Turbulent two-phase flows are multi-scale and multi-dimensional phenomena, i.e. which depends on:
– the region within the flow,
– the scales of the involved liquid structures
– the fluid/flow physical parameters.
When breaking-up, these liquid structures have complex
– geometry (surface area, curvatures),
– morphology (sphericity, ligamentarity)
– topology (connectedness)
– dynamics (singularity at finite time)
Tackling the complexity and the multi-scale facets of liquid atomization requires elaborating a new theory:
– which draws explicit links with the geometry, morphology and topology of the liquid-gas interface,
– which allows scrutinizing the transport of liquid in both scale- and flow-position-space
– which allows some degree of predictibility and regularity to be recovered (in the statistical sense).
Schematic representation of the phase indicator field . Gray zones represent the liquid phase (), white zones the gas phase (). The two points and together with the mid-point and the separation vector are also sketched.
We proposed using the machinery of two-point statistical equations which originates from the single-phase turbulence community which is adapted here to a relevant scalar in two-phase flows: the phase indicator function . This field variable is simply defined as:
The transport equation for reads
Writing Eq. (2) at two points and , arbitrarily separated by a distance , one obtains after some manipulations:
In Eq. (3),
The litterature of porous media makes extensive usage of two-point statistics of the phase indicator. Some analytical results allows to be related to some geometrical properties of the liquid-gas interface. In particular, at small scales, the expansion of the second-order increments up to third order writes:
In Eq. (4)
Hence, the limit at small scales of Eq. (3) naturally degenerates to the transport equation for the surface density which reads
where is the stretch rate. At larger scales, is expected to provide insights into the tortuousness of the interface.
(top) Simulation of homogeneous decaying liquid-gaz turbulence. (bottom) Simulation of liquid-gas shear flow. Both were carried out using the ARCHER code
The framework has been first tested in homogeneous decaying liquid-gas turbulence. Homogeneity implies that the gradient w.r.t vanishes, thereby reducing the problem to the analysis of the scale/time evolution of the system (a 4D problem which depends on and ).
– There exists a characteristic length-scale based on the surface density and liquid volume that allows characterizing the scale evolution of
– The stretch rate drives the cascade process of the liquid phase and hence plays the same role as the scalar dissipation rate for diffusive scalars.
Results have been published in Journal of Fluid Mechanics:
We then extended the analysis to a temporally evolving liquid-gas shear layer. In this situation, homogeneity holds only along two directions and one has to resort to the 5D version of the scale/space/time budget (3 dimensions for , 1 inhomogeneity direction, 1 dimension of time).
– The complexity of liquid transport in the combined space of scales and flow positions is exemplified.
– Some range of scales and flow positions comply with a direct transfer of ‘energy’ (from large to small scales), some others with an inverse cascade.
Results have been presented at the ILASS conference: