Hello, welcome to my talk, on physical modelling, Froude similitude
for the physical modeling of marine structures, other than the physical
modeling of the aerodynamic structures. This is the first part of 3 talks on
this topic, Froude similitude, on why and how we can carry out physical
modeling of marine structures.
Let's watch a video of the Pelamis model
test in the Plymouth wave tank. This is the model test for the five segment
Pelamis wave energy converter. We can see the large amplitude motions of the
device in such a large wave. In the model test, we can also see the fluorescent
balls on the structure for measuring the motion of the device,
we have many different types of marine structures, including: the wave generation
by a boat, traveling on the sea, we can see the shape of the wave, which is called
as Kelvin ship wave. Kelvin ship wave could have a similar pattern, especially
in the deep water. Another example is the wave generation
by a duck, which has a similar wave pattern as that of the ship; For ships, we
may need to study the drag of the ship travelling at different speeds, and
the motion due to the wave-structure interaction;
For a floating structure either in measuring the wave or extracting energy
from the waves, it is basically a problem of wave and structure interaction;
the ocean platform under the actions of wind, wave and currents; and many more
examples. The question is: how we can carry out the study more accurately and reliably?
For instance, take a design of a shape as an example. In the development of the
ship design, we need to understand the performance of the ship reliably
before the real construction of the ship. Here the USS Gerald R. Ford the aircraft
carrier (the information is from the internet). The aircraft carrier has an overall
length of 333 meter, and the full load of 100,000 long tons,
that is 101.6 million kilograms; the maximum speed is about
30 knots, that is, 56 kilometers per hour or 15.56 m/s.
The question we are looking at the design would be: can we design a ship
with a desired speed under a given power?
Would the motion of the ship be small enough in waves, for instance, to allow
aircrafts to takeoff or landing, or to carry out the installation and the
maintenance. The most important question would
be: how we can carry out this study most accurately and reliably?
The dynamics problem of the marine structures are mainly for the performance
and motions of the marine structure under the fluid-structure interaction or wave-
structure interaction. For solving the dynamics problem of the
marine structure, we have three different approaches:
we can use numerical modeling, but can we solve the navier-stokes equation
for this? or can we use the simplified method to study the problem with enough
confidence? it is accurate? or it is reliable?
We can use the scale model test. In the scale model test, can we model dynamics
correctly? or how we can reduce the scaling effect? how we can extrapolate the
experiment data to the full scale structures?
Or we can use the full scale test, but this is a very expensive way to check the
design: whether it satisfies all the requirements. However, this would be
important for collecting the data for further development,
for calibrating the numerical tools, or for correlating the scale model test.
In this talk I will focus on the scale model test,
and in this part, the fundamental issues on why and how we can carry out the
scale model test are introduced.
In the scale model test, the similitude laws play a very important role, and
these similitude laws could guarantee the success of a scale physical modeling.
For a successful scale modeling, we have three similitudes: the first is the
geometric similitude, for which the scale model must be similar geometrically to
the full scale structure, which means in all directions the lengths of the
structure and the model must be proportional correctly, and in some
special circumstances, the structure surface must be scaled;
the second similitude is the kinematic similitude, this corresponds to
the motion similarities, for instance, if we have a laminar flow in the small
scaled model, but the flow for the full structure is turbulent. In that case the
kinematic similitude is not satisfied; the third similitude is the dynamic
similitude, meaning all the forces acting on the structure and the model are similar,
this dynamics similitude can be guaranteed by ensuring the same non-
dimensional numbers for the full scale and the scale model, for instance,
Reynolds number, which is the ratio of the inertia force over the viscous force,
or the Froude number, a ratio of the inertia force over the gravitational force.
For the study of the conventional marine
structures, these two non-dimensional numbers would be used.
In the physical modeling, it is useful to estimate the orders of magnitude of
the forces. Basically these forces would be the
functions of the following physical parameters, include: the physical length, L;
the fluid velocity, U; the fluid viscosity MU; the fluid density Rho, and
the gravitational acceleration g. For marine structures, three types of
forces are considered: inertia force, which corresponding to the fluid dynamic
pressure and its order of magnitude is given as Rho*U^2*L^2, and
the gravitational force is actually the weight of the structure or the fluid
displaced by the structure, according to the Archimedes buoyancy principle, which
is given in a form of Rho*g*L^3; the viscous force due to the fluid
viscosity is given in a form, MU*U*L.
Based on the orders of the magnitudes of the forces, we can see if we have two
different scale models, for instance, 3m and 6m in length, so
according to the relevant similitude, for instance, the Froude similitude, we
have the corresponding changes in the forces, as the inertia force would be
8 times of the force for the large model over that of the small model; and
the gravitation force would be 8 times; but the viscous force would be
2.83 times.
This fundamental balance among three types of the forces must be
considered for a successful physical modelling, for instance, what are the
main force we are focused on the scale model, for example, the viscous force is
important? and what is the appropriate size for a
scaled model testing, so we may think about the question: is this model good
for testing? is good for the tank size? because we must consider the
scaling effect, the blocking effect and etc.
Are the forces properly modelled? since the forces would be significantly
different in laminar and in turbulent flow.
Can the measurements be appropriately
made in the lab? so the appropriate sensors are important for the test data,
too large or too small sensors would not be good for the test.
In the next few slides, we will use dimensional analysis to study the
hydrodynamic force acting on a floating structure under the action of the wave,
and we will see how the dimensional analysis could simplify the problem.
so the first step, we need to identify the physical variables, which might have
effects on the hydrodynamic force, F_hyd. Obviously the hydrodynamic force would
be very relevant to the fluid velocity;
Fluid viscosity is surely important for the hydrodynamic force;
Fluid density: for different fluids, the hydrodynamic force would be
different; Structure length would be a very
important factor for the hydrodynamic force;
Gravitational acceleration might be an important factor for the hydrodynamic
force acting on the structure, that's because the ocean wave is a
gravitational wave, which is very relevant to the gravitational (hydrodynamic) force.
So in a general form, the hydrodynamic force can be expressed as the function
as this, here we can see there are total 6 variables or parameters in the
problem: the variable of the hydrodynamic force and five variables in
the function, therefore, we have the number n of 6.
For such a problem, there are 3 base units: Length, L, unit meter; mass M, unit
kg and time T, unit second. so we have the base unit number m
equalling to 3. However in the analysis for these base
units, the representatives are chosen as: the structure length for the base length,
and the fluid density for the base unit, mass.
because the mass can be expressed as the fluid density times the structure length cubic,
and the velocity for the base unit,
time, the time can be calculated as the length divided by the velocity.
so based on the principle of dimensional analysis, the Buckingham PI theorem, we can
calculate the number of non-dimensional groups (n-m) = 3,
if you want to see more details on the dimensional analysis, you can watch
my talk, titled 'Dimensional Analysis'. So the first group of the non-dimensional
parameter PI_1, given as this, so we can define a coefficient of the
hydrodynamic force, C_hyd, given as this; the second non-dimensional parameter
would be PI_2, given as this, so the corresponding non-dimensional number
would be the Reynolds number, Re, defined in this formula; and the third non-
dimensional parameter PI_3, given as this, and from which we can deduce the Froude
number as this, Fr.
After the dimensional analysis, the hydrodynamic force coefficient can be
expressed as a function of the Reynolds number and the Froude number, given in
this form. So if we compare with the original form of the hydradynamic force,
F_hyd, which is a function of five parameters, we can see the problem is
much simplified. As such, we can study the relation of the hydrodynamic force
coefficient with the Reynolds number and Froude number, so the problem becomes
problem of for non-dimensional parameters, Reynolds number and Froude
number, two variables, vesus the function of five physical variables:
velocity, viscosity, density, length and gravitational acceleration. And once
the hydrodynamic force coefficient is determined, the hydrodynamic force can
be easily calculated from the definition of the hydrodynamic force coefficient.
Another advantage for this simplified expression is: in many turbulent
flows of a very large Reynolds number, the viscous force becomes
relatively small when compared to the gravitational force, and thus it is can
be neglected, so we can have the expression simply as this: the
hydrodynamic force coefficient is only dependent on the Froude number.
This relation has been served as the principle for physical modelling test in
the wave tank.
So for the physical modeling, the advantages include:
The full physical phenomena can be included if the modelling is appropriate;
The control lab conditions to isolate some dynamic effect or create the
extreme condition in minutes. For instance, use the regular wave or create
in minutes the extreme events of, for instance
once-in-50 or -100 years; Easy and accurate measurements using the
corret sensors, with good signal connection, good reference for measuring;
It could provide the fast optimization process: the model modification can be
made relatively easy; the installation and the removal of the
device would be easy because the model would not be too large and too heavy.
and the cost for the physical model test would be cheaper, when compared
to the full scale sea trials.
Similarly, we have also several disadvantages for the physical modeling,
include: model manufacturing, is the model is the fully downscaled?
the model surface roughness is scaled? and etc; The scaling effect: how large
the model is appropriate?
The block effects: whether a large tank is available for testing a large test model?
Some long-term effects would be
difficult, for example, biofouling on the marine structure, on wave energy
converter, which would affect the performance of the marine structure
and wave energy converters; For the wave energy converter, the
power take-off system and the control system would be difficult to be modelled,
for instance, in many small wave energy converters, their (its)  wave power is about
1W, so this is difficult for the practical power take-off system or the control
system, hence we have feasibility problems here.
And the cost could be expensive when compared to the conventional numerical modeling.
