An adaptive Cartesian grid approach for fluidstructure interactions

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An adaptive Cartesian grid approach for
fluid-structure interactions

• Petter Andreas Berthelsen
• Corner office meeting
• March 29, 2006
• http//www.cesos.ntnu.no/petterab/

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Research

• Dynamical behaviour of a damaged floating vessel
• Modelling of violent, strongly nonlinear, free
  surface - solid body interaction.
• Current work
• Develop a suitable CFD methodology capable of
  solving violent fluid-structure interactions with
  moving boundaries
• Sharp immersed boundary (IB) technique for
  solid boundaries
• Level set method for free surface/interface
  capturing
• This talk focus on the immersed boundary
  technique
• Computational Fluid Dynamics

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Mathematical formulation

• Navier-Stokes equations Incompressible,
  Newtonian fluid
• Assume flow to be laminar

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Numerical method

• Higher order projection method
• Runge-Kutta time-integration
• Trapezoid rule/Heuns method (2nd order)
• Finite difference method on 2D Cartesian grid
• WENO upwind scheme for convective term (3rd-5th
  order)
• Central scheme for viscous term (2nd order)
• Solid boundaries treated by a ghost-cell approach
• Block structured adaptive grid refinement

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Projection method (Euler step)

• Obtain a tentative velocity
• Solve a Poisson Equation for pressure
• Update velocity to satisfy divergence-free
  constraint

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Immersed boundary methods

• Cartesian grid immersed boundary methods
• Avoids the difficulty of generating body-fitted
  grids
• Physical domain intersects with the underlying
  Cartesian grid
• Two categories
• Diffuse methods
• Sharp methods

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Immersed boundary methods

• Diffuse methods
• The effect of boundary is distributed over
  several grid points near the boundary
• Robust and easy to use
• Less accurate
• Sharp methods
• The effect of boundary is only considered at the
  exact location of the boundary
• Complex implementation
• More accurate
• Example
• Ghost cell (finite difference)
• Cut cell (finite volume)

Immersed boundary thickness
Sharp boundary
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Ghost cell approach

• Inactive cells (solid domain) are defined as
  ghost cells
• Standard finite difference schemes can be used
  with ghost cells
• Boundary conditions are enforced by extrapolating
  values from fluid and boundary into ghost cells
• Values can be extrapolated
• normal direction
• Smooth surfaces only
• x- and y-direction
• Suitable for complex geometries

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Numerical example

• Steady, uniform flow past a circular cylinder
• Two cases
• Re 40 (Steady solution)
• Re 100 (Unsteady solution)

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Flow past a cylinder, Re 40

• Cd 1.59 (1.50-1.61)
• L/D 2.26 (2.13-2.35)
• a/D 0.71 (0.71-0.76)
• b/D 0.59 (0.59-0.60)
• q 52.3o (51.2o-55.6o)
• Typical range reported by others

L
a
q
b
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Flow past a cylinder, Re 100

• Cd 1.370.0098 (1.35-1.40, 0.009-0.012)
• Cl 0.333 (0.333-0.339)
• St 0.167 (0.16-0.18)
• Typical range reported by others

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Numerical example

• Oscillating flow past a ship cross section
• KC 3, 7, 12
• Re 200, 400, 1000
• With and without bilge keel
• Experimental data by Sortland (1986)

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Oscillating flow past a ship cross section
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Oscillating flow past a ship cross section
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Moving boundaries

• Moving boundaries across the Cartesian grid
• Allows for elastic boundaries
• Freshly-cleared cells must be updated
• Creates unphysical oscillations in pressure field
• Difficult to obtain correct forces on body
• Moving, overlapping grid
• No oscillation in pressure field
• Rigid bodies only

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Numerical example

• Moving circular cylinder in quiescent fluid in a
  channel with slip walls
• Re 100
• Two cases
• Moving boundary
• Moving overlapping grid

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Moving circular cylinder Pressure field
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Moving circular cylinderDrag force
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Colourful Fluid Dynamics
The end!
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Summary and conclusion

• A Cartesian grid method with immersed boundaries
  is presented
• Can handle complex geometries
• Simulation of flow past fixed bodies gives
  acceptable results
• Work remains for moving boundaries