The Navier-Stokes-Fourier model for a 3D thermoconducting viscous fluid, where the evolution equation for the temperature T contains a term proportional to the rate of energy dissipation, is investigated analitically at the light of the rotational invariance property. Two cases are considered: the Couette flow and a flow with a radial velocity between two rotating impermeable and porous coaxial cylinders, respectively. In both cases, we show the existence of a maximum value of T, T_max, when the difference of temperature Delta T=T_2-T_1 on the surfaces of the cylinders is assigned. The role of T_max is discussed in the context of different physical situations.
In this work, we provide two novel approaches to show that incompressible fluid flow in a finite domain contains at most a finite number vortices. We use a recently developed geometric theory of incompressible viscous flows along with an existing mathematical analysis concept to establish the finiteness. We also offer a second proof of finiteness by roping in the Kolmogorov's length scale criterion in conjunction with the notion of diametric disks.
Through the Ginzburg-Landau and the Navier-Stokes equations, we study turbulence phenomena for viscous incompressible and compressible fluids by a second order phase transition. For this model, the velocity is defined by the sum of classical and whirling components. Moreover, the laminar-turbulent transition is controlled by rotational effects of the fluid. Hence, the thermodynamic compatibility of the differential system is proved. The same model is used to understand the origins of tornadoes and their behavior and the birth of the vortices resulting from the fall of water in a vertical tube.…
Reconnection plays a significant role in the dynamics of plasmas, polymers and macromolecules, as well as in numerous laminar and turbulent flow phenomena in both classical and quantum fluids. Extensive studies in quantum vortex reconnection show that the minimum separation distance δ between interacting vortices follows a 1/2 scaling. Due to the complex nature of the dynamics (e.g., the formation of bridges and threads as well as successive reconnections and avalanche), such scaling has never been reported for (classical) viscous vortex reconnection. Using the direct numerical simulation of t…
Till date, the sequence of vortices present in the solid corners of steady internal viscous incompressible flows, widely known as Moffatt vortices was thought to be infinite. However, the already existing and most recent geometric theories on incompressible viscous flows that express vortical structures in terms of critical points in bounded domains, indicate a strong opposition to this notion of infiniteness. In this study, we endeavor to bridge the gap between the two opposing stream of thoughts by addressing what might have gone wrong and pinpoint the shortcomings on the assumptions of the …
In this paper we investigate the asymptotic validity of boundary layer theory. For a flow induced by a periodic row of point-vortices, we compare Prandtl's solution to Navier-Stokes solutions at different $Re$ numbers. We show how Prandtl's solution develops a finite time separation singularity. On the other hand Navier-Stokes solution is characterized by the presence of two kinds of viscous-inviscid interactions between the boundary layer and the outer flow. These interactions can be detected by the analysis of the enstrophy and of the pressure gradient on the wall. Moreover we apply the comp…
In the present note, we solved numerically the viscous vorticity equation of the initial-boundary value problem describing the classic Helmholtz phenomena of vortex interaction. In the leapfrogging of vortex pairs, we demonstrate the fact that there exists a variety of initial vortex configurations, such as the initial vortex core structures, the starting speeds, the lateral and longitudinal separations as well as the fluid viscosity. To simulate leapfrogging appears to be a straightforward task, based on the vorticity calculations for a number of initial vortices with different core structure…
Through Ginzburg-Landau and Navier-Stokes equations, we study turbulence phenomena for viscous incompresible and compressible fluids by a second order phase transition. For this model, the velocity is defined by the sum of classical and whirling components. Moreover, the laminar-turbulent transition is controlled by rotational effects of the fluid. Hence, the thermodynamic compatibility of the differential system is proved. This model can explain the turbulence by instability effects motivated by a double well potential of the Ginzburg-Landau equation. The same model is used to understand the …
This report addresses an important question discussed by the quantum turbulence community during the last decade: do quantized vortices form, in zero-temperature superfluids, coherent structures similar to vortex tubes in ordinary, viscous turbulence? So far the evidence provided by numerical simulations is that bundles of quantized vortices appear in finite-temperature superfluids, but from the interaction with existing coherent structures in the turbulent (viscous) normal fluid, rather than due to the intrinsic superfuid dynamics. In this report we show that, in very intense quantum turbulen…
Numerical simulations using the Lattice Boltzmann Method are presented of the following two-dimensional incompressible flow problem. Starting from configurations corresponding to translating inviscid equilibria, namely, the translating Föppl equilibria (counter-rotating point vortex pair, either fore or aft of a translating circular cylinder) and the translating Hill equilibria (counter-rotating point vortex pair, either fore or aft of an elliptic cylinder), viscosity is turned on for $t >0$ and the subsequent viscous interaction is simulated. The interaction is in a dynamically coupled settin…
Electron transport in two-dimensional conducting materials such as graphene, with dominant electron-electron interaction, exhibits unusual vortex flow that leads to a nonlocal current-field relation (negative resistance), distinct from the classical Ohm's law. The transport behavior of these materials is best described by low Reynolds number hydrodynamics, where the constitutive pressure-speed relation is Stoke's law. Here we report evidence of such vortices observed in a viscous flow of Newtonian fluid in a microfluidic device consisting of a rectangular cavity$-$analogous to the electronic s…
The initial-boundary value problem of the vorticity equation has been solved numerically by an iterative method. A variety of initial vorticity distributions is specified. All of them can be described by simple mathematical functions: there are a vorticity source-sink pair, circular shears out of a localised monopole, a cat's-eye topology and a few flows originating from single or multiple vortices. Our computational results show diverse flow phenomena, such as roll-ups of shear layers, vortex merging or impingement, as well as birth of spiral structures.
We explore the velocity fluctuations in a fluid due to a dilute suspension of randomly-distributed vortex rings at moderate Reynolds number, for instance those generated by a large colony of jellyfish. Unlike previous analysis of velocity fluctuations associated with gravitational sedimentation or suspensions of microswimmers, here the vortices have a finite lifetime and are constantly being produced. We find that the net velocity distribution is similar to that of a single vortex, except for the smallest velocities which involve contributions from many distant vortices; the result is a trunca…
The ability of vortices to induce aerodynamic loads on proximal surfaces plays a significant role in a wide variety of flows. However, most studies of vortex-induced effects primarily focus on analyzing the influence of the rotation-dominated cores of vortices. In this work, we show that not only are vortices in viscous flows surrounded by strain-dominated regions, but that these regions are dynamically important and can sometimes even dictate the induced aerodynamic loads. We demonstrate this for a pitching airfoil, which exhibits dynamic stall and generates several force-inducing vortices. U…
In the present paper, we examine the viscous flow evolution in a square cavity. Coupled with the stream function, the initial-boundary value problem of the vorticity is numerically solved by a method of iteration. The only boundary condition is the wall velocity which in turn defines the Dirichlet value for the stream function. We assert that the corner singularity in the cavity flow is in fact a theoretical artefact. By adopting suitably regulated lid velocity, excellent comparison with experiments is found because of the converged palinstrophy field. Our calculations demonstrate that asymmet…
The flow in a shock tube is extremely complex with dynamic multi-scale structures of sharp fronts, flow separation, and vortices due to the interaction of the shock wave, the contact surface, and the boundary layer over the side wall of the tube. Prediction and understanding of the complex fluid dynamics is of theoretical and practical importance. It is also an extremely challenging problem for numerical simulation, especially at relatively high Reynolds numbers. Daru & Tenaud (Daru, V. & Tenaud, C. 2001 Evaluation of TVD high resolution schemes for unsteady viscous shocked flows. Computers & …
We consider the mixing of a viscous fluid by the rotation of a pitched blade turbine inside an open, cylindrical tank, with air as the lighter fluid above. To examine the flow and interfacial dynamics, we utilise a highly-parallelised implementation of a hybrid front-tracking/level-set method that employs a domain-decomposition parallelisation strategy. Our numerical technique is designed to capture faithfully complex interfacial deformation, and changes of topology, including interface rupture and dispersed phase coalescence. As shown via transient, three-dimensional direct numerical simulati…
A new class of exact solutions of hydrodynamic equations for an incompressible fluid (gas) at the presence of a bulk sink and uprising vertical flows of matter is considered. The acceleration of the rotation velocity of classical non-stationary vortices is conditioned by the joint action of the convective and Coriolis hydrodynamic forces (accelerations), which appear due to the converging radial flows of the matter in the region of a bulk sink. It is shown that there exist velocity profiles that nullify viscous terms in the Navier-Stokes equations and represent a vortex structure with a "rigid…
Numerical calculations of Helium-II hydrodynamics show that a dense tangle of superfluid vortices induces in an initially stationary normal fluid a highly dissipative, complex, vortical flow pattern ("turbulence") with a -2.2 energy spectrum scaling exponent and fluctuations Reynolds number of order unity. In this normal fluid flow the effects of mutual friction excitation from the superfluid vortices and those of viscous stresses are of the same order. The results suggest that in previous experiments the dynamics of decaying, high Reynolds number, quantum turbulent flows could only weakly be …
The temporal evolution of the fluid circulation generated by a buoyancy force when two-dimensional (2D) arrays of 2D thermals are released into a quiescent incompressible fluid is studied through the results of numerous lattice Boltzmann simulations. It is observed that the circulation magnitude grows to a maximum value in a finite time. When both the maximum circulation and the time at which it occurs are non-dimensionalised by appropriately defined characteristic scales, it is shown that two simple Prandtl number (Pr) dependent scaling relations can be devised that fit these data very well o…
We report experiments on the deformation and transport of an elastic fiber in a viscous cellular flow, namely a lattice of counter-rotative vortices. We show that the fiber can buckle when approaching a stagnation point. By tuning either the flow or fiber properties, we measure the onset of this buckling instability. The buckling threshold is determined by the relative intensity of viscous and elastic forces, the elasto-viscous number Sp. Moreover we show that flexible fibers escape faster from a vortex (formed by closed streamlines) compared to rigid fibers. As a consequence, the deformation …
In this work we are interested in extreme vortex states leading to the maximum possible growth of palinstrophy in 2D viscous incompressible flows on periodic domains. This study is a part of a broader research effort motivated by the question about the finite-time singularity formation in the 3D Navier-Stokes system and aims at a systematic identification of the most singular flow behaviors. We extend the results reported in Ayala & Protas (2013) where extreme vortex states were found leading to the growth of palinstrophy, both instantaneously and in finite-time, which saturates the estimates …
This work investigates intensive vortices, which are characterised by the existence of a converging radial flow that significantly intensifies the flow rotation. Evolution and amplification of the vorticity present in the flow play important roles in the formation of the vortex. When rotation in the flow becomes sufficiently strong - and this implies validity of the strong swirl approximation developed by Einstein and Li (1951), Lewellen (1962), Turner (1966) and Lundgren (1985) - the analysis of Klimenko (2001a-c) and of the present work determine that further amplification of vorticity is mo…
The off-center collision of binary bouncing droplets of equal size was studied numerically by a volume-of-fluid (VOF) method with two marker functions, which has been validated by comparing with available experimental results. A non-monotonic kinetic energy recovery with varying impact parameters was found based on the energy budget analysis. This can be explained by the prolonged entanglement time and the enhanced internal-flow-induced viscous dissipation for bouncing droplets at intermediate impact parameters, compared with those at smaller or larger impact parameters. The universality of th…
Viscous heating can play an important role in the dynamics of fluids with strongly temperature-dependent viscosities because of the coupling between the energy and momentum equations. The heat generated by viscous friction produces a local temperature increase near the tube walls with a consequent decrease of the viscosity and a strong stratification in the viscosity profile. The problem of viscous heating in fluids was investigated and reviewed by Costa & Macedonio (2003) because of its important implications in the study of magma flows. Because of the strong coupling between viscosity and te…
Université de Caen Normandie（法国诺曼底大学）；CNRS UMR 7190, Institut Jean Le Rond d'Alembert（法国国家科学研究中心联合研究实验室7190，达朗贝尔研究所）；Indian Institute of Technology Bombay（印度孟买理工学院）
We derive a relationship for the vortex aspect ratio $α$ (vertical half-thickness over horizontal length scale) for steady and slowly evolving vortices in rotating stratified fluids, as a function of the Brunt-Vaisala frequencies within the vortex $N_c$ and in the background fluid outside the vortex $\bar{N}$, the Coriolis parameter $f$, and the Rossby number $Ro$ of the vortex: $α^2 = Ro(1+Ro) f^2/(N_c^2-\bar{N}^2)$. This relation is valid for cyclones and anticyclones in either the cyclostrophic or geostrophic regimes; it works with vortices in Boussinesq fluids or ideal gases, and the backg…
One of the most prominent dynamical features of turbulent, rapidly-rotating convection is the formation of large-scale coherent structures, driven by Reynolds stresses resulting from the small-scale convective flows. In spherical geometry, such structures consist of intense zonal flows that are invariant along the rotation axis. In planar geometry, long-lived, depth-invariant structures also form at large scales, but, in the absence of horizontal anisotropy, they consist of vortices that grow to the domain size. In this work, through the introduction of horizontal anisotropy into a numerical m…
In this paper, we examine the dimensionality of a single electrically driven vortex bounded by two no-slip and perfectly insulating horizontal walls distant by $h$. The study was performed in the weakly inertial limit by means of an asymptotic expansion, which is valid for any Hartmann number. We show that the dimensionality of the leading order can be fully described using the single parameter $l_z^ν/ h$, where $l_z^ν$ represents the distance over which the Lorentz force is able to act before being balanced by viscous dissipation. The base flow happens to introduce inertial recirculations in …
In classical viscous fluids, turbulent eddies are known to be responsible for the rapid spreading of embedded particles. But in an inviscid quantum fluid where the turbulence is induced by a chaotic tangle of quantized vortices, dispersion of the particles is achieved via a non-classical mechanism, i.e., their binding to the evolving quantized vortices. However, there is limited existing knowledge on how the vortices diffuse and spread in turbulent quantum fluids. Here we report a systematic numerical study of the apparent diffusion of vortices in a random vortex tangle in superfluid helium-4 …
The study focuses on the 3D electro-hydrodynamic (EHD) instability for flow between to parallel electrodes with unipolar charge injection with and without cross-flow. Lattice Boltzmann Method (LBM) with two-relaxation time (TRT) model is used to study flow pattern. In the absence of cross-flow, the base-state solution is hydrostatic, and the electric field is one-dimensional. With strong charge injection and high electrical Rayleigh number, the system exhibits electro-convective vortices. Disturbed by different perturbation patterns, such as rolling pattern, square pattern, and hexagon pattern…
We investigate superfluid flow around an airfoil accelerated to a finite velocity from rest. Using simulations of the Gross--Pitaevskii equation we find striking similarities to viscous flows: from production of starting vortices to convergence of airfoil circulation onto a quantized version of the Kutta-Joukowski circulation. We predict the number of quantized vortices nucleated by a given foil via a phenomenological argument. We further find stall-like behavior governed by airfoil speed, not angle of attack, as in classical flows. Finally we analyze the lift and drag acting on the airfoil.
The merger of two like-signed vortices is a well-studied problem, but in a turbulent flow, we may often have more than two like-signed vortices interacting. We study the merger of three or more identical co-rotating vortices initially arranged on the vertices of a regular polygon. At low to moderate Reynolds numbers, we find an additional stage in the merger process, absent in the merger of two vortices, where an annular vortical structure is formed and is long-lived. Vortex merger is slowed down significantly due to this. Such annular vortices are known at far higher Reynolds numbers in studi…
We investigate a mechanism to manipulate wall-bounded flows whereby wave-like undulations of the wall topography drives the creation of bespoke longitudinal vortices. A resonant interaction between the ambient vorticity of the undisturbed shear flow and the undulation of streamlines enforced by the wall topography serves to slightly rotate the spanwise vorticity of the mean flow into the streamwise direction, creating a swirling motion, in the form of regular streamwise rolls. The process is kinematic and essentially identical to the `direct drive' CL1 mechanism for Langmuir circulation (LC) p…
We validate a new law for the aspect ratio $α= H/L$ of vortices in a rotating, stratified flow, where $H$ and $L$ are the vertical half-height and horizontal length scale of the vortices. The aspect ratio depends not only on the Coriolis parameter f and buoyancy (or Brunt-Vaisala) frequency $\bar{N}$ of the background flow, but also on the buoyancy frequency $N_c$ within the vortex and on the Rossby number $Ro$ of the vortex such that $α= f \sqrt{[Ro (1 + Ro)/(N_c^2- \bar{N}^2)]}$. This law for $α$ is obeyed precisely by the exact equilibrium solution of the inviscid Boussinesq equations that …
In addition to mass, energy, and momentum, classical dissipationless flows conserve helicity, a measure of the topology of the flow. Helicity has far-reaching consequences for classical flows from Newtonian fluids to plasmas. Since superfluids flow without dissipation, a fundamental question is whether such a conserved quantity exists for superfluid flows. We address the existence of a "superfluid helicity" using an analytical approach based on the the symmetry underlying classical helicity conservation: the particle relabeling symmetry. Furthermore, we use numerical simulations to study wheth…
When considering flows in biological membranes, they are usually treated as flat, though more often than not, they are curved surfaces, even extremely curved, as in the case of the endoplasmic reticulum. Here, we study the topological effects of curvature on flows in membranes. Focusing on a system of many point vortical defects, we are able to cast the viscous dynamics of the defects in terms of a geometric Hamiltonian. In contrast to the planar situation, the flows generate additional defects of positive index. For the simpler situation of two vortices, we analytically predict the location o…
The paper reports the recent results on application and extension of the matrix formulation of lagrangian hydrodynamic equations. The matrix approach is based on the notion of continuous deformation of infinitesimal material elements and treats the Jacobi matrix of their derivatives with respect to lagrangian variables as the fundamental quantity completely describing fluid motion. We begin with brief review of the governing matrix equation (for the detailed discussion see Yakubovich, E.I. & Zenkovich, D.A. 2001 Matrix approach to Lagrangian fluid dynamics. J. Fluid Mech. vol. 443, 167-196). T…
We perform numerical simulations of a turbulent channel flow over an hyper-elastic wall. In the fluid region the flow is governed by the incompressible Navier-Stokes (NS) equations, while the solid is a neo-Hookean material satisfying the incompressible Mooney-Rivlin law. The multiphase flow is solved with a one-continuum formulation, using a monolithic velocity field for both the fluid and solid phase, which allows the use of a fully Eulerian formulation. The simulations are carried out at Reynolds bulk $Re=2800$ and examine the effect of different elasticity and viscosity of the deformable w…
The gravity-driven motion of rigid particles in a viscous fluid is relevant in many natural and industrial processes, yet this has mainly been investigated for spherical particles. We therefore consider the sedimentation of non-spherical (spheroidal) isolated and particle pairs in a viscous fluid via numerical simulations using the Immersed Boundary Method. The simulations performed here show that the critical Galileo number for the onset of secondary motions decreases as the spheroid aspect ratio departs from 1. Above this critical threshold, oblate particles perform a zigzagging motion where…
Viscous fluid exiting a long horizontal circular pipe develops a complex structure comprising a primary jet above and a smaller secondary jet below with a thin fluid curtain connecting them. We present here a combined experimental, theoretical and numerical study of this 'Torricelli's curtain' phenomenon, focusing on the factors that control its morphology. We propose a theoretical model for the curtain in which particle trajectories result from the composition of two motions: a horizontal component corresponding to the evolving axial velocity profile of an axisymmetric viscous jet, and a vert…
A commonplace view of pressure-driven turbulence in pipes and channels is as "cascades" of streamwise momentum toward the viscous layer at the wall. We present in this paper an alternative picture of these flows as "inverse cascades" of spanwise vorticity, in the cross-stream direction but away from the viscous sublayer. We show that there is a constant spatial flux of spanwise vorticity, due to vorticity conservation, and that this flux is necessary to produce pressure-drop and energy dissipation. The vorticity transport is shown to be dominated by viscous diffusion at distances closer to the…
In the fluid transport of particles, it is generally expected that heavy particles carried by a laminar fluid flow moving downward will also move downward. We establish a theory to show, however, that particles can be dynamically levitated and lifted by interacting vortices in such flows, thereby moving against gravity and the asymptotic direction of the flow, even when they are orders of magnitude denser than the fluid. The particle levitation is rigorously demonstrated for potential flows and supported by simulations for viscous flows. We suggest that this counterintuitive effect has potenti…
We study the steady state motion of incompressible and viscous fluid flow in a rotating reference frame where vortices may take place. An approximated analytic solution of the Stokes flow problem is proposed for situations where the vorticity is highly concentrated along a given direction. The approximation disconnects the component of velocity along the axis of rotation from the momentum equation. This enables to find the vorticity from the solution of the Poisson's equation. We use the Green's function approach to provide solutions of the three-dimensional flow on cylindrical domains with ra…
Using the previously developed model to describe laminar/turbulent states of a viscous fluid flow, which treats the flow as a collection of coherent structures of various size (Chekmarev, Chaos, 2013, 013144), the statistical temperature of the flow state is determined as a function of the Reynolds number. It is shown that at small Reynolds numbers, associated with laminar states, the temperature is positive, while at large Reynolds numbers, associated with turbulent states, it is negative. At intermediate Reynolds numbers, the temperature changes from positive to negative as the size of the c…
