I. Amidror, Scattered data interpolation methods for electronic imaging systems : A survey, Journal of electronic imaging, vol.11, pp.157-176, 2002.

B. Andreianov, M. Bendahmane, F. Hubert, and S. Krell, On 3D DDFV discretization of gradient and divergence operators. I. Meshing, operators and discrete duality, IMA Journal of Numerical Analysis, vol.32, issue.4, pp.1574-1603, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00355212

, Ansys Fluent Ansys. 14.0 Theory Guide. ANSYS inc, pp.218-221, 2011.

A. Arakawa, Computational design for long-term numerical integration of the equations of fluid motion : two-dimensional incompressible flow, part I, J. Comput. Phys, vol.1, pp.119-143, 1966.

F. Babik, R. Herbin, W. Kheriji, and J. Latché, Discretization of the viscous dissipation term with the MAC scheme, Finite Volumes for Complex Applications VI Problems & Perspectives, pp.571-579, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00576268

E. Bayraktar, , 2016.

E. Bayraktar, O. Mierka, and S. Turek, Benchmark computations of 3D laminar flow around a cylinder with CFX, OpenFOAM and FeatFlow, International Journal of Computational Science and Engineering, vol.7, issue.3, pp.253-266, 2012.

M. Bergmann and A. Iollo, Modeling and simulation of fish-like swimming, Journal of Computational Physics, vol.230, issue.2, pp.329-348, 2011.
URL : https://hal.archives-ouvertes.fr/inria-00546358

E. Bertolazzi and G. Manzini, A cell-centered second-order accurate finite volume method for convection-diffusion problems on unstructured meshes, Mathematical Models and Methods in Applied Sciences, vol.14, pp.1235-1260, 2004.

E. Bertolazzi and G. Manzini, On vertex reconstructions for cell-centered finite volume approximations of 2D anisotropic diffusion problems, Mathematical Models and Methods in Applied Sciences, vol.17, pp.1-32, 2007.

R. Bird, R. C. Armstrong, O. Hassager, and C. F. Curtiss, Dynamics of polymeric liquids, vol.1, 1977.

J. Blazek, Computational fluid dynamics : principles and applications, 2015.

O. Botella, M. Ait-messaoud, A. Pertat, C. Rigal, and Y. Cheny, The LS-STAG immersed boundary method for non-Newtonian flows in irregular geometries : Flow of shear-thinning liquids between eccentric rotating cylinders, Theoretical and Computational Fluid Dynamics, vol.29, pp.93-110, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01417331

O. Botella and Y. Cheny, The LS-STAG method for viscous incompressible flows in irregular geometries : Basics of the discretization and application to viscoelastic flows, In American Society of Mechanical Engineers, Fluids Engineering Division (Publication) FEDSM, vol.1, pp.2441-2451, 2010.

O. Botella, F. Nikfarjam, M. Stoica, and Y. Cheny, Entry flow computations of shearthinning and viscoelastic liquids with the LS-STAG immersed boundary method, Eighth International Conference on Computational Fluid Dynamics (ICCFD8), 2014.

O. Botella, Y. Cheny, F. Nikfarjam, and M. Stoica, Application of the LS-STAG immersed boundary/cut-cell method to viscoelastic flow computations, Communications in Computational Physics, vol.20, issue.4, pp.870-901, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01408974

M. Braack and T. Richter, Solutions of 3D Navier-Stokes benchmark problems with adaptive finite elements. Computers & fluids, vol.35, pp.372-392, 2006.

K. Bühler, Symmetric and asymmetric Taylor vortex flow in spherical gaps, Acta Mechanica, vol.81, issue.1-2, pp.3-38, 1990.

Y. Cheny, La méthode LS-STAG : une nouvelle approche de type frontière immergée/levelset pour la simulation d'écoulements visqueux incompressible en géométries complexes. Application aux fluides Newtonians et viscoélastiques, 2009.

Y. Cheny and O. Botella, The LS-STAG method : A new immersed boundary / level-set method for the computation of incompressible viscous flows in complex moving geometries with good conservation properties, J. Comput. Phys, vol.229, pp.1043-1076, 2010.

N. Pramod, A. G. Chivate, and . Jablokow, Review of surface representations and fitting for reverse engineering, Computer Integrated Manufacturing Systems, vol.8, issue.3, pp.193-204, 1995.

S. Cieslak, S. Ben-khelil, I. Choquet, and A. Merlen, Cut cell strategy for 3-D blast waves numerical simulations, Shock Waves, vol.10, issue.6, pp.421-429, 2001.

D. Clarke, H. A. Hassan, and M. D. Salas, Euler calculations for multielement airfoils using Cartesian grids, AIAA journal, vol.24, issue.3, pp.353-358, 1986.

G. Constantinescu and K. Squires, LES and DES investigations of turbulent flow over a sphere, 38th Aerospace Sciences Meeting and Exhibit, p.540, 1999.

Y. Coudière, J. Vila, and P. Villedieu, Convergence rate of a finite volume scheme for a two dimensional convection-diffusion problem, ESAIM : Mathematical Modelling and Numerical Analysis, vol.33, pp.493-516, 1999.

Y. Coudière and G. Manzini, Benchmark 3D : The cell-centered finite volume method using least squares vertex reconstruction ("diamond scheme"), Finite Volumes for Complex Applications VI Problems & Perspectives, pp.985-992, 2011.

M. Crouzeix and P. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations I. Revue française d'automatique, informatique, recherche opérationnelle, Mathématique, vol.7, issue.3, pp.33-75, 1973.

I. Demird?i?, ?. Lilek, and M. Peri?, A collocated finite volume method for predicting flows at all speeds, International Journal for Numerical Methods in Fluids, vol.16, issue.12, pp.1029-1050, 1993.

J. Droniou and R. Eymard, A mixed finite volume scheme for anisotropic diffusion problems on any grid, Numerische Mathematik, vol.105, issue.1, pp.35-71, 2006.
URL : https://hal.archives-ouvertes.fr/hal-00005565

J. Droniou, R. Eymard, T. Gallouët, C. Guichard, and R. Herbin, The gradient discretisation method, vol.82, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01382358

R. Eymard, T. Gallouet, and R. Herbin, Finite volume methods, Handbook for Numerical Analysis, pp.715-1022, 2000.
URL : https://hal.archives-ouvertes.fr/hal-02100732

. Ea-fadlun, . Verzicco, J. Orlandi, and . Mohd-yusof, Combined immersed-boundary finite-difference methods for three-dimensional complex flow simulations, Journal of computational physics, vol.161, issue.1, pp.35-60, 2000.

D. Robert, U. M. Falgout, and . Yang, Hypre : A library of high performance preconditioners, International Conference on Computational Science, pp.632-641, 2002.

Z. Feng and E. E. Michaelides, Heat transfer in particulate flows with direct numerical simulation (DNS), International Journal of Heat and Mass Transfer, vol.52, issue.3, pp.777-786, 2009.

J. H. Ferziger and M. Peric, Computational Methods for Fluid Dynamics, 1996.

B. Fornberg, Steady viscous flow past a sphere at high Reynolds numbers, Journal of Fluid Mechanics, vol.190, pp.471-489, 1988.

R. Franke, Scattered data interpolation : tests of some methods. Mathematics of computation, vol.38, pp.181-200, 1982.

F. Gibou, R. P. Fedkiw, L. Cheng, and M. Kang, A second-order-accurate symmetric discretization of the Poisson equation on irregular domains, J. Comput. Phys, vol.176, pp.205-227, 2002.

A. Gilmanov, F. Sotiropoulos, and E. Balaras, A general reconstruction algorithm for simulating flows with complex 3D immersed boundaries on Cartesian grids, Journal of Computational Physics, vol.191, issue.2, pp.660-669, 2003.

. Ansys-fluent-user-guide, , 2011.

E. Guyon, Physical hydrodynamics, 2001.

F. H. Harlow and J. E. Welch, Numerical calculation of time-dependent viscous incompressible flow of fluid with free surfaces, Physics of fluids, vol.8, issue.12, pp.2181-2189, 1965.

D. Hartmann, M. Meinke, and W. Schröder, An adaptive multilevel multigrid formulation for Cartesian hierarchical grid methods, Computers & Fluids, vol.37, issue.9, pp.1103-1125, 2008.

D. Hartmann, M. Meinke, and W. Schröder, A strictly conservative Cartesian cut-cell method for compressible viscous flows on adaptive grids, Computer Methods in Applied Mechanics and Engineering, vol.200, issue.9, pp.1038-1052, 2011.

R. Herbin, W. Kheriji, and J. Latche, Staggered schemes for all speed flows, ESAIM : Proceedings, vol.35, pp.122-150, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00636173

C. Hirsch, Numerical computation of internal and external flows : The fundamentals of computational fluid dynamics, 2007.

K. Holmström and J. Petersson, A review of the parameter estimation problem of fitting positive exponential sums to empirical data, Applied Mathematics and Computation, vol.126, issue.1, pp.31-61, 2002.

K. Hormann, Barycentric interpolation, Approximation Theory XIV, pp.197-218, 2013.

F. Jacon and D. D. Knight, A Navier-Stokes algorithm for turbulent flows using an unstructured grid and flux difference splitting, 1994.

J. Volker, Higher order finite element methods and multigrid solvers in a benchmark problem for the 3D Navier-Stokes equations, International Journal for Numerical Methods in Fluids, vol.40, issue.6, pp.775-798, 2002.

T. A. Johnson and V. C. Patel, Flow past a sphere up to a Reynolds number of 300, Journal of Fluid Mechanics, vol.378, pp.19-70, 1999.

. Da, D. B. Jones, and . Clarke, Simulation of flow past a sphere using the Fluent code, Defense Science and Technology Organization, 2008.

D. Kim and H. Choi, Laminar flow past a sphere rotating in the streamwise direction, Journal of Fluid Mechanics, vol.461, pp.365-386, 2002.

M. P. Kirkpatrick, S. W. Armfield, and J. H. Kent, A representation of curved boundaries for the solution of the Navier-Stokes equations on a staggered three-dimensional Cartesian grid, Journal of Computational Physics, vol.184, pp.1-36, 2003.

W. Xiu-quan-lu, W. X. Cai, Y. Ma, W. Shi-wu, and . Xu, Interpolation Methods used for one-way Fluid Structure Interaction of hydrodynamic coupling, In Applied Mechanics and Materials, vol.513, pp.4298-4301, 2014.

M. Mahloul, A. Mahamdia, and M. Kristiawan, The spherical Taylor-Couette flow
URL : https://hal.archives-ouvertes.fr/hal-01343235

, European Journal of Mechanics-B/Fluids, vol.59, pp.1-6, 2016.

. Nn-mansour, . Moin, J. H. Reynolds, and . Ferziger, Improved methods for large-eddy simulations of turbulence, Symposium on Turbulent Shear Flows, vol.1, p.14, 1977.

P. S. Marcus and L. S. Tuckerman, Simulation of flow between concentric rotating spheres. Part 1. Steady states, Journal of Fluid Mechanics, vol.185, pp.1-30, 1987.

S. Marella, S. Krishnan, H. Liu, and H. S. Udaykumar, Sharp interface Cartesian grid method I : an easily implemented technique for 3D moving boundary computations, Journal of Computational Physics, vol.210, issue.1, pp.1-31, 2005.

L. Martinelli, Calculations of viscous flows with a multigrid method, 1987.

L. Martinelli and A. Jameson, Validation of a multigrid method for the Reynolds averaged equations, 1988.

S. R. Mathur and J. Y. Murthy, A pressure-based method for unstructured meshes, Numerical Heat Transfer, vol.31, pp.195-215, 1997.

. The-mathworks and . Inc, MATLAB version 9, 2019.

C. Mimeau, G. Cottet, and I. Mortazavi, Direct numerical simulations of threedimensional flows past obstacles with a vortex penalization method, Computers & Fluids, vol.136, pp.331-347, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01855265

R. Mittal and G. Iaccarino, Immersed boundary methods, Annu. Rev. Fluid Mech, vol.37, pp.239-261, 2005.

R. Mittal, H. Dong, M. Bozkurttas, F. M. Najjar, A. Vargas et al., A versatile sharp interface immersed boundary method for incompressible flows with complex boundaries, Journal of computational physics, vol.227, issue.10, pp.4825-4852, 2008.

Y. Morinishi, T. S. Lund, O. V. Vasilyev, and P. Moin, Fully conservative higher order finite difference schemes for incompressible flow, J. Comput. Phys, vol.143, pp.90-124, 1998.

F. Moukalled, L. Mangani, and M. Darwish, The Finite Volume Method in Computational Fluid Dynamics, 2016.

B. Muralidharan and S. Menon, A high-order adaptive Cartesian cut-cell method for simulation of compressible viscous flow over immersed bodies, Journal of Computational Physics, vol.321, pp.342-368, 2016.

F. Nicoud, Conservative high-order finite-difference schemes for low-Mach number flows, Journal of Computational Physics, vol.158, pp.71-97, 2000.
URL : https://hal.archives-ouvertes.fr/hal-00910303

F. Nikfarjam, Y. Cheny, and O. Botella, The LS-STAG immersed boundary/cut-cell method for non-Newtonian flows in 3D extruded geometries, Computer Physics Communications, vol.226, pp.67-80, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01874475

F. Nikfarjam, Extension de la méthode LS-STAG de type frontière immergée/cut-cell aux géométries 3D extrudées : applications aux écoulements newtoniens et non newtoniens, 2018.

H. Oertel, Fundamental equations of fluid mechanics, Prandtl-Essentials of Fluid Mechanics, pp.265-343, 2010.

S. Osher and R. P. Fedkiw, Level set methods : An overview and some recent results, Journal of Computational Physics, vol.169, pp.463-502, 2001.

S. Osher and R. P. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, 2003.

S. Osher and J. A. Sethian, Fronts propagating with curvature dependent speed : Algorithms based on Hamilton-Jacobi formulations, Journal of Computational Physics, vol.79, pp.12-49, 1988.

. Jr, A. Pacheco, T. Pacheco-vega, R. E. Rodi?, and . Peck, Numerical simulations of heat transfer and fluid flow problems using an immersed-boundary finite-volume method on nonstaggered grids, Numerical Heat Transfer, vol.48, issue.1, pp.1-24, 2005.

. Charles-s-peskin, The immersed boundary method. Acta numerica, vol.11, pp.479-517, 2002.

C. S. Peskin, Flow patterns around heart valves : a numerical method, Journal of computational physics, vol.10, pp.252-271, 1972.

V. Puzikova and . Marchevsky, Application of the LS-STAG immersed boundary method for numerical simulation in coupled aeroelastic problems, Proceedings of the 11th World Congress on Computational Mechanics (WCCM XI, pp.20-25, 2014.

V. Puzikova and . Marchevsky, Extension of the LS-STAG cut-cell immersed boundary method for RANS-based turbulence models, Proceedings of the International Summer School-Conference, 2014.

V. Valeria and . Puzikova, On generalization of the LS-STAG immersed boundary method for Large Eddy Simulation and Detached Eddy Simulation, Advanced Problems in Mechanics International Summer School-Conference (Inst. for Problems in Mechanical Engineering, pp.411-417, 2015.

M. Schäfer, S. Turek, F. Durst, E. Krause, and R. Rannacher, Benchmark computations of laminar flow around a cylinder, Flow simulation with high-performance computers II, pp.547-566, 1996.

H. Jung, R. Seo, and . Mittal, A sharp-interface immersed boundary method with improved mass conservation and reduced spurious pressure oscillations, Journal of computational physics, vol.230, issue.19, pp.7347-7363, 2011.

D. Shepard, A two-dimensional interpolation function for irregularly-spaced data, Proceedings of the 1968 23rd ACM national conference, pp.517-524, 1968.

F. Sotiropoulos and X. Yang, Immersed boundary methods for simulating fluidstructure interaction, Progress in Aerospace Sciences, vol.65, pp.1-21, 2014.

K. Takizawa, T. Yabe, Y. Tsugawa, E. Tayfun, H. Tezduyar et al., Computation of free-surface flows and fluid-object interactions with the cip method based on adaptive meshless soroban grids, Computational Mechanics, vol.40, issue.1, pp.167-183, 2007.

A. Tomboulides and S. Orszag, Numerical investigation of transitional and weak turbulent flow past a sphere, Journal of Fluid Mechanics, vol.416, pp.45-73, 2000.

H. J. Lubbertus-van-der-heiden, Modelling viscous effects in offshore flow problems : A numerical study, 2019.

P. Van-der-plas, . Van-der-heiden, . Veldman, R. Luppes, and . Verstappen, Efficiently simulating viscous flow effects by means of regularization turbulence modeling and local grid refinement, Seventh International Conference on Computational Fluid Dynamics (ICCFD7), p.2503, 2012.

J. Van-kan, C. Vuik, and P. Wesseling, Fast pressure calculation for 2D and 3D time dependent incompressible flow, Numer. Linear Algebra Appl, vol.7, pp.429-447, 2000.

O. V. Vasilyev, High order finite difference schemes on non-uniform meshes with good conservation properties, Journal of Computational Physics, vol.157, pp.746-761, 2000.

R. W. Verstappen and A. E. Veldman, Symmetry-preserving discretization of turbulent flow, J. Comput. Phys, vol.187, pp.343-368, 2003.

R. Verstappen and . Veldman, Preserving symmetry in convection-diffusion schemes, Turbulent flow computation, pp.75-100, 2002.

H. Wendland, Scattered data approximation, vol.17, 2004.

J. Wu and G. M. Faeth, Sphere wakes in still surroundings at intermediate Reynolds numbers, AIAA journal, vol.31, issue.8, pp.1448-1455, 1993.

G. Yang, D. M. Causon, and D. M. Ingram, Calculation of compressible flows about complex moving geometries using a three-dimensional Cartesian cut cell method, International Journal for Numerical Methods in Fluids, vol.33, issue.8, pp.1121-1151, 2000.

J. Yang, An embedded-boundary formulation for large-eddy simulation of turbulent flows interacting with moving boundaries, 2005.

T. Ye, R. Mittal, H. S. Udaykumar, and W. Shyy, Résumé La méthode LS-STAG est une méthode cartésienne pour le calcul d'écoulements incompressibles en géométries complexes, qui propose une discrétisation précise des équations de Navier-Stokes dans les cut-cells, cellules polyédriques de forme complexe créées par l'intersection du maillage cartésien avec la frontière du solide immergé. Originalement développée pour les géométries 2D, où seuls trois types de cut-cells génériques sont présents, son extension aux géométries 3D se, J. Comput. Phys, vol.156, pp.209-240, 1999.

L. Récemment and . Méthode, Cette étude a notamment souligné deux points à élucider pour le développement d'une méthode totalement 3D : premièrement, le calcul des flux diffusifs par un simple schéma à deux points s'est révélé insuffisamment précis dans les cut-cells 3D-extrudées du fait de la non orthogonalité. Ensuite, l'implémentation de ces flux à la paroi, qui s'effectue en imposant une discrétisation distincte pour chaque type de cut-cell extrudée, se révèle trop complexe pour être étendue avec succès aux nombreux types supplémentaires de cut-cells 3D, STAG a été étendue aux géométries complexes 3D dont la frontière est parallèle à l'un des axes du repère cartésien, où sont uniquement présentes les contreparties extrudées des cut-cells 2D

T. Dans-cette, En outre, les schémas diamants ont permis de revisiter intégralement la discrétisation du tenseur des contraintes des équations de Navier-Stokes, où disparaît le traitement au cas par cas selon la disposition de la frontière solide dans les cut-cells. Cela a permis d'aboutir à une discrétisation systématique, précise et algorithmiquement efficace pour les écoulements en géométries totalement 3D. La validation numérique de la méthode LS-STAG avec schémas diamants est présentée pour une série de cas tests en géométries complexes 2D et 3D. Sa précision est d'abord évaluée par comparaison avec des solutions analytiques en 2D, puis en 3D par la simulation d'un écoulement de Stokes entre deux sphères concentriques. La robustesse de la méthode est notamment mise en évidence par l'étude d'écoulements autour d'une sphère en rotation, le premier point est résolu en utilisant l'outil des schémas diamants, d'abord étudié en 2D pour l'équation de la chaleur puis les équations de Navier-Stokes dans l'approximation de Boussinesq, puis étendu en 3D

. Mots-clés, Mécanique des fluides numérique, Méthode de cut-cell, Discrétisation de gradient, p.3