Infinite Reynolds Number Research Articles

Stability analysis of viscous multi-layer shear flows with interfacial slip

Abstract One of the most fundamental interfacial instabilities in ideal, immiscible, incompressible multifluid flows is the celebrated Kelvin–Helmholtz (KH) instability. It predicts short-wave instabilities that, in the absence of other mollifying physical mechanisms (e.g. surface tension, viscosity), render the nonlinear problem ill-posed and lead to finite-time singularities. The crucial driving mechanism is the jump in tangential velocity across the liquid–liquid interface, i.e. interfacial slip, that can occur since viscosity is absent. The purpose of the present work is to analyse analogous instabilities for viscous flows at small or moderate Reynolds numbers as opposed to the infinite Reynolds numbers that underpin KH instabilities. The problem is physically motivated by both experiments and simulations. The fundamental model considered consists of two superposed viscous, incompressible, immiscible fluid layers sheared in a plane Couette flow configuration, with slip present at the deforming liquid–liquid interface. The origin of slip in viscous flows has been observed in experiments and molecular dynamics simulations, and can be modelled by employing a Navier-slip boundary condition at the liquid–liquid interface. The emerging novel instabilities are studied in detail here. The linear stability of the system is addressed asymptotically for long- and short-waves, and for arbitrary wavenumbers using a combination of analytical and numerical calculations. Slip is found to be capable of destabilising perturbations of all wavelengths. In regimes where the flow is stable to perturbations of all wavelengths in the absence of slip, its presence can induce a Turing-type instability by destabilization of a small band of finite wavenumber perturbations. In the case where the underlying layer is asymptotically thin, the results are found to agree with the linear properties of a weakly non-linear asymptotic model that is also derived here. The weakly nonlinear model extends previous work by the authors that had a thin overlying layer that produces a different evolution equation.

Theory and simulations of confined periodic turbulence

Confined Periodic Turbulence (CPT) is numerical homogeneous turbulence with periodic conditions, over an extended time until the eddy size is only limited by the period L. With large or infinite Reynolds number, this allows a self-similar decaying state with a constant spectrum shape and turbulent kinetic energy k asymptotically equal to C CPT L 2 / t 2 where t is time and C CPT is a constant, as predicted by Skrbek & Stalp. This setting may provide a long inertial range for a given resolution and is free of inputs such as initial spectra or forcing devices. Outside the viscous range, it generates the same spectra as the Linear Forcing proposed by Lundgren and exercised by Rosales & Meneveau. We conduct DNS, with the viscosity artificially decreasing in time to keep the Reynolds number Re ≡ L k / ν approximately constant, and LES at infinite Reynolds number, with resolution 10243. The solutions indeed lose memory of initial conditions. Both agree well with Rosales & Meneveau and with the L 2 / t 2 conjecture although with modulations; C CPT is about 0.5. Kovasznay's extension of Kolmogorov's theory, based on the local energy-transfer rate across wavenumbers, predicts the spectrum well even for intermediate wavenumbers, with the Kolmogorov constant C K at 1.65.

A robust phase-field method for two-phase flows on unstructured grids

A phase-field method for unstructured grids that is accurate, conservative, and robust is proposed in this work. The proposed method also results in bounded transport of volume fraction, and the interface thickness adapts automatically to local grid size. In addition to this, we present a novel formulation for two-phase flows on collocated grids that is provably energy stable, which is a critical feature for robust simulations of two-phase turbulent flows. The proposed method is first evaluated on canonical test cases, including scenarios like drop advection and drop in a shear flow. The accuracy of the proposed method in the presence of grid transitions, which is important for simulations in complex geometries using unstructured grids, is also evaluated. We assess the robustness of our method by performing simulations of a drop in homogeneous isotropic turbulence at infinite Reynolds number with varying density ratios. Furthermore, as a step toward verification and validation, simulations of test cases in complex geometries and conditions, such as, a damped surface wave, infinitesimal interface distortion on a liquid jet, and the atomization of a liquid jet from the engine combustion network's Spray A nozzle are presented. These simulations illustrate the accuracy, robustness, and applicability of the proposed method in various kinds of complex two-phase flow applications of engineering interest.

Fractional and tempered fractional models for Reynolds-averaged Navier–Stokes equations

Turbulence is a non-local phenomenon and has multiple-scales. Non-locality can be addressed either implicitly or explicitly. Implicitly, by subsequent resolution of all spatio-temporal scales. However, if directly solved for the temporal or spatially averaged fields, a closure problem arises on account of missing information between two points. To solve the closure problem in Reynolds-averaged Navier–Stokes equations (RANS), an eddy-viscosity hypotheses has been a popular modelling choice, where it follows either a linear or nonlinear stress–strain relationship. Here, a non-constant diffusivity is introduced. Such a non-constant diffusivity is also characteristic of non-Fickian diffusion equation addressing anomalous diffusion process. An alternative approach, is a fractional derivative-based diffusion equations. Thus, in the paper, we formulate a fractional stress–strain relationship using variable-order Caputo fractional derivative. This provides new opportunities for future modelling effort. We pedagogically study of our model construction, starting from one-sided model and followed by two-sided model applied to channel, couette and pipe flow. Non-locality at a point is the amalgamation of all the effects, thus we find the two-sided model is physically consistent. Further, our construction can also addresses viscous effects, which is a local process. Thus, our fractional model addresses the amalgamation of local and non-local process. We also show its validity at infinite Reynolds number limit. An expression for the fractional order is also found, thereby solving the closure problem for the considered cases. This study is further extended to tempered fractional calculus, where tempering ensures finite jump lengths, this is an important remark for unbounded flows. Within the context of this paper, we limit ourselves to wall bounded flow. Two tempered definitions are introduced with a smooth and sharp cutoff, by the exponential term and Heaviside function, respectively and we also define the horizon of non-local interactions. We further study the equivalence between the two definitions, as truncating the domain has computational advantages. For the above investigations, we carefully designed algorithms, notably, the pointwise version of fractional physics-informed neural network to find the fractional order as an inverse problem.

Intermittency across Reynolds numbers – the influence of large-scale shear layers on the scaling of the enstrophy and dissipation in homogenous isotropic turbulence

Direct numerical simulations up to Reλ = 1445 show that the scaling exponents for the enstrophy and the dissipation rate extrema are different and depend on the Reynolds number. A similar Reynolds number dependence of the scaling exponents is observed for the moments of the dissipation rate, but not for the moments of the enstrophy. Significant changes in the exponents occur at approximately Reλ ≈ 250, where Reλ is the Taylor based Reynolds number, which coincides with structural changes in the flow, in particular the development of large-scale shear layers. A model for the probability density functions (PDFs) of the enstrophy and dissipate rate is presented, which is an extension of our existing model (Proc. R. Soc. A, vol. 476, 2020, p. 20200591) and is based on the mentioned development of large-scale layer regions within the flow. This model is able to capture the observed Reynolds number dependencies of the scaling exponents, in contrast to the existing theories which yield constant exponents. Moreover, the model reconciles the scaling at finite Reynolds number with the theoretical limit, where the enstrophy and dissipation rate scale identically at infinite Reynolds number. It suggests that the large-scale shear layers are vital for understanding the scaling of the extrema. Furthermore, to reach the theoretical limit, the scaling exponents must remain Reynolds number dependent beyond the present Reλ range.

Spontaneous Stochasticity in the Presence of Intermittency.

Spontaneous stochasticity is a modern paradigm for turbulent transport at infinite Reynolds numbers. It suggests that tracer particles advected by rough turbulent flows and subject to additional thermal noise, remain nondeterministic in the limit where the random input, namely, the thermal noise, vanishes. Here, we investigate the fate of spontaneous stochasticity in the presence of spatial intermittency, with multifractal scaling of the lognormal type, as usually encountered in turbulence studies. In principle, multifractality enhances the underlying roughness, and should also favor the spontaneous stochasticity. This letter exhibits a case with a less intuitive interplay between spontaneous stochasticity and spatial intermittency. We specifically address Lagrangian transport in unidimensional multifractal random flows, obtained by decorating rough Markovian monofractal Gaussian fields with frozen-in-time Gaussian multiplicative chaos. Combining systematic MonteCarlo simulations and formal stochastic calculations, we evidence a transition between spontaneously stochastic and deterministic behaviors when increasing the level of intermittency. While its key ingredient in the Gaussian setting, roughness here surprisingly conspires against the spontaneous stochasticity of trajectories.

The Wall-Jet Region of a Turbulent Jet Impinging on Smooth and Rough Plates

The study reports direct numerical simulations of a turbulent jet impinging onto smooth and rough surfaces at Reynolds number Re = 10,000 (based on the jet mean bulk velocity and diameter). Surface roughness is included in the simulations using an immersed boundary method. The deflection of the flow after jet impingement generates a radial wall-jet that develops parallel to the mean plate surface. The wall-jet is structured into an inner and an outer layer that, in the limit of infinite local Reynolds number, resemble a turbulent boundary layer and a free-shear flow. The investigation assesses the self-similar character of the mean radial velocity and Reynolds stresses profiles scaled by inner and outer layer units for varying size of the roughness topography. Namely the usual viscous units u_tau and delta _nu are used as inner layer scales, while the maximum radial velocity u_m and its wall-normal location z_m are used as outer layer scales. It is shown that the self-similarity of the mean radial velocity profiles scaled by outer layer units is marginally affected by the span of roughness topographies investigated, as outer layer velocity and length reference scales do not show a significantly modified behavior when surface roughness is considered. On the other hand, the mean radial velocity profiles scaled by inner layer units show a considerable scatter, as the roughness sub-layer determined by the considered roughness topographies extends up to the outer layer of the wall-jet. Nevertheless, the similar character of the velocity profiles appears to be conserved despite the profound impact of surface roughness.

Accurate conservative phase-field method for simulation of two-phase flows

In this work, we propose a novel phase-field model for the simulation of two-phase flows that is accurate, conservative, bounded, and robust. The proposed model conserves the mass of each of the phases, and results in bounded transport of the volume fraction. We present results from the canonical test cases of a drop advection and a drop in a shear flow, showing significant improvement in the accuracy over the commonly used conservative phase-field method.Moreover, the proposed model imposes a lesser restrictive Courant-Friedrichs-Lewy condition, and hence, is less expensive compared to other conservative phase-field models. We also propose improvements on computation of surface tension forces and show that the proposed improvement significantly reduces the magnitude of spurious velocities at the interface.We also derive a consistent and conservative momentum transport equation for the proposed phase-field model and show that the proposed model when coupled with the consistent momentum transport equation results in discrete conservation of kinetic energy, which is a sufficient condition for the numerical stability of incompressible flows, in the absence of dissipative mechanisms. To illustrate the robustness of the method in simulating high-density ratio turbulent two-phase flows, we present the numerical simulations of high-density ratio droplet-laden isotropic turbulence at finite and infinite Reynolds numbers.

Two-time Lagrangian velocity correlation function for particle pairs in two-dimensional inverse energy-cascade turbulence

We numerically investigate a two-time Lagrangian velocity correlation function (TTLVCF) for particle pairs in two-dimensional energy inverse-cascade turbulence. We consider self similarity of the correlation function by means of incomplete similarity. In this framework, we propose a self-similar form of the correlation function, whose scaling exponents cannot be determined by only using the dimensional analysis based on the Kolmogorov's phenomenology. As a result, the scaling laws of the correlation function can depend on the initial separation. This initial-separation dependency is frequently observed in laboratory experiments and direct numerical simulations of the relative dispersion, which is directly related to the correlation function, at moderate Reynolds numbers. We numerically verify the self-similar form by direct numerical simulations of two-dimensional energy inverse-cascade turbulence. The involved scaling exponents and the dependencies on finite Reynolds number effects are determined empirically. Then, we consider implication of the scaling laws of the correlation function on the relative dispersion, i.e. the Richardson-Obukhov $t^3$ law. Our results suggest a possibility not to recover the Richardson-Obukhov $t^3$ law at infinite Reynolds number.

Vortex stretching and enstrophy production in stationary homogeneous isotropic turbulence

On average, the length of a material line element at any given time is longer than that at the initial time. Because a vortex line is frozen into the fluid at an infinite Reynolds number, it seems that the relation between the mean enstrophies (half the squared vorticity) at two time instances can be easily derived using a method similar to that for a material line element. However, this relation has not been analytically derived because a vortex line depends on the velocity even at the initial time. In this paper, we analytically show, using the conditional average, that the mean enstrophy normalized by the initial enstrophy at any given time is larger than one. We also analytically show that the mean enstrophy production is positive in stationary homogeneous isotropic turbulence for high Reynolds numbers.

Exact Eddy-Viscosity Equation for Turbulent Wall Flows—Implications for Computational Fluid Dynamics Models

For turbulent channel flow, pipe flow, and zero-pressure gradient boundary layer, Heinz yielded recently analytical formulas for the eddy viscosity as a product of a function of (the wall-normal distance scaled in inner units) and a function of (the same scaled in outer units). By calculating the eddy-viscosity turbulent diffusion term, an exact high-Reynolds-number equation with one production and two dissipation terms is constructed for those flows. One dissipation term is universal, peaks near the wall, and scales mainly with . The second, smaller one, is flow dependent, peaks in the wake, and scales mainly with . The production term is flow dependent, peaks in between, and scales similarly. The universal dissipation term implies a length scale analogous to the von Karman length scale used in the scale-adaptive simulation models of Menter. This length scale also appears in the production term. This confirms the relevance of these length scales. An asymptotic analysis of all terms in the budget in the limit of infinite Reynolds numbers is provided. This yields a test bench of existing Reynolds-averaged Navier–Stokes models with a similar equation. It is shown that some models, e.g., the one of Spalart and Allmaras, do not respect the flow physics: they display a production peak in the near-wall region. The most promising model, a scale-adaptive simulation model, is modified. As a step forward toward a solution to the wall damping problem, the equation of our model behaves much more correctly in the near-wall region.

Linear instability of lid- and pressure-driven flows in channels textured with longitudinal superhydrophobic grooves

It is known that an increased flow rate can be achieved in channel flows when smooth walls are replaced by superhydrophobic surfaces. This reduces friction and increases the flux for a given driving force. Applications include thermal management in microelectronics, where a competition between convective and conductive resistance must be accounted for in order to evaluate any advantages of these surfaces. Of particular interest is the hydrodynamic stability of the underlying basic flows, something that has been largely overlooked in the literature, but is of key relevance to applications that typically base design on steady states or apparent-slip models that approximate them. We consider the global stability problem in the case where the longitudinal grooves are periodic in the spanwise direction. The flow is driven along the grooves by either the motion of a smooth upper lid or a constant pressure gradient. In the case of smooth walls, the former problem (plane Couette flow) is linearly stable at all Reynolds numbers whereas the latter (plane Poiseuille flow) becomes unstable above a relatively large Reynolds number. When grooves are present our work shows that additional instabilities arise in both cases, with critical Reynolds numbers small enough to be achievable in applications. Generally, for lid-driven flows one unstable mode is found that becomes neutral as the Reynolds number increases, indicating that the flows are inviscidly stable. For pressure-driven flows, two modes can coexist and exchange stability depending on the channel height and slip fraction. The first mode remains unstable as the Reynolds number increases and corresponds to an unstable mode of the two-dimensional Rayleigh equation, while the second mode becomes neutrally stable at infinite Reynolds numbers. Comparisons of critical Reynolds numbers with the experimental observations for pressure-driven flows of Daniello et al. (Phys. Fluids, vol. 21, issue 8, 2009, p. 085103) are encouraging.

Asymptotics of streamwise Reynolds stress in wall turbulence

The scaling of different features of streamwise normal stress profiles$\langle uu\rangle ^+(y^+)$in turbulent wall-bounded flows is the subject of a long-running debate. Particular points of contention are the scaling of the ‘inner’ and ‘outer’ peaks of$\langle uu\rangle ^+$at$y^+\approxeq ~15$and$y^+ ={O}(10^3)$, respectively, their infinite Reynolds number limit, and the rate of logarithmic decay in the outer part of the flow. Inspired by the thought-provoking paper of Chen & Sreenivasan (J. Fluid Mech., vol. 908, 2021, p. R3), two terms of an inner asymptotic expansion of$\langle uu\rangle ^+$in the small parameter$Re_{\tau }^{-1/4}$are constructed from a set of direct numerical simulations (DNS) of channel flow. This inner expansion is for the first time matched through an overlap layer to an outer expansion, which not only fits the same set of channel DNS within 1.5 % of the peak stress, but also provides a good match of laboratory data in pipes and the near-wall part of boundary layers, up to the highest$Re_{\tau }$values of$10^5$. The salient features of the new composite expansion are first, an inner$\langle uu\rangle ^+$peak, which saturates at 11.3 and decreases as$Re_{\tau }^{-1/4}$. This inner peak is followed by a short ‘wall log law’ with a slope that becomes positive for$Re_{\tau }$beyond${O}(10^4)$, leading up to an outer peak, followed by the logarithmic overlap layer with a negative slope going continuously to zero for$Re_{\tau }\to \infty$.

Josephson-Anderson Relation and the Classical D’Alembert Paradox

Generalizing prior work of P. W. Anderson and E. R. Huggins, we show that a "detailed Josephson-Anderson relation" holds for drag on a finite body held at rest in a classical incompressible fluid flowing with velocity ${\bf V}.$ The relation asserts an exact equality between the instantaneous power consumption by the drag, $-{\bf F}\cdot{\bf V},$ and the vorticity flux across the potential mass current, $-(1/2)\int dJ\int \epsilon_{ijk}\Sigma_{ij}\,d\ell_k.$ Here $\Sigma_{ij}$ is the flux in the $i$th coordinate direction of the conserved $j$th component of vorticity and the line-integrals over $\ell$ are taken along streamlines of the potential flow solution ${\bf u}_\phi=\nabla\phi$ of the ideal Euler equation, carrying mass flux $dJ=\rho\,{\bf u}_\phi\cdot d{\bf A}.$ The results generalize the theories of M. J. Lighthill for flow past a body and, in particular, the steady-state relation $(1/2)\epsilon_{ijk}\langle\Sigma_{jk}\rangle =\partial_i\langle h\rangle,$ where $h=p+(1/2)|{\bf u}|^2$ is the generalized enthalpy or total pressure, extends Lighthill's theory of vorticity generation at solid walls into the interior of the flow. We use these results to explain drag on the body in terms of vortex dynamics, unifying the theories for classical fluids and for quantum superfluids. The results offer a new solution to the "D'Alembert paradox" at infinite Reynolds numbers and imply the necessary conditions for turbulent drag reduction.

Modeling the third-order velocity structure function in the scaling range at finite Reynolds numbers

A model for the third-order velocity structure function, S3, is proposed for closing the transport equation of the second-order velocity structure function, S2, in the scaling range of finite Reynolds number homogeneous and isotropic turbulence (HIT). The model is based on a gradient type hypothesis with an eddy-viscosity formulation. The present model differs from previous ones in that no assumptions are made with regard to the behaviors of S2 and S3 in the scaling range. This allows S3 to be modeled whether the intermittency of the energy dissipation ɛ (as modeled by the intermittency phenomenology) is considered or not. In both cases, the model predicts the same (infinite Reynolds number) asymptotic behavior for S2. This corresponds to the K41 prediction, i.e., S2 ∼ r2/3. The model yields good agreement against direct numerical simulation data for forced HIT at all scales of motion except in the transition region between the dissipative and scaling ranges. The introduction of a simple bridging function in the model improves significantly the agreement in this region. Furthermore, the model illustrates the effect of the finite Reynolds number on the scaling range and shows that this effect is responsible for the deviation from a power-law behavior.

Integral measures of the zero pressure gradient boundary layer over the Reynolds number range ≤Rτ<∞

A recently developed mixing length model of the turbulent shearing stress has been shown to generate a universal velocity profile that provides an accurate approximation to incompressible pipe flow velocity profiles over a wide Reynolds number range [B. J. Cantwell, “A universal velocity profile for smooth wall pipe flow,” J. Fluid Mech. 878, 834–874 (2019)]. More recently, the same profile was shown to accurately approximate velocity profiles in channel flow, the zero pressure gradient boundary layer, and the boundary layer in an adverse pressure gradient [M. A. Subrahmanyam, B. J. Cantwell, and J. J. Alonso, “A universal velocity profile for turbulent wall flows,” AIAA Paper No. 2021-0061, 2021 and M. A. Subrahmanyam, B. J. Cantwell, and J. J. Alonso, “A universal velocity profile for turbulent wall flows including adverse pressure gradient boundary layers,” J. Fluid Mech. (unpublished) (2021)] The universal velocity profile is uniformly valid from the wall to the free stream at all Reynolds numbers from zero to infinity. At a low Reynolds number, the profile approaches the laminar channel/pipe flow limit. The primary measure of the Reynolds number used in this work is the friction Reynolds number Rτ=uτδ/ν. It is a little unusual to use Rτ for the boundary layer since it requires that the velocity profile be cutoff using an arbitrarily defined overall boundary layer thickness, δ. Because of the slow approach of the velocity to the free stream, different conventions used to define the thickness lead to different values of Rτ assigned to a given flow. It will be shown in this paper that, through its connection to channel/pipe flow, the universal velocity profile can be used to define a practically useful, unambiguous, measure of overall boundary layer thickness, called here the equivalent channel half height, δh. For Rτ>≈5000, the universal velocity profile defines a Reynolds number independent shape function that can be used to generate explicit expressions for the infinite Reynolds number behavior of all the usual integral boundary layer measures; displacement thickness, momentum thickness, energy thickness, overall boundary layer thickness, and skin friction. The friction coefficient Cf(Rδ2) generated by the universal velocity profile accurately approximates data over a wide range of momentum thickness Reynolds numbers collected by Nagib et al. [“Can we ever rely on results from wall-bounded turbulent flows without direct measurements of wall shear stress?,” AIAA Paper No. 2004-2392, 2004]. The universal velocity profile is used to integrate the von Kaŕmań boundary layer integral equation [T. von Kármán, “Uber laminaire und turbulente reibung,” Z. Angew. Math. Mech. 1, 233–252 (1921)] in order to generate the various thicknesses and friction velocity as functions of the spatial Reynolds number, Rx=uex/ν.

Entropy generation and curvature effect on peristaltic thrusting of (Cu–Al2O3) hybrid nanofluid in resilient channel: Nonlinear analysis

AbstractThe pharyngeal stage of bolus transport or blood motion in the arteries cannot ignore the effect of curvature and inertial forces in peristaltic channels. Thus, the nonlinear curvature of peristaltic pumping becomes significant when the moving wall induces a curvilinear fluid motion at infinite Reynolds numbers, also the intrauterine fluid flow in the case of curvature nonpregnant uterus is an application of it. Hybrid nanofluids are new types of nanofluids, which can be produced by the suspension of multiple kinds of nanoparticles in base fluids. A hybrid nanomaterial is a substance that contains chemical and physical features of multiple materials together and gives these features in a homogeneous phase. The synthetic hybrid nanomaterial reveals significant physio‐chemical features that do not present in the individual components. In this investigation, the flow of an incompressible viscous fluid driven by sinusoidal waves moves on boundaries of the symmetric channel is studied when inertia and streamline curvature effects are not negligible, where the flow of nanofluid and hybrid nanofluid has been utilized to investigate the axial and normal behavior of velocities and heat transfer. An approximate solution to the second‐order in the wavenumber (giving the curvature effect) is obtained. To transform the variable cross‐section of the channel to a uniform cross‐section, we used a domain transformation, and this facilitates an easy way of getting a closed‐form of solutions to higher orders. Also, the pressure gradient and time flow rate relations are obtained. The influence of various biophysical parameters has been demonstrated through the graphs. All flow properties, temperature profile, and rate of heat transfer at the walls are greatly influenced by the presence of copper nanofluid. Furthermore, we observed that the case of spherical‐shaped nanoparticles provides a better heat transfer ability as compared to the case of cylindrically shaped nanoparticles. The effects of inertia and curvature are discussed and compared with some previous results in the literature.

A Subgrid Stabilized Method for Lid-driven Cavity Flow at Higher Reynolds Number

A subgrid stabilized method based on two local Gauss integrations is presented to the numerical simulation of 2-D steady incompressible lid-driven cavity flow at higher Reynolds numbers. The main idea of this method is to use the subgrid model based on two local Gauss integrations as a stabilized term on the finite element discretization. In this method, the discretization equation is solved by Oseen iterative scheme. It is shown that steady flow simulations of the lid-driven cavity problem are computable up to Re = 45000. This maximum Reynolds number has not been reached by other stabilized finite element methods reported. Moreover, the computed vorticity values at the center of the primary vortex agree well with previous analytical solutions in the limit of infinite Reynolds number, and the numerical results for the properties of the primary vortex and the velocity components are also in agreement with the benchmark data in earlier studies.

Modelling Lagrangian velocity and acceleration in turbulent flows as infinitely differentiable stochastic processes

Abstract

Flow of Spatiotemporal Turbulentlike Random Fields

We study the Lagrangian trajectories of statistically isotropic, homogeneous, and stationary divergence-free spatiotemporal random vector fields. We design this advecting Eulerian velocity field such that it gets asymptotically rough and multifractal, both in space and time, as it is demanded by the phenomenology of turbulence at infinite Reynolds numbers. We then solve numerically the flow equations for a differentiable version of this field. We observe that trajectories get also rough, characterized by nearly the same Hurst exponent as the one of our prescribed advecting field. Moreover, even when considering the simplest situation of the advection by a fractional Gaussian field, we evidence in the Lagrangian framework additional intermittent corrections. The present approach involves properly defined random fields, and asks for a rigorous treatment that would explain our numerical findings and deepen our understanding of this long lasting problem.