## Final Report Summary - VISCELTURBFLOW (Computational study of macro- and microscopic turbulence controlled by polymer additives)

The goal of this project is the study of a multiscale challenge of fluid mechanics, polymer dynamics in flow, one that spans from molecular to macroscale. In spite of decades of research on the dynamics of single polymers in flow, a comprehensive and predictive knowledge of polymer-flow interactions had eluded us. Consequently, industrial applications of polymers' abilities to control turbulence, heat transfer, or enhancing mixing at microscopic scales have yet to exploit the full potential of polymer dynamics. The project aims to develop numerical tools to address the shortcomings of current numerical methods used for polymeric fluids and to further our understanding of the polymer dynamics in turbulent, or turbulent-like, flows. Its broader impacts are (1) to advance the state of the art of predictive theories and models of polymer dynamics in flow, (2) to contribute to the general knowledge of turbulence control and turbulent transport and mixing, (3) to demonstrate the benefits of the proposed research in turbulence drag control, heat transfer manipulation and mixing applied to practical problems.

The dilution at very low concentration of polymers into a Newtonian solvent changes the rheology of the solvent. The fluid then shows a macroscopic viscoelastic behavior originating in the microscopic stretching and coiling dynamics of the polymer chains, which modifies the flow dynamics. In particular, it can lead to two seemingly contradictory macroscopic effects: at large Reynolds numbers, polymers reduce turbulence by damping vortices (drag reduction), while at very low to transitional Reynolds numbers, polymers can induce instabilities and a chaotic flow (elastic turbulence, early turbulence).

The drag reducing effect has applications for instance in the transport of oil in pipelines. It has been shown that this effect is bounded by the so-called maximum drag reduction (MDR) asymptote, so that the flow never relaminarizes. On the other hand, the use of polymers in microchannel flows can find applications in mixing enhancement. Although the mechanisms of drag reduction at large Reynolds number and of elastic instabilities at very low Reynolds are now mostly understood, numerous questions remain open, mostly regarding the nature of the MDR regime and the link between elastic turbulence and polymer drag reduction. The present project aims thus at providing a better understanding of these phenomena and at developing a robust and accurate numerical infrastructure for simulating them.

The methodology relies on direct numerical simulations of polymeric flows in simple canonical configurations to focus on the key physical processes. More specifically, most of the simulations are done in periodic channel flows although other canonical cases are also considered (e.g. Kolmogorov flow, natural convection in a periodic box). The viscoelastic properties of the fluid are represented by an additional stress appearing in the incompressible Navier-Stokes equations. This polymeric stress depends on the polymer configuration. Polymers are here represented through a mesoscopic model by simple nonlinear bead-spring dumbbells. The problem is then controlled by four parameters: the Reynolds number, the Weissenberg number (the ratio of the polymer longest relaxation time to a characteristic flow time), the polymer concentration parameter and the polymer maximum extensibility parameter. Two modeling strategies are considered: the FENE-P transport equation for the polymer configuration tensor is solved in an Eulerian framework, or both the stochastic FENE and the constitutive FENE-P models are solved in a Lagrangian framework based on particle tracking.

The major numerical challenge stems from the low diffusivity of polymers, whose transport equation is mostly hyperbolic. Artificial diffusion is thus typically added (locally or globally) to ensure numerical stability. The corresponding Schmidt number for polymer diffusion should be of the order of 500 or larger, so that the appearance of sub-Kolmogorov scales is expected. Nonetheless, most works in the literature consider values of the Schmidt number lower than or of the order of one, and thus miss most of the small-scale dynamics. Therefore, significant effort has been devoted to analyzing and implementing accurate and robust numerical schemes. Moreover, simulations were performed either on highly resolved meshes with a global diffusion at high Schmidt number, or with artificial diffusion added only locally where stability requires it. In this way, most of the relevant small-scale dynamics can be captured. In addition, the Lagrangian approach provides a natural framework to treat such advection-dominated problems.

The main accomplishment of the project has been its critical contribution to the discovery and the characterization of a new state of turbulence, called elasto-inertial turbulence (EIT). This new regime has been both observed experimentally and reproduced numerically, and could represent the missing link between elastic turbulence and polymer drag reduction. EIT is driven by both inertial and elastic instabilities at large Weissenberg number and extensibility parameter. Once triggered, EIT is self-sustained since the elastic instability creates the very velocity fluctuations it feeds upon. EIT is most likely a two-dimensional phenomenon (i.e. it is observed in 2D simulations) characterized by very thin and elongated sheets of large polymer stretch in the vicinity of the wall, and weak spanwise vortical structures above them. Because this dynamics is driven by small scales, it can only be reproduced with an adequate numerical approach.

The observation of EIT structures in MDR flows suggests that their dynamics is not purely of Newtonian nature, but rather that it may be driven by EIT at finite Reynolds number and increasing Weissenberg number. The sheer magnitude of extensional viscosity is likely to prevent the emergence of any vortical structures, thus leaving MDR to be sustained by near-wall spanwise structures similar to the ones observed at low Reynolds numbers. The flow is therefore stuck in a transitional state, specifically the stage of breakdown of nonlinear flow instabilities, which does not support a logarithmic mean velocity profile. EIT also provides support to de Gennes' picture that drag reduction derives from two-way energy transfers between turbulent kinetic energy of the flow and elastic energy of polymers at small scales, resulting in an overall modification of the turbulence energy cascade at high Reynolds numbers. On the other hand, EIT is not confined to low Reynolds numbers as for elastic turbulence. Nonetheless, the exact role of inertia is still unclear. Whether EIT is simply a manifestation of elastic turbulence in inertial flows or a separate regime remains an open question.

Another accomplishment of the project has also been to provide new insight in the modification of heat transfer by polymer additives in weakly turbulent natural convection. Depending on the polymer parameters, two different regimes can be achieved: the polymers can either increase or decrease heat transfer. In both cases, viscoelasticity reduces the size of the convection cells, and thus increases their number. At moderate values of the Weissenberg number and extensibility parameter, this is sufficient to increase heat transfer. However, at larger values of these parameters, the rotation speed of the convection cells is drastically reduced, which provides an overall lower heat transfer. In two dimensions, the Newtonian simulations do not lead to a weakly turbulent flow, but to a quasi-steady state. The addition of polymers with moderate elasticity then stabilizes the flow. However, a higher elasticity produces a chaotic flow, in a similar fashion to EIT in channel flow forced convection.

The discovery of EIT has demonstrated that our previous understanding of the interaction between turbulence and polymers was only very partial. EIT has thus provided a first set of answers, but it has also led to a series of new questions that need now to be answered. Moreover, this research provides further suggestions on how to leverage the effect of polymers, or more generally viscoelastic rheology, for the manipulation of flows, such as for mixing or heat transfer.

In a broader context, the impact of this research is also the advancement of our understanding of fundamental aspects of turbulence. On the one hand, it demonstrates the possible existence of turbulence at subcritical Reynolds number (subcritical turbulence). This represents an important aspect for transition, which remains one of the major challenges in turbulence modeling. It also contributes to the understanding of backward energy transfer at small scales that is also observed in Newtonian turbulent flows. Finally, it suggests possible new strategies, beyond polymers, for controlling turbulent flows.

Although not all initial objectives have been yet achieved, the project has been very successful. Moreover, it is seen in a long-term context and research will continue beyond the present project.

The dilution at very low concentration of polymers into a Newtonian solvent changes the rheology of the solvent. The fluid then shows a macroscopic viscoelastic behavior originating in the microscopic stretching and coiling dynamics of the polymer chains, which modifies the flow dynamics. In particular, it can lead to two seemingly contradictory macroscopic effects: at large Reynolds numbers, polymers reduce turbulence by damping vortices (drag reduction), while at very low to transitional Reynolds numbers, polymers can induce instabilities and a chaotic flow (elastic turbulence, early turbulence).

The drag reducing effect has applications for instance in the transport of oil in pipelines. It has been shown that this effect is bounded by the so-called maximum drag reduction (MDR) asymptote, so that the flow never relaminarizes. On the other hand, the use of polymers in microchannel flows can find applications in mixing enhancement. Although the mechanisms of drag reduction at large Reynolds number and of elastic instabilities at very low Reynolds are now mostly understood, numerous questions remain open, mostly regarding the nature of the MDR regime and the link between elastic turbulence and polymer drag reduction. The present project aims thus at providing a better understanding of these phenomena and at developing a robust and accurate numerical infrastructure for simulating them.

The methodology relies on direct numerical simulations of polymeric flows in simple canonical configurations to focus on the key physical processes. More specifically, most of the simulations are done in periodic channel flows although other canonical cases are also considered (e.g. Kolmogorov flow, natural convection in a periodic box). The viscoelastic properties of the fluid are represented by an additional stress appearing in the incompressible Navier-Stokes equations. This polymeric stress depends on the polymer configuration. Polymers are here represented through a mesoscopic model by simple nonlinear bead-spring dumbbells. The problem is then controlled by four parameters: the Reynolds number, the Weissenberg number (the ratio of the polymer longest relaxation time to a characteristic flow time), the polymer concentration parameter and the polymer maximum extensibility parameter. Two modeling strategies are considered: the FENE-P transport equation for the polymer configuration tensor is solved in an Eulerian framework, or both the stochastic FENE and the constitutive FENE-P models are solved in a Lagrangian framework based on particle tracking.

The major numerical challenge stems from the low diffusivity of polymers, whose transport equation is mostly hyperbolic. Artificial diffusion is thus typically added (locally or globally) to ensure numerical stability. The corresponding Schmidt number for polymer diffusion should be of the order of 500 or larger, so that the appearance of sub-Kolmogorov scales is expected. Nonetheless, most works in the literature consider values of the Schmidt number lower than or of the order of one, and thus miss most of the small-scale dynamics. Therefore, significant effort has been devoted to analyzing and implementing accurate and robust numerical schemes. Moreover, simulations were performed either on highly resolved meshes with a global diffusion at high Schmidt number, or with artificial diffusion added only locally where stability requires it. In this way, most of the relevant small-scale dynamics can be captured. In addition, the Lagrangian approach provides a natural framework to treat such advection-dominated problems.

The main accomplishment of the project has been its critical contribution to the discovery and the characterization of a new state of turbulence, called elasto-inertial turbulence (EIT). This new regime has been both observed experimentally and reproduced numerically, and could represent the missing link between elastic turbulence and polymer drag reduction. EIT is driven by both inertial and elastic instabilities at large Weissenberg number and extensibility parameter. Once triggered, EIT is self-sustained since the elastic instability creates the very velocity fluctuations it feeds upon. EIT is most likely a two-dimensional phenomenon (i.e. it is observed in 2D simulations) characterized by very thin and elongated sheets of large polymer stretch in the vicinity of the wall, and weak spanwise vortical structures above them. Because this dynamics is driven by small scales, it can only be reproduced with an adequate numerical approach.

The observation of EIT structures in MDR flows suggests that their dynamics is not purely of Newtonian nature, but rather that it may be driven by EIT at finite Reynolds number and increasing Weissenberg number. The sheer magnitude of extensional viscosity is likely to prevent the emergence of any vortical structures, thus leaving MDR to be sustained by near-wall spanwise structures similar to the ones observed at low Reynolds numbers. The flow is therefore stuck in a transitional state, specifically the stage of breakdown of nonlinear flow instabilities, which does not support a logarithmic mean velocity profile. EIT also provides support to de Gennes' picture that drag reduction derives from two-way energy transfers between turbulent kinetic energy of the flow and elastic energy of polymers at small scales, resulting in an overall modification of the turbulence energy cascade at high Reynolds numbers. On the other hand, EIT is not confined to low Reynolds numbers as for elastic turbulence. Nonetheless, the exact role of inertia is still unclear. Whether EIT is simply a manifestation of elastic turbulence in inertial flows or a separate regime remains an open question.

Another accomplishment of the project has also been to provide new insight in the modification of heat transfer by polymer additives in weakly turbulent natural convection. Depending on the polymer parameters, two different regimes can be achieved: the polymers can either increase or decrease heat transfer. In both cases, viscoelasticity reduces the size of the convection cells, and thus increases their number. At moderate values of the Weissenberg number and extensibility parameter, this is sufficient to increase heat transfer. However, at larger values of these parameters, the rotation speed of the convection cells is drastically reduced, which provides an overall lower heat transfer. In two dimensions, the Newtonian simulations do not lead to a weakly turbulent flow, but to a quasi-steady state. The addition of polymers with moderate elasticity then stabilizes the flow. However, a higher elasticity produces a chaotic flow, in a similar fashion to EIT in channel flow forced convection.

The discovery of EIT has demonstrated that our previous understanding of the interaction between turbulence and polymers was only very partial. EIT has thus provided a first set of answers, but it has also led to a series of new questions that need now to be answered. Moreover, this research provides further suggestions on how to leverage the effect of polymers, or more generally viscoelastic rheology, for the manipulation of flows, such as for mixing or heat transfer.

In a broader context, the impact of this research is also the advancement of our understanding of fundamental aspects of turbulence. On the one hand, it demonstrates the possible existence of turbulence at subcritical Reynolds number (subcritical turbulence). This represents an important aspect for transition, which remains one of the major challenges in turbulence modeling. It also contributes to the understanding of backward energy transfer at small scales that is also observed in Newtonian turbulent flows. Finally, it suggests possible new strategies, beyond polymers, for controlling turbulent flows.

Although not all initial objectives have been yet achieved, the project has been very successful. Moreover, it is seen in a long-term context and research will continue beyond the present project.