Hydromechanics 7.1 Dimensional Analysis, Basic Estimation, and Model Testing 135
Part A | 7.1
b) Dynamic similitude.
Forces acting on model masses must have the same
ratio everywhere in the flow field. To achieve this,
relevant dimensionless numbers are equated so that
each type of force (e.g., viscous, momentum, gravitational, etc.) that is considered to be dominant in
determining the flow will act in a proportion determined by the ˘ -groups.
The relevant ˘ -groups are those which are most
likely to play a role in governing a particular flow
phenomenon. For example, when examining the flow
in a completely enclosed space with no free surface,
one would not generally need to consider the possible effects of either gravity or capillary waves. Hence
the Froude number and Weber number are unlikely to
play a role. The interplay between viscous and momentum forces is likely to dominate the flow. Hence, the
Reynolds number would be likely to play a role.
When determining which ˘ groups might play
a role in governing a flow, it is often helpful to examine the magnitude of the ˘ groups. When a ˘ group
is much larger than unity ˘ 1 or much smaller than
unity ˘ 1, the effects of the ˘ group can sometimes
be neglected.
Solving similitude problems:
i. Think about the dominant ˘ groups that are likely
to apply. When a free surface is present, the Froude
number, Weber number, and Reynolds number are
likely to play a role. When there is no free surface,
the Reynolds number is likely to be an important
parameter. When vortex shedding is present, the
Strouhal number will play role.
ii. Equate the ˘ groups that are felt to be important
and use geometric similitude to solve for the related
length and velocity scales.
iii. Equate force and pressure coefficients to find the
corresponding forces on the prototype, when model
forces are known.
Example 7.5
An offshore wind turbine is supported with a cylindrical
monopile. The system is deployed in currents of U p D
2:5 [m s
1 ] and waves with a T p D 8:8 [s] period and
H p D 1:9 [m] height. If a 1 W 40 scale model is tested
in a wave flume, what are the model current speed U m ,
model wave period T m and model wave height H m ?
Solution 7.5
The geometry of the model wave can be scaled by
L m =L p D 1=40, where L m and L p are characteristic
length scales for the model and prototype flows, respectively
H m D
H p
60
D 4:75 cm :
As the flow past the monopile will create surface
gravity waves, the Froude number scaling can be used
to determine the model current speed and wave period.
Using the definition of the Froude number,
Fr Á
U
p
gL
;
we can develop relations for the model current U m
U m
p
gL m
D
U p
p
gL p
! U m D U p
s
L m
L p
D 0:40 m s
1
:
The celerity (phase speed) of a wave is its wavelength
divided by its period T. Froude scaling can also be used
to relate the celerity of the prototype and model waves
m =T m
p
gL m
D
p =T p
p
gL P
! T m D T p
s
L m
L p
D 1:39 s ;
where we have taken m == p D L m =L p .
7.1.4 Skin-Friction Drag
Breakdown of a Laminar Boundary Layer
The transition to turbulence over a flat plate is mainly
governed by the Reynolds number, but can be affected
by the amount of freestream turbulence present in the
flow and the surface roughness of the plate. Above
a critical Reynolds number of about Re x Á Ux==
510
5 the flow becomes turbulent. Here the length scale
for the Reynolds number is based on the distance from
the leading edge of the plate x so that the Reynolds
number increases with an increase in the downstream
distance from the leading edge. The sequence of events
that happen in the transition to turbulence in a flat plate
boundary layer is shown in Fig. 7.5.
Near the leading edge of the flat plate, the flow is
laminar and the full Navier–Stokes relations hold and
Re x 1. Downstream when Re x 1, the flow is still
laminar, but the approximate boundary layer equations
hold [7.16]. A similarity solution for the Blasius boundary layer can be found (Fig. 7.6).
Part A | 7.1
b) Dynamic similitude.
Forces acting on model masses must have the same
ratio everywhere in the flow field. To achieve this,
relevant dimensionless numbers are equated so that
each type of force (e.g., viscous, momentum, gravitational, etc.) that is considered to be dominant in
determining the flow will act in a proportion determined by the ˘ -groups.
The relevant ˘ -groups are those which are most
likely to play a role in governing a particular flow
phenomenon. For example, when examining the flow
in a completely enclosed space with no free surface,
one would not generally need to consider the possible effects of either gravity or capillary waves. Hence
the Froude number and Weber number are unlikely to
play a role. The interplay between viscous and momentum forces is likely to dominate the flow. Hence, the
Reynolds number would be likely to play a role.
When determining which ˘ groups might play
a role in governing a flow, it is often helpful to examine the magnitude of the ˘ groups. When a ˘ group
is much larger than unity ˘ 1 or much smaller than
unity ˘ 1, the effects of the ˘ group can sometimes
be neglected.
Solving similitude problems:
i. Think about the dominant ˘ groups that are likely
to apply. When a free surface is present, the Froude
number, Weber number, and Reynolds number are
likely to play a role. When there is no free surface,
the Reynolds number is likely to be an important
parameter. When vortex shedding is present, the
Strouhal number will play role.
ii. Equate the ˘ groups that are felt to be important
and use geometric similitude to solve for the related
length and velocity scales.
iii. Equate force and pressure coefficients to find the
corresponding forces on the prototype, when model
forces are known.
Example 7.5
An offshore wind turbine is supported with a cylindrical
monopile. The system is deployed in currents of U p D
2:5 [m s
1 ] and waves with a T p D 8:8 [s] period and
H p D 1:9 [m] height. If a 1 W 40 scale model is tested
in a wave flume, what are the model current speed U m ,
model wave period T m and model wave height H m ?
Solution 7.5
The geometry of the model wave can be scaled by
L m =L p D 1=40, where L m and L p are characteristic
length scales for the model and prototype flows, respectively
H m D
H p
60
D 4:75 cm :
As the flow past the monopile will create surface
gravity waves, the Froude number scaling can be used
to determine the model current speed and wave period.
Using the definition of the Froude number,
Fr Á
U
p
gL
;
we can develop relations for the model current U m
U m
p
gL m
D
U p
p
gL p
! U m D U p
s
L m
L p
D 0:40 m s
1
:
The celerity (phase speed) of a wave is its wavelength
divided by its period T. Froude scaling can also be used
to relate the celerity of the prototype and model waves
m =T m
p
gL m
D
p =T p
p
gL P
! T m D T p
s
L m
L p
D 1:39 s ;
where we have taken m == p D L m =L p .
7.1.4 Skin-Friction Drag
Breakdown of a Laminar Boundary Layer
The transition to turbulence over a flat plate is mainly
governed by the Reynolds number, but can be affected
by the amount of freestream turbulence present in the
flow and the surface roughness of the plate. Above
a critical Reynolds number of about Re x Á Ux==
510
5 the flow becomes turbulent. Here the length scale
for the Reynolds number is based on the distance from
the leading edge of the plate x so that the Reynolds
number increases with an increase in the downstream
distance from the leading edge. The sequence of events
that happen in the transition to turbulence in a flat plate
boundary layer is shown in Fig. 7.5.
Near the leading edge of the flat plate, the flow is
laminar and the full Navier–Stokes relations hold and
Re x 1. Downstream when Re x 1, the flow is still
laminar, but the approximate boundary layer equations
hold [7.16]. A similarity solution for the Blasius boundary layer can be found (Fig. 7.6).