Liquid transport in scale space

International audienceWhen a liquid stream is injected into a gaseous atmosphere, it destabilizes and continuously passes through different states characterized by different morphologies. Throughout this process, the flow dynamics may be different depending on the region of the flow and the scales of the involved liquid structures. Exploring this multi-scale, multi-dimensional phenomenon requires some new theoretical tools, some of which need yet to be elaborated. Here, a new analytical framework is proposed on the basis of two-point statistical equations of the liquid volume fraction. This tool, which originates from single phase turbulence, allows us notably to decompose the fluxes of liquid in flow–position space and scale space. Direct numerical simulations of liquid–gas turbulence decaying in a triply periodic domain are then used to characterize the time and scale evolution of the liquid volume fraction. It is emphasized that two-point statistics of the liquid volume fraction depend explicitly on the geometrical properties of the liquid–gas interface and in particular its surface density. The stretch rate of the liquid–gas interface is further shown to be the equivalent for the liquid volume fraction (a non-diffusive scalar) of the scalar dissipation rate. Finally, a decomposition of the transport of liquid in scale space highlights that non-local interactions between non-adjacent scales play a significant role

Liquid transport in scale space

When a liquid stream is injected into a gaseous atmosphere, it destabilizes and continuously passes through different states characterized by different morphologies. Throughout this process, the flow dynamics may be different depending on the region of the flow and the scales of the involved liquid structures. Exploring this multi-scale, multi-dimensional phenomenon requires some new theoretical tools, some of which need yet to be elaborated. Here, a new analytical framework is proposed on the basis of two-point statistical equations of the liquid volume fraction. This tool, which originates from single phase turbulence, allows notably to decompose the fluxes of liquid in flow-position space and scale space. Direct Numerical Simulations of liquid-gas turbulence decaying in a triply periodic domain are then used to characterize the time and scale evolution of the liquid volume fraction. It is emphasized that two-point statistics of the liquid volume fraction depend explicitly on the geometrical properties of the liquid-gas interface and in particular its surface density. The stretch rate of the liquid-gas interface is further shown to be the equivalent for the liquid volume fraction (a non diffusive scalar) of the scalar dissipation rate. Finally, a decomposition of the transport of liquid in scale space highlights that non-local interactions between non adjacent scales play a significant role

The illusion of a Kolmogorov cascade

International audienceThe theory of Kolmogorov, enunciated for very large Reynolds numbers, has progressively been shown to be inoperative for characterizing flows of practical relevance. Yet, in a recent study by Alves Portela et al. (2020), the turbulence statistics in the very near wake of a square prism at modest Reynolds numbers, reveal a significant portion of scales complying with a cascade of Kolmogorov-type. By resorting to a generalized version of the Kármán-Howarth-Kolmogorov equation, this intriguing observation is shown to be an illusion, hiding a measurable influence of coherent structures and statistical inhomogeneity. This striking conclusion highlights that a complete statistical theory of turbulence cannot dispense with the influence of large scales, possibly coherent, motions

Morphology of contorted fluid structures

International audienceMultiphase flows reveal contorted fluid structures which cannot be described in terms of drop/bubble diameter distribution. Here we use a morphological descriptor which originates from the field of heterogeneous materials that was proved to be nicely tailored for characterizing the microstructure of e.g. porous media. It is based on the Minkowski Functionals-an erudite expression which simply designates the integrated volume, surface, mean and Gaussian curvatures-of all surfaces parallel to the liquid-gas interface. We here apply this framework to different multiphase flow systems and prove that the Minkowski Functionals are effective for providing insights into their morphodynamical behavior

Space-scale-time dynamics of liquid–gas shear flow

International audienceTwo-point statistical equations of the liquid phase indicator function are used to appraise the physics of liquid-gas shear flows. The contribution of the different processes in the combined scale/physical space is quantified by means of direct numerical simulations of a temporally liquid-gas shear layer. Light is first shed onto the relationship between twopoint statistics of the phase indicator and the geometrical properties of the liquid/gas interface, namely its surface density, mean and Gaussian curvatures. Then, the theory is shown to be adequate for highlighting the preferential direction of liquid transport in either scale or flow positions space. A direct cascade process, i.e. from large to small scales, is observed for the total phase indicator field, while the opposite applies for the randomly fluctuating part suggesting a transfer of 'energy' from the mean to the fluctuating component. In the space of positions within the flow, the flux tend to redistribute energy from the centreline to the edge of the shear layer. The influence of the mean shear rate and statistical inhomogeneities on the different scales of the liquid field are revealed

Dynamical interactions between the coherent motion and small scales in a cylinder wake

Most turbulent flows are characterized by coherent motion (CM), whose dynamics reflect the initial and boundary conditions of the flow and are more predictable than that of the random motion (RM). The major question we address here is the dynamical interaction between the CM and the RM, at a given scale, in a flow where the CM exhibits a strong periodicity and can therefore be readily distinguished from the RM. The question is relevant at any Reynolds number, but is of capital importance at finite Reynolds numbers, for which a clear separation between the largest and the smallest scales may not exist. Both analytical and experimental tools are used to address this issue. First, phase-averaged structure functions are defined and further used to condition the RM kinetic energy at a scale <i>r</i> on the phase ϕ of the CM. This tool allows the dependence of the RM to be followed as a function of the CM dynamics. Scale-by-scale energy budget equations are established on the basis of phase-averaged structure functions. They reveal that energy transfer at a scale <i>r</i> is sensitive to an additional forcing mechanism due to the CM. Second, these concepts are tested using hot-wire measurements in a cylinder wake, in which the CM is characterized by a well-defined periodicity. Because the interaction between large and small scales is most likely enhanced at moderate/low Reynolds numbers, and is also likely to depend on the amplitude of the CM, we choose to test our findings against experimental data at Rλ∼102 and for downstream distances in the range 10≤x/D≤40. The effects of an increasing Reynolds number are also discussed. It is shown that: (i) a simple analytical expression describes the second-order structure functions of the purely CM. The energy of the CM is not associated with any single scale; instead, its energy is distributed over a range of scales. (ii) Close to the obstacle, the influence of the CM is perceptible even at the smallest scales, the energy of which is enhanced when the coherent strain is maximum. Further downstream from the cylinder, the CM clearly affects the largest scales, but the smallest scales are not likely to depend explicitly on the CM. (iii) The isotropic formulation of the RM energy budget compares favourably with experimental results

Consequences of self-preservation on the axis of a turbulent round jet

On the basis of a two-point similarity analysis, the well-known power-law variations for the mean kinetic energy dissipation rate ϵ¯ and the longitudinal velocity variance u²¯ on the axis of a round jet are derived. In particular, the prefactor for ϵ¯∝(x−x₀)⁻⁴, where x₀ is a virtual origin, follows immediately from the variation of the mean velocity, the constancy of the local turbulent intensity and the ratio between the axial and transverse velocity variance. Second, the limit at small separations of the two-point budget equation yields an exact relation illustrating the equilibrium between the skewness of the longitudinal velocity derivative S and the destruction coefficient G of enstrophy. By comparing the latter relation with that for homogeneous isotropic decaying turbulence, it is shown that the approach towards the asymptotic state at infinite Reynolds number of S+2G/Rλ in the jet differs from that in purely decaying turbulence, although +2G/R_λ∝R⁻¹_λ in each case. This suggests that, at finite Reynolds numbers, the transport equation for ϵ¯ imposes a fundamental constraint on the balance between S and G that depends on the type of large-scale forcing and may thus differ from flow to flow. This questions the conjecture that S and G follow a universal evolution with R_λ; instead, S and G must be tested separately in each flow. The implication for the constant C_ϵ2 in the k−ϵ¯ model is also discussed

A look at the turbulent wake using scale-by-scale energy budgets

cited By 0International audienceIt is now well established that coherent structures exist in the majority of turbulent flows and can affect various aspects of the dynamics of these flows, such as the way energy is transferred over a range of scales as well as the departure from isotropy at the small scales. Reynolds and Hussain (J Fluid Mech 54:263–288, 1972) were first to derive one-point energy budgets for the coherent and random motions respectively. However, at least two points must be considered to define a scale and allow a description of the mechanisms involved in the energy budget at that scale. A transport equation for the second-order velocity structure function, equivalent to the Karman-Howarth (1938) equation for the two-point velocity correlation function, was written by Danaila et al. (1999) and tested in grid turbulence, which represents a reasonable approximation to (structureless) homogenous isotropic turbulence. The equation has since been extended to more complicated flows, for example the centreline of a fully developed channel flow and the axis of a selfpreserving circular jet. More recently, we have turned our attention to the intermediate wake of a circular cylinder in order to assess the effect of the coherent motion on the scale-by-scale energy distribution. In particular, energy budget equations, based on phase-conditioned structure functions, have revealed additional forcing terms, the most important of which highlights an additional cascade mechanism associated with the coherent motion. In the intermediate wake, the magnitude of the maximum energy transfer clearly depends on the nature of the coherent motion. © Springer-Verlag Berlin Heidelberg 2014

Restricted scaling range models for turbulent velocity and scalar energy transfers in decaying turbulence

The effect of finite Reynolds numbers and/or internal intermittency on the total kinetic energy and scalar energy transfers is examined in detail. For this purpose, two distinct models for velocity and scalar energy transfer are proposed in the specific context of freely decaying isotropic turbulence. The first one extends the already existing dynamical models (hereafter DYM, i.e. based on transport equations originated in Navier–Stokes and advection-diffusion transport equations). The second one relies on the characteristic time of the strain at a specific scale (hereafter SBM). Both models account for the Reynolds number dependence of the scaling exponent of the second-order structure functions, over a range of scales where such exponents may be defined, i.e. a restricted scaling range (RSR). Therefore, the models developed aim at reproducing the energy transfer over the RSR. The predicted energy transfer is very sensible to variations of the scaling exponent, especially at low Reynolds numbers. The approach towards the asymptotic 4/3 law is closely reproduced by the two models. The dynamical model reproduces the experimental results accurately especially in the vicinity of the Taylor microscale, while the SBM agrees almost perfectly with measurements at nearly all scales