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 wall than the peak Reynolds stress, well into the classical log-layer. The Perry-Chong model based on "representative" hairpin/horsehoe vortices predicts a single sign of the turbulent vorticity flux over the whole log-layer, whereas the actual flux must change sign at the location of the Reynolds-stress maximum. Sign-reversal may be achieved by assuming a slow power-law decay of the Townsend "eddy intensity function" for wall-normal distances greater than the hairpin length-scale. The vortex-cascade picture presented here has a close analogue in the theory of quantum superfluids and superconductors, the "phase slippage" of quantized vortex lines. Most of our results should therefore apply as well to superfluid turbulence in pipes and channels. We also discuss issues about drag-reduction from this perspective.
We present a statistical analysis of the two-point vorticity probability density of the vorticity field generated in the inverse cascade of stationary two-dimensional turbulence.
We report the first extensive experimental observation of the two-dimensional enstrophy cascade, along with the determination of the high order vorticity statistics. The energy spectra we obtain are remarkably close to the Kraichnan Batchelor expectation. The distributions of the vorticity increments, in the inertial range, deviate only little from gaussianity and the corresponding structure functions exponents are indistinguishable from zero. It is thus shown that there is no sizeable small scale intermittency in the enstrophy cascade, in agreement with recent theoretical analyses.
The tendency of turbulent flows to produce fine-scale motions from large-scale energy injection is often viewed as a scale-wise cascade of kinetic energy driven by vorticity stretching. This has been recently evaluated by an exact, spatially-local relationship [Johnson, P. L. 2020, Phys. Rev. Lett., \textbf{124}, 104501], which also highlights the contribution of strain self-amplification. In this paper, the role of these two mechanisms is explored in more detail. Vorticity stretching and strain amplification interactions between velocity gradients filtered at the same scale account for about …
In this paper, the scaling property of the inverse energy cascade and forward enstrophy cascade of the vorticity filed $ω(x,y)$ in two-dimensional (2D) turbulence is analyzed. This is accomplished by applying a Hilbert-based technique, namely Hilbert-Huang Transform, to a vorticity field obtained from a $8192^2$ grid-points direct numerical simulation of the 2D turbulence with a forcing scale $k_f=100$ and an Ekman friction. The measured joint probability density function $p(C,k)$ of mode $C_i(x)$ of the vorticity $ω$ and instantaneous wavenumber $k(x)$ is separated by the forcing scale $k_f$ …
We investigate incompressible, inviscid vorticity dynamics on a rotating unit sphere using a Discrete Exterior Calculus (DEC) scheme. For a prescribed initial vorticity distribution, we vary the rate of rotation of the sphere from zero (non-rotating case, which corresponds to infinite Rossby number (Ro)) to 320 (which corresponds to Ro = $1.30 \times 10^{-3}$), and investigate the evolution with time of the vorticity field. For the non-rotating case, the vortices evolve into thin filaments due to so-called forward/direct enstrophy cascade. At late times an oscillating quadrupolar vortical fiel…
We investigate the dynamics of energy cascades in quantum turbulence by directly observing the vorticity distributions in numerical simulations of the Gross-Pitaevskii equation. By Fourier filtering each scale of the vorticity distribution, we find that antiparallel vortex tubes at a large scale generate small-scale vortex tubes orthogonal to those at the large scale, which is a manifestation of the energy cascade from large to small scales. We reveal the dynamics of quantized vortex lines in these processes.
In this paper, we show that the spatio-temporal evolution of incompressible flows in a long circular pipe can be described by vorticity dynamics. The principal techniques to obtain solutions are similar to those used for flows in the whole space. As the consideration of the Navier-Stokes equations is given in a cylindrical co-ordinates system, two aspects of complication arise. One is the interaction of the velocity components in the radial and azimuthal directions, due to the fictitious centrifugal force in the equations of motion. The rate of the vorticity production at the pipe wall depends…
Existence of 2D enstrophy cascade in a suitable mathematical setting, and under suitable conditions compatible with 2D turbulence phenomenology, is known both in the Fourier and in the physical scales. The goal of this paper is to show that the same geometric condition preventing the formation of singularities - 1/2-Hölder coherence of the vorticity direction - coupled with a suitable condition on a modified Kraichnan scale, and under a certain modulation assumption on evolution of the vorticity, leads to existence of 3D enstrophy cascade in physical scales of the flow.
We present results of numerical simulation of the direct cascade in two-dimensional hydrodynamic turbulence (with spatial resolution up to $16384 \times 16384$). If at the earlier stage (at the time of order of the inverse pumping growth rate $τ\simΓ_{max}^{-1}$), the turbulence develops according to the same scenario as in the case of a freely decaying turbulence \cite{KNNR-07, KKS}: quasi-singular distributions of di-vorticity are formed, which in $k$-space correspond to jets, leading to a strong turbulence anisotropy, then for times of the order of $10τ$ turbulence becomes almost isotropic.…
We generalize Kirchhoff's point vortex model of two-dimensional fluid motion to a rotor model which exhibits an inverse cascade by the formation of rotor clusters. A rotor is composed of two vortices with like-signed circulations glued together by an overdamped spring. The model is motivated by a treatment of the vorticity equation representing the vorticity field as a superposition of vortices with elliptic Gaussian shapes of variable widths, augmented by a suitable forcing mechanism. The rotor model opens up the way to discuss the energy transport in the inverse cascade on the basis of dynam…
A vorticity surge event that could be a paradigm for a wide class of bursting events in turbulence is studied to examine how the energy cascade is established and how this event could serve as a new test of LES turbulence models. This vorticity surge event is tied to the formation of the energy cascade in a direct numerical simulation by the traditional signatures of a turbulent energy cascade such as spectra approaching -5/3 and strongly Beltramized vortex tubes. A coherent mechanism is suggested by the nearly simultaneous development of a maximum of the peak vorticity $\|ω\|_\infty$, growth …
Spin hydrodynamic coupling is a recently discovered method to directly generate electricity from an electrically conducting fluid flow in the absence of Lorentz forces. This method relies on a collective coupling of electron spins - the internal quantum mechanical angular momentum of the electrons - with the local vorticity of a fluid flow. In this work, we experimentally investigate the spin hydrodynamic coupling in circular and non-circular capillary pipe flows and extend a previously obtained range of Reynolds numbers to smaller and larger values, 20<Re<21,500, using the conducting liquid m…
The aim of this paper is to understand the tendency to organization of the turbulence in two-dimensional ideal fluids. We show that nonlinear processes as inverse cascade of the energy and vorticity concentration are essentially determined by trajectory trapping or eddying. The statistics of the trajectories of the vorticity elements is studied using a semianalytic method. The separation of the positive and negative vorticity is due to the attraction produced by a large scale vortex on the small scale vortices of the same sign. More precisely, a large scale velocity is shown to determine avera…
The transfer of kinetic energy from large to small scales is a hallmark of turbulent flows. Yet, a precise mechanistic description of this transfer, which is expected to occur via an energy cascade, is still missing. Several conceptually simple configurations with vortex tubes have been proposed as a testing ground to understand the energy cascade. Here, we focus on incompressible flows and compare the energy transfer occurring in a statistically steady homogeneous isotropic turbulent (HIT) flow with the generation of fine-scale motions in configurations involving vortex tubes. We start by fil…
As a minimal mathematical model generating cascade analogous to that of the Navier-Stokes turbulence in the inertial range, we propose a one-dimensional partial-differential-equation model that conserves the integral of the squared vorticity analogue (enstrophy) in the inviscid case. With a large-scale forcing and small viscosity, we find numerically that the model exhibits the enstrophy cascade, the broad energy spectrum with a sizable correction to the dimensional-analysis prediction, peculiar intermittency and self-similarity in the dynamical system structure.
The results of an analysis of turbulent pipe flow based on a Karhunen-Lo`eve decomposition are presented. The turbulent flow is generated by a direct numerical simulation of the Navier-Stokes equations using a spectral element algorithm at a Reynolds number Re_τ=150. This simulation yields a set of basis functions that captures 90% of the energy after 2,453 modes. The eigenfunctions are categorised into two classes and six subclasses based on their wavenumber and coherent vorticity structure. Of the total energy, 81% is in the propagating class, characterised by constant phase speeds; the rema…
We report a linear scaling law for an electrical voltage as a function of the pressure drop in capillary pipes and ducts. This voltage is generated by a process which is termed spin hydrodynamic generation (SHDG), a result of the collective electron spin--coupling to the vorticity field in the laminar flow in combination with an inverse spin-Hall effect. We study this phenomenon in laminar duct flows with different width-to-height aspect ratios ranging from 1 (square ducts) to infinite (two dimensional channels). First, we analytically solve the governing Valet-Fert spin diffusion equations fo…
In this Letter we show that a bifurcation cascade and fully sustained turbulence can share the phase space of a fluid flow system, resulting in the presence of competing stable attractors. We analyse the toroidal pipe flow, which undergoes subcritical transition to turbulence at low pipe curvatures and supercritical transition at high curvatures, as was previously documented. We provide decisive evidence that the nature of the supercritical transition is of Ruelle--Takens type and that, in a narrow range of intermediate curvatures, its dynamics competes with that of sustained turbulence emergi…
We present direct numerical simulation of a mechanism for creating longitudinal vortices in pipe flow, compared with a simple model theory. By furnishing the pipe wall with a pattern of crossing waves secondary flow in the form of spanwise vortex pairs is created. The mechanism `CL1' is kinematic and known from oceanography as a driver of Langmuir circulation. CL1 is strongest when the `wall wave' vectors make an accute angle with the axis, $φ=10^\circ$ - $20^\circ$ (a `contracted eggcarton'), changes sign near $45^\circ$ and is weak and opposite beyond this angle. A competing, dynamic mechani…
We present a study on the nonlinear dynamics of small long-wave disturbances to the laminar state in non-rotating axisymmetric Poiseuille pipe flows. At high Reynolds numbers, the associated Navier-Stokes equations can be reduced to a set of coupled Korteweg-de Vries-type (KdV) equations that support inviscid and smooth travelling waves numerically computed using the Petviashvili method. In physical space they correspond to localized toroidal vortices concentrated near the pipe boundaries (wall vortexons) or that wrap around the pipe axis (centre vortexons), in agreement with the analytical so…
We study numerically a succession of transitions in pipe Poiseuille flow that leads from simple travelling waves to waves with chaotic time-dependence. The waves at the origin of the bifurcation cascade possess a shift-reflect symmetry and are both axially and azimuthally periodic with wave numbers κ = 1.63 and n = 2, respectively. As the Reynolds number is increased, successive transitions result in a wide range of time dependent solutions that includes spiralling, modulated-travelling, modulated-spiralling, doubly-modulated-spiralling and mildly chaotic waves. We show that the latter spring …
We investigate the vorticity-preserving properties of the compressible, second-order residual-based scheme, "RBV2". The scheme has been extensively tested on hydrodynamical problems, and has been shown to exhibit remarkably accurate results on the propagation of inviscid compressible vortices, airfoil-vortex interactions on a curvilinear mesh, vortex mergers in an astrophysical accretion disk, and the establishment of a two-dimensional inverse cascade in high-resolution turbulent simulations. Here, we demonstrate that RBV2 sustains the analytic solution for a one-dimensional shear flow. We ass…
We first summarize briefly several properties concerning the dynamics of two-dimensional (2D) turbulence, with an emphasis on the inverse cascade of energy to the largest accessible scale of the system. In order to study a similar phenomenon in three-dimensional (3D) turbulence undergoing strong solid-body rotation, we test a previously developed Large Eddy Simulation (LES) model against a high-resolution direct numerical simulation of rotating turbulence on a grid of $3072^3$ points. We then describe new numerical results on the inverse energy cascade in rotating flows using this LES model an…
It is well known that an inverse turbulent cascade in a finite ($2 π\times 2 π$) two-dimensional periodic domain leads to the emergence of a system-sized coherent vortex dipole. We report a numerical hyperviscous study of the spatial vorticity profile inside one of the vortices. The exciting force was shortly correlated in time, random in space, and had a correlation length $l_f = 2π/k_f$ with $k_f$ ranging from $100$ to $12.5$. Previously, it was found that in the asymptotic limit of small-scale forcing, the vorticity exhibits the power-law behavior $Ω(r) = (3 ε/α)^{1/2} r^{-1}$, where $r$ is…
We investigate the non-linear equilibration of a two-layer quasi-geostrophic flow in a channel forced by an imposed unstable zonal mean flow, paying particular attention to the role of bottom friction. In the limit of low bottom friction, classical theory of geostrophic turbulence predicts an inverse cascade of kinetic energy in the horizontal with condensation at the domain scale and barotropization on the vertical. By contrast, in the limit of large bottom friction, the flow is dominated by ribbons of high kinetic energy in the upper layer. These ribbons correspond to meandering jets separat…
Vorticity distributions in axisymmetric vortex rings produced by a piston-pipe apparatus are numerically studied over a range of Reynolds numbers, $\mathrm{Re}$, and stroke-to-diameter ratios, $L/D$. It is found that a state of advective balance, such that $ζ\equiv ω_ϕ/r \approx F(ψ, t)$, is achieved within the region (called the vortex ring bubble) enclosed by the dividing streamline. Here $ζ\equivω_ϕ/r$ is the ratio of azimuthal vorticity to cylindrical radius, and $ψ$ is the Stokes streamfunction in the frame of the ring. Some but not all of the $\mathrm{Re}$ dependence in the time evolutio…
Kelvin's Theorem on conservation of circulations is an essential ingredient of G. I. Taylor's theory of turbulent energy dissipation by the process of vortex-line stretching. In previous work, we have proposed a nonlinear mechanism for the breakdown of Kelvin's Theorem in ideal turbulence at infinite Reynolds number. We develop here a detailed physical theory of this ``cascade of circulations''. Our analysis is based upon an effective equation for large-scale ``coarse-grained'' velocity, which contains a turbulent-induced ``vortex-force'' that can violate Kelvin's Theorem. We show that singula…
Most of the turbulent flows appearing in nature (e.g. geophysical and astrophysical flows) are subjected to strong rotation and stratification. These effects break the symmetries of classical, homogenous isotropic turbulence. In doing so, they introduce a natural decomposition of phase space in terms of wave modes and potential vorticity modes. The appearance of a new time scale associated to the propagation of waves, in addition to the eddy turnover time, increases the complexity of the energy transfers between the various scales; nonlinearly interacting waves may dominate at some scales whil…
Exploring 2D turbulent properties in anisotropic and disordered Fourier space: Insights into inverse cascades and universal superdiffusion from randomly sampled triadic interaction
National Research Council, Institute of Atmospheric Pollution Research（国家研究委员会，大气污染研究所）；University of Calabria（卡拉布里亚大学）；Dipartimento di Fisica, University of Calabria（卡拉布里亚大学物理系）
We use momentum transfer arguments to predict the friction factor $f$ in two-dimensional turbulent soap-film flows with rough boundaries (an analogue of three-dimensional pipe flow) as a function of Reynolds number Re and roughness $r$, considering separately the inverse energy cascade and the forward enstrophy cascade. At intermediate Re, we predict a Blasius-like friction factor scaling of $f\propto\textrm{Re}^{-1/2}$ in flows dominated by the enstrophy cascade, distinct from the energy cascade scaling of $\textrm{Re}^{-1/4}$. For large Re, $f \sim r$ in the enstrophy-dominated case. We use …
Results from a new series of experiments on turbulent flows in a rotating circular container are presented. Electromagnetic forcing is applied to induce flow in a layer of fluid of constant depth. Continuously forced as well as decaying flows are investigated. Optical altimetry is used to measure the gradient of the surface elevation field and to obtain the velocity and vorticity fields with high temporal and spatial resolution. Spectral analysis of the flows demonstrates the formation of dual cascade with energy and enstrophy intervals although the corresponding spectral fluxes of energy and …
In this paper, we study the nonlinear dynamics of an axisymmetric disturbance to the laminar state in non-rotating Poiseuille pipe flows. In particular, we show that the associated Navier-Stokes equations can be reduced to a set of coupled Camassa-Holm type equations. These support inviscid and smooth localized travelling waves, which are numerically computed using the Petviashvili method. In physical space they correspond to localized toroidal vortices that concentrate near the pipe boundaries (wall vortexons) or wrap around the pipe axis (centre vortexons) in agreement with the analytical so…
We present experimental evidence of statistical conformal invariance in isocontours of fluid thickness in experiments of two-dimensional turbulence using soap films. A Schlieren technique is used to visualize regions of the flow with constant film thickness, and association of isocontours with Schramm-Löwner evolution (SLE) is used to identify conformal invariance. In experiments where an inverse energy cascade develops, statistical evidence is consistent with such an association. The diffusivity of the associated one-dimensional Brownian process is close to 8/3, a value previously identified …
This study focuses on a parametric study of the laminar fast transient flow of non-Newtonian fluids through helical pipes. Classical simulations of fluid hammer do not deal with the pipeline helicity and non-Newtonian characteristics of the fluid, while the present work addresses those features. To this end, the power-law model is employed to accommodate the non-Newtonian behavior of the fluid. Effects of the pipe wall elasticity and compressibility of the working fluid are taken into account through a modified bulk modulus elasticity of the fluid. The results of the three-dimensional numerica…
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…
Helicity is a quadratic invariant of the Euler equation in three dimensions. As the energy, when present helicity cascades to smaller scales where it dissipates. However, the role played by helicity in the energy cascade is still unclear. In non-helical flows, the velocity and the vorticity tend to align locally creating patches with opposite signs of helicity. Also in helical flows helicity changes sign rapidly in space. Not being a positive definite quantity, global studies considering its spectral scaling in the inertial range are inconclusive, except for cases where one sign of helicity is…
The role of short-wave instabilities on geostrophic turbulence is studied in a simplified model consisting of three layers in the quasi-geostrophic approximation. The linear stability analysis shows that short-wave instabilities are created by the interplay between the shear in the upper and the lower layers. If the stratification is non-uniform, in particular surface intensified, the linear growth rate is larger for short-wave instabilities than for long-wave instabilities and the layers are essentially decoupled, with the small scales growing independently. The fully developed homogeneous tu…
