Polymer additives can substantially reduce the drag of turbulent flows and the upper limit, the so called "maximum drag reduction" (MDR) asymptote is universal, i.e. independent of the type of polymer and solvent used. Until recently, the consensus was that, in this limit, flows are in a marginal state where only a minimal level of turbulence activity persists. Observations in direct numerical simulations using minimal sized channels appeared to support this view and reported long "hibernation" periods where turbulence is marginalized. In simulations of pipe flow we find that, indeed, with increasing Weissenberg number (Wi), turbulence expresses long periods of hibernation if the domain size is small. However, with increasing pipe length, the temporal hibernation continuously alters to spatio-temporal intermittency and here the flow consists of turbulent puffs surrounded by laminar flow. Moreover, upon an increase in Wi, the flow fully relaminarises, in agreement with recent experiments. At even larger Wi, a different instability is encountered causing a drag increase towards MDR. Our findings hence link earlier minimal flow unit simulations with recent experiments and confirm that the addition of polymers initially suppresses Newtonian turbulence and leads to a reverse transition. The MDR state on the other hand results from a separate instability and the underlying dynamics corresponds to the recently proposed state of elasto-inertial-turbulence (EIT).
The velocity and friction properties of laminar pipe flow of a viscoelastic solution are bounded by the corresponding values for two Newtonian fluids, namely, the solvent and a fluid with a viscosity identical to the total viscosity of the solution. The lower friction factor for the flow of the solution when compared to the latter is tracked to an increased strain rate needed to enhance viscous dissipation. Lastly, we show analytically that the effective viscosity varies similarly to the radial diagonal component of the conformation tensor as observed numerically in turbulent flows, and give a…
Polymer additives are commonly utilized to manipulate bubbly flows in various applications. Here, we investigate the effects of clean and contaminated bubbles driven upwards (upflow) in Newtonian and viscoelastic turbulent channel flows. Interface-resolved direct numerical simulations are performed to examine sole and combined effects of soluble surfactant and viscoelasticity using an efficient 3D finite-difference/front-tracking method. The incompressible flow equations are solved fully coupled with the FENE-P viscoelastic model and the equations governing interfacial and bulk surfactant conc…
We investigate the effects of dissipative air drag on the dynamics of electrified jets in the initial stage of the electrospinning process. The main idea is to use a Brownian noise to model air drag effects on the uniaxial elongation of the jets. The developed numerical model is used to probe the dynamics of electrified polymer jets at different conditions of air drag force, showing that the dynamics of the charged jet is strongly biased by the presence of air drag forces. This study provides prospective beneficial implications for improving forthcoming electrospinning experiments.
Long-chained, viscoelastic surfactant solutions (VES) have been widely employed in the oil and gas industry, particularly in hydraulic fracturing and gravel-packing operations, where turbulence is commonly reached due to high pumping rates. With this motivation, we experimentally investigate the turbulent duct flow of an under-studied class of wormlike micellar solutions that forms a gel at room temperature. The fluid is characterized via rotational rheometry, and the turbulent velocity and Reynolds stress profiles are measured via Laser Doppler Anemometry (LDA). Three surfactant concentration…
We experimentally investigate the influence of finite-size spherical particles in turbulent flows of a Newtonian and a drag reducing viscoelastic fluid at varying particle volume fractions and fixed Reynolds number. Experiments are performed in a square duct at a Reynolds number $Re_{2H}$ of nearly $1.1\times10^4$, Weissenberg number $Wi$ for single-phase flow is between 1-2 and results in a drag-reduction of 43% compared to a Newtonian flow (at the same $Re_{2H}$). Particles are almost neutrally-buoyant hydrogel spheres having a density ratio of 1.0035$\pm$0.0003 and a duct height ${2H}$ to p…
Recently, the nature of viscoelastic drag-reducing turbulence (DRT), especially maximum drag reduction (MDR) state, has become a focus of controversy. It has long been regarded as polymers-modulated inertial turbulence (IT), but is challenged by the newly proposed concept of elasto-inertial turbulence (EIT). This study is to re-picture DRT in parallel plane channels by introducing dynamics of EIT based on statistical and budget analysis for a series of flow regimes from the onset of DR to EIT. Energy conversion between velocity fluctuations and polymers as well as polymeric pressure redistribu…
Creeping flow of polymeric fluid without inertia exhibits elastic instabilities and elastic turbulence accompanied by drag enhancement due to elastic stress produced by flow-stretched polymers. However, in inertia-dominated flow at high $\mbox{Re}$ and low fluid elasticity $El$, a reduction in turbulent frictional drag is caused by an intricate competition between inertial and elastic stresses. Here, we explore the effect of inertia on the stability of viscoelastic flow in a broad range of control parameters $El$ and $(\mbox{Re}, \mbox{Wi})$. We present the stability diagram of observed flow r…
Nontrivial steady flows have recently been found that capture the main structures of the turbulent buffer layer. We study the effects of polymer addition on these "exact coherent states" (ECS) in plane Couette flow. Despite the simplicity of the ECS flows, these effects closely mirror those observed experimentally: Structures shift to larger length scales, wall-normal fluctuations are suppressed while streamwise ones are enhanced, and drag is reduced. The mechanism underlying these effects is elucidated. These results suggest that the ECS are closely related to buffer layer turbulence.
The flow of viscoelastic fluids in channels and pipes remain poorly understood, particularly at low Reynolds numbers. Here, we investigate the flow of polymeric solutions in straight channels using pressure measurements and particle tracking. The law of flow resistance is established by measuring the flow friction factor $f_η$ versus flow rate. Two regimes are found: a transitional regime marked by rapid increase in drag, and a turbulent-like regime characterized by a sudden decrease in drag and a weak dependence on flow rate. Lagrangian trajectories show finite transverse modulations not seen…
An alternative step in understanding the flows of near wall drag-reducing turbulence can be examining the flow in a well-organized streamwise vortex with a laminar background. Herein, we studied the flow behaviors of the Giesekus viscoelastic fluid in a rotating plane Couette flow system, which is accompanied by steady roll cell structures. By fixing the Reynolds and rotation numbers at values that provide as steady wavy roll cell, i.e., an array of meandering streamwise vortices, the Weissenberg number was increased up to 2000 (where the relaxation time was normalized by the wall speed and th…
A modal stability analysis shows that pressure-driven pipe flow of an Oldroyd-B fluid is linearly unstable to axisymmetric perturbations, in stark contrast to its Newtonian counterpart which is linearly stable at all Reynolds numbers. The dimensionless groups that govern stability are the Reynolds number, the elasticity number, and the ratio of solvent to solution viscosity. The unstable mode has a phase speed close to the base-state maximum over the entire unstable region in the relevant parameter space, implying that the unstable mode belongs to a class of viscoelastic center modes. Unlike t…
We address the phenomenon of drag reduction by dilute polymeric additive to turbulent flows, using Direct Numerical Simulations (DNS) of the FENE-P model of viscoelastic flows. It had been amply demonstrated that these model equations reproduce the phenomenon, but the results of DNS were not analyzed so far with the goal of interpreting the phenomenon. In order to construct a useful framework for the understanding of drag reduction we initiate in this paper an investigation of the most important modes that are sustained in the viscoelastic and Newtonian turbulent flows respectively. The modes …
Axisymmetric viscoelastic pipe flow of Oldroyd-B fluids has been recently found to be linearly unstable by Garg et al. Phys. Rev. Lett., 121.024502 (2018). From a nonlinear point of view, this means that the flow can transition to turbulence supercritically, in contrast to the subcritical Newtonian pipe flows. Experimental evidences of subcritical and supercritical bifurcations of viscoelastic pipe flows have been reported, but these nonlinear phenomena have not been examined theoretically. In this work, we study the weakly nonlinear stability of this flow by performing a multiple-scale expans…
Accurately resolving the coupled momentum transfer between the liquid and solid phases of complex fluids is a fundamental problem in multiphase transport processes, such as hydraulic fracture operations. Specifically we need to characterize the dependence of the normalized average fluid-particle force $\langle F \rangle$ on the volume fraction of the dispersed solid phase and on the rheology of the complex fluid matrix. Here we use direct numerical simulations (DNS) to study the creeping flow ($Re\ll 1$) of viscoelastic fluids through static random arrays of monodisperse spherical particles us…
This work explains a scaling law of the first Landau coefficient of the derived Ginzburg-Landau equation (GLE) in the weakly nonlinear analysis of axisymmetric viscoelastic pipe flows in the large-Weissenberg-number ($Wi$) limit, recently reported in Wan et al. J. Fluid Mech. (2021), vol. 929, A16. Using an asymptotic method, we derive a reduced system, which captures the characteristics of the linear centre-mode instability near the critical condition in the large-$Wi$ limit. Based on the reduced system we then conduct a weakly nonlinear analysis using a multiple-scale expansion method, which…
We make time resolved velocity measurements of steel spheres in free fall through liquid using a continuous ultrasound technique. We explore two different ways to induce large changes in drag on the spheres: 1) a small quantity of viscoelastic polymer added to water and 2) altering the surface of the sphere. Low concentration polymer solutions and/or a pattern of grooves in the sphere surface induce an early drag crisis, which may reduce drag by more than 50 percent compared to smooth spheres in pure water. On the other hand, random surface roughness and/or high concentration polymer solutions…
Recent direct numerical simulations of the FENE-P model of non-Newtonian hydrodynamics revealed that the phenomenon of drag reduction by polymer additives exists (albeit in reduced form) also in homogeneous turbulence. We introduce here a simple shell model for homogeneous viscoelastic flows that recaptures the essential observations of the full simulations. The simplicity of the shell model allows us to offer a transparent explanation of the main observations. It is shown that the mechanism for drag reduction operates mainly on the large scales. Understanding the mechanism allows us to predic…
We report experimental results on elastic instability in a viscoelastic channel shear flow due to only a natural non-smoothed inlet and small holes along the channel for pressure measurements. We show that non-normal mode instability results in elastic waves and chaotic flow self-organized into periodically cycled stream-wise streaks synchronized by elastic wave frequency. The chaotic flow persists above the transition with increasing $Wi$ further into elastic turbulence and drag reduction regimes. Thus, we resolve the recent puzzle whether strong prearranged perturbations are necessary to get…
Newtonian pipe flow is known to be linearly stable at all Reynolds numbers. We report, for the first time, a linear instability of pressure driven pipe flow of a viscoelastic fluid, obeying the Oldroyd-B constitutive equation commonly used to model dilute polymer solutions. The instability is shown to exist at Reynolds numbers significantly lower than those at which transition to turbulence is typically observed for Newtonian pipe flow. Our results qualitatively explain experimental observations of transition to turbulence in pipe flow of dilute polymer solutions at flow rates where Newtonian …
We show that viscoelastic plane Poiseuille flow becomes linearly unstable in the absence of inertia, in the limit of high elasticities, for ultra-dilute polymer solutions. While inertialess elastic instabilities have been predicted for curvilinear shear flows, this is the first ever report of a purely elastic linear instability in a rectilinear shear flow. The novel instability continues upto a Reynolds number ($Re$) of $O(1000)$, corresponding to the recently identified elasto-inertial turbulent state believed to underlie the maximum-drag-reduced regime. Thus, for highly elastic ultra-dilute …
This work introduces a mathematical approach to analysing the polymer dynamics in turbulent viscoelastic flows that uses a new geometric decomposition of the conformation tensor, along with associated scalar measures of the polymer fluctuations. The approach circumvents an inherent difficulty in traditional Reynolds decompositions of the conformation tensor: the fluctuating tensor fields are not positive-definite and so do not retain the physical meaning of the tensor. The geometric decomposition of the conformation tensor yields both mean and fluctuating tensor fields that are positive-defini…
The study of creeping motion of viscoelastic fluid around a rotating rigid torus is investigated. The analysis of the problem is performed using a second-order viscoelastic model. The study is carried out in terms of the bipolar toroidal system of coordinates where the toroid is rotating about its axis of symmetry (z-axis). The problem is solved within the frame of slow flow approximation. Therefore, all variables in the governing equations are expanded in a power series of angular velocity. A set of successive partial differential equations is obtained. The equations of motion governing the f…
The current theories on the mechanism for polymer drag-reduction (DR) are generally applicable for long-chain flexible polymers that form viscoelastic solutions. Rigid polymer solutions that generate DR seemingly lack prevalent viscoelastic characteristics. They do, however, demonstrate larger viscosities and a noticeable shear-thinning trend, well approximated by generalized Newtonian models. The following experimental investigation scrutinized the flow statistics of an aqueous xanthan gum solution in a turbulent channel flow, with friction Reynolds numbers ($Re_τ$) between 160 and 680. The a…
In most settings, from international pipelines to home water supplies, the drag caused by turbulence raises pumping costs many times higher than if the flow were laminar. Drag reduction has therefore long been an aim of high priority. In order to achieve this end, any drag reduction method must modify the turbulent mean flow. Motivated by minimization of the input energy this requires, linearly optimal forcing functions are examined. It is shown that the forcing mode leading to the greatest response of the flow is always of m=1 azimuthal symmetry. Little evidence is seen of the second peak at …
The drag reduction induced by superhydrophobic surfaces is investigated in turbulent pipe flow. Wetted superhydrophobic surfaces are shown to trap gas bubbles in their asperities. This stops the liquid from coming in direct contact with the wall in that location, allowing the flow to slip over the air bubbles. We consider a well defined texture with streamwise grooves at the walls in which the gas is expected to be entrapped. This configuration is modelled with alternating no-slip and shear-free boundary conditions at the wall. With respect to classical turbulent pipe flow, a substantial drag …
The interaction of flexible structures with viscoelastic flows can result in very rich dynamics. In this paper, we present the results of the interactions between the flow of a viscoelastic polymer solution and a cantilevered beam in a confined microfluidic geometry. Cantilevered beams with varying length and flexibility were studied. With increasing flow rate and Weissenberg number, the flow transitioned from a fore-aft symmetric flow to a stable detached vortex upstream of the beam, to a time-dependent unstable vortex shedding. The shedding of the unstable vortex upstream of the beam imposed…
In this study, the OpenFOAM platform, based on the finite volume method, is applied to investigate the two-dimensional viscoelastic flow past a circular cylinder. The FENE-P model, which considers the bounded elongation of polymer molecules, is chosen to describe the elastic constitutive relationship of the polymer solution. The maximum molecular chain lengths of L = 10, 50, 100, and 200 are considered, which describe the molecular conformation characteristics of the polymer solution. To improve the numerical instability of the viscoelastic flow simulation, three different methods, i.e., the t…
The perturbations of the laminar shear-thinning viscoelastic pipe flow under Finitely Extensible Nonlinear Elastic model with Peterlin approximation (FENE-P) are shown to exhibit leading-order power-law behaviours, and the expected odd-even parities with respect to the radial coordinate that depend on the azimuthal wave\-number, $n$. The analysis helps regularizing the governing system of equations at the centreline, and allows for a complete stability analysis of three-dimensional perturbations for a general integer value of $n$, which has hitherto remained a challenge for FENE-P models. It i…
The results of a comparative analysis based upon a Karhunen-Loève expansion of turbulent pipe flow and drag reduced turbulent pipe flow by spanwise wall oscillation are presented. The turbulent flow is generated by a direct numerical simulation at a Reynolds number $Re_τ= 150$. The spanwise wall oscillation is imposed as a velocity boundary condition with an amplitude of $A^+ = 20$ and a period of $T^+ = 50$. The wall oscillation results in a 27% mean velocity increase when the flow is driven by a constant pressure gradient. The peaks of the Reynolds stress and root-mean-squared velocities shi…
We present an analytical study of oscillatory laminar shear flow over a compliant viscoelastic layer on a rigid base. This problem relates to oscillating blood flow in viscoelastic vessels. The deeper motivation for this study, however, is the possible use of compliant coatings for turbulent drag reduction. An analytical solution of the fluid and solid velocity is presented, and five dimensionless parameters emerge. The interaction between fluid and solid appears to be determined by a single combined dimensionless parameter, which we call the shear interaction parameter $χ$. The fluid satisfie…
Dynamic properties of elasto-inertial turbulence (EIT) are studied in a Taylor-Couette geometry. EIT is a chaotic flow state that develops upon both non-negligible inertia and viscoelasticity. A combination of direct flow visualisation and torque measurement allows to verify the earlier onset of EIT compared to purely inertial instabilities (and inertial turbulence). The scaling of the pseudo-Nusselt number with inertia and elasticity is discussed here for the first time. Variations in the friction coefficient, temporal frequency spectra, and spatial power density spectra highlight that EIT un…
Lagrangian dynamics unveil polymer conformation in viscoelastic flows
Tufts University（塔夫茨大学）；Institute for Mechanobiology, Department of Bioengineering, College of Engineering, Northeastern University（东北大学生物工程学院机制生物学研究所）；Department of Mechanical Science and Engineering, University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校机械科学与工程系）；Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania（宾夕法尼亚大学机械工程和应用力学系）
Turbulent channel flow of drag-reducing polymer solutions is simulated in minimal flow geometries. Even in the Newtonian limit, we find intervals of "hibernating" turbulence that display many features of the universal maximum drag reduction (MDR) asymptote observed in polymer solutions: weak streamwise vortices, nearly nonexistent streamwise variations and a mean velocity gradient that quantitatively matches experiments. As viscoelasticity increases, the frequency of these intervals also increases, while the intervals themselves are unchanged, leading to flows that increasingly resemble MDR.
Elasto-inertial turbulence (EIT) is a new, two-dimensional chaotic flow state observed in polymer solutions with possible connections to inertialess elastic turbulence and drag-reduced Newtonian turbulence. In this Letter, we argue that the origins of EIT are fundamentally different from Newtonian turbulence by finding a dynamical connection between EIT and an elasto-inertial linear instability recently found at high Weissenberg numbers (Garg et al. Phys. Rev. Lett. 121, 024502, 2018). This link is established by isolating the first known exact coherent structures in viscoelastic parallel flow…
Considered here is the derivation of partial differential equations arising in pulsatile flow in pipes with viscoelastic walls. The equations are asymptotic models describing the propagation of long-crested pulses in pipes with cylindrical symmetry. Additional effects due to viscous stresses in bio-fluids are also taken into account. The effects of viscoelasticity of the vessels on the propagation of solitary and periodic waves in a vessel of constant radius are being explored numerically.
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…
Hydrodynamic problems with stagnation points are of particular importance in fluid mechanics as they allow study and investigation of elongational flows. In this article, the uniaxial elongational flow appearing at the surface of a viscoelastic drop and its role on the deformation of the droplet at low inertial regimes is studied. In studies related to viscoelastic droplets falling/raising in an immiscible Newtonian fluids, it is well known that by increasing the Deborah number (the ratio of the relaxation time of the interior fluid to a reference time scale) the droplet might lose its spheric…
We experimentally assess the capabilities of an active, open-loop technique for drag reduction in turbulent wall flows recently introduced by Quadrio et al. [J. Fluid Mech., v.627, 161, (2009)]. The technique consists in generating streamwise-modulated waves of spanwise velocity at the wall, that travel in the streamwise direction. A proof-of-principle experiment has been devised to measure the reduction of turbulent friction in a pipe flow, in which the wall is subdivided into thin slabs that rotate independently in the azimuthal direction. Different speeds of nearby slabs provide, although i…
The problem of oscillating flows inside pipes under periodic forcing of viscoelastic fluids is addressed here. Starting from the linear Oldroyd-B model, a generalized Darcy's law is obtained in frequency domain and an explicit expression for the dependence of the dynamic permeability on fluid parameters and forcing frequency is derived. Previous results in both viscoelastic and Newtonian fluids are here shown to be particular cases of our results. On the basis of our calculations, a possible explanation for the observed damping of local dynamic response as the forcing frequency increases is gi…
Drag reduction and subcritical turbulence in controlled pipe flows
Politecnico di Milano（米兰理工大学）；Univ. Lille, CNRS, ONERA, Arts et Metiers Institute of Technology, Centrale Lille（里尔大学、法国国家科学研究中心、法国航空发展局、巴黎国立高等工艺学院、里尔中央理工学院）；Massachusetts Institute of Technology（麻省理工学院）
The dynamics of dense finite-size particles in vertical channel flows of Newtonian and viscoelastic carrier fluids are examined using particle resolved simulations. Comparison to neutrally buoyant particles in the same configuration highlights the effect of settling. The particle volume fraction is $5\%$, and a gravity field acts counter to the flow direction. Despite a modest density ratio ($ρ_r = 1.15$), qualitative changes arise due to the relative velocity between the particle and fluid phases. While dense particles are homogeneously distributed in the core of the channel, the mean concent…
This paper is focused on the fundamental mechanism(s) of viscoelastic turbulence that lead to polymer induced turbulent drag reduction phenomenon. A great challenge in this problem is the computation of viscoelastic turbulent flows, since the understanding of polymer physics is restricted to mechanical models. An effective state-of-the-art numerical method to solve the governing equation for polymers modelled as non-linear springs, without using any artificial assumptions as usual, was implemented here for the first time on a three-dimensional channel flow geometry. The capability of this algo…
We study numerically two-dimensional creeping viscoelastic flow past a biperiodic square array of cylinders within the Oldroyd B, FENE-CR and FENE-P constitutive models of dilute polymer solutions. Our results capture the initial mild decrease then dramatic upturn ('thickening') seen experimentally in the drag coefficient as a function of increasing Weissenberg number. By systematically varying the porosity of the flow geometry, we demonstrate two qualitatively different mechanisms underpinning this thickening effect: one that operates in the highly porous case of widely spaced obstacles, and …
A modal stability analysis shows that plane Poiseuille flow of an Oldroyd-B fluid becomes unstable to a `center mode' with phase speed close to the maximum base-flow velocity, $U_{max}$. The governing dimensionless groups are the Reynolds number $Re = ρU_{max} H/η$, the elasticity number $E = λη/(H^2ρ)$, and the ratio of solvent to solution viscosity $η_s/η$; here, $λ$ is the polymer relaxation time, $H$ is the channel half-width, and $ρ$ is the fluid density. For experimentally relevant values (e.g., $E \sim 0.1$ and $β\sim 0.9$), the predicted critical Reynolds number, $Re_c$, for the center…
Nonlinear traveling waves that are precursors to laminar-turbulent transition and capture the main structures of the turbulent buffer layer have recently been found to exist in all the canonical parallel flow geometries. We study the effect of polymer additives on these "exact coherent states" (ECS), in the plane Poiseuille geometry. Many key aspects of the turbulent drag reduction phenomenon are found, including: delay in transition to turbulence; drag reduction onset threshold; diameter and concentration effects. Furthermore, examination of the ECS existence region leads to a distinct predic…
