3.1. Statistical analysis

We begin by presenting key findings from our flow statistical analysis. Figure 1 illustrates the mean streamwise velocity profiles at various $\textit{Wi}$ . Here, $u^+$ and $y^+$ are the dimensionless parameters normalised based on the inner scale, where skin-friction velocity $u_{\tau } = \sqrt {\tau _w/\rho }$ is used as the reference velocity, and $\tau _w$ is wall shear stress. Previous investigations (Dubief et al. Reference Dubief, Terrapon and Julio2013; Zhang et al. Reference Zhang, Zhang, Li, Yu and Li2021a ; Wang et al. Reference Wang, Zhang, Wang, Li, Zhang and Li2023) have established that 3-D flows exhibit a systematic upward shift in velocity profiles with enhanced elasticity, ultimately approaching the MDR asymptote ( $u^+ = 11.7 \ln y^+ - 17.0$ ; Virk et al. Reference Virk, Mickley and Smith1970) from the IT baseline ( $u^+ = 2.5 \ln y^+ + 5.5$ ). In stark contrast, our 2-D flow results reveal an inverse behaviour: velocity profiles progressively deviate downwards from the laminar state with increasing $\textit{Wi}$ , ultimately converging to a distinct asymptotic regime ( $u^+ = 6 \ln y^+ + 1.5$ ) when $\textit{Wi}=200$ . This counterintuitive trend aligns with recent friction factor observations (Zhu et al. Reference Zhu and Xi2021), collectively suggesting fundamentally different underlying mechanisms between dimensional configurations. This dimensional discrepancy stems from contrasting dynamical frameworks. In 2-D flows, turbulence generation is exclusively governed by elasto-inertial instability (Sid et al. Reference Sid, Terrapon and Dubief2018; Zhu et al. Reference Zhu and Xi2021), resulting in pure EIT devoid of IT components. The enhanced elasticity amplifies the intensity of EIT, driving the steady transition of velocity profile from laminar to turbulent characteristics. Conversely, the 3-D flows exhibit dual-regime turbulence (Wang et al. Reference Wang, Zhang, Wang, Li, Zhang and Li2023), where EIT and IT dynamics coexist in a competitive interplay. The dominance shift from IT to EIT with increasing $\textit{Wi}$ suppresses inertial contributions, ultimately steering the flow towards MDR asymptotics. Notably, the 3-D flows demonstrate progressive suppression of IT due to the enhancement of EIT, while the 2-D flows manifest the pure EIT enhancement. This fundamental distinction inevitably produces divergent elastic responses in flow statistical characteristics. Regarding the asymptotes, neither the MDR nor IT asymptote appears to influence the convergence of velocity profiles in 2-D EIT. Additionally, we speculate that this new asymptote within this parameter space is likely to be altered by the changes in $\textit{Re}$ , $\beta$ and $L^2$ , suggesting potential avenues for future investigation.

Figure 2 shows the distributions of root mean square (r.m.s.) of streamwise velocity $u_{\textit{rms}}$ and wall-normal velocity $v_{\textit{rms}}$ at different $\textit{Wi}$ . As is shown, $u_{\textit{rms}}$ and $v_{\textit{rms}}$ both tend to increase with $\textit{Wi}$ and eventually converge, which is similar to the trend observed in figure 1, and opposite to trends of the 3-D flows (Wang et al. Reference Wang, Zhang, Wang, Li, Zhang and Li2023). This still reflects the differences in dynamical processes of 2-D and 3-D flows. For $u_{\textit{rms}}$ , with the increase of $\textit{Wi}$ , the peak is increasing in value, and gradually moves towards the channel wall. This indicates increasing velocity fluctuations near the wall and the existence of strong extension structures. For $v_{\textit{rms}}$ , the maximum values are at the channel centre, which is consistent with the 3-D EIT case with high $\textit{Wi}$ (Dubief et al. Reference Dubief, Terrapon and Julio2013; Wang et al. Reference Wang, Zhang, Wang, Li, Zhang and Li2023). In fact, for IT and VDRT before the MDR state, the peak of the $v_{\textit{rms}}$ profile is located far from the centre, which can be considered as a feature that distinguishes EIT from IT and non-MDR states. Moreover, the $v_{\textit{rms}}$ profile has significantly smaller values than the $u_{\textit{rms}}$ profile, and nearly converges when $\textit{Wi}\gt60$ , indicating the significant anisotropy of EIT.

Figure 2.

Distributions of r.m.s. of velocity fluctuations at different $\textit{Wi}$ .

In channel flow, the total flow drag can be composed of the Reynolds stress $\tau _R$ , elastic stress $\tau _e$ , and viscous stress $\tau _v$ , governed by the stress balance equation

(3.1) \begin{equation} {\underbrace {-\overline {u^{\prime }\nu ^{\prime }}}_{\tau _{R}}+\underbrace {\overline {\tau ^p_{xy}}}_{\tau _{e}}+\underbrace {\frac {\beta }{\textit{Re}}\frac {{\rm d}\overline {u}}{{\rm d}y}}_{\tau _{v}}=\tau _{\textit{total}},} \end{equation}

where $\overline {({\cdot })}$ represents the ensemble-averaged variable, $({\cdot })^{\prime }$ is the fluctuating variable, and $({\cdot })^{\prime } = ({\cdot })-\overline {({\cdot })}$ .

Figure 3.

Distributions of three kinds of stress at different $\textit{Wi}$ .

Figure 3 illustrates the distributions of mean $\tau _v$ , $\tau _e$ and $\tau _R$ at different $\textit{Wi}$ . As shown in figure 3(a), there is an approximate critical $y$ -position that separates two regions with opposing trends for $\tau _v$ . Near the wall, $\tau _v$ increases with $\textit{Wi}$ , while closer to the channel centre, it decreases. The reason for this evolution behaviour can be seen in figure 1: under the premise of the unchanged bulk average velocity, a significant reduction in velocity near the channel centre (decreasing velocity gradient) must be balanced by an acceleration in the near-wall region (increasing velocity gradient). Since $\tau _v$ is positively correlated with the velocity gradient, it changes accordingly. Figure 3(b) demonstrates a monotonic increase in $\tau _e$ with rising $\textit{Wi}$ , with peak magnitudes consistently located between $y = 0.2$ and $y = 0.3$ . This behaviour aligns with 3-D flow observations (Wang et al. Reference Wang, Zhang, Wang, Li, Zhang and Li2023), reflecting the role of elasticity in intensifying EIT. However, contrary to the positive $\tau _v$ and $\tau _e$ values, $\tau _R$ in figure 3(c) consistently exhibits negative values, although significantly lower than the other stresses. The value of $\tau _R$ becomes more negative with increasing $\textit{Wi}$ , eventually approaching convergence. This finding deviates fundamentally from experimental measurements (Warholic et al. Reference Warholic, Massah and Hanratty1999; Samanta et al. Reference Samanta, Dubief, Holzner, Schafer, Morozov, Wagner and Hof2013) and 3-D simulations (Dubief et al. Reference Dubief, Terrapon and Julio2013; Wang et al. Reference Wang, Zhang, Wang, Li, Zhang and Li2023), where the Reynolds stress remains positive in the MDR state, albeit minimal and negligible. The anomalous and negative $\tau _R$ further distinguishes 2-D EIT from IT where the Reynolds stress dominantly governs total flow drag. The detailed mechanistic insights into this phenomenon will be discussed in § 3.2.

Similar to the variation in elastic stress, figure 4 shows the distributions of mean polymer extension and the maximum mean polymer extension at different $\textit{Wi}$ . Specifically, in figure 4(a), the extension levels within the central range are consistently lower than those in the near-wall range, due to the smaller shear rate away from the wall. However, the extension saturation range has been obviously closer to the centre as $\textit{Wi}$ increases. In the absence of suppression by Newtonian turbulence, the polymer extension in 2-D EIT can easily reach a much higher degree as compared to 3-D numerical simulations (Dubief et al. Reference Dubief, Terrapon and Julio2013). At the same time, compared with the polymer extension in laminar flow ( $\textit{Wi} = 2$ ), the local peak of the turbulent profile away from the wall can be observed, which indicates the existence of the extensional flow topologies described by Dubief et al. (Reference Dubief, Terrapon and Julio2013). In figure 4(b), by comparing with the maximum mean polymer extension of laminar flow at corresponding $\textit{Wi}$ , it is observed that EIT can provide an additional amount of extension. The additional extension is randomly distributed in the turbulent extension field (see figure 8 below), manifested as the characteristic polymer extension sheet structures of EIT. Moreover, as $\textit{Wi}$ increases, the extension exhibits a convergent trend, which is precisely the embodiment of the effective elasticity as previously reported (Zhang et al. Reference Zhang, Cheng, Wang, Zhang, Li and Li2024). At $\textit{Wi} = 200$ , the maximum mean polymer extension reaches approximately $90 \,\%$ , and will continue to slowly approach $100 \,\%$ at ultra-high $\textit{Wi}$ (e.g. $\textit{Wi} = 1000$ , not shown).

Figure 4.

(a) Distributions of mean polymer extension $\overline {{\rm Tr}(\boldsymbol{C})}/L^2$ (the $\textit{Wi} = 2$ case is laminar flow). (b) The maximum mean polymer extension at different $\textit{Wi}$ (where the dashed line represents the maximum mean polymer extension of corresponding laminar solutions).

3.2. Anomalous Reynolds stress

The anomalous Reynolds stress behaviour in 2-D EIT, as evidenced by figure 3(c), reveals fundamentally distinct momentum transport mechanisms compared to classical Newtonian turbulence. This naturally raises the question of whether this small and negative Reynolds stress originates from the lack of correlation between the streamwise $u^{\prime }$ and wall-normal $v^{\prime }$ velocity fluctuations. To address this, figure 5 presents the correlation between $u^{\prime }$ and $v^{\prime }$ , defined as $C_{uv} = -\tau _{R}/(u_{\textit{rms}} v_{\textit{rms}})$ , for cases at high $\textit{Wi}$ , alongside a comparison with our previous IT simulation (Wang et al. Reference Wang, Zhang, Wang, Li, Zhang and Li2023). In classical IT, a moderate negative correlation ( $C_{uv} \approx -0.5$ ) between $u^{\prime }$ and $v^{\prime }$ is a well-known feature. This negative correlation is also characteristic of VDRT, as concluded by Foggi Rota et al. (Reference Foggi Rota, Amor, Le Clainche and Rosti2024) from the downward tilt of the major axis of the probability density distribution outline. In stark contrast, the peak $C_{uv}$ for 2-D EIT is positive, ranging from $0.2$ to $0.3$ . This signifies a moderate positive correlation between $u^{\prime }$ and $v^{\prime }$ , a finding that aligns with the upward tilt of the outline major axis shown in figure 7 below. Therefore, the negative Reynolds stress in 2-D EIT is not an artefact of uncorrelated fluctuations. This provides compelling evidence for the validity of the negative Reynolds stress.

Figure 5.

Correlation $C_{uv}$ between $u^{\prime }$ and $v^{\prime }$ in 2-D EIT and IT. The IT data come from Wang et al. (Reference Wang, Zhang, Wang, Li, Zhang and Li2023).

To systematically deconstruct this phenomenon, we employ the quadrant analysis (Lu et al. Reference Lu and Willmarth1973) that is frequently employed to study the Reynolds stress. Through quadrant analysis, the Reynolds stress $\tau _R$ is resolved into four components based on the signs of velocity fluctuations $u^{\prime }$ and $v^{\prime }$ : Q1 (outward high-speed fluid motion, $u^{\prime } \gt 0$ , $v^{\prime } \gt 0$ ), Q2 (low-speed fluid ejection, $u^{\prime } \lt 0$ , $v^{\prime } \gt 0$ ), Q3 (inward low-speed motion, $u^{\prime } \lt 0$ , $v^{\prime } \lt 0$ ), and Q4 (high-speed sweep motion, $u^{\prime } \gt 0$ , $v^{\prime } \lt 0$ ). In IT, the classical dominance of Q2 and Q4 motions drives substantial Reynolds stress production through intense bursting events (Lu et al. Reference Lu and Willmarth1973). For VDRT, Cai et al. (Reference Cai, Li, Zhang, Li, Yu, Wei, Kawaguchi and Hishida2009) found experimentally that the existence of surfactant additives in viscoelastic fluid can suppress turbulent bursting motions, namely the ejection and sweep motions of Q2 and Q4. This led to drag reduction, or rather the extremely small Reynolds stress. Similarly, in polymer solution flows, Guan, Yao & Jiang (Reference Guan, Yao and Jiang2013) reported that the occurrence of small Reynolds stress is due to the significant decrease of ejections and sweeps of Q2 and Q4 near the wall. We speculate that this phenomenon must also exist in the pure 2-D EIT. Therefore, figure 6 illustrates the distributions of Reynolds stress quadrant components at different $\textit{Wi}$ . With the increase of $\textit{Wi}$ , the Q1/Q3 motions promote the formation of obvious peak of negative Reynolds stress at $y \approx 0.4{-}0.6$ . Moreover, especially at high $\textit{Wi}$ , the motions of Q1 and Q3 are obviously stronger than those of Q2 and Q4, which is the direct cause of the negative Reynolds stress. The characteristic bursting events (Q2/Q4 motions) of classical IT still retain the weak dominance in 3-D viscoelastic flows, but are overwhelmingly overshadowed in 2-D EIT, resulting in a unique negative contribution to the overall stress. Therefore, the EIT-related dynamics with elasticity nature tend to facilitate the co-increasing and co-decreasing motions of Q1 and Q3, whether in 2-D or 3-D flow.

Figure 6.

Quadrant distributions of Reynolds stress at different $\textit{Wi}$ : (a) 10, (b) 40, (c) 100, (d) 200.

In order to elucidate the spatial variations of quadrant components, figure 7 presents the relevant probability density distributions across four quadrants at different wall-normal positions. As is shown, the outlines of distribution regions are obliquely distributed along the first and third quadrants, with their major axes inclined upwards. This distribution is in stark contrast to the diagonal distribution along the second and fourth quadrants in IT (Guan et al. Reference Guan, Yao and Jiang2013) and VDRT (Foggi Rota et al. Reference Foggi Rota, Amor, Le Clainche and Rosti2024). It can be explained that the distributions in Q1 and Q3 are at the dominant level, suppressing the positive contributions of Q2 and Q4 to Reynolds stress, which well explains the emergence of the negative peaks of Reynolds stress profiles in figure 6. Therefore, in contrast to the dominance of ejection and sweep motions in IT, the above results demonstrate the significant dominance of the Q1/Q3 motions in 2-D EIT, implying the different dynamic mechanisms.

Figure 7.

Probability density distribution contours of four quadrant components at different positions away from the wall (from left to right, $y$ values are 0.2, 0.4, 0.6 and 1.0): (a–d) $\textit{Wi} = 10$ , (e–h) $\textit{Wi} = 40$ , (i–l) $\textit{Wi} = 100$ , (m–p) $\textit{Wi} = 200$ .

Figure 8.

Superposition of the instantaneous extension and the fluctuating velocity vector field at $\textit{Wi}$ values 10, 40, 100 and 200 (the $x$ direction is compressed by half for improved display).

Instead of focusing on the frequency and amplitude of the fluctuating velocity at the single position, a more comprehensive understanding can be obtained in the whole flow field. Figure 8 depicts the superposition of instantaneous polymer extension and fluctuating velocity vector field at $\textit{Wi}$ values 10, 40, 100 and 200. These flow fields are characterised by numerous vortices composed of the fluctuating velocity vectors of the four quadrant components, with areas of concentrated Q1/Q3 motions highlighted (thick white/black arrows). Combined with figure 6, the dominance of Q1/Q3 motions can be confirmed, more significantly as $\textit{Wi}$ increases. In figure 8(c), the black ellipse region approximates a large vortex structure, within which the upper and lower parts are undergoing two opposing processes of stretching and contraction of the polymer extension sheet-like structures. In the viscoelastic flow, this mechanical response primarily arises from the intrinsic coupling between the dynamics of the polymer chains and the velocity field. At a macro level, the stretching and contraction processes are directly and closely related to the spatial direction of fluctuating velocity vectors. Thus it is clear that the vast majority of fluctuating velocity vectors in the first and third quadrants, namely the Q1/Q3 motions, are arranged along and against the polymer extension sheet-like structures, respectively. For the viscoelastic fluid flow, a typical characteristic is the presence of numerous elongated polymer extension sheet-like structures in the flow field, which exhibit the inclined distribution from the near-wall region towards the channel centre along the streamwise direction. As $\textit{Wi}$ increases, these structures tend to overlap or merge, showing an increasingly intense extension distribution, accompanied by the intensification of the anisotropy of EIT. Thus this distribution indicates that the fluctuating velocity vectors accompanying the polymer extension sheet-like structures are preferentially in the first or third quadrants. This is also consistent with the fact that the main axis of the probability density distribution outline in figure 7 is inclined along the first and third quadrants. It indicates that the negative Reynolds stress is closely related to the polymer extension sheet-like structures.

Finally, when comparing the results of 2-D EIT with numerical (Dubief et al. Reference Dubief, Terrapon and Julio2013; Wang et al. Reference Wang, Zhang, Wang, Li, Zhang and Li2023) and experimental (Samanta et al. Reference Samanta, Dubief, Holzner, Schafer, Morozov, Wagner and Hof2013; Choueiri et al. Reference Choueiri, Lopez and Hof2018) results of 3-D MDR state, it is evident that the Reynolds stress values in both cases are significantly lower than other stresses. However, the Reynolds stress in 3-D flow is not negative. Based on the above understanding, we can make the following two speculations. On the one hand, the 3-D MDR state may not be pure EIT, as it contains weak IT-related dynamics with positive Reynolds stress, which supports the viewpoint of Zhu et al. (Reference Zhu and Xi2021). On the other hand, the 3-D MDR state may essentially be an EIT state modulated by fluid inertia, and its weak positive Reynolds stress may be an IT-related dynamics behaviour temporarily organised by turbulent fluctuations in EIT, but it cannot be further developed due to elastic suppression. Future work needs to be conducted to clarify this issue.

3.3. Budgets of turbulent kinetic energy and elastic energy

In the MDR state, the negligible Reynolds stress cannot dominate the production of turbulent kinetic energy (TKE), and polymers provide almost all the energy required for turbulence maintenance (Min et al. Reference Min, Choi and Yoo2003; Dubief et al. Reference Dubief, Terrapon and Julio2013; Wang et al. Reference Wang, Zhang, Wang, Li, Zhang and Li2023). Similarly, in the previous subsection, the discovery of minimal and anomalous Reynolds stress also proves that the main source of TKE in EIT cannot possibly be the production term related to the Reynolds stress. Therefore, the energetic properties in 2-D EIT are well worth exploring and comparing with those of 3-D flow. The transformation between the turbulent elastic energy (TEE) and TKE greatly distinguishes the viscoelastic fluid flow from the Newtonian fluid flow. The integral form of the budget of TKE is

(3.2) \begin{align}&\qquad\quad\,\,\, {\int _0^2{P_k\mathrm{d}y}-\int _0^2{\varepsilon _k\mathrm{d}y}-\int _0^2{G}\mathrm{d}y=0,} \end{align}

(3.3) \begin{align}& {P_k = -\frac {1}{2} \overline {u^{\prime } v^{\prime }} \frac {\partial \bar {u}}{\partial y}, \quad \varepsilon _k = \frac {\beta }{\textit{Re}} \overline {\frac {\partial u_i^{\prime }}{\partial x_{\!j}} \frac {\partial u_i^{\prime }}{\partial x_{\!j}}}, \quad G = \overline {\tau _{\textit{ij}}' \frac {\partial u_i'}{\partial x_{\!j}},}} \end{align}

where $P_k$ is the production term of TKE, $\varepsilon _k$ is the dissipation term, and $G$ is the energy transfer term between TKE and TEE. (A positive value of $G$ indicates transfer from TKE to TEE, while a negative value signifies the reverse process.)

Similarly, the budget of TEE can be obtained by integrating the elastic energy transport equation:

(3.4) \begin{align}& {\int _0^2P_e\mathrm{d}y-\int _0^2\varepsilon _e\mathrm{d}y+\int _0^2G\mathrm{d}y=0,} \end{align}

(3.5) \begin{align}&\quad P_e = {\overline {\tau _e}\frac {\partial \overline {u}}{\partial y}},\quad \varepsilon _e = \frac {\overline {f(\mathrm{tr}(\boldsymbol{C}))\,\tau _{ii}}}{2\,Wi}, \end{align}

where $P_e$ is the production term dominated by elastic stress of TEE, and $\varepsilon _e$ is the dissipation term.

Figure 9 illustrates the distributions of the budget of TKE under varying $\textit{Wi}$ . In all cases, the primary source of TKE is consistently the energy transfer term − $G$ : TEE is converted into TKE through the interaction between polymers and turbulent flow. Also, while the magnitude of − $G$ varies with $\textit{Wi}$ , its dominant role in TKE production remains unchanged. However, this situation, in which the generation of TKE is overwhelmingly dominated by − $G$ , occurs only in the asymptotic MDR state or EIT of 3-D flow at high $\textit{Wi}$ (Dubief et al. Reference Dubief, Terrapon and Julio2013; Wang et al. Reference Wang, Zhang, Wang, Li, Zhang and Li2023). Therefore, it can be inferred that EIT-related dynamics is likely to have occured in 3-D VDRT, but its partial features are obscured due to the coexistence with IT. Moreover, − $G$ is inverted within a narrow region near the wall, and this phenomenon also exists in 3-D flow (Dubief et al. Reference Dubief, Terrapon and Julio2013; Wang et al. Reference Wang, Zhang, Wang, Li, Zhang and Li2023). The only difference lies in the fact that the production term $P_k$ exhibits a negative value in 2-D EIT, which is attributable to the aforementioned abnormal Reynolds stress. Nonetheless, $P_k$ constitutes a small proportion in the budget of TKE, exerting a negligible impact on the self-sustaining mechanism of 2-D EIT.

Figure 9.

Profiles of TKE budget at different $\textit{Wi}$ .

Similar to figure 9, figure 10 shows the distributions of the budget of TEE at different $\textit{Wi}$ . The trend of each curve is consistent with the 3-D simulation results (Zhang et al. Reference Zhang, Shao, Li, Ma, Zhang and Li2021b ). The generation and dissipation of TEE are mainly concentrated in the near-wall region, resembling the distribution of the budget of TKE. However, unlike the generation of TKE derived mainly from − $G$ , $P_e$ provides the vast majority of the generation of TEE, indicating the dominant role of the work done by the elastic stress. As $\textit{Wi}$ increases, the enhancement of the turbulent intensity of EIT is driven primarily by the increase of − $G$ , yet there is a saturation trend, which is likely attributed to the effective elasticity (Zhang et al. Reference Zhang, Cheng, Wang, Zhang, Li and Li2024). The unique TEE characteristics of EIT show remarkable consistency in the budget analysis of between 2-D and 3-D flow, which is also strong evidence of the 2-D nature of EIT.

Figure 10.

Profiles of TEE budget at different $\textit{Wi}$ .

3.4. Redistributive role of pressure

The above content provides a detailed analysis of the TKE budget of EIT. In fact, the majority of studies, including this paper, choose to use the pressure-driven flow, where the pressure plays a critical role in the redistribution of energy among the components of momentum (Terrapon et al. Reference Terrapon, Dubief and Soria2015). Moving forward, this paper will decompose the pressure into several parts, and assess their roles in redistributing TKE.

Many scholars have analysed the redistributive role of pressure in 3-D turbulent flow of viscoelastic fluid (Ptasinski et al. Reference Ptasinski, Boersma, Nieuwstadt, Hulsen, Van den Brule and Hunt2003; Terrapon et al. Reference Terrapon, Dubief and Soria2015; Zhang et al. Reference Zhang, Zhang, Wang, Li, Yu and Li2022). As shown in (3.6)–(3.12), the pressure fluctuations can be decomposed into four contributing terms. As reported by Kim (Reference Kim1989) and Terrapon et al. (Reference Terrapon, Dubief and Soria2015), the rapid part $p^{\prime }_r$ is a direct and linear response to the change and deformation in the mean flow field, while the slow part $p^{\prime }_s$ is the response through nonlinear turbulent interactions. The distinctive polymer contribution $p^{\prime }_{\!p}$ exists in the viscoelastic flow due to the polymer stress. Notably, the first two terms ( $p^{\prime }_r$ and $p^{\prime }_s$ ) and $p^{\prime }_{\!p}$ are identified as the inertial and elastic contributions, respectively (Terrapon et al. Reference Terrapon, Dubief and Soria2015; Zhang et al. Reference Zhang, Zhang, Wang, Li, Yu and Li2022). The inertia and nonlinear elasticity of the fluid play varying roles in the pressure redistribution effect and the self-sustaining process of EIT:

(3.6) \begin{align}&\qquad\qquad {p^{\prime }(\boldsymbol{x})=p(\boldsymbol{x})-\overline {p}(\boldsymbol{x}),} \end{align}

(3.7) \begin{align}& {p^{\prime }(\boldsymbol{x})=p^{\prime }_{r}(\boldsymbol{x})+p^{\prime }_{\!s}(\boldsymbol{x})+p^{\prime }_{\!p}(\boldsymbol{x})+p^{\prime }_{\!St}(\boldsymbol{x}),} \end{align}

(3.8) \begin{align}&\qquad\qquad\,\, {{\nabla} ^{2}p_{r}^{\prime }=-2\frac {\mathrm{d}\overline {u}}{\mathrm{d}y}\frac {\partial v^{\prime }}{\partial x},} \end{align}

(3.9) \begin{align}&\qquad {{\nabla} ^{2}p_{\!s}^{\prime }=-\frac {\partial u_{i}^{\prime }}{\partial x_{\!j}}\frac {\partial u_{\!j}^{\prime }}{\partial x_{i}}+\frac {\mathrm{d}^{2}\overline {{{v^{\prime 2}}}}}{\mathrm{d}y^{2}},} \end{align}

(3.10) \begin{align}&\qquad\qquad {{\nabla} ^{2}p_{\!p}^{\prime }=\frac {\partial ^{2}\tau ^{\prime }}{\partial x_{i}\,\partial x_{\!j}},} \end{align}

(3.11) \begin{align}&\qquad\qquad\quad {{\nabla} ^{2}p_{\!St}^{\prime }=0,} \end{align}

(3.12) \begin{align}&\qquad\qquad\,\, {\varPhi _{ii}=2\overline {p^{\prime }\frac {\partial u_i^{\prime }}{\partial x_i}},} \end{align}

where $p^{\prime }_r, p^{\prime }_s, p^{\prime }_{\!p},p^{\prime }_{\!St}$ represent rapid, slow, polymer and Stokes pressures, respectively, and the negligible Stokes pressure $p^{\prime }_{\!St}$ reflects the role of the wall boundary condition (Terrapon et al. Reference Terrapon, Dubief and Soria2015). Here, $\boldsymbol{x} = (x,y)$ , $(\cdot) ^{\prime }$ represents fluctuations, and $\overline {(\cdot)}$ represents average quantities.

Figure 11.

Pressure r.m.s. distribution along the wall-normal direction with different $\textit{Wi}$ : (a) total, (b) rapid, (c) slow and (d) polymeric contributions.

Figure 11 shows the r.m.s. distribution of various pressure contributions at different $\textit{Wi}$ . As is shown, these different kinds of profiles in 2-D EIT are generally similar to the results of the MDR state (Terrapon et al. Reference Terrapon, Dubief and Soria2015), characterised by smoothness but not complete flatness. However, in other research with the Oldroyd-B model (Zhang et al. Reference Zhang, Zhang, Wang, Li, Yu and Li2022), the r.m.s. profiles of 3-D EIT exhibits a quasi-flat distribution. This difference likely arises from the effect of the maximum extension length $L$ ( $L$ is $\infty$ for the Oldroyd-B model, 100 in this paper and 200 in Terrapon et al. Reference Terrapon, Dubief and Soria2015). In detail, with the increase of $\textit{Wi}$ , $p_{\textit{rms}}$ gradually increases and approaches convergence, which fits with the effective elasticity effect that we proposed (Zhang et al. Reference Zhang, Cheng, Wang, Zhang, Li and Li2024). Moreover, the $p^{\prime }_s$ r.m.s. values reflect that the contribution of slow pressure is distinctly smaller than the other contributions, in line with the findings by Terrapon et al. (Reference Terrapon, Dubief and Soria2015) in 3-D simulations: polymers have a strong suppression on the slow pressure contribution $p^{\prime }_s$ . However, for the contribution of rapid pressure $p^r_{\textit{rms}}$ , there exists a decreasing trend at high $\textit{Wi}$ ( $\textit{Wi} \gt 80$ ), which is actually also slightly reflected in the slow pressure. Unlike the effect of relaminarisation in the 3-D flows with low $\textit{Re}$ (Zhang et al. Reference Zhang, Zhang, Wang, Li, Yu and Li2022), the reason here is the suppression effect of strong elasticity. Furthermore, figure 12 illustrates the distribution of $p^s_{\textit{rms}}/(p^r_{\textit{rms}} + p^s_{\textit{rms}})$ along the wall-normal direction. The increasing trend of this ratio indicates that the contribution of $p^{\prime }_s$ in the inertial pressure gradually approaches saturation. Consequently, the rapid pressure contribution $p^{\prime }_r$ is more significantly suppressed by polymers in 2-D EIT. Note the visible bump or peak of $p_{\textit{rms}}$ within the $y$ range 0.2–0.4 in figure 11(a), unequivocally originating from the polymer pressure contribution $p^{\prime }_{\!p}$ . This observation is in stark contrast to the buffer layer of IT where the obvious peak in the pressure fluctuations arises from the slow pressure (Terrapon et al. Reference Terrapon, Dubief and Soria2015). Therefore, especially at high $\textit{Wi}$ , the increase in total pressure is attributed primarily to the dominant modulating role of polymer pressure $p^{\prime }_{\!p}$ in the pressure redistribution effect.

Figure 12.

Distribution of the ratio of $p_{\textit{rms}}^{s}$ to $p_{\textit{rms}}^{r}+p_{\textit{rms}}^{s}$ along the wall-normal direction.

According to Dubief et al. (Reference Dubief, Terrapon and Julio2013), pressure plays a crucial role in redistributing energy, especially TKE, across the momentum components. Through a comprehensive analysis of Reynolds stress transport, Terrapon et al. (Reference Terrapon, Dubief and Soria2015) decomposed the velocity–pressure gradient correlation term into the pressure–strain and pressure–diffusion terms, and pointed out that the pressure–strain term $\mathit{\varPhi }_{\textit{ij}}$ transports TKE from the streamwise direction to other directions. The trace of the pressure–strain term is given by the equality ${\rm Tr}(\mathit{\varPhi}_{\textit{ij}})=\mathit{\varPhi }_{xx} + \mathit{\varPhi }_{yy} + \mathit{\varPhi }_{zz} = 0$ ( $\mathit{\varPhi }_{zz}$ is not present in 2-D simulations of this paper, and $\mathit{\varPhi }_{xx} = - \mathit{\varPhi }_{yy}$ ). Figure 13 shows the distribution of the wall-normal pressure–strain component of total, rapid, slow and polymer pressure contributions at different $\textit{Wi}$ (streamwise pressure–strain component $\mathit{\varPhi }_{xx}$ is just the opposite, not shown). Overall, the distribution profiles of different pressure contributions show great consistency with the results of the MDR state as well as 3-D EIT (Terrapon et al. Reference Terrapon, Dubief and Soria2015; Zhang et al. Reference Zhang, Zhang, Wang, Li, Yu and Li2022). Upon comparing the order of magnitude, we can intuitively discern the weak contribution of slow pressure in the energy transport of EIT in figure 13(c). However, unlike the inertial pressure, the polymer pressure contribution does not show an obvious convergence trend due to its elasticity nature, which is also the main source of the continuous enhanced TKE transport effect of pressure. And the $\mathit{\varPhi }^p_{yy}$ peak starts to be larger than the $\mathit{\varPhi }^r_{yy}$ peak at $\textit{Wi} \gt 40$ , which means that the polymer pressure dominates the TKE transport in 2-D EIT.

As is shown in figure 13, at approximately $y \lt 0.1$ , all pressure contributions have negative wall-normal pressure–strain component, meaning that the TKE is redistributed from the $y$ direction to the $x$ direction. Concurrently, in this range, the negative $-G$ shown in figure 9 indicates the conversion from TKE to TEE, thereby contributing to the characteristic turbulence streaks of viscoelastic flow (Zhang et al. Reference Zhang, Zhang, Wang, Li, Yu and Li2022). And the positive $\mathit{\varPhi }_{yy}$ at approximately $0.1 \lt y \lt 0.4$ indicates the formation of wall-normal TKE within this range. Far from the wall (approximately $0.5 \lt y \lt 1.0$ ), the streamwise TKE is formed again, with the rapid pressure and polymer pressure dominating this process. In addition, near the channel centre, the positive $\mathit{\varPhi }^r_{yy}$ and $\mathit{\varPhi }^s_{yy}$ indicate that the inertial pressure seems to have a promotion effect on the formation of wall-normal TKE, although offset by the negative effect of the polymer pressure. Notably, in fact, Terrapon et al. (Reference Terrapon, Dubief and Soria2015) found that the polymer pressure made a greater contribution to $\mathit{\varPhi }_{yy}$ than $\mathit{\varPhi }_{zz}$ , in the analysis of 3-D flow. They have argued that the elastic contribution promotes the 2-D flow to a certain extent, while the inertial part mainly drives the 3-D flow. Combined with the dominant role of polymer or elastic pressure on TKE transport of 2-D EIT in figure 13, this further demonstrates that the 2-D nature of elasto-inertial instability is dominated by the elasticity in EIT and the objective existence of 2-D EIT (Sid et al. Reference Sid, Terrapon and Dubief2018).

Figure 13.

Distribution of wall-normal pressure–strain component $\varPhi _{yy}$ from different pressure contributions under different $\textit{Wi}$ .

3.5. Energy spectrum analysis

The transformation between TEE and TKE, as illustrated in figures 9 and 10, reveals that the most intense energy transfer occurs within the $y$ range 0.1–0.2, which aligns perfectly with the characteristic position of energy transformation observed in 3-D numerical simulations (Wang et al. Reference Wang, Zhang, Wang, Li, Zhang and Li2023). To further investigate the dynamics of 2-D EIT, the energy spectrum analysis is performed at this specific range, as well as at $y=0.4$ and the channel centreline. Figure 14 illustrates the energy spectrum of TKE along the streamwise direction under different $\textit{Wi}$ at different wall-normal positions, with the corresponding power-law spectral decay $k^{\alpha }$ of interest. A key observation is the monotonic decrease in $-\alpha$ with increasing $\textit{Wi}$ across these four different positions, ultimately converging to distinct slope values at very high $\textit{Wi}$ (e.g. $\textit{Wi} \gt 100$ ). Near the wall ( $y=0.1-0.2$ , figure 14 a,b), the energy spectrum exhibits a converged $k^{-11/3}$ power-law decay within $k \gt 10$ , a feature reminiscent of 3-D EIT. Notably, in 3-D flows, Dubief et al. (Reference Dubief, Terrapon and Julio2013) reported that the power-law decay at high wavenumber seems to be closer to $k^{-11/3}$ and deviates from $k^{-14/3}$ , supporting the analogy between 2-D and 3-D EIT dynamics. In the energy spectrum of $y=0.4$ in figure 14(c), it is observed that the $k^{-19/6}$ power-law decay is present within $k \approx 4-80$ , demonstrating the mighty presence and activity of EIT dynamics in this region. However, for 3-D EIT, the $k^{-19/6}$ power-law decay occurs due to the gradual conversion from $-5/3$ power-law decay with the enhanced elasticity in the low wavenumber region (Zhang et al. Reference Zhang, Zhang, Li, Yu and Li2021a ). Its appearence represents the prominent presence of EIT dynamics that gradually replaces IT dynamics. In the energy spectrum of the channel centre in figure 14(d), there is also a special power law with $-13/3$ at $k \approx 5-60$ , which is the intermediate between $-11/3$ and $-14/3$ . This fits the reported power-law decay of 3-D EIT (Dubief et al. Reference Dubief, Terrapon and Julio2013; Steinberg Reference Steinberg2021), further bridging the dynamics of 2-D and 3-D EIT.

Figure 14.

Energy spectra at different $\textit{Wi}$ and wall-normal positions.

Theoretical studies by Fouxon et al. (Reference Fouxon and Lebedev2003) establish the theoretical rate of decay $\alpha \lt -3$ of ET (Steinberg Reference Steinberg2021), which is also considered to be a hallmark of viscoelastic turbulent flows (Lellep et al. Reference Lellep, Linkmann and Morozov2024). Foggi Rota et al. (Reference Foggi Rota, Amor, Le Clainche and Rosti2024) discovered a common power-law decay in the TKE spectrum, connecting ET and EIT. Therefore, $\alpha \lt -3$ also appears to apply to EIT, in both experiments and numerical simulations, and regardless of channel flow or other flows. Yamani et al. (Reference Yamani, Keshavarz, Raj, Zaki, McKinley and Bischofberger2021) discovered a power-law decay of $-3$ in jet experiments, and a similar invariant measure exists in Taylor–Couette flow experiments (Boulafentis et al. Reference Boulafentis, Lacassagne, Cagney and Balabani2024). Song et al. (Reference Song, Lin, Liu, Lu and Khomami2021) reported the decay exponent $-14/3$ in Taylor–Couette flow through numerical studies, consistent with that in channel flow (Dubief et al. Reference Dubief, Terrapon and Julio2013). All the above power-law decays for 2-D EIT in figure 14 as well as 3-D EIT satisfy $\alpha \lt -3$ , which is a remarkable similarity. However, under the same parameters, $\alpha$ for 3-D flow exhibits a quantitative difference compared to that for 2-D EIT (Sid et al. Reference Sid, Terrapon and Dubief2018), which means that 2-D EIT requires higher $\textit{Wi}$ to achieve the spectral convergence. And this discrepancy highlights that the quantitative analysis of both 2-D and 3-D EIT is likely to encounter challenges and cause unsatisfactory situations, necessitating the qualitative approach. Therefore, to some extent, we can still identify the shared power-law characteristics at high $\textit{Wi}$ , underscoring the fundamental similarities between 2-D and 3-D EIT. This observation reinforces the perspective of Sid et al. (Reference Sid, Terrapon and Dubief2018).