Discussion panel 1: Enforcing incompressibility in SPH

Transcription

1 Discussion panel 1: Enforcing incompressibility in SPH Weakly compressible or not weakly compressible, this is the question. Introduction: Andrea Colagrossi Chairman : Antonio Souto Iglesias,

2 Physical/Mathematical modelling of liquid phases: It is well known that entrophy has a large influence on the pressure field for gaseous phases while it is generally negligible for liquids(at least for pressure regime less than 1 GPa). Generally the liquid can be treated as a barotropicfluid, which means that pressure p and internal energy e are both singlevalued functions of density. This implies also that the energy equation is decoupled from the other governing equations. p = f ( ρ, S); ρ p = 221 atm 22 MPa; 0.01 ρ

3 Physical/Mathematical modelling of liquid phases: A) weakly-compressible p = f ( ρ) ρ 10 ρ 2 c p =, Ma ρ = U c Explicit scheme «easy» to implement, dependency from the speed of sound adopted. B) Incompressible div u p L u u 2 ( ) = 0 = ρ (, ) The momentum equation becomes a Poisson equation for the pressure. An algebraic system has to be solved numerically.

4 N.S. Solution in the limit of Ma 0 C.-D. Munz, S. Roller, R. Klein, K.J. Geratz, The extension of incompressible flow solvers to the weakly compressible regime, Computers & Fluids, Volume 32, Issue 2, February 2003, Pages JungH. Seo, Young J. Moon, Linearizedperturbedcompressibleequationsforlow Mach number aeroacoustics, Journal of Computational Physics, Volume 218, Issue 2, 1 November 2006, Pages , Incompressible Linear Acoustic Weakly- Compressible

5 Getting incompressible from weakly-compressible Artificial compressibility A. J. Chorin(1967) Steady N.S. Pseudo compressibility C.L. Merkle& M. Athavale, (1987). Unsteady N.S. Weaklycompressibility G.L. Browning etal. (1989), Incompressible oceanographic flows. J.J. Monaghan, (1994), Simulating free surface flows with SPH. P.A. Madsen, H.A. Schäffer, A discussion of artificial compressibility, Coastal Engineering, Volume 53, Issue 1, Pages 93-98, (2006)

6 B) Incompressible div u p ρl u u 2 ( ) = 0 = (, ) S. J. Cummins, M. Rudman, An SPH Projection Method, Journal of Computational Physics, Volume 152, Issue 2, July (1999) Koshizuka S., Tamako H., Oka Y. A Particle Method for Incompressible Viscous Flow with Fluid Fragmentation, Computational Fluid Dynamics Journal, Vol. 4, No. 1, 29 46, (1995). MPS = SPH A. Souto-Iglesias, F. Macià, L. M. González, J.L.Cercos-Pita, On the consistencyofmps, Computer PhysicsCommunications, Volume 184, Issue3, March 2013, Pages

7 B) Incompressible div u p ρl u u 2 ( ) = 0 = (, ) Koshizuka S., Tamako H., Oka Y. A Particle Method for Incompressible Viscous Flow with Fluid Fragmentation, Computational Fluid Dynamics Journal, Vol. 4, No. 1, 29 46, (1995). S. J. Cummins, M. Rudman, An SPH Projection Method, Journal of Computational Physics, Volume 152, Issue 2, July (1999) E. Y.M. Lo, S. Shao, Simulation of near-shore solitary wave mechanics by an incompressible SPH method, Applied Ocean Research, Volume 24, Issue 5, pp , October (2002)

8 1. Hu, X. Y., and NikolausA. Adams. An incompressible multi-phase SPH method. Journal of computational physics (2007): E.-S. Lee, C. Moulinec, R. Xu, D. Violeau, D. Laurence, P. Stansby, Comparisons of weaklycompressibleand trulyincompressiblealgorithmsforthe SPH meshfree particle method, Journal of Computational Physics, Volume 227, Issue 18, 10 September 2008, Pages Khayyer, A., H. Gotoh, and S. D. Shao. Corrected incompressible SPH method for accurate water-surface tracking in breaking waves. Coastal Engineering 55.3 (2008): Solenthaler, Barbara, and Renato Pajarola. Predictive-correctiveincompressibleSPH. ACM transactions on graphics(tog). Vol. 28. No. 3. ACM, Xu, Rui, Peter Stansby, and Dominique Laurence. "Accuracy and stability in incompressible SPH (ISPH) based on the projection method and a new approach." Journal of Computational Physics (2009): Hughes, Jason P., and David I. Graham. Comparison of incompressible and weaklycompressible SPH models for free-surface water flows. Journal of Hydraulic Research 48.S1 (2010): Shadloo, MostafaSafdari, et al. A robust weakly compressible SPH method and its comparison with an incompressible SPH, International Journal for Numerical Methods in Engineering 89.8 (2012):

9 The three myths of the weakly compressible 1. Use of a stiff equation of state 2. Small time steps because of the speed of sound 3. Pressure solution disturbed by the acoustic waves p( ) K γ 0 ρ = ρ P p ρ 0 2 γ c ρ 0 ρ ( ) = 1 γ ρ 0 For water this EoS is called Stiffned EoS (γ=6.1-7 and c0= 1497 m/s) weakly compressible regime ρ 2 1 = ε 10 ρ0 p( ) 2 c ρ = c γ γ ρ ε ρ 0 0ε γ playsa negligible role!!!

10 The three myths of the weakly compressible 1. Use of a stiff equation of state 2. Small time step because of the speed of sound 3. Pressure solution disturbed by the acoustic waves Wendland C2 50 neighbours (2D) 2ε - Radius Support R-K 4 th order ISPH MPS, Poisson Solver needs a number of iterations!!!

11 The three myths of the weakly compressible 1. Use of a stiff equation of state 2. Small time step because of the speed of sound 3. Pressure solution disturbed by the acoustic waves wc-sph Pressure!!!

12 WCCH (Weakly-Compressible Cartesian Hydrodynamics) Features: Finite volume method Basedon a Cartesian grid Incompressible flow treated as weakly-compressible Fully explicit High-order spatial scheme (WENO 5) Adaptive Mesh Refinement (AMR) Massively parallel (distributed memory) Arbitrary-complex geometries Multi-fluid interfaces

13 Weakly Compressible SPH, need of numerical diffusion Riemann-SPH (SPH-Flow Solver) LS-FVM (χ-navis Solver)

14 Free-Surface flow 1. Weakly-compressible SPH, the dynamic boundary condition is implicitly enforced (in a weak sense). Colagrossi, Andrea, Matteo Antuono, and David Le Touzé. Theoretical considerations on the freesurfacerolein the smoothed-particle-hydrodynamicsmodel. PhysicalReview E 79.5 (2009): Colagrossi, A., Antuono, M., Souto-Iglesias, A., & Le Touzé, D. (2011). Theoreticalanalysisand numerical verification of the consistency of viscous smoothed-particle-hydrodynamics formulations in simulating free-surface flows. Physical Review E, 84(2), (2011) Incompressible-SPH, free-surface particles need to be detected for enforcing the dynamic boundary How the Poisson solver is affected by the detection free-surface particle algorithm? For highly violent flows leading to 3D complex fragmented fluid domain, the Poisson solver may requires a large number of iterations to reach the convergence?

15 The issue of the Volume Conservation 1. Weakly-Compressible SPH, the mass conservation is ensured but not the volume one V i 1 = W 2. Incompressible SPH, j ij SPH Kernel does not give a Partition of Unity at discrete level Particle methods that compute a divergence-free velocity field to achieve incompressibility suffer from a volume conservation issue when a finite timestep position update scheme is used Prapanch Nair, Gaurav Tomar, Volume conservation issues in incompressible smoothed particle hydrodynamics, Journal of Computational Physics, Volume 297,(2015), Pages

16 Water Impact Problems: Is it better incompressible or weakly compressible? U U S. Marrone, A. Colagrossi, A. Di Mascio, D. Le Touzé, Prediction of energy losses in water impacts usingincompressibleand weaklycompressiblemodels, Journal offluidsand Structures, Volume 54, April2015

17 Water Impact Problems: Is it better incompressible or weakly compressible? Peregrine, D. H. Water-wave impact on walls. Annual review of fluid mechanics 35.1 (2003): The effect of even a small volume fraction of air in water greatly increases its compressibility. From a fluid dynamic view point this is best demonstrated in the variation of the velocity of sound. At atmospheric pressure, just 1% of air gives a velocity of sound of 120 m/s, and velocities as low as 30 m/s for 20% of air.

18 HPC context: Is it better incompressible or weakly compressible? The cosmological simulation code gadget-2 (2005!!! ) SPH-cosmological SPH simulations total particle numbers of more than 250 million. 144 million particles, 4,096 cores (EU-funded PRACE project 2014) More than 1000 Million particles simulated with DualSPHysics (SPH on Multi-GPU) 2012 Very Large I-SPH simulations???

19 Summarizing: 1. Physical/Mathematical modelling of liquid phases 2. N.S. Solution in the limit of Ma 0 3. Getting incompressible from weakly-compressible 4. Solution of a Pressure Poisson equation in SPH framework 5. The three weakly-compressible myths: EoS, time-steps, pressure noise 6. Role of the free-surface on the two schemes ISPH wcsph 7. Volume Conservation issue 8. Water Impact Problems 9. HPC algorithms for ISPH and wcsph

20 Weakly compressible SPH enforcing no-slip B.C. M. De Leffe, D. Le Touzé, B. Alessandrini, A modified no-slip condition in weakly-compressible SPH, In 6th ERCOFTAC SPHERIC workshop on SPH applications (2011), pp Velocity-divergence operator Reverse the the velocity normal component leaving the tangential one unaltered - Viscous term Reverse the tangential component, leaving the normal one unaltered