F. C. G. A. Nicolleau

Turbulent diffusion in rapidly rotating flows with and without stable stratification

Three different approaches are used for evaluating some Lagrangian properties of homogeneous turbulence containing anisotropy due to the application of a stable stratification and a solid-body rotation. The two external frequencies are the magnitude of the system vorticity 2Q, chosen vertical here, and the Brunt-Vaisala frequency N, which gives the strength of the vertical stratification. Analytical results are derived using linear theory for the Eulerian velocity correlations (single-point, two-time) in the vertical and the horizontal directions, and Lagrangian ones are assumed to be equivalent, in agreement with an additional Corrsin assumption used by Kaneda. They are compared with results from the kinematic simulation model (KS) by Nicolleau & Vassilicos, which also incorporates the wave-vortex dynamics inherited from linear theory, and directly yields Lagrangian correlations as well as Eulerian ones. Finally, results from direct numerical simulations (DNS) are obtained and compared for the rotation-dominant case B = 2Ω/N = 10, the stratification-dominant case B = 1/10, the non-dispersive case B = 1, and pure stratification B = 0 and pure rotation N = 0

Turbulent diffusion in stably stratified non-decaying turbulence

We develop a Lagrangian model of both one-particle In this paper ‘particle’ and ‘fluid element’ are synonymous. and two-particle turbulent diffusion in high Reynolds number and low Froude number stably stratified non- decaying turbulence. This model is a kinematic simulation (KS) that obeys both the linearized Boussinesq equations and incompressibility. Hence, turbulent diffusion is anisotropic and is studied in all three directions concurrently with incompressibility satisfied at the level of each and every trajectory. Horizontal one-particle and two-particle diffusions are found to be independent of the buoyancy (Brunt–Vaissala) frequency N . For one-particle diffusion we find that formula here and formula here where i = 1,2 and u ′ and L are a r.m.s. velocity and a length-scale of the energy-containing motions respectively, and formula here This capping of one-particle vertical diffusion requires the consideration of the entire three-dimensional flow, and we show that each and every trajectory is vertically bounded for all times if the Lagrangian vertical pressure acceleration a 3 is bounded for all times. Such an upper bound for a 3 can be derived from the linearized Boussinesq equations as a consequence of the coupling between vertical pressure acceleration and the horizontal and vertical velocities. Two-particle vertical diffusion exhibits two plateaux. The first plateaus scaling is different according to whether the initial separation Δ 0 between the two particles is larger or smaller than η, the smallest length-scale of the turbulence: formula here The second plateau is reached when the two particles become statistically independent, and therefore formula here The transition between the two plateaux coincides with the time when the two particles start moving significantly apart in the horizontal plane.

Experimental study of a turbulent pipe flow through a fractal plate

We study flows forced through plates in a circular wind-tunnel. The plates are based on different iterations of a pattern which would correspond to the generation of a fractal object. Two types of patterns are considered: one leading to an orifice with a fractal perimeter, the other to a fractal equivalent of a perforated plate. We propose this approach as a systematic way to introduce scales forcing and construct a scale by scale multiscale flow. For the sake of comparison we also consider simple orifice plates having a different number of sharp edges but which are based on a single scale geometry (monoscale orifices). In terms of analysis we focus on the pressure drop across the plate as in terms of engineering application this is the most salient effect of the plate multiscale generation. We measure the pressure recovery as a function of the distance from the plate. We also use hotwire anemometry to understand the action of the fractal-based plate on the flow and measure velocity statistics on the pipes axis. We found that as far as pressure drop recovery is concerned, it is not always necessary to choose large numbers of iteration for the fractal pattern. However, the velicity statistics can remember the order of the fractal generation over a long distance. In certain cases it would require a very long pipe to quantify the full recovery from the fractal-generated turbulent flow.

Two-particle diffusion and locality assumption

A three-dimensional kinematic simulation (KS) model is used to study one- and two-particle diffusion in turbulent flows. The energy spectrum E(k) takes a power law form E(k)∼k−p. The value of this power p is varied from 1.2 to 3, so that its effects on the diffusion of one and two particles can be studied. The two-particle diffusion behaves differently depending on whether the two-particle separation is larger or smaller than the smallest scale of turbulence (Kolmogorov length scale η). When the two-particle mean square separation 〈Δ2(t)〉 is smaller than η2 it experiences a time exponential growth 〈Δ2(t)〉=Δ02eγ(t/tη) but for a very short time. For longer times, when η2<〈Δ2(t)〉<L2, the locality assumption is revisited in terms of two-particle mean diffusivity d/dt〈Δ2(t)〉. In this inertial range we observe that d/dt〈Δ2(t)〉={a ln(〈Δ(t)2〉1/2/η)+b}u′L(η/L)(p+1)/2(〈Δ(t)2〉1/2/η)(p+1)/2 for p⩽3. For Δ0/η≫1 a=0, but a≠0 for Δ0/η⩽1 and as a consequence the pair diffusion cannot have lost its dependence on the initi...

The fractional Laplacian as a limiting case of a self-similar spring model and applications to n-dimensional anomalous diffusion

AbstractWe analyze the “fractional continuum limit” and its generalization to n dimensions of a self-similar discrete spring model which we introduced recently [21]. Application of Hamilton’s (variational) principle determines in rigorous manner a self-similar and as consequence non-local Laplacian operator. In the fractional continuum limit the discrete self-similar Laplacian takes the form of the fractional Laplacian

Turbulent Pair Diffusion

- ( - \Delta )^{\tfrac{\alpha } {2}}

Wavelets for the study of intermittency and its topology

with 0 < α < 2. We analyze the fundamental link of fractal vibrational features of the discrete self-similar spring model and the smooth regular ones of the corresponding fractional continuum limit model in n dimensions: We find a characteristic scaling law for the density of normal modes ∼

Wave Propagation in Quasi-continuous Linear Chains with Self-similar Harmonic Interactions: Towards a Fractal Mechanics

\omega ^{\tfrac{{2n}} {\alpha } - 1}

Numerical determination of turbulent fractal dimensions

with a positive exponent

Eddy break-up model and fractal theory: comparisons with experiments

\tfrac{{2n}} {\alpha } - 1 > n - 1