Elastic Turbulence Research Articles

Announcement: PRFluids Publishes Invited Perspective on Viscoelastic flow instabilities and elastic turbulence

Announcement: PRFluids Publishes Invited Perspective on Viscoelastic flow instabilities and elastic turbulence

Perspectives on viscoelastic flow instabilities and elastic turbulence

Viscoelastic fluids can exhibit striking flow instabilities under conditions where ordinary Newtonian fluids are stable, owing to the nonlinear coupling of the elastic and viscous stresses. This article provides perspectives on viscoelastic flow instabilities by integrating the input from speakers at a recent international workshop: historical remarks, characterization of fluids and flows, discussion of experimental and simulation tools, and modern questions and puzzles that motivate further studies of this fascinating subject. The materials here will be useful for researchers and educators alike, especially as the subject continues to evolve in both fundamental understanding and applications in engineering and the sciences.

Experimental study on the interaction of solitary wave with elastic submerged plate

This study aimed to understand the effect of stiffness under the interaction of a thin plate with a solitary wave. For this purpose, two-dimensional experiments were carried out with three different plates. Surface elevation, elastic plate motion, velocity field, and turbulence field were measured using the particle image velocimetry (PIV) technique. We first illustrated the evolution of the flow fields. Furthermore, a comparison study of different materials was carried out which showed that the more rigid plate could generate the larger velocity magnitude around the core, but the smaller velocity magnitude at the bottom in the end. By comparing the time histories of surface elevation, the x direction displacement at the upper-left corner of the plate, and the acting force on the plate, it was found that the maximum force occurred first, and the maximum deformation of the plate occurred last. The result showed the higher stiffness the greater wave attenuation behind the plate. Interestingly, it was also seen that the force varied linearly with the wave nonlinearity for different elastic plates. Not only can the detailed flow hydrodynamics and motion of elastic plate be obtained and analyzed, but the newly measured data can be used for validating numerical models.

Relaminarization of elastic turbulence

We report frictional drag reduction and a complete flow re-laminarization of elastic turbulence (ET) at vanishing inertia in a viscoelastic channel flow past an obstacle. We show that intensity of observed elastic waves and wall-normal vorticity correlate well with the measured drag above the ET onset. Moreover, we find that the elastic wave frequency grows with Weissenberg number, and at sufficiently high frequency it causes decay of the elastic waves, resulting in ET attenuation and drag reduction. Thus, this allows us to substantiate a physical mechanism, involving interaction of elastic waves with wall-normal vorticity fluctuations, leading to the drag reduction and re-laminarization phenomena at low Reynolds number.

Flow of wormlike micelles: From shear banding to elastic turbulence

Shear banding is a ubiquitous phenomenon in complex fluids flows. Most often it corresponds to a flow state in which a fluid splits into layers of differing apparent viscosities supporting different local shear rates. Shear banding has been observed, either transiently or at steady state, in different classes of complex systems including polymeric fluids and soft glassy materials. Its description and understanding is particularly advanced in surfactant wormlike micelles, for which the experimental evidence of its mechanical signature dates back to the 1990s. Over the past 10 years a strong effort was made to combine global rheology with local time-resolved techniques to probe both the velocity field and the structural properties in various macro- and micro-fluidic geometries. Overall, complex flows have been observed to emerge on top of the shear-banding flow and have been connected to the development of viscoelastic instabilities.In this talk I will try to describe the complete picture of the shear-banding transition in wormlike micelles flowing in Taylor-Couette geometry, from the onset of banding to the development of elastic turbulence. The response of shear-banding wormlike to time-dependent flow protocols such as step stress and shear startup will be examined. The mechanical signatures of the onset of banding will be discussed and compared with general criteria derived by Moorcroft and Fielding. Then I will describe the spatio-temporal dynamics over longer time scales showing the development of secondary flows on top of shear banding. I will also address the transition towards a disordered flow state reminiscent of elastic turbulence in regular polymers. Finally, we will see that non-shear banding wormlike micelles also exhibit elastic instabilities and turbulence, providing a way to explore a large range of elasticity numbers.

Complex fluids with visco-elastic rheology for applications in oil & gas industry

<h2>Abstract</h2> During the oil and gas recovery from subsurface, the recovery efficiency is mainly limited by two factors: pore-scale trapping resulting in irreducible hydrocarbon saturations and bypassing because of limited sweep efficiency due to rock heterogeneity, viscosity contrast and viscous instabilities. In order to improve the sweep efficiency, hydro-soluble polymer solutions with Non-Newtonian rheology are applied. The difficulty is in predicting the in-situ effective viscosity in Darcy-scale flow in the porous medium, based on bulk rheology provided for instance by shear rheometers (Berg and van Wunnik, 2017). Most of the relevant polymer systems, however, also exhibit viscoelasticity which can lead to elastic turbulence during flow through porous medium (De et al., 2022a). Here we show progress on the fundamental side, how to visualize the respective pore scale flow fields and understand the fundamental origins using a system that contains one single pore throat (Eguagie et al., 2020) and multiple pore throats and its effect on residual oil saturation (De et al., 2022b). In addition, we demonstrate how we numerically simulate the flow of viscoelastic fluids in porous medium and characterize elastic turbulence (De et al., 2022c).

Patches of patches: Spatial fluctuations of elastic turbulence in porous media

Polymer solutions are often injected in porous media for applications such as oil recovery and groundwater remediation. As the fluid navigates the tortuous pore space, elastic stresses build up, causing the flow to become unstable at sufficiently large flow rates—a phenomenon often known as “elastic turbulence”. However, what physical factors determine the onset of this instability, and what its spatial and temporal characteristics are, remain unknown. In this talk, I describe some of our work using direct visualization in model porous media to address this gap in knowledge. First, we show that the spacing between pores strongly influences the flow: when the pore spacing is sufficiently small, the unstable flow in the different pores exhibits a surprising bistability due to the interplay between elongation and relaxation of polymers as they are advected through the pore space (Browne et al., J. Fluid Mech. 2020). Second, using experiments in full three-dimensional porous media, we show that the onset of unstable flow in each pore is akin to a second-order phase transition, arising due to the persistence of discrete patches of instability. Thus, unstable flow is patchy across the different pores of the medium. Guided by these findings, we directly link the energy dissipated by pore-scale fluctuations to the flow resistance through the entire medium, enabling prediction of the macroscopic transport behavior (Browne and Datta, Science Advances 2021). Together, these results reveal the rich array of behaviors that can arise during the unstable flow of polymer solutions through porous media, and provide a general framework by which flow fluctuations can be predicted and controlled.

Numerical simulations of interfacial and elastic instabilities

<h2>Abstract</h2> The interplay of elasticity and capillarity in free-surface flows of viscoelastic fluids gives rise to intriguing flow instabilities. We present novel numerical simulations and analyse the physics related to the appearance of surface distortions on polymer melt extrudates, often referred to as "sharkskin" instability, and the abrupt increase in the rise velocity of a bubble in a viscoelastic solution. Regarding the extrusion process, the transition from smooth to wavy extrudate with increasing flow rate is attributed to a Hopf bifurcation, followed by a sequence of period-doubling bifurcations, which eventually lead to elastic turbulence under creeping flow. We suggest that the inception of these waves is related with intense stretch of polymer chains and subsequent recoil at the region where the melt detaches from the solid wall of the die. The ‘velocity discontinuity' of the rising bubble is attributed to a hysteresis loop, triggered by the change of shape of the rear pole into a tip that favors the formation of intense stress gradients, which reduce the fluid viscosity and facilitate the bubble translation. Finally, we discuss the numerical methodology that has allowed us to obtain stable numerical solutions in highly deformed meshes and for high values of the Weissenberg number.

Conserving elastic turbulence numerically using artificial diffusivity.

To simulate elastic turbulence, where viscoelasticity dominates, numerical solvers introduce an artificial stress diffusivity term to handle the steep polymer stress gradients that ensue. This has recently been shown [A. Gupta and D. Vincenzi, J. Fluid Mech. 870, 405 (2019)0022-112010.1017/jfm.2019.224; V. Dzanic et al., J. Fluid Mech. 937, A31 (2022)0022-112010.1017/jfm.2022.103] to introduce unphysical artifacts with a detrimental impact on simulations. In this Letter, we propose that artificial diffusion is limited to regions where stress gradients are steep instead of seeking the zero-diffusivity limit. Through the cellular forcing and four-roll mill problem, we demonstrate that this modified artificial diffusivity is devoid of unphysical artifacts, allowing all features of elastic turbulence to be retained. Results are found to conform with direct simulations, reducing the impact of artificial diffusivity from a qualitative scale to a quantitative scale while only requiring a fraction of the numerical resolution.

Characterizing purely elastic turbulent flow of a semi-dilute entangled polymer solution in a serpentine channel

Polymer solutions in the semi-dilute regime are of considerable industrial importance. The complex rheological properties of such highly viscoelastic fluids and the complexity of their flow characteristics, especially in curved geometries, necessitate a thorough experimental characterization of the dynamics of such fluid flows. We apply statistical, spectral, and structural analyses to the experimentally obtained velocity fields of a semi-dilute entangled polymer solution in a serpentine channel to fully characterize the corresponding flow. Our results show that at high Weissenberg numbers, yet vanishing Reynolds numbers, the flow resistance is significantly increased, which indicates the emergence of a purely elastic turbulent flow. Spatial flow observations and statistical analysis of temporal flow features show that this purely elastic turbulent flow is non-homogeneous, non-Gaussian, and anisotropic at all scales. Moreover, spectral analysis indicates that compared to elastic turbulence in the dilute regime, the range of present scales of the excited fluctuations is narrower. This is partly due to the entanglement of the polymers in this concentration regime, which restricts their movement, and partly due to the mixed flow type inherent in the serpentine geometry, which can reduce the extent of polymer stretching and, thus, reduce the intensity of the fluctuations in the flow. Furthermore, proper orthogonal decomposition analysis is applied to directly extract the turbulent flow structure and reveals the activity of the counter-rotating vortices associated with secondary flow, which significantly contribute to the total kinetic energy of the flow.

Coherent Structures in Plane Channel Flow of Dilute Polymer Solutions with Vanishing Inertia.

When subjected to sufficiently strong velocity gradients, solutions of long, flexible polymers exhibit flow instabilities and chaotic motion, often referred to as elastic turbulence. Its mechanism differs from the familiar, inertia-driven turbulence in Newtonian fluids and is poorly understood. Here, we demonstrate that the dynamics of purely elastic pressure-driven channel flows of dilute polymer solutions are organized by exact coherent structures that take the form of two-dimensional traveling waves. Our results demonstrate that no linear instability is required to sustain such traveling wave solutions and that their origin is purely elastic in nature. We show that the associated stress profiles are characterized by thin, filamentlike arrangements of polymer stretch, which is sustained by a solitary pair of vortices. We discuss the implications of the traveling wave solutions for the transition to elastic turbulence in straight channels and propose ways for their detection in experiments.

New direction and perspectives in elastic instability and turbulence in various viscoelastic flow geometries without inertia

We shortly describe the main results on elastically driven instabilities and elastic turbulence in viscoelastic inertialess flows with curved streamlines. Then we describe a theory of elastic turbulence and prediction of elastic waves Re ≪ 1 and Wi ≫ 1, which speed depends on the elastic stress similar to the Alfvén waves in magneto-hydrodynamics and in a contrast to all other, which speed depends on medium elasticity. Since the established and testified mechanism of elastic instability of viscoelastic flows with curvilinear streamlines becomes ineffective at zero curvature, so parallel shear flows are proved linearly stable, similar to Newtonian parallel shear flows. However, the linear stability of parallel shear flows does not imply their global stability. Here we switch to the main subject, namely a recent development in inertialess parallel shear channel flow of polymer solutions. In such flow, we discover an elastically driven instability, elastic turbulence, elastic waves, and drag reduction down to relaminarization that contradict the linear stability prediction. In this regard, we discuss briefly normal versus non-normal bifurcations in such flows, flow resistance, velocity and pressure fluctuations, and coherent structures and spectral properties of a velocity field as a function of Wi at high elasticity number.

Corrigendum to “The Oldroyd-B fluid in elastic instabilities, turbulence and particle suspensions”;

Corrigendum to “The Oldroyd-B fluid in elastic instabilities, turbulence and particle suspensions”;

Towards Predicting the Onset of Elastic Turbulence in Complex Geometries

Flow of complex fluids in porous structures is pertinent in many biological and industrial processes. For these applications, elastic turbulence, a viscoelastic instability occurring at low Re—arising from a non-trivial coupling of fluid rheology and flow geometry—is a common and relevant effect because of significant over-proportional increase in pressure drop and spatio-temporal distortion of the flow field. Therefore, significant efforts have been made to predict the onset of elastic turbulence in flow geometries with constrictions. The onset of flow perturbations to fluid streamlines is not adequately captured by Deborah and Weissenberg numbers. The introduction of more complex dimensionless numbers such as the M-criterion, which was meant as a simple and pragmatic method to predict the onset of elastic instabilities as an order-of-magnitude estimate, has been successful for simpler geometries. However, for more complex geometries which are encountered in many relevant applications, sometimes discrepancies between experimental observation and M-criteria prediction have been encountered. So far these discrepancies have been mainly attributed to the emergence from disorder. In this experimental study, we employ a single channel with multiple constrictions at varying distance and aspect ratios. We show that adjacent constrictions can interact via non-laminar flow field instabilities caused by a combination of individual geometry and viscoelastic rheology depending (besides other factors) explicitly on the distance between adjacent constrictions. This provides intuitive insight on a more conceptual level why the M-criteria predictions are not more precise. Our findings suggest that coupling of rheological effects and fluid geometry is more complex and implicit and controlled by more length scales than are currently employed. For translating bulk fluid, rheology determined by classical rheometry into the effective behaviour in complex porous geometries requires consideration of more than only one repeat element. Our findings open the path towards more accurate prediction of the onset of elastic turbulence, which many applications will benefit.Article HighlightsWe demonstrate that adjacent constrictions “interact” via the non-laminar flow fields caused by individual constrictions, implying that the coupling of rheological effects and fluid geometry is more complex and implicit.The concept of characterizing fluid rheology independent of flow geometry and later coupling back to the geometry of interest via dimensionless numbers may fall short of relevant length scales, such as the separation of constrictions which control the overlap of flow fields.By providing direct experimental evidence illustrating the cause of the shortcoming of the status-quo, the expected impact of this work is to challenge and augment existing concepts that will ultimately lead to the correct prediction of the onset of elastic turbulence.

Characterizing elastic turbulence in the three-dimensional von Kármán swirling flow using the Oldroyd-B model

We present a comprehensive three-dimensional numerical investigation of the von Kármán swirling flow between two parallel plates using the Oldroyd-B model and characterize the onset and development of elastic turbulence. We quantify the flow state with the secondary-flow strength, a measure of the average strength of the velocity fluctuations, and then define an order parameter as the time average of the secondary-flow strength. The order parameter displays a subcritical transition from the laminar to a bistable flow that switches between weakly chaotic flow and elastic turbulence. The transition to the bistable flow occurs at the critical Weissenberg number Wic=12. In the elastic turbulent state, we observe a strong increase in velocity fluctuations and flow resistance which we define as the total work performed on the fluid. Upon starting simulations in the turbulent state and subsequently lowering Wi below its critical value, we observe hysteretic behavior in the order parameter and the flow resistance, which is a common feature of a subcritical transition. Hysteresis has also been found in experiments. Additionally, we find power-law scaling in the spatial and temporal power spectra of the velocity fluctuations, a characteristic for elastic turbulence. The maximum values of the power-law exponents in our simulations are αt=3.69 for the temporal exponent and αs=3.18 for the spatial exponent, which are remarkably close to the values obtained in experiments.

The effect of periodicity in the elastic turbulence regime

In this work, we show that the double-periodic boundary conditions typically applied in numerical simulations of elastic turbulence can lead to significantly incorrect results if not treated properly. This is demonstrated by simulating elastic turbulence using the popular four-roll mill benchmark at different levels of periodicity, namely, 16, 36 and 64 rolls using the popular Oldroyd-B model with added artificial diffusivity. We find that the initial onset of elastic turbulence causes a breakdown in symmetry independent of periodicity, which is characterised by a leading vortex and is known to be attributed to artificial diffusivity. Beyond this initial transition, the standard four-roll mill case transitions into a periodic state, a well-known characteristic from the literature. On the other hand, the cases with higher levels of periodicity quickly overcome the effects of a leading vortex and experience purely chaotic flow fluctuations, characterised by a broadband spectrum and steep power law behaviour. Certain qualities of the flow at higher levels of periodicity are reminiscent of the true solutions of elastic turbulence obtained numerically without any artificial diffusivity (Gupta &amp; Vincenzi, J. Fluid Mech., vol. 870, 2019). These results suggest that the well-known periodic states observed for the four-roll mill are due to insufficient periodicity as the problem transitions into the elastic turbulence regime, leading to a dominant vortex cycling around all four quadrants of the unit cell throughout time unable to recover the initial symmetry. This work demonstrates the importance and caution required when applying periodic boundary conditions in numerical experiments of the elastic turbulence regime and further emphasises the impact and care required for artificial diffusivity.

Flow resistance and convective heat transfer by elastic turbulence in 1D/2D/3D geometries

Inducing elastic turbulence is a promising method to intensify flow and heat transfer at low Reynolds number Re. Different geometries shall have great impacts on the occurrence of elastic turbulence and its heat transfer effect, which are however seldom investigated especially for flow in 3D geometry. This work conducts a comparative experimental study of the dimension effect on elastic turbulence on 3D, 2D and 1D geometries with similar hydraulic diameters. Polyacrylamide is used to induce elastic turbulence and the corresponding flow resistance and heat transfer performance are obtained for different geometries under laminar flow conditions. The results provide strong evidence that curvatures are necessary for the onset of elastic turbulence in 1D geometry as shown by similar flow resistance and heat transfer coefficient between the Newtonian and viscoelastic fluids. Comparing to the 2D geometry, flow in 3D geometry achieves both reduced pressure drop gradient and increased heat transfer at the same time due to the unique interaction of elastic stress with the torsion and rotation effects. New correlations of Nusselt number Nu and Weissenberg number Wi are proposed in curved geometries as Nu∝Wi2.2 and Wi1.9 for the 3D and 2D flow structures, respectively, under the boundary condition of constant heat flux. The mechanisms of flow and heat transfer of elastic turbulence in different geometries are also discussed to help further optimization.

Influence of elastic instability and elastic turbulence on mixed convection of viscoelastic fluids in a lid-driven cavity

The addition of a minute amount of solid polymers into a Newtonian solvent like water can originate an elastic instability as the Weissenberg number (ratio of the elastic forces to that of the viscous forces) gradually increases. These instabilities further transit to elastic turbulence (ET) regime at high Weissenberg numbers. In this study, we investigate how these elastic instabilities and elastic turbulence would tend to influence the mixed convective heat transfer phenomena in a lid-driven cavity, which is one of the widely studied problems in the domain of transport phenomena. In doing so, extensive two-dimensional time-resolved numerical simulations have been conducted by using an open-source CFD code OpenFOAM. The Oldroyd-B viscoelastic constitutive model is used in this study to realize the rheological behaviour of a constant shear-viscosity viscoelastic fluid or the so-called Boger fluid. To investigate the effect of the shear-thinning behaviour of a viscoelastic fluid on the elastic instability and elastic turbulence phenomena, we have further conducted limited simulations using the FENE-P (finitely extensible non-linear elastic spring with the Peterlin’s approximation) viscoelastic fluid model. A detailed discussion is presented on how the flow dynamics and heat transfer mechanism inside the cavity can be influenced by the Weissenberg number along with the Reynolds, Prandtl, and Richardson numbers. We show that the heat transfer rate inside the cavity can be increased by more than 100% under appropriate conditions if one uses a constant shear-viscosity Oldroyd-B fluid instead of a simple Newtonian fluid due to the presence of elastic instability and elastic turbulence in the former fluid. Furthermore, we have found that the shear-thinning behaviour of a FENE-P viscoelastic fluid tends to suppress these instabilities and turbulence and hence decreases the heat transfer rate inside the cavity. Therefore, one can use the present approach of using a constant shear-viscosity Oldroyd-B fluid as an efficient way to increase the rate of heat and mass transport processes in various systems. Our approach can be served as an alternative possibility to the existing approaches such as the application of an electric or a magnetic field or the use of a nanofluid. However, one needs to be careful in the selection of the polymer type and its degradation behaviour as well as its appropriate concentration in the solution when choosing this approach.

The Oldroyd-B fluid in elastic instabilities, turbulence and particle suspensions

The Oldroyd-B fluid has become the starting point for almost all complex flow calculations and analysis involving the behavior of dilute polymer solutions. The reasons for this are clear: based on the kinetic theory of high molecular weight “phantom” chains near equilibrium, the resulting model, in its simplest form, produces a single mode description of the solution constitutive equation in terms of a single symmetric dyad order parameter. The physical connection to the individual chains is therefore unambiguous through the end-to-end configuration tensor. The flow of the fluid resulting from the mathematical solution of the constitutive equation coupled to Cauchy’s equation of motion and continuity has been examined for decades and its strengths and weaknesses in predicting the features of non-linear viscometric and non-viscometric polymer solution flows are well known. Moreover, “simple” extensions (e.g. the FENE-P model) can ameliorate the most serious of these shortcomings. For example, the multi-mode Oldroyd-B model can provide quantitative predictions of the linear viscoelasticity of a wide range of polymer solutions. Remarkably, even with all its shortcomings, the Oldroyd-B fluid has been invaluable in predicting at least qualitatively, features as complex as purely elastic instabilities, turbulent drag modification, and even the effective rheology of particles suspended in polymer solutions. While remaining mindful of the model’s shortcomings, any student of these diverse fields may find the Oldroyd-B fluid an important starting point.

Elastic turbulence generates anomalous flow resistance in porous media.

Many energy, environmental, industrial, and microfluidic processes rely on the flow of polymer solutions through porous media. Unexpectedly, the macroscopic flow resistance often increases above a threshold flow rate in a porous medium, but not in bulk solution. The reason why has been a puzzle for over half a century. Here, by directly visualizing flow in a transparent 3D porous medium, we demonstrate that this anomalous increase is due to the onset of an elastic instability in which the flow exhibits strong spatiotemporal fluctuations reminiscent of inertial turbulence, despite the small Reynolds number. Our measurements enable us to quantitatively establish that the energy dissipated by pore-scale fluctuations generates the anomalous increase in the overall flow resistance. Because the macroscopic resistance is one of the most fundamental descriptors of fluid flow, our results both help deepen understanding of complex fluid flows and provide guidelines to inform a broad range of applications.