1. Numerical Study of Strong Free surface
Flow and Wave Breaking
Yi Liu
Department of Civil Engineering
Johns Hopkins University

2. Numerical Method
air
water
interface
,a a 
,w w 
Fixed Eulerian grid
Coupled air-water system

 Interface is represented implicitly
 Fixed Cartesian grid
 Automatically handle surface overturning, merging, and pinching
off
Variable density and viscosity NS equation
0 
0 
0  Level set equation
and

3. Coupled Level Set/VOF (CLSVOF) Method
0




u
t

Cartesian grid
interface
water
air ,a a 
,w w 
Breaking of 3rd order Stokes wave (ak=0.55)
Pure LS method CLSVOF method
0


Fu
t
F 
Level Set Method Volume-of-Fluid Method
Mass is not exactly
conserved
Calculation of surface
normal and curvature is
precise and relatively easy
Accurate calculation of
surface normal and
curvature is challenging
Mass is accurately
conserved
0F 
0 1F 
1F 
VOF update
,n n
F
1 1
,n n
F  
LS reinitialize
0




u
t

VOF reinitialize
n
 n
F
1n
 
Construct interface
using PLIC
1n
F 
,n n    
v v
g
*

**

Volume Flux calc
1n
F 
,n
F f
0 
0 
0 
(Sussman & Puckett 1998)

4. [.]
Interface Jump Conditions
Stress discontinuity
 
































0
0
2
1

 T
NIp
T
T
N




Numerical simulation of a static air bubble without gravity effect
Density and viscosity discontinuity
[ ]
[ ]
w a
w a
  
  
 
 
bubble
u
Numerical simulation of multi-layer Couette flow

Continuous Surface Force Method
Ghost Fluid Method
[.]
1
1




1
0.1





5. High Performance Computing (HPC) on Supercomputers
 Large-scale parallel computing is necessary for CPU- and memory-intensive
simulations of wave-turbulence-body interactions.
 Message Passing Interface (MPI) is used for parallelization.
 Our parallel codes show excellent performance on supercomputers.
Cray XE6
20,224 cores, 192.4T Flops
SGI Altix ICE
15,360 cores, 172T Flops
Cray XT4
8,584 cores, 72.3T Flops
Cray XE6
11648 cores, 107.2T Flops
# of cores MPI+MPI_SYNC I/O Imbalance%
16 1.7% 1.1% 0.5
32 4.5% 1.0% 0.8
64 9.4% 1.3% 1.8
128 8.6% 2.3% 1.5
256 15.2% 4.1% 2.4
Profiling result of the CLSVOF code
 Computing resources provided by DoD High Performance
Computing Modernization Program (HPCMP).

6. Research Topics
1. Numerical study of breaking waves with different
intensity.
2. Numerical study of the interaction between wind
turbulence and wave breaking.
3. Numerical study of the wind wave generation and
growth.
4. Mechanistic study of strong free surface turbulence.
5. Hybrid Euler-Lagrangian method for the numerical
simulation of wave breaking.
6. Multi-scale simulation of wind-wave-structure
interaction.

7. (ak)0=0.3 (ak)0=0.35 (ak)0=0.4 (ak)0=0.44 (ak)0=0.55
Topic 1: Breaking Waves without Wind Effect
 To investigate the breaking criteria and the energy dissipation

8. Energy Evolution during Wave Breaking
(ak)0=0.55
(ak)0=0.44
(ak)0=0.40
(ak)0=0.35
(ak)0=0.3
 For all the breaking cases, there are three regimes of energy evolution: (a) initial
slow decay; (b) strong decay; and (c) slow decay afterward.
 The duration of initial slow decay decreases as the wave steepness increases.
 The strong decay lasts for approximately 2 wave periods.
 The total energy loss increases with wave steepness.
 For steep waves, the relative loss of wave energy is independent of the wave
steepness.
2 2
2
total k p
u v
E E E dxdy gydxdy 

    

9. Topic 2: Wind Turbulence over Breaking Waves
Problem Setup
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
 To investigate the interaction between wind turbulence and the breaking
wave
 Velocity field at both air and
water side
 Wind stress and drag coefficient
 Air flow separation
 Turbulence and current
generation by breaking
 Energy dissipation rate
Simulation Results

10. Wave Breaking under Different Wind Speeds
U10=5.06m/s U10=11.08m/s
U10=14.97m/s U10=19.70m/s
 Breaking associated
with high wind appears
more violent.
 The breaking affects
the turbulence in the
wind.
 Splash-up enhances
turbulent mixing in the
airflow.
spume
jet

11. Plunging Breaking and Vortex Generation
T=1.33T
T=1.78T
T=2.67T
T=2.22T
 Plunging breaker generate
large mean vortex structure.
 Small co-rotating vortices
coalesce into larger ones.
 Spilling breaker only
generate mean shear.

12. Topic 3: Wave Evolution from Flat under Turbulent Wind
Amplitude spectrumEvolution of rms of surface elevation
 To investigate the wave generation and growth under turbulent wind
 Wave field evolution and growth rate
 Spectral characteristics and its
evolution
 Frequency downshifting
 Comparison with JONWAP spectrum

13. Topic 4: Mechanistic Study of Strong Free-Surface Turbulence
 To investigate the interaction between free surface and underlying turbulence
 Features of free surface in different flow
regimes.
 Thickness of intermittency layer and the
distribution of intermittency factor.
 Scale dependence of surface structure on
Froude and Weber numbers.
 Effect of Froude and Weber numbers on
turbulence kinetic energy.

14. Instantaneous Surface Features
Small surface
elevation
Gravity
dominated
Surface tension
dominated
Very strong
turbulence
Breaking
surface
Marginal
breaking
 Dimples and scars are
observed on free surface.
 Dimples are generated due
to low pressure at the core
of surface-connected
vortices.
 Scars are associated with
near-surface horizontal
vortices.
 Knobs are observed on free
surface.
 The surface is smooth and
dominated by the large-scale
structures.
 Breaking waves and complex
structures are observed on free
surface.
2 2
Fr U gL 2
We U L 

15. Splat and Anti-Splat
 Strong vertical motion towards the surface;
 Radial horizontal flow motion;
 Induces strong pressure at the surface;
 Accompanied by horizontal vortex pair;
 Generates vortex in the air.
Splat:
Anti-splat:
 Formed when radial motion encounters;
 Downward flow motion;
 Has long and thin shape.
Vortex pair
Splat
Vortex in the water
Splat-induced vortex in air
Splat
Anti-splat

16. Level Set
SPH
Topic 5: Level Set-SPH Coupled Simulation for Wave Breaking
 To improve the resolution locally and capture fine scale droplets formed by
breaking

17. Smoothed Particle Hydrodynamics (SPH) Method
( ) ( ') ( ' ) 'f x f x x x dx 
SPH interpolation:
1 1
( ) ( , )
N N
j j
i j i j j ij
j jj j
m m
f x f W x x h f W
  
   
1),( 
xdhxxW )'(),(lim
0
xxhxxW
h



where kernel function w satisfies
Continuity equation:
 
1 1
N N
i
j i j ij j ij ij
j j
d
m v v W m v W
dt

 
     
v v v
Momentum conservation equations:
2 2 2 2
j ij j j iji i i i
j j
j ji j i i j i
p W Wdv p
m m
dt x x
 
 
  
   
    
              
 
strain rate
1 1 1
2
3
N N N
j ij j ij j
i ji ji ji i ij
j j jj i j i j
m W m W m
v v v W
x x
   
 
 
    
  
        
  
v
0
1p B



  
   
   
Equation of state (EOS):
Weakly compressible for ca>10cp
ca
cp
Acoustic wave speed
Surface wave speed

18. Breaking Wave Simulation with SPH
(ak)0=0.55, 3rd-order Stokes wave
 Dispersed water parcels are
generated in the breaking region.
 Particle located far from the
breaking wave crest has an orbital
motion.
 Particle located at the breaking
crest starts with a circular motion.
After it reaches the wave crest, it
moves forward with the breaking
jet, falls down to the water, and
then bounces up with the splash. Particle trajectory
Breaking onset Jet touchdown

19. ap
BC
Inflow BC
object
v
air
wave
water
HOS simulation
of wave fields
coupled LS/VOF/GFM for
air-water simulation
LES of wind
turbulence
IBM for structure
Topic 6: Multi-scale Simulation of Wind-Wave-Structure Interaction
 To investigate the wave effect on the wind forcing over structures

20. Immersed Boundary Method for Flow-Structure Interaction
n
n b
b
u u
f RHS
t

  

v vr
0
b
u
RHS f
int
u

 

 
r r
r
In immersed boundary method, the structure is represented by
adding a force term into the momentum equation. Then the
governing equations become
Direct discrete forcing approach is used to calculate the
boundary force
forcing points
fluid
solid
x
x
boundary points
f b
Immersed
Structure
f
b

Fluid
,
u

where
  
 2
1 1
2
( ) ( )Re
1
( )
RHS p D
k
Fr We
 
   
  
 
    
  
g
r
is interpolated on the forcing point from its nearby
flow points and the corresponding boundary point.
bu
v
An immersed boundary method is used to simulate
the flow-structure interaction.
fluid points

21. weaker horseshoe vortexstronger horseshoe vortex
Dependence of Wind Load on Wave Phase
The wave phase dependence of wind load may be induced by:
 Phase dependence inherits from inflow field
 Variation of horseshoe vortex in front of the object
wave crest reaches the frontal face wave trough reaches the frontal face

22. Angle Effect to Force and Moment Coefficients
 0  30  60
 For 0°attack angle case, the strongest flow separation happens on the two
side walls and the lowest pressure happens on those two side walls. The
pressure in the wake is a little bit higher than the other two cases.
 For 30°attack angle case, the strongest flow separation happens on the back
and one side walls.
 For 60°attack angle case, one side wall faces the inflow and the pressure on
the original front is not so high. The wake region is larger than the other two
cases.