ${\it\beta}{\it\delta}$ as a result of increased vorticity stretching related to the axial gradient of circulation. It decreases with increase of ${\it\beta}{\it\delta}$ at a fixed ${\it\omega}$ as a result of increased baroclinic effects related to axial temperature gradient. This variation of $y_{s}$ is a result of the swirl difference ${\it\omega}-{\it\omega}_{0{\it\delta}}$ . Since ${\it\omega}_{0{\it\delta}}$ increases with heat release, for a given swirl level, this swirl difference becomes smaller, making the breakdown zone smaller in size at the outlet and all over in the domain. We also observe in figure 6 the nonlinear interaction between the azimuthal vorticity stretching by axial gradient of temperature and the baroclinic terms at swirl level around ${\it\omega}=1.916$ (the critical swirl level for the non-reacting flow) for low amount of exothermicity $0<{\it\beta}{\it\delta}<0.5$ . This nonlinear interaction was predicted in the discussion in Choi et al. (Reference Choi, Rusak and Kapila2007). It is manifested by the results that $w(x_{0},0)$ is less than unity around the critical swirl and first decreases when heat release, ${\it\beta}{\it\delta}$ , is increased from 0 to around 0.1 and then increases as ${\it\beta}{\it\delta}$ increases from around 0.1 to 0.5.

Figure 7.

Variation of the critical swirl ${\it\omega}_{0{\it\delta}}$ as a function of ${\it\beta}{\it\delta}$ for an inlet solid-body rotation flow with $Ma_{0}=0.001$ .

Figure 8.

Variation of stagnation zone size, $y_{s}$ , at the pipe outlet as a function of ${\it\omega}$ at various ${\it\beta}{\it\delta}$ for an inlet solid-body rotation flow with $Ma_{0}=0.001$ .

5.2 Lamb–Oseen vortex model

Let the inlet flow describe a Lamb–Oseen vortex with a uniform axial speed, uniform temperature and constant inlet reactant mass fraction, i.e.

(5.18a−c ) $$\begin{eqnarray}v_{0}(y)=\frac{1}{\sqrt{2y}}(1-\exp (-2by)),\quad w_{0}(y)=T_{0}(y)=1,\quad Y_{0}(y)={\it\delta},\quad \text{for}~0\leqslant y\leqslant 1/2.\end{eqnarray}$$

Here, $b$ is the vortex core size parameter where the vortex core radius is $r_{c}=1.12/\sqrt{b}$ . Then, the base flow pressure obeys $p_{0y}={\it\gamma}Ma_{0}^{2}{\it\omega}^{2}p_{0}({v_{0}}^{2}(y))/(2yT_{0}(y))={\it\gamma}Ma_{0}^{2}{\it\omega}^{2}p_{0}((1-\exp (-2by))^{2})/4y^{2}$ with centreline condition $p_{0}(0)=1$ . This equation can be integrated numerically to determine $p_{0}(y)$ . Since the base flow temperature is $T_{0}(y)=1$ , the base flow density is ${\it\rho}_{0}(y)=p_{0}(y)$ . Then, the base flow stream function obeys ${\it\Psi}_{0y}=p_{0}(y)$ with ${\it\Psi}_{0}(0)=0$ , and ${\it\Psi}_{0}(y)=\int _{0}^{y}p_{0}(\bar{y})\,\text{d}\bar{y}$ . This equation can again be integrated numerically in the range $0\leqslant y\leqslant 1/2$ . Therefore, ${\it\Psi}_{w}={\it\Psi}_{0}(1/2)$ . The function ${\it\Psi}_{0}(y)$ is monotonically increasing and can be inverted to give $y({\it\Psi}_{0})$ . Using a curve fit formula, a ninth-order polynomial expansion for $y({\it\Psi}_{0})$ at every ${\it\omega}$ was determined. Then, replacing ${\it\Psi}_{0}$ with ${\it\Psi}$ , it is found that,

(5.19) $$\begin{eqnarray}\displaystyle & \displaystyle K_{0}({\it\Psi})={\it\omega}(1-\exp (-2by({\it\Psi}))), & \displaystyle\end{eqnarray}$$

(5.20) $$\begin{eqnarray}\displaystyle & \displaystyle K_{0}^{\prime }({\it\Psi})=\frac{2b{\it\omega}\exp (-2by({\it\Psi}))}{p_{0}(y({\it\Psi}))}, & \displaystyle\end{eqnarray}$$

(5.21) $$\begin{eqnarray}\displaystyle & \displaystyle H_{0}({\it\Psi})=1+\frac{({\it\gamma}-1)Ma_{0}^{2}}{2}\left(1+\frac{{\it\omega}^{2}}{2y({\it\Psi})}[1-\exp (-2by({\it\Psi}))]^{2}\right), & \displaystyle\end{eqnarray}$$

(5.22) $$\begin{eqnarray}\displaystyle & \displaystyle H_{0}^{\prime }({\it\Psi})=\frac{({\it\gamma}-1)Ma_{0}^{2}{\it\omega}^{2}v_{0}(y({\it\Psi}))v_{0y}(y({\it\Psi}))}{p_{0}(y({\it\Psi}))}, & \displaystyle\end{eqnarray}$$

(5.23) $$\begin{eqnarray}\displaystyle & \displaystyle s_{0}({\it\Psi})=-\frac{{\it\gamma}({\it\gamma}-1)Ma_{0}^{2}{\it\omega}^{2}}{2}\int _{0}^{y({\it\Psi})}\frac{(1-\exp (-2b\bar{y}))^{2}}{2\bar{y}^{2}}\,\text{d}\bar{y} & \displaystyle\end{eqnarray}$$

(5.24) $$\begin{eqnarray}\displaystyle & \displaystyle s_{0}^{\prime }({\it\Psi})=-\frac{{\it\gamma}({\it\gamma}-1)Ma_{0}^{2}{\it\omega}^{2}(1-\exp (-2by({\it\Psi})))^{2}}{4y^{2}({\it\Psi})p_{0}(y({\it\Psi}))}. & \displaystyle\end{eqnarray}$$

To determine the outlet state, we use total enthalpy and entropy equations, (2.28) and (2.29), to formulate the outlet functions $H({\it\Psi})$ and $s({\it\Psi})$ . Since in the present example the inlet reactant mass fraction is constant, $H^{\prime }({\it\Psi})=H_{0}^{\prime }({\it\Psi})$ and $s^{\prime }({\it\Psi})=s_{0}^{\prime }({\it\Psi})$ . The effect of reaction appears only in computing $T$ and ${\it\rho}$ from $H({\it\Psi})$ and $s({\it\Psi})$ . Then, at the outlet, the simplified ODEs (3.14) and (3.15), temperature and density become

(5.25) $$\begin{eqnarray}\displaystyle & \displaystyle w_{y}=\frac{1}{({\it\gamma}-1)Ma_{0}^{2}}{\it\rho}\left({H^{\prime }}_{0}({\it\Psi})-({\it\gamma}-1)Ma_{0}^{2}\frac{K({\it\Psi})K^{\prime }({\it\Psi})}{2y}-\frac{1}{{\it\gamma}}T{s^{\prime }}_{0}({\it\Psi})\right), & \displaystyle\end{eqnarray}$$

(5.26) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\Psi}_{y}={\it\rho}w, & \displaystyle\end{eqnarray}$$

(5.27) $$\begin{eqnarray}\displaystyle & \displaystyle T=H_{0}({\it\Psi})-\frac{({\it\gamma}-1)Ma_{0}^{2}}{2}\left(w^{2}+\frac{K^{2}({\it\Psi})}{2y}\right)+{\it\beta}{\it\delta}, & \displaystyle\end{eqnarray}$$

(5.28) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\rho}=\left(\frac{H_{0}({\it\Psi})-{\displaystyle \frac{({\it\gamma}-1)Ma_{0}^{2}}{2}}\left(w^{2}+{\displaystyle \frac{K^{2}({\it\Psi})}{2y}}\right)+{\it\beta}{\it\delta}}{\exp (s_{0}({\it\Psi}))(1+{\it\beta}{\it\delta})^{{\it\gamma}}}\right)^{1/({\it\gamma}-1)}, & \displaystyle\end{eqnarray}$$

and with boundary conditions,

(5.29a,b ) $$\begin{eqnarray}{\it\Psi}(0)=0,\quad {\it\Psi}(1/2)={\it\Psi}_{w}=\int _{0}^{1/2}p_{0}(\overline{y})\,\text{d}\overline{y}.\end{eqnarray}$$

In the present example, we focus on an inlet vortex with core size parameter $b=4$ ( $r_{c}=0.56$ ). Solutions similar to the type 0 (at ${\it\omega}=0$ ), 1 and 2 of the solid-body rotation case are also found in this case. Numerically computed examples from Choi et al. (Reference Choi, Rusak and Kapila2007) of flow and temperature patterns for type 1 reacting flow solutions are shown in figure 5(a–c), and an example of these patterns for a type 2 reacting flow solution is shown in figure 5(d). However, the manifold of solutions for the Lamb–Oseen inlet vortex is more complicated and exhibits nonlinearity as the heat release, ${\it\beta}{\it\delta}$ , is increased. The bifurcation diagrams of the outlet centreline axial velocity, $w(x_{0},0)$ , as a function of the inlet swirl level, ${\it\omega}$ , for various small values of ${\it\beta}{\it\delta}$ in the range $0<{\it\beta}{\it\delta}\leqslant 0.2$ are first studied and shown in figure 9. For the inert flow case of ${\it\beta}{\it\delta}=0$ , the base type 0 flow is a solution of the problem for all inlet swirl level, ${\it\omega}$ , and the outlet centreline axial velocity is constant, $w(x_{0},0)=1$ for all ${\it\omega}$ . There is a transcritical bifurcation of type 1 non-columnar decelerated flow states along the centreline with $0<w(x_{0},0)<1$ as ${\it\omega}$ is decreased from a critical swirl level ${\it\omega}_{1}=0.882$ to ${\it\omega}=0.74$ , see the solid black line in figure 9. With respect to the solid-body rotation, the critical swirl level decreases with reduction of core size. This is consistent with Rusak et al. (Reference Rusak, Whiting and Wang1998). Type 2 states appear as the inlet swirl level, ${\it\omega}$ , is further decreased from 0.74 to a second critical swirl level ${\it\omega}_{0}=0.71$ . At ${\it\omega}_{0}=0.71$ , the type 2 solutions fold and develop with the increase of ${\it\omega}$ above ${\it\omega}_{0}=0.71$ .

Figure 9.

Bifurcation diagram of the outlet centreline axial velocity $w(x_{0},0)$ as a function of ${\it\omega}$ at various values of $0\leqslant {\it\beta}{\it\delta}\leqslant 0.2$ for an inlet Lamb–Oseen vortex with $b=4$ and $Ma_{0}=0.001$ .

With a small increase of heat release, ${\it\beta}{\it\delta}$ , from 0 to 0.002, the transcritical bifurcation of solutions is modified into one branch of type 1 solutions that starts at ${\it\omega}=0$ with the outlet centreline axial velocity of $w(x_{0},0)=1.002$ . The value of $w(x_{0},0)$ of this branch decreases with the increase of ${\it\omega}$ up to a modified critical swirl level ${\it\omega}_{1{\it\delta}}=0.86$ . The branch folds at ${\it\omega}_{1{\it\delta}}=0.86$ and $w(x_{0},0)$ decreases with the decrease of ${\it\omega}$ down to 0.74. A branch of type 2 solutions develops between 0.74 and a second modified critical swirl level ${\it\omega}_{0{\it\delta}}=0.71$ and then folds to form breakdown states as ${\it\omega}$ increases above ${\it\omega}_{0{\it\delta}}$ . Further increase of ${\it\beta}{\it\delta}$ up to 0.2 results in similar behaviour, but with the decrease of the first fold point ${\it\omega}_{1{\it\delta}}$ and the increase of the second fold point ${\it\omega}_{0{\it\delta}}$ . At ${\it\beta}{\it\delta}=0.15$ , ${\it\omega}_{1{\it\delta}}$ achieves a minimum value with increase of heat release, ${\it\beta}{\it\delta}$ . This result is supported by the asymptotic analysis of Rusak et al. (Reference Rusak, Kapila and Choi2002) for weak reaction in swirling flows. In this range of relatively small heat release, the flow is dominated by the interaction between the convection of azimuthal vorticity and the stretching of vorticity by axial gradient of circulation. The small heat release creates small baroclinic effects due to the coupling between the small axial temperature gradient and the circulation. These effects support the flow mechanism of reduction of azimuthal vorticity along the centreline to form a decelerated flow at lower swirl levels and a transition to breakdown at a lower critical swirl level with respect to the inert flow.

The bifurcation diagrams of the outlet centreline axial velocity, $w(x_{0},0)$ , as a function of the inlet swirl, ${\it\omega}$ , for larger values of heat release, ${\it\beta}{\it\delta}>0.2$ , are shown in figure 10. At ${\it\beta}{\it\delta}=0.4$ , we find that ${\it\omega}_{1{\it\delta}}={\it\omega}_{0{\it\delta}}$ , and the fold behaviour of the branch of solutions disappears with increase of heat release, ${\it\beta}{\it\delta}$ , above 0.4, i.e. for a fixed ${\it\beta}{\it\delta}$ , $w(x_{0},0)$ develops as one uniform branch and breakdown appears when ${\it\omega}>{\it\omega}_{0{\it\delta}}$ . The values of $w(x_{0},0)$ in the range $0\leqslant {\it\omega}\leqslant {\it\omega}_{0{\it\delta}}$ increase monotonically with increase of ${\it\beta}{\it\delta}$ at a fixed ${\it\omega}$ . We find that the first appearance of vortex breakdown zone is delayed with increase of ${\it\beta}{\it\delta}$ . This result is supported by the discussion in Choi et al. (Reference Choi, Rusak and Kapila2007) on reacting swirling flows. In this range of higher heat release, the flow is dominated by the interaction between the convection of azimuthal vorticity and the stretching of vorticity by axial gradient of circulation. The higher values of heat release create large baroclinic effects due to the coupling between the gradients of temperature and the gradients of the axial and radial velocities. These effects overcome the baroclinic term due to the coupling between axial temperature gradient and circulation. Thereby, the net baroclinic effects make a smaller decrease of the azimuthal vorticity, which results in the axial flow acceleration with respect to case with ${\it\beta}{\it\delta}<0.15$ , and the increase of the first critical swirl with increase of heat release above 0.15.

Figure 10.

Bifurcation diagram of the outlet centreline axial velocity $w(x_{0},0)$ as a function of ${\it\omega}$ at various values of ${\it\beta}{\it\delta}\geqslant 0.2$ for an inlet Lamb–Oseen vortex with $b=4$ and $Ma_{0}=0.001$ .

The size, $y_{s}$ , of the breakdown zone at the outlet as a function of the inlet swirl level, ${\it\omega}$ , and for various heat release, ${\it\beta}{\it\delta}$ , is described in figure 11. For small ${\it\beta}{\it\delta}$ between 0 and 0.4, the lines of $y_{s}$ first increase with the decrease of ${\it\omega}$ . This corresponds to the appearance of small breakdown zones as swirl is decreased from 0.74 to ${\it\omega}_{0{\it\delta}}$ . The lines of $y_{s}$ fold at ${\it\omega}_{0{\it\delta}}$ and continue to increase with ${\it\omega}>{\it\omega}_{0{\it\delta}}$ . This part of lines corresponds to the appearance of larger breakdown zones as swirl is increased above ${\it\omega}_{0{\it\delta}}$ . However, for higher values of heat release, ${\it\beta}{\it\delta}>0.4$ , the lines monotonically increase as ${\it\omega}$ increases above ${\it\omega}_{0{\it\delta}}$ . This corresponds to the disappearance of the fold behaviour in the bifurcation diagram when ${\it\beta}{\it\delta}>0.4$ . We also find that the breakdown zone size, $y_{s}$ , at the outlet decreases with the increase of heat release, ${\it\beta}{\it\delta}$ , at a fixed inlet swirl level, ${\it\omega}$ . This behaviour of $y_{s}$ is dominated by the interaction between convection of azimuthal vorticity, vorticity stretching by circulation gradients and various baroclinic effects as mentioned above.

Figure 11.

Variation of stagnation zone size, $y_{s}$ , at the pipe outlet as a function of ${\it\omega}$ at various ${\it\beta}{\it\delta}$ for an inlet Lamb–Oseen vortex with $b=4$ and $Ma_{0}=0.001$ .

Figure 12.

Variation of critical swirls, ${\it\omega}_{0{\it\delta}}$ (solid line) and ${\it\omega}_{1{\it\delta}}$ (line with filled triangles), as a function of ${\it\beta}{\it\delta}$ for an inlet Lamb–Oseen vortex with $b=4$ and $Ma_{0}=0.001$ .

The variation of the first critical swirl level, ${\it\omega}_{1{\it\delta}}$ , (line with filled triangles) and of the second critical swirl level, ${\it\omega}_{0{\it\delta}}$ , (solid line) with heat release, ${\it\beta}{\it\delta}$ , for the Lamb–Oseen inlet vortex with $b=4$ are shown in figure 12. It can be seen that for small values of ${\it\beta}{\it\delta}$ , ${\it\omega}_{1{\it\delta}}$ decreases and ${\it\omega}_{0{\it\delta}}$ increases with increase of ${\it\beta}{\it\delta}$ . However, once ${\it\beta}{\it\delta}$ is increased beyond a certain level of about 0.4, ${\it\omega}_{0{\it\delta}}$ coincides with ${\it\omega}_{1{\it\delta}}$ and only ${\it\omega}_{0{\it\delta}}$ exists and increases with further increase of ${\it\beta}{\it\delta}$ . This indicates again that fold behaviour exists only for small values of heat release and disappears for larger values. Also shown in figure 12 are the results of numerical simulations from Choi et al. (Reference Choi, Rusak and Kapila2007) for the first appearance of vortex breakdown in premixed reacting flows in a straight open pipe (line with filled squares). In the numerical simulation, the inlet flow is also given by the Lamb–Oseen vortex with $b=4$ . It is understood now that the transition to breakdown in the simulations at low values of ${\it\beta}{\it\delta}$ below 0.4 is related to the jump of solutions from a near-columnar state to a breakdown state when swirl was increased from below to above the first critical value at ${\it\omega}_{1{\it\delta}}({\it\beta}{\it\delta})$ . On the other hand, the appearance of breakdown in the simulations at high values of ${\it\beta}{\it\delta}$ above 0.4 is related to the gradual change of nature of the states when swirl level increases above ${\it\omega}_{0{\it\delta}}({\it\beta}{\it\delta})$ . The computed results from the present approach capture the major trend of the nonlinear behaviour predicted by the numerical simulations of Choi et al. (Reference Choi, Rusak and Kapila2007) although the numerical results differ by small amount of swirl. This difference can be explained by the possible effect of pipe length. The present study assumes a relatively long pipe such that a fully developed columnar state is established ahead of the outlet. On the other hand, the numerical simulations used a relatively short pipe of two radii length. It is known from the inert flow simulations of Rusak et al. (Reference Rusak, Whiting and Wang1998) that pipe length increases the critical swirl level for vortex breakdown from ${\it\omega}_{0}$ by an amount that is inversely proportional to the pipe length. Specifically for short pipes that amount of increase is not negligible and creates a finite small difference between the predicted line by the present approach and the numerical simulation results.