58
CHAPTER 3. PRINCIPLES OF SIMILITUDE
Expressing in terms of scale ratios, and rearranging yields
„
Nh
Na ~ Nx2'3
For a geometrically undistorted model, the vertical and horizontal length scales are
the same (i.e., Nh — Nx = Nl), so finally we get
Na = NLl/3 = (25)1/3 = 2.92
Therefore, the model shape factor can be found from Ap/Am — 2.92. (Note in this
example that the equation for the equilibrium beach profile is nonhomogeneous in
nature, therefore, the same length dimensions used in the prototype must also be
used in the model, e.g., if Ap [=] m1/3 in the prototype, then Am [=] m1/3 in the
model.)
3.2.3 Kinematic Similarity
The term kinematic refers to the motion of a system. This motion can be
the motion of a solid body or the motion of fluid particles within a flow
regime. Motion is defined as any order differential of length with respect
to time3.
Kinematic similarity indicates a similarity of motion between particles in model and prototype. Kinematic similarity is achieved when the
ratio between the components of all vectorial motions for the prototype and
model is the same for all particles at all times (Hudson, et al. 1979). In a
geometrically similar model, kinematic similarity gives particles paths that
are geometrically similar to the prototype.
Example 3.3. Kinematically Similar Wave Motion
What is the scaling criterion necessary to have kinematically similar wave motion
for gravity waves whose length (small amplitude wave theory) is given as the following
equation?
L =
tanh
(3.2)
where
L
- wavelength
g
- gravity
T
- wave period
h
- water depth
Velocity is given by the first order differential, dX/dt, acceleration by the second
order differential, d2X/dt2, etc.
CHAPTER 3. PRINCIPLES OF SIMILITUDE
Expressing in terms of scale ratios, and rearranging yields
„
Nh
Na ~ Nx2'3
For a geometrically undistorted model, the vertical and horizontal length scales are
the same (i.e., Nh — Nx = Nl), so finally we get
Na = NLl/3 = (25)1/3 = 2.92
Therefore, the model shape factor can be found from Ap/Am — 2.92. (Note in this
example that the equation for the equilibrium beach profile is nonhomogeneous in
nature, therefore, the same length dimensions used in the prototype must also be
used in the model, e.g., if Ap [=] m1/3 in the prototype, then Am [=] m1/3 in the
model.)
3.2.3 Kinematic Similarity
The term kinematic refers to the motion of a system. This motion can be
the motion of a solid body or the motion of fluid particles within a flow
regime. Motion is defined as any order differential of length with respect
to time3.
Kinematic similarity indicates a similarity of motion between particles in model and prototype. Kinematic similarity is achieved when the
ratio between the components of all vectorial motions for the prototype and
model is the same for all particles at all times (Hudson, et al. 1979). In a
geometrically similar model, kinematic similarity gives particles paths that
are geometrically similar to the prototype.
Example 3.3. Kinematically Similar Wave Motion
What is the scaling criterion necessary to have kinematically similar wave motion
for gravity waves whose length (small amplitude wave theory) is given as the following
equation?
L =
tanh
(3.2)
where
L
- wavelength
g
- gravity
T
- wave period
h
- water depth
Velocity is given by the first order differential, dX/dt, acceleration by the second
order differential, d2X/dt2, etc.