2.1 Fluid flow equations

We consider a swirling flow of a single-phase van der Waals fluid in a finite-length straight circular pipe. The gas thermo-physical specific heat capacity at constant volume $C_{v}$ is assumed to be constant. The thermodynamic equation of state and the continuity, linear momentum and energy equations describing the motion of a steady, inviscid-limit, non-heat-conducting and compressible flow of the fluid are given respectively by Thompson (Reference Thompson1982)

(2.1a ) $$\begin{eqnarray}\displaystyle & \displaystyle \bar{p}=\frac{\bar{\unicode[STIX]{x1D70C}}R\bar{T}}{1-b\bar{\unicode[STIX]{x1D70C}}}-a\bar{\unicode[STIX]{x1D70C}}^{2}, & \displaystyle\end{eqnarray}$$

(2.1b ) $$\begin{eqnarray}\displaystyle & \displaystyle \bar{\unicode[STIX]{x1D735}}\boldsymbol{\cdot }\left(\bar{\unicode[STIX]{x1D70C}}\bar{\boldsymbol{V}}\right)=0, & \displaystyle\end{eqnarray}$$

(2.1c ) $$\begin{eqnarray}\displaystyle & \displaystyle \bar{\unicode[STIX]{x1D70C}}\bar{\boldsymbol{V}}\boldsymbol{\cdot }\bar{\unicode[STIX]{x1D735}}\bar{\boldsymbol{V}}=-\bar{\unicode[STIX]{x1D735}}\bar{p}, & \displaystyle\end{eqnarray}$$

(2.1d ) $$\begin{eqnarray}\displaystyle & \displaystyle \bar{\unicode[STIX]{x1D70C}}\bar{\boldsymbol{V}}\boldsymbol{\cdot }\bar{\unicode[STIX]{x1D735}}\bar{h}=\bar{\boldsymbol{V}}\boldsymbol{\cdot }\bar{\unicode[STIX]{x1D735}}\bar{p}. & \displaystyle\end{eqnarray}$$

$\bar{p}$

$\bar{\unicode[STIX]{x1D70C}}$

$\bar{T}$

$\bar{h}$

$\bar{\boldsymbol{V}}$

$R={\mathcal{R}}/M$

${\mathcal{R}}=8.3145~\text{J}~(\text{mol}~\text{K})^{-1}$

$M$

$a$

$b$

$(\unicode[STIX]{x2202}\bar{p}/\unicode[STIX]{x2202}\bar{\unicode[STIX]{x1D70C}})_{c}=0$

$(\unicode[STIX]{x2202}^{2}\bar{p}/\unicode[STIX]{x2202}\bar{\unicode[STIX]{x1D70C}}^{2})_{c}=0$

$a=(27/64)(R^{2}\bar{T}_{c}^{2})/\bar{p}_{c}$

$b=(R\bar{T}_{c})/(8\bar{p}_{c})$

$\bar{p}_{c}$

$\bar{\unicode[STIX]{x1D70C}}_{c}$

$\bar{T}_{c}$

$\bar{p}_{c}=\bar{\unicode[STIX]{x1D70C}}_{c}R\bar{T}_{c}/(1-b\bar{\unicode[STIX]{x1D70C}_{c}})-a\bar{\unicode[STIX]{x1D70C}_{c}}^{2}$

$Z_{c}=\bar{p}_{c}/(\bar{\unicode[STIX]{x1D70C}}_{c}R\bar{T}_{c})=3/8$

$a=3\bar{p}_{c}/\bar{\unicode[STIX]{x1D70C}}_{c}^{2}$

$b=1/(3\bar{\unicode[STIX]{x1D70C}}_{c})$

$0<\bar{\unicode[STIX]{x1D70C}}/\bar{\unicode[STIX]{x1D70C}}_{c}\ll 1$

$\bar{p}=\bar{\unicode[STIX]{x1D70C}}R\bar{T}$

$0\leqslant \bar{\unicode[STIX]{x1D70C}}/\bar{\unicode[STIX]{x1D70C}}_{c}<3$

Following the results in the appendix in Rusak & Wang (Reference Rusak and Wang1997), the specific internal energy $\bar{e}$ of the van der Waals fluid is given by $\bar{e}=~C_{v}\bar{T}-a\bar{\unicode[STIX]{x1D70C}}$ , where we assume that $\bar{e}=0$ at $\bar{T}=\bar{\unicode[STIX]{x1D70C}}=0$ . Therefore, the specific enthalpy $\bar{h}=\bar{e}+\bar{p}/\bar{\unicode[STIX]{x1D70C}}$ is given by

(2.2) $$\begin{eqnarray}\displaystyle \bar{h}=C_{v}\bar{T}-2a\bar{\unicode[STIX]{x1D70C}}+\frac{R\bar{T}}{1-b\bar{\unicode[STIX]{x1D70C}}}, & & \displaystyle\end{eqnarray}$$

where $\bar{h}=0$ at $\bar{T}=\bar{\unicode[STIX]{x1D70C}}=0$ . The Gibbs equation is used to define the specific entropy $\bar{s}$ (Thompson Reference Thompson1982),

(2.3) $$\begin{eqnarray}\displaystyle \bar{T}\bar{\unicode[STIX]{x1D735}}\bar{s}=\bar{\unicode[STIX]{x1D735}}\bar{h}-\frac{1}{\bar{\unicode[STIX]{x1D70C}}}\bar{\unicode[STIX]{x1D735}}\bar{p}. & & \displaystyle\end{eqnarray}$$

The entropy of the van der Waals fluid is given by (Rusak & Wang Reference Rusak and Wang1997)

(2.4) $$\begin{eqnarray}\displaystyle \bar{s}=\bar{s}_{c}+C_{v}\ln \left(\frac{\bar{T}}{\bar{T}_{c}}\right)+R\ln \left[\frac{3}{2}\frac{\bar{\unicode[STIX]{x1D70C}}_{c}}{\bar{\unicode[STIX]{x1D70C}}}\left(1-\frac{\bar{\unicode[STIX]{x1D70C}}}{3\bar{\unicode[STIX]{x1D70C}}_{c}}\right)\right]. & & \displaystyle\end{eqnarray}$$

We assume that the critical point specific entropy is $\bar{s}_{c}=-R\ln (3/2)$ . We choose this constant to eliminate the constant term $R\ln (3/2)$ from the formula for $\bar{s}$ . We emphasize that the specific value of $\bar{s}_{c}$ does not affect the calculations as long as it is consistently used. In addition, the isentropic speed of sound $\bar{c}$ of the van der Waals fluid is given by (see details in the appendix of Rusak & Wang (Reference Rusak and Wang1997)):

(2.5) $$\begin{eqnarray}\displaystyle \bar{c}^{2}=\left(\frac{\unicode[STIX]{x2202}\bar{p}}{\unicode[STIX]{x2202}\bar{\unicode[STIX]{x1D70C}}}\right)_{\bar{s}=\text{const.}}=\frac{\bar{p}+a\bar{\unicode[STIX]{x1D70C}}^{2}}{\bar{\unicode[STIX]{x1D70C}}(1-b\bar{\unicode[STIX]{x1D70C}})}\left(1+\frac{R}{C_{v}}\right)-2a\bar{\unicode[STIX]{x1D70C}}. & & \displaystyle\end{eqnarray}$$

Non-dimensionalization of (2.1)–(2.4) is carried out as follows. Lengths are normalized against the pipe inlet radius $\bar{r}_{0}$ . Then, $\unicode[STIX]{x1D735}=(1/\bar{r}_{0})\bar{\unicode[STIX]{x1D735}}$ . Pressure, density and temperature are normalized with respect to the fluid thermodynamic critical properties $\bar{p}_{c}$ , $\bar{\unicode[STIX]{x1D70C}}_{c}$ and $\bar{T}_{c}$ , respectively, i.e. the reduced pressure, density and temperature are $p=\bar{p}/\bar{p}_{c}$ , $\unicode[STIX]{x1D70C}=\bar{\unicode[STIX]{x1D70C}}/\bar{\unicode[STIX]{x1D70C}}_{c}$ and $T=\bar{T}/\bar{T}_{c}$ , respectively. Also, we scale the specific enthalpy by $C_{v}\bar{T}_{c}$ and the specific entropy by $C_{v}$ , i.e. $h=\bar{h}/(C_{v}\bar{T}_{c})$ and $s=\bar{s}/C_{v}$ . The velocity vector is normalized with respect to the inlet flow centreline axial speed $\bar{U}_{0}$ , i.e. $\boldsymbol{V}=\bar{\boldsymbol{V}}/\bar{U}_{0}$ . Then, (2.1)–(2.4) become

(2.6a ) $$\begin{eqnarray}\displaystyle & \displaystyle p=\frac{8}{3}\frac{\unicode[STIX]{x1D70C}T}{1-\frac{1}{3}\unicode[STIX]{x1D70C}}-3\unicode[STIX]{x1D70C}^{2}, & \displaystyle\end{eqnarray}$$

(2.6b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\left(\unicode[STIX]{x1D70C}\boldsymbol{V}\right)=0, & \displaystyle\end{eqnarray}$$

(2.6c ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70C}\boldsymbol{V}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{V}=-\frac{1}{Ma_{c}^{2}}\unicode[STIX]{x1D735}p, & \displaystyle\end{eqnarray}$$

(2.6d ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70C}\boldsymbol{V}\boldsymbol{\cdot }\unicode[STIX]{x1D735}h=\frac{3}{8}\frac{R}{C_{v}}\boldsymbol{V}\boldsymbol{\cdot }\unicode[STIX]{x1D735}p, & \displaystyle\end{eqnarray}$$

(2.6e ) $$\begin{eqnarray}\displaystyle & \displaystyle h=T+\frac{R}{C_{v}}\frac{T}{1-\frac{1}{3}\unicode[STIX]{x1D70C}}-\frac{9}{4}\frac{R}{C_{v}}\unicode[STIX]{x1D70C}, & \displaystyle\end{eqnarray}$$

(2.6f ) $$\begin{eqnarray}\displaystyle & \displaystyle T\unicode[STIX]{x1D735}s=\unicode[STIX]{x1D735}h-\frac{3}{8}\frac{R}{C_{v}}\frac{1}{\unicode[STIX]{x1D70C}}\unicode[STIX]{x1D735}p, & \displaystyle\end{eqnarray}$$

(2.6g ) $$\begin{eqnarray}\displaystyle & \displaystyle s=\ln T+\frac{R}{C_{v}}\ln \left(\frac{1}{\unicode[STIX]{x1D70C}}-\frac{1}{3}\right). & \displaystyle\end{eqnarray}$$

$Ma_{c}$

$Ma_{c}=\bar{U}_{0}/\sqrt{\bar{p}_{c}/\bar{\unicode[STIX]{x1D70C}}_{c}}$

$R/C_{v}$

$R/C_{v}=2/3$

$\text{H}_{2},\text{N}_{2},\text{O}_{2}$

$R/C_{v}=2/5$

$300$

$2/7$

$1000$

$R/C_{v}$

$0.1$

$0.01$

Let $\bar{p}_{0}$ , $\bar{\unicode[STIX]{x1D70C}}_{0}$ and $\bar{T}_{0}$ be the flow centreline inlet pressure, density and temperature, respectively, and $p_{0}=\bar{p}_{0}/\bar{p}_{c}$ , $\unicode[STIX]{x1D70C}_{0}=\bar{\unicode[STIX]{x1D70C}}_{0}/\bar{\unicode[STIX]{x1D70C}}_{c}$ and $T_{0}=\bar{T}_{0}/\bar{T}_{c}$ be the respective non-dimensional centreline inlet reduced variables. Then, the centreline inlet isentropic speed of sound $\bar{c}_{0}$ is given from (2.5) by

(2.7) $$\begin{eqnarray}\displaystyle \bar{c}_{0}^{2}=\frac{\bar{p}_{c}}{\bar{\unicode[STIX]{x1D70C}}_{c}}\left[\frac{p_{0}+3\unicode[STIX]{x1D70C}_{0}^{2}}{\unicode[STIX]{x1D70C}_{0}(1-\unicode[STIX]{x1D70C}_{0}/3)}\left(1+\frac{R}{C_{v}}\right)-6\unicode[STIX]{x1D70C}_{0}\right]. & & \displaystyle\end{eqnarray}$$

We also define the inlet flow Mach number $Ma_{0}=\bar{U}_{0}/\bar{c}_{0}$ as

(2.8) $$\begin{eqnarray}Ma_{0}=\frac{Ma_{c}}{\sqrt{{\displaystyle \frac{p_{0}+3\unicode[STIX]{x1D70C}_{0}^{2}}{\unicode[STIX]{x1D70C}_{0}(1-\unicode[STIX]{x1D70C}_{0}/3)}}\left(1+{\displaystyle \frac{R}{C_{v}}}\right)-6\unicode[STIX]{x1D70C}_{0}}}.\end{eqnarray}$$

In the present paper we focus on subsonic flows with $0<Ma_{0}<1$ .

Using the flow linear momentum equation in (2.6c ), the Gibbs formula (2.6f ) and the vector identity, $\boldsymbol{V}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{V}=\unicode[STIX]{x1D735}(\boldsymbol{V}\boldsymbol{\cdot }\boldsymbol{V}/2)-\boldsymbol{V}\times \unicode[STIX]{x1D6FA}$ (where $\unicode[STIX]{x1D6FA}=\unicode[STIX]{x1D735}\times \boldsymbol{V}$ is the non-dimensional vorticity vector), Crocco’s formula is found (Thompson Reference Thompson1982)

(2.9) $$\begin{eqnarray}\frac{3}{8}\frac{R}{C_{v}}Ma_{c}^{2}\left(\boldsymbol{V}\times \unicode[STIX]{x1D6FA}\right)=\unicode[STIX]{x1D735}H-T\unicode[STIX]{x1D735}s,\end{eqnarray}$$

where $H$ is the non-dimensional specific total enthalpy (scaled by $C_{v}\bar{T}_{c}$ ) and given by

(2.10) $$\begin{eqnarray}\displaystyle H=h+\frac{3}{16}\frac{R}{C_{v}}Ma_{c}^{2}(\boldsymbol{V}\boldsymbol{\cdot }\boldsymbol{V})=T+\frac{R}{C_{v}}\left(\frac{T}{1-\frac{1}{3}\unicode[STIX]{x1D70C}}-\frac{9}{4}\unicode[STIX]{x1D70C}+\frac{3}{16}Ma_{c}^{2}(\boldsymbol{V}\boldsymbol{\cdot }\boldsymbol{V})\right). & & \displaystyle\end{eqnarray}$$

Note that there is less exchange between thermal energy (specific enthalpy) and kinetic energy when $R/C_{v}$ is low.

From the dot product of $\boldsymbol{V}$ with the linear momentum equation (2.6c ), the identity $\boldsymbol{V}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{V}=\unicode[STIX]{x1D735}(\boldsymbol{V}\boldsymbol{\cdot }\boldsymbol{V}/2)-\boldsymbol{V}\times \unicode[STIX]{x1D6FA}$ , the identity $\boldsymbol{V}\boldsymbol{\cdot }(\boldsymbol{V}\times \unicode[STIX]{x1D6FA})=0$ and the energy equation (2.6d ), we find that

(2.11) $$\begin{eqnarray}\unicode[STIX]{x1D70C}\boldsymbol{V}\boldsymbol{\cdot }\unicode[STIX]{x1D735}H=0.\end{eqnarray}$$

Then, from the dot product of $\boldsymbol{V}$ with Crocco formula (2.9), we find that

(2.12) $$\begin{eqnarray}\unicode[STIX]{x1D70C}\boldsymbol{V}\boldsymbol{\cdot }\unicode[STIX]{x1D735}s=0.\end{eqnarray}$$

This means that in a steady flow the specific total enthalpy and the specific entropy are constant along a streamline.

We consider a flow in a pipe with a cylindrical coordinate system $(x,r,\unicode[STIX]{x1D703})$ , where the $x$ -axis measures non-dimensional distance from the inlet along the centreline, $0\leqslant x\leqslant x_{0}$ , $r$ measures radial distance from the centreline, $0\leqslant r\leqslant 1$ , and $\unicode[STIX]{x1D703}$ is the azimuthal angle, $0\leqslant \unicode[STIX]{x1D703}<2\unicode[STIX]{x03C0}$ (see figure 1). Here $x_{0}$ is the non-dimensional pipe length. The corresponding non-dimensional velocity components are the axial velocity $w$ , the circumferential velocity $v$ and the radial velocity $u$ . In the entire paper, indices $x$ and $r$ represent partial derivatives with respect to $x$ and $r$ . Then, assuming axisymmetry, the continuity equation in (2.6b ) is

(2.13) $$\begin{eqnarray}\left(\unicode[STIX]{x1D70C}ru\right)_{r}+\left(\unicode[STIX]{x1D70C}rw\right)_{x}=0.\end{eqnarray}$$

From (2.13), streamfunction $\unicode[STIX]{x1D6F9}(x,y)$ is defined, where $y=r^{2}/2$ is the Stokes classical coordinate for flow problems in cylindrical coordinates, $0\leqslant y\leqslant 1/2$ , $u=-\unicode[STIX]{x1D6F9}_{x}/(\unicode[STIX]{x1D70C}\sqrt{2y})$ and $w=\unicode[STIX]{x1D6F9}_{y}/\unicode[STIX]{x1D70C}$ . Here, index $y$ represents partial derivative with respect to $y$ . Also, the circumferential momentum equation in (2.1) is

(2.14) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70C}u\left(rv\right)_{r}+\unicode[STIX]{x1D70C}w\left(rv\right)_{x}=0 & & \displaystyle\end{eqnarray}$$

Figure 1.

Schematic diagram of a compressible swirling flow of a supercritical fluid in a pipe.

or

(2.15) $$\begin{eqnarray}\displaystyle -\unicode[STIX]{x1D6F9}_{x}\left(\sqrt{2y}v\right)_{y}+\unicode[STIX]{x1D6F9}_{y}\left(\sqrt{2y}v\right)_{x}=0. & & \displaystyle\end{eqnarray}$$

Equation (2.15) shows that the circulation function $K=\sqrt{2y}v$ is given by $K=K(\unicode[STIX]{x1D6F9})$ . This means that in a steady and axisymmetric flow the circulation function is constant along a $\unicode[STIX]{x1D6F9}=\text{const}.$ surface and that $v=K(\unicode[STIX]{x1D6F9})/\sqrt{2y}$ . In addition, equations (2.11) and (2.12) become, respectively,

(2.16) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\unicode[STIX]{x1D70C}uH_{r}+\unicode[STIX]{x1D70C}wH_{x}=0,\\ \unicode[STIX]{x1D70C}us_{r}+\unicode[STIX]{x1D70C}ws_{x}=0,\end{array}\right\} & & \displaystyle\end{eqnarray}$$

or

(2.17) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}-\unicode[STIX]{x1D6F9}_{x}H_{y}+\unicode[STIX]{x1D6F9}_{y}H_{x}=0,\\ -\unicode[STIX]{x1D6F9}_{x}s_{y}+\unicode[STIX]{x1D6F9}_{y}s_{x}=0.\end{array}\right\} & & \displaystyle\end{eqnarray}$$

We find that

(2.18a,b ) $$\begin{eqnarray}H=H(\unicode[STIX]{x1D6F9})\quad \text{and}\quad s=s(\unicode[STIX]{x1D6F9}).\end{eqnarray}$$

This means that in a steady and axisymmetric flow the total enthalpy and specific entropy are constant along a $\unicode[STIX]{x1D6F9}=\text{const}.$ surface.

From (2.10) and (2.18) and the definition of the velocity components in terms of $\unicode[STIX]{x1D6F9}$ and its derivatives, we find that

(2.19) $$\begin{eqnarray}\displaystyle H(\unicode[STIX]{x1D6F9})=T+\frac{R}{C_{v}}\left[\frac{T}{1-\frac{1}{3}\unicode[STIX]{x1D70C}}-\frac{9}{4}\unicode[STIX]{x1D70C}+\frac{3}{16}Ma_{c}^{2}\left(\frac{\unicode[STIX]{x1D6F9}_{y}^{2}}{\unicode[STIX]{x1D70C}^{2}}+\frac{\unicode[STIX]{x1D6F9}_{x}^{2}}{2y\unicode[STIX]{x1D70C}^{2}}+\frac{K^{2}(\unicode[STIX]{x1D6F9})}{2y}\right)\right], & & \displaystyle\end{eqnarray}$$

and from (2.6g ) and (2.18),

(2.20) $$\begin{eqnarray}\displaystyle \exp \left(s(\unicode[STIX]{x1D6F9})\right)=T\left(\frac{1}{\unicode[STIX]{x1D70C}}-\frac{1}{3}\right)^{R/C_{v}}. & & \displaystyle\end{eqnarray}$$

The $x$ -component of the Crocco’s equation (2.9) is $(3/8)(R/C_{v})Ma_{c}^{2}(u\unicode[STIX]{x1D702}-v\unicode[STIX]{x1D701})=H_{x}-Ts_{x}$ , where the radial and circumferential components of the non-dimensional vorticity are $\unicode[STIX]{x1D701}=-v_{x}=-(1/\sqrt{2y})K_{x}$ and $\unicode[STIX]{x1D702}=u_{x}-w_{r}=u_{x}-\sqrt{2y}w_{y}$ , respectively. With $v=(K/\sqrt{2y})$ , we find that

(2.21) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle \frac{3}{8}\frac{R}{C_{v}}Ma_{c}^{2}u\left(u_{x}-\sqrt{2y}w_{y}\right)=H_{x}-\frac{3}{8}\frac{R}{C_{v}}Ma_{c}^{2}\frac{KK_{x}}{2y}-Ts_{x},\quad \text{or}\\ \displaystyle -\frac{3}{8}\frac{R}{C_{v}}Ma_{c}^{2}\frac{\unicode[STIX]{x1D6F9}_{x}}{\unicode[STIX]{x1D70C}\sqrt{2y}}\left(\!u_{x}\!-\!\sqrt{2y}w_{y}\!\right)=\left[H^{\prime }\left(\unicode[STIX]{x1D6F9}\right)-\frac{3}{8}\frac{R}{C_{v}}Ma_{c}^{2}\frac{KK^{\prime }\left(\unicode[STIX]{x1D6F9}\right)}{2y}-Ts^{\prime }\left(\unicode[STIX]{x1D6F9}\right)\right]\unicode[STIX]{x1D6F9}_{x},\quad \text{or}\\ \displaystyle \frac{3}{8}\frac{R}{C_{v}}Ma_{c}^{2}\left[\frac{1}{\unicode[STIX]{x1D70C}}\left(\frac{\unicode[STIX]{x1D6F9}_{y}}{\unicode[STIX]{x1D70C}}\right)_{y}+\frac{1}{2y\unicode[STIX]{x1D70C}}\left(\frac{\unicode[STIX]{x1D6F9}_{x}}{\unicode[STIX]{x1D70C}}\right)_{x}\right]=H^{\prime }\left(\unicode[STIX]{x1D6F9}\right)-\frac{3}{8}\frac{R}{C_{v}}Ma_{c}^{2}\frac{K(\unicode[STIX]{x1D6F9})K^{\prime }\left(\unicode[STIX]{x1D6F9}\right)}{2y}-Ts^{\prime }\left(\unicode[STIX]{x1D6F9}\right).\end{array}\right\} & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Here, $(\,)^{\prime }$ represents the derivative with respect to $\unicode[STIX]{x1D6F9}$ . The left-hand side of (2.21) represents the azimuthal vorticity in an axisymmetric compressible flow while the right-hand side includes production terms of azimuthal vorticity related to the derivatives of total enthalpy, circulation and entropy.

Equation (2.21) is a PDE for the solution of $\unicode[STIX]{x1D6F9}(x,y)$ . It generalizes the classical Squire–Long equation (Long Reference Long1953; Squire Reference Squire, Batchelor and Davies1956) for steady incompressible swirling flows to the case of steady compressible swirling flows of the van der Waals fluid. In (2.21), given $\unicode[STIX]{x1D6F9}(x,y)$ , $K(\unicode[STIX]{x1D6F9})$ , $H(\unicode[STIX]{x1D6F9})$ and $s(\unicode[STIX]{x1D6F9})$ , the non-dimensional density $\unicode[STIX]{x1D70C}(x,y)$ and temperature $T(x,y)$ are found from the solution of coupled system of equations (2.19) and (2.20) at all points in the flow domain.

2.2 Boundary conditions

For a flow in a rotating, finite-length, straight circular pipe of non-dimensional length $x_{0}$ , we consider the following boundary conditions that support (2.21) – see set-up in figure 1. At the pipe inlet at $x=0$ , we prescribe the profiles of the streamfunction, the circulation function, the axial derivative of radial velocity, the total head and the specific entropy by the following continuous and smooth functions, which are consistent with (2.21),

(2.22) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}rl@{}}\unicode[STIX]{x1D6F9}(0,y) & =\unicode[STIX]{x1D6F9}_{0}(y),\quad K(0,y)=\unicode[STIX]{x1D714}K_{0}(y),\quad u_{x}(0,y)=0,\quad H(0,y)=H_{0}(y),\!\\ & \quad s(0,y)=s_{0}(y)\quad \text{for}~0\leqslant y\leqslant 1/2.\!\end{array}\right\} & & \displaystyle\end{eqnarray}$$

Here $\unicode[STIX]{x1D714}$ is the swirl ratio of the incoming flow, as defined in the introduction and $K_{0}(0)=0$ . We also assume that pressure at the inlet centreline, $p(0,0)=p_{CL}$ is given. Let $T_{CL}=T(0,0)$ and $\unicode[STIX]{x1D70C}_{CL}=\unicode[STIX]{x1D70C}(0,0)$ , then the non-dimensional equation of state $p_{CL}=(8/3)\unicode[STIX]{x1D70C}_{CL}T_{CL}/(1-(1/3)\unicode[STIX]{x1D70C}_{CL})-3\unicode[STIX]{x1D70C}_{CL}^{2}$ gives $p_{CL}$ in terms of the given $\unicode[STIX]{x1D70C}_{CL}$ and $T_{CL}$ . These inlet conditions may describe a compressible swirling flow that is generated by a vortex generator ahead of the pipe at a continuous and steady operation, see the experiments of Umeh et al. (Reference Umeh, Rusak, Gutmark, Villalva and Cha2010) and Umeh, Rusak & Gutmark (Reference Umeh, Rusak and Gutmark2012) who used a triple annular swirler to generate the inlet flow. The inlet condition on the axial derivative of the radial velocity allows the inlet state a degree of freedom to develop a radial velocity to account for the upstream influence of steady disturbances in the flow. In appendix A, we comment on the relationship between the inlet profiles of $H_{0}(y)$ and $s_{0}(y)$ and the inlet temperature, density, and axial and circumferential velocity profiles. The inlet radial velocity $u(0,y)$ is not needed for the calculation of these functions.

At the pipe outlet at $x=x_{0}$ , we assume that the radial velocity vanishes, i.e. $u(x_{0},y)=0$ , or

(2.23) $$\begin{eqnarray}\unicode[STIX]{x1D6F9}_{x}(x_{0},y)=0\quad \text{for}~0\leqslant y\leqslant 1/2.\end{eqnarray}$$

This outlet condition may represent the case of a flow in a long open pipe where the flow settles on a parallel (columnar) state ahead of the outlet. This assumption describes an apparatus with a passive discharge device. The experiments of Umeh et al. (Reference Umeh, Rusak, Gutmark, Villalva and Cha2010) show that the flow exhibits a parallel flow state at distances of six radii and higher from the inlet. The pipe non-dimensional length $x_{0}$ needs to be sufficiently long, say $x_{0}>6$ , but not too long and must be less than the entrance length for a fully developed viscous flow, $x_{0}\ll Re^{1/2}$ where $Re$ is the Reynolds number based on pipe radius $\bar{r}_{0}$ and characteristic velocity $\bar{U}_{0}$ . On the other hand, for pipes that are shorter, with $x_{0}<4$ , the outlet zero radial velocity may be a too strong condition on the convective flow and may fail to describe a non-columnar flow in a short pipe turning from the given inlet conditions to a passive device.

Along the pipe centreline at $y=0$ , we assume the axisymmetry condition, $\unicode[STIX]{x1D6F9}(x,0)=0$ for $0\leqslant x\leqslant x_{0}$ . Along the rotating pipe wall at $y=1/2$ , we assume constant flux, $\unicode[STIX]{x1D6F9}(x,1/2)=\unicode[STIX]{x1D6F9}_{w}\equiv \unicode[STIX]{x1D6F9}_{0}(1/2)$ .

By inverting $\unicode[STIX]{x1D6F9}(0,y)=\unicode[STIX]{x1D6F9}_{0}(y)$ to $y=y(\unicode[STIX]{x1D6F9})$ , we find the functions $K(\unicode[STIX]{x1D6F9})=\unicode[STIX]{x1D714}K_{0}(y(\unicode[STIX]{x1D6F9}))$ , $H(\unicode[STIX]{x1D6F9})=H_{0}(y(\unicode[STIX]{x1D6F9}))$ and $s(\unicode[STIX]{x1D6F9})=s_{0}(y(\unicode[STIX]{x1D6F9}))$ are defined by the inlet flow profiles in the range of $0\leqslant \unicode[STIX]{x1D6F9}\leqslant \unicode[STIX]{x1D6F9}_{w}$ . In addition, to accommodate possible solutions with separation zones, we assume that

(2.24) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}K(\unicode[STIX]{x1D6F9})=K(0)=0,\quad H(\unicode[STIX]{x1D6F9})=H(0)=H_{0},\quad s(\unicode[STIX]{x1D6F9})=s(0)=s_{0},\\ K^{\prime }(\unicode[STIX]{x1D6F9})=H^{\prime }(\unicode[STIX]{x1D6F9})=s^{\prime }(\unicode[STIX]{x1D6F9})=0\end{array}\right\} & & \displaystyle\end{eqnarray}$$

when $\unicode[STIX]{x1D6F9}<0$ to describe centreline breakdown zones, and

(2.25) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}K(\unicode[STIX]{x1D6F9})=K\left(\unicode[STIX]{x1D6F9}_{w}\right)=K_{w},\quad H(\unicode[STIX]{x1D6F9})=H\left(\unicode[STIX]{x1D6F9}_{w}\right)=H_{w},\quad s(\unicode[STIX]{x1D6F9})=s\left(\unicode[STIX]{x1D6F9}_{w}\right)=s_{w},\\ K^{\prime }(\unicode[STIX]{x1D6F9})=H^{\prime }(\unicode[STIX]{x1D6F9})=s^{\prime }(\unicode[STIX]{x1D6F9})=0\end{array}\right\} & & \displaystyle\end{eqnarray}$$

when $\unicode[STIX]{x1D6F9}>\unicode[STIX]{x1D6F9}_{w}$ to describe wall-separation zones.

The PDE (2.21) with the assumed boundary conditions and conditions (2.24) and (2.25) forms a well-defined problem for the solution of the flow streamfunction $\unicode[STIX]{x1D6F9}(x,y)$ in the domain $0\leqslant x\leqslant x_{0}$ and $0\leqslant y\leqslant 1/2$ .

A base solution of this problem for all values of $R/C_{v}$ , $Ma_{c}$ and $\unicode[STIX]{x1D714}$ is the columnar flow state given by

(2.26) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\unicode[STIX]{x1D6F9}(x,y)=\unicode[STIX]{x1D6F9}_{0}(y),\quad K(x,y)=\unicode[STIX]{x1D714}K_{0}(y),\quad H(x,y)=H_{0}(y),\quad s(x,y)=s_{0}(y)\\ \text{for}~0\leqslant x\leqslant x_{0}\quad \text{and}\quad 0\leqslant y\leqslant 1/2.\end{array}\right\}\quad & & \displaystyle\end{eqnarray}$$

This solution is denoted as type 0 state of the swirling flow problem in a finite-length straight circular pipe. We seek to determine the possible existence of additional steady states of swirling flows in the finite-length straight circular pipe with the same $R/C_{v}$ , $Ma_{c}$ , $\unicode[STIX]{x1D714}$ , $H(\unicode[STIX]{x1D6F9})$ , $K(\unicode[STIX]{x1D6F9})$ , $s(\unicode[STIX]{x1D6F9})$ and assumed boundary conditions.

4.1 Case 1: the calorically perfect-gas limit ( $\unicode[STIX]{x1D70C}_{CL}=10^{-6}$ )

In this case (figure 3), the base flow density $\unicode[STIX]{x1D70C}$ , pressure $p$ and streamfunction $\unicode[STIX]{x1D6F9}$ increase monotonically with $y$ for all $\unicode[STIX]{x1D714}$ and increase with $\unicode[STIX]{x1D714}$ at all $y$ (figure 3(a); note that the scale of these functions is on the order of $10^{-6}$ ) and flow is a calorically perfect gas for all $y$ and $\unicode[STIX]{x1D714}$ studied. The function $(3/8)(R/C_{v})Ma_{c}^{2}K(\unicode[STIX]{x1D6F9})K^{\prime }(\unicode[STIX]{x1D6F9})$ is positive and increases monotonically with $\unicode[STIX]{x1D6F9}$ , while $H^{\prime }(\unicode[STIX]{x1D6F9})$ is positive but decreases monotonically with $\unicode[STIX]{x1D6F9}$ , and $s^{\prime }(\unicode[STIX]{x1D6F9})$ is negative and increases monotonically with $\unicode[STIX]{x1D6F9}$ (figure 3(b); note that the range of $\unicode[STIX]{x1D6F9}$ increases with $\unicode[STIX]{x1D714}$ since $\unicode[STIX]{x1D6F9}_{w}=\unicode[STIX]{x1D6F9}(1/2)$ increases as a function of $\unicode[STIX]{x1D714}$ , see figure 3(a)). The resulting bifurcation diagram (figure 3 c) shows that type 0 solution with $w_{CL}=1$ exists for all $\unicode[STIX]{x1D714}$ . At $\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}_{c}=2.218$ , type 1 and type 2 solutions bifurcate from the type 0 branch of solutions. The type 1 branch of solutions with $0<w_{CL}<1$ develops with the decrease of $\unicode[STIX]{x1D714}$ from $\unicode[STIX]{x1D714}_{c}=2.218$ to $\unicode[STIX]{x1D714}_{\ast }=2.012$ . The type 2 solutions with $w_{CL}>1$ develop as $\unicode[STIX]{x1D714}$ increases above $\unicode[STIX]{x1D714}_{c}=2.218$ . Type 3 breakdown solutions with $w_{CL}=0$ develop with the increase of $\unicode[STIX]{x1D714}$ above $\unicode[STIX]{x1D714}_{\ast }=2.012$ . The related size $y_{0}$ of the type 3 solutions increases with the increase of $\unicode[STIX]{x1D714}$ from $\unicode[STIX]{x1D714}_{\ast }=2.012$ to $\unicode[STIX]{x1D714}=3.3$ and then decreases with further increase of $\unicode[STIX]{x1D714}$ (figure 3 d).

Figure 3.

Case 1: Perfect-gas limit, $\unicode[STIX]{x1D70C}_{CL}=10^{-6}$ , $T_{0}=1.02$ , $Ma_{0}=0.5$ , $R/C_{v}=0.01$ . In panels (a,b) $\unicode[STIX]{x1D714}=2.1$ (solid lines), $\unicode[STIX]{x1D714}=2.5$ (dashed lines) and $\unicode[STIX]{x1D714}=3$ (dotted lines): (a) solutions of the base columnar flow density $\unicode[STIX]{x1D70C}$ (black lines), pressure $p$ (blue lines) and streamfunction $\unicode[STIX]{x1D6F9}$ (red lines) as a function of $y$ ; (b) the functions $H^{\prime }(\unicode[STIX]{x1D6F9})$ (blue lines), $s^{\prime }(\unicode[STIX]{x1D6F9})$ (black lines) and $(3/8)(R/C_{v})Ma_{c}^{2}K(\unicode[STIX]{x1D6F9})K^{\prime }(\unicode[STIX]{x1D6F9})$ (red lines) as a function of $\unicode[STIX]{x1D6F9}$ ; (c) the bifurcation diagram of solutions of $w_{CL}$ against $\unicode[STIX]{x1D714}$ ; (d) the size of the breakdown zone $y_{0}$ as a function of $\unicode[STIX]{x1D714}$ for type 3 solutions.

The swirling flow behaviour in the perfect-gas limit case is similar to that reported in Rusak et al. (Reference Rusak, Choi, Bourquard and Wang2015a ) where the dynamics of swirling flows of a calorically perfect gas in a straight pipe was studied. Moreover, the present results match with the previous results when $R/C_{v}=0.4$ (or the ratio of specific heat is $\unicode[STIX]{x1D6FE}=1.4$ ) and $Ma_{0}=0.5$ are used.

The results of the calorically perfect-gas case in Rusak et al. (Reference Rusak, Choi and Lee2007, Reference Rusak, Choi, Bourquard and Wang2015a ) show for a long pipe that the columnar flow (type 0) states are asymptotically stable when $0<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{c}$ and unstable when $\unicode[STIX]{x1D714}>\unicode[STIX]{x1D714}_{c}$ . Similarly, the centreline-decelerated flow (type 1) states are also unstable in the range $\unicode[STIX]{x1D714}_{\ast }<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{c}$ , while the centreline-accelerated flow (type 2) states are asymptotically stable when $\unicode[STIX]{x1D714}>\unicode[STIX]{x1D714}_{c}$ . The breakdown (type 3) states are found to be attractors of flow dynamics when $\unicode[STIX]{x1D714}>\unicode[STIX]{x1D714}_{\ast }$ . Therefore, the columnar states attract the long-term flow evolution of a perturbed columnar state when $0<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{c}$ while the breakdown states dominate the long-term evolution of a perturbed columnar flow as $\unicode[STIX]{x1D714}$ is increased above $\unicode[STIX]{x1D714}_{\ast }$ . In the overlap range $\unicode[STIX]{x1D714}_{\ast }<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{c}$ , the evolution is determined by the initial perturbations and the domains of attraction of the columnar and breakdown states, separated by the branch of type 1 states.

4.2 Case 2: $\unicode[STIX]{x1D70C}_{CL}=0.1$

With the increase of the base flow centreline density $\unicode[STIX]{x1D70C}_{CL}$ from near-zero values to 0.1, the flow behaviour changes from that of the calorically perfect gas found in Rusak et al. (Reference Rusak, Choi, Bourquard and Wang2015a ) since the base flow density and pressure achieve at $\unicode[STIX]{x1D714}>3.3$ values around and above 1 (the critical thermodynamic point) near the pipe wall at $y=1/2$ (figure 4 a). The streamfunction $\unicode[STIX]{x1D6F9}_{0}(y)$ remains monotonic with $y$ but increases its slope near the wall. It is important to also note that the density $\unicode[STIX]{x1D70C}_{0}(y)$ has a positive curvature for all $y$ when $\unicode[STIX]{x1D714}\leqslant 3.3$ . However, the high swirl causes $\unicode[STIX]{x1D70C}_{0}(y)$ to increase rapidly within the domain as it varies around 1 and to change its curvature when it is above 1, as $\unicode[STIX]{x1D714}$ increases above $3.3$ . This behaviour of $\unicode[STIX]{x1D70C}_{0}(y)$ can be seen directly from (3.1). The term $(8/3)(T_{0}/(1-(1/3)\unicode[STIX]{x1D70C}_{0})^{2})-6\unicode[STIX]{x1D70C}_{0}$ is positive for all $0<\unicode[STIX]{x1D70C}_{0}<3$ but decreases as $\unicode[STIX]{x1D70C}_{0}$ increases, reaches a minimum at $\unicode[STIX]{x1D70C}_{0}=1$ with near-zero value and then increases with increase of $\unicode[STIX]{x1D70C}_{0}$ above 1. As a result, at near-critical thermodynamic conditions, the derivative $\unicode[STIX]{x1D70C}_{0y}$ is positive, increases with $y$ as $\unicode[STIX]{x1D70C}_{0}$ tends towards 1, reaches a large maximum value when $\unicode[STIX]{x1D70C}_{0}=1$ , and then decreases with increase of $\unicode[STIX]{x1D70C}_{0}$ above 1. This behaviour of $\unicode[STIX]{x1D70C}_{0}$ strongly affects the functions $H^{\prime }(\unicode[STIX]{x1D6F9})$ and $s^{\prime }(\unicode[STIX]{x1D6F9})$ .

Figure 4.

Case 2: $\unicode[STIX]{x1D70C}_{CL}=0.1$ , $T_{0}=1.02$ , $Ma_{0}=0.5$ , $R/C_{v}=0.01$ . In panels (a,b) $\unicode[STIX]{x1D714}=2.5$ (solid lines), $\unicode[STIX]{x1D714}=3$ (dashed lines), $\unicode[STIX]{x1D714}=3.25$ (dotted lines) and $\unicode[STIX]{x1D714}=3.4$ (dash-dotted lines). In panels (c,d) the calorically perfect-gas bifurcation diagram is shown for reference (dotted line). (a) Solutions of the base columnar flow density $\unicode[STIX]{x1D70C}$ (black lines), pressure $p$ (blue lines) and streamfunction $\unicode[STIX]{x1D6F9}$ (red lines) as a function of $y$ ; (b) the functions $H^{\prime }(\unicode[STIX]{x1D6F9})$ (blue lines), $s^{\prime }(\unicode[STIX]{x1D6F9})$ (black lines) and $(3/8)(R/C_{v})Ma_{c}^{2}K(\unicode[STIX]{x1D6F9})K^{\prime }(\unicode[STIX]{x1D6F9})$ (red lines) as a function of $\unicode[STIX]{x1D6F9}$ ; (c) the bifurcation diagram of solutions of $w_{CL}$ against $\unicode[STIX]{x1D714}$ ; (d) the size of the breakdown zone $y_{0}$ as a function of $\unicode[STIX]{x1D714}$ for type 3 solutions.

The related function $(3/8)(R/C_{v})Ma_{c}^{2}K(\unicode[STIX]{x1D6F9})K^{\prime }(\unicode[STIX]{x1D6F9})$ is positive for the range of $\unicode[STIX]{x1D714}$ studied but reaches a maximum value around $\unicode[STIX]{x1D6F9}=0.06$ and then decreases with increase of $\unicode[STIX]{x1D6F9}$ towards $\unicode[STIX]{x1D6F9}_{w}$ (figure 4 b). The function $H^{\prime }(\unicode[STIX]{x1D6F9})$ decreases monotonically with $\unicode[STIX]{x1D6F9}$ for $\unicode[STIX]{x1D714}\leqslant 3.3$ and becomes negative in a certain range of $\unicode[STIX]{x1D6F9}$ . However, when $\unicode[STIX]{x1D714}=3.4$ , near- and above-critical thermodynamic properties exist near the wall (figure 4 a). Then, the function $H^{\prime }(\unicode[STIX]{x1D6F9})$ decreases significantly to negative values, reaches a minimum at $\unicode[STIX]{x1D6F9}=0.122$ , and increases as $\unicode[STIX]{x1D6F9}$ increases towards $\unicode[STIX]{x1D6F9}_{w}$ , yet exhibiting negative values near the wall. The function $s^{\prime }(\unicode[STIX]{x1D6F9})$ is negative for all $\unicode[STIX]{x1D714}$ studied and increases monotonically with $\unicode[STIX]{x1D6F9}$ when $\unicode[STIX]{x1D714}<3$ . For $3<\unicode[STIX]{x1D714}\leqslant 3.3$ , this function reaches a maximum value and then decreases with $\unicode[STIX]{x1D6F9}$ . However, when $\unicode[STIX]{x1D714}=3.4$ and near- and above-critical thermodynamic properties exist near the wall (figure 4 a), the function $s^{\prime }(\unicode[STIX]{x1D6F9})$ decreases significantly to negative values, reaches a minimum at $\unicode[STIX]{x1D6F9}=0.122$ , and then increases as $\unicode[STIX]{x1D6F9}$ increases towards $\unicode[STIX]{x1D6F9}_{w}$ , yet exhibiting negative values near the wall (figure 4 b). At the specific inlet conditions ( $T_{0}(y)=1.02=\text{const}.$ , $w_{0}(y)=1=\text{const}.$ ) and at near-critical thermodynamic conditions ( $\unicode[STIX]{x1D70C}_{0}\sim 1$ ), equations (3.6) and (3.7) show that the behaviour of the functions $H^{\prime }(\unicode[STIX]{x1D6F9})$ and $s^{\prime }(\unicode[STIX]{x1D6F9})$ is directly related to $-\unicode[STIX]{x1D70C}_{0y}$ . Therefore, when $\unicode[STIX]{x1D714}$ is increased and above-critical thermodynamic conditions appear near the wall, the functions $H^{\prime }(\unicode[STIX]{x1D6F9})$ and $s^{\prime }(\unicode[STIX]{x1D6F9})$ decrease as $\unicode[STIX]{x1D70C}_{0}$ tends towards 1, reach a large negative minimum when $\unicode[STIX]{x1D70C}_{0}=1$ , and then increase with increase of $\unicode[STIX]{x1D70C}_{0}$ above 1.

This significant change of the behaviour of the functions $(3/8)(R/C_{v})Ma_{c}^{2}K(\unicode[STIX]{x1D6F9})$ $K^{\prime }(\unicode[STIX]{x1D6F9})$ , $H^{\prime }(\unicode[STIX]{x1D6F9})$ and $s^{\prime }(\unicode[STIX]{x1D6F9})$ at $\unicode[STIX]{x1D714}>3.3$ affects the computed solutions of (3.11)–(3.13). The results are shown in the bifurcation diagram in figure 4(c). Again, the base type 0 branch of solutions exists for all $\unicode[STIX]{x1D714}$ . The type 1 and 2 solutions bifurcate at a slightly higher swirl level than that in the calorically perfect-gas case, $\unicode[STIX]{x1D714}_{c}=2.235$ . The type 3 breakdown solutions also start at a higher swirl ratio $\unicode[STIX]{x1D714}_{\ast }=2.02$ . However, there is a clear change in the behaviour of type 2 and type 3 solutions when $3.2<\unicode[STIX]{x1D714}<3.5$ , when near- and above-critical thermodynamic pressure and density are achieved near the wall. The type 2 states exhibit a slight decrease in $w_{CL}$ with a minimum at $\unicode[STIX]{x1D714}=3.4$ . Above $\unicode[STIX]{x1D714}=3.5$ , the minima of $H^{\prime }(\unicode[STIX]{x1D6F9})$ and $s^{\prime }(\unicode[STIX]{x1D6F9})$ move away from the wall into the bulk and become more negative. Then, the branch of type 2 states continues monotonically with the increase of $\unicode[STIX]{x1D714}$ above $3.5$ . The type 3 solutions exhibit a size of $y_{0}$ that increases with $\unicode[STIX]{x1D714}$ in the range $2.02<\unicode[STIX]{x1D714}<2.87$ , decreases in the range $2.87<\unicode[STIX]{x1D714}<3.4$ and then increases again when $\unicode[STIX]{x1D714}>3.4$ , see figure 4(d).

In order to shed additional physical insight into the nature of the various types of solutions, we describe in the phase diagram in figure 5 the variations of the pressure ( $p$ ) as a function of the specific volume ( $1/\unicode[STIX]{x1D70C}$ ) for the type 0, type 2 and type 3 solutions computed at a representative high swirl ratio, $\unicode[STIX]{x1D714}=3.4$ . Also shown in this figure is the saturation line as computed from Maxwell construction condition (Maxwell Reference Maxwell1875). Owing to the relatively low $R/C_{v}=0.01$ , the change of temperature from the constant $T_{0}=1.02$ is not significant for all three types of solutions. However, the changes of pressure and specific volume induced by the axial and circumferential velocities are visible, specifically at the high level of swirl ratio. The base type 0 solution (black solid line) ranges along the isothermal curve $T_{0}=1.02$ from $1/\unicode[STIX]{x1D70C}=10$ and $p=0.2514$ at the centreline, that is well inside the vapour region, to $1/\unicode[STIX]{x1D70C}=0.6681$ and $p=1.404$ at the wall, that is, in the thermodynamically supercritical dense-flow region. The type 2 solution (red solid line), with accelerated axial velocity near the centreline and decelerated velocity near the wall, exhibits as a result a decrease of pressure and an increase of specific volume near the centreline and an increase of pressure and slight decrease of specific volume near the wall. The type 3 solution (blue solid line), with decelerated axial velocity near the centreline and accelerated velocity near the wall, exhibits as a result an increase of pressure and a decrease of specific volume near the centreline and a decrease of pressure and slight increase of specific volume near the wall. We note that solutions for high swirl levels with $\unicode[STIX]{x1D714}>3.4$ exhibit a similar behaviour.

Figure 5.

Variations of pressure ( $p$ ) as a function of fluid specific volume ( $1/\unicode[STIX]{x1D70C}$ ) for type 0 (black solid line), type 2 (red solid line) and type 3 (blue solid line) solutions at inlet conditions of case 2: $\unicode[STIX]{x1D70C}_{CL}=0.1$ , $T_{0}=1.02$ , $Ma_{0}=0.5$ , $R/C_{v}=0.01$ and at constant $\unicode[STIX]{x1D714}=3.4$ . Also shown for reference is the saturation line (black dotted line) determined by the Maxwell construction condition.

Although not proven analytically, it is conjectured that the stability characteristics of the various types of the corresponding flow states in the finite-length pipe according to PDE (2.21) are similar to those found in the calorically perfect-gas case, proven analytically in Rusak et al. (Reference Rusak, Choi and Lee2007). To prove the same for the case of real gases requires a dedicated long study but we expect that results are continuous from the calorically perfect-gas limit to the behaviour of real gases. The critical state at $\unicode[STIX]{x1D714}_{c}$ is a change-of-stability point for the columnar (type 0) states, as well as between the centreline-decelerated (type 1) and the centreline-accelerated (type 2) states. Again, it is expected that the breakdown (type 3) states are attractors of flow dynamics. Therefore, for the case with $\unicode[STIX]{x1D70C}_{CL}=0.1$ , the columnar states attract the long-term flow evolution of a perturbed columnar state when $0<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{c}$ while the breakdown states dominate the long-term evolution of a perturbed columnar flow as $\unicode[STIX]{x1D714}$ is increased above  $\unicode[STIX]{x1D714}_{\ast }$ .

4.3 Case 3: $\unicode[STIX]{x1D70C}_{CL}=0.19$

With further increase of the base flow centreline density to $\unicode[STIX]{x1D70C}_{CL}=0.19$ , we find that the flow achieves near- and above-critical pressure and density conditions at swirl levels $\unicode[STIX]{x1D714}>2.7$ that are lower than those found in case 2 (see figure 6 a). The behaviour of the functions $(3/8)(R/C_{v})Ma_{c}^{2}K(\unicode[STIX]{x1D6F9})K^{\prime }(\unicode[STIX]{x1D6F9})$ , $H^{\prime }(\unicode[STIX]{x1D6F9})$ and $s^{\prime }(\unicode[STIX]{x1D6F9})$ is similar to those in case 2 but occurs at lower swirl levels (see figure 6 b). For example, the sharp minima of $H^{\prime }(\unicode[STIX]{x1D6F9})$ and $s^{\prime }(\unicode[STIX]{x1D6F9})$ at $\unicode[STIX]{x1D714}=2.8$ occur at $\unicode[STIX]{x1D6F9}=0.163$ . The behaviour of $\unicode[STIX]{x1D70C}_{0}(y)$ and its derivative $\unicode[STIX]{x1D70C}_{0y}$ with the increase of $\unicode[STIX]{x1D714}$ dominates the functions $H^{\prime }(\unicode[STIX]{x1D6F9})$ and $s^{\prime }(\unicode[STIX]{x1D6F9})$ in a way similar to that explained in case 2.

Figure 6.

Case 3: $\unicode[STIX]{x1D70C}_{CL}=0.19$ , $T_{0}=1.02$ , $Ma_{0}=0.5$ , $R/C_{v}=0.01$ . In panels (a,b) $\unicode[STIX]{x1D714}=2.3$ (solid lines), $\unicode[STIX]{x1D714}=2.5$ (dashed lines), $\unicode[STIX]{x1D714}=2.6$ (dotted lines) and $\unicode[STIX]{x1D714}=2.8$ (dash-dotted lines). In panels (c,d) the calorically perfect-gas bifurcation diagram is shown for reference (dotted line). (a) Solutions of the base columnar flow density $\unicode[STIX]{x1D70C}$ (black lines), pressure $p$ (blue lines) and streamfunction $\unicode[STIX]{x1D6F9}$ (red lines) as a function of $y$ ; (b) the functions $H^{\prime }(\unicode[STIX]{x1D6F9})$ (blue lines), $s^{\prime }(\unicode[STIX]{x1D6F9})$ (black lines) and $(3/8)(R/C_{v})Ma_{c}^{2}K(\unicode[STIX]{x1D6F9})K^{\prime }(\unicode[STIX]{x1D6F9})$ (red lines) as a function of $\unicode[STIX]{x1D6F9}$ ; (c) the bifurcation diagram of solutions of $w_{CL}$ against $\unicode[STIX]{x1D714}$ ; (d) the size of the breakdown zone $y_{0}$ as a function of $\unicode[STIX]{x1D714}$ for type 3 solutions.

As a result of the changes in these functions, the critical swirl levels increase to $\unicode[STIX]{x1D714}_{c}=2.262$ and $\unicode[STIX]{x1D714}_{\ast }=2.048$ (see figure 6 c). As before, the type 0 solutions are a base solution of (3.11)–(3.13) for all $\unicode[STIX]{x1D714}$ . The type 1 solutions develop in the range $\unicode[STIX]{x1D714}_{\ast }<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{c}$ . The type 2 solutions exhibit a further decrease of $w_{CL}$ with a minimum at a lower swirl, $\unicode[STIX]{x1D714}=2.686$ . The type 3 solutions start at $\unicode[STIX]{x1D714}_{\ast }=2.048$ , develop in the range $\unicode[STIX]{x1D714}_{\ast }<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{\ast 1}=2.726$ , then turn into a short branch of type 1 solutions in the range $\unicode[STIX]{x1D714}_{\ast 1}<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{\ast 2}=2.818$ and return to type 3 solutions when $\unicode[STIX]{x1D714}>\unicode[STIX]{x1D714}_{\ast 2}$ . The resulting size of $y_{0}$ of type 3 solutions is a function of $\unicode[STIX]{x1D714}$ in the ranges $\unicode[STIX]{x1D714}_{\ast }<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{\ast 1}$ and $\unicode[STIX]{x1D714}>\unicode[STIX]{x1D714}_{\ast 2}$ and is presented in figure 6(d).

The nature of the types 0, 2 and 3 of solutions at $\unicode[STIX]{x1D714}=2.7$ is described in the phase diagram in figure 7. Also shown in this figure is the saturation line according to Maxwell (Reference Maxwell1875). The base type 0 solution (black solid line) ranges along the isothermal curve $T_{0}=1.02$ from $1/\unicode[STIX]{x1D70C}=5.263$ and $p=0.4434$ at the centreline, that is well inside the vapour region, to $1/\unicode[STIX]{x1D70C}=0.7947$ and $p=1.145$ at the wall. The type 2 solution (red solid line), with accelerated axial velocity near the centreline and decelerated velocity near the wall, exhibits as a result a decrease of pressure and an increase of specific volume near the centreline and an increase of pressure and slight decrease of specific volume near the wall. The type 3 solution (blue solid line), with decelerated axial velocity near the centreline and accelerated velocity near the wall, exhibits as a result an increase of pressure and a decrease of specific volume near the centreline and a decrease of pressure and slight increase of specific volume near the wall. In this case, the type 3 breakdown solution is entirely in the vapour region since the swirl level is less than that in the previous case shown in figure 5.

Figure 7.

Variations of pressure ( $p$ ) as a function of fluid specific volume ( $1/\unicode[STIX]{x1D70C}$ ) for type 0 (black solid line), type 2 (red solid line) and type 3 (blue solid line) solutions at inlet conditions of case 3: $\unicode[STIX]{x1D70C}_{CL}=0.19$ , $T_{0}=1.02$ , $Ma_{0}=0.5$ , $(R/C_{v})=0.01$ and at constant $\unicode[STIX]{x1D714}=2.7$ . Also shown for reference is the saturation line (black dotted line) determined by the Maxwell construction condition.

Again, it is expected that the stability characteristics of the various types of the corresponding flow states in the finite-length pipe according to PDE (2.21) are similar to those found in the calorically perfect-gas case. The critical state at $\unicode[STIX]{x1D714}_{c}$ is a change-of-stability point for the columnar (type 0) states, as well as between the centreline-decelerated (type 1) and the centreline-accelerated (type 2) states. It is expected that the branch of breakdown (type 3) states, turning into the short centreline-decelerated (type 1) states and back into breakdown (type 3) states, are attractors of flow dynamics. Therefore, for the case with $\unicode[STIX]{x1D70C}_{CL}=0.19$ , the columnar states attract the long-term flow evolution of a perturbed columnar state when $0<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{c}$ while the branch of breakdown into centreline-decelerated into breakdown states dominates the long-term evolution of a perturbed columnar flow as $\unicode[STIX]{x1D714}$ is increased above  $\unicode[STIX]{x1D714}_{\ast }$ .

4.4 Case 4: $\unicode[STIX]{x1D70C}_{CL}=0.206$

With a small increase of the base flow centreline density to $\unicode[STIX]{x1D70C}_{CL}=0.206$ , the flow continues to exhibit near- and above-critical pressure and density at slightly lower swirl levels than those in case 3, when $\unicode[STIX]{x1D714}>2.57$ (see figure 8 a). The behaviour of the functions $(3/8)(R/C_{v})Ma_{c}^{2}K(\unicode[STIX]{x1D6F9})K^{\prime }(\unicode[STIX]{x1D6F9})$ , $H^{\prime }(\unicode[STIX]{x1D6F9})$ and $s^{\prime }(\unicode[STIX]{x1D6F9})$ is similar to those in case 3 but occurs at slightly lower swirl levels (see figure 8 b). For example, the minima of $H^{\prime }(\unicode[STIX]{x1D6F9})$ and $s^{\prime }(\unicode[STIX]{x1D6F9})$ at $\unicode[STIX]{x1D714}=2.57$ occur at the wall, and the sharp minima of these functions at $\unicode[STIX]{x1D714}=2.8$ occur in the bulk at $\unicode[STIX]{x1D6F9}=0.160$ . Again, as explained in case 2, the behaviour of $\unicode[STIX]{x1D70C}_{0}(y)$ and its derivative $\unicode[STIX]{x1D70C}_{0y}$ with the increase of $\unicode[STIX]{x1D714}$ dominates the functions $H^{\prime }(\unicode[STIX]{x1D6F9})$ and $s^{\prime }(\unicode[STIX]{x1D6F9})$ .

Figure 8.

Case 4: $\unicode[STIX]{x1D70C}_{CL}=0.206$ , $T_{0}=1.02$ , $Ma_{0}=0.5$ , $R/C_{v}=0.01$ . In panels (a,b) $\unicode[STIX]{x1D714}=2.3$ (solid lines), $\unicode[STIX]{x1D714}=2.5$ (dashed lines), $\unicode[STIX]{x1D714}=2.57$ (dotted lines) and $\unicode[STIX]{x1D714}=2.8$ (dash-dotted lines). In panels (c,d) the calorically perfect-gas bifurcation diagram is shown for reference (dotted line). (a) Solutions of the base columnar flow density $\unicode[STIX]{x1D70C}$ (black lines), pressure $p$ (blue lines) and streamfunction $\unicode[STIX]{x1D6F9}$ (red lines) as a function of $y$ ; (b) the functions $H^{\prime }(\unicode[STIX]{x1D6F9})$ (blue lines), $s^{\prime }(\unicode[STIX]{x1D6F9})$ (black lines) and $(3/8)(R/C_{v})Ma_{c}^{2}K(\unicode[STIX]{x1D6F9})K^{\prime }(\unicode[STIX]{x1D6F9})$ (red lines) as a function of $\unicode[STIX]{x1D6F9}$ ; (c) the bifurcation diagram of solutions of $w_{CL}$ against $\unicode[STIX]{x1D714}$ ; (d) the size of the breakdown zone $y_{0}$ as a function of $\unicode[STIX]{x1D714}$ for type 3 solutions.

As a result of the changes in these functions, the critical swirl levels increase to $\unicode[STIX]{x1D714}_{c}=2.273$ and $\unicode[STIX]{x1D714}_{\ast }=2.054$ (see figure 8 c). As before, the type 0 solutions form a base solution of (3.11)–(3.13) for all $\unicode[STIX]{x1D714}$ . The type 1 solutions develop in the range  $\unicode[STIX]{x1D714}_{\ast }<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{c}$ . The type 2 solutions exhibit a further decrease of $w_{CL}$ with respect to case 3. The centreline axial velocity $w_{CL}$ of type 2 solutions increases in the range $\unicode[STIX]{x1D714}_{c}<\unicode[STIX]{x1D714}<2.500$ , then decreases in the range $2.500<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{c1}=2.585$ , turns into type 1 solutions in the range $\unicode[STIX]{x1D714}_{c1}<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{c2}=2.646$ with a minimum at $\unicode[STIX]{x1D714}=2.621$ , and then turns back into type 2 solutions and increases when $\unicode[STIX]{x1D714}>\unicode[STIX]{x1D714}_{c2}$ . The type 3 solutions start at $\unicode[STIX]{x1D714}_{\ast }=2.054$ , develop in the range $\unicode[STIX]{x1D714}_{\ast }<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{\ast 1}=2.634$ , then turn into a short branch of type 1 solutions in the range $\unicode[STIX]{x1D714}_{\ast 1}<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{\ast 2}=2.774$ and return to type 3 solutions when $\unicode[STIX]{x1D714}>\unicode[STIX]{x1D714}_{\ast 2}$ . With respect to case 3, shown in figure 6(c), the minimum and maximum of $w_{CL}$ for type 1 solutions approach each other. The resulting size of $y_{0}$ of type 3 solutions is a function of $\unicode[STIX]{x1D714}$ in the ranges $\unicode[STIX]{x1D714}_{\ast }<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{\ast 1}$ and $\unicode[STIX]{x1D714}>\unicode[STIX]{x1D714}_{\ast 2}$ and is presented in figure 8(d).

The following stability characteristics of the various types of flow states in the finite-length pipe in case 4 are expected. The critical state at $\unicode[STIX]{x1D714}_{c}$ is a change-of-stability point for the columnar (type 0) states, as well as between the unstable centreline-decelerated (type 1) and the asymptotically stable centreline-accelerated (type 2) states bifurcating at $\unicode[STIX]{x1D714}_{c}$ . The critical state at $\unicode[STIX]{x1D714}_{c1}$ is also a point of change of stability where the unstable columnar (type 0) states become asymptotically stable as $\unicode[STIX]{x1D714}$ is increased from below to above $\unicode[STIX]{x1D714}_{c1}$ in the range $\unicode[STIX]{x1D714}_{c1}<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{c2}$ . In addition, the state at $\unicode[STIX]{x1D714}_{c1}$ is a change-of-stability point from the asymptotically stable centreline-accelerated (type 2) flow states at $\unicode[STIX]{x1D714}_{c}<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{c1}$ to the unstable centreline-decelerated (type 1) flow states in the range $\unicode[STIX]{x1D714}_{c1}<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{c2}$ . Similarly, the state at $\unicode[STIX]{x1D714}_{c2}$ is another of point of change of stability where the columnar (type 0) states again become unstable when $\unicode[STIX]{x1D714}>\unicode[STIX]{x1D714}_{c2}$ and the unstable centreline-decelerated (type 1) states turn into asymptotically stable centreline-accelerated (type 2) flow states as $\unicode[STIX]{x1D714}$ increases above $\unicode[STIX]{x1D714}_{c2}$ . It is also expected that the branch of breakdown (type 3) states, turning into the short branch of centreline-decelerated (type 1) states and back into breakdown (type 3) states, are attractors of flow dynamics. Therefore, for the case with $\unicode[STIX]{x1D70C}_{CL}=0.206$ , the columnar states attract the long-term flow evolution of a perturbed columnar state when $0<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{c}$ while the branch of breakdown into centreline-decelerated into breakdown states dominates the long-term evolution of a perturbed columnar flow as $\unicode[STIX]{x1D714}$ is increased above  $\unicode[STIX]{x1D714}_{\ast }$ .

4.5 Case 5: $\unicode[STIX]{x1D70C}_{CL}=0.22$

With a further small increase of the base flow centreline density to $\unicode[STIX]{x1D70C}_{CL}=0.22$ , the flow continues to exhibit near- and above-critical pressure and density at slightly lower swirl levels than those in case 4, when $\unicode[STIX]{x1D714}>2.5$ (see figure 9 a). The function $\unicode[STIX]{x1D70C}_{0y}$ exhibits an increase to a large maximum value when near-critical thermodynamic conditions are achieved. This density derivative dominates the behaviour of the functions $(3/8)(R/C_{v})Ma_{c}^{2}K(\unicode[STIX]{x1D6F9})K^{\prime }(\unicode[STIX]{x1D6F9})$ , $H^{\prime }(\unicode[STIX]{x1D6F9})$ and $s^{\prime }(\unicode[STIX]{x1D6F9})$ in a similar way to those in cases 2, 3 and 4 but occurs at slightly lower swirl levels (see figure 9 b). For example, the minima of $H^{\prime }(\unicode[STIX]{x1D6F9})$ and $s^{\prime }(\unicode[STIX]{x1D6F9})$ at $\unicode[STIX]{x1D714}=2.5$ occur at the wall, and the sharp minima of these functions at $\unicode[STIX]{x1D714}=2.6$ occur in the bulk at $\unicode[STIX]{x1D6F9}=0.182$ .

Figure 9.

Case 5: $\unicode[STIX]{x1D70C}_{CL}=0.22$ , $T_{0}=1.02$ , $Ma_{0}=0.5$ , $R/C_{v}=0.01$ . In panels (a,b) $\unicode[STIX]{x1D714}=2.2$ (solid lines), $\unicode[STIX]{x1D714}=2.4$ (dashed lines), $\unicode[STIX]{x1D714}=2.5$ (dotted lines) and $\unicode[STIX]{x1D714}=2.6$ (dash-dotted lines). In panels (c,d) the calorically perfect-gas bifurcation diagram is shown for reference (dotted line). (a) Solutions of the base columnar flow density $\unicode[STIX]{x1D70C}$ (black lines), pressure $p$ (blue lines) and streamfunction $\unicode[STIX]{x1D6F9}$ (red lines) as a function of $y$ ; (b) the functions $H^{\prime }(\unicode[STIX]{x1D6F9})$ (blue lines), $s^{\prime }(\unicode[STIX]{x1D6F9})$ (black lines) and $(3/8)(R/C_{v})Ma_{c}^{2}K(\unicode[STIX]{x1D6F9})K^{\prime }(\unicode[STIX]{x1D6F9})$ (red lines) as a function of $\unicode[STIX]{x1D6F9}$ ; (c) the bifurcation diagram of solutions of $w_{CL}$ against $\unicode[STIX]{x1D714}$ ; (d) the size of the breakdown zone $y_{0}$ as a function of $\unicode[STIX]{x1D714}$ for type 3 solutions.

As a result of the changes in these functions, the critical swirl levels continue to increase to $\unicode[STIX]{x1D714}_{c}=2.289$ and $\unicode[STIX]{x1D714}_{\ast }=2.062$ (see figure 9 c). As before, the type 0 solutions form a base solution of (3.11)–(3.13) for all $\unicode[STIX]{x1D714}$ . The type 1 solutions develop in the range $\unicode[STIX]{x1D714}_{\ast }<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{c}$ . The centreline axial velocity $w_{CL}$ of type 2 solutions increases in the range $\unicode[STIX]{x1D714}_{c}<\unicode[STIX]{x1D714}<2.424$ , then decreases in the range $2.424<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{c1}=2.505$ and turns into type 1 solutions in the range $\unicode[STIX]{x1D714}_{c1}<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{\ast 1}=2.558$ . Additional bifurcation of type 1 and type 2 solutions occurs at $\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}_{c2}=2.616$ . The additional type 1 solutions develop with the decrease of $\unicode[STIX]{x1D714}$ from $\unicode[STIX]{x1D714}_{c2}$ to a fold point at $\unicode[STIX]{x1D714}_{f}=2.611$ and then turn back and develop with a decreasing $w_{CL}$ to zero in the range $\unicode[STIX]{x1D714}_{f}<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{\ast 2}=2.734$ . The additional type 2 solutions develop with an increasing $w_{CL}$ above unity in the range $\unicode[STIX]{x1D714}>\unicode[STIX]{x1D714}_{c2}$ . The type 3 solutions start at $\unicode[STIX]{x1D714}_{\ast }=2.062$ , develop in the small range $\unicode[STIX]{x1D714}_{\ast }<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{\ast 1}=2.558$ , and then reappear from the additional type 1 solutions when $\unicode[STIX]{x1D714}>\unicode[STIX]{x1D714}_{\ast 2}$ . The resulting size of $y_{0}$ of type 3 solutions is a function of $\unicode[STIX]{x1D714}$ in the ranges $\unicode[STIX]{x1D714}_{\ast }<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{\ast 1}$ and $\unicode[STIX]{x1D714}>\unicode[STIX]{x1D714}_{\ast 2}$ and is presented in figure 9(d). Essentially, the two separated branches of type 1 solutions in case 4, both at $\unicode[STIX]{x1D714}$ around 2.6 (see figure 8 c), are merged in case 5 to form two distinct branches of solutions with a small but finite gap of swirl ratio between them, where only the columnar states exist in the range $2.558<\unicode[STIX]{x1D714}<2.611$ .

It is expected that the various types of flow states in the finite-length pipe in case 5 are characterized by the following stability characteristics. The columnar type 0 states are asymptotically stable in the ranges $0<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{c}$ and $\unicode[STIX]{x1D714}_{c1}<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{c2}$ and unstable when $\unicode[STIX]{x1D714}_{c}<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{c1}$ and $\unicode[STIX]{x1D714}>\unicode[STIX]{x1D714}_{c2}$ . The centreline-decelerated (type 1) states are unstable in the ranges $\unicode[STIX]{x1D714}_{\ast }<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{c}$ , $\unicode[STIX]{x1D714}_{c1}<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{\ast 1}$ and $\unicode[STIX]{x1D714}_{f}<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{c2}$ and are asymptotically stable after the fold point when $\unicode[STIX]{x1D714}_{f}<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{\ast 2}$ . The centreline-accelerated (type 2) states in the ranges $\unicode[STIX]{x1D714}_{c}<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{c1}$ and $\unicode[STIX]{x1D714}>\unicode[STIX]{x1D714}_{c2}$ are asymptotically stable. The breakdown (type 3) states that appear when $\unicode[STIX]{x1D714}_{\ast }<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{\ast 1}$ and when $\unicode[STIX]{x1D714}>\unicode[STIX]{x1D714}_{\ast 2}$ are expected to be attractors of the flow dynamics. Therefore, for the case with $\unicode[STIX]{x1D70C}_{CL}=0.22$ , the columnar states attract the long-term flow evolution of a perturbed columnar state when $0<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{c}$ and $\unicode[STIX]{x1D714}_{c1}<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{c2}$ . The branch of centreline-decelerated flow states in the range $\unicode[STIX]{x1D714}_{f}<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{\ast 2}$ and the branches of breakdown states when $\unicode[STIX]{x1D714}_{\ast }<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{\ast 1}$ and $\unicode[STIX]{x1D714}>\unicode[STIX]{x1D714}_{\ast 2}$ dominate the long-term evolution of a perturbed columnar flow.

4.6 Case 6: $\unicode[STIX]{x1D70C}_{CL}=0.26$

For the case with $\unicode[STIX]{x1D70C}_{CL}=0.26$ , the flow exhibits near- and above-critical pressure and density at slightly lower swirl levels than those in case 5, when $\unicode[STIX]{x1D714}>2.3$ (see figure 10 a). As in previous cases, the behaviour of the functions $(3/8)(R/C_{v})Ma_{c}^{2}K(\unicode[STIX]{x1D6F9})K^{\prime }(\unicode[STIX]{x1D6F9})$ , $H^{\prime }(\unicode[STIX]{x1D6F9})$ and $s^{\prime }(\unicode[STIX]{x1D6F9})$ is dominated by $\unicode[STIX]{x1D70C}_{0y}$ and is similar to those in cases 2–5 but occurs at even lower swirl levels (see figure 10 b). For example, the minima of $H^{\prime }(\unicode[STIX]{x1D6F9})$ and $s^{\prime }(\unicode[STIX]{x1D6F9})$ at $\unicode[STIX]{x1D714}=2.33$ occur at the wall, and the sharp minima of these functions at $\unicode[STIX]{x1D714}=2.55$ occur in the bulk at $\unicode[STIX]{x1D6F9}=0.180$ .

Figure 10.

Case 6: $\unicode[STIX]{x1D70C}_{CL}=0.26$ , $T_{0}=1.02$ , $Ma_{0}=0.5$ , $R/C_{v}=0.01$ . In panels (a,b) $\unicode[STIX]{x1D714}=2.2$ (solid lines), $\unicode[STIX]{x1D714}=2.3$ (dashed lines), $\unicode[STIX]{x1D714}=2.33$ (dotted lines) and $\unicode[STIX]{x1D714}=2.55$ (dash-dotted lines). In panels (c,d) the calorically perfect-gas bifurcation diagram is shown for reference (dotted line). (a) Solutions of the base columnar flow density $\unicode[STIX]{x1D70C}$ (black lines), pressure $p$ (blue lines) and streamfunction $\unicode[STIX]{x1D6F9}$ (red lines) as a function of $y$ ; (b) the functions $H^{\prime }(\unicode[STIX]{x1D6F9})$ (blue lines), $s^{\prime }(\unicode[STIX]{x1D6F9})$ (black lines) and $(3/8)(R/C_{v})Ma_{c}^{2}K(\unicode[STIX]{x1D6F9})K^{\prime }(\unicode[STIX]{x1D6F9})$ (red lines) as a function of $\unicode[STIX]{x1D6F9}$ ; (c) the bifurcation diagram of solutions of $w_{CL}$ against $\unicode[STIX]{x1D714}$ ; (d) the size of the breakdown zone $y_{0}$ as a function of $\unicode[STIX]{x1D714}$ for type 3 solutions.

As a result of the changes in these functions, the critical swirl level of the columnar state jumps to $\unicode[STIX]{x1D714}_{c}=2.514$ and there are no $\unicode[STIX]{x1D714}_{c1}$ and $\unicode[STIX]{x1D714}_{c2}$ in this case. Actually, $\unicode[STIX]{x1D714}_{c}$ in this case is connected to $\unicode[STIX]{x1D714}_{c2}$ from case 5. Also, $\unicode[STIX]{x1D714}_{\ast }=2.096$ , $\unicode[STIX]{x1D714}_{\ast 1}=2.353$ , $\unicode[STIX]{x1D714}_{\ast 2}=2.624$ and $\unicode[STIX]{x1D714}_{f}=2.511$ (see figure 10 c). The type 0 solutions form a base solution of (3.11)–(3.13) for all $\unicode[STIX]{x1D714}$ . The type 1 solutions develop with a strict $w_{CL}<1$ in the range $\unicode[STIX]{x1D714}_{\ast }<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{\ast 1}$ . Additional type 1 solutions bifurcate at $\unicode[STIX]{x1D714}_{c}$ to a lower $\unicode[STIX]{x1D714}$ with a fold at $\unicode[STIX]{x1D714}_{f}$ . Then, the branch of type 1 solutions continues with decreased values of $w_{CL}$ to zero when $\unicode[STIX]{x1D714}$ increases from $\unicode[STIX]{x1D714}_{f}$ to $\unicode[STIX]{x1D714}_{\ast 2}$ . Type 2 solutions also bifurcate at $\unicode[STIX]{x1D714}_{c}$ and develop with an increased $w_{CL}>1$ when $\unicode[STIX]{x1D714}>\unicode[STIX]{x1D714}_{c}$ . The type 3 solutions start at $\unicode[STIX]{x1D714}_{\ast }$ , develop in the small range $\unicode[STIX]{x1D714}_{\ast }<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{\ast 1}$ , and then reappear from the additional type 1 solutions when $\unicode[STIX]{x1D714}>\unicode[STIX]{x1D714}_{\ast 2}$ . The resulting size of $y_{0}$ of type 3 solutions is a function of $\unicode[STIX]{x1D714}$ in the ranges $\unicode[STIX]{x1D714}_{\ast }<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{\ast 1}$ and $\unicode[STIX]{x1D714}>\unicode[STIX]{x1D714}_{\ast 2}$ and is presented in figure 10(d). Essentially, the gap between the two separated branches in case 5 (see figure 9 c) is increased while the first branch is suppressed and there are only the columnar flow states in the range $\unicode[STIX]{x1D714}_{\ast 2}<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{f}$ .

The expected stability characteristics of the various types of flow states in the finite-length pipe in case 6 are as follows. The columnar type 0 states are asymptotically stable in the range $0<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{c}$ and unstable when $\unicode[STIX]{x1D714}>\unicode[STIX]{x1D714}_{c}$ . The centreline-decelerated (type 1) states are unstable in the ranges $\unicode[STIX]{x1D714}_{\ast }<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{\ast 1}$ and $\unicode[STIX]{x1D714}_{f}<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{c}$ and are asymptotically stable after the fold point when $\unicode[STIX]{x1D714}_{f}<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{\ast 2}$ . The centreline-accelerated (type 2) states in the range $\unicode[STIX]{x1D714}>\unicode[STIX]{x1D714}_{c}$ are asymptotically stable. The breakdown (type 3) states that appear when $\unicode[STIX]{x1D714}_{\ast }<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{\ast 1}$ and when $\unicode[STIX]{x1D714}>\unicode[STIX]{x1D714}_{\ast 2}$ are expected to be attractors of the flow dynamics. Therefore, for the case with $\unicode[STIX]{x1D70C}_{CL}=0.26$ , the columnar states attract the long-term flow evolution of a perturbed columnar state when $0<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{c}$ . The branch of centreline-decelerated flow states in the range $\unicode[STIX]{x1D714}_{f}<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{\ast 2}$ and the branches of breakdown states when $\unicode[STIX]{x1D714}_{\ast }<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{\ast 1}$ and $\unicode[STIX]{x1D714}>\unicode[STIX]{x1D714}_{\ast 2}$ dominate the long-term evolution of a perturbed columnar flow.

4.7 Case 7: $\unicode[STIX]{x1D70C}_{CL}=0.3$

With the increase of $\unicode[STIX]{x1D70C}_{CL}$ to 0.3, the flow exhibits near- and above-critical pressure and density inside the pipe at similar swirl levels to those in case 6, when $\unicode[STIX]{x1D714}>2.3$ (see figure 11 a). The functions $(3/8)(R/C_{v})Ma_{c}^{2}K(\unicode[STIX]{x1D6F9})K^{\prime }(\unicode[STIX]{x1D6F9})$ , $H^{\prime }(\unicode[STIX]{x1D6F9})$ and $s^{\prime }(\unicode[STIX]{x1D6F9})$ are similar to those in cases 2–6 (see figure 11 b) and again they are dominated by the base density derivative $\unicode[STIX]{x1D70C}_{0y}$ . For example, the minima of $H^{\prime }(\unicode[STIX]{x1D6F9})$ and $s^{\prime }(\unicode[STIX]{x1D6F9})$ shift from the wall into the bulk with increase of  $\unicode[STIX]{x1D714}$ .

Figure 11.

Case 7: $\unicode[STIX]{x1D70C}_{CL}=0.3$ , $T_{0}=1.02$ , $Ma_{0}=0.5$ , $R/C_{v}=0.01$ . In panels (a,b) $\unicode[STIX]{x1D714}=2.4$ (solid lines), $\unicode[STIX]{x1D714}=2.6$ (dashed lines) and $\unicode[STIX]{x1D714}=2.8$ (dotted lines). In panels (c,d) the calorically perfect-gas bifurcation diagram is shown for reference (dotted line). (a) Solutions of the base columnar flow density $\unicode[STIX]{x1D70C}$ (black lines), pressure $p$ (blue lines) and streamfunction $\unicode[STIX]{x1D6F9}$ (red lines) as a function of $y$ ; (b) the functions $H^{\prime }(\unicode[STIX]{x1D6F9})$ (blue lines), $s^{\prime }(\unicode[STIX]{x1D6F9})$ (black lines) and $(3/8)(R/C_{v})Ma_{c}^{2}K(\unicode[STIX]{x1D6F9})K^{\prime }(\unicode[STIX]{x1D6F9})$ (red lines) as a function of $\unicode[STIX]{x1D6F9}$ ; (c) the bifurcation diagram of solutions of $w_{CL}$ against $\unicode[STIX]{x1D714}$ ; (d) the size of the breakdown zone $y_{0}$ as a function of $\unicode[STIX]{x1D714}$ for type 3 solutions.

As a result of the changes in these functions, the critical swirl level of the columnar state decreases with respect to case 6 to $\unicode[STIX]{x1D714}_{c}=2.421$ . Also, $\unicode[STIX]{x1D714}_{\ast }=2.523$ and decreases with respect to $\unicode[STIX]{x1D714}_{\ast 2}=2.624$ in case 6. There is a fold point at $\unicode[STIX]{x1D714}_{f}=2.417$ that is slightly below $\unicode[STIX]{x1D714}_{c}$ (see figure 11 c). The type 0 solutions form a base solution of (3.11)–(3.13) for all $\unicode[STIX]{x1D714}$ . The type 1 solutions bifurcate at $\unicode[STIX]{x1D714}_{c}$ to a lower $\unicode[STIX]{x1D714}$ with a fold at $\unicode[STIX]{x1D714}_{f}$ and then turn to higher swirl levels with $w_{CL}$ decreasing to zero in the range $\unicode[STIX]{x1D714}_{f}<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{\ast }$ . Type 2 solutions also bifurcate at $\unicode[STIX]{x1D714}_{c}$ and develop with increased values of $w_{CL}>1$ when $\unicode[STIX]{x1D714}>\unicode[STIX]{x1D714}_{c}$ . The type 3 solutions develop when $\unicode[STIX]{x1D714}>\unicode[STIX]{x1D714}_{\ast }$ . The resulting size of $y_{0}$ of type 3 solutions as a function of $\unicode[STIX]{x1D714}$ is presented in figure 11(d). The values of $y_{0}$ grow with $\unicode[STIX]{x1D714}>\unicode[STIX]{x1D714}_{\ast }$ . Figure 12 shows examples of type 0, 2 and 3 solutions at $\unicode[STIX]{x1D714}=3$ in the phase diagram $p$ versus $1/\unicode[STIX]{x1D70C}$ . Solutions show a similar behaviour to those described in figures 5 and 7.

Figure 12.

Variations of pressure ( $p$ ) as a function of fluid specific volume ( $1/\unicode[STIX]{x1D70C}$ ) for type 0 (black solid line), type 2 (red solid line) and type 3 (blue solid line) solutions at inlet conditions of case 7: $\unicode[STIX]{x1D70C}_{CL}=0.3$ , $T_{0}=1.02$ , $Ma_{0}=0.5$ , $R/C_{v}=0.01$ and at constant $\unicode[STIX]{x1D714}=3$ . Also shown for reference is the saturation line (black dotted line) determined by the Maxwell construction condition.

The various types of flow states in the finite-length pipe in case 7 are characterized by the following expected stability characteristics. The columnar type 0 states are asymptotically stable in the ranges $0<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{c}$ and unstable when $\unicode[STIX]{x1D714}>\unicode[STIX]{x1D714}_{c}$ . The centreline-decelerated (type 1) states bifurcating at $\unicode[STIX]{x1D714}_{c}$ are unstable in the range $\unicode[STIX]{x1D714}_{f}<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{c}$ and are asymptotically stable after the fold point when $\unicode[STIX]{x1D714}_{f}<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{\ast }$ . The centreline-accelerated (type 2) states in the range $\unicode[STIX]{x1D714}>\unicode[STIX]{x1D714}_{c}$ are also asymptotically stable. The breakdown (type 3) states that appear when $\unicode[STIX]{x1D714}>\unicode[STIX]{x1D714}_{\ast }$ are expected to be attractors of the flow dynamics. Therefore, for the case with $\unicode[STIX]{x1D70C}_{CL}=0.3$ , the columnar states attract the long-term flow evolution of a perturbed columnar state when $0<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{c}$ while the branches of centreline-decelerated flow states turning into breakdown states dominate the long-term evolution of a perturbed columnar flow when $\unicode[STIX]{x1D714}>\unicode[STIX]{x1D714}_{f}$ .

4.8 Case 8: $\unicode[STIX]{x1D70C}_{CL}=0.5$

The case of $\unicode[STIX]{x1D70C}_{CL}=0.5$ exhibits a similar behaviour to that of $\unicode[STIX]{x1D70C}_{CL}=0.3$ . The pressure and density become near- and above-critical thermodynamic values inside the bulk at even lower swirl levels (figure 13 a). The various functions in (3.11) are described in figure 13(b). The minima of $H^{\prime }(\unicode[STIX]{x1D6F9})$ and $s^{\prime }(\unicode[STIX]{x1D6F9})$ shift correspondingly away from the wall into the bulk with the increase of swirl level. In this case, the critical swirl levels decrease with respect to case 7, $\unicode[STIX]{x1D714}_{c}=2.202$ , $\unicode[STIX]{x1D714}_{\ast }=2.135$ and $\unicode[STIX]{x1D714}_{f}=2.095$ (see figure 13 c,d). The expected stability characteristics of the various types of flow states in the finite-length pipe in case 8 are also similar to those in case 7.

Figure 13.

Case 8: $\unicode[STIX]{x1D70C}_{CL}=0.5$ , $T_{0}=1.02$ , $Ma_{0}=0.5$ , $R/C_{v}=0.01$ . In panels (a,b) $\unicode[STIX]{x1D714}=2.3$ (solid lines), $\unicode[STIX]{x1D714}=2.6$ (dashed lines) and $\unicode[STIX]{x1D714}=3$ (dotted lines). In panels (c,d) the calorically perfect-gas bifurcation diagram is shown for reference (dotted line). (a) Solutions of the base columnar flow density $\unicode[STIX]{x1D70C}$ (black lines), pressure $p$ (blue lines) and streamfunction $\unicode[STIX]{x1D6F9}$ (red lines) as a function of $y$ ; (b) the functions $H^{\prime }(\unicode[STIX]{x1D6F9})$ (blue lines), $s^{\prime }(\unicode[STIX]{x1D6F9})$ (black lines) and $(3/8)(R/C_{v})Ma_{c}^{2}K(\unicode[STIX]{x1D6F9})K^{\prime }(\unicode[STIX]{x1D6F9})$ (red lines) as a function of $\unicode[STIX]{x1D6F9}$ ; (c) the bifurcation diagram of solutions of $w_{CL}$ against $\unicode[STIX]{x1D714}$ ; (d) the size of the breakdown zone $y_{0}$ as a function of $\unicode[STIX]{x1D714}$ for type 3 solutions.

We also note that the cases of $\unicode[STIX]{x1D70C}_{CL}=0.6,0.7$ exhibit a similar behaviour to that of $\unicode[STIX]{x1D70C}_{CL}=0.5$ but with slightly decreasing values of $\unicode[STIX]{x1D714}_{c}$ and $\unicode[STIX]{x1D714}_{\ast }$ with the increase of $\unicode[STIX]{x1D70C}_{CL}$ .

4.9 Case 9: $\unicode[STIX]{x1D70C}_{CL}=1$

With the increase of $\unicode[STIX]{x1D70C}_{CL}$ above 0.7 the flow dynamics changes; the base flow thermodynamic properties are near and above critical in the entire pipe domain. Then, the various functions $(3/8)(R/C_{v})Ma_{c}^{2}K(\unicode[STIX]{x1D6F9})K^{\prime }(\unicode[STIX]{x1D6F9})$ , $H^{\prime }(\unicode[STIX]{x1D6F9})$ and $s^{\prime }(\unicode[STIX]{x1D6F9})$ become nearly monotonic for most of $0<\unicode[STIX]{x1D6F9}<\unicode[STIX]{x1D6F9}_{w}$ . This is similar to the behaviour of these functions in the calorically perfect-gas case, compare figures 14(b) and 3(b). This results in the merging of the fold point in the bifurcation diagram with $\unicode[STIX]{x1D714}_{\ast }$ and both $\unicode[STIX]{x1D714}_{c}$ and $\unicode[STIX]{x1D714}_{\ast }$ decrease with the increase of $\unicode[STIX]{x1D70C}_{CL}$ above 0.7. For example, the case of $\unicode[STIX]{x1D70C}_{CL}=1$ is shown in figure 14. In this case, the base flow density and pressure are above thermodynamic critical values all across the pipe from the centreline to the wall and for all swirl ratio levels (figure 14 a). The functions $(3/8)(R/C_{v})Ma_{c}^{2}K(\unicode[STIX]{x1D6F9})K^{\prime }(\unicode[STIX]{x1D6F9})$ , $H^{\prime }(\unicode[STIX]{x1D6F9})$ and $s^{\prime }(\unicode[STIX]{x1D6F9})$ are monotonic for all $\unicode[STIX]{x1D6F9}$ and $\unicode[STIX]{x1D714}$ (figure 14 b). The bifurcation diagrams of $w_{CL}$ and $y_{0}$ as a function of $\unicode[STIX]{x1D714}$ are similar to that of the calorically perfect-gas case (figure 14 c,d). Figure 15 shows examples of type 0, 2 and 3 solutions at $\unicode[STIX]{x1D714}=2.5$ in the phase diagram $p$ versus $1/\unicode[STIX]{x1D70C}$ . Solutions show a similar behaviour to those described in figures 5 and 7. However, in this case, pressure and density vary close to the critical thermodynamic point and pressure is above critical for the three solutions.

Figure 14.

Case 9: $\unicode[STIX]{x1D70C}_{CL}=1$ , $T_{0}=1.02$ , $Ma_{0}=0.5$ , $R/C_{v}=0.01$ . In panels (a,b) $\unicode[STIX]{x1D714}=2.3$ (solid lines), $\unicode[STIX]{x1D714}=3$ (dashed lines) and $\unicode[STIX]{x1D714}=4$ (dotted lines). In panels (c,d) the calorically perfect-gas bifurcation diagram is shown for reference (dotted line). (a) Solutions of the base columnar flow density $\unicode[STIX]{x1D70C}$ (black lines), pressure $p$ (blue lines) and streamfunction $\unicode[STIX]{x1D6F9}$ (red lines) as a function of $y$ ; (b) the functions $H^{\prime }(\unicode[STIX]{x1D6F9})$ (blue lines), $s^{\prime }(\unicode[STIX]{x1D6F9})$ (black lines) and $(3/8)(R/C_{v})Ma_{c}^{2}K(\unicode[STIX]{x1D6F9})K^{\prime }(\unicode[STIX]{x1D6F9})$ (red lines) as a function of $\unicode[STIX]{x1D6F9}$ ; (c) the bifurcation diagram of solutions of $w_{CL}$ against $\unicode[STIX]{x1D714}$ ; (d) the size of the breakdown zone $y_{0}$ as a function of $\unicode[STIX]{x1D714}$ for type 3 solutions.

Figure 15.

Variations of pressure ( $p$ ) as a function of fluid specific volume ( $1/\unicode[STIX]{x1D70C}$ ) for type 0 (black solid line), type 2 (red solid line), and type 3 (blue solid line) solutions at inlet conditions of case 9: $\unicode[STIX]{x1D70C}_{CL}=1.0$ , $T_{0}=1.02$ , $Ma_{0}=0.5$ , $R/C_{v}=0.01$ and at constant $\unicode[STIX]{x1D714}=2.5$ . Also shown for reference is the saturation line (black dotted line) determined by the Maxwell construction condition.

4.10 Summary of computed results for flows with $R/C_{v}=0.01$ , $T_{0}(y)=1.02$ and $Ma_{0}=0.5$

We conducted additional similar computations for flow cases with $0<\unicode[STIX]{x1D70C}_{CL}\leqslant 1$ and identified the various critical swirl levels of a compressible swirling flow of a van der Waals fluid in a straight pipe when $R/C_{v}=0.01$ , $Ma_{0}=0.5$ , and the inlet is given by a solid-body rotation $K_{0}(y)=2y$ with a uniform axial velocity $w_{0}(y)=1$ and temperature $T_{0}(y)=1.02$ . The results are summarized in figure 16. Figure 16(a) describes the combined line of increasing $\unicode[STIX]{x1D714}_{c}$ with $\unicode[STIX]{x1D70C}_{CL}$ in the range $0<\unicode[STIX]{x1D70C}_{CL}<0.237$ , followed by the line of increasing $\unicode[STIX]{x1D714}_{c1}$ with decrease of $\unicode[STIX]{x1D70C}_{CL}$ from 0.237 to 0.199, connected to the line of decreasing $\unicode[STIX]{x1D714}_{c2}$ with $\unicode[STIX]{x1D70C}_{CL}$ in the range from 0.199 to 0.237, and then connected to the line of decreasing $\unicode[STIX]{x1D714}_{c}$ with $\unicode[STIX]{x1D70C}_{CL}$ when $\unicode[STIX]{x1D70C}_{CL}>0.237$ . Along this combined complicated line, the columnar (type 0) state in a finite-length straight pipe is expected to be neutrally stable. This neutral stability line separates regions in the $\unicode[STIX]{x1D714}$ versus $\unicode[STIX]{x1D70C}_{CL}$ operational diagram where the columnar state is asymptotically stable and where it is unstable and must evolve to another state; see again figure 16(a).

Figure 16.

The critical swirl ratios $\unicode[STIX]{x1D714}_{c}$ and $\unicode[STIX]{x1D714}_{\ast }$ as a function of $\unicode[STIX]{x1D70C}_{CL}$ for the case of $T_{0}=1.02$ , $Ma_{0}=0.5$ , $R/C_{v}=0.01$ : (a)  $\unicode[STIX]{x1D714}_{c}$ versus $\unicode[STIX]{x1D70C}_{CL}$ and the regions of asymptotic stability and instability of the columnar state; (b)  $\unicode[STIX]{x1D714}_{\ast }$ versus $\unicode[STIX]{x1D70C}_{CL}$ and the regions of no breakdown states and breakdown states; (c) the correlation between $\unicode[STIX]{x1D714}_{c}$ and $\unicode[STIX]{x1D714}_{\ast }$ as a function of $\unicode[STIX]{x1D70C}_{CL}$ .

Figure 16(b) presents the combined lines of increasing $\unicode[STIX]{x1D714}_{\ast }$ with $\unicode[STIX]{x1D70C}_{CL}$ in the range $0<\unicode[STIX]{x1D70C}_{CL}<0.284$ , connected to the line of increasing $\unicode[STIX]{x1D714}_{\ast 1}$ with decreasing values of  $\unicode[STIX]{x1D70C}_{CL}$ from 0.284 to 0.175, followed by the line of decreasing $\unicode[STIX]{x1D714}_{\ast 2}$ with the increase of $\unicode[STIX]{x1D70C}_{CL}$ from 0.175 to 0.284, and then the line of decreasing $\unicode[STIX]{x1D714}_{\ast }$ with $\unicode[STIX]{x1D70C}_{CL}$ in the range $\unicode[STIX]{x1D70C}_{CL}>0.284$ . Along this line, the type 3 solution appears for the first time and $y_{0}=0$ . This line divides regions in the $\unicode[STIX]{x1D714}$ versus $\unicode[STIX]{x1D70C}_{CL}$ operational diagram where a breakdown state cannot exist and where it may exist and is expected to be an attractor of flow dynamics.

Figure 16(c) shows the two lines presented in figure 16(a,b). We observe six regions of flow behaviour with the increase of inlet centreline density $\unicode[STIX]{x1D70C}_{CL}$ .

1. (1) In the range of $0<\unicode[STIX]{x1D70C}_{CL}<0.175$ (region I), we find that $\unicode[STIX]{x1D714}_{\ast }<\unicode[STIX]{x1D714}_{c}$ and this behaviour is similar to that of the calorically perfect-gas limit case (case 1, figure 3). In this range of $\unicode[STIX]{x1D70C}_{CL}$ , the bifurcation of type 1 and type 2 states at $\unicode[STIX]{x1D714}_{c}$ is transcritical, and there is a range of $\unicode[STIX]{x1D714}$ between $\unicode[STIX]{x1D714}_{\ast }$ and $\unicode[STIX]{x1D714}_{c}$ where both the columnar and breakdown states may coexist. In this range of swirl, the flow dynamics of a perturbed columnar state evolves according to the initial perturbations and domains of attraction of these states (see case 2, figure 4, where $\unicode[STIX]{x1D70C}_{CL}=0.1$ ). With increase of swirl above $\unicode[STIX]{x1D714}_{c}$ , the columnar states are unstable and the flows are attracted to the breakdown states.

2. (2) In the range $0.175<\unicode[STIX]{x1D70C}_{CL}<0.237$ (region II), the critical swirl $\unicode[STIX]{x1D714}_{c}$ is greater than the swirl for appearance of breakdown $\unicode[STIX]{x1D714}_{\ast }$ . However, at higher swirl levels, there is a split along the branch of breakdown states where at certain swirl levels between $\unicode[STIX]{x1D714}_{\ast 1}$ and $\unicode[STIX]{x1D714}_{\ast 2}$ type 1 states appear (see cases 3, 4 and 5, figures 6, 8 and 9, where $\unicode[STIX]{x1D70C}_{CL}=0.19,~0.206$ and $0.22$ ). Also, when $0.199<\unicode[STIX]{x1D70C}_{CL}<0.237$ , there is a change of the type 2 states into type 1 states that may affect the stability of the columnar states in a certain range of swirl between $\unicode[STIX]{x1D714}_{c1}$ and $\unicode[STIX]{x1D714}_{c2}$ (see cases 4 and 5, figures 8 and 9, where $\unicode[STIX]{x1D70C}_{CL}=0.206,~0.22$ ).

3. (3) In the range $0.237<\unicode[STIX]{x1D70C}_{CL}<0.284$ (region III), the critical swirls at $\unicode[STIX]{x1D714}_{c1}$ and $\unicode[STIX]{x1D714}_{c2}$ disappear and the transcritical bifurcation at the first critical swirl along the branch of columnar states is at $\unicode[STIX]{x1D714}_{c}$ . The branch of type 1 states exhibits a fold at $\unicode[STIX]{x1D714}_{f}$ while the branch of breakdown states splits into two regions $\unicode[STIX]{x1D714}_{\ast }<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{\ast 1}$ and $\unicode[STIX]{x1D714}>\unicode[STIX]{x1D714}_{\ast 2}$ , with only stable columnar states between $\unicode[STIX]{x1D714}_{\ast 1}$ and $\unicode[STIX]{x1D714}_{f}$ (see case 6, figure 10, where $\unicode[STIX]{x1D70C}_{CL}=0.26$ ).

4. (4) In the range $0.284<\unicode[STIX]{x1D70C}_{CL}<0.452$ (region IV), the columnar state becomes unstable when $\unicode[STIX]{x1D714}>\unicode[STIX]{x1D714}_{c}$ and may evolve into either stable type 1 states when $\unicode[STIX]{x1D714}_{f}<\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{\ast }$ or breakdown states at swirl levels above $\unicode[STIX]{x1D714}_{\ast }$ (see case 7, figure 11, where $\unicode[STIX]{x1D70C}_{CL}=0.3$ ).

5. (5) When $0.452<\unicode[STIX]{x1D70C}_{CL}<0.7$ (region V), we find that $\unicode[STIX]{x1D714}_{f}<\unicode[STIX]{x1D714}_{\ast }<\unicode[STIX]{x1D714}_{c}$ and there is a range of swirl between $\unicode[STIX]{x1D714}_{f}$ and $\unicode[STIX]{x1D714}_{c}$ where the columnar and centreline-decelerated flow states connected to breakdown states may coexist (see case 8, figure 13, where $\unicode[STIX]{x1D70C}_{CL}=0.5$ ).

6. (6) When $\unicode[STIX]{x1D70C}_{CL}>0.7$ (region VI), the fold swirl level disappears and the flow dynamical behaviour exhibits similarity to that of a calorically perfect gas (see case 9, figure 14, where $\unicode[STIX]{x1D70C}_{CL}=1$ ).

4.11 Additional flow cases

In addition to the previous case of flows with a low $R/C_{v}$ and base temperature slightly above the thermodynamic critical value, we also investigated the effect of increasing the values of $R/C_{v}$ , such that the flows exhibit more significant exchange between enthalpy and kinetic energy, see (2.10). However, with the increase of $R/C_{v}$ at base temperature slightly above the thermodynamic critical value, we observed breakdown (type 3) solutions that tend to penetrate into the saturation dome because the variation in temperature is higher. Such solutions require an additional model for the liquid and vapour mixture within the saturation zone. To avoid such complex situations and remain within the model assumption of a single-phase flow, we focus on cases with somewhat higher values of the base temperature. For example, we studied the behaviour of flows with $R/C_{v}=0.1$ and $T_{0}(y)=1.05$ , flows with $R/C_{v}=0.1$ and $T_{0}(y)=1.1$ , and flows with $R/C_{v}=0.4$ and $T_{0}(y)=1.1$ . For all cases, $Ma_{0}=0.5$ . The increase of base temperature in these cases results in a less sharp increase of the base density $\unicode[STIX]{x1D70C}_{0}(y)$ when the swirl is increased and near thermodynamic critical conditions are achieved (see the term $(8/3)(T_{0}/(1-(1/3)\unicode[STIX]{x1D70C}_{0})^{2})-6\unicode[STIX]{x1D70C}_{0}$ in (3.1) that increases from a near-zero value to higher positive values with the increase of $T_{0}(y)$ from near 1). As a result, the density derivative $\unicode[STIX]{x1D70C}_{0y}$ achieves lower maximum values with the increase of $\unicode[STIX]{x1D714}$ and $T_{0}(y)$ . This induces smaller minima of the functions $H^{\prime }(\unicode[STIX]{x1D6F9})$ and $s^{\prime }(\unicode[STIX]{x1D6F9})$ , which affect the behaviour of the bifurcation diagram of solutions. The computed results show bifurcation diagrams that exhibit some similarity to the calorically perfect-gas case for all $0<\unicode[STIX]{x1D70C}_{CL}<1$ and with none of the complexity that was found in the case of $R/C_{v}=0.01$ studied above. The critical swirl $\unicode[STIX]{x1D714}_{c}$ and the swirl for appearance of breakdown $\unicode[STIX]{x1D714}_{\ast }$ for the three additional flow cases are presented in figure 17. In comparison with figure 16(c), it can be seen that the increase of $R/C_{v}$ from a small value (0.01) to higher values (0.1 and 0.4), accompanied with a small increase in the base temperature, results in $\unicode[STIX]{x1D714}_{\ast }<\unicode[STIX]{x1D714}_{c}$ for all $0<\unicode[STIX]{x1D70C}_{CL}<1$ . Also, there are no folds in the curves of these swirl levels. Moreover, in comparison between figures 16(c) and 17(a–c), we observed that the peaks of both $\unicode[STIX]{x1D714}_{c}$ and $\unicode[STIX]{x1D714}_{\ast }$ around $\unicode[STIX]{x1D70C}_{CL}\sim 0.25$ decrease with either the increase of the fluid $R/C_{v}$ or the increase of the base temperature.

Figure 17.

The correlation between the critical swirl ratios $\unicode[STIX]{x1D714}_{c}$ and $\unicode[STIX]{x1D714}_{\ast }$ as a function of $\unicode[STIX]{x1D70C}_{CL}$ for the cases: (a)  $T_{0}=1.05$ , $Ma_{0}=0.5$ , $R/C_{v}=0.1$ ; (b)  $T_{0}=1.1$ , $Ma_{0}=0.5$ , $R/C_{v}=0.4$ ; (c)  $T_{0}=1.1$ , $Ma_{0}=0.5$ , $R/C_{v}=0.4$ .