Development of an Incompressible SPH Method through SPARTACUS-2D Eun-Sug Lee E.Lee-2@postgrad.manchester.ac.uk D. Laurence, C. Moulinec, P. Stansby, D. Violeau, Developing of truly incompressible method in SPH 1 / 28
1. Introduction: The need of Incompressible SPH method 2. Theory of Incompressible SPH method 3. Boundary conditions for pressure 4. ISPH applications - Dam breaking - Bluff body - Lid Driven Cavity Flow Developing of truly incompressible method in SPH 2 / 28
Scheme Limitations of the of configuration weakly compressible SPH Periodic condition in x-direction h = 0.01m, <u> = 0.0002 m s -1, Re = 10 Developing of truly incompressible method in SPH 3 / 28
By increasing the numerical speed of sound Developing of truly incompressible method in SPH 4 / 28
Pressure field p a γ ρ = B 0 1 ρa c B = ρ γ 2 0 0 γ = 7 Developing of truly incompressible method in SPH 5 / 28
Theory of Incompressible SPH method Developing of truly incompressible method in SPH 6 / 28
Du Dt u = 0 1 ρ p ν 2 = + u + F e Incompressibility is solved by projection method u * u t n ν 2 n = u + u F e u t n+ 1 * 1 = p ρ Developing of truly incompressible method in SPH 7 / 28 u 1 = u p t ρ n+ 1 * n+ 1 n+ 1
t ρ n+ 1 * n+ 1 u = u = p 0 t u = ρ * n 1 p + 1 n+ 1 u p = ρ t = constant * 1 dρ 0 ρ dt + u = 1 n+ 1 ρ0 ρ* p = 2 ρ* ρ* t Developing of truly incompressible method in SPH 8 / 28
! Exact Projection m m = a b ρb b c ( B A) B A w ( ) w ( ) r r c c b h bc a h ab c ρc Approximate Projection ρ B + ρ B A r r a 2 b ρaρb rab a a b b ab ( B A) m w ( ) b ab a h ab Developing of truly incompressible method in SPH 9 / 28
"#$% ISPH : Incompressible SPH WCSPH: Weakly Compressible SPH Method Pressure CFL condition Density ISPH Semi-implicit Pressure Poisson equation Maximal fluid velocity Constant WCSPH Fully explicit State equation Numerical speed of sound 1% fluctuation Developing of truly incompressible method in SPH 10 / 28
Study of Boundary conditions Developing of truly incompressible method in SPH 11 / 28
! # Open channel p = 0 at surface (Dirichlet BC) dp/dn = 0 at the wall (Neumann BC) Use of dummy particles Developing of truly incompressible method in SPH 12 / 28
&'!# 1 n+ 1 u p = ρ t * Lp = B L; Laplacian B; Source term Initial condition p = 0 B = -1 ; fluid particles B = 0 ; edge and dummy particles Developing of truly incompressible method in SPH 13 / 28
##$(''! # Boundary: p = 0 Developing of truly incompressible method in SPH 14 / 28
##&''! # Extended domain in axial direction (1 by 4) Boundary: p = 0 (top, bottom), dp/dn=0 (left, right) p edge = p fluid Developing of truly incompressible method in SPH 15 / 28
! # - top, bottom p = 0 - left, right dp/dn=0 Fictitious point M in the domain m p p w r = ( ) b M b h ab b ρb dp dn p a p M p M p a Developing of truly incompressible method in SPH 16 / 28
##&''! # Developing of truly incompressible method in SPH 17 / 28
##)&''! # B=0 for edge particles, -1 for fluid particles Boundary: p = 0 (top, bottom), Lp = 0 (left, right) Developing of truly incompressible method in SPH 18 / 28
##*&''! # B=0 for edge particles, -1 for fluid particles Boundary: p = 0 (top, bottom), Lp = -1 (left, right) Developing of truly incompressible method in SPH 19 / 28
Applications: - dam breaking - bluff body - lid driven cavity flow Developing of truly incompressible method in SPH 20 / 28
##+!'!,- ' ## ## Developing of truly incompressible method in SPH 21 / 28
##+!'!,- ##!./# WCSPH (left), Smoothed WCSPH (centre), ISPH (right) Developing of truly incompressible method in SPH 22 / 28
##+!'!,- ##!.0*# WCSPH (left), Smoothed WCSPH (centre), ISPH (right) Developing of truly incompressible method in SPH 23 / 28
! # Constrains of using dummy particles for pressure Developing of truly incompressible method in SPH 24 / 28
## ' ## ## Re=10, physical time = 31min. Developing of truly incompressible method in SPH 25 / 28
##)+1+ "%!"23 Weakly compressible SPH (left) Incompressible SPH (right) Developing of truly incompressible method in SPH 26 / 28
2! '!,# Conclusions Smoother pressure field in Incompressible SPH Numerical stability increased and the code is more robust Future work Improve iteration solver (preconditioning..) Numerical study at boundaries Apply the incompressibility to the wave set-up on an idealised coral reef case Developing of truly incompressible method in SPH 27 / 28
2 3, 3!"#!!#! Scheme of the configuration based on Gourlay (1996) (unit: m) Developing of truly incompressible method in SPH 28 / 28