Numerical Simulation of Cavitation Flows around NACA66 ...cases like supercavitation torpedo. At...
Transcript of Numerical Simulation of Cavitation Flows around NACA66 ...cases like supercavitation torpedo. At...
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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