Numerical Methods in Aerodynamics. Fluid Structure Interaction. Lecture 4: Fluid Structure Interaction

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1 Fluid Structure Interaction Niels N. Sørensen Professor MSO, Ph.D. Department of Civil Engineering, Alborg University & Wind Energy Department, Risø National Laboratory Technical University of Denmark 1

2 Outline of lecture Introduction to moving geometries Moving frame Derivation of the governing equations Implementation issues Moving mesh Derivation of the governing equations Implementation issues Structural Models Coupling Strategies Parallelization and Timestep Considerations Applications 2

3 Examples of moving geometries Non-deforming geometries Airplanes, Submarines cars, trains Pitching/plunging airfoils/bridge decks, rotors Turbo machinery Deforming geometries Airfoil with moving flap, rotor during pitching motion Deforming bridge section/turbine blades, high rise buildings Free surface flows, arterial flows 3

4 Geometries in motion x v 0 x v 0 y y x v 0 y 4

5 Frame of reference Non-inertial frame of reference (Moving Frame) Additional acceleration terms, changed boundary conditions Can be used both steady/unsteady Don t need re-computations of geometrical quantities Need s careful treatment of source term implementation Don t allow deformation of the geometry Inertial frame of reference (Moving Mesh) Mesh fluxes, changed boundary conditions The standard formulation is unsteady or transient Allows deformation of the geometry Need re-computations of metrics (geometrical quantities) 5

6 Reynolds Averaged Navier-Stokes equations (RANS) 6

7 Moving Frame of Reference 7

8 Computing the velocity in the moving frame (1) Taking the time derivative of the position vector we get Which can be determined as 8

9 Computing the velocity in the moving frame (1) Taking the time derivative of the position vector we get Which can be determined as 9

10 Computing the velocity in the moving frame (1) Taking the time derivative of the position vector we get Which can be determined as 10

11 Velocity seen from moving frame The velocity in the moving coordinate frame can now be expressed by inserting the derived expressions and rearranging 11

12 Components of moving velocity The cartesian components of the velocity can be determined by the inner products 12

13 Components of moving velocity x-component 13

14 Acceleration in moving coordinate frame 14

15 Moving Frame Implementation Issues The moving frame method implies that: The boundary conditions for the velocities are time dependent That additional time dependent volume sources arises The problem is fully specified by: The inflow velocity in the Initial system The position of the origin of the coordinate system as function of time, either prescribed or found as part of the solution The rotation of the moving coordinate frame 15

16 Moving Mesh The governing equation for the moving mesh option can be derived by applying a generalized version of the Leibnitz rule to the Navier- Stokes equations Leibnitz rule: 16

17 1D continuity for moving mesh 17

18 1D continuity 18

19 Navier-Stokes equations 19

20 Geometrical Conservation Law How can the mesh fluxed be determined in order to assure that no artificial mass sources are genrated Discrete version of the continuity equation 20

21 21

22 Geometrical conservation law 22

23 Moving Mesh Implementation Issues The moving mesh method implies that: Transient computations are performed Additional terms in the form of mesh fluxes arises in the convective terms The problem is fully specified by: The inflow velocity in the Initial system The position of the vertices of the mesh as function of time The vertex positions could for a non-deforming case be given in the same fashion as for the Moving Frame method Alternatively the vertex position could be determined by some additional equations. (deforming geometry, free surface flows) 23

24 Application issues Time step restrictions Moving Frame The method can be used both steady and unsteady. The expressions for the velocities at the boundaries and the accelerations terms are in both cases given by the analytical expressions for the prescribed motion Moving Mesh The M.M. method is in the standard form unsteady The mesh fluxes are computed from the discrete location of the vertices, and the approximation will therefore depend on the timestep 24

25 Test cases Rotating mesh/frame over stagnant fluid Constant translatoric movement of mesh over stagnant fluid Mesh moving with free stream velocity Deforming mesh over flow with constant velocity Moving airfoil/airfoil with inflow 25

26 Mesh types Moving frame Structured single/multi block meshes Unstructured meshes Moving Mesh Structured single/multi block meshes Unstructured meshes Chimera/overlapping meshes Dynamic remeshing Cartesian cut cells 26

27 Typical aeroelastic computation Forced motion (2D) Perform forced oscillations at different reduced frequencies Study the influence on the aerodynamic damping Forced motion (3D) Derive mode shapes from structural models Force the structure to move/vibrate in the mode shapes Compute the aerodynamic work/damping Determine critical wind speeds Fully coupled simulations Time simulations at performed at selected conditions (2D/3D), that are found from analysis using simpler models or forced motion Static aeroelasticity 27

28 Structural equations, two dimensions Time stepping Newmark Runge Kutta Crank-Nicolson 28

29 Structural model HAWC/HAWC2 3/N sub-structures Timoshenko beams 6 DOFs per node Bending - torsion Nonlinear kinematics Linear elasticity Typically nodes Newmark integration scheme 29

30 Fluid forces Flow solver EllipSys Navier-Stokes solver (2D/3D) General non-orthogonal curvilinear coordinats Incompressible fluid Turbulence modeling (k-omega RANS/DES) Flow field simulation in complex geometries Second order accurate in time and space Multiblock Parallel implementation 30

31 Coupling of aerodynamics and structure Beam element Aerodynamic force 31

32 Coupling procedure Conventional serial staggered (CSS) procedure Fluid Structure tn 4 tn+1 Only first order time accurate 32

33 Coupling procedure Improved Serial Staggered (ISS) procedure Fluid tn-1/2 1 2 tn+1/2 3 Structure 4 Second order time accurate 33

34 Parallelization issues Fluid simulation may often have ~50 million DOF Needs large scale computer Parallelization of fluid problem essential Structural model for typical turbine ~1000 DOF Can be solved on standard PC The same problem can be solved simultaneously on all processors Typical approach Compute the structural problem (full problem on all processors) Deform computational mesh on each processor Computed flow problem using domain decomposition Assemble the fluid forces from all processors (AllGather) Redo the loop 34

35 Timestep considerations for rotor computation NS-computations needs in the order of > 1000 time steps per rev. Speed of sound ~ 330 m/s Max tip speed < 0.2 c ~ 70 m/s (To avoid compressibility effects) Typical rotational speed for D=120 m rotor ~ 5 sec/revolution Typical tip chord 0.5 meter Δt = Chord U N Tip TP 0.001s Time integration procedure Number of timesteps per revolution

36 Numerical Instabilities Often models without structural damping is used Vibrations may result Add structural damping Use dissipative numerical method Structural damping 36

37 Computational grid Regenerate the mesh after the deformation Control is needed to assure that the new mesh is not to far from the old May be time consuming for 3D simulations To exploit the full potential some kind of interpolation procedure is needed Deformation of mesh Can be done using simple algebraic approach Can not accommodate all kind of deformation Is efficient for 3D domains Overlapping mesh Valve motion the first step is shown in cyan (light blue) the second is overlayed in magenta, and grey denotes no change to the mesh between the steps 37

38 Static aeroelasticity of airplane wing 38

39 2D dynamic stall of pitching airfoil 2π θ ( t) = θ 0 + θ Ampl sin( t + ϕθ ) T θ Computation of aerodynamic work dw dw = M ( t) dα = Fy ( t) dy 39

40 Results, 2 DOF classical flutter (2) 40

41 Rotor Tower Interaction (F. Zahle) 41

42 Today's lecture The difference between moving frame/mesh How to derive the additional terms for the two formulations Which mesh approaches couples naturally with the different formulations A series of applications of the methods The method of determining the aerodynamic damping from forced motion of the structure The main components of fluid structure have been introduced Coupling, grid deformation, and numerical instabilities have been discussed Finally a few applications have been shown 42

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