No-Slip Boundary Conditions in Smoothed Particle Hydrodynamics

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No-Slip Boundary Conditions in Smoothed Particle
Hydrodynamics

• by
• Frank Bierbrauer

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Updating Fluid Variables

• In SPH fluid variables f are updated through
  interpolation about a given point (xa,ya) using
  information from surrounding points (xb,yb) .
• Each surrounding point is given a weight Wab with
  respect to the distance between point a and b.

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Particle Deficiency

• Near a no-slip boundary there is a particle
  deficiency
• Any interpolation carried out in this region will
  produce an incorrect sum

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Three Ways to Resolve the Particle Deficiency
Problem

• Insert fixed image particles outside the boundary
  a distance dI away from the boundary c.f. nearest
  fluid particle at distance dF
• Insert fixed virtual particles within the fluid
  and in a direct line to the fixed image particles
• Avoids creation of errors when fluid and image
  particles are not aligned
• Co-moving image particles with dI dF

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1 2 3
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Velocity Update Using Image Particles

• Fixed image approach
• uI uF(1dI /dF)(uW - uF)
• 2. Virtual image approach
• uI uV(1dI /dV)(uW - uV)
• Virtual velocities uV are created through
  interpolation

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Velocity Update Using the Navier-Stokes Equations

• Update the velocity using the Navier-Stokes
  equations and a second order finite difference
  approximation to the velocity derivatives

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At the no-Slip Wall (W)
Navier-Stokes Equations
Finite-Difference Approximation at the wall
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Velocity Update
Much of this reduces down as, in general, a
no-slip wall has condition uW(U0,0). Therefore,
at the wall, ut ux uxx v vx vxx 0
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The Viscoelastic Case
The equations are (a,b 1,2)
where
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Further Reduction
Using
giving
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At the Wall
As well as
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Non-Newtonian (elastic) Stress
Only have the velocity condition uW (U0,0) as
well as ry0
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Must Solve

• Need ub and vb and rW
• Need as well as St and
• e.g.

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Density Update Equation
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Polymeric Stress Update Equations
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Velocity Update Equations
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Solution for ub and vb
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If rx 0, l21 1, h m, Sab 0
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Equivalent Newtonian Update Equations
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Giving