3. Results and discussion

Rotation-driven flow transition from a DR to a DE flow regime is realized by varying $Ro$ from $0$ to $0.9$ in the viscoelastic RPCF (see figure 2). A comparison of the viscoelastic RPCF friction Reynolds number ($Re_\tau$) with its Newtonian counterpart ($Re_{\tau 0}$) is used to quantify the flow drag modification due to polymer additives. Specifically, at $Ro<0.1$ polymer-induced DR is observed and this is attributed to suppression of QSVs and Reynolds stress production. At $Ro>0.1$ (see curves in figure 2), DE is observed. In this regime, the drag and vortical structures have similar rotation dependence in the Newtonian and viscoelastic RPCF. Specifically, increasing $Ro$ from 0.1 to 0.2 results in a gradual increase of $Re_{\tau }$ and $Re_{\tau 0}$, and development and enhancement of very well-defined large-scale roll cells (see insets $A$, $B$ and $F$, $G$ of figure 2 and figure 3a,b,e,f). Prior studies have demonstrated that the formation of highly organized roll cells results in amplified transverse momentum transport and a commensurate DE as compared to the non-rotating case (Salewski & Eckhardt Reference Salewski and Eckhardt2015; Gai et al. Reference Gai, Xia, Cai and Chen2016). Upon further increase of $Ro$ from 0.2 to 0.9 the large-scale roll cells progressively deteriorate and give way to small-scale vortices generated by the Coriolis forces (see insets $C$–$E$ and $H$–$J$ of figure 2 and figure 3d,d,g,h). This vortical change is accompanied by a continuous decrease of $Re_{\tau 0}$ and $Re_{\tau }$. Finally, both Newtonian and viscoelastic RPCF (see inset $K$ of figure 2) fully relaminarize at $Ro=1$ with two slightly different drag values because of their different viscosity, i.e. $Re_{\tau 0}=36.1$ and $Re_{\tau }=36.0$. Therefore, this rotation-driven transition of viscoelastic RPCF is accomplished by obvious suppression of Newtonian vortical structures caused by polymer additives. Since vortical circulations play an important role in momentum exchange between the near-wall and core regions, such polymer-induced suppression of vortical structures is expected to result in a commensurate decrease of the Reynolds stress, as discussed below. This phenomenon is commonly referred to as the polymer-induced DR effect (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).

Figure 2.

Frictional Reynolds numbers of the viscoelastic ($Re_{\tau }$) and Newtonian ($Re_{\tau 0}$) RPCF are determined based on the frictional velocity $u_\tau$ and the total wall shear stress $\tau _{w}$, for example, $Re_\tau =\rho u_\tau h/\eta _t$ with $u_\tau =\sqrt {\tau _{w}/\rho }$. Here, DR and DE denote $Re_{\tau }-Re_{\tau 0}<0$ and $>0$, respectively. Insets $A$–$E$ depict the averaged streamwise vorticity ${\langle \omega _x\rangle }_{x,t}$ of Newtonian RPCF in the ($y, z$) plane at $Ro=0.02, 0.2, 0.4, 0.7, 0.9$ and insets $F$–$J$ depict their viscoelastic counterparts. For $0.02\leqslant Ro\leqslant 0.4$, 10 contours are plotted within the range ($-0.3, 0.3$) and for the other $Ro$ the range is ($-0.05$, 0.05).

Figure 3.

Instantaneous vortical structures identified by $Q$-criterion with $Q=0.2$ and coloured by distance to the lower wall for typical cases. The upper and lower plots are the full and front ($x$-direction) views, respectively; (a–d)  represent Newtonian and (e–h) viscoelastic RPCF, respectively; (a,e)  $Ro=0.02$, (b,f)  $Ro=0.2$, (c,g)  $Ro=0.7$ and (d,h)  $Ro=0.9$.

The DM of viscoelastic RPCF at various $Ro$, in comparison with its Newtonian counterpart, is inherently linked to the aforementioned vortical changes, which result from two distinct effects, namely, polymer stretch and anticyclonic rotation. These changes and their underlying mechanism can be examined via the conserved momentum flux (Salewski & Eckhardt Reference Salewski and Eckhardt2015; Brauckmann et al. Reference Brauckmann, Salewski and Eckhardt2016; Zhu et al. Reference Zhu, Song, Liu, Lu and Khomami2020):

(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}

where $\langle \ \rangle ={\langle {\langle {\langle \ \rangle }_{x}\rangle }_{z}\rangle }_{t}$ denotes averaging in the $x$-direction (${\langle \ \rangle }_{x}$), $z$-direction (${\langle \ \rangle }_{z}$) and time (${\langle \ \rangle }_{t}$), and $\boldsymbol {u}'=\boldsymbol {u}-\langle \boldsymbol {u}\rangle$. Evidently, $J^u$ is evaluated by summing the three terms on the right-hand side of (3.1), namely, 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 $J^u$ is recast in wall units, i.e. $J^u={Re^2_\tau }/{Re^2}$, polymer-induced DM can be directly related to the changes of $J^u$ through variation of its components in the core flow region, i.e. $J^u_R$ and $J^u_p$. Note that, $J^u_R$ is calculated from the fluctuating velocity $\boldsymbol {u}'$ that results from both the large-scale roll cells and the small-scale turbulent vortices; thus it can be used to ascertain changes in transverse momentum flux by vortical motions.

The underlying physics associated with the DM discussed above is scrutinized by examining the variation of Reynolds stress ($J^u_R$) and polymer stress ($J^u_p$) at the gap centre. As expected for all $Ro$ (see figure 4a), $J^u_R$ exhibits a tremendous decrease as compared to its Newtonian counterpart $J^u_{R0}$. This is in concert with the expected polymer-induced suppression of turbulent vortices leading to weakened transverse momentum transport (see insets in figures 2 and 3). For the viscoelastic RPCF, $J^u_p$ is increased as $Ro$ varies from 0 to 0.5 and in turn diminishes as $Ro$ is increased further to 1. Therefore, the DM realized in viscoelastic RPCF results from the competition between the polymer-induced decrease of $J^u_R$ and the rotation-rendered increase of $J^u_p$. As indicated in figure 4(a), for $Ro<0.1$, the polymer-induced decrease of $J^u_R$ accounts for the DR of viscoelastic RPCF. This is due to the suppression of vortical structures by polymer additives. While, for $Ro>0.1$, the DE observed in the viscoelastic RPCF is related to the variation of $J^u_p$, which, as shown below, has two different generation mechanisms: one is associated with the large-scale roll cells and the other arises due to Coriolis-force-generated small-scale vortices.

Figure 4.

(a) The Reynolds stress term $J^u_R$ and polymer stress term $J^u_p$ obtained at the gap centre versus $Ro$. (b)  The turbulent kinetic energy ($E_t$) and elastic potential energy ($E_p$) calculated by $E_t=({1}/(2h)){{\int }_{-h}^h}{\langle u'^2_i\rangle }\,{\textrm {d} y}$ and $E_p=({1}/(2h)){{\int }_{-h}^h}{\langle (L^2-3)(1-\beta )\ln (f(\boldsymbol{\mathsf{C}}))\rangle }/(2ReWi)\,{\textrm {d} y}$, respectively; their Newtonian counterparts are denoted by $J^u_{R0}$ and $E_{t0}$. The inset in panel (b) depicts the one-dimensional spectra of $E_t$ in the EIT regime ($Ro=0.5, 0.7, 0.9$) at $y=0.95h$, i.e. near the wall.

Interestingly, $J^u_p$ becomes larger than $J^u_R$ at $Ro\gtrsim 0.3$. Commensurate with this change, the elastic potential energy ($E_p$) exceeds the turbulent kinetic energy ($E_t$) (see figure 3b). As $J^u_R$ and $E_t$ are both calculated based on $\boldsymbol {u}'$, they capture the influence of both large- and small-scale fluid motions. This observation further underscores the intrinsic competition between elastic and inertial forces in determining flow transitions and the commensurate DM in viscoelastic RPCF. Specifically, at $Ro\approx 0.3$ a transition from an inertia-dominated to an elasto-inertial flow regime is observed. In fact, the flow at $Ro=0.5 - 0.9$ is a novel elasto-inertial turbulent flow state, comprising streamwise-elongated small-scale vortices attached to the walls (see insets $I$ and $J$ of figure 2 and figure 3g,h). Evidently, in this flow state, the kinetic energy spectrum decays with a $-14/3$ slope (see inset of figure 4b) as for other EIT flow states such as planar DR viscoelastic WBT flows (Dubief et al. Reference Dubief, Terrapon and Soria2013; Samanta et al. Reference Samanta, Dubief, Holzner, Schäfer, Morozov, Wagner and Hof2013).

The polymer-induced suppression of $J^u_R$ and $E_t$ compared to their Newtonian counterparts (see figure 4a,b) further highlights the universality of polymer–turbulence interaction. This interaction is quantified by the energy exchange between turbulent motions and polymer chains, denoted here by $-P^t_{p}$ (Teng et al. Reference Teng, Liu, Lu and Khomami2018; Zhu et al. Reference Zhu, Song, Liu, Lu and Khomami2020). Specifically, $-P^t_{p}$ has similar profiles for all DR and DE viscoelastic RPCF, namely, two elastic-energy-storing ($-P^t_{p}<0$) regions located close to the wall and in the core, and a transition region between the two, i.e. the buffer layer where elastic energy is released back to the flow($-P^t_{p}>0$) (see figure 5a). Hence, irrespective of polymer-induced DM, polymer chains stretch near the wall, namely, they extract kinetic energy from turbulent motions and in turn this stored elastic energy is released back to the flow as the chains relax in the buffer layer (Zhu et al. Reference Zhu, Song, Liu, Lu and Khomami2020).

Figure 5.

(a) Energy exchange between turbulent motions and polymer chains denoted by $-P^t_{p}=-\langle {\tau ^{p}_{ik}}'(\partial {u'_i} / \partial {x_k})\rangle$. (b) The work done by the polymer-induced torque $W_{x,p}$. Here, ${\boldsymbol {\tau }^p}' = \boldsymbol {\tau }^p - \langle \boldsymbol {\tau }^p\rangle$. The results for the laminar flow state at $Ro=1$ are not included hereafter.

Kim et al. (Reference Kim, Li, Sureshkumar, Balachandar and Adrian2007) have demonstrated that polymer–turbulence interactions can also be examined by evaluating the work done by the polymer-induced torque, denoted by $W_{x,p}$ in the following budget equation for the streamwise enstrophy $\langle \omega ^2_x\rangle$:

(3.2)\begin{align} \frac{1}{2}\frac{{D\langle\omega^2_x\rangle}}{\partial t} & = \underbrace{\left\langle{\omega_x}\left(\frac{\partial f_z}{\partial y}-\frac{\partial f_y}{\partial z}\right)\right\rangle}_{W_{x,p}}+ \underbrace{\left\langle{\omega^2_x}\frac{\partial u}{\partial x}\right\rangle- \left\langle{\omega_x}\frac{\partial w}{\partial x}\frac{\partial u}{\partial y}\right\rangle+ \left\langle{\omega_x}\frac{\partial v}{\partial x}\frac{\partial u}{\partial z}\right\rangle}_{W_{x,c}} \nonumber\\ & \quad +\underbrace{\left\langle{\omega_x}Ro\frac{\partial u}{\partial z}\right\rangle}_{W_{x,C}} +\left\langle{\omega_x}\frac{\beta}{Re}\nabla^2{\omega_x}\right\rangle. \end{align}

Here $f_i=(1-\beta )(\tau _{ik}/ \partial {x_k})/Re$ denotes the polymer body force; and $W_{x,p}$ represents the correlation between streamwise vorticity $(\boldsymbol {\nabla }\times \boldsymbol {u})_x$ and the torque generated by the polymer body force $(\boldsymbol {\nabla }\times \boldsymbol {f})_x$; $W_{x,p}$ is negative for all $Ro$ (see figure 5b), which indicates that the polymer-induced torque acts in the opposite direction to vorticity and thus suppresses turbulent vortices and Reynolds stress production. Commensurate with above-mentioned vortical changes (see figures 2 and 3), $W_{x,p}$, is enhanced when $Ro\leqslant 0.2$ and it diminishes for $Ro>0.2$.

The origin of small-scale vortices in the EIT flow regime at $Ro=0.5 - 0.9$ depicted in figures 2 and 3 is also examined via the combined production term $W_{x,c}$ (see figure 6a) and the Coriolis production term $W_{x,C}$ (see figure 6b) defined in (3.2). In this representation, $W_{x,c}$ combines the contributions from stretching, tilting and twisting of vortical structures (Kim et al. Reference Kim, Li, Sureshkumar, Balachandar and Adrian2007), and $W_{x,C}$ represents vorticity generation by Coriolis forces that arise due to spanwise rotation. In the buffer layer ($y^+\approx 10$), $W_{x,c}$ has a positive value. This indicates a slight rotation dependence and serves as the main production of $\langle \omega ^2_x\rangle$, as it is much larger than $W_{x,C}$. For $Ro=0.3 - 0.7$, $W_{x,C}$ monotonically increases as $Ro$ is enhanced; hence, it becomes the dominant production term for $\langle \omega ^2_x\rangle$ in the buffer layer. In fact, this term is approximately six times larger than $W_{x,c}$, even at $Ro=0.9$, where it has been greatly diminished. Therefore, at $Ro\geqslant 0.3$, Coriolis forces deteriorate large-scale roll cells and give rise to the formation of small-scale turbulent vortices attached to the walls (see figures 2 and 3). This vortical feature highlights the mechanistic difference between the present rotation-driven EIT flow state and other EIT or elasticity-dominated turbulent flow states (EDT). Specifically, prior findings underscore that EDT in Taylor–Couette flows is signatured by small-scale elastic Görtler vortices (Song et al. Reference Song, Teng, Liu, Ding, Lu and Khomami2019) while EIT in channel and pipe flows is composed of two-dimensional sheets of highly stretched polymers (Dubief et al. Reference Dubief, Terrapon and Soria2013) and streamwise-elongated streaks (Choueiri et al. Reference Choueiri, Lopez and Hof2018; Lopez et al. Reference Lopez, Choueiri and Hof2019), respectively.

Figure 6.

(a) The combined production term $W_{x,c}$ and (b) the Coriolis production term $W_{x,C}$.

A close examination of the polymer stress term $J^u_p$ and its spatial relation to vortical structures, shown in figure 7, clearly points to two different DE mechanisms. Specifically, at $Ro=0 - 0.2$, a significant increase in the polymer stress is observed near the walls (see figure 7a). This is attributed to the formation of roll cells as $Ro$ is increased. The extensional flow generated between those adjacent roll cells produces significant polymer stress, which gives rise to large polymer body forces (see figure 7b,c). Specifically, the transition from polymer-induced DR to DE results from the gradual enhancement of roll cells at $Ro=0 - 0.2$ and has an onset at $Ro\approx 0.1$ where the enhanced roll cells become energetic enough to produce equivalent polymer stress that complements the decrease of Reynolds stress. For $Ro=0.2 - 0.5$, the deterioration of the large roll cells leads to significant reduction of polymer stress that arises from polymer chain interactions with the roll cells. Meanwhile, the appearance of Coriolis-force-generated small-scale vortices much like the small-scale elastic Görtler vortices in Taylor–Couette flows (Song et al. Reference Song, Teng, Liu, Ding, Lu and Khomami2019) gives rise to incoherent transport and homogenization of polymer stress via vortical circulations. That is, highly stretched polymer chains at the walls are transported to the bulk where intense turbulent mixing occurs, resulting in larger polymer stress. This new DE mechanism is more clearly seen at $Ro=0.5 - 0.9$ where large-scale roll cells are entirely absent (see figure 7d,e). Although the gradual suppression and final elimination of small-scale vortices in this regime leads to a decrease in $J^u_p$, the polymer body forces remain sufficiently large to cause DE. Hence, this intricate flow–microstructure coupling gives rise to a distinct DE mechanism at high rotation.

Figure 7.

(a) Ensemble-averaged polymer stress term $J^u_p$ and (b–e) contour plots of $J^u_p$ superimposed with the streamlines of the ensemble-averaged velocity vector ($\boldsymbol {v}, \boldsymbol {w}$), for (b)  $Ro=0.02$, (c)  $Ro=0.2$, (d)  $Ro=0.7$ and (e)  $Ro=0.9$.