2 Problem formulation and computational details

The present simulations are performed via a fully spectral, three-dimensional parallel algorithm (Teng et al. Reference Teng, Liu, Lu and Khomami2018), as used in our prior studies of high-order nonlinear transitions (Thomas et al. Reference Thomas, Khomami and Sureshkumar2006, Reference Thomas, Khomami and Sureshkumar2009) and elastically induced turbulence (Liu & Khomami Reference Liu and Khomami2013a ,Reference Liu and Khomami b ) in the viscoelastic TC flow. We have chosen $d=R_{o}-R_{i}$ , $d/(\unicode[STIX]{x1D6FA}R_{i})$ , $\unicode[STIX]{x1D6FA}R_{i}$ , $\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D6FA}R_{i})^{2}$ and $\unicode[STIX]{x1D702}_{p}\unicode[STIX]{x1D6FA}R_{i}/d$ as scales for length, time, velocity $\boldsymbol{u}$ , pressure $P$ and polymer stress $\unicode[STIX]{x1D749}^{p}$ , respectively. $\unicode[STIX]{x1D6FA}$ denotes the inner cylinder angular velocity, $\unicode[STIX]{x1D70C}$ represents the solution density. The polymer stress $\unicode[STIX]{x1D749}^{p}$ is related to the polymer conformation tensor $\unicode[STIX]{x1D63E}$ through the FENE-P (finitely extensible nonlinear elastic-Peterlin) constitutive relation (Bird et al. Reference Bird, Curtiss, Armstrong and Hassager1987). The dimensionless governing equations for the incompressible flow of FENE-P fluid are as follows:

(2.1) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}=0, & \displaystyle\end{eqnarray}$$

(2.2) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{u}}{\unicode[STIX]{x2202}t}+\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}=-\unicode[STIX]{x1D735}P+\frac{\unicode[STIX]{x1D6FD}}{Re}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{u}+\frac{1-\unicode[STIX]{x1D6FD}}{Re}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D749}^{p}, & \displaystyle\end{eqnarray}$$

and

(2.3) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D63E}}{\unicode[STIX]{x2202}t}+\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D63E}=\unicode[STIX]{x1D63E}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}+(\unicode[STIX]{x1D735}\boldsymbol{u})^{\text{T}}\boldsymbol{\cdot }\unicode[STIX]{x1D63E}-\unicode[STIX]{x1D749}^{p}+\unicode[STIX]{x1D705}\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D63E},\end{eqnarray}$$

where polymer molecules are modelled as dumb-bells composed of two beads and a nonlinear spring and the polymer stress $\unicode[STIX]{x1D749}^{p}$ can be related to the conformation tensor $\unicode[STIX]{x1D63E}$ as

(2.4) $$\begin{eqnarray}\unicode[STIX]{x1D749}^{p}=\frac{f(\unicode[STIX]{x1D63E})\unicode[STIX]{x1D63E}-\unicode[STIX]{x1D644}}{Wi}.\end{eqnarray}$$

The Peterlin function $f(\unicode[STIX]{x1D63E})$ is defined as

(2.5) $$\begin{eqnarray}f(\unicode[STIX]{x1D63E})=\frac{L^{2}-3}{L^{2}-\text{trace}(\unicode[STIX]{x1D63E})},\end{eqnarray}$$

where $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D702}_{s}/\unicode[STIX]{x1D702}_{t}$ , and $L$ is the maximum chain extensibility, the total solution viscosity $\unicode[STIX]{x1D702}_{t}$ is the sum of the solvent ( $\unicode[STIX]{x1D702}_{s}$ ) and polymeric ( $\unicode[STIX]{x1D702}_{p}$ ) contributions, the Weissenberg number is $Wi=\unicode[STIX]{x1D706}\unicode[STIX]{x1D6FA}R_{i}/d$ with $\unicode[STIX]{x1D706}$ being the elastic relaxation time, the Reynolds number is $Re=\unicode[STIX]{x1D70C}\unicode[STIX]{x1D6FA}R_{i}d/\unicode[STIX]{x1D702}_{t}$ . The governing equations are supplemented by no-slip boundary conditions at walls, as well as periodic boundary conditions in the axial direction.

The diffusive term $\unicode[STIX]{x1D705}\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D63E}$ in (2.3) is added in the bulk flow region for numerical stabilization (Sureshkumar, Beris & Avgousti Reference Sureshkumar, Beris and Avgousti1995; Sureshkumar et al. Reference Sureshkumar, Beris and Handler1997). The original constitutive equation without this diffusive term is applied at the cylinder walls, where thus no boundary conditions are imposed for $\unicode[STIX]{x1D63E}$ . Note that, this strategy has been proved successful in predicting accurate TC flow dynamics of elasto-inertia turbulent states (Liu & Khomami Reference Liu and Khomami2013a ,Reference Liu and Khomami b ). It has also been established in the vast literature on DNS of turbulent flows of dilute polymeric solution that a numerical diffusivity of $O(0.01)$ does not significantly impact the flow dynamics at high $Re$ (Sureshkumar et al. Reference Sureshkumar, Beris and Avgousti1995, Reference Sureshkumar, Beris and Handler1997; Li et al. Reference Li, Sureshkumar and Khomami2006, Reference Li, Sureshkumar and Khomami2015; Kim et al. Reference Kim, Li, Balachandar, Sureshkumar and Adrian2007). Based on a study of $\unicode[STIX]{x1D705}$ sensitivity analysis of the overall flow dynamics (see appendix A), a small $\unicode[STIX]{x1D705}$ value of $8\times 10^{-4}$ corresponding to a Schmidt number $Sc[=(Re\unicode[STIX]{x1D705})^{-1}]$ of $0.42$ has been proved to produce accurate results. It should be noted that this value is close to the value of $0.5$ typically used in DNS of high- $Re$ turbulence (Graham Reference Graham2014; Zhu et al. Reference Zhu, Schrobsdorf, Schneider and Xi2018; Lopez et al. Reference Lopez, Choueiri and Hof2019). Moreover, Gupta & Vincenzi (Reference Gupta and Vincenzi2019) have shown that such a small diffusivity will not modify the essential features of the velocity and polymer conformation fields as the TC flows considered in this study are chaotic prior to the addition of polymers.

All the simulations for viscoelastic TC flow are started from their fully developed Newtonian turbulent states, i.e., turbulent Taylor vortex flow at $Re=3000$ (Dutcher & Muller Reference Dutcher and Muller2009). We choose $Wi=60$ corresponding to a low elasticity number $E=Wi/Re=0.02$ , with $\unicode[STIX]{x1D6FD}=0.8$ and $L=100$ to investigate the effect of polymer additives. A small time step ( $10^{-3}$ ) and a large mesh size ( $128\times 256\times 256$ for $r\times \unicode[STIX]{x1D703}\times z$ directions) are used based on test calculations (see appendix A). As Gauss–Lobatto–Chebyshev polynomials are applied in the wall normal ( $r$ -) direction and Fourier series in the periodic ( $\unicode[STIX]{x1D703}$ - and $z$ -) directions, mesh grids are clustered near the inner and outer walls in the $r$ -direction, and uniform in the $\unicode[STIX]{x1D703}$ - and $z$ -directions. Sufficiently long simulations (typically of an order of $300T$ , $T=d/(\unicode[STIX]{x1D6FA}R_{i})$ ) are performed to ensure that the statistically steady flow states have been realized. Moreover, ensemble averaging is performed for time periods of ${\sim}80T$ .

3 Results and discussion

The curvature effect on the polymer-induced drag can be examined by the torque experienced by the cylinder walls and fluid across the gap, which is quantified as the conserved angular momentum current ( $J^{\unicode[STIX]{x1D714}}$ ) from the inner to the outer cylinder (Eckhardt, Grossmann & Lohse Reference Eckhardt, Grossmann and Lohse2007). For the viscoelastic TC flow, $J^{\unicode[STIX]{x1D714}}$ is obtained as

(3.1) $$\begin{eqnarray}J^{\unicode[STIX]{x1D714}}=r^{3}\left[\langle u_{r}\unicode[STIX]{x1D714}\rangle -\frac{\unicode[STIX]{x1D6FD}}{Re}\frac{\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D714}\rangle }{\unicode[STIX]{x2202}r}-\frac{(1-\unicode[STIX]{x1D6FD})}{Re}\frac{\langle \unicode[STIX]{x1D70F}_{r\unicode[STIX]{x1D703}}^{p}\rangle }{r}\right].\end{eqnarray}$$

$\langle \,\rangle =\langle \langle \langle \,\rangle _{\unicode[STIX]{x1D703}}\rangle _{z}\rangle _{t}$ , denotes hereafter averaging in the $\unicode[STIX]{x1D703}$ -direction ( $\langle \,\rangle _{\unicode[STIX]{x1D703}}$ ), the $z$ -direction ( $\langle \,\rangle _{z}$ ) and time ( $\langle \,\rangle _{t}$ ). The right-hand terms of (3.1) represent in sequence the convective flux ( $J_{c}^{\unicode[STIX]{x1D714}}$ ), the diffusive flux ( $J_{d}^{\unicode[STIX]{x1D714}}$ ) and the polymeric source/sink term ( $J_{p}^{\unicode[STIX]{x1D714}}$ ) to angular momentum. In analogy to the dimensionless heat flux of turbulent Rayleigh–Bénard (RB) convection, $J^{\unicode[STIX]{x1D714}}$ is rescaled as a quasi-Nusselt number $Nu_{\unicode[STIX]{x1D714}}$ by its Newtonian laminar value $J_{lam}^{\unicode[STIX]{x1D714}}$ as $Nu_{\unicode[STIX]{x1D714}}=J^{\unicode[STIX]{x1D714}}/J_{lam}^{\unicode[STIX]{x1D714}}$ (Eckhardt et al. Reference Eckhardt, Grossmann and Lohse2007). A detailed derivation of (3.1) is given in appendix B.

Figure 1. Profiles of (a) mean azimuthal velocity $\langle u_{\unicode[STIX]{x1D703}}\rangle$ and (b) mean polymer stress component $\langle \unicode[STIX]{x1D70F}_{r\unicode[STIX]{x1D703}}^{p}\rangle$ . Here, $\tilde{r}=(r-R_{i})/d$ is the dimensionless distance to the inner cylinder wall.

Table 1. Nusselt number of Newtonian ( $Nu_{\unicode[STIX]{x1D714}}^{N}$ ) and viscoelastic ( $Nu_{\unicode[STIX]{x1D714}}^{p}$ ) TC flows for different radius ratios and curvatures. The DE is calculated as $(Nu_{\unicode[STIX]{x1D714}}^{p}-Nu_{\unicode[STIX]{x1D714}}^{N})/Nu_{\unicode[STIX]{x1D714}}^{N}$ .

Drastic DE is observed for the viscoelastic TC flow ( $Nu_{\unicode[STIX]{x1D714}}^{p}$ ) (see table 1), as compared to the Newtonian case ( $Nu_{\unicode[STIX]{x1D714}}^{N}$ ). Similar to Newtonian flows (Ostilla-Mońico et al. Reference Ostilla-Mońico, Huisman, Jannink, Van Gils, Verzicco, Grossmann, Sun and Lohse2014), $Nu_{\unicode[STIX]{x1D714}}^{p}$ also has a non-monotonic $\unicode[STIX]{x1D702}$ -dependence, namely, a great decrease is observed from 171 % for $\unicode[STIX]{x1D702}=0.5$ to 62 % for $\unicode[STIX]{x1D702}=0.833$ followed by a slight increase to 72 % for $\unicode[STIX]{x1D702}=0.912$ . This variation clearly points to the poor efficiency of angular momentum transport at intermediate $\unicode[STIX]{x1D702}$ in the viscoelastic TC flow. Correspondingly, the wall shear stress that arises due to the mean flow has a significant enhancement compared to the Newtonian cases, as indicated by the $\langle u_{\unicode[STIX]{x1D703}}\rangle$ profiles (see figure 1 a). In particular, a positive $\langle u_{\unicode[STIX]{x1D703}}\rangle$ -gradient is observed in the bulk of viscoelastic flows for $\unicode[STIX]{x1D702}>0.72$ , while for $\unicode[STIX]{x1D702}\leqslant 0.72$ a negative one is seen. The positive $\langle u_{\unicode[STIX]{x1D703}}\rangle$ -gradient of $\unicode[STIX]{x1D702}>0.72$ in the bulk points to reverse mean shear flow which results in polymer stretch and orientation in a manner opposite to that in the wall regions where the $\langle u_{\unicode[STIX]{x1D703}}\rangle$ -gradient has negative values. As a consequence, the mean polymer stress component $\langle \unicode[STIX]{x1D70F}_{r\unicode[STIX]{x1D703}}^{p}\rangle$ for $\unicode[STIX]{x1D702}>0.72$ obtains a local maximum in the bulk bridging the two near-wall local minima; while for $\unicode[STIX]{x1D702}\leqslant 0.72$ , $\langle \unicode[STIX]{x1D70F}_{r\unicode[STIX]{x1D703}}^{p}\rangle$ monotonically increases between its minimum in the inner wall region and the local minimum of markedly higher value in the outer wall region (see figure 1 b). For $\unicode[STIX]{x1D702}>0.72$ , this prediction points to an excess angular momentum transport across the gap and the fluid physics associated with this phenomenon is discussed below.

Figure 2. Time and $\unicode[STIX]{x1D703}$ -direction averaged vectors of radial ( $\langle u_{r}\rangle _{\unicode[STIX]{x1D703},t}$ ) and axial ( $\langle u_{z}\rangle _{\unicode[STIX]{x1D703},t}$ ) velocities and contour plots of streamwise vorticity $\langle \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D703}}\rangle _{\unicode[STIX]{x1D703},t}$ in ( $r,z$ ) plane.

Figure 3. Instantaneous vectors of radial ( $u_{r}$ ) and axial ( $u_{z}$ ) velocities and contour plots of streamwise vorticity $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D703}}$ in ( $r,z$ ) plane with $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2$ .

Figure 4. One-dimensional spectra of (a) the streamwise turbulent kinetic energy ( $\langle u_{\unicode[STIX]{x1D703}}^{\prime }u_{\unicode[STIX]{x1D703}}^{\prime }\rangle$ ) and (b) the Reynolds stress ( $\langle u_{r}^{\prime }u_{\unicode[STIX]{x1D703}}^{\prime }\rangle$ ) sampled at the middle of the gap for $\unicode[STIX]{x1D702}=0.912$ . Here, the fluctuating part of variable $v$ is obtained as $v^{\prime }=v-\langle v\rangle$ . Note that, the convective flux holds $J_{c}^{\unicode[STIX]{x1D714}}=r^{2}\langle u_{r}u_{\unicode[STIX]{x1D703}}\rangle =r^{2}\langle u_{r}^{\prime }u_{\unicode[STIX]{x1D703}}^{\prime }\rangle$ .

The curvature dependence of flow structure modification in viscoelastic TC flows is highlighted by the intriguing streamwise (azimuthal) vortices visualized in figures 2 and 3. Similar to our previous findings (Liu & Khomami Reference Liu and Khomami2013b ), for small $\unicode[STIX]{x1D702}$ values ( ${<}0.72$ ) the well-organized large-scale TV identified in Newtonian cases become unstable in the presence of polymer additives and break down into small-scale vortices, which are larger in size and fewer in number near the outer wall (see figures 2 a,b; 3 a,b). In the near-outer-wall region, the observed small-scale vortices are inertio-elastic Görtler vortices (IEGV) that arise due to the so-called IEGI (Liu & Khomami Reference Liu and Khomami2013b ). As shown below, the near-inner-wall vortices are generated mainly due to the elastic stresses and therefore are dubbed ‘elastic Görtler vortices’ (EGV). The increase in number of streamwise vortices for TC flow is usually commensurate with a drag (and torque) increase (Martińez-Arias et al. Reference Martińez-Arias, Peixinho, Crumeyrolle and Mutabazi2014; Teng et al. Reference Teng, Liu, Lu and Khomami2015). For the largest $\unicode[STIX]{x1D702}$ (of 0.912), the large-scale TV stack axially and occupy the entire gap, however, great differences in contour patterns of $\langle \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D703}}\rangle _{\unicode[STIX]{x1D703},t}$ are observed between Newtonian and viscoelastic cases (see figure 2 e). These large-scale TV become more energetic and have their spanwise scales unaltered in the presence of polymer additives, as indicated in figure 4(a,b) by the highest spectral peaks at a wavenumber of $k_{z}=5$ which is the same for the Newtonian and viscoelastic cases. In terms of ensemble-averaged and instantaneous flow patterns (see figures 2 and 3), three main flow regimes are identified as: (1) small-scale GV only ( $\unicode[STIX]{x1D702}=0.5$ ), (2) co-existence of small-scale GV and large-scale TV ( $\unicode[STIX]{x1D702}=0.72$ ) and (3) large-scale TV only ( $\unicode[STIX]{x1D702}=0.912$ ). The appearance of vortical structures is qualitatively consistent with the prediction given by the extended PM criterion (Schäfer et al. Reference Schäfer, Morozov and Wagner2018), i.e., increasing curvature $\unicode[STIX]{x1D716}$ (decreasing radius ratio $\unicode[STIX]{x1D702}$ ) enhances purely elastic instability in TC flows.

Figure 5. Balance of angular momentum current across the gap for (a) Newtonian flow of $\unicode[STIX]{x1D702}=0.5$ and viscoelastic flows of (b)  $\unicode[STIX]{x1D702}=0.5$ , (c)  $\unicode[STIX]{x1D702}=0.6$ , (d)  $\unicode[STIX]{x1D702}=0.72$ , (e)  $\unicode[STIX]{x1D702}=0.833$ and (f)  $\unicode[STIX]{x1D702}=0.912$ . Note that, the balance is similar for Newtonian flows of all $\unicode[STIX]{x1D702}$ considered, i.e., $J_{c}^{\unicode[STIX]{x1D714}}$ is dominant in the bulk while $J_{d}^{\unicode[STIX]{x1D714}}$ in the wall regions.

For viscoelastic TC flows, the $\unicode[STIX]{x1D702}$ -dependent vortical structures underscore two coexisting DE mechanisms that could be interpreted separately in the following via the cases of $\unicode[STIX]{x1D702}=0.5$ and 0.912. First, the dramatic DE of the smallest $\unicode[STIX]{x1D702}$ has been ascribed to the occurrence of the small-scale GV near the inner and outer walls (Liu & Khomami Reference Liu and Khomami2013b ). These GV as shown in figures 2(a) and 3(a) enhance transverse momentum exchange and turbulent mixing of $\unicode[STIX]{x1D70F}_{r\unicode[STIX]{x1D703}}^{p}$ via their vortical circulations. Specifically, the fluid parcels with $\unicode[STIX]{x1D70F}_{r\unicode[STIX]{x1D703}}^{p}$ of much higher magnitude are carried from the near-inner-wall region to the bulk by the elastic Görtler vortices’ circulations and subsequently to the near-outer-wall region by the inertio-elastic Görtler vortices’ circulations, and vice versa. Consequently, $\langle \unicode[STIX]{x1D70F}_{r\unicode[STIX]{x1D703}}^{p}\rangle$ monotonically increases in the bulk (see figure 1 b) and has the most significant contribution to the angular momentum current away from the walls (see $J_{p}^{\unicode[STIX]{x1D714}}$ in figure 5 b, which is positive and thus a source term). In contrast, for the Newtonian case (see figure 5 a) the convective flux ( $J_{c}^{\unicode[STIX]{x1D714}}$ ) has the dominant contribution in the bulk whereas the diffusive flux ( $J_{d}^{\unicode[STIX]{x1D714}}$ ) is dominant in the wall regions. This situation is similar for Newtonian flows of all $\unicode[STIX]{x1D702}$ considered. This DE mechanism is clearly depicted in figure 6(a), where the viscoelastic TC flow with $\unicode[STIX]{x1D702}=0.5$ has positive $J_{r,z}^{\unicode[STIX]{x1D714}}$ values almost in the entire gap and the contour patterns are very reminiscent of those small-scale vortical structures shown in figures 2(a) and 3(a).

For the largest $\unicode[STIX]{x1D702}$ of 0.912, the DE mechanism originates from in the persistence of large-scale Taylor vortex circulations that facilitate a more efficient angular momentum transport, as evidenced by the following co-supportive facts. Specifically, the fluid parcels of high/low angular momentum are carried effectively by the intense outflows/inflows at the narrow boundaries of these Taylor vortex cells from the inner/outer to the outer/inner walls, before a comprehensive mixing occurs in the bulk (see figures 2 e and 3 e). This efficient angular momentum transport leads to a striking decrease/increase of angular momentum in the inner/outer wall regions, evinced by a positive $\langle u_{\unicode[STIX]{x1D703}}\rangle$ -gradient (reverse mean flow shear) that occurs in the bulk flow (see figure 1 a). Such a transverse transport is also applied to the polymer stress $\unicode[STIX]{x1D70F}_{r\unicode[STIX]{x1D703}}^{p}$ as indicated by figure 1(b). The most convincing evidence for this DE mechanism is provided in figure 5(f), which demonstrates that in the bulk $J_{c}^{\unicode[STIX]{x1D714}}$ has a positive contribution at $\unicode[STIX]{x1D702}=0.912$ that is even higher than $J^{\unicode[STIX]{x1D714}}$ and the negative $J_{p}^{\unicode[STIX]{x1D714}}$ acts as a sink term for the angular momentum. Also, the spectrum of the Reynolds stress ( $\langle u_{r}^{\prime }u_{\unicode[STIX]{x1D703}}^{\prime }\rangle$ ) has a higher polymer-induced basic peak at $k_{z}=5$ (see figure 4 b), pointing to the enhanced transverse momentum flux due to the large-scale TV. The well-organized axially stacked contour patterns of positive and negative $J_{r,z}^{\unicode[STIX]{x1D714}}$ shown in figure 6(e) for the viscoelastic TC flow present further evidence that these large-scale TV are the main driving force for the polymer-enhanced angular momentum transport.

Figure 6. Time and $\unicode[STIX]{x1D703}$ -direction averaged angular momentum current $J_{r,z}^{\unicode[STIX]{x1D714}}$ in ( $r,z$ ) plane for (a)  $\unicode[STIX]{x1D702}=0.5$ , (b)  $\unicode[STIX]{x1D702}=0.6$ , (c)  $\unicode[STIX]{x1D702}=0.72$ , (d)  $\unicode[STIX]{x1D702}=0.833$ and (e)  $\unicode[STIX]{x1D702}=0.912$ , where $J_{r,z}^{\unicode[STIX]{x1D714}}=r^{3}\langle u_{r}\unicode[STIX]{x1D714}-\unicode[STIX]{x1D6FD}(\unicode[STIX]{x2202}\unicode[STIX]{x1D714}/\unicode[STIX]{x2202}r)/Re-(1-\unicode[STIX]{x1D6FD})(\unicode[STIX]{x1D70F}_{r\unicode[STIX]{x1D703}}^{p}/r)/Re\rangle _{\unicode[STIX]{x1D703},t}$ .

Considering the analogy between the turbulent TC flow and the turbulent RB convection (Eckhardt et al. Reference Eckhardt, Grossmann and Lohse2007), the DE mechanism of $\unicode[STIX]{x1D702}=0.912$ has a physical origin akin to the polymer-enhanced heat transport in the bulk of turbulent RB convection. Specifically, the polymer-enhanced heat transport in turbulent RB convection is realized by enhancing the coherent heat fluxes related to thermal plumes and suppressing the incoherent heat fluxes related to small-scale turbulent fluctuations (Xie et al. Reference Xie, Huang, Funfschilling, Li, Ni and Xia2015). Similarly, in viscoelastic TC flow at $\unicode[STIX]{x1D702}=0.912$ , DE is realized via enhancing the coherent angular momentum fluxes driven by the large-scale Taylor vortex circulations and weakening the incoherent transport by the small-scale GV (see figure 4 b). Further, the DE mechanism at $\unicode[STIX]{x1D702}=0.5$ can be ascribed to a polymer-induced enhancement of incoherent transport and homogenization of polymer stress, $\unicode[STIX]{x1D70F}_{r\unicode[STIX]{x1D703}}^{p}$ , by the small-scale GV.

The transition between the aforementioned DE mechanisms is clearly illustrated by detailed examination of figures 2, 3, 5 and 6. As shown in figures 2, 3 and 7, the viscoelastic TC flows present a striking transition in vortical structures with increasing $\unicode[STIX]{x1D702}$ ; specifically this increase leads to weakening and elimination of the small-scale GV and the development and better organization (occupying the entire gap) of large-scale TV. The small-scale GV circulations evidently have a decreasing contribution to the angular momentum transport, while the contribution of large-scale TV to $J^{\unicode[STIX]{x1D714}}$ becomes more and more significant in the bulk (see $J_{p}^{\unicode[STIX]{x1D714}}$ , $J_{c}^{\unicode[STIX]{x1D714}}$ and $J^{\unicode[STIX]{x1D714}}$ in figure 5 b–f). An $\unicode[STIX]{x1D702}$ -dependent transition of contour patterns of $J_{r,z}^{\unicode[STIX]{x1D714}}$ which is consistent with the aforementioned vortical structures is clearly shown in figure 6. Note that the distorted large-scale TV appear at $\unicode[STIX]{x1D702}=0.833$ (see figures 2 d, 3 d and 6 d) where the smallest DE is observed in the present study. It should be caused by the weakening of small-scale GV (thus $J_{p}^{\unicode[STIX]{x1D714}}$ ) and the enhancement in coherent transport of large-scale TV (thus $J_{c}^{\unicode[STIX]{x1D714}}$ ). Although, the large-scale TV observed at $\unicode[STIX]{x1D702}=0.912$ are reminiscent of the secondary flow patterns driven by purely elastic instabilities as observed in Taylor–Dean flows (Joo & Shaqfeh Reference Joo and Shaqfeh1992) and serpentine channel flows (Zilz et al. Reference Zilz, Poole, Alves, Bartolo, Levaché and Lindner2012); however, it is evident that these large-scale TV have a different driving mechanism due to the presence of significant inertial force.

Figure 7. Instantaneous vortical structures visualized by $Q$ -criterion with $Q=0.001$ and coloured by the distance to the inner wall for the three main regimes of viscoelastic TC flow. The flow structures in the region $\unicode[STIX]{x1D703}\in [3/2\unicode[STIX]{x03C0},2\unicode[STIX]{x03C0}]$ and $\tilde{r}\in [1/4,1]$ are not shown to clearly display the small-scale vortical structures near the inner wall.

Figure 8. Production terms of mean streamwise enstrophy ( $\langle \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D703}}\rangle ^{2}$ ) near (a) the inner and (b) the outer wall for Newtonian and viscoelastic TC flows. Here, $S_{\unicode[STIX]{x1D714}}$ denotes the Newtonian production including the mean $[(\langle \unicode[STIX]{x1D74E}\rangle \boldsymbol{\cdot }\unicode[STIX]{x1D735}\langle \boldsymbol{u}\rangle \boldsymbol{\cdot }\langle \unicode[STIX]{x1D74E}\rangle )_{\unicode[STIX]{x1D703}}]$ and fluctuating $[(\langle \unicode[STIX]{x1D74E}^{\prime }\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}^{\prime }\rangle \boldsymbol{\cdot }\langle \unicode[STIX]{x1D74E}\rangle )_{\unicode[STIX]{x1D703}}]$ strain stretch as well as the fluctuating enstrophy $[(\langle \boldsymbol{u}^{\prime }\unicode[STIX]{x1D74E}^{\prime }\rangle \boldsymbol{ : }\unicode[STIX]{x1D735}\langle \unicode[STIX]{x1D74E}\rangle )_{\unicode[STIX]{x1D703}}]$ , and $T_{\unicode[STIX]{x1D714}}$ represents the elastic production $[(1-\unicode[STIX]{x1D6FD})(\unicode[STIX]{x1D735}\times \langle \unicode[STIX]{x1D749}^{p}\rangle \boldsymbol{ : }\unicode[STIX]{x1D735}\langle \unicode[STIX]{x1D74E}\rangle )_{\unicode[STIX]{x1D703}}/Re]$ , where $\unicode[STIX]{x1D74E}=\unicode[STIX]{x1D735}\times \boldsymbol{u}$ . The insets show $S_{\unicode[STIX]{x1D714}}$ of the Newtonian cases. Here, plots are shown only for the three main regimes for clarity.

The mechanism that gives rise to the EGV is scrutinized by comparing the production terms of the balanced $\langle \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D703}}\rangle ^{2}$ -budget equations (Kim et al. Reference Kim, Li, Balachandar, Sureshkumar and Adrian2007) near the inner wall for the three main flow regimes. As seen in figure 8(a), the elastic production $T_{\unicode[STIX]{x1D714}}$ acts as the source term of increasing importance to the $\langle \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D703}}\rangle ^{2}$ budgets versus the decreasing $\unicode[STIX]{x1D702}$ , and at $\unicode[STIX]{x1D702}=0.5$ overwhelms the contribution of $S_{\unicode[STIX]{x1D714}}$ that is the sole generation mechanism for streamwise vortical structures in the Newtonian TC flow. To this end, it is reasonable to describe the near-inner-wall small-scale vortices in viscoelastic cases of small $\unicode[STIX]{x1D702}$ , as EGV as they are reminiscent of those classical Newtonian GV (Wei et al. Reference Wei, Kline, Lee and Woodruff1992). In contrast, $T_{\unicode[STIX]{x1D714}}$ and $S_{\unicode[STIX]{x1D714}}$ have comparable contributions in the outer wall region (see figure 8 b), where, consequently, the IEGV are generated (Liu & Khomami Reference Liu and Khomami2013b ). This is evidenced by the EGV and IEGV observed near the inner and outer walls, respectively, especially for $\unicode[STIX]{x1D702}=0.5$ (see figure 7 a).

4 Concluding remarks

In summary, a mechanism for curvature dependence of polymer-induced DE in TC flow has been proposed and substantiated for the first time via quantification of transverse angular momentum transport. Specifically, the drag enhancement is facilitated for small $\unicode[STIX]{x1D702}$ (large $\unicode[STIX]{x1D716}$ ) by the small-scale GV that occur near the walls and result in much higher incoherent transport and homogenization of the polymer stress $\unicode[STIX]{x1D70F}_{r\unicode[STIX]{x1D703}}^{p}$ ; and for large $\unicode[STIX]{x1D702}$ (small $\unicode[STIX]{x1D716}$ ) by the large-scale TV that persist and account for much higher coherent angular momentum fluxes across the gap. The DE change as $\unicode[STIX]{x1D702}$ is increased is commensurate with a transition in vortical structures characterized by weakening and elimination of the small-scale GV and development and better organization (occupying the entire gap) of large-scale TV. Of particular interest is the viscoelastic TC flow with high curvature (small $\unicode[STIX]{x1D702}$ ), where an enhanced elastic instability is observed as evinced by the excitation of small-scale streamwise vortices and a commensurate drastic enhancement of flow resistance. This suggests that elastic turbulence could be realized in TC flow by increasing the curvature over a certain critical value even at high inertia ( $Re$ ). To this end, this study has provided a viable road map for future research focused on the nature of localized elastic turbulence structures via coordinated experiments and DNS in curvilinear flows.