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APPLICATION DE LA MÉTHODE DE MAILLAGES DYNAMIQUES POUR LA

PRÉDICTION D’ÉCOULEMENTS AUTOUR D’UN PROFIL D’AILEOSCILLANT DANS LE CONTEXTE DE L’INTERACTION

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

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BiomechanicBlood and breath flows

Civil engineeringFlutter on the Tacoma bridge

(1940)Nuclear engineering : cooling system.

Naval architecture : dykes construction, offshore petroleum platforms.

Naval hydrodynamic : ship hulls conception.

IntroductionIntroduction

Applications

Aeronautical field

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Aeronautical field

IntroductionIntroduction

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)

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IntroductionIntroduction

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

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DiffusionpressureConvection

RGFE

t

q

Re

1

•Unsteady, Viscous, Compressible equation system

•Dimensionless, under strong conservative form

•General, non-orthogonal, curvilinear coordinates system

iep )1(

e

v

uJq

Equations & Numerical SchemesEquations & Numerical Schemes

Navier-Stokes equation

Spatial scheme

Finite Differences

Convective term

Diffusion term Centered differences

Precision O(2)

1Monotonic Upstream Scheme for Conservation

Law

Roe Upwind SchemeMUSCL1 Approach

Temporal scheme

Explicit

Three-Stages Runge-Kutta

Precision O(3)

Uc

Re

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Flow domain configurationFlow domain configuration

Flow parameters :

Re 100005000

M [0.1,0.4] M= 0.4,0.5Incidence 0° variable

Meshes parameters :Structured C-Type grid (2D)

NACA0012 AirfoilInflow and Outer boundariesFree stream conditions

Outflow boundaryFirst order extrapolation for unknown variables

Wake lineAveraging 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

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Dynamic mesh methodDynamic mesh method

Instant t0Instant t0+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

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Geometric conservation lawGeometric conservation law (GCL)(GCL)

Conservative character

Conservative character of continuous equations

Numerical conservation ?

Thomas & Lombard (1979)

2D local form :

Consistent schemeNumerical discretisation of the GCL ?

Injection of a constant solution in the numerical scheme

: Contravariant mesh velocities

p : Roe’s scheme constant1

2 Metrics compatibilityrelations :

Centered, second order derivative

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Mesh actualizationMesh actualization

Spring analogy

Computational mesh movement• Compatibility nodes-walls• Mesh integrity (avoid ill-conditioned cells)

Linear tension springs

: global parameter : local functionOn each node :

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Mesh actualizationMesh actualization

Spring analogy

Torsional springs

Stiffness :

Flat plate oscillation

Iterative solver

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ValidationValidation

Geometric conservation law

Oscillation of a fictitious flat plate

Re=104

M∞=0.5= 2max =+/- 15 °

Constant solution for fluid

Comparison of two simulations

Longitudinal velocity fieldWithout GCL With GCL

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Pitching case

ValidationValidation

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)

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CL, CM coefficients

ValidationValidation

Barakos & Drikakis

Present study

Dynamic stall : 19,3°

Coherent amplitude, hysteresisDifferent stallVortex dynamic

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ValidationValidation

Vortex dynamicStreamlines

Temporal evolution of the Lift coefficient

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ValidationValidation

Dynamic Stall Vortex Convection Velocity

Q-criterion, present study

density contoursBarakos & Drikakis

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ValidationValidation

Pitching Simulation

Vorticity contoursWhite: positive vorticity, black:

negative vorticity

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Dynamic mesh

Conclusions - PerspectivesConclusions - Perspectives

Perspectives

• Others test-cases, experimental datas • Second step … Two degrees of freedomNumerical coupling

Conclusion

• Numerical code using dynamic mesh• Mesh actualization• Independence of physical results on mesh motion (GCL)• Realistic vortex dynamic