High Order Moment Model for Polydisperse Evaporating Sprays towards Interfacial Geometry Description

Abstract

In this paper we propose a new Eulerian model and related accurate and robust numerical methods, describing polydisperse evaporating sprays, based on high order moment methods in size. The main novelty of this model relies on the use of fractional droplet surface moments and their ability to predict some geometrical variables of the droplet-gas interface, by analogy with the liquid-gas interface in interfacial flows. Evaporation is evaluated by using a Maximum Entropy (ME) reconstruction. The use of fractional moments introduces some theoretical and numerical difficulties. First, relying on a study of the moment space, we extend the ME reconstruction to the case of fractional moments. Then, we propose a new high order and robust algorithm to solve the moment evolution due to evaporation, which preserves the structure of the moment space. It involves some negative order fractional moments for which a novel treatment is introduced. The present model and numerical schemes yield an accurate and stable evaluation of the moment dynamics with minimal number of variables, as well as computational cost, but also provides an additional capacity of coupling with diffuse interface model and transport equation of averaged geometrical interface variables, which are essential in order to describe atomization.

Statistical modeling of the gas–liquid interface using geometrical variables: Toward a unified description of the disperse and separated phase flows

“…In this case, the geometrical quantities are averaged on the surface of isolated droplets or bubbles. We show that this new formalism can be related to the high-order moment model proposed in [24,40] for sprays of spherical droplets. Indeed, by considering fractional moments of a NDF, this model uses the same geometrical quantities to describe the polydispersity of the droplets.…”

“…This reconstruction has been already used in the EMSM model [20,22] and CSVM model [23] and shows a high capacity to model the polydispersion effect on the evaporation and the drag force response of droplets compared to the Multi-fluid approach [17]. In the case of fractional moments (54), the reconstruction by entropy maximization reads: The existence and uniqueness of such a reconstruction have been proved in [24] and the reconstructed NDF has the following form:…”

“…The transport part is solved using a kinetic scheme [16] adapted for fractional moments. A new solver for the source terms (evaporation and drag force) is proposed in [24]. The proposed numerical schemes ensure the realizability of the moments and show a high accuracy for evaporation effects at 0D and 1D transport [24].…”

“…multi-fluid models, where the size phase space is discretized into sections. Then moments up to order two can be used in each section [16,17], B. quadrature-moment methods such as QMOM, DQMOM or EQMOM [18,19], consider the NDF as a sum of Dirac-delta functions, potentially extended to kernels, C. high order moments with continuous reconstruction developed in [20,21,22,23,24]. At each step, a continuous NDF maximizing the Shannon entropy is reconstructed from the high order moments [25], with a complete coverage of the whole moment space.…”

Where does the drop size distribution come from?

“…This quantity can be defined everywhere whatever is the liquid phase topology and, combined with the liquid volume fraction , it gives the mean Sauter diameter once the spray is formed: 32 = 6 Σ . Recently, Essadki et al [7] used high order fractional moments of the DSD for disperse phase, where the size is given by the surface area of droplet, to recover some interface geometrical quantities already used in describing the gas-liquid interface in [9]. These quantities are the volume fraction, the mean surface density and the two averaged Gauss = 1 * 2 and mean = 1 + 2 2 curvatures, where 1 and 2 are the two principal curvatures of the surface.…”