2.1. Flow geometry

Figure 1 is a schematic drawing of the problem geometry. We consider the flow of a viscous incompressible fluid in a cylindrical container of length $H$ and radius $R$ driven by the end disk at $z=0$ , which rotates with angular velocity ${\it\Omega}$ . The two main control parameters are the aspect ratio $h=H/R$ and the Reynolds number, $\mathit{Re}={\it\Omega}R^{2}/{\it\nu}$ , characterizing the swirl strength, where ${\it\nu}$ is the fluid kinematic viscosity. We also consider the flow with additional co-rotation of the sidewall (§ 7). The co-rotation parameter, ${\it\Omega}_{s}$ , is the sidewall-to-disk angular velocity ratio.

Figure 1. Schematic of the problem.

2.2. Governing equations

Using $R,1/{\it\Omega},{\it\Omega}R$ , and ${\it\rho}{\it\Omega}^{2}R^{2}$ as scales for length, time, velocity, and pressure, respectively, renders all variables dimensionless. The flow is governed by the Navier–Stokes equations for a viscous incompressible fluid, whose dimensionless form is:

where ${\rm\nabla}^{2}\equiv (1/r)(\partial /\partial r)(r(\partial /\partial r))+(1/r^{2})(\partial ^{2}/\partial {\it\phi}^{2})+(\partial ^{2}/\partial z^{2})$ is the Laplace operator for a scalar field, ( $u,v,w$ ) are the velocity components in cylindrical coordinates ( $r,{\it\phi},z$ ), $t$ is time, and $p$ is pressure. We denote the list ( $u,v,w,p$ ) as $\boldsymbol{V}$ , and look for a solution of the system (2.1)–(2.4) in the form

where subscripts ‘ $b$ ’ and ‘ $d$ ’ denote the base flow and a disturbance, respectively; $\text{c.c.}$ denotes the complex conjugate of the preceding term; ${\it\varepsilon}\ll 1$ is an amplitude; integer $m$ is an azimuthal wavenumber; and ${\it\omega}={\it\omega}_{r}+\text{i}{\it\omega}_{i}$ is a complex number to be found, with frequency ${\it\omega}_{r}$ and growth rate of disturbance ${\it\omega}_{i}$ . For a decaying (growing) disturbance, ${\it\omega}_{i}$ is negative (positive). The equations governing the base flow result from substituting (2.5) in system (2.1)–(2.4) and setting ${\it\varepsilon}=0$ . The terms of order $O({\it\varepsilon})$ constitute equations governing infinitesimal disturbances.

2.3. Boundary conditions

The conditions for the base flow are the following.

1. (i) No-slip at the walls:

   $u_{b}=v_{b}=w_{b}=0$ at $z=h$ and $0<r<1$ (still disk),

   $u_{b}=v_{b}=w_{b}=0$ at $r=1$ and $0<z<h$ (sidewall),

   $u_{b}=w_{b}=0$ , $v_{b}=r$ at $z=0$ and $0<r<1$ (rotating disk).

2. (ii) Regularity at the axis: $u_{b}=v_{b}=0$ , $\partial w_{b}/\partial r=0$ at $r=0$ and $0<z<h$ .

For the problem where the sidewall also rotates, the conditions at the sidewall are modified to $u_{b}=w_{b}=0$ and $v_{b}={\it\Omega}_{s}$ at $r=1$ and $0<z<h$ .

The conditions for disturbances are the following.

1. (i) No-slip at all walls: $u_{d}=v_{d}=w_{d}=0$ .

2. (ii) Regularity at the axis:

   $u_{d}=v_{d}=\partial w_{d}/\partial r=0$ at $m=0$ ;

   $w_{d}=0$ , $\partial u_{d}/\partial r=0$ , and $u_{d}+mv_{d}=0$ at $|m|=1$ ;

   $u_{d}=v_{d}=w_{d}=0$ for $|m|>1$ .

Since the equations and boundary conditions are uniform for disturbances, there is a trivial (zero) solution. Non-zero solutions exist at eigenvalues of ${\it\omega}$ . A disturbance with ${\it\omega}_{i}=0$ is neutral. Neutral disturbances corresponding to the minimal Re at prescribed $m$ are marginal. The marginal disturbance corresponding to the minimal Re among all $m$ values is critical. Corresponding parameters ${\it\omega}_{r}$ and $m$ are also named as marginal and critical (Sorensen et al. Reference Sorensen, Gelfgat, Naumov and Mikkelsen2009).

4.1. Formation of global counterflow

It is known that a swirling flow in a long cylindrical container with one rotating end disk is multicellular for $\mathit{Re}\ll 1$ (Hills Reference Hills2001). We show here how this multicellular flow transforms into a global counterflow as Re increases. Figure 2 represents streamlines, i.e. contours of ${\it\Psi}=\text{const.}$ , of the meridional motion driven by the rotating disk located at $z=0$ in a cylinder with $h=10$ .

Figure 2. Development of global counterflow as Re increases at $h=10$ ; $\mathit{Re}=1$ (a), 2000 (b), 4000 (c), and 5000 (d).

Figure 2(a) depicts the six flow cells existing at $\mathit{Re}=1$ . Our simulations show that the magnitude of ${\it\Psi}$ rapidly decays as $z$ increases. This agrees with the known feature that ${\it\Psi}$ decays at least as $\exp (-{\it\lambda}_{r}z)$ and oscillates as $\sin ({\it\lambda}_{i}z)$ for small $\mathit{Re}$ ; ${\it\lambda}_{r}=4.466$ and ${\it\lambda}_{i}=1.468$ (Shankar Reference Shankar1998). In order to better observe the slow motion near the stationary disk, located at $z=10$ , ${\it\Psi}$ is multiplied by $\exp ({\it\lambda}_{r}z)$ in figure 2(a).

As Re increases, the cell adjacent to the rotating disk (left-hand cell in figure 2 a–d) extends while the other cells move toward the stationary disk, shrink, and disappear. There are three cells at $\mathit{Re}=2000$ (figure 2 b), one large and two small cells for $\mathit{Re}=4000$ (figure 2 c). The second cell shown in figure 2(b) separates from the axis and shrinks near the sidewall as Re further increases, as figure 2(c) illustrates for $\mathit{Re}=4000$ . The near-sidewall cell disappears, the large and small cells merge, and the flow becomes one-cellular at $\mathit{Re}=5000$ (figure 2 d).

The solid curve in figure 3 depicts the dependence on Re of maximal axial extent $L$ of the cell adjacent to the rotating disk. Here, $L$ is the distance from the rotating disk to the point where the velocity at the axis changes its sign the first time (see figure 2 a). The dependence $L(\mathit{Re})$ can be approximated by the linear relation, $L=4+0.00134\,\mathit{Re}$ , depicted by the dots in figure 3. In the range $2000<\mathit{Re}<4480$ , the curve and dots merge within the accuracy displayed in figure 3.

Figure 3. Dependence of $L$ (see figure 2 a) on the Reynolds number. The dotted discrete points represent the linear approximation of the numerical data (solid curve).

Below we discuss an important feature for the instability development: the appearance of a local maximum of the swirl vorticity as Re increases.

4.2. Formation of a local maximum of swirl vorticity

At the axis, the swirl vorticity is ${\it\omega}_{z}=2{\it\Omega}_{a}$ , where ${\it\Omega}_{a}$ is the angular velocity at $r=0$ , which is prescribed at the rotating disk ( $z=0$ ) as ${\it\Omega}_{a}=1$ . Figure 4 presents the distribution of ${\it\Omega}_{a}$ along the axis in the vicinity of $z=0$ for the Re values shown near the curves. Curve $\mathit{Re}=1$ in figure 4 depicts the distribution mostly governed by viscous diffusion of ${\it\omega}_{z}$ from the rotating disk; the role of meridional circulation (figure 2) is negligible for $\mathit{Re}\leqslant 1$ . As Re increases, the circulation contribution becomes remarkable. The backflow near the axis transports ${\it\omega}_{z}$ toward the disk thus diminishing the diffusion of ${\it\omega}_{z}$ away from the disk. Comparison of curves $\mathit{Re}=1$ and $\mathit{Re}=100$ in figure 4 illustrates this effect: ${\it\Omega}_{a}$ is significantly reduced near the disk at $\mathit{Re}=100$ compared with that at $\mathit{Re}=1$ .

Figure 4. Development of a local maximum of angular velocity ${\it\Omega}_{a}$ on the axis as Re (shown near the curves) increases; the dashed curve corresponds to the Kármán boundary layer.

For larger Re, the boundary layer develops near the rotating disk (von Kármán Reference von Kármán1921). The dashed curve in figure 4 depicts the distribution of ${\it\Omega}_{a}$ , corresponding to Kármán’s solution at $\mathit{Re}=500$ . This curve and those for $\mathit{Re}=1$ and 100 illustrate the formation of the boundary layer as Re increases.

The distribution of ${\it\Omega}_{a}$ , depicted by solid curve $\mathit{Re}=500$ in figure 4, qualitatively differs from those shown by the other curves: there is a local maximum of ${\it\Omega}_{a}$ near $z=0.8$ . The maximum appears due to the intensified meridional circulation, as discussed below.

The circulatory flow (left-hand cell in figure 2 a–d) transports the angular momentum, $rv$ , from the disk periphery along the sidewall and then toward the axis, thus increasing the swirl vorticity there. As figure 5 illustrates, the backflow velocity magnitude, $-w_{a}$ , at the axis, $r=0$ , is maximal in the vicinity of the local maximum of ${\it\Omega}_{a}$ (figure 4). The dashed curve in figure 5 depicts Kármán’s solution for $-w_{a}$ , which saturates to $0.885/\mathit{Re}^{1/2}$ outside the boundary layer as $z$ increases.

Figure 5. Velocity on the axis near the rotating disk at $\mathit{Re}=500$ ; the dashed curve corresponds to the Kármán boundary layer.

The difference between the solid and dashed curves in figures 4 and 5 is provided by the meridional circulation which develops due to the presence of the sidewall. This flow transformation causes an important change in the pressure pattern discussed next.

4.3. Alteration of local minimum of pressure

The local maximum of swirl vorticity results in a local minimum of pressure. The cyclostrophic balance, $\partial p/\partial r=v^{2}/r$ , and the solid-body swirl distribution, $v={\it\Omega}_{a}r$ , occurring near the axis, yield that $\partial p/\partial r={\it\Omega}_{a}^{2}r$ , i.e. the pressure is minimal on the axis at the $z$ value where ${\it\Omega}_{a}$ is maximal. Figure 6 illustrates that the location of the pressure minimum at the axis, $r=0$ , shifts away from the rotating disk, $z=0$ , as Re increases. This shift occurs because the pressure drop across the Kármán boundary layer, ${\rm\Delta}p_{K}=0.3825/\mathit{Re}$ , decreases while the pressure drop due to the meridional circulation, ${\rm\Delta}p_{c}$ , increases, as Re grows. For example, ${\rm\Delta}p_{K}=0.000765$ and ${\rm\Delta}p_{c}=0.005$ at $\mathit{Re}=500$ . The $p_{min}$ location at $\mathit{Re}=500$ (figure 6 b) is close to the ${\it\Omega}_{amax}$ location (figure 4). As Re further increases, the pressure minimum location moves away from the rotating disk.

Figure 6. Pressure contours near the rotating disk for $\mathit{Re}=1$ (a) and 500 (b). The pressure minimum location on the axis, $r=0$ , shifts for larger $z$ as Re increases.

4.4. Base-flow features at $\mathit{Re}=3100$ and $h=8$

Figure 7(a) depicts contours of azimuthal vorticity ${\it\omega}_{{\it\phi}}=\partial u/\partial z-\partial w/\partial r$ , which is positive (dark contours) near the rotating disk and the sidewall, and negative (light contours) near the axis. It is important to remark for the following stability study that the negative vorticity has a local minimum at $r=0.46$ and $z=2.95$ (bold black point in figure 7 a).

Figure 7. Azimuthal vorticity (a) and stream function (b) contours at $\mathit{Re}=3100$ and $h=8$ . The dark contours are positive and the light contours negative.

Figure 7(b) depicts streamlines, ${\it\Psi}=\text{const.}$ , at these Re and $h$ . The rotating disk ( $z=0$ ) pushes the fluid to the periphery near the disk and develops the clockwise circulation. The streamlines are packed in the Kármán boundary layer near the rotating disk, form the annular jet near the sidewall ( $r=1$ ), turn around, and constitute the backflow near the axis ( $r=0$ ). Here, ${\it\Psi}$ reaches its maximal magnitude at $r=0.68$ and $z=1.58$ (bold black point in figure 7 b).

Figure 8 depicts the $z$ -dependence at the axis, $r=0$ , of $w_{a}$ , ${\it\Omega}_{a}$ , and $p$ . Pressure $p$ is multiplied by 15 for convenient observation. Functions ${\it\Omega}_{a}(z)$ and $w_{a}(z)$ have large-magnitude derivatives near $z=0$ , related to the Kármán boundary layer (see the dashed curves in figures 4 and 5). In this layer, ${\it\Omega}_{a}$ drops from 1 at $z=0$ down to its local minimum 0.033 at $z=z_{K}=0.17=9.2/\mathit{Re}^{1/2}$ .

Figure 8. Distribution of velocity $w_{a}$ , angular velocity ${\it\Omega}_{a}$ , and pressure $p$ , as labelled, at the axis for $\mathit{Re}=3100$ .

Pressure $p$ and axial velocity $w_{a}$ reach their minimal values at $z=2.5$ and 2.7 respectively, while ${\it\Omega}_{a}$ reaches its local maximum at $z=2.7$ . Therefore, near these $z$ values, the circulation flow is strong. All quantities presented in figure 8 are negligibly small for $z>z_{c}\approx 7$ where ${\it\Omega}_{a}<1/\mathit{Re}$ , i.e. the flow is creeping in the region, $z_{c}<z<h$ .

4.5. Three sub-regions of base flow

Summarizing the above results, we can distinguish the following flow regions for $h\gg 1$ :

1. (i) a Kármán boundary layer in the range $0<z<z_{K}=9.2/\mathit{Re}^{1/2}$ ;

2. (ii) circulation flow in the range $z_{K}<z<z_{c}$ ;

3. (iii) creeping flow in the range $z_{c}<z<h$ .

The strength of the circulation flow (ii) can be characterized by the maximal magnitude of velocity at the axis, $|w_{a}|_{max}$ , or by $\mathit{Re}_{a}=|w_{a}|_{max}\mathit{Re}$ . For example, we have $|w_{a}|_{max}=0.0734$ and $\mathit{Re}_{a}=227$ at $\mathit{Re}=3100$ . As shown below, the instability emerges in the middle of the circulation region (ii).

5.1. Critical parameters for $h>3$

Figure 9 depicts numerical data for the dependence of (a) critical Reynolds number, $\mathit{Re}_{cr}$ , and (b) critical disturbance frequency, ${\it\omega}$ , on cylinder aspect ratio $h$ . The filled square symbols represent our results, which are also denoted by letter ‘M’. The other symbols, denoted by letter ‘S’, correspond to the results of Sorensen et al. (Reference Sorensen, Gelfgat, Naumov and Mikkelsen2009). Numbers next to ‘M’ and ‘S’ in figure 9 indicate the values of $m$ for the critical disturbances. Our results agree with results of Sorensen et al. (Reference Sorensen, Gelfgat, Naumov and Mikkelsen2009) for $h\leqslant 5.5$ and reveal that $\mathit{Re}_{cr}$ , ${\it\omega}_{cr}$ , and $m_{cr}$ are nearlly $h$ -independent for $h>5.5$ , as can be checked with the numerical values shown in table 1.

Figure 9. Critical values of (a) Reynolds number ( $\mathit{Re}_{cr}$ ) and (b) disturbance frequency ( ${\it\omega}$ ) for cylinder aspect ratio $3.2<h<8$ . Letters mark our (M) and Sorensen et al.’s (S) results, respectively. Numbers next to ‘M’ and ‘S’ indicate helix value $m$ .

5.2. Spatial distribution of disturbance energy

To help understand why the critical parameters, presented in figure 9 and table 1, are nearly $h$ -independent, figure 10 depicts contours $E_{d}=\text{const.}$ on the $r{-}z$ plane for $h$ varying from 5.5 to 8 at $\mathit{Re}=3100$ ; $E_{d}$ is the time-averaged kinetic energy of critical disturbance normalized by its maximal value, $E_{dm}$ . The outermost contours in figure 10 correspond to $E_{d}=0.1E_{dm}$ and show that $E_{d}$ is localized in the middle of circulation flow region (ii). The maximum $E_{d}$ is located at $r=r_{E}=0.52$ and $z=z_{E}=2.3$ . These values and the $E_{d}$ contours are nearly the same for all $h$ in figure 10.

Figure 10. Contours of constant kinetic energy $E_{d}$ of critical disturbance for $h=5.5$ (a), 6 (b), 6.5 (c), 7 (d), 7.5 (e), and 8 (f) at $\mathit{Re}=3100$ .

Figure 11 depicts the $z$ -dependence of $E_{d}$ and the base-flow velocity at the axis, $-w_{a}$ . Both $E_{d}$ and $-w_{a}$ are normalized by their maximal values. Here, $E_{d}$ is shown at the radial coordinate of the $E_{d}$ peak, i.e. $r=0.52$ (figure 10). Figure 11 confirms that the disturbance energy is localized in the middle of circulatory flow region (ii) and shows that $E_{d}$ vanishes both in the Kármán boundary layer region (i) and in the region (iii) near $z=h$ where the flow is creeping. This feature helps understand why the critical parameters are invariant for $5.5<h<8$ (figure 9), and indicates that the invariance can be expected for larger $h$ as well.

Figure 11. $z$ -dependence of base flow velocity, $w_{a}$ , at axis $r=0$ , and kinetic energy of critical disturbance, $E_{d}$ , at $r=0.52$ ; for $h=8$ and $\mathit{Re}=\mathit{Re}_{cr}$ .

Figure 12 depicts the $r$ -dependence of base-flow velocities $u,v,w$ , and disturbance energy $E_{d}$ at $z=2.3$ , which is the axial coordinate of the $E_{d}$ peak, for $h=8$ and $\mathit{Re}=\mathit{Re}_{cr}$ (table 1). The velocities are normalized by $|w(0)|$ and $E_{d}$ is normalized by its maximal value. Figure 12 reveals that the radial velocity $u$ of the base flow is very small compared to the axial $w$ and swirl $v$ velocities, while maximal magnitudes of $w$ and $v$ are comparable. Figure 12 also shows that $E_{d}$ is localized in the radial direction as well.

Figure 12. $r$ -dependence of base-flow velocities $u,v,w$ , and kinetic energy of critical disturbance $E_{d}$ at $z=2.3$ , for $h=8$ and $\mathit{Re}=\mathit{Re}_{cr}$ .

The peak location of $E_{d}$ (figure 10) is close to the location of minimum base-flow azimuthal vorticity (see the bold black point in figure 7 a). This feature implies that the instability could be of the shear-layer type, caused by the presence of an inflection point in the radial profile of axial velocity (see curve for $w$ in figure 12).

Since $E_{d}$ is localized in $1<z<4$ (figure 11), we guess that the instability is a result of the strong counterflow developing in the central part of circulatory region (ii). The central part of the flow also remains nearly invariant as $h$ increases from 5.5 to 8. A suitable characteristic of the strong counterflow is $\mathit{Re}_{a}=|w_{a}|_{max}\,\mathit{Re}$ , whose critical value is 239 at $h=8$ .

7.1. Shear-layer instability

Table 3 shows how the critical parameters depend on sidewall-to-disk angular velocity ratio ${\it\Omega}_{s}$ at $h=8$ . There $z_{w}~(z_{E})$ is the axial coordinate where the magnitude of base-flow axial velocity $w$ (disturbance energy $E_{d}$ ) reaches its maximal value. Table 3 reveals that even a weak co-rotation causes a significant increase in $\mathit{Re}_{cr}$ . In contrast to $\mathit{Re}_{cr}$ , the critical $\mathit{Re}_{a}$ is nearly invariant and even slightly decreases as ${\it\Omega}_{s}$ grows.

Note that the $z$ -location of the $E_{d}$ peak ( $z_{E}$ ) is smaller than that for the maximum of $-w_{a}$ ( $z_{w}$ ) for all ${\it\Omega}_{s}$ values in table 3. Therefore, the instability develops where the base flow decelerates near the axis. This feature agrees with the destabilizing effect of base-flow deceleration typical of swirling and swirl-free jets (Shtern Reference Shtern2012).

Figure 14 depicts the $z$ -dependence of (a) velocity $w_{a}$ , (b) angular velocity ${\it\Omega}_{a}$ , and (c) pressure $p$ along the axis, $r=0$ , of the base flow for the parameter values listed in table 3. Figure 14 reveals the following effects of the sidewall co-rotation.

1. (a) The co-rotation kills creeping flow in the region (iii) and induces a tornado-like motion near the stationary disk. The swirling flow converges to the axis near the disk and forms a strong near-axis jet directed away from the disk. The motion mechanism is similar to that driving the Bödewadt flow. According to the Bödewadt (Reference Bödewadt1940) solution, the flow characteristics are ‘wavy’ at the axis. The dotted lines in figure 14(b,c) are also ‘wavy’, having local minima and maxima, near the stationary disk located at $z=h=8$ .

2. (b) The $w$ maximal magnitude decreases as the co-rotation increases. This feature follows from the swirl decay mechanism, SDM, which drives the meridional circulation (§ 1). The co-rotation increases angular velocity ${\it\Omega}_{a}$ near the stationary disk (figure 14 b) and thus reduces the swirl decay and the magnitude of pressure variations in the $z$ direction (figure 14 c). This weakens circulation flow in the region (ii).

3. (c) The location of the $w$ maximal magnitude shifts toward the stationary disk as ${\it\Omega}_{s}$ increases. A local maximum of $w_{a}$ emerges near the rotating disk (dotted and dashed curves in figure 14 a). This is a precursor of a VB developing as ${\it\Omega}_{s}$ further increases (dash-dotted curve in figure 14 a).

Figure 14. Effect of sidewall co-rotation on $z$ -dependence of (a) base-flow velocity $w_{a}$ , (b) angular velocity ${\it\Omega}_{a}$ , and (c) pressure $p$ on axis $r=0$ ; for ${\it\Omega}_{s}=0$ , 0.01, 0.02 and 0.025 at critical parameters listed in table 3.

It is known that as Re increases in elongated ( $h>2.7$ ) cylinders with one rotating disk and the other walls being stationary, a VB first develop near the rotating disk (Iwatsu Reference Iwatsu2005). Even for the unsteady 3D flow in the $h=4.5$ cylinder, a VB first develops near the rotating disk at $\mathit{Re}_{VB1}=3200$ and then near the stationary disk at $\mathit{Re}_{VB2}=3800$ (Kulikov et al. Reference Kulikov, Mikkelsen, Naumov and Okulov2014). It is interesting that these numbers are only slightly smaller than those for the steady axisymmetric flow.

The physical reason for VB development as ${\it\Omega}_{s}$ increases is also the SDM. The solid curve in figure 14(b) shows that ${\it\Omega}_{a}$ sharply drops from its maximal value ${\it\Omega}_{a}=1$ at $z=0$ , as $z$ increases within the Kármán boundary layer. Figure 14(b) also shows that ${\it\Omega}_{a}$ drops from its local maximum at the centre of the circulation flow region (ii) (at $z=2.7$ in figure 14 b) as $z$ decreases. For larger Re, these ${\it\Omega}_{a}$ maxima become more separated and the in-between minimal ${\it\Omega}_{a}$ decreases. Accordingly, pressure has two local minima separated by the local maximum. The fluid moves against the increasing pressure as $z$ decreases from 3.52 down to 0.4 (see dotted line in figure 14 c) which decelerates the backflow (weakened by effect b) and causes its reversal as Re further increases.

This emergence of a VB does not contradict the result of Shtern et al. (Reference Shtern, Torregrosa and Herrada2012) that the sidewall co-rotation suppresses VBs, because as ${\it\Omega}_{s}$ increases in that work Re is fixed, while here Re significantly grows. This growth occurs because increasing ${\it\Omega}_{s}$ weakens circulation flow (ii) and thus suppresses the shear-layer instability. For this instability, $\mathit{Re }=7361$ is marginal at ${\it\Omega}_{s}=0.025$ (table 3), but the flow is centrifugally unstable at these Re and ${\it\Omega}_{s}$ as discussed below.

7.2. Centrifugal instability

The square symbols and curve K4 in figure 15 represent the marginal and critical Reynolds numbers of the shear-layer instability listed in table 3. The other symbols and curves in figure 15 represent the marginal and critical Reynolds numbers related to a different kind of instability. They are marked by letter ‘T’ followed by the corresponding value of $m$ . The letters denote the Kelvin (K) and Taylor (T) instabilities. The critical and marginal Re values, related to the new (T) instability, are smaller than those related to the shear-layer (K) instability for ${\it\Omega}_{s}>0.0212$ .

Figure 15. Dependence on sidewall angular velocity ${\it\Omega}_{s}$ of marginal and critical Reynolds numbers for the shear-layer (K) and centrifugal (T) instabilities at $h=8$ . Azimuthal wavenumbers $m$ of marginal disturbances are shown next to T and K.

To clarify the nature of T-instability, it is helpful to explore the spatial distribution of time-averaged kinetic energy $E_{d}$ for marginal disturbances. As an example, figure 16 depicts the $E_{d}$ contours of the disturbance at ${\it\Omega}_{s}=0.03$ , $m=4$ and $\mathit{Re}=4633$ . This Re is the smallest among those corresponding to the diamond symbols in figure 15.

Figure 16. Energy contours for marginal disturbance at ${\it\Omega}_{s}=0.03$ , $\mathit{Re}=4633$ , and $m=4$ .

The energy distribution, depicted in figure 16, indicates that this disturbance is localized near the sidewall. The maximum of disturbance energy $E_{d}$ is located at $r=r_{E}=0.91$ and $z=z_{E}=1.85$ (compare with $r_{E}=0.52$ and $z_{E}=2.3$ for the shear-layer instability in figure 10). Therefore, it is instructive to analyse $r$ -profiles of base-flow velocity components at $z=z_{E}=1.85$ , depicted in figure 17.

Figure 17. $r$ -dependence of base-flow velocities $u$ , $v$ and $w$ at $z=1.85$ for $h=8$ , ${\it\Omega}_{s}=0.03$ , and $\mathit{Re}=4633$ .

It is clear from figure 17 that the instability is not due to radial velocity $u$ , since $u$ is negligibly small compared with $v$ and $w$ . The maximal swirl velocity, $v_{m}=0.109$ , is about three times the maximal axial velocity, $w_{m}=0.0367$ . The fact that $v_{m}$ is significantly larger than swirl velocity $v=0.03$ of the sidewall, $r=1$ , indicates that the angular momentum at $z=1.85$ is mostly provided by the flow transport from the rotating disk.

The dominance of $v$ over $u$ and $w$ points out that the swirl velocity distribution might be responsible for the instability. The instability can be centrifugal, similar to that occurring in the Taylor–Couette flow (Chandrasekhar Reference Chandrasekhar1961). To check this guess, figure 18 depicts the $r$ -profiles of the base-flow squared angular momentum $(rv)^{2}$ and marginal-disturbance energy $E_{d}$ at $z=z_{E}=1.85$ . Both $(rv)^{2}$ and $E_{d}$ are normalized by their maximal values in figure 18.

Figure 18. $r$ -dependence of base-flow squared angular momentum $(rv)^{2}$ and energy $E_{d}$ of marginal disturbance at $z=1.85$ for ${\it\Omega}_{s}=0.03$ , $\mathit{Re}=4633$ , $m=4$ , and $h=8$ .

According to the Rayleigh criterion, no centrifugal instability can develop if $(rv)^{2}$ grows with $r$ (Chandrasekhar Reference Chandrasekhar1961). The necessary condition for the centrifugal instability is the decaying of $(rv)^{2}$ as $r$ increases (as occurs in the Taylor–Couette flow where the inner (outer) cylinder is rotating (stationary)). The pattern in figure 18 exactly matches the Rayleigh criterion: $E_{d}$ peaks in the middle of the region where $(rv)^{2}$ reduces as $r$ increases; and $E_{d}$ becomes negligibly small in the region where $(rv)^{2}$ grows with $r$ .

In addition, the centrifugal instability induces Taylor cells in the circular Couette flow (Chandrasekhar Reference Chandrasekhar1961). Similarly, figure 16 shows two peaks of $E_{d}$ , indicative of two cells of disturbance motion. The Taylor cells are steady, but become unsteady if the base flow includes an axial stream. The frequency of time oscillations increases with the base-flow axial velocity. This feature agrees with that found here: the above discussed marginal disturbance has low frequency, ${\it\omega}_{i}=-0.096$ , compared to ${\it\omega}_{i}=0.405$ for the shear-layer mode. It is reasonable that the frequency is low because the axial flow is weak in relation to the swirl (compare figures 12 and 17).

Based on the above analysis, we conclude that the low-frequency instability is centrifugal. The centrifugal instability also develops in the Vogel–Escudier flow, i.e. at ${\it\Omega}_{s}=0$ (see curves T3 and T4 in figure 15 and data in table 4 at ${\it\Omega}_{s}=0$ ), but for larger Re than $\mathit{Re}_{cr}=3100$ at which the shear-layer instability emerges. As ${\it\Omega}_{s}$ increases, the azimuthal wavenumber $m$ of critical disturbances also increases, as figure 15 and table 4 show.

The critical Re values correspond to the shear-layer disturbance modes with $m=4$ for $0<{\it\Omega}_{s}<0.0212$ , and to the centrifugal disturbance modes with $m=3$ for $0.0212<{\it\Omega}_{s}<0.0505$ , $m=4$ for $0.0505<{\it\Omega}_{s}<0.085$ , $m=5$ for $0.085<{\it\Omega}_{s}<0.1$ (and so on).

Figure 19 depicts the energy distribution of critical disturbance at $\mathit{Re}=5192$ , ${\it\Omega}_{s}=0.1$ , and $m=5$ . The pattern of $E_{d}$ contours in figure 19, being similar to that in figure 16, indicates that the instability remains centrifugal as ${\it\Omega}_{s}$ and $m$ increase (figure 15). There are three $E_{d}$ peaks in figure 19, while figure 16 depicts only two $E_{d}$ peaks, i.e. the number of critical-disturbance cells grows as ${\it\Omega}_{s}$ and $m$ increase.

Figure 19. Energy contours for critical disturbance at $\mathit{Re}=5192$ , ${\it\Omega}_{s}=0.1$ , and $m=5$ .

The sidewall co-rotation suppresses the centrifugal instability as well: critical Re grows with ${\it\Omega}_{s}$ (as figure 15 and table 4 show). The stabilizing effect is stronger (weaker) for the shear-layer (centrifugal) instability. The difference is due to the sidewall co-rotation more strongly (weakly) suppressing the meridional motion near the axis (wall) as ${\it\Omega}_{s}$ increases.

Comparison of $w$ profiles in figures 12 and 17, shows that the velocity at the axis, $r=0$ , is drastically reduced at ${\it\Omega}_{s}=0.03$ compared to that at ${\it\Omega}_{s}=0$ . The significant shear-layer characteristic, $w_{max}-w_{min}$ , is also reduced, being 0.067 (0.11) at ${\it\Omega}_{s}=0.03$ (0). These reductions, occurring despite $\mathit{Re}=4633$ (3100) being larger (smaller) at ${\it\Omega}_{s}=0.03$ (0), explain the strong suppression of the shear-layer instability by the sidewall co-rotation.

In contrast, $w_{max}=0.0329$ (0.0367) at ${\it\Omega}_{s}=0$ (0.03) increases, i.e. the near-wall flow, transporting the angular momentum from the rotation disk along the sidewall, remains rather strong as ${\it\Omega}_{s}$ grows. This explains the relatively weak suppression of the centrifugal instability by the sidewall co-rotation.