Computational Modelling of Free Surface Flows: wave interaction with fixed and floating bodies

Computational Modelling of Free

Surface Flows: wave interaction with

fixed and floating bodies

Derek Causon, Clive Mingham, Ling Qian

Zheng Zheng Hu and Hanbin Gu

Department of Computing & Mathematics

CMMFA Team Photo

B Wang, ZZ Hu, Y Zhang, D Causon, J Armesto, K Bennett, L Qian (Back) S Higgins, N Subramaniam, F Gao, J Shiach and C Mingham (Front)

Outline

• Shallow water code

AMAZON-CC

• Surface capturing Navier Stokes code

AMAZON-SC

• Wave overtopping and wave energy converters

• Further methods development

• High Performance Computing – the GPU

• Future work

Manchester Bobber

The Pelamis

AMAZON two-fluid solver

• Incompressible Navier-Stokes solver — Based on an artificial compressibility solver.

• Surface-capturing method

– Treats the free surface as a contact discontinuity in the density field, allowing the use of modern high resolution “shock capturing” methods.

• Fully two phase approach which solves in both the air

and the water fluid regions.

overlay Cartesian grid

(adaptive) cut cell grid for an island

AMAZON-CC: generation of oblique

waves using cut cells

3D incompressible, Euler equations with variable density: T I T I T I T z I y I x I T S w p w vw uw w v vw p v uv v u uw uv p u u n n n p w v u s t 0 g 0 0 0 , , where . 2 2 2 h g f B h g f F Q B n F Q

is the coefficient of artificial compressibility

• The convective flux (F_k) is evaluated using Roe’s approximate Riemann solver.

• To ensure second order accuracy, MUSCL reconstruction is used

where (x,y,z) is a point inside the cell (i,j,k), r is the vector from

the cell centre to the point (x,y,z), Q_i,j,k is the cell centre data

and is the slope limited gradient.

k k k I k I I k

F

Q

F

Q

R

LQ

Q

F

1₂

r

Q

Q

Q

(

x

,

y

,

z

)

_{i,j,k i,j,k}

Convective fluxes

k j i

Q

_{, ,}

Time discretisation

The implicit backward Euler scheme is used together with an

artificial time variable (to ensure a divergence free velocity field) and a linearised RHS. 1 1 1 1 1 1 1 , 1 , , 1 , 1 1 , 1 , 1 where ) ( t t t m m n m n m n ta m n m n m n m diag I R t V I R V I Q Q Q Q Q Q Q

The resulting system is solved using an approximate LU factorisation. A Jameson-type dual time iteration is used to eliminate at each real (outer) iteration.

AMAZON-SC 2D simulation

(air/water)

• Overtopping event occurred 142 seconds into the experiment.

• Seaward boundary located 2m from wall. • Landward boundary is transmissive.

Merged cell

_{Cut cell}

Computational domain and

boundary conditions

Body Air

Reflection boundary (the fictional cell R): Wave maker Water Numerical Tank Non-reflecting Non-reflecting Non-reflecting Wall RO gn p p v v v v z ijk ijk R b ijk ijk R ijk R n n n n) 2( ) ( 2 Free surface

Results

A. Fixed Body

B. Moving Body

C. Wave propagation and Extreme

waves

1. A vertical cylinder in a 1st _{order wave maker}

2

. A horizontal cylinder in a 1st _{order wave maker}

3. The Dam break flow past various bodies (a right cube, a skewed cube, a bottom mounted and truncated cylinder) 4. A bottom mounted and truncated cylinder in regular waves

5

. A Pelamis-type Geometry in regular waves

• A regular 1st _{order wave imposed at the inlet:}

• Experiments and Theoretical analyses conducted by

A.G. Dixon et al. (1979) and W.J. Easson et al. (1985)

as well as used the STAR CCM by J. Westphalen

2. Horizontal Cylinder

)

cosh(

)

cos(

))

(

cosh(

k h

t

k x

h

z

k

gAk

u

)

cosh(

)

sin(

))

(

sinh(

k h

t

k x

h

z

k

gAk

w

)

cos(

kx

t

A

-0.40 -0.30 -0.20 -0.10 0.00 0.10 0.20 0.30 0.40 0.00 0.50_t/T 1.00 R e la ti v e v e rt ic a l fo rc e Theoretical force Experimental force present result -0.60 -0.40 -0.20 0.00 0.20 0.40 0.00 0.50 1.00 t/T R e la ti v e v e rti c a l fo rc e Theoretical force Experimental force Present result Case1: d= 0.0m Case 2: d= -0.075m Case 3: d= -0.15m -0.40 -0.30 -0.20 -0.10 0.00 0.10 0.20 0.30 0.40 0.00 0.50 1.00 t/T R e la ti v e v e rti c a l fo rc e Theoretical force Experimental force present result

2. Horizontal Cylinder

• Pelamis: 0.6 × 0.2 × 0.2m

• Tank: 6.86 × 1 × 1m

• Water depth: 0.45m

• Grid size: 256 × 39 × 39

• Wave gauges: at front of Pelamis

• CPU about 25 days for 8s • Input velocity = Asin t

A=-0.1, =2 , T =1s

5. Fixed Pelamis-type geometry in

regular waves

1 2 3 4 5 6 7

front _bac

k

•Tank: 13.0m×1.0m×3.5m and the water depth:h=2.8m

•The front end of the cylinder is placed at 5.0m from the wave makers • Input amplitude: A=0.025,0.05,0.1 & 0.15, wave period: T=1.78s, wave maker number: k=1.277, wave length: L=5.0m

the angular frequency: w=3.534 and frequency: f=0.5625

the radius of cylinder: a=0.095m, the total length of Pelamis=1.40m

Sketch of the Pelamis split to 9

sections

A horizontal position of

Pelamis

5. Pelamis Tests

Sketch of the reference of

B. Moving Body Cases

1. Oscillating cone describing a vertical Gaussian wave packet motion (blind test validation)

2. Water entry of various rigid bodies (a wedge; Bobber-type shape; sphere shape; cone shape)

at a prescribed velocity

• The vertical position of the cone followed the form of a Gaussian wave packets

• Experiments conducted by K. Drake et al.(2008)

• Test case: A=50mm and m=9 • The dead-rise angle is 45º

• Tank: 2.0m×1.6m, the water depth: h=1.02m and the initial

draught of the cone is z=0.148m

• Dimensions: a =0.228m, d=0.2281m, b=0.05m • The time step=0.00005

the CPU for coarse mesh (dx=dy=0.02m) 15 hours the CPU for fine mesh (dx=dy=0.01m) 41 hours

1. Oscillating Cone using

AMAZON-2D (axisymmetric) code

d z

a

Comparison with experimental data (K. Drake et al. 2008) and numerical result for non-dimensional vertical forces

on cone against time

-6 -4 -2 0 2 4 6 t/T -0.8 -0.4 0 0.4 0.8 1.2 n o n -d im e n s io n a lis e d v e rt ic a l fo rc e m=9 AMAZON 001 AMAZON 002 experimental data CV-FE -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 -6 -4 -2 0 2 4 6 t/T N o n -d im e n s io n a l v e rt ic a l fo rc e s AMAZON (Δ=0.01) AMAZON (Δ=0.02) Experimental data

• Body moves vertically downwards towards an initially calm water surface at a velocity V=1.19m/s

• Experiments conducted by Tveitnes et al.(2008)

• Tank: 2.0m×0.4m×2.0m and the water depth: h=1.0m

• The dead-rise angle is 45º

• Uniform mesh: 80×16×80=102,400 (dx=dy=dz=0.025m)

• Dimensions: breadth=0.6m, length=0.3m, height=0.3m

• A time step=0.0002s and total CPU about 3 days and 6 hours (t=1s)

Comparison with experimental data (Tveitnes et al. 2008) and numerical result for water entry forces

0 100 200 300 400 500 600 700 0 0.5 1 1.5 z/d F o rc e ( N ) Experimental data Present result d z

Beam sea direction Head sea direction

wave maker number : k=6.63 K=8.36

Wave period: T=0.78s Wave length: L=0.95m

T=0.69s

Wave length: L=0.75m

frequency: f=1.28 f=1.44 amplitude: A=0.013 A=0.013

Head sea direction

Beam sea direction Sx

Radius a

mf=1.2 kg (float mass)

mc=0.4 kg (counterweight mass) Pulley radius=0.0175m

Bobber geometry: a=0.074m, b=0.06m and c=0.07m, Sx = 4a, the water depth =0.46m

Initial free surface

b a c

3. Single and

multiple floating

Bobbers in regular

waves

Manchester Bobber

3a. Free-decay for the Manchester Bobber

flat-bottomed and conical-topped

float

3 6 1 i i i i i

U

f

mg

M



g Mf Mc ds z p a Mf Mc b s ) ( * ) ( 3 n float mass=2.1kg Counterweight mass=1.0kg Mf Mc ds z p b s n m= 1.1kg M = 3.1kg 3

a

a. The drop test

-0.10 -0.05 0.00 0.05 0.10 0.00 3.00 6.00 9.00 time (s) V e rt ica l d isp la ce m e n t (m) Experimental data Present result Case 1: mf=2.1kg, mc=1.0kg

Comparison: experimental data (T. Stallard (2008) ) and current results for decay rate

Case 2: mf=2.1kg, mc=1.2kg -0.10 -0.05 0.00 0.05 0.10 0.00 3.00 6.00 9.00 time (s) V e rt ica l d isp la ce m e n t (m ) Experimental data Present result

b. The rise test

Comparison: experimental data (T. Stallard (2008) ) and current results for decay rate

Case 1: mf=2.1kg, mc=1.0kg Case 2: mf=2.1kg, mc=1.2kg -0.15 -0.10 -0.05 0.00 0.05 0.10 0.00 3.00 6.00 9.00 time (s) ve rt ica l d isp la ce m e n t (m ) Experimental data Present result -0.15 -0.10 -0.05 0.00 0.05 0.10 0.00 3.00 6.00 9.00 time (s) V e rt ica l d isp la ce m e n t (m ) Experimental data present result

Beam sea direction Head sea direction

wave maker number : k=6.63 K=8.36

Wave period: T=0.78s Wave length: L=0.95m

T=0.69s

Wave length: L=0.75m

frequency: f=1.28 f=1.44 amplitude: A=0.013 A=0.013

Initial free surface

a

b

Single and multiple hemispherical Bobbers

in regular waves

• mf=1.2kg,mc=0.4kg

• the water depth =0.46m

• The radius of a=0.075m,

b=0.07m

• The floats separated by Sx = 4a

Head sea direction

Beam sea direction

Sx

a. Head seas

direction

Isolation

Displacement vs time -0.04 -0.02 0.00 0.02 0.04 0.06 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 time (s) V e rt ic a l d is p la c e m e n t (m ) Bobber 1 Isolation -0.40 -0.20 0.00 0.20 0.40 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 time (s) V e rt ica l ve lo ci ty (m /s) _{Bobber 1} Isolation -6.00 -4.00 -2.00 0.00 2.00 4.00 6.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 time (s) V e rt ic a l fo rc e ( N ) _{Bobber 1} Isolation

Vertical force vs time Vertical velocity vs time

Comparison: The Bobber at same position in isolation and array

-4.00 -2.00 0.00 2.00 4.00 34.00 34.50 35.00 35.50 36.00 36.50 t/T d (t )/ A Exp.data

Present result (float 3)

-4.00 -2.00 0.00 2.00 4.00 34.00 34.50 35.00 35.50 36.00 36.50 t/T d (t )/ A Exp. data

Present result (float 1)

-4.00 -2.00 0.00 2.00 4.00 34.00 34.50 35.00 35.50 36.00 36.50 t/T d (t )/ A Exp.data

Present result (float 5)

-4.00 -2.00 0.00 2.00 4.00 34.00 34.50 35.00 35.50 36.00 36.50 t/T d (t )/ A Exp. data

Present result (float 4)

Displacement

in head seas direction

-4.00 -2.00 0.00 2.00 4.00 34.00 34.50 35.00 35.50 36.00 36.50 t/T d (t )/ A Exp. data

present result (float 2)

Comparison: experimental data (T. Stallard (2008) ) and

b. Beam seas

direction

Isolation

Displacement vs time

-0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 time (s) V e rt ic a l d is p la c e m e n t (m ) Bobber 1 Isolation -4.00 -2.00 0.00 2.00 4.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 time (s) V e rt ic a l fo rc e ( N ) Bobber 1 Isolation

Vertical force vs time

Vertical velocity vs time

-0.25 -0.13 0.00 0.13 0.25 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 time (s) V e rt ica l ve lo ci ty (m /s) Bobber 1 Isolation

Comparison: The Bobber at same position in isolation and array

-4.00 -2.00 0.00 2.00 4.00 30.00 30.50 31.00 31.50 32.00 32.50 t/T d (t )/ A Exp.data

Present result (float 3)

-4.00 -2.00 0.00 2.00 4.00 30.00 30.50 31.00 31.50 32.00 32.50 t/T d (t )/ A Exp.data

Present result (float 1)

Displacement

in beam seas direction

-4.00 -2.00 0.00 2.00 4.00 30.00 30.50 31.00 31.50 32.00 32.50 t/T d (t )/ A Exp. data

Present result (float 5)

Comparison: experimental data (T. Stallard (2008) ) and present results for displacement vs wave period

-4.00 -2.00 0.00 2.00 4.00 30.00 30.50 31.00 31.50 32.00 32.50 t/T d (t )/ A Exp. data

present result (float 4)

-4.00 -2.00 0.00 2.00 4.00 30.00 30.50 31.00 31.50 32.00 32.50 t/T d (t )/ A Exp.data

C. Devices in Extreme waves

1. Wave propagation in empty tank for comparison of the experiments conducted by Gao (2003)

2. Extreme wave in empty tank for comparison of the

experiments conducted by Ning et al. (2008)

Inlet condition for ‘New Wave’ extreme wave:

) ) ( ) ( sin( ) cosh( )) ( sinh( 0 0 1 ) 1 ( i i i i i N i i i i _{k x x t t} h k h z k k gA w N i i i i i k x x t t A 1 0 0 ) 1 ( ) ) ( ) ( cos( _N i i i i i N i j i i j i j i j i j j i j i j i j i t t x x k A t t x x k k A A t t x x k k A A 1 0 0 2 1 0 0 0 0 ) 2 ( ) ) ( ) ( ( 2 cos ) ( ) )( ( ) )( ( cos ) ( ) )( ( ) )( ( cos ) ) ( ) ( cos( ) cosh( )) ( cosh( 0 0 1 ) 1 ( i i i i i N i i i i _{k x x t t} h k h z k k gA u ) ) ( ) ( ( 2 cos 2 cosh ) ( 2 cosh ) ( ) ( )] ( ) )( ( ) )( cos[( ) cosh( ) )( cosh( ) ( ) ( ) ( )] ( ) )( ( ) )( cos[( ) cosh( ) )( cosh( ) ( ) ( ) ( 0 0 1 2 1 0 0 ) 2 ( i i i i i N i i j j i i i N i j i j i i j i j i j i j i j i j i j i j i j i i j i j i j i j i j i j i j i j i t t x x k h k h z k D G A k t t x x k k h k k h z k k D G A A k k t t x x k k h k k h z k k D G A A k k u ) ) ( ) ( ( 2 sin 2 cosh ) ( 2 sinh ) ( ) ( 2 )] ( ) )( ( ) )( sin[( ) cosh( ) )( sinh( ) ( ) ( ) ( )] ( ) )( ( ) )( sin[( ) cosh( ) )( sinh( ) ( ) ( ) ( 0 0 1 2 1 0 0 ) 2 ( i i i i i N i i j j i i i N i j i j i i j i j i j i j i j i j i j i j i j i i j i j i j i j i j i j i j i j i t t x x k h k h z k D G A k t t x x k k h k k h z k k D G A A k k t t x x k k h k k h z k k D G A A k k w

Comparison with experimental data by Ning et al. (2008) and present numerical results for free surface elevation at x=3.0 m

The focus point at x =3m maximum elevation minimum elevation Physical Exp. by Ning et al.

(2008) 0.5653 (t=9.2s)

Present result (1st _{order) 0.5707 (t =9.147s) 0.4444 (t=8.557s)}

Present result (1st _{+ 2}nd order) 0.5717 (t =9.147s) 0.4439 (t=8.557s) time (s) S u rf a c e e le v a ti o n (m ) 6 7 8 9 10 11 12 0.44 0.46 0.48 0.5 0.52 0.54 0.56

0.58 present result (1st order)

present result (1st+2nd order) Physical Exp.

Bobber Device in

Extreme Waves

• Input velocity under case 2 (1st ₊₂nd _order)

• Tank: 13.0m×0.48m×1m with the water depth: h=0.5m

• The initial position of the apex of Bobber in the tank is

3.0m(=the focus point)×0.24m×0.35m

• The radius of the hemispherical base: a=0.15m, the height of cylindrical section: b=0.15m and non-uniform mesh:

425×22×40=374,000 and the refined regions=0.02m

• The mass of the Bobber = the volume of the hemispherical (m=ρ2 a3/3=7.068kg),which is a little larger than the initial immersed volume

• Wave gauges: 3.48m,3.68m and the front side of the Bobber • A time step=0.0003s and CPU near 25 days up to t=12s

Initial free surface

b

Time history of the heave force on the Bobber

Time history of wave run-up on the front side of the Bobber

Case 2:

Case 3:

Case 2 0.40 0.45 0.50 0.55 0.60 0.00 2.00 4.00 6.00 8.00 10.00 12.00 time (s) S u rf a c e e le v a ti o n ( m ) _{Case 3} 0.40 0.45 0.50 0.55 0.60 0.65 0.00 4.00 8.00 12.00 time (s) S u rf a c e e le v a ti o n ( m ) Case 2 -20.00 -10.00 0.00 10.00 20.00 0.00 4.00 8.00 12.00 time (s) H e a v e f o rc e ( N ) Case 3 -30.00 -20.00 -10.00 0.00 10.00 20.00 30.00 0.00 4.00 _{time (s)} 8.00 12.00 H e a v e f o rc e ( N )

Case 3:

Case 2:

Case 2 -0.40 -0.20 0.00 0.20 0.40 0.00 4.00 8.00 12.00 time (s) V e rt ic a l v e lo c it y (m /s ) Case 3 -0.60 -0.30 0.00 0.30 0.60 0.00 4.00 8.00 12.00 time (s) V e rt ic a l v e lo c it y (m /s )

Time history of the vertical velocity of the Bobber

Case 2 -0.08 -0.04 0.00 0.04 0.08 0.00 4.00 8.00 12.00 time (s) V e rt ic a l d is p la c e m e n t (m ) Case 3 -0.10 -0.05 0.00 0.05 0.10 0.00 4.00 8.00 12.00 time (s) V e rt ic a l d is p la c e m e n t (m )

Case 2:

Case 3:

The Manchester

Bobber

Improved Resolution of Free Surface

Lin et. al.

Two step projection method

Finite difference

Fast marching

particle level

set

Partial cell + LRS (local relative stationary method)

Non-uniform structured rectangular mesh

Violent Sloshing: Model Requirements

• Finite Volume

• Inviscid

• Compressible Flow

• Change of State Thermodynamics

• Air/Water

Compressible Flow Equations

Finite Volume Euler Equations

1D Shock Tube Problem with Cavitation

Initial Conditions

LEFT STATE RIGHT STATE

ρ = 995.5450 kg/m3 ρ = 995.5450 kg/m3

u = -10 m/s u = 10 m/s

p = 0.9 bar p = 0.9 bar

T = 303.15 K T = 303.15 K

1D Shock Tube with Cavitation: Flow solver with

no

1D Shock Tube with Cavitation: Flow solver with

no

1D Shock Tube with Cavitation: Flow solver with

Change of State Thermodynamics

• Equation of State for

Water (Tait)

• Equation of State for Air

(Ideal Gas)

Change of State Thermodynamics

Properties of Mixtures

• Oldenbourg Polynomials

Modelling Change of State Thermodynamics

• For p: replace EoS by Call Thermo(ρ, e, p, c, α)

• Within

SUBROUTINE

Thermo:

• Enter with ρ, e

• Solve saturated mixture equation for T using Secant Method • Calculate ρsat_liquid and ρsat_vapour

• If (ρ > ρsat_liquid) then use Tait EoS for water • If(ρ < ρsat_vapour) then use Ideal gas EoS

• Else use saturated mixture relations with p = psat • Calculate mixture speed of sound, c

• Calculate water vapour traction, α • Return with p, c and α

1D Shock Tube with Cavitation: Flow solver with

change of state thermodynamics – 10 Δt

1D Shock Tube with Cavitation: Flow solver with

change of state thermodynamics – 10 Δt

1D Shock Tube with Cavitation: Flow solver with

change of state thermodynamics – 120Δt

1D Shock Tube with Cavitation: Flow solver with

change of state thermodynamics – 120Δt

1D Shock Tube with Cavitation: Flow solver with

change of state thermodynamics – 120Δt

High Performance Computing

• Multi-core high spec desk-top PC

• NEC SX-8 vector supercomputer

• UK National Supercomputing facilities

• Graphical Processor Unit (GPU)

• NVIDIA Tesla C-870 with 2 GPU boards

• NVIDIA CUDA software development kit (SDK)

• Rack of NVIDIA Tesla S1070 systems

• url:

Execution time in seconds for

20,000 time steps

32x32 64x64 128x128 RADIX 36.36 53.97 110.46 TILED 40 142.41 1140.96 UNTILED 41.52 164.92 1895.97 SX8 219.03 1746.27

•Three problem sizes with number of real particles used being 32x32, 6x64, 128x128

•Code : SX8: means executed on 1 processor of the NEC SX8, all others executed on

2D Dambreak with Obstacle

using a Smoothed Particle Hydrodynamics

Code

Conclusions

• 2D shallow water and fully 3D cut cell free surface

capturing codes have been developed for the simulation of violent wave loadings on fixed and floating bodies

such as seawalls and wave energy devices.

• A number of test cases have been used to validate the codes.

• The underlying method is generic and can be applied to any application area involving free surfaces and

stationary/moving bodies including compliant bodies (e.g. LPG carrier in steep waves).

• Full fluid-structure-interaction capability with

wave loads and derived motions under regular

and extreme wave conditions including moving

cargo and resolve violent sloshing issues in both

physics and numerics.

• Extensions of our work on wave energy

converters, greenwater overtopping of fixed and

floating vessels and scour. More use of GPUs.

Further Details and

Publication References

http://www.docm.mmu.ac.uk/cmmfa/

[email protected]