Assessment of LES sub-grid models for turbulent reacting flows

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Numerical Analysis of a New Mixed Formulation for Eigenvalue Convection-Diffusion Problems

A mixed formulation is proposed and analyzed mathematically for coupled convection-diffusion in heterogeneous medias. Transfer in solid parts driven by pure diffusion is coupled with convection-diffusion transfer in fluid parts. This study is carried out for translation-invariant geometries (general infinite cylinders) and unidirectional flows. This formulation brings to the fore a new convection-diffusion operator, the properties of which are mathematically studied: its symmetry is first shown using a suitable scalar product. It is proved to be self-adjoint with compact resolvent on a simple Hilbert space. Its spectrum is characterized as being composed of a double set of eigenvalues: one converging towards −∞ and the other towards +∞, thus resulting in a nonsectorial operator. The decomposition of the convection-diffusion problem into a generalized eigenvalue problem permits the reduction of the original three-dimensional problem into a two-dimensional one. Despite the operator being nonsectorial, a complete solution on the infinite cylinder, associated to a step change of the wall temperature at the origin, is exhibited with the help of the operator’s two sets of eigenvalues/eigenfunctions. On the computational point of view, a mixed variational formulation is naturally associated to the eigenvalue problem. Numerical illustrations are provided for axisymmetrical situations, the convergence of which is found to be consistent with the numerical discretization

A model for the investigation of the second-order structure of caustic formations in dispersed flows

The formation of caustics by inertial particles is distinctive of dispersed flows. Their pressureless nature allows crossing trajectories resulting in singularities that cannot be captured accurately by standard Lagrangian approaches due to their fine spatial scale. A promising method for the investigation of caustics is the Osiptsov method or fully Lagrangian approach (FLA). The FLA has the advantage of identifying caustics, but its applicability is hindered by the occurrence of singularities. We present an original robust framework based on the FLA that provides an explicit expression of the dispersed phase structure that does not degenerate in the vicinity of caustics, using a single representative particle. The FLA is extended to account for the Hessian of the dispersed continuum (DC). It demonstrates the integrability of the FLA number density and allows for the calculation of the number density on a given length scale, retaining the functionality of the FLA. Number density models based on the second-order representation of the DC and on the one-dimensional structure of the particle distribution, that account for the anisotropy of the DC on caustics, are derived and applied for analytical flows. The number density is linked to a finite length scale, needed for the introduction of the FLA to spatially filtered flow fields. Finally, the method is used for the calculation of the interparticle separation on caustics. The identification of the structure of caustics presented in this work paves the way to a robust understanding of the mechanisms of particle accumulation

The Generalized Graetz Problem in Finite Domains

We consider the generalized Graetz problem associated with stationary convection-diffusion inside a domain having any regular three-dimensional translationally invariant section and finite or semi-infinite extent. Our framework encompasses any previous “extended” and “conjugated” Graetz generalizations and provides theoretical bases for computing the orthogonal set of generalized two-dimensional Graetz modes. The theoretical framework includes both heterogeneous and possibly anisotropic diffusion tensors. In the case of semi-infinite domains, the existence of a bounded solution is shown from the analysis of two-dimensional operator eigenvectors which form a basis of L2 . In the case of finite domains a similar basis can be exhibited, and the mode’s amplitudes can be obtained from the inversion of newly defined finite domain operator. Our analysis includes both the theoretical and practical issues associated with this finite domain operator inversion as well as its interpretation as a multireflection image method. Error estimates are provided when numerically truncating the spectrum to a finite number of modes. Numerical examples are validated for reference configurations and provided in nontrivial cases. Our methodology shows how to map the solution of stationary convection-diffusion problems in finite three-dimensional domains into a two-dimensional operator spectrum, which leads to a drastic reduction in computational cost

An efficient Adaptive Mesh Refinement (AMR) algorithm for the Discontinuous Galerkin method: Applications for the computation of compressible two-phase flows

We present an Adaptive Mesh Refinement (AMR) method suitable for hybrid unstructured meshes that allows for local refinement and de-refinement of the computational grid during the evolution of the flow. The adaptive implementation of the Discontinuous Galerkin (DG) method introduced in this work (ForestDG) is based on a topological representation of the computational mesh by a hierarchical structure consisting of oct- quad- and binary trees. Adaptive mesh refinement (h-refinement) enables us to increase the spatial resolution of the computational mesh in the vicinity of the points of interest such as interfaces, geometrical features, or flow discontinuities. The local increase in the expansion order (p-refinement) at areas of high strain rates or vorticity magnitude results in an increase of the order of accuracy in the region of shear layers and vortices. A graph of unitarian-trees, representing hexahedral, prismatic and tetrahedral elements is used for the representation of the initial domain. The ancestral elements of the mesh can be split into self-similar elements allowing each tree to grow branches to an arbitrary level of refinement. The connectivity of the elements, their genealogy and their partitioning are described by linked lists of pointers. An explicit calculation of these relations, presented in this paper, facilitates the on-the-fly splitting, merging and repartitioning of the computational mesh by rearranging the links of each node of the tree with a minimal computational overhead. The modal basis used in the DG implementation facilitates the mapping of the fluxes across the non conformal faces. The AMR methodology is presented and assessed using a series of inviscid and viscous test cases. Also, the AMR methodology is used for the modelling of the interaction between droplets and the carrier phase in a two-phase flow. This approach is applied to the analysis of a spray injected into a chamber of quiescent air, using the Eulerian–Lagrangian approach. This enables us to refine the computational mesh in the vicinity of the droplet parcels and accurately resolve the coupling between the two phases

Formation number of confined vortex rings

This paper investigates the formation number of vortex rings generated by a piston-cylinder mechanism in a confined tube. We use Direct Numerical Simulations (DNS) of axisymmetric confined vortex rings to study the influence of different parameters on the separation (or pinch-off) of the vortex ring from the trailing jet. It is shown that the structure of the vortex ring at pinch-off depends on the type of injection program (pulse dominated by either positive or negative acceleration ramps) and the confinement ratio D w /D , where D w is the inner diameter of the tube and D the diameter of the cylinder). For low confinement ratios ( D w /D ≤ 2), a vortex of opposite sign generated at the lateral wall strongly interacts with the vortex ring and the pinch-off is not clearly observed. The pinch-off is observed and analysed for confinement ratios D w /D ≥ 2 . 5. DNS data are used to estimate the value of the formation time, which is the time necessary for the vortex generator to inject the same amount of circulation as carried by the detached vortex ring. The confined vortex ring at pinch-off is described by the model suggested by Danaila, Kaplanski and Sazhin [A model for confined vortex rings with elliptical core vorticity distribution, Journal of Fluid Mechanics, 811 :67-94, 2017]. This model allows us to take into account the influence of the lateral wall and the elliptical shape of the vortex core. The value of the formation time is predicted using this model and the slug-flow model

Modelling of the evolution of a droplet cloud in a turbulent flow

The effects of droplet inertia and turbulent mixing on the droplet number density distribution in a turbulent flow field are studied. A formulation of the turbulent convective diffusion equation for the droplet number density, based on the modified Fully Lagrangian Approach, is proposed. The Fully Lagrangian Approach for the dispersed phase is extended to account for the Hessian of transformation from Eulerian to Lagrangian variables. Droplets with moderate inertia are assumed to be transported and dispersed by large scale structures of a filtered field in the Large Eddy Simulation (LES) framework. Turbulent fluctuations, not visible in the filtered solution for the droplet velocity field, induce an additional diffusion mass flux and hence additional dispersion of the droplets. The Lagrangian formulation of the transport equation for the droplet number density and the modified Fully Lagrangian Approach (FLA) make it possible to resolve the flow regions with intersecting droplet trajectories in the filtered flow field. Thus, we can cope successfully with the problems of multivalued filtered droplet velocity regions and caustic formation. The spatial derivatives for the droplet number density are calculated by projecting the FLA solution on the Eulerian mesh, resulting in a hybrid Lagrangian–Eulerian approach to the problem. The main approximations for the method are supported by the calculation of droplet mixing in an unsteady one-dimensional flow field formed by large-scale oscillations with an imposed small-scale modulation. The results of the calculations for droplet mixing in decaying homogeneous and isotropic turbulence are validated by the results of Direct Numerical Simulations (DNS) for several values of the Stokes number

Solution of cavitating compressible flows using Discontinuous Galerkin discretisation

A methodology for modelling cavitating flows using a high-order Adaptive Mesh Refinement (AMR) approach based on the Discontinuous Galerkin method (DG) is presented. The AMR implementation used features on-the-fly adaptive mesh refinement for unstructured hybrid meshes. The specific implementation has been developed for the resolution of complex multi-scale phenomena where high accuracy p-adaptive discretisations are combined with an h-adaptive data structure. This approach accommodates the fine spatial resolution for the interface discontinuities and the shock waves observed in compressible cavitating flows. The Tait equation of state is used for the modelling of the liquid phase while an isentropic path is assumed for the liquid/vapour mixture. Second order spatial and a third order non-oscillatory temporal discretisation are used for the integration of the mass and momentum conservation equations, in order to resolve the flow structures responsible for the formation of cavitation bubbles and the resulting compression waves. Assessment of the developed methodology is performed for the one-dimensional advancement of a compressible liquid-vapour interface and the symmetric collapse of a spherical vapour bubble. Following, results obtained with the developed multi-scale modelling AMR approach has revealed a complex bubble collapse mechanism near a rigid wall, providing evidence of processes that have been unknown before due to reduced resolution and dissipative nature of past simulations. The impinging jet accompanying the collapse of a bubble near a wall, was found to induce vortical structures, which result to the formation of a secondary cavitation of a wall-attached bubble at the vicinity of the impingement jet shear layer. At the final stages of the initial bubble collapse, the impinging jet was found to penetrate the centre-line of the wall bubble inducing its partial collapse. This secondary collapse results to a rich spatial structure of shock waves, interacting with the secondary bubbles. Moreover, the calculated pressure level are found to be much higher than those reported from previous methodologies

A Reference Architecture for Management of Security Operations in Digital Service Chains

Modern computing paradigms (i.e., cloud, edge, Internet of Things) and ubiquitous connectivity have brought the notion of pervasive computing to an unforeseeable level, which boosts service-oriented architectures and microservices patterns to create digital services with data-centric models. However, the resulting agility in service creation and management has not been followed by a similar evolution in cybersecurity patterns, which still largely rest on more conventional device- and infrastructure-centric models. In this Chapter, we describe the implementation of the GUARD Platform, which represents the core element of a modern cybersecurity framework for building detection and analytics services for complex digital service chains. We briefly review the logical components and how they address scientific and technological challenges behind the limitations of existing cybersecurity tools. We also provide validation and performance analysis that show the feasibility and efficiency of our implementation

Droplet nuclei caustic formations in exhaled vortex rings

Vortex ring (VR) structures occur in light or hoarse cough configurations. These instances consist of short impulses of exhaled air resulting to a self-contained structure that can travel large distances. The present study is the first implementation of the second order Fully Lagrangian Approach (FLA) for three-dimensional realistic flow-fields obtained by means of Computational Fluid Dynamics (CFD) and provides a method to calculate the occurrence and the intensity of caustic formations. The carrier phase flow field is resolved by means of second order accurate Direct Numerical Simulation (DNS) based on a Finite Difference approach for the momentum equations, while a spectral approach is followed for the Poisson equation using Fast Fourier Transform (FFT). The effect of the undulations of the carrier phase velocity due to large scale vortical structures and turbulence is investigated. The evaluation of the higher order derivatives needed by the second order FLA is achieved by pre-fabricated least squares second order interpolations in three dimensions. This method allows for the simulation of the clustering of droplets and droplet nuclei exhaled in ambient air in conditions akin to light cough. Given the ambiguous conditions of vortex-ring formation during cough instances, three different exhale (injection) parameters n are assumed, i.e. under-developed ([Formula: see text]), ideal ([Formula: see text]) and over-developed ([Formula: see text]) vortex rings. The formation of clusters results in the spatial variance of the airborne viral load. This un-mixing of exhumed aerosols is related to the formation of localised high viral load distributions that can be linked to super-spreading events