On the effect of finite-time correlations on the turbulent mixing in smooth chaotic compressible velocity fields

Abstract

Most theoretical results about turbulent mixing have been obtained for ideal flows that are delta-correlated in time. As is often believed, those ideal flows are, with regard to mixing, very similar to real flows with a finite correlation time. However, recent results show that these two cases may differ considerably. In this article we study the effects of finite correlation time in a chaotic smooth statistically isotropic two-dimensional velocity field. As mixing is predominantly determined by the statistics of the stretching of material elements (e.g. lines "painted" onto a liquid), in this article we focus on the characteristics describing such stretching: finite-time Lyapunov exponents and the Lyapunov dimension. For these quantities, we derive analytical expressions as functions of the correlation time and the compressibility of the velocity field, and we investigate these expressions numerically. The results agree well with numerical results of other authors, and are useful for understanding several physical phenomena, e.g. patchiness of pollution spreading on an ocean and kinematic magnetic dynamos.

Implications of the theory of turbulent mixing for wave propagation in media with fluctuating coefficient of refraction

“…Unfortunately, the exact value for the parameter u, describing the underlying velocity fields, is not available. However, an approximate value can be deduced from general considerations: while for 2-dimensional Kraichnan flows u = 2, in the case of moderate Kubo numbers, values u ∈ [2.2, 2.5] can be expected [19]. While the data are provided here without a proper error analysis, a nearly linear trend implies a qualitative agreement with the theory, and the degree of fluctuations is indicative of the statistical uncertainties.…”

“…Apart from the evolution time t, this distribution has only one parameter, u, which, in the case of incompressible ideal Kraichnan flows (delta-correlated in time), equals to the dimensionality of the flow. Recently it has been shown that in the case of real flows, this parameter can take slightly larger values [19]. In the case of non-smooth velocity fields, Eq.…”

“…1), c.f. [5], for real timecorrelated flows it has somewhat larger values, typically increased by few decimal digits [19]. The phase space dimensionality is six, but the flow is constrained to a fivedimensional subspace defined by the energy conservation law: as long as the refraction index fluctuations can be considered to be constant in time, the wave frequency remains constant during the wave propagation.…”

Turbulent mixing: matching real flows to Kraichnan flows

“…The case of compressible fields has been studied preliminarily in a technical report [30]; here we use our model to explain the challenging findings of Ref. [10] and provide a qualitative explanation of this phenomenon.…”

“…(3) has been tabulated in [29, Eq. 2.6.38.4], and derived in [30]; the one of Eq. ( 4) is derived in the Supplementing Material [27].…”