Elastic turbulence can lead to to increased flow resistance, mixing and heat transfer. Its control -- either suppression or promotion -- has significant potential, and there is a concerted ongoing effort by the community to improve our understanding. Here we explore the dynamics of uncertainty in elastic turbulence, inspired by an approach recently applied to inertial turbulence in Ge et al. (2023) \textit{J. Fluid Mech.} 977:A17. We derive equations for the evolution of uncertainty measures, yielding insight on uncertainty growth mechanisms. Through numerical experiments, we identify four regimes of uncertainty evolution, characterised by I) rapid transfer to large scales, with large scale growth rates of $τ^{6}$ (where $τ$ represents time), II) a dissipative reduction of uncertainty, III) exponential growth at all scales, and IV) saturation. These regimes are governed by the interplay between advective and polymeric contributions (which tend to increase uncertainty), viscous, relaxation and dissipation effects (which reduce uncertainty), and inertial contributions. In elastic turbulence, reducing Reynolds number increases uncertainty at short times, but does not significantly influence the growth of uncertainty at later times. At late times, the growth of uncertainty increases with Weissenberg number, with decreasing polymeric diffusivity, and with the logarithm of the maximum length scale, as large flow features adjust the balance of advective and relaxation effects. These findings provide insight into the dynamics of elastic turbulence, offering a new approach for the analysis of viscoelastic flow instabilities.
We demonstrate through numerical solutions of the Oldroyd-B model in a two-dimensional Taylor-Couette geometry that the onset of elastic turbulence in a viscoelastic fluid can be controlled by imposed shear-rate modulations. While for slow modulations elastic turbulence is still present, it vanishes for fast modulations and a laminar response with the Taylor-Couette base flow is recovered. We find that the transition from the laminar to the turbulent state is supercritical and occurs at a critical Deborah number. In the state diagram of both control parameters, Weissenberg versus Deborah numbe…
I shortly describe the main results on elastically driven instabilities and elastic turbulence in viscoelastic inertia-less flows with curved streamlines. Then I describe a theory of elastic turbulence and prediction of elastic waves at Re<<1 and Wi>>1, which speed depends on the elastic stress similarly to the Alfven waves in magneto-hydrodynamics and in in a contrast to all other fluid flows with wave speed depending on medium elasticity. Since the established and testified mechanism of elastic instability of viscoelastic flows with curvilinear streamlines becomes ineffective at zero curvatu…
We investigate theoretically the near-wall region in elastic turbulence of a dilute polymer solution in the limit of large Weissenberg number. As it was established experimentally, elastic turbulence possesses a boundary layer where the fluid velocity field can be approximated by a steady shear flow with relatively small fluctuations on the top of it. Assuming that at the bottom of the boundary layer the dissolved polymers can be considered as passive objects, we examine analytically and numerically statistics of the polymer conformation, which is highly nonuniform in the wall-normal direction…
We report the onset of elastic turbulence in a two-dimensional Taylor-Couette geometry using numerical solutions of the Oldroyd-B model, also performed at high Weissenberg numbers with the program OpenFOAM. Beyond a critical Weissenberg number, an elastic instability causes a supercritical transition from the laminar Taylor-Couette to a turbulent flow. The order parameter, the time average of secondary-flow strength, follows the scaling law $Φ\propto (\mathrm{Wi} -\mathrm{Wi}_c)^γ$ with $\mathrm{Wi}_c=10$ and $γ= 0.45$. The power spectrum of the velocity fluctuations shows a power-law decay wi…
Transition from a weak-elastic to an elastic-activated regime in two-dimensional polymeric turbulence
Huazhong University of Science and Technology（华中科技大学）；Hubei Key Laboratory of Engineering Structural Analysis and Safety Assessment（湖北省工程结构分析与安全评估重点实验室）
We present the full three-dimensional numerical investigation of the von Karman 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 …
Nonhomogeneous elastic turbulence in the two-dimensional Taylor-Couette flow
Univ. Lille, CNRS, ONERA, Arts et Métiers Institute of Technology, Centrale Lille（里尔大学、法国国家科学研究中心、法国航空国防实验室、巴黎国立高等工艺学院、里尔中央理工学院）；Univ. Lille（里尔大学）；Laboratoire de Thermique et Energie de Nantes, CNRS, Nantes Université（南特热能与实验室、法国国家科学研究中心、南特大学）
We study single-phase and particulate turbulent channel flows, bounded by two incompressible hyper-elastic walls. Different wall elasticities are considered with and without a 10% volume fraction of finite-size rigid spherical particles, while elastic walls are modelled as a neo-Hookean material. We report a significant drag increase and an enhancement of the turbulence activity with growing wall elasticity for both single-phase and particulate cases in comparison with the single-phase flow over rigid walls. A drag reduction and a turbulence attenuation is obtained for the particulate cases wi…
Small flexible fibers in a turbulent flow are found to be most of the time as straight as stiff rods. This is due to the cooperative action of flexural rigidity and fluid stretching. However, fibers might bend and buckle when they tumble and experience a strong-enough local compressive shear. Such events are similar to an activation process, where the role of temperature is played by the inverse of Young's modulus. Numerical simulations show that buckling occurs very intermittently in time. This results from unexpected long-range Lagrangian correlations of the turbulent shear.
We numerically investigate the spatial and temporal statistical properties of a dilute polymer solution in the elastic turbulence regime, i.e., in the chaotic flow state occurring at vanishing Reynolds and high Weissenberg numbers. We aim at elucidating the relations between measurements of flow properties performed in the spatial domain with the ones taken in the temporal domain, which is a key point for the interpretation of experimental results on elastic turbulence and to discuss the validity of Taylor's hypothesis. To this end, we carry out extensive direct numerical simulations of the tw…
We study the fully-coupled dynamics between a fully-developed turbulent flow and an ensemble of immersed flexible fibers. We vary the concentration of the suspension, the mechanical properties and the length of the fibers in a vast parametric range. For all configurations, the fiber dynamics falls in only two possible dynamical states: (i) the fiber manifests its natural response to the flow forcing or (ii) its motion fully synchronizes to the hydrodynamic timescales of the turbulent flow. This scenario holds for both a dilute condition, where the carrier flow is not affected by the fluid-stru…
Recent progress in the study of Cahn-Hilliard Navier-Stokes (CHNS) turbulence is summarized. This is an example of \textit{elastic turbulence}, which can occur in elastic (i.e. self-restoring) media. Such media exhibit memory due freezing-in laws, as does MHD, which in turn constrains the dynamics. We report new results in the theory of CHNS turbulence in 2D, with special emphasis on the role of structure (i.e. `blob') formation and its interaction with the dual cascade. The evolution of a concentration gradient in response to a single eddy -- analogous to flux expulsion in MHD -- is analyzed.…
We use direct numerical simulations to study homogeneous, and isotropic turbulent flows of dilute polymer solutions at high Reynolds and Deborah numbers. We find that for small wavenumbers $k$, the kinetic energy spectrum shows Kolmogorov--like behavior which crosses over at a larger $k$ to a novel, elastic scaling regime, $E(k) \sim k^{-ξ}$, with $ξ\approx 2.3$. We study the contribution of the polymers to the flux of kinetic energy through scales, and find that it can be decomposed into two parts: one increase in effective viscous dissipation, and a purely elastic contribution that dominates…
Viscoelastic fluids are a subclass of complex fluids used in widespread applications ranging from biological to large-scale industrial settings. These fluids are often associated with various complex flow phenomena due to the presence of non-linear elastic stresses, originating due to the stretching and relaxation phenomena of microstructure (such as polymer molecules in the case of a viscoelastic polymer solution) in a deformed flow field. One such phenomenon is elastic instability (EI) which emerges due to the interaction between elastic stresses and streamline curvature present in a flow sy…
In the present study, we investigated flow structures and properties of elastic turbulence in straight 2D channel viscoelastic fluid flow and tested earlier observations. We discovered self-organized cycling process of weakly unstable coherent structures (CSs) of co-existing streaks and stream-wise vortices, with the former being destroyed by Kelvin-Helmholtz-like instability resulting in chaotic structures. The sequence periodically repeats itself leading to stochastically steady state. This self-sustained process (SSP) remarkably resembles one investigated theoretically and experimentally fo…
With the advent of improved computational resources, aerospace design has testing-based process to a simulation-driven procedure, wherein uncertainties in design and operating conditions are explicitly accounted for in the design under uncertainty methodology. A key source of such uncertainties in design are the closure models used to account for fluid turbulence. In spite of their importance, no reliable and extensively tested modules are available to estimate this epistemic uncertainty. In this article, we outline the EQUiPS uncertainty estimation module developed for the SU2 CFD suite that …
Elastic turbulence is a chaotic regime that emerges in polymer solutions at low Reynolds numbers. A common way to ensure stability in numerical simulations of polymer solutions is to add artificially large polymer-stress diffusion. In order to assess the accuracy of this approach in the elastic-turbulence regime, we compare numerical simulations of the two-dimensional Oldroyd-B and FENE-P models sustained by a cellular force with and without artificial diffusion. We find that artificial diffusion can have a dramatic effect even on the large-scale properties of the flow and we show some of the …
In a viscoelastic flow over a microfluidic canopy of polymeric pillars, we report the spontaneous emergence of waves in the form of propagating regions of low flow velocity compared to the surrounding flow. The occurrence of the wave is chaotic and shows characteristics of elastic turbulence. We systematically study the coupling between the low velocity wave and the microfluidic canopy by combining flow velocimetry experiments and high speed tracking of the pillars. The waves form an angle $β$ with the primary flow direction that depends on the geometry of the pillar array. If the canopy is co…
Minute amount of long chain flexible polymer dissolved in a turbulent flow can drastically change flow properties, such as reducing the drag and enhancing mixing. One fundamental riddle is how these polymer additives interact with the eddies of different spatial scales existing in the turbulent flow and in turn alter the turbulence energy transfer. Here we show how turbulent kinetic energy is transferred through deferent scales in the presence of the polymer additives. In particular, we observed experimentally the emerging of a new scaling range, referred to as the elastic range, where increas…
White paper: The aim of this work is to apply and analyze machine learning methods for uncertainty quantification of turbulence models. In this work we investigate the classical and data-driven variants of the eigenspace perturbation method. This methodology is designed to estimate the uncertainties related to the shape of the modeled Reynolds stress tensor in the Navier-Stokes equations for Computational Fluid Dynamics (CFD). The underlying methodology is extended by adding a data-driven, physics-constrained machine learning approach in order to predict local perturbations of the Reynolds str…
Speed of sound waves in gases and liquids is governed by medium compressibility. There exists another type of non-dispersive waves which speed depends on stress instead of medium elasticity. A well-known example is the Alfven wave propagating, with a speed determined by a magnetic tension, in plasma permeated by a magnetic field. Later, an elastic analog of the Alfven waves has been predicted in a flow of dilute polymer solution, where elastic stress engendered by polymer stretching determines the elastic wave speed. Here, we present quantitative evidence of elastic Alfven waves observed in el…
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 [Gupta & Vincenzi, J. Fluid Mech. 870, 405-418 (2019); Dzanic, From & Sauret, J. Fluid Mech. 937, A31 (2022)] 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…
The broken link between space and time in elastic turbulence
Okinawa Institute of Science and Technology Graduate University（冲绳科学技术大学院大学）；Politecnico di Milano（米兰理工大学）；Nordita（北欧理论天体物理学研究所）；KTH Royal Institute of Technology（瑞典皇家理工学院）；Stockholm University（斯德哥尔摩大学）
Reynolds-averaged Navier-Stokes (RANS)-based transition modeling is widely used in aerospace applications but suffers inaccuracies due to the Boussinesq turbulent viscosity hypothesis. The eigenspace perturbation method can estimate the accuracy of a RANS model by injecting perturbations to its predicted Reynolds stresses. However, there lacks a reliable method for choosing the strength of the injected perturbation, while existing machine learning models are often complex and data craving. We examined two light-weighted machine learning models to help select the strength of the injected pertur…
We report evidence of irregular unsteady flow of two-dimensional polymer solutions in the absence of inertia in cross-slot geometry using numerical simulations of Oldroyd-B model. By exploring the transition to time-dependent flow versus both the fluid elasticity and the polymer concentration, we find periodic behaviour close to the instability threshold and more complex flows at larger elasticity, in agreement with experimental findings. For high enough elasticity we obtain dynamics pointing to elastic turbulence, with temporal spectra of velocity fluctuations showing a power-law decay, of ex…
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 wa…
The prediction of extreme uncertainty-production events in three-dimensional Navier-Stokes turbulence
Univ. Lille, CNRS, ONERA, Arts et Métiers ParisTech, Centrale Lille, UMR 9014 - LMFL - Laboratoire de Mécanique des Fluides de Lille - Kampé de Feriet（里尔大学、法国国家科学研究中心、法国航空发展局、巴黎工艺美术学院、里尔中央理工学院）
