Numerical Studies of Problems in Turbulence : 1) Fluid Films with Polymer Additives; 2) Fluid Films with Inertial and Elliptical Particles; 3) Scaled Vorticity Moments in Three and Twodimensional Turbulence
Abstract
In this thesis we study a variety of problems in fluid turbulence, principally in two dimensions. A summary of the main results of our studies is given below; we indicate the Chapters in which we present these.
In Chapter 1, we provide an overview of several problems in turbulence with special emphasis on background material for the problems we study in this thesis. In particular, we give (a) natural and laboratory examples of fluid turbulence, (b) and introductory accounts of the equations of hydrodynamics, without and with polymer additives, Eulerian and Lagrangian frameworks, and the equations of motion of inertial particles in fluid flows. We end with a summary of the problems we study in subsequent Chapters of this thesis.
In Chapter 2, we carry out the most extensive and highresolution direct numerical simulation, attempted so far, of homogeneous, isotropic turbulence in twodimensional fluid films with airdraginduced friction and with polymer additives. Our study reveals that the polymers (a) reduce the total fluid energy, enstrophy, and palinstrophy, (b) modify the fluid energy spectrum both in inverse and forwardcascade regimes, (c) reduce smallscale intermittency, (d) suppress regions of large vorticity and strain rate, and (e) stretch in straindominated regions. We compare our results with earlier experimental studies and propose new experiments.
In Chapter 3, we perform a direct numerical simulation (DNS) of the forced, incompressible twodimensional NavierStokes equation coupled with the FENEP equations for the polymer conformation tensor. The forcing is such that, without polymers and at low Reynolds numbers Re, the lm attains a steady state that is a square lattice of vortices and antivortices. We nd that, as we increase the Weissenberg number (Wi), this lattice undergoes a series of nonequilibrium phase transitions, first to spatially distorted, but temporally steady, crystals and then to a sequence of crystals that oscillate in time, periodically, at low Wi, and quasiperiodically, for slightly larger Wi. Finally, the system becomes disordered and displays spatiotepmoral chaos and elastic turbulence. We then obtain the nonequilibrium phase diagram for this system, in the Wi − Re plane, and show that (a) the boundary between the crystalline and turbulent phases has a complicated, fractaltype character and (b) the OkuboWeiss parameter provides us with a natural measure for characterizing the phases and transitions in this diagram.
In Chapter 4, our study is devoted to heavy, inertial particles in twodimensional (2D) tur bulent, but statistically steady, flows that are homogeneous and isotropic. The inertial particles are distributed uniformly in our simulation domain when St = 0; they start to cluster as St increases; this clustering tendency reaches a maximum at St 1 and decreases thereafter. We then obtain PDFs of and show that their left tails, which come from extensional regions, do not depend sensitively on St; in contrast, their right tails, from the vortical regions of the flow, are consistent with the exponential form ∼ exp ‰− + Ž; and we nd that the scale + decreases with St until St _0:1 and then saturates at a value _0:75. Our persistencetype studies yield the following results, when we consider forcing that leads to an energy spectrum that is dominated by a forwardcascade regime: In straindominated or extensional regions of the flow, wend that the cumulative PDF of the persistence time decays exponentially; this decay yields a time scale T−, which increases rapidly with St, at low values of St, but more slowly after St _0:75. By contrast, in vortical regions of the flow, this cumulative PDF displays a tail that has powerlaw and exponential parts; the powerlaw part yields the persistence exponent _ and the exponential tail gives a time scale T−; _ increases with St, whereas T− decreases with St; _ and T− reach saturation values as St increases. From the cumulative PDF of the particle meansquare displacement r2, we obtain the time scale Ttrans at which there is a crossover from ballistic to diffusive behavior; we _nd that Ttrans increases with St. The PDFs of v2, the square of the particle velocity, and v2 ejected, the square of the velocity of a particle just as it is ejected from a region with _ > 0 (vortical region) to one that has _ < 0 (extensional region), do not show a significant dependence on St; the tails of these PDFs are characterized by powerlaw decays with exponents _1 and _5~3, respectively. Our next set of results deal with statistical properties of special combinations of the acceleration a =dv~dt and the velocity v. For instance, the curvature of the trajectory is _ =aÙ~v2, where the subscript Ù denotes the component perpendicular to the particle trajectory; we obtain PDFs of _ and _nd there from that particles in regions of elongational flow have, on average, trajectories with a lower curvature than particles in vortical regions; this . We also determine how the number of number of points NI , at which a ×v changes sign along a particle trajectory, as time increases; we _nd that the increase of NI with time and decrease as St increases. Our ninth set of results show that the characteristic decay time T_ for decreases with St.
In Chapter 5, we study the statistical properties of orientation and rotation dynamics of elliptical tracer particles in twodimensional, homogeneous and isotropic turbulence by direct numerical simulations. We consider both the cases in which the turbulent flow is generated by forcing at large and intermediate length scales. We show that the two cases are qualitatively different. For the largescale forcing, the spatial distribution of particle orientations forms large scale structures, which are absent for the intermediatescale forcing. The alignment with the local directions of the flow is much weaker in the latter case than in the former. For the intermediate scale forcing, the statistics of rotation rates depends weakly on the Reynolds number and on the aspect ratio of particles. In contrast with what is observed in threedimensional turbulence, in two dimensions the meansquare rotation rate decreases as the aspect ratio increases.
In Chapter 6, we study the issue of intermittency in numerical solutions of the 3D NavierStokes equations on a periodic box [0; L]3. This is addressed through four sets of numerical simulations that calculate a new set of variables defined by Dm(t) = where All four simulations unexpectedly show that the Dm are ordered for m =1 ….,9 such that Dm+1 <Dm. Moreover, the Dm squeeze together such that Dm+1/Dm 1 as m increases. The values of D1 lie far above the values of the rest of the Dm, giving rise to a suggestion that a depletion of nonlinearity is occuring which could be the cause of Navier{Stokes regularity. The first simulation, by R. Kerr, is of very anisotropic decaying turbulence ; the second and third, which have been carried out by me, are of decaying isotropic turbulence from random initial conditions and forced isotropic turbulence at fixed Grashof number, respectively ; the fourth, by D. Donzis, is of veryhighReynoldsnumber forced, stationary, isotropic turbulence at resolutions up to 40963 collocation points. For the sake of completeness and for a comparison of the data from all these four simulations, all the results are presented; however, in the Sections that deal with the simulations, I indicate who carried out the calculations reported there. I also present an extension of this work to twodimensional fluid turbulence; this has not been submitted for publication so far.
We hope our in silico studies of 2D and 3D turbulence will stimulate new experimental, numerical, and theoretical studies.
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 Physics (PHY) [413]
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