1 / 23

No-Slip Boundary Conditions in Smoothed Particle Hydrodynamics

No-Slip Boundary Conditions in Smoothed Particle Hydrodynamics. by Frank Bierbrauer. Updating Fluid Variables. In SPH fluid variables f are updated through interpolation about a given point (x a ,y a ) using information from surrounding points (x b ,y b ) .

fauve
Download Presentation

No-Slip Boundary Conditions in Smoothed Particle Hydrodynamics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. No-Slip Boundary Conditions in Smoothed Particle Hydrodynamics by Frank Bierbrauer

  2. 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.

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

  4. 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

  5. 1 2 3

  6. Velocity Update Using Image Particles • Fixed image approach: uI = uF+(1+dI /dF)(uW - uF) 2. Virtual image approach: uI = uV+(1+dI /dV)(uW - uV) • Virtual velocities uV are created through interpolation

  7. 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

  8. At the no-Slip Wall (W) Navier-Stokes Equations Finite-Difference Approximation at the wall

  9. 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

  10. The Viscoelastic Case The equations are (a,b = 1,2) where

  11. Further Reduction Using giving

  12. At the Wall As well as

  13. Non-Newtonian (elastic) Stress Only have the velocity condition uW = (U0,0) as well asry=0

  14. Must Solve • Need ub’ and vb’andrW • Needas well as St and • e.g.

  15. Density Update Equation

  16. Polymeric Stress Update Equations

  17. Velocity Update Equations

  18. Solution for ub’ and vb’

  19. If rx = 0, l21 = 1, h = m, Sab =0

  20. Equivalent Newtonian Update Equations

  21. Giving

More Related