Solutions des équations de Navier-Stokes

Le problème de l'existence et de la régularité des solutions des équations de Navier-Stokes constitue l'un des problèmes du prix du millénaire de l'Institut de mathématiques Clay. Les équations de Navier-Stokes décrivent la dynamique des fluides liquides ou gazeux. Leur étude n'a pas permis à ce jour de montrer l'existence de solution régulières dans le cas général. La solution de ce problème peut constituer une étape dans la compréhension des phénomènes de turbulence. Le problème est de prouver ou de trouver un contre-exemple à la proposition suivante qui résume l'énoncé qui porte sur le problème incompressible. Dans un problème temporel en dimension 3 d'espace pour lequel on spécifie une condition initiale, il existe des champs de vitesses et de pression scalaire réguliers qui sont solutions des équations de Navier-Stokes. Problème Pour un milieu incompressible, les équations de Navier-Stokes s'écrivent pour les champs de vitesse V et de pression p :

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Numerical simulation of randomly forced turbulent flows

Michel Deville

Several authors have proposed studying randomly forced turbulent hows (e.g., E. A. Novikov, Soviet Physics JETP, 20(5), 1290 1965). More recently, theoretical investigations (e.g., renormalization group) have focused on whim-noise forced Navier-Stokes equations (V. Yakhot and S. A. Orszag, J.Sci.Comput. 1(1), 3 1986), The present article aims to provide an appropriate numerical method for the simulation of randomly forced turbulent systems. The spatial discretization is based on the classical Fourier spectral method. The time integration is performed by a second-order Runge-Kutta scheme. The consistency of an extension of this scheme to treat additive noise stochastic differential equations is proved. The random number generator is based on lagged Fibonacci series. Results are presented for two randomly forced problems: the Burgers and the incompressible Navier-Stokes equations with a white-noise in time forcing term characterized by a power-law correlation function in spectral space. A variety of statistics are computed for both problems, including the structure functions, The third-order structure functions are compared with their exact expressions in the inertial subrange. The influence of the dissipation mechanism (viscous or hyperviscous) on the inertial subrange is discussed. In particular, probability density functions of velocity increments are computed for the Navier-Stokes simulation, Finally, for both Burgers and Navier-Stokes problems, our results support the view that random sweeping is the dominant effect of the large-scale motion on the small-scales. (C) 1998 Academic Press.

1998

Simulation of benchmark and industrial unsteady compressible turbulent fluid flows

Michele Mossi

Large-eddy simulation (LES) is a very promising technique for the numerical computation of unsteady turbulent flows, and seems to be the perfect tool to simulate the compressible air flow around a high-speed train in a tunnel, providing unsteady results for aerodynamic and aeroacoustic analysis. To look into this possible future application of LES, two major lines of investigation are pursued in this thesis: first, the study of the effective ability of shock-capturing schemes to predict fundamental turbulent phenomena; second, the analysis of the aerodynamic phenomena induced by a high-speed train in a tunnel. The numerical simulation of compressible flows requires the use of shock-capturing schemes. These schemes can be relatively dissipative and mask the subgrid-scale contribution introduced in a large-eddy simulation to account for the unresolved turbulence scales. To estimate their effective dissipation and their ability to resolve turbulence phenomena, shock-capturing schemes widely used for aeronautical applications, from second- to fifth-order space accuracy, are employed here for simulating well-known fundamental flows in subsonic and supersonic regimes. Direct and large-eddy numerical simulations are undertaken for the inviscid and viscous Taylor-Green vortex decay problem, the freely decaying homogeneous and isotropic turbulence, and the fully developed channel flow. For all the turbulent flows investigated, several turbulence statistics are computed and the numerical dissipation of the schemes tested is interpreted in terms of subgrid-scale dissipation in a LES context, yielding an equivalent Smagorinsky or dynamic coefficient. This coefficient is for all schemes of the same order of magnitude as the commonly accepted values in LES for the subgrid-scale term. On the grounds of this analysis and of the comparisons of the turbulence statistics with accurate data obtained in the literature, the addition of explicit subgrid-scale models to the shock-capturing schemes tested is not recommended. It is thus concluded that the use of the LES technique for compressible turbulent flows is not yet suitable for industrial applications. The aerodynamic phenomena generated by a high-speed train travelling in a tunnel are also discussed, their importance on the design of high-speed lies is pointed out, and the analysis tools commonly employed for their study are reviewed. To study numerically the three-dimensional, compressible and turbulent air flow around a high-speed train accelerating in a tunnel, by accounting for the unsteady effects at inlet and outlet boundaries due to the propagation of pressure waves generated at the train departure, new coupling conditions between one-dimensional and three-dimensional domains are developed. These conditions are applied successfully to the numerical simulation of the unsteady wake developing behind two- and three-dimensional vehicles, where the averaged Navier-Stokes equations are solved with the turbulence modelling approach. The influence on the wake of the length of the vehicle tail is also discussed and results of multi-dimensional simulations are compared with one-dimensional data.

EPFL1999

Interaction fluide-structure par la méthode des éléments spectraux

Nicolas Bodard

We elaborate in this thesis the numerical simulation of the fluid-structure interaction by the spectral element method. To this end we consider the Navier-Stokes equations for a viscous Newtonian incompressible fluid with an elastic solid the movement of which being described by the equations of the dynamics. The arbitrary Lagrangian Eulerian (ALE) formulation is introduced in the fluid governing equations to deal with the structure movement that is described in Lagrangian representation. The geometrical motion in each domain is built up by the mesh deformation. The space-time discretization of the full mathematical model rests upon the spectral element method. The solid is discretized in the space of polynomials of degree N, PN, while the fluid uses the PN - PN-2 approach. The efficiency of the ALE formulation is tested and validated through various applications. The full algorithm is based on a particular case of the staggered method. The simplified case of a solid immersed in a plane channel closes the thesis. It is then possible to draw the conclusions about the pros and cons of the proposed methodology.

EPFL2006

Simulation of benchmark and industrial unsteady compressible turbulent fluid flows

Michele Mossi

EPFL1999