Using a multi-scaled, chaotic flow known as the KS model of turbulence, we investigate the dependence of Lyapunov exponents on various characteristics of the flow. We show that the KS model yields a power law relation between the Reynolds number and the maximum Lyapunov exponent, which is similar to that for a turbulent flow with the same energy spectrum. Our results show that the Lyapunov exponents are sensitive to the advection of small eddies by large eddies, which can be explained by considering the Lagrangian correlation time of the smallest scales. We also relate the number of stagnation points within a flow to the maximum Lyapunov exponent, and suggest a linear dependence between the two characteristics.
In three dimensional turbulence there is on average a cascade of kinetic energy from the largest to the smallest scales of the flow. While the dominant idea is that the cascade occurs through the physical process of vortex stretching, evidence for this is debated. In the framework of the Karman-Howarth equation for the two point turbulent kinetic energy, we derive a new result for the average flux of kinetic energy between two points in the flow that reveals the role of vortex stretching. However, the result shows that vortex stretching is in fact not the main contributor to the average energy…
Interscale energy transfers at the vicinity of the turbulent/non-turbulent interface are from small to large scales in directions close to the interface's tangent plane where motions are predominantly stretching, but from large to small scales in the other directions where motions are predominantly compressive and significantly correlated with square angular momentum. An important role in this predominance is played by the extreme compressive motions which can be significantly more likely than extreme stretching motions even where motions are on average stretching. The most intense interscale …
Energy dissipation and pressure fluctuations are features inherent to turbulent fluid motion, and the internal origin of spatial and temporal temperature fluctuations. By measuring local velocity, pressure or temperature, it is possible to detect the presence of coherent structures ---like concentrated vortices--- and quantify their intensity. Here we report measurements of speed and temperature in a filamentary vortex lying in a strongly stretched turbulent airflow. The largest core temperature drop measured in our experiment was $ΔT = 9$ K, while the fastest measured air speed was $v_{max} =…
Small flexible fibers in a turbulent flow are found to be most of the time as straight as stiff rods. This is due to the cooperative action of flexural rigidity and fluid stretching. However, fibers might bend and buckle when they tumble and experience a strong-enough local compressive shear. Such events are similar to an activation process, where the role of temperature is played by the inverse of Young's modulus. Numerical simulations show that buckling occurs very intermittently in time. This results from unexpected long-range Lagrangian correlations of the turbulent shear.
Turbulent flows play an important role in many aspects of nature and technics from sea storms to transport of particles or chemicals. Transport of energy from large scales to small fluctuations is the essential feature of three-dimensional turbulence. What mechanism is responsible for this transport and how do the small fluctuations appear? The conventional conception implies a cascade of breaking vortices. But it faces crucial problems in explaining the mechanism of the breaking, and fails to explain the observed long-living structures in turbulent flows. We suggest a new concept based on rec…
We performed two-dimensional numerical simulations of passive stirring of two liquids which generated two spatially distinguishable paradigmatic velocity flows, viz. few large extended vortices and rapid oscillations. Using the Okubo- Weiss criterion, we mapped regions of stretching and rotation in the flow field. We find that an elementary oscillatory sequence of stretching-rotation-stretching induces spatial redistribution and steepening of concentration gradients, and the aggregation of such sequences determines fluid mixing at a downstream position. Furthermore, we quantify the paradigms o…
The equations governing the magnetohydrodynamic stagnation point flow toward a non-conducting, thermally insulated, nonporous, linearly stretching sheet are cast in a self similar form. Consistent boundary conditions on the velocity, magnetic field and temperature are invoked. The flow problem involves three parameters- the magnetic Prandtl number, the magnetic interaction number, and the ratio of the stretching rate to the strength of the stagnation point flow. The energy equation includes viscous dissipation and Joule heating, and introduces the Prandtl number as a fourth parameter. Numerica…
Large eddy simulation allows to incorporate the important driving physics of turbulent flow through forest- or vegetation-like canopies. In this paper we investigate the effects of vortex stretching and coherent structures on the subgrid-scale (SGS) turbulence kinetic energy (TKE). We present three simulations (SGS-d/s/w) of turbulence-canopy interactions. SGS-d assumes a local, dynamic balance of SGS production with SGS dissipation. SGS-s averages the SGS contributions of coherent structures over Lagrangian pathlines. SGS-w assumes that an average cascade of TKE from large- to small-scales oc…
We study the relation between flow structure and fluid deformation in steady two-dimensional random flows. Beyond the linear (shear flow) and exponential (chaotic flow) elongation paradigms, we find a broad spectrum of stretching behaviors, ranging from sub- to superlinear, which are dominated by intermittent shear events. We analyze these behaviors from first principles, which uncovers stretching as a result of the non-linear coupling between Lagrangian shear deformation and velocity fluctuations along streamlines. We derive explicit expressions for Lagrangian deformation and demonstrate that…
Viscoelastic flows are pervasive in a host of natural and industrial processes, where the emergence of nonlinear and time-dependent dynamics regulate flow resistance, energy consumption, and particulate dispersal. Polymeric stress induced by the advection and stretching of suspended polymers feeds back on the underlying fluid flow, which ultimately dictates the dynamics, instability, and transport properties of viscoelastic fluids. However, direct experimental quantification of the stress field is challenging, and a fundamental understanding of how Lagrangian flow structure regulates the distr…
In this article we consider the linear stability of the two-dimensional flow induced by the linear stretching of a surface in the streamwise direction. The basic flow is a rare example of an exact analytical solution of the Navier-Stokes equations. Using results from a large Reynolds number asymptotic study and a highly accurate spectral numerical method we show that this flow is linearly unstable to disturbances in the form of Tollmien-Schlichting waves. Previous studies have shown this flow is linearly stable. However, our results show that this is only true for Görtler-type disturbances.
The full MHD problem of the flow and heat transfer due to a linearly stretching sheet in the presence of a transverse magnetic field is put in a self-similar form. Traditionally ignored physical processes such as induced magnetic field, viscous dissipation, Joule heating, and work shear are included and their importance is established. Cases of prescribed surface temperature, prescribed heat flux, surface feed (injection or suction), velocity slip, and thermal slip are also considered. The problem is shown to admit self similarity. Sample numerical solutions are obtained for chosen combination…
The solution for the MHD flow, due to a linearly stretching sheet, has a simple form for the velocity field, with a companion simple form for the induced magnetic field. The associated thermal problem, including viscous dissipation and Joule heating, involves three temperature constituents, the solutions for two of which are obtained in terms of Kummer's function. The solution for the third temperature constituent is obtained in a convergent series form.
Lagrangian Phase-Lag and Geometric Precedence in Turbulent Vortex Stretching
Mechanical Engineering Deparatment, College of Engineering and Technology（机械工程系，工程学院与技术学院）；Arab Academy for Science, Technology and Maritime Transport（阿拉伯科学、技术与海运学院）
We propose a natural stretching function for DNS of wall-bounded flows, which blends uniform near-wall spacing with uniform resolution in terms of Kolmogorov units in the outer wall layer. Numerical simulations of pipe flow are used to educe optimal value of the blending parameter and of the wall grid spacing which guarantee accuracy and computational efficiency as a results of maximization of the allowed time step. Conclusions are supported by DNS carried out at sufficiently high Reynolds number that a near logarithmic layer is the mean velocity profile is present. Given a target Reynolds num…
The full MHD equations, governing the flow due to the axisymmetric stretching of a sheet in the presence of a transverse magnetic field, can be cast in a self similar form. This allows evaluation of the induced magnetic field and its effect on the flow and heat transfer. The problem involves three parameters- the magnetic Prandtl number, the magnetic interaction number, and the Prandtl number. Numerical solutions are obtained for the velocity field, the magnetic field, and the temperature, at different values of the magnetic Prandtl number and the magnetic interaction number. The contributions…
At the point of pinch-off of an underwater air bubble, the speed of water rushing in diverges. Previous studies that assumed radial flow throughout showed that the local axial shape is two smoothly connected, slender cones that transition very slowly (logarithmically) to a cylindrical segment. Our simulations show that even with initially radial flow, a transient vertical flow develops with comparable speeds. Bernoulli pressure draws water into the singularity region while incompressibility forces it away from the neck minimum, generating significant vertical flows that rapidly slenderize and …
How pore-scale disorder controls fluid stretching in two-dimensional porous media
University of Oslo（奥斯陆大学）；University of Rennes（雷恩大学）；Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, IRD, Univ. Gustave Eiffel, ISTerre（格勒诺布尔阿尔卑斯大学、萨瓦蒙布朗克大学、法国国家科学研究中心、法国发展研究所、居斯塔夫埃菲尔大学、ISTerre研究所）；Norwegian University of Science and Technology（挪威科技大学）
The current study is a pioneering work in presenting the boundary layer equations for the two-dimensional flow and heat transfer of the Cross fluid over a linearly stretching sheet. The system of partial differential equations is turned down into highly non-linear ordinary differential equations by applying suitable similarity transformations. The stretching sheet solutions are presented via. a numerical technique namely the shooting method and graphs are constructed for the shear-thinning as well as shear-thickening regime. The impact of the emerging parameters namely the power-law index , th…
An analysis has been investigated for the effects of thermal radiation and Hall current on magnetohydrodynamic free-convective flow and mass transfer over a stretching sheet with variable viscosity in the presence of heat generation/absorption. The fluid viscosity is assumed to vary as an inverse linear function of temperature. The boundary-layer equations governing the flow problem under consideration have been reduced to a system of non-linear ordinary differential equations by employing a similarity transformation. Using the finite difference scheme, numerical solutions to the transform ord…
How hydrodynamic interactions alter polymer stretching in turbulence
IITB-Monash Research Academy, Indian Institute of Technology Bombay（印度孟买理工学院-蒙纳士大学联合研究院）；Indian Institute of Technology Bombay（印度孟买理工学院）；Monash University（蒙纳士大学）；Université Côte d’Azur（蔚蓝海岸大学）
In this paper we present a mathematical analysis of thin film flow and heat transfer to a laminar liquid film from a horizontal stretching sheet. The flow of thin liquid film and subsequent heat transfer from the stretching surface is investigated with the aid of similarity transformations. Similarity transformations are used to convert unsteady boundary layer equations to a system of non-linear ordinary differential equations. The resulting non-linear differential equations are solved numerically using Runge-kutta-Fehlberg and Newton-Raphson schemes. A relationship between film thickness $β$ …
The effects of MHD boundary layer flow of non-linear thermal radiation with convective heat transfer and non-uniform heat source/sink in presence of thermophortic velocity and chemical reaction investigated in this study. Suitable similarity transformation are used to solve the partial ordinary differential equation of considered governing flow. Runge-Kutta fourth fifth order Fehlberg method with shooting techniques are used to solved non-dimensional governing equations. The variation of different parameters such as thermophoretic parameter, chemical reaction parameter, non- uniform heat sourc…
In homogeneous isotropic turbulence, slender rods are known to align with the Lagrangian stretching direction. However, how the degree of alignment depends on the aspect ratio of the rod is not understood. Moreover, many flows of practical interest are anisotropic and inhomogeneous. Here we study the alignment of rods with the Lagrangian stretching direction in a channel flow, which is approximately homogeneous and isotropic near the center but inhomogeneous and anisotropic near the walls. Our main question is how the distribution of relative angles between a rod and the Lagrangian stretching …
Polymers in a turbulent flow are stretched out by the fluctuating velocity gradient; the stationary probability distribution function (p.d.f.) of extensions $R$ has a power-law tail with an exponent that increases with the Weissenberg number $Wi$, a nondimensional measure of polymer elasticity. This study addresses the following questions: (i) What is the role of the non-Gaussian statistics of the turbulent velocity gradient on polymer stretching? (ii) How does the p.d.f. of $R$ evolve to its asymptotic stationary form? Our analysis is based on simulations of the dynamics of finitely-extensibl…
In this paper we present an extension of the \emph{Recent Fluid Deformation (RFD)} closure introduced by Chevillard and Meneveau (2006) which was developed for modeling the time evolution of Lagrangian fluctuations in incompressible Navier-Stokes turbulence. We apply the RFD closure to study the evolution of magnetic and passive scalar fluctuations. This comparison is especially interesting since the stretching term for the magnetic field and for the gradient of the passive scalar are similar but differ by a sign such that the effect of stretching and compression by the turbulent velocity fiel…
In the present study, MHD boundary layer flow with heat and mass transfer of a nanofluid with viscous dissipation over a non-linear stretching sheet embedded in a porous medium is studied. The governing non-linear partial differential equations are converted into coupled non-linear ordinary differential equations which are solved by hybrid technique consisting of finite element method with symbolic computation. The effect of Viscous dissipation, magnetic influence parameter, permeability parameter, Brownian motion, Eckert number, Thermophoresis parameter, Prandtl number parameter, Dufour Lewis…
Of concern in this paper is an investigation of heat and mass transfer over a stretching sheet under the influence of an applied uniform magnetic field and the effects of Hall current are taken into account. The non-linear boundary layer equations together with the boundary conditions are reduced to a system of non-linear ordinary differential equations by using the similarity transformation. The system of non-linear ordinary differential equations are solved by developing a suitable numerical techniques such as finite difference scheme and Newton's method of linearization. The numerical resul…
The three dimensional optimal energy growth mechanism, in plane parallel shear flows, is reexamined in terms of the role of vortex stretching and the interplay between the span-wise vorticity and the planar divergent components. For high Reynolds numbers the structure of the optimal perturbations in Couette, Poiseuille, and mixing layer shear profiles is robust and resembles localized plane-waves in regions where the background shear is large. The waves are tilted with the shear when the span-wise vorticity and the planar divergence fields are in (out of) phase when the background shear is pos…
The problem of natural convection in a laterally heated three-dimensional cubic cavity under the action of an externally imposed magnetic field is revisited. Flows at the Rayleigh number Ra=10^6 and the Hartmann number Ha=100, and three different orientations of the magnetic field are considered. The problem is solved using two independent numerical methods based on the second order finite-volume discretization schemes on structured Cartesian grids. Convergence toward grid-independent results is examined versus the grid refinement and near-wall grid stretching. Converged benchmark-quality resu…
Computational Exploration of Inclined Magnetic Fields and Variable Thermal Flux Effects on the Flow of Dusty Hybrid Nanofluid around Stretching/Shrinking Wedge
The present investigation discusses the flow and heat transfer characteristics of a steady three dimensional Sisko fluid. The flow is induced due to bidirectional stretching sheet. The influence of power-law index and stretching ratio on flow and heat transfer is studied thoroughly. Governing partial differential equations are reduced to coupled ordinary differential equations by suitable similarity variable. The resulting equations are then solved numerically by shooting method using adaptive Runge Kutta algorithm in combination with Broyden's method in the domain . The numerical results for …
This paper presents a method for calculating the wall shear rate in pipe turbulent flow. It collapses adequately the data measured in laminar flow and turbulent flow into a single flow curve and gives the basis for the design of turbulent flow viscometers. Key words: non-Newtonian, wall shear rate, turbulent, rheometer
In this work, we study the role of viscoelastic instability in the mechanical dispersion of fluid flow through porous media at high Peclet numbers. Using microfluidic experiments and numerical simulations, we show that viscoelastic instability in flow through a hexagonally ordered (staggered) medium strongly enhances dispersion transverse to the mean flow direction with increasing Weissenberg number (Wi). In contrast, preferential flow paths can quench the elastic instability in disordered media, which has two important consequences for transport: First, the lack of chaotic velocity fluctuatio…
In this paper we have studied the flow and heat transfer in a viscous fluid by a horizontal sheet. The stretching rate and temperature of the sheet vary with time. The governing equations for momentum and thermal energy are reduced to ordinary differential equations by means of similarity transformation. These equations are solved approximately by means of the Optimal Homotopy Asymptotic Method (OHAM) which provides us with a convenient way to control the convergence of approximation solutions and adjust convergence rigorous when necessary. Some examples are given and the results obtained reve…
The invariants of the velocity gradient tensor, $R$ and $Q$, and their enstrophy and strain components are studied in the logarithmic layer of an incompressible turbulent channel flow. The velocities are filtered in the three spatial directions and the results analyzed at different scales. We show that the $R$--$Q$ plane does not capture the changes undergone by the flow as the filter width increases, and that the enstrophy/enstrophy-production and strain/strain-production planes represent better choices. We also show that the conditional mean trajectories may differ significantly from the ins…
We present new experimental results on the development of turbulent spots in channel flow. The internal structure of a turbulent spot is measured, with Time Resolved Stereoscopic Particle Image Velocimetry. We report the observation of travelling-wave-like structures at the trailing edge of the turbulent spot. Special attention is paid to the large-scale flow surrounding the spot. We show that this large-scale flow is an asymmetric quadrupole centred on the spot. We measure the time evolution of the turbulent fluctuations and the mean flow distortions and compare these with the predictions of …
One of the most enigmatic science question concerning inertial particle transport by a turbulent boundary layer flow is the value of the turbulent Schmidt number as the ratio of particle diffusivity and turbulent eddy viscosity. Using direct acoustic measurement of turbulent particle flux profile, and two-phase flow turbulence-resolving numerical simulation, it is demonstrated that turbulent dispersion of particles is reduced rather than enhanced when predicted with existing literature model. The explanation lies in the misleading assumption of settling velocity in quiescent water to estimate …
We investigate the spatio-temporal quantity of coherence for turbulent velocity fluctuations at spatial distances of the order or larger than the integral length scale $l_{0}$. Using controlled laboratory experiments, an exponential decay as a function of distance is observed with a decay rate which depends on the flow properties. The same law is observed in two different flows indicating that it can be a generic property of turbulent flows.
We discuss the possibility of dual local and non-local cascades in a 3D turbulent Beltrami flow, with inverse energy cascade and direct helicity cascade, by analogy with 2D turbulence. We discuss the corresponding energy spectrum in both local and non-local case. Comparison with a high Reynolds number turbulent von Karman flow is provided and discussed.
A simple analytical solution for turbulent plane Couette flow is obtained from a subset of the Navier-Stokes equations. This approach analyses the effect of the unsteady state Lagrangian diffusion of viscous momentum on the smoothed phase velocity. Three initial velocity profiles at the start of a turbulent wall cycle are considered. The predictions adequate identify the existence of three layers usually observed in turbulent flow fields, in particular a viscosity dominated wall layer and a log-law layer. Experimental measurements of time-averaged velocity are also well correlated. Key words: …
