Numerical Analysis of Two-Dimensional Turbulent Super- Cavitating Flow around a Cavitator Geometry

8/16/2019 Numerical Analysis of Two-Dimensional Turbulent Super- Cavitating Flow around a Cavitator Geometry

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American Institute of Aeronautics and Astronautics

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Numerical Analysis of Two-Dimensional Turbulent Super-

Cavitating Flow around a Cavitator Geometry

Sunho Park 1

Department of Naval Architecture and Ocean Engineering, Seoul National University, Seoul, Korea

and

Shin Hyung Rhee2

Department of Naval Architecture and Ocean Engineering, Research Institute of Marine Systems Engineering,

Seoul National University, Seoul, Korea

With applications to high speed military under-water vehicles in mind, super-cavitating

flow around two-dimensional body is being studied by many researchers in various countriesfor many years. In the present study, high speed super-cavitating flow around a two-

dimensional symmetric wedge-shaped cavitator was studied using an unsteady Reynolds-

averaged Navier-Stokes equations solver based on a cell-centered finite volume method.

Various computational conditions, such as different wedge angles and cavitation numbers,

were considered for super-cavitating flow around the wedge-shaped cavitator. Super-

cavitation begins to form in the low pressure region and propagates downstream. The

computed cavity length was compared with analytic solution and computational results

using a potential flow solver. Fairly good agreement was observed in the three-way

comparison. The computed velocity on the cavity interface was also predicted quite closely to

that derived from the Bernoulli equation. Finally, the super-cavitating flow around the body

with a cavitator was simulated and validated by comparing with existing experimental data.

Nomenclature

k = turbulent kinetic energy

l body = body length

l c = cavity length

l w = wedge length

no = nuclei concentration per unit volume of pure liquid

P o = reference pressure

P v = vapor pressure

R = radius of bubble

Re = Reynolds number

U c = total velocity along the cavity boundary

U ¥ = free stream velocity

yo = wedge heighta = volume fraction

e = turbulent dissipation rate

f = wedge angle

m t = turbulent viscosity

s = Cavitation number

1 Ph.D. student, Naval Architecture and Ocean Engineering, [email protected] Professor, Naval Architecture and Ocean Engineering, [email protected]

20th AIAA Computational Fluid Dynamics Conference27 - 30 June 2011, Honolulu, Hawaii

AIAA 2011-3209

Copyright © 2011 by Sunho Park and Shin Hyung Rhee. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.


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Figure 1. Category of cavitation models

I.

Introduction

avitation is a physical phenomenon of liquid not to withstand the external stress. Especially, water vaporizes

easily under certain vapor pressure. Cavitation is observed in many hydrodynamic mechanical devices such as

pump, turbine, nozzle and marine propeller, and could have an intensive effect on the performance of them.Cavitation causes erosion and abrasion of a metal surface, and vibration and noise of a system. However, for

military purposes such as torpedoes, it is necessary to generate partial- or super-cavitation to reduce a viscous drag

intentionally. To prevent and use cavitation technically, its understanding and prediction are essential.

Cavitation phenomenon is divided into three stages. Initial cavitation is the bubble stage which causes bad

influence on mechanical systems. Partial cavitation is the stage at which cavity covers the body partially. Super-

cavitation is the stage where cavity length exceeds the characteristic length of the body. The velocity of underwater

body moving in super-cavitating flow could reach higher than 200 knots. This paper deals with super-cavitation

around a two-dimensional cavitator.

Thanks to the rapid advancement of

computational fluid dynamics (CFD) and

computing resources, numerical studies on

mathematical modeling of cavitation have been

widely popular recently. Cavitation models aredivided into the Lagrangian or discrete bubble,

and the Eulerian or continuum approaches by

computational framework, as summarized in

Figure 1. The Lagrangian approach focuses on

the behavior of discrete bubbles using bubble

tracking and bubble dynamics equations.

Chahine1 computed cavitation inception and

noise of a marine propeller using the Lagrangian

approach. The Eulerian approach is based on the

homogeneous mixture approximation. The

mixture flow moves at the same velocity and

each mixture phase is determined by solving a volume fraction transport equation. The Eulerian approach was

divided into the barotropic relation model and two phase mixture model. The barotropic relation model solves a

single continuity equation with the barotropic relation equation between pressure and density1,2. On the other hand,

the two phase mixture model deals with two phase continuity equations with the volume fraction transport equation3-

10. The transport equations are different by different developers and show reasonable results against existing

experimental data. Park and Rhee11 summarized characteristics of various ad-hoc cavitation models for CFD. Along

with mathematical cavitation models, researches on numerical scheme for cavitating flow have been done. Kunz et

al.12 suggested preconditioning strategy with favorable eigen-system characteristics and a block implicit dual-time

solution strategy for high density ratio cavitating flow. Senocak and Shyy10 developed pressure-velocity-density

scheme into the pressure correction equation and applied upwind density interpolation in the cavity region. Bilanceri

et al.13 suggested an implicit low-diffusive Harten-Lax-van Leer (HLL) scheme with implicit time advancing for

barotropic cavitating flow.

Many studies on super-cavitating flow have been done using experimental and numerical methods. Experimental

methods rely mostly on image process technology. Hrubes14 studied super-cavitating underwater projectiles using

images and analyzed flight behavior, stability mechanism, cavity shape, and in-barrel launch characteristics.

Savchenko15 investigated a motion of bodies in super-cavitating flow experimentally. Young and Kinnas16 analyzedunsteady super-cavitating flow using a potential flow based three dimensional boundary element method (BEM).

Varghese et al.17 modeled super-cavitation using a steady potential flow BEM and studied various conditions

including cavity closure on the conical/cylindrical portions of a vehicle, variants in the cone angle, and variation in

the radius of the cylindrical section. Nouri and Eslamdoost18 developed iterative algorithm to obtain cavity length

and applied it to super-cavitaing potential flow by BEM. Ahn et al.19 computed for a cavitator using a potential flow

solver by BEM and validated by experimental observation. Serebryakov20 predicted super-cavitating flows based on

slender body theory. Savchenko21 summarized problems and perspectives on ad-hoc super-cavitation models.

For the present study, an unsteady Reynolds-averaged Navier-Stokes (RANS) equations solver based on a cell-

centered finite volume method was developed that couples velocity, phase compositions, and pressure. Turbulence

was considered using the RANS approach. Thus developed solver, called SNUFOAM-Cavitation, was based on the

OpenFOAM platform, an object-oriented, open source CFD tool-kit, and validated by applying it to symmetric body.

C


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Figure 2. Problem description of wedge-shaped cavitator

SNUFOAM-Cavitation included the barotropic relation and two phase mixture models for cavitating flows. Kim and

Brewton22 also built multiphase computational fluid dynamics (MCFD) solver using the OpenFOAM platform and

validated it for cavitating flows on hydrofoils.

The present study focused on the cavitating flow around a wedge-shaped cavitator. The objectives were therefore,(1) to validate the developed solver, SNUFOAM-Cavitation, (2) to understand the super-cavitating flow around the

wedge-shaped cavitator, and (3) to apply the cavitator to the under-water vehicle and confirm the cavitating

performance by comparing with existing experimental data. Two dimensional (2D) flow computations were done

with the RANS and volume fraction transport equations for phase changing multi-phase flow around the wedge-

shaped cavitator for various wedge angles and cavitation numbers. A computed cavity length was compared with

analytic solutions and the results of potential flow solver. To apply the cavitator to the under-water vehicle, thesuper-cavitating flow around the body with cavitator was simulated and compared with existing experimental data.

The paper is organized as follows. The description of the physical problem is presented in section II, followed by

computational methods in section III. The computational results are presented and discussed in section IV. Finally,

section V provides the summary and conclusions.

II.

Problem Description

The 2D wedge geometry for the present studywith the wedge angle, f , length, l w, and height,

2yo, is shown in Figure 1. Two wedge angles of

15° and 45° were considered. The wedge height

was determined by the wedge angle, because the

wedge length was fixed as unity. The Reynolds

number ( Re) based on the free stream velocity

(U ¥ ) of 1 m/s and the wedge length of 1 m was

8.8x105, and the cavitation number (s ) based on

the free stream velocity and reference pressure

( P o) was in the range of 0.2 to 0.41 and defined as

2

21 ¥

-=

U

P P vo

r

s (1)

where P v was the vapor pressure and r was the density of the fluid. Test conditions are summarized in Table 1.

Cavitation begins to form in the low pressure region behind the wedge like the backward facing step flow and

propagates downstream as shown in Figure 2.

Table 1. Test Conditions

Wedge Angle

(degree)

Wedge Height

(m)

U ∞

(m/s)

Re

(×105)

P o

(Pa)s

15 0.2632 1 8.8 2624.63 0.41

15 0.2632 1 8.8 2579.71 0.32

15 0.2632 1 8.8 2554.76 0.27

15 0.2632 1 8.8 2534.79 0.23

15 0.2632 1 8.8 2519.82 0.20

45 0.8284 1 8.8 2594.69 0.35


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III.

Computational Methods

A. Mathematical Modeling

The equations for mass and momentum conservation were solved to obtain the velocity and the pressure flowfields. The equation for conservation of mass, or continuity equation, can be written as

( ) 0=×Ñ+¶

¶m

m vt

r

r r

(2)

where vr

is the velocity vector. The subscript m indicates mixture phase.

The equation for conservation of momentum, or the Navier-Stokes equations, can be written as

( ) ( )t r r ×Ñ+-Ñ=×Ñ+¶

¶ P vv

t

vmm

mm rr

r

(3)

where P is the static pressure and the turbulent stress tensor, t , is given by

( ) úûù

êë

é×Ñ-Ñ+Ñ= I vvv m

T

mmeff

rrr

3

2m t (4)

with the second term on the right hand side for the volume dilation effect. where m eff =m +m t , with m is the viscosity

and the subscripts eff and t denote effective and turbulent. I is the unit tensor.

The mixture properties such as the density and the viscosity are computed as function of a from

l l vvm r a r a r += (5)

l l vvm m a m a m += (6)

where a is the volume fraction, subscripts v and l indicate vapor and liquid, respectively.

Once the Reynolds averaging approach for turbulence modeling is applied, the unknown term, i.e., the Reynolds

stress term, is related to mean velocity gradients by the Boussinesq hypothesis as

( ) ( ) I vk vvvv mt T mmt mmm úûù

êë

éÑ+-Ñ+Ñ=-

rrr

m r m r 3

2 (7)

The standard k-e turbulence model23, which is based on the Boussinesq hypothesis with transport equations for

the turbulent kinetic energy, k , and its dissipation rate, e , was adopted for turbulence closure. The turbulent viscosity,

m t , is computed by combining k and e as m t = r C m k 2 / e , and the turbulence kinetic energy and its rate of dissipation

are obtained from the transport equations, which are

( ) ( ) M bk t

t mmm Y GGk vk k

t --++ú

û

ùêë

éÑ÷÷

ø

öççè

æ +×Ñ=×Ñ+

¶

¶ re

s

m m r r

r (8)

( ) ( ) ( )k

C GC Gk

C vt

cbk

t

t mmm

2

31

e r

e e

s

m m e r e r e e e -++ú

û

ùêë

éÑ÷÷

ø

öççè

æ +×Ñ=×Ñ+

¶

¶ r (9)


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where C m is an empirical constant of 0.09. Here, the model constants C 1e , C 2e , s k , and s e are 1.44, 1.92, 1.0, and 1.3,

respectively. The turbulent viscosity was used to calculate the Reynolds stresses to close the momentum equations.

The wall function was used for the near wall treatment.

The volume fraction transport equation to account for the cavitation dynamics was considered. The cavitation process was governed by the thermodynamics and the kinetics of the phase change dynamics in the system. In

SUNFOAM-Cavitation, several cavitation models5,6,8,9,24 were implemented. In this paper, the cavitation model of

Singhal et al.9 was used and expressed as,

v

l

vl l

ch Prod v

l

vl v

chcond v

v

t mmv

mv f P P MIN v

C f P P MIN v

C f v f t

f

r r r

g r r r

g s

m r

r )0,(

3

2)1(

)0,(

3

2)(

---

-+÷÷

ø

öççè

æ Ñ×Ñ=×Ñ+

¶

¶ r

(10)

where the model constant C dest for condensation and C prod for production of bubbles are set to 0.01 and 0.02,

respectively. The standard k-e turbulence model23 with cavitation model of Singhal et al.9 were selected to solve for

the the wedge-shaped cavitator and body with cavitator.

B.

Numerical Methods

A cell-centered finite volume method was employed along with a linear reconstruction scheme that allows the

use of computational cells of arbitrary shapes. Time derivative terms were discretized using the first-order accurate

backward implicit scheme, which has proved sufficient for engineering accuracy with carefully chosen time step

sizes. The solution gradients at cell centers could be evaluated by the least-square method. The convection terms

were discretized using a quadratic upwind interpolation scheme, and the diffusion terms using a central differencing

scheme. The velocity-pressure coupling and overall solution procedure were based on a pressure-implicit with

splitting order (PISO) type segregated algorithm25 adapted to an unstructured grid. The van Leer scheme26 was used

for interface capturing. The discretized algebraic equations were solved using a pointwise Gauss-Seidel iterative

algorithm, while an algebraic multi-grid method was employed to accelerate solution convergence.

IV.

Results and Discussion

In the Cartesian coordinate system adopted. The positive x-axis was in the streamwise direction, and the positive

y-axis was in the vertical direction. The solution domain extent shown in Figure 3 was -30 £ x/l w £ 41, and 0 £ y/l

w £

30~40 in the streamwise and vertical directions, respectively. The left and top inlet boundary was specified as the

Dirichlet boundary condition, i.e., the fixed value of the velocity. On the right exit boundary, the reference pressure

with extrapolated velocity and the volume fraction was applied. The reference pressure was taken from the exit

boundary. No-slip condition was applied on the wedge surface, and symmetric condition was applied on the bottom

boundaries.

Figure 4 shows the computational mesh. A single-block mesh of 0.2 million hexahedral cells was used. On the

inclined slope of the wedge, 70 cells were used, while along the back of the wedge, 40 cells were applied.

Figure 3. Boundary condition and domain extent Figure 4. Mesh and domain extents


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In order to tame computational instability caused by large difference in density and high rate of mass transfer, the

cavitating flow around the wedge-shaped cavitator was computed with the converged single phase solution of the

same condition. Figures 5 and 6 show the nondimensionalized streamwise velocity component and the pressure

coefficient contours for the non-cavitating flow around the 45° angled wedge. The streamwise velocity componentincreased by 50% at the wedge tip and low pressure region was observed behind the wedge. Note that the location of

the maximum streamwise velocity component was different from that of the minimum pressure location.

Figure 5. Streamwise velocity component Figure 6. Pressure contours without cavitation

contours without cavitation

Figure 7 shows initial shape of the cavity, which takes place at the same location of the low pressure region

shown in Figure 6. The fully developed cavity and streamlines are presented in Figure 8. At the Cavitation number

of 0.3, it is observed that a 45° angled wedge-shaped cavitator could generate a cavity with a length of 9.2 times the

wedge length. Note that the fully developed cavity shape was entirely different from that of the low pressure region

shown in Figure 6. Therefore, it can be said that the cavitation inception could be predicted from the non-cavitating

flow solution. However, the fully developed cavity shape could not be predicted from single phase flow solutions.

Figure 7. Initial cavity Figure 8. Fully developed cavity

Figures 9 and 10 show the nondimensionalized streamwise velocity component and pressure coefficient contours

of the fully developed cavitating flow. The maximum streamwise velocity component was 10% higher than the free

stream one, but 40% lower than that obtained from the non-cavitating flow. The velocity and pressure fields around

the wedge-shaped cavitator were completely changed by the existence of cavitation.

Figure 9. Streamwise velocity component Figure 10. Pressure contours in cavitating flow

contours in cavitating flow


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Figure 11. Comparison of cavity lengths computed by analytic

solution, potential flow solver and present viscous flow solver.

The computed cavity length was compared

with an analytic solution27 and potential flow

solutions19. Newman27 derived the relation

between the cavitation number, s , the cavitylength, l c, and the wedge height, yo, for 2D

symmetric bodies.

ò =÷ ø

öçè

æ ++

1

0

/

2/1

0)(1

4dt t y

t

l l o

ccs

p (11)

where, yo/

was a derivative for the inclined

slope.

Ahn et al.19 developed a potential flow

solver and computed the cavity length of the

selected wedge-shaped cavitator. Cavity

lengths for various cavitation number

computed by present viscous flow solver, the potential flow computation19 and the analytic

solutions27 are plotted together in Figure 11.

Cavity lengths were calculated using the

vapor volume fraction value of 0.1. The

computed cavity length by SNUFOAM-

Cavitation was a little smaller than that of

experimental data, and the cavity length calculated by the analytic solution was at the half way between the potential

and viscous flow solutions at high cavitation numbers. Better agreement is seen for the cavity length with decreasing

the cavitation number.

A dynamic condition on the cavity boundary was derived from the Bernoulli’s equation, which could be used to

derive the following expression for the total velocity along the cavity interface,

s += ¥ 1U U c (12)

The total velocities along the cavity interface are compared in Table 2. The present computational solution was

predicted quite closely to that derived from the Bernoulli’s equation.

Table 2. Velocity along the cavity interface

s U ∞

(m/s)

s += ¥ 1U U c

(m/s)

Present U c

(m/s)

Error

(%)

0.20 1 1.095 1.099 0.3

0.23 1 1.109 1.152 3.8

0.27 1 1.127 1.190 5.6

0.32 1 1.149 1.207 5.1

0.41 1 1.187 1.243 4.6


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To apply the cavitator to the under-water vehicle, the super-cavitating flows around the body with wedge-shaped

cavitator were simulated. The body with cavitator geometry and tunnel extent are described in Figure 12. The body

with the length of 80 mm and half height of 10 mm, and the cavitator with the wedge angle of 25.6o and length of 20

mm were designed. The half height of the computation domain was set to 60 mm conforming to the experimentalfacility28.

Figure 12. Body with cavitator geometry and cavitation tunnel extent (unit: mm)

(a) Cavitation number of 1.2

(b)

Cavitation number of 1.0

Figure 13. Volume fraction contours of body with cavitator


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Table 3. Cavity length of body with cavitator

s

l c /l body

Present computation Experiment28

1 2.500 2.50

1.2 0.937 0.875

1.35 0.550 0.500

2.0 0.275 0.250

The solution domain extent was -10 £ x/l w £ 20, and 0 £ y/l w £ 3 in the streamwise and vertical directions,

respectively. The left inlet and right exit boundaries were specified as for the computation of the wedge-shapedcavitator. No-slip condition was applied on the cavitator and body surface, and symmetric condition was applied on

the bottom boundaries. A single-block mesh of 24,000 hexahedral cells was used. On the inclined slope of the

wedge, 60 cells were used, while along the back of the wedge, 20 cells were applied.

Unsteady computations were done for the the cavitation numbers of 1, 1.2, 1.35, and 2.0. The cavitation number

from 1.2 to 2.0 implied the cavity covering the body partially, and the cavitation number of 1.0 implied the cavity

covering the entire body. The liquid volume fraction contours for cavitation number of 1.0 and 1.2 are shown in

Figure 13. The cavity partially covering the body showed steady behavior once fully developed. However, the cavity

covering the entire body showed unsteady behavior at the body end. The cavity separated at the body end and

progressed downstream. Special care is required when

the cavity reaches the body end. Figure 13 (b) shows the

time averaged volume fraction contours for several

cavity sheddings at the body end. Table 3 summarizes

the cavity length nondimensionalized by the bodylength for various the cavitation numbers.

The findings from the computational results were

confirmed by comparing with the experimental

observations in the cavitation tunnel at Chungnam

National University28. Figure 14 is a snap shot of the

cavity around the body with cavitator in the cavitation

number of 1.016 and Reynolds number of 2x105. Table

3 presents the comparison of the cavity length between

the present computation and experiment28. The cavity

covered the entire body with the cavitation number over

1.0, which was confirmed by both the computation and

experiment.

V.

Concluding Remarks

Mostly for military purposes, which require high speed and low drag, super-cavitating flows around under-water

bodies have been an interesting research topic. In this paper, SNUFOAM-Cavitation has been developed and

validated for super-cavitation around the wedge-shaped cavitator.

Super-cavitating flow around the wedge-shaped cavitator was simulated for various wedge angles and cavitation

numbers. The cavity length behind the wedge-shaped cavitator was compared with the analytic and the potential

flow solutions, and showed good agreement in a three-way comparison. The total velocity on the cavity interface

was close to that derived from the Bernoulli equation.

To apply the wedge-shaped cavitator to the under-water body, the turbulent super-cavitating flow around the

body with cavitator was simulated. The developed cavity covered the entire body at low cavitation numbers and

showed unsteady behavior at the body end. Special care is recommended when the cavity propagates over the body

Figure 14. Snap shot of cavity around the cavitator

with body in experiment31


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end. The computed cavity lengths of the body with cavitator were compared with experimental data carried out in

the cavitation tunnel at Chungnam National University28 to confirm the cavitating performance of the body with

wedge-shaped cavitator. CFD solutions for the cavity length were consistent with experimental results, showing

good agreement.

Acknowledgments

This work was supported by Incorporative Research on Super-cavitating Underwater Vehicles funded by Agency

for Defense Development (09-01-05-26), and by the World Class University Project (R32-2008-000-10161-0) and

Multi-Phenomena CFD Research Center (20090093103) funded by the Ministry of Education, Science and

Technology of the Korea government.

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01-05-26, Chungnam National University, 2010.

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Numerical Analysis of Two-Dimensional Turbulent Super- Cavitating Flow around a Cavitator Geometry