Post on 16-Apr-2018
International Journal of Multiphase Flow 90 (2017) 102–117
Contents lists available at ScienceDirect
International Journal of Multiphase Flow
journal homepage: www.elsevier.com/locate/ijmulflow
Multiscale tow-phase flow modeling of sheet and cloud cavitation
Chao-Tsung Hsiao
∗, Jingsen Ma , Georges L. Chahine
Dynaflow, Inc., 10621-J Iron Bridge Road, Jessup, MD, USA
a r t i c l e i n f o
Article history:
Received 20 July 2016
Revised 20 October 2016
Accepted 25 December 2016
Available online 5 January 2017
a b s t r a c t
A multiscale two-phase flow model based on a coupled Eulerian/Lagrangian approach is applied to cap-
ture the sheet cavitation formation, development, unsteady breakup, and bubble cloud shedding on a
hydrofoil. No assumptions are needed on mass transfer. Instead natural free field nuclei and solid bound-
ary nucleation are modelled and enable capture of the sheet and cloud dynamics. The multiscale model
includes a micro-scale model for tracking the bubbles, a macro-scale model for describing large cavity dy-
namics, and a transition scheme to bridge the micro and macro scales. With this multiscale model small
nuclei are seen to grow into large bubbles, which eventually merge to form a large scale sheet cavity. A
reentrant jet forms under the sheet cavity, travels upstream, and breaks the cavity, resulting in the emis-
sion of high pressure peaks as the broken pockets shrink and collapse while travelling downstream. The
method is validated on a 2D NACA0015 foil and is shown to be in good agreement with published exper-
imental measurements in terms of sheet cavity lengths and shedding frequencies. Sensitivity assessment
of the model parameters and 3D effects on the predicted major cavity dynamics are also discussed.
© 2017 Elsevier Ltd. All rights reserved.
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1. Introduction
Various types of cavitation occur over a propeller blade. These
include traveling bubble cavitation, tip vortex cavitation, sheet cav-
itation, cloud cavitation, supercavitation, and combinations of the
above. Among these, the transition from sheet to cloud cavitation
is recognized as the most erosive and is quite difficult to accurately
model. The sheet cavity starts near the leading edge, effectuates
large amplitude length/volume oscillations, and breaks up periodi-
cally releasing clouds of bubbles and cavitating vortical structures,
which collapse violently generating high local pressures on the
blade. Accurate simulation and modeling of the physics involved
is challenging since the problems of sheet cavity dynamics, cloud
cavity dynamics, and sheet-to-cloud transition involve length and
time scales of many different orders of magnitude.
Experimental observations have indicated weak or little de-
pendence of the sheet cavity inception and dynamics on the nu-
clei distribution in the test facility, which has led many numer-
ical modelers to believe that sheet cavitation is not connected
with bubble dynamics. Actually, over the years, several researchers
(e.g. Kodama et al., 1981 ; Kuiper, 2010 ) have observed that the
presence of free field nuclei affected significantly the repeata-
bility of the experimental results. In experiments where nuclei
count was very low the value of the incipient cavitation number
∗ Corresponding author.
E-mail addresses: ctsung@dynaflow-inc.com (C.-T. Hsiao), jingsen@dynaflow-
inc.com (J. Ma), glchahine@dynaflow-inc.com (G.L. Chahine).
(
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b
http://dx.doi.org/10.1016/j.ijmultiphaseflow.2016.12.007
0301-9322/© 2017 Elsevier Ltd. All rights reserved.
howed large scatter, whereas in water rich in nuclei the results
ere more repeatable and were not affected by further addition
f nuclei. One potential explanation to the above observations is
hat nucleation from the solid surface of the blades could play a
uch more important role than the free field nuclei, and this may
ave been overlooked in previous studies of sheet cavitation even
hough bubble nucleation has been extensively investigated (e.g.
arvey et al., 1944 ; Yount, 1979; Mørch, 2009 ).
Conventional numerical approaches for modeling sheet cavi-
ies cannot address travelling bubble cavitation or sheet cavita-
ion inception. Instead, they assume that the cavity already exits,
s filled with vapor at the liquid vapor pressure and thus arbitrar-
ly define as a liquid-cavity interface the liquid iso-pressure sur-
ace with the vapor pressure value. The flow solution in the liquid
s obtained using either a Navier–Stokes solver ( Deshpande et al.,
993 ; Chen and Heister, 1994 ; Hirschi et al., 1998 ) or a potential
ow solver using a boundary element method ( Kinnas and Fine,
993 ; Chahine and Hsiao, 20 0 0 ; Varghese et al., 20 05 ). Since the
eveloped sheet cavity is very unsteady and dynamic, numerical
ethods requiring a moving grid to resolve the flow field are ac-
urate but encounter numerical difficulties to describe folding or
reaking up interfaces. Other approaches have thus been devel-
ped to avoid using moving grids to track the liquid-gas inter-
ace. They are mainly in two categories: Volume of Fluid methods
VoF) ( Merkle et al., 1998 ; Kunz et al., 20 0 0 ; Yuan et al., 2001 ;
inghal et al., 2002 ) and Level-Set Methods (LSM) ( Kawamura and
akoda, 2003 ; Dabiri et al., 2007 , 2008, Kinzel et al., 2009 ). While
oth interface capturing methods solve the Navier–Stokes equa-
C.-T. Hsiao et al. / International Journal of Multiphase Flow 90 (2017) 102–117 103
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ions using a fixed computational grid, the liquid/gas interface is
aptured differently. In the VoF method, an advection equation
or the volume fraction of the vapor is solved and the two-phase
edium density is computed knowing the volume fraction of each
hase. This method applies a mass transfer model between the liq-
id and the cavity to model liquid/vapor exchange at the interface
sing ‘calibration’ parameters to match experimental results. In the
evel set method, the interface location is determined by solving
n advection equation for the level set function with the interface
epresented by the zero level (or iso-surface) of the level-set func-
ion. In this method boundary conditions similar to those used in
n interface tracking scheme (e.g. a constant pressure value and
ero shear stress) can be imposed at the zero level set surface if a
host Fluid Method is used ( Fedkiw et al., 1999 ).
Interface capturing approaches have been successfully applied
o describe the dynamic evolution of a cavitation sheet and its
reakup but not to the dispersed bubble phase and to the res-
lution of the transition from sheet to bubbles since this would
equire an extremely fine spatial resolution relative to bubble size
Fedkiw et al., 1999 ; Osher and Fedkiw, 2001 ). Resolving sheet cav-
ty together with the microbubbles involves characteristic lengths
f different orders of magnitude and using the smallest scale re-
uires computational resources not currently available.
Instead of resolving all individual bubbles, continuum-based Eu-
erian approaches ( Biesheuvel and Van Wijngaarden, 1984 ; Zhang
nd Prosperetti, 1994a,b ; Druzhinin and Elghobashi, 1998, 1999,
001 ; Ma et al., 2011, 2012 ) assuming the bubbly phase to be a
ontinuum are typically used for bubbles much smaller than the
haracteristic lengths associated with the motion of the overall
ixture. In this case, the precise location and properties of indi-
idual bubbles are not directly apparent at the global mixture flow
cale. As a result, this approach is not able to address appropri-
tely the bubble dynamics and interactions which are important in
avitation and more so in cloud cavitation.
Approaches combining both continuum and discrete bub-
les have been proposed ( Chahine and Duraiswami, 1992 ;
hahine, 2009 ; Balachandar and Eaton, 2010 ; Raju et al., 2011 ;
hams et al., 2011 ; Hsiao et al., 2013 ). In these approaches the
arrier fluid is modeled using standard continuum fluid methods
Eulerian approach) and dispersed bubbles are tracked directly (La-
rangian approach). The interaction between the two phases is
chieved through a locally distributed force field imposed in the
omentum equation ( Ferrante and Elghobashi, 2004 ; Xu et al.,
002 ) or through the inclusion of local density variation due to
ubble motion and dynamics ( Chahine, 2009 ; Raju et al., 2011 ;
siao et al., 2013 ; Ma et al., 2015a ).
We have recently developed a multiscale method, which aims
t addressing all the various scales of the problem; notably a
acro-scale at the order of the blade length, a micro-scale for the
ubble nuclei, an intermediary scale for the bubble clouds, and
ransition and coupling procedures between these scales. To in-
lude sheet cavitation inception in the study, we use the same
odel and code that we have pioneered to study travelling bub-
le or tip vortex cavitation inception ( Chahine, 2004 ; Hsiao and
hahine, 2004 ; Hsiao and Chahine, 20 08 ; Chahine, 20 09 ) and en-
ance it to include a two-phase medium as well as large cavi-
ies. This starts from the precept that water always contains micro-
copic bubble nuclei and solid liquid interfaces do not have perfect
ontact and thus entrap nanoscopic or microscopic gas pockets,
hich are a source of nucleation (freeing of nuclei into the liquid)
hen appropriate local conditions occur. Cavitation inception of all
ypes is initiated through explosive growth of these nuclei. With
uch nucleation models, the multiscale two-phase flow model, as
resented in this paper, not only can simulate the sheet cavita-
ion accurately without the assumption of mass transfer, it can
lso allow us to simulate different nuclei conditions which may be
ncountered in different experimental applications. Another ma-
or advantage of the current multiscale two-phase flow model is
ts use of an Eulerian-Lagrangian coupled approach, which allows
ne to account for the unresolved phase. While most of the pre-
ious studies have investigated statistically averaged dynamics ig-
oring the slip velocity between the bubble and liquid phases, the
ulerian–Lagrangian approach is also able to include slip velocity
etween phases, which can be large and is essential for accuracy.
. Numerical model
The sketch in Fig. 1 presents an overview of the multiscale
odel developed in this study. This model aims at addressing the
hysics at the various scales involved in sheet and cloud cavitation:
• At the micro-scale, transport of nuclei and microbubbles, nucle-
ation from solid surfaces, and corresponding bubble dynamics
are considered. The model addresses dispersed pre-existing nu-
clei in the liquid, nuclei originating from solid boundaries, and
micro-bubbles resulting from disintegration of sheet and cloud
cavities or bubbles breaking up from vaporous/gaseous cavities.
At this scale, the model tracks the bubbles in a Lagrangian fash-
ion. • At the macro-scale, a two-phase continuum-based flow is
solved on an Eulerian grid. At this scale gas/liquid interfaces of
large bubbles, air pockets, and cavities are directly discretized
and resolved through tracking the gas-liquid interfaces with a
Level Set method. • Inter-scale transition schemes are used to bridge micro and
macro scales as bubbles grow or merge to form large cavi-
ties, or as bubbles shrink or break up from a large cavity. At
this scale, model transition between micro-scale and macro-
scale and vice-versa is realized by using information from both
the Discrete Singularity Model (DSM) and the Level Set method
(LSM).
To handle the interaction between the unresolved bubbles
nd the two-phase medium, a coupled Eulerian/Lagrangian two-
hase flow model is used. The Eulerian approach addresses the
ontinuum-based two-phase model while the Lagrangian approach
racks sub-grid bubble motion and dynamics. This method was ini-
ially developed to simulate directly cavitation inception and noise
or traveling and tip vortex cavitation ( Chahine, 2004 ; Hsiao and
hahine, 2004 ; Hsiao and Chahine, 2008 ), bubble dynamics in a
ubbly medium ( Raju et al., 2011 ; Ma et al., 2015a ), and bubble
loud dynamics ( Ma et al., 2015b ).
.1. Eulerian continuum-based two-phase model
The two-phase medium continuum model uses a viscous flow
olver to solve the Navier–Stokes equations and satisfy the conti-
uity and momentum equations:
∂ ρm
∂t + ∇ · ( ρm
u ) = 0 , (1)
m
D u Dt
= −∇ p + ∇
{τi j − 2
3 μm
( ∇ · u ) δi j
},
τi j = μm
(∂ u i ∂ x j
+
∂ u j ∂ x i
),
(2)
here the subscript m refers to the mixture properties. u is the
ixture velocity, p is the pressure, and δij is the Kronecker delta.
he mixture density, ρm
, and the mixture viscosity, μm
, can be ex-
ressed as functions of the void volume fraction α:
m
= ( 1 − α) ρl + αρg , μm
= ( 1 − α) μl + αμg , (3)
here the subscript l refers to the liquid and the subscript g refers
o the gas.
104 C.-T. Hsiao et al. / International Journal of Multiphase Flow 90 (2017) 102–117
Fig. 1. Overview of the various scales involved in the problem of cavitation on a foil and sketch of the modeling strategy.
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The equivalent medium has a time and space dependent den-
sity since the void fraction α varies in both space and time. This
makes the overall flow field problem similar to a compressible flow
problem, but wave phenomena such as shocks emitted by collaps-
ing bubble will not be accurately predicted as the present approach
is not aiming for such flow details. In our approach, which cou-
ples the continuum medium with the discrete bubbles, the mix-
ture density is not an explicit function of the pressure through an
equation of state. Instead, tracking the bubbles and knowing their
concentration provides α and ρm
as functions of space and time.
This is achieved by using a Gaussian distribution scheme which
smoothly “spreads” each bubble’s volume over neighboring cells
within a selected radial distance while conserving the total bubble
volumes ( Ma et al., 2015a ). Specifically, in a fluid cell i , the void
fraction is computed using:
αi =
N i ∑
j=1
f i, j V
b j ∑ N cel l s
k f k, j V
cell k
, (4)
where V b j and V cell
k are the volumes of a bubble j and cell k re-
spectively, N i is the number of bubbles which are influencing cell
i . N cells is the total number of cells “influenced” by a bubble j. f i,j is the weight of the contribution of bubble j to cell i and is deter-
mined by the Gaussian distribution function. This scheme has been
found to significantly increase numerical stability and to enable the
handling of high-void bubbly flow simulations.
The system of equations is closed by an artificial compressibil-
ity method ( Chorin, 1967 ) in which a pseudo-time derivative of
the pressure multiplied by an artificial-compressibility factor, β , is
added to the continuity equation as
1
β
∂ p
∂τ+
∂ ρm
∂t + ∇ · ( ρm
u ) = 0 . (5)
As a consequence, a hyperbolic system of equations is formed
and can be solved using a time marching scheme. This method
can be marched in pseudo-time to reach a steady-state solution.
To obtain a time-dependent solution, a Newton iterative procedure
is performed at each physical time step in order to satisfy the con-
tinuity equation. It is noted that using the artificial compressibil-
ity method alters the speed of sound and thus enables to use the
scheme in the low Mach number regime such as the current prob-
lem for the flow over a hydrofoil.
The Navier–Stokes solver uses a finite volume formulation.
First-order Euler implicit differencing is applied to the time deriva-
tives. The spatial differencing of the convective terms uses the flux-
ifference splitting scheme based on Roe’s method ( Roe, 1981 ) and
an Leer’s MUSCL method ( van Leer, 1979 ) for obtaining the first-
rder and the third-order fluxes, respectively. A second-order cen-
ral differencing is used for the viscous terms which are simplified
sing the thin-layer approximation. The flux Jacobians required in
n implicit scheme are obtained numerically. The resulting system
f algebraic equations is solved using the Discretized Newton Re-
axation method ( Vanden and Whitfield, 1995 ) in which symmetric
lock Gauss–Seidel sub-iterations are performed before the solu-
ion is updated at each Newton iteration.
.2. Level-set approach
In order to enable the multiscale code to simulate large free
urface deformations such as folding and breakup in the sheet
avity problem, a Level-Set method using the Ghost Fluid Method
Fedkiw et al., 1999 ; Kang et al., 20 0 0 ) is used. A smooth function
( x,y,z,t ), whose zero level coincides with the liquid/gas interface
hen the level set is introduced, is defined in the whole physical
omain (i.e. in both liquid and gas phases) as the signed distance
( x,y,z ) from the interface:
( x, y, z, 0 ) = d ( x, y, z ) . (6)
This function is enforced to be a material surface at each time
tep using:
dϕ
dt =
∂ϕ
∂t + u ∇ϕ = 0 , (7)
here u is the velocity of interface. Integration of Eq. (7) does not
nsure that the thickness of the interface region remains constant
n space and time during the computations due to numerical dif-
usion and to distortion by the flow field. To avoid this problem, a
ew distance function is constructed by solving a “re-initialization
quation” in pseudo time iterations ( Sussman, 1998 ):
∂ϕ
∂τ= S ( ϕ 0 ) [ 1 − | ∇ϕ | ] , (8)
here τ is the pseudo time, ϕ0 is the initial distribution of ϕ, and
( ϕ0 ) is a sign function, which is zero at the interface.
Eqs. (7) and (8) are solved with a Ghost Fluid Method impos-
ng boundary conditions at the interface after identifying to which
hase a concerned cell belongs. Fig. 2 illustrates the identification
f the cells. When one of the following six inequalities is satisfied
C.-T. Hsiao et al. / International Journal of Multiphase Flow 90 (2017) 102–117 105
Fig. 2. Cell location identification in the computational domain.
f
ϕ
ϕ
ϕ
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“
or two adjacent cells:
i, j,k ϕ i ±1 , j,k < 0 ,
i, j,k ϕ i, j±1 ,k < 0 ,
i, j,k ϕ i, j,k ±1 < 0 ,
(9)
he zero level-set interface is identified.
In this approach two ghost cells are used to impose the bound-
ry conditions at the interface cells without use of smoothing func-
ions. If the shear due to the air is neglected, the dynamic bound-
ry conditions (balance of normal stresses and zero shear) can be
ritten:
p =
n � τ � n
ρl
+
gz
ρl
+
γ κ
ρl
, n � τ � t = 0 , (10)
here τ is the stress tensor, g is the acceleration of gravity, and γs the surface tension.
=
∇ �∇ϕ
| ∇ϕ | (11)
s the surface curvature. n and t are the surface normal and tangen-
ial vectors, respectively.
Although the Ghost Fluid Method can help maintain a sharp in-
erface, additional computational cost is required for the cells in
he ignored phase, e.g. air. To further reduce the computational
ost we apply a single-phase level-set approach in which only the
iquid phase of the fluid is solved, while the cells belonging to
he gas phase are deactivated during the computations. To solve
q. (7) we use the velocity information from the gas cells which
re near the interface. To obtain this information, we assume that
he following condition across the interface applies:
�∇u = 0 , with n = ∇ϕ / |∇ϕ | , (12)
hich means there is no normal component of the velocity gra-
ient at the interface. With this assumption we can extend the
elocity from the liquid phase along the normal direction of the
istance function to the gas phase by solving the pseudo time it-
ration equation:
∂u
∂τ+ n �∇u = 0 . (13)
.3. Lagrangian discrete bubble model
The Lagrangian discrete bubble model, which enables tracking
nitially sub-grid nuclei in the flow, identifying when explosive
rowth (inception) and collapse (erosion) occur, is based on the
iscrete singularity model which uses a Surface Average Pressure
SAP) approach ( Hsiao et al., 20 03 ; Chahine, 20 04 ; Choi et al.,
004 ) to average fluid quantities along the bubble surface. This
odel has been shown to produce accurate results when com-
ared to full 3D two-way interaction computations ( Hsiao et al.,
003 ). The averaging scheme allows one to consider only a spheri-
al equivalent bubble and use a modified Rayleigh-Plesset equation
o describe the bubble dynamics,
l
(R ̈R +
3
2
˙ R
2 )
= p v + p g0
(R 0
R
)3 k
− p enc − 2 γ
R
− 4 μ ˙ R
R
+ ρl
| u s | 2 4
,
(14)
here the slip velocity, u s = u enc − u b , is the difference between
he encountered (SAP averaged) liquid velocity, u enc , and the bub-
le translation velocity, u b . R and R 0 are the bubble radii and time
and 0, p v is the liquid vapor pressure, p g0 is the initial bubble gas
ressure, k is the gas polytropic compression constant, and p enc is
he average SAP pressure “seen” by the bubble during its travel.
ith the SAP model, u enc and p enc are respectively the averages of
he liquid velocities and pressures over the bubble surface.
The bubble trajectory is obtained from the following bubble
otion equation:
d u b
dt =
(ρl
ρb
)[3
8 R
C D | u s | u s +
1
2
(d u enc
dt − d u b
dt
)
+
3 ̇
R
2 R
u s − ∇p
ρl
+
( ρb − ρl )
ρl
g +
3 C L 4 π
√
ν
R
u s × √ | |
]
, (15)
here ρb is the density of the bubble, C D is the drag coefficient
iven by an empirical equation, e.g. from Haberman and Mor-
on (1953) , C L is the lift coefficient and is the vorticity vector.
he first right hand side term is a drag force. The second and third
erms account for the added mass. The fourth term accounts for
he presence of a pressure gradient, while the fifth term accounts
or gravity and the sixth term is a lift force ( Saffman 1965 ).
.4. Transition model between micro- and macro-scale
The transition between micro- and macro-scale models includes
wo scenarios:
a. transition from collected coalescing microbubbles into relatively
large two-phase cavities,
b. transition from a large cavity into a set of micro-scale dispersed
bubbles.
In the first scenario a criterion based on bubble / cell relative
ize is set to “activate” bubbles for computation of a level set func-
ion:
≥ max ( R thr , m thr �L ) , (16)
here R thr is a threshold bubble radius, �L is the size of the local
rid which hosts the bubble, and m thr is a threshold grid factor.
herefore a bubble interface is activated and is represented by the
ero level set when the bubble radius exceeds a threshold value
nd becomes larger than a multiple of the local grid size.
At the beginning of the simulation where no cavity exists, all
ells have the same level set value, ϕLS 0 , fixed to be a very large
egative value. When bubbles are detected to satisfy Eq. (16) , the
evel set value of each cell in the computation domain is modified
ased on the distance from each cell center to the bubbles’ sur-
aces, using the following equation:
i = min ( ϕ LS0 , ϕ b, j ) ; j = 1 , N i , (17)
here ϕb, j is the local distance function computed based on the
urface of bubble j , and N i is the number of bubbles which are
activated” around cell i .
106 C.-T. Hsiao et al. / International Journal of Multiphase Flow 90 (2017) 102–117
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s
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3
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s
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a
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b
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2
(
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v
This scheme allows multiple bubbles to merge together into a
large cavity, or isolated bubbles to coalesce with a previously de-
fined large cavity.
In the second scenario, we use the zero level-set to identify
the gas/liquid interface of the cavity. The volume of the cavity is
tracked at each time step to determine when the cavity collapse
occurs. As the cavity collapses, surface instabilities and cavity dis-
integration are accounted for using empirical criteria based on ex-
perimental observations such as described in Ma et al. (2011) and
Hsiao et al. (2013) . The model assumes that turbulence at the free
surface produces a rough gas/liquid interface, which consists of a
bubbly layer with microbubbles of size, a . These cavities will be
entrained into the liquid once the local downward liquid velocity
of this mixture layer, u n ( a /2), exceeds the mean downward veloc-
ity of the free surface, u n (0). An amount of micro-bubbles of the
same volume is initiated around the cavity to replace volume loss
between time steps. The procedure is similar to that used to simu-
late bubble entrainment in breaking waves. Micro-bubbles are en-
trained into the liquid once the local normal velocity pointing to-
wards the liquid exceeds the speed of the gas/liquid interface. This
leads to a simple expression for the location and rate of bubble
generation that is proportional to the liquid’s local turbulent ki-
netic energy times the gradient of the liquid velocity in the liquid
normal direction.
2.5. Nucleation model
The simulations presented below use free field nuclei as mea-
sured experimentally by many researchers such as Medwin (1977) ,
Billet (1985) , Franklin (1992) and Wu et al. (2010) . In our modeling
the bubbles are distributed randomly in the liquid following the
measured bubble size distribution, as more detailed in our previ-
ous work in Chahine (20 04, 20 09) and Hsiao et al. (2013) . Another
important source of nuclei for the present problem is nuclei peri-
odically released from the foil surface. Boundary nucleation is eas-
ily observed in conditions where gas diffusion (beer or champagne
glass) and/or boiling are dominant; bubbles grow out of distributed
fixed spots. This phenomenon has been studied extensively in the
past. Harvey et al. (1944) suggested that gas pockets, trapped in
hydrophobic conical cracks and crevices of solid surfaces, act as
cavitation nuclei. More recently Mørch (2009) investigated this ef-
fect in more depth and suggested that a skin model similar to that
proposed by Yount (1979) for dispersed nuclei can also be used to
describe the wall gas bubbles because solid surfaces submerged in
a liquid tend to absorb amphiphilic molecules (molecules having a
polar water-soluble group attached to a nonpolar water-insoluble
hydrocarbon chain) and capture nanoscopic air pockets. One can
expect that clusters of such molecules form weak spots, which are
the source for bubble nucleation from the solid surface when low
pressure conditions are achieved. Briggs (2004) , for instance, found
that scrupulous cleaning was decisive for obtaining a high tensile
strength of water, and that contamination of interfaces is a primary
factor in cavitation. Atchley and Prosperetti (1989) modeled nucle-
ation out of crevices with liquid pressure drop. They identified un-
stable motion of the liquid-solid-gas contact line in the crevice and
unstable growth of the nucleus volume due to loss of mechani-
cal stability (force balance). Specifically, they related the threshold
pressure to crevice aperture, surface tension, and gas concentration
etc.
Based on existing experimental observations and theoretical
studies, the present work uses a nucleation model with the fol-
lowing characteristics parameters, which all are physical properties
characteristic of the foil material and should be deduced from im-
proved future characterization of material surfaces:
• P : a nucleation pressure threshold,
thr• N s: a number density of nucleation sites per unit area, • f n: a nucleation time emission rate, function of the local flow
conditions, • R 0 : initial nuclei size, function of the local surface conditions .
In this nucleation model, at each time step, nuclei are released
rom the foil boundary cell when the pressure at the cell center
rops below a threshold value, P thr . The cell of surface area �A ,
hen releases N nuclei during the time interval �t , obtained as fol-
ows:
= N s f n �t�A. (18)
Since the Lagrangian solver (bubble tracking) usually requires
ime steps much smaller than the Eulerian flow solver, the N nuclei
an be released randomly in space from the area �A and in time
uring the Eulerian time step, �t .
It is expected that the above parameters are also functions of
urface roughness, S, temperature, T , and other physico-chemical
arameters not explicitly considered here. In this study, we use
his simplified nucleation model to investigate whether it enables
ith the above described overall multiscale approach to recover
he main characteristics of sheet cavitation.
. Sheet cavitation on a 2D hydrofoil
.1. Case setup
We study first the development of a sheet cavity over a 2D
ACA0015 hydrofoil to study the sensitivity of the solution to the
alues of the parameters used in the numerical model. An H–H
ype grid is generated with 181 × 101 grid points in the stream
ise and normal direction, respectively. Only 2 grid points are used
long the span wise direction to enforce two-dimensionality. 101
rid points are used on each side of the hydrofoil surface and non-
lip wall boundary conditions are imposed there. The grid is sub-
ivided into 12 blocks for a computational domain which extends
.0 chord-length upstream and 5.0 chord-length downstream. The
rid also extends 2.0 chord-length vertically away from the foil sur-
ace where far-field boundary conditions are applied. The grid is
esigned to provide high clustering around the hydrofoil surface
s shown in Fig. 3 . In particular, the first grid above the hydrofoil
urface is located within y + < 1 in order to properly capture the
oundary layer. The unsteady viscous flow field is simulated di-
ectly using the Navier–Stokes equations without applying any tur-
ulence modeling. Grid resolution was determined following grid
onvergence studies in previous work.
In this section, the foil is considered for the angles of attack
f 8 ° and 10 °. The chord length is 0.12 m, and the incoming flow
rom far-field upstream has a mean velocity of 10 m/s, resulting in
Reynolds number of 1.2 × 10 6 . The nuclei are dispersed in the
ulk of the liquid and are also released from the surface of the foil
ccording to the solid surface nucleation model described above.
he field nuclei are here selected to have radii ranging from 20 μm
o 60 μm, and correspond to an average initial void fraction of
.5 × 10 −5 .
Nucleation from the solid wall is initiated when the local pres-
ure drops below a critical value, here selected to be the vapor
ressure. The initial size of the nuclei emitted is selected to be
0 μm and the number of bubbles is determined by Eq. (18) with
he number of nucleation sites per unit area selected to be N s =24 /c m
2 and the nucleation frequency selected to be f n = 22 kHz
these choices are further discussed later). The microbubbles be-
ome a cavity described by a Level Set surface when they grow
arger than the size of the local grid and exceed 350 μm ( ∼10
imes the median initial nuclei size). The effects of these selected
alues of the parameters on the solution are investigated below.
C.-T. Hsiao et al. / International Journal of Multiphase Flow 90 (2017) 102–117 107
Fig. 3. Zoomed-in view of the multi-block grid used in this study for the simulation of sheet cavitation on a NACA0015 hydrofoil.
Fig. 4. Importance of the inclusion of wall nucleation on the proper modeling of
the sheet cavity using the Eulerian–Lagrangian and level set methods. Field nu-
clei alone do not reach the wall or grow enough to fill the cavity. σ = 1.65, α = 8 °, U ∞ = 10 m/s, L = 0.12 m, N s = 224 cm
−2 , F n = 22 kHz.
v
i
i
t
o
c
a
t
i
t
v
p
i
g
T
d
3
c
N
Fig. 5. Time sequence of sheet cavitation formation and cloud shedding with pres-
sure contours. ( σ = 1.65, α = 8 °, U ∞ = 10 m/s, L = 0.12 m). The white dots are discrete
bubbles and the white lines are the zero level-set lines denoting the liquid-vapor
interface.
Comparison of the numerical results with experimental obser-
ations are very satisfactory only when wall nucleation is taken
nto account, as described in more detail in the next section. This
s illustrated in Fig. 4 , which clearly shows a major difference be-
ween time average cavity sizes when wall nucleation is included
r when it is ignored. Accounting for field nuclei only is unable to
apture the sheet length properly unless the void fraction is rel-
tively high (above 10 −3 ). This is very much in agreement with
he experimental observations. Including the wall nucleation alone
s capable of capturing the sheet length but is insufficient to cap-
ure properly the events behind the cavity closure such as the de-
elopment of cloud cavities and cavitating vortical structures. This
rovide important insight into the physics of sheet cavities, imply-
ng that sheet inception and dynamics is mostly controlled by the
rowth and dynamics of nucleated bubbles from the foil surface.
his physics is in addition slightly effected by the presence and
ynamics of free field nuclei.
.2. Dynamics of the sheet cavity
A time sequence of sheet cavity shape oscillation and cloud
avitation shedding for one cycle is shown in Fig. 5 for σ = 1.65.
ucleation initiates at the hydrofoil surface near the leading edge
108 C.-T. Hsiao et al. / International Journal of Multiphase Flow 90 (2017) 102–117
Fig. 6. Zoom views of the velocity field and pressure contours around the sheet
cavity (Top) and a zoom view of the red-boxed region of the streamlines in the
reverse re-entrant jet flow region (Bottom).
Fig. 7. Time histories of the pressures and stream wise velocities at x = 0.4 L at the
solid wall near the sheet cavity trailing edge. σ = 1.65.
3
3
p
e
t
e
o
i
c
t
N
c
s
f
i
w
u
b
f
d
f
w
l
t
t
e
t
s
c
t
a
l
F
b
p
3
s
where the pressure drops below the selected nucleation threshold,
p thr = p v . Initially, bubble nuclei from the bulk of the liquid being
convected downstream, as well as nuclei released from the solid
surface, grow, coalesce and form a discrete small cavity described
with a level set function. Later, free field nuclei have difficulty to
enter in the cavity just move downward, while wall nucleation
keeps feeding vapor and gas into the sheet cavity which grows to
become a fully developed sheet cavity.
At the back of the cavity a stagnation flow forms ( Shen and Di-
motakis, 1989 ) as shown in the zoomed view in Fig. 6 . The reverse
flow results in a re-entrant jet, which moves upstream under the
sheet cavity and ultimately contributes to breaking the cavity into
two. The shed cavity collapses as it travels downstream generat-
ing high pressures. At the same time a new sheet cavity forms and
expands, and the cycle repeats.
This periodic shedding phenomenon can be further quantified
by analyzing the time histories of the instantaneous pressures and
stream wise velocities in the flow. For example, Fig. 7 shows the
pressures and velocities monitored at the first grid point in the liq-
uid on the suction side of the solid surface at the location x = 0.4 L
downstream of the leading edge. This is at about the same location
as the maximal extent of the trailing edge of the fully developed
sheet cavity. This location alternatively experiences positive and
negative stream-wise velocities with magnitudes as high as the in-
coming flow, U ∞
. The time interval between consecutive negative
velocity maxima is repeatable and has a value close to 1.15 L / U ∞
.
The corresponding oscillation frequency is about 0.86 U ∞
/ L , which
matches very well with the experimentally measured shedding fre-
quency for the same cavitation number ( Berntsen et al., 2001 ).
Fig. 7 also shows that each cycle of flow velocity oscilla-
tions coincides with the alternation of peaks and minimum pres-
sure values with the same oscillation frequency. This illustrates a
strong correlation between reentrant jet initiation and the develop-
ment of pressure fluctuations during each cycle of the sheet cavi-
tation.
.3. Sensitivity study of model parameters
.3.1. Nucleation model parameters
Since we do not have actual measured values of the nucleation
arameter, which we hope will become accessible in the future, we
xamine here the effects of the selection of the numerical values of
hese parameters on the simulation results. The nucleation param-
ters f n and N s , which express the number density and frequency
f the nuclei emitted from the wall, as expressed by Eq. (18) , are
nvestigated.
The left column of Fig. 8 compares the time-averaged sheet
avity shapes for different assumed nucleation rates, f n , and for
he same value of the number of nucleation sites per unit area,
s = 224 cm
−2 . The right column of Fig. 8 , on the other hand,
ompares the averaged shapes for different assumed nucleation
ites per unit area, N s , for the same value of the nucleation rate,
n = 22 kHz. The same information is shown in quantitative form
n Fig. 9 . It is seen that the time-averaged cavity length increases
ith the nucleation rate when f n is small, but then remains almost
nchanged for values of f n larger than 22 kHz. A similar trend can
e seen when N s is increased for a constant nucleation frequency,
n .
The time-averaged cavity length increases with the number
ensity N s when N s is small, but then remains almost unchanged
or values of N s larger than 112 cm
−2 . These trends imply that
hen the nucleation rate and nucleation sites per unit area are
arger than some values, the cavity length reaches a saturation sta-
us and shows negligible dependency on those parameters. Since
hese two parameters are associated with the solid surface prop-
rties, the implication of the results is that surface treatment of
he lifting surface can significantly influence the development of
heet cavitation. Also, with common metallic surfaces with no spe-
ial treatment the number of available nuclei sites is large enough
o be in the saturation region. This picture should change with new
nd future materials with special surface treatment.
The same can be seen in Fig. 9 concerning the effect of the se-
ection of the radius, R 0 , of the bubbles nucleated from the wall.
or the whole range of radii tested between 10 μm and 100 μm,
oth the cavity length and the frequency of oscillations remain ap-
roximately the same.
.3.2. Transition model parameters
In the model for transition between micro scale and macro
cale, criteria are introduced to switch discrete singularity bub-
C.-T. Hsiao et al. / International Journal of Multiphase Flow 90 (2017) 102–117 109
Fig. 8. Effect of varying f n (left column) and N s (right column) on the time-averaged cavity shape for σ = 1.65.
Fig. 9. Effects of varying the wall nucleation parameters f n , N s and R 0 , on the normalized a) time-averaged cavity length, and b) cavity shedding frequency for σ = 1.65.
( f n,0 = 22 kHz, N s,0 = 224/cm
2 , R 0,0 = 50 μm).
b
l
r
t
F
d
w
t
r
c
l
R
d
s
b
n
3
a
b
i
c
a
t
c
l
s
b
s
s
i
les to non-spherical deformable cavities represented by the zero
evel set. In order to demonstrate the generality of these crite-
ia, a parametric study is conducted on the effects on the solu-
ion of the choice of the threshold values R thr and m thr in Eq. (16) .
ig. 10 compares the time-averaged sheet length and the shed-
ing frequency for different assumed threshold bubble radii, R thr ,
ith the same value of the mesh factor, m thr = 1.0. It is seen that
he time-averaged cavity length increases as the threshold bubble
adius increases when R thr is small, but then remains almost un-
hanged when R thr exceeds 200 μm. The same figure also shows
ittle dependency on the threshold grid factor, m thr , for a constant
thr = 350 μm. The same conclusions apply for the predicted shed-
ing frequency. This strongly indicates little dependency of the re-
ults on the switching criteria as long as the switching of the bub-
les to a cavity is not done too early, i.e. when the bubbles have
ot grown significantly.
e
.4. Validation against experimental measurements
In the following, the results of the present model are compared
gainst published experimental data for different cavitation num-
ers and angles of attack. To do so, the cavitation number is varied
n the computations by varying the far field pressure, p ∞
. Fig. 11
ompares the instantaneous pressures and stream wise velocities
t x = 0.4 L for σ = 1.65 and σ = 1.50. Similar periodic oscillations of
he local pressure and reversal of the jet velocity are seen in both
ases. However, for σ = 1.50 most of the time the stream wise ve-
ocity remains negative, but from time to time it is interrupted by
harp positive peaks. This is because, at this low cavitation num-
er, the cavity extends further downstream than x = 0.4 L . As a re-
ult, during most of the time this location is below the cavity and
ees the reentrant jet. The sharp peaks appear when the sheet cav-
ty breaks down due to the periodic shedding. This can be further
xplained by the pressure curves in Fig. 11 b. The pressure mostly
110 C.-T. Hsiao et al. / International Journal of Multiphase Flow 90 (2017) 102–117
Fig. 10. Effects of varying the threshold values, R thr and m thr , in the switching from singular bubbles to large cavities, on the normalized (a) time-averaged cavity length, and
(b) cavity shedding frequency σ = 1.65, R thr,0 = 350 μm, and m thr,0 = 1.
Fig. 11. Comparison for σ = 1.65 and σ = 1.50 of the time histories of (a) the stream wise velocity and (b) the instantaneous pressure monitored at x = 0.4 L on the NACA
0015 foil. α = 8 °, U ∞ = 10 m/s, L = 0.12 m.
Fig. 12. Comparison between the cases of σ = 1.65 and σ = 1.50 of the frequency
spectra of the temporal variations of the stream wise velocity (a) and of the pres-
sure (b) monitored at x = 0.4 L at the solid wall on the suction side.
Fig. 13. Comparison of the shedding frequencies of the sheet cavity on the
NACA0015 for different angles of attack, U ∞ = 10 m/s, L = 0.12 m with the experi-
mental results summarized in Berntsen et al. (2001 ).
(
o
i
w
a
r
stays at the vapor pressure, C p = −1 . 5 , with repeated peak pressure
increases at a nearly constant frequency. The observation point for
the σ = 1.65 case does not see vapor pressure but still observes the
repeated pressure peaks at a frequency slightly higher than that
for σ = 1.50. This is made apparent in Fig. 12 , which shows the
frequency spectrum of the signal through a Fast Fourier Transform
FFT) analysis of both the re-entrant jet velocity and the pressure
scillations.
Quantitative comparisons of these frequencies with the exper-
mental data, which measured cloud shedding frequencies for a
hole set of cavitation numbers between σ = 0.85 and σ = 1.80
nd for two different angle of attacks are displayed in Fig. 13 . The
esults are presented in terms of the non-dimensional shedding
C.-T. Hsiao et al. / International Journal of Multiphase Flow 90 (2017) 102–117 111
Fig. 14. Comparison of the time-averaged cavity shapes between σ = 1.65 and
σ = 1.50 for the NACA 0015 hydrofoil. α = 8 °, U ∞ = 10 m/s, L = 0.12 m.
Fig. 15. Comparison of the time-averaged cavity length on the NACA0015 for
α = 8 ° and 10 °, U ∞ = 10 m/s, L = 0.12 m with the experimental results summarized
in Berntsen et al. (2001 ).
f
t
t
m
i
p
t
a
i
t
a
i
c
4
m
i
o
4
t
t
c
t
d
a
b
w
b
i
i
c
r
b
p
s
F
t
i
o
a
2
4
4
f
c
a
(
i
i
1
d
a
a
a
i
o
t
4
a
a
α
w
t
g
l
s
i
e
c
H
r
p
g
i
requency, Lf sheet /U ∞
, versus the ratio of the cavitation number and
he angle of attack, σ /2 α. A very good agreement can be seen be-
ween the numerical predictions and the experimental measure-
ents in the simulated range.
In addition to the shedding frequency, the time-averaged cav-
ty lengths deduced from the unsteady simulations are also com-
ared to the experimental measurements. Fig. 14 illustrate the
ime-averaged cavity shape deduced for σ = 1.65 and σ = 1.50 by
veraging the solution from the unsteady simulations after exclud-
ng the initial transient period. Fig. 15 shows a comparison with
he results summarized in Berntsen et al. (2001) . The figure shows
very good agreement between numerical predictions and exper-
mental measurements for the time-averaged cavity length for all
avitation numbers considered and for the two angle of attacks.
. Application to 3D lifting surfaces
To test capability of the developed multiscale two-phase flow
odel for simulations of sheet and cloud cavitation on 3D lift-
ng surface we present 3D simulation cases with different levels
f complexity.
.1. 3D computation of the NACA0015 foil
3D computations on a hydrofoil having the same foil shape as
he 2D NACA0015 shown above, but with a span equal to one quar-
er of the chord length, are considered. The foil uses the same
omputational domain and grid resolution as for the 2D compu-
ation except that 20 grid points are now used in the span wise
irection which is bounded by two side walls. The same bound-
ry conditions as in previous sections are applied for the far field
oundaries and a slip boundary condition is imposed on both side
alls in the 3D simulation.
A time sequence of sheet cavity development, oscillations, and
ubble cloud shedding on the foil for σ = 1.65 and α = 8 ° is shown
n Figs. 16 and 17 . It is seen that the cloud shedding dynamics
s similar to that observed in the 2D computations, i.e. the sheet
avity starts near the leading edge and gradually evolves until a
eentrant jet breaks the cavity into parts and forms a detached
ubble cloud. However, the 3D computations exhibit a more com-
licated breakup, i.e. the cavity breakup location varies along the
pan wise direction due to the 3D nature of the unsteady flow.
ig. 18 shows the time-averaged cavity shape deduced by averaging
he solution from the 3D unsteady simulations after excluding the
nitial transient period. Average lengths and oscillation frequencies
f the cavity for two 3D computations at two cavitation numbers
re included in Figs. 13 and 15 and compare quite well with the
D results.
.2. 3D oscillating finite-span hydrofoil
.2.1. Case setup
In this section we consider an oscillating finite-span hydrofoil
or which experimental data is available from Boulon (1996) . This
oncerns an elliptic hydrofoil having a NACA16020 cross-section
nd a root chord length of L = 6 cm with an aspect ratio of 1.5
based on semi-span). In order to simulate the hydrofoil oscillat-
ng in an overall fixed fluid frame, a moving overset grid scheme
s applied to the neighborhood of the foil. As illustrated in Fig. 19 a
2-block grid is generated to encompass the finite-span foil in a
omain bounded by boundaries located at about one chord length
way from the foil surface. This 12-block grid is then overset over
fixed background grid which has domain boundaries located at
bout 5 times of chord length away from the foil surface as shown
n Fig. 20 . A total of 0.25 million grid points was generated for the
verset grids with a first grid spacing of 1 × 10 −4 chord length in
he normal direction to the foil surface.
.2.2. Single-phase flow computations
We present the case of the hydrofoil in a uniform flow field at
velocity of 3.33 m/s oscillating with a time-dependent angle of
ttack, α( t ), prescribed by the sinusoidal function:
(t) = α0 + �α sin (2 π f t) , (19)
here the initial angle of attack α0 is 10 °. The oscillations ampli-
ude is �α = 5 ° and the oscillations frequency is f = 23 Hz. The sin-
le phase flow field was first simulated until the solution reached
imit cycle oscillations. Fig. 21 shows the time histories of the pres-
ure coefficient monitored at three locations, x = 0.04 L (near lead-
ng edge) and x = 0.5 L (middle chord) and x = L (at the trailing
dge). It is seen that the pressure has reached a repeated limit
ycle oscillation at the leading edge and the mid chord locations.
owever, the pressure at the trailing edge location still shows ir-
egular vortex pairing occurring upstream.
Fig. 22 shows pressure contours and the iso-pressure surface
= p v for one oscillating cycle. It is seen that a low pressure re-
ion where the flow separation occurs starts to form near the lead-
ng edge as the angle of attack increases during the cycle. The
112 C.-T. Hsiao et al. / International Journal of Multiphase Flow 90 (2017) 102–117
Fig. 16. Time sequence of the 3D sheet cavity dynamics on a NACA0015 hydrofoil for σ = 1.65, α = 8 °, U ∞ = 10 m/s, L = 0.12 m.
Fig. 17. 2D projected side views of a time sequence of 3D simulations of an oscillating sheet cavity on a NAC0015 hydrofoil for σ = 1.65, α = 8 °, U ∞ = 10 m/s, L = 0.12 m.
Fig. 18. Time-averaged cavity shapes on the NACA 0015 hydrofoil from 3D simula-
tion for σ = 1.650 α = 8 °, U ∞ = 10 m/s, L = 0.12 m.
Fig. 19. Overset 12-block grid generated for a finite-span NACA16020 elliptic hy-
drofoil.
t
s
d
t
low pressure region extends downstream and reaches a maximum
length before the hydrofoil reaches the highest angle of attack. The
length of the low pressure region reduces but its height increases
as the hydrofoil reaches the maximum angle of attack. The low
pressure region starts to shrink as the angle of attack is decreased
from the highest value and the low pressure region reduced to the
smallest size when the angle of attack reaches the lowest value.
Strong well defined vortical regions are seen to propagate down-
stream form the leading edge.
4.2.3. Simulation of sheet and cloud cavitation
Two-phase simulation to capture sheet and cloud cavitation
was initiated once the single phase flow solution reached limit cy-
cle oscillation. Free nuclei were allowed to propagate in the liq-
uid, while nucleation was also activated at the foil boundary as in
he 2D study shown earlier. Fig. 23 shows a time sequence of the
heet cavitation evolving on the suction side of the finite-span hy-
rofoil for σ = 0.9 after the two-phase simulation was started at
he non-dimensional time T = 21.42. The unsteady two-phase sim-
C.-T. Hsiao et al. / International Journal of Multiphase Flow 90 (2017) 102–117 113
Fig. 20. Overset grids with a 12-block moving grid over a fixed background grid
used for the simulation of sheet cavitation and cloud shedding on an oscillating
hydrofoil.
Fig. 21. Time histories of the pressure coefficient monitored at three locations,
x = 0.04 L (near the leading edge) and x = 0.5 L (at middle chord) and x = L (at the
trailing edge).
u
2
c
n
s
fi
d
s
t
c
s
f
c
e
c
T
a
b
Fig. 22. A time sequence of the pressure contours and an isopressure surface
( p = p v ) during one oscillating cycle of the finite span NACA16020 elliptic hydrofoil
after the unsteady solution reached limit cycle oscillation.
c
(
c
t
i
l
t
l
t
a
lation was continued up to the end of the third cycle. Figs. 24 and
5 show time sequences of the solutions for the second and third
ycles. The oscillatory behavior of the sheet cavity is observed to
ot change significantly between the second and the third cycle
ince numerical starting transition phase is completed during the
rst cycle.
A zoomed view of the root sheet cavity near the leading edge
uring the second cycle is shown in Fig. 26 . It is seen that the
heet cavity starts to form from the leading edge near the blade
ip. It then extends toward the root as the angle of attack in-
reases. However, unlike for the single phase flow solution, the
heet cavity does not reach its maximum extent before the hydro-
oil reaches the highest angle of attack. On the contrary, the sheet
avity continues to grow beyond the time the foil has the high-
st angle of attack. After reaching its maximum extent, the sheet
avity subsides without showing clear cloud shedding in this case.
his could be due to the corase grid resolution. Simulations with
much finer grid should be conducted in the future to clarify this
ehavior.
Fig. 27 shows a comparison of the time histories of the pressure
oefficient monitored at x = 0.04L (near leading edge) and x = 0.5L
middle chord) between the single-phase and the two-phase flow
omputations up to three cycles after the two-phase model was
urned on. The modifications of the pressure due to the sheet cav-
ty can be clearly seen at both locations. For the location near the
eading edge, the pressure at the hydrofoil surface becomes equal
o the vapor pressure ( C p = −0.9) once the sheet cavity reaches this
ocation. Pressure spikes are also observed as in the 2D computa-
ions due to the local breakup of the sheet cavity. For the location
t the mid-chord, the pressure is also changed by the presence of
114 C.-T. Hsiao et al. / International Journal of Multiphase Flow 90 (2017) 102–117
Fig. 23. A time sequence of sheet cavity evolving on the suction side of an oscillat-
ing finite-span hydrofoil obtained for σ = 0.9 for the first oscillation cycle.
Fig. 24. A time sequence of sheet cavity evolving on the suction side of an os-
cillating finite-span hydrofoil obtained for σ = 0.9 during the second oscillation
cycle.
c
m
d
w
p
a
p
m
f
n
the unsteady cavity. However, the pressure does not reach vapor
pressure because the sheet cavity length does not reach this loca-
tion.
5. Conclusions
A multiscale framework was developed for the simulation of
unsteady sheet cavitation on a hydrofoil by smoothly bridging a
Level Set Method (LSM) with a Ghost Fluid approach for large
size cavities and a Discrete Singularity Model (DSM) for small
bubbles. Starting with the micro-scale physics of nucleation from
the solid surface and dispersed nuclei in the bulk liquid, sheet
avitation along a NACA0015 hydrofoil is well captured by the
odel. The predicted sheet lengths and oscillation frequencies un-
er several cavitation numbers match the experimental data very
ell.
Parametric studies indicate that when the presently unknown
hysical parameters nucleation rate and nucleation sites per area
re not too small, the cavity characteristics show negligible de-
endency on those parameters. This is probably the case for com-
only machined material surfaces, however, newly developed sur-
ace treatments may result in future more sensitive results to the
ucleation rates.
C.-T. Hsiao et al. / International Journal of Multiphase Flow 90 (2017) 102–117 115
Fig. 25. A time sequence of sheet cavity evolving on the suction side of an oscillat-
ing finite-span hydrofoil obtained for σ = 0.9 for the third oscillation cycle.
t
l
f
s
p
T
w
Fig. 26. Zoomed view of the root sheet cavity near the leading edge obtained for
σ = 0.9.
c
fl
s
s
w
p
i
t
A
d
s
Sensitivity assessment of the threshold values of the switch cri-
eria between discreet microbubbles and large cavities also shows
ittle dependency on the parameters as long as the size criterion
or transforming a nuclei bubble into a large cavity is not too
mall.
The developed multiscale two-phase flow model was also ap-
lied to simulate sheet and cloud cavitation on 3D lifting surface.
hree-dimensional simulations for the 2D NACA0015 foil capture
ell sheet cavitation dynamics and cloud shedding with a more
omplicated breakup observed due to the 3D unsteadiness of the
ow.
Sheet cavitation on an oscillating finite-span hydrofoil was also
imulated using the developed model and a moving overset grid
cheme. Although the unsteady development of the sheet cavity
as also captured, the result do not show yet clear cloud shedding
robably due to inadequate grid resolution. This will be improved
n future effort s by refining the grids and conducting longer dura-
ion simulations.
cknowledgments
This work was supported by the Office of Naval Research un-
er contract N0 0 014-12-C-0382 monitored by Dr. Ki-Han Kim. This
upport is highly appreciated.
116 C.-T. Hsiao et al. / International Journal of Multiphase Flow 90 (2017) 102–117
Fig. 27. Comparison of time histories of the pressure coefficients monitored at: (a) x = 0.04 L (near the leading edge), and (b) x = 0.5 L (mid-chord) for single phase and
two-phase flow computations.
H
H
H
H
H
K
K
K
K
KK
M
M
M
M
M
O
R
R
S
References
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