1. Riemann Techniques for the Simulation of Compressible Liquid Flows
with Phase-transition at all Mach numbers – Shock and Wave Dynamics
in Cavitating Micro and Macro Systems
Steffen J. Schmidt, Ismail H. Sezal, Günter H. Schnerr, Matthias Thalhamer
46th AIAA – ASME Reno, Nevada

2. Outline Motivation - Cavitating liquid 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∞ = 10-4
- 3-D bubble collapse - comparison with Rayleigh-Plesset-Eqn.
- 3-D simulation “Obernach experiment” - comparison of erosion areas
Conclusion & Outlook

3. Cavitation Definition:
„Cavitation“ is the flow-induced evaporation of a liquid.
No external heat addition ( boiling…)!
Typical properties of cavitating flows within hydraulic machinery:
uref = O(10) m/s, pref = O(1) bar, exept for fuel injection systems -> pref= O(100) bar,
ρref = O(1000) kg/m³, Tref = O(300) K, cref = O(1000) m/s, psat(Tref)=0.025 bar.
Mref = – 0.01  Low Mach number flow as long as no vapor
content exists!

4. Sheet and cloud cavitation
Cavitation phenomena
Vortex
cavitation
Bubble and cloud cavitation
Sheet and cloud cavitation
Cavitation erosion
Supercavitation

5. 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∞ = 10-4
- 3-D bubble collapse - comparison with Rayleigh-Plesset-Eqn.
- 3-D simulation “Obernach experiment” - comparison of erosion areas
Conclusion & Outlook

6. Numerical method - CATUM
To solve the balance laws of mass, momentum and energy we apply:
3-D unsplit finite volume method (semi-discrete)
Convective fluxes: modified flux function + TVB (WENO-3) / TVD (VanLeer)
Diffusive fluxes: central discretization  not used here!
Turbulence models: k-ω, EASM, ω-RSM  not used here!
Explicit 4-stage Runge-Kutta scheme (2nd order accuracy, enlarged stability region)
Semi-implicit time integration for source terms (turbulence models)  not used here!
No subiterations required (as for pressure based approaches), therefore low cost per time step.
Well suited to simulate hydrodynamic and wave dynamic features with time steps down to nanoseconds.

7. „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

8. „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

9. Two-phase flow properties via integral averages per cell
Instead of modelling small scale structures we investigate average thermodynamic properties:
Conservative averaging (filtering) leads to:
Consider stable thermodynamic conditions only:
 constitutive relations (EOS) determine cell variables p, T

10. - 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 ∙108 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)

11. Thermodynamic model - EOS
Combined EOS contains relations for:
pure water → modified Tait equation
vapor phase → ideal gas law
two-phase region → saturation conditions
psat(Tsat)
Tsat

12. 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∞ = 10-4
- 3-D bubble collapse - comparison with Rayleigh-Plesset-Eqn.
- 3-D simulation “Obernach experiment” - comparison of erosion areas
Conclusion & Outlook

13. Numerical results – validation: single phase flow
Case 1: 2-D steady state liquid flow around circular cylinder at M∞=10-4, grid 128 x 32 cells.
Pressure coefficient cp – isolines;
Drag coefficient cD,p=1.5·10-5.

14. Numerical results – validation: two-phase flow
Case 2: 3-D bubble collapse – comparison with Rayleigh-Plesset solution
Bubble radius R0=0.4 mm, time step ΔtCFD=6.5 nanoseconds, collapse time 37 microseconds,
initial pressures pliquid=1.0 bar, pbubble=0.023 bar , T=20° C, liquid: water, bubble: water vapor.

15. Numerical results – application: erosive two-phase flow
Case 3: 3-D simulation of the “Obernach-experiment” on cavitation erosion
0,85 m
0,3 m
uin=11 m/s
Tin=300 K
σref=1.8
Pout,mix=1.12 bar
3.1·106 cells
106 time steps, Δt ≈ 3·10-7 s
64 CPU (SGI AltixBx2)  240 h.

16. Numerical results – application: erosive two-phase flow
Case 3: Dynamic phase-transition and related pressure field
Top view: Two-phase regions
Perspective view: Two-phase regions and static pressure at the walls

17. Numerical results – application: erosive two-phase flow
Case 3: Comparison of two-phase structures experiment/simulation
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, 2004.
Simulation CATUM: Isosurfaces α=0.01, one instant in time.

18. Numerical results – application: erosive two-phase flow
Case 3: Fragmentation of two-phase structure, collapse, shock formation 1 2
p [bar]
> 2 1
0.02 3
pmax = 65 bar

19. Numerical results – application: erosive two-phase flow
Case 3: Areas of intense erosion (experiment) - maximum pressures (simulation)
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, 2004.
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.

20. Conclusion and outlook
CFD-Tool CATUM enables the simulation of compressible cavitating liquid flows including the
formation and propagation of collapse induced shocks.
Numerically predicted areas of intense impulsive loads agree well with experimentally
observed areas of cavitation erosion.
CATUM is used to simulate wave dynamics and cavitation phenomena in fuel injectors and
around hydrofoils as well (previous investigations and AIAA ).
Effects of non-condensable gas content within the liquid fluid are currently investigated.

21. Thank you for your attention!

22. Discussion…

23. Two-phase flow properties via integral averages per cell
physical situation 1
average behaviour
physical situation 2

24. Shock formation and cavitation erosion
Shock front
Liquid embedded vapor bubble, pbubble = psat < pliquid
Shock front
Solid wall
Erosion
Clusters (clouds) of liquid embedded bubbles show comparable behavior!
Idea: Resolution of large scale structures could be sufficient to predict erosive shocks.