Direct Numerical Simulation of Wind-Wave Generation Processes - - PowerPoint PPT Presentation
Direct Numerical Simulation of Wind-Wave Generation Processes Mei-Ying Lin Taiwan Typhoon and Flood Research Institute Direct Numerical Simulation of Wind-Wave Generation Processes Collaborators : Chin-Hoh Moeng (National Center for
Mei-Ying Lin
Taiwan Typhoon and Flood Research Institute
Direct Numerical Simulation of Wind-Wave Generation Processes Collaborators : Chin-Hoh Moeng (National Center for Atmospheric Research, USA) Wu-Ting Tsai (National Central University, Taiwan) Peter P. Sullivan (National Center for Atmospheric Research, USA) Stephen E. Belcher (University of Reading, UK)
Wind Waves : Wind-generated waves are the most visible signature of air-sea interaction and play a major influence on the momentum and energy transfer across the interface. The system of atmosphere and ocean is not independent
Jaync Douccllo W1101
The mechanisms that generate these surface waves are still
(1) Difficulties in obtaining a dataset from laboratory and field measurements that records the time evolution of motions in both atmosphere and ocean domains (2) Mathematical difficulties in dealing with highly turbulent flows over complex moving surfaces (3) Lack of a suitable coupled model to simulate turbulent flows in both atmosphere and ocean simultaneously
Develop an air-water coupled model Study the wind-wave generation processes (laboratory waves)
air water
) 65 , 64 , 64 ( 2 : points Grid ×
3
cm 8 24 24 : size Domain × ×
i
x
j
y
k
z
DNS numerically solves the Navier-Stokes equation subject to boundary conditions and hence such simulated flow fields contain no uncertainties other than numerical errors.
Spatial Differencing :
horizontal: pseudo-spectral method vertical: second order finite differencing
Time Differencing :
second order Runge-Kutta scheme
Grid System :
stretching grid system high resolution near interface
i
x
j
y
k
z
Boundaries & Boundary Conditions For 4 side walls :
periodic boundary conditions
lower boundary :
free-slip boundary conditions
upper boundary :
a constant velocity is imposed
interfacial boundary : (at air-water interface)
p u u t u u
2
Re 1 ∇ + −∇ = ∇ ⋅ + ∂ ∂ = ⋅ ∇
Governing Equations : Interfacial Boundary Conditions : ( linearized )
( ) ( )
y v x u w t y w z v y w z v x w z u x w z u w v u f w v u f P P
w w a w a a w w a w a a w w w w a a a a a w a w
w w a w w v a v w u a u
∂ ∂ − ∂ ∂ − = ∂ ∂ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ − = −
= = =
η η η μ μ μ μ η η ρ ρ , , , , , , , ,
w a = =
water
Problem Formulation of Two-Phase Coupled Flow
continuity of velocity continuity of shear stress continuity of normal stress Kinematic free surface B. C.
( )
v u u , , =
= t start
≠ η = η > t < t ) (z U w ( )
B , , = ′ ′ ′ +
a a a a
p u U θ
)
B , , = ′ ′ ′ +
w w w w
p u U θ
(z U a
( )
B , , ≠ ′ ′ ′ +
a a a a
p u U θ
)
B , , ≠ ′ ′ ′ +
w w w w
p u U θ
shear-driven turbulent flow
s 70 = t
t (s) τs (dyn cm
10 20 30 40 50 60 70 0.06 0.08 0.1 0.12
air water
s
τ
s
τ
reached a statistically quasi-steady state t < 50 s : τs ~ constant increases due to the growth of surface waves t > 50 s : τs increases with time
Wind-Wave Generation Processes (t=0~70 s)
When wave amplitude changes, what will be the behavior of the flow fields above and below the interface?
(surface wave elevation)
Waves & Streamwise Velocity at the Interface
0.024 0.037y (cm)
5 10 15 20
x (cm) y (cm)
5 10 15 20 5 10 15 20
x (cm)
5 10 15 20
0.032 0.045s 6 . 2 = t s 64 = t
( )
y x, η
( )
, , = z y x uw
x (cm)
5 10 15 20
y ( c m )
5 10 15 20
z (cm)
(a)
x (cm)
5 10 15 20
y (cm)
5 10 15 20
z (cm)
(b)
in the Water
s 16 = t s 68 = t
At shear-dominated stage (t=16 s) : the distribution of updrafts and downdrafts is irregular
At wave-dominated stage (t=68 s) : the vertical velocity field aligns with waves
Waves, Surface Pressure of the Air & Shear Stress Fluctuations
x (cm) y (cm)
5 10 15 20 5 10 15 20
x (cm) y (cm)
5 10 15 20 5 10 15 20
x (cm)
y (cm)
5 10 15 20
5 10 15 20
x (cm)
y (cm)
5 10 15 20
5 10 15 20
x (cm) y (cm)
5 10 15 20 5 10 15 20
x (cm) y (cm)
5 10 15 20
s 16 = t s 68 = t
( )
y x, η
( )
, , = ′ z y x pa
( )
, , = ′ z y x
s
τ
Waves & Pressure Fluctuations (a vertical cross-section)
At early stage
Waves & Pressure Fluctuations (a vertical cross-section)
At early stage :
cm . ., 1 , : component wave
For =
y x k
k
Waves & Pressure Fluctuations (a vertical cross-section)
At late stage
Both domains are strongly influenced by waves
Waves & Pressure Fluctuations (a vertical cross-section) At late stage :
cm . ., 1 , : component wave
For =
y x k
k
kx (rad/cm) ky (rad/cm)
0.5 1 1.5 2 0.5 1 1.5 2
0.003 0.03kx (rad/cm) ky (rad/cm)
0.5 1 1.5 2 0.5 1 1.5 2
0.005 0.06kx (rad/cm) ky (rad/cm)
0.5 1 1.5 2 0.5 1 1.5 2
0.001 0.08( )
y x k
k , = κ
( )
2
η η Φ
s 16 ~ t
s 66 ~ t
Wave number (cm-1) (0.26, 1.05) (1.05, 1.05) (1.05, 0.) (0.52, 0.52) (0.52, 0.) 7.1 % 5.6 % 5.3 % 4.5 % 4.1 %
(0.78, 0.) (0.78, 0.26) (0.52, 0.) (0.52,0.26) (1., 0.26) 28 % 24.7 % 24.5 % 7.2% 3.3%
Wavelength ~ 8-12 cm Wave frequency ~ 37 s-1 Satisfy the dispersion relationship
Dominated waves are different
⎩ ⎨ ⎧ > < quickly grow waves s 40 slowly grow waves s 40 t t
t (s) <η2>1/2 (cm)
10 20 30 40 50 60 70 0.02 0.04
(a)
Time Evolution of Some Parameters at the Interface
Us (cm s
10 12
(b)
<pa'
2> 1/2 (dyn cm0.2 0.3 0.4 0.5
(c)
<τs'
2> 1/2 (dyn cm0.02 0.04 0.06
(d)
Dp (dyn cm
0.01
(e)
t (s) z0
+ (air)10 20 30 40 50 60 70
0.5 1
(f)
s
U
2 1 2 a
p′
2 1 2 s
τ′
p
D
+
z
Linear : t < 16 s Exponential : t > 40 s
t (s)
4 8 12 16 0.0002 0.0004 0.0006 0.0008 0.001
(0.26, 1.) (1., 1.) (1., 0.) (0.52, 0.52) (0.52, 0.)
(kx, ky) (cm
(a)
t (s)
40 44 48 52 56 60 64 68 0.02 0.04 0.06
(0.78, 0.) (0.52, 0.) (0.78, 0.26)
(kx, ky) (cm
(b)
(cm)
( )
y x k
k , ˆ η
( )
y x k
k , = κ
( )
2
η η Φ
Wave number (cm-1)
(0.26, 1.) (1., 1.) (1., 0.) (0.52, 0.52) (0.52, 0.) 7.1 % 5.6 % 5.3 % 4.5 % 4.1 %
(0.78, 0.) (0.52, 0.) (0.78, 0.26) 32 % 24 % 21 %
s 16 < t
s 40 t >
x t t p L L D
y x
L L a y x p
∂ ∂ ′ =
, , 1 κ η κ
Some theoretical studies suggest form stress plays an important role in exponential wave growth stage
t (s)
4 8 12 16
4E-05 8E-05
(a) t (s)
40 44 48 52 56 60 64 68 10
10
10
10
D p
(b) (dyn cm
Linear Growth Stage
Phillips (1957) : the turbulence-induced pressure fluctuations in the air are responsible for the birth and early growth of waves
t gU p
c w a
2 2 ~
2 2 2
ρ ξ ′
∗ a c
t (s)
4 8 12 16
0. 2E-06 4E-06 6E-06
(a)
(cm
2)
Exponential Growth Stage
Belcher & Hunt (1993) : (Non-separated sheltering mechanism)
dt dE E dt da a 1 2 σ σ β = =
( )
2 2
1 2 1 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = = c ak D dt dE E
p w
ρ σ β
k c a E dy dx x p L L D
w L L a y x p
y x
2 2
5 . ; 1 where ρ η = ∂ ∂ ′ =
the form stress dominates the contribution of energy input from air to waves at the exponential wave growth stage Wave growth rate
+ + +
ua
* / c2πβ
10
10 10
10
10
10
10
(a)
0. 2E-06 4E-06 6E-06
(a)
(cm
2)t (s)
4 8 12 16 0. 2E-06 4E-06 6E-06
(d)
0. 2E-06 4E-06 6E-06
(c)
0. 2E-06 4E-06 6E-06
(b)
control case Turbulence in the water Surface tension Domain size
+ ++
2πβ
10
10
10
10
10
(a) ++ + (b) + + +
ua
* / c2πβ
10
10 10
10
10
10
10
(c) ++ +
ua
* / c10
10
(d)
control case Turbulence in the water Surface tension Domain size
A new air- water coupled model is developed The initial wind-wave generation processes is simulated The characteristics of flow fields are different at early and late stages Wave growth types : linear & exponential The wavelengths found here (8-12 cm) are close to those found in laboratory at low wind speed. Some of the simulated wave growth rates are close to previous studies’ results, but some of them are about 1~3 times larger than their prediction or measurements.