2 Axisymmetric vortices in narrow-gap cylinders

Consider a Taylor–Couette flow governed by the incompressible Navier–Stokes equations in infinitely long rotating concentric cylinders with radius ratio $\unicode[STIX]{x1D702}<1$ . Following convention, the gap and fluid density are scaled to be unity. The viscous velocity scale is chosen for the velocity components, and thus the Reynolds numbers appear in the boundary conditions rather than in the governing equations. Hence, in cylindrical coordinates $(r,\unicode[STIX]{x1D703},z)$ , the normalised azimuthal velocity $V$ must satisfy the no-slip conditions on the cylinder walls:

where $R_{i}$ and $R_{o}$ are the cylinder speeds non-dimensionalised by the kinematic viscosity of the fluid and the gap.

In this section we assume that the disturbance is axisymmetric. Defining the streamfunction $\unicode[STIX]{x1D711}$ , where the radial and axial velocity components $U$ and $W$ are written as $-\unicode[STIX]{x1D711}_{z}$ and $r^{-1}(r\unicode[STIX]{x1D711})_{r}$ , respectively, the Navier–Stokes equations become

Here note that $-\unicode[STIX]{x0394}\unicode[STIX]{x1D711}=-(\unicode[STIX]{x1D711}_{rr}+r^{-1}\unicode[STIX]{x1D711}_{r}-r^{-2}\unicode[STIX]{x1D711}+\unicode[STIX]{x1D711}_{zz})$ describes the streamwise vorticity. In order to perform the stability analysis, we linearise the governing equations around the basic flow $V=V_{b}(r)$ ,

We further simplify the equations by taking the narrow-gap limit $\unicode[STIX]{x1D702}\rightarrow 1$ . Since we have fixed the gap to be unity, the limit corresponds to $r_{i},r_{o}\sim (1-\unicode[STIX]{x1D702})^{-1}\rightarrow \infty$ . Thus we define an $O((1-\unicode[STIX]{x1D702})^{0})$ gap variable $y=r-\bar{r}$ with some representative radius $\bar{r}\in [r_{i},r_{o}]$ , which is of course $O((1-\unicode[STIX]{x1D702})^{-1})$ . In order to take the limit, it is convenient to write $V=V_{b}+2Av$ in (2.2) with the scaled velocity $v$ , to get

where $\unicode[STIX]{x1D6EF}(r)\equiv -4AV_{b}/r$ is called the Rayleigh discriminant. In order to keep the terms on the right-hand side of (2.4a ), we see that $\unicode[STIX]{x2202}_{z}\sim O(\overline{r}^{0})$ , and that $v$ and $\unicode[STIX]{x1D711}$ are comparable in size. Then, when $\bar{r}\gg y$ , the Rayleigh discriminant on the right-hand side of (2.4b ) expands to

where $\overline{V}_{b}\equiv V_{b}(\overline{r})=\unicode[STIX]{x1D6FE}R_{o}+(1-\unicode[STIX]{x1D6FE})R_{i}$ and $\unicode[STIX]{x1D6FE}\equiv \overline{r}-r_{i}$ . Here $\unicode[STIX]{x1D6FE}$ is chosen to be $1/2$ for later convenience. If we assume $T\equiv -4A\overline{V}_{b}/\overline{r}$ and $G\equiv 8A^{2}/\overline{r}$ are $O(\overline{r}^{0})$ , then $\unicode[STIX]{x1D6EF}\rightarrow [T-Gy]\sim O(\overline{r}^{0})$ as $\unicode[STIX]{x1D702}\rightarrow 1$ . Therefore, assuming an infinitesimally small disturbance of normal mode form $(v,\unicode[STIX]{x1D711})=(\widetilde{v},\widetilde{\unicode[STIX]{x1D711}})\exp [\unicode[STIX]{x1D70E}t+\text{i}kz]$ with $\widetilde{v}=\widetilde{\unicode[STIX]{x1D711}}=\unicode[STIX]{x2202}_{y}\widetilde{\unicode[STIX]{x1D711}}=0$ on the cylinder walls, we have the eigenvalue problem for a complex growth rate $\unicode[STIX]{x1D70E}$ :

This problem can be solved numerically by the Chebyshev collocation method to obtain the stability curve in terms of $(T,G)$ . In order to compare the stability to that for the wide-gap cases, it is convenient to rescale $R_{i}$ and $R_{o}$ as These scaled Reynolds numbers are $O(\overline{r}^{0})$ if $T,G\sim O(\overline{r}^{0})$ and therefore, as shown in figure 2, in the $R_{i}^{+}$ – $R_{o}^{+}$ plane the neutral curve for the wide-gap case converges to the narrow-gap result as $\unicode[STIX]{x1D702}\rightarrow 1$ .

Now, let us perform a rapid-rotation asymptotic analysis of (2.6). First we focus on the corotating case. For this case, we may assume that $|B|$ is large but $|A|$ is small, since we already know that the neutral curve is close to the Rayleigh line where $A=0$ . In the context of the narrow-gap limit, this means that we are concerned with the limit of small $G/T$ . This special case corresponds to the rotating plane Couette flow (RPCF) investigated, for example, by Nagata (Reference Nagata1986). For the RPCF, $\bar{r}$ is taken at the midgap, and thus we have chosen $\unicode[STIX]{x1D6FE}$ to be $1/2$ . It is noteworthy that, as is well known, streamwise independent RPCF has an exact analogy to Rayleigh–Bénard convection, and the Rayleigh discriminant, which now becomes a constant called the Taylor number $T$ , plays the role of the Rayleigh number (Veronis Reference Veronis1970). In this analogy, a positive/negative Rayleigh discriminant corresponds to an unstable/stable stratification. In figure 2, the results for RPCF are plotted as the blue solid curve using the well-known critical value of the Taylor number $T=1707.8=R_{i}^{+2}-R_{o}^{+2}$ . Near the Rayleigh line $R_{i}^{+}\approx R_{o}^{+}$ , and the asymptotic stability curve is written as

It is now clear that the Taylor number describes the leading-order effect of the viscous correction.

Figure 2. The stability of Taylor–Couette flow under axisymmetric disturbances. The value of the axial wavenumber is optimised. The rescaled parameters $R_{i}^{+}$ and $R_{o}^{+}$ are defined in (2.7). The magenta dot-dashed curve represents the results for $\unicode[STIX]{x1D702}=5/7,0.9,0.99$ . The result for $\unicode[STIX]{x1D702}=0.99$ is almost indistinguishable from the narrow-gap results indicated by the black thick dashed curve. The red and blue solid curves are asymptotic results for the narrow-gap limit.

Figure 3. The solid curve is the neutral disturbance of the narrow-gap limit equations (2.6) at $(G,T,k)=(1.6\times 10^{6},-1.22\times 10^{6},8.6)$ . The dashed line is the corresponding normalised Rayleigh discriminant $\unicode[STIX]{x1D6EF}/G=T/G-y$ .

Figure 4. The thick dashed curves are the neutral curve of the narrow-gap problem (2.6) and the optimised value of $k$ . These results are rescaled by the near-wall-layer thickness $G^{-1/5}$ . The dot-dashed lines are the large- $G$ asymptotic results $Q_{0}=4.11$ and $k_{0}=0.494$ computed using (2.9).

On the other hand, the neutral curve for the counter-rotating case corresponds to the limit of large $G/T$ ; also recall that it appears near the other boundary of the Rayleigh stable region, $R_{i}=0$ . The normalised Rayleigh discriminant at sufficiently large $G$ along the neutral curve of (2.6) is shown in figure 3. This time, the discriminant is not uniform, but a monotonically decreasing function in $r$ with the positive part concentrated near the inner cylinder. As expected from the analogy to the convection problem, we see that it is this positive part that drives the near-wall instability shown by the streamfunction. This observation motivates the asymptotic analysis of the limiting neutral curve for the case of counter-rotation. Assuming that $G$ is asymptotically large and all the disturbances are concentrated near the inner cylinder, we analyse the stability in the near-wall layer where the stretching variable $Y=\unicode[STIX]{x1D6FF}^{-1}(y+1/2)$ is $O(1)$ . The thickness of the boundary layer $\unicode[STIX]{x1D6FF}\ll 1$ is to be fixed in terms of $G$ . If we assume that the positive part of $\unicode[STIX]{x1D6EF}=G[(T/G+1/2)-\unicode[STIX]{x1D6FF}Y]$ is concentrated in the wall layer, the small positive constant part and the term proportional to $Y$ must be comparable in size. Thus we define an $O(1)$ quantity $Q_{0}=\unicode[STIX]{x1D6FF}^{-1}(T/G+1/2)$ to write $\unicode[STIX]{x1D6EF}=\unicode[STIX]{x1D6FF}G[Q_{0}-Y]$ . We further define the scaled wavenumber $k_{0}=\unicode[STIX]{x1D6FF}k$ to balance the wall-normal and spanwise diffusion. From the scaling, $k$ is asymptotically large, and this is indeed the case in view of the computational result. Likewise, we write the scaled growth rate as $\unicode[STIX]{x1D70E}_{0}=\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D70E}$ . Substituting the scaled variables and the expansions $\widetilde{\unicode[STIX]{x1D711}}=\unicode[STIX]{x1D6FF}^{-1}\widetilde{\unicode[STIX]{x1D711}}_{0}+\cdots \,$ , $\widetilde{v}=\widetilde{v}_{0}+\cdots \,$ into (2.6), we get the leading-order problem

The balance in the second equation gives the layer thickness $\unicode[STIX]{x1D6FF}=G^{-1/5}$ . The problem (2.9) can be solved by a standard linear eigenvalue solver to seek the neutral point where the real part of $\unicode[STIX]{x1D70E}_{0}$ vanishes. Here the equations are discretised by the Chebyshev collocation method in $Y$ , mapping $[0,\infty ]$ to $[-1,1]$ ; see Boyd (Reference Boyd2001), for example. The accuracy of the computation has been checked by changing the number of collocation points, typically taking more than 200. Along the computed neutral curve $Q_{0}(k_{0})$ , we minimise the value of $Q_{0}$ to yield the critical point $(Q_{0},k_{0})=(4.11,0.494)$ . As shown in figure 4, the scaled Navier–Stokes result computed by (2.2) quickly converges to the asymptotic result when increasing the value of $G$ . The asymptotic result can also be confirmed in figure 2, where the result is presented in terms of $R_{i}^{+}$ and $R_{o}^{+}$ based on the critical value transformed using the definition of $Q_{0}$ and (2.7). We can see that the asymptotic result shown by the red solid curve excellently approximates the counter-rotating limit of the Navier–Stokes result.

3 General case

On the basis of the basic physics of the instability given in the previous section, we now generalise the theory to a wide gap and arbitrary linear disturbances. In order to generalise the asymptotic results we introduce the toroidal–poloidal potential decomposition

where $\unicode[STIX]{x1D6E5}_{2}\equiv r^{-2}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D703}}^{2}+\unicode[STIX]{x2202}_{z}^{2}$ is the two-dimensional Laplace operator. The linearised Navier–Stokes equations can then be written in the form (see Deguchi & Nagata Reference Deguchi and Nagata2011, for example)

where

If the flow is axisymmetric, i.e. $\unicode[STIX]{x2202}_{\unicode[STIX]{x1D703}}=0$ , we recover (2.2) with the replacement of $(\unicode[STIX]{x1D711},V)$ by $(\unicode[STIX]{x1D719}_{z},V_{b}+\unicode[STIX]{x1D713}_{z})$ .

Figure 5. The asymptotic results for the corotating case. (a) The coefficient $C_{1}$ in (3.3). The filled circle is the narrow-gap result $C_{1}=1707.8$ . (b) Comparison of the neutral curve for $\unicode[STIX]{x1D702}=5/7$ (black dashed curve) and the asymptotic prediction (3.4) truncated at the second term (blue dot-dashed curve). The magenta solid line is the Rayleigh line $R_{i}=\unicode[STIX]{x1D702}^{-1}R_{o}$ .

It is known that the most unstable mode for the corotating case is axisymmetric and the neutral curve approaches the Rayleigh line, where the value of $|A|$ is small and that of $|B|$ is large. It is reasonable to assume the size of $A$ is $O(B^{-1})$ , because by substituting the Rayleigh discriminant and the basic flow (2.3), we see that the right-hand side of (2.4) depends on $A^{2}$ and $(AB)$ . Then we can numerically seek the critical value of $(AB)$ , neglecting the terms proportional to $A^{2}$ . Assuming $\unicode[STIX]{x1D711}$ and $v$ are $O(1)$ , no other term drops from (2.4) at the asymptotic limit. Thus the vortices are fully viscous, in the sense that the leading-order perturbation at the limit can be determined only by the viscous problem. Using the critical value we can calculate the coefficient

As shown in figure 5(a), reducing $\unicode[STIX]{x1D702}$ from the narrow-gap value 1, the value of $C_{1}$ decreases from $1707.8$ (shown by the filled circle, the same value as the critical Taylor number at the RPCF limit) and then reaches a minimum at $\unicode[STIX]{x1D702}\approx 0.33$ . Upon further decreasing $\unicode[STIX]{x1D702}$ , the curve shown in the figure tends to infinity as $\unicode[STIX]{x1D702}\rightarrow 0$ . The value of $C_{1}$ gives the asymptotic prediction

in $R_{i}$ – $R_{o}$ parameter space; this is a generalisation of (2.8). In figure 5(b), the Navier–Stokes result and the asymptotic prediction are compared for $\unicode[STIX]{x1D702}=5/7$ . As in the last section, the asymptotic result describes the gap between the neutral curve and the Rayleigh line.

Figure 6. The asymptotic results for the counter-rotating case. (a) The neutral solution of the asymptotic problem (3.7). The contours indicate $Q_{0}=4.3$ , 4.2, 4.1, 4.0 and 3.9. The value of $Q_{0}$ decreases in the direction of the arrow and takes the minimum value $Q_{0}=3.85$ at $(k_{0},m_{0})=(0.300,0.315)$ as indicated by the circle. The square corresponds to the minimum of $Q_{0}$ for the axisymmetric case. (b) The comparison of the asymptotic and full Navier–Stokes stability. The solid curve is the asymptotic result $C_{2}$ given in (3.8). The crosses are the neutral point at $R_{o}=-2\times 10^{5}$ based on the Navier–Stokes computations. The arrows indicate the values of the optimum azimuthal wavenumber $m$ used in the computation.

Next we shall analyse the counter-rotating case, writing the potentials in their normal mode form $(\unicode[STIX]{x1D719},\unicode[STIX]{x1D713})=(\widetilde{\unicode[STIX]{x1D719}},\widetilde{\unicode[STIX]{x1D713}})\exp [\unicode[STIX]{x1D70E}t+\text{i}m\unicode[STIX]{x1D703}+\text{i}kz]$ . As in the previous section, we may seek an asymptotic problem near the inner cylinder with asymptotically large axial wavenumber $k$ . From numerical observations, the azimuthal wavenumber $m$ also increases asymptotically (see figure 1) but at a smaller rate than $k$ . Thus we assume $\unicode[STIX]{x1D6FF}^{-1}\sim k\gg m\gg 1$ in (3.2) and neglect small terms, to get

from which we find the balance $m(V_{b}/r)\sim \unicode[STIX]{x1D6FF}^{-2}$ for non-axisymmetric disturbances. Because we already know that the instability appears near the inner wall, it is convenient to choose $\overline{r}=r_{i}$ and seek an asymptotic structure in a layer where $Y=\unicode[STIX]{x1D6FF}^{-1}(r-r_{i})\sim O(1)$ , where the layer thickness is $\unicode[STIX]{x1D6FF}=G^{-1/5}$ with $G=r_{i}^{-1}8A^{2}\gg 1$ , similar to the previous section. Substituting the scaled quantities $Q_{0}=-V_{b}|_{r=r_{i}}/(2\unicode[STIX]{x1D6FF}A)$ , $m_{0}=2Am\unicode[STIX]{x1D6FF}^{3}/r_{i}$ , $\unicode[STIX]{x1D70E}_{0}=\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D70E}$ , $k_{0}=\unicode[STIX]{x1D6FF}k$ and the potential expansions $\widetilde{\unicode[STIX]{x1D719}}=\unicode[STIX]{x1D6FF}^{-1}\widetilde{\unicode[STIX]{x1D719}}_{0}+\cdots \,$ , $\widetilde{\unicode[STIX]{x1D713}}=2A\widetilde{\unicode[STIX]{x1D713}}_{0}+\cdots \,$ into (3.2), we get the leading-order equations Here it is important to notice that (3.6) does not explicitly depend on $\unicode[STIX]{x1D702}$ , meaning that the solution of this canonical problem describes the rapidly counter-rotating asymptotic limit of the neutral curve for all $\unicode[STIX]{x1D702}$ . This system is of course a natural generalisation of (2.9) and we can solve it numerically to seek neutral values $Q_{0}(m_{0},k_{0})$ . The global minimum of $Q_{0}(m_{0},k_{0})$ is the critical value that gives the asymptotic prediction of the neutral curve. The computational result shown in figure 6(a) reveals that the minimum is at $(Q_{0},k_{0},m_{0})=(3.85,0.300,0.315)$ , as shown by the circle. The fact that the minimum is at $m_{0}\neq 0$ explains why we observe spiral disturbances near the neutral curve in the counter-rotating case. Having determined the critical value, then by definition of the scaled parameters we have the leading-order asymptotic results

where

describe the non-trivial dependence of the asymptotic stability on $\unicode[STIX]{x1D702}$ . Here we have used the fact that $|R_{o}|\gg |R_{i}|$ in the limit.

In figure 1 the asymptotic result is compared with the full Navier–Stokes result for $\unicode[STIX]{x1D702}=5/7$ . The asymptotic prediction given by (3.7), the red solid curve, is in excellent agreement with the rapid counter-rotation limit of the full neutral curve. For other values of the radius ratio, the convergence has been checked by comparing the full Navier–Stokes neutral value of $R_{i}|R_{o}|^{-3/5}$ at sufficiently large $-R_{o}$ with its asymptotic value $C_{2}$ in (3.8). We can see in figure 6(b) that the asymptotic result derived here predicts well the full computational result over a wide range of $\unicode[STIX]{x1D702}$ , although the approximation gets slightly worse for small $\unicode[STIX]{x1D702}$ because the optimum value of $m$ is restricted to be an integer value in the full computation.

4 Conclusion and discussion

An asymptotic theory is developed for the neutral curve of Taylor–Couette flow. The analysis is mathematically consistent with the rapid-rotation limit of the Navier–Stokes equations, in contrast to previous works based on a fully empirical model approach (Esser & Grossmann Reference Esser and Grossmann1996; Dutcher & Mullar Reference Dutcher and Mullar2007).

Our theory is first derived for the axisymmetric disturbances for the narrow-gap limit ( $\unicode[STIX]{x1D702}\rightarrow 1$ ) and then extended to the general case. Here we briefly discuss the implications of the results in § 2 since, although the limiting equations (2.6) were derived and solved in, for example, Drazin & Reid (Reference Drazin and Reid1981), the convergence of the wide-gap neutral curve to this limit has not been discussed. In order to observe the convergence, we found that the rescaled Reynolds numbers (2.7) must be used, as seen in figure 2. Here we further remark that the convergence can be observed even when the disturbances are non-axisymmetric. For the non-axisymmetric case, we must scale the azimuthal wavenumber $m$ so that the azimuthal convective terms balance with the viscous terms, namely $Am/\overline{r}\sim O(\overline{r}^{0})\sim O((1-\unicode[STIX]{x1D702})^{-1})$ . In the limit, the problem reduces to the Cartesian problem and the streamwise wavenumber $\unicode[STIX]{x1D6FC}\equiv m/\overline{r}$ is $O(\overline{r}^{-1/2})$ since $A\sim O(\overline{r}^{1/2})$ from (2.3) and (2.7). This means that the streamwise scale is much longer than the gap, and thus the limiting equations are similar to the Görtler vortex equations derived by Hall (Reference Hall1983), i.e. the streamwise derivatives in the pressure and viscous terms are dropped from the Navier–Stokes equations for unit Reynolds number; these equations are also called boundary region equations, see Deguchi, Hall & Walton (Reference Deguchi, Hall and Walton2013), for example. When the neutral curve is close to the Rayleigh line, the narrow-gap limit becomes RPCF, and this is the special case where there is a degree of freedom in choosing the scaling of the Reynolds numbers because we do not need to balance the term proportional to $G$ in (2.6). The freedom can be removed by further assuming that the Navier–Stokes equations are recovered in the limit, and this is in fact the case studied by Nagata (Reference Nagata1986). This assumption requires $\unicode[STIX]{x1D6FC}\sim O(\overline{r}^{0})$ , and thus $\overline{V}/\overline{r}$ , $R_{i}$ , $R_{o}$ are $O(\overline{r}^{0})$ quantities as well.

The high-Reynolds-number asymptotic analysis is performed assuming the size of $A$ and $B$ , which appear in the basic flow (2.3). From observation of the Rayleigh discriminant, it is obvious that a neutral asymptotic structure is possible when $|A|$ is small and $|B|$ is large, as long as $|AB|$ is $O(1)$ . The smallness of $|A|$ means that we are concerned with the neutral curve near the Rayleigh line, where the cylinders are corotating and the most unstable disturbance is known to take an axisymmetric form. In the asymptotic limit, we can compute the critical value of $AB$ , which gives a viscous correction of the neutral curve from the Rayleigh line (3.3). Although the correction is asymptotically small, it is important, because our analysis showed that the neutral disturbance is fully viscous even when the cylinders rotate asymptotically fast.

For the counter-rotating case, the neutral curve can be described by the asymptotic problem obtained when $|A|$ is large and $|R_{i}/R_{o}|$ is small. The asymptotic problem for the counter-rotating case can be derived by considering the thin layer near the inner cylinder. It is only in that layer that the Rayleigh discriminant remains positive and thus the vortices are localised. This asymptotic vortex structure is quite different from that for the corotating limit, occupying the whole gap. Within the layer, asymptotic balance gives the non-trivial asymptotic behaviour $R_{i}\propto R_{o}^{3/5}$ , $k\propto R_{o}^{2/5}$ and $m\propto R_{o}^{1/5}$ . Further analysis gives an analytic expression for the neutral curve, with the coefficients fixed by the numerical computation of the canonical asymptotic problem, valid for all $\unicode[STIX]{x1D702}$ , as shown in (3.6). The asymptotic result predicts that the most unstable mode is non-axisymmetric, consistent with the fact that spiral vortices are observed in the counter-rotating Taylor–Couette experiments.

The present linear results may also be important in building asymptotic theories for nonlinear vortices in Taylor–Couette flow, where the torque scaling is a matter of great concern for recent experiments. When the amplitude of the disturbance is $O(1)$ or less, a nonlinear theory can be found by a simple extension of the viscous linear theory here. For larger amplitudes, the viscous nonlinear asymptotic structure may break down to produce inviscid vortices at the asymptotic limit. It is noteworthy that, from the form of (2.2), such inviscid vortices naturally develop the zero-absolute vorticity states observed, for example, in Johnston, Halleen & Lezius (Reference Johnston, Halleen and Lezius1972); see appendix A. In that inviscid case, the viscous layer must appear around the cell perimeter to satisfy the boundary conditions, as seen in the high-Reynolds-number direct numerical simulations (see Ostilla et al. Reference Ostilla, Stevens, Grossmann, Verzicco and Lohse2013, for example). The three-dimensional extension of such a fully nonlinear theory is also of interest since it may describe the nonlinear structures that could even reach the Rayleigh stable region (Deguchi et al. Reference Deguchi, Meseguer and Mellibovsky2014).