Aymeric Vié

Size-velocity correlations in hybrid high order moment/multi-fluid methods for polydisperse evaporating sprays: Modeling and numerical issues

Kah et al. (2010) [30,33] recently developed the Eulerian multi-size moment model (EMSM) which tackles the modeling and numerical simulation of polydisperse multiphase flows. Using a high order moment method in a compact interval, they suggested to reconstruct the number density function (NDF) by entropy maximization, which leads to a unique and realizable NDF, potentially in several size intervals, thus leading to an hybrid method between Multifluid and high order. This reconstruction is used to simulate the evaporation process, by an evaluation of the flux of droplet disappearance at zero size, the fluxes of droplets between size intervals, and an accurate description of the size shift induced by evaporation Massot et al. (2010) [15]. Although this method demonstrated its potential for evaporating polydisperse flows, two issues remain to be addressed. First, the EMSM only considers one velocity for all droplets, thus decoupling size from velocity, which is too restrictive for distributions with a large size spectrum. In most applications size-conditioned dynamics have to be accounted for. Second, the possibility to have separated dynamics for each size can lead to quasi-monodisperse distributions, which corresponds to a hard limiting case for the EM algorithm. So the behavior of the algorithm needs to be investigated, in order to reproduce the entire moment space with a reasonable accuracy. The aim of this paper is thus twofold. The EM and its related algorithm are enhanced by using a more accurate integration method in order to handle NDF close to the frontier of the moment space associated with an adaptive number of parameters to reconstruct the NDF accurately and efficiently, as well as tabulated initial guess to optimize the computational time. Then, a new model called CSVM (coupled size-velocity moments model) is introduced. Size-velocity correlations are addressed either in the evaporation and drag processes, or in the convective transport. To reach this goal, a velocity reconstruction for each size is suggested, using only one additional moment per dimension, and which can be directly applied to several size intervals. Thus, this method is a direct generalization of EMSM. To handle the convective transport, a flux splitting scheme is proposed, based on the underlying kinetic description of the disperse phase. Comparing to existing approaches, a main novelty of the CSVM is that our kinetic approach ensures built-in realizability conditions, no additional corrections of the moments being needed at each time step. The full strategy is first evaluated in 0D and 1D cases, which either demonstrates the ability to reproduce both evaporation, drag force and convection with size-velocity correlations, or the possible extension to several size intervals. Finally, the method is applied on 2D cases with only one section, showing the ability of the CSVM and its related algorithms to capture the main physics of polydisperse evaporating sprays with a minimal number of moments.

On the Development of High Order Realizable Schemes for the Eulerian Simulation of Disperse Phase Flows: A Convex-State Preserving Discontinuous Galerkin Method

In the present work, a high order realizable scheme for the Eulerian simulation of disperse phase flows on unstructured grids is developed and tested. In the Eulerian modeling framework two approaches are studied: the monokinetic (MK) [1] and the Gaussian closures [2, 3]. The former leads to a pressureless gas dynamics system (PGD). It accurately reproduces the physics of such flows at low Stokes number, but is challenging for numerics since the resulting system is weakly hyperbolic. The latter deals with higher Stokes numbers by accounting for particle trajectory crossings (PTC) [4]. Compared to the MK closure, the resulting system of equation is hyperbolic but has a more complex structure; realizability conditions are satisfied at the continuous level, which imply a precise framework for numerical methods. To achieve the goals of accuracy, robustness and realizability, the Discontinuous Galerkin method (DG) is a promising numerical approach [5, 6, 7, 8]. Based on the recent work of Zhang et al. [6], the...

On the Eulerian Large Eddy Simulation of Disperse Phase Flows: An Asymptotic Preserving Scheme for Small Stokes Number Flows

In the present work, the Eulerian large eddy simulation (LES) of dilute disperse phase flows is investigated. By highlighting the main advantages and drawbacks of the available approaches in the literature, a choice is made in terms of modeling: a Fokker--Planck-like filtered kinetic equation proposed by Zaichik, Simonin, and Alipchenkov [L. I. Zaichik, O. Simonin, and V. M. Alipchenkov, J. Turbul., 10 (2009), N4] and a kinetic-based moment method based on a Gaussian closure for the number density function proposed by Vie, Doisneau, and Massot [A. Vie, F. Doisneau, and M. Massot, Comm. Comput. Phys., 17 (2015), pp. 1--46]. The resulting Euler-like system of equations is able to reproduce the dynamics of particles for small to moderate Stokes number flows, given a LES model for the gaseous phase, and is representative of the generic difficulties of such models. Indeed, it encounters strong constraints in terms of numerics in the small Stokes number limit, which can lead to a degeneracy of the accuracy of s...

Comparison of Realizable Schemes for the Eulerian Simulation of Disperse Phase Flows

In the framework of fully Eulerian simulation of disperse phase flows, the use of a monokinetic closure for the kinetic based moment method is of high importance since it accurately reproduces the physics of low inertia particles with a minimum number of moments. The free transport part of this model leads to a pressureless gas dynamics system which is weakly hyperbolic and can generate \(\delta \)-shocks. These singularities are difficult to handle numerically, especially without globally degenerating the order or disrespecting the realizability constraints. A comparison between three second order schemes is conducted in the present work. These schemes are: a realizable MUSCL/HLL finite volume scheme, a finite volume kinetic scheme, and a convex state preserving Runge-Kutta discontinuous Galerkin scheme. Even though numerical computations have already been led in 2D and 3D with this model and numerical methods, the present contribution focuses on 1D results for a full understanding of the trade off between robustness and accuracy and of the impact of the limitation procedures on the numerical dissipation. Advantages and drawbacks of each of these schemes are eventually discussed.

Multivariate Gaussian extended quadrature method of moments for turbulent disperse multiphase flow

The present contribution introduces a fourth-order moment formalism for particle trajectory crossing (PTC) in the framework of multiscale modeling of disperse multiphase flow. In our previous work, the ability to treat PTC was examined with direct-numerical simulations (DNS) using either quadrature reconstruction based on a sum of Dirac delta functions denoted as Quadrature-Based Moment Methods (QBMM) in order to capture large scale trajectory crossing, or by using low order hydrodynamics closures in the Levermore hierarchy denoted as Kinetic-Based Moment Methods (KBMM) in order to capture small scale trajectory crossing. Whereas KBMM leads to well-posed PDEs and has a hard time capturing large scale trajectory crossing for particles with enough inertia, QBMM based on a discrete reconstruction suffers from singularity formation and requires too many moments in order to capture the effect of PTC at both small scale and large scale both to small-scale turbulence as well as free transport coupled to drag in an Eulerian mesoscale framework. The challenge addressed in this work is thus twofold: first, to propose a new generation of method at the interface between QBMM and KBMM with less singular behavior and the associated proper mathematical properties, which is able to capture both small scale and large scale trajectory crossing, and second to limit the number of moments used for applicability in 2-D and 3-D configurations without losing too much accuracy in the representation of spatial fluxes. In order to illustrate its numerical properties, the proposed Gaussian extended quadrature method of moments (Gaussian-EQMOM) is applied to solve 1-D and 2-D kinetic equations representing finite-Stokes-number particles in a known turbulent fluid flow.

Large Eddy Simulations of a Liquid Fuel Swirl Burner: Flame Characterization for Pilot and Multipoint Injection Strategies

A laboratory-scale swirling burner fueled with dodecane is studied numerically using large eddy simulations. The burner is composed of two stages, allowing the use of two different injector types (namely pilot and multipoint) in order to study the influence of droplet size and initial position with fixed geometry and delivered power. For a chosen lean operating point, the two liquid injection types are tested, highlighting a dramatic influence on the flame stabilization process. When fuel is injected through the multipoint stage, evaporation and mixing are enhanced and a partially premixed mixture enters the combustion chamber. The flame then takes an ‘M’ shape, mainly controlled by the large inner and outer recirculation zones associated with this highly swirling flow and in which trapped burnt gases guarantee permanent ignition of the fresh mixture entering the chamber. The situation is much more complex when fuel is solely injected through the pilot nozzle. Due to the large amount of liquid fuel present in the pilot zone, premixing is not achieved and the flame must stabilize itself mainly in a hybrid combustion regime. This is only possible thanks to a very complex situation in this region, where hot evaporated fuel is trapped in front of the nozzle and oxygen is mainly coming from the large central recirculation zone. In that case, the flame takes a ‘tulip’ shape, with a stabilization point inside the injection device. Both flame shapes are compared using scatterplots and flame dynamics are analyzed.Copyright

Conditional Hyperbolic Quadrature Method of Moments for Kinetic Equations

The conditional quadrature method of moments (CQMOM) was introduced by Yuan and Fox [J. Comput. Phys. 230 (22), 8216–8246 (2011)] to reconstruct a velocity distribution function (VDF) from a finite set of its integer moments. The reconstructed VDF takes the form of a sum of weighted Dirac delta functions in velocity phase space, and provides a closure for the spatial flux term in the corresponding kinetic equation. The CQMOM closure for the flux leads to a weakly hyperbolic system of moment equations. In subsequent work [Chalons et al., Proceed. CTR Sum. Prog. 2010, 347–358 (2010)], the Dirac delta functions were replaced by Gaussian distributions, which make the moment system hyperbolic but at the added cost of dealing with continuous distributions. Here, a hyperbolic version of CQMOM is proposed that uses weighted Dirac delta functions. While the moment set employed for multi-Gaussian and conditional HyQMOM (CHyQMOM) are equivalent, the latter is able to access all of moment space whereas the former cannot (e.g. arbitrary values of the fourth-order velocity moment in 1-D phase space with two nodes). By making use of the properties of CHyQMOM in 2-D phase space, it is possible to control a symmetrical subset of the optimal moments [Fox, Indust. & Engng. Chem. Res. 48 (21), 9686–9696 (2009)]. Furthermore, the moment sets for 2-D problems are smaller for CHyQMOM than in the original CQMOM thanks to a judicious choice of the velocity abscissas in phase space.

Large Eddy Simulation of Swirling Kerosene/Air Spray Flame Using Tabulated Chemistry

Accurate characterization of swirled flames is a key point in the development of more efficient and safer aeronautical engines. The task is even more challenging for spray injection systems. On the one side, spray interacts with both turbulence and flame, eventually affecting the flame dynamics. On the other side, spray flame structure is highly complex due to equivalence ratio inhomogeneities caused by the evaporation process. Introducing detailed chemistry in numerical simulations, necessary for the prediction of flame stabilization, ignition and pollutant concentration, is then essential but extremely expensive in terms of CPU time. In this context, tabulated chemistry methods, expressly developed to account for detailed chemistry at a reduced computational cost in Large Eddy Simulation of turbulent gaseous flames, are attractive. The objective of this work is to propose a first computation of a swirled spray flame stabilized in an actual turbojet injection system using tabulated chemistry. A Large Eddy Simulation of an experimental benchmark, representative of an industrial swirl two-phase air/kerosene injection system, is performed using a standard tabulated chemistry method. The numerical results are compared to the experimental database in terms of mean and fluctuating axial velocity. The reactive two-phase flow is deeper investigated focusing on the flame structure and dynamics.Copyright