Numerical Simulation of Cavitation Flows around NACA66 Hydrofoil by Adaptive Mesh Refinement

Minsheng Zhao1, Decheng Wan1*, Zhonghua Li2

1 Computational Marine Hydrodynamics Lab (CMHL), State Key Laboratory of Ocean Engineering,

School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, China 2 Nanjing Hydraulic Research Institute, Nanjing, China

*Corresponding author

ABSTRACT

The main attention of present work is to investigate the ability of a

adaptive mesh ganeration method in accurate cavitation simulation. A

numerical study of the cavitation around NACA66 hydrofoil has been

carried out. The simulation results such as cavitation shape, the lift

coefficient CL and the drag coefficient CD are analyzed and compared

with each other. It is found that the adaptive mesh refinement can

effectively capture the change of cavitation morphology, especially the

cavitation shedding process, while the the calculation cost is generally

lowered.

KEY WORDS: OpenFOAM; cavitation flow; adaptive mesh

refinement; NACA66 hydrofoil

INTRODUCTION

Cavitation is a physical phenomenon which occurs when the local

pressure of liquid drops below the limit of static vapor pressure. This

severe vaporization has been a research hotspot in fluid mechanics field

for it leads to problems such as pressure fluctuation, increased drag

force, vibration, and noise( Ghorbanim et al., 2015;WU J et al., 2005).

On the other hand, it is required to be able to utilize cavitation in some

cases like supercavitation torpedo. At present, the speed of ships is

generally increased, and the occurrence of cavitation appearing in

hydraulic devices including pumps, turbines, and propellers is

inevitable. Therefore it is important to study cavitation through

numerical methods.

Cavitation around hydraulic machineries have been studied for decades.

A number of experiments were carried out to investigate cavitation

phenomena, because it was the most efficient way, and possibly the

only way. Attached cavitation that formed on stationary hydrofoils with

significant sweep was tested by Crimi(1970), Bark(1986), and Ihara,

Watanabe & Shizukuishi(1989).The unsteady cavitation in mixed-flow

pumps have been experimentally and numerically investigated by

Kobayashi and Chiba(2010) with data close to the ones obtained with

other researchers. Among the different types of cavitation around

hydraulic machineries, the tip vortex cavitation and the sheet cavity are

the focus of propeller cavitation study. (Oprea and Bulten, 2009;

Bensow and Bark, 2010). When the ship hull is also concerned, the

impacts of the irregular wake behind the ship to the propeller cavitation

is a research emphasis with experimental and numerical methods. The

unsteady cavitation characteristics have been studied through

experimental research with a simplified hydrofoil in the cavitation

tunnel, from standard NACA hydrofoils (Wu et al., 2015; Zhang et al.,

2014) or Clark-Y (Huang et al., 2014; Kato, 2011) to special types of

profiles like a plano-convex hydrofoils (Coutier et al., 2007; Le et al.,

1993).

Experimental method is the most direct way to study cavitation,

however, this method has disadvantages that can not be ignored. Firstly,

the scale effect is one of the technical constraints of the experimental

results. Secondly, the re-entrant jets can not be described through high

speed cameras or other experimental equipments, yet it is critical to the

periodical shedding of the cavitation and causes the collapse of a sheet

cavity (Foeth, 2008). In addition to the above, the most important

reason to use numerical methods instead of experimental methods is

that the latter can be extremly resources consuming. Hence the

numerical simulation of cavitation is becoming more popular in recent

years.

At first the potential flow was used in computational fluid dynamics

areas, including cavitation study. This approach, however, has

difficulties dealing with vertical structures. In the last years, the viscous

CFD method is being used in numerical simulations. In recent years,

the numerical simulation of cavitating flow by solving viscous fluid

dynamics control equation (N-S) is the development trend. Solving the

governing equations of viscous fluid mechanics numerically can not

only consider the viscosity effect of the flow process, but also save the

trouble of including the nonphysical cavity closure hypothesis.

In the development of this method, the research of numerical cavitation

model is always the focus of it. There are three popular cavitation

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models. One is the equation of state model (Delannoy and Kueny

1990) . The second is bubble dynamics model (Kubota et al. 1992), and

the third is the transport equation model. The equation of state model

treats density as a function of pressure, this method has been further

developed by Song and He (1998). They use a quintic polynomial to

define the relationship between density and pressure. The relationship

changes abruptly at the saturated vapor pressure, indicating the

cavitation process.

In CFD framework of turbulent flow computation, the flows around

typical hydraulic devices like turbines, fuel injectors and pumps, have

been widely studied. The Navier-Stokes equations for a circular

cylinder and NACA0015 airfoil in air using the collocated grid finite

volume method was solved by Shen et al. (2004). The transport

equation model (TEM), in which a governing equation for the liquid or

vapor phase fraction is solved, has been used for the one-fluid model.

Many cavitation models are derived from different mathematical

methods. The cavitation model based on the simplified Rayleigh

Plesset equation includes Singhal model(2002) and Zwart model(2004),

the Kunz model and Merkle model are based on experience; cavitation

models obtained from cavitation force analysis, such as Tamura

model(2009), etc. Martynov(2006) equates the cavitation radius to the

cavitation length scale, and they are expressed by the number density of

vacuoles; In the compressible cavitation model of Hosangadi(2001) and

the cavitation model of Kunz(2000), there is no expression of cavity

radius or cavity scale, instead the time scale is used to measure the inter

phase mass exchange. The flow around a two-dimensional hydrofoil

cavity under turbulent conditions was further studied by Rebound and

Delannoy (1994).The Euler solution of the flow around an

axisymmetric cavity was calculated by Chen and Heister(1996).

Feneno(2002) useed the RANS method to investigate the

hydrodynamic performance of the large skew propeller, the steady and

unsteady results are in good agreement with the experimental results.

Based on Rhee's method(2003), Watanabe (2003) calculated the

unsteady hydrodynamic force and unsteady cavitation of propeller, and

the calculated results are in good agreement with the experimental

results. Hoeijmakers et al. (1998) assumed that the density and pressure

in the water vapor mixing region conform to the sine function change

law, and simulated the two-dimensional hydrofoil cavitation flow with

the Euler equation. Merkle et al. (1998) assume that the density and

pressure in the water vapor mixing region are in accordance with the

change rule of cubic spline curve. The numerical simulation of

NACA66 (MOD) two-dimensional hydrofoil cavitation flow shows

that the maximum density ratio can only reach 10. Qin et al. (2003)

also used the weak compressible model and the 5th degree polynomial,

and the influence of non condensable gas on cavitation was considered.

The cloud cavitation with an artificial reduction of the turbulent

viscosity in the water-vapor mixing zone was predicted (Reboud et al.

1998).

Adaptive grid method refers to a technique that the grid is adjusted

continuously in the iterative process to refine the grid and achieve the

coupling between the grid point distribution and object understanding

in some areas with more drastic changes, such as large deformation,

shock surface, contact interface and sliding surface, so as to improve

the accuracy and resolution of the solution (Kamkar S J et al, 2011).

The adaptive grid hopes that the grid will be automatically dense in the

region with large changes, while the grid will be relatively sparse in the

region with gentle changes, so that the high-precision solution can be

obtained while maintaining the high efficiency of calculation.

In the present paper, the adaptive grid technique has been used to

localize the cavitation region, especially the mesh near the two-phase

interface. The numerical simulations are obtained by

InterPhaseChangeFoam solver in the open source CFD software

platform OpenFOAM with Schnerr-Sauer cavitation model. The

numerical results above are basically in accordance with experimental

ones. Simulation results show that the hydrodynamic coefficients CL

and CD are predicted accurately with different mesh generation

methods under same cavitation number. Research shows that adaptive

mesh refinement for cavitation regions can effectively improve the

simulation accuracy of cavitation flow with a certain amount of

computation.

NUMERICAL METHODS

Governing Equations

The governing equations for cavitation flow are based on a single phase

flow approach, regarding the mixture of fluid and vapor as a single

phase whose density can change according to the pressure. The flow

field is solved by the mixture continuity and momentum equations plus

a volume fraction transport equation to model the cavitation dynamics.

As for RANS turbulence model, the equations are presented below.

0

m jm

j

u

t x

(1)

( ) ( )

[ ( )]

m j m i j

j j

j iji

j j i j

u u u p

t x x

uu

x x x x

(2)

( ) ( ) /ll j c v l

j

u m mt x

(3)

The mixture density and the viscosity are defined as follows.

(1 )m l l v l (4)

(1 )m l l v l (5)

In the above equations, ,l v are the liquid and vapor density, ,l v

are the liquid fraction and the vapor fraction, t is the turbulent

viscosity,+m ,

-m represent the condensation and evaporation rates.

ij is the subgrid stress (SGS), representing the influence of small scale

vortex on the momentum equation.

( ) (2 )( ) 1i j ij iji

j j j j

u u Su p

t x x x x

(6)

Modified Turbulence Model

Turbulence model plays an important role in the numerical simulation

of cavitation flows. The SST k-ω turbulence model which developed by

Menter is mixed with the k-ω model in the near-wall area and the k-

epsilon model in the far field.

*ik

i

k t

i i

k kukP

t x

k

x x

(7)

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2 2

21

12 1

ik

i

t

i i i i

uS P

t x

kF

x x x x

(8)

Reboud gave the suggestion that an artificial reduction of the turbulent

viscosity of this model can predict a more accurate frequency of the

periodical shedding of cavitation. So a serial of modified SST k-ω

models are applied following his idea.

( )t

kf C

(9)

1

( )( ) ; 1

( )

n

m vv n

l v

f n

(10)

Mass Transfer Model of Schnerr-Sauer

The mass transfer model which also called cavitation model adopted

here was developed by Schnerr and Sauer. In their papers, the vapor

fraction is related to the number of gas nucleus per unit volume and the

average radius of gas nucleus. The condensation and evaporation rates

are defined as follows.

3 3

v 0 0

4 4/ ( 1)

3 3n R n R (11)

23 (1 )sgn( )

3

vv l v vc c v

l

P Pm C P P

R

(12)

v

23 (1 )sgn( )

3

vv l v vv v

l

P Pm C P P

R

(13)

R is the average radius of gas nucleus expressed as:

1/3

0

3=( )

1 4

v

v

Rn

(14)

Adaptive Mesh Refinement

The quadtree structure is used to generate sub grid based on the original

grid, and the local area is encrypted adaptively. The corresponding

relationship between adaptive mesh and quadtree is shown in Figure 1.

In each iteration, the region is encrypted according to the volume

fraction defined in the VOF method. When volume fraction is 0 or 1,

the encryption is not performed. When the volume fraction is between

0 and 1, the phase interface area is encrypted. Every time the

encryption level is increased, a layer of sub grid will be generated

based on the existing refined grid. The computing time of adaptive grid

is mainly spent on interpolation and information exchange between

grids.

Fig.1 Adaptive mesh and quadtree structure

CASE DESCRIPTION

Boundary Condition

For the investigations in the computational domain, the test geometry is

a NACA66 hydrofoil at 5 degree angle of attack with chord length C =

100mm. The computational domain is shown in Fig. 2. The size of the

domain is 500 × 300mm. All distances are based on the chord length C.

The inlet of the computational domain is 1C upstream from the leading

edge and the outlet of the computational domain is 4C downstream

from the leading edge. Front and back boundaries are considered as

empty and symmetry, and other detailed boundary conditions are

described in Table 1.

Fig.2 the grid in lengthwise section and cross section

Table 1. Boundary condition

Hydrofoil Wall

Inlet Velocity

Outlet Pressure

Top and Bottom Wall

Front and back Symmetry

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The inlet velocity is 5m/s, the pressure gradient and the outlet velocity

is zero. Reference pressure is obtained by the cavitation number. The

cavitation number in the present work is 0.8. All parameters of the

work condition in this paper are listed in Table 2.

2

=1

2

vP P

U

(15)

Table 2. Parameters of work condition

Chord length 100 mm

Attack angel α 4 deg.

Freestream velocity U∞ 5 m/s

cavitation number σ 1.07

Saturation vapor pressure 2338.8 Pa

Outlet pressure 12952.32 Pa

Mesh Size

The influence of mesh size is considered in the present paper. To find

out the appropriate mesh size for the numerical simulation, four kinds

of mesh are calculated. Specific grid information and corresponding

calculation results are listed in Table.3. As can bee seen that when the

mesh size increases, the life and drag coefficients tend to be convergent.

When the mesh size reaches 150×90, the changes of coefficients are

relatively subtle.

Table 3. Effect of mesh size on lift and drag coefficients

Mesh size Lift coefficient Drag coefficient

150×90 0.8741 0.04384

120×72 0.8732 0.04373

105×63 0.8695 0.0435

90×54 0.8607 0.0431

Fig. 3 Lift and drag coefficients variations along with the mesh

resolution

Computational Domain

During mesh generation, a region-wide background grid region is

established first. Then the grids are encrypted by snappyHexMesh

method in OpenFOAM. This process is to better capture the force on

NACA66 two-dimensional hydrofoil and the cavitation on the back

side of the foil.

The sketch map of the grids in the computational domain is shown in

Fig. 4. The mesh system contains three parts. The first part is the outer

mesh which is the initial mesh for the whole flow field. In the adaptive

mesh generation method, the difference of encryption levels is reflected

in the grid density in the calculation process, but there is no difference

in the initial grid.

(ⅰ) NACA66 hydrofoil with static grids

(ⅱ) NACA66 hydrofoil with adaptive grids

Fig. 4 Computational mesh around NACA66 hydrofoil

RESULT AND DISSCUSION

Unsteady Cavitation Simulation

Figure 5 shows the cloud pattern of different mesh generation methods,

and six typical moments in a cycle to compare with the experiment

have been selected. Under the action of re-entrant jet, the velocity

gradient and pressure gradient of the hydrofoil surface are very large,

and the deformation of the phase interface is very fast. At this time, the

traditional static grid often cannot provide enough grid resolution to

reflect the local details at the moment of cavitation. In this case, the use

of adaptive mesh can not only significantly improve the mesh size, but

also simulate the shape of the sheet cavity more precisely. It can be

seen from the comparison of the figure below that both the static grid

and the adaptive grid can simulate the process of cavitation

development and shedding, but in the calculation results of the adaptive

grid, the overall length and shedding phenomenon of cavitation are

more close to the experimental results, especially in capturing the water

vapor mixing interface.

(ⅰ) Simulation of cavitation with static grids

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(ⅱ) Simulation of cavitation with adaptive grids in 3 level

(ⅲ) Simulation of cavitation with adaptive grids in 4 level

(ⅳ) Experimental results

Fig. 5 Change of cavitation morphology with different mesh generation

methods at 4 degree of attack

Comparing Fig. 6 (a) and Fig. 6 (b), it can be seen that in the local area

where the sheet cavitation is about to shed, the shape of the cavitation

interface is not only complex but also has a large change rate, which is

more sensitive to grid scale and density. At this time, it is particularly

important to capture the deformation process through local encryption.

Upgrading the encryption level does not cause a significant increase in

the calculation cost, but the effect of improving the simulation accuracy

is relatively large.

In order to capture the collapse and deformation process of cloud

cavitation, the traditional static grid usually sets the encryption area

near the wall. It needs to extend the encryption area to a considerable

distance from the back of the hydrofoil, which will also lead to a

significant increase in the calculation cost. By tracking the deformation

process of cloud cavitation interface with adaptive grid, it is

unnecessary to set global encryption area at the back of hydrofoil.

Adaptive grid automatically encrypts the grid with volume fraction

between 0 and 1 in the coarse grid area. In this way, the simulation

accuracy is improved and the calculation cost is generally lowered.

Figure 6 shows the process of cavitation during breaking away from the

hydrofoil surface to deformation and collapse. In the interior (gas phase)

and the exterior (liquid phase) of the cavity, there is no grid

densification, and the grid is relatively sparse. On the surface of the

cavity, the volume fraction is between 0 and 1. The adaptive grid

densification of this part of the area can better capture the whole

deformation process of the cavity after falling off. After leaving the

surface of the hydrofoil, the cavitation enters the high-pressure area and

deforms under the action of external pressure. The volume of the

cavitation decreases rapidly. Figure 6 (ⅱ) (d) shows the local grid at the

moment of the collapse of the cavitation. From Figure 6 (ⅱ) (c) to

figure 6 (ⅱ) (d), it shows that after the collapse of the cavitation, the

two-phase interface cannot be distinguished completely and becomes a

mixture of water and steam. The adaptive grid encrypts the whole part

of the area.

(ⅰ) Shedding of cavitation with static grids

(ⅱ) Shedding of cavitation with adaptive grids

Fig. 6 Interface tracking of cavity shedding

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Hydrodynamic Characteristics

The results of numerical simulation of hydrofoil hydrodynamic

coefficients by static grid and adaptive grid are as follows. The lift

coefficient and drag coefficient obtained by the two methods are in

good agreement with the experimental results. The accuracy of the

hydrodynamic performance calculated by the adaptive grid is slightly

different with different encryption levels, though the difference is not

obvious.

Fig. 7 Variation of lift and drag coefficient with cavitaion number.

The SST Turbulence model using RANS method has some advantages

in the calculation of the average lift resistance coefficient because it

treats different volume vortices equally and takes the time average.

However, it has disadvantages in capturing the structure of small

vortices, and the sudden change of lift coefficient is not obvious. This

has been introduced in the previous research, and will not be described

here.

Analysis about the Mechanism of Periodical Change in

Cavitation

The contraction of the joint between the end of the inner cavity and the

hydrofoil indicates that there is a re-entry jet, which causes the small

bubbles at the end of the inner cavity to fall off. The inflow air flows

into the middle of the hydrofoil, resulting in unstable cloud cavitation,

which moves downward from the hydrofoil to form obvious cloud

droplets. The results of the calculation basically describe the fracture

and shedding behavior in the cavitation process, which is in good

agreement with the experimental results.

(ⅰ) Re-entrant jet

(ⅱ) Experimental results

Fig. 8 Velocity vector of the flow fields.

The velocity vector diagram during cavitation separation is shown in

Figure 8. Based on the analysis of the velocity vector diagram, the

vortices of cavitation and wall edge re-entering the tail region during

the development of sheet cavitation are predicted accurately. The

vortex structure leads to the re-entrant jet, and cavitation clouds are

produced due to the shear effect during the collision. Therefore, vortex

is actually the cause of cavitation.

CONCLUSIONS

In the present paper, a numerical simulation of cloud cavitation around

NACA66 hydrofoil based on an adaptive mesh ganeration method is

carried out. The calculated results have been analyzed and compared

with experomental results and the static mesh generation method.

(1) Generally, both methods can simulate the growth, shedding and

collapse of cloud cavitation. The calculated results including

cavitation shape and hydrodynamic characteristics are in agreement

with the experimental ones.

(2) By analyzing the numerical results in the cavitation area at the tail

of hydrofoil, it is found that the adaptive grid method is a more

efficient way to encrypt the water-vapor interface and save more

computation in other areas

(3) Further studies on the re-entrant jet show that the vortex structure at

the tail of the hydrofoil is the main cause of cavitation shedding.

ACKNOWLEDGEMENTS

This work is supported by the National Natural Science Foundation of

China (51879159), The National Key Research and Development

Program of China (2019YFB1704200, 2019YFC0312400), Chang

Jiang Scholars Program (T2014099), Shanghai Excellent Academic

Leaders Program (17XD1402300), and Innovative Special Project of

Numerical Tank of Ministry of Industry and Information Technology

of China (2016-23/09), to which the authors are most grateful.

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