* Corresponding author. Tel.: 00-49-721-608-2120; fax: 00-49-721- 608-4820. E-mail address: bockhorn@ict.uni-karlsruhe.de (H. Bockhorn). Chemical Engineering Science 55 (2000) 4255 } 4269 Numerical simulation of the mixing of passive and reactive scalars in two-dimensional #ows dominated by coherent vortices Wolfgang Gerlinger!, Kai Schneider!, Laurent Falk!,", Henning Bockhorn!,* !Institut fu( r Chemische Technik, Universita( t Karlsruhe (TH), Kaiserstra}e 12, 76128 Karlsruhe, Germany "Laboratoire des Sciences du Ge& nie Chimiques, Ecole Nationale Supe& rieure des Industries Chimiques, 1 rue Grandville, BP 451, 54001 Nancy CEDEX, France Received 3 March 1999; accepted 21 February 2000 Abstract The present paper comprises direct numerical simulation of the mixing of passive and reactive scalars in two-dimensional #ows dominated by coherent vortices. By means of highly accurate pseudo-spectral methods the instationary Navier}Stokes and convection}di!usion}reaction equations are numerically integrated on the torus. We present the evolution of typical vortex arrangements and analyze their ability to mix scalars. Furthermore, we attribute the in#uence of vortices on the mixing to the formation of spirals and to the merging of vortices. As a way of quantifying the mixing, we consider global and local mixing time scales which describe the overall and the early variance decay, respectively. Moreover, the in#uence of the Schmidt number and of a chemical reaction on mixing processes are investigated. ( 2000 Elsevier Science Ltd. All rights reserved. Keywords: Fluid dynamics; Simulation; Two-dimensional turbulence; Mixing; Di!usion; Nonlinear dynamics 1. Introduction Turbulent mixing is ubiquitous. It can be observed in many natural phenomena, and has a wide range of ap- plications. It is of primordial importance in many tech- nical devices, in particular, in chemical engineering science. For safety reasons, especially in exothermic sys- tems, the chemical species are initially segregated. Before any chemical reaction takes place, the species have to be mixed at the molecular level. For optimization and con- trol of chemical processes, the role of convection, di!u- sion, mixing, chemical reactions, and their coupling is of pivotal interest. In many practically relevant devices, turbulent #ow conditions are present which guarantee accelerated mix- ing of all transported quantities like momentum, mass and energy, and lead to an enhanced di!usion of all quantities (Ottino, 1990). Turbulent #ows are irregular, unsteady and rotational and show chaotic behaviour, further complicated when chemistry is involved and in- teracting. Although the governing equations are well known, i.e. the Navier}Stokes equations are determinis- tic and for a given initial condition the #ow can, in principle, be computed due to causality, there are still many open questions. One fundamental problem goes with the fact that there is a wide range of space and time scales involving an enormous number of degrees of freedom. Hence, for practical applications models are unalterable. So far many di!erent ad hoc models specifying the mixing from phenomenological point of view have been developed. The classical, statistical approach describes the evolution of the mean #ow and the second-order moments. These methods are not appropriate to discuss the in#uence of coherent structures as these are averaged out and, hence, they do not account for intermittency of the #ow. Permanent advances in scienti"c computing allow to handle large-scale problems with increasing complexity so that the coupling of #uid dynamics with mixing and chemical reaction can be examined. In direct numerical simulation (DNS), the governing equations are solved directly without any subgrid scale model. Per de"nition, DNS is model free (Givi, 1989; Poinsot, Candel & TrouveH , 1996). Apart from laboratory experiments the 0009-2509/00/$ - see front matter ( 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 0 0 ) 0 0 0 6 4 - 6