1. Introduction
As a curvature-free limit of Taylor–Couette flow (TCF), the spanwise-rotating plane Couette flow (RPCF) has been known to share many dynamical properties with TCF (Tsukahara, Tillmark & Alfredsson Reference Tsukahara, Tillmark and Alfredsson2010; Brauckmann, Salewski & Eckhardt Reference Brauckmann, Salewski and Eckhardt2016), namely, analogous transitions from laminar to turbulent flow at different combinations of the same key driving forces, i.e. shear rate gradient and the system rotation quantified as the shear Reynolds (
$Re$) and rotation (
$Ro$) numbers, respectively (Dubrulle et al. Reference Dubrulle, Dauchot, Daviaud, Longaretti, Richard and Zahn2005; Grossmann, Lohse & Sun Reference Grossmann, Lohse and Sun2016). In fact, several similar turbulent flow regimes have been identified in the 
$(Re, Ro)$ parameter space in RPCF and TCF (Andereck, Liu & Swinney Reference Andereck, Liu and Swinney1986; Tsukahara et al. Reference Tsukahara, Tillmark and Alfredsson2010). These flow regimes are essentially characterized by large-scale vortical structures (roll cells for RPCF that are reminiscent of Taylor vortices) overlaid by small-scale turbulent vortices (Bech & Andersson Reference Bech and Andersson1996, Reference Bech and Andersson1997; Gai et al. Reference Gai, Xia, Cai and Chen2016; Xia et al. Reference Xia, Shi, Wan, Sun, Cai and Chen2019). These large-scale vortical structures have a cross-stream size of approximately the gap width and are streamwise-oriented and spanwise-organized; these structures are also known to play an important role in transverse momentum transport and in turn in the drag/torque features of the flow (Bech & Andersson Reference Bech and Andersson1997; Martínez-Arias et al. Reference Martínez-Arias, Peixinho, Crumeyrolle and Mutabazi2014; Salewski & Eckhardt Reference Salewski and Eckhardt2015; Brauckmann et al. Reference Brauckmann, Salewski and Eckhardt2016; Gai et al. Reference Gai, Xia, Cai and Chen2016). For example, in the turbulent RPCF at 
$Re=1300$, the roll cells are found to facilitate a substantial enhancement of transverse momentum transport as they become more well-defined and energetic; specifically, the maximum drag is realized at an anticyclonic spanwise rotation 
$Ro\approx 0.2$ (Brauckmann et al. Reference Brauckmann, Salewski and Eckhardt2016; Gai et al. Reference Gai, Xia, Cai and Chen2016).
Of interest, the RPCF at 
$Re=1300$ and 
$Ro=0.2$ has been identified as a flow regime termed ‘contained turbulence in roll cells (CNT)’ (Tsukahara et al. Reference Tsukahara, Tillmark and Alfredsson2010; Gai et al. Reference Gai, Xia, Cai and Chen2016). It is characterized by turbulence that seems to be contained in the roll cells and not connected to turbulence in neighbouring roll cells. According to Tsukahara et al. (Reference Tsukahara, Tillmark and Alfredsson2010), two typical transition pathways have been indicated for the RPCF of anticyclonic rotation to access this intriguing flow regime. One is realized by increasing 
$Ro$ at a fixed 
$Re$, for example 800. It undergoes in sequence the flow regimes labelled ‘featureless turbulence’, ‘turbulence with roll cells (TRC)’ and, finally, CNT. The other is realized by decreasing 
$Re$ while fixing 
$\varOmega =Re\,Ro$ as a constant, for example 20. If started from the TRC, it goes through the CNT to the laminar Couette flows with stable roll cells of wavy three-dimensional (3-D) structure (COU3D), to that of meandering 2-D structure (COU2Dm) and then to that of straight 2-D structure (COU2Dh/COU2D). The sub-regimes COU3D, COU2Dm and COU2Dh/COU2D are indicated to have strong, weak and absent interplays between the neighbouring roll cells, respectively.
For viscoelastic turbulent TCF of a dilute polymeric solution, significant curvature dependence of drag enhancement (DE) has been reported due to flow transitions driven by significant changes in large-scale Taylor vortices (TV) (Liu & Khomami Reference Liu and Khomami2013b; Song et al. Reference Song, Teng, Liu, Ding, Lu and Khomami2019). Specifically, in the high curvature limit (evaluated as the inner-to-outer radius ratio of 
$\eta =0.5$), and in presence of polymer additives the large-scale Newtonian TV break down into small-scale Görtler vortices (GV) near the walls. These small-scale GV lead to much higher incoherent transport and homogenization of the polymer stress component 
$\tau ^p_{r\theta }$ which in turn acts as a dominant source term in the angular momentum transport. Consequently, a drastic DE of 
$171\,\%$ is realized. In contrast, at a low curvature (
$\eta =0.912$), the large-scale TV persist and become better organized and more energetic, resulting in much higher coherent angular momentum flux (equivalent to the Reynolds stress) across the gap; hence, a much lower DE 
${\sim }72\,\%$ is realized. Undoubtedly, the role of vortical structures on momentum transport in viscoelastic TCF is intimately coupled with the flow curvature. To fundamentally understand this coupling the following question must be addressed: what is the flow response (including drag and vortical structure dynamics) to polymer additives in turbulent RPCF (zero curvature analogue of TCF) where only large-scale roll cells are present? In fact, the primary motivation of the current work is to answer this question.
In recent years, intriguing flow features have also been reported in turbulent viscoelastic plane Couette flows (PCF). Specifically, much weaker roll cells are observed as compared to their Newtonian counterparts (Pereira et al. Reference Pereira, Mompean, Thais and Soares2017a,Reference Pereira, Mompean, Thais, Soares and Thompsonb; Teng et al. Reference Teng, Liu, Lu and Khomami2018). Commensurate with this change, as the polymeric elastic forces (commonly quantified in terms of the Weissenberg number 
$Wi$) increase, much weaker transverse momentum transfer is realized. Consequently, drag reduction (DR) of up to 
${\sim }34\,\%$ is observed in viscoelastic PCF at O(1) 
$Wi$. This is in contrast to the DE observed in turbulent viscoelastic TCF (Liu & Khomami Reference Liu and Khomami2013b; Song et al. Reference Song, Teng, Liu, Ding, Lu and Khomami2019). The extent of the aforementioned DR is commensurate with formation of weaker quasi-streamwise vortices (QSV) in the near-wall regions and elongated vortices away from the walls. This observation supports earlier findings that indicate polymer-induced DR has the same physical origin in wall-bounded flows, namely, the mean flow shear gradient renders significant polymer stretch in the near-wall region that acts to suppress the self-sustaining process of wall turbulence and in turn the Reynolds stress production (Sureshkumar, Beris & Handler Reference Sureshkumar, Beris and Handler1997; Dubief et al. Reference Dubief, White, Terrapon, Shaqfeh, Moin and Lele2004; Li, Sureshkumar & Khomami Reference Li, Sureshkumar and Khomami2006; Kim et al. Reference Kim, Li, Sureshkumar, Balachandar and Adrian2007; White & Mungal Reference White and Mungal2008). Moreover, recent studies have shed light on the origin of the maximum drag reduction (MDR), namely, in this regime the flow dynamics is driven by an elasto-inertial instability (Dubief, Terrapon & Soria Reference Dubief, Terrapon and Soria2013; Samanta et al. Reference Samanta, Dubief, Holzner, Schäfer, Morozov, Wagner and Hof2013; Li, Sureshkumar & Khomami Reference Li, Sureshkumar and Khomami2015; Sid, Terrapon & Dubief Reference Sid, Terrapon and Dubief2018) that could even eliminate Newtonian turbulence (NT). Furthermore, flow relaminarization has recently been observed in channel and pipe flows of polymeric solutions as polymer concentration (elasticity level) is increased at the transition 
$Re$ (Choueiri, Lopez & Hof Reference Choueiri, Lopez and Hof2018; Lopez, Choueiri & Hof Reference Lopez, Choueiri and Hof2019; Shekar et al. Reference Shekar, McMullen, Wang, McKeon and Graham2019; Chandra, Shankar & Das Reference Chandra, Shankar and Das2020). Specifically, the MDR state is achieved via a reverse transition pathway from NT through a relaminarized state. These intriguing new observations bring up yet another interesting question: can elasticity-driven flow relaminarization occur in a turbulent RPCF where the flow dynamics is dominated by large-scale roll cells? Answering this question is the secondary goal of this study.
To answer the aforementioned two fundamentally questions, extensive high-fidelity direct numerical simulations (DNS) of viscoelastic RPCF (see figure 1) have been performed at 
$Re=1300$, 
$Ro=0.2$, 
$0<Wi<40$. To this end, we report for the first time polymer-induced flow relaminarization of turbulent RPCF which results in DE. The elastically driven fluid physics behind these major kinematics and frictional changes are also discussed.
Figure 1. Sketch of spanwise-rotating plane Couette flow.
2. Problem formulation and computational details
Similar to our earlier studies of polymer-induced DR in planar wall-bounded turbulent flows and elasto-inertial turbulence in TCF, DNS is performed via a fully spectral, three-dimensional parallel algorithm (Li et al. Reference Li, Sureshkumar and Khomami2006, Reference Li, Sureshkumar and Khomami2015; Liu & Khomami Reference Liu and Khomami2013a,Reference Liu and Khomamib; Teng et al. Reference Teng, Liu, Lu and Khomami2018; Song et al. Reference Song, Teng, Liu, Ding, Lu and Khomami2019). As indicated in figure 1, we have chosen 
$h, h/U_w, U_w$ and 
$\rho {U_w}^2$ as scales for length, time, velocity 
$\boldsymbol {u}$ and pressure 
$P$, respectively. Here, 
$2h$ denotes the gap width, 
$U_w$ is the wall translation velocity, 
$\rho$ represents the solution density. The polymer stress 
$\boldsymbol {\tau }^p$ is related to the polymer conformation tensor 
${\boldsymbol{\mathsf{C}}}$ through the FENE-P (finitely extensible nonlinear elastic-Peterlin) constitutive model. The governing equations for an incompressible flow of FENE-P fluids are non-dimensionalized as
\begin{gather} \boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{u}=0, \end{gather}
\begin{gather} \frac{\partial \boldsymbol{u}}{\partial t}+\boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{\nabla}\boldsymbol{u} = -\boldsymbol{\nabla} P+\frac{\beta}{Re}\nabla^2\boldsymbol{u}+\frac{1-\beta}{Re}\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{\tau}^p-{Ro}{\boldsymbol{e_z}}\times\boldsymbol{u}, \end{gather}
\begin{gather} \frac{\partial {\boldsymbol{\mathsf{C}}}}{\partial t} = -\boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{\nabla}{\boldsymbol{\mathsf{C}}}+{\boldsymbol{\mathsf{C}}}\boldsymbol{\cdot}\boldsymbol{\nabla}\boldsymbol{u} +(\boldsymbol{\nabla}\boldsymbol{u})^T\boldsymbol{\cdot}{\boldsymbol{\mathsf{C}}}-\boldsymbol{\tau}^p. \end{gather}
The polymer stress tensor 
$\boldsymbol {\tau }^p$ is obtained as
\begin{equation} \boldsymbol{\tau}^p=\frac{1}{Wi}\left(\frac{L^2-3}{L^2-\textrm{trace}({\boldsymbol{\mathsf{C}}})}{\boldsymbol{\mathsf{C}}}-{\boldsymbol{\mathsf{I}}}\right), \end{equation}
where 
$\beta =\eta _s / \eta _t$, with 
$\eta _s$ and 
$\eta _p$ being respectively the solvent and polymeric contributions to the total viscosity 
$\eta _t$, 
$L$ signifies the maximum polymer chain extension and the Reynolds and rotation numbers are defined as 
$Re=\rho U_w h/\eta _t$ and 
$Ro=2{\varOmega _z}h/U_w$, respectively, with 
$\varOmega _z$ being the spanwise system rotation. Weissenberg number is defined as 
$Wi=\lambda U_w/h$ where 
$\lambda$ is the polymer relaxation time.
Following the numerical strategy for the spectral-based algorithm (Sureshkumar & Beris Reference Sureshkumar and Beris1995; Sureshkumar et al. Reference Sureshkumar, Beris and Handler1997), a small diffusive term 
$\kappa \nabla ^2{\boldsymbol{\mathsf{C}}}$ is added to (2.3) in the bulk flow region for numerical stabilization (Li et al. Reference Li, Sureshkumar and Khomami2006, Reference Li, Sureshkumar and Khomami2015; Teng et al. Reference Teng, Liu, Lu and Khomami2018; Song et al. Reference Song, Teng, Liu, Ding, Lu and Khomami2019). Based on simulations performed at various 
$Sc$ (see appendix), a 
$\kappa$ value corresponding to a Schmidt number 
$Sc=1/(\kappa Re)=0.15$ has been used for all calculations. It is verified that such a 
$Sc$ value does not modify the essential features of the velocity and polymer conformation fields. This 
$Sc$ is of the same order as the value of 0.5 used to capture the elasticity-driven reverse transition occurring in pipe flows from the NT to elasto-inertial turbulence (EIT) regime via a flow relaminarization (Lopez et al. Reference Lopez, Choueiri and Hof2019), and close to 
$Sc=0.2$ used to obtain the EIT developed in the PCF (Pereira et al. Reference Pereira, Mompean, Thais and Soares2017a,Reference Pereira, Mompean, Thais, Soares and Thompsonb; Pereira, Thompson & Mompean Reference Pereira, Thompson and Mompean2019). Note that the original constitutive equation without the diffusive term is applied at the walls; thus no boundary conditions are imposed for 
${\boldsymbol{\mathsf{C}}}$ at solid boundaries.
All the simulations for viscoelastic RPCF are started from a statistically steady Newtonian turbulent state at 
$Re=1300$ and 
$Ro=0.2$ (Tsukahara et al. Reference Tsukahara, Tillmark and Alfredsson2010; Gai et al. Reference Gai, Xia, Cai and Chen2016). Large values of 
$\beta =0.9$ and 
$L=120$ are used to ensure the dilute polymeric solution has a nearly shear-independent viscosity and a significant elongational viscosity (scales with 
$L^2$). A comprehensive examination of elasticity-driven flow transition is performed by considering a broad range of 
$Wi=0\sim 40$. The computational domain is set as 
$L_x\times L_y\times L_z=10{\rm \pi} \times 2\times 4{\rm \pi}$ (Bech & Andersson Reference Bech and Andersson1996, Reference Bech and Andersson1997; Gai et al. Reference Gai, Xia, Cai and Chen2016; Xia et al. Reference Xia, Shi, Wan, Sun, Cai and Chen2019) for the streamwise, wall-normal and spanwise directions, respectively. According to the grid independence check (see appendix), we use a grid size of 
$N_x\times N_y\times N_z=256\times 129\times 256$, which is much larger than those used for the Newtonian RPCF (Gai et al. Reference Gai, Xia, Cai and Chen2016; Xia et al. Reference Xia, Shi, Wan, Sun, Cai and Chen2019) but is required for accurate viscoelastic simulations. Following our prior study (Teng et al. Reference Teng, Liu, Lu and Khomami2018), a small time step, namely, 
$\Delta t=0.01$ is used to ensure accuracy; calculations of 
${\sim }1000h/U_w$ are performed to ensure that a statistically steady flow state has been realized.
3. Results and discussion
The elasticity-driven flow relaminarization pathway is depicted in figure 2 through visualization of instantaneous vortical structures of the viscoelastic RPCF. Specifically, in the Newtonian flow (see figure 2a), large-scale roll cells are identified in the flow domain; they are essentially 3-D structures superimposed by small-scale turbulent vortices (QSV) in the near-wall region. According to Tsukahara et al. (Reference Tsukahara, Tillmark and Alfredsson2010), this flow corresponds to the flow regime labelled CNT. Evidently, once the polymer additive is introduced, the small-scale QSV near the walls are suppressed, giving rise to a great reduction in their number and intensity (see figure 2b). With the increase of 
$Wi$, further suppression of small-scale turbulent vortices is observed, while the roll cells become more regular and meander in the streamwise direction (see figure 2c–e). Finally, at 
$Wi=40$ (see figure 2f), the QSV usually referred to as turbulent vortices are totally eliminated and the roll cells occupy the entire flow domain. The roll cells have a 2-D structure, i.e. they have little to no variation in the streamwise direction. It corresponds to the flow sub-regime termed ‘laminar Couette flow with straight 2-D roll cells’, i.e. COU2Dh/COU2D (Tsukahara et al. Reference Tsukahara, Tillmark and Alfredsson2010). This evinces that the viscoelastic RPCF has been relaminarized. A similar polymer-induced destruction of QSV or turbulent vortices has also been documented in the viscoelastic PCF where substantial DR is realized (Teng et al. Reference Teng, Liu, Lu and Khomami2018). However, great DE is obtained in the viscoelastic RPCF (see figure 2a–f). Here, DE factor is evaluated as 
$(Re_\tau ^2-Re_{{\tau }0}^2)/Re_{{\tau }0}^2$, where 
$Re_{\tau }$ and 
$Re_{\tau 0}$ are the frictional Reynolds numbers of the viscoelastic and Newtonian flows, respectively. The DE features and the fluid physics related to roll cell dynamics are discussed in detail below.
Figure 2. Flow relaminarization pathway visualized by instantaneous vortical structures that are identified by 
$Q$-criterion and coloured by the 
$Q$ value (red/blue for positive/negative values). Q is defined as the difference between the magnitudes of fluid rotation and strain. 
$(a)\, {\rm Newtonian}, Q = \pm 0.5, Re_{\tau} = 106.2; (b)\, Wi = 5, Q = \pm 0.1, Re_{\tau} =114.9,$
$DE = 17\,\%; (c)\ Wi = 10, Q = \pm 0.1, Re_{\tau} =117.9, DE = 23\,\%;$
$(d)\ Wi = 20, Q = \pm 0.1,$
$Re_{\tau} =120, DE\,{=}\,28\,\%; (e)\, Wi\,{=}\,30, Q\,{=}\,\pm 0.1, Re_{\tau}\,{=}\,127.8, DE\,{=}\,45\,\%;$
$(f)\, Wi\,{=}\,40, Q\,{=}\,\pm 0.1, Re_{\tau} = 130.5, DE = 51\,\%.$
Polymer-induced changes leading to flow relaminarization can be quantified via the one-dimensional spectra (
$E_{uu}$) of the streamwise turbulent kinetic energy (TKE) (see figure 3a). The most distinct variation in 
$E_{uu}$ is observed at high wavenumbers. Specifically, a continuous decrease of several orders of magnitude over a broad range of wavenumbers occurs as 
$Wi$ is increased. This observation is consistent with the enhanced suppression of turbulent vortices shown in figure 2 as well as a great reduction in contribution of small-scale fluctuating motions to TKE. As the viscoelastic RPCF becomes fully relaminarized at 
$Wi=40$, 
$E_{uu}$ shows a remarkable increase in its spectral peak value at wavenumber 
$k_z=3$. This points to the fact that the roll cells have become more energetic and have a spanwise wavelength of approximately 
$L_z/3$ (see figures 2 and 4). As shown below, these large-scale roll cells greatly influence the transverse momentum transport across the gap.
Figure 3. One-dimensional spectra of (a) the streamwise turbulent kinetic energy (
$\langle u'u'\rangle$) at the channel centre and (b) time series of the near-wall streamwise velocity 
$u$ sampled at 
$y=0.97h$. Here, the fluctuating part of 
$u$ is obtained as 
$u'=u-\langle u\rangle$; 
$\langle \,\rangle$ denotes hereafter the ensemble averaging.
Figure 4. Instantaneous vortical structures visualized by streamline plots of the velocity vector (
$v$, 
$w$) in the 
$(y, z)$ plane at 
$x=L_x/2$ for (a) Newtonian flow, (b) 
$Wi=5$, (c) 
$Wi=10$, (d) 
$Wi=20$, (e) 
$Wi=30$ and (f) 
$Wi=40$. The contour patterns are plotted based on the value of polymer stress term 
$J^u_p$.
Flow relaminarization is further evinced by the polymer-induced changes of unsteady characteristics of turbulent RPCF, illustrated in terms of the time series of near-wall streamwise velocity 
$u$ (see figure 3b). For the Newtonian case, 
$u$ is highly random in time, namely, high frequency fluctuations and significant intermittency are observed in the time series. Increasing 
$Wi$ leads to a notable decrease of the fluctuating part of 
$u$ especially at high frequencies as 
$Wi\geq 20$. When the viscoelastic turbulent RPCF transitions to a laminar flow at 
$Wi=40$, 
$u$ becomes steady, i.e. no temporal fluctuations are seen after a transition period, i.e. 
$t\sim 800h/U_w$. Moreover, as flow goes through a transition to a laminar flow state, the pairing of the large-scale counter-rotating roll cells arranged in the spanwise direction is not altered, i.e. it remains at three (see figure 4). Similar findings have been reported in our recent study of viscoelastic PCF when DR is observed (Teng et al. Reference Teng, Liu, Lu and Khomami2018). Hence, the answer to the second question posed in the introduction is a resounding yes.
The elasticity-driven transition to a laminar flow is accompanied with a monotonic increase in DE, i.e. up to 
$51\,\%$ at 
$Wi=40$ (see figure 2). This is in stark contrast to viscoelastic channel and pipe flows where flow relaminarization gives rise to drastic DR (Choueiri et al. Reference Choueiri, Lopez and Hof2018; Lopez et al. Reference Lopez, Choueiri and Hof2019; Shekar et al. Reference Shekar, McMullen, Wang, McKeon and Graham2019; Chandra et al. Reference Chandra, Shankar and Das2020). To this end, the mechanism of DE is ascribed to momentum exchange modified by roll cells in the presence of polymers that evidently influence the force experienced at the channel walls and even between fluid layers across the gap (Teng et al. Reference Teng, Liu, Lu and Khomami2018). This force can be quantified in the context of the conserved momentum flux (Salewski & Eckhardt Reference Salewski and Eckhardt2015; Brauckmann et al. Reference Brauckmann, Salewski and Eckhardt2016) as follows,
\begin{equation} J^u=-\langle u'v'\rangle+\frac{\beta}{Re} \frac{\textrm{d}\langle u\rangle}{\textrm{d}y}+\frac{1-\beta}{Re}\langle\tau_{xy}^p\rangle. \end{equation}
The terms on the right-hand side of (3.1) in sequence are, the momentum flux associated with the Reynolds stress (
$J^u_R$; convective flux), the viscous stress (
$J^u_v$; diffusive flux) and the polymer stress (
$J^u_p$; polymeric source/sink term). When 
$J^u={Re^2_\tau }/{Re^2}$ is recast in wall units, DE can be directly related to polymer-induced changes of 
$J^u$ through its primary contributions (i.e. 
$J^u_R$ and 
$J^u_p$) in the core region (see below for details).
Clearly, introducing polymer additive leads to a reduction in 
$J^u_R$ (see figure 5a). This reduction becomes more pronounced as 
$Wi$ is increased from 0 to 20. As 
$Wi$ is further increased from 20 to 40, 
$J^u_R$ is significantly enhanced, particularly in the core region. Once the viscoelastic RPCF becomes fully laminar at 
$Wi=40$, the maximum in 
$J^u_R$ which occurs in the core region becomes greater than the Newtonian momentum flux 
$J^u$. Meanwhile, in the range of 
$Wi=20\sim 40$, 
$J^u_p$ continuously increases near the walls while slightly decreasing in the core region (see figure 5b). Evidently, the monotonic increase of polymer-induced DE as a function of 
$Wi$ for 
$Wi<20$ is facilitated by the monotonic and significant increase of 
$J^u_p$ which is a source term in the momentum flux; at 
$Wi>20$ 
$J^u_R$ markedly increases, leading to efficient transverse momentum transport via convection.
Figure 5. (a) Reynolds stress term 
$J^u_R$ and (b) polymer stress term 
$J^u_p$.
In contrast to polymer-induced DR in wall-bounded flows that are devoid of transverse roll cells (Li et al. Reference Li, Sureshkumar and Khomami2006; Kim et al. Reference Kim, Li, Sureshkumar, Balachandar and Adrian2007; White & Mungal Reference White and Mungal2008; Pereira et al. Reference Pereira, Mompean, Thais and Soares2017a,Reference Pereira, Mompean, Thais, Soares and Thompsonb; Teng et al. Reference Teng, Liu, Lu and Khomami2018), in viscoelastic RPCF, two maxima appear in 
$J^u_p$ near the walls; these peak values increase continuously as 
$Wi$ is enhanced. In fact, at high 
$Wi$ these peak values approach the magnitude of the Newtonian momentum flux 
$J^u$ (see figure 5b). The changes in 
$J^u_p$ are a direct consequence of the extensional flows created at boundaries of counter-rotating transverse roll cells as depicted in figure 4. Coupled with the polymer stretching driven by the mean flow shear, these extensional flows result in enhanced polymer stress production as 
$Wi$ is increased (Groisman & Steinberg Reference Groisman and Steinberg1998; Kumar & Graham Reference Kumar and Graham2001; Liu & Khomami Reference Liu and Khomami2013a). Consequently, the increase in the polymer stress in the near-wall region gives rise to a monotonic increase in DE in the viscoelastic RPCF. Evidently, the magnitude of polymer stress production in this region exceeds the polymer-induced Reynolds stress reduction close to the walls (see figure 5b); hence, significant DE is realized.
The specific role of roll cells on polymer-induced DE can be ascertained by close inspection of figure 6. A decomposition of 
$J^u_R$ into 
$J^u_{r}$ and 
$J^u_{t}$, i.e. the convective momentum fluxes due to roll cells and turbulent vortices (Bech & Andersson Reference Bech and Andersson1996), respectively, demonstrates that 
$J^u_{r}$ has an elastic response akin to that of 
$J^u_R$, i.e. a great decrease in the core region at 
$Wi<20$ followed by a significant increase at 
$Wi>20$ (see figure 6a). This underscores the fact that the roll cells have a dominant role in convective momentum exchange across the gap. Correspondingly, the related transverse momentum transport is weakened for 
$Wi<20$ while enhanced for 
$Wi>20$.
Figure 6. Reynolds stress decomposed into contributions of (a) roll cells (
$J^u_{r}=-\langle {u^r}{v^r}\rangle$) and (b) turbulent vortices (
$J^u_{t}=-\langle u''v''\rangle$), as 
$-{\langle u'v'\rangle }=-{\langle {u^r}{v^r}\rangle }-{\langle u''v''\rangle }$. Here, the variable of the roll cells is defined as 
$\phi ^r={\langle \phi \rangle }_{x,t}-\langle \phi \rangle$, and that of the turbulent vortices as 
$\phi ''=\phi -{\langle \phi \rangle }_{x,t}$, with 
${\langle \rangle }_{x,t}$ denoting averaging in the 
$x$-direction and in time (Bech & Andersson Reference Bech and Andersson1996; Gai et al. Reference Gai, Xia, Cai and Chen2016; Xia et al. Reference Xia, Shi, Wan, Sun, Cai and Chen2019).
The turbulent momentum flux 
$J^u_{t}$ shows intriguing variation versus 
$Wi$ (see figure 6b). As 
$Wi$ is enhanced, turbulent vortices exhibit a strikingly different response when compared to roll cells. Specifically, the near-wall Newtonian peaks of 
$J^u_{t}$ become invisible upon introduction of polymer additives. This behaviour of 
$J^u_{t}$ corresponds to the great suppression of near-wall QSV in viscoelastic RPCF (see figure 2). Interestingly, 
$J^u_{t}$ has a small polymer-induced enhancement in the core region at 
$Wi=5$ and exhibits a dome-shaped profile indicative of a very small decrease at 
$Wi=5\sim 20$. This trend underscores the fact that the momentum exchange of turbulent vortices is only slightly affected in the core region at 
$Wi\leq 20$. Interestingly, 
$J^u_{t}$ becomes almost equal to 
$J^u_{r}$ in the core region at 
$Wi=20$, highlighting the fact that turbulent vortices and roll cells make equivalent contributions to the momentum exchange. In contrast, at 
$Wi>20$, 
$J^u_{t}$ becomes very small as the flow transitions to a laminar state. This observation further supports the lack of turbulent vortices in the laminar viscoelastic RPCF at 
$Wi=40$ (see figure 7).
Figure 7. Turbulent vortices identified by 
$Q''$ and coloured by the 
$J^u_t$ value. Here, 
$Q''$ is calculated by the 
$Q$-criterion based on the instantaneous fluctuating velocity 
$(u'',v'',w'')$. The upper and lower plots are the full and front (
$x$-direction) views, respectively. 
$(a)\ {\rm Newtonian}, Q'' = \pm 0.5; (b)\ Wi = 5, Q'' = \pm 0.1; (c)\ Wi = 10, Q'' = \pm 0.1;$
$(d)\ Wi = 20, Q'' = \pm 0.1; (e)\ Wi = 30, Q'' = \pm 0.1; (\,f)\ Wi = 40, Q'' = \pm 0.1.$
The flow physics behind the elastic response of 
$J^u_{t}$ is further scrutinized by the polymer-induced changes of turbulent vortices depicted in figure 7. At 
$Wi<20$, elongated turbulent vortices are visible and active in the core region. As 
$Wi$ is increased above 20, these vortices slowly decay and finally disappear at 
$Wi=40$, giving way to a laminar viscoelastic RPCF. This further rationalizes the behaviours of 
$J^u_{t}$ in the core region, namely, momentum exchange facilitated by elongated turbulent vortices (see figure 6b). Furthermore, these polymer-induced changes of turbulent vortices are quite similar to that of polymer-induced transition toward MDR (Li et al. Reference Li, Sureshkumar and Khomami2006, Reference Li, Sureshkumar and Khomami2015; Kim et al. Reference Kim, Li, Sureshkumar, Balachandar and Adrian2007; White & Mungal Reference White and Mungal2008; Xi & Graham Reference Xi and Graham2010) via the relaminarization route (Choueiri et al. Reference Choueiri, Lopez and Hof2018; Lopez et al. Reference Lopez, Choueiri and Hof2019; Shekar et al. Reference Shekar, McMullen, Wang, McKeon and Graham2019; Chandra et al. Reference Chandra, Shankar and Das2020). This underscores the existence of a universal coupling dynamics between polymer chains and turbulent vortices in wall-bounded viscoelastic flows, irrespective of DR or DE. Clearly, the vortical changes illustrated in figures 2 and 7 and associated drag modifications succinctly answer the first question posed in the introduction.
The universal mechanism of the polymer–turbulence interaction can be quantified via the energy exchange terms 
$P^t_m/-P^t_{p}$ that represent energy exchange between turbulent motions and mean flow/polymer chains (Dallas, Vassilicos & Hewitt Reference Dallas, Vassilicos and Hewitt2010; Tsukahara et al. Reference Tsukahara, Ishigami, Yu and Kawaguchi2011; Thais, Gatski & Mompean Reference Thais, Gatski and Mompean2013; Teng et al. Reference Teng, Liu, Lu and Khomami2018). Specifically, two important features observed in the polymer-induced DR flows are demonstrated by 
$P^t_m$ and 
$-P^t_{p}$ (see figure 8). First, 
$P^t_m$ is positive across the gap, this corresponds to energy extraction from the main flow to the turbulent motions. Its peak value decreases gradually as 
$Wi$ increases, indicating an enhanced suppression of the energy extracting process of turbulent motions. Second, the vertical flow domain is divided into three typical regions, i.e. two elastic energy storing (
$-P^t_p<0$) regions that are located in the sublayer (
$y^+\lesssim 7$) and in the core (
$y^+\gtrsim 40$) and connected by an elastic energy releasing (
$-P^t_p>0$) region in the buffer layer (
$7\lesssim y^+\lesssim 40$). This is in accordance with the consensus established in the polymer-induced DR flows that the elastic energy stored into the stretched polymers in the sublayer and in the core is released to the turbulent motions in the buffer layer (Dallas et al. Reference Dallas, Vassilicos and Hewitt2010; Xi & Graham Reference Xi and Graham2010; Tsukahara et al. Reference Tsukahara, Ishigami, Yu and Kawaguchi2011; Thais et al. Reference Thais, Gatski and Mompean2013; Teng et al. Reference Teng, Liu, Lu and Khomami2018). The above findings provide a substantial evidence that a universal coupling dynamics exists between polymer chains and turbulent vortices irrespective of DR or DE.
Figure 8. Energy exchange between turbulent motions and mean flow (a) quantified as 
$P^t_{m}=-\langle {u''}{v''}\rangle (\textrm {d}\langle u\rangle /\textrm {d}y)$, and that between turbulent motions and polymer chains (b) quantified as 
$-P^t_{p}=-\langle \tau '_{ik}(\partial {u''_i} / \partial {x_k})\rangle$.
4. Concluding remarks
In summary, polymer-induced flow relaminarization has been reported for the first time for turbulent RPCF. Specifically, a reverse transition from the Newtonian turbulent RPCF to a fully relaminarized viscoelastic flow of DE occurs. The transition occurs gradually leading to elimination of small-scale vortices as 
$Wi$ is enhanced, paving the way for a 2-D laminar flow consisting of large-scale and highly organized roll cells. The underlying mechanism of polymer-induced DE is elucidated in terms of transverse momentum transport, namely, the polymer stress source term (
$J^u_p$) and the convective momentum exchange due to roll cells (
$J^u_r$). Specifically, in the core region the increase in DE at 
$Wi<20$ is ascribed to 
$J^u_p$ that increases at 
$Wi=5\sim 20$, and at 
$Wi>20$ to 
$J^u_r$. In the near-wall region, 
$J^u_p$ increases continuously and compensates the polymer-induced near-wall reduction of convective momentum exchange. Finally, a universal mechanism of coupling dynamics between polymer chains and turbulent vortices in DR or DE inertio-elastic flows is proposed and substantiated. In addition, we are aware of the limitations of the simulation with finite 
$Sc$; nevertheless, we feel that the results obtained from the present simulations will be helpful in a physical understanding of the mechanisms of polymer-induced relaminarization.
Acknowledgements
We are grateful to Dr H. Teng and Dr Z. Xia for many helpful discussions and valuable suggestions. This work was supported by the NSFC grant 91752110, 11621202, 11572312, Science Challenge Project (no. TZ2016001) and NSF grant CBET0755269.
Declaration of interests
The authors report no conflict of interest.
Appendix
The influence of 
$Sc$ and mesh size used on the reported results has been performed. To this end, test calculations were run for the RPCF of 
$Wi = 40$ which is of key importance as flow relaminarization is realized at this 
$Wi$. In our test calculations, the value of 
$Sc$ is varied from 0.19 and 0.38 for mesh sizes of 
$256\times 129\times 256$ and 
$512\times 129\times 512$, respectively.
Overall, flow relaminarization has been realized at 
$Wi=40$ for various 
$Sc$ and different mesh sizes (see figure 9a–d). This laminarized RPCF with highly organized 2-D roll cells corresponds to a flow sub-regime labelled ‘laminar Couette flow with straight 2-D roll cells’, i.e. COU2Dh/COU2D (Tsukahara et al. Reference Tsukahara, Tillmark and Alfredsson2010). Moreover, the momentum flux associated with the polymer stress and Reynolds stress of roll cells and that of turbulent vortices, i.e. 
$J^u_p$, 
$J^u_r$ and 
$J^u_t$ (see figure 10a,b), show no qualitative modifications with respect to variation in 
$Sc$. Note that 
$J^u_t$ becomes practically converged, while 
$J^u_p$ and 
$J^u_r$ demonstrate somewhat quantitative differences when 
$Sc$ is increased to 0.38 from 0.26. Nevertheless, figure 10(b) indicates that the quantitative influence of 
$Sc$ would vanish for larger-
$Sc$ calculations performed on larger mesh sizes. This finding is consistent with the previous findings (Sureshkumar & Beris Reference Sureshkumar and Beris1995; Sureshkumar et al. Reference Sureshkumar, Beris and Handler1997) that the solution convergence is expected when 
$Sc$ increases linearly with the mesh size. Hence, the present calculations can reliably reveal the underlying flow physics of the polymer-induced relaminarization.
Figure 9. Comparison of (a–d) instantaneous vortical structures identified by 
$Q$-criterion at different 
$Sc$ and mesh sizes. The test calculations are performed with mesh size 
$256\times 129\times 256$ (in the 
$x\times y\times z$ directions) for 
$Sc=0.15$ and 0.19, and 
$512\times 129\times 512$ for 
$Sc=0.26$ and 
$Sc=0.38$.
Figure 10. (a) The momentum flux associated with the polymer stress (
$J^u_p$) and the Reynolds stress of roll cells (
$J^u_r$) and that of turbulent vortices (
$J^u_t$) at different 
$Sc$. (b) The deviation of the maximum value of 
$J^u_i$ for different 
$Sc$ from that for 
$Sc=0.15$, where 
$i=r, p, t$ for 
$J^u_r, J^u_p, J^u_t$, respectively.
Based on the recent work of Gupta & Vincenzi (Reference Gupta and Vincenzi2019) and Lopez et al. (Reference Lopez, Choueiri and Hof2019), the high-
$Re$ results discussed in the present study will not be significantly affected by a further increase in 
$Sc$. Specifically, Gupta & Vincenzi (Reference Gupta and Vincenzi2019) claimed that for 
$Sc$ values that are not excessively small quantitative rather than qualitative changes in the numerical solutions of the high-
$Re$ or laminar flows are expected. Furthermore, Lopez et al. (Reference Lopez, Choueiri and Hof2019) have shown that polymer-induced relaminarization in their simulations of pipe flows using 
$Sc=0.5$ is realized and have verified that this relaminarization phenomenon is a robust feature with an onset 
$Wi$ threshold that is not altered by changing 
$Sc$. These previous findings taken together with our test calculations assure that the present calculations produce the essential flow features of viscoelastic RPCF.
