A compressible fluid particle model based on the Voronoi ...

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How can we incorporate in the <strong>model</strong> the dissipative dynamics in order to keep as close as possible to the continuum Navier Stokes equations? dv ∂v = ∑ dt ∂R i i j j dV ∂v m = p + i j i dt j ∂Ri j dS i Ti dt dR dt i Irreversible dynamics = = 0 + V i -A e ∑ ? i. V j ? We requiere expressions for the second derivatives, ij ij dv = v∇⋅v dt dv 2 ⎛ d −2 ⎞ m =− v∇P+ ηv ∇ ∇ v+ ⎜η + ζ⎟v ∇(∇ ∇(∇⋅v) dt ⎝ d ⎠ dS 2 T = κv∇∇ ∇ ∇T+ 2 ηv ∇ ∇v: ∇ ∇ v+ ζv(∇ (∇ (∇⋅v) dt ⎣ ⎦ ∑ v ⎣ ⎦ ⎡ f⎤ ⎣ ∇∇ ∇∇ ⎦ ≈− ∑ f 11 ⎡ f ⎤ =− A ⎡ ij ij f ⎤ ⎣ =− ∇i ⎦ ∑ ijeij e i v ⎣ ⎦ Divergence ⎡ theorem ∇∇ ∇∇f⎤ A ⎡ ∇ ∇f⎤ fij i i i j i j ij ij i 1 vi j Aij ee ij ij Rij ij ≈ 2 R ij e ij
Spatial derivatives SDPD/Voronoi “Soft” volume SDPD rc vi i e ij ( { } ) ∂v ∂ R j v j 1 ∑≠ i j i Volume [ ] [ ( ) ] µ ν ' µ ν µ ∇ ∇ f i= −∑ v jWij d + 2 eij eij −δ ij fij j “Sharp” volume Ri VORONOI Volume v R =∫ χ ( r) dr ( { } ) v0 i i ∂v ⎛ c i ij e ⎞ ⎛ ij cij e ⎞ ij =− Aij⎜ + + δ ij Aij − ⎜ ⎟ j Rij2 ⎟ ∑ ⎜ ⎟ ∂R ⎜ i≠j Rij 2 ⎟ ⎝ ⎠ ⎝ ⎠ 1 ⎛ cij e ⎞ ij ∇f i= − ∑ A ⎜ ij − ⎟ v ⎜ i j i R ⎟ ≠ ⎝ ij 2 ⎠ 1 Aij ⎡ f ⎤ ⎣ ∇∇ ∇∇ ⎦ ≈− ∑ ee f i v R [ ] ( 2 2 ) ' ∇ f i= − vi fi + v j f j Wije [ ] ij fij v 1 v i ⎛ ⎜W ⎝ ( { R } ) k = ∑ j W i Rk ' ' = ij eij −δ ij∑W 2 jk j vi k ≠i r c 1 ( R − R ) Español, Revenga, Phys. Rev. E, 2003 i e jk ⎞ ⎟ ⎠ j i j ij ij ij ij Serrano, Physica A, 2005 i
Page 1 and 2: A compressible <st
Page 3 and 4: Outline 1. Motivacion: Lagrangian t
Page 5 and 6: SPH Discretization of continuum hyd
Page 7 and 8: Mesoscopic fluid <
Page 11 and 12: m i dVi dt This dynamics can be und
Page 13 and 14: Voronoi Ri Aij Rij Rj cij eij lim
Page 15: Outline 1. Motivacion: Lagrangian t
Page 19 and 20: 2 m i Ex ( ) = ∑ + ε ( mi, Si, v
Page 21 and 22: Including thermal fluctuations ⎛
Page 23 and 24: y Comparing the performance of SDPD
Page 25 and 26: Stationary velocity profiles Disord
Page 29: Conclusions -The discrete m
Page 33 and 34: 2d implementation Structure of the
Page 35 and 36: OLD TOPOLOGY Fik Fil v1 i j v2 Time
Page 37 and 38: Properties of the non dissipative V
Page 39: SDPD Overlapping Input theory s=3 s

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