BEHAVIOR OF AN INERT SCALAR UNDER THE DIFFERENT SMALL SCALE TOPOLOGIES OF A TURBULENT FLOW

Abstract

The physical phenomena that determine the evolution and characteristics of a scalar in a turbulent flow are studied usin g direct numerical simulation (DNS) of both the scalar and th e velocity fields . The statistics of essential magnitudes relate d to the mixing, such as the scalar diffusion, the scalar dissipation or the variance is analysed for the different small scale dynamical structures in the flow. This study is conducted in the framework of the Topological Methodology introduced b y Chong et al (1990), which classifies the different small scale dynamics in terms of velocity gradient invariants . From thi s investigation, the scalar properties display different feature s for each of the four topologies considered. INTRODUCTION Scalar mixing in a turbulent flow is a complex process characterized by a wide range of time and length scales, over which a variety of physical mechanisms take place (Ottino , 1989; Dopazo, 1994) . The scalar evolution, driven by the turbulent velocity field, involves convection, random strainin g and rotation, and molecular transport . Scalar heterogeneitie s are smeared out by molecular diffusion enhanced by stretching and folding of isoscalar surfaces . The rational formulation of stochastic molecular mixin g modeis crucially depends on the ability to parameterize the scalar mixing mechanisms in terms of the scalar fluctuations , the scalar gradient vector, and knowable information pertaining both to the scalar and to the turbulence fields . It is , thus, logical searching for some correlation of the molecula r transport terms in the conservation equations of the scalar related magnitudes and the properties of the velocity field . Th e statistics of a passive scalar in turbulence has been widel y investigated in the past years using DNS (See, for example , Kerr (1985), Ashurst et al (1987) ; Ruetsch and Maxey (1991 ) or Pumir (1994), among others) . The aim of the present work is to investigate this statistics considering the different pattems or motions that appear in the small scales of a turbulent flow using the topological classification introduced by Chong et al (1990). Specifically, regions in the flow with high kineti c energy dissipation rates (dissipative motions) and regions o f high vorticity (focal motions) are of special interest here . THEORETICAL BACKGROUN D For an inert scalar C the mean value, , is a constant and the scalar fluctuations, e, obey the equation (Dopazo , 1994) where c is statistically homogeneous, D is the Fickian diffusion coefficient of c in the mixture and u is a zero-mea n statistically homogeneous solenoidal random velocity field . The transport equation for the scalar gradient fluctuation c, ; is 2 ac,i +uac, : = —c, j + DV c, i . at 2 ax; The dynamics of the scalar gradient is essential in the scala r field evolution. This vector is perpendicular to the loca l isoscalar surface, at each point in the flow ; the geometry and characteristics of these isosurfaces are determined by the spatial distributions of the scalar gradient vector (Ottino, 1989) . Equation (1) can be altematively rephrased as at +u.ax = DV 2 c2 2e, . where ec = Dc, ;c,i is the locallinstantaneous scalar fluctuation dissipation rate . Apart from the pressure term, the transport equation for c2 is analogous to that for the instantaneou s turbulent kinetic energy . ac + uJ ac = Dp2c. at ax; (1 )