Open boundary conditions for SPH representing free surface flows A - PowerPoint PPT Presentation

Open boundary conditions for SPH representing free surface flows A big (?) step to handle SPH-Boussinesq cou- pling Christophe Kassiotis School of MACE, University of Manchester e Paris-Est (´ EDF R&D, ´ LHSV, Universit´ Ecole des Ponts ParisTech, CETMEF) SPH Meeting – June 1 st , 2010

Outline 1 Introduction & Context 2 First attempt: imposing velocity/pressure Description Examples 3 Second attempt: boundary for SWE Description Example: steady flow over a bump 4 Conclusion C. Kassiotis 2 / 21 OBC for SPH

Introduction Waves model (Boussinesq, Saint-Venant) Unable to represent complex free surface (multi-connected domains) Can represent sloshing with damping [Benoit 02, Yu 99] Studies shows the necessity of more physical models [Duthyk 10] Sloshing representation (VOF, SPH) Waves damping (can be handled by ad-hoc treatment) Computational coast (3D computations un-reachable) Coupling is a natural choice C. Kassiotis 3 / 21 OBC for SPH

Wave propagation Boussinesq model – results C. Kassiotis 4 / 21 OBC for SPH

Breaking wave Using SPH model Usual boundary condition: solid wavemaker We want to avoid it and implement Open Boundary Condition C. Kassiotis 5 / 21 OBC for SPH

Outline 1 Introduction & Context 2 First attempt: imposing velocity/pressure Description Examples 3 Second attempt: boundary for SWE Description Example: steady flow over a bump 4 Conclusion C. Kassiotis 6 / 21 OBC for SPH

Open boundary condition Using a buffer zone To complete particle kernels Where we can impose pressure/velocity Handle deleting/creating particles Problems No-free surface at the boundary If other quantities should be imposed? C. Kassiotis 7 / 21 OBC for SPH

Open boundary condition Using a buffer zone To complete particle kernels Where we can impose pressure/velocity Handle deleting/creating particles Problems No-free surface at the boundary If other quantities should be imposed? Riemman invariant C. Kassiotis 7 / 21 OBC for SPH

Open boundary condition C. Kassiotis 8 / 21 OBC for SPH

Outline 1 Introduction & Context 2 First attempt: imposing velocity/pressure Description Examples 3 Second attempt: boundary for SWE Description Example: steady flow over a bump 4 Conclusion C. Kassiotis 9 / 21 OBC for SPH

Open boundary condition based on shallow water equations [Vacondio et al 10] Shallow water equations D t ρ h = − ρ h ∇ · v D t v = − h ∇ ρ h + g ( ∇ b + S f ) C. Kassiotis 10 / 21 OBC for SPH

Open boundary condition based on shallow water equations Shallow water equation boundary conditions Inflow v · n out < 0 � Subcritical � v � < gh Outflow v · n out > 0 � Supercritical � v � > gh Use Riemann invariants at OB according to local Froude number C. Kassiotis 11 / 21 OBC for SPH

Open boundary condition based on shallow water equations Shallow water equation boundary conditions Inflow v · n out < 0 � Subcritical � v � < gh Outflow v · n out > 0 � Supercritical � v � > gh Use Riemann invariants at OB according to local Froude number Subcritical outflow: h is imposed √ v 1 = v i , 1 + 2 √ g �� � h i − h v 3 = 0 C. Kassiotis 11 / 21 OBC for SPH

Open boundary condition based on shallow water equations Shallow water equation boundary conditions Inflow v · n out < 0 � Subcritical � v � < gh Outflow v · n out > 0 � Supercritical � v � > gh Use Riemann invariants at OB according to local Froude number Subcritical inflow: v is imposed � 2 � 1 � h = 2 √ g ( v i , 1 − v 1 ) + h i C. Kassiotis 11 / 21 OBC for SPH

Open boundary condition based on shallow water equations Shallow water equation boundary conditions Inflow v · n out < 0 � Subcritical � v � < gh Outflow v · n out > 0 � Supercritical � v � > gh Use Riemann invariants at OB according to local Froude number Supercritical outflow: v 1 = v i , 1 h = h i C. Kassiotis 11 / 21 OBC for SPH

Open boundary condition based on shallow water equations Shallow water equation boundary conditions Inflow v · n out < 0 � Subcritical � v � < gh Outflow v · n out > 0 � Supercritical � v � > gh Use Riemann invariants at OB according to local Froude number Supercritical inflow: v is imposed h is imposed C. Kassiotis 11 / 21 OBC for SPH

OBC based on shallow water equations Compute SWE unknowns with SPH Compute water depth using at x = ( x 1 , x 3 ): ρ j W ( x − x j ) m j � ρ ( x 1 , x 3 ) = ρ j j Water for x 3 ∈ [ b , h ] with ρ ( x 1 , x 3 ) > αρ w Here α = 0 . 5 Mean velocity computed as: � h 1 v 1 ( x 1 ) = v 1 ( x 1 , x 3 )d x 3 h − b b C. Kassiotis 12 / 21 OBC for SPH

OBC based on shallow water equations Compute SWE unknowns with SPH Compute water depth using at x = ( x 1 , x 3 ): ρ j W ( x − x j ) m j � ρ ( x 1 , x 3 ) = ρ j j Water for x 3 ∈ [ b , h ] with ρ ( x 1 , x 3 ) > αρ w Here α = 0 . 5 Mean velocity computed as: h − b � h ∆ x 3 1 1 � v 1 ( x 1 ) = v 1 ( x 1 , x 3 )d x 3 ≃ v 1 ( x 1 , k ∆ x 3 )∆ x 3 h − b h − b b k =1 C. Kassiotis 12 / 21 OBC for SPH

OBC based on shallow water equations Compute SWE unknowns with SPH 0.75 0.5 Height ( m ) 0.25 0 0 200 400 600 800 1000 1200 Density ( kg . m − 3 ) C. Kassiotis 13 / 21 OBC for SPH

OBC based on shallow water equations Remarks Initialize buffer zone Generate initial state using normal SphysicGEN OBC localization and default state in obc.dat Create buffer particle Delete out of bounds particle Buffer zone behaviour Values imposed (for kernel completeness) Delete fluid particle entering buffer zone Delete and create particles Deleted particles stored in trash Generate a particle using trash if possible C. Kassiotis 14 / 21 OBC for SPH

Water over a bump Problem description Problem used in [Vacondio et al , 10] to validate SPH-SWE Fluid domain x 1 ∈ [0 m , 10 m ] Fluid bottom 1 − ( x 1 − 5) 2  � � if x 1 ∈ [3 m , 8 m ] b 0  b ( x 1 ) = 4 0 elsewhere  with b 0 = 20 cm Analytical solution Subcritical or supercritical case C. Kassiotis 15 / 21 OBC for SPH

Water over a bump Problem description Problem used in [Vacondio et al , 10] to validate SPH-SWE Fluid domain x 1 ∈ [0 m , 10 m ] Fluid bottom 1 − ( x 1 − 5) 2  � � if x 1 ∈ [3 m , 8 m ] b 0  b ( x 1 ) = 4 0 elsewhere  with b 0 = 20 cm Analytical solution Subcritical or supercritical case C. Kassiotis 15 / 21 OBC for SPH

Water over a bump Large overview C. Kassiotis 16 / 21 OBC for SPH

Water over a bump Zoom on inlet C. Kassiotis 17 / 21 OBC for SPH

Water over a bump Zoom on outlet C. Kassiotis 18 / 21 OBC for SPH

Outline 1 Introduction & Context 2 First attempt: imposing velocity/pressure Description Examples 3 Second attempt: boundary for SWE Description Example: steady flow over a bump 4 Conclusion C. Kassiotis 19 / 21 OBC for SPH

Conclusions and Outlooks Conclusion: Open Boundary Condition based on Shallow Water Equations Use of Riemann invariants Problems: If the flow does not verify SWE hypothesis? (open question) Automatic boundary condition switching (easy) Accurate computation of water depth (less easy) Outlook Implementing SPH component Coupling with Boussinesq solver C. Kassiotis 20 / 21 OBC for SPH

Conclusions and Outlooks Thank you for attention C. Kassiotis 21 / 21 OBC for SPH