B O S Z. - Boussinesq Ocean & Surf Zone model - International Research Institute of Disaster Science (IRIDeS), Tohoku University, JAPAN

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1 B O S Z - Boussinesq Ocean & Surf Zone model - Volker Roeber 1 Troy W. Heitmann 2, Kwok Fai Cheung 2, Gabriel C. David 3, Jeremy D. Bricker 1 1 International Research Institute of Disaster Science (IRIDeS), Tohoku University, JAPAN 2 Department of Ocean & Resources Engineering, University of Hawaii, USA 3 Franzius Institute for Hydraulics and Coastal Engineering, University of Hanover, GERMANY 1

2 B O S Z in a nutshell - Developed for energetic breaking waves - Applications in: Flood hazard mapping, Disaster management, Studying nearshore processes Coastal engineering design. - Requirements: Cover wide range of wave scenarios, Numerical robustness, Efficient computation. As complex as necessary, as lean as possible! 2

3 BOSZ Key Features Depth-integrated Boussinesq-type Equations Conservative form of Nwogu s (1993) Boussinesq equations, see: Roeber & Cheung (2012) Shallow water equations as subset Extension with nonlinear dispersion Momentum Conservation Imbedded conservation laws for sub- and supercritical flows Finite Volume - Finite Difference scheme Explicit 2 nd or 4 th order Runge-Kutta time integration, adaptive time step, Riemann solver for wave breaking Parallelization Full OpenMP parallelization to handle large flow problems 3

4 BOSZ Governing Equations Conserved variables: ( Hu) t + ( Hu 2 ) + Huv x ( Hv) t + ( Hv 2 ) + Huv y ( ) + H u t + uu x + vu y ( ) y = u H t + ( Hu) x + ( Hv) y ( ) ( ) + H v t + vv y + uv x ( ) x = v H t + ( Hv) y + ( Hu) x ( ) H t + ( Hu) x + ( Hv) y + z α, local flux * $ 2 2 h 2 ' & ) h ( u xx + v ) xy % 6 ( + $ z + h ' & % α 2 ) h hu + ( + z dispersion (" 2 α 2 % $ h2 'h ( u xy + v ) yy # 6 & + " z + h % * $ # α 2 'h hu ) & (( ) xx + ( hv) xy ) (( ) xy + ( hv) yy ) - /. x + -, y = 0 Continuity: ( Hu) t + ( Hu 2 ) + Huv x ( ) y + ghη x + H z 2 α local flux + bottom slope [ ] + Hz α hu t [( ) xx + ( hv t ) xy ] 2 u xxt + v xyt + + u z 2 α 2 h dispersion % 2 ( ' * h ( u xx + v ) xy & 6 ) + % z + h ( - ' & α 2 * h hu, ) * $ +u z 2 α 2 h 2 ' & ) h ( u xy + v ) yy % 6 ( + $ z + h ', & % α 2 ) h hu + (. 0 / - /. (( ) xx + ( hv) xy ) (( ) xy + ( hv) yy ) y x = 0 Momentum x: 4

5 BOSZ Equation Restructuring Vector form: U t + F( U) x + G( U) y + S( U) = 0 U = [H, P, Q] T F = # Hu & % Hu ( % 2 gη2 + gηh( % $ Huv ( ' # & % Hv ( G = % Hvu ( % Hv ( 2 gη2 + gηh $ % '( S= & ψ C ) ( + ( gηh x H t ψ P + uψ C + τ 1 + '( gηh y H t ψ Q + vψ C + τ 2 * + Nonlinear Shallow-Water Equations With 2D TVD reconstruction and HLLC Riemann solver Dispersion terms only containing spatial derivatives In Finite Difference - 2D 5 th order TVD reconstruction (Kim & Kim 2005) - Wet/dry boundary based on Audusse et al., 2004, and Liang, Frictional drag based on Manning s n - Indexing of only wet cells to save computation time 5

6 Equation Solution Runge-Kutta time integration of 2 nd or 4 th order with adaptive time step H i,! n+1 j = H n i, j + Δt n n n ( E 1i, j + E 2i, j ) P i,! n+1 j = P n i, j + Δt n F n n ( 1i, j + F 2i, j ) Q i,! n+1 j = Q n i, j + Δt n n n ( G 1i, j + G 2i, j ) ( ) + hu P = Hu + Hz! α " 0.5z α u xx + v xy ( ) xx + ( hv) xy [ ( ) + ( hv) yy + ( hu) xy ] Q = Hv + Hz α 0.5z α v yy + u xy # $ H n+1 i, j = 1 2 P n+1 i, j = 1 2 Q n+1 i, j = 1 2 H! n+1 + H n ( ) + Δt n i, j i, j 2 P! n+1 + P n ( ) + Δt n i, j i, j 2 Q! n+1 +Q n ( ) + Δt n i, j i, j 2 E! n+1 + E n+1 (! ) 1i, j 2i, j F! n+1 + F n+1 (! ) 1i, j 2i, j G! n+1 + G n+1 (! ) 1i, j 2i, j Flow velocities U and V from series of one-dimensional systems of equations Favorable to parallelization in the x and the y direction without data dependency Temporal derivatives in cross-terms are evaluated with one-sided derivatives φ P = Hz! α " 0.5z α v txy φ Q = Hz! α " 0.5z α u txy ( ) + hv t ( ) xy ( ) + hu t ( ) xy # $ # $ 6

7 Horizontal velocity at any depth: ( ) = z α h( u α ) + ( h u α ) u x, z,t Velocity Variables ( ) #$ % & z 2 #$ u α α % & Vertical velocity at any depth: ( ) = u α h + z + h w x, z,t $% ( ) u & α ' For range h < z < η Evaluation of velocity variables at approximately mid depth z α = 0.53h z α = h z α = 0.333h 7

8 Wave Breaking in BOSZ Potentially arising instability Momentum conserved for sub- and supercritical flow without additional treatment Two strategies: 1. Problem Buffering 2. Problem Prevention Eddy viscosity Neglecting dispersion 8

9 Wave Breaking in BOSZ Celerity gh Momentum Momentum (HU) Gradient Gradient > 0.5 gh x (HU) (HU) x x Ignore dispersion locally and momentarily if: (HU) x > 0.5 gh, (HV ) y > 0.5 gh 9

10 Previous BOSZ Validation US National Tsunami Hazard Mitigation Program Model verification/validation with analytical solutions and laboratory experiments 10

11 Previous Model Validation 2. Solitary wave reflection Test C: A/h = BOSZ experimental data (Briggs et al., 1995) BOSZ performance: Reasonable results even for conditions beyond model s range of applicability Runup experiment: cm Runup BOSZ: cm 81 %

12 Previous Model Validation 3. Solitary wave around conical island BOSZ, no breaking BOSZ, with breaking experimental data (Briggs et al., 1995) Basin configuration 0.32 m Wave input

13 Previous Model Validation 3. Solitary wave around conical island BOSZ, no breaking BOSZ, with breaking experimental data (Briggs et al., 1995) Free Inundation surface Basin configuration 0.32 m Wave input

14 Previous Model Validation 4. N-wave at Okushiri Island, Monai BOSZ experimental data (Matsuyama & Tanaka 2001) x [m] Wavemaker y [m]

15 Previous Model Validation 4. N-wave at Okushiri Island, Monai BOSZ experimental data (Matsuyama & Tanaka 2001) 0 Inundation Line y [m] x [m]

16 Additional Benchmark Tests Large Wave Flume, Oregon State University 16

17 BOSZ - validation, real reef Southshore Oahu, HI Waikiki Aquadopp ADCP

18 BOSZ flow velocity, Waikiki Mean flow velocity Wave Setup

19 Nearshore Waves on top of storm surge 19

20 BM1, flow over cone Flume: Water depth: 5.4 cm Flow speed : 11.5 cm/s Courant : BOSZ settings: Δx : 1-3 cm n : s/m 1/3 Boundary conditions: Discharge at left boundary by adding mass Iterative adjustment of flow speed (control point ) Subtraction of extra flow depth from bathymetry 20

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22 Comparison at Gauge 1 n= 0.01 s/m 1/3 22

23 Comparison at Gauge 2 n= 0.01 s/m 1/3 23

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25 Comparison at Gauge 1 n= 0.01 s/m 1/3 n= 0.02 s/m 1/3 25

26 Comparison at Gauge 2 n= 0.01 s/m 1/3 n= 0.02 s/m 1/3 26

27 Grid size dependency dx = m dx = m dx = m dx = m 27

28 Grid size dependency dx = m dx = m dx = m dx = m 28

29 dispersive vs. hydrostatic dispersive hydrostatic 29

30 dispersive vs. hydrostatic dispersive hydrostatic 30

31 BM2, Tohoku tsunami in Hilo Bay Δx Δy Boundary conditions: BOSZ settings: Δx,Δy : 18.7 m / 20.6 m 9.3 m / 10.3 m 4.7 m / 5.1 m n : s/m 1/3 Change in continuity equation according to tsunami time series Input only along North boundary Radiation boundary conditions at North and East side 31

32 Initial test with input wave signal over flat bathymetry - uniform bathymetry of 30 m - no friction - open radiation boundary conditions Data BOSZ 32

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34 Dispersive vs. hydrostatic Dispersive hydrostatic 34

35 Free surface comparison Δx = 20 m Δx = 10 m Δx = 5 m 35

36 HA velocity Δx = 20 m Δx = 10 m Δx = 5 m 36

37 HA velocity Δx = 20 m Δx = 10 m Δx = 5 m 37

38 Maximum Velocities dx = 20 m dx = 10 m dx = 5 m 38

39 BM5, reef with cone Top view Wavemaker x Wave gauges ADV (velocity) Laboratory Facility Tsunami Wave Basin, Oregon State University Tsunami Wave Basin Wave height: 0.39 m Water depth: 0.78 m A/h: 0.5 3D reef with apex 39

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41 BM5 free surface BOSZ OSU data 41

42 BM5 velocity BOSZ OSU data 42

43 Conclusions for BM tests Results are not very grid dependent. BOSZ resolves current flow structures reasonably well. Friction is a critical factor in shallow flows. Even for fairly hydrostatic problems, use dispersive model. Volker Roeber roeber@irides.tohoku.ac.jp 43