Pressure-based Solver for Incompressible and Compressible Flows with Cavitation

1. Pressure-based Solver for Incompressible and Compressible Flows with Cavitation Sunho Park1, Shin Hyung Rhee1, and ByeongRog Shin2 1 Seoul National University, 2 Changwon National University 8thInternational Symposium on Cavitation 13-16 August 2012, Singapore

2. Introduction • Cavitation • Cavitation : the liquid phase changes to vapor phase under the certain pressure • The liquid phase is usually treated as an incompressible flow • The vapor phase is treated as a compressible flow  To understand and predict cavitating flows correctly, incompressible and compressible flows should be considered at the same time vapor liquid Bark et al. (2009)

3. Introduction • Incompressible flows • Pressure based method • Pressure is a primary variable • Advantage: liquid phase • Disadvantage: vapor phase Incompressible flows with compressibility - Pressure based methods for compressible flows (Rincon and Elder, Issa and Javareshkian, Darbandi et al.) • Compressible flows • Density based method • Density is a primary variable • Advantage: vapor phase • Disadvantage: liquid phase

4. Introduction • Pressure-based compressible computation method • Shock waves • Underwater explosions • Cavitations • The objectives were • to develop pressure-based incompressible and isothermal compressible flow solvers, termed SNUFOAM-Cavitation • to understand compressibility effects in the cavitating flow around the hemispherical head-form body

5. Governing Equations • Mass conservation equation • Momentum conservation equation • Standard k-turbulence model

6. Cavitation Model • Singhal et al. (2002)

7. Code Development Incompressible cavitating flow solver Start Momentum equation Turbulence Equations Update Properties take the divergence Increase t Momentum Equation using continuity equation Correct Velocity & Flux yes no Converge? Finish

8. Code Development Isothermal compressible cavitating flow solver Start Momentum equation Turbulence Equations take the divergence Continuity Equation using continuity equation Update Properties Increase t Momentum Equation Substitute density to pressure Correct Velocity & Flux yes no Converge? Finish

9. Numerical Methods • Unsteady state RANS equation solvers were used • A cell-centered finite volume method was employed • PISO type algorithm adapted for velocity-pressure coupling • Convection terms were discretized using a TVD MUSCL scheme • Diffusion terms were discretized using a central differencing scheme • Gauss-Seidel iterative algorithm, while an algebraic multi-grid method was employed • CFD code: SNUFOAM-Cavitation (Developed using OpenFOAM platform)

10. Validation • Developed CFD code • Apply non-cavitating flow around the hemispherical head-form body • Apply cavitating flow around the hemispherical head-form body • Problem description (Hemispherical head-form body) • Experiment • Rouse, H. and McNown, J. S., 1948 • hemispherical (0.5 caliber ogive), blunt (0.0 caliber ogive) and conical (22.5o cone half-angle) cavitator shapes • Cavitation number: 0.2, 0.3, 0.4, 0.5 & Reynolds number > 105

11. Results and Discussion • Domain size and boundary condition • Mesh • 70 x (70+100) =11,900 • Axi-sym x (Hemisphere+cylinder)

12. Results and Discussion • Uncertainty Assessment • To evaluate the numerical uncertainties in the computational results, the concept of grid convergence index (GCI) was adopted • Three levels of mesh resolution were considered for the solution convergence of the drag coefficient, and cavity length. • The solutions show good mesh convergence behavior with errors from the corresponding RE less than 0.5 %.

13. Validation of Non-Cavitating Flow • Non-cavitating flow was simulated and validated against existing experimental data in a three-way comparison • the compressible flow solver well predicted the incompressible flow • Both almost same

14. Validation of Cavitating Flow • Cavitating flow was simulated and validated against existing experimental data in a three-way comparison • the incompressible flow solver showed the earlier cavity closure, while the pressure overshoot was more prominent by the isothermal compressible flow solver • Overall, both results showed quite a close to the existing experimental data.  validate developed incompressible and compressible flow solvers

15. Results and Discussion • The volume fraction contours when the cavity is fully developed • Overall cavity behavior was almost same for both solvers. • Noteworthy is the undulation of the cavity interface • Variations of the vapor volume fraction due to a re-entrant jet caused the change of the vapor volume, and then the cavity interface showed unsteady undulation Fully developed cavity Incompressible flow solver Compressible flow solver

16. Results and Discussion • The volume fraction contours when re-entrant jet was greatest developed to its length • Incompressible flow solution: the cavity shedding was seen near the cavity closure due to a short re-entrant jet • Compressible flow solution: the cavity shedding was observed up to the middle of the cavity due to a relatively longer re-entrant jet • Unsteady undulation of the cavity interface was observed continuously Re-entrant jet Compressible flow solver Incompressible flow solver

17. Results and Discussion • Cavity shedding cycle • Observed in the results with compressible flow solution • Developed cavity dynamics (shedding) repeats below two figures Compressible flow solver cavity is fully developed re-entrant jet is developed longest

18. Results and Discussion • Streamwisevelocity contours when the re-entrant jet was fully developed • relatively strong and long re-entrant jet, which was in the reverse direction to the freestream flow, was observed Incompressible flow solver Compressible flow solver

19. Results and Discussion • Nondimensionalizedturbulent eddy viscosity contours when the re-entrant jet is fully developed • The turbulent viscosity was large near the cavity closure in both cases • In the result of the compressible flow solution, the large turbulent viscosity was seen because of the stronger re-entrant jet. Incompressible flow solver Compressible flow solver

20. Results and Discussion • Time history of the drag coefficient • 0  x/D  2 • Incompressible flow solution: converged to a certain constant value • Compressible flow solution : showed fluctuation behavior due to the unsteady cavity shedding.

21. Results and Discussion • Strouhalnumber • Stinebring et al. (1983) carried out experiments on natural cavitation around axi-symmetric body with Re of 0.35 105 to 0.55105 • The Strouhal number (St) was calculated using the obtained cavity shedding frequency • St =0 by incompressible flow solver • The overall trend well captured by the compressible flow solver

22. Concluding Remarks • To simulate the compressibility in the cavitating flow, the incompressible and compressible flow solvers were developed, and validated by applying it to a hemispherical head-form body • In the compressible flowsolver • The re-entrant jet was appeared to be relatively longer and the cavity interface showed unsteady undulation due to the re-entrant jet • The drag coefficient of the incompressible flow solver was converged to a certain value, while, one of the compressible flow solver showed fluctuation behavior due to the cavity shedding frequency • The Strouhal number, calculated using the drag coefficient history, shows quite a close agreement between experiments and computations by the compressible flow solver • From the results, the compressible flow computations, which including compressibility effects, were recommended for the computation of cavitatingflows

23. Acknowledgement • This work was supported by the Incorporative Research on Super-Cavitating Underwater Vehicles funded by Agency for Defense Development (09-01-05-26), the World Class University Project (R32-2008-000-10161-0) and the Research Foundation of Korea (2010-0022835) funded by the Ministry of Education, Science and Technology of the Korea government.

24. Thank you for attention