Technische Universität München Lehrstuhl für Fluidmechanik - Fachgebiet Gasdynamik Univ. Professor Dr.-Ing.habil. G.H. Schnerr FLM Lehrstuhl für Aerodynamik TUM - Seminar Strömungstechnik Steffen Schmidt, Ismail Sezal, Günter H. Schnerr, Matthias Thalhamer p max =60 bar On the computation of compressible liquid flows with dynamic phase-change: Physical model, numerical challenges and technical relevance

Technische Universität München Lehrstuhl für Fluidmechanik - Fachgebiet Gasdynamik Univ. Professor Dr.-Ing.habil. G.H. Schnerr FLM Outline Motivation - Liquid flows with phase-change: Basic aspects and phenomena Physical model and numerical method - CFD-Tool CATUM, “modified” flux function - Thermodynamic model for water / water vapor Validation, Results - 2-D steady state liquid flow around circular cylinder at M ∞ = D liquid shock tube - 3-D bubble collapse - comparison with Rayleigh-Plesset-Eqn. - 3-D twisted wing - dynamic shedding - 3-D simulation “Obernach experiment” - comparison of erosion areas - 3-D multi-hole injector - wave dynamics and flow development Conclusion & Outlook

Technische Universität München Lehrstuhl für Fluidmechanik - Fachgebiet Gasdynamik Univ. Professor Dr.-Ing.habil. G.H. Schnerr FLM Cavitation - Aspects Typical properties of cavitating flows within hydraulic machinery (fluid water): u ref = O(10) m/s,p ref = O(1) bar, except for fuel injection systems -> p ref = O(100) bar, ρ ref = O(1000) kg/m³,T ref = O(300) K, c ref = O(1000) m/s, p sat (T ref )=0.023 bar. M ref = – 0.01  Low Mach number flow as long as no vapor content exists! „Cavitation“ is the flow-induced evaporation/recondensation of a liquid  No external heat addition (boiling…)! Definition: Cavitation number σ:

Technische Universität München Lehrstuhl für Fluidmechanik - Fachgebiet Gasdynamik Univ. Professor Dr.-Ing.habil. G.H. Schnerr FLM Cavitation phenomena Cavitation erosion Vortex cavitation Bubble and cloud cavitation Sheet and cloud cavitation Supercavitation

Technische Universität München Lehrstuhl für Fluidmechanik - Fachgebiet Gasdynamik Univ. Professor Dr.-Ing.habil. G.H. Schnerr FLM Outline Motivation - Cavitating flows: Aspects and phenomena Physical model and numerical method - CFD-Tool CATUM, “modified” flux function - Thermodynamic model for water / water vapor Validation, Results - 2-D steady state liquid flow around circular cylinder at M ∞ = D liquid shock tube - 3-D bubble collapse - comparison with Rayleigh-Plesset-Eqn. - 3-D twisted wing - dynamic shedding - 3-D simulation “Obernach experiment” - comparison of erosion areas - 3-D multi-hole injector - wave dynamics and flow development Conclusion & Outlook

Technische Universität München Lehrstuhl für Fluidmechanik - Fachgebiet Gasdynamik Univ. Professor Dr.-Ing.habil. G.H. Schnerr FLM Numerical method - CATUM To solve the balance laws of mass, momentum and energy for compressible flows: - Diffusive fluxes: central discretization  not used here - Turbulence models: k-ω, EASM, ω-RSM  not used here - Semi-implicit time integration for source terms (turbulence models)  not used here No subiterations required - in contrast to pressure based approaches, therefore low cost per time step. Well suited to simulate hydrodynamic and wave dynamic features with time steps down to nanoseconds. - 3-D unsplit finite volume method (semi-discrete) - Convective fluxes: (density based solution strategy)  modified flux function + TVB (WENO-3) / TVD (VanLeer) - Explicit 4-stage Runge-Kutta scheme (2nd order accuracy, enlarged stability region)

Technische Universität München Lehrstuhl für Fluidmechanik - Fachgebiet Gasdynamik Univ. Professor Dr.-Ing.habil. G.H. Schnerr FLM „Classical“ flux function Euler equations (1-D): Approximation of the states * at the shared surface of the cells L,R – „classical“:  Solve local Riemann problem between adjacent cells!

Technische Universität München Lehrstuhl für Fluidmechanik - Fachgebiet Gasdynamik Univ. Professor Dr.-Ing.habil. G.H. Schnerr FLM „Classical“ flux function Euler equations (1-D): Approximation of the states * at the shared surface of the cells L,R – „classical“: define numerical flux:

Technische Universität München Lehrstuhl für Fluidmechanik - Fachgebiet Gasdynamik Univ. Professor Dr.-Ing.habil. G.H. Schnerr FLM Consistency of the „Classical“ flux function for M  0? Investigate incompressible 2-D potential flow around circular cylinder and apply the „classical“ pressure flux… Potential solution along cylinder wall - S: Approximate average velocity differences: Apply „classical“ formula for p * : S

Technische Universität München Lehrstuhl für Fluidmechanik - Fachgebiet Gasdynamik Univ. Professor Dr.-Ing.habil. G.H. Schnerr FLM „Classical“ flux function Euler equations (1-D): Approximation of the states * at the shared surface of the cells L,R – „classical“: Inconsistent for M  0

Technische Universität München Lehrstuhl für Fluidmechanik - Fachgebiet Gasdynamik Univ. Professor Dr.-Ing.habil. G.H. Schnerr FLM „Modified“ flux function Euler equations (1-D): Approximation of the states * at the shared surface of the cells L,R – „modified“: Consistent for M  0

Technische Universität München Lehrstuhl für Fluidmechanik - Fachgebiet Gasdynamik Univ. Professor Dr.-Ing.habil. G.H. Schnerr FLM Outline Motivation - Cavitating flows: Aspects and phenomena Physical model and numerical method - CFD-Tool CATUM, “modified” flux function - Thermodynamic model for water / water vapor Validation, Results - 2-D steady state liquid flow around circular cylinder at M ∞ = D liquid shock tube - 3-D bubble collapse - comparison with Rayleigh-Plesset-Eqn. - 3-D twisted wing - dynamic shedding - 3-D simulation “Obernach experiment” - comparison of erosion areas - 3-D multi-hole injector - wave dynamics and flow development Conclusion & Outlook

Technische Universität München Lehrstuhl für Fluidmechanik - Fachgebiet Gasdynamik Univ. Professor Dr.-Ing.habil. G.H. Schnerr FLM Instead of modelling small scale structures we investigate average thermodynamic properties: Conservative averaging (filtering) leads to: Two-phase flow properties via integral averages per cell Consider stable thermodynamic conditions only:  constitutive relations (EOS) determine cell variables p, T

Technische Universität München Lehrstuhl für Fluidmechanik - Fachgebiet Gasdynamik Univ. Professor Dr.-Ing.habil. G.H. Schnerr FLM - Equation of state for liquid water: modified Tait EOS (thermal and caloric EOS for pure liquids) - EOS of pure water vapour: perfect gas law (thermal and caloric description of pure vapour) For water: B ≈ 3.3 ∙10 8 Pa, n ≈ 7.15, reference state ref.: expected mean temperature ( K). - EOS for saturated water/vapour: saturation conditions (conditions for saturated mixture of water and water vapour for a void fraction α  Here: Oldenbourg-polynomials) Thermodynamic model - EOS

Technische Universität München Lehrstuhl für Fluidmechanik - Fachgebiet Gasdynamik Univ. Professor Dr.-Ing.habil. G.H. Schnerr FLM Combined EOS contains relations for: - pure water → modified Tait equation - vapor phase → ideal gas law - two-phase region→ saturation conditions Thermodynamic model - EOS p sat (T sat ) T sat

Technische Universität München Lehrstuhl für Fluidmechanik - Fachgebiet Gasdynamik Univ. Professor Dr.-Ing.habil. G.H. Schnerr FLM Outline Motivation - Cavitating flows: Aspects and phenomena Physical model and numerical method - CFD-Tool CATUM, “modified” flux function - Thermodynamic model for water / water vapor Validation, Results - 2-D steady state liquid flow around circular cylinder at M ∞ = D liquid shock tube - 3-D bubble collapse - comparison with Rayleigh-Plesset-Eqn. - 3-D twisted wing - dynamic shedding - 3-D simulation “Obernach experiment” - comparison of erosion areas - 3-D multi-hole injector - wave dynamics and flow development Conclusion & Outlook

Technische Universität München Lehrstuhl für Fluidmechanik - Fachgebiet Gasdynamik Univ. Professor Dr.-Ing.habil. G.H. Schnerr FLM Numerical results – validation: single phase flow Case 1a: 2-D steady state liquid flow around circular cylinder at M ∞ =10 -4, grid 128 x 32 cells. Pressure coefficient c p – isolines; Drag coefficient c D,p =1.5·  NO Low Mach number problem!

Technische Universität München Lehrstuhl für Fluidmechanik - Fachgebiet Gasdynamik Univ. Professor Dr.-Ing.habil. G.H. Schnerr FLM Numerical results – validation: single phase flow Case 1b: 1-D liquid shock tube, grid 100 cells. - No oscillations, time accurate wave propagation. - High pressure difference but (only) weakly nonlinear behavior (shock Mach number M S ≈1.17).

Technische Universität München Lehrstuhl für Fluidmechanik - Fachgebiet Gasdynamik Univ. Professor Dr.-Ing.habil. G.H. Schnerr FLM Bubble radius R 0 =0.4 mm, time step Δt CFD =6.5·10 -9 s, collapse time 37·10 -6 s, initial pressures p liquid =1.0 bar, p bubble =0.023 bar, T=20° C, liquid: water, bubble: water vapor. Numerical results – validation: two-phase flow Case 2a: 3-D bubble collapse – comparison with Rayleigh-Plesset solution

Technische Universität München Lehrstuhl für Fluidmechanik - Fachgebiet Gasdynamik Univ. Professor Dr.-Ing.habil. G.H. Schnerr FLM Reference: C.E. Brennen: „Cavitation and Bubble Dynamics“ Simulation CATUM – top view Initial radius R 0 =0.5 mm, time step Δt CFD =6.0·10 -9 s, collapse time: 17·10 -6 s, Δt Movie =24.0·10 -6 s. Pressures: p liquid =10.0 bar, p bubble =0.023 bar, T=20°C, water/vapor. Numerical results – validation: two-phase flow Case 2b: 3-D bubble collapse with wall interaction

Technische Universität München Lehrstuhl für Fluidmechanik - Fachgebiet Gasdynamik Univ. Professor Dr.-Ing.habil. G.H. Schnerr FLM Numerical results – sheet and cloud cavitation Case 3a: 3-D simulation of the “Foeth-experiment: Twist NACA 0009” (TU Delft) 3·10 5 cells (one half of the domain) 3 ·10 6 time steps with Δt CFD =4.5·10 -8 s

Technische Universität München Lehrstuhl für Fluidmechanik - Fachgebiet Gasdynamik Univ. Professor Dr.-Ing.habil. G.H. Schnerr FLM Numerical results – sheet and cloud cavitation Case 3a: Cloud shedding and cloud collapse Dynamic shedding, blue iso-surfaces indicate α≥5%, Δt Movie =3.5·10 -2 s. Cloud collapse and shock formation, Δt Movie =4.6·10 -4 s.

Technische Universität München Lehrstuhl für Fluidmechanik - Fachgebiet Gasdynamik Univ. Professor Dr.-Ing.habil. G.H. Schnerr FLM Numerical results – sheet and cloud cavitation Experiment, E.J. Foeth et al., TU Delft Simulation CATUM.

Technische Universität München Lehrstuhl für Fluidmechanik - Fachgebiet Gasdynamik Univ. Professor Dr.-Ing.habil. G.H. Schnerr FLM 0,85 m 0,3 m u in =11 m/s T in =300 K σ ref =1.8 P out,mix =1.12 bar 3.1·10 6 cells 10 6 time steps, Δt ≈ 3·10 -7 s 64 CPU (SGI AltixBx2)  240 h. Numerical results – application: erosive two-phase flow Case 3b: 3-D simulation of the “Obernach-experiment” on cavitation erosion

Technische Universität München Lehrstuhl für Fluidmechanik - Fachgebiet Gasdynamik Univ. Professor Dr.-Ing.habil. G.H. Schnerr FLM Perspective view: Two-phase regions and static pressure at the walls, Δt Movie =0.17 s. Top view: Two-phase regions, Δt Movie =0.17 s. Numerical results – application: erosive two-phase flow Case 3b: Dynamic phase-transition and related pressure field

Technische Universität München Lehrstuhl für Fluidmechanik - Fachgebiet Gasdynamik Univ. Professor Dr.-Ing.habil. G.H. Schnerr FLM Experiment: Huber R., Geschwindigkeitsmaßstabseffekte bei der Kavitationserosion in der Scherschicht nach prismatischen Kavitatoren, Berichte des Lehrstuhls und der Versuchsanstalt für Wasserbau und Wasserwirtschaft, Hrsg. Univ.-Prof. Dr.-Ing. Th. Strobl, Nr. 102, Simulation CATUM: Isosurfaces α=0.01, one instant in time. Numerical results – application: erosive two-phase flow Case 3b: Comparison of two-phase structures experiment/simulation

Technische Universität München Lehrstuhl für Fluidmechanik - Fachgebiet Gasdynamik Univ. Professor Dr.-Ing.habil. G.H. Schnerr FLM p [bar] > Numerical results – application: erosive two-phase flow Case 3b: Fragmentation of two-phase structure, collapse, shock formation 12 3 p max = 65 bar Δt 1  2 =1.17·10 -4 s Δt 2  3 =0.58·10 -4 s

Technische Universität München Lehrstuhl für Fluidmechanik - Fachgebiet Gasdynamik Univ. Professor Dr.-Ing.habil. G.H. Schnerr FLM Experiment: Huber R., Geschwindigkeitsmaßstabseffekte bei der Kavitationserosion in der Scherschicht nach prismatischen Kavitatoren, Berichte des Lehrstuhls und der Versuchsanstalt für Wasserbau und Wasserwirtschaft, Hrsg. Univ.-Prof. Dr.-Ing. Th. Strobl, Nr. 102, Simulation CATUM: Collapse induced maximum pressure at the bottom wall of the numerical test-section, analysis interval seconds. Stars indicate the barycenters (experimental) of the erosion ares. Numerical results – application: erosive two-phase flow Case 3b: Areas of intense erosion (experiment) - maximum pressures (simulation)

Technische Universität München Lehrstuhl für Fluidmechanik - Fachgebiet Gasdynamik Univ. Professor Dr.-Ing.habil. G.H. Schnerr FLM 3-D 6-hole injector, needle position at maximum lift. Computational domain, 4∙10 5 cells. p in = 600 bar p out = 26 bar d = 0.2 mm Numerical results – application: fuel injector Case 3c: 6-hole injector – CAD model and computational grid

Technische Universität München Lehrstuhl für Fluidmechanik - Fachgebiet Gasdynamik Univ. Professor Dr.-Ing.habil. G.H. Schnerr FLM t ≥ s t = 5.6·10 -6 s p in = 600 bar, p out = 26 bar Movie, s Numerical results – application: fuel injector

Technische Universität München Lehrstuhl für Fluidmechanik - Fachgebiet Gasdynamik Univ. Professor Dr.-Ing.habil. G.H. Schnerr FLM Dominating behavior during: ≤ t ≤10 -4 s, p in = 600 bar, p out = 26 bar, T init = 333 K, Δt CFD = s. Numerical results – application: fuel injector Case 3c: Cavitation pattern, fragmentation and collapse - p in = 600 bar, p out = 26 bar.

Technische Universität München Lehrstuhl für Fluidmechanik - Fachgebiet Gasdynamik Univ. Professor Dr.-Ing.habil. G.H. Schnerr FLM t ≥ s t = 5.6·10 -6 s p in = 600 bar, p out = 26 bar Numerical results – application: fuel injector Movie pressure, s Movie void frac., s

Technische Universität München Lehrstuhl für Fluidmechanik - Fachgebiet Gasdynamik Univ. Professor Dr.-Ing.habil. G.H. Schnerr FLM Simulation CATUM. t ≥ s. p in = 600 bar, p out = 26 bar, T init = 333 K, Δt CFD = s. Experiment R. Busch 2001 (*). p in = 600 bar, p out = 20 bar (*) Busch, R., ‘Untersuchung von Kavitationsphänomenen in Dieseleinspritzdüsen’, Ph.D. Thesis, University of Hannover, Numerical results – application: fuel injector Case 3c: Steady state cavitation pattern - comparison with experiment of Busch (*)

Technische Universität München Lehrstuhl für Fluidmechanik - Fachgebiet Gasdynamik Univ. Professor Dr.-Ing.habil. G.H. Schnerr FLM Simulation CATUM t ≥ s, p in = 600 bar, p out = 26 bar, T init = 333 K, Δt CFD = s. Experiment R. Busch p in = 600 bar, p out = 1 bar α = Numerical results – application: fuel injector Case 3c: Steady state cavitation pattern - comparison with experiment of Busch (*)

Technische Universität München Lehrstuhl für Fluidmechanik - Fachgebiet Gasdynamik Univ. Professor Dr.-Ing.habil. G.H. Schnerr FLM p in (bar) (g s -1 ) (%) Numerical results – application: fuel injector Case 3c: Mass flow defect due to cavitation, p out = 26 bar = const. Investigation of mass flow rates of various inlet pressures 100 ≤ p in ≤ 1400 bar.

Technische Universität München Lehrstuhl für Fluidmechanik - Fachgebiet Gasdynamik Univ. Professor Dr.-Ing.habil. G.H. Schnerr FLM Conclusion and outlook - CFD-Tool CATUM enables the simulation of compressible cavitating liquid flows including the formation and propagation of collapse induced shocks. - Modified flux function is consistent for M  0 and allows for stable integration of the governing equations if the time integration scheme contains parts of the imaginary axis. - Thermodynamic modelling of phase transition provides reasonable results:  no empirical parameters (nuclei concentration, bubble radius distribution) required! - Effects of non-condensable gas content within the liquid fluid are currently investigated. - Turbulence modelling of compressible two-phase flows …

Technische Universität München Lehrstuhl für Fluidmechanik - Fachgebiet Gasdynamik Univ. Professor Dr.-Ing.habil. G.H. Schnerr FLM Thank you for your attention!

Technische Universität München Lehrstuhl für Fluidmechanik - Fachgebiet Gasdynamik Univ. Professor Dr.-Ing.habil. G.H. Schnerr FLM Discussion…

Technische Universität München Lehrstuhl für Fluidmechanik - Fachgebiet Gasdynamik Univ. Professor Dr.-Ing.habil. G.H. Schnerr FLM Numerical results – application: fuel injector Case 3c: Mass flow at nozzle inlet and at bore hole exit - p in = 600 bar, p out = 26 bar.

Technische Universität München Lehrstuhl für Fluidmechanik - Fachgebiet Gasdynamik Univ. Professor Dr.-Ing.habil. G.H. Schnerr FLM Shock formation and cavitation erosion Shock front Liquid embedded vapor bubble, p bubble = p sat < p liquid Clusters (clouds) of liquid embedded bubbles show comparable behavior! Idea: Resolution of large scale structures could be sufficient to predict erosive shocks. Shock front Solid wall Erosion

Technische Universität München Lehrstuhl für Fluidmechanik - Fachgebiet Gasdynamik Univ. Professor Dr.-Ing.habil. G.H. Schnerr FLM physical situation 1 average behaviour physical situation 2