Shell Model for Drag Reduction with Polymer Additive in Homogeneous Turbulence

Recent direct numerical simulations of the FENE-P model of non-Newtonian hydrodynamics revealed that the phenomenon of drag reduction by polymer additives exists (albeit in reduced form) also in homogeneous turbulence. We introduce here a simple shell model for homogeneous viscoelastic flows that recaptures the essential observations of the full simulations. The simplicity of the shell model allows us to offer a transparent explanation of the main observations. It is shown that the mechanism for drag reduction operates mainly on the large scales. Understanding the mechanism allows us to predict how the amount of drag reduction depends of the various parameters in the model. The main conclusion is that drag reduction is not a universal phenomenon, it peaks in a window of parameters like Reynolds number and the relaxation rate of the polymer

Universal long-time properties of Lagrangian statistics in the Batchelor regime and their application to the passive scalar problem

We consider transport of dynamically passive quantities in the Batchelor regime of smooth in space velocity field. For the case of arbitrary temporal correlations of the velocity we formulate the statistics of relevant characteristics of Lagrangian motion. This allows to generalize many results obtained previously for the delta-correlated in time strain, thus answering the question of universality of these results.Comment: 11 pages, revtex; added references, typos correcte

Scaling Exponents in Anisotropic Hydrodynamic Turbulence

In anisotropic turbulence the correlation functions are decomposed in the irreducible representations of the SO(3) symmetry group (with different "angular momenta" $\ell$). For different values of $\ell$ the second order correlation function is characterized by different scaling exponents $\zeta_2(\ell)$. In this paper we compute these scaling exponents in a Direct Interaction Approximation (DIA). By linearizing the DIA equations in small anisotropy we set up a linear operator and find its zero-modes in the inertial interval of scales. Thus the scaling exponents in each $\ell$-sector follow from solvability condition, and are not determined by dimensional analysis. The main result of our calculation is that the scaling exponents $\zeta_2(\ell)$ form a strictly increasing spectrum at least until $\ell=6$, guaranteeing that the effects of anisotropy decay as power laws when the scale of observation diminishes. The results of our calculations are compared to available experiments and simulations.Comment: 10 pages, 4 figures, PRE submitted. Fixed problems with figure

Anomalous and dimensional scaling in anisotropic turbulence

We present a numerical study of anisotropic statistical fluctuations in homogeneous turbulent flows. We give an argument to predict the dimensional scaling exponents, (p+j)/3, for the projections of p-th order structure function in the j-th sector of the rotational group. We show that measured exponents are anomalous, showing a clear deviation from the dimensional prediction. Dimensional scaling is subleading and it is recovered only after a random reshuffling of all velocity phases, in the stationary ensemble. This supports the idea that anomalous scaling is the result of a genuine inertial evolution, independent of large-scale behavior.Comment: 4 pages, 3 figure

Stabilization of Hydrodynamic Flows by Small Viscosity Variations

Motivated by the large effect of turbulent drag reduction by minute concentrations of polymers we study the effects of a weakly space-dependent viscosity on the stability of hydrodynamic flows. In a recent Letter [Phys. Rev. Lett. {\bf 87}, 174501, (2001)] we exposed the crucial role played by a localized region where the energy of fluctuations is produced by interactions with the mean flow (the "critical layer"). We showed that a layer of weakly space-dependent viscosity placed near the critical layer can have a very large stabilizing effect on hydrodynamic fluctuations, retarding significantly the onset of turbulence. In this paper we extend these observation in two directions: first we show that the strong stabilization of the primary instability is also obtained when the viscosity profile is realistic (inferred from simulations of turbulent flows with a small concentration of polymers). Second, we analyze the secondary instability (around the time-dependent primary instability) and find similar strong stabilization. Since the secondary instability develops around a time-dependent solution and is three-dimensional, this brings us closer to the turbulent case. We reiterate that the large effect is {\em not} due to a modified dissipation (as is assumed in some theories of drag reduction), but due to reduced energy intake from the mean flow to the fluctuations. We propose that similar physics act in turbulent drag reduction.Comment: 10 pages, 17 figs., REVTeX4, PRE, submitte

Drag Reduction by Polymers in Turbulent Channel Flows: Energy Redistribution Between Invariant Empirical Modes

We address the phenomenon of drag reduction by dilute polymeric additive to turbulent flows, using Direct Numerical Simulations (DNS) of the FENE-P model of viscoelastic flows. It had been amply demonstrated that these model equations reproduce the phenomenon, but the results of DNS were not analyzed so far with the goal of interpreting the phenomenon. In order to construct a useful framework for the understanding of drag reduction we initiate in this paper an investigation of the most important modes that are sustained in the viscoelastic and Newtonian turbulent flows respectively. The modes are obtained empirically using the Karhunen-Loeve decomposition, allowing us to compare the most energetic modes in the viscoelastic and Newtonian flows. The main finding of the present study is that the spatial profile of the most energetic modes is hardly changed between the two flows. What changes is the energy associated with these modes, and their relative ordering in the decreasing order from the most energetic to the least. Modes that are highly excited in one flow can be strongly suppressed in the other, and vice versa. This dramatic energy redistribution is an important clue to the mechanism of drag reduction as is proposed in this paper. In particular there is an enhancement of the energy containing modes in the viscoelastic flow compared to the Newtonian one; drag reduction is seen in the energy containing modes rather than the dissipative modes as proposed in some previous theories.Comment: 11 pages, 13 figures, included, PRE, submitted, REVTeX

A simple model for drag reduction

Direct Numerical Simulations established that the FENE-P model of viscoelastic flows exhibits the phenomenon of turbulent drag reduction which is caused in experiments by dilute polymeric additives. To gain analytic understanding of the phenomenon we introduce in this Letter a simple 1-dimensional model of the FENE-P equations. We demonstrate drag reduction in the simple model, and explain analytically the main observations which include (i) reduction of velocity gradients for fixed throughput and (ii) increase of throughput for fixed dissipation.Comment: submitted to PR

Generation of Large-Scale Vorticity in a Homogeneous Turbulence with a Mean Velocity Shear

An effect of a mean velocity shear on a turbulence and on the effective force which is determined by the gradient of Reynolds stresses is studied. Generation of a mean vorticity in a homogeneous incompressible turbulent flow with an imposed mean velocity shear due to an excitation of a large-scale instability is found. The instability is caused by a combined effect of the large-scale shear motions (''skew-induced" deflection of equilibrium mean vorticity) and ''Reynolds stress-induced" generation of perturbations of mean vorticity. Spatial characteristics, such as the minimum size of the growing perturbations and the size of perturbations with the maximum growth rate, are determined. This instability and the dynamics of the mean vorticity are associated with the Prandtl's turbulent secondary flows. This instability is similar to the mean-field magnetic dynamo instability. Astrophysical applications of the obtained results are discussed.Comment: 8 pages, 3 figures, REVTEX4, submitted to Phys. Rev.

Comparison of some Reduced Representation Approximations

In the field of numerical approximation, specialists considering highly complex problems have recently proposed various ways to simplify their underlying problems. In this field, depending on the problem they were tackling and the community that are at work, different approaches have been developed with some success and have even gained some maturity, the applications can now be applied to information analysis or for numerical simulation of PDE's. At this point, a crossed analysis and effort for understanding the similarities and the differences between these approaches that found their starting points in different backgrounds is of interest. It is the purpose of this paper to contribute to this effort by comparing some constructive reduced representations of complex functions. We present here in full details the Adaptive Cross Approximation (ACA) and the Empirical Interpolation Method (EIM) together with other approaches that enter in the same category

Distortion in a 7xxx aluminum alloy during liquid phase sintering

The distortion in a sintered 7xxx aluminum alloy, Al-7Zn-2.5Mg-1Cu (wt. pct), has been investigated by sintering three rectangular bars in each batch at 893 K (620 °C) for 0 to 40 minutes in nitrogen, followed by air or furnace cooling. They were placed parallel to each other, equally spaced apart at 2 mm, with their long axes being perpendicular to the incoming nitrogen flow. Pore evolution in each sample during isothermal sintering was examined metallographically. The compositional changes across sample mid-cross section and surface layers were analyzed using energy dispersive X-ray spectroscopy and X-ray photoelectron spectroscopy depth profiling, respectively. The two outer samples bent toward the middle one, while the middle sample was essentially distortion free after sintering. The distortion in the outer samples was a result of differential shrinkage between their outer and inner surfaces during isothermal sintering. The porous outer surface showed an enrichment of oxygen around the large pores as well as lower magnesium and zinc contents than the interior and inner surface of the same sample, while the inner surface was distinguished by the presence of AlN. The differential shrinkage was caused by different oxygen contents in local sintering atmosphere and unbalanced loss of magnesium and zinc between the outer and inner surfaces