On the alignment dynamics of a passive scalar gradient in a two-dimensional flow

Abstract

A Lagrangian study of the statistical properties of the orientation of a passive scalar gradient is performed using experimental data and a simple, numerical analysis. It is shown that, in a low-Reynolds number Bénard-von Kármán street, the temperature gradient downstream of a heated line source does not align with the asymptotic orientation predicted by the Lapeyre et al. model ͓Phys. Fluids 11, 3729 ͑1999͔͒ in the hyperbolic regions. This result is ascribed to fluctuations of strain persistence along Lagrangian trajectories. A numerical analysis of the scalar gradient alignment properties shows that these fluctuations, together with a low level of the rate of strain, may lead to preferential orientations that are different from the theoretical one predicted by an asymptotic model.

Nonstationary aspects of passive scalar gradient behaviour

“…The study of Garcia et al [11] reveals that the above approach remains valid as long as the response time scale of the scalar gradient is short enough compared to the time scale of the Lagrangian fluctuations of r. They also put forward that in the opposite case, namely when the gradient does not keep up with r fluctuations and its response is poor, the alignment of the scalar gradient is determined by the mean value of r, r . In the following we try to generalize and support these results by the study of regimes that have not been originally addressed.…”

Analysis of passive scalar gradient alignment in a simplified three-dimensional case

“…8 It is precisely in the two-dimensional case that statistical alignment with the equilibrium orientation depending on local strain persistence has been clearly revealed. 8,9 Even so, Garcia et al 10 have recently shown that the Lagrangian variations of strain persistence r have to occur on a time scale larger than the response time scale of the scalar gradient ͑which is of the order of −1 where is the strain intensity͒ for the latter to be constantly close to the equilibrium orientation defined by the instantaneous r. In the opposite case, when strain persistence is fluctuating rapidly with respect to the gradient response, the preferential alignment is determined by ͗r͘ as ͗r͘ = −arccos͗r͘. Then, if ͗r͘Ӎ0, the preferential orientation turns out to be the compressional direction; this is not because it corresponds to any equilibrium, but just for the reason that the scalar gradient does not keep up with r fluctuations and only feels ͗r͘.…”

“…This direction is known as a simple function of the strain persistence parameter. 8 Nevertheless, the study of Garcia et al 10 suggests that statistical alignment with either the compressional or the equilibrium direction is actually governed by the response of the scalar gradient to Lagrangian fluctuations of strain persistence.…”

Influence of molecular diffusion on alignment of vector fields: Eulerian analysis

“…This formalism uses invariants -the vector orientation in the strain eigenframe and the vector norm -, and is connected to the structure of the vector field. This approach was used to study the fine structure of the scalar gradient field and small-scale mixing mechanisms [16,17]. Vector B is defined by its norm, B, and its orientation, θ, in the fixed frame of reference as: B = B(cos θ, sin θ).…”