Governing Equations and Setup

WORKSHOP ON LONG-WAVE RUNUP MODELS

Khairil Irfan Sitanggang and Patrick LynettDept of Civil & Ocean Engineering, Texas A&M University

Governing Equations and Setup

)(),,,(

less)(DimensionEquation Momentum

)(),,,()(1

less)(DimensionEquation Continuity

4232222

4232222

OOuuu

OOHuH

t

t

= a/ho

= (ho/lo)

h

ho

lo

x

z

ao

h +

Governing Equations – Highly Nonlinear Boussinesq

)(2

1

2

1

6

1

)(1

42

22222

Oh

huzhhH

uzhhH

HuH

t

t

Continuity Equation (Dimensionless)

Governing Equations – Highly Nonlinear Boussinesq

t

t

t

ttt

t

hhuQ

Ouuu

uQQuu

uuz

uzuz

QuzQzuQQQ

QzuzzQzuz

uuu

:where

)(2

2

2

2

42

2

23

2

22

2

2

2

2

2

Momentum Equation (Dimensionless)

Numerical Algorithm

• Predictor - Corrector Scheme– 3rd-order explicit Adams-Bashforth (Press et al.,

1989) predictor step - (t)3

– 4th -order implicit Adams-Moulton corrector step - (t)4

– Finite difference spatial derivatives to 4th-order accuracy - (x)4

– Scheme developed to solve the Boussinesq equations in 2HD

• Not a computationally efficient solver for the NLSW equations or 1HD setups

Practical Simulation• Bottom friction based on the quadratic friction law

• Wave breaking with an eddy viscosity model

Comparison with data from Hansen & Svendsen (1979)

• Runup/rundown uses the extrapolation moving shoreline procedure (Lynett et al, 2002)

Moving Boundary Algorithm

• The underlying assumption is that very near the wet-dry boundary, the wave is linear in slope:

Linear extrapolation- imaginary points in dry

region

• The free surface and velocity are linearly extrapolated through the wet-dry boundary, creating “imaginary” points in the dry region.

• Wet nodal points near the wet-dry boundary use the extrapolated points when calculating finite difference derivates (5-point centered differences)

t1

t7

t6t5

t4t3

t2

t8

Validation of Runup algorithm

• Runup of solitary waves

– Comparison with experimental data taken from Synolakis (1987)

• Numerical simulation parameters:

– Wave height / water depth = 0.04

– Beach slope = 1:20

Comparison with experimental data:

Numerical results

Experimental data

Validation of Runup algorithm

• Runup of solitary wave around a circular island

– Experimental data taken from Liu et al. (1995)

Inundation comparisons

Black dots represent the maximum experimental runup, while the light

red shows the inundated area

Benchmark #1

• Initial free surface given

– Run with:

• NLSW

• Boussinesq

• Will examine the dispersive effect & required grid resolution near shoreline for numerical convergence

• Simulations have no bottom friction, wave breaking not included

Benchmark #1• Numerical

convergence

– Nearshore grid ~1.5 m is required for numerical convergence

– Error much larger in velocity predictions

Benchmark #1• CPU requirements:

– Numerical model uses a constant dx and dt• dt is chosen based on a Courant formulation, where the

characteristic velocity is the long wave velocity in the deepest water depth in the domain.

• With constant slope to offshore boundary (and very deep water) a small time step is required.

– NLSW simulation:• dx=1.5 m, dt=0.0018 s• nx~32,000, nt~170,000 (endx~50 km, endt~300s)• 51 MB of RAM• 40 hrs of CPU time on a 1.8 Ghz desktop!

– ~0.9 seconds per time step– Numerical implementation not developed for large 1HD problems or

NLSW equations

Benchmark #1• Shoreline elevation and velocity time histories Boussinesq numerical vs NLSW numerical vs Analytical

• NLSW numerical result matches solution very well

• Boussinesq solution indicates that dispersive effects during shoaling may play a significant role

Benchmark #1• Snapshots of free surface and velocity at various times

Benchmark #1• Conclusions

– Required small grid for convergent runup results leads to very large CPU times for a 1HD problem

• Large grid error more significant for velocity comparisons

– NLSW numerical results match well for both shoreline and spatial profiles

– Boussinesq predicts higher maximum runup, but lower maximum speeds

• Frequency dispersion may be important during shoaling, as the wave becomes very steep

Benchmark #2 Khairil Irfan Sitanggang, Patrick Lynett

Dept of Civil & Ocean Engineering, Texas A&M University, U.S.A

Alejandro OrfilaIMEDEA (CSIC-UIB), Spain

• CPU requirements:

– Numerical model uses a constant dx and dt

• To capture the oscillations inside the cove, a relatively small global grid size was used

– Boussinesq simulation:

• dx~0.012 m, dt~0.005 s

• nx~450, ny~300, nt~4,500 (endt=25s)

• 200 MB of RAM

• 5.5 hrs of CPU time on a 1.8 Ghz desktop

– ~4.5 seconds per time step

• Tsunami generation by internal source generator, using specified input wave time series

Benchmark #2• Tsunami Approach on Complex Bathymetry

– Bottom friction included (quadratic friction law)• Friction coefficient = 0.0025

– Breaking model used• Wave begins to “break” when t>0.65[g(h+)]0.5

• Energy dissipation (eddy viscosity model) is added at the breaking locations


– Animation from Boussinesq simulation (wide view)

Benchmark #2

• Tsunami Approach on Complex Bathymetry– Animation from

Boussinesq simulation (shoreline closeup)

– Max predicted runup in cove ~ 6.5 cm.


– Comparison w/ video data

• Angle of approach of the positive wave is different, leading to different runup patterns

Benchmark #2• Free surface time series comparisons


– Model does an OK job at recreating the experiment

– Primary differences due to different approach angle of the positive wave

• Possible causes:

– Breaking not predicted correctly

– Bottom friction underestimated

– Input wave not exact

– No significant differences between NLSW and Boussinesq for this case

Benchmark #3 Khairil Irfan Sitanggang, Patrick Lynett

Dept of Civil & Ocean Engineering, Texas A&M University, U.S.A

• CPU requirements:– Case A:

• dx=10 m, dt=0.16 s (small dt for deep water stability)• nx=470, nt=450• Boussinesq CPU time= 35 seconds on 1.8 Ghz desktop

– Case B:• dx=0.25 m, dt=0.008 s • nx=400, nt=1400• Boussinesq CPU time= 90 seconds on 1.8 Ghz desktop

– For reference, numerical integration of the analytical solution required:

• nx=1300, nt=2000• CPU time = 6 hours on 1.8 Ghz desktop (lots of integration with Bessel &

exponential functions)

Benchmark #3• Subaerial slide

– Animation of Case A from Boussinesq Simulation

No bottom friction

Breaking model not implemented

Slide is shown as the yellow area

Slide very thin

Benchmark #3• CASE A comparisons (large tan/)

Benchmark #3• Subaerial slide

– Animation of Case B from Boussinesq Simulation

No bottom friction

Breaking model not implemented

Benchmark #3• CASE B comparisons (small tan/)

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

x (m)

z (

m)

Free Surface at t*= 0.5

Analytical SolutionNLSW Numerical Bous Numerical

0 10 20 30 40 50-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

x (m)

z (

m)



0 20 40 60 80 100-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x (m)

z (

m)



0 50 100 150

-0.5

0

0.5

1

1.5

x (m)

z (

m)




– Case A:

• Except for very early time, agreement excellent

• Shoreline does not move much

• NLSW and Boussinesq identical

– Case B:

• Agreement with analytical solution OK at early time, but differences grow quickly

• The analytical solution, due to assumptions on the slide forcing, is not adequate for this case

• Some difference between NLSW and Boussinesq at later times, but relatively minor

• Indications that turbulence and/or dispersive effects become important at later times

Governing Equations and Setup

Documents