1. 1 APPLICATION DE LA MÉTHODE DE MAILLAGES DYNAMIQUES POUR LA PRÉDICTION DÉCOULEMENTS AUTOUR DUN PROFIL DAILE OSCILLANT DANS LE CONTEXTE DE LINTERACTION FLUIDE-STRUCTURE Sébastien Bourdet, Marianna Braza Institut de Mécanique des Fluides de Toulouse, Unité Mixte de Recherche CNRS/INPT UMR N° 5502, Allée du Prof. Camille Soula, 31400 Toulouse GDR 2902, 26-27 Septembre, Sophia-Antipolis

2. 2 Biomechanic Blood and breath flows Civil engineering Flutter on the Tacoma bridge (1940) Nuclear engineering : cooling system. Naval architecture : dykes construction, offshore petroleum platforms. Naval hydrodynamic : ship hulls conception.Introduction Applications Aeronautical field

3. 3 Introduction Flutter phenomenonBuffeting Drag increase Vibrations materials fatigue Reduction of the range of operation Structure destruction Structure enforcement Velocity reduction Dynamic stall Sudden lift loss Manoeuvrability limitation Velocity reduction (helicopter)

4. 4Introduction Unsteady flows Natural unsteadiness Forced unsteadiness Spontaneous development : von Kármán rows alley. Local injection of perturbations. Boundary motion : deformation, pitching, plunging etc. Understanding of unsteady phenomenon. Appearance mechanisms. Major interest

5. 5 Unsteady, Viscous, Compressible equation system Dimensionless, under strong conservative form General, non-orthogonal, curvilinear coordinates system Equations & Numerical Schemes Navier-Stokes equation Spatial scheme Finite Differences Convective term Diffusion term Centered differences Precision O(2) 1 Monotonic Upstream Scheme for Conservation Law Roe Upwind Scheme MUSCL 1 Approach Temporal scheme Explicit Three-Stages Runge-Kutta Precision O(3)

6. 6 Flow domain configuration Flow parameters : Re 100005000 M [0.1,0.4]M = 0.4,0.5 Incidence 0° variable Meshes parameters : Structured C-Type grid (2D) NACA0012 Airfoil Inflow and Outer boundaries Free stream conditions Outflow boundary First order extrapolation for unknown variables Wake line Averaging of variables above and below the wake line Wall Non-slip condition Neumann condition for temperature,density and energy Pressure : Resolution of NS equations with non-slip condition Initial conditions: Uniform fields from inflow conditions

7. 7 Dynamic mesh method Instant t 0 Instant t 0 + t Static mesh Lagrangian or Eulerian formulation Dynamic mesh Generalized formulation Displacement field Continuity equation : J(t) : time dependent Jacobian Equation formulation : Mesh velocity field

8. 8 Geometric conservation law(GCL) Geometric conservation law (GCL) Conservative character of continuous equations Numerical conservation ? Thomas & Lombard (1979) 2D local form : Consistent scheme Numerical discretisation of the GCL ? Injection of a constant solution in the numerical scheme : Contravariant mesh velocities p : Roes scheme constant 1 2 Metrics compatibility relations : Centered, second order derivative

9. 9 Mesh actualization Spring analogy Computational mesh movement Compatibility nodes-walls Mesh integrity (avoid ill-conditioned cells) Linear tension springs : global parameter : local function On each node :

10. 10 Mesh actualization Spring analogy Torsional springs Stiffness : Flat plate oscillation Iterative solver

11. 11Validation Geometric conservation law Oscillation of a fictitious flat plate Re=10 4 M =0.5 = 2 max =+/- 15 ° Constant solution for fluid Comparison of two simulations Longitudinal velocity field Without GCLWith GCL

12. 12 Pitching caseValidation Barakos & Drikakis (1999) No mesh motion Harmonic oscillation of the airfoil : Comparison of lift and moment coefficients Comparison of the Dynamic Stall Vortex (DSV) convection velocity (Guo et al, 1994 ; Chandrasekhara & Carr, 1990)

13. 13 C L, C M coefficientsValidation Barakos & Drikakis Present study Dynamic stall : 19,3° Coherent amplitude, hysteresis Different stall Vortex dynamic

14. 14Validation Vortex dynamic Streamline s Temporal evolution of the Lift coefficient

15. 15Validation Dynamic Stall Vortex Convection Velocity Q-criterion, present study density contours Barakos & Drikakis

16. 16Validation Pitching Simulation Vorticity contours White: positive vorticity, black: negative vorticity

17. 17 Dynamic mesh Conclusions - Perspectives Perspectives Others test-cases, experimental datas Second step … Two degrees of freedom Numerical coupling Conclusion Numerical code using dynamic mesh Mesh actualization Independence of physical results on mesh motion (GCL) Realistic vortex dynamic