# 1 Pressure-based Solver for Incompressible and Compressible Flows with Cavitation Sunho Park 1, Shin Hyung Rhee 1, and Byeong Rog Shin 2 1 Seoul National.

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1 Pressure-based Solver for Incompressible and Compressible Flows with Cavitation Sunho Park 1, Shin Hyung Rhee 1, and Byeong Rog Shin 2 1 Seoul National University, 2 Changwon National University 8 th International Symposium on Cavitation 13-16 August 2012, Singapore

2 □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 Bark et al. (2009) vapor liquid

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

4 □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 ○Mass conservation equation ○Momentum conservation equation ○Standard k-  turbulence model

6 ○Singhal et al. (2002)

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

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

9 ○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 □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 (22.5 o cone half-angle) cavitator shapes  Cavitation number: 0.2, 0.3, 0.4, 0.5 & Reynolds number > 10 5

11 □Domain size and boundary condition □Mesh ○70 x (70+100) =11,900 ○Axi-sym x (Hemisphere+cylinder)

12 □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 %. CoarseMediumFinep/RE CDCD 0.08840.09010.09045.155/0.0905  0.01890.0033 GCI 0.00400.0007 l C /R2.1592.1832.1894.120/2.1910  0.01100.0027 GCI 0.00370.0009

13 □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 □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 □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 Compressible flow solver Fully developed cavity Incompressible flow solver

16 □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 Incompressible flow solver Compressible flow solver

17 □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 developedre-entrant jet is developed longest

18 □Streamwise velocity 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 solverCompressible flow solver

19 □Nondimensionalized turbulent 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 solverCompressible flow solver

20 □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 □Strouhal number ○Stinebring et al. (1983) carried out experiments on natural cavitation around axi-symmetric body with Re of 0.35  10 5 to 0.55  10 5 ○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 □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 flow solver ○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 cavitating flows

23 □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.

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