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Pressure-based Solver for Incompressible and Compressible Flows

with Cavitation

Sunho Park1, Shin Hyung Rhee1, and Byeong Rog Shin2

1 Seoul National University, 2 Changwon National University

8th International Symposium on Cavitation13-16 August 2012, Singapore

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□ 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 com-pressible flows should be considered at the same time

Introduction

Bark et al. (2009)

vapor

liquid

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Introduction

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 com-

pressibility

- Pressure based methods for

compressible flows

(Rincon and Elder, Issa and

Javareshkian, Darbandi et al.)

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□ 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

Introduction

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Governing Equa-tions

○ Mass conservation equation

○ Momentum conservation equation

○ Standard k- turbulence model

0 vt

v vv pt

k

CGCGk

Cvt cbk

t

t2

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Mbkt

t YGGkvkkt

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Cavitation Model

○ Singhal et al. (2002)

evapcondvv

tmmv

mv RRfvftf

)(

vl

vll

chcondcond fPPvCR

32

vl

vlv

chevapevap fPPvCR

1

32

vPP

PPv

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Code Develop-ment

Incompressible cavitating flow solver

PSvava NNPP

P)v(Hva PP

01)(1

P

avH

a PP

0 v

Momentum equation

using continuity equation

take the divergence

Turbulence Equations

UpdateProperties

MomentumEquation

Correct Velocity & Flux

Converge?

Start

Finish

Increaset

yes no

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Code Develop-mentIsothermal compressible cavitating flow solver

P)v(Hva PP

01

vvt

Momentum equation

using continuity equation

take the divergence

Substitute density to pressure

0Pa1vH

a1vPPv

tP

P1

PP

P

Turbulence Equations

UpdateProperties

MomentumEquation

Correct Velocity & Flux

Converge?

Start

Finish

Increaset

yes no

ContinuityEquation

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○ 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)

Numerical Methods

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□ 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

Validation

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□ Domain size and boundary condition

□ Mesh○ 70 x (70+100) =11,900○ Axi-sym x (Hemisphere+cylinder)

Results and Discus-sion

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□ 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 %.

Results and Dis-cussion

  Coarse Medium Fine p/RE

CD 0.0884 0.0901 0.0904 5.155/0.0905

  0.0189 0.0033  

GCI   0.0040 0.0007  

lC/R 2.159 2.183 2.189 4.120/2.1910

  0.0110 0.0027  

GCI   0.0037 0.0009  

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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

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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

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Results and Discus-sion

□ 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

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Results and Discus-sion

□ 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

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Results and Discus-sion

□ 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

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Results and Discus-sion

□ 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 solver Compressible flow solver

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Results and Discus-sion

□ 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 solver Compressible flow solver

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Results and Discus-sion

□ 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.

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Results and Dis-cussion

□ Strouhal number○ Stinebring et al. (1983) carried out

experiments on natural cavitation around axi-symmetric body with Re of 0.35 105 to 0.55105

○ 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

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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 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

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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.

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Thank you for

attention