On the role of unsteady forcing of tracer gradient in local stirring

Abstract

Gonzalez, M. Paranthoen, P.Local stirring properties in two basic mixing flows - namely, the blinking vortex and the sine flow are studied through the tracer gradient approach. The velocity gradient tensor and related quantities such as the strain persistence parameter are derived from the analytical velocity fields. Numerical Lagrangian tracking of the gradient of a tracer shows how local stirring is affected by forcing experienced through strain persistence. In both flows Lagrangian variations of strain persistence occuring on a time scale shorter than the response time scale of the tracer gradient lead the latter to align close to the direction determined by the mean strain persistence. It is the special alternating behaviour of strain persistence resulting from flow operation that makes this direction coincide with the local compressional strain direction for both the sine flow and the clockwise/counterclockwise blinking vortex. The rise of the tracer gradient and thus local stirring are in turn promoted by this statistical alignment. (C) 2009 Elsevier Masson SAS. All rights reserved

Modelling the growth rate of a tracer gradient using stochastic differential equations

“…This suggests the favoured orientation is close to the compressive direction. This situation corresponds to that described by Gonzalez and Paranthoën [10], where in rapidly-varying flows, the tracer gradient aligns with a direction fixed by r , which coincides with the compressive direction only when r = 0.…”

“…This flow is amenable to a comparison with our stochastic model because the randomisation of the phases breaks the invariant tori, induces homogeneity, and promotes mixing [21]. This is in contrast to the unrandomised sine flow studied elsewhere [10].…”

Effect of variable mass density on the kinematics of scalar gradient

“…Restricting the analysis to non-diffusive scalars as already done in previous studies 24,26 leads to a simpler equation for the components of the scalar gradient,…”

Analysis of a cross-channel micromixer through the dynamics of tracer gradient

“…As a result, the mean growth rate decays at large Strouhal numbers. On a more general level, this mechanism is clearly reminiscent of the nonadiabatic regime of the scalar gradient response to unsteady forcing in which the scalar gradient does not keep up with fast changes in strain persistence -or in strain principal axes direction (Garcia et al, 2008;Gonzalez and Paranthoën, 2010). The probability density function (p.d.f) of the tracer gradient orientation (Fig.…”

“…The role of alignment may be especially critical in non-stationary regimes in which the dynamics of the tracer gradient -through its response to unsteady mechanical action -drives the growth rate. Previous studies addressed the dynamics of the scalar gradient orientation (Lapeyre et al, 2001; Garcia et al, 2005) and its role in the mechanism of the gradient growth was pointed out (Garcia et al, 2008;Gonzalez and Paranthoën, 2010).…”