3.1. Statistical properties

At first, we present some results for the flow statistics. Figure 1 illustrates friction coefficient $f$, global TKE, global TEE, global TKE production rate $P_{k}=\int _0^2\bar \tau _{R}^* ({\textrm {d} \bar u^{*}}/{{{\textrm {d} y}^ * }})\, {\textrm {d} y}^*$ by Reynolds shear stress (RSS), global TEE production rate $P_{e}=\int _0^2\bar \tau _{E2}^* ({\textrm {d} \bar u^{*}}/{{{\textrm {d} y}^*}})\,{\textrm {d} y}^*$ by NESS as well as the global energy transfer rate $G=\int _0^2\overline {\tau '^*_{ij}({\partial {u'^*_i}}/{\partial {x^*_j}})}\,{\textrm {d} y}^*$ between TEE and TKE. The datasets qualitatively reproduce the interesting observations in Choueiri's experiments, which can be read from the evolutions of $f$ with $Wi$ in figure 1(a). The specific analysis is as follows.

Figure 1.

Evolution of statistics with $Wi$: (a) friction coefficient $f$; (b) global TKE $e_k$ and TEE $e_e$; (c) global TKE production ${P_{{k}} = - \int _0^2 {\overline {{{u'}^{* } }{{v'}^{* } }} ({{\partial {{\bar u}^ * }}}/{{\partial {y^ * }}})\, \textrm {d} {y^ * }} }$ and TEE production rate ${P_{{e}} = - \int _0^2 {\bar \tau _{E2}^{{ \ast }}({{\partial {{\bar u}^ * }}}/{{\partial {y^ * }}})\, \textrm {d} {y^ * }} }$; (d) global energy transformation rate ${G = \int _0^2 {\overline {\tau '^{*}_{ij} ({{\partial u'^{*}_i}}/{{\partial x_j^*}})} \, \textrm {d} {y^ * }} }$. The MDR asymptote of Virk (Reference Virk1975) in (a) is $f=0.58 {Re}^{-0.58}$. Here $G^* >0$ represents the energy that is globally transferred from polymers to the flow structures, otherwise from the flow structures to the polymers.

At moderate $Re$ (e.g. 6000 or 20 000), $f$ continuously declines and finally converges to a certain value approaching the MDR asymptote of Virk (Reference Virk1975), $f=0.58{Re}^{-0.58}$. To be more specific, the viscoelastic turbulence experiences the onset of DR, LDR, HDR, MDR and EIT with the increase of $Wi$ according to the calculation results of $DR\,\%$ in table 1. In figure 1(b), global TKE gradually decreases before the flow enters the MDR stage, but begins to increase at a large $Wi$ ($Wi \approx 15$ for $Re=6000$ and $Wi \approx 10$ for $Re=20 000$); global TEE also turns to increase rapidly after the flow enters the MDR stage, compared with the slow increase before MDR. The above observations support the viewpoint that the underlying dynamics do not converge even after the flow enters the MDR stage reported by Zhu & Xi (Reference Zhu and Xi2021). We believe that the non-convergence results from the continuous evolution of EIT dynamics with $Wi$. Correspondingly, evolutions of ${P_{{k}}}$, ${P_{{e}} }$ and ${G }$ in figures 1(c) and 1(d) share similar trends for moderate $Re$. Since the onset of DR, ${P_{{k}}}$ starts to decrease, while ${P_{{e}}}$ starts to increase. When the flow enters the MDR and EIT regime, ${P_{{k}}}$ becomes negligible, whereas ${P_{{e}}}$ becomes dominant and saturated. During this process, $G$ first decreases to a valley and then starts to increase again with $Wi$. More strikingly, the value of $G$ is negative below the critical $Wi$ of MDR indicating the energy is globally transferred from TKE to TEE, but positive above the critical $Wi$ indicating the globally inverted energy transfer. Moreover, the evolution of $G$ after the valley point for the two $Re$ collapse to each other showing a power-law relationship with $Wi$ implying a scaling for $G$ independent of $Re$, which needs further investigations.

At a small $Re$ (e.g. 2500), the flow experiences the regimes of the onset of DR, LDR, HDR, relaminarization and EIT ($Wi =30, 60$) regime as shown in figure 1(a). The existence of relaminarization for moderate $Wi$ ($\approx 6\text{--} 15$ in the current work) which distinguishes the dynamics of DRT in two different stages. Global TKE and TEE show similar evolution behaviours to those at moderate $Re$, except that they all disappear in the relaminarization window. Evolutions of ${P_{{k}} }$, ${P_{{e}}}$ and ${G}$ in figures 1(c) and 1(d) show different behaviours for low $Re$ compared with that for moderate $Re$. Before relaminarization, ${P_{{k}}}$ is dominant and the value of $G$ is always negative, indicating the turbulence is always IT-dominated which is inhibited by polymers; when the flow enters laminar regime for $Wi \approx 6\text{--} 15$, ${P_{{k}} }$, ${P_{{e}}}$ and ${G}$ become eliminated. In contrast, during the process of entering EIT, ${P_{{e}}}$ is dominant and the value of $G$ is positive for $Wi =30, 60$ indicating the turbulence is sustained by polymers.

The above results demonstrate a global picture for the elastic effect on the viscoelastic turbulence: (1) at small $Wi$ (LDR for moderate $Re$ or before relaminarization for small $Re$), the flow is IT-dominated and in these cases the traditional TKE production by RSS is suppressed and thus IT-related SSP is modulated; (2) increasing $Wi$ above a critical value, a new TKE generation term induced by elastic effects appears, which grows with $Wi$ and changes the SSP in DRT; (3) further increasing $Wi$, the turbulence is changed from being sustained by $P_{k}$ to being sustained by $P_{e}$, and the nature of turbulence changes from polymer-modulated IT to EIT.

The characteristics of RSS attract the most attention in the comparison analysis between IT and DRT. What is unusual is that although there exist obvious velocity fluctuations (especially $u'^*$) in EIT, $-\overline {{{u'}^{* }}{{v'}^{*}}}$ is extremely small, implying somehow the decoupling between $u'^*$ and $v'^*$. From our recent demonstrated SSP in EIT (Zhang et al. Reference Zhang, Zhang, Li, Yu and Li2021b), the energy picture including the TKE production in different directions and the embedded TKE transfer amongst them clearly differs from that of IT, which is possible to delay or decouple the generation of $u'^*$ from $v'^*$. Figure 2 presents the negative cross-correlation between $u'^*$ and $v'^*$ over $x$, $z$ and $t$, $C_{uv}(y^*, \xi ^*)=-{\langle {u'^*( {x^*,y^*,z^*,t^*} )v'^*( {x^*+\xi ^*,y^*,z^*,t^*} )} \rangle }$. Strikingly, unlike that in IT, there exists an obvious phase shift between $u'^*$ and $v'^*$ in the HDR and EIT state: the maximum values of $C_{uv}(y^*, \xi ^*)$ all locate at $\xi ^*=0$ in IT but at some locations where $\xi ^*$ is less than zero, approximately $2h$ to $4h$ in figure 2(a). This means that the local cross-correlation between $u'^*$ and $v'^*$ is the strongest in IT, but $u'^*$ and upstream $v'^*$ is most relevant in EIT. Applying Taylor's frozen hypothesis, it further indicates $u'^*$ lags behind $v'^*$, somehow indicating that $v'^*$ induces the generation of $u'^*$. Moreover, in EIT, the lagging distance $\xi _{{max}}^*$ can reach several times the channel half-height $h$. For example, at $Re = 1000, Wi = 60, \xi _{{max}}^*$ can reach more than $5h$ in figure 2(b). In addition, $\xi _{{max}}^*$ gradually increases with an approximately linear relationship with $Wi$. The above results once again imply different pictures of the TKE transfer process between velocity components in IT and DRT.

Figure 2.

Distributions of $C_{uv}(y^*,\xi ^*)$: (a) in the $\xi$–$y$ plane; (b) at $y^* \approx 0.5$ and normalized by RSS here. Inset is $\xi _{{max}}^*$ versus $Wi$ at different $Re$. Here $\xi _{{max}}^*$ corresponds to the position where $C_{uv}(\xi ^*)$ reaches the maximum.

In the following, in-depth investigations on the link between EIT and DRT are carried out through analysis on structures, budgets and pressure fluctuations. Note that structures, budgets and pressure fluctuations are not shown at $Re=20\,000$ for brevity because similar conclusions as those obtained at $Re = 6000$ can be drawn.

3.2. Structural analysis

Figures 3–6 show the structure characteristics (sheet-like structures, spanwise vortexes and the interaction between polymers and turbulence) in the $x$–$y$ plane of EIT and their performance in DRT, including instantaneous contours of streamwise and wall-normal polymer extension ($R_x=\sqrt {c_{xx}}$ and $R_y=\sqrt {c_{yy}}$, respectively), superimposed by the isolines of $G_{xx}$, $G_{yy}$ and $Q$, where $Q=(-1/2){\partial {_iu^*_j}}{\partial {_ju^*_i}}$ is the second invariant of the velocity gradient tensor. Note that these instantaneous results are extracted at high flow drag moment (namely, the active state in Xi & Graham (Reference Xi and Graham2010) and Dubief et al. (Reference Dubief, Terrapon and Soria2013)). The abscissa and ordinate of the subgraphs are $x^*$ and $y^*$, respectively, and the planes are properly compressed in the streamwise direction. Here $y^*$ uniformly ranges from 0 to 2, and $x^*$ ranges from 0 to $L_x$ (see table 1 for different channel lengths).

Figure 3.

Instantaneous contours of streamwise polymer extension ($R_x=\sqrt {c_{xx}}$) in the $x$–$y$ plane at active state. The superimposed isolines represent $Q$ in different subgraphs: green, negative; red, positive. The abscissa and ordinate of the subgraphs are $x^*$ and $y^*$ (from 0 to 2), respectively: (a) $x^*\in (0, 5{\rm \pi} )$; (b) $x^*\in (0, 7{\rm \pi} )$; (c) $x^*\in (0, 11{\rm \pi} )$; (d) $x^*\in (0, 16{\rm \pi} )$.

It can be observed from the instantaneous contours of $R_x$ and $R_y$ at moderate $Re$ in these figures that (1) polymers are increasingly stretched with the increase of $Wi$ in both the streamwise and wall-normal directions; (2) sheet-like structures of streamwise polymer extension featured by EIT are activated at $Wi=6$, and further expanded to channel centre at $Wi=15$; (3) wall-normal polymer extension structures are also inclined from channel wall to channel centre along the streamwise direction but are not sheet-like, and these high extension structures shift from the channel wall at the LDR condition to the channel centre at the HDR and MDR conditions; (4) polymer extension structures between the streamwise and wall-normal directions are interrelated, and the relevance strengthens with the increase of $Wi$. At low $Re$, although streamwise polymer extension structures also incline from channel wall to channel centre at $Wi=5$ before laminarization, which are weakly correlated to wall-normal structures, they are less like thin sheets, which is more or less related to the plane compression in streamwise direction. It is notable that the sheet-like structures of polymer extension (especially in the streamwise direction) under the numerical conditions presented in this study ($Wi \le 15$) seem like they are originating from the near-wall region rather than the channel centre. This supports that the wall-mode instability (TS attractor) evolves to EIT, as reported by Shekar et al. (Reference Shekar, Mcmullen, Wang, Mckeon and Graham2019, Reference Shekar, Mcmullen, Mckeon and Graham2020, Reference Shekar, Mcmullen, Mckeon and Graham2021) in the present parameter space. Moreover, compared with that of streamwise polymer extension, the structures of wall-normal extension appear earlier near the channel centre (e.g. at $Wi = 15$), indicating the vortex structures are formed locally as shown in figure 5.

Two-dimensional projection of $Q$ in the $x$–$y$ plane shows the spanwise vortex structures of DRT in figure 3 (or figure 5). At moderate $Re$, the LDR flow is filled with abundant small-scale spanwise vortex structures near the wall; these positive and negative vortexes are not related to polymer extension structures as in EIT; nearly circular and ellipse red isolines inclined from channel wall to channel centre are likely to be the two-dimensional projection of the three-dimensional hairpin vortex heads and legs in IT (see figure 3a). For low $Re$, these spanwise vortex structures are also weakly related to the polymer extension structures before laminarization at $Wi = 5$, but the scale of the vortex becomes larger (see figure 3d). After the flow enters the HDR state, the situation changes significantly (see figure 3b for the HDR state and figure 3c for the MDR state): it can be clearly found that the vortex structures are gradually lifted from the near wall to the centre region with the increase of $Wi$; at the near-wall region, small-scale spanwise vortex structures alternate sign around thin sheets of large polymer extension, indicating that the dynamics of EIT appear long before MDR; at the turbulent core region, large-scale spanwise vortex structures are found to be associated with the wall-normal extension structures, meaning that wall-normal polymer extension structures and the interaction with turbulent structures play important roles in EIT-related SSP.

Here, $G_{xx}$ and $G_{yy}$ represent the energy conversion from TKE to TEE in the streamwise and wall-normal directions due to the interaction between polymer extension structures and turbulent structures, respectively. At $Re=6000$ and $Wi=2$, large energy conversion is mainly located at the near-wall region and plying-up with large polymer extension as shown in figures 4(a) and 6(a). Moreover, an obvious feature lies in the following: when large polymer extension structures begin to appear in the high shear region near the wall, related energy conversion is positive and polymers absorb energy from turbulent fluctuations leading to polymers’ coil-stretch transition; after extension structures are lifted to the turbulent core area, $G_{xx}$ turns to negative, that is, polymers relax and release energy to turbulence, which is consistent with the findings of Min, Choi & Yoo (Reference Min, Choi and Yoo2003). These phenomena can be also observed in figures 4(d) and 6(d), excepting that the structures of large $G_{yy}$ are lifted towards the channel centre likely due to broadened transition regions. After the flow enters the HDR state, the distribution pattern of $G_{xx}$ and $G_{yy}$ changes significantly in panels (b) and (c) of figures 4 and 6: $G_{xx}$ alternates its sign around sheet-like structures of high polymer extension, which is closely related to small-scale spanwise vortexes of $Q$ at the near-wall region; structures of large $G_{yy}$ gradually shift from the near-wall region to the turbulent core region and show strong correlation with large-scale spanwise vortexes of $Q$ there.

Figure 4.

Instantaneous contours of streamwise polymer extension ($R_x=\sqrt {c_{xx}}$) in the $x$–$y$ plane at active state. The superimposed isolines represent $G_{xx}$ in different subgraphs: green, negative; red, positive. The abscissa and ordinate of the subgraphs are $x^*$ and $y^*$ (from 0 to 2), respectively: (a) $x^*\in (0, 5{\rm \pi} )$; (b) $x^*\in (0, 7{\rm \pi} )$; (c) $x^*\in (0, 11{\rm \pi} )$; (d) $x^*\in (0, 16{\rm \pi} )$.

Figure 5.

Instantaneous contours of wall-normal polymer extension in the $x$–$y$ plane at active state. The superimposed isolines represent $Q$ in different subgraphs: green, negative; red, positive. The abscissa and ordinate of the subgraphs are $x^*$ and $y^*$ (from 0 to 2), respectively: (a) $x^*\in (0, 5{\rm \pi} )$; (b) $x^*\in (0, 7{\rm \pi} )$; (c) $x^*\in (0, 11{\rm \pi} )$; (d) $x^*\in (0, 16{\rm \pi} )$.

Figure 6.

Instantaneous contours of wall-normal polymer extension in the $x$–$y$ plane at active state. The superimposed isolines represent $G_{yy}$ in different subgraphs: green, negative; red, positive. The abscissa and ordinate of the subgraphs are $x^*$ and $y^*$ (from 0 to 2), respectively: (a) $x^*\in (0, 5{\rm \pi} )$; (b) $x^*\in (0, 7{\rm \pi} )$; (c) $x^*\in (0, 11{\rm \pi} )$; (d) $x^*\in (0, 16{\rm \pi} )$.

The above observations indicate that characteristic structures of EIT appear long before MDR, and that wall-normal polymer behaviours are also important in EIT dynamics. In order to further quantitatively reveal the correlation of the above important turbulent and polymer structures in DRT-related SSP, budget analysis is carried out in the next section.

3.3. Budget analysis

This section re-evaluates the energy transfer process in DRT by budget analysis of TKE, TEE and shear stresses between different components at moderate $Re$ ($Re = 6000$) and low $Re$ ($Re = 2500$), as illustrated in figures 7 and 8 considering whether the EIT-related SSP is formed. As is well known, in IT, RSS absorbs energy from the mean motion and produces streamwise TKE, which is transformed into wall-normal TKE through fast and slow pressure redistribution, and then regenerates RSS. The above process forms a cycle to maintain the turbulence against dissipative effects. As demonstrated in Zhang et al. (Reference Zhang, Zhang, Li, Yu and Li2021b), when the flow enters EIT, it follows a very different SSP. The most striking differences lie in: (1) rather than by RSS, NESS absorbs energy from the mean motion to sustain the turbulence; (2) the conversion of wall-normal TKE into TEE and streamwise TEE into TKE are added; (3) the interaction between polymers and turbulence suppresses both RSS and NESS. In addition, polymers also contribute to the energy redistribution in EIT which generates wall-normal TKE in the SSP cycle of EIT (Terrapon et al. Reference Terrapon, Dubief and Soria2015). Moreover, EIT-related SSP is possibly formed and survives from being dissipated once the above features appear. In the following, special attention is paid to the performance of terms related with these features.

Figure 7.

Budgets analysis for $\overline {{{u'}^{* }}{{u'}^{*}}}$, $\overline {{{v'}^{* }}{{v'}^{*}}}$, $\overline {{{u'}^{* }}{{v'}^{*}}}$, $\overline {\tau _{xx}^*}$, $\overline {\tau _{yy}^*}$ and $\overline {\tau _{xy}^*}$ at $Re=6000$, where ${P_{ij}^{{R}}}$ is Reynolds production rate; ${P_{ij}^{{E}}}$ is elastic production rate; ${\varepsilon _{ij}}$ is dissipation rate; ${G_{ij}}$ is elastic stress–strain term. Especially, ${P_{xx}^{{E2}}}=2\tau _{E2}\partial \bar u^*/\partial y^*$ is the elastic production by NESS.

Common with earlier studies (Xi Reference Xi2019), production and dissipation terms of Reynolds stresses such as $P_{xx}^{{R}}$, $P_{xy}^{{R}}$ and $\epsilon _{xx}^{{R}}$, $\epsilon _{yy}^{{R}}$, $\epsilon _{xy}^{{R}}$ in panels (a), (c) and (e) of figures 7 and 8 gradually decrease with the increase of $Wi$, indicating the suppression of elasticity on the IT-related SSP in DRT. Moreover, it is observed from figures 7 and 8 that (1) at the moderate $Re$, $P_{xx}^{{E2}}$ continuously grows with $Wi$ and exceeds $P_{xx}^{{R}}$ almost everywhere at $Wi \approx 9$ if compared together, indicating that for cases at $Wi > 9$, more energy is transferred from the mean motion to polymers by NESS than that to turbulent structures and the turbulence is dominated by NESS; (2) at the small $Re$, the flow is dominated by $P_{xx}^{{R}}$ before the relaminarization while by $P_{xx}^{{E2}}$ when EIT is excited.

Figure 8.

Budgets analysis for $\overline {{{u'}^{* }}{{u'}^{*}}}$, $\overline {{{v'}^{* }}{{v'}^{*}}}$, $\overline {{{u'}^{* }}{{v'}^{*}}}$, $\overline {\tau _{xx}^*}$, $\overline {\tau _{yy}^*}$ and $\overline {\tau _{xy}^*}$ at $Re=2500$, where ${P_{ij}^{{R}}}$ is Reynolds production rate; ${P_{ij}^{{E}}}$ is elastic production rate; ${\varepsilon _{ij}}$ is dissipation rate; ${G_{ij}}$ is elastic stress–strain term. Especially, ${P_{xx}^{{E2}}}=2\tau _{E2}\partial \bar u^*/\partial y^*$ is the elastic production by NESS.

Interesting to notice from profiles of $G_{xx}$ in figures 7(b) and 8(b) is that it experiences a morphological change with the increase of $Wi$. For a small $Wi$ (e.g. at $Re=6000, Wi = 2$ and $Re=2500, Wi = 5$), energy is transferred from streamwise TKE to TEE in the bulk region, which increases with $Wi$ and introduces additional energy loss. This effect modulates the IT-related SSP. For moderate and large $Wi$ (e.g. at $Re=6000, Wi > 4$ and $Re=2500, Wi =30, 60$ after the emergence of the EIT regime), the energy transfer direction turns over with the increase of $Wi$. The negative energy transfer region (from TEE to TKE) shifts from the near-wall region to the bulk region, and enlarges with $Wi$. Meanwhile, the positive energy transfer region (from TKE to TEE) shifts from the bulk region to the near-wall region, and shrinks with $Wi$. Notable is that the region where polymers absorb energy shifts upward with $Wi$, which probably originates from turbulent streaks lifting, as reported in Zhang et al. (Reference Zhang, Zhang, Li, Yu and Li2021b). The core area is mainly characterized by the energy transfer of TEE to TKE, which indicates that EIT dynamics can be self-sustained, i.e. EIT-related SSP involved. With the increase of $Wi$, more TEE is transferred from polymers to TKE until the flow completely develops into EIT. Compared with $G_{xx}$, the energy transfer rate $G_{yy}$ between $\overline {{{v'}^{* }}{{v'}^{*}}}$ and $\bar \tau _{yy}^*$ is negative for all the covered viscoelastic flow regimes (see figures 7d and 8d), indicating that polymers always suppress the wall-normal turbulence. With the increase of $Wi$, as $\overline {{{v'}^{* }}{{v'}^{*}}}$ is suppressed by polymers, less energy is transferred from TKE to TEE.

The evolutions of $P^{R}_{xy}$ are illustrated in figures 7(e) and 8(e). Originating from the production of wall-normal TKE and the mean shear, this term dominates the gain effect of RSS. Similarly, it decreases with $Wi$ due to suppression of wall-normal TKE in DRT. Interestingly, the term $G_{xy}$ is always negative throughout all DR stages as shown in figures 7( f) and 8( f), implying that the interaction between polymers and turbulence shows a dual suppressing effect on the RSS and also NESS. At last, the budget of NESS is analysed by focusing on the term of $P^{E}_{xy}$ as shown in figures 7(h) and 8(h), induced by the production of wall-normal extension and the mean shear similar to the RSS production term $P^{R}_{xy}$. Unlike $P^{R}_{xy}$, $P^{E}_{xy}$ first increases with $Wi$ and then dampens in DRT. This can be attributed to the behaviour of $\tau _{yy}$, induced by the interaction between polymers and turbulence, which on the one hand is enlarged by the increase of elasticity, and on the other is reduced due to the suppression of the turbulence.

3.4. Pressure and its redistribution effect

Pressure fluctuations in wall turbulence play an important role in the redistribution of TKE in different directions. As is shown in equation (2.5), the rapid and slow pressure fluctuations are induced by the deformation of the mean flow field and turbulent interaction, respectively. As Terrapon et al. (Reference Terrapon, Dubief and Soria2015) reported, the rapid contribution represents the immediate response to a change imposed on the mean flow field, while the slow part feels this change through nonlinear interactions. In addition, the polymeric pressure fluctuations are due to the polymeric stress fluctuations. Therefore, in Ptasinski et al. (2003) and Terrapon et al. (Reference Terrapon, Dubief and Soria2015), the first two parts (rapid and slow contributions) and the third part are intuitively identified to be inertial and elastic, respectively. This is necessary to be reconfirmed here since it is related to determining which dynamics dominates the pressure redistribution effect, IT or EIT. To this end, the root mean square (r.m.s.) values of total, fast, slow and polymeric pressure at the same moderate $Re (= 6000)$ with Terrapon et al. (Reference Terrapon, Dubief and Soria2015) as well as low $Re (= 2500)$ across the bottom channel height are shown in figure 9.

Figure 9.

Pressure r.m.s. as a function of $y^*$: (a–d) total, rapid, slow and polymeric contribution (see panel (a) for legend) at $Re=6000$; (e–h) total, rapid, slow and polymeric contribution (see panel (e) for legend) at $Re=2500$.

For moderate $Re$, compared with Newtonian IT, when $Wi$ increases to a certain value (e.g. $Wi = 9$), $p_{rms}$ of DRT turns into quasi-flat distribution (see figure 9a), which well reproduces the observation in the simulation by Terrapon et al. (Reference Terrapon, Dubief and Soria2015). It can be also found that $p_{rms}$ generally shows a downward trend with the increase of $Wi$ at first, and then begins to rise when the flow enters the MDR state (e.g. $Wi = 9$), which greatly implies the transformation of dominant dynamics in DRT. Comparing the distribution patterns of fast, slow and polymeric contributions, it can be seen that the fast contribution shows a similar development law with the total contribution, indicating that the pattern change of $p_{rms}$ is mainly affected by the fast contribution. In Newtonian IT abundant turbulent fluctuations and coherent structures locate in the near-wall region (especially in the buffer layer), which can well explain the single hump distribution pattern of $p_{rms}$ with the near-wall peak. In contrast, EIT is excited from the centre-mode instability and is affected by both the wall-mode and the centre mode at moderate and high $Re$ (see Choueiri et al. Reference Choueiri, Lopez, Varshney, Sankar and Hof2021), which can qualitatively explain the reason for the quasi-flat distribution pattern of $p_{rms}$. The fast contribution represents the interaction between turbulent fluctuations and mean motion, while the slow contribution reflects the turbulence–turbulence interaction. From figures 9(b) and 9(c), it appears that (1) both distributions of fast contribution and slow contribution become more and more flat with the enhancement of the DR effect; (2) they all completely become the quasi-flat distribution featured by EIT after the flow enters the MDR state. The above phenomena show that the inertial nature of fast and slow pressure fluctuations are likely to be gradually replaced by an elastic nature with the increase of $Wi$, because elastic nonlinearity gradually replaces inertial nonlinearity to generate turbulent fluctuations during this process. For $p_{rms}^{P}$ in figure 9(d), the distribution also shows a quasi-flat pattern after the flow enters MDR state (e.g. $Wi = 9$). Interestingly, at the LDR state (e.g. $Wi = 1$, 2), it shows a single hump distribution with near-wall peak similar to fast and slow contributions. The evolution behaviour shows that with the increase of $Wi$, polymeric pressure fluctuations also experience the transformation from inertial nature to elastic nature, considering that polymers can be significantly stretched by interacting with coherent structures in IT near the wall.

For low $Re$, compared with the cases at moderate $Re$, the peak positions of $p_{rms}$, $p_{rms}^R$ and $p_{rms}^S$ of IT move away from the wall; the r.m.s. value decreases at the channel wall and rises at the channel centre. With the increase of $Wi$, the total pressure fluctuations also gradually decrease and become flatter and flatter, and finally show a quasi-flat distribution at $Wi = 5$ before laminarization, similar to that in EIT. We speculate that the elasto-inertial instability may be triggered intermittently at $Wi = 5$, but cannot sustain itself. Similar to the cases at moderate $Re$, slow pressure fluctuations are gradually suppressed. When $Wi \ge 4$, the r.m.s. value of rapid pressure fluctuations begins to increase abnormally, which may be a unique phenomenon of slugs in the flows, because they are higher than the r.m.s. value in EIT at $Wi = 30$; moreover, the r.m.s. value of fast pressure fluctuations is even higher than that of total pressure fluctuations, indicating that part of the fast pressure fluctuations offsets part of the slow ones. Before entering the laminar state, polymeric pressure fluctuations are very weak and r.m.s. values are almost zero. For EIT at $Wi = 30$, the r.m.s. values of total, fast, slow and polymeric pressure fluctuations all show a quasi-flat pattern and lower than those at moderate $Re$. Prominently, $p_{rms}^R$ at $Wi=30$ is much higher than that in the flow before the window of relaminarization.

In short, through the above analysis of the distribution pattern of the r.m.s. value of pressure, we speculate that the fast, slow and polymeric pressure fluctuations can behave with both inertial and elastic nature, which depends on the dominating nonlinear dynamics in DRT. In the following, we focus on the pressure redistribution term related with fast, slow and polymeric effects in the budgets to understand the role of pressure redistribution in the link between DRT and EIT.

Different from $\overline {{{u'}^{* }}{{u'}^{*}}}$, the generation of $\overline {{{v'}^{* }}{{v'}^{*}}}$ is from pressure redistribution. Figures 10 and 11 show the evolutions of total, rapid, slow and polymeric pressure redistribution terms under different flow conditions. In IT both $\varPhi _{xx}^{{R}}$ and $\varPhi _{xx}^{{S}}$ behave as negative in the turbulent core area, i.e. TKE is redistributed to other directions from streamwise direction through slow and fast pressure fluctuations; the behaviour of $\varPhi _{yy}^{{S}}$ is opposite to that of $\varPhi _{xx}^{{S}}$ but this is not the case for $\varPhi _{yy}^{{R}}$: near the wall (at approximately $y^* < 0.1$), $\varPhi _{yy}$ is negative, indicating the energy is redistributed to the streamwise direction forming turbulent streaks and in turbulent core region (at approximately $y^* > 0.1$), $\varPhi _{yy}$ is positive, indicating the energy is redistributed to the wall-normal direction and forms $\overline {{{v'}^{* }}{{v'}^{*}}}$.

Figure 10.

Profiles of total pressure distribution at $Re=6000$ on $\overline {{{u'}^{* }}{{u'}^{*}}}$, $\overline {{{v'}^{* }}{{v'}^{*}}}$ and the divided rapid, slow and elastic contributions (see panel (h) for legend), where ${\varPhi _{xx} =2{\overline {{{p'}^{* } }({{{\partial u'^{* } }}/{{\partial x^{* } }}} )} }}$; ${\varPhi _{yy} =2{\overline {{{p'}^{* } }( {{{\partial v'^{* } }}/{{\partial y^{* } }}} )} }}$; ${{{p'}^{* } }( \boldsymbol {x} ) = {p'}_R^{* } ( \boldsymbol {x} ) +{p'}_S^{* } ( \boldsymbol {x} ) +{p'}_P^{* } ( \boldsymbol {x} )}$.

In DRT at moderate $Re$ in figure 10, for small $Wi$ when the turbulence is IT-dominated, both $\varPhi ^{{R}}$ and $\varPhi ^{{S}}$ are far larger than $\varPhi ^{{P}}$ in both the streamwise and wall-normal directions, and $\varPhi ^{{P}}_{yy}$ does not generate but consumes $\overline {{{v'}^{* }}{{v'}^{*}}}$. With the increase of $Wi$, $\varPhi ^{{P}}$ of an active elastic nature is gradually excited and generates $\overline {{{v'}^{* }}{{v'}^{*}}}$ on one hand, and $\varPhi ^{{R}}$ and $\varPhi ^{{S}}$ are dampened on the other hand. When $Wi$ is large enough, $\varPhi ^{{P}}$ catches up with those two terms, and can affect the total pressure redistribution. Strikingly, the evolutions of $\varPhi ^{{P}}$ demonstrate a similar trend with that of $G_{xx}$ and $P^{E2}_{xx}$ with the increase of $Wi$. For cases of small $Wi$, e.g. $Wi = 2$ at $Re = 6000$, $\varPhi _{yy}^{{P}}$ is small and negative, i.e. $\overline {{{v'}^{* }}{{v'}^{*}}}$ is transferred to other directions through $\varPhi _{yy}^{{P}}$. However, for large $Wi$, e.g. $Wi \ge 4$ at $Re = 6000$, there is an obvious positive polymer effect near the wall, i.e. TKE in other directions is transferred to the wall-normal direction through $\varPhi _{yy}^{{P}}$. Polymers play a gain effect on the normal-wise TKE generation which sustains $\overline {{{v'}^{* }}{{v'}^{*}}}$ and turbulent SSP. Therefore, polymers introduce a new pressure redistribution mechanism in DRT. It plays a modulation effect on pressure redistribution when $Wi$ is small, but can reshape the pressure redistribution effect when $Wi$ is large enough, making the correlation between ${u'}^{*}$ and ${v'}^{*}$ in DRT more complex. When entering EIT, the characteristics of the pressure redistribution effect are obviously different from that in IT, and the polymeric contribution plays an indispensable role in generating ${v'}^{*}$. Although it is speculated in the previous section that the slow and fast contributions are likely to be driven by elastic nonlinearity rather than inertial nonlinearity at high $Wi$, in any case, the morphological transformation of pressure redistribution by polymers is the most notable feature for EIT-related SSP.

In DRT at low $Re$ in figure 11, there exist some distinctions from the cases at moderate $Re$. Before the relaminarization phenomenon occurs (e.g. $Wi \leq 5$), both $\varPhi ^{{R}}$ and $\varPhi ^{{S}}$ in the $x$ and $y$ directions are also far larger than $\varPhi ^{{P}}$, and polymeric pressure fluctuations redistribute wall-normal TKE to other directions. But the morphological transformation of $\varPhi ^{{P}}$ does not occur until $Wi=5$. With the increase of $Wi$, $\varPhi ^{{R}}$ and $\varPhi ^{{S}}$ are dampened, but always much higher than $\varPhi ^{{P}}$, which is also broadly suppressed with the decline of IT in the wall-normal direction. At $Wi = 5$, the polymeric pressure redistribution effect still does not catch up with slow and rapid ones, and consequently does not show morphological change. In other words, the pressure redistribution effect featured by EIT is not excited before relaminarization, and polymeric pressure fluctuations at this stage are a passive response to IT dynamics, similar to these under LDR conditions at moderate $Re$ (e.g. $Wi=1,2$ for $Re=6000$). When an instability appears and the flow enters EIT from the laminar regime, pressure fluctuations induced by polymers begin to generate wall-normal velocity fluctuations and play an indispensable role in energy SSP (see figures 11g and 11h). Moreover, $\varPhi ^{{R}}_{yy}$ in figure 11(d) extraordinarily shows morphological inversion: it is positive at near-wall region (at approximately $y^*<0.12$) and negative at part of turbulent core region (at approximately $0.12< y^*<0.55$) in Newtonian IT, but show the opposite effects in roughly the same regions in EIT. Again, this transformation from consumption to generation of $\overline {{{v'}^{* }}{{v'}^{*}}}$ implies rapid pressure fluctuations experience the change from inertial nature to elastic nature.

Figure 11.

Profiles of total pressure distribution at $Re=2500$ on $\overline {{{u'}^{* }}{{u'}^{*}}}$, $\overline {{{v'}^{* }}{{v'}^{*}}}$ and the divided rapid, slow and elastic contributions (see panel (h) for legend), where ${\varPhi _{xx} =2{\overline {{{p'}^{* } }( {{{\partial u'^{* } }}/{{\partial x^{* } }}} )} }}$; ${\varPhi _{yy} =2{\overline {{{p'}^{* } }( {{{\partial v'^{* } }}/{{\partial y^{* } }}} )} }}$; ${{{p'}^{* } }( \boldsymbol {x} ) = {p'}_R^{* } ( \boldsymbol {x} ) +{p'}_S^{* } ( \boldsymbol {x} ) +{p'}_P^{* } ( \boldsymbol {x} )}$.

From the above budget analysis, there exists a distinct SSP from IT in viscoelastic DRT. On the one hand, the existence of elastic effect suppresses the self-sustaining mechanism related to IT. On the other hand, when $Wi$ is large enough, a cycle of EIT-related SSP is introduced, which finally replaces that of IT.

3.5. Repicturing viscoelastic DRT

Based on the above analysis, we repicture the energy process for viscoelastic DRT from IT to EIT by combing energy and stress budgets as shown in Figure 12. First, the cycles of energy SSP for IT and EIT are illustrated by the cyan paths in figure 12(a) and the blue paths attached by cyan dashed paths in figure 12(d), respectively. Classical understanding about energy SSP of IT is that RSS absorbs energy from the mean motion and converts it into streamwise TKE, part of which is further redistributed to wall-normal TKE through pressure fluctuations, and then regenerates RSS. In this study, the energy SSP of EIT is further updated relative to our previous work (see Zhang et al. Reference Zhang, Shao, Li, Ma, Zhang and Li2021a). Specifically, a similar energy SSP of IT is added with cyan dashed paths, considering these paths still exist weakly in the so-called EIT regime. Here, we speculate these paths are a passive response to the SSP of EIT, similar to the passive response of polymers to the energy SSP of IT at the LDR state (see discussion about the energy process at the LDR state in figure 12b, below). These paths may be responsible for the three-dimensional structures in EIT, such as the velocity streaks (see figure 13) distributed in the $x$–$z$ plane similar to streaks in ET (see Jha & Steinberg Reference Jha and Steinberg2021). The difference lies in that these paths do not seem to inhibit EIT, but are likely to boost EIT. For example, fast and slow pressure fluctuations can help to regenerate wall-normal velocity fluctuations, although it is uncertain whether these velocity fluctuations will participate in the EIT-related SSP. In this case, the converged friction coefficient independent of the increase of $Wi$ in EIT needs to find other reasons, such as the negative feedback regulation of suppression effect on NESS due to the interaction between polymers and turbulence, $G_{xy}$. The energy SSP in two-dimensional EIT will provide an important reference to solve these problems, which is a research work we are carrying out.

Figure 12.

Energy picture of viscoelastic DRT: —— (cyan) and —— (blue), SSP cycles in IT and EIT, respectively; —— (orange), the bypass of energy process caused by the passive response of polymers on IT; —— (red), energy source and dissipation.

Figure 13.

Instantaneous contours of streamwise velocity in the $x$–$z$ plane close to the channel centre. The abscissa and ordinate of the subgraphs are $x^*$ (from 0 to 20${\rm \pi}$) and $z^*$ (from 0 to 5${\rm \pi}$), respectively.

The energy process in DRT from onset of DR to MDR, is illustrated in figure 12(b) for early LDR conditions and in figure 12(c) for later LDR, HDR and early MDR conditions. Under early LDR conditions (e.g. $Wi = 1, 2$ for $Re=6000$), polymers passively respond to the SSP of IT with orange paths. The TKE is partially transformed into TEE through the interaction between polymers and turbulence in both the streamwise and wall-normal directions. In addition, partial streamwise TEE in a very small region near the channel walls is transformed into TKE. The NESS is formed due to a wall-normal polymer extension. The NESS absorbs energy from mean motion and turns into streamwise TEE. The above process is a passive response to the SSP of IT, which cannot support but significantly interferes its SSP. Pressure fluctuations induced by polymers distribute TKE from the wall-normal direction to other directions such as the streamwise, opposing that of fast and slow pressure distributions although with almost negligible magnitude. Moreover, there is an orange path showing the interaction between polymers and turbulence, which inhibits both RSS and NESS and is another important factor for polymers to interfere the SSP of IT. When $Wi$ is high enough (e.g. $Wi=4$ at $Re=6000$), the energy process of EIT begins to intervene and coexist with that of IT disturbed by polymers in figure 12(c). This coexistent phase sustains from later LDR conditions to early MDR conditions at least. With the increase of $Wi$, the energy process of DRT gradually transits to that of the EIT state. In this process, the IT-related SSP weakens and the EIT-related SSP gradually dominates the flow.

In Choueiri et al. (Reference Choueiri, Lopez and Hof2018), it was believed that elasto-inertial instability sets in before complete relaminarization at moderate $Re$, resulting in a mixed state which then eventually approaches MDR. With the above energy picture in mind, the different DR phenomena at low $Re$ and moderate $Re$ can be well unified and the underlying mechanism can be further confirmed. As shown in figure 12, the two important links related with energy SSP cycle of EIT described by $G_{xx}$ (see figures 7b and 8b) and $\varPhi ^P_{yy}$ (see figures 10h and 11h) can be well hold at $Wi = 4$ for $Re = 6000$ but are not formed even at $Wi = 5$ for $Re = 2500$. That is to say, with the increase of $Re$, the triggering of EIT becomes easier since smaller fluid elasticity is required at higher $Re$. On the one hand, IT becomes more difficult to be suppressed as $Re$ increases since the streamwise vortices are stronger, so it takes larger fluid elasticity (or higher $Wi$) to suppress them (see Li, Xi & Graham Reference Li, Xi and Graham2006; Li & Graham Reference Li and Graham2007). In short, whether laminarization will occur or not needs to comprehensive consideration of the above two effects. That is, with the increase of $Re$, EIT becomes easier to trigger but IT becomes more difficult to suppress. Thus, consistent with Choueiri et al. (Reference Choueiri, Lopez and Hof2018), when $Wi$ is further increased, the flow is laminarized at $Re = 2500$ since the weaker IT disappears before EIT (the so-called ‘barrier to laminarization’ in Zhu & Xi (Reference Zhu and Xi2021) is excited, whereas, the flow directly enters EIT regime at $Re = 6000$ without a window of relaminarization as EIT-related SSP can be well sustained before IT-related SSP completely disappears).

Looking back at figure 1(c), the following explanation about the valley of $G$ distribution at low $Wi$ ($Wi \approx 2$ for $Re=6000$ and $Wi \approx 1$ for $Re=20\,000$) can be given based on the proposed energy process in figure 12. For the cases before the valley, the EIT-related SSP has not been formed because the two necessary links of $G_{xx}$ (see figure 7b for $Re=6000$) and $\varPhi ^P_{yy}$ (see figure 10h for $Re=6000$) are not yet formed, and polymers passively participate in the IT-related SSP, absorbing energy from turbulence and converting it into TEE. Here $G$ denotes the energy conversion between TKE and TEE through the interaction between polymers and turbulence, that is, work done by fluctuating polymer stress depending on both the strength of fluid elasticity and turbulence intensity. On the one hand, the enhancement of fluid elasticity (the increase of $Wi$) is conducive to enhanced interaction, which increases the magnitude of $G$. On the other hand, with the enhancement of the DR effect due to the increase of fluid elasticity, IT is significantly inhibited, which suppresses the interaction before EIT is formed. In addition, EIT begins to intervene in DRT almost at the end of the valley, which introduces reverse energy transfer from TEE to TKE. As a result, the valley is formed under the confrontation of the above three effects.

We can also further understand the reason why $u'^*$ lags behind $v'^*$ in DRT observed in figure 2. In IT, DRT and EIT, the formation of $u'^*$ are all inseparable with $v'^*$ but with a different routine. In IT, the energy absorbed by RSS sustains the formation of $u'^*$. The formed TKE in the streamwise direction is then redistributed to $v'^*$ directly through the pressure redistribution effect. The formation of $u'^*$ and the energy redistribution to $v'$ are direct processes and rapid terms. However, in the DRT and EIT regime, polymers even change the formation mechanism of $u'^*$, as illustrated by the orange path and blue path in figure 8. Different from that in IT, $u'^*$ is mainly derived from the conversion of streamwise TEE in the EIT regime, and the generation of streamwise TEE is from the wall-normal TEE caused by $v'^*$. Due to the polymer-memory effect, the formation and development of this process are indirect and takes a certain time, related to the fluid relaxation time, which results in the phenomenon $u'^*$ lags behind $v'^*$. For DRT, with the increase of $Wi$, the IT-related SSP gradually weakens, while the EIT-related SSP enhances and the time memory effect increases, resulting in the phenomenon that the lag distance increases.