Proceedings of the Seventh International Conference on Hydroscience and Engineering, Philadelphia, PA, September 2006.  http://hdl.handle.net/1860/732This paper proposes a simple numerical technique to impose free-surface boundary conditions
(FSBC) for free-surface flows governed by surface-vortex interactions. Accuracy and efficiency of
this technique are examined through the analytical and experimental comparisons. The deformations
of a free-surface and evolution of vortices at droplet impacts onto a water pool is reasonably
reproduced by properly prescribing FSBC via the proposing method. Typical vortex rings are found
to be initiated at the contacts between the droplet and receiving water-surface and to threedimensionally
develop through a splashing process for interacting with a free-surface. The
dependencies of flow field on an impact angle of the droplet are also discussed in this study

Saruwatari, Ayumi

Watanabe, Yasunori

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Proceedings of the 7th International Conference on HydroScience and Engineering 
Philadelphia, USA September 10-13, 2006 (ICHE 2006) 
ISBN: 0977447405 
 
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College of Engineering 
 
 
 
 
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The 7th Int.  Conf. on Hydroscience and Engineering (ICHE-2006),Sep 10 –Sep 13,Philadelphia,USA 1 
 
 
 
 
 
 
NUMERICAL METHOD TO IMPOSE FREE-SURFACE BOUNDARY CONDITIONS 
FOR LOCAL FREE-SURFACE FLOWS  
 
 
Ayumi Saruwatari1 and Yasunori Watanabe2 
 
 
ABSTRACT 
 
This paper proposes a simple numerical technique to impose free-surface boundary conditions 
(FSBC) for free-surface flows governed by surface-vortex interactions. Accuracy and efficiency of 
this technique are examined through the analytical and experimental comparisons. The deformations 
of a free-surface and evolution of vortices at droplet impacts onto a water pool is reasonably 
reproduced by properly prescribing FSBC via the proposed method. Typical vortex rings are found 
to be initiated at the contacts between the droplet and receiving water-surface and to three-
dimensionally develop through a splashing process for interacting with a free-surface. The 
dependencies of flow field on an impact angle of the droplet are also discussed in this study.  
 
 
1. INTRODUCTION 
 
Breaking waves produce numbers of vortices at wide ranging scales through a splashing process of 
jets in a surf zone. Although major fluid motion under breaking waves is driven by the large-scale 
roller vortices, local flows may be predominated by small-scale surface deformations and vortex 
structures underneath the surface via surface-vortex interactions. While there have been some 
computational investigations for the dynamic splash-up process and resulting large-scale vortices in 
breaking waves (e.g. Watanabe & Saeki 1999, Watanabe et al. 2005), local vortices induced within a 
curved surface boundary layer and local surface deformations due to capillary effects, which is also 
an important factor to describe wave breaking process, have not been understood. The final goal of 
this study is to develop a numerical model to describe wave breaking flows involving small- to 
large-scale turbulence and three-dimensional surface deformations. This study presents a numerical 
technique to compute local free-surface flows and sub-surface vorticity field governed by the 
surface-vortex interactions. 
There have been some studies dealing with the interaction between a surface and sub-surface 
vortex. The vorticity is generated on a curved surface where tangential shear must be vanished 
(Longuet-Higgins 1992). When there is a horizontally located vortex beneath a surface, the surface 
is involved in the rotating motion and is entrained into an inner fluid region, resulting in a local 
surface deformation so-called "scar" (Sarpkaya 1996, Brocchini & Peregrine 2001). In order to 
properly compute the local surface deformation and vorticity field adjacent to the surface via the 
                                                 
1 Graduate Student, Hydraulic Engineering Laboratory, Hokkaido University, N-13 W-8, Sapporo, Japan 
(saruwata@eng.hokudai.ac.jp) 
2 Assistant Professor, Hydraulic Engineering Laboratory, Hokkaido University, N-13 W-8, Sapporo, Japan 
(yasunori@eng.hokudai.ac.jp) 
  2 
surface-vortex interactions, the dynamic jump conditions must be appropriately given at the surface. 
In this study, a numerical technique to satisfy the free-surface boundary conditions (FSBC) in 
a fixed Cartesian grid system was developed for reproducing the evolving free-surface and vortices 
produced near the surface under jets. 
This paper is organized as follows. The computational method for our large eddy simulation 
(LES) and proposing technique to impose FSBC are described in section 2, and numerical 
conditions are explained in section 3. Section 4 presents numerical tests to examine accuracies of 
our methods. Local vortices and surface deformations under impacting spherical droplet are 
investigated in section 5. The results are summarized in section 6. 
 
 
2. COMPUTATIONAL METHOD 
 
Three-dimensional LES for incompressible viscous fluid was performed for the local free-surface 
flows in the same manner as Watanabe et al. (2005). The following Navier-Stokes equation 
performed by a filtering operation is used as a governing equation. 
 guu +∇+∇−= 21 TpDt
D
ν
ρ
, (1) 
where u denotes the filtered velocity, p is the pressure, g is the gravity acceleration and Tν  is the 
sub-grid viscosity based on the renormarization group theory (Yakhot 1986). All variables are non-
dimensionalized by the fluid density ρ , representative length d and velocity v. Equation (1) is split 
into linear and non-linear finite difference equations on the basis of fractional step method: 
 ,1 2
1
22
12
1
guuu +∇+∇−=
∆
−
++
+
m
T
mm
m
p
t
ν
ρ
 (2) 
 ( )[ ] 0grad 212
1
1
=+
∆
− +
+
+
m
mm
t
uuuu , (3) 
where m denotes the time step. Equation (2) is solved by the predictor-corrector method, and the 
pressure equation is solved by the multigrid method to obtain intermediate variables at m+1/2 (see 
Watanabe et al. 2006). After the velocity is updated by eq. (2), eq. (3) is solved by the cubic 
interpolation polynomial (CIP) method (Yabe et al. 1991). 
Level-set method (Sethian and Smereka 2003) is adopted to capture a free-surface. The level-
set function φ , which indicates a distance to the nearest surface, is defined at computational grids 
near the surface. The reinitializing method proposed by Sussman and Fatemi (1999) is used to 
maintain the level-set function as a distance function. An unit normal vector on a surface n and 
surface curvature κ  can be computed by 
 φ
φ
∇
∇
=n , (4) 
 φ
φ
κ
∇
∇
⋅∇= . (5) 
A dynamic FSBC appropriately imposed on a surface in the proposing technique will be 
explained in the following sections.  
 
2.1 Free-surface boundary conditions 
 
Normal and tangential components of the dynamic FSBC are expressed by 
  3 
 
n
u
p n
∂
∂
+= µτκ 22 , (6) 
 01
1
=



∂
∂
+
∂
∂
n
u
t
u tnµ , 02
2
=



∂
∂
+
∂
∂
n
u
t
u tnµ , (7) 
where µ  is the viscosity, and ui is the fluid velocity in the x, y and z axes denoted by i= 1, 2 and 3, 
respectively. n and ti (i = 1, 2) are the normal and tangential unit vectors of the free-surface and τ  is 
the surface tension coefficient. Equation (6) represents a discontinuous jump of pressure across the 
surface due to the surface tension and viscosity and eq. (7) represents zero tangential shear 
conditions on the free-surface. 
 
2.2 Numerical technique to impose FSBC 
 
The zero tangential shear conditions (eq. (7)) are numerically achieved at the free-surface in a fixed 
Cartesian grid system on the basis of first-order extrapolations of velocity gradients from the inner 
fluid to the free-surface. 
Assuming that the local velocity gradients at an arbitrary position ( ξx +s ) of inner fluid near 
the free-surface can be determined by the sum of the velocity gradients at the surface location ( sx ) 
and correction function fij: 
 ),(
)()(
ξ
xξx
ij
j
s
s
i
j
si f
x
u
x
u
+
∂
∂
=
∂
+∂
 (8) 
where xs is the position vector on the surface and ξ  is the relative position vector from xs. In our 
method, a first-order polynomial iijij af ξ=)(ξ  is chosen as a correction function. The coefficient 
tensor aij is computed by the method of least squares for the zero shear conditions (eq. (7)) to which 
eq. (8) is substituted. In this way, appropriate surface velocity gradients j
s
i xu ∂∂ and surface 
velocity siu  can be obtained by using the inner velocity near the surface.  
In order to compute the advection term of the momentum equation (eq. (3)) in the fixed grid 
system, the fluid velocity needs to be extrapolated to empty grids. The local velocity on the outside 
of the surface is assumed to be described by keeping the first order of Taylor series for the velocity 
with respect to the surface location: 
 ( ) j
j
s
is
ii ξx
u
uu
∂
∂
+=ξ  (9) 
The outer grid velocity can be extrapolated by using the surface velocity and its gradients that have 
been known through the above procedure. In this way, the zero shear conditions (eq. (7)) can be 
approximately satisfied at the surface. 
The pressure jump condition (eq. (6)) is given as a boundary condition for the pressure 
equation.  
 
  4 
(a) 
x
y
z
d
v
r
 (b) 
v
d
θ
x
y
z
h
 
 
Figure 1 Computational domain for a rotating water cylinder in the numerical test (a). d is the 
cylinder diameter and v is the surface velocity. The droplet impacting at an angle θ  onto the target 
water pool of the depth h (b). d is the droplet diameter and v is the impacting velocity. 
 
 
3. COMPUTATIONAL CONDITIONS 
 
3.1 Numerical test for "sold-body rotation" 
 
In numerical tests for examining accuracies of our method, surface shears on a rotating water 
cylinder is investigated. Figure 1-(a) shows the computational domain for this test. The cylinder 
diameter d and axisymmetric linear distribution of fluid velocity in the cylinder (v = ar, a: constant, 
r: distance from the cylinder center) are given. 
 
3.2 Impacting droplet 
 
Figure 1-(b) shows the computational domain and impact condition for vertically and obliquely 
dropping droplet. A spherical droplet of the diameter d with the impacting velocity v at impacting 
angle θ  is dropped onto the still water pool of the depth h. Periodic boundary and non-slip boundary 
conditions are given at the sidewalls and bottom of the computational domain. 
 
(a) 
Grid interval dx
Ta
ng
en
tia
l 
su
rf
ac
e 
sh
ea
r
Ta
ng
en
tia
l 
su
rf
ac
e 
sh
ea
r
 
(b) 
qκω 2=
Su
rf
ac
e 
vo
rti
ci
ty
Su
rf
ac
e 
vo
rti
ci
ty
 
 
Figure 2 Computational error of the tangential surface shear in the rotating water cylinder (a), and 
comparison of the computed vorticity on the cylinder surface and the analytical solution (b). 
 
 
 
  5 
4. NUMERICAL TESTS 
 
4.1 Surface shear and vorticity on a rotating water cylinder 
 
The numerical test of the "solid-body rotation" (Longuet-Higgins 1992) is carried out to confirm 
whether the tangential surface shear properly vanishes on the surface and to examine the numerical 
accuracies. The tangential surface shear and surface vorticity on a rotating water cylinder are 
computed using the present technique (see Fig. 1-(a)). 
Figure 2-(a) shows the tangential shear on the cylinder surface, which should be zero, against 
the computational grid interval dx. The surface shear is found to be less than O(10-13) for any 
computational grid interval, which is less than the machine precision and is therefore negligible.  
Figure 2-(b) shows the computed surface vorticity and analytical solution qκω 2=  on the 
cylinder surface, where ω  is the tangential surface vorticity and q is the tangential surface velocity. 
Our results present very good agreements with the solutions, demonstrating a high level of accuracy 
of our method. 
 
Dimesionless time
D
im
en
si
on
le
ss
 c
av
ity
 d
ep
th
D
im
en
si
on
le
ss
 c
av
ity
 d
ep
th
 
 
Figure 3 Dimensionless cavity depth against dimensionless time from the droplet impact. 
(1) v = 1.29 [ms-1], d = 1.86 [mm]. (2) v = 1.61 [ms-1], d = 2.09 [mm]. 
 
4.2 Droplet impacts 
 
The local free-surface flow under impacting droplets is computed to compare the surface 
deformation with experimental measurements of Liow (2001). A droplet of which the diameter 
2≈d  [mm] is dropped onto a still water surface at the impacting velocity v (see Fig. 1-b). The 
computational grid interval dx was d/10. 
Figure 3 shows the evolution of the cavity depth after the impact. The computational cavity 
depth was defined as a distance from a still water level to the deepest level of the cavity. There are 
good agreements in the computed cavity depths with the experimental results, demonstrating high 
capability of the present method for practical computations of free-surface flows. 
 
 
5. COMPUTATIONAL RESULTS 
 
The local surface deformation and vortex formation beneath the surface are significantly affected by 
the surface-vortex interactions (Sarpkaya 1996). A typical formation of a vortex ring under a 
  6 
plunging droplet is also due to the surface-vortex interactions and the proposed method is applied to 
this problem for understanding the jet-induced vortex structure. A spherical droplet of which the 
diameter d = 0.1 [m] is dropped onto a still water at impacting velocity v = 1.0 [ms-1] at two 
different impacting angles θ  (see Fig. 1-(b)). The computational domain is a cube 5d on a side and 
the computational grid interval dx is d/20.  
 
Initial vortex ring
Cavity crater Surface vorticity
Secondary jet Second 
vortex rings
First vortex 
ring
Second vortex 
rings
(a) (b) (c)
(d) (e) (f)
 
 
Figure 4 Evolution of the free-surface and vortex structure under the vertical impact of the droplet 
(d = 0.1 [m], v = 1.0 [m/s], 2/πθ = ). (a) t = 0.1d/v, (b) t = 1.2d/v, 
 (c) t = 2.6d/v, (d) t = 4.3d/v, (e) t = 6.4d/v, (f) t = 7.8d/v. 
 
5.1 Vertical impact of a droplet 
 
Figure 4 shows the evolution of the free-surface and vortex cores after the vertical impact ( 2/πθ = ). 
The vortex core is computed on the basis of 2λ -method (Jeong and Hussain 1995). The primary 
vortex ring is initiated at contacts between the droplet and target fluid (Fig. 4-(a)) since the vorticity 
must be induced on the curved surface. A cavity crater is formed by the impact, and the vortex ring 
is developed under the cavity bottom (Fig. 4-(b)). In Fig. 4-(c), the cavity bounces back due to the 
pressure gradient towards the center of the cavity, and then the vortex ring is ejected downwards 
from the cavity bottom. It is also seen that the surface vorticity is generated on a radically 
propagating cavity front and near the secondary jet projecting upward. The jet finally plunges again 
into water to produce another vortex ring (Fig. 4-(e)). The second vortex ring is rapidly displaced 
downward faster than the previous adjacent ring due to vortex-vortex interaction and pass through 
the previous vortex ring (Fig. 4-(f)). 
 
  7 
(a) (b)
(c) (d)
Initial vortex ring Jet Cavity crater
Vortex tube
Vortex 
stretch
Secondary jet Second 
plunging point
 
 
Figure 5 Evolution of the free-surface and vortex structure under the oblique impact of the droplet 
(d = 0.1 [m], v = 1.0 [m/s], 4/πθ = ). (a) t = 0.2d/v, (b) t = 1.3d/v, (c) t = 4.0d/v, (d) t = 6.2d/v. 
 
5.2 Oblique impact of a droplet 
 
Figure 5 shows the evolving free-surface and vortex cores produced by the droplet impacting at 
4/πθ = . More complicated flow field is formed in this case. A bore-like jet projects ahead of the 
crater and produces a radical vortex tube underneath (Fig. 5-(b)). The vortex initiated at the contacts 
between the droplet and receiving water surface (Fig. 5-(a)) develops beneath a rear part of the 
cavity for being the vortex tube (Fig. 5-(b)). This vortex tube is obliquely stretched and intensified 
within the secondary jet through the stretch-and-intensification process, resulting in a formation of 
three-dimensional vortex structure. After the secondary jet touches down onto the forward surface, 
another three-dimensional vortices appear at the plunging point (Fig. 5-(d)). 
 
 
6. CONCLUSIONS 
 
The numerical technique to impose FSBC in a fixed Cartesian grid system for local surface flows 
governed by the surface-vortex interactions was proposed. It is confirmed that the tangential FSBC 
can be satisfied at high accuracy and the surface flow under impacting droplet can be reproduced 
using the present technique. The numerical experiments about the plunging droplets were performed 
to investigate the evolution of the free-surface and vortex structure under the impacting droplets. 
The proposed method to impose the tangential FSBC appropriately reproduces local vortex 
structures on a curved surface in plunging droplets. The surface vorticity generated at contacts 
between the droplet and water surface is developed underneath the surface of a cavity and is 
released from the cavity to displace downward. The surface vortices work as sources of the 
turbulence in free-surface flows.  
  8 
It is also found that a structure of the vortices depends on an impacting angle of the plunging 
jet with respect to a target fluid surface. While an axisymmetric vortex ring is formed around the 
cavity and displaced downward in the case of the vertical drop impact, in the obliquely plunging 
case, a vortex ring is highly deformed within a secondary jet projecting obliquely upward due to 
strong stretches along the jet. 
 
 
REFERENCES 
 
Brocchini, M. and Peregrine, D. H. (2001) “The Dynamics of Strong Turbulence at Free Surfaces. 
Part 1. Description”, Journal of Fluid Mech. Cambridge University Press, Vol. 449, pp.225-
254. 
Jeoung, J. and Hussain, F. (1995) “On the Identification of a Vortex”, Journal of Fluid Mech. 
Cambridge University Press, Vol. 285, pp.69-94. 
Liow, J. L. (2001) “Splash Formation by Spherical Drops”, Journal of Fluid Mech. Cambridge 
University Press, Vol. 427, pp.73-105. 
Longuet-Higgins, M. S. (1992) “Capillary Rollers and Bores”, Journal of Fluid Mech. Cambridge 
University Press, Vol. 240, pp. 658-679. 
Sarpkaya, T. (1996) “Vorticity, Free-Surface, and Surfactants”, Ann. Rev. Fluid Mech. Annual 
Reviews, Vol. 28, pp. 83-128. 
Sethian, J. A. and Smereka, P. (2003) “Level Set Method for Fluid Interfaces”, Annu. Rev. Fluid 
Mech. Annual Reviews, Vol. 35, pp. 341-372. 
Sussman, M. and Fatemi, E. (1999) “An Efficient, Interface-Preserving Level Set Redistancing 
Algorithm and Its Application to Interfacial Incompressible Fluid Flow”, Journal of Scientific 
Computing, Springer Netherlands, Vol. 20, No. 4, pp. 1165-1191. 
Watanabe, Y. and Saeki, H. (1999) “Three-Dimensional Large Eddy Simulation of Breaking 
Waves”, Coastal Engineering Journal, World Scientific, Vol. 41, Nos. 3 & 4, pp. 281-301. 
Watanabe, Y., Saeki, H., and Hosking, R. J. (2005) “Three-Dimensional Vortex Structure under 
Breaking Waves”, Journal of Fluid Mech. Cambridge University Press, Vol. 545, pp. 291-328. 
Yabe, T., Ishikawa, T., Wang, P. Y. and Aoki, T. (1991) “A Universal Solver for Hyperbolic 
Equations by Cubic-Polynomial Interpolation I. Two- and Three-Dimensional Solvers”, Comp. 
Phys. Comm. Vol. 66, pp. 233-242. 
Yakhot, V. and Orszag, S. A. (1986) “Renormarization Group Analysis of Turbulence. I. Basic 
Theory”, Journal of Scientific Computing, Springer Netherlands, Vol. 1, No. 1, pp. 3-51. 


Numerical method to impose free-surface boundary conditions for local free-surface flows

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Numerical method to impose free-surface boundary conditions for local free-surface flows

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