3.1. Drag modification characteristics and mechanisms

Polymer additives significantly modify turbulent drag in TCF (Song et al. Reference Song, Zhu, Lin, Liu and Khomami2023). This modification is quantified by variations in the global response of the flow system, i.e. the pseudo-Nusselt number $Nu_\omega$ , which quantifies the overall transport of angular momentum and has been used in experiments to examine the torque scaling of viscoelastic TCF (Moazzen et al. Reference Moazzen, Lacassagne, Thomy and Bahrani2023). Specifically, we define $Nu_\omega$ as $J^\omega /J^{\omega }_{lam}$ (Lathrop et al. Reference Lathrop, Fineberg and Swinney1992a , Reference Lathrop, Fineberg and Swinneyb ; Lewis & Swinney Reference Lewis and Swinney1999). Here, $J^\omega$ is the conserved angular momentum flux, analogous to the heat flux in Rayleigh–Bénard convection, with its definition given by

(3.1) \begin{equation} J^\omega = r^3[\langle u_r \omega \rangle -\beta \partial _r \langle \omega \rangle /Re -(1-\beta )\langle \tau _{r\theta } \rangle /(rRe)], \end{equation}

where $\omega =u_\theta /r$ and $\langle \boldsymbol{\cdot }\rangle =\langle \langle \langle \boldsymbol{\cdot }\rangle _{\theta }\rangle _z\rangle _t$ denotes the averaging in the $\theta$ and $z$ directions and in time. In addition, the laminar value of $J^{\omega }$ is calculated as $J^{\omega }_{lam}=2\omega _i R_i^2 R_o^2/Re(R_o^2 -R_i^2)$ . We define the extent of DM as $DM=(Nu_{\omega } - Nu_{\omega }^{N})/Nu_{\omega }^{N}$ , where $Nu_{\omega }^{N}$ is the Newtonian value at $Re=6000$ . Figure 1(a) shows a complete diagram of the ADM states, where an intriguing DM phenomenon is observed as $\beta$ and $Wi$ vary, that is, asymptotic flow states of MDR and MDE are realised. Specifically, continuously increasing $Wi$ at higher $\beta \geqslant 0.98$ leads to MDR, while MDE is achieved at lower $\beta \leqslant 0.97$ . For a given $\beta$ , $DM$ monotonically changes with increasing $Wi$ , except for lower $Wi=10$ at $\beta =0.98$ and $0.97$ . Note that $DM$ becomes almost saturated at sufficiently high $Wi\geqslant 100$ , indicating the realisation of asymptotic flow states. It should be noted that different $\beta$ leads to asymptotic flow states to varying degrees of $DM$ . For example, an MDR state with $DM=-11.7\,\%$ is observed at $\beta =0.99$ , which is comparable to recent experimental findings in viscoelastic TCF with $\eta =0.714$ and $Re=1.3\times 10^4$ (Zhang et al. Reference Zhang, Fan, Su, Xi and Sun2025). This observed MDR value is considerably lower than that typically found in the channel/pipe flow (which is typically around $DM\sim -70\,\%$ ) (Procaccia et al. Reference Procaccia, L’vov and Benzi2008; White & Mungal Reference White and Mungal2008). In contrast, at the smallest $\beta =0.90$ considered, an MDE state with $DM=34.6\,\%$ is realised. This clearly highlights a continuous state transition pathway from MDR to MDE as $\beta$ decreases. In particular, the transition threshold is approximately $\beta =0.98$ , for which a nearly zero $DM$ ( $-2.4\,\%$ ) is observed. This suggests that in TCF, the appropriate selection of $\beta$ (corresponding to the polymer concentration in experiments) can lead to ineffective drag modulation by polymer addition.

Figure 1.

( $a$ ) Global response $Nu_{\omega }$ versus $Wi$ at various $\beta$ . The extent of DM for the highest $Wi=150$ realised in our simulations is reported. To distinguish MDR and MDE states, a horizontal reference line corresponding to $Nu_{\omega }$ of the Newtonian flow is depicted. ( $b$ ) Dynamical evolution cycles of a rescaled system response $4Nu_{\omega }/(1+\eta )^2$ versus total dissipation $\varepsilon _v + \varepsilon _p$ for different ADM states ( $Wi=150$ ) at various $\beta$ . Here, $\varepsilon _v = \langle (\partial _i u_j + \partial _j u_i)^2 \rangle _V /2$ and $\varepsilon _p = \langle \tau _{\textit{ij}} \partial _j u_i \rangle _V$ are the viscous and elastic dissipation rates, respectively, with $\langle \boldsymbol{\cdot } \rangle _V$ denoting the volume average. The result of Newtonian flow is plotted for comparison. The dashed line in panel ( $b$ ) corresponds to the exact relation $\varepsilon _v+\varepsilon _p=4Nu_\omega /(1+\eta )^2$ .

We further illustrate this $\beta$ -dependent state transition pathway in figure 1( $b$ ) by plotting the dynamical evolution cycles for different ADM states with $Wi=150$ . Specifically, we examine the exact relation $\varepsilon _v+\varepsilon _p=4Nu_\omega /(1+\eta )^2$ (Lin et al. Reference Lin, Song, Liu, Liu, Lu and Khomami2024a ), using both sides as axes to explore the property of dynamical evolution. This choice is based on the fact that the left-hand side characterises energy dissipation, while the right-hand side represents angular momentum transport. In fact, the latter corresponds directly to the rate of energy input through the rotating cylinder wall (Eckhardt, Grossmann & Lohse Reference Eckhardt, Grossmann and Lohse2007). As shown in figure 1( $b$ ), the dynamical evolution of ADM states traces elliptical-like envelopes whose principal axes align with the exact relation. This alignment confirms that statistically converged states are achieved, signifying a balance between energy input and dissipation. The continuous state transition is evident as the evolution cycle of the ADM states gradually shifts to encompass that of the Newtonian flow as  $\beta$ decreases. Intriguingly, while the evolution cycle at $\beta =0.98$ overlaps greatly with that of the Newtonian fluid (consistent with their comparable $Nu_\omega$ values), the shapes of their elliptical-like envelopes exhibit distinct topological characteristics. Specifically, the evolution cycle at $\beta =0.98$ occupies a smaller area, which can be attributed to the polymer-induced suppression of turbulent motions, i.e. weakened fluctuations in energy dissipation and angular momentum transport (Dallas, Vassilicos & Hewitt Reference Dallas, Vassilicos and Hewitt2010; Zhang et al. Reference Zhang, Fan, Su, Xi and Sun2025).

It is important to note that, although various DM phenomena have been observed, the transition path from MDR to MDE has been reported for the first time. Figure 2 summarises the DM studied to date in high- $Re$ ( ${\gt } 1000$ ) viscoelastic TCFs, highlighting the significance and novelty of our present findings. As illustrated, the wide parameter space, encompassing variations in $Re$ , $\eta$ , $Wi$ , $\beta$ , and the presence of rotation of the outer cylinder, captures the intriguing phenomena of DM, covering a wide range of DR and DE. At comparable $Re$ and $\eta$ values, we find good agreement between numerical data (Lin et al. Reference Lin, Wan, Zhu, Liu, Lu and Khomami2022) and experimental data (Zhang et al. Reference Zhang, Fan, Su, Xi and Sun2025), which demonstrates the effectiveness of the FENE-P model and our numerical method in faithfully predicting polymer-induced DM in viscoelastic TCF. Although asymptotic flow states have also been identified numerically in Lin et al. (Reference Lin, Song, Liu, Wan, Lu and Khomami2024b , Reference Lin, Song, Liao, Liu, Lu and Khomami2025) and experimentally in Zhang et al. (Reference Zhang, Fan, Su, Xi and Sun2025), these previous studies observed MDR or MDE separately. As figure 2 shows, two critical considerations contribute to the transition of the ADM state observed in our study: (i) we chose a moderate curvature with $\eta =0.6$ , which is weaker than that leading to the MDE state (at $\eta =0.5$ ) but stronger than the value leading to the MDR state (at $\eta =0.99,0.714$ ); (ii) we consider a wide range of values of $\beta$ , allowing an extensive examination of elastic effects. We will discuss later how these considerations lead to a subtle competition between elastic and inertial forces, which, in turn, modifies angular momentum transport and flow structures. A recent experiment on TCF of a dilute surfactant solution observed a transition from DE to DR by increasing $Re$ or switching the outer cylinder rotation from co-rotating to counter-rotating (Horimoto & Okuyama Reference Horimoto and Okuyama2025). However, they did not achieve an asymptotic flow state, and the physical mechanisms for the DM transition in their experiments remain unclear, as the direct measurement of polymer stress is experimentally intractable. Our numerical approach overcomes this limitation by directly resolving the stress fields to uncover the underlying physics.

Figure 2.

A summary of the DM induced by the elastic effect in high- $Re$ viscoelastic TCF. For consistency, here the definition of DM is based on the friction factor $C_f$ , namely, $DM^*=C_f/C_f^N-1$ , with $C_f^N$ being the value for the corresponding Newtonian flow. For TCF, $C_f$ is calculated as $[(1-\eta )^2/\pi ]G/Re^2$ , with $G$ being the dimensionless torque (Lathrop et al. Reference Lathrop, Fineberg and Swinney1992a , Reference Lathrop, Fineberg and Swinneyb ; Grossmann et al. Reference Grossmann, Lohse and Sun2016). Both experimental data (Exp.) and numerical data (Num.) are shown. In addition to $Re$ , another remarkable difference between these studies is the curvature effect: $\eta =0.5$ for Rajappan & McKinley (Reference Rajappan and McKinley2020) and Lin et al. (Reference Lin, Wan, Zhu, Liu, Lu and Khomami2022, Reference Lin, Song, Liu, Wan, Lu and Khomami2024b ); $\eta =0.6$ for the present study; $\eta =0.714$ for Zhang et al. (Reference Zhang, Fan, Su, Xi and Sun2025); $\eta =0.731$ for Horimoto & Okuyama (Reference Horimoto and Okuyama2025); $\eta =0.95$ for Barbosa et al. (Reference Barbosa, Cussuol, Soares, Andrade and Khalil2022); $\eta =0.99$ for Lin et al. (Reference Lin, Song, Liao, Liu, Lu and Khomami2025). For Barbosa et al. (Reference Barbosa, Cussuol, Soares, Andrade and Khalil2022), only data obtained with polyacrylamide polymers are shown here, as they exhibit the most pronounced DR. Data obtained from Zhang et al. (Reference Zhang, Fan, Su, Xi and Sun2025) correspond to polyacrylamide polymers with two different molecular weights. Moreover, only Horimoto & Okuyama (Reference Horimoto and Okuyama2025) have considered the outer cylinder rotation, and all the other studies are focused on TCF of a purely inner cylinder rotation.

Figure 3 illustrates how the mean velocity profile $u_\theta ^+$ responds to increments in $Wi$ . We present results of two representative values of $\beta$ : $\beta =0.99$ for MDR states and $\beta =0.90$ for MDE states. For higher $\beta =0.99$ , the $u_\theta ^+$ profiles immediately collapse onto a single curve at low $Wi=10$ . The introduction of polymers leads to a boost of $u_\theta ^+$ in the core region, a common observation in polymer drag-reducing flows (Li et al. Reference Li, Sureshkumar and Khomami2006; White & Mungal Reference White and Mungal2008). Interestingly, the $u_\theta ^+$ profile for the MDR state exhibits a von Kármán-like logarithmic law for $r^+\gt 30$ , closely following $u_\theta ^+=(1/\kappa ) \text{ln} r^+ + C$ with $\kappa = 1.06, C=7.6$ (see figure 3 $a$ ). Similarly to the observations in Zhang et al. (Reference Zhang, Fan, Su, Xi and Sun2025),  $\kappa$ is not significantly modified compared with its Newtonian counterpart ( $\kappa =0.94$ ). In contrast, the $u_\theta ^+$ profile in $\beta =0.90$ demonstrates a more pronounced variation due to the stronger elastic effect. Specifically, an increase in $Wi$ within the range of $10{\sim}80$ leads to a gradual decrease in $u_\theta ^+$ in the core region, consistent with the increase in DE reported in figure 1. As the MDE state is realised for $Wi\geqslant 100$ , the $u_\theta ^+$ profile shows a very weak dependence on further increases in $Wi$ . For the MDE state at $\beta =0.90$ , we obtain $\kappa =1.85$ , which clearly deviates from its Newtonian counterpart.

Figure 3.

The log plots of mean velocity profile $u_\theta ^+$ in the inner half of the gap versus wall-normal distance $r^+$ for ( $a$ ) $\beta =0.99$ and ( $b$ ) $\beta =0.90$ . Here, $u_\theta ^+$ and $r^+$ are calculated as $(1-\langle u_\theta \rangle )/u_\tau$ and $(r-R_i)/(u_{\tau }\eta _t/\rho )$ , respectively. The log law plot of $u_\theta ^+=(1/\kappa ) \text{ln} r^+ + C$ is obtained by fitting the mean velocity profile at $Wi=150$ in the region $r^+ \geqslant 30$ . Here $\kappa =1.06, C=7.6$ for the MDR state at $\beta =0.99$ , and $\kappa =1.85, C=8.1$ for the MDE state at $\beta =0.90$ .

Figure 4.

( $a$ ) Comparison of the mean velocity profile $u_\theta ^+$ for different ADM states in viscoelastic TCF. The present data show $u_\theta ^+$ of drag-limiting states at $Wi=150$ with various $\beta$ . Available numerical and experimental data are plotted for comparison: $\eta =0.5,Re=5000$ , Lin et al. (Reference Lin, Song, Liu, Wan, Lu and Khomami2024b ); $\eta =0.714,Re=1.3\times 10^4$ , Zhang et al. (Reference Zhang, Fan, Su, Xi and Sun2025); $\eta =0.99,Re=5000$ , Lin et al. (Reference Lin, Song, Liao, Liu, Lu and Khomami2025). The grey dashed line corresponds to the Virk law in the MDR state of parallel shear turbulent flows, namely, $u_\theta ^+=(1/\kappa ) \text{ln} r^+ + C$ with $\kappa =1/11.7, C=-17$ (Virk et al. Reference Virk, Mickley and Smith1970). ( $b$ ) The diagnostic function $r^+ \text{d}u_\theta ^+/\text{d}r^+$ for the ADM states achieved in this study. The inset shows the change of $\kappa$ in ADM states with increasing $1-\beta$ .

To further highlight the properties of the mean velocity profile, figure 4( $a$ ) shows $u_\theta ^+$ as a function of $r^+$ for various ADM states realised in this study. Data available from previous studies are also added for comparison (Virk et al. Reference Virk, Mickley and Smith1970; Lin et al. Reference Lin, Song, Liu, Wan, Lu and Khomami2024b , Reference Lin, Song, Liao, Liu, Lu and Khomami2025; Zhang et al. Reference Zhang, Fan, Su, Xi and Sun2025). Evidently, the transition to the ADM state is accompanied by a corresponding transition in the $u_\theta ^+$ profile. Specifically, as $\beta$ decreases, the asymptotic flow state transitions from MDR to MDE, and the corresponding profile $u_\theta ^+$ decreases in the core region. It is also found that, although the ADM state at $\beta =0.98$ exhibits a turbulent flow drag similar to its Newtonian counterpart (see figure 1 $a$ ), their $u_\theta ^+$ profiles are not identical, indicating significant modifications introduced by polymers. Interestingly, comparing the present results with previous data also reveals an ADM state transition driven by decreasing $\eta$ . To highlight this, table 1 summarises the corresponding degree of DM for the $u_\theta ^+$ profile depicted in figure 4 ( $a$ ). Specifically, a gradual increase in the curvature effect (corresponding to a decrease in $\eta$ ) leads to a sequence of transitions in the ADM state: from the classical MDR state ( $\eta \to \infty$ ) to MDR states with significantly lower DR rates ( $\eta =0.99, 0.95, 0.714$ ), followed by the MDR/MDE states with low/moderate DR/DE observed in this study ( $\eta =0.6$ ), and eventually to the MDE state with significantly enhanced DE ( $\eta =0.5$ ). This confirms our earlier argument that the observed state transition is a result of the appropriate choice of $\eta$ and $\beta$ in this study. In fact, the curvature effect is closely related to elastic forces and instabilities (Larson et al. Reference Larson, Shaqfeh and Muller1990; Shaqfeh & Khomami Reference Shaqfeh and Khomami2021; Datta et al. Reference Datta2022), which, as we show later, are critical in determining the flow dynamics of the ADM states.

Table 1.

The extent of DM corresponding to the $u_\theta ^+$ profile shown in figure 4( $a$ ). Here, the parallel shear (channel/pipe) flow is presented as a curvature-free case, in which exceedingly high DR can be achieved (Procaccia et al. Reference Procaccia, L’vov and Benzi2008; White & Mungal Reference White and Mungal2008; Xi Reference Xi2019). The MDR rate for Zhang et al. (Reference Zhang, Fan, Su, Xi and Sun2025) is experimentally obtained by increasing the polymer concentration at $Re=1.3\times 10^4$ using polyacrylamide solutions. Moreover, the MDR state at comparable $Re=6200$ , realised in Barbosa et al. (Reference Barbosa, Cussuol, Soares, Andrade and Khalil2022) using 500 ppm polyacrylamide solutions, is also included for comparison. The geometric curvature $\epsilon =d/R_i$ is provided to clearly illustrate the variations in curvature effects.

We have also examined the logarithmic dependence of the $u_\theta ^+$ profiles for the ADM states achieved in this study. As shown in figure 4( $b$ ), the diagnostic function indicates a clear logarithmic behaviour, especially for the MDR state realised at higher $\beta$ . This observation aligns with the clear logarithmic law reported for the MDR state in Zhang et al. (Reference Zhang, Fan, Su, Xi and Sun2025). The transition of the ADM state is accompanied by a change in $\kappa$ . Specifically, $\kappa =0.94$ is obtained for the Newtonian flow, which is significantly larger than $\kappa =0.4$ obtained for canonical pipe and channel flows (Marusic et al. Reference Marusic, Monty, Hultmark and Smits2013). This discrepancy is attributed to the boundary layer of the non-zero pressure gradient in the TCF (Ostilla-Mónico et al. Reference Ostilla-Mónico, Van Der Poel, Verzicco, Grossmann and Lohse2014b ). Decreasing $\beta$ results in a gradual increase in $\kappa$ , as shown by the inset in figure 4( $b$ ). For $\beta \geqslant 0.95$ , a plateau is found in $\kappa$ , for which the underlying mechanisms are unclear. Further exploration of the $\kappa$ dependence on the ADM states would be an interesting question for future study.

As mentioned previously, the ADM state transition stems from a subtle competition between the inertial and elastic forces, which can be closely examined by the Reynolds and polymer stresses. As illustrated in figure 5( $a$ ), the ADM states always have a reduced Reynolds stress compared with their Newtonian counterpart, regardless of the extent of DM. This observation suggests a universal nature of polymer-induced suppression of velocity fluctuations in wall-bounded turbulent flows. Furthermore, Reynolds stress monotonically decreases with decreasing $\beta$ , due to the enhanced elastic effect. For a detailed examination of the Reynolds stress in ADM states for TCF, we compared our current data with the previously available results. For $\eta =0.5$ and $0.6$ , the maximum value of $\langle u'_r u'_\theta \rangle$ occurs in the inner wall region. This has been attributed to the ‘elastic Görtler vortices’ that induce intense azimuthal velocity fluctuations (Bilson & Bremhorst Reference Bilson and Bremhorst2007; Lin et al. Reference Lin, Song, Liu, Wan, Lu and Khomami2024b ). In contrast, the radial distributions of $\langle u^{\prime}_r u^{\prime}_\theta \rangle$ depict a qualitative difference for the MDR states realised at $\eta =0.99,0.714$ , where the peak value occurs in the core region. This, in contrast, is attributed to the large-scale Taylor vortices that persist in asymptotic flow states, inducing intense radial velocity in the bulk region (Lin et al. Reference Lin, Song, Liao, Liu, Lu and Khomami2025; Zhang et al. Reference Zhang, Fan, Su, Xi and Sun2025). These intriguing structural changes and their correspondence to the ADM state transition will be detailed in § 3.2. To accurately evaluate the influence of polymer stress on flow dynamics, we calculated the rescaled polymer stress $-(1-\beta )\langle \tau _{r\theta }\rangle /Re$ , allowing for a direct comparison with Reynolds stress. Figure 5( $b$ ) delineates a monotonic increase in the magnitude of rescaled polymer stress as the MDR state gradually transitions to the MDE state. In a commensurate manner, the contribution of polymer stress to angular momentum transport (turbulent flow drag) becomes more significant. This argument is further confirmed by the large polymer stress obtained at $\eta =0.5$ (Lin et al. Reference Lin, Song, Liu, Wan, Lu and Khomami2024b ), which facilitates an MDE state with a dramatic DE of ${\sim}73.8\,\%$ (see table 1).

Figure 5.

Radial distribution of ( $a$ ) Reynolds stress and ( $b$ ) rescaled polymer stress for different ADM states in viscoelastic TCF. The present data correspond to the drag-limiting states at $Wi=150$ with various $\beta$ . Previous numerical data (Lin et al. Reference Lin, Song, Liu, Wan, Lu and Khomami2024b , Reference Lin, Song, Liao, Liu, Lu and Khomami2025) and experimental data (Zhang et al. Reference Zhang, Fan, Su, Xi and Sun2025) at different $\eta$ are added for comparison. Note that Zhang et al. (Reference Zhang, Fan, Su, Xi and Sun2025) did not provide the polymer stress due to the lack of experimental techniques for characterising polymer deformation.

Figure 6.

( $a$ ) Competition of the convective and elastic contributions to the angular momentum transport. Only four typical values of $\beta$ are shown for simplicity. ( $b$ ) The contributions of diffusive ( $J_d^v$ ), convective ( $J_c^v$ ) and elastic ( $J_p^v$ ) fluxes to the angular momentum transport at various $\beta$ . The values have been scaled by the angular momentum flux in the Newtonian flow state. The superscript ‘ $v$ ’ denotes the volume average calculated by the integral across the gap, e.g. $J_{c}^{v}=({1}/{d})\int _{R_i}^{R_o}{ J_{c}^{\omega } {\textrm{d}}r}$ .

Evidently, the competition between the reduction of Reynolds stress and the development of polymer stress gives rise to the observed ADM state transition. To highlight this, we examine the contribution of convective fluxes ( $J_c^\omega$ ) and elastic ( $J_p^\omega$ ) fluxes to angular momentum transport that correspond to Reynolds and polymer stresses, respectively. As depicted in (3.1), these fluxes are calculated as $J^{\omega }_c=r^3\langle u_r\omega \rangle$ and $J^{\omega }_p= -r^2( 1-\beta )\langle \tau _{r\theta }\rangle /Re$ . The remaining part of (3.1) represents the diffusive flux $J_d^\omega$ . To illustrate the general contribution of different angular momentum fluxes, we present in figure 6 the volume-averaged values of $J_c^\omega$ , $J_p^\omega$ and $J_d^\omega$ . Clearly, the ADM state is realised as a consequence of the saturation of different angular momentum transport mechanisms (see figure 6 $a$ ), consistent with our previous observations (Zhu et al. Reference Zhu, Wan, Lin, Liu, Lu and Khomami2023; Lin et al. Reference Lin, Song, Liu, Wan, Lu and Khomami2024b , Reference Lin, Song, Liao, Liu, Lu and Khomami2025). Furthermore, the underlying mechanism for the occurrence of the ADM state transition is substantiated by the competition of $J_c^v$ reduction and $J_p^v$ development illustrated in figure 6( $b$ ). Specifically, the addition of polymers to the turbulent TCF inevitably leads to the suppressed Reynolds stress and to the introduction of polymer stress. In the ADM state at $\beta =0.98$ , the reduction $J_c^v$ is perfectly compensated by the development $J_p^v$ , leading to an asymptotic drag-limiting state of nearly zero DM extent. The $J_c^v$ – $J_p^v$ competition in TCF drastically differs from observations in parallel shear turbulent flow, where the Reynolds stress nearly vanishes as turbulent motions are greatly suppressed (Warholic, Massah & Hanratty Reference Warholic, Massah and Hanratty1999; Li et al. Reference Li, Sureshkumar and Khomami2006; Wang et al. Reference Wang, Zhang, Wang, Li, Zhang and Li2023), but the polymer stress remains small because the weak and few turbulent vortices cannot stretch polymers sufficiently (Xi Reference Xi2019; Zhu & Xi Reference Zhu and Xi2021; Lin et al. Reference Lin, Song, Liao, Liu, Lu and Khomami2025). This is analogous to a large reduction $J_c^v$ and a small development $J_p^v$ , which explains why only the MDR state is observed for channel/pipe flows. The above argument regarding the competition between $J_c^v$ reduction and $J_p^v$ development also provides a consistent view to understand the intriguing DM phenomena in viscoelastic TCF (see figure 2). That is, variations in $Re,\eta ,Wi,\beta$ result in varying degrees of reduction in $J_c^v$ and $J_p^v$ growth, leading to different degrees of DM.

In general, the above findings provide clear answers to the questions posed in the introduction. That is, direct transition paths exist between MDR and MDE, while they each represent distinctive regimes in the phase space of various ADM states. To achieve this state transition pathway, our study highlights two critical parameters, namely $\eta$ and $\beta$ , which quantify the curvature effect and polymer concentration, respectively. We speculate that varying other parameters that can effectively manipulate the $J_c^v$ – $J_p^v$ competition may also lead to a similar transition pathway from MDR to MDE. In fact, in the following section we demonstrate that changes in control parameters indeed modify the elastic instabilities, which in turn result in structural changes and corresponding turbulent transport modifications.

3.2. Flow structures and regime identification

An overview of the flow structures for various ADM states is depicted by the space–time plots in figure 7. As $\beta$ decreases, the dominant flow structure in the asymptotic drag-limiting flow state undergoes a structural change. At $\beta =0.99$ , the highly intermittent inflow/outflow streaks signify the persistence of large-scale Taylor vortices in the flow field (Dutcher & Muller Reference Dutcher and Muller2009), whereas the strong SSS characteristic of Newtonian TCF are markedly suppressed. This is consistent with previous numerical and experimental observations in the MDR state with a weaker curvature effect $\eta =0.714,0.99$ (Zhang et al. Reference Zhang, Fan, Su, Xi and Sun2025; Lin et al. Reference Lin, Song, Liao, Liu, Lu and Khomami2025). In contrast, the MDE state at $\beta =0.90$ is populated by small-scale flow structures, which manifest themselves as random inflow/outflow spots in the space–time plot (see figure 7 $f$ ). Interestingly, for intermediate $\beta$ values, a transition of flow structures is observed in the ADM state. In particular, transitional flow patterns are attained for the ADM states at $\beta =0.98,0.97$ , for which only a slight DM rate is achieved (see figure 1 $a$ ). As shown in figure 7( $b,\!c$ ), although stronger randomness has been excited, the imprints of LSS can still be discerned within these transitional flow patterns.

Figure 7.

Space–time plots of radial velocity $u_r$ along the axial sampling line at the middle of the gap ( $\tilde {r}=0.5$ ) for the ADM states ( $Wi=150$ ) at ( $a$ ) $\beta =0.99$ , ( $b$ ) $\beta =0.98$ , ( $c$ ) $\beta =0.97$ , ( $d$ ) $\beta =0.95$ , ( $e$ ) $\beta =0.93$ and ( $f$ ) $\beta =0.90$ .

In contrast to the classical MDR in parallel shear channel/pipe flows, here the MDR states at higher $\beta$ do not exhibit hibernating/active behaviours. This is due to the persistence of centrifugal-force-induced LSS, which contribute significantly to global angular momentum transport (Zhang et al. Reference Zhang, Fan, Su, Xi and Sun2025). It is also worth noting that the flow structures observed in the ADM states are consistent with previous findings in the EIT or elastically dominated turbulence (EDT) regime of viscoelastic TCF. Specifically, the LSS at higher $\beta$ are reminiscent of the lower- $Re$ solitary pairs of vortices (Boulafentis et al. Reference Boulafentis, Lacassagne, Cagney and Balabani2024), which display chaotic merging and splitting dynamics (Lacassagne et al. Reference Lacassagne, Cagney, Gillissen and Balabani2020). By contrast, the SSS observed in the MDE at lower $\beta$ correspond to similar structures documented in the EDT regime (Song et al. Reference Song, Lin, Liu, Lu and Khomami2021), highlighting the relevance of EIT/EDT to ADM states. Overall, the above observation demonstrates that the extent of DM for an ADM state is closely related to its dominant flow structure. That is, ADM states exhibiting a stronger dominance of small-scale flow structures are more likely to experience higher turbulent flow drag, underscoring the intriguing competition between large- and small-scale flow structures in the ADM state transition.

Although the SSS are present in both Newtonian flow and the MDE state, their generation mechanisms are strikingly different. This is demonstrated by scrutinising the budget equation for streamwise enstrophy $\langle \omega _\theta ^2 \rangle$ (detailed descriptions are given in Appendix B) (Kim et al. Reference Kim, Li, Sureshkumar, Balachandar and Adrian2007). Two key production terms are examined: (i) the elastic production term $W_p$ , which represents the work performed by the polymer-induced torque; and (ii) the inertial production term $W_c$ , which combines the contribution of stretching, tilting and twisting of vortical structures (Zhu et al. Reference Zhu, Song, Lin, Liu, Lu and Khomami2022). Figure 8 demonstrates the mechanistic differences in the generation of SSS between Newtonian flow and polymer-induced ADM states. Specifically, $W_c$ , which drives the formation of inertial Görtler vortices, is greatly suppressed in the ADM states. This finding is consistent with experimental observations that polymer additives effectively inhibit the generation of inertial Görtler vortices (Lee, Sengupta & Wei Reference Lee, Sengupta and Wei1995). On the other hand, $W_p$ increases monotonically with decreasing $\beta$ , and at $\beta =0.98$ it becomes greater than $W_c$ in its Newtonian counterpart. This demonstrates the emergence of distinctive SSS, which have been referred to as elastic Görtler vortices due to their elastically driven nature (Song et al. Reference Song, Teng, Liu, Ding, Lu and Khomami2019, Reference Song, Lin, Liu, Lu and Khomami2021). In particular, the positive $W_p$ observed presents a striking contrast to the negative $W_p$ reported in viscoelastic channel flow (Kim et al. Reference Kim, Li, Sureshkumar, Balachandar and Adrian2007), where the effects of polymer-induced torque were shown to suppress shear-energised vortices.

Figure 8.

The generation mechanism for the SSS in the near inner wall region for the ADM states ( $Wi=150$ ) at various $\beta$ , which is quantified as the competition between the combined inertial production term $W_c$ and the work done by the polymer-induced torque $W_p$ .

Figure 9.

One-dimensional spanwise spectra of the TKE ( $\langle \boldsymbol{u}' \boldsymbol{\cdot }\boldsymbol{u}' \rangle /2$ ) at the middle of the gap ( $\tilde {r}=0.5$ ) for the ADM states ( $Wi=150$ ) achieved in this study. For a more reasonable comparison, the values have been scaled by the corresponding $u_\tau ^2$ at various $\beta$ . The decaying law $k_z^{-2.6}$ is obtained by fitting the spectral curve at $\beta =0.90$ in the $8\leqslant k_z\leqslant 40$ regime (Boulafentis et al. Reference Boulafentis, Lacassagne, Cagney and Balabani2024). The classical decaying laws for Newtonian turbulence $k_z^{-5/3}$ and for MDR in parallel shear turbulent flows $k_z^{-14/3}$ (Dubief et al. Reference Dubief, Terrapon and Soria2013) are added for comparison. The vertical dashed line indicates the wavenumber corresponding to the gap width, namely, $k_z^d=2\pi /d\approx 6$ .

To examine the interaction between these flow structures and polymers, we analyse the spectral properties of the ADM states first, which also allows for a quantitative characterisation of the polymer-induced modification on LSS. As depicted in figure 9, varying $\beta$ significantly modulates the spectral characteristics of the TKE, especially in the large-scale regime. Specifically, the appearance of the spectral peak shifts from $k_z=3$ for Newtonian flow to $k_z=2$ for $\beta =0.99$ . The dominance of large-scale modes gradually decreases as $\beta$ is decreased, and eventually no obvious spectral peak can be seen for MDE states at lower $\beta$ . This provides direct evidence for our observation depicted in figure 7, namely, the decrease of $\beta$ gradually modifies the LSS in the ADM states. In the following, we quantitatively demonstrate that these modifications lead to a smaller portion of large-scale TKE; equivalently, to a larger portion of small-scale TKE. However, the power-law decay for small-scale TKE of ADM states is not significantly altered, as demonstrated by the spectral curves that obey the scaling $-2.6$ in the $8\leqslant k_z \leqslant 40$ regime. This scaling exponent exhibits remarkable agreement with the experimental findings reported by Boulafentis et al. (Reference Boulafentis, Lacassagne, Cagney and Balabani2024) in the EIT regime of viscoelastic TCF. Additionally, Boulafentis et al. (Reference Boulafentis, Lacassagne, Cagney and Balabani2024) showed that this decay exponent is independent of fluid elasticity ( $Wi/Re$ ). In the present case, it is also independent of $\beta$ , indicating that the scaling $k_z^{-2.6}$ may be unaffected by fluid rheology and flow parameter (Yamani et al. Reference Yamani, Keshavarz, Raj, Zaki, McKinley and Bischofberger2021). The power-law decay $k_z^{-14/3}$ , commonly observed in the EIT of channel flow (Dubief et al. Reference Dubief, Terrapon and Soria2013; Wang et al. Reference Wang, Zhang, Wang, Li, Zhang and Li2023), has also been added for comparison. The TKE spectrum deviates from $k_z^{-14/3}$ (for a more rigorous examination, see Appendix C), indicating the distinctive polymer-induced dynamics for viscoelastic TCF. It is important to note that, due to the wide parameter space involved, different decay exponents for the spatial spectra have been reported (Latrache et al. Reference Latrache, Kelai, Bai, Crumeyrolle and Mutabazi2025). For example, experimental findings have reported values of $-7/3$ (Moazzen et al. Reference Moazzen, Lacassagne, Thomy and Bahrani2023) and $-2.6$ (Boulafentis et al. Reference Boulafentis, Lacassagne, Cagney and Balabani2024), while $-3$ (Lopez Reference Lopez2022) or $\lt -3$ (Song et al. Reference Song, Zhu, Lin, Liu and Khomami2023; Lin et al. Reference Lin, Song, Liao, Liu, Lu and Khomami2025) are found in numerical simulations. This highlights the need for a thorough investigation to ascertain the intriguing dependence of spectral characteristics for viscoelastic TCF across different parameter regimes.

Given the substantial modification of the flow structures during the transition from MDR to MDE of the ADM state, an important question arises: How do these structural changes influence polymer-flow interactions? To answer this question and illustrate the relevant polymer dynamics, we examined the polymer stretch for different ADM states. As shown in figure 10( $a$ ), all ADM states exhibit large polymer stretch ( $\langle \textrm{tr}(\unicode{x1D63E}\,)\rangle /L^2\gt 0.8$ ) across the gap. This is in stark contrast to the small polymer stretch (e.g. $\langle \textrm{tr}(\unicode{x1D63E}\,)\rangle /L^2\lt 0.1$ for $L=120$ ) typically observed in classical MDR for channel/pipe flows (Li et al. Reference Li, Sureshkumar and Khomami2006; Dallas et al. Reference Dallas, Vassilicos and Hewitt2010). This observation is consistent with the significant polymer stress development shown in figures 5( $b$ ) and 6, which facilitates the occurrence of MDE or low-DR MDR states. Comparison of different ADM states reveals a smaller and more asymmetric polymer stretch for smaller $\beta$ , particularly in the bulk region. However, a larger polymer stretch at larger $\beta$ does not necessarily guarantee larger effective elastic contributions to angular momentum transport (see the rescaled polymer stress in figures 5 and 6), which takes into account the factor $(1-\beta )$ in the definition of $J_p^\omega$ in (3.1).

Figure 10.

( $a$ ) Ensemble-averaged polymer chain extension rate $\langle {\textrm {tr}}(\unicode{x1D63E}\,)\rangle / L^2$ and ( $b$ ) interaction between polymers and vortical structures for the ADM states ( $Wi=150$ ) at various $\beta$ . In panel ( $b$ ) the contour plots illustrate the fluctuations of polymer chain extension, quantified as the r.m.s. values of $\textrm{tr}(\unicode{x1D63E}\,)$ , and the superimposed solid and dashed lines respectively represent the positive and negative ensemble-averaged streamwise vorticity $\langle \omega _\theta \rangle$ .

As clearly captured in figure 10( $b$ ), distinct radial distributions of polymer stretch in various $\beta$ suggest unique polymer dynamics induced by different flow structures in ADM states. Specifically, we observe a strong correlation between the fluctuating polymer stretch and vortical structures. In fact, the pattern of polymer stretch changes as the ADM state transitions from MDR to MDE. For MDRs with higher $\beta$ , polymers exhibit a high fluctuating stretch in the core region of large-scale Taylor vortices; that is, the stretch and relaxation of polymers are mainly confined within LSS. This feature is still discernible for the transitional $\beta =0.98,0.97$ , where remnants of large-scale Taylor vortices persist in these two ADM states. The polymer dynamics induced by LSS occupying the entire gap also explain the relatively stronger symmetry of $\langle \textrm{tr}(\unicode{x1D63E}\,)\rangle /L^2$ at $\beta =0.99$ (see figure 10 $a$ ), where the LSS result in more uniform polymer stretch in the bulk region. As $\beta$ further decreased, the irregular pattern of $\langle \omega _\theta \rangle$ demonstrates that in the ADM states the SSS are dominant (see figure 7). In this case, strong fluctuations of polymer stretch occur in the near-wall region where elastic Görtler vortices are most intense (Song et al. Reference Song, Lin, Liu, Lu and Khomami2021; Lin et al. Reference Lin, Song, Liu, Wan, Lu and Khomami2024b ). Such correlations between polymers and small-scale vortical structures are also highlighted in the wall-mode-dominated EIT of channel flows, for which pronounced polymer stretch correlates with shear-energised spanwise vortices in the near-wall regions (Dubief et al. Reference Dubief, Terrapon and Soria2013; Zhang et al. Reference Zhang, Shao, Li, Ma, Zhang and Li2021b ). Moreover, the polymer stretch fluctuations become stronger for smaller $\beta$ , as indicated by the higher values of $\textrm{tr}(\unicode{x1D63E}\,)'_{rms}/L^2$ in $\beta =0.90$ . This underscores the increasingly dominant role of polymer dynamics and associated elastic effects on the transition path from MDR to MDE.

To quantify the strength of the elastic effect at various $\beta$ and $Wi$ , we evaluated the Pakdel–McKinley (PM) criterion, which was originally proposed to identify the primary elastic instability in curvilinear flows (McKinley, Pakdel & Öztekin Reference McKinley, Pakdel and Öztekin1996; Pakdel & McKinley Reference Pakdel and McKinley1996; Schäfer et al. Reference Schäfer, Morozov and Wagner2018). Specifically, motivated by the fact that elastic forces are closely related to the curvature of streamlines and polymer hoop stresses, McKinley et al. (Reference McKinley, Pakdel and Öztekin1996) proposed the following formulation for the onset of an elastic instability:

(3.2) \begin{equation} M=\sqrt {Wi^{\textit{eff}}\epsilon \frac {N_1}{\varSigma _{r\theta }}}. \end{equation}

Here $Wi^{\textit{eff}}$ is the effective Weissenberg number, $N_1=\langle (1-\beta )(\tau _{\theta \theta }-\tau _{\textit{rr}})/Re\rangle$ is the first normal stress difference and $\varSigma _{r\theta }$ is the total shear stress. As this study focuses on the viscoelastic TCF of the FENE-P fluid, the effective Weissenberg number is calculated as $Wi^{\textit{eff}}=Wi[1-\textrm{tr}(\unicode{x1D63E}\,)/L^2]$ to account for the finite extensibility of the polymer chain (Li et al. Reference Li, Sureshkumar and Khomami2015). In general, the parameter $M$ can be viewed as a viscoelastic Görtler number (Pakdel & McKinley Reference Pakdel and McKinley1996), which combines dimensionless measures of the radius of curvature and elasticity of the fluid. Evidently, $Wi^{\textit{eff}}$ , $N_1$ and $\varSigma _{r\theta }$ depend on the flow conditions; therefore, $M$ is a dynamic measure of the strength of elastic instability. We note that More et al. (Reference More, Patterson, Pashkovski and McKinley2024) recently proposed a unified PM criterion to predict the onset of elasto-inertial instability and validated it in a cone-plate geometry. Here, we do not choose this form, as we focus on the primary elastic instability resulting in small-scale elastic Görtler vortices observed at smaller $\beta$ . In fact, the mechanism for the generation of these vortices is of elastic (instead of elasto-inertial) origin, as demonstrated in figure 8.

Figure 11( $a$ ) illustrates the radial dependence of $M$ as the flow reaches the ADM state at $\beta =0.99$ . For all $Wi$ considered, two local maxima of $M$ are observed in the regions near the wall. This results from the pronounced polymer stretch and the associated substantial $N_1$ generated in the high-shear wall regions (see figure 10 $a$ ). Although the polymer stretch is slightly larger in the near inner wall region, the implementation of $Wi^{\textit{eff}}$ yields a larger local maximum of $M$ in the near outer wall region. In general, the magnitude of $M$ exhibits a monotonic increase across the entire gap as $Wi$ increases, demonstrating the progressive enhancement of the elastic instability. This growth, however, diminishes at sufficiently high $Wi$ , indicating imbibition of elastic instabilities at a given $\beta$ . To quantify the variation in the strength of elastic instability, we have extracted the maximum value of $M$ (the one near the outer wall) for each ( $\beta , Wi$ ). Figure 11( $b$ ) reveals that elastic instabilities intensify with decreasing $\beta$ and increasing $Wi$ . In particular, $M_{\textit{max}}$ exhibits a saturating trend with increasing $Wi$ , consistent with the trend observed in $Nu_\omega$ (see figure 1 $a$ ). This correspondence underscores that asymptotic saturation of turbulent DM is fundamentally related to the imbibition of elastic instability, implying the elastically driven nature of drag-limiting states (Zhu et al. Reference Zhu, Wan, Lin, Liu, Lu and Khomami2023; Lin et al. Reference Lin, Song, Liu, Wan, Lu and Khomami2024b ). The higher values of $M$ are obtained in the smaller $\beta$ , demonstrating that decreasing $\beta$ is obviously an efficient way to enhance the elastic instability.

Figure 11.

( $a$ ) Radial distribution of the parameter $M$ obtained from the Pakdel–McKinley criterion for various $Wi$ at $\beta =0.99$ . The radial dependence of $M$ for other $\beta$ exhibits similar behaviour, and thus, is not shown here. ( $b$ ) The maximum value of $M$ as a function of $Wi$ at various $\beta$ .

Figure 12.

( $a$ ) Percentage of large-scale TKE ( ${\textit{TKE}}^L$ ) to total TKE for each pair of ( $\beta , Wi$ ). Three regimes are roughly identified: LSS-dominated regime I, $55\,\% \leqslant {\textit{TKE}^L}$ ; transitional regime II, $45\,\%\lt {\textit{TKE}^L}\lt 55\,\%$ ; SSS-dominated regime III, ${\textit{TKE}^L}\leqslant 45\,\%$ . ( $b$ ) Phase map based on the percentage of ${\textit{TKE}^L}$ . Hollow, half-filled and solid stars represent regimes I, II and III, respectively. In panel ( $b$ ) the colour illustrates the $M_{\textit{max}}$ obtained in figure 11( $b$ ). The dashed line indicates a contour line corresponding to $M_{\textit{max}}=6.8$ (at $\beta =0.97, Wi=150$ ) with the grey shaded region showing a $10\,\%$ deviation, i.e. $M_{\textit{max}}=6.8\pm 0.68$ .

In fact, the elastic instability modulates the ADM states by suppressing large-scale Taylor vortices while promoting the formation of small-scale elastic Görtler vortices. To establish this relationship, we quantify the relative contributions of LSS and SSS in different ADM states by calculating the fraction of large-scale TKE ( ${\textit{TKE}^L}$ ) in the total TKE. Specifically, we define LSS as flow structures with an axial length scale comparable to the width of the gap, such as Taylor vortices (Barcilon & Brindley Reference Barcilon and Brindley1984; Dong Reference Dong2007). This criterion corresponds to a cutoff wavenumber of $k_z^d=2\pi /d\approx 6$ as the spectral demarcation between large and small scales (see figure 9). Consequently, ${\textit{TKE}}^L$ is evaluated by integrating the TKE in all spatial modes that satisfy $k_z\leqslant k_z^d$ , while the remainder is the small-scale contribution. As shown in figure 12( $a$ ), the energy fraction ${\textit{TKE}^L}$ varies significantly with both $\beta$ and $Wi$ . At low $Wi$ , this fraction decreases significantly, consistent with the known polymer-induced suppression and stabilisation of LSS (Lin et al. Reference Lin, Song, Liao, Liu, Lu and Khomami2025; Zhang et al. Reference Zhang, Fan, Su, Xi and Sun2025). As $Wi$ increases further, ${\textit{TKE}^L}$ approaches a fluctuating plateau, indicating that the ADM states are closely related to the saturation of energy at either large or small scales. Based on this energy fraction, we quantitatively classify the flow into three regimes: an LSS-dominated regime for $55\,\% \leqslant {\textit{TKE}^L}$ , a transitional regime for $45\,\%\lt {\textit{TKE}^L}\lt 55\,\%$ and an SSS-dominated regime for ${\textit{TKE}^L}\leqslant 45\,\%$ . This classification reveals that decreasing $\beta$ drives the flow state from an LSS-dominated MDR to an SSS-dominated MDE, corroborating the observations in figure 7. Although the precise demarcations of flow regimes may depend on specific threshold values, the fundamental physics governing the transition in the dominant flow structure as $\beta$ decreases is insensitive to threshold choice.

Figure 12( $b$ ) illustrates the relationship between elastic instability and the dominant flow structure in viscoelastic TCF. At lower values of $M_{\textit{max}}$ (corresponding to low $Wi$ and high $\beta$ ), the weak elastic instability is insufficient to destroy Taylor vortices (Liu & Khomami Reference Liu and Khomami2013), so the flow state is dominated by LSS. As $M_{\textit{max}}$ increases, the flow enters the transitional regime dominated by LSS and SSS. In particular, this transition regime approximates the grey-shaded region, $M_{\textit{max}}=6.8\pm 0.68$ , indicating that this value may be taken as a reasonable critical threshold for the transition of predominance between LSS and SSS. In fact, $M_{\textit{max}}=6.8$ corresponds to the state of the ADM in $\beta =0.97, Wi=150$ , where ${\textit{TKE}^L}$ is approximately $50\,\%$ (see figure 12 $a$ ). For comparison, Schäfer et al. (Reference Schäfer, Morozov and Wagner2018) reported a significantly lower critical elastic instability threshold of $M=1.8\pm 0.3$ in a purely elastic ( $Re\to 0$ ) TCF. This discrepancy is attributed to the following two factors. (i) In this study, conducted at the Reynolds number of $Re=6000$ , stronger elastic forces (higher $M$ ) are required to overcome inertial stabilisation and excite different elastic structures. This is supported by previous findings that fluid inertia monotonically suppresses elasticity-driven flow modifications (Song et al. Reference Song, Lin, Liu, Lu and Khomami2021). (ii) The nature of the elastic instability differs between the purely elastic and EIT regimes. Whereas Schäfer et al. (Reference Schäfer, Morozov and Wagner2018) observed flow patterns reminiscent of large-scale diwhirls, our study reveals small-scale vortical structures, indicating a distinct instability mechanism.

At sufficiently large $M_{\textit{max}}$ , the flow becomes SSS dominated, as shown by the solid stars in figure 12( $b$ ). The phase map demonstrates that a stronger elastic instability promotes the SSS dominance; for example, $M_{\textit{max}} \approx 10$ for the ADM state at $\beta =0.90, Wi=150$ leads to a ${\textit{TKE}^L}$ of merely $40\,\%$ . Thus, these enhanced elastic Görtler vortices facilitate incoherent mixing of polymer stress, which, in turn, leads to efficient angular momentum transport and significant DE. This correlation between flow structures and elastic instability also explains the underlying mechanisms of the ADM state transition pathway with varying geometric curvature (see table 1). According to the PM criterion (3.2), a lower curvature effect results in a lower value of $M$ and, thus, weaker elastic instability. In low-curvature TCF ( $\eta =0.99,0.714$ ), the instability is too weak to generate elastic Görtler vortices, while polymer additives still suppress the inertial Taylor vortices, resulting in an asymptotic MDR state (Lin et al. Reference Lin, Song, Liao, Liu, Lu and Khomami2025; Zhang et al. Reference Zhang, Fan, Su, Xi and Sun2025). In contrast, the stronger elastic instability in the high-curvature case ( $\eta =0.5$ ) easily eliminates Taylor vortices and generates elastic Görtler vortices, driving the flow to an asymptotic MDE state (Lin et al. Reference Lin, Song, Liao, Liu, Lu and Khomami2024b ). These findings suggest a promising strategy for control of the flow structure in TCF, namely, by tuning the elastic effects and their associated instabilities, one can control the balance between LSS/SSS to manipulate turbulent transport, as indicated by Xu et al. (Reference Xu, Liu, Li and Xia2025). Moreover, our findings highlight the enhancement of elastic instability as $\beta$ decreases (i.e. as concentration increases), which may induce a degree of shear thinning in real polymer solutions. Previous experimental studies (Cagney, Lacassagne & Balabani Reference Cagney, Lacassagne and Balabani2020; Lacassagne, Cagney & Balabani Reference Lacassagne, Cagney and Balabani2021) have demonstrated that significant shear thinning can suppress or even completely eliminate viscoelastic instabilities. However, within the range of $\beta$ values considered in this study, the selected value of $L=100$ is sufficiently large such that shear-thinning effects are minimal in our simulations, as demonstrated in Appendix D. Based on these prior experiments, it is anticipated that, if significant shear thinning occurs, it could inhibit the onset of the MDE state, as the elastic instability responsible for the formation of elastic Görtler vortices will be suppressed.

By further considering the elastic instabilities in different parameter spaces, we summarise the mechanism governing the ADM state transition as follows. At high values of $\beta$ (e.g. 0.99), weak elastic effects suppress but do not eliminate Taylor vortices, resulting in a significant reduction of Reynolds stress and, hence, an asymptotic MDR state. As $\beta$ decreases, the enhanced elastic forces and instabilities promote the formation and progressive increase of elastic Görtler vortices. The incoherent transport driven by these SSS leads to efficient mixing and a significant development of polymer stress, ultimately establishing an asymptotic MDE state. In fact, elastic effects and associated instabilities depend on multiple key parameters, including $Re,\beta ,\eta ,L,Wi$ . This dependence underscores the potential for the realisation of diverse ADM transition pathways and points to the existence of distinct phase diagrams of ADM states with varying degrees of DM.

3.3. Energy transfers and scale interactions

The intriguing competition between LSS and SSS in the ADM states of viscoelastic TCF motivates fundamental questions regarding the mechanisms of TKE production at different scales and the process of interscale energy transfer. To investigate these interscale interactions, we derive and examine the scale-decomposed TKE transport equation for viscoelastic TCF, following the methodology proposed by Kawata & Alfredsson (Reference Kawata and Alfredsson2018). Specifically, the fluctuating velocity and polymer stress fields, denoted by $\phi '$ , are decomposed into large-scale ( $\phi '_L$ ) and small-scale ( $\phi '_S$ ) components by spatial filtering at an axial cutoff wavenumber, $k_z^{\textit{filter}}$ , such that $\phi '=\phi '_L+\phi '_S$ . Because this spatial filtering is based on an orthogonal-mode (Fourier-mode) decomposition, cross-correlation terms between the large- and small-scale quantities vanish. This property enables a direct decomposition of the TKE transport equation into large- and small-scale budgets:

(3.3) \begin{align} \frac {\partial e_L}{\partial t} &=P_{t}^{L}+D_{p}^{L}+D_{t}^{L}-T_r+D_{d}^{L}+\varepsilon _t^{L}+D_{e}^{L}+P_{e}^{L}, \end{align}

(3.4) \begin{align} \frac {\partial e_S}{\partial t}& =P_{t}^{S}+D_{p}^{S}+D_{t}^{S}+T_r+D_{d}^{S}+\varepsilon _t^{S}+D_{e}^{S}+P_{e}^{S}. \\[9pt] \nonumber \end{align}

Basically, the scale-decomposed transport equation of TKE has a similar form to the transport equation of the total TKE (Bilson & Bremhorst Reference Bilson and Bremhorst2007; Song et al. Reference Song, Lin, Liu, Lu and Khomami2021). Using (3.3) as an example, the left-hand side is merely the total time derivative of large-scale TKE $e_L$ following a mean flow fluid parcel, while the terms on the right-hand side represent the various large-scale mechanisms that produce, dissipate or transport large-scale TKE. Specifically, $P_{t}^{L}$ is the turbulence shear production term that arises due to the interaction of mean flow shear and large-scale Reynolds stress, and $P_{e}^{L}$ represents the production of TKE associated with polymer stretch. As the primary TKE generation terms, they will be examined in detail. The terms $D_{p}^{L}$ , $D_{t}^{L}$ , $D_{d}^{L}$ and $D_{e}^{L}$ denote the transport and redistribution of large-scale TKE caused by pressure, turbulent stress, viscous diffusion and polymer stress, respectively. Here $\varepsilon _t^{L}$ represents the direct viscous dissipation of large-scale TKE through its conversion to heat. A key difference from the total TKE budget is the term $T_r$ , which represents the turbulent interscale transport of TKE arising from the nonlinear interaction between the large- and small-scale parts of the velocity field (Kawata & Alfredsson Reference Kawata and Alfredsson2018, Reference Kawata and Alfredsson2019; Kawata & Tsukahara Reference Kawata and Tsukahara2021). A positive $T_r$ indicates an interscale energy flux from large to small scales along the cutoff wavenumber $k_z^{filter}$ , and vice versa. A detailed description of the scale-decomposed transport equation of TKE is given in Appendix E.

Following our earlier definition of LSS, we set the filter scale at $k_z^{filter} = k_z^d$ to scrutinise the transport and generation of TKEs within large- and small-scale regimes. To highlight polymer-induced modifications, we analyse the AMD state that corresponds to the largest elastic effect ( $\beta =0.90, Wi=150$ ). The transport budgets illustrated in figure 13 show good total balances for (3.3) and (3.4), confirming the reliability of the analytical derivation of budget equations and numerical solution to obtain the data set. A key finding is that in both budgets, TKE redistribution is overwhelmingly dominated by the polymer stress term ( $D_e^{L},D_e^{S}$ ), whereas in the Newtonian counterpart, pressure and turbulent stress transport are the primary mechanisms (Bilson & Bremhorst Reference Bilson and Bremhorst2007). The viscous transport term ( $D_d^{L},D_d^{S}$ ) is significant only near the walls. This dominance of polymer stress transport is consistent with the efficient angular momentum transport observed for the MDE state in figure 6.

Figure 13.

The transport budgets for ( $a$ ) large-scale and ( $b$ ) small-scale TKE of the ADM state at $\beta =0.90, Wi=150$ . Here, only the inner half of the gap is presented for clarity. The values have been scaled by $u_{\tau }^4/\nu$ (Teng et al. Reference Teng, Liu, Lu and Khomami2018). We implement a filtered scale of $k_z^{filter}=k_z^d$ , corresponding to the gap width.

Figure 14.

The radial distributions of scale-decomposed ( $a$ ) inertial and ( $b$ ) elastic production terms in various flow states. The values have been scaled by $u_{\tau }^4/\nu$ . For different ADM states, the same filtered scale of $k_z^{filter}=k_z^d$ is implemented.

To further investigate the mechanisms of TKE generation in large- and small-scale regimes, we delve into the inertial and elastic production terms. Figure 14 compares $P_t^{L},P_t^{S}$ and $P_e^{L},P_e^{S}$ in various $\beta$ to highlight the differences in TKE production for various ADM states. Evidently, $P_t^{L}$ and $P_t^{S}$ are positive across the gap, with their peaks occurring in the near-wall region, for example, $r^+=10$ for $P_t^S$ and $r^+=14$ for $P_t^L$ at $\beta =0.99$ . This behaviour can be attributed to the significant mean flow shear (see figures 3 and 4). As the ADM state transitions from MDR to MDE, these peaks are slightly away from the walls, indicating enlargement of the buffer layer (White & Mungal Reference White and Mungal2008; Xi Reference Xi2019). Based on the values of $P_e^{L}$ and $P_e^{S}$ , the radial distributions of $P_e^{L}$ and $P_e^{S}$ demonstrate three typical energy exchange regions (Dallas et al. Reference Dallas, Vassilicos and Hewitt2010; Tsukahara et al. Reference Tsukahara, Ishigami, Yu and Kawaguchi2011; Teng et al. Reference Teng, Liu, Lu and Khomami2018) for elastic TKE production at large and small scales (see figure 14 $b$ ), indicating that the universal mechanism of polymer–turbulence interaction remains valid in a scale-decomposed framework. Specifically, in the vicinity of the wall and in the core regions, the negative values of $P_e^{L}$ and $P_e^{S}$ signify that the kinetic energy carried by fluctuating large- and small-scale motions is absorbed and stored as elastic potential energy in the stretched polymer chains. In contrast, the positive values of $P_e^{L}$ and $P_e^{S}$ in the near-wall region indicate that elastic potential energy is released from the stretched polymer chains back to turbulent flow. As the ADM state transitions with decreasing $\beta$ , we observe a weakening of inertial production and an increase of elastic production, consistent with the progressive enhancement of elastic instability depicted in figure 11. Nevertheless, the inertial production does not vanish at $\beta =0.90$ , and turbulent motion arises from both inertial and elastic forces in various $\beta$ . Hence, the ADM states observed in this study are in the EIT/EDT regime (Zhu et al. Reference Zhu, Wan, Lin, Liu, Lu and Khomami2023; Lin et al. Reference Lin, Song, Liu, Wan, Lu and Khomami2024b ).

Although the ADM state at higher $\beta$ possesses a larger fraction of large-scale TKE (see figure 12 $a$ ), the production of TKE arises primarily from SSS, as evidenced by the higher values of $P_t^S$ and $P_e^S$ (see figure 14). This observation is physically reasonable because the TKE generated by the small-scale dynamics can still be transferred to the large-scale regime through the interaction between LSS and SSS, which can be quantified by examining the interscale energy transport term $T_r$ . In figure 15( $a$ ), pronounced negative values of $T_r$ , which signify the inverse energy transfer from small to large scales, are observed in the near-wall region (around $r^+=10$ ). This finding is consistent with observations in Newtonian channel flow, where the inverse energy cascade occurs in the buffer layer ( $6\lt y^+\lt 37$ , where $y^+$ is the wall distance at the inner scale in channel flow) (Saikrishnan et al. Reference Saikrishnan, De Angelis, Longmire, Marusic, Casciola and Piva2012). As the ADM state transitions from MDR to MDE, the magnitude of $T_r$ decreases, indicating weak-scale interactions for the transfer of TKE. This reduction is attributed to the polymer-induced suppression of velocity fluctuations (see figure 5 $a$ ), which facilitates smaller Reynolds stresses on large and small scales (see (E6)). The introduction of polymer additives produces a striking difference in energy transfer in the core region. Although Newtonian flow exhibits a classical forward energy transfer (positive $T_r$ ) from larger to smaller scales in $r^+\gt 20$ , the inverse energy cascade persists for the asymptotic ADM states at various $\beta$ , as highlighted in the inset of figure 15( $a$ ).

Figure 15.

( $a$ ) The radial distribution of interscale transfer term $T_r$ at the filter scale $k_z^{filter}=k_z^d$ for various ADM states. The inset depicts a zoom in of the core region. ( $b$ ) Space–wavelength diagrams of $T_r$ in the Newtonian flow state (left), MDR state at $\beta =0.99, Wi=150$ (middle) and MDE state at $\beta =0.90, Wi=150$ (right). The horizontal dashed line indicates the filter scale $k_z^{filter}=k_z^d$ , corresponding to the radial distribution in panel ( $a$ ). Following the same style in figure 13, the values are also scaled by $u_{\tau }^{4}/\nu$ .

It should be noted that the above discussions are based on energy transfer on the cutoff scale $k_z^{filter}=k_z^d$ . To provide a more comprehensive picture of the energy transfer at different scales, we examine the space–wavelength diagrams of $T_r$ for three typical cases: Newtonian flow, MDR at $\beta =0.99$ and MDE at $\beta =0.90$ . As depicted in figure 15( $b$ ), for the Newtonian flow, the interscale flux mostly indicates energy transfer from larger to smaller scales, while inverse energy transfer is only observed in the near-wall regions for relatively larger scales ( $\lambda _z \gt 0.3d$ ). Of interest, Kawata & Alfredsson (Reference Kawata and Alfredsson2019) reported a similar space–wavelength diagram for the interscale flux of wall-normal velocity fluctuations in turbulent plane Couette flow under strong rotation (cf. figure 7 in their paper). This underscores the similarity of flow dynamics between TCF and the rotating plane Couette flow (Brauckmann, Salewski & Eckhardt Reference Brauckmann, Salewski and Eckhardt2016). In contrast, for the polymer-induced ADM states, striking differences emerge in the core region, consistent with the observation in figure 15( $a$ ). Specifically, while $T_r$ in the near-wall region exhibits a behaviour similar to its Newtonian counterpart, negative values of $T_r$ are observed across a wide range of $\lambda _z$ for the core region ( $0.2\lt \tilde {r}\lt 0.8$ ). To the best of our knowledge, such inverse TKE transfer in wall-bounded viscoelastic turbulent flows has not been previously reported. The underlying mechanism for the inverse TKE transfer is related to the distinctive polymer dynamics, which we discuss in detail through the following scale-by-scale analysis.

To further examine how TKE is generated and transferred between different scales in the ADM states, we examine the scale-by-scale production and interscale transport of TKE. Specifically, differentiating (3.3) with respect to $k_z$ produces the scale-by-scale transport equation of the TKE (Kawata & Alfredsson Reference Kawata and Alfredsson2018). We focus in particular on scale-by-scale inertial production $p_t$ , elastic production $p_e$ and interscale transport $t_r$ . A detailed description of these terms is provided in Appendix E. Here, positive and negative values of $p_t$ , $p_e$ , $t_r$ represent the local gain and loss of TKE on scale $k_z$ , respectively. Given that the above discussions have highlighted the occurrence of inverse energy transfer in the core region, we specifically inspect the scale-by-scale TKE production and transfer in the middle of the gap ( $\tilde {r}=0.5$ ).

Figure 16( $a$ ) shows the premultiplied scale-by-scale inertial production $k_z p_t$ . For Newtonian flow, $k_z p_t$ exhibits two local maxima, one at $\lambda _z=2.09d$ corresponding to the axial length of Taylor vortices of a large scale (see figure 9), and another at a relatively smaller scale $\lambda _z=0.7d$ , demonstrating that mean flow shear generates TKE across a wide range of scales (Kawata & Alfredsson Reference Kawata and Alfredsson2019). Consistent with previous observations (figure 14), the addition of polymers results in the suppression of inertial production at different scales, with the suppression effect becoming more significant as $\beta$ decreases. For the ADM state at $\beta =0.99$ , a pronounced local maximum at $\lambda _z=3.14d$ remains observable in the large-scale regime, as turbulent motions of persistent LSS contribute to TKE production. However, this peak disappears as $\beta$ decreases, because the enhanced elastic forces have broken the regular pattern of LSS and reinforced the dominance of SSS. These findings align with the structural changes shown in figure 7.

Figure 16.

Profiles at the middle of the gap of the premultiplied scale-by-scale ( $a$ ) inertial production $k_z p_t$ , ( $b$ ) elastic production $k_z p_e$ , and ( $c$ ) interscale transport $k_z t_r$ for the TKE of various flow states. The values have been scaled by $u_{\tau }^{3}/d$ , and the vertical dashed line indicates the filtered scale implemented in figure 13. Here, positive and negative values represent the local gain and loss of TKE at scale $\lambda _z$ , respectively.

The premultiplied scale-by-scale elastic production $k_z p_e$ in the middle of the gap is depicted in figure 16( $b$ ). As polymers can absorb energy from turbulent motions through their extensions, $k_z p_e$ has negative values in the core region. Based on the value of $k_z p_e$ , two distinct regimes are identified: polymers extract TKE on relatively larger scales (around $\lambda =d$ ) and release energy to turbulent motion on small scales. Thus, the effect of polymers on turbulent fluctuations is a redistribution of energy among different scales. Interestingly, this can be viewed as an effective polymer-induced energy cascade, in which the elastic energy flux plays a significant role (Xi, Bodenschatz & Xu Reference Xi, Bodenschatz and Xu2013; Zhang et al. Reference Zhang, Bodenschatz, Xu and Xi2021a ). This scenario is consistent with the early cascade picture proposed by Tabor & de Gennes (Reference Tabor and de Gennes1986), wherein polymer elasticity reshapes turbulent interscale energy transfer. Similar findings have been observed in viscoelastic homogeneous isotropic turbulence (Valente, Da Silva & Pinho Reference Valente, Da Silva and Pinho2014), where polymers can remove more energy from the large scales than they can dissipate and transfer the excess energy to turbulent dissipative scales. As polymers continuously pump energy to small scales, the following mechanism emerges: if the accumulated energy on the small scales cannot be dissipated directly into heat, it would be transferred back to the large-scale regime via the interaction between large- and small-scale velocity fluctuations (i.e. term $T_r$ ), causing the inverse energy transfer observed in the core region. A recent observation in Newtonian TCF has confirmed this postulated mechanism: Zhou et al. (Reference Zhou, Dou, Niu and Xu2025) found that small-scale vortices with high turbulent energy coexist with large-scale vortices and cannot be dissipated by low shear stress, resulting in an inverse energy cascade.

The premultiplied interscale transport scale-by-scale $k_z t_r$ is shown in figure 16( $c$ ), further highlighting the distinction between the scale interactions in Newtonian and viscoelastic flows. For Newtonian flow, $t_r$ transfers energy from large scales to smaller scales, consistent with the classical view of the turbulent energy cascade. In contrast, $t_r$ in viscoelastic flow exhibits an opposite trend due to the polymer-induced energy redistribution discussed above among different scales. That is, positive and negative values of $t_r$ are observed in the large- and small-scale regimes, respectively, demonstrating that the interaction between large- and small-scale turbulent motions results in inverse energy transfer. Exploring whether such a reverse energy cascade exists in other viscoelastic flows and its correlation with the asymptotic saturation flow states represent intriguing topics for future research.