Session X38: Focus Session: Chaotic Flows in Polymeric Solutions

Instabilities in rectilinear, sheared polymer flows and their relationship with elastic and elasto-inertial turbulence

Adding polymers to a Newtonian solvent introduces elasticity to the solution and it is now known can give rise to new forms of turbulence. `Elastic Turbulence’ (ET) arises when the elastic forces are large enough in the absence of inertia, and `Elasto-Inertial Turbulence’ (EIT) appears to need both inertia and elasticity to exist. Many questions exist about these two forms of viscoelastic turbulence: in particular, whether they are dynamically connected, what triggers them and how EIT interacts with the presence of Newtonian turbulence at low levels of elasticity (e.g. Datta et al. Phys. Rev Fluids, 7, 080701, 2022). I will attempt to review recent work seeking some answers largely stimulated by a recently-discovered viscoelastic centre mode instability (Garg et al. Phys. Rev. Lett., 121, 024502, 2018) and a newly-discovered `polymeric diffusive instability’ (Beneitez et al. arXiv:2210.09961, 2022).   

Polymeric diffusive instability (PDI) leading to viscoelastic chaos in planar wall-bounded flows

Elastic turbulence is a chaotic flow state observed in dilute polymer solutions in the absence of inertia. It was discovered experimentally in circular geometries and has long been thought to require a finite amplitude perturbation in wall-bounded parallel flows. In this talk we will discuss, within the commonly-used FENE-P model, that a self-sustaining chaotic state can be initiated via a linear instability in a simple inertialess shear flow caused by the presence of small but non-zero diffusivity of the polymer stress. Numerical simulations show that the instability leads to a three-dimensional self-sustaining chaotic state. We will discuss how this instability might present a generic pathway to experimentally observed viscoelastic chaotic states, i.e. elastic turbulence and elasto-inertial turbulence.   

Small-scale dynamics of elasto-inertial and elastic turbulence: Role and resolution of large polymer stress sheets

Polymer additives in liquid flows may trigger chaos in a variety of flows at various Reynolds numbers (Re). Under favorable conditions, polymers fully sustain flow perturbations in elastic turbulence (ET) when Re << 1. Elasto-inertial turbulence (EIT) is the general category of flows affected by polymer-driven perturbations at inertial Reynolds numbers. In some range of subcritical Reynolds numbers, polymer dynamics may drive chaos like ET, whereas, in some upper range of subcritical $Re$ and in a range of supercritical Re, EIT may coexist and interact with inertial instabilities and coherent structures with direct consequences to drag and heat transfer. In numerical simulations, a universal structure of ET and EIT is the thin sheet of large polymer stress, large polymer stress sheet (LPSS). It is surmised that sheet-to-sheet and/or sheet-to-wall interactions are at the center of chaos generation in ET and EIT. We conduct a systematic spectral analysis of fluctuations of polymer stress and flow variables for various flow configurations to discover the relation between Re, viscoelastic parameters, and polymer conformation tensor diffusion, characterized by a Peclet number Pe. Our study highlights the strong dependence of the LPSS’ thickness and intensity with Pe, which in turn has significant impact on the small and large scale dynamics of the flow. One of the flows isolates the sheet-to-sheet interactions, and the energy fluctuations they cause.   

Influence of geometric ordering on viscoelastic flow instabilities in 3D porous media

Many applications involve flow of viscoelastic polymer solutions in geometrically complex 3D porous media. Polymers accumulate elastic stresses as they navigate the pore space, leading to a flow instability characterized by spatiotemporally chaotic flow fluctuations. Our previous studies in disordered 3D media suggested that this instability onset is highly sensitive to medium geometry; however, how exactly geometry influences the flow instability remains unclear. We address this gap by directly imaging flow in microfabricated 3D porous media with precisely controlled geometries consisting of body-centered cuboid or simple-cubic arrays of spheres. Unexpectedly, in both cases, the flow instability is generated upstream of the contact regions between spheres rather than at sphere surfaces—suggesting that the consolidation of solid grains, inherent in naturally-occurring media, may play a pivotal role in establishing the flow instability in field settings. Further, the characteristics of the flow instability strongly depend on the unit cell geometry, and we quantify how the pore-scale flow features control the macroscopic flow resistance across the entire medium. Our work thus provides a key step towards elucidating how porous medium geometry shapes viscoelastic flow behavior.   

Accelerating mixing and reaction kinetics in porous media using an elastic instability

A wide range of environmental, industrial, and energy processes rely on reactive transport in disordered 3D porous media, but laminar flow under strong geometric confinement (Re«1) imposes a fundamental limit on reagent transport. Here, we report a novel technique to mimic turbulent-enhanced reactivity using by the addition of dilute high molecular weight polymers, which induce an elastic flow instability. Micro-scale imaging within a transparent porous medium reveals dynamic chaotic fluctuations that stretch and fold solute gradients exponentially in time—analogous to turbulent Batchelor mixing, despite the low Re. We observe a dramatic reduction in the required mixing length and improvement in the dispersion of concentration gradient, suggesting a cooperation between the elastic instability and the laminar chaotic advection inherent to the disordered 3D porous media. We show these two mixing mechanisms can be modeled with additive independent mixing rates, representing a dramatic conceptual simplification. We then extend these results to reactive mixing, accelerating a model reaction by 75% while simultaneously increasing throughput by 20×—circumventing the traditional trade-off between throughput and reactor length. Our results thus provide the first demonstration, to our knowledge, that elastic flow instabilities can provide turbulent-like enhancements in chemical reaction rates, which can operate cooperatively with laminar chaotic advection in industrially-relevant geometries.   

Elastoinertial nonlinear traveling waves

Polymeric additives are commonly used in pipeline transport of liquids such as crude oil transport, water heating and cooling systems, and airplane tank filling because the addition of a tiny amount of polymers dramatically reduces turbulent drag and ultimately the pumping cost. However, the turbulent drag cannot be reduced beyond a certain limit through the polymers. Because the polymeric addition can lead to the emergence of elasticity-induced a new type of turbulence known as elastoinertial turbulence, which is suspected to limit the drag reduction due to polymer additives. We investigate nonlinear traveling waves in the elastoinertial regime in viscoelastic channel flows using a parallelized nonlinear solver based on Dedalus and explore the possibility of pure traveling waves underlying the chaotic dynamics of elastoinertial turbulence. One family of such waves emerges from the nonlinear self-sustaining Tollmien–Schlichting wave of Newtonian flow. This study also sheds light on the transition route to elastoinertial turbulence in channel flow.   

Purely Elastic Instabilities in Channel Flows at Low Re: Dynamics, Structure & Resistance

Recent evidence suggests that the viscoelastic parallel shear flows may be unstable to finite amplitude perturbations. This type of instability is akin to the transition from laminar to turbulent flows in ordinary Newtonian fluids where the control parameter is the Reynolds number (Re). Here, on the hand, the control parameter is the Weissenberg number (Wi). In this talk, we present evidence of a subcritical nonlinear instability for the flow of a dilute polymeric solution in a parallel channel flows using a microfluidic device. The flow is investigated using dye advection, particle tracking velocimetry, and pressure sensors. Results show sustained velocity fluctuations far downstream away from the initial perturbation for strong enough disturbances, while small disturbances decay quickly under the same flow conditions. A hysteresis loop, characteristic of subcritical instabilities, is observed. The flow is characterized by non-periodic velocity fluctuation showing common signatures of elastic turbulence. Pressure measurements show rapid increase in drag as Wi is increased followed by and a turbulent-like regime characterized by a sudden decrease in drag and a weak dependence on Wi. Finally, we explore the mechanisms for these instabilities using 3D, holographic particle tracking.    

A blessing of viscoelastic narwhals

Recent years have seen significant progress in our understanding of chaotic motion of dilute polymer solutions. The key component of these advances is the discovery of a linear instability in a viscoelastic pressure-driven channel flows [1]. While the linear instability itself only exists at ultralow polymer concentrations and rather large values of the Weissenberg number, the ensuing 2D non-linear states have been shown to extend sub-critically to a wide range of experimentally relevant material parameters, both in the elasto-inertial [2] and purely-elastic regimes [3]. In 2D, these 'narwhal' states -  named to reflect the spatial organisation of the associated polymer stress - take the form of stable travelling-wave solutions, while in 3D they are linearly unstable [4] and are implicated in organising the dynamics of purely elastic turbulence [5].

Here, we investigate the spatial extent of narwhal states in the absence of inertia. We demonstrate that they are localised along the flow direction and we map their typical size for various material parameters. We further probe their localised nature by studying a group of travelling-wave states simultaneously occupying the same channel - a blessing of narwhals. We show that beyond a particular distance, narwhal states are insensitive to each other's presence and can co-exist. We speculate how these findings are connected to the dynamics of localised disturbances in 3D purely-elastic turbulence.

[1] M. Khalid et al., Phys. Rev. Lett. 127, 134502 (2021)

[2] J. Page et al., Phys. Rev. Lett. 125, 154501 (2020)

[3] A. Morozov, Phys. Rev. Lett. 129, 017801 (2022)

[4] M. Lellep et al., J. Fluid Mech. 959, R1 (2023).

[5] M. Lellep et al. (under review)   

Elastic range scaling in polymeric turbulence at high and low Reynolds number

Turbulent flows containing modest amounts of long-chained polymers have remained an intriguing area of research since the discovery of turbulent drag reduction. Here, we perform direct numerical simulations of statistically stationary, homogeneous, and isotropic turbulent flows of dilute solutions of polymers at both high and low Reynolds numbers. We show that the interplay of fluid inertia, viscous dissipation and polymer elasticity results in multi-scaling statistics. We explain the mechanism of the multi-scaling behavior by studying the contribution of the polymers to the flux of kinetic energy through scales, and further show how energy dissipation rate shows strong departures from a log-normal behavior. Finally, we extend the results to canonical shear flows.   

Hibernating states in elastoinertial turbulence

We report on the spatiotemporal evolution of the flow structures observed in a jet of dilute polymer solution entering a quiescent bath of either a Newtonian or a viscoelastic fluid. We study elastoinertial turbulence (EIT) in both Lagrangian and Eulerian form using high-speed digital Schlieren imaging and laser doppler velocimetry (LDV). Despite the spectral universality of the velocity and concentration fluctuations observed at fixed locations along the centerline of the turbulent jet in the EIT state, the Lagrangian structure can be tuned by varying the viscoelasticity of the surrounding bath or the extensibility of the polymer chains. The fluctuations far from the nozzle switch between active and hibernating states for jets in the viscoelastic bath. The percentage of time that the flow spends in the active and hibernating states, respectively, is set by the Reynolds number and the elasticity mismatch between the jet and the bath.   

Measuring flow-polymer misalignment fluctuations that drive instability

Viscoelastic flow instabilities limit polymer processing rates. Spatially and time-resolved measurements of stress and flow, and calculations of the same, may help clarify the mechanisms of hydrodynamic fluctuations and instability. We therefore studied flow of high-molar-mass polyethylene oxide solutions through a cross-slot geometry. At sufficiently high speeds, flow switches its asymmetric flow direction aperiodically. Data was acquired by synchronized velocimetry (2D piv and 3D holographic tracking) and stress measurements (polarization imaging) at sub-ms resolution. 3D numerical simulations (of the Giesekus model) also demonstrate similar flow switching behavior and confirm buildup of flow-polymer misalignment prior to switching of flow asymmetry. Extending this work to explore how molecular characteristics, fluid composition, and channel design might moderate susceptibility to misalignment may lead to more efficient processing.