We study transition from circular Couette flow to Taylor–Couette flow in the narrow-gap limit. We find that subcritical bifurcations of a nonlinear axisymmetric flow occur at the linear critical point when the angular velocity ratio for the cylinders, $μ$ , is less than approximately −0.5. We also demonstrate that two nonlinear axisymmetric flow branches with the same wavenumber as that for the critical mode exist for $μ = 0$ . Introducing the modified Taylor number $τ$ , analyses for any $μ < 1$ are made possible, so that this study supplements the recent analysis of Nagata M. 2023 Taylor-couette flow in the narrow-gap limit. Philos. Trans. R. Soc. A 381, 20220134. (doi:10.1098/rsta.2022.0134), in which the case $μ = - 1$ was excluded, because the critical value of the conventional Taylor number $T a$ used therein approaches zero as $μ \to - 1$ . The present analysis takes into account all the contributions from the inertia term, including a component which was neglected in [1]. The effect of the inclusion of this previously neglected component is surveyed at the end.

1. Introduction

Taylor–Couette flow is one of the most heavily explored flow configurations in fluid mechanics that serves as a theoretical and experimental testing ground to analyse the problem of flow instabilities and the transition to turbulence. In the seminal experimental work [1] by Andereck et al. [2], various types of flow state were identified in the parameter space based on the Reynolds numbers, $R_{o}$ and $R_{i}$ , which are defined with respect to the outer and the inner cylinder angular velocities and their radii. The radius ratio used in this experiment was 0.883, and one necessary question to answer is how the experimental observations of the transitions from one flow to another change when the gap is reduced to zero, that is, when the radii are increased to infinity. In this so-called narrow-gap limit, the circular geometry of the cylinder walls approaches a plane geometry.

The similarity between Taylor–Couette flow and Rayleigh–Bénard convection is well-known ([3,4] and [5]) and, in fact, axisymmetric Taylor–Couette flow in the narrow-gap limit becomes mathematically equivalent to two-dimensional Rayleigh–Bénard convection of a fluid with a unit Prandtl number (see, for instance, [6]).

Recently, while large-Rayleigh-number thermal convection has been investigated numerically [7] and experimentally [8], high-Reynolds-number Taylor–Couette turbulence has also gathered attention ([9] and [10], also see references in [11]).

The analogy between Taylor–Couette flow and Rayleigh–Bénard convection is utilized for the comparison of physical quantities, such as momentum transport and energy dissipation, occurring in the turbulent states of the two flows. In particular, the similarity has been investigated using plane Couette flow and rotating plane Couette flow, as these flows are believed to correspond to Taylor–Couette flow in the limit of a narrow gap between the cylinders ([12] and [13]).

Noting that there exists a distinct difference between Taylor–Couette flow and plane Couette flow, namely, that Taylor–Couette flow becomes unstable to axisymmetric disturbances, whereas plane Couette flow is always stable to infinitesimal disturbances, Nagata [1] reviewed the way of representing the Taylor–Couette system in the narrow gap in a Cartesian coordinate system.

The experimental observations by [2,14] and [15] for $η = 0.883, 0.912$ and $0.959$ , respectively, where $η \equiv r_{i} / r_{o}$ is the radius ratio of the inner cylinder of radius $r_{i}$ to the outer cylinder of radius $r_{o}$ , show that the onset of instability of circular Couette flow occurs at a Reynolds number (based on the inner cylinder motion) of order 100 when the outer cylinder is at rest, for example. Given such moderate Reynolds numbers, the study of [1] neglected the nonlinear perturbation term, ${\tilde{u}}_{φ}^{2} / r$ , resulting from the inertia term of the Navier–Stokes equations expressed in the original cylindrical coordinate system (see the last paragraph of this section). This is in contrast to [16,17], who considered the case when this inertia term remains large in this coordinate system. Furthermore, the assumption of high-Reynolds-number or rapid-rotation flow considered by [18] were also not applied in [1]. With this modelling assumption, the order of magnitude of the nonlinear perturbation term resulting from the inertia in the original cylindrical coordinate system, which influences the development of the flow structures, is determined only after the linear stability analysis as described below.

The equations presented in [1] are reproduced in §§2 and 3 below. The difference from the conventional representation of rotating plane Couette flow, where the control parameters are the Reynolds number $R e$ and the system rotation $Ω$ , is the existence of an additional parameter $Q$ , which results from the consideration of the inertia term in the equations of motion expressed in a cylindrical coordinate system. It is shown that both $Ω$ and $Q$ are $(1 - η)$ -order smaller than $R_{e}$ . At first glance this may appear to suggest all Taylor–Couette flows in the narrow-gap limit, $η \to 1$ , are reduced to plane Couette flow. This is, however, true only when $R_{e}$ is of order 1. As pointed out in [1] , when $R_{e}$ is of order $1 / \sqrt{1 - η}$ so that $Ω$ is of order $\sqrt{1 - η}$ , the product of $R_{e}$ and $Ω$ can be finite, and the Taylor number defined by $T a = Ω (R_{e} - Ω)$ can take a finite critical value at the onset of axisymmetric instabilities. The numerical results for the critical Taylor number $T a_{c}$ calculated by [1] were found to be in excellent agreement with previous studies.

It was also shown in [1] that $Q$ may be expressed by $Q = 2 (μ - 1) Ω / (1 + μ)$ , where $μ = ω_{o} / ω_{i}$ is the ratio of the angular velocities of the outer to the inner cylinders, with angular velocities $ω_{o}$ and $ω_{i}$ , respectively. Therefore, $Q$ is not determined when $μ = - 1$ , that is, when $ω_{o} = - ω_{i}$ . If $Ω$ is instead written in terms of $Q$ , we have that $Ω = 0$ for $μ = - 1$ , and so $T a = 0$ , making it impossible to determine its critical state $T a_{c}$ , and so this case was excluded from the analysis of [1]. We stress here that the case $μ = - 1$ does not correspond to an inherent physical singularity of this flow. The inability to obtain $T a_{c}$ for $μ = - 1$ is due to an artefact where $Q$ is expressed by means of $Ω$ . We add here that Davey et al. [16] (page 22) note that the replacement of $T a$ by $T^{'} \equiv \frac{2}{1 + μ} T a$ is more appropriate when the cylinders rotate in the opposite direction, so that $T^{'}$ may take a finite value for $μ = - 1$ .

In this study, we consider the cases for $μ < 1$ , which supplements the analysis of Nagata [1], by including the case $μ = - 1$ . We find that the critical state for the onset of axisymmetric instability for $μ < 1$ is determined by the product of $Q$ and $R e$ , i.e. the instability sets in at $τ \equiv Q R e = τ_{c}$ . This means that although our system approaches that of plane Couette flow as $Q \to 0$ , the critical state exists even in the limit as $R e \to \infty$ . We call $τ$ the modified Taylor number; its relation with $T a$ is given by $T a = C (μ) τ$ where $C (μ) = (1 + μ) / 2 (1 - μ)$ , when $μ \neq \pm 1$ .

The aim of the current paper is to investigate transition from circular Couette flow to Taylor-vortex flow in the narrow-gap limit for the for $μ < 1$ case, including the case which was excluded in the study by [1]. This investigation contributes towards understanding further transitions to non-axisymmetric flows.

We review briefly the analysis of the narrow gap limit by Nagata (2023) in §§2 and 3, followed by an analysis in §4, where we show that among two contributions, $2 U {\tilde{u}}_{φ} / r$ and ${\tilde{u}}_{φ}^{2} / r$ , derived from the inertia term, $(U + {\tilde{u}}_{φ})^{2} / r$ , the latter, which was neglected in Nagata (2023) in the limit as $r \to \infty$ , becomes important in the region of $R e \sim O (1 / \sqrt{1 - η})$ , and so is here taken into account in the nonlinear analysis after proper scalings for variables are introduced. (The former contribution leads to the term with $Q$ in the equation of motion.) After describing the numerical procedures in §§5 and 6, we present the results for the counter-rotating case $μ = - 1$ and the case with the outer cylinder at rest in §7; discussion is provided in §8. Finally, an appendix devoted to Nagata (2023) is included.

2. Formulation of the problem

We consider incompressible motion of a fluid with kinematic viscosity $ν$ and density $ρ$ , which is confined between two coaxial cylinders of infinite extent in the axial direction, where the inner and outer cylinders, with radii $r_{i}$ and $r_{o}$ , rotate with angular velocities, $ω_{i}$ and $ω_{o}$ , respectively. We assume that $ω_{i} \neq 0$ .

The equations of motion in a cylindrical coordinate frame $(r, φ, z)$ that rotates with an angular velocity $Ω_{r f} = (ω_{i} + ω_{o}) / 2$ are given by

\begin{matrix} \frac{\partial u_{r}}{\partial r} + \frac{u_{r}}{r} + \frac{1}{r} \frac{\partial u_{φ}}{\partial φ} + \frac{\partial u_{z}}{\partial z} = 0, \end{matrix}

(2.1)

\begin{matrix} \frac{D u_{r}}{D t} - \frac{u_{φ}^{2}}{r} - 2 u_{φ} Ω_{r f} = - \frac{1}{ρ} \frac{\partial p}{\partial r} + ν (\nabla^{2} u_{r} - \frac{u_{r}}{r^{2}} - \frac{2}{r^{2}} \frac{\partial u_{φ}}{\partial φ}), \end{matrix}

(2.2)

\begin{matrix} \frac{D u_{φ}}{D t} + \frac{u_{r} u_{φ}}{r} + 2 u_{r} Ω_{r f} = - \frac{1}{ρ} \frac{1}{r} \frac{\partial p}{\partial φ} + ν (\nabla^{2} u_{φ} - \frac{u_{φ}}{r^{2}} + \frac{2}{r^{2}} \frac{\partial u_{r}}{\partial φ}), \end{matrix}

(2.3)

\begin{matrix} \frac{D u_{z}}{D t} = - \frac{1}{ρ} \frac{\partial p}{\partial z} + ν \nabla^{2} u_{z}, \end{matrix}

(2.4)

where $p$ is the pressure and $u = (u_{r}, u_{φ}, u_{z})$ is the total velocity with components in the corresponding rotating coordinates $(r, φ, z)$ .

In the above equations, $D / D t$ denotes the material time derivative,

\begin{matrix} \frac{D}{D t} = \frac{\partial}{\partial t} + u_{r} \frac{\partial}{\partial r} + \frac{u_{φ}}{r} \frac{\partial}{\partial φ} + u_{z} \frac{\partial}{\partial z}, \end{matrix}

(2.5)

while $\nabla^{2}$ denotes the Laplacian operator,

\begin{matrix} \nabla^{2} = \frac{1}{r} \frac{\partial}{\partial r} (r \frac{\partial}{\partial r}) + \frac{1}{r^{2}} \frac{\partial^{2}}{\partial φ^{2}} + \frac{\partial^{2}}{\partial z^{2}} . \end{matrix}

(2.6)

The basic angular velocity distribution in the rotating frame of reference, $Ω_{r o t} (r)$ , satisfying no-slip boundary conditions at $r = r_{i}$ and $r_{o}$ , takes the form

\begin{matrix} Ω_{r o t} (r) = A + \frac{B}{r^{2}} - Ω_{r f}, \end{matrix}

(2.7)

where

A = \frac{ω_{o} r_{o}^{2} - ω_{i} r_{i}^{2}}{r_{o}^{2} - r_{i}^{2}} = ω_{i} \frac{μ - η^{2}}{1 - η^{2}} and B = \frac{r_{i}^{2} r_{o}^{2}}{r_{o}^{2} - r_{i}^{2}} (ω_{i} - ω_{o}) = ω_{i} \frac{r_{i}^{2} (1 - μ)}{1 - η^{2}} .

(2.8)

To analyse the stability of the basic state and the ensuing nonlinear states, we superimpose disturbances $({\tilde{u}}_{r}, {\tilde{u}}_{φ}, {\tilde{u}}_{z}; \tilde{p})$ , where $\tilde{p}$ includes the factor $1 / ρ$ , to the basic state $(0, U (r), 0; P (r))$ , where $U (r) = r Ω_{r o t}$ . The equations of motion for these disturbances are obtained by substituting $({\tilde{u}}_{r}, U + {\tilde{u}}_{φ}, {\tilde{u}}_{z}; P + \tilde{p})$ for $(u_{r}, u_{φ}, u_{z}; p)$ into equations (2.1)–(2.4) and by recalling that the basic state $(0, U (r), 0; P (r))$ solves these equations:

\begin{matrix} \frac{\partial {\tilde{u}}_{r}}{\partial r} + \frac{{\tilde{u}}_{r}}{r} + \frac{1}{r} \frac{\partial {\tilde{u}}_{φ}}{\partial φ} + \frac{\partial {\tilde{u}}_{z}}{\partial z} = 0, \end{matrix}

(2.9)

(\frac{\partial}{\partial t} - ν Δ_{*}) {\tilde{u}}_{r} + ν \frac{2}{r^{2}} \frac{\partial {\tilde{u}}_{φ}}{\partial φ} = - \frac{\partial \tilde{p}}{\partial r} - \frac{U}{r} \frac{\partial {\tilde{u}}_{r}}{\partial φ} + 2 Ω_{r o t} {\tilde{u}}_{φ} + \frac{{\tilde{u}}_{φ}^{2}}{r} + 2 Ω_{r f} {\tilde{u}}_{φ} - \tilde{u} \cdot \nabla {\tilde{u}}_{r},

(2.10)

(\frac{\partial}{\partial t} - ν Δ_{*}) {\tilde{u}}_{φ} - ν \frac{2}{r^{2}} \frac{\partial {\tilde{u}}_{r}}{\partial φ} = - \frac{1}{r} \frac{\partial \tilde{p}}{\partial φ} - \frac{U}{r} \frac{\partial {\tilde{u}}_{φ}}{\partial φ} - \frac{d U}{d r} {\tilde{u}}_{r} - (\frac{U}{r} - \frac{{\tilde{u}}_{φ}}{r}) {\tilde{u}}_{r} - 2 Ω_{r f} {\tilde{u}}_{r} - \tilde{u} \cdot \nabla {\tilde{u}}_{φ},

(2.11)

(\frac{\partial}{\partial t} - ν Δ) {\tilde{u}}_{z} = - \frac{\partial \tilde{p}}{\partial z} - \frac{U}{r} \frac{\partial {\tilde{u}}_{z}}{\partial φ} - \tilde{u} \cdot \nabla {\tilde{u}}_{z},

(2.12)

where

Δ_{*} = Δ - \frac{1}{r^{2}} and Δ = \frac{\partial^{2}}{\partial r^{2}} + \frac{1}{r} \frac{\partial}{\partial r} + \frac{1}{r^{2}} \frac{\partial^{2}}{\partial φ^{2}} + \frac{\partial^{2}}{\partial z^{2}} .

(2.13)

The angular velocity $Ω_{r f}$ and the azimuthal velocity $U$ of the basic state in the rotating frame take the form

\begin{matrix} Ω_{r f} = \frac{ω_{i} + ω_{o}}{2} = \frac{ω_{i} (1 + μ)}{2} \end{matrix}

(2.14)

and

\begin{matrix} U (r) = r Ω_{r o t} (r) = (A - Ω_{r f}) r + \frac{B}{r}, \end{matrix}

(2.15)

respectively.

The third and fourth terms on the right-hand side of equation (2.10) result from $u_{φ}^{2} / r = (U + {\tilde{u}}_{φ})^{2} / r = (U^{2} + 2 U {\tilde{u}}_{φ} + {\tilde{u}}_{φ}^{2}) / r$ with $U / r = Ω_{r o t}$ , where $U^{2} / r$ has been absorbed in the pressure term $P (r)$ , whereas the third and fourth terms on the right-hand side of equation (2.11) result from $(u_{r} \frac{\partial}{\partial r}) u_{φ}$ in $D u_{φ} / D t = D (U + {\tilde{u}}_{φ}) / D t$ and from $u_{r} u_{φ} / r = {\tilde{u}}_{r} (U + {\tilde{u}}_{φ}) / r$ , respectively.

3. Transformation to the Cartesian system

We now focus on the narrow gap limit. It can be found from equations (2.8) and (2.14) that

Ω_{r o t} \to - ω_{i} \frac{1 - μ}{2} ξ as η \to 1,

(3.1)

where $ξ = \frac{r - r}{(r_{o} - r_{i}) / 2}$ with $r = (r_{o} + r_{i}) / 2$ . This shows that $Ω_{r o t} = 0$ for all $ξ \in [- 1, 1]$ when $μ = 1$ , and that $Ω_{r o t} = ω_{i} \frac{1 - μ}{2} = ω_{i} - Ω_{r f}$ at $r = r_{i}$ or $ξ = - 1$ , whereas $Ω_{r o t} = - ω_{i} \frac{1 - μ}{2} = ω_{o} - Ω_{r f}$ at $r = r_{o}$ or $ξ = 1$ , when $μ \neq 1$ .

From equations (2.15), (2.8) and (2.14) we obtain

\begin{matrix} \frac{d U}{d r} + \frac{U}{r} = 2 (A - Ω_{r f}) = 2 ω_{i} ((μ - η^{2}) / (1 - η^{2}) - (1 + μ) / 2) = - ω_{i} \frac{(1 - μ) (1 + η^{2})}{1 + η} \frac{r_{o}}{2 d}, \end{matrix}

(3.2)

where $d$ is a half-gap between the cylinders.

The transformation of the equations of motion for the narrow gap limit in the cylindrical coordinates $(r, φ, z)$ to those in non-dimensional Cartesian coordinates $(\hat{z}, \hat{x}, \hat{y})$ is accomplished by the identification $\hat{x} = r φ / d$ and $\hat{y} = z / d$ and $\hat{z} = (r - r) / d = ξ$ , as $r \to \infty$ . Here, we have chosen $d = (r_{o} - r_{i}) / 2$ as the length scale, $d^{2} / ν$ as the time scale and $ν / d$ as the velocity scale. Then, the non-dimensional form of the equations of motion for non-dimensional disturbances $\hat{u} = ({\hat{u}}_{x}, {\hat{u}}_{y}, {\hat{u}}_{z})$ is transformed from equation (2.9) to equation (2.12) as $r \to \infty$ by neglecting $ν \frac{2}{r^{2}} \frac{\partial {\tilde{u}}_{φ}}{\partial φ}$ and ${\tilde{u}}_{φ}^{2} / r$ in equation (2.10), and $ν \frac{2}{r^{2}} \frac{\partial {\tilde{u}}_{r}}{\partial φ}$ and ${\tilde{u}}_{r} {\tilde{u}}_{φ} / r$ in equation (2.11), followed by multiplying both sides of equations (2.10)–(2.12) by $d^{3} / ν^{2}$ , as follows.

\begin{matrix} \frac{\partial {\hat{u}}_{x}}{\partial \hat{x}} + \frac{\partial {\hat{u}}_{y}}{\partial \hat{y}} + \frac{\partial {\hat{u}}_{z}}{\partial \hat{z}} = 0, \end{matrix}

(3.3)

(\frac{\partial}{\partial \hat{t}} - {\hat{\nabla}}^{2}) {\hat{u}}_{z} = - \frac{\partial \hat{p}}{\partial \hat{z}} - \hat{U} \frac{\partial {\hat{u}}_{z}}{\partial \hat{x}} - 2 \hat{ω} \frac{(1 - μ)}{2} ξ {\hat{u}}_{x} + 2 \hat{ω} \frac{1 + μ}{2} {\hat{u}}_{x} - \hat{u} \cdot \hat{\nabla} {\hat{u}}_{z},

(3.4)

(\frac{\partial}{\partial \hat{t}} - {\hat{\nabla}}^{2}) {\hat{u}}_{x} = - \frac{\partial \hat{p}}{\partial \hat{x}} - \hat{U} \frac{\partial {\hat{u}}_{x}}{\partial \hat{x}} + \hat{ω} (1 - μ) \frac{r_{o}}{2 d} {\hat{u}}_{z} - 2 \hat{ω} \frac{1 + μ}{2} {\hat{u}}_{z} - \hat{u} \cdot \hat{\nabla} {\hat{u}}_{x},

(3.5)

(\frac{\partial}{\partial \hat{t}} - {\hat{\nabla}}^{2}) {\hat{u}}_{y} = - \frac{\partial \hat{p}}{\partial \hat{y}} - \hat{U} \frac{\partial {\hat{u}}_{y}}{\partial \hat{x}} - \hat{u} \cdot \hat{\nabla} {\hat{u}}_{y},

(3.6)

where

\hat{\nabla} = (\frac{\partial}{\partial \hat{x}}, \frac{\partial}{\partial \hat{y}}, \frac{\partial}{\partial \hat{z}}), {\hat{\nabla}}^{2} = \frac{\partial^{2}}{\partial {\hat{x}}^{2}} + \frac{\partial^{2}}{\partial {\hat{y}}^{2}} + \frac{\partial^{2}}{\partial {\hat{z}}^{2}} .

(3.7)

Here, $\hat{U}$ and $\hat{ω}$ are the non-dimensional forms of $U$ and $ω_{i}$ , that is, $\hat{U} = U / (ν / d)$ and $\hat{ω} = (d^{2} / ν) ω_{i}$ , respectively. We have transformed $Ω_{r o t}$ in equation (2.10) by using equation (3.1) in equation (3.4), and $\frac{d U}{d r} + \frac{U}{r}$ in equation (2.11) by using equation (3.2) in equation (3.5). For clarity, we drop the hat (^∧) on $t, x, y, z$ , hereafter. In addition, we identify $ξ$ in equation (3.4) as $z$ .

Then in a vector form, the above system of equations are written as

\nabla \cdot \hat{u} = 0,

(3.8)

(\partial_{t} - \nabla^{2}) \hat{u} = - \hat{\nabla} \hat{p} - \hat{U} \partial_{x} \hat{u} + R e {\hat{u}}_{z} \hat{i} - \frac{1}{2} Q z {\hat{u}}_{x} \hat{k} - Ω \hat{j} \times \hat{u} - (\hat{u} \cdot \nabla) \hat{u},

(3.9)

where $\hat{i}$ , $\hat{j}$ and $\hat{k}$ are the unit vectors in the $x$ -, $y$ - and $z$ -directions, and

R e = \frac{ω_{i} (1 - μ) \bar{r} d}{2 ν}, Ω = \frac{ω_{i} (1 + μ) d^{2}}{ν} and Q = \frac{2 ω_{i} (1 - μ) d^{2}}{ν} .

(3.10)

It can be found that $Ω$ is the non-dimensional form of $2 Ω_{r f}$ by equation (2.14). The non-dimensional form of $U$ is found to be $\hat{U} \to - R e ξ$ as $η \to 1$ , since $\hat{U} = r Ω_{r o t} / (ν / d) = r (1 + (d / r) x) Ω_{r o t} / (ν / d) \to - r ω_{i} \frac{1 - μ}{2} ξ / (ν / d)$ by equation (3.1).

The reason why the ${\tilde{u}}_{φ}^{2} / r$ term has been neglected in equation (2.10) is that the non-dimensional form of $1 / r$ using the length scale $d$ is of order $(1 - η)$ , and it is supposed that $\tilde{u}$ is not as large as $1 / \sqrt{1 - η}$ . However, we show in the next section that this model, which was also used in [1], cannot express the general case. Note however, that since this term is nonlinear the above system of equations, equations (3.3)–(3.6) or equations (3.8)–(3.9), remain valid for linear analysis.

4. Analysis

From equation (3.10), the values of $Ω$ and $Q$ are constrained by $R e$ as

Ω = 2 \frac{1 + μ}{1 - μ} \frac{d}{\bar{r}} R e \approx \frac{1 + μ}{1 - μ} (1 - η) R e, (μ \neq 1) and Q = \frac{4 d}{\bar{r}} R e \approx 2 (1 - η) R e,

(4.1)

where $d / r = (1 - η) / (1 + η) \approx \frac{1 - η}{2}$ is used.

The constraint (equation (4.1)) suggests that the system, equations (3.8) and (3.9), is governed by a single parameter $R e$ for a given $μ$ .

The choice of

T \propto R e^{2} with Ω = Ω (R e) and Q = Q (R e) .

(4.2)

is used by Krueger [19] for $μ < 0$ .

One can make an alternative choice for this single parameter.

The choice of

T a = Ω (R e - Ω) with Q = Q (Ω),

(4.3)

corresponds to the conventional Taylor number, used by, for instance, Lin [20] and Chandrasekhar [21] for $μ \geq 0$ . This choice of parameter was also made in [1]. Since $Q = 2 Ω \frac{1 - μ}{1 + μ}$ , this choice would be unsuitable when $μ = - 1$ . It should be stressed, however, that the case with $μ = - 1$ is not a physical singularity as discussed below. The Reynolds numbers,

R_{i} = \frac{ω_{i} r_{i} d}{ν} and R_{o} = \frac{ω_{o} r_{o} d}{ν},

(4.4)

based on the inner and the outer cylinders in the real space, can be expressed by $R e$ , as follows:

R_{i} = \frac{2 η}{1 - μ} R e and R_{o} = \frac{2 μ}{1 - μ} R e .

(4.5)

Therefore, we find that a physical singularity occurs only when $μ = 1$ , not when $μ = - 1$ .

When $μ \approx 1$ , the ratio, $Ω / R e$ , can only be of $O (1)$ as $η \to 1$ by equation (4.1), so that $R e$ and $Ω$ can take finite values, but $Q$ should be neglected. So, this case reduces to rotating plane Couette flow in Cartesian geometry. Here, $μ$ does not appear in the analysis. This case corresponds to the study by [6] of flow between almost co-rotating cylinders. Axisymmetric flow in this case is governed by the Taylor number, $T a$ , (equation (4.3)) with $Q = 0$ .

When $μ$ is not close to 1, $Ω$ is $(1 - η)$ -order smaller than $R e$ . Since $Q$ is also $(1 - η)$ -order smaller than $R e$ , one might suppose that the system is reduced to that of plane Couette flow. However, this is true only when ${\hat{u}}_{x}$ is of $O (1)$ . In this case, $R e$ is of $O (1)$ , because ${\hat{u}}_{x} \sim O (U_{B}^{'}) = O (R e)$ by the $x$ -component of equation (3.9). On the other hand, when ${\hat{u}}_{x}$ is as large as $O (1 / Q)$ or $O (1 / Ω)$ , the $O (1)$ balance is achieved in the $z$ -component of equation (3.9) with ${\hat{u}}_{z} \sim O (1)$ , provided $U_{B} \partial_{x}$ is of $O (1)$ or zero, i.e. the axisymmetric case ( $\partial_{x} \equiv 0$ ). Then, we find ${\hat{u}}_{x} \sim O (\frac{1}{(1 - η) R e})$ by equation (4.1). Since ${\hat{u}}_{x} \sim O (R e)$ , as described above, we find $R e \sim O (1 / \sqrt{1 - η}$ ) by comparing the two order evaluations for ${\hat{u}}_{x}$ above. In this case, $Q \sim O (\sqrt{1 - η})$ and $Ω \sim O (\sqrt{1 - η})$ ; however these parameters are retained since even though they are $(1 - η)$ -order smaller than $R e$ the products of $R e Ω$ and $R e Q$ are of $O (1)$ . In fact, the linear stability analysis conducted by [1] showed that the product of $R e Ω$ , and so the Taylor number (equation (4.3)), takes a finite value at the linear criticality in good agreement with [20] and [21]. Thus, the scaling discussion below follows as a consequence of the linear stability results rather than being imposed as a premise, such as has been used for high-rotation Taylor–Couette flow [18].

In any case, we stress here that Taylor–Couette flow in the narrow gap limit must be distinguished from plane Couette flow ( $Ω \equiv 0$ ) by the fundamental difference between them: the former experiences axisymmetric instability at $R e \sim O (1 / \sqrt{1 - η})$ , and $Ω \sim O (\sqrt{1 - η})$ as $η \to 1$ , whereas the latter is linearly stable for all $R e$ .

The choice of parameter in the present study will be given in §6.

In the region above the axisymmetric linear criticality at $R e \sim O (1 / \sqrt{1 - η})$ and $Ω \sim O (\sqrt{1 - η})$ , the nonlinear disturbance velocity is $({\hat{u}}_{x}, {\hat{u}}_{y}, {\hat{u}}_{z}) \sim (O (1 / \sqrt{1 - η}), O (1), O (1))$ . Then, the curvature disturbance term in the wall-normal direction, which has been ignored so far, is $- \frac{1}{2} (1 - η) {\hat{u}}_{x}^{2} k \sim O (1)$ , becoming comparable with the other quadratic nonlinear terms, $({\hat{u}}_{y} \partial_{y} + {\hat{u}}_{z} \partial_{z}) {\hat{u}}_{z} \sim O (1)$ . Hence, we need to include $- \frac{1}{2} (1 - η) {\hat{u}}_{x}^{2} k$ on the left-hand side of equation (3.9).

It can be found that the following scaling is adequate for this higher $R e$ number region,

\begin{aligned} R e^{*} = \sqrt{1 - η} R e, Ω^{*} = (1 \sqrt{1 - η}) R e, Q^{*} = (1 / \sqrt{1 - η}) Q, \\ U_{B}^{*} (z^{*}) = \sqrt{1 - η} \hat{U} (z), p^{*} = \hat{p}, \\ (x^{*}, y^{*}, z^{*}) = (\sqrt{1 - η} x, y, z), ({\hat{u}}_{x^{*}}^{*}, {\hat{u}}_{y^{*}}^{*}, {\hat{u}}_{z^{*}}^{*}) = (\sqrt{1 - η} {\hat{u}}_{x}, {\hat{u}}_{y}, {\hat{u}}_{z}), \end{aligned}

(4.6)

where $U_{B}^{*} (z^{*}) = - R e^{*} z^{*}$ . From the above scalings, we see that

\begin{matrix} R e^{*} Q^{*} = R e Q, U_{B}^{*} / R e^{*} = \hat{U} / R e . \end{matrix}

(4.7)

Then, the equations to be solved become

\nabla^{*} \cdot {\hat{u}}^{*} = 0,

(4.8)

\begin{aligned} (\partial_{t} - {\nabla^{*}}^{2}) {\hat{u}}^{*} = & - \nabla^{*} p^{*} - U_{B}^{*} \partial_{x^{*}} {\hat{u}}^{*} + R e^{*} {\hat{u}}_{z^{*}}^{*} \hat{i} - \frac{1}{2} Q^{*} z^{*} {\hat{u}}_{x^{*}}^{*} \hat{k} - \frac{1}{2} ({\hat{u}}_{x^{*}}^{*})^{2} k \\ - Ω^{*} \hat{j} \times {\hat{u}}^{*} - ({\hat{u}}^{*} \cdot \nabla^{*}) {\hat{u}}^{*} . \end{aligned}

(4.9)

Note that some unnecessary terms with $x^{*}$ -derivatives have been included in the above for the sake of formality, such as $\partial_{x^{*} x^{*}}^{2}$ in ${\nabla^{*}}^{2}$ which is $(1 - η) \partial_{x x}^{2}$ before scaling and should not be included. For clarity, asterisks $(\cdot)^{*}$ on the spatial coordinates and their derivatives are omitted hereafter.

The nonlinear analysis of [1] ignored the term, $\frac{{\tilde{u}}_{φ}^{2}}{r}$ in equation (2.10). The short-coming owing to this neglection is rectified in the appendix .

5. Numerical analysis

We outline the numerical procedure to describe the axisymmetric case in this section. We separate the velocity perturbation ${\hat{u}}^{*}$ into the modification, ${\overset{ˇ}{U}}^{*} (z, t) i$ , of the mean flow and a periodic (in space) part, ${\overset{ˇ}{u}}^{*}$ . Then, we express the latter solenoidal vector field in terms of poloidal and toroidal parts,

{\overset{ˇ}{u}}^{*} = ({\overset{ˇ}{u}}^{*}, {\overset{ˇ}{v}}^{*}, {\overset{ˇ}{w}}^{*}) = \nabla \times (\nabla \times k ϕ) + \nabla \times k ψ = (\partial_{x z}^{2} ϕ + \partial_{y} ψ, \partial_{y z}^{2} ϕ - \partial_{x} ψ, - Δ_{2} ϕ),

(5.1)

where $Δ_{2} = \partial_{x x}^{2} + \partial_{y y}^{2}$ . By doing so, the continuity equation is satisfied automatically.

Operations $k \cdot \nabla \times \nabla \times$ and $k \cdot \nabla \times$ on equation (4.9) lead to

\begin{aligned} \partial_{t} \nabla^{2} Δ_{2} ϕ = & (\nabla^{4} + (U_{B}^{*} + {\overset{ˇ}{U}}^{*})^{^{″}} \partial_{x} - (U_{B}^{*} + {\overset{ˇ}{U}}^{*}) \partial_{x} \nabla^{2}) Δ_{2} ϕ \\ + \frac{1}{2} Q^{*} z Δ_{2} (\partial_{x z}^{2} ϕ + \partial_{y} ψ) + \frac{1}{2} Δ_{2} (2 {\overset{ˇ}{U}}^{*} \partial_{y} ψ + \partial_{y} ψ \partial_{y} ψ) \\ - Ω^{*} \partial_{y} Δ_{2} ψ - k \cdot \nabla \times \nabla \times [{\overset{ˇ}{u}}^{*} \cdot \overset{ˇ}{u^{*}}], \end{aligned}

(5.2)

\begin{aligned} \partial_{t} Δ_{2} ψ = & (\nabla^{2} - (U_{B}^{*} + {\overset{ˇ}{U}}^{*}) \partial_{x}) Δ_{2} ψ \\ + (U_{B}^{*} + {\overset{ˇ}{U}}^{*})^{'} \partial_{y} Δ_{2} ϕ + k \cdot \nabla \times [{\overset{ˇ}{u}}^{*} \cdot {\overset{ˇ}{u}}^{*}], \end{aligned}

(5.3)

where the dash ( $'$ ) denotes the partial differentiation with respect to $z$ . The two terms of the second line in equation (5.2) corresponds to the correction to the previous formulation of [1]. Since $z$ is an odd function in $z$ , the first term breaks the $z$ -symmetry.

The equation for $\overset{ˇ}{U} (z)$ is obtained by averaging the $x$ -component of equation (4.9) in $x y$ -space:

\partial_{t} {\overset{ˇ}{U}}^{*} = {\overset{ˇ}{U}}^{*^{″}} + \partial_{z} \bar{Δ_{2} ϕ (\partial_{x z}^{2} ϕ + \partial_{y} ψ}) .

(5.4)

This shows that ${\overset{ˇ}{U}}^{*}$ is created by the Reynolds stress. Note that $Δ_{2} ϕ = - \overset{ˇ}{w}$ and $(\partial_{x z}^{2} ϕ + \partial_{y} ψ) = \overset{ˇ}{u}$ .

The no-slip boundary conditions become

ϕ = \partial_{z} ϕ = ψ = {\overset{ˇ}{U}}^{*} = 0 at z = \pm 1.

(5.5)

Note again that unnecessary terms with $x^{*}$ -derivatives have been included in equations (5.1)–(5.4).

To examine an axisymmetric perturbation we let $\partial_{x} \equiv 0$ in equations (5.2), (5.3) and (5.4), and thus obtain

\begin{array}{ll} \partial_{t} (\partial_{y y}^{2} + \partial_{z z}^{2}) \partial_{y y}^{2} ϕ = & (\partial_{y y}^{2} + \partial_{z z}^{2})^{2} \partial_{y y}^{2} ϕ + (\frac{1}{2} z Q^{*} - Ω^{*}) \partial_{y y y}^{3} ψ \\ + \frac{1}{2} \partial_{y y}^{2} (2 {\overset{ˇ}{U}}^{*} \partial_{y} ψ + \partial_{y} ψ \partial_{y} ψ) \\ - \partial_{y z}^{2} [(\partial_{y z}^{2} ϕ \partial_{y} - \partial_{y y}^{2} ϕ \partial_{z}) \partial_{y z}^{2} ϕ] - \partial_{y y}^{2} [(\partial_{y z}^{2} ϕ \partial_{y} - \partial_{y y}^{2} ϕ \partial_{z}) \partial_{y y}^{2} ϕ], \end{array}

(5.6)

\begin{aligned} _{t} \partial_{y y}^{2} ψ = & (\partial_{y y}^{2} + \partial_{z z}^{2}) \partial_{y y}^{2} ψ + (- R e^{*} + {\overset{ˇ}{U}}^{*'} (t, z)) \partial_{y y y}^{3} ϕ \\ - \partial_{y} [(\partial_{y z}^{2} ϕ \partial_{y} - \partial_{y y}^{2} ϕ \partial_{z}) \partial_{y} ψ], \end{aligned}

(5.7)

\partial_{t} {\overset{ˇ}{U}}^{*} (t, z) = {\overset{ˇ}{U}}^{*^{″}} (t, z) + \partial_{z} \bar{(\partial_{y y}^{2} ϕ \partial_{y} ψ)} .

(5.8)

It is clear from equations (5.6), (5.7) and (5.8) that the axisymmetric nonlinear solutions satisfy the following symmetry:

ϕ (t, - y, z) = ϕ (t, y, z) and ψ (t, - y, z) = - ψ (t, y, z) .

(5.9)

To study the stability of the axisymmetric nonlinear solutions we superimpose infinitesimal perturbations on them. Here, we consider only axisymmetric perturbations, which satisfy

\begin{aligned} _{t} (\partial_{y y}^{2} + \partial_{z z}^{2}) \partial_{y y}^{2} \tilde{ϕ} = & (\partial_{y y}^{2} + \partial_{z z}^{2})^{2} \partial_{y y}^{2} \tilde{ϕ} + (\frac{1}{2} z Q^{*} - Ω^{*}) \partial_{y y y}^{3} \tilde{ψ} \\ + \frac{1}{2} \partial_{y y}^{2} (2 {\overset{ˇ}{U}}^{*} \partial_{y} \tilde{ψ} + \partial_{y} \tilde{ψ} \partial_{y} ψ + \partial_{y} ψ \partial_{y} \tilde{ψ}) \\ - \partial_{y z}^{2} [(\partial_{y z}^{2} \tilde{ϕ} \partial_{y} - \partial_{y y}^{2} \tilde{ϕ} \partial_{z}) \partial_{y z}^{2} ϕ] - \partial_{y y}^{2} [(\partial_{y z}^{2} \tilde{ϕ} \partial_{y} - \partial_{y y}^{2} \tilde{ϕ} \partial_{z}) \partial_{y y}^{2} ϕ] \\ - \partial_{y z}^{2} [(\partial_{y z}^{2} ϕ \partial_{y} - \partial_{y y}^{2} ϕ \partial_{z}) \partial_{y z}^{2} \tilde{ϕ}] - \partial_{y y}^{2} [(\partial_{y z}^{2} ϕ \partial_{y} - \partial_{y y}^{2} ϕ \partial_{z}) \partial_{y y}^{2} \tilde{ϕ}], \end{aligned}

(5.10)

\begin{aligned} _{t} \partial_{y y}^{2} \tilde{ψ} = & (\partial_{y y}^{2} + \partial_{z z}^{2}) \partial_{y y}^{2} \tilde{ψ} + (- R e^{*} + {\overset{ˇ}{U}}^{*'} (t, z)) \partial_{y y y}^{3} \tilde{ϕ} \\ - \partial_{y} [(\partial_{y z}^{2} \tilde{ϕ} \partial_{y} - \partial_{y y}^{2} \tilde{ϕ} \partial_{z}) \partial_{y} ψ] - \partial_{y} [(\partial_{y z}^{2} ϕ \partial_{y} - \partial_{y y}^{2} ϕ \partial_{z}) \partial_{y} \tilde{ψ}] . \end{aligned}

(5.11)

Taking into account the symmetry (equation (5.9)) of the nonlinear solutions $ϕ$ and $ψ$ , we can find from equations (5.10) and (5.11) that the axisymmetric perturbations satisfy the same symmetry as

\tilde{ϕ} (t, - y, z) = \tilde{ϕ} (t, y, z) and \tilde{ψ} (t, - y, z) = - \tilde{ψ} (t, y, z) .

(5.12)

6. Numerical method

The perturbations $ϕ$ , $ψ$ and ${\overset{ˇ}{U}}^{*}$ are expanded by Fourier series in the $y$ (spanwise) directions for $ϕ$ and $ψ$ , and by Chebyshev polynomials $T_{l} (z)$ , in the wall-normal direction $z$ , which are multiplied by $(z^{2} - 1)^{2}$ for $ϕ$ and $(z^{2} - 1)$ for $ψ$ and ${\overset{ˇ}{U}}^{*}$ so as to satisfy no-slip boundary conditions.

Then, the perturbations, truncated at $(L, N)$ , are expressed as

ϕ (y, z, t) = \sum_{l = 0}^{L} \sum_{\begin{matrix} n = - N \\ n \neq 0 \end{matrix}}^{N} ϕ_{l n} (t) \exp [i n β y] (z^{2} - 1)^{2} T_{l} (z),

(6.1)

ψ (y, z, t) = \sum_{l = 0}^{L} \sum_{\begin{matrix} n = - N \\ n \neq 0 \end{matrix}}^{N} ψ_{l n} (t) \exp [i n β y] (z^{2} - 1) T_{l} (z),

(6.2)

{\overset{ˇ}{U}}^{*} (z, t) = \sum_{l = 0}^{L} {\overset{ˇ}{U}}_{l}^{*} (t) (z^{2} - 1) T_{l} (z),

(6.3)

where $β$ is the wavenumber in the spanwise directions.

Similarly, the infinitesimal axisymmetric perturbations, $\tilde{ϕ}$ and $\tilde{ψ}$ , imposed on the nonlinear axisymmetric solutions (equations (6.1) and (6.2)) are expanded as follows:

\tilde{ϕ} (y, z, t) = \sum_{l = 0}^{L} \sum_{\begin{matrix} n = - N \end{matrix}}^{N} {\tilde{ϕ}}_{l n} (t) \exp [i (n β + b) y] (z^{2} - 1)^{2} T_{l} (z),

(6.4)

\tilde{ψ} (y, z, t) = \sum_{l = 0}^{L} \sum_{\begin{matrix} n = - N \end{matrix}}^{N} {\tilde{ψ}}_{l n} (t) \exp [i (n β + b) y] (z^{2} - 1) T_{l} (z),

(6.5)

where $b$ is the Floquet exponent (note that the present system is periodic in $y$ ). Also note that the case for $n = 0$ is included in equations (6.4) and (6.5), which act as a perturbation imposed on ${\overset{ˇ}{U}}^{*}$ .

Upon discretizing the equations (5.6), (5.7) and (5.8) in the wall-normal direction using the collocation method, the resulting algebraic equations for the amplitudes, $q_{l n} (t)$ , are solved as an eigenvalue problem for the complex growth rate $σ$ of perturbation $q_{l n} (t) \propto \exp [σ t]$ in the stability analysis, and by the Newton method for nonlinear analysis.

For calculations, the parameters, $R e^{*}$ , $Ω^{*}$ and $Q^{*}$ , are varied with the constraints

Ω^{*} = \frac{1 + μ}{1 - μ} R e^{*} and Q^{*} = 2 R e^{*},

(6.6)

which corresponds to the relations in equation (4.1) for the unscaled $R e$ , $Ω$ and $Q$ .

Since equation (4.3) can be written as

T a = \sqrt{1 - η} Ω^{*} (\frac{R e^{*}}{\sqrt{1 - η}} - \sqrt{1 - η} Ω^{*}) = Ω^{*} (R e^{*} - (1 - η) Ω^{*}) \approx Ω^{*} R e^{*},

(6.7)

and expressing $Ω^{*}$ in terms of $Q^{*}$ from equation (6.6), we obtain

\begin{matrix} Ω^{*} = \frac{1}{2} (\frac{1 + μ}{1 - μ}) Q^{*} = C (μ) Q^{*}, \end{matrix}

(6.8)

where

\begin{matrix} C (μ) \equiv \frac{1}{2} (\frac{1 + μ}{1 - μ}) . \end{matrix}

(6.9)

Then, equation (6.7) is simply rewritten as

\begin{matrix} T a = C (μ) Q^{*} R e^{*} . \end{matrix}

(6.10)

The fact that the axisymmetric instabilities depend on only $T a (μ)$ for a given $μ$ ( $\neq - 1$ ) implies here that the flows ought to depend only on

\begin{matrix} τ (μ) = Q^{*} R e^{*} \end{matrix}

(6.11)

for that given $μ$ . Now, it is possible to determine the critical state owing to axisymmetric instability for $μ = - 1$ , the case excluded in [1], by $τ$ . The involvement of $μ$ in $τ (μ)$ results from $Ω^{*} (= C (μ) Q^{*})$ in equation (5.6). We call $τ$ the modified Taylor number. Since $Q^{*} R e^{*} = Q R e$ , the linear stability analysis for the original system, equations (3.8) and (3.9), gives the same critical value of the modified Taylor number as the one for the scaled system, equations (4.8) and (4.9). Note that one drawback is that the case $μ = 1$ must be excluded for this choice of parameter, $τ (μ)$ .

7. Results

(a) The counter rotating case $(μ = - 1)$

Let us start describing linear and nonlinear results when the cylinders are rotating in the opposite directions with $ω_{o} = - ω_{i}$ , which has been excluded in [1].

(i) Stability analysis

Table 1 lists the convergence of the real part of the eigenvalue $σ = σ^{R} + i σ^{I}$ for axisymmetric perturbations with $μ = - 1$ , with respect to the truncation level $L$ of the Chebyshev polynomials. The parameter values, $τ = 2400$ and $β = 2.0$ , are chosen. The imaginary part, $σ^{I}$ , is found to be zero in this case, and also for other cases with different values of $τ$ and $β$ . The critical value, $τ_{c}$ , is plotted against $β$ in figure 1a. Detailed calculations show the minimum $τ_{c} = 2332.824$ occurs at $β = 1.999$ . This value can be compared with that of Krueger et al. [19]. The linear critical value they obtained¹ by solving an eigenvalue problem for a relevant six-order system by the Runge–Kutta method was 18 669 ( $\times \frac{1 - μ}{16} = 2333.625$ ) for the critical wavenumber 3.999 (/2 = 1.9995) (their figures, based on our scaling are shown in the parentheses).

Figure 1. (a) The neutral curve for the critical value $τ_{c}$ against $β$ for $μ = - 1$ . The minimum $τ_{c}$ occurs at 2332.824 for $β = 1.999$ . (b) The branch of the nonlinear axisymmetric solution with $β = 1.999$ bifurcating from the basic state with $M_{T}^{*} / R e^{*} (= M_{T} / R e) = - 1.0$ at $τ = 2332.824$ .

Table 1. Convergence of the real part of the eigenvalue $σ$ with respect to $L$ at $τ = 2400$ . $β = 2.0$ and $μ = - 1$ .
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$L$	$σ^{R}$
7	0.157362382950
9	0.157356630439
11	0.157355610920
13	0.157355638457
15	0.157355638527
17	0.157355638539
19	0.157355638532

The critical values $τ_{c}$ for some given $β$ are compared with $\frac{T a_{c}}{C (μ)}$ for $- 1 < μ < 1$ at $L = 19$ in table 2. Note that $τ_{c}$ is not determined when $μ = 1$ , whereas $T a_{c}$ is not determined when $μ = - 1$ . It is clear from the consideration described for equations (4.3) and (6.11) that $τ_{c}$ is identical to $\frac{T a_{c}}{C (μ)}$ for $- 1 < μ < 1$ . The important thing is that $τ_{c} (μ)$ is continuous for $- 1 \leq μ < 1$ , and so the linear critical state for $μ = - 1$ exists, and the existence of a nonlinear solution branch bifurcating from the critical state is assured by the implicit function theorem.

Table 2. Comparison of $\frac{T a_{c}}{C (μ)}$ and $τ_{c}$ with respect to $μ$ for a given $β$ at $L = 19$ .
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$μ$	$T a_{c}$	$β$	$C (μ) \equiv \frac{1}{2} \frac{1 + μ}{1 - μ}$	$\frac{T a_{c}}{C (μ)}$	$τ_{c}$
1.0	106.7346	1.5585	$\infty$	$1 / \infty$	—
0.5	106.6455	1.56	3/2	159.9683	159.9683
0.0	105.9345	1.56	1/2	211.8708	211.8708
−0.5	100.2996	1.56	1/6	601.7976	601.7976
−0.9	48.3777	1.56	1/38	1838.3526	1838.3526
−0.99	6.1335	1.56	1/398	2441.133	2441.133
−1.0	0	1.56	0	—	2518.388

(ii) Nonlinear analysis

The convergence of the momentum transport $M_{T}^{*}$

\begin{matrix} M_{T}^{*} = \frac{d (U_{B}^{*} + {\overset{ˇ}{U}}^{*})}{d z} |_{z = - 1} \end{matrix}

(7.1)

at $z = - 1$ is checked with respect to the truncation levels $(L, N)$ for $τ = 3200$ $(R e^{*} = 40, Q^{*} = 80)$ , and the optimal wavenumber $β = 1.999$ in table 3. The critical value $τ_{c}$ is $2332.824$ , and so this $τ = 3200$ is approximately $40 %$ higher than the critical value. For consistency with the linear analysis, the truncation level $(L, N) = (19, 11)$ is chosen in the following nonlinear calculations.

Table 3. Convergence of the momentum transport $M_{T}^{*} / R e^{*}$ $(= M_{T} / R e)$ at $τ = 3200$ , $(R e^{*} = 40, Q^{*} = 80)$ for $β = 1.999$ with respect to $(L, N)$ .
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$(L, N)$	$M_{T}^{} / R e^{}$
(9, 6)	−0.1426510720228113D+01
(11, 7)	−0.1427378424590911D+01
(13, 8)	−0.1426697986553903D+01
(15, 9)	−0.1426804715949070D+01
(17, 10)	−0.1426807574622539D+01
(19, 11)	−0.1426799709544118D+01

It is confirmed that the bifurcation of the nonlinear state takes place subcritically at $τ_{c}$ as shown in figure 1b.

As described above, the two terms of the second line in equation (5.2) are due to the coordinate transformation. Since $z$ is an odd function in $z$ , the first term breaks the $z$ -symmetry. As a result, an even component ${\overset{ˇ}{U}}_{s y m}^{*}$ of the mean flow distortion ${\overset{ˇ}{U}}^{*}$ is produced. Both anti-symmetric and symmetric components, ${\overset{ˇ}{U}}_{a n t i}^{*} / R e^{*}$ and ${\overset{ˇ}{U}}_{s y m}^{*} / R e^{*}$ , of the mean flow increase gradually as $τ$ is increased as displayed in figure 2.

Figure 2. Mean flow distortions for $μ = - 1$ at $τ = 2450, (R e^{*} = 35, Q^{*} = 70)$ (orange), $τ = 3528, (R e^{*} = 42, Q^{*} = 84)$ (blue) and $τ = 5000, (R e^{*} = 50, Q^{*} = 100)$ (green). $β = 1.999$ . (a) Anti-symmetric part and (b) Symmetric part.

The total mean flow is plotted in figure 3a. The degree of asymmetry is strengthened as $τ$ increases. The asymmetry also appears in the flow field: the Taylor vortices move closer to the inner cylinder as $τ$ increases, as seen from the stream function on the $(z, y)$ -cross-section in figure 3b.

(b) The case when the outer cylinder is at rest $(μ = 0)$

There is experimental evidence [2] for the case of $η = 0.883$ that stable axisymmetric Taylor-vortex flow exists near the $R_{i}$ axis in the physical $(R_{o}, R_{i})$ -space, where $R_{o}$ and $R_{i}$ are Reynolds numbers based on the outer and the inner cylinder motions, respectively. When these Reynolds numbers are scaled in the same way as $R e$ , they can be expressed by $R e^{*}$ , as follows:

R_{i}^{*} = \frac{2 η}{1 - μ} R e^{*} and R_{o} * = \frac{2 μ}{1 - μ} R e^{*}

(7.2)

(see equation (4.5)). To compare the results in this section with the experimental observations, it would be convenient to switch the controlling parameter from $τ$ to $R e^{*}$ .

The basic state becomes unstable with respect to an axisymmetric perturbation with $β \equiv β_{c} = 1.5635$ at $R e_{c}^{*} = 10.292$ as seen from the neutral curve in figure 4a. Figure 4b shows the supercritical nonlinear solution branch with $β = β_{c}$ which bifurcates from the basic state at this $R e_{c}^{*}$ . Also shown in figure 4b is another supercritical nonlinear solution branch for $β = 3.217 (= 2 β_{c})$ , bifurcating at $R e^{*} = 15.099$ . We confirm that both branches are stable against perturbations with the same wavenumbers, that is, with the Floquet parameter $b = 0$ in equations (6.4) and (6.5), as those of the nonlinear solutions. This observation is consistent with the so called exchange of stability.

Figure 4. (a) The critical value $R e_{c}^{*}$ against $β$ for $μ = 0$ . The minimum value $R e_{c}^{*} = 10.292$ $(τ = 211.87)$ occurs for $β = 1.5635 (= β_{c})$ . (b) The branches of the nonlinear axisymmetric solution with $β = 1.5635$ and $β = 2 β_{c} = 3.127$ bifurcating from the basic state with $M_{T}^{*} / R e^{*} (= M_{T} / R e) = - 1.0$ at $R e^{*} = 10.292$ and $R e^{*} = 15.099$ $(τ = 455.96)$ , respectively.

We call the axisymmetric flows on the nonlinear solution branches with $β = β_{c}$ and $β = 2 β_{c}$ the primary mode and the harmonic mode, respectively. The reason why the branch with $β = 2 β_{c}$ is focused on here is that the wavenumber $β_{c}$ of the prime mode could be regarded as subharmonic, i.e. $1 / 2 (2 β_{c})$ , from the view point of the harmonic mode, and that we anticipate that the harmonic mode might become unstable to subharmonic perturbations giving rise to solution branches with the same wavenumber as that of the primary mode. In fact, this anticipation is realized as shown later in this section.

The stability of the axisymmetric solutions against axisymmetric perturbations with other Floquet parameter values $b \neq 0$ is examined. Instability caused by axisymmetric perturbations, in our case, for flow near the neutral curve is the well-known as Eckhaus instability [22]. The domain of the stable axisymmetric flow is bounded towards larger or smaller wavenumbers away from the neutral curve. Figure 5 shows the largest real part, $σ^{R}$ , of the eigenvalues $σ$ as a function of $b / β$ for various $β$ at $R e^{*} = 15.2$ . Recall that the value of the marginal $R e^{*}$ for $β = 3.127$ is given by $R e^{*} = 15.099$ , so that the neutral stability point for $R e^{*} = 15.2$ takes place at a wavenumber slightly larger than $β = 3.127$ . As is clear, the axisymmetric flow is unstable at this value of $R e^{*}$ when $β \geq 2.6$ , with $σ^{R} \geq 0$ for all $0 \leq b / β \leq 1$ . Along the line of $R e^{*} = 15.2$ , $σ^{R}$ decreases as $β$ is decreased. Figure 5 shows that the axisymmetric flow gains stability as $β$ decreases through $β \approx 2.5$ . Thus, the stable domain within the neutral curve for the axisymmetric flow is reduced wavenumber-wise. (A similar reduction of the stable domain near the neutral curve for smaller wavenumbers should occur.)

Figure 5. Eckhaus instability. The largest real part of the eigenvalue value, $σ^{R}$ , as a function of $b / β$ at $R e^{*} = 15.02$ . $μ = 0$ . $β = 3.127$ , 3.0, 2.8, 2.6, 2.4, 2.2 and 2.0 from top to bottom.

It should be recognized that the eigenvalues are symmetric about $b / β = \frac{1}{2}$ . We see from the symmetries (equations (5.9) and (5.12)) that the system of the perturbation (equations (5.10) and (5.11)) is symmetric in $y$ . With this symmetry, and the fact that the system is periodic in $y$ , we can conclude that the system is symmetric about $b / β = \frac{1}{2}$ . Therefore, the eigenvalues as a function of $b / β$ are also symmetric about $b / β = \frac{1}{2}$ , which is verified numerically as seen in figure 5. Also, it should be recognized that the $σ^{R}$ reaches a local minimum or maximum at $b / β = \frac{1}{2}$ . When it takes a positive local maximum, perturbations with the wavenumber $\frac{1}{2} β$ , that is, growing subharmonic perturbations, would be created.

The largest and the second (and the third, if necessary) largest real parts of the eigenvalues are plotted as a function of $b / β$ for two supercritical states for the primary mode in figure 6a and for two supercritical states for the harmonic mode in figure 6b. It is seen that the largest real parts start from zero at $b / β = 0$ and end at zero at $b / β = 1$ for the two critical states for both modes. In fact, the largest real part is zero at $b / β = 0$ for any supercritical states. It is a well-known fact that the supercritical nonlinear solutions are always stable against infinitesimal translation, $δ ≪ 1$ , in the periodic direction, i.e. in the $y$ -direction for our nonlinear solution, that is, $ϕ (y + δ, z) = ϕ (y, z) + δ \partial ϕ / \partial y \approx ϕ (y, z)$ .

The second largest real part at $b / β = 0$ starts from zero when $R e^{*}$ is critical and gradually decreases in the negative direction as $R e^{*}$ increases. This second largest eigenvalue determines the stability of the nonlinear solutions against axisymmetric perturbations with the same wavenumber as that of the nonlinear solutions. Figure 6a,b indicates that both primary mode and harmonic mode are stable against an axisymmetric perturbation of the same wavenumber as that of each mode.

The real parts of the two least stable eigenvalues are negative when $b = \frac{1}{2} β$ for the primary mode near the critical state (see figure 6a), while they are positive for the harmonic mode (see figure 6b). They decrease as $R e^{*}$ is increased from the critical value. This means that the stable domain of the axisymmetric primary mode is shifted towards the neutral curve for larger $R e^{*}$ .

For the harmonic mode, the sign of the second largest real part of the eigenvalue at $b = \frac{1}{2} β$ changes from positive to negative as $R e^{*}$ is increased from 18 to 19 as seen in figure 7a, while the change of the sign of the largest real part of the eigenvalue from positive to negative takes place as $R e^{*}$ is increased from 22 to 23 as shown in figure 7b.

Figure 7. The largest and the second largest real parts of the eigenvalue values, $σ^{R}$ , as a function of $b / β$ for the harmonic mode ( $β = 3.127$ ). $μ = 0$ . (a) $R e^{*} = 18$ (black). $R e^{*} = 19$ (blue). (b) $R e^{*} = 22$ (black). $R e^{*} = 23$ (blue).

The reason why both the largest and the second largest parts of the eigenvalue at $b = \frac{1}{2} β$ start from positive values, actually the same positive values, at the onset of the harmonic mode is due to the fact that the basic flow is already unstable against those perturbations, and that the wavenumber of the perturbations can be regarded as subharmonic, $b = \frac{1}{2} (2 β_{c})$ , from the view point of the harmonic mode, $β = 2 β_{c}$ . Note that the amplitude of the harmonic mode vanishes at its onset, so that the state under consideration is on the neutral curve, while below the neutral curve flow is the basic state.

Figure 8a shows that the values of the positive real eigenvalue for $b = \frac{1}{2} β$ at the onset of the harmonic mode are exactly the same as those from the stability result of the basic state against perturbation with $β = 1.5635$ at this point.

The eigenvalues for $β$ and $- β$ for the subharmonic perturbation to the harmonic mode with $β = 1.5635$ are connected to those for perturbations with $n (2 β) + \frac{1}{2} (2 β)$ , ( $n = 0$ ), and for perturbations with $n (2 β) + \frac{1}{2} (2 β)$ , ( $n = - 1$ ) in figure 8a.

Both the largest and the second largest eigenvalues for subharmonic perturbations to the harmonic mode decrease as $R e^{*}$ is increased and they cross the line of $σ^{R} = 0$ at $R e^{*} = 18.91$ and at $R e^{*} = 22.54$ , respectively, as seen in figure 8a. At these values of $R e^{*}$ the Jacobian of the eigenvalue equations is singular and therefore the uniqueness of the solution branches is not assured by the implicit function theorem. As a result a solution branch with the same wavenumber as that of the primary mode bifurcates, as can be seen in figure 8b. It is found that these two branches continue to exist without merging for larger $R e^{*}$ . We stopped the numerical continuation at $R e^{*} = 50$ . We call the solution on the branch bifurcating from the harmonic mode at $R e^{*} = 18.91$ the subharmonic mode-I and the one bifurcating at $R e^{*} = 22.54$ mode-II.

The cross-sectional flow patterns, in terms of the stream function $Ψ \equiv \partial_{y} ϕ$ , for the subharmonic mode-I and mode-II together with that for the primary mode are compared at the same value of $R e^{*} = 30$ in figure 9. From the values of $Ψ_{m a x}$ , the vortex of the primary mode is stronger than those of both subharmonic mode-I and mode-II, although its asymmetric (in $z$ ) nature is weaker. The vortex strength does not differ significantly between the two subharmonic modes I and II. A secondary weak vortex pair can be seen near the inner cylinder wall, $z = - 1$ , for the subharmonic mode-I, while a similar secondary weak vortex pair is seen to exist near the outer cylinder wall, $z = 1$ , for the subharmonic mode-II. The weaker and the stronger vortex pairs inside one periodic box for the subharmonic mode-I and mode-II originate from the two vortex pairs of equal strength inside the same periodic box size for the harmonic mode.

Figure 9. The stream function, $Ψ \equiv \partial_{y} ϕ$ , on the $(z, y)$ -plane for $μ = 0$ at $R e^{*} = 30$ . $β = 1.5635$ . (a) The primary mode. $Ψ_{m a x} = 22.5464$ (yellow) : $Ψ_{m i n} = - 22.5464$ (blue). (b) The subharmonic mode-I. $Ψ_{m a x} = 15.2234$ (yellow) : $Ψ_{m i n} = - 15.2234$ (blue). (c) The subharmonic mode-II. $Ψ_{m a x} = 15.4205$ (yellow) : $Ψ_{m i n} = - 15.4205$ (blue).

8. Summary

The transition from circular Couette flow to axisymmetric Taylor-vortex flow in the limit of a narrow gap is examined by applying the Cartesian representation of the Taylor–Couette flow system which has recently been studied by Nagata (2023), to the counter-rotating case $μ < 0$ . An extension is also made to examine the stability of the nonlinear axisymmetric flow to axisymmetric perturbations.

The narrow-gap Taylor–Couette flow experiments conducted by Andereck et al. [2] for $η = 0.883$ , Dutcher et al. [14] for $η = 0.912$ and Snyder [15] for $η = 0.959$ report the transition from circular Couette flow to axisymmetric Taylor-vortex flow for the case of the outer cylinder at rest ( $μ = 0$ ) when $R_{i} \approx 60$ , 70 and 108, respectively, where $R_{i}$ is the Reynolds number based on the inner cylinder motion and a half gap between the cylinders (see figures 2 and 18 in [2], figure 8 in [14] and figure 18 in [15], respectively). Although $R_{i}$ may increase drastically as $η \to 1$ , these cited values do not seem sufficiently large as to make a premise for introducing scaled analyses based on high rotation rate or high-Reynolds-number flows, although some such asymptotic analyses ([17,18]) surprisingly succeeded in obtaining simplified equations of motion similar to equation (4.9). In the present study, the following steps were taken to reach equation (4.9) without requiring any parameter modelling assumptions such as a high rotation rate.

Reduction of the equations of motion from the cylindrical coordinate system to the Cartesian coordinate system. Identification of controlling parameters $R e$ , $Ω$ and $Q$ . The definitions of the parameter shows $R e$ is of $O (\frac{1}{1 - η})$ larger than $Q$ or $Ω$ .
Stability of the basic flow shows that the linear critical state is determined by the product of $R e$ and $Q$ irrespective of the individual values of $R e$ or $Q$ .
The linear critical state is applied for $R e ≫ 1$ and $Q ≪ 1$ , so that $R e \sim O (\frac{1}{\sqrt{1 - η}})$ and $Q \sim O (\sqrt{1 - η})$ .
Introduction of scaled analysis for high-Reynolds-number Taylor–Couette flow by $R e^{*} = \sqrt{1 - η} R e$ and $Q^{*} = \frac{1}{\sqrt{1 - η}} Q$ .

In the present system, the curvature effect $u_{φ}^{2}$ is expressed by $2 U {\tilde{u}}_{φ} / r + {\tilde{u}}_{φ}^{2} / r$ , where $U$ is the circular Couette solution and ${\tilde{u}}_{φ}$ is the azimuthal velocity disturbance. The analysis of [1] took into account only the first term, by assuming that the second term could be neglected as $r \to \infty$ . This is true in the linear theory and would be valid only when ${\tilde{u}}_{φ}$ is not very large. When ${\tilde{u}}_{φ} \sim O (1 / \sqrt{1 - η})$ , however, it is found that the second term becomes important in the region where the linear instability sets in. The current study also includes this second term in the nonlinear analysis.

Unfortunately, the analysis of Nagata (2023) had to exclude the case $μ = - 1$ , owing to the expression $Q$ as a function of $Ω$ , $Q = 2 Ω (1 - μ) / (1 + μ)$ , which was used in that study. It should be stressed again that the inability to determine the critical state by means of the ordinary Taylor number $T a$ is not due to a physical singularity, but the problem is simply a result of the substitution of $2 Ω (1 - μ) (1 + μ)$ in place of $Q$ . To study the case for $μ = - 1$ , or $Ω = 0$ we instead express $Ω$ in terms of $Q$ , that is, $Ω = (1 + μ) / (2 (1 - μ)) Q$ , and introduce the modified Taylor number $τ$ in the current study, where $τ = 2 (1 - μ) / (1 + μ) T a$ . Our linear calculations show that the critical values of $τ$ are continuous across $μ = - 1$ , and they reproduce the criticalities in terms of $T a$ for other $μ (\neq \pm 1)$ .

Our nonlinear analysis for $μ = - 1$ based on the scaled parameters, $R e^{*}$ , $Ω^{*}$ and $Q^{*}$ , shows a supercritical bifurcation of axisymmetric flows at $τ = 2332.824$ . The mean flows are asymmetrically distorted by the additionally created symmetric components of the mean flow distortion. The mean flows would be anti-symmetric about the channel centreplane if $Q^{*}$ were absent, so $Q^{*}$ is the symmetry-breaking parameter. Moreover, the Taylor vortices are distributed asymmetrically, with the centres of the vortices positioned, not on the mid-plane ( $z = 0$ ) between the gap as for the ordinary two-dimensional roll-cells with $Q^{*} = 0$ in the rotating plane Couette flow system, but closer to the inner cylinder wall ( $z = - 1$ ).

In the case of $μ = 0$ , which describes the case when the outer cylinder is at rest, a supercritical bifurcation of nonlinear axisymmetric Taylor–Couette flow (the primary mode) from the basic flow is found at $R e^{*} = 10.292$ with the critical wavenumber $β_{c} = 1.5635$ . This value corresponds to the scaled Reynolds number based on the inner cylinder motion, $R i_{c}^{*} = 20.584$ by equation (4.5), and should be compared with those of the aforementioned experimental results by [2,14] and [15] for $η = 0.883, 0.921$ and $0.959$ , respectively. Those values are $R i_{c}^{*} = 20.52, 20.76$ and $21.84$ , respectively, where $R i^{*} = \sqrt{1 - η} R_{i}$ .

Furthermore, we examined the stability of the nonlinear axisymmetric Taylor-vortex flow to axisymmetric perturbations when $μ = 0$ . We found that the domain of the stable Taylor-vortex flow above the neutral curve is bounded towards smaller wavenumbers away from the neutral curve by the Eckhaus instability. During the course of this investigation, two new solution branches were also found. One of them (mode-I) is characterized by the appearance of a weak secondary vortex pair near the inner cylinder wall, while for the other (mode-II) a similar weak secondary vortex pair appears on the outer cylinder wall. The wavenumber of both modes is the same as the primary mode. Even if the two modes are found to be unstable to three-dimensional perturbations, their existence should affect the appearance of successive flow states.

We now map the flow from $(R e, Ω)$ -space, or $(R e, Q)$ -space, to the physical space $(R o, R i)$ . The solution for a fixed $μ$ is represented by a point on the straight line $R i = \frac{1}{μ} R o$ in the $(R o, R i)$ -space for $η \to 1$ . Once the flow is determined by either $R e^{*}$ and $Ω^{*}$ at $T a = R e^{*} Ω^{*}$ , or $R e^{*}$ and $Q^{*}$ at $τ = R e^{*} Q^{*}$ , with the constraint of equation (6.6), the point on the line $R i^{*} = (1 / μ) R o^{*}$ with the scaled coordinates, $(R i^{*}, R o^{*}) = (\frac{2}{1 - μ} R e^{*}, \frac{2 μ}{1 - μ} R e^{*})$ in the $(R o^{*}, R i^{*})$ -space represents the physical flow state.

The bifurcation nature of Taylor-vortex flow for a given $μ$ is shown in the $(R o^{*}, R i^{*}) -$ space in figure 10.

The figure shows subcritical bifurcations for $μ = - 0.75, - 1$ and $- 1.5$ with the saddle-node points indicated by red open circles. The subcritical region seems to expand as $μ$ is decreased further from −1.5. The existence of the subcritical branches is consistent to the numerical results by [23] for $η = 0.883$ . Their figure 3 shows a subcritical branch of the axisymmetric Taylor-vortex flow in a small interval of $R i$ near $R i \approx 490$ at $R o = - 1200$ $(μ = (R o / R i) η \approx - 2.16)$ , although their primary interest is in non-axisymmetric flows.

It is known that the critical modes change at $μ = - 0.78$ from the axisymmetric steady mode to a non-axisymmetric oscillatory mode in the narrow gap limit of Taylor–Couette flow ([19]). It may then be the case the solution obtained in the current study cannot be realized. Also, turbulent states prevail at large Reynolds number. However, the author believes that the recognition of the existence of axisymmetric flows as an underlying nonlinear state is still important.

For the current axisymmetric case, we set $\partial_{x} \equiv 0$ in equations (5.6), (5.7) and (5.8). We can confirm that the scalings introduced in §4 are adequate, i.e. $O (1)$ balance for equation (5.6) and $O (1 / \sqrt{1 - η})$ balance for equations (5.7) and (5.8) are attained with $R e \sim O (1 / \sqrt{1 - η})$ and $Ω, Q \sim O (\sqrt{1 - η})$ , when $ϕ \sim O (1)$ and ${\hat{u}}_{x}, ψ \sim O (1 / \sqrt{1 - η})$ . For non-axisymmetric cases, if we consider a long azimuthal length scale of order $(1 / \sqrt{1 - η})$ , then, $\partial_{x} \sim O (\sqrt{1 - η})$ , and the same order balances as the axisymmetric case are still attained with $(U_{B} + \overset{ˇ}{U}) \sim O (1 / \sqrt{1 - η})$ . In fact, the choice of this longer azimuthal length scale was chosen by [19], and used later in the study of a Ginzburg–Landau type of equation to investigate transition between axisymmetric and non-axisymmetric modes by [24–26] and [27]. Furthermore, this choice of a longer length scale was recommended by [28] for analysing flow dynamics in the narrow gap Taylor–Couette flow.

With this choice of a longer azimuthal length scale, three-dimensional investigation, such as the stability of axisymmetric Taylor-vortex flow against three-dimensional perturbations, and bifurcating three-dimensional flows, is accomplished easily in our Cartesian representation, equations (5.2)–(5.5).

Finally, it would be beneficial to provide narrow-gap limit studies based on the Cartesian coordinate system, similar to equation (3.9), by [18] and [17], especially for those with interest in asymptotic analysis. The aim of the former study was to obtain a rational analytic description for a linear neutral curve based on asymptotic theory, while an eigenvalue problem is solved numerically and compared with wide gap cases with $η = 5 / 7, 0.9$ and $0.99$ , not with those of Lin [20] or Chandrasekhar [21]. Nonlinear steady solutions were obtained in the latter study by restricting the computation to the $μ = 0$ case at a few selected values of very large Taylor numbers of order $10^{5}$ and a large axial wavenumber, $β = 16$ , to support the high-wavenumber asymptotic analysis. Note, however, that neither study is sufficient to describe the transition process from circular Couette flow to Taylor-vortex flow in the narrow-gap limit, which is the main objective of [1] and the current paper. We have shown that the transition proceeds through a supercritical or subcritical bifurcation depending on the angular velocities of the inner and the outer cylinder. As described above, the numerical study for $η = 0.883$ by [23] supports our finding of the subcritical branches for $μ < - 0.75$ .

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We have not used AI-assisted technologies in creating this article.

Authors’ contributions

M.N.: writing—original draft.

Conflict of interest declaration

We declare we have no competing interests.

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No funding has been received for this article.

Appendix Corrigendum

First, the reason why the case $μ = - 1$ was excluded was not due to a physical singularity, as has been described in the main text above, but simply an artefact related to how $Q$ was expressed in terms of $Ω$ . The inability to obtain a critical Taylor number $T a_{c}$ for $μ = - 1$ cannot be resolved by the introduction of scaled analysis, contrary to the wishful expectation described in the summary of [1].

Second, as described in §4, one of the inertia terms, ${\tilde{u}}_{φ}^{2} / r$ , which was omitted in [1], becomes important in the region of $R e \sim O (1 / \sqrt{1 - η})$ and $Ω \sim O (\sqrt{1 - η})$ , where the linear criticality occurs, and so should have been retained. With its inclusion the equations to be solved become

\begin{aligned} \nabla \cdot {\overset{ˇ}{u}}^{*} & = 0, \\ \partial_{t} {\overset{ˇ}{u}}^{*} + (u^{*} \cdot \nabla) u^{*} + Ω^{*} \frac{1 - μ}{1 + μ} z {\overset{ˇ}{u}}_{x}^{*} k - \frac{1}{2} ({\overset{ˇ}{u}}_{x}^{*})^{2} k & = - \nabla {\overset{ˇ}{p}}^{*} - Ω^{*} j \times {\overset{ˇ}{u}}^{*} + Δ {\overset{ˇ}{u}}^{*}, \end{aligned}

(A 1)

in place of eqn (5.1) of [1], where asterisks indicate the scaled quantities as expressed in §4.

For the axisymmetric case, equation (A 1) leads to

\begin{aligned} _{t} (\partial_{y y}^{2} + \partial_{z z}^{2}) \partial_{y y}^{2} ϕ = & (\partial_{y y}^{2} + \partial_{z z}^{2})^{2} \partial_{y y}^{2} ϕ + (\frac{1 - μ}{1 + μ} z - 1) Ω^{*} \partial_{y y y}^{3} ψ \\ + \frac{1}{2} \partial_{y y}^{2} (2 {\overset{ˇ}{U}}^{*} \partial_{y} ψ + \partial_{y} ψ \partial_{y} ψ) \\ - \partial_{y z}^{2} [(\partial_{y z}^{2} ϕ \partial_{y} - \partial_{y y}^{2} ϕ \partial_{z}) \partial_{y z}^{2} ϕ] - \partial_{y y}^{2} [(\partial_{y z}^{2} ϕ \partial_{y} - \partial_{y y}^{2} ϕ \partial_{z}) \partial_{y y}^{2} ϕ], \end{aligned}

(A 2)

\begin{aligned} _{t} \partial_{y y}^{2} ψ = & (\partial_{y y}^{2} + \partial_{z z}^{2}) \partial_{y y}^{2} ψ + (- R e^{*} + {\overset{ˇ}{U}}^{*'} (t, z)) \partial_{y y y}^{3} ϕ \\ - \partial_{y} [(\partial_{y z}^{2} ϕ \partial_{y} - \partial_{y y}^{2} ϕ \partial_{z}) \partial_{y} ψ], \end{aligned}

(A 3)

\begin{aligned} _{t} {\overset{ˇ}{U}}^{*} (t, z) = {\overset{ˇ}{U}}^{*^{″}} (t, z) + \partial_{z} \bar{(\partial_{y y}^{2} ϕ \partial_{y} ψ)} . \end{aligned}

(A 4)

The difference from eqns (5.10), (5.11) and (5.12) of [1] is the second line of (A 2), resulting from the inclusion of $\frac{1}{2} ({\overset{ˇ}{u}}_{x}^{*})^{2} \overset{ˇ}{k}$ in (A 1), which is nonlinear, and the absence of $Ω \partial_{y y y}^{3} ϕ$ in eqn (5.1) of [1] in (A 3). The deletion of $Ω \partial_{y y y}^{3} ϕ$ is necessary because $Ω$ is $O (1 - η)$ smaller than $(- R e + \overset{ˇ}{U},^{'} (t, z))$ in this high-Reynolds-number region. It can be shown (and numerically confirmed) that a linear analysis based on (A 2) and (A 3) does not change the results of [1].

As for the nonlinear analysis, the results differ, as can be seen by comparison of figures 11 and 12 with figures 1 and 2 of [1]. We followed the bifurcation branch for each $μ$ with the constraint

Figure 11. The bifurcations of the axisymmetric Taylor vortex flow from the basic state ( $M_{T}^{*} / R e^{*} = - 1$ ) for $μ = 0.5$ (purple), $μ = 0.0$ (green), $μ = - 0.5$ (light blue) and $μ = - 0.75$ (orange); $T a$ is given in equation (6.7).

Figure 12. Mean flow distortions. $T a = 147$ . $μ = 0.5$ (purple): $R e = 7, Ω = 21$ , $μ = 0.0$ (green): $R e = Ω = \sqrt{147} = 12.1243557$ , $μ = - 0.5$ (light blue): $R e = 21, Ω = 7$ according to $R e = (1 - μ) / (1 + μ) Ω$ . (a) Symmetric part and (b) anti-symmetric part.

\begin{matrix} \frac{Ω^{*}}{R e^{*}} = \frac{1 + μ}{1 - μ}, \end{matrix}

(A 5)

which is the scaled version of equation (4.1). Figure 11 shows the change in bifurcation nature from supercritical to subcritical at approximately $μ = - 0.75$ as $μ$ is decreased from 1, instead of the linear increases of the amplitude of the supercritical branches in [1]. Recall that we have seen the subcritical bifurcation of axisymmetric flow for $μ = - 1$ in figure 1b.

Footnotes

¹ Their definition of the Taylor number is given by $T \sim 2 (1 - μ) (ω_{i} r_{i} D / ν)^{2} D / r_{i})$ where $D = 2 d$ (see (6) in [19]). From equations (6.6) and (6.11) we get $τ = 2 (R e^{*})^{2}$ . Substitution of (4.4) leads to $T = 16 τ / (1 - μ)$ .

Footnotes

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Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences cover image

December 2024

Volume 480Issue 2304

Article Information

DOI:https://doi.org/10.1098/rspa.2023.0890
Published by:Royal Society
Online ISSN:1471-2946

History:

Manuscript received03/12/2023
Manuscript accepted17/10/2024
Published online18/12/2024

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Taylor–Couette system in the narrow-gap limit, revisited, with a corrigendum to the paper by Nagata (2023, Philosophical Transaction A.)

Abstract

1. Introduction

2. Formulation of the problem

3. Transformation to the Cartesian system

4. Analysis

5. Numerical analysis

6. Numerical method

7. Results

(a) The counter rotating case (μ=−1)

(i) Stability analysis

(ii) Nonlinear analysis

(b) The case when the outer cylinder is at rest (μ=0)

8. Summary

Data accessibility

Declaration of AI use

Authors’ contributions

Conflict of interest declaration

Funding

Appendix Corrigendum

Footnotes

Footnotes

This Issue

(a) The counter rotating case $(μ = - 1)$

(b) The case when the outer cylinder is at rest $(μ = 0)$