Recently, the nature of viscoelastic drag-reducing turbulence (DRT), especially the maximum drag reduction (MDR) state, has become a focus of controversy. It has long been regarded as polymer-modulated inertial turbulence (IT), but is challenged by the newly proposed concept of elasto-inertial turbulence (EIT). This study is to repicture DRT in parallel plane channels by introducing dynamics of EIT through statistical, structural and budget analysis for a series of flow regimes from the onset of drag reduction to EIT. Some underlying mechanistic links between DRT and EIT are revealed. Energy conversion between velocity fluctuations and polymers as well as pressure redistribution effects are of particular concern, based on which a new energy self-sustaining process (SSP) of DRT is repictured. The numerical results indicate that at low Reynolds number ( $Re$ ), weak IT flow is replaced by a laminar regime before the barrier of EIT dynamics is established with the increase of elasticity, whereas, at moderate $Re$ , EIT-related SSP can get involved and survive from being relaminarized. This further explains the reason why relaminarization phenomenon is observed for low $Re$ while the flow directly enters MDR and EIT at moderate $Re$ . Moreover, with the proposed energy picture, the newly discovered phenomenon that streamwise velocity fluctuations lag behind those in the wall-normal direction can be well explained. The repictured SSP certainly justifies the conjecture that IT nature is gradually replaced by that of EIT in DRT with the increase of elasticity.

The essence of the maximum drag reduction (MDR) state of viscoelastic drag-reducing turbulence (DRT) is still under debate, which mainly holds two different types of views: the marginal state of inertial turbulence (IT) and elasto-inertial turbulence (EIT). To further promote its understanding, this paper conducts a large number of direct numerical simulations of DRT at a modest Reynolds number Re with $Re = 6000$ for the FENE-P model that covers a wide range of flow states and focuses on the problem of how nonlinear extension affects the nature of MDR by varying the maximum extension length $L$ of polymers. It demonstrates that the essence of the MDR state can be both IT and EIT, where $L$ is somehow an important parameter in determining the dominant dynamics. Moreover, there exists a critical $L_c$ under which the minimum flow drag can be achieved in the MDR state even exceeding the suggested MDR limit. Systematic analyses on the statistical properties, energy spectrum, characteristic structures and underly dynamics show that the dominant dynamics of the MDR state gradually shift from IT-related to EIT-related dynamics with an increase of $L$ . The above effects can be explained by the effective elasticity introduced by different $L$ at a fixed Weissenberg number (Wi) as well as the excitation of pure EIT. It indicates that larger $L$ introduces more effective elasticity and is favourable to EIT excitation. Therefore, we argue that the MDR state is still dominated by IT-related dynamics for the case of small $L$ , but replaced by EIT-related dynamics at high $L$ . The obtained results can harmonize the seemingly controversial viewpoints on the dominant dynamics of the MDR state and also provide some ideas for breaking through the MDR limit, such as searching for a polymer solution with a proper molecular length and concentration.

The high Weissenberg number (Wi) problem (HWNP) has long been a challenge of high-Wi viscoelastic fluid flow simulation. This letter points out that the tensor interpolation method during solving the differential constitutive equations is the main origin of the loss of symmetric positive-definite (SPD) property of the conformation tensor which is the trigger of the HWNP. Instead of component-based interpolation, we propose a tensor-based interpolation method and the results show that this method is very effective in tackling the HWNP by guaranteeing the SPD property of the conformation tensor as well as numerical accuracy. Moreover, the high-order total variation diminishing (TVD) schemes can also be easily constructed under the proposed framework.

The effect of viscoelasticity on the flow and heat transport in the Rayleigh-Bénard convection (RBC) in a rectangular with horizontal periodic boundary is investigated via direct numerical simulation. The working fluid is described by the finitely extensible nonlinear elastic-Peterlin (FENE-P) constitutive model that is able to capture some of the most important polymeric flow behaviors. Numerical simulations are conducted at a low concentration β = 0.9, where β = μs/μ0, μs is the solvent viscosity, μ0 = μs + μp is sum of μs and the polymer viscosity μp. A parametric analysis is performed to understand the influence of the Weissenberg number Wi, the viscosity ratio β and the extension length L on the oscillating mode of the viscoelastic RBC. It is found that both Wi and β weakly inhibit the convection onset and the transition from steady to oscillatory convection. The amplitude and frequency of the oscillations in the oscillatory flow regime are both suppressed. However, the elastic nonlinearity may make the flow transition irregular and even may bring about the relaminarization or lead to the convection cells traveling in the horizontal direction. The extension length L may induce multiple pairs of roll flow patterns at a specific setting of (Ra, Wi). Heat transport is reduced by elasticity but still obeys the power law with Ra if the flow pattern has one pair of rolls. However, heat transfer enhancement occurs if multiple pairs of rolls are induced.

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