Similitude in Hydrosimilitude in hydrodynamicsdynamics
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Transcript of Similitude in Hydrosimilitude in hydrodynamicsdynamics
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Similitude in Hydrodynamic
Problems
Department of Ocean EngineeringIndian Institute of Technology Madras
Chennai 600 036
Prof.V.G.Idichandy
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Introduction
Hydrodynamic problems need to be separately treated.
Difficult to achieve complete similitude
There are many more problems that require solution using
an empirical approaches based on experimental data. Background for deriving appropriate scaling relationship
for hydrodynamic model tests.
Complete similarity models are models in which the values
of all relevant dimension less parameters in the prototypeare maintained in the model.
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Geometric similarity
Geometrically similar models are undistorted models
Horizontal and vertical scales are the same
The model is a true geometric reproduction.
Definite objectives can be achieved by departing formgeometric similarity.
Such models are called distorted models.
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Kinematic similarity
In hydrodynamic model kinematic similarity is achieved
when the ratio between the components of all vertical
motions for the prototype and the model are the same forall particles at all times.
In a geometrically similar model, the kinematic similarity
gives particles paths that are geometrically similar toprototype.
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Kinematically similar wave motion
What is the scaling criteria necessary for kinematically
similar wave motion for gravity waves whose length is
given by
L is the wave lengthg acceleration due to gravity
T wave period
h water depth
2 2tanh
2
gT hL
L
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Kinematically similar wave motion
2h/L is dimension lessThis ratio for model and prototype should be an invariant
for geometrically similar model.
2h/L is the same, then the hyperbolic tangent will also bethe same.
The scale relationship between L and T is found from theprototype to model ratio of wave length.
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Kinematically similar wave
motion
2
2
2
2
2
tanh(2 / )2
tanh(2 /2
p p
m
m
p p p
m m m
e g T
gTh L
L
L gT h L
L g T
L g T
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Kinematically similar wave motion
g= 1
Therefore the kinematic similarity of wave motions can
be achieved by making
T l
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Dynamic Similarity
Kinematic similarity is achieved without considering any
other properties of the fluid
But this is not so for dynamic similarity.
Dynamic similarity between two geometrically and
kinematically similar system requires that the ratio's of all
vectorial forces in the two systems be the same.
In other words, there must be constant prototype to model
ratio's of all masses and forces acting on the system.
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Dynamic Similarity
For fluid mechanics problems Newton's second law can be written as,
Fi= Fg+ F+ F + Fe+ Fpr
Where Fi inertial force, mass x accelaeration
Fg gravitational force
F viscous forceF surface tension forceFe elastic compression force
pr pressure force
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Dynamic Similarity
Forces being vectors both magnitude and directionmust be represented.
Overall dynamic similarity requires that the ratio of
the inertial forces between prototype and model be
equal to the ratio of the sum of all the active forces
expressed as,
g e pri p p
i g e pr m m
F F F F FF
F F F F F F
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Dynamic Similarity
Perfect similitude requires in addition to the above that all
force ratios between the prototype and model be equal or,
or in terms of scale ratios,
g pri ep p p p p p
i eg prm m m mm m
F FF F F F
F F F FF F
i g e pr F F F F F F
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Dynamic Similarity
No fluid is known that will satisfy all force ratio
requirements, if the model is smaller than the prototype.
So an important task in scale model design is to relate
the important force ratios and to provide justification for
neglecting the others
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Practical aspects
Dynamic similarity is practically unachievable forscale models other than for scale factor 1
For model tests the similarity requirements have
to be violated
In model tests, it should be possible to justifydepartures and where required apply theoretical
corrections
It should also be possible to predict which of theforces are important for a given situation.
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Practical aspects
Almost any major problem can besimplified into the interplay of two majorforces.
Inertia forces are always present in flowproblems. So inertia needs to be balancedby one or more of the other forces.
The first step is to express each of theforces in terms of their physical units.
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Practical aspects
3 22 2
3
2
e
L acceleration L
accleration due to gravity = L g
Velocity VF cos area = V Ldistance L
F unit surface tension length = L
F modules of elastic
i
g
VF mass V
L
F mass
Vis ity L
2
2
ity area = E L
unit pressure area = p LprF
The ratio of the inertial force to any other force
provides the relative influence of the two forces in
hydrodynamic problems. Each force ratio leads
similitude criterion as given below.
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Froude Criterion
It is the parameter that express the relative
influence of inertial and gravitational
forces,expressed as the square root of theratio of forces
2 2
r3Froude Number ( F )
L V V
L g Lg
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Froude Criterion
The Froude criterion for modeling
inertial forces are balanced primarily by gravitational forces,which is true in most cases involving free surface.
Majority of problems in hydrodynamics are scaled accordingto Froude model law and therefore it is the most important
model law .
p m
V V
Lg Lg
p p p
m m m
V g L
V g L
1
1v
g
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Reynolds Criterion
When viscous forces dominate, the important parameter is
the ratio of inertial to viscous forces
Reynolds used this number to distinguish between laminar
and turbulent flows.
Reynolds similitude is achieved when,
2 2
e L V Reynolds Number (R )
L VV L
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Reynolds Criterion
Reynolds law is intended for flows where viscous forces
dominate.
Examples are laminar boundary layer, forces on cylinder for
low values of Re
Lv
1
p m
p p p p
m m m m
v L p
LV
V L
V L
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Weber Criterion
Relative importance of surface tension is given by inertia to surface tension forces
Weber model criteria is used when surface tension forces dominates
Surface tension effects are seldom encountered in ocean engineering problems in
prototype. In very small models surface tension may play some role.
2 2 2int forces vL Weber Number
surface tension forces
ertial V V L
L
2
1p v L
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Cauchy Criterion
An index of the relative importance of inertial forces to
compressive forces
This is of importance where inertial forces are large enough to cause
changes in fluid compressibility. Cauchy number is related to Machnumber (v/c) because the speed of sound c in a fluid is given by
2 2 2
2
inertial forcesCauchy number
elastic forces
L V V
E L E
E 2
2 V
Ea aM C
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Cauchy Criterion
Mach number is used in studies of air flow having high
velocities. Cauchy modeling criterion is,
This criterion has little application in Ocean engineering
problems because the fluid is considered incompressible.
There could application in breaking waves.
2
1v
E
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Euler Criterion
When the pressure forces are dominant Euler
criterion is used
Euler model criterion is
2
2 2 2
Pressure force P L
intertial force L
P
V V
2
v
1
p
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Strouhal Number
Inertial forces in fluid can be caused by two types of accelerations. Convective
accelerations are due to different fluid velocities at different locations in the flow field
and they are represented mathematically as,
Temporal accelerations are changes in flow velocity at a point that occur in time. They
represent the unsteadiness of the flow and can be expressed mathematically as,
/u u x
/u t 3
22
Tenporal inertial force = L
Connective inertial force = L
V
t
V
L
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Strouhal Number
The relative importance of temporal inertia force to connectiveinertial force is given as
which is represented as Strouhal Number .
This dimensional parameter is important in unsteady, oscillatory flows.Where the period of oscillation is given by variable it. Often StrouhalNumber is expressed as (L/V) or (fL/V)
2
22
L
L
V
Lt
V VT
L
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Strouhal Number
For Strouhal Criteria in the model
which states that the velocity scalar ratio is equal to the length scale ratio divided by
time scale ratio. In unsteadyoscillating flows it is important to maintain similarity of
Strouhal Numbers
1
p m
p p p
m m m
L
v t
L L
Vt Vt
L V t
L V t
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Importance of Froude Scaling
Practically most of the ocean engineering problems and fluid flowproblems, where the forces due to surface tension, pressure andelastic compressionare relatively small and can be safely neglected.
This leaves an appropriate hydrodynamic scaling law to an evaluationof whether gravity or viscousforces are dominant.
Therefore either Froude or Reynolds similarity combined with
geometric and Kinematic similarity provides the condition forhydrodynamic similitude in almost all ocean related problems.
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Importance of Froude Scaling
Reynolds similitude is seldom
involved for most models as gravity
forces dominate in free surface flowsand consequently most models are
designed for Froude criteria.
However, the viscous effects must bereducedotherwise dissimilar viscous
forces will constitute scale effect.
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Scaling of A Froude Model
In the study of wave mechanics three non-
dimensional parameters are most
important. They are Froude, Reynolds andStrouhal. Keulegan-Carpenter number also
becomes important in showing the
dependence of inertia and dragcoefficients.
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Scaling of A Froude Model
Except Cm and Cdall terms follow Froude scaling.
The hydrodynamic coefficients Cd& Cmare non-
dimensional and are function of Keulegan-
Carpenter number KC defined as (uT/D).According to Froude's law the velocity and wave
period scale are square root of the scale factor
while linear dimension scale linearly
p m
KC KC
3
2
ewhere as R ep mR
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Scaling of A Froude Model
For current drag, the drag force is proportional to the
square of the velocity. Experiments have shown that flow
characteristics in the boundary layer are most likely to be laminar
at Re < 106 but turbulent for Re > 106. In most model of Froude
scale boundary layer is laminar whereas for prototype it is
turbulent. Thus two scaling laws have to be applied which is ,not
possible. It is convenient to employ Froude scaling in a laboratary
and apply correction for violating Reynolds number.
Once the flow is turbulent the drag coefficient is only
weakly dependent as Re. As a result in many experiments, the
laminar flow is deliberately tripped by some kind of roughness
near the bow of the structure.