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 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)
Outline 1. Why study wind-wave generation processes? 2. How to develop an air-water coupled model? 3. What we observe? 4. Wave growth types 5. Compare with previous studies
Overview The system of atmosphere and ocean is not independent 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. Jaync Douccllo W1101
Overview The mechanisms that generate these surface waves are still open issue due to (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
The Purpose of this Research Develop an air-water coupled model Study the wind-wave generation processes (laboratory waves) air water
Outline 1. Why study wind-wave generation processes? 2. How to develop an air-water coupled model? 3. What we observe? 4. Wave growth types 5. Compare with previous studies
Direct Numerical Simulation DNS numerically solves the Navier-Stokes equation subject to boundary conditions and hence such simulated flow fields contain no uncertainties other than numerical errors. z k y j x i × × × 3 Domain size : 24 24 8 cm Grid points : 2 ( 64 , 64 , 65 )
Differencing Schemes Spatial Differencing : horizontal: pseudo-spectral method vertical: second order finite differencing Time Differencing : second order Runge-Kutta scheme Grid System : stretching grid system z k y j high resolution near interface x i
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)
The conditions for interfacial boundary are 1. Velocity is continuous 2. Stress is continuous
Problem Formulation of Two-Phase Coupled Flow Governing Equations : � ∇ ⋅ = = � air a 0 u � = � water w � ∂ � � � u ( ) 1 � ( ) + ⋅ ∇ = −∇ + ∇ � 2 = u u p u u u , v , w � � � � ∂ � � � � t Re � Interfacial Boundary Conditions : ( linearized ) = = = u u v v w w , , a w a w a w continuity of velocity ρ ( ) ( ) − = η − η a P P f u , v , w , f u , v , w , continuity of ρ w a a a a a w w w w w normal stress ∂ ∂ μ ∂ ∂ ⎛ ⎞ ⎛ ⎞ u w u w + = + ⎜ ⎟ ⎜ ⎟ a a w w w μ ∂ ∂ ∂ ∂ ⎝ ⎠ ⎝ ⎠ z x z x continuity of a shear stress ⎛ ⎞ ⎛ ⎞ ∂ ∂ μ ∂ ∂ v w v w ⎜ + ⎟ = ⎜ + ⎟ a a w w w ⎜ ⎟ ⎜ ⎟ ∂ ∂ μ ∂ ∂ ⎝ ⎠ ⎝ ⎠ z y z y a Kinematic free ∂ η ∂ η ∂ η u v = − − surface B. C. w ∂ ∂ ∂ t x y
Interfacial boundary conditions are linearized Limitation of the air-water coupled model only for small amplitude waves
Initialization start = t 0 < > t 0 t 0 = t 70 s � � ( ) ( ) Fully-developed, ′ ′ ′ ′ ′ ′ + θ + θ U u , p , U u , p , U a ( z ) shear-driven a a a a a a a a ≠ = B 0 B 0 turbulent flow t � � ( ) ( ) ′ ′ ′ + θ ′ ′ ′ + θ U u , p , U u , p , w w w w U w ( z ) w w w w ≠ = B 0 B 0 η ≠ η = 0 0
Outline 1. Why study wind-wave generation processes? 2. How to develop an air-water coupled model? 3. What we observe? 4. Wave growth types 5. Compare with previous studies
Mean wind stress at the interface t < 50 s : τ s ~ constant reached a statistically quasi-steady state t > 50 s : τ s increases with time increases due to the growth of surface waves 0.12 τ s 0.1 -2 ) τ τ s (dyn cm air s 0.08 water 0.06 0 10 20 30 40 50 60 70 t (s)
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? η ↔ u , v , w , p (surface wave elevation)
Waves & Streamwise Velocity at the Interface ( ) ( ) η = x , y u w x , y , z 0 -0.0006 0.0006 0.024 0.037 20 = t 2 . 6 s 15 y (cm) 10 5 0 -0.02 0.02 0.032 0.045 20 = 15 y (cm) t 64 s 10 5 0 0 5 10 15 20 0 5 10 15 20 x (cm) x (cm)
w ′ in the Water 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 (a) (b) 20 20 0 0 15 15 y y (cm) z (cm) z (cm) ( c -2 -2 m 10 10 ) 5 5 -4 -4 20 20 15 15 0 0 10 10 5 5 0 0 x (cm) x (cm) = = t 16 s t 68 s
Waves, Surface Pressure of the Air & Shear Stress Fluctuations ( ) ( ) ( ) ′ ′ η = τ = , , 0 x , y p a x y z x , y , z 0 s 20 20 20 15 15 15 = y (cm) y (cm) y (cm) t 16 s 10 10 10 5 5 5 0 0 0 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 x (cm) x (cm) x (cm) 20 20 = t 68 s 15 15 y (cm) y (cm) y (cm) 10 10 5 5 0 0 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 x (cm) x (cm) x (cm)
Waves & Pressure Fluctuations (a vertical cross-section) s 5 . 16 ~ 16 = t At early stage
Waves & Pressure Fluctuations (a vertical cross-section) ( ) ( ) = -1 For one wave component : k x k , 1 ., 0 . cm y = 16 ~ 16 . 5 s t At early stage : air
Waves & Pressure Fluctuations (a vertical cross-section) = 50 . 5 ~ 51 s t At late stage Both domains are strongly influenced by waves
Waves & Pressure Fluctuations (a vertical cross-section) ( ) ( ) = -1 For one wave component : k x k , 1 ., 0 . cm y = At late stage : 50 . 5 ~ 51 s t air
0.003 0.03 Spectra of Wave Energy 2 1.5 = Wavelength ~ 8-12 cm k y (rad/cm) t 3 s 1 Wave frequency ~ 37 s -1 0.5 Satisfy the dispersion relationship 0 0 0.5 1 1.5 2 k x (rad/cm) 0.005 0.06 2 Wave number ( ) Φ η η 2 (cm -1 ) ( ) 1.5 κ = k , x k = k y (rad/cm) y 16 s t 1 (0.26, 1.05) 7.1 % (1.05, 1.05) 5.6 % 0.5 t ~ 16 s (1.05, 0.) 5.3 % (0.52, 0.52) 4.5 % 0 0 0.5 1 1.5 2 k x (rad/cm) (0.52, 0.) 4.1 % 0.001 0.08 2 (0.78, 0.) 28 % (0.78, 0.26) 24.7 % 1.5 = t ~ 66 s k y (rad/cm) (0.52, 0.) 24.5 % 64 s t 1 (0.52,0.26) 7.2% (1., 0.26) 3.3% 0.5 0 0 0.5 1 1.5 2 Dominated waves are different k x (rad/cm)
Outline 1. Why study wind-wave generation processes? 2. How to develop an air-water coupled model? 3. What we observe? 4. Wave growth types 5. Compare with previous studies
Some theoretical studies suggest that wave growth process can be separated into 1. Linear (waves grow slowly) 2. Exponential (waves grow quickly)
Time Evolution of Wave Amplitude < ⎧ t 40 s waves grow slowly ⎨ > ⎩ t 40 s waves grow quickly 0.04 < η 2 > 1/2 (cm) (a) 0.02 0 10 20 30 40 50 60 70 t (s)
Time Evolution of Some Parameters at the Interface (b) U 12 -1 ) U s (cm s s 10 0.5 -2 ) 1/2 (dyn cm (c) 1 2 p ′ 2 0.4 a 0.3 2 > <p a ' 0.2 0.06 -2 ) 1/2 (dyn cm (d) 1 2 τ ′ 2 0.04 s 2 > < τ s ' 0.02 (e) -2 ) D 0.01 D p (dyn cm p 0 (f) + 1 z + (air) 0 0.5 z 0 0 10 20 30 40 50 60 70 t (s)
Waves Growth Types Wave number ( ) (cm -1 ) Φ η η 2 ( ) κ = k , x k y (0.26, 1.) 7.1 % Linear : t < 16 s < (1., 1.) 5.6 % 16 s t (1., 0.) 5.3 % (0.52, 0.52) 4.5 % Exponential : t > 40 s (0.52, 0.) 4.1 % (0.78, 0.) 32 % t > 40 s (0.52, 0.) 24 % (0.78, 0.26) 21 % (cm) (a) (b) 0.001 0.06 -1 ) 0.04 (k x , k y ) (cm 0.0008 (0.26, 1.) 0.02 (1., 1.) 0.0006 (1., 0.) ( ) -1 ) (k x , k y ) (cm η (0.52, 0.52) ˆ k , x k y 0.0004 (0.52, 0.) (0.78, 0.) (0.52, 0.) 0.0002 (0.78, 0.26) 0 4 8 12 16 40 44 48 52 56 60 64 68 t (s) t (s)
Form Stress Some theoretical studies suggest form stress plays an important role in exponential wave growth stage ( ) dxdy ∂ η κ ( ) 1 , t ∫ ∫ L L ′ = κ y x D p , t ∂ p a L L x 0 0 x y -2 ) (a) (b) (dyn cm -2 8E-05 10 4E-05 -3 10 D p 0 -4 10 -4E-05 -5 10 -8E-05 4 8 12 16 40 44 48 52 56 60 64 68 t (s) t (s) < > t 16 s t 40 s
Outline 1. Why study wind-wave generation processes? 2. How to develop an air-water coupled model? 3. What we observe? 4. Wave growth types 5. Compare with previous studies
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