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CHAPTER 4. HYDRODYNAMIC MODELS
Long-Wave Model Similarity Conditions
There are four independent dimensionless coefficients in the long-wave continuity equation and equations of motion; one related to the temporal accelerations, one related to the pressure terms, one related to the viscous
shear stress terms, and one related only to the vertical viscous shear stress
terms.
Just as in the case of short-wave models, complete similitude would be
achieved for any free surface long-wave hydrodynamic phenomena governed by the above form of the Navier-Stokes equations if the value of each
dimensionless coefficient in Eqns. 4.102-4.104 remained constant between
prototype and model.
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 pressure terms in the two equations comes the criterion given as
Ny
(4.105)
which is the prototype-to-model scale ratio of a Froude
number based on a horizontal velocity and a vertical length.
Therefore, maintaining the same value of this Froude number between prototype and model is a similitude criterion
arising from consideration of the pressure terms.
• Condition 2.
requirement
The temporal acceleration terms give the
Ny Nt
(4.106)
which means the prototype Strouhal number (based on
horizontal length and velocity) must be conserved in the
model. This is achieved by selecting the time scale as the
prototype-to-model ratio of wave periods. The inverse of
Eqn. 4.106 can be equated to Eqn. 4.105 to get the time
scale for long-wave models, i.e.,
Nt —
(4.107)
CHAPTER 4. HYDRODYNAMIC MODELS
Long-Wave Model Similarity Conditions
There are four independent dimensionless coefficients in the long-wave continuity equation and equations of motion; one related to the temporal accelerations, one related to the pressure terms, one related to the viscous
shear stress terms, and one related only to the vertical viscous shear stress
terms.
Just as in the case of short-wave models, complete similitude would be
achieved for any free surface long-wave hydrodynamic phenomena governed by the above form of the Navier-Stokes equations if the value of each
dimensionless coefficient in Eqns. 4.102-4.104 remained constant between
prototype and model.
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 pressure terms in the two equations comes the criterion given as
Ny
(4.105)
which is the prototype-to-model scale ratio of a Froude
number based on a horizontal velocity and a vertical length.
Therefore, maintaining the same value of this Froude number between prototype and model is a similitude criterion
arising from consideration of the pressure terms.
• Condition 2.
requirement
The temporal acceleration terms give the
Ny Nt
(4.106)
which means the prototype Strouhal number (based on
horizontal length and velocity) must be conserved in the
model. This is achieved by selecting the time scale as the
prototype-to-model ratio of wave periods. The inverse of
Eqn. 4.106 can be equated to Eqn. 4.105 to get the time
scale for long-wave models, i.e.,
Nt —
(4.107)