Download Modelling of 3D Net Structures Exposed to Waves and Current

Modelling of 3D Net Structures Exposed to Waves and Current
Pål F. Lader, Birger Enerhaug, Arne Fredheim and Jørgen Krokstad
SINTEF Fisheries and Aquaculture, Trondheim, Norway
ABSTRACT: A dynamic model for 3D net structures exposed to waves and current is proposed. The model is based on a
super-element formulation where the net structure is divided into four-sided super-elements interconnected in each corner
node. The hydrodynamic and structural forces are calculated on each super-element, and the total forces are collected in
each node where the equation of motion are solved. The hydrodynamic forces on the net is calculated using formulas based
on earlier empirical results for net panels in steady current, and are assumed to be drag dominated. The structural forces are
calculated by assuming that each element consists of six nonlinear springs, interconnecting each node to the other three. The
model is partially validated by comparing it with experimental measurements of the drag force on a cylindrical net structure
exposed to accelerating and steady uniform current. The comparison show that the agreement is within +/- 20% for Reynolds numbers in the same range as for which the hydrodynamic load model was derived. For lower Reynolds numbers the
discrepancy increases.
1
INTRODUCTION
The fish farming and aquaculture industry are expanding
and the demand for suitable locations for fish farms is
increasing. In Norway, as well as in other countries, this
calls for new technological challenges, as fish farms are
being installed at locations that are more exposed to
waves, wind and current. In the future more of the fish
farms will be located offshore, as the number of suitable
near-shore locations are limited. A move towards the use
of offshore locations are also motivated by environmental
and aesthetic aspects of the industry, and future fish-farms
will most likely be in the form of large scale offshore
installations, rather than small near-shore farms. Before
installation of such structures it is however necessary to
assess the behaviour of the structure when it is exposed to
the environmental forces typical of the location. In contrast
to large floating oil installations which can be assumed to
be rigid, the fish farms are complex flexible structures with
an infinite number degrees of freedom. This makes it necessary to develop new numerical tools for simulating the
behaviour of such structure.
2
MODEL DESCRIPTION
The model proposed her is a fully 3D dynamic model
where the equations of motion are solved in the time
domain. The present model is based on a model of a single
net panel (Lader et. al., 2001a and Lader et. al., 2001b),
and is a direct extension of this work.
Net model
Waves
Elements
Current
Nodes
Fig. 1: Model overview
2.1 Overview
The net structure is modelled by using elements and nodes
(Fig. 1). The elements are four-sided with a node in each
corner. Hydrodynamic and structural forces are calculated
for each element based on the wave particle velocity, current velocity, structural velocity, and node positions. A
mass (inertia) and a weight (gravity) are associated with
each node, and the nodes can be either free, fixed or have
their motion prescribed. This enables, for example,
selected nodes to be fixed or connection to other structures
which dominates the motion. Movement of the free nodes
are not restricted, and the node points are allowed full 3D
movement. The forces on each element are distributed
equally to its four nodes, and the equation of motion
( F = ma ) is evaluated in each node, yielding the acceleration. The position and the velocity of the nodes are then
found by time integrating the acceleration.
2.2 Hydrodynamic Forces
The hydrodynamic forces are divided into drag ( F D )1 and
lift ( F L ) force, and are calculated on each individual element. The drag force is parallel to the velocity direction
( F D || U ), and the lift force is perpendicular to the flow
direction ( F L ⊥ U ). The drag and lift force are calculated
from:
1
F D = --- ρC D A U 2 n D
2
(1)
1
F L = --- ρCL A U 2 n L
2
(2)
where ρ is the mass density of water, C D and C L are the
drag and lift coefficients respectively, A is the element
area and U is the relative velocity between the fluid and
the element. n D and nL are the unit vectors in the direction of the drag and lift force respectively.
x1
x center
A1
A4
x2
A2
A3
x3
x4
(a)
(3)
U = U w + Uc – Us
where U w is the wave particle velocity, U c is the current
velocity and U s is the structural velocity (the velocity of
the element). The wave particle velocity is calculated
using linear, regular wave theory. The velocity U is taken
to be constant over the whole area of the element, and U is
evaluated in the element centre. The wave and current is
assumed not to be disturbed by the structure, and this
means that shielding effects are not taken into account in
the model. The position of the element centre x center
(Fig. 2a) is calculated by taking the mean value of the node
positions of the elements four nodes ( x 1 , x 2 , x 3 and x 4 ):
1
x center = --- ( x 1 + x 2 + x 3 + x 4 )
4
(4)
The elements are four sided, and since the nodes are
free to move in three dimensions the elements are not necessarily plane. To calculate the element area the element is
therefore divided into four triangles, and the total element
area is take to be the sum of the areas of the four triangles:
A = A 1 + A 2 + A3 + A 4 (Fig. 2a).
The element normal unit vector ( nen ), which is used to
calculate the lift unit vector ( nL ), is found by taking the
cross product of two unit vectors parallel to the diagonals
in the element. The diagonal unit vectors ( n13 and n24 ) are
given by:
x 1 – x3
n 13 = ------------------x 1 – x3
(5)
x 2 – x4
n 24 = ------------------x 2 – x4
(6)
and the element normal unit vector are then given by:
n 13 × n 24
n en = -----------------------n 13 × n 24
(7)
The direction of n en is by definition in the positive flow
direction.
The unit vectors for drag and lift are then given by:
x1
n en
n 13
x2
n 24
x4
x3
(b)
U
n D = ------U
(8)
( U × n en ) × U
n L = -----------------------------------( U × n en ) × U
(9)
The drag and lift coefficients ( CD and C L ) are calculated using formulas found by Aarsnes et al. (1990). These
formulas are based on both theoretical work and comprehensive model tests (Rudi et al., 1988). C D and C L are
given by:
2
The velocity U is the relative velocity between the
Bold type faces indicates vectors
3
CD = 0.04 + ( – 0.04 + S – 1.24S + 13.7S ) cos ( α )
2
Fig. 2: Element area (a) and element normal vector (b)
calculation for an element
1.
fluid and the net:
3
C L = ( 0.57S – 3.54S + 10.1S ) sin ( 2α )
(10)
(11)
where S is the solidity of the net, which is the ratio
between the solid area of the net ( As ) and the total area
enclosed by the net ( A ): S = As ⁄ A . α is the angle of
attack, and is defined to be the angle between the direction
of U and the normal vector of the element (Fig. 3). The
1
F 21
F 41
12
F 31
n
n en
14
α
F 12
13
2
F 32
F 42
U
23
24
F 24
F 13
F14
Fig. 3: The angle of attack ( α ).
4
formulas should not be used for S higher than 0.35.
The model tests on witch the formulas are based were
done with Reynolds numbers in the range 1400 to 1800,
and the Reynolds number dependency of the drag and lift
forces are thus not included in the formulas. Also note that
these formulas are strictly valid for stationary flow only.
The element area A is not constant, but varies with the
deformation of the element. Consequently the solidity S
also varies accordingly since the solid area of the net element ( A s ) is constant. The solidity used to calculate the
hydrodynamical forces on each net elements must therefore be corrected for on each individual element based on
the instantaneous area of that element:
A0
S = S 0 -----A
F 34
2.3
Structural Forces
The elements are modelled structurally as shown in
Fig. 4. Each node is connected with the other nodes
through a non-linear spring. Each spring is assumed to
have the following force - elongation relationship:
2
Fs = C 2 ε + C 1 ε
ε>0
Fs = 0
ε≤0
(13)
where F s [N] is the structural force, and ε [-] is the global
elongation. The global elongation is given by
ε = ( l – l0 ) ⁄ l 0 where l0 is the un-deformed length and l is
the deformed length of the panel. C 2 and C 1 are constants
describing the force/elongation characteristics of each
spring. In each four sided element there are six springs,
yielding twelve force contributions, three in each node.
The main reason for introducing the diagonal springs is to
be able to model both square and diamond oriented meshes
F 43
34
3
Fig. 4: The structural model of the element. Node numbers
are indicated in circles, and spring numbers in squares.
without having to re-grid the net. The force in the spring
between node n and node m ( Fnm ) are calculated from
the following equations:
(14)
F nm = F nm n nm
here F nm is the force magnitude, and n nm is the unit directional vector for the force. The force magnitude is calculated from:
2
F nm = C 2 nm εnm + C 1 nm εnm
ε nm > 0
F nm = 0
ε nm ≤ 0
(12)
where S and A is the instantaneous solidity and area of the
element respectively, and S 0 and A 0 is the solidity and
area of the net when it is undeformed.
The hydrodynamical forces acting on each element is
distributed equally to each element node, where the equation of motion is solved.
F 23
(15)
where the elongation ε nm is given by:
lnm – l 0nm
x n – x m – x 0n – x 0m
εnm = ----------------------- = ---------------------------------------------------l 0nm
x 0n – x 0m
(16)
where x 0n and x 0m is the position of the nodes when the
element is undeformed, and no internal forces are acting in
the element. x n and x m are the instantaneous position of
the nodes as indicated previously (Fig. 2a). The unit directional vector is given by:
xn – xm
n nm = -------------------xn – xm
(17)
2.4 Equation of Motion
The structural and hydrodynamical forces are calculated in
each element, and the forces are then collected in each
node, where the equation of motion is evaluated. For one
node surrounded by the four elements e 1 , e 2 , e 3 and e 4
the total structural and hydrodynamical forces in the node
is given by (Figs. 5 and 6):
e1
e1
e1
e2
e2
e2
e3
F 21
e3
F31
e3
F 41
e4
F 12
e4
F 32
e4
F42
F structural = F 13 + F23 + F 43 + F 14 + F 24 + F34
+
+
+
+
+
+
e1
e2
e2
1 e1
Fhydrodynamic = --- ( F L + F D + F L + F D
4
e3
e3
e4
e4
+ F L + F D + FL + F D )
(18)
(19)
3
The equation of motion now becomes:
(20)
Fstructural + F hydrodynamic + wgn g = ma
where w and m is the weight and mass respectively associated with the node, and ng is the gravity unit vector
equal [ 0, 0, – 1 ] .
Element
e1
e1
F 13
e2
F 14
e4
F 12
e3
e4
F42
Element
e4
e4
F32
Element
e2
e2
F24
e1
e1
F 43 F 23
The numerical model was validated by comparing numerical simulations with physical experiments. The behaviour
of a cylindrical net structure exposed to accelerating and
steady current was simulated using the numerical model,
and physical experiments were carried out at the North Sea
Centre Flume Tank in Hirtshals, Denmark. The tank, which
is the second largest flume tank in the world, is filled with
fresh water and has an observation/measuring section of
21.3x 2.7 x 8 m (LxHxW). This size has made it possible
to use rather large models with full-sized netting panels in
the tests.
3.1
Laboratory Setup
e2
F 34
F 21
e3
F 41
EXPERIMENTAL VALIDATION
e3
F 31
Load cells
Element
e3
1.435m
Fig. 5: Structural forces in one node.
e1
FL
e1
FD
Element
e1
1.435m
Hoop
e2
FL
e2
FD
e2
1 e1 1--- F
--- F
Element
4 L 4 L 1 e2 e
1--F
2
--- F
4 D
4 D
e4
1
Element
--- F
1 e3
--- F
4 D1--- e4
e3
F 1 e3 4 L
4 L --- F
e3
Element 4 D
FL
e4
e1
e4
FD
e4
FL
Bottom weights
e3
FD
Fig. 6: Hydrodynamical forces in one node.
Fig. 7: Model and general setup.
2.5 Time Integration
From the equation of motion the acceleration of each element is found. To calculate the movement of the node, the
acceleration is integrated twice:
The model is composed of a hoop (or ring), a net and a
number of weights attached to bottom of the net. The top
of the net is mounted on the hoop, which is kept in a fixed
position during each test. The weights are suspended
around the bottom opening of the net, in order to stretch
the net and maintain its shape under the influence of water
flow (current).
The hoop is made of stainless steel and has the following dimensions: hoop diameter 1.435 m (centre line to centre line) and rod diameter 0.025 m (diameter of the crosssection).
The net is made up of two panels which are joined
together in the centre line of the cage. Each panel is 81
meshes high and 125 meshes wide. With joining meshes
included, the total number of meshes in circumferential
direction is 252. The netting material is nylon with a mass
u =
x =
∫ a dt
∫ u dt =
∫ ∫ a dt dt
(21)
The numerical integration is performed by using the
ode15s routine implemented in the MATLAB (v 6.1)
software package (Shampine and Reichelt, 1997).
ode15s is a variable order solver based on the numerical
differentiation formulas (NDFs), and this method is chosen
since the problem has a stiff characteristic due to the high
structural eigenfrequency, much higher than the common
wave frequency.
Table 1: Weight modes.
Weight
Mode
Weight size
WM1
16 x 400g
WM2
16 x 600g
WM3
16 x 800g
Weight positions
0.10
Case 1: Usteady= 0.04 [m/s]
Rn= 53 [-]
0.05
0.00
0.10
Case 2: Usteady= 0.13 [m/s]
Rn= 172 [-]
0.05
0.00
Velocity [m/s]
0.2
Case 3: Usteady= 0.21 [m/s]
Rn= 278 [-]
0.1
0.0
0.2
Case 4: Usteady= 0.26 [m/s]
Rn= 344 [-]
0.1
0.0
Case 5: Usteady= 0.33 [m/s]
Rn= 437 [-]
0.2
0.0
0.6
0.4
0.2
0.0
Case 6: Usteady= 0.52 [m/s]
Rn= 688 [-]
0
40
80
120
Time [s]
Fig. 8: The transient start-up phase of the six different
velocity cases. The Reynolds numbers ( R n ) are calculated
using the twine diameter (1.8 mm) as characteristic size.
3.2 Numerical model
In the numerical model the vertical net cylinder was
divided into 16 elements around the circumference, and 4
elements over its height. The measured current velocity
(Fig. 8), and net elasticity (Fig. 9) were used as input to the
numerical model.
80
Force [N]
density of 1130 kg/m3. The netting is knotless with a mesh
size of 32 mm and twine thickness of 1.8 mm. Mounted as
square meshes, the solidity ratio ( S ) of the netting is 0.225.
The net then forms an open vertical cylinder with diameter
1.435m and height 1.435m.
Three sets of weights with nominal mass values of 400,
600 and 800 g are used in the tests. The weights are made
of steel and have cylindrical shapes. Three different weight
modes (WM) where tested (see Table 1)
The hoop with net was positioned in the centre line of
the tank, approximately 0.9 - 1.0 m below the surface. The
hoop was held in place with four pair of lines to balance
the forces from weights and hydrodynamic loads (see
Fig. 7). The equilibrium position of hoop and net was
achieved by minutely adjustments of the length and tension in each of the eight lines. Eight load cells using strain
gauge technology were used to measure the strain in each
line, and the global forces on the hoop were calculated
from these measurements.
Flow speed was measured with a Høntzsch Vane wheel
A, which is commonly referred to as a “propeller log”. The
principle of measurement is that a vane wheel rotates at a
speed proportionally to the flow velocity. Because of its
little weight, the vane wheel rotational speed adapts
quickly to velocity increases. A transducer (Høntzsch U1a)
transforms the signal from the vane wheel into a proportional analogue signal that is sent to the data acquisition
system. Since the vane wheel is mounted in a nozzle, it
measures what can be considered as the horizontal component of the actual flow speed.
The model was subjected to six different current velocity cases ( U steady = 0.04, 0.13, 0.21, 0.26, 0.33, 0.52 [m/s] ).
The whole transient start-up phase of the flow is measured
(see Fig. 8), so that the exact same velocity development
can be modelled in the numerical simulations.
60
Least squares curve fit
Measured
40
20
0
0.00
0.05
0.10
Elongation [-]
Fig. 9: Force - elongation characteristics in the net used in
the physical model. The force - elongation tests were
conducted on a 4 meshes wide and 68 meshes long
segment.
3.3
Results and Discussion
Measurements
Simulations
10
3
Measurements
Simulations
2
5
Usteady= 0.13 [m/s]
1
0
0
Usteady= 0.04 [m/s]
20
10
Usteady= 0.21 [m/s]
10
5
Usteady= 0.13 [m/s]
0
30
20
Usteady= 0.21 [m/s]
10
Drag force [N]
Drag force [N]
0
40
Usteady= 0.26 [m/s]
20
0
80
0
60
40
Usteady= 0.26 [m/s]
Usteady= 0.33 [m/s]
40
20
20
0
0
80
150
100
60
Usteady= 0.52 [m/s]
Usteady= 0.33 [m/s]
40
50
20
0
0
0
0
40
80
120
Time [s]
Fig. 10: Time series of drag force for weight mode 1 (16 x
400g).
The combined results from the physical measurements and
the numerical simulations are shown in Figs. 10 to 14.
Time series of the drag force measured and simulated
for different steady state velocities and weight configurations are shown in Figs. 10 to 12. The transient phase of
the experiments are used for partial validation of the
dynamic capabilities in the numerical model. In Fig. 13 the
measured and simulated steady state drag force are shown
as a function of velocity for the different weight configurations. The steady state phase is taken to be at t=100-140s,
and the steady state drag force is the mean value of the
drag force over this period. The error of steady state drag
force from the simulations relative to the measurements
are shown in Fig. 14.
40
80
Time [s]
120
Fig. 11: Time series of drag force for weight mode 2 (16 x
600g).
The best agreement between measurements and simulations are for velocities in the range 0.2 - 0.4 m/s. The
twine diameter in the net is 1.8mm. If the twine diameter is
defined to be the characteristic dimension, the corresponding Reynolds number range is 260 - 530. In this range the
relative error of the steady state drag force is +/- 0.2
(Fig. 14). For lower velocities the error increases, and the
simulations under-predicts the drag. For higher velocities
the simulations tends to over-predict the drag.
The main reason for the under-prediction of drag for
low velocities is due to the calculations of the lift and drag
coefficients (Eq. 10 and 11) used in the simulations. As
can be seen from the equations, the lift and drag coefficients are assumed to be independent of Reynolds number.
The coefficients are derived from measurements of drag
and lift of net meshes towed for Reynolds numbers in the
Measurements
Simulations
Measurements
Simulations
150
Drag force [N] (steady state phase: t=100-140s)
20
Usteady= 0.21 [m/s]
10
Drag force [N]
0
40
Usteady= 0.26 [m/s]
20
0
80
60
40
Usteady= 0.33 [m/s]
100
Weight mode 1
(16 x 0.4kg)
50
0
150
Weight mode 2
(16 x 0.6kg)
100
50
0
150
Weight mode 3
(16 x 0.8kg)
100
50
0
0.0
20
0.1
0
0
40
80
0.2
0.3
0.4
0.5
0.6
Velocity [m/s]
120
Time [s]
Fig. 13: Drag force at steady state (t=100-140s)
range l400 - 1800 (Rudi et al., 1988). The drag coefficient
on a net structure is approximately independent of Reynolds number for R n > 500 (Fridman et al., 1986), but the
drag increases with decreasing Reynolds numbers for
R n < 500 . Therefor the drag coefficient found for Reynolds
numbers in the range l400 - 1800 will under-predict the
drag for Reynolds numbers below 500.
The over-prediction of the drag force for higher flow
velocities can be due to the shielding effect. The part of the
net that is on the aft part of the net cylinder relative to the
incoming current experiences a decrease in current velocity due to the shielding of the front side of the cylinder. As
the current is assumed to be undisturbed by the structure,
this effect is not taken into account in the model. The simulations will thus over-predict the drag force.
Also note that the over-prediction is larger for weight
mode 3 (16 x 800g). This is the weight mode with the largest weights, and will thus be the mode with less deformation. The over-prediction because of the shielding effect
will be larger with less deformation of the net. The overprediction will be largest for a stiff structure, and will
decrease for an increasing flexible structure as the exposed
area decreases with increasing current.
The same discrepancies as in the steady state phase are
also present in the transient phase. For low velocities the
numerical model under-predicts the drag force, while it
tends to over-predict the drag force for higher velocities.
Steady state drag force relative error [-]
Fig. 12: Time series of drag force for weight mode 3 (16 x
800g).
0.2
0.0
-0.2
-0.4
Weight mode 1 (16 x 0.4kg)
Weight mode 2 (16 x 0.6kg)
Weight mode 3 (16 x 0.8kg)
-0.6
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Velocity [m/s]
Fig. 14: Steady state drag force relative error. The relative
error is given by: R= ( D simulated – D measured ) ⁄ D measured ,
where D simulated and D measured are the steady state drag
force from the simulations and measurements respectively.
4
FURTHER WORK
The numerical model proposed here include potential for
improvement. The authors will over the next three years
continue to improve the numerical model. Further work
will include the following aspects:
• A more detailed validation will include comparison of
the net structure deformation, and the numerical model
will also be validated against experimental measure-
•
•
•
•
5
ments of net structures in waves.
The hydrodynamics load model will be adjusted to
include the Reynolds number dependency. Experiments on net panels in regular waves are currently
under work, and will subsequently be used to refine the
load model.
Shielding effects with respects to current reduction and
wave damping will be included.
For a real fish cages the movement of the floating
device to which the net is attached will to a great extent
influence the tension in the net structure. The floater
movement will thus be modelled and coupled with the
net.
A wave kinematics description based on Gerstner wave
theory will be included. Multidirectional irregular
waves will also be modelled with emphasis on correct
free surface modelling.
SUMMARY AND CONCLUSIONS
A dynamic model of a 3D net structure is proposed.
The model is partially validated against measurements of
drag force on a cylindrical net structure in accelerating and
steady current. The comparison between the drag force
from experiments and simulations show that:
• Best agreement between the simulations and the experimental results are found for velocities in the range 0.2
- 0.4 m/s.
• The model under-predict the drag for velocities below
this range, and tends to over-predict the drag for higher
velocities.
• The discrepancies can partly be explained by the Reynolds number independency of the hydrodynamic load
model used in the simulations and neglection of shielding effects.
• The results indicate that the steady state assumption is
valid for the transient phase.
6
ACKNOWLEDGEMENT
This work was funded by the Norwegian Research Council
(NFR) through the research programs 3 Dimensional netpen movement and mooring of fishfarms, and Modelling
and simulation of interaction between fluid and deformable net structures and aquacultural installations.
7
REFERENCES
Aarsnes, J. V., Løland, G., and Rudi, H. (1990). “Current Forces on Cage, Nett Deflection.” Engineering for offshore fish farming, Glasgow, United Kingdom.
Fridman, A. L., Carrothers, P. J. G., and FAO. (1986).
Calculations for fishing gear designs, Published by
arrangement with the Food and Agriculture Organization
of the United Nations by Fishing News Books, Farnham.
Lader, P., Fredheim, A. and Lien, E. (2001a).
“Dynamic Behavior of 3D Nets Exposed to Waves and
Current.” The 20th International Conference on Offshore
Mechanics and Arctic Engineering, Rio de Janeiro, Brazil.
Lader, P. and Fredheim, A. (2001b). “Modeling of Net
Structures Exposed to 3D Waves and Current.” OPEN
OCEAN AQUACULTURE IV, St. Andrews, Canada.
Shampine, L. F., and Reichelt, M. W. (1997). “The
MATLAB ODE Suite.” SIAM Journal on Scientific Computing, 18, 1-22.
Rudi, H., Løland, G., and Furunes, I. (1988). “Experiments with nets; Forces on and flow trough net panels and
cage systems.” MT 51 F88-0215, MARINTEK, Trondheim, Norway.