Solving the 3D Landslide Generated Tsunamis by

Implicit Velocity Method 利用隱式速度法，求解三維山崩海嘯

South China Sea Tsunami Workshop 6 Day 2: 7 November 2013 1530-1600

Prof. Tso-Ren Wu （吳祚任）

柯利鴻、莊美惠

tsoren@ncu.edu.tw

National Central University (Taiwan)

中央大學（台灣）

2

An overturning RC building （陳慧慈，2011）

We adopted the Splash3D numerical model to solve for the breaking wave

problems (Wu, 2004; Liu et al., 2005). This model solves 3-dimensional

incompressible flow with Navier-Stokes equations. The free-surface is tracked by

Volume-of-Fluid (VOF) method. The domain is discretized by finite volume

method (FVM). The turbulent effect is closed by large eddy simulation (LES) with

Smagorinsky model.

Splash 3D: A breaking wave model

0u

0

( ) 1 1( )

uuu g FP

t

Incompressible continuity equation:

Navier-Stokes Equation

3

The fluid density is presented in fluid fraction, and the transport equation is used to

describe the fluid movement.

0

m m

m

f

( ) 0umi m

ff

t

Volume of Fluid (VOF) method

4

( ) 0m m m m mm

t t x y z

u u v w

0p pN x C

Piecewise linear interface

calculation (PLIC)

( ) ( ) 0p tr p mF C V C f

LES (Large Eddy Simulation) Filtering

D : the filter width

h : the radius

A low-pass filtering operation is performed so that the resulting filtered velocity can be adequately resolved on a relatively coarse grid.

Filtered Conservation Equations

• Continuity equation:

• Conservation of Momentum: ( )

'

0

0

ii

i i

iii

i i

U U

x x

uU U

x x

æ ö¶ ¶÷ç ÷= =ç ÷ç ÷ç ¶ ¶è ø

¶ ¶= - =

¶ ¶

2

2

1

Let

1

where

j i j j

i i i j

i j i j

R

ij i j i j

r

j j ij

i i j i

U U UU P

t x x x x

U U U U

U U U U

DU U p

x x x xDt

D

tDt

nr

t

tn

r

¶ ¶¶ ¶+ = -

¶ ¶ ¶ ¶ ¶

¹

º -

¶ ¶¶= - -

¶ ¶ ¶ ¶

¶º + ×Ѷ

U

Q

1

2

2

3

the anisotropic residual-stress tensor is:

2

3

2

3

R

r ii

r R

ij ij r ij

r R

ij ij r ij

r

k

k

k

p P k

t

t t d

t t d

º

º -

º -

º +

Smagorinsky Closure Model

2i jr

ijij t t

j i

u uS

x xt n n

æ ö¶ ¶ ÷ç ÷ç= - + = -÷ç ÷÷¶ ¶çè ø

( )22

t S SCS Sn = = Dl

( )1 2

2 ij ijS SºS : the characteristic filtered rate of strain

: Smagorinsky length scale Sl

( )1/3

1 2 3x x xD = D ´ D ´ D

: Smagorinsky coefficient SC

: filter width D

8

淚望大海「無處去」 葛西先生說，當天強震後，海嘯警報響起，他趕緊與老婆爬到高處避難，起初第一、二波海嘯襲來時，就與漲潮無異，「只是第三波後越來越強，海水激烈衝撞，有一股無形力量將海水都吸到遠方去，幾乎能看到海底，第四波海嘯累積能量後，衝過來打向岸邊發出『砰』的聲音，威力就像一顆炸彈。」 他們親眼目睹岸邊房子瞬間被「炸」毀，2、300人罹難

0 0.5 1 1.5 2 2.5-20

-10

0

10

20

30

40

Time (s)

Forc

e (

N)

Computed

Lab1

Lab2

Lab3Lab4

Total force comparison: (on the square cylinder)

9

0 0.5 1 1.5 2 2.5-0.5

0

0.5

1

1.5

2

2.5

3

Time (s)

Velo

city (

m/s

)

On centerline, 14.6 cm from upstream face, 2.6 cm from floor

nonlinear k-Lab data 1

Lab dara 2

Velocity comparisons

10

(e)

(f)

(g)

(h)

(i)

Validation on the Free-Surface

Base maps is experiment snapshot ; Blue

is water; purple is structure 11

Partial-Cell treatment

1eff solidf

0

m

m

ff V

t

0

VVV p g F

t

If a cell contains partial volume of solid material, the flow solver has to deal with

it. Cell faces are defined either to be entirely closed, or not. Cell faces are “closed”

only if at least one of the two immediately neighboring cells is entirely occupied

by solid material. If the cell faces are “closed”, the face velocity of the cell is set

to zero, and the face pressure is no longer calculated in the pressure solution. On

the other hand, if any face between two cells, containing at least a partial cell

volume of fluid, is “open”, the code solves the velocities and pressure gradients.

12

DEM topography module

The real topography can be easily constructed in the Splash3D by using PCT.

13

Example of DEM topography module

Topography of Toce River Valley Dam-

break BenchMark problem.

Dx=dy=0.05 m, dz=0.01m.

No.3 Nuclear Power Plant

DEM topography module

3D Simulation (BC is obtained from

COMCOT) Domain 650×650

dx,dy 10m

dz 0.5m

Cells 1,376,000

Jiupeng (九棚) Tsunami Boulders Tsunami Boulders were found in Southern Taiwan

16

One of them presents a huge scour hole

Motivated from the real sediment transport

18

(2002 at Cornell)

Suspended load 懸浮載

Bed load 底床載

Firm bed 固態底床

Bingham and Bi-viscous Fluids Bi-viscous becomes Bingham as Mu_inf goes infinite

r

0

Newtonian Fluid

Bingham Fluid

Bi-viscous Fluid

B

Approaching to Bingham Fluid

(Strain rate)

(Shear stress)

19

切變率

剪力

Bingham Constitutive Model

2 ( )D D

200

2

0

1 if :

212 :

2

1 and 0 if :

2

B

D DD

D

strain rate

Bingham viscosity

Yield stress (Bingham yield)

Shear stress

A large number indicating the solid behavior

20

There are three unknown variables only.

(1) (2)

(3)

is just a huge value to keep the rigidity

Validation 1. Pressure Gradient Channel Flow (Bird et al. 1983)

Newtonian Fluid

Bingham Fluid

22

0 00

0 0

1 12

0

L

B B

M

P P B By yu y y y B

L B B

u y u y u y y

Analytical Solution of Bingham Fluid in a Channel

Plug Area Liquefied Area

5.0 Pa sB 0 0.5 Pa 1 6 Pa se 21

High Pressure

Low Pressure

(Flat in the Plug area)

(Accurate turning point)

Validation 2. Spreading of Bingham fluid on an inclined plane The analytical solution of Bingham flow was derived by Liu and Mei (1989)

Experimental set-up for gravity currents down a dry bed 22

Spreading of Bingham fluid on an inclined plane

23

Validation 3. Failure of Gypsum Tailings Dam East Texas, 1966

Flow of Liquefied Tailings from Gypsum Tailings Impoundment (1966)

Elevation ( t = 0 ~ 180 s )

Velocity Magnitude ( t = 0 ~ 180 s )

Velocity Distribution: The flow motion vanishes at the end.

Flow surface after freezing time computed by Splash3D model 27

Result Competition

Inundation distance

（m）

Freezing time

（s）

Mean velocity

（m/s）

Observed values 300 60-120 2.5-5.0

Theoretical results from charts 550 132 4.2

Result using TFLOW

（Jeyapalan, 1983）

470 85 5.5

Result computed by Pastor et

al.（2004）

170 1.4

Result computed by Chen

（2006）

200 1.7

Result using NS-VOF model 320 2.3

28

Validation 4. Simulation on THE FAILURE OF SHUAN-YUAN BRIDGE in the event of 2009 Typhoon Morakot

29

The undular waves imply the uneven soft bottom

Local scour induced by the strong flood mud_vof =0.05

30 （陳孟志製）

Maximum Scour Depth: Right in front of the bridge piers: Field survey: about 23 m. Numerical: 23 m.

30 m upstream away from the bridge piers: Field survey: 15 m Numerical: 15 m

Compare with the Field Survey Data

地電阻法 （Electrical Resistivity Tomography, ERT）

31

•With floating logs

32

無掛淤

入流

有掛淤

掛淤型態比較

In between On the upstream

face

Among the

piles

Rheology for Grains

The relationships of strain rate and shear stress for Newtonian, traditional Bingham, and hyperconcentration fluids. 33

• O’Brien and Julien (1985) 提出方程式，可同時考慮黏滯性及顆粒碰撞特性

• Liu (2005) developed Debris-2D

To simulate the debris flow.

Hyperconcentrated Sediment Flow

一次項

34

Plug into our rheological model

35

剪應力 Shear stress 切變率 strain rate

賓漢黏滯係數

Bingham viscosity

降伏應力

Yield stress (Bingham yield)

二次項 Quadratic term (Here is the new guy)

（液態）

（固態）

Snapshot of the experiment：

Numerical result without particle collision term：

Numerical result with practical collision term：

(Not much difference in the scour hole: Controlled by Yield Stress )

(Significant improvement in the wake area: Controlled by the quadratic term)

36

This year, this talk, moving solid.

Volume-averaged velocity for each numerical cell:

1 f ob

i ob i ob iv f v f v

fob = 1.0

0.0 < fob < 1.0

fob = 0.0 ・ f = 0.0

fob = 0.0 ・ f = 1.0

Water

Air

Obstacle

( ) 1 1( ) P

t

uuu g F

38

Adding an extra body force term in the NS equation to enforce the velocity.

Governing equations

( ) ( )

( )

0

1 1

0kk

pt

ff

t

T

u

uuu u u g F

u

　mr r

Ñ× =

¶+ Ñ× = - Ñ + Ñ× Ñ + Ñ + +

¶

¶+ Ñ× =

¶gwhere : gravity force

kf : volume fractions of fluid k

1,

0, 1,

0,

kf

inside fluid k;

At the fluid k interface;

outside fluid k;

VOF Equation

Projection Methods

1 *

1 1 1 *1 1 1

1 *

1

1 11 *

1 1

n n nn n n n n T n

n n nn n n

n

n

n nn

n n

t

Pt

P

t

Pt

u uu u u u

u ug F

ug

Fu u g

Momentum:

Predictor:

Projection:

Pressure Poisson Equation Iterate until both of the velocity and pressure converged.

Volume-averaged velocity for each numerical cell:

1 f ob

i ob i ob iv f v f v

Taking divergence to both side

Implicit Velocity Correction Method

• Avoid using partial cell treatment

• Avoid tweaking the Pressure Poisson Equation

• Adding a body force in the correction step to ensure the n+1 pressure matches the n+1 velocity

• Ensure no numerical oscillation in the pressure field

Before applying this method to the practical cases, we need to calculate the synthetic forces on the moving obstacles.

To do so, the first step is know volume fraction of the obstacles.

41

),,( 000 zyxO

),,( 1111 zyxP

),,( 2222 zyxP

x

y

z

1R2R

0Ri

圖(1) 球體體積分率計算。

Calculation for the volume fraction of a sphere obstacle on each grid cell

（王仲宇、邱佳聖）

),,( 000 zyxO

),,( 2222 zyxP

x

y

z

2d

i

),,( 000 zyxO1d

i

N

M

K

L

N

M

K

L

),,( 1111 zyxP

圖(2) 矩形固體體積分率計算。

Calculation for the volume fraction of a square obstacle on each grid cell

（王仲宇、邱佳聖）

Integrate the stress tensor

• 如圖(3)所示，梯形部分為表面積之積分微元，其中CP即積分微元之形狀中心；在流固耦合程序中，固體受力即CP所在位置之流體網格應力與面積的乘積之總和。另外，流場網格之應力張量可以表示如下

11 12 13

21 22 23

31 32 33

z

x

y

),,( zyxCp

Area

（王仲宇、邱佳聖）

3

2

1

333231

232221

131211

3

2

1

ˆ

ˆ

ˆ

n

n

n

T

T

T

n

n

n

Calculate the Force vectors based on the Cauchy formula

（王仲宇、邱佳聖）

X

Z

r

nk nc

n

ij

Discrete Element Method (DEM) for Collision A spring with stiffness kn is installed. From kn and displacement δ, the rebounding movement can be calculated.

（王仲宇、邱佳聖）

47

Verification of surface pressure

• Sphere properties : 1. Density : 800.52 kg/ m3

2. Radius : 0.015 m 3. Mass : 0.0113 kg

• Numerical model :

1. Ncell : 30x28x30 (25200) 2. Coord : 0.15 x 0.14 x 0.1 (m) 3. CPUs total : 1 4. CPU time : 1hr 5. BCs : free-slip & pressure dirichlet

Unit : m

（王仲宇、邱佳聖）

48

Velocity profile Freesurface profile

Result

（王仲宇、邱佳聖）

49

（王仲宇、邱佳聖）

Contact with boundary 自由落體測試 ; 真空 ; 球密度2000 kg/m3

（王仲宇、邱佳聖）

Contact with boundary in fluid

水密度 1000 kg/m3 ; 球密度2000 kg/m3

（王仲宇、邱佳聖）

（王仲宇、邱佳聖）

（王仲宇、邱佳聖）

Simulation on a Floating Obstacle

• 渠槽大小：15 × 14 cm、30 × 30 cm

• 網格大小：0.33 × 0.33 × 0.25 cm

• 楔形體：4.8 × 4.9 × 2.4 cm

• 變動條件：渠槽大小 54

（莊美惠製）

55

Floating and sinking balls

56

Fast moving ball with collapsing water

57

Floating obstacle in the river current

Rotation )( 100

srad

y

Dam-break bore interacting with a movable ball

1.6 m

0.6 m

0.96 m

0.12 m

0.4 m

0.4 m

0.3 m

0.02 m

0.6 m

[Side view]

[Top view]

Water density 1000 kg/m3 Ball density 1200 kg/m3

Resolution 70 X 30 X 30; Computational time: 3hrs 19min; CPU:i5-2500 CPU @ 3.30GHZ

Contact with incline slope

計算域(0.15,0.14,0.1) 水面高度 0.05 斜坡角度10.30993247 球初始位置(0.75,0.7,0.04) 球半徑 0.015 水密度 1000 kg/m3 球密度2600 kg/m3

Velocity

Overview

Fully coupled 3D landslide Simulation

Trajectory comparison

Dynamic Pressure

RUN40 Gauge comparisons (2/1)

RUN40 gauge comparisons (2/2)

Conclusion • Bingham constitutive model is able to describe mud and sediment motions. • Combining with VOF model, we are able to simulate the complex local

scour problem with only 3 property parameters. • Very accurate results are presented. • This model can be used on many practical problems, such as landslide,

mudslide, local scour problems. • The moving solid algorithm produced accurate and stable result. • Because there is no partial-cell treatment for the moving obstacle, the time

step can be reasonable large. • We have done the parallel version. Larger domain and higher resolution

can be applied to the future studies.

• Thanks for listening. Any questions for Prof. WU (吳祚任)？

70