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  • Numerical solution of 2D unsteady integral boundary layer equations with

    a discontinuous Galerkin method

    H. zdemir

    European Conference on High Order Nonlinear Numerical Methods for Evolutionary PDEs: Theory and Applications, HONOM 2011, April 11-15, 2011, University of Trento and

    CIRM-FBK, Trento, Italy

    Mei 2005

  • www.ecn.nl

    Numerical solution of 2D unsteady integral boundary layer

    equations with a discontinuous Galerkin method

    Hseyin zdemir

  • EFMC8 - 16.09.2010 HONOM 2011 11.04.2011 2/20

    Introduction: motivation Detailed wind turbine dynamics simulation

    Local aerodynamic forces, structural stresses and deformations

    Introduction Governing equations Numerical method Results Conclusions

    127 m

    85 87 89 91 93 95 97 99 01 03 05 07

    .05 . 3 . 5 2 4.5 5 6 MW

    33 m

    A380

    Airbus

    Van Kuik, 2007

  • EFMC8 - 16.09.2010 HONOM 2011 11.04.2011 3/20

    Engineering tools: not accurate enough

    - 3D, steady state methods: Blade Element,

    Momentum (BEM), Vortex line method (AWSM),

    - 2D, steady state methods: XFOIL, RFOIL

    CFD tools (CFX): too expensive, too much time

    - Axial-symmetric (1/3rd of the domain)

    - 2.7 M elements

    - ~2 weeks on 16 node cluster

    Introduction: available tools

    Introduction Governing equations Numerical method Results Conclusions

  • EFMC8 - 16.09.2010 HONOM 2011 11.04.2011 4/20

    Van Dyke, An Album of Fluid Motion,

    11th ed., Parabolic Press, 2007, USA

    Introduction: approach

    Introduction Governing equations Numerical method Results Conclusions

    Integral boundary layer method (IBL) + Panel method + Strong

    quasi-simultaneous viscous inviscid interaction

    Navier-Stokes equations

    Boundary layer equations

    Integral boundary layer equations

  • EFMC8 - 16.09.2010 HONOM 2011 11.04.2011 5/20

    Governing equations

    Introduction Governing equations Numerical method Results Conclusions

    2D, unsteady integral boundary layer eqns: Closure set:

    Shape factors

    Friction coefficient

    Viscous diffusion coefficient

    Slip velocity

    Shear stress coefficient

    i.e.:

  • EFMC8 - 16.09.2010 HONOM 2011 11.04.2011 6/20

    Governing equations: some analysis

    Eigenvalues and + for laminar closure relations:

    system is hyperbolic

    Introduction Governing equations Numerical method Results Conclusions

    Separation point becomes negative

    Closure relations for the H dependent variables

  • EFMC8 - 16.09.2010 HONOM 2011 11.04.2011 7/20

    Introduction Governing equations Numerical method Results Conclusions

    Numerical method: Discontinuous Galerkin (DG) Method

    Weak formulation

    Solution vector:

    Approximate solution

    Discrete equation

  • EFMC8 - 16.09.2010 HONOM 2011 11.04.2011 8/20

    Introduction Governing equations Numerical method Results Conclusions

    Numerical method: DG Method

    Local Lax-Friedrich flux formula:

    Explicit multi-stage Runge-Kutta time integration: [Cockburn & Shu]

    Slope limiter: [Cockburn & Shu]

  • EFMC8 - 16.09.2010 HONOM 2011 11.04.2011 9/20

    Introduction Governing equations Numerical method Results Conclusions

    Results model problem: transport equation, continuous initial condition

    periodic b.c.

    x = 1/25

  • EFMC8 - 16.09.2010 HONOM 2011 11.04.2011 10/20

    Introduction Governing equations Numerical method Results Conclusions

    Results

    model problem: accuracy

  • EFMC8 - 16.09.2010 HONOM 2011 11.04.2011

    Introduction Governing equations Numerical method Results Conclusions

    Results

    model problem: transport equation, discontinuous initial condition

    - periodic b.c. - slope limiter applied

    11/20

  • EFMC8 - 16.09.2010 HONOM 2011 11.04.2011

    Introduction Governing equations Numerical method Results Conclusions

    Results

    model problem: Burgers equation

    12/20

    t = 0.4 t = 1/

    x = 1/160

  • EFMC8 - 16.09.2010 HONOM 2011 11.04.2011 13/20

    Introduction Governing equations Numerical method Results Conclusions

    Results: IBL equations

    Laminar flow over a flat plate, Re=1e5

  • EFMC8 - 16.09.2010 HONOM 2011 11.04.2011 14/20

    Introduction Governing equations Numerical method Results Conclusions

    Turbulent flow over a flat plate, Re=1e7

    Results: IBL equations

  • EFMC8 - 16.09.2010 HONOM 2011 11.04.2011 15/20

    Introduction Governing equations Numerical method Results Conclusions

    Results: IBL equations

    Laminar and turbulent flows over NACA profiles

    Prescribed edge velocity Ue (extracted from XFOIL) Dirichlet boundary conditions used Initial condition is set Only the suction side is considered Converging to a steady state problem NACA0009 and NACA0012 profiles are used

  • EFMC8 - 16.09.2010 HONOM 2011 11.04.2011 16/20

    Introduction Governing equations Numerical method Results Conclusions

    Results: IBL equations Laminar flow over NACA0009 profile, Re=1e5

  • EFMC8 - 16.09.2010 HONOM 2011 11.04.2011 17/20

    Introduction Governing equations Numerical method Results Conclusions

    Results: IBL equations Turbulent flow over NACA0012 profile, Re=1e5

  • EFMC8 - 16.09.2010 HONOM 2011 11.04.2011 18/20

    Introduction Governing equations Numerical method Results Conclusions

    Results: unsteady simulation Laminar flow over a flat plate, Re=1e5,

    with small perturbation in time Ue(x,t)

  • EFMC8 - 16.09.2010 HONOM 2011 11.04.2011 19/20

    Introduction Governing equations Numerical method Results Conclusions

    Results: unsteady simulation Laminar flow over a cylinder

    ,

  • EFMC8 - 16.09.2010 HONOM 2011 11.04.2011 20/20

    Fully laminar and turbulent flows over a flat plate

    show good agreement with the literature

    Flow over NACA profiles are in good agreement

    up to separation point

    Non-conservative implementation

    TVBM slope limiter will be implemented

    More experimental data needed for 3D unsteady

    boundary layers and also for rotational effects

    Introduction Governing equations Numerical method Results Conclusions

    Conclusions and Outlook

  • EFMC8 - 16.09.2010 HONOM 2011 11.04.2011

    Referenses

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