4.2. SHORT-WAVE HYDRODYNAMIC MODELS
91
z-Direction
/ L \ dw
„ dw
„ dw
„ dw
( P \ dp
\VTJ ~dî+ud£+v~d^ + w~dï
di
(\ d2™
^2w
d2w
~
/ +
J[dF + ~d^ + Ji^
- ^7*7)+â(^)+^S)
(4.18)
Short-Wave Model Similarity Conditions
The number of independent dimensionless coefficients in the equations of
motion has been reduced to four by constraining the model to being a geometrically undistorted version of the prototype. In addition, the coefficients
are recognized as important dimensionless products previously developed
through dimensionless analysis (see Table 2.2 or Example 2.5, Chapter 2).
The coefficient of the temporal acceleration terms (L/VT) is recognized
as the Strouhal number; the coefficient of the pressure terms (P/pV2) is the
Euler number; the coefficient of the viscous stress terms (v/LV) is the inverse of the Reynolds number; and the singular coefficient representing the
gravity forces (gL/V2} is the inverse square of the Froude number. Thus, we
see how these important flow parameters occur naturally in the mathematical representation of the physical hydrodynamic system. Using governing
equations to establish similarity requirements is superior to dimensional
analysis because the variables are known and the associated assumptions
are clearly identified (Munson, et al. 1990). Also, physical interpretation
of each similitude criterion can be often be given.
As mentioned, the requirement for complete similitude in an undistorted
short-wave model is that each dimensionless coefficient must remain the
same in the model as in the prototype. Equating prototype coefficient to
model coefficient and forming the prototype-to-model ratios of each variable
contained in the coefficient yields the following conditions for complete
similitude.
Condition 1. From the gravity term in the z-direction
equation
Ny
_ 1
(4-19)
which means that the Froude number must be the same in
the model as in the prototype.
91
z-Direction
/ L \ dw
„ dw
„ dw
„ dw
( P \ dp
\VTJ ~dî+ud£+v~d^ + w~dï
di
(\ d2™
^2w
d2w
~
/ +
J[dF + ~d^ + Ji^
- ^7*7)+â(^)+^S)
(4.18)
Short-Wave Model Similarity Conditions
The number of independent dimensionless coefficients in the equations of
motion has been reduced to four by constraining the model to being a geometrically undistorted version of the prototype. In addition, the coefficients
are recognized as important dimensionless products previously developed
through dimensionless analysis (see Table 2.2 or Example 2.5, Chapter 2).
The coefficient of the temporal acceleration terms (L/VT) is recognized
as the Strouhal number; the coefficient of the pressure terms (P/pV2) is the
Euler number; the coefficient of the viscous stress terms (v/LV) is the inverse of the Reynolds number; and the singular coefficient representing the
gravity forces (gL/V2} is the inverse square of the Froude number. Thus, we
see how these important flow parameters occur naturally in the mathematical representation of the physical hydrodynamic system. Using governing
equations to establish similarity requirements is superior to dimensional
analysis because the variables are known and the associated assumptions
are clearly identified (Munson, et al. 1990). Also, physical interpretation
of each similitude criterion can be often be given.
As mentioned, the requirement for complete similitude in an undistorted
short-wave model is that each dimensionless coefficient must remain the
same in the model as in the prototype. Equating prototype coefficient to
model coefficient and forming the prototype-to-model ratios of each variable
contained in the coefficient yields the following conditions for complete
similitude.
Condition 1. From the gravity term in the z-direction
equation
Ny
_ 1
(4-19)
which means that the Froude number must be the same in
the model as in the prototype.