SURFZONE WAVES II (PHASERESOLVING)

SURF-ZONE WAVES II (PHASE-RESOLVING) Surf-Zone Wave Models II Phase-Resolving Models Waves in Shallow Water Airy (1845):     ( h  )u   t x conservation of mass u u  u  g 0 t x x momentum equation non-linear shallow-water equations (non-dispersive waves) Boussinesq (1872):     ( h  )u   t x u u  h  3 u  g  0 t x x xt original, 1D form of Boussinesq equations (linear variation in vertical velocity, pressure not hydrostatic) Scaling Parameters and Limits (long-wave theory) Non-linear effects   h Dispersive effects  h L Non-dimensional equations:  '    (1   ')u '  t ' x ' u ' u '  '  3 '  u '   0 t ' x ' x ' x ' t '2 Different forms of the Boussinesq Equations Causes for differences: • simplifying assumptions • integration procedures • dependent variables Different Types of Boussinesq Equations Abbott (1979): u u  h    3 u  g  0 t x x xt I u u  h    3  3  u  g  u  0 t x x  xt x t  II u u  h     3  3  3  u  g   2u  u   t x x  xt x t x  III Peregrine (1967): u u  h  3u u  g  0 t x x x 2t (different integration procedure) Other Velocity Variables Nwogu (1993):      2u 2   ( h  )u    a1h3 2  a2 h 2  hu    t x x  x x  u  3u   u  u   b1h 2   b2 h  h    u   g 0 t x t x  t  x x use velocity at level z=z Madsen et al (1991):  P  0 t x P   P     gh  1  t x  h  x 1 3 P  3 h   P 2   1    B   h 2  Bgh3  h   Bgh   x t x x  xt x   P is the depth-averaged volume flux (B is a linear dispersion coefficient) Applications for the Boussinesq Equations • nearshore wave transformation • wave-current interaction • nearshore currents • interaction waves/structures Application of Boussinesq Equation to Simulate Waves in the Surf Zone Boussinesq Equations Mass conservation:  P  0 t x Momentum conservation: P   P   3  1 3P     gD  Bgh3   B   h2  t x  D  x x   x t h  2 P     h  Bgh   M D ,b  x  x xt  (Madsen et el 1991) Momentum Loss due to Wave Breaking Watanabe and Dibajnia (1988): M D ,b   f DQ   D tan  g Qˆ  Qr Q d Qs  Qr D=2.5 Wave-induced flow: Qs  0.4  0.57  5.3 tan   gd Flow in reformed waves: Qr  0.135 gd Sato et al (1992): M D,b  b b   2Q x  D gd tan  g Qˆ  Qr 2 d Qs  Qr Random waves: f D,ir   r  D tan  g Qˆ  Qr d Qs  Qr Roller model (Schäffer et al 1993) Evolution of roller (geometry):  t  tB  tan (t )  tan o  (tan B  tan o ) exp   ln  t1/   : angle defining roller (b indicates breaking and o terminal conditions) Momentum loss: M D ,b  1    P      c   1    x   D   D   Model Validation: Duck Data Date Profile Type Hmo (m) Tp(s) S0 Sbr 0 br 0(10-3) Sept 5, 1985, 09.55 Shelf 0.61 11.4 0.81 1.12 0.16 0.70 1.68 Sept 5, 1985, 10.15 Shelf 0.42 13.1 0.93 1.29 0.11 0.66 1.27 Sept 5, 1985, 13.52 Shelf 0.64 10.9 0.77 1.00 0.17 0.68 1.88 Sept 5, 1985, 15.25 Shelf 0.53 11.1 0.83 1.13 0.12 0.68 1.61 Sept 15, 1986,14.45 Shelf 1.07 10.1 0.62 0.72 0.15 0.48 2.84 Sept 19,1986, 11.00 Bar 0.70 11.2 0.79 1.00 0.10 0.47 2.5 slope parameter S= hxL/d dispersion parameter  =d/gT2 non-linearity parameter  =H/2d Model Agreement with Data (Hrms) (H1/3) (H1/10) Date Roller Mod W-D Mod Roller Mod W-D Mod Roller Mod W-D Mod Sept 5, 1985, 09.55 19.0 25.0 18.0 23.0 18.0 24.0 Sept 5, 1985, 10.15 16.8 27.4 27.0 21.0 30.0 23.0 Sept 5, 1985, 13.52 26.0 28.0 27.0 28.0 24.0 28.0 Sept 5, 1985, 15.25 17.0 23.0 18.0 19.0 12.0 13.0 Sept.15, 1986, 14.45 12.5 15.0 13.0 16.0 16.7 22.3 Sept.19, 1986, 11.00 8.8 13.7 6.3 15.1 13.8 19.4 Duck85 Beach Profile Shape shelf-type profile SuperDuck Beach Profile Shape bar-type profile Calculated and Measured Water Surface Elevation 1985-09-05 09:55 Calculated and Measured Water Surface Elevation 1986-09-19 11:00 Measured and Computed Skewness and Kurtosis Shelf profile Measured and Computed Statistical Wave Properties Shelf profile ... 3  3  u  g  u  0 t x x  xt x t  II u u  h     3  3  3  u  g   2u  u   t x x  xt x t x  III Peregrine (1967): u u  h  3u u  g  0 t... transformation • wave-current interaction • nearshore currents • interaction waves/ structures Application of Boussinesq Equation to Simulate Waves in the Surf Zone Boussinesq Equations Mass conservation: ... 0.4  0.57  5.3 tan   gd Flow in reformed waves: Qr  0.135 gd Sato et al (1992): M D,b  b b   2Q x  D gd tan  g Qˆ  Qr 2 d Qs  Qr Random waves: f D,ir   r  D tan  g Qˆ  Qr d