2.1. Axisymmetric TCF and boundary conditions

Focussing on the axisymmetric case ($\partial /\partial \phi (\cdot )=0$), we multiply the governing equations (2.2a)–(2.2c) by $r$ throughout. The resulting continuity, $r$-momentum, $\phi$-momentum, $z$-momentum and energy equations are then rewritten as

(2.6a)\begin{equation} \frac{\partial \bar{\rho}}{\partial t}+\frac{\partial}{\partial r}(\bar{\rho} u)+ \frac{\partial}{\partial z}(\bar{\rho}w)=0, \end{equation}

(2.6b)\begin{align} & \frac{\partial (\bar{\rho} u)}{\partial t} + \frac{\partial}{\partial r} [ u (\bar{\rho} u)] + \frac{\partial}{\partial z}[w(\bar{\rho} u)] -\frac{\bar{\rho} v^{2}}{r} \nonumber\\ &\quad =-\frac{Re_{i}^{2}}{Ma^{2}}\left( r \frac{\partial p}{\partial r}\right) + r\left[\left( \frac{\partial^{2} u}{\partial r^{2}}+\frac{\partial^{2} u}{\partial z^{2}} + \frac{\partial}{\partial r}\left(\frac{u}{r}\right)\right) + \frac{1}{3} \frac{\partial}{\partial r}\left( \frac{1}{r}\frac{\partial}{\partial r}(r u)+ \frac{\partial}{\partial z}(w)\right)\right], \end{align}

(2.6c)\begin{equation} \frac{\partial (\bar{\rho} v)}{\partial t}+\frac{\partial}{\partial r} [u (\bar{\rho} v)]+ \frac{\partial}{\partial z}[w(\bar{\rho} v)]+\frac{\bar{\rho} uv}{r} = r\left[\left(\frac{\partial^{2} v}{\partial r^{2}}+\frac{\partial^{2} v}{\partial z^{2}} + \frac{\partial}{\partial r}\left(\frac{v}{r}\right)\right)\right], \end{equation}

(2.6d)\begin{align} & \frac{\partial (\bar{\rho} w)}{\partial t}+ \frac{\partial}{\partial r} [u(\bar{\rho} w)]+ \frac{\partial}{\partial z}[w(\bar{\rho} w)] \nonumber\\ &\quad =-\frac{Re_{i}^{2}}{Ma^{2}}\left( r \frac{\partial p}{\partial z}\right) + r\left[\left(\frac{\partial^{2} w}{\partial r^{2}}+\frac{\partial^{2} w}{\partial z^{2}} + \frac{1}{r}\frac{\partial w}{\partial r} \right) +\frac{1}{3} \frac{\partial}{\partial z} \left( \frac{1}{r}\frac{\partial}{\partial r}(r u)+\frac{\partial}{\partial z}(w)\right)\right], \end{align}

(2.6e)\begin{align} & \frac{\partial (\bar{\rho} T)}{\partial t}+\frac{\partial}{\partial r} [ (\bar{\rho} u) T]+ \frac{\partial}{\partial z}[(\bar{\rho} w) T] \nonumber\\ &\quad =-(\gamma-1)p r\left( \frac{1}{r}\frac{\partial}{\partial r}(r u)+ \frac{\partial}{\partial z}(w)\right) +r\left[\frac{\gamma}{Pr} \left(\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial T}{\partial r}\right)+ \frac{\partial^{2}T}{\partial z^{2}}\right)\right] \nonumber\\ &\qquad + r(\gamma -1)\frac{Ma^{2}}{Re_{i}^{2}}\left\{\left[2\left(\frac{\partial u}{\partial r}\right)^{2}+ 2\left(\frac{u}{r} \right)^{2}+2\left(\frac{\partial w}{\partial z}\right)^{2}+ \left(\frac{\partial v}{\partial r} -\frac{v}{r} \right)^{2} \right.\right.\nonumber\\ &\qquad \left.\left.+\left(\frac{\partial u}{\partial z}+\frac{\partial w}{\partial r}\right)^{2}+ \left(\frac{\partial v}{\partial z} \right)^{2}\right]-\frac{2}{3}\left[\frac{1}{r^{2}} \left(\frac{\partial (ru)}{\partial r} \right)^{2}+\left(\frac{\partial w}{\partial z}\right)^{2}+ \frac{2}{r}\frac{\partial (ru)}{\partial r}\frac{\partial w}{\partial z}\right]\right\}, \end{align}

where

(2.7)\begin{equation} \bar{\rho}=\rho r \end{equation}

is the rescaled density, and the equation of state is $p=\bar {\rho } T/r$. Writing equations in this way (Harada Reference Harada1980) amounts to making the convective derivatives look as if (2.6a)–(2.6e) are written in Cartesian coordinates, which is advantageous for numerical discretization.

The boundary conditions are zero-slip at the cylinder walls as well as at stationary top and bottom lids,

(2.8a)\begin{gather} u=w=0 \quad \textrm{at}\ r= \frac{\eta}{1-\eta} \quad \mbox{and} \quad r= \frac{1}{1-\eta}, \end{gather}

(2.8b)\begin{gather}v=Re_i \quad \textrm{at}\ r= \frac{\eta}{1-\eta}, \quad v=Re_o \quad \textrm{at}\ r= \frac{1}{1-\eta}, \end{gather}

(2.8c)\begin{gather}u=v=w=0 \quad \textrm{at}\ z=0 \quad \textrm{and} \quad \varGamma, \end{gather}

and the Dirichlet boundary conditions for temperature are

(2.8d)\begin{equation} T=1 \quad \textrm{at}\ r= \frac{\eta}{1-\eta} \quad \mbox{and} \quad T=\frac{T_o}{T_i}=\chi \quad \textrm{at}\ r= \frac{1}{1-\eta}. \end{equation}

Note that the inner and outer cylinders are located at $r=\eta /(1-\eta )$ and $1/(1-\eta )$, respectively, with $\eta =R_i/R_o$ being the radius ratio. The temperature ratio $\chi = T_o/T_i$ is taken to be $1$, unless stated otherwise, i.e. both cylinders are kept at the same temperature.

2.2. Numerical method and the code validation

A finite difference code has been developed to numerically solve (2.6) along with boundary conditions (2.8). The details of the numerical scheme (Harada Reference Harada1980) are documented in appendix A.1 (see supplementary material). To validate the present code, the onset of Taylor vortices with periodic boundary conditions along the axial direction was analysed.

The simulation data on the axial variation of the radial-averaged radial velocity, i.e. $u(z) =\langle u(r,z)\rangle _r$, are shown in figure 3(a) for a range of inner-cylinder Reynolds number $Re$; the Mach number is set to $Ma=1$, the Prandtl number is $Pr=1$ and the aspect ratio of the TC-cell is $\varGamma =h/\delta =1.98$ with a radius ratio of $\eta =R_i/R_o=1/2$. For the idealized case of purely azimuthal CCF that occurs at small values of $Re$, the radial velocity is zero, see the blue horizontal line for $Re=71$ in figure 3(a). However, when the flow bifurcates to TVF, the radial velocity is non-zero due to presence of Taylor rolls, see the velocity profiles for $Re\geq 72$ in figure 3(a); the related velocity vectors and the streamline patterns at $Re=75$ are displayed in figures 3(c) and 3(d), respectively. To quantify bifurcation from CCF to TVF, the maximum of the radial-averaged radial velocity,

(2.9)\begin{equation} {\rm \Delta} u =max\, u(z), \end{equation}

is used as a metric. The variation of (2.9) with $Re$ is shown in figure 3(b); the critical/minimum Reynolds number for the onset of Taylor rolls, extracted from figure 3(b) is $Re_{cr}\approx 71$ – this value is very close to $Re_{cr} = 71.635$ reported by Welsh et al. (Reference Welsh, Kersal'e and Jones2014) from the linear stability analysis of compressible TCF.

Figure 3.

(a) Radial-averaged axial profiles of the radial velocity $u(z)=\langle u(r,z)\rangle _r$ for different values of $Re$. (b) Here ${\rm \Delta} u=max\, u (z)$ versus $Re$ quantifying the bifurcation from CCF to Taylor vortex flow (TVF). Blue circles are numerical values from the present code and the green-line represents a square root fit of the form ${\rm \Delta} u = 1.14(Re-71)^{1/2}$. (c) Velocity vectors and (d) streamline patterns of TVF in the meridional plane at ${Re} = 75$. Periodic boundary conditions are imposed along the axial direction, with parameter values of $\varGamma = h/\delta =1.98$, $\eta =R_i/R_0=1/2$, $Ma=1$ and $Pr=1$ and the outer cylinder is stationary ($Re_o=0$).

Another validation of the present code was done by reproducing the results of Harada (Reference Harada1980) for the two test cases of the spin-up problem in a compressible gas: (i) thermally driven flow and (ii) mechanically driven flow. Our results agree well with those of Harada (Reference Harada1980) – the related details are omitted, but can be obtained from the authors.

3.1. Multiroll ($2k\to 2(k+1)$) transition: normal Taylor vortices

The panels from the left- to the right-hand side in figure 4(a) display the streamline patterns in the ($r,z$)-plane at specific values of the aspect ratio $\varGamma$ when the height ($h$) of the TC-cell is first increased and then decreased in a quasistatic manner at a Reynolds number of $Re=100$; other parameters are $Ma=1$, $Pr=1$ and $\eta =0.5$. Note that the streamline plots have been drawn using the surface line-integral convolution (known as surface-LIC) module of the ParaView software, see Ahrens et al. (Reference Ahrens, Geveci and Law2005) – this helps in visualization by indicating an apparent motion along the direction of the vector field.

Figure 4.

(a) Surface line integral (known as LIC) (Ahrens, Geveci & Law Reference Ahrens, Geveci and Law2005) plots in the ($r, z$)-plane at different $\varGamma$ during upsweeping and downsweeping runs at a fixed inner Reynolds number of $Re=100$ with radius ratio $\eta =1/2$; see the text for details. (b) Time series of the radial kinetic energy $KE_{u}$, (3.1): the aspect ratio $\varGamma$ was increased from $3.1$ to $6.4$ and decreased back to $3.1$, at a ramp rate of $\textrm {d}\varGamma /\textrm {d} t=10^{-2}$. (c) Bifurcation diagram in the ($u, \varGamma$)-plane, where $u$ is the midplane, midheight radial velocity; note that the dashed lines joining two limit points, representing unstable states, are drawn to guide the eye. Other parameters are $Ma=1$, $Pr=1$ and $Re_{o} = 0$ (i.e. stationary outer cylinder), with axial and radial grids of $N_{z}\times N_{r} = 41 \times 41$.

The corresponding variation of the average kinetic energy based on radial velocities, defined via

(3.1)\begin{equation} {KE}_u = \left\langle \tfrac{1}{2}\rho u^2 \right\rangle_{(r,\theta,z)} = 2{\rm \pi} \int_{{\eta}/{(1-\eta)}}^{{1}/{(1-\eta)}}\int_{0}^{h} (\rho u^2/2) r \,\textrm{d} r\,\textrm{d} z, \end{equation}

is shown in figure 4(b). For these simulations, $\varGamma$ is increased from $3.1$ to $6.4$ and then decreased back to $3.1$ in steps of $\textrm {d}\varGamma =0.05$ with a ramp rate of ${\textrm {d}\varGamma }/{\textrm {d} t} = 0.01$, where $t$ is the dimensionless time in terms of the viscous time scale (see table 1). The finite jumps seen in figure 4(b) correspond to sudden changes in the number of Taylor vortices with varying $\varGamma$ using above protocol. During the upsweep from $\varGamma =3.1$, the initial decrease of $KE_{u}$ in figure 4(b) is tied to the fact that the vortices in the 2-roll state are elongated in the axial direction (compare the two left-hand panels in figure 4a). The first jump in $KE_u$ implies the transition from two rolls to four rolls since the net radial velocity should increase in the 4-roll state; the second jump in $KE_u$ denotes a transition from four to six rolls. During the downsweep from $\varGamma =6.4$, the flow structure collapses from 6-rolls to 4-rolls and then to 2-rolls. The evolution of Taylor vortices and their structural changes with quasistatic increase/decrease of the height ($\varGamma =h/\delta$) of the TC-cell can be ascertained from different panels in figure 4(a). Indeed, the long-time stability of these patterns have been ascertained by separately running the code for each ($\varGamma , Re$).

The above roll transitions are made clearer in figure 4(c) which shows the related bifurcation diagram in the ($u, \varGamma$)-plane, where the order parameter,

(3.2)\begin{equation} u \equiv u\left(r=r_i+ \frac{\delta}{2}, z=\frac{h}{2} \right), \end{equation}

is the radial velocity at the midgap and midheight of the coaxial cylinders; note that the open circles correspond to the $2$-roll branch, open diamonds represent $4$-roll branch, while open hexagrams represent the $6$-roll branch. It is seen that the 2-roll state jumps to the 4-roll configuration at around $\varGamma =4.5$; as the aspect ratio is increased, the 4-roll state jumps to the 6-rolls at $\varGamma \approx 6.1$; while reducing the aspect ratio from $\varGamma =6.4$, the 6-roll configuration collapses to a 4-roll configuration at $\varGamma \approx 5$ which subsequently jumps back to the original 2-roll state at $\varGamma \leq 3.6$. The locations at which above jump-transitions occur correspond limit or turning points; two successive limit points are connected via unstable orbits, denoted by the dashed lines in figure 4(c); note that these dashed lines are drawn by hand to guide the eye since our time-marching code is unable to track the unstable states.

It is clear from figure 4(c) that with increasing/decreasing $\varGamma$ the transitions among ‘$2\leftrightarrow 4\leftrightarrow 6$’-rolls occur via subcritical pitchfork bifurcations such that there are different ranges of $\varGamma$ over which (i) 2-roll and 4-roll states and (ii) 4-roll and 6-roll states coexist with each other. We further note in figure 4(c) that the radial velocity at the midheight and midgap location is negative ($u<0$) on the 4-roll branch but is positive ($u>0$) on both 2-roll and 6-roll branches, indicating the existence of an ‘outward-jet’ at the midheight of the TC-cell for 2-roll and 6-roll solutions as depicted in various panels in figure 4(a) and a midheight ‘inward-jet’ for 4-roll solutions. In either case, however, there are two inward-jets near the top and bottom stationary end walls and thus satisfying the no-slip conditions at both axial ends – hence the solutions depicted in figure 4 refer to ‘normal’ Taylor vortices, according to the classification of Benjamin & Mullin (Reference Benjamin and Mullin1981).

3.2. Phase diagram and the coexistence of different states

Several numerical experiments were conducted to reveal hysteresis between $2k$- and $2(k+1)$-rolls, such as in figure 4(c), by increasing and decreasing $\varGamma$ quasistatically while keeping the inner-cylinder Reynolds number $Re$ constant for a specified radius ratio of $\eta =1/2$. This procedure was repeated for a range of Reynolds numbers, the results of which are summarized as a phase diagram in figure 5. In this figure, different marker-lines represent different transitions observed during the up($\uparrow \varGamma$)- or down($\downarrow \varGamma$)-sweep protocols in $\varGamma$ – the red circles, blue squares and green triangles refer to transition locations during upsweep runs, whereas the transitions marked by the purple inverted-triangles and light-green squares are found during the downsweep runs. The regions of coexistence between different numbers of Taylor-vortices are shaded by different colours; as discussed in figure 4(c), the $2$-rolls were found during the upsweep and $4$-rolls were observed during the downsweep in the yellow shaded region, whereas the $4$- and $6$-rolls were found during upsweep and downsweep, respectively, in the grey-shaded region. The ‘cusp-like’ regions displayed in the left- and right-hand insets of figure 5 represent zoomed versions of ‘$2\leftrightarrow 4$’-rolls and ‘$4\leftrightarrow 6$’-rolls transition lines that nearly touch each other at smaller values of $Re< 80$. The above transition lines terminate at $Re=Re_{cr}\approx 71$ (i.e. the critical Reynolds number for linear instability, viz. figure 3b), and only 2-roll ‘Ekman-vortices’ (Coles Reference Coles1965; Benjamin Reference Benjamin1978a,Reference Benjaminb) are possible states at $Re<Re_{cr}$ in axially bounded TCFs.

Figure 5.

Phase diagram of even-roll (‘$2k\rightarrow 2(k+1)$’) transition in the ($Re, \varGamma$)-plane for $\eta =1/2$ and $Ma=1$. Forward-bifurcations are denoted by squares ($2\to 4$), circles ($2\to 6$) and triangles ($4\to 6$), and the backward-bifurcations are denoted by diamonds ($6\to 4$) and down triangles ($4\to 2$). The insets represent zoomed versions of two cusp-regions at $Re\sim 80$. These data have been obtained using $\varGamma$-increase/decrease protocol – a change of $|\textrm {d}\varGamma |=0.05$ after every $10^{6}$ time steps (with filtering done at every $20$ steps) corresponds to a ramp rate of $|{\textrm {d}\varGamma }/{\textrm {d} t}| =10^{-2}$, where $t$ is the dimensionless time. The black and red circles represent experimental (Benjamin Reference Benjamin1978b) and numerical (Cliffe Reference Cliffe1988) data, respectively, for ‘$2\leftrightarrow 4$’-roll transition in incompressible ($Ma=0$) TCF with $\eta =0.6$; the black and red stars denote experimental (Mullin, Pfister & Lorenzen Reference Mullin, Pfister and Lorenzen1982) and numerical (Cliffe Reference Cliffe1988) data, respectively, for ‘$4\leftrightarrow 6$’-roll transition in incompressible ($Ma=0$) TCF with $\eta =0.5$.

In figure 5, we have superimposed the experimental (Mullin et al. Reference Mullin, Pfister and Lorenzen1982) and numerical (Cliffe Reference Cliffe1988) data, marked by the black and red stars, respectively, which refer to the ‘$4\leftrightarrow 6$’-roll transition in incompressible ($Ma=0$) TCF with the same radius ratio of $\eta =0.5$ as in our calculations. Although the location of the cusp at $Ma=0$ agrees closely with that of $Ma=1$, there are significant quantitative differences for the loci of ‘$4\leftrightarrow 6$’-roll transitions between the present results ($Ma=1$) and its incompressible limit. Similar inferences can be made about the ‘$2\leftrightarrow 4$’-roll transition for which we have made a comparison with the experiments (black circles) and numerics (red circles) of Benjamin (Reference Benjamin1978b) and Cliffe (Reference Cliffe1988), respectively; note that the latter data correspond to a slightly larger radius ratio of $\eta =0.6$ at $Ma=0$. For both cases, the quantitative differences between $Ma=1$ and $Ma=0$ cases about the locations of limit and bifurcation points may be attributed to nonlinear and thermal effects, with the latter being absent in the incompressible limit.

The green-shaded region in figure 5 (at $Re>120$) corresponds to a 3-state coexistence of 2-, 4- and 6-rolls, and the underlying bifurcation scenario is discussed next. Figure 6(a) is the analogue of figure 4(a) at a higher value of the inner-cylinder Reynolds number $Re=150$, with other parameters being the same for both plots. For this case, the aspect ratio was increased from $3$ to $7$ and subsequently decreased back to $3$ with a ramp rate of ${\textrm {d}\varGamma }/{\textrm {d} t} = 0.01$. It is seen that the number of rolls/vortices are increased from $2$ to $4$ and $6$ during the upsweep protocol and a reverse transition scenario ($6\to 4\to 2$ rolls) occurs during the downsweep protocol. The corresponding bifurcation diagram in the ($u, \varGamma$)-plane is displayed in figure 6(b); we have also shown a schematic in figure 6(c) to identify the underlying bifurcation scenario by treating the number of rolls as an order parameter. Here too the subcritical pitchfork bifurcations lead to roll transitions among different number of Taylor rolls and we have verified that they all represent ‘normal’ modes (with inward jets near two stationary end walls) as explained above.

Figure 6.

(a,b) Same as figure 4(a,c), but for $Re=150$ at a ramp rate of $\textrm {d}\varGamma /\textrm {d} t=10^{-2}$. (c) The same bifurcation diagram of panel (b) is redrawn in terms of the number of rolls versus $\varGamma$; the ranges of $\varGamma$ over which ($2+4$)-, ($2+4+6$)- and ($4+6$)-rolls coexist are marked. Note that the dashed lines joining two limit points in panel (b,c) are drawn to guide the eye.

A noteworthy feature of figure 6(b,c) is that there is a range of aspect ratio $\varGamma \in (4.85,5.3)$ over which 2-, 4- and 6-roll states coexist with each other. The meridional-plane temperature ($T$), density ($\rho$) and azimuthal velocity ($v$) fields of three coexisting roll states are displayed in figures 7(a), 7(b) and 7(c), respectively, for $(\varGamma , Re) = (5, 150)$; the 2-, 4- and 6-roll states are stacked from the left- to the right-hand side in each panel. It is seen that the density maxima is located near the stationary outer cylinder, but the temperature maxima is located within the annular gap since both cylinders are kept at the same temperature and the added shear work due to the rotation of the inner cylinder is responsible for the increased temperature of the gas. Note that the red blobs in each panel of figure 7(a) correspond to the ‘outward’ jet through which the relatively hotter and rarefied gas (viz. figure 7b) is thrown out from the inner towards the outer cylinder. Comparing the density fields among different roll states in figure 7(b) we find that the density field is relatively more homogeneous in the 6-roll state than in the 2-roll state; the same observation holds for the temperature field too due to the enhanced mixing with increasing number of vortices. The azimuthal velocity fields in figure 7(c), along with the related colourmaps of the specific angular momentum,

(3.3)\begin{equation} \mathcal{L}(r,z) =\langle \rho(r,z) v(r,z) r\rangle, \end{equation}

in figure 7(d), further clarify the locations of the outward and inward jets, characterized by higher and lower values, respectively, of both azimuthal velocity and ${\mathcal {L}}$ for each case. All three states represent stable configurations which have been verified by running long-time simulations for each solution branch at $\varGamma =5$.

Figure 7.

Colourmaps in the ($r, z$)-plane of (a) temperature $T(r,z)$, (b) density $\rho (r,z)$, (c) azimuthal velocity $v(r,z)$ and (d) the specific angular momentum $\mathcal {L}(r,z) =\langle \rho v r\rangle$; on each panel, 2-roll (left-hand side), 4-roll (middle) and 6-roll (right-hand side) states are shown. The Reynolds number is $Re=150$ and the aspect ratio is $\varGamma =5$, with other parameters as in figure 6.

With reference to the phase diagram of figure 5, three representative bifurcation diagrams in the ($u, \varGamma$)-plane are displayed in figures 8(a), 8(b) and 8(c) for Reynolds numbers of $Re=75$, $120$ and $170$, respectively. In each panel, the lower branch with $u<0$ corresponds to the ‘normal’ 4-roll state for which there is an inward jet at the midheight of the TC-cell. Note that the 4-roll state represents the primary bifurcation (from the CCF state) at $\varGamma =5$, see figure 5. The upper branches in figure 8(a–c) refer to 2-roll (left-hand branch) and 6-roll (right-hand branch) states which are characterized by an outward jet ($u>0$) at the midheight of the TC-cell.

Figure 8.

Evolution of bifurcation diagrams with Reynolds number in the ($u, \varGamma$)-plane, where $u$ is the radial velocity at the midheight and mid gap: (a) $Re=75$, (b) $Re=120$ and (c) $Re=170$. Note that the dashed lines joining two limit points in panel (b,c) are drawn to guide the eye.

Collectively, the bifurcation diagrams in figures 8(a) ($Re=75$), 4(c) ($Re=100$), 8(b) ($Re=120$), 6(c) ($Re=150$) and 8(c) ($Re=170$) can be stacked together to construct a three-dimensional bifurcation diagram in the ($u, Re, \varGamma$)-plane (not shown). The projection of the latter in the ($u, Re$)-plane is shown in figure 9(a) for selected values of $\varGamma$; while the lower branch of figure 9(a) with $u<0$ belongs to the 4-roll state, the upper branches with $u>0$ correspond to 2- and 6-roll states. To identify the possible existence of limit points in figure 9(a), additional computations were performed at $\varGamma = 5$ by varying $Re$ starting with the initial state from both upper and lower branches of figure 9(a). More specifically, starting from the $2$-roll branch at $Re = 130$ (see the line marked by blue circles in figure 9b), the Reynolds number was decreased in steps of ${\rm \Delta} Re = 2$, while starting from the $4$-roll branch at $Re=110$ (see the line marked by orange diamonds), the Reynolds number was increased in the steps of ${\rm \Delta} Re = 2$. The results are summarized in figure 9(b) which confirms discrete jumps between $4$- and $2$-/$6$-roll states in the ($u, Re$)-plane.

Figure 9.

Bifurcation diagrams in ($u, Re$)-plane for (a) various $\varGamma$ and (b) $\varGamma =5$. In panel (b), the 2-roll and 4-roll branches have been continued by varying $Re$ to verify the existence of any limit point; expectedly, the 2-roll and 6-roll branches are disconnected in the ($u, Re$)-plane.

3.3. Role of compressibility on bifurcation structure and patterns

Recall from figure 5 that we found noticeable differences between the compressible (${Ma=1}$) and incompressible ($Ma=0$, Benjamin (Reference Benjamin1978b); Mullin et al. (Reference Mullin, Pfister and Lorenzen1982); Cliffe (Reference Cliffe1988)) TCF regarding the ‘$2\leftrightarrow 4$’ and ‘$4\leftrightarrow 6$’-roll transition loci in the ($\varGamma , Re$)-plane; the quantitative differences were seen to grow at larger $Re$, away from the linear stability threshold. The above results indicate that our nonlinear results differ from the linear stability results of Welsh et al. (Reference Welsh, Kersal'e and Jones2014) who showed that the neutral stability boundary, characterizing the transition between CCF and TVF states, is nearly insensitive to Mach numbers of ${Ma}\leq 1$, i.e. $Re_{cr}(Ma\leq 1) \approx Re_{cr}(Ma=0)$. On the other hand, they also showed that the critical Reynolds number increases with increasing ${Ma}>1$, implying the overall stabilizing role of compressibility on ‘$\textrm {CCF}\to \textrm {TVF}$’ transition in axially periodic TCF. In the following, we address the role of compressibility in detail by analysing results over a range of Mach numbers $0.2\leq Ma\leq 3$ in the context of axially bounded compressible TCF.

Figure 10(a,b) shows the effects of Mach number on the bifurcation diagram in the ($u, \varGamma$)-plane; the Reynolds number is set to (a) $Re=100$ as in figure 4(c) and 4(b) $Re=150$ as in figure 6(b). While the left- and right-hand branches in figure 10(a,b) refer to 2-roll (viz. figure 11c) and 6-roll states, respectively, the middle branch corresponds to a $4$-roll state (viz. figure 12c). Comparing three solution branches in figure 10(a), we find that the extent of the subcritical/hysteretic regions increases when the Mach number is decreased to $Ma=0.2$, but the transition among different roll states seems to be continuous over $\varGamma$ at ${Ma=3}$; with increasing $Re$, however, the hysteretic transition reappears at $Ma=3$, see figure 10(b). A general trend in figure 10(a,b) is a reduction in the magnitude of the midgap, midheight radial velocity, $u(r=r_i+\delta /2, z=h/2)$, implying a relatively weaker outward and inward jet (in the bulk) with increasing $Ma$.

Figure 10.

Effect of Mach number on the bifurcation diagram in ($u, \varGamma$)-plane for (a) $Re=100$ and (b) $Re=150$. Note that the dashed lines joining two limit points are drawn to guide the eye.

Figure 11.

Effects of Mach number on the meridional-plane (a) specific angular momentum ${\mathcal {L}}(r,z)$, (b) temperature $T(r,z)$ and (c) streamlines, corresponding to the left-hand upper branch of figure 10(a). On each panel, two snapshots correspond to peripheral Mach number of ${Ma} = 0.2$ (left-hand side) and $1$ (right-hand side). The Reynolds number is $Re=100$ and the aspect ratio is $\varGamma =4$.

Figure 12.

Same as figure 11, but for the lower (4-roll) branch of figure 10(a). On each panel, three snapshots correspond to ${Ma} = 0.2$ (left-hand side), $1$ (middle) and $3$ (right-hand side).

Referring to different solution branches of figure 10(a) and setting the aspect ratio to $\varGamma =4$, the meridional plane distributions of (i) specific angular momentum ${\mathcal {L}}(r,z)$, (ii) temperature $T(r,z)$ and (iii) streamlines are compared in figures 11(a–c) and 12(a–c) for different values of Mach number; while the left- and right-hand snapshots in each panel of figure 11(a–c) correspond to 2-roll states at $Ma=0.2$ (left-hand side) and $1$ (right-hand side), those in figure 12(a–c) correspond to 4-roll states at $Ma=0.2$ (left-hand side), $1$ (middle) and $3$ (right-hand side). Although the roll structures (panel c) look similar at all $Ma$, the temperature fluctuations (panel b) decay, expectedly, for nearly incompressible flows (${Ma}=0.2$) which should vanish at ${Ma}=0$. On the other hand, the angular momentum field in figures 11(a) and 12(a) is seen to decay with increasing ${Ma}$, implying that the compressibility diminishes angular momentum transport from the inner to the outer cylinder at given $Re$.

The right-hand-most snapshots in figure 12(a,c) at ${Ma}=3$ indicate that the two central vortices are much weaker compared with Ekman vortices (i.e. those located near the end plates). The related evolution of roll structures with increasing aspect ratio can be ascertained from figure 13 which displays streamline patterns at different $\varGamma$; this refers to the solution branch marked by triangles in figure 10(a). It is clear that, with increasing $\varGamma$, the transition from a 2-roll state (the first panel in figure 13, $\varGamma =3.5$, whose origin is tied to Ekman vortices) to a 4-roll state (second, third, fourth and fifth panels at $\varGamma =3.7, 4, 5$ and $5.5$) is accompanied by the genesis of two daughter modes around the midheight of the annulus, which grow in size with increasing $\varGamma$ and subsequently gives birth to a 6-roll state at $\varGamma \approx 5.6$. The streamline patterns of two 6-roll states at $\varGamma =5.7$ can be ascertained from the two right-hand-most panels in figure 13. The latter two states belong to the hysteresis loop on the solution branch for ${Ma}=3$, see figure 10(a) at around $\varGamma =5.7$.

Figure 13.

Streamline patterns on the meridional-plane for different aspect ratio: $\varGamma =3.5, 3.7, 4, 5, 5.5, 5.7$ and $5.7$ (from left to right); the last two snapshots are for $\varGamma =5.7$ that belong to the hysteretic loop for the triangled-curve in figure 10(a). The Mach number is $Ma=3$ and the Reynolds number is $Re=100$.

Juxtaposing the bifurcation diagrams in figure 10(a,b) at different ${Ma}$ with those in figure 8 (for $Ma=1$), we infer that the increasing Mach number would lead to a shift in the locations of multiroll transitions ($2\to 4\to 6$ rolls) and related hysteresis to higher values of Reynolds number. This implies that the overall phase diagram in figure 5 in the ($\varGamma , Re$)-plane would be shifted to higher values of $Re$ when the Mach number is increased from the incompressible limit. In summary, the primary effects of compressibility ($Ma\uparrow$) on symmetric roll states are: (i) a delayed onset of multiroll transitions to larger $Re$; (ii) measurable changes in the successive higher-order (‘$2 \leftrightarrow 4$’ and ‘$4 \leftrightarrow 6$’-roll) transition loci in the ($\varGamma , Re$)-plane (viz. figure 5) in comparison to what is known in incompressible TCF (Benjamin Reference Benjamin1978b; Mullin et al. Reference Mullin, Pfister and Lorenzen1982; Cliffe Reference Cliffe1988); (iii) the emergence of weaker outward jets (figure 10a,b and 12a,c); and (iv) relatively larger (and hence stronger) Ekman vortices (viz. figure 12c) near the two end walls. Collectively, the weakening of outward jets and the strengthening of Ekman vortices with increasing $Ma$ (for given $Re$) are responsible for the decreased angular momentum transport (figure 12a), from the rotating inner cylinder to the outer cylinder in the same limit, and consequently to the overall stabilizing role of compressibility on nonlinear Taylor vortices.

4.1. Tracking asymmetric 2-roll modes via a homotopy method

An example of an asymmetric 2-roll state, with a smaller vortex lying over a larger one, is shown in figure 14(b–d) which was obtained using the following procedure. The simulation is started with an axial temperature gradient which is switched on at $t=0$ by setting the top and bottom wall temperatures to

(4.1a,b)\begin{equation} T(z=h) = 1 + \epsilon \quad \mbox{and} \quad T(z=0)=1- \epsilon,\end{equation}

respectively; this initial condition breaks the midplane symmetry of the temperature field, see figure 14(e). The time evolution of this initial condition, with a value of $\epsilon =-0.1$, yields an asymmetric 2-roll state as confirmed by figure 14(b,f), at time $t=8$; this asymmetric state (with asymmetric end-wall conditions (4.1a,b)) remained stable with time, which was verified by continuing simulations down to $t=16$ (not shown); other parameters are $\varGamma =1$, $Ma=1$ and $Re=190$. After reaching the steady state the temperature gradient is switched off by setting $\epsilon =0$ at $t=16$, and the simulation is further continued in time. Figures 14(c,g) confirm that this asymmetric 2-roll state is indeed stable with ‘symmetric’ temperature boundary conditions ($\epsilon =0$) at two axial ends. Note that the location of the larger/smaller vortex depends on the sign of the homotopy parameter $\epsilon$ in (4.1a,b); for example, starting the same simulations with $\epsilon =0.1$ yielded figure 14(d,h) with the larger vortex being on the top of the smaller one, i.e. a reflection of the mode shown in figure 14(c,g) about its midheight.

Figure 14.

Stable asymmetric 2-roll state obtained with asymmetric temperature boundary condition (4.1a,b): meridional velocity field (panels a–d) and temperature contours (panels e–h). (a,e) Axial temperature gradient ($\epsilon =-0.1$) switched on at $t=0$; (b,f) asymmetric mode settles at $t = 8$ which is continued up to $t=16$ (not shown) at which the temperature gradient is switched off ($\epsilon =0$); (c,g) final state $t=80$ (with $\epsilon =0$) which survives at larger times. (d,h) Starting with $\epsilon =0.1$ leads to a ‘mirror-image’ asymmetric 2-roll state with the larger vortex lying on top of the smaller one. Parameter values are $Re=190$, $Ma=1$ and $\varGamma =1$.

In the context of compressible TCF, the asymmetric temperature protocol (4.1a,b) can be used as a computationally inexpensive alternative to Schaeffer's homotopy method (Schaeffer Reference Schaeffer1980) to track anomalous modes. It should be noted that the mathematical framework of Schaeffer (Reference Schaeffer1980) was numerically implemented by Cliffe (Reference Cliffe1983, Reference Cliffe1988) and others to track both normal and anomalous solution branches that helped in seamlessly tying the ‘periodic’ and ‘axially bounded’ models of incompressible TCF. The latter technique uses mixed Dirichlet–Neumann axial boundary conditions on the velocity field,

(4.2a,b)\begin{equation} (1-\epsilon) \frac{\partial (u,v)}{\partial z} \pm \epsilon (u,v) = 0 \quad \mbox{and} \quad w=0 \quad \mbox{at}\ z=0, h, \end{equation}

that interpolates between no-slip and periodic/free-slip end walls via a homotopy parameter $\epsilon \in (0,1)$. The grid refinement needed to accurately discretize velocity derivatives at the end walls is likely to increase the computational cost of the Schaeffer (Reference Schaeffer1980) method in comparison to the present method (4.1a,b). While (4.2a,b) has been extensively used to track both symmetric and asymmetric anomalous modes, the non-isothermal and compressible nature of the gas offer two alternative axial boundary conditions ($T=1\pm \epsilon$, or, $\rho =1\pm \epsilon$ at $z=0$ and $h$) to track ‘asymmetric’ modes in compressible TCF as demonstrated in this work. It should be noted that the ‘symmetric’ anomalous modes can, in principle, be constructed via the present technique by stacking an even number of asymmetric solutions via an appropriate mapping of related control parameters – the related issues are left for a future work.

Returning to the TCF with symmetric axial boundary conditions, an example of ‘$1\to 2$’-roll transition with decreasing $Re$ is shown in figure 15(a–c), with the aspect ratio being set to $\varGamma =1.2$. The asymmetric 2-roll state at $Re=240$ (figure 15a) was realized by running the simulation using (4.1a,b) as the initial condition as described in the preceding paragraph. The resulting equilibrium solution (with $\epsilon =0$) at $Re=240$ was continued subsequently by decreasing $Re$, in steps of ${\rm \Delta} Re=2$, up to $Re=180$. The corresponding bifurcation diagram in the ($|w|, Re$)-plane, shown in figure 15(d), confirms the discontinuous nature of ‘$1\to 2$’-roll transition at $\varGamma =1.2$; while the inverted triangles denote the above downsweep data, the up triangles represent an ‘upsweep’ run starting from $Re=180$ (figure 15c) up to $Re=240$ which tracked the symmetric 2-roll ($|w|=0$) branch. From a comparison between figures 15(a) and 15(b) we find that the location of the outward jet is shifted up with increasing $Re$, implying an increased size of the larger vortex, and the smaller vortex is further squeezed towards the top left-hand corner. In comparison to the bifurcation diagram in figure 15(d) for $\varGamma =1.2$, the bifurcation becomes subcritical when the aspect ratio of the TC-cell is decreased to $\varGamma =0.9$, see figure 15(e); the related details are discussed in appendix B (supplementary materials). The nature of bifurcation over a range of $\varGamma$, and the phase diagram of symmetric and asymmetric 2-roll states, are analysed below.

Figure 15.

Transition from ‘asymmetric’ to ‘symmetric’ 2-roll states with decreasing Reynolds number: velocity vectors in the ($r,z$)-plane at (a) $Re=240$, (b) $Re=200$ and (c) $Re=180$; the aspect ratio is $\varGamma =1.2$ and $Ma=1$. (d,e) Bifurcation diagrams in the ($|w|, Re$)-plane at (d) $\varGamma =1.2$ and (e) $\varGamma =0.9$, where $w\equiv w(r=r_i+\delta /2,z=h/2)$ is the midgap, midheight axial velocity and the ramp rate is $\textrm {d} Re/\textrm {d} t=2\times 10^{-2}$; the dashed line (postulated) in each panel denotes unstable states.

4.2. Phase diagram and its comparison with incompressible limit

Figure 16(a) displays the phase diagram in the ($\varGamma , Re$)-plane, depicting the transition between symmetric and asymmetric 2-roll (i.e. the single-roll mode) states; the ‘open’ symbols were obtained by following the ‘$\varGamma$ increase/decrease’ protocol (see figure 17). The black lines, representing approximate phase boundaries between ‘$1\leftrightarrow 2$’-roll transitions, are drawn to guide the eye; other parameters are $Ma=1$ and $Pr=1$. Note that the tracking of the midplane asymmetric modes was time consuming since the ramp rate ($\textrm {d}\varGamma /\textrm {d} t$) had to be extremely small; for example, $\textrm {d}\varGamma /\textrm {d} t$ was an order of magnitude smaller than that used to track transitions among symmetric rolls in figure 5.

Figure 16.

(a) Phase diagram in the ($\varGamma , Re$)-plane, obtained from the ‘$\varGamma$ increase/decrease’ protocol, for $\eta =1/2$, ${Ma}=1$ and ${Pr=1}$; the ramp rate is ${\textrm {d}\varGamma }/{\textrm {d} t} = 2.5\times 10^{-4}$ (with an increment of $\textrm {d}\varGamma =0.01$ after every $4\times 10^{6}$ time steps). While the ‘open’ squares and up triangles were extracted from the upsweep run in $\varGamma$, the diamonds and down triangles were extracted from the downsweep run in $\varGamma$; the ‘filled’ squares and circles (at $\varGamma =0.9, 0.95, 1.0, 1.05, 1.1, 1.15$ and $1.2$) were obtained from the ‘$Re$ increase/decrease’ protocol described in § 4.1 (see appendix B in the supplementary materials). The lines, postulated phase boundaries, are drawn to guide the eye; the yellow shaded region represents the coexistence of asymmetric and symmetric 2-roll states. (b) Phase diagram for incompressible TCF in the ($\varGamma , Re$)-plane, adapted from Pfister et al. (Reference Pfister, Schmidt, Cliffe and Mullin1988), that holds for a radius ratio of $\eta =1/2$. The blue shaded region, over $\varGamma ^{sub}_c \leq \varGamma \leq \varGamma _c^{LP}$, represents subcritical bifurcations with varying $Re$ (say, in the ($u, Re$)-plane); the green-shaded region at $\varGamma >\varGamma _c^{LP}$ represent disconnected single-roll solutions coexisting with symmetric 2-roll solutions, with the loci of limit points being marked by the black circled-dashed line.

Figure 17.

Effect of Mach number on bifurcation diagrams in the ($|w|, \varGamma$)-plane, where $w$ is the midheight, midgap axial velocity: (a) $Re=180$, (b) $Re=200$, (c) $Re=220$ and (d) $Re=240$. The aspect ratio $\varGamma$ was changed in steps of ${\rm \Delta} \varGamma =0.01$ after every $4\times 10^6$ time steps, resulting in a ramp rate of $\textrm {d}\varGamma /\textrm {d} t=2.5\times 10^{-4}$ and the grid size is $21\times 21$. Panel (e) is a zoomed version of panel (c) for $Ma=3$ using two ramp rates of $\textrm {d}\varGamma /\textrm {d} t=2.5\times 10^{-4}$ (green triangles) and $2.5\times 10^{-3}$ (black triangles). The curly red arrow in panel (e) indicates the continuous nature of the solution branch during the ‘upsweep’ run; see the text for details.

To check the robustness of figure 16(a), the ‘$Re$ increase/decrease’ protocol of § 4.1 was employed to identify the limit point and bifurcation points for ‘$1\leftrightarrow 2$’-roll transition at selected values of $\varGamma =0.9, 0.95, 1.0, 1.05, 1.1, 1.15$ and $1.2$, refer to figure 15(d) and 15(e) for the bifurcation diagrams in the ($|w|, Re$)-plane at $\varGamma =1.2$ and $0.9$, respectively. These additional data are marked by the ‘filled’ squares (bifurcation point) and circles (limit point) in figure 16(a). It is seen that both sets of data agree closely with each other, confirming the robustness of the phase diagram in figure 16(a), including the yellow shaded region that corresponds to the coexistence of (i) symmetric and (ii) asymmetric 2-roll states. While the white region bounded by the yellow-shaded region in figure 16(a) refers to ‘stable’ asymmetric 2-roll states, the symmetric 2-roll states exist on the exterior of the limit point branch. Comparing figure 16(a) with the phase diagram for the ‘higher-order’ symmetric rolls in figure 5, we find that the asymmetric 2-roll states appear at much larger values of the Reynolds number $Re$ for specified $Ma$.

Figure 16(a) should be contrasted with figure 16(b) that represents the phase diagram of ‘$1\leftrightarrow 2$’-roll transition for incompressible TCF which has been adapted from the experimental and numerical study of Pfister et al. (Reference Pfister, Schmidt, Cliffe and Mullin1988). The left-hand branch of figure 16(b) represents supercritical bifurcation, whereas a part of the right-hand branch (the blue dotted line over $\varGamma _c^{sub}\leq \varGamma \leq \varGamma _c^{LP}$) represents subcritical bifurcations; the green-shaded region at $\varGamma \geq \varGamma _c^{LP}$ represents disconnected single-roll solutions that coexist with stable symmetric 2-roll solutions at any $Re$ above the limit point branch marked by the circled dashed line. Therefore, in contrast to the incompressible TCF (figure 16b), the ‘$1\leftrightarrow 2$’-roll transition in compressible TCF (viz. figure 16a at $Ma=1$) is subcritical/hysteretic at any aspect ratio $\varGamma$ of the TC-cell. The above general finding holds also at higher values of $Ma>1$ as we demonstrate below, via an analysis of the underlying bifurcation diagrams and the hydrodynamic fields.

4.3. Hysteretic transition and the role of compressibility

The subcritical/hysteretic bifurcations uncovered in figure 16(a) are further confirmed from the bifurcation diagrams in figure 17(a–d) which display the variations of the midheight, midgap axial velocity,

(4.3)\begin{equation} w \equiv w\left(r=r_i+\frac{\delta}{2}, z=\frac{h}{2} \right),\end{equation}

as a function of $\varGamma$ for (a) $Re=180$, (b) $Re=200$, (c) $Re=220$ and (d) $Re=240$; while the blue circles denote the data for $Ma=1$, the green triangles represent $Ma=3$. Note that the axial velocity at the midheight and midgap vanishes (i.e. $w=0$) for symmetric rolls and $|w|>0$ for asymmetric rolls. The red arrows in each panel of figure 17 indicate abrupt jumps from the 2-roll state to the asymmetric 2-roll state and vice versa during upsweep (increasing $\varGamma$) and downsweep (decreasing $\varGamma$) protocols. Juxtaposing the bifurcation diagrams for $Ma=1$ in figure 17(a–d) with the phase diagram in figure 16(a), we find that the outer phase boundary in figure 16(a) represents the loci of limit points, while its inner boundary refers to the loci of (subcritical) bifurcation points.

Comparing the bifurcation diagrams between $Ma=1$ and $Ma=3$ in each panel of figure 17(a–d), we find that the single-roll mode persists at $Re=180$ and $200$ for $Ma=1$ but are absent at $Ma=3$; the single-roll solution starts appearing at $Ma=3$ when the Reynolds number is increased to $Re\geq 220$ (panel c). This suggests that the phase diagram of ‘$1\leftrightarrow 2$’-roll transitions in the ($\varGamma , Re$)-plane, such as in figure 16(a), would shift to larger values of $Re$ with increasing Mach number.

Returning to figure 17, we confirmed that the left- and right-hand hysteresis loops are robust features of axially bounded TCF since both loops remained intact with changing ramp rate $\textrm {d}\varGamma /\textrm {d} t$ (see appendices B and C in the supplementary material for details). A further confirmation in this regard can be ascertained from figure 17(e) that shows the near invariance of the shape of the bifurcation diagram at $Ma=3$ with two ramp rates $d\varGamma /dt=2.5\times 10^{-4}$ (green triangles) and $2.5\times 10^{-3}$ (black triangles) that differ by an order of magnitude. Recall from figure 16(b) that the left-hand hysteresis loop of figure 17 is completely absent in the incompressible limit (figure 16b, $Ma=0$) and therefore its existence at $Ma>0$ is solely due to the compressibility of the gas. The unequivocal support for the latter observation is ascertained from figure 17(c,d) that compares the bifurcation diagrams for $Ma=1$ and $3$; ‘the range of $\varGamma$ spanning the left-hand hysteresis loop increases with increasing $Ma$ at given $Re$’. Moreover, our data at $Ma=0.2$ (not shown) indicate a decreased size of the left-hand hysteresis loop, thus confirming its disappearance in the incompressible limit $Ma\to 0$. From the above discussion, we conclude that the ‘$1\leftrightarrow 2$’-roll transition in compressible TCF is hysteretic at any $\varGamma$ (figure 16a) in contrast to the super-critical/subcritical nature of bifurcation at smaller/larger values of $\varGamma$, respectively, in its incompressible counterpart (figure 16b).

An interesting feature of the bifurcation diagram in figure 17(e) is that the order parameter (4.3) is close to zero at $\varGamma \sim 1.07$, resulting in an apparent kink/discontinuity in the ($|w|, \varGamma$)-plane – this occurs within the range of $\varGamma$ over which the single-roll solution should be a stable state. The reason for the appearance of this kink in the ($|w|, \varGamma$)-plane can be explained as follows: the topological changes in the velocity patterns with increasing $\varGamma$, see figure 18(a–c), may render the midgap, midheight location, at which the order parameter is measured, to coincide with the centre of the larger vortex, resulting in $w(r_i+\delta /2, h/2)\approx 0$ at some value of $\varGamma$. The corresponding pattern still represents an ‘asymmetric’ state (not a symmetric 2-roll for which $w=0$ as well) as confirmed in figure 18(b). The continuous nature of the solution branch can be better appreciated if we replot the bifurcation diagrams, such as those in figure 17(c,d), in the ($w, \varGamma$)-plane.

Figure 18.

Meridional velocity field at $Re = 220$ and $Ma=3$: (a) $\varGamma =0.93$, (b) $1.07$ and (c) $1.17$; the red circle in each panel denotes the midgap, midheight location. Refer to figure 17(e) for the corresponding bifurcation diagram in the ($|w|, Re$)-plane.

Lastly, to establish the connection of the asymmetric 2-roll mode found in the present study to those in incompressible TCF (Benjamin & Mullin Reference Benjamin and Mullin1981; Cliffe Reference Cliffe1983; Pfister et al. Reference Pfister, Schmidt, Cliffe and Mullin1988), we numerically continued the steady solution found at $Ma = 1$ (see figure 14 that holds for $Re = 190$, $\varGamma = 1$ and $Pr = 1$) to $Ma = 0.1$ in steps of ${\rm \Delta} Ma = 0.1$. A sample calculation is summarized in figure 19 which displays the meridional velocity field (a–c), the temperature (d–f) and the specific angular momentum (g–i), with other parameters as in figure 14. The asymmetric mode manifested using (4.1a,b) was allowed to settle at $Ma = 1$, see the left-hand column of figure 19; after reaching the steady state, $Ma$ was lowered to $Ma = 0.9$ and the flow was allowed to stabilize again. The grid size is $21 \times 21$ and the requirement on time-step (${\rm \Delta} t$) was severe towards the incompressible limit; for example, a time step of ${\rm \Delta} t = 10^{-5}$ is used for $Ma\geq 0.7$ and ${\rm \Delta} t=10^{-7}$ for smaller $Ma$ to obtain converged solutions. The above procedure was checked for both positive and negative values of $\epsilon$, and, in both cases, the asymmetric 2-roll modes survived until $Ma = 0.1$. We conclude that the single-roll modes uncovered in the present study are indeed connected to those found in its incompressible counterpart.

Figure 19.

Asymmetric 2-roll (i.e. ‘single-roll’) modes at $Ma = 1$ (a,d,g), $Ma = 0.5$ (b,e,h) and $Ma = 0.1$ (c,f,i); meridional velocity vectors (a–c), temperature contours (d–f) and angular momentum contours (g–i). Parameter values are $Re = 190$, $Pr=1$ and $\varGamma = 1$.

It is clear from the middle row of figure 19 that the magnitudes of temperature fluctuations decrease and approach zero as we move towards the incompressible limit ($Ma\to 0$). The bottom row of figure 19 confirms that the angular momentum transport via asymmetric 2-roll mode is enhanced in the incompressible limit – this finding is similar to that for higher-order symmetric rolls (viz. figures 10–12) as summarized in the last paragraph of § 3.3. Collectively, therefore, the compressibility aids in stabilizing both asymmetric and symmetric Taylor vortices and the underlying multiroll transitions.