A TWO-FLUID NUMERICAL MODEL
OF THE LIMPET OWC
CG Mingham, L Qian, DM Causon and DM Ingram
Centre for Mathematical Modelling and Flow Analysis
Manchester Metropolitan University, Chester Street,
Manchester M1 5GD, U.K.
M Folley and TJT Whittaker
School of Civil Engineering,
Queen’s University, Belfast

• EPSRC (UK) for funding the project
(grant number GR/S12333)

• LIMPET: a wave energy converter based on the Oscillating Water
Column (OWC) principle.
• LIMPET installation on Islay,
Scotland (75kw).
• Small scale experimental trials at
Queen’s University, Belfast.

• The problem involves both water and air flows, wave breaking, non-sinusoidal waves, vortex formation and air entrainment.
• Linear wave theory is not suitable for modelling such flow problems.
• A two-fluid (water/air) non-linear model
(Qian,Causon,Ingram and Mingham,
Journal of Hydraulic Engineering,
Vol.129, no.9, 2003) has been applied in the present study.

• Two fluid (air/water), boundary conforming, time accurate, conservation law based, flow code utilising the surface capturing approach.
• Cartesian cut cell techniques are used to represent solid static or moving boundaries.

• 2D incompressible, Euler equations with variable density.

 t


Q
   
S
F .
n
 s
 

B
  where
Q



 f
I 

 u
 u
 v
 u
2  p p



T
, F
 uv
 f
I n x u
T

 g
I n y
, B and g
I 

 v


0
 uv
0
  g
 v
2  p
0

T v

T
 is the coefficient of artificial compressibility

• The equations are discretised using a finite volume formulation

Q i
V i
 t
  
B V ij
 
R
  i j

 k ( i )
F ij
D l j
Where Q i is the average value of Q in cell i (stored at the cell centre), V volume of the cell, F ij i is the is the numerical flux across the interface between cells i and j and and D l j is the length of side j.

• The convective flux (F
Riemann solver. ij
) is evaluated using Roe’s approximate
F ij
I 
1
2

F
I
    ij ij

R

L ( Q
 ij

Q
 ij
)
• To ensure second order accuracy,
MUSCL reconstruction is used

Q ( x , y )

Q ij
 D
Q ij
 r where (x,y) is a point inside the cell ij, r is the coordinate vector of
(x,y) relative to ij and D Q ij slope limited gradient.
is the

The implicit backward Euler scheme is used together with an artificial time variable t (to ensure a divergence free velocity field) and a linearised RHS.




I m
V



I ta where


R ( Q n

1 , m

Q n

1 , m

Q

)
Q n , m



I m

D t diag

D
1 t


Q n

1 , m

1
V

1
D t

R
D
1 t

Q n

1 , m

Q n

1 , m





1
D t
1
D t



1
D t
D
1 t

The resulting system is solved using an approximate LU factorisation.

• A Jameson-type dual time iteration is used to eliminate t at each real (outer) iteration.
• The code vectorises efficiently with simulations typically taking about three hours to run on an NEC SX6i deskside supercomputer.

• Seaward boundary – a solid moving paddle (boundary) is used to generate waves (wave-maker).
• Atmospheric boundary – a constant atmospheric pressure gradient is applied.
Spray and water passing out of this boundary are lost from the computation.
• Landward boundary – a solid wall boundary condition is used for the landward end of the domain.
• Bed and wave power device – modelled using Cartesian cut cell techniques.

• Automatic mesh generation
• Boundary fitted
• Extends to moving boundaries

• Input vertices of solid boundary (and domain)

• Input vertices of solid boundary (and domain)
• Overlay Cartesian grid

• Input vertices of solid boundary (and domain)
• Overlay Cartesian grid
• Identify Cut Cells and compute intersection points.

• Waves are generated using a moving paddle with prescribed velocity:
U=-0.2sin(2  t)
• 0.3m Still Water Level; 6.0m long wave tank
• Using 120x40 grid cells
• 10 waves simulated, starting from still water

• Comparison with experimental results for free surface elevation at two locations

• Wave Conditions: Regular waves with wave length L  1.5m , period T=1.0s and still water level H = 0.15m.
• Device located at about 2 wave lengths from the moving paddle
• 5 seconds simulated, starting from still water

Free surface position and velocity vectors at T=4.0s

Free surface position and velocity vectors at T=4.2s

Free surface position and velocity vectors at T=4.4s

Free surface position and velocity vectors at T=4.6s

Free surface position and velocity vectors at T = 4.8s

Free surface position and velocity vectors at T = 5.0s.

• Some initial results have been presented for simulation of LIMPET OWC device using a surface capturing method in a
Cartesian cut cell framework.
– The method is computationally efficient,
– Capable of modelling both water and air, as well as their interface
– Can handle both static and moving boundary easily.
• Detailed comparisons with the small scale test from QUB using the same wave conditions are in progress.
• The numerical model is generic and can be used to model a wide range of wave energy devices.