Numerical simulations are presented for three dimensional viscous incompressible free shear flows. The numerical method is based on solving the vorticity equation using Vortex-In-Cell method. In this method, the vorticity field is discretized into a finite set of Lagrangian elements (particles) and the computational domain is covered by Eulerian mesh. Velocity field is computed on the mesh by solving Poisson equation. The solution proceeds in time by advecting the particles with the flow. Second order Adam-Bashford method is used for time integration. Exchange of information between Lagrangian particles and Eulerian grid is carried out using the M’4 interpolation scheme. The classical inviscid scheme is enhanced to account for stretching and viscous effects. For that matter, two schemes are used. The first one used periodic remeshing of the vortex particles along with fourth order finite difference approximation for the partial derivatives of the stretching and viscous terms. In the second scheme, derivatives are approximated by least squares polynomial. The novelty of this work is signified by using the moving least squares technique within the framework of the Vortex-in-Cell method and implementing it to a three dimensional temporal mixing layer. Comparisons of the mean flow and velocity statistics are made with experimental studies. The results confirm the validity of the present schemes. Both schemes also demonstrate capability to qualitatively capture significant flow scales, and allow gaining physical insight as to the development of instabilities and the formation of three dimensional vortex structures. The two schemes show acceptable low numerical diffusion as well.

Numerical simulations are presented for three dimensional viscous incompressible free shear flows. The numerical method is based on solving the vorticity equation using Vortex-In-Cell method. In this method, the vorticity field is discretized into a finite set of Lagrangian elements (particles) and the computational domain is covered by Eulerian mesh. Velocity field is computed on the mesh by solving Poisson equation. The solution proceeds in time by advecting the particles with the flow. Second order Adam-Bashford method is used for time integration. Exchange of information between Lagrangian particles and Eulerian grid is carried out using the M’4 interpolation scheme. The classical inviscid scheme is enhanced to account for stretching and viscous effects. For that matter, two schemes are used. The first one used periodic remeshing of the vortex particles along with fourth order finite difference approximation for the partial derivatives of the stretching and viscous terms. In the second scheme, derivatives are approximated by least squares polynomial. The novelty of this work is signified by using the moving least squares technique within the framework of the Vortex-in-Cell method and implementing it to a three dimensional temporal mixing layer. Comparisons of the mean flow and velocity statistics are made with experimental studies. The results confirm the validity of the present schemes. Both schemes also demonstrate capability to qualitatively capture significant flow scales, and allow gaining physical insight as to the development of instabilities and the formation of three dimensional vortex structures. The two schemes show acceptable low numerical diffusion as well.

Numerical simulations are presented for three dimensional viscous incompressible free shear flows. The numerical method is based on solving the vorticity equation using Vortex-In-Cell method. In this method, the vorticity field is discretized into a finite set of Lagrangian elements (particles) and the computational domain is covered by Eulerian mesh. Velocity field is computed on the mesh by solving Poisson equation. The solution proceeds in time by advecting the particles with the flow. Second order Adam-Bashford method is used for time integration. Exchange of information between Lagrangian particles and Eulerian grid is carried out using the M’4 interpolation scheme. The classical inviscid scheme is enhanced to account for stretching and viscous effects. For that matter, two schemes are used. The first one used periodic remeshing of the vortex particles along with fourth order finite difference approximation for the partial derivatives of the stretching and viscous terms. In the second scheme, derivatives are approximated by least squares polynomial. The novelty of this work is signified by using the moving least squares technique within the framework of the Vortex-in-Cell method and implementing it to a three dimensional temporal mixing layer. Comparisons of the mean flow and velocity statistics are made with experimental studies. The results confirm the validity of the present schemes. Both schemes also demonstrate capability to qualitatively capture significant flow scales, and allow gaining physical insight as to the development of instabilities and the formation of three dimensional vortex structures. The two schemes show acceptable low numerical diffusion as well.

Depuis les travaux pionniers de Brown et Roshko portant sur les effets des variations de masse volumique au sein de l’écoulement de couche de mélange, plusieurs autres études tant théoriques, expérimentales ou numériques se sont attelées à étudier finement cet écoulement. Les motivations sont d’ordre pratiques (industrie de la chimie, l’aérodynamique, la combustion . . .) ou alors purement théoriques (rôle des structures cohérentes, instabilités secondaires. . .). Ces études se sont intéressées, entre autres, aux effets de compressibilité et/ou de masse volumique variable. A notre connaissance, les effets des variations de viscosité dans la configuration de couche de mélange sont peu abordés dans la littérature. L’objectif de ces travaux de recherche est l’exploration théorique et numérique de l’écoulement de couche de mélange temporelle à viscosité variable, plus particulièrement durant sa phase initiale de développement. D’un point de vu numérique, les équations de Navier-Stokes sont résolues,en formulation faiblement compressible, au moyen du solveur CHOC-WAVES, basé sur le schéma WENO. L’approche DNS est justifiée par l’absence, dans la littérature, de modèles de sous-maille capables de prendre en compte les effets de la viscosité variable. Les équations de transport des différentes grandeurs moyennes et fluctuantes en un point et en chaque échelle (bilan d’énergie cinétique) sont réécrites en formulations incompressible et à viscosité variable. Des termes supplémentaires, engendrés par les variations spatio-temporelles de la viscosité, apparaissent dans ces équations. Celles-ci sont utilisées comme outil, afin d’explorer l’écoulement de couche de mélange et d’étudier le développement de la turbulence dans un milieu hétérogène. Les rapports de viscosité simulés sont Rv = [1−18]. Les résultats numériques montrent que l’épaisseur de la zone de mélange δθ évolue plus rapidement lorsque le rapport de viscosité Rv est élevé. De même, les gradients verticaux de la vitesse longitudinale sont amplifiés par les gradients de viscosité, un gain de près de 60%, par rapport aux valeurs initiales, est observé. La production de l’énergie cinétique turbulente est également amplifiée.L’évolution temporelle des fluctuations des vitesse est accélérée, celles-ci sont augmentées de près de 120% par rapport à l’écoulement à viscosité constante. Le régime autosimilaire du tenseur de Reynolds est atteint plus rapidement par l’écoulement à viscosité variable et l’isotropie des fluctuations de vitesse est améliorée. / Since the pioneering work of Brown and Roshko on the effects of density variations within the mixed layer flow, several other theoretical, experimental and numerical studies harnessed to finely investigate this flow. The motivations are of practical order (chemical industry, aerodynamics, combustion. . .) or purely theoretical (the role of coherent structures,secondary instabilities). These studies have focused on, among others, the effects of compressibility and/or variable density. To our knowledge, the effects of viscosity variations in the mixing layer configuration are not discussed in the literature. The objective of this researchis the theoretical and numerical exploration of the variable viscosity temporal mixedlayer flow, especially during its initial phase of development. From a numerical viewpoint, the Navier-Stokes equations are solved in weakly compressible formulation, using the solver CHOC-WAVES, based on WENO scheme. The DNS approach is justified by the absence in the literature of subgrid models that account for the effects of variable viscosity. The transport equations of different mean and fluctuating quantities at a point and each scale (scale-by-scale energy budget) are rewritten in incompressible and variable-viscosity formulation. Additional terms, generated by the spatial and temporal variations of viscosity occur in these equations. These are used as a tool to explore the mixed layer flow and study the development of turbulence in a heterogeneous environment. The simulated viscosity ratios are Rv = [1 − 18]. The numerical results show that the mixing layer thickness δθ growsfaster when the viscosity ratio Rv is high. The vertical gradients of the longitudinal mean velocity are amplified by the viscosity gradients, a gain of almost 60 %, compared to initial values was observed. The production of turbulent kinetic energy is also amplified. The temporal evolution of the velocity fluctuations is accelerated, they are increased to nearly 120 % with respect to the constant viscosity flow. The self-similar regime of the Reynolds tensor is reached more quickly by the variable viscosity flow and the isotropy of the velocity fluctuations is improved.