Non-modal amplification of disturbances in streamwise-constant channel flows of Oldroyd-B fluids is studied from an input-output point of view by analyzing the responses of the velocity components to spatio-temporal body forces. These inputs into the governing equations are assumed to be harmonic in the spanwise direction and stochastic in the wall-normal direction and in time. An explicit Reynolds number scaling of frequency responses from different forcing to different velocity components is developed, showing the same $Re$-dependence as in Newtonian fluids. It is found that some of the frequency response components peak at non-zero temporal frequencies. This is in contrast to Newtonian fluids, where peaks are always observed at zero frequency, suggesting that viscoelastic effects introduce additional timescales and promote development of flow patterns with smaller time constants than in Newtonian fluids. The temporal frequencies, corresponding to the peaks in the components of frequency response, decrease with an increase in viscosity ratio (ratio of solvent viscosity to total viscosity) and show maxima for non-zero elasticity number. Our analysis of the Reynolds-Orr equation demonstrates that the energy-exchange term involving the streamwise/wall-normal polymer stress component $τ_{xy}$ and the wall-normal gradient of the streamwise velocity $\partial_y u$ becomes increasingly important relative to the Reynolds stress term as the elasticity number increases, and is thus the main driving force for amplification in flows with strong viscoelastic effects.
We correct certain errors and ambiguities in the recent pedagogical article by Hopkins and de Bruyn. The early-time asymptotics of the solution to the transient version of Stokes' second problem for an Oldroyd-B fluid in a half-space is presented, as an Appendix, to complement the late-time asymptotics given by Hopkins and de Bruyn.
We present a novel distributed Lagrange multiplier/fictitious domain (DLM/FD) method for simulating fluid-particle interaction in Oldroyd-B fluids under creeping conditions. The results concerning two ball interaction in a three dimensional (3D) bounded shear flow are obtained for Weissenberg numbers up to 1 . The pass and return trajectories of the two ball mass centers are similar to those in a Newtonian fluid; but they lose the symmetry due to the effect of elastic force arising from viscoelastic fluids. A tumbling chain of two balls (a dipole) may occur, depending on the value of the Weiss…
The Saffman-Taylor instability occurs when { a} Stokes fluid is displaced by a less viscous one { in} a Hele-Shaw cell. { This model is useful to study the secondary oil recovery from a pororus medum.} Since 1960, { polymer solutions} were used as displacing fluids; moreover, the oil in a porous reservoir can often be considered a non-Newtonian fluid. Motivated by this fact, in this paper we study the linear stability of the displacement of two Oldroyd-B fluids in a rectilinear Hele-Shaw cell, even if the direct relevance for the flow through porous media is not so evident. We get an { approxi…
The motion of incompressible fractional Oldroyd-B fluids between two parallel walls perpendicular to a plate that applies time-dependent shear stresses to the fluid is studied by means of integral transforms. In the special cases of Newtonian and second grade fluids, these shear stresses reduces to fH(t)sin(wt) or fH(t)cos(wt). General solutions for velocity are presented as a sum of Newtonian solutions and the corresponding non-Newtonian contributions. They reduce to the similar solutions corresponding to the motion over an infinite plate if the distance between walls tend to infinity and can…
When a drop of fluid containing long, flexible polymers breaks up, it forms threads of almost constant thickness, whose size decreases exponentially in time. Using an Oldroyd-B fluid as a model, we show that the thread profile, rescaled by the thread thickness, converges to a similarity solution. Using the correspondence between viscoelastic fluids and non-linear elasticity, we derive similarity equations for the full three-dimensional axisymmetric flow field in the limit that the viscosity of the solvent fluid can be neglected. A conservation law balancing pressure and elastic energy permits …
We point out that an erroneous derivation in the recent paper [Fetecau et al., Nonlinear Anal. RWA 12 (2011) 1] yields a correct solution by accident. Additionally, a number of misrepresentations and inaccuracies in the latter recent paper are identified, corrected and/or clarified in this Comment. Finally, a listing of recent papers in this journal that make a mistake applying the Fourier sine transform, and thus present erroneous solutions, is given as an Appendix.
We analyze the steady non-Newtonian fluid-structure interaction between the flow of an Oldroyd-B fluid and a deformable channel. Specifically, we provide a theoretical framework for calculating the leading-order effect of the fluid's viscoelasticity on the flow rate-pressure drop relation and on the deformation of the channel's elastic wall. We first identify the characteristic scales and dimensionless parameters governing the fluid-structure interaction in slender and shallow channels. Applying the lubrication approximation for the flow and employing a perturbation expansion in powers of the …
Most biological fluids are viscoelastic, meaning that they have elastic properties in addition to the dissipative properties found in Newtonian fluids. Computational models can help us understand viscoelastic flow, but are often limited in how they deal with complex flow geometries and suspended particles. Here, we present a lattice Boltzmann solver for Oldroyd-B fluids that can handle arbitrarily-shaped fixed and moving boundary conditions, which makes it ideally suited for the simulation of confined colloidal suspensions. We validate our method using several standard rheological setups, and …
We analyze the pressure-driven flow of a viscoelastic fluid in arbitrarily shaped, narrow channels and present a theoretical framework for calculating the relationship between the flow rate $q$ and pressure drop $Δp$. We utilize the Oldroyd-B model and first identify the characteristic scales and dimensionless parameters governing the flow in the lubrication limit. Employing a perturbation expansion in powers of the Deborah number ($De$), we provide analytical expressions for the velocity, stress, and the $q-Δp$ relation in the weakly viscoelastic limit up to $O(De^2)$. Furthermore, we exploit…
In this article we present a numerical study of the dynamics of two disks settling in a narrow vertical channel filled with Oldroyd-B fluid. Two kinds of particle dynamics are obtained: (i) periodic interaction between two disks and (ii) the chain formation of two disks. For the periodic interaction of two disks, two different motions are obtained: (a) two disks stay far apart and interact periodically and (b) two disks interact closely and then far apart in a periodic way, like the drafting, kissing and tumbling of two disks sedimenting in Newtonian fluid, due to the lack of strong enough ela…
Energy theory for incompressible Newtonian fluids is, in many cases, capable of producing strong absolute stability criteria for steady flows. In those fluids the kinetic energy naturally defines a norm in which perturbations decay monotonically in time at sufficiently low (but non-zero) Reynolds numbers. There are, however, at least two obstructions to the generalization of such methods to Oldroyd-B fluids. One previously recognized problem is the fact that the natural energy does not correspond to a proper functional norm on perturbations. Another problem, original to this work, is the fact …
Recently, detailed experiments on visco-elastic channel flow have provided convincing evidence for a nonlinear instability scenario which we had argued for based on calculations for visco-elastic Couette flow. Motivated by these experiments we extend the previous calculations to the case of visco-elastic Poiseuille flow, using the Oldroyd-B constitutive model. Our results confirm that the subcritical instability scenario is similar for both types of flow, and that the nonlinear transition occurs for Weissenberg numbers somewhat larger than one. We provide detailed results for the convergence o…
A semi-analytical solution for the lubrication force between two spheres approaching in viscoelastic fluids described by the Oldroyd-B model under small Deborah numbers
Nonmodal amplification of stochastic disturbances in elasticity-dominated channel flows of Oldroyd-B fluids is analyzed in this work. For streamwise-constant flows with high elasticity numbers $μ$ and finite Weissenberg numbers $We$, we show that the linearized dynamics can be decomposed into slow and fast subsystems, and establish analytically that the steady-state variances of velocity and polymer stress fluctuations scale as $O (We^2)$ and $O (We^4)$, respectively. This demonstrates that large velocity variance can be sustained even in weakly inertial stochastically driven channel flows of …
The Oldroyd-B model has been used extensively to predict a host of instabilities in shearing flows of viscoelastic fluids, often realized experimentally using polymer solutions. The present review, written on the occasion of the birth centenary of James Oldroyd, provides an overview of instabilities found across major classes of shearing flows. These comprise (i) the canonical rectilinear shearing flows including plane Couette, plane and pipe Poiseuille flows; (ii) viscometric shearing flows with curved streamlines such as those in the Taylor-Couette, cone-and-plate and parallel-plate geometri…
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.
Although the equations governing fluid flow are well known, there are no analytical expressions that describe the complexity of turbulent motion. A recent proposition is that in analogy to low dimensional chaotic systems, turbulence is organized around unstable solutions of the governing equations which provide the building blocks of the disordered dynamics. We report the discovery of periodic solutions which just like intermittent turbulence are spatially localized and show that turbulence arises from one such solution branch.
Channel flow, the pressure driven flow between parallel plates, has exact coherent structures that show various degrees of localization. For states which are localized in streamwise direction but extended in spanwise direction, we show that they are exponentially localized, with decay constants that are different on the upstream and downstream side. We extend the analysis of Brand and Gibson [J. Fluid Mech. 750, R1 (2014)] for stationary states to the case of advected structures that is needed here, and derive expressions for the decay in terms of eigenvalues and eigenfunctions of certain seco…
A spherical droplet is given an initial rotation in the streamwise direction and is impulsively accelerated by a uniform free stream. Numerical results for the deformation and dynamics of the droplet are obtained by utilising a finite volume staggered mesh method with a moving mesh interface tracking scheme. The fluid dynamics videos of the droplet are presented in the Gallery of Fluid Motion, 2010. By initially rotating the droplet in the streamwise direction, the droplet deforms differently depending on the non-dimensional rotation rate, $Ω^*$. The families of droplet shape are, in ascending…
In wall-bounded turbulence, a multitude of coexisting turbulence structures form the streamwise velocity energy spectrum from the viscosity- to the inertia-dominated range of scales. Definite scaling-trends for streamwise spectra have remained empirically elusive, although a prominent school of thought stems from the works of Perry and Abell (J. Fluid Mech., vol. 79, 1977, pp. 785-799) and Perry et al. (J. Fluid Mech., vol. 165, 1986, pp. 163-199), which were greatly inspired by the attached-eddy hypothesis of Townsend (The Structure of Turbulent Shear Flow, Cambridge University Press, 1976). …
We extend the procedure outlined in [Basse, "Scaling of global properties of fluctuating and mean streamwise velocities in pipe flow: Characterization of a high Reynolds number transition region," Phys. Fluids Vol. 33, 065127 (2021)] to study global, i.e. radially averaged, scaling of streamwise velocity fluctuations. A viscous term is added to the log-law scaling which leads to the existence of a mathematical abstraction which we call the "global peak". The position and amplitude of this global peak are characterized and compared to the inner and outer peaks. A transition at a friction Reynol…
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 …
The steady flow of three viscoelastic fluids (Oldroyd-B, FENE-P, and Owens model for blood) in a two-dimensional channel, partly bound by a deformable, finite thickness neo-Hookean solid, is computed. The limiting Weissenberg number beyond which computations fail to converge is found to increase with increasing dimensionless solid elasticity parameter Γ, following the trend Owens > FENE- P > Oldroyd-B. The highly shear thinning nature of Owens model leads to the elastic solid always collapsing into the channel, for the wide range of values of Γ considered here. In the case of the FENE-P and Ol…
In this article we present a numerical method for simulating the sedimentation of circular particles in two-dimensional channel filled with a viscoelastic fluid of FENE-CR type, which is generalized from a domain/distributed Lagrange multiplier method with a factorization approach for Oldroyd-B fluids developed in [J. Non-Newtonian Fluid Mech. 156 (2009) 95]. Numerical results suggest that the polymer extension limit L for the FENE-CR fluid has no effect on the final formation of vertical chain for the cases of two disks and three disks in two-dimensional narrow channel, at least for the value…
In this work, we revisit the temporal stability of slip channel flow. Lauga & Cossu (Phys. Fluids 17, 088106 (2005)) and Min & Kim (Phys. Fluids 17, 108106 (2005)) have investigated both modal stability and non-normality of slip channel flow and concluded that the velocity slip greatly suppresses linear instability and only modestly affects the non-normality. Here we study the stability of channel flow with streamwise and spanwise slip separately as two limiting cases of anisotropic slip and explore a broader range of slip length than previous studies did. We find that, with sufficiently large…
The Betz limit expresses the maximum proportion of the kinetic energy flux incident on an energy conversion device that can be extracted from an unbounded flow. The derivation of the Betz limit requires an assumption of steady flow through a notional actuator disk that is stationary in the streamwise direction. The present derivation relaxes the assumptions of steady flow and streamwise actuator disk stationarity, which expands the physically realizable parameter space of flow conditions upstream and downstream of the actuator disk. A key consequence of this generalization is the existence of …
We study the global, i.e. radially averaged, high Reynolds number (asymptotic) scaling of streamwise turbulence intensity squared defined as ${I^2=\overline{u^2}/U^2}$, where $u$ and $U$ are the fluctuating and mean velocities, respectively (overbar is time averaging). The investigation is based on the mathematical abstraction that the logarithmic region in wall turbulence extends across the entire inner and outer layers. Results are matched to spatially integrated Princeton Superpipe measurements [Hultmark M, Vallikivi M, Bailey SCC and Smits AJ. Logarithmic scaling of turbulence in smooth- a…
The well-known problem of unidirectional plane flow of a fluid in a half-space due to the impulsive motion of the plate it rests upon is discussed in the context of the second-grade and the Oldroyd-B non-Newtonian fluids. The governing equations are derived from the conservation laws of mass and momentum and three correct known representations of their exact solutions given. Common mistakes made in the literature are identified. Simple numerical schemes that corroborate the analytical solutions are constructed.
Turbulent flows of viscoplastic fluids at high Reynolds numbers have been investigated recently with direct numerical simulations (DNS) but experimental results have been limited. For this reason, we carry out an experimental study of fully turbulent flows of a yield stress fluid in a rectangular aspect ratio channel with a high-resolution laser doppler velocimetry (LDA) setup. We employ aqueous Carbopol solutions, often considered to be a simple yield stress fluid. We formulate different concentrations to address the effect of the rheology of the fluid on the turbulence statistics at an appro…
Many biological fluids have polymeric microstructures and display non-Newtonian rheology. We take advantage of such nonlinear fluid behavior and combine it with geometrical symmetry-breaking to design a novel small-scale propeller able to move only in complex fluids. Its propulsion characteristics are explored numerically in an Oldroyd-B fluid for finite Deborah numbers while the small Deborah number limit is investigated analytically using a second-order fluid model. We then derive expressions relating the propulsion speed to the rheological properties of the complex fluid, allowing thus to i…
This fluid dynamics video, presented at the 66th Annual Meeting of the American Physical Society Division of Fluid Dynamics (Pittsburgh, PA), shows how dielectric colloidal polystyrene (PS) particles with a negative surface charge in a dilute suspension flowing through a microchannel can be manipulated and assembled into streamwise structures by using a combination of shear and an applied negative electric field along the streamwise direction. Interestingly, these structures are only observed within about 1 um of the wall, which also has a negative surface charge, and are not present in the bu…
We show that a rather simple, steady modification of the streamwise velocity profile in a pipe can lead to a complete collapse of turbulence and the flow fully relaminarizes. Two different devices, a stationary obstacle (inset) and a device to inject additional fluid through an annular gap close to the wall, are used to control the flow. Both devices modify the streamwise velocity profile such that the flow in the center of the pipe is decelerated and the flow in the near wall region is accelerated. We present measurements with stereoscopic particle image velocimetry to investigate and capture…
A generalised quasilinear (GQL) approximation (Marston \emph{et al.}, \emph{Phys. Rev. Lett.}, vol. 116, 104502, 2016) is applied to turbulent channel flow at $Re_τ\simeq 1700$ ($Re_τ$ is the friction Reynolds number), with emphasis on the energy transfer in the streamwise wavenumber space. The flow is decomposed into low and high streamwise wavenumber groups, the former of which is solved by considering the full nonlinear equations whereas the latter is obtained from the linearised equations around the former. The performance of the GQL approximation is subsequently compared with that of a QL…
To elucidate the key factor for the quantitative prediction of the shear-thickening in suspensions in viscoelastic fluids, direct numerical simulations of many-particle suspensions in a multi-mode Oldroyd-B fluid are performed using the smoothed profile method. Suspension flow under simple shear flow is solved under periodic boundary conditions by using Lees--Edwards boundary conditions for particle dynamics and a time-dependent oblique coordinate system that evolves with mean shear flow for fluid dynamics. Semi-dilute many-particle suspensions up to a particle volume fraction of 0.1 are inves…
As the first endeavour, we have analyzed the pulsatile flow of Oldroyd-B viscoelastic fluid where the combined effects of fluid elasticity and pulsation parameters on the flow characteristics are numerically studied at a low Reynolds number. Computations are performed using a finite-volume based open-source solver OpenFOAM by appending the log-conformation tensor approach to stabilize the numerical solution at high Deborah number. Significant flow velocity enhancement is achieved by increasing the viscoelastic behaviour of the fluid. High-velocity gradient zones and high polymeric stress regio…
Direct numerical simulations (DNSs) of turbulent pipe flow subjected to streamwise-varying wall rotation are performed. This control method is able to achieve drag reduction and even relaminarize the flow under certain control parameters at low Reynolds number. Two control parameters, which are velocity amplitude and wavelength, are considered. An annular boundary layer, called Spatial Stokes Layer (SSL), is formed by the wall rotation. Based on the thickness of SSL, two types of drag reduction scenarios are identified. When the thickness is low, the SSL acts as a spacer layer, obstructing the…
In molecular dynamics simulations, nanochannel flows are usually driven by a constant force, that aims to represent a pressure difference between inlet and outlet, and periodic boundary conditions are applied in the streamwise direction resulting in an homogeneous flow. The homogeneity hypothesis can be eliminated adding reservoirs at the inlet and outlet of the channel which permits to predict streamwise variation of flow properties. It also opens the door to drive the flow by applying pressure gradient instead of a constant force. We analyze the impact of these modeling modifications in the …
The effect of streamwise-traveling waves of spanwise wall velocity (StTW) on the quasi-streamwise vortices (QSV) populating the near-wall region of turbulent channels is studied via a conditional averaging technique applied to flow snapshots obtained via Direct Numerical Simulation. The analysis by A. Yakeno, Y. Hasegawa, N. Kasagi, "Modification of quasi-streamwise vortical structure in a drag-reduced turbulent channel flow with spanwise wall oscillation", Phys. Fluids 26, 085109 (2014), where the special case of spatially uniform wall oscillation (OW) was considered, is extended to the gener…
The background of this work is the problem of reducing the aerodynamic turbulent friction drag, which is an important source of energy waste in innumerable technological fields. We develop a theoretical framework aimed at predicting the behaviour of existing drag reduction techniques when used at the large values of Re which are typical of applications. We focus on one recently proposed and very promising technique, which consists in creating at the wall streamwise-travelling waves of spanwise velocity. A perturbation analysis of the Navier-Stokes equations that govern the fluid motion is carr…
It has been recently shown that the energy-containing motions in turbulent channel flow exist in the form of Townsend's attached eddies by a numerical experiment which simulates the energy-containing motions only at a prescribed spanwise length scale using their self-sustaining nature (Hwang, 2015, J. Fluid Mech., 767, p254). In the present study, a detailed investigation of the self-sustaining process of the energy-containing motions at each spanwise length scale (i.e. the attached eddies) in the logarithmic and outer regions is carried out with an emphasis on its relevance to 'bursting', whi…
The no-slip boundary condition results in a velocity shear forming in fluid flow near a solid surface. This shear flow supports the turbulence characteristic of fluid flow near boundaries at Reynolds numbers above $\approx1000$ by making available to perturbations the kinetic energy of the externally forced flow. Understanding the physical mechanism underlying this energy transfer poses a fundamental question. Although qualitative understanding that this transfer involves nonlinear destabilization of the roll-streak coherent structure has been established, identification of this instability ha…
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…
We present numerical simulations of laminar and turbulent channel flow of an elastoviscoplastic fluid. The non-Newtonian flow is simulated by solving the full incompressible Navier-Stokes equations coupled with the evolution equation for the elastoviscoplastic stress tensor. The laminar simulations are carried out for a wide range of Reynolds numbers, Bingham numbers and ratios of the fluid and total viscosity, while the turbulent flow simulations are performed at a fixed bulk Reynolds number equal to 2800 and weak elasticity. We show that in the laminar flow regime the friction factor increas…
It has often been proposed that the formation of large-scale motion (or bulges) is a consequence of successive mergers and/or growth of near-wall hairpin vortices. In the present study, we report our direct observation that large-scale motion is generated by an instability of an `amplified' streaky motion in the outer region (i.e. very-large-scale motion). We design a numerical experiment in turbulent channel flow up to $Re_τ\simeq 2000$ where a streamwise-uniform streaky motion is artificially driven by body forcing in the outer region computed from the previous linear theory (Hwang \& Cossu,…
Amplification of deterministic disturbances in inertialess shear-driven channel flows of viscoelastic fluids is examined by analyzing the frequency responses from spatio-temporal body forces to the velocity and polymer stress fluctuations. In strongly elastic flows, we show that disturbances with large streamwise length scales may be significantly amplified even in the absence of inertia. For fluctuations without streamwise variations, we derive explicit analytical expressions for the dependence of the worst-case amplification (from different forcing to different velocity and polymer stress co…
The present experimental study addresses the flow of a Yield Stress Fluid with some elasticity (Carbopol gel) in a square duct. The behaviour of two fluids with lower and higher yield stress is investigated at multiple Reynolds numbers $Re^* \in$ (1, 200) and Bingham numbers $Bi \in$ (0.01, 0.35). A secondary flow consisting of eight vortices is observed to recirculate the fluid from the corners to the core. Their extent and intensity grows with increasing $Re^*$. The second objective of this study is to explore the change in flow in the presence of particles. Almost neutrally-buoyant finite-s…
