Positivity of flux vector splitting schemes

Over the last ten years, robustness of schemes has raised an increasing interest among the CFD community. One mathematical aspect of scheme robustness is the positivity preserving property. At high Mach numbers, solving the conservative Euler equations can lead to negative densities or internal energy. Some schemes such as the flux vector splitting (FVS) schemes are known to avoid this drawback. In this study, a general method is detailed to analyze the positivity of FVS schemes. As an application, three classical FVS schemes (Van Leer's, Hänel's variant, and Steger and Warming's) are proved to be positively conservative under a CFL-like condition. Finally, it is proved that for any FVS scheme, there is an intrinsic incompatibility between the desirable property of positivity and the exact resolution of contact discontinuities

Two Dimensional Model of an Electro-Thermal Ice Protection System

In this communication we shall focus on the main governing equations and building blocks of the M.A.D (Anti-icing Deicing Modelling) numerical tool, which is now renamed as INUIT (Integrated NUmerical model of Ice protection sysTems) and part of the new generation of ONERA icing codes. The code simulates the functioning of an electro-thermal de-icing system. We shall also discuss the various improvements and new features we have added, especially a mechanical model of the ice block in order to improve the ice-shedding criterion

Monte-Carlo simulation of colliding particles or coalescing droplets transported by a turbulent flow in the framework of a joint fluid–particle pdf approach

The aim of the paper is to introduce and validate a Monte-Carlo algorithm for the prediction of an ensemble of colliding solid particles, or coalescing liquid droplets, suspended in a turbulent gas flow predicted by Reynolds Averaged Navier Stokes approach (RANS). The new algorithm is based on the direct discretization of the collision/coalescence kernel derived in the framework of a joint fluid–particle pdf approach proposed by Simonin et al. (2002). This approach allows to take into account correlations between colliding inertial particle velocities induced by their interaction with the fluid turbulence. Validation is performed by comparing the Monte-Carlo predictions with deterministic simulations of discrete solid particles coupled with Direct Numerical Simulation (DPS/DNS), or Large Eddy Simulation (DPS/LES), where the collision/coalescence effects are treated in a deterministic way. Five cases are investigated: elastic monodisperse particles, non-elastic monodisperse particles, binary mixture of elastic particles and binary mixture of elastic settling particles in turbulent flow and finally coalescing droplets. The predictions using the new Monte-Carlo algorithm are in much better agreement with DPS/DNS results than the ones using the standard algorithm

A CONSERVATIVE SAINT-VENANT TYPE MODEL TO DESCRIBE THE DYNAMICS OF THIN PARTIALLY WETTING FILMS WITH REGULARIZED FORCES AT THE CONTACT LINE

This paper deals with the numerical simulation of thin liquid films flowing on partially wetting solid substrates. A 2D Saint-Venant like model is proposed. Its originality lies in the conservative formulation of the capillary forces and in the model used for the disjoining pressure that accounts for the contact line capillary forces. A finite volume scheme is proposed for the resolution of the system and various numerical examples are presented and discussed. In particular, when the mesh resolution is fine enough, the model is proved to be able to predict correctly the spreading of a film with the exact contact angle in the vicinity of the contact line. When the mesh size is larger than the film thickness (which could be the case for many industrial applications), it is of course no longer possible to recover the contact angle. However, the model is proved to correctly predict the spreading of the film. This important feature is related to the thermodynamic consistency of the model in the sense that the latter ensures by construction the decrease of the film total free energy in the absence of external driving forces

Theory and validation of a 2D Finite-Volume integral boundary layer method intended for icing applications

International audienceA two-dimensional integral boundary layer method is developed to enable fast and economical computations of boundary layer flows. The ultimate goal is to provide some experience for the extension of this method in three dimensions. In this study, the unsteady momentum and kinetic energy integral equations are solved numerically, together with a set of closure relations based on assumed velocity profiles for laminar and turbulent flows. The robustness of the method is ensured by a Finite-Volume formulation based on an upwind scheme and a semi-implicit time discretization. The accuracy of the numerical method in the vicinity of the stagnation point is strongly improved by introducing a consistent corrective source term in the right-hand side of the equation system. The chosen closure relations are validated with test cases of self-similar flows. Numerical results are also compared with those of a full Prandtl equations code for NACA0012, GLC305 and MS317 airfoils test cases to demonstrate the capabilities of the method. Finally, preliminary results are shown proving the ability of the method to deal with iced airfoils even for complex glaze ice shapes.Une méthode bidimensionnelle de couche limite intégrale est développée pour permettre des calculs rapides et économiques des écoulements de couche limite. L'objectif est de fournir une certaine expérience pour l'extension de cette méthode en trois dimensions. Dans cette étude, les équations intégrales instationnaires de la quantité de mouvement et de l'énergie cinétique sont résolues numériquement, ainsi qu'un ensemble de relations de fermeture basées sur des profils de vitesse supposés pour des écoulements laminaires et turbulents. La robustesse de la méthode est assurée par une formulation aux Volumes Finis, basée sur un schéma de décentré amont et une discrétisation semi-implicite du temps. De plus, une méthode de contrôle a été développée afin d'éviter la singularité de Goldstein. La précision de la méthode numérique au voisinage du point d'arrêt est très élevée et améliorée par l'introduction d'un terme source correctif au second membre du système d'équations. Les relations de fermeture choisies sont validées avec des cas-test d'écoulements auto-similaires. Les résultats numériques sont également comparés à ceux d'une résolution des équations de Prandtl pour des cas-tests sur profils NACA0012, GLC305 et MS317 afin de démontrer les capacités de la méthode. Enfin, des résultats préliminaires sont présentés prouvant la capacité de la méthode à traiter des profils givrés, même pour les formes complexes de givre

On the use of a 2D Finite-Volume Integral Boundary Layer Method for Ice Accretion Calculations

In this paper, a two-dimensional integral boundary layer method developed in a recent work is applied to ice accretion computations. The method has already been validated in terms of boundary layer dynamic effects in another article. It is here validated for its ability to capture ice shapes, once the method is included in an icing suite. To be more specific, results using the new boundary layer method are compared against experimental ice shapes and simulated ones with the widely-used simplified integral method. The validation is carried out at an aggregated level because icing databases generally provide access to final ice shapes only. But since the simplified integral method is used in many icing numerical tools, this comparison makes it possible to investigate the benefits of introducing the new method for calculating the boundary layer. The main outcome of the new method is an improvement of the prediction of the boundary layer prediction under smooth-wall assumption, which in turn improves ice shape prediction. It is shown that, overall, the ice shapes are indeed either better predicted with the new method than with the baseline approach, or equally predicted with both methods. In addition, since the heat transfer coefficient tends to be underestimated by simplified integral methods, the new approach tends to predict lower horn angles than the baseline approach. Finally, the consequences of these results on current and future developments of ice accretion solvers are discussed. In particular, the new method is better suited to a 3D extension than the simplified integral method

Numerical simulation and modeling of ice shedding: Process initiation

In aeronautics, the issue of ice shedding prediction is of prime importance in the assessment of electro-thermal ice protection systems. In this paper, an ice shedding mechanism based on pressure redistribution in the water film formed at the ice/airfoil interface is proposed. This pressure distribution induces a stress concentration that leads to crack propagation in the ice. To determine whether this mechanism is relevant or not, two numerical experiments are performed. The results of these numerical experiments and the influence of a few material parameters are discussed, as well as their limitations and possible consequences arising from some of the hypotheses. The numerical modeling is based on recent works on damage/fracture mechanics which provide a general framework for fracture mechanics computation. The effects of numerical parameters and mesh size are discussed. A mixed mode test case based on experimental data is also performed. This test case had not been attempted before on this particular numerical method, which therefore serves as further validation

Lagrangian Point Force regularization for dispersed two-phase flows

International audienceThe present paper presents a regularization procedure of the Lagrangian point-particle approach for the simulation of dispersed two-phase flows in a statistical framework. The aim is to regularize the probability presence of a particle, written as a Dirac delta function centered on the particle position in the standard formulation, by a Gaussian like distribution. The associated regularization length scale is obtained by solving additional transport equations in the Lagrangian framework. The regularization itself is then achieved by solving two non-linear diffusion equations. The first diffusion equations allows to spread the field of spatially varying diffusion coefficients required for regularization over the computational mesh. Once this field is defined, regularization of the Lagrangian fields to be projected on the Eulerian grid such as particle density, particle velocity, etc... is performed. These ideas are then tested on simplified one-dimensional test cases. While preliminary results seem encouraging as the dispersed phase fields projected on the Eulerian grid appear much less sensitive to the initial sampling of the spray, further tests on more realistic test cases are necessary to conclude on precision gains with repect to the additional computational expense resulting from the regularization procedure

Regularization of the Lagrangian point force approximation for deterministic discrete particle simulations

International audienceThe current article presents a regularization procedure of the Lagrangian point-force approach commonly used to account for the perturbation of a fluid phase by a dispersed particle phase. The regularization procedure is based on a nonlinear diffusion equation to naturally ensure parallel efficiency when the regularization length scale extends over several grid cells. The diffusion coefficient thus becomes a function of the particle source term gradient and expressions allowing to approximately adjust the regularization length scale according to the local particle to mesh size ratio are proposed, so that mesh refinement or polydisperse sprays may be handled. Elementary numerical test cases confirm the convergence of the present procedure under mesh refinement and its ability to locally adapt the regularization length scale. Furthermore, the chosen regularization length scale allows to match the leading order term of the perturbation flow field set by the particle beyond approximately two particle diameters in the Stokes regime. When applying the presented source term regularization procedure, the terminal velocity of a particle settling under gravity in the Stokes regime becomes relatively insensitive to mesh refinement. However, errors with respect to the theoretical settling velocity remain substantial and removal of the particle's self induced velocity appears necessary to recover the undisturbed fluid velocity at the particle location and correctly evaluate the drag force. As the current regularization procedure yields source terms that are close to c 2019. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/ Gaussian, an analytic expression from the literature is used to estimate the particle's self induced velocity. When combining source term regularization and removal of the particle's self induced velocity, good results are obtained for the terminal settling speed in the Stokes regime. Results obtained for horizontally separated particle pairs settling under gravity in the Stokes regime show equally good agreement with theoretical results. Because analytic expressions for the particle's self-induced velocity are no longer available at finite particle Reynolds numbers, correlations recently proposed in the literature are used to obtain correct settling velocities beyond the Stokes regime