2. Problem formulation and computational details

2.1. Governing equations

We seek a time-dependent solution of the Navier–Stokes equation coupled with the finitely extensible nonlinear elastic-Peterlin (FENE-P) constitutive equation. It should be noted that this constitutive model has been widely used in numerical simulations of various types of polymer-induced and/or modified turbulence due to its close connection with the kinetic theory of polymer solutions, and has faithfully reproduced experimental observations (Yang & Khomami Reference Yang and Khomami1999; Somasi & Khomami Reference Somasi and Khomami2000; Li, Sureshkumar & Khomami Reference Li, Sureshkumar and Khomami2006; Thomas, Sureshkumar & Khomami Reference Thomas, Sureshkumar and Khomami2006; Alves, Oliveira & Pinho Reference Alves, Oliveira and Pinho2021; Dubief et al. Reference Dubief, Terrapon and Hof2023; Song et al. Reference Song, Zhu, Lin, Liu and Khomami2023b ). We have chosen a reference frame that co-rotates with the outer cylinder, which in turn allows quantifying the effect of the outer cylinder rotation as a Coriolis force (Ostilla et al. Reference Ostilla, Stevens, Grossmann, Verzicco and Lohse2013). In addition, this simplifies the boundary condition at both cylinder walls, reverting to that for ICRO TCF. Moreover, we have chosen $d=R_o-R_i$ , $d/U_i$ , $U_i$ and $\eta _p U_i/d$ as scales for length, time, velocity $\boldsymbol{u}$ and polymer stress $\boldsymbol{\tau }$ , respectively, where $U_i=R_i(\varOmega _i - \varOmega _o)$ is the inner cylinder velocity in the chosen frame of reference. Here, $\rho$ represents the density of the solution, and the total viscosity of the solution $\eta _t$ is the sum of the solvent ( $\eta _s$ ) and polymeric ( $\eta _p$ ) contributions . The resulting non-dimensional governing equations for the incompressible flow of FENE-P fluid are

(2.1) \begin{align} \boldsymbol {\nabla }\boldsymbol{\cdot }\boldsymbol{u}=0 , \\[-28pt] \nonumber \end{align}

(2.2) \begin{align} \frac {\partial \boldsymbol{u}}{\partial t}+\boldsymbol{u}\boldsymbol {\cdot } \boldsymbol{\nabla }{\boldsymbol{u}}=-\boldsymbol {\nabla }P+\frac {\beta }{\textit{Re}} {\nabla} ^2\boldsymbol{u}+\frac {1-\beta }{\textit{Re}}\boldsymbol {\nabla }\boldsymbol {\cdot }\boldsymbol{\tau }-Ro^{-1}\,\boldsymbol{e}_z\times \boldsymbol{\boldsymbol{u}} \\[0pt] \nonumber \end{align}

and

(2.3) \begin{align} \frac {\partial \boldsymbol{C}}{\partial t}+\boldsymbol{u}\boldsymbol{\cdot } \boldsymbol{\nabla }\boldsymbol{C}=\boldsymbol{C} \boldsymbol{\cdot } \boldsymbol{\nabla }\boldsymbol{u}+(\boldsymbol {\nabla }\boldsymbol{u})^{T}\boldsymbol {\cdot }\boldsymbol{C}-\boldsymbol{\tau } , \end{align}

where $\beta ={\eta _s}/{\eta _t}$ is the solvent to total zero-shear viscosity ratio, and $\boldsymbol{e}_z$ is the unit vector in the axial direction. Moreover, ${\boldsymbol{u}}=(u_r,u_\theta ,u_z)$ represents the velocity vector field in a cylindrical coordinate system $(r,\theta ,z)$ with the $z$ -axis coinciding with the cylinder axis. It should be noted that here the generalised pressure $P=p-\rho \varOmega _o^2r^2/2$ , where $p$ is the pressure, and $r$ is the radial distance to the cylinder axis, has included the effect of the centrifugal force $r\varOmega _o^2 \boldsymbol{e_r}$ (Dubrulle et al. Reference Dubrulle, Dauchot, Daviaud, Longaretti, Richard and Zahn2005; Dong Reference Dong2008). The conformation tensor $\boldsymbol{C}$ represents the ensemble average of the second moment of the end-to-end vector of the polymer chain, thus quantifying the polymer dynamics including elongation and orientation. Polymer stress $\boldsymbol{\tau }$ is related to conformation tensor $\boldsymbol{C}$ through the relationship

(2.4) \begin{align} \boldsymbol{\tau }=\frac {f(\boldsymbol{C})\,\boldsymbol{C}-\boldsymbol{I}}{\textit{Wi}}, \end{align}

where $\boldsymbol{I}$ is the unit tensor. The Peterlin function $f(\boldsymbol{C})$ provides a nonlinear force law that prevents over-extension of the polymer chain (Shaqfeh & Khomami Reference Shaqfeh and Khomami2021), and is defined as

(2.5) \begin{align} f(\boldsymbol{C})=\frac {L^2-3}{L^2-\text{tr}(\boldsymbol{C})}, \end{align}

where $\text{tr}(\boldsymbol{C})=C_{\textit{rr}}+C_{\theta \theta }+C_{zz}$ is the trace of the conformation tensor, and $L$ is the maximum extensibility of the polymer chain. The no-slip boundary conditions on walls, i.e. $(u_r,u_\theta ,u_z)=(0,U_{w\textit{all}},0)$ , as well as periodic boundary conditions along the homogeneous $z$ -direction, are supplemented to close the governing equation of fluid flow. It should also be noted that boundary conditions are not imposed on the walls for the constitutive equation (2.3) due to its hyperbolic nature (Yang & Khomami Reference Yang and Khomami1999; Alves et al. Reference Alves, Oliveira and Pinho2021). In fact, the wall values of $\boldsymbol{C}$ are obtained by solving a reduced form of (2.3), which effectively simplifies the convection term by implementing the constant wall velocity and the continuity equation (Lin et al. Reference Lin, Wan, Zhu, Liu, Lu and Khomami2022).

2.3. Computational set-ups

The flow geometry and the Taylor–Couette system configuration employed in our simulations are schematically illustrated in figure 3. We perform thorough DNS of viscoelastic TCF in radius ratio $\eta =5/7$ , which corresponds to a moderate curvature effect. To facilitate comparison with Newtonian flows, this curvature has been chosen because it is commonly used for investigations of rotation-induced turbulence dynamics in Newtonian flows (Ostilla et al. Reference Ostilla, Stevens, Grossmann, Verzicco and Lohse2013). The axial length of the computational domain is set as $L_z=2\pi d$ , which allows for at least three pairs of large-scale TV, demonstrating that the flow structures exhibit reasonable axial wavelengths, and ensuring the reliability of the turbulence statistics (Ostilla-Mónico et al. Reference Ostilla-Mónico, Verzicco and Lohse2015, Reference Ostilla-Mónico, Lohse and Verzicco2016). Here, $ \textit{Re}$ is fixed at $1600$ , for which a flow state containing turbulent TV is realised for ICRO (Andereck et al. Reference Andereck, Liu and Swinney1986; Grossmann et al. Reference Grossmann, Lohse and Sun2016). To fix the elastic effect, we have chosen $ \textit{Wi}=30$ . Therefore, the elasticity number, which quantifies the relative importance of elastic to inertial effects, is obtained as $El=\textit{Wi}/\textit{Re}=0.019$ , corresponding to a ‘weak elasticity’ regime in the previous experimental study (Dutcher & Muller Reference Dutcher and Muller2011, Reference Dutcher and Muller2013). In addition, $\beta$ , a polymer concentration measurement, is set at $0.9$ ; this choice, combined with a large value $L=120$ , can effectively rule out shear-thinning effects. To comprehensively examine the flow dynamics of viscoelastic TCF with rotation of the outer cylinder, a wide range $ \textit{Ro}^{-1}=-0.6$ to $ \textit{Ro}^{-1}=1$ is simulated, covering both counter- and co-rotating regimes. A summary of the control parameters is provided in table 1.

Figure 3.

Schematic of the Taylor–Couette system. Here, $R_i$ and $R_o$ denote the inner and outer cylinder radii, respectively, with $\eta = R_i/R_o$ being the radius ratio and $d = R_o - R_i$ the gap width, $\varOmega _i$ and $\varOmega _o$ are the respective angular velocities, and $L_z$ represents the axial length of the periodic computational domain.

Table 1.

The rotation effect considered in this study, which is quantified by the Rossby number $ \textit{Ro}^{-1}$ and covers both counter- and co-rotating regimes. To facilitate comparison with existing studies, the control parameters in the rotating frame of reference have also been expressed as two Reynolds numbers corresponding to the inner and outer cylinders, $ \textit{Re}_{i,o}=\rho R_{i,o}\varOmega _{i,o} d/\eta _t$ .

Figure 4.

Profiles of root mean square values of $(a)$ three velocity components and $(b)$ the typical polymer stress component $\tau _{\mathit{r\theta }}$ at $ \textit{Ro}^{-1}=0$ with various mesh sizes. Here, $\tilde {r}=(r-R_i)/d$ is the dimensionless distance to the inner cylinder wall.

A comprehensive grid convergence is conducted for the baseline case, i.e. viscoelastic TCF with purely inner cylinder rotation ( $ \textit{Ro}^{-1}=0$ ). Specifically, three mesh sizes are implemented for test simulations: ${N_{r}\times N_{\theta }\times N_z}={64\times 128\times 192}$ , $128\times 192\times 256$ and $192\times 192\times 512$ , with $N_{r,\theta ,z}$ being the grid points in the radial, azimuthal and axial directions, respectively. As shown in figure 4, both the velocity and the polymer stress fields attain excellent grid convergence with increasing spatial resolution. For the two finer grids, the results for the predicted velocity and polymer stress fields are nearly identical, i.e. only a small quantitative difference can be observed as the grid is successively refined. Hence the grid with resolution $128\times 192\times 256$ has been chosen because it balances computational efficiency and precision. The simulation protocol is as follows. We first perform the simulation at $ \textit{Ro}^{-1}=0$ , and start all other simulations at a new $ \textit{Ro}^{-1}$ from the statistically steady state obtained at the previous $ \textit{Ro}^{-1}$ . To ensure that we obtain statistically stationary flows, simulations are performed with a small time step $\Delta t=0.001$ and sufficiently long calculations of $O(1000T)$ , where $T=d/({\varOmega _i-\varOmega _o})R_i$ is the convective time unit. In addition, to obtain reliable turbulence statistics, 200 snapshots in a time period of approximately $1000T$ are used when performing the ensemble averaging $\langle {\cdot }\rangle =\langle {\cdot }\rangle _{\theta ,z,t}$ in the $\theta $ - and $z$ - directions, and in time. This allows for the evaluation of the fluctuating part of the variable $\phi$ as $\phi '=\phi -\langle \phi \rangle$ .

3. Rotation-driven transition pathways

A complete rotation-driven transition pathway for viscoelastic TCF is shown in figure 5( $a$ ). Moreover, to highlight the polymer-induced modifications, the Newtonian counterpart with varying $ \textit{Ro}^{-1}$ is shown in figure 5( $b$ ). For a stationary outer cylinder at $ \textit{Ro}^{-1}=0$ , while large-scale TV induced by centrifugal force still occupy the entire gap, they are significantly modified by polymer additives, as evidenced by the varying axial wavelengths and distorted patterns. Introducing outer cylinder rotation produces intriguing flow patterns in both counter-rotating ( $ \textit{Ro}^{-1}=-0.2\text{ to } {-0.6}$ ) and co-rotating ( $ \textit{Ro}^{-1}=0.5\text{ to } 1$ ) regimes. Specifically, counter-rotation inhibits the TV from penetrating the entire gap, consequently a relaminarised layer develops in the near-outer-wall region, which is consistent with the experimental observations in Dutcher & Muller (Reference Dutcher and Muller2013) for the lower $ \textit{Re}$ transitional TCF. As the strength of counter-rotation is enhanced, the turbulent vortices become weaker and the relaminarised layer becomes larger, delineating a neutral surface that separates turbulent and non-turbulent regions (Dong Reference Dong2008; Ostilla et al. Reference Ostilla, Stevens, Grossmann, Verzicco and Lohse2013). Although a stable relaminarised layer is also observed for counter-rotating Newtonian TCF (see figure 5 $b$ ), the addition of polymers and commensurate elastic forces has resulted in two important modifications. (i) Chaotic vortical structures are found in the inner-wall turbulent region, while the Newtonian counterpart demonstrates the persistence of regular TV with diminished radial length. This underscores the differences in the vortex generation mechanism for viscoelastic TCF. (ii) The shape and radial position of the neutral surface are modified, which is particularly striking for the case $ \textit{Ro}^{-1}=-0.5$ depicted in figure 6( $a$ ). Specifically, we extract the neutral surface, which is defined as the surface where, in the inertial reference frame, the angular velocity is zero, and compare it with its Newtonian counterpart. To that end, the neutral surface in the viscoelastic flow exhibits a weaker dependence on the axial height, while its radial location is slightly altered. These observations are highlighted by the black solid lines in figure 6( $a$ ).

Figure 5.

Time and $\theta$ -direction averaged vectors of radial $(\langle u_r\rangle _{\mathit{\theta ,t}})$ and axial $(\langle u_z\rangle _{{\theta ,t}})$ velocities and contour plots of streamwise vorticity $\langle \omega _{\theta }\rangle _{{\theta ,t}}$ in the $(r,z)$ plane for ( $a$ ) viscoelastic and ( $b$ ) Newtonian TCF at various  $ \textit{Ro}^{-1}$ .

Moreover, we evaluate the axially averaged radial position $\tilde {r}_N$ to quantitatively capture the variation of the neutral surface, which shows that the elastic effect leads to a narrower relaminarised layer (see figure 6 $b$ ). As this study is conducted with a relatively weak elasticity effect ( $El=0.019$ ), it can be expected that increasing the elasticity may further narrow the relaminarised layer, substantially widening the turbulent region. Interestingly, due to the emergence of the relaminarised layer, the effective gap $d_{\textit{eff}}$ reduces from $d$ to $\tilde {r}_N$ as counter-rotation is introduced, therefore altering the effective elasticity number $El_{\textit{eff}}=\lambda \nu /d_{\textit{eff}}^2$ (Dutcher & Muller Reference Dutcher and Muller2013). As demonstrated in figure 6( $b$ ), $El_{\textit{eff}}$ increases from 0.024 to 0.214 as $ \textit{Ro}^{-1}$ decreases from $-0.2$ to $-0.6$ ; this stronger ‘effective elastic effect’ may be responsible for the more chaotic vortical structures observed in the inner-wall turbulent region (figure 5). Nevertheless, the maximum $El_{\textit{eff}}=0.214$ remains below the transition threshold for EDT ( $El_{\textit{eff}}=0.371$ ) documented in Dutcher & Muller (Reference Dutcher and Muller2013), indicating that the inner-wall turbulence falls within the elasto-inertial regime.

Evidently, the variations of the neutral surface originate from the elastic forces that modify the balance between the pressure gradient and the Coriolis force in Newtonian flow. Inspired by the research of neutral surface in Newtonian TCF (Ostilla et al. Reference Ostilla, Stevens, Grossmann, Verzicco and Lohse2013), the following new relation that embeds the elastic force is derived in viscoelastic TCF:

(3.1) \begin{eqnarray} \frac {\langle {u_\theta }\rangle ^2_{\theta ,t}}{r}=\frac {\partial {\langle {p}\rangle }_{\theta ,t}}{\partial {r}}-{\underbrace {\frac {1-\beta }{\textit{Re}}\left(\frac {\langle {\tau _{\textit{rr}}}\rangle _{\theta ,t}}{r}-\frac {\langle {\tau _{\theta \theta }}\rangle _{\theta ,t}}{r}+\left\langle \frac {\partial {\tau _{\textit{rr}}}}{\partial {r}}\right\rangle _{\theta ,t}\right)}_{f_{pr}}}-Ro^{-1}\,\langle {u_{\theta }}\rangle _{\theta ,t}. \end{eqnarray}

A detailed derivation of this relation is given in Appendix A. The radial position of the neutral surface in the rotating reference corresponds to the location where $\langle {u_{\theta }}\rangle _{\theta ,t}$ , once transformed to the stationary frame, equals zero. As demonstrated by (3.1), the radial polymer body force $f_{pr}$ , which includes the first normal stress difference $N_1=(1-\beta ) \langle \tau _{\textit{rr}}-\tau _{\theta \theta }\rangle /\textit{Re}$ and the polymer radial stress gradient $(1-\beta )\langle {\partial \tau _{\textit{rr}}/\partial {r}}\rangle /\textit{Re}$ , contributes to the modification of the neutral surface. As expected, a theoretical prediction of the neutral surface obtained from the right-hand side of (3.1), depicted as red dashed lines in figure 6( $a$ ), exhibits excellent agreement with the ‘real’ neutral surface (black lines) obtained from the simulated velocity field.

The new balance (3.1) also indicates that the radial polymer body force $f_{pr}$ is closely related to the development of the relaminarised layer. Hence we further examine the radial distributions of $f_{pr}$ for the counter-rotating regime. As shown in figure 7( $a$ ), $f_{pr}$ is much more significant in the inner-wall region, facilitating the destabilisation of the flow in this area. In the outer-wall region, $f_{pr}$ approaches zero, especially for $ \textit{Ro}^{-1}=-0.6$ , which contributes to the formation of a relaminarised layer. Moreover, enhancement of counter-rotation ( $ \textit{Ro}^{-1}=0\to -0.6$ ) suppresses the magnitude of $f_{pr}$ , which explains the weaker vortical structure observed in figure 5( $a$ ). That is, the suppressed $f_{pr}$ and associated weaker torque cannot guarantee a sufficiently strong elastic vortex generation, which will be demonstrated in the following.

Figure 6.

( $a$ ) Contour plots of the time and $\theta$ -direction averaged angular velocity $\langle {\omega }\rangle _{\theta ,t}$ , with neutral surface indicated by the black solid line for various $ \textit{Ro}^{-1}$ . For a discernible comparison, the left- and right-hand halves are shown for Newtonian and viscoelastic flows, respectively. The red dashed lines shown in the viscoelastic flows correspond to the right-hand side of (3.1). Here, for each specific $ \textit{Ro}^{-1}$ , an identical scaling factor for vector length has been applied in both the Newtonian and viscoelastic cases to ensure a fair and meaningful comparison. ( $b$ ) The variation of the axially averaged neutral surface position $\tilde {r}_N$ and the corresponding effective elasticity number $El_{\textit{eff}}$ as a function of $ \textit{Ro}^{-1}$ .

Introducing co-rotation of the outer cylinder results in strikingly different flow physics. As shown in figure 5( $a$ ), for $ \textit{Ro}^{-1}=0.5$ , large-scale TV break down into vortical structures exhibiting much smaller axial length scales, which have been named ‘penetrating vortices’ in our previous study (Lin et al. Reference Lin, Song, Liu, Liu, Lu and Khomami2024a ). The enhancement of co-rotation leads to a further reduction of the axial length scale of the penetrating vortices, and a larger population of small-scale near-wall vortices. These observations indicate the importance of the rotation effect in the generation of vortical structures in the co-rotation regime of viscoelastic TCF. In stark contrast, the large-scale TV persist in the Newtonian TCF at $ \textit{Ro}^{-1}=0.5$ , and they completely disappear as co-rotation is increased (see figure 5 $b$ ), and the flow enters the Rayleigh stable regime. The distinction in rotation-driven transition sequence is attributed to the existence of the so-called ERI in viscoelastic TCF, which originates from the combination of elasticity and rotation effects (Bai et al. Reference Bai, Latrache, Kelai, Crumeyrolle and Mutabazi2023). Specifically, two distinct radial forces, namely, the centrifugal force and the aforementioned $f_{pr}$ , can destabilise the flow in a viscoelastic TCF (Bai et al. Reference Bai, Vieu, Crumeyrolle and Mutabazi2021, Reference Bai, Latrache, Kelai, Crumeyrolle and Mutabazi2023). The fact that centrifugal force is insufficient to trigger turbulent flow in the stable Rayleigh regime highlights the important role of radial elastic forces. Therefore, we have also examined the radial distributions of $f_{pr}$ in the co-rotation regime depicted in figure 7( $b$ ). In contrast to counter-rotating cases, the co-rotation effect leads to substantial development of $f_{pr}$ , especially in the near-wall regions. This observation is consistent with the emergence of small-scale vortices near the wall for $ \textit{Ro}^{-1}\ge 0.5$ , which highlights the critical role of $f_{pr}$ in the generation of vortices. In particular, the magnitude of $f_{pr}$ exhibits saturation behaviour in the strong co-rotation regime, indicating that the polymer chains approach full extension under these intense flow conditions. In fact, the torque induced by the force of the polymer body $f_{pr}$ introduces a novel mechanism for the generation of an elastic vortex, which can be quantified by reviewing the flow-wise enstrophy budget $\langle \omega ^2_{\theta }\rangle$ (Kim et al. Reference Kim, Li, Sureshkumar, Balachandar and Adrian2007; Zhu et al. Reference Zhu, Song, Lin, Liu, Lu and Khomami2022). To that end, we have derived the corresponding budget equation in cylindrical coordinates, detailed in Appendix B.

Figure 7.

Ensemble-averaged radial polymer body force $f_{pr}$ for ( $a$ ) counter-rotating cylinders and ( $b$ ) co-rotating cylinders.

Figure 8.

The various production terms in the streamwise enstrophy budget for (a,b) counter-rotating and (c,d) co-rotating outer cylinders. Here, only the near-wall regions, where these production terms are most intense, are shown.

We closely examine the following production terms for streamwise enstrophy: the combined production term $W_{\theta ,c}$ , comprising the contributions of stretching, tilting and twisting of vortical structures; the rotation term $W_{\theta ,R}$ , which represents the generation of vorticity by Coriolis forces that arise due to rotation of the outer cylinder; and the polymer-induced torque $W_{\theta ,p}$ , which represents the correlation between streamwise vorticity and the torque generated by the force of the polymer body. As shown in figure 8, the classical generation term in Newtonian flow, $W_{\theta ,c}$ , is relatively small in viscoelastic TCF with various $ \textit{Ro}^{-1}$ , which aligns with the observations in previous studies showing that inertial vortices are suppressed by polymer additives (Teng et al. Reference Teng, Liu, Lu and Khomami2018; Dubief et al. Reference Dubief, Terrapon and Hof2023). Here, the elastic term $W_{\theta ,p}$ makes a large contribution to the generation of turbulent vortices, especially in the wall regions. This is consistent with the intense polymer body forces observed near the wall (see figure 7), confirming that the substantial magnitude of $f_{pr}$ in this region generates the crucial torque that facilitates the formation of a vortex. Elastically modified turbulent flows further indicate the mechanistic differences in rotation-driven transition pathways between Newtonian and viscoelastic TCFs.

A comparison between the co- and counter-rotating cases reveals the contrasting effects of rotation. Specifically, $W_{\theta ,R}$ inhibits the vortex generation when the outer cylinder counter-rotates, as evidenced by the negative values of $W_{\theta ,R}$ in the inner-wall region (see figure 8 $a$ ). In contrast, as demonstrated in figures 8(c,d), $W_{\theta ,R}$ becomes positive throughout most of the gap for $ \textit{Ro}^{-1}\gt 0$ , indicating that rotation facilitates the formation of vortical structures. This supports the ERI mechanism proposed by Bai et al. (Reference Bai, Latrache, Kelai, Crumeyrolle and Mutabazi2023), which requires the combination of elastic and rotation effects to trigger turbulent flow. Moreover, strengthening the rotation effect leads to further suppression of various vortex generation mechanisms for $ \textit{Ro}^{-1}\lt 0$ , whereas these mechanisms are progressively enhanced for $ \textit{Ro}^{-1}\gt 0$ . It is also interesting to compare the present findings with curvature-free cases, i.e. the rotation-driven transition pathway in viscoelastic PRCF observed by Zhu et al. (Reference Zhu, Song, Lin, Liu, Lu and Khomami2022). Specifically, two notable distinctions emerge. (i) The viscoelastic PRCF fully relaminarises at $ \textit{Ro}^{-1}=1$ , whereas in the present TCF system, turbulence can still be sustained under strong co-rotation. (ii) In viscoelastic PRCF, the polymer-induced torque $W_{\theta ,p}$ , which is negative across the gap, suppresses the generation of turbulent vortices. We attribute these distinctions to the presence of substantial hoop stress and the associated radial polymer body force $f_{pr}$ (figure 7) induced by curvature ( $f_{pr}$ is directly related to the polymer-induced torque; see (B1)), which is absent in turbulent unidirectional shear flows.

Overall, the observations presented above have identified a novel rotation-driven transition pathway that is unique to viscoelastic TCF. Specifically, counter-rotation produces a modified neutral surface and an inner layer of turbulent vortices whose dynamics are strongly modulated by elasticity, whereas co-rotation facilitates the breakdown of large-scale TV into small-scale vortical structures sustained by polymer-induced torque. These findings highlight the central role of elastic forces in altering the rotation-driven transition sequence, thereby fundamentally distinguishing viscoelastic flows from their Newtonian counterparts. It is also interesting to discuss how this transition pathway will be modified under a stronger elastic effect: since larger elastic forces may trigger the elastic Görtler instability at $ \textit{Ro}^{-1}=0$ (Song et al. Reference Song, Teng, Liu, Ding, Lu and Khomami2019, Reference Song, Lin, Liu, Lu and Khomami2021), a transition from small-scale elastic Görtler to small-scale penetrating vortices can be observed for $ \textit{Ro}^{-1}=0\to 1$ . In the following, we further examine the corresponding turbulence dynamics arising from these topological changes in the flow structure to provide mechanistic insights into the rotation-driven transition pathway in viscoelastic TCF.

4. Turbulence dynamics

Based on the observations of flow structures depicted in § 3, significant differences in the turbulence dynamics between the co-rotating and counter-rotating regimes are anticipated. This is first demonstrated by examining the global response of TCF using the Nusselt number ( $ \textit{Nu}_{\omega }$ ), which quantifies the efficiency of the angular momentum transport. In particular, $ \textit{Nu}_{\omega }$ is defined as $J^{\omega }/J^{\omega }_{\textit{lam}}$ , where the conserved transverse angular momentum current $J^{\omega }$ is calculated as (Song et al. Reference Song, Teng, Liu, Ding, Lu and Khomami2019)

(4.1) \begin{align} J^{\omega }=r^3\left[\langle {u_r\omega }\rangle -\frac {\beta }{\textit{Re}}\frac {\partial \langle {\omega }\rangle }{\partial r}-\frac {1-\beta }{\textit{Re}}\frac {\langle {\tau _{r\theta }}\rangle }{r}\right] \!, \end{align}

with $J^{\omega }_{\textit{lam}}$ being the laminar value of $J^{\omega }$ . As shown in figure 9( $a$ ), $ \textit{Nu}_{\omega }$ in viscoelastic TCF is larger than its Newtonian counterpart for all $ \textit{Ro}^{-1}$ considered, demonstrating the occurrence of drag enhancement. The degree of drag enhancement is found to be strongly influenced by the rotation of the outer cylinder. Specifically, counter-rotation produces only mild drag enhancement (approximately $20\,\%$ ), whereas co-rotation produces a pronounced drag enhancement, especially at large positive $ \textit{Ro}^{-1}$ . To this end, the optimal transport documented in Newtonian TCF, i.e. the local maxima of $ \textit{Nu}_{\omega }$ at $ \textit{Ro}^{-1}=0$ (Ostilla et al. Reference Ostilla, Stevens, Grossmann, Verzicco and Lohse2013), has been eradicated by polymer additives. The underlying mechanism for the disappearance of optimal transport can be attributed to the efficient angular momentum transport induced by polymer stresses.

Figure 9.

( $a$ ) Plots of $ \textit{Nu}_{\omega }$ as a function of $ \textit{Ro}^{{-1}}$ . Here, $ \textit{Nu}_{\omega }$ is obtained as ${J^{\omega }}/{J^{\omega }_{\textit{lam}}}$ , where $J^{\omega }_{\textit{lam}}$ is the laminar value of $J^{\omega }$ . Previous DNS results for a Newtonian TCF at the same $ \textit{Re}$ (Ostilla et al. Reference Ostilla, Stevens, Grossmann, Verzicco and Lohse2013) are included for comparison. ( $b$ ) Volume-averaged angular momentum fluxes as a function of $ \textit{Ro}^{-1}$ . Here, the volume-averaged values are obtained by the integral across the gap, e.g. $J^{v}_c = (1/d) {\int _{R_i}^{R_o}{J^{\omega }_c}\,{\rm d}r}$ and $J^{v}_p = (1/d) {\int _{R_i}^{R_o}{J^{\omega }_p}\,{\rm d}r}$ . (c,d) The balance of angular momentum flux for viscoelastic TCF at two representative values of $ \textit{Ro}^{-1}$ : $-0.5$ and $0.83$ , respectively.

Hence we examine $J^{\omega }$ that encompasses various mechanisms for angular momentum transport. Specifically, the right-hand side of (4.1) in sequence represents the contributions of Reynolds stress (convective flux $J^{\omega }_{c}$ ), viscous stress (diffusive flux $J^{\omega }_d$ ) and polymer stress (polymeric source/sink term $J^{\omega }_p$ ). Figure 9( $b$ ) illustrates the competition between $J^{\omega }_{c}$ and $J^{\omega }_p$ as $ \textit{Ro}^{-1}$ varies. Due to the polymer-induced suppression of Reynolds stress,  $J^{\omega }_{c}$ in viscoelastic TCF is much smaller than in its Newtonian counterpart. Interestingly, $J^{\omega }_{c}$  still exhibits optimal transport at $ \textit{Ro}^{-1}=-0.2$ , as $J^{\omega }_p$ increases monotonically when the outer cylinder changes from counter-rotating to co-rotating and, in turn, dominates angular momentum transport at large positive $ \textit{Ro}^{-1}$ . As a result, the optimal transport in $J^{\omega }_{c}$ is overshadowed by the large and monotonic increase of $J^{\omega }_p$ , thus eradicating the Newtonian optimal transport peak. It raises a new question that will be considered in our future study, namely whether there is optimal transport in viscoelastic TCF at high $ \textit{Re}$ for which $J^{\omega }_c$ is sufficiently strong.

We further elucidate the mechanism responsible for the occurrence of large $J^{\omega }_p$ , by examining the balance of the angular momentum flux and its connection to flow structures. As shown in figures 9(c,d), the spatial distribution of angular momentum fluxes differs markedly between counter-rotating and co-rotating regimes. For counter-rotation in $ \textit{Ro}^{-1}=-0.5$ , the formation of a relaminarised layer causes the outer-wall region to be dominated by the diffusive $J_d^{\omega }$ . In contrast, within the inner-wall region, chaotic Taylor-like vortices drive notable convective transport (Zhang et al. Reference Zhang, Fan, Su, Xi and Sun2025), producing a local maximum in $J_c^{\omega }$ at approximately $\tilde {r}=0.3$ . These vortices also effectively stretch polymer chains, resulting in substantial $J_p^{\omega }$ (Lin et al. Reference Lin, Song, Liao, Liu, Lu and Khomami2025b ). For co-rotation at $ \textit{Ro}^{-1}=0.83$ , the turbulent flow develops in the entire bulk region, and correspondingly both $J_p^{\omega }$ and $J_c^{\omega }$ are non-trivial across the gap. Nevertheless, $J_p^{\omega }$ is much greater than $J_c^{\omega }$ , and therefore governs the efficient transport of angular momentum. This pronounced $J_p^{\omega }$ originates from the distinct penetrating vortices generated by ERI (Lin et al. Reference Lin, Song, Liu, Liu, Lu and Khomami2024a ), which promote efficient incoherent transport and homogenisation of polymer stress (Lin et al. Reference Lin, Liu, Zhao, Lu and Khomami2025a ). It should be noted that enhanced polymer-stress-related transport has also been observed in viscoelastic PRCF subjected to strong rotation (Zhu et al. Reference Zhu, Song, Lin, Liu, Lu and Khomami2022), for which wall-attached Coriolis-force-generated vortices play a dominant role.

Figure 10.

Ensemble-averaged Reynolds stress $\langle {u'_r{u'_\theta }}\rangle$ and polymer shear stress $-(1-\beta )\langle {\tau _{r\theta }}\rangle /\textit{Re}$ for ( $a$ ) counter-rotation and ( $b$ ) co-rotation. Here, the polymer stress has been rescaled by $(1-\beta )/\textit{Re}$ for a more reasonable comparison with the Reynolds stress.

Figure 10 allows for a direct examination of the competition between inertial and elastic forces, further highlighting the above observations in the transport of angular momentum. Inertial Reynolds stress $\langle {u'_ru'_{\theta }}\rangle$ is suppressed by improving co-rotation or counter-rotation. An exception occurs at $ \textit{Ro}^{-1}=-0.2$ , for which the formation of more structured large-scale TV (see figure 5 $a$ ) enhances coherent convective transport; this also explains the non-monotonic trend of the convective flux $J_c^{\omega }$ observed in figure 9( $b$ ). In contrast, the polymer stress gradually increases as the outer cylinder changes from counter-rotating to co-rotating ( $ \textit{Ro}^{-1}=-0.6\to 1$ ), leading to a monotonic increase in elastic flux $J_p^{\omega }$ . Furthermore, for negative $ \textit{Ro}^{-1}$ , Reynolds and elastic stresses are of the same order of magnitude, making inertia and elastic effects comparable, while for positive $ \textit{Ro}^{-1}$ , the elastic force becomes dominant. The above competition of driving forces indicates the variations in the self-sustaining mechanisms of turbulent motions as a function of the rotation of the outer cylinder.

Hence the commensurate turbulence dynamics associated with the rotation-driven transition pathway is further elucidated by scrutinising the production mechanism of TKE. The complete expression of the TKE budget, along with a detailed analysis of the physical significance of each term, is provided in Appendix C. To elucidate the rotation-induced modifications in the self-sustaining mechanism of viscoelastic TCF, the elastic and inertial production terms, i.e. $P_e$ and $P_{t}$ , are extracted and compared across the full range of $ \textit{Ro}^{-1}$ in figure 11. In general, enhancing counter-rotation suppresses the overall production of TKE, whereas co-rotation significantly promotes the elastic production mechanism. Specifically, in the counter-rotation regime $ \textit{Ro}^{-1} \le -0.5$ , the magnitudes of $P_t$ and $P_e$ are found to be comparable, as shown in figure 11 $(a)$ . This suggests that for strong counter-rotation, the two production mechanisms are both important in sustaining turbulence, characterising a classical elasto-inertial regime that is distinct from the EDT state observed by Dutcher & Muller (Reference Dutcher and Muller2013) at the ‘moderate elasticity’ regime ( $El\approx 0.1{-} 0.2$ ). Furthermore, as $ \textit{Ro}^{-1}$ increases from $-0.5$ towards the co-rotation regime, $P_e$ exhibits a monotonic and pronounced intensification that progressively exceeds $P_t$ . This dominance of $P_e$ signifies a fundamental transition in the self-sustaining cycle, where the flow evolves from elasto-inertial to EDT. A particularly notable feature, clearly seen in figure 11( $b$ ), is that $P_e$ becomes positive in the bulk region ( $\tilde {r}=0.2 {-} 0.7$ ) within the co-rotation regime ( $ \textit{Ro}^{-1}=0.5{-} 1$ ), indicating that polymers release energy in turbulent motions. This behaviour is strikingly different from the negative $P_{e}$ typically observed in the bulk of various wall-bounded polymeric turbulent flows (Teng et al. Reference Teng, Liu, Lu and Khomami2018; Zhu & Xi Reference Zhu and Xi2020; Song et al. Reference Song, Lin, Liu, Lu and Khomami2021). We attribute the emergence of positive $P_e$ to the ERI that triggers turbulence in the co-rotating regime (Bai et al. Reference Bai, Latrache, Kelai, Crumeyrolle and Mutabazi2023). This mechanism not only intensifies the polymer stretch but also enables the polymers to feed energy back in turbulent motions, giving rise to the observed EDT.

Figure 11.

Rotation-induced modifications in the elastic production $P_{e}$ and the inertial production $P_{t}$ of TKE for $(a)$ counter-rotation and $(b)$ co-rotation.

As noted above, the introduction of the rotation effect results in a delicate competition between the Reynolds and polymer stresses, accompanied by corresponding modifications to the turbulence sustaining mechanism. Although a nearly vanishing Reynolds stress caused by polymer suppression is a prominent feature of EIT in pipe and channel flows (Xi Reference Xi2019; Dubief et al. Reference Dubief, Terrapon and Hof2023), previous studies have shown that elastic forces enhance the production of turbulent shear stress in the bulk and outer-wall regions of TCF (Song et al. Reference Song, Lin, Liu, Lu and Khomami2021). However, a comprehensive understanding of how the combined effects of rotation and elasticity influence the generation of Reynolds shear stress is still lacking. In this study, we examine the $\langle u'_r u'_{\theta }\rangle$ budget to elucidate the mechanistic response of the flow to variations in $ \textit{Ro}^{-1}$ . For the sake of clarity and completeness, the governing budget equation and a comprehensive interpretation of the physical significance of its constituent terms are detailed in Appendix D.

We first concentrate on the impact of system rotation on the generation of Reynolds shear stress, which is explicitly captured by the Coriolis term $Co_{r\theta }$ . As seen in figure 12, $Co_{r\theta }$ reverses its role from suppressing to facilitating $\langle u'_r u'_\theta \rangle$ production as the outer cylinder changes from counter-rotating to co-rotating, according to the transition pathway observed in figure 5( $a$ ). However, the response of $Co_{r\theta }$ to increasing rotation is strongly direction-dependent. Specifically, the magnitude of $Co_{r\theta }$ decreases with the enhancement of counter-rotation ( $ \textit{Ro}^{-1}=-0.2\to -0.6$ ), which can be attributed to the gradual attenuation of turbulent vortices. On the other hand, $Co_{r\theta }$ increases monotonically with stronger co-rotation ( $ \textit{Ro}^{-1} = 0.5 \to 1$ ), with the most pronounced enhancement occurring in the near-wall regions. This behaviour can be ascribed to the strengthening of ERI under stronger co-rotation. Nevertheless, the increase in $Co_{r\theta }$ is offset by the reduction in $P_{r\theta }$ and $C_{r\theta }$ (see figures 19(a,b) below), which eventually leads to a net decrease in $\langle {u'_ru'_{\theta }}\rangle$ as $ \textit{Ro}^{-1}$ increases from 0.5 to 1 (figure 10 $a$ ).

Figure 12.

Rotation-induced modifications in the production of Reynolds stress: ( $a$ ) the Coriolis force term $Co_{r\theta }$ , and ( $b$ ) the elastic production term $E_{r\theta }$ .

The elastic force term $E_{r\theta }$ has been shown to play a non-trivial role in promoting/suppressing $\langle {u'_ru'_{\theta }}\rangle$ production (Song et al. Reference Song, Lin, Liu, Lu and Khomami2021). Here, we further examine how $E_{r\theta }$ responds to variations in $ \textit{Ro}^{-1}$ . As illustrated in figure 12( $b$ ), $E_{r\theta }$ retains a similar near-wall signature for all $ \textit{Ro}^{-1}$ , but exhibits a markedly different profile in the bulk. That is, $E_{r\theta }$ facilitates the production of $\langle {u'_ru'_{\theta }}\rangle$ for $ \textit{Ro}^{-1}\geq 0$ , but inhibits it for $ \textit{Ro}^{-1}\leq -0.5$ . This is again in line with the fact that the ERI in the co-rotation regime results from the combination of elastic and rotation effects. However, an exception occurs at $ \textit{Ro}^{-1}=-0.2$ , for which $E_{r\theta }$ still contributes positively to $\langle {u'_ru'_{\theta }}\rangle$ production. This is reasonable because at $ \textit{Ro}^{-1}=-0.2$ , the counter-rotation is relatively weak, and a flow pattern similar to that at $ \textit{Ro}^{-1}=0$ is attained (see figure 5 $a$ ).

In general, the above discussions have thoroughly examined the turbulence dynamics commensurate with the rotation-driven transition pathway observed in § 3, demonstrating a transition from elasto-inertial turbulence to EDT as the outer cylinder transitions from counter-rotation to co-rotation ( $ \textit{Ro}^{-1}=-0.6 \to 1$ ). These mechanistic changes have highlighted the important role of polymer stress, which develops monotonically with increasing $ \textit{Ro}^{-1}$ , and eradicates the optimal transport observed in Newtonian TCF. Indeed, the polymer stress originates from the dynamics of polymer chains, i.e. their stretch and orientation in turbulent flows, which will be carefully examined in the following.

5. Polymer dynamics

Polymer molecules deform through their intricate coupling with coherent turbulent vortices, thereby generating elastic stress. Examining these intriguing polymer behaviours provides a mesoscopic perspective to understand the mechanisms underlying diverse viscoelastic wall-bounded turbulence, ranging from polymer-induced drag reduction in channels (White & Mungal Reference White and Mungal2008; Pereira et al. Reference Pereira, Mompean, Thais and Thompson2017) to the self-sustaining cycle of elastic turbulence (Song et al. Reference Song, Lin, Zhu, Wan, Liu, Lu and Khomami2023a ). Nevertheless, the behaviour of the polymer in viscoelastic TCF with rotation of the outer cylinder remains an open question. We therefore investigated how rotation influences polymer dynamics (i.e. deformation and orientation) and relations to the resultant elastic stress.

Figure 13.

Ensemble-averaged ( $a$ ) polymer chain extension rate $\langle {\text{tr}(\boldsymbol{C})}\rangle /L^{\textrm {2}}$ and its ( $b$ ) azimuthal, ( $c$ ) radial and ( $d$ ) axial components at various $ \textit{Ro}^{-1}$ values. Note that the vertical axis in ( $c$ ) has been adjusted to clearly show the variations in $\langle {C_{\textit{rr}}}\rangle /L^{\textrm {2}}$ .

We examine the overall elongation of the polymer by evaluating the polymer chain extension rate $\langle {\text{tr}(\boldsymbol{C})}\rangle /L^2$ . As shown in figure 13( $a$ ), the increase in $ \textit{Ro}^{-1}$ from $-0.6$ to $1$ leads to a monotonic enhancement of $\langle {\text{tr}(\boldsymbol{C})}\rangle /L^2$ . Similarly to viscoelastic wall-bounded parallel shear turbulent flows (Dallas et al. Reference Dallas, Vassilicos and Hewitt2010), the most significant polymer elongation occurs near the wall, as the near-wall turbulence dynamics plays a crucial role in polymer deformation. This feature is particularly evident in the co-rotation cases ( $ \textit{Ro}^{-1}\gt 0$ ), where the population of near-wall vortices increases. Interestingly, strengthening the counter-rotation pushes the outer-wall peak inwards, as the relaminarised layer, which acts as an equivalent outer cylinder, becomes thicker. Examination of different components of $\langle {\text{tr}(\boldsymbol{C})}\rangle /L^2$ provides additional insight into polymer elongation in viscoelastic TCF with rotation of the outer cylinder. As evidenced by the high values of the azimuthal component $\langle C_{\theta \theta }\rangle /L^2$ (figure 13 $b$ ), the primary elongation occurs in the streamwise direction due to the mean shear, a phenomenon that is well documented in a host of viscoelastic wall-bounded turbulence (Sureshkumar, Beris & Handler Reference Sureshkumar, Beris and Handler1997; Dubief et al. Reference Dubief, White, Terrapon, Shaqfeh, Moin and Lele2004; Li et al. Reference Li, Sureshkumar and Khomami2006). The normal (radial) component of the wall $\langle C_{\textit{rr}}\rangle /L^2$ is considerably smaller compared to $\langle C_{\theta \theta }\rangle /L^2$ , yet exhibits a pronounced peak in the centre of the gap (see figure 13 $c$ ). As shown in the following, this behaviour is related to the vortex-induced variation in polymer orientation. It should be noted that both $\langle C_{\theta \theta }\rangle /L^2$ and $\langle C_{\textit{rr}}\rangle /L^2$ exhibit a similar trend in response to variations in $ \textit{Ro}^{-1}$ , both increasing monotonically as $ \textit{Ro}^{-1}$ increases from $-0.6$ to $1$ . In contrast, altering the rotation of the outer cylinder induces a more complicated response in $\langle C_{zz}\rangle /L^2$ . As shown in figure 13( $d$ ), the most pronounced axial polymer elongation occurs in the ICRO case $ \textit{Ro}^{-1}=0$ , while both counter-rotation and co-rotation suppress $\langle C_{zz}\rangle /L^2$ . This observation emphasises the need for a closer examination of the relationship between polymer behaviour and vortical structures.

Figure 14.

Ensemble-averaged polymer normal stresses: ( $a$ ) $\langle {\tau _{\theta \theta }}\rangle$ , ( $b$ ) $\langle {\tau _{\textit{rr}}}\rangle$ , ( $c$ ) $\langle {\tau _{zz}}\rangle$ for various $ \textit{Ro}^{-1}$ values. Here, the polymer stress has been rescaled by $(1-\beta )/\textit{Re}$ for a more reasonable comparison with the Reynolds stress.

The normal stresses of the polymer (i.e. $\tau _{\textit{rr}}$ , $\tau _{\theta \theta }$ and $\tau _{zz}$ ) are shown in figure 14, which demonstrates the obvious correspondence with the components of the extension of the polymer chain (figures 13 $b$ – $d$ ). Hence we observe a substantial enhancement of $\langle \tau _{\theta \theta }\rangle$ as $ \textit{Ro}^{-1}$ changes from $-0.6$ to $1$ (see figure 14 $a$ ). This increase in $\langle \tau _{\theta \theta }\rangle$ in turn dictates changes in the hoop stress, since the normal stress of the polymer in the radial direction is much smaller than $\langle \tau _{\theta \theta }\rangle$ , resulting in large hoop stresses in the co-rotation regime. Eventually, the radial polymer body force exhibits two near-wall local minima (figure 7); this facilitates polymer-induced vortex generation as discussed in § 3. Similarly to $\langle C_{zz}\rangle$ , the axial polymer normal stress $\langle \tau _{zz} \rangle$ is also suppressed by both counter-rotation and co-rotation. To this end, the strong coupling between the polymer elongation and associated hoop stresses and the vortex generation mechanism is supported, highlighting the importance of intricate flow microstructure coupling in determining turbulence dynamics.

The above discussion, based on $\langle \text{tr}(\boldsymbol{C}) \rangle /L^2$ , is restricted to the overall stretch of the polymer and its relationship to elastic normal stresses, characterising only the elongational behaviour of the polymer chains. However, a complete description of the polymer dynamics also requires quantification of the polymer orientation induced by the coherent turbulent vortices. In fact, in channel flow, significant alignment between the conformation tensor and the Reynolds stress tensor has been shown to provide a supplementary contribution to polymer extension beyond mean shear (Pereira et al. Reference Pereira, Mompean, Thais and Thompson2017). To this end, we further investigate the orientation of the polymer in viscoelastic TCF following the tensor analysis framework proposed in Pereira et al. (Reference Pereira, Mompean, Thais and Thompson2017). Specifically, the orientation of the polymer can be characterised by the first principal direction of the conformation tensor $\boldsymbol{C}$ , which allows us to examine the alignment between the polymer chain and other flow quantities of interest.

Figure 15.

Snapshots for the alignments between the first principal direction of $\boldsymbol{C}$ , denoted as $\boldsymbol{e^{C}}_1$ , and the three unit vectors $\boldsymbol{e}_\theta$ , $\boldsymbol{e}_r$ and $\boldsymbol{e}_z$ . For brevity, the angles between $\boldsymbol{e^{C}}_1$ and $\boldsymbol{e}_\theta$ , $\boldsymbol{e}_r$ , $\boldsymbol{e}_z$ are represented as $\varphi , \gamma, \psi$ , respectively. Here, contours illustrate the cosine values of these angles in (a–c) counter-rotation case $ \textit{Ro}^{-1}=-0.5$ , (d–f) ICRO case $ \textit{Ro}^{-1}=0$ , and (g–i) co-rotation case $ \textit{Ro}^{-1}=0.83$ . The $(r,z)$ plane streamlines are plotted using the $(u_r,u_z)$ velocities. To clearly show the vortex-induced polymer orientation, the visualisation is limited to the axial region $z=1.0{-}2.0$ , within which the results are normalised by their respective maximum values to facilitate comparison.

The first key question concerns the influence of vortical structures on polymer orientation within a turbulent flow field. To address this, we investigate the alignments between the first principal direction of $\boldsymbol{C}$ and the three unit vectors in cylindrical coordinates. As anticipated, the most significant alignment occurs in the streamwise direction due to the mean shear flow (see figures 15 a,d,g). This behaviour is particularly noticeable in the relaminarised layer at $ \textit{Ro}^{-1}=-0.5$ (figure 15 $a$ ), where the value of $\cos\phi$ is almost 1. Furthermore, consistent with observations from Brownian dynamics simulations of FENE dumbbells in wall-bounded and spatially periodic turbulence (Gupta, Sureshkumar & Khomami Reference Gupta, Sureshkumar and Khomami2004; Koide & Goto Reference Koide and Goto2024), turbulent vortices significantly modulate the instantaneous orientation of polymer chains in TCF, resulting in alignments between the polymer orientation and $\boldsymbol{e}_{r},\boldsymbol{e}_{z}$ . In particular, the alignment between polymer orientation and radial direction (i.e. $\cos {\gamma }$ ) is enhanced in the boundary region that separates two adjacent vortical structures, particularly when large-scale Taylor vortices are the dominant flow structure (i.e. $ \textit{Ro}^{-1}=-0.5, 0$ , figures 15 b,e). In fact, this is attributed to the intensified radial inflow/outflow induced by two adjacent vortices. On the other hand, we observe significant alignment between polymer orientation and axial direction (i.e. $\cos {\psi }$ ) in the near-wall boundaries of vortical structures, as demonstrated in figures 15(f,i). Note that at $ \textit{Ro}^{-1}=-0.5$ , the large value of $\cos {\psi }$ is located around $\tilde {r}=0.5$ due to the emergence of the relaminarised layer in the outer-wall region. This observation aligns with the pronounced axial flows induced by turbulent vortices, as depicted by the corresponding streamlines. Moreover, well-organised Taylor vortices in $ \textit{Ro}^{-1}=0$ give rise to a large region of high $\cos {\psi }$ (figure 15 $f$ ), consistent with the higher values of $\langle C_{zz}\rangle /L^2$ shown in figure 13( $c$ ). This suggests the following vortex-mediated mechanism of polymer elongation and orientation: although the overall polymer stretch $\langle \text{tr}(\boldsymbol{C})\rangle /L^2$ at $ \textit{Ro}^{-1}=0$ is relatively small compared to the co-rotation cases, the significant alignment of the $z$ -direction induced by turbulent vortices leads to a higher $\langle C_{zz}\rangle /L^2$ .

Figure 16.

Ensemble-averaged alignments between $\boldsymbol{e^{C}}_1$ and the three unit vectors for various $ \textit{Ro}^{-1}$ values: (a,b) $\langle \cos {\varphi }\rangle$ , (c,d) $\langle \cos {\gamma }\rangle$ and (e, f) $\langle \cos {\psi }\rangle$ for (a,c,e) $ \textit{Ro}^{-1} \leqslant 0$ and (b,d, f) $ \textit{Ro}^{-1}\geqslant 0$ , thereby highlighting the effect of the outer cylinder’s rotation direction.

As demonstrated in the preceding analysis, a direct relation between turbulent vortices and polymer orientation is established. We further quantitatively examine the alignment between polymer orientation and unit vectors in cylindrical coordinates to explore how polymer dynamics correlates with variations in the rotation effect. As shown in figure 16, the peak values of the alignments corroborate the observations in figure 15. Moreover, the polymer orientation exhibits a clear dependence on the direction of rotation of the outer cylinder. Although enhancing both counter-rotation and co-rotation results in more significant alignment between the polymer orientation and the streamwise direction, the underlying mechanisms are distinct. For counter-rotation cases ( $ \textit{Ro}^{-1}\lt 0$ ), the turbulent vortices are weakened, so the mean shear becomes the dominant factor that governs the orientation of the polymer. However, the larger $\langle \cos {\varphi }\rangle$ in the co-rotation cases ( $ \textit{Ro}^{-1}\gt 0$ ) is attributed to the more significant polymer elongation in the streamwise direction (see figures 13 a,b). The alignment between the polymer orientation and the radial direction is strongly influenced by the rotation of the outer cylinder. As demonstrated in figures 16(c,d), counter-rotation suppresses $\langle \cos {\gamma }\rangle$ , whereas co-rotation enhances it. This behaviour arises because $\langle \cos {\gamma }\rangle$ is closely related to the radial inflow/outflow generated by the turbulent vortices (as demonstrated in figure 15). Consequently, the increased intensity of turbulent vortices, which increases monotonically as $ \textit{Ro}^{-1}$ increases from $-0.6$ to $1$ , gives rise to larger values of $\langle \cos {\gamma }\rangle$ . As expected, $\langle \cos {\psi }\rangle$ decreases when the outer cylinder is co-rotating or counter-rotating (figures 16 e, f), which can be attributed to the fact that rotation-induced suppression of Taylor vortices reduces the alignment between polymer orientation and axial direction in near-wall regions.

Figure 17.

The correlation between the polymer shear stress $\tau _{r\theta }$ (contour) and polymer behaviour $\cos \gamma$ (line, the alignment between $\boldsymbol{e^{C}}_{1}$ and $\boldsymbol{e}_{r}$ ) for $(a)$ counter-rotation case $ \textit{Ro}^{{-1}}={-0.5}$ , $(b)$ ICRO case $ \textit{Ro}^{ {-1}}=\textrm {0}$ , and $(c)$ co-rotation case $ \textit{Ro}^{{-1}}=\textrm {0.83}$ , with dark grey frames marking the regions with large values of $\cos {\gamma }$ .

The relation between polymer elongation and polymer normal stresses is well established (see figures 13 and 14). However, the relationship between polymer dynamics and the resulting polymer shear stresses (especially $\tau _{r\theta }$ , which directly contributes to the transport of angular momentum) remains poorly understood. On the basis of the observations discussed above, we now demonstrate that the polymer orientation is a critical factor in determining the polymer shear stress. Specifically, we examine the spatial distribution of the polymer shear stress $\tau _{r\theta }$ , which is shown by the contour plots in figure 17. Since $\tau _{r\theta }$ is negative throughout the domain, a red–blue colour scale is used, in which blue denotes negative values of large magnitude; thus the blue regions correspond to strong polymer shear stress. These strong polymer shear stress regions are located primarily in bulk flow. Interestingly, we observe a strong correlation between the spatial distribution of $\tau _{r\theta }$ and $\cos {\gamma }$ . This reveals the following mechanism for the generation of polymer shear stress: even when polymer chains experience substantial elongation, significant polymer shear stress arises only when their orientation undergoes a pronounced change. In § 4, the large values of $\tau _{r\theta }$ and the associated $J_{p}^{\omega }$ were attributed to the formation of different penetrating vortices. Here, by examining the polymer behaviour, we further show that the occurrence of significant polymer shear stress is closely linked to the vortex-mediated polymer orientation. As illustrated in figures 15 and 16, the radial alignment that typically occurs in the boundaries between two adjacent vortices is enhanced as $ \textit{Ro}^{-1}$ increases from $-0.6$ to 1; this enhancement is attributed to the strengthened vortices as the outer cylinder changes from strong counter-rotation to strong co-rotation (demonstrated by the vortex generation in figure 8). This improved alignment eventually results in a more substantial polymer shear stress, and in turn, alters the turbulence dynamics.