60
CHAPTER 3. PRINCIPLES OF SIMILITUDE
Thus, we see that the motion of water waves and simple pendulums scale according
to the same relationship. This arises from the fact that gravity is the major restoring
(external) force in both cases.
3.2.4 Dynamic Similarity
In the above example, the derived scale law for wave motion did not depend
on the properties of the prototype or model fluid. Therefore, the kinematic
law will hold if the model fluid has a different density than the prototype
fluid (provided that the fluid density is still within the assumptions invoked
in deriving the wavelength equation, particularly the inviscid flow assumption). However, the forces exerted by the wave motion on an object or
boundary may not be in similitude in the model unless additional requirements are met related to the prototype and model fluid properties. These
additional requirements stem from the necessity of maintaining dynamic
similarity.
Dynamic similarity between two geometrically and kinematically
similar systems requires that the ratios of all vectorial forces in
the two systems be the same (Warnock 1950).
This definition means that there must be constant prototype-to-model ratios of all masses and forces acting on the system.
The requirement for dynamic similarity arises from Newton’s second law
that equates the vector sum of the external forces acting on an element to
the element’s mass reaction to those forces, i.e.,
dV
(33)
n
For fluid mechanics problems Newton’s second law can be written as
A = Fg + Fp + Fa + Fe + Fpr
(3.4)
where
Fi - inertial force (mass x acceleration)
Fg - gravitational force
Fp - viscous force
Fa - surface tension force
Fe - elastic compression force
Fpr - pressure force
CHAPTER 3. PRINCIPLES OF SIMILITUDE
Thus, we see that the motion of water waves and simple pendulums scale according
to the same relationship. This arises from the fact that gravity is the major restoring
(external) force in both cases.
3.2.4 Dynamic Similarity
In the above example, the derived scale law for wave motion did not depend
on the properties of the prototype or model fluid. Therefore, the kinematic
law will hold if the model fluid has a different density than the prototype
fluid (provided that the fluid density is still within the assumptions invoked
in deriving the wavelength equation, particularly the inviscid flow assumption). However, the forces exerted by the wave motion on an object or
boundary may not be in similitude in the model unless additional requirements are met related to the prototype and model fluid properties. These
additional requirements stem from the necessity of maintaining dynamic
similarity.
Dynamic similarity between two geometrically and kinematically
similar systems requires that the ratios of all vectorial forces in
the two systems be the same (Warnock 1950).
This definition means that there must be constant prototype-to-model ratios of all masses and forces acting on the system.
The requirement for dynamic similarity arises from Newton’s second law
that equates the vector sum of the external forces acting on an element to
the element’s mass reaction to those forces, i.e.,
dV
(33)
n
For fluid mechanics problems Newton’s second law can be written as
A = Fg + Fp + Fa + Fe + Fpr
(3.4)
where
Fi - inertial force (mass x acceleration)
Fg - gravitational force
Fp - viscous force
Fa - surface tension force
Fe - elastic compression force
Fpr - pressure force