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,

(3.1)\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 $-P^t_{p}=-\langle \tau '_{ik}(\partial {u''_i} / \partial {x_k})\rangle$.