4.2. SHORT-WAVE HYDRODYNAMIC MODELS
121
where Ny = NpNg. If the model bottom roughness is reproduced in geometric proportion to the prototype, then Nkf = NL and (7VTo)wave = NyNL,
which is the same as the stress scale for a Froude-scaled model.
In summary we can state -
The rough turbulent boundary layer formed beneath
waves in short-wave models will be in similitude provided the bottom roughness scale is the same as the
geometric length scale.
Unidirectional Current Shear Stress (Tidal Currents)
When the relative roughness, a.b/ks, gets very large (as in long-wave motion or tidal currents), the friction factor relationship given by Eqn. 4.71 is
no longer valid (Kamphuis 1973). Therefore, a different shear stress scaling requirement is needed to represent unidirectional currents9 and tidal
motions.
9 The term unidirectional is this context means fluid flow that tends to move from one
point to another in a reasonably smooth path and without an oscillatory component. It
does not mean the flow is limited strictly to one vectorial direction.
Yalin (1971) gave the following expression relating maximum bottom
shear stress and bottom roughness for unidirectional flow.
(^o)mar
7W
(4-77)
where Uc is the maximum value of average current velocity and ho is the
vertical width of the boundary layer (which is considered to be extended
up to the free surface).
Yalin approximated the logarithmic profile over the range
10 < ^ < 105
ks
as a power curve so that
This power curve approximation simplifies the shear stress in Eqn. 4.77 to
P (W
,(h-V114
= const —
k «5 7
(4.78)
121
where Ny = NpNg. If the model bottom roughness is reproduced in geometric proportion to the prototype, then Nkf = NL and (7VTo)wave = NyNL,
which is the same as the stress scale for a Froude-scaled model.
In summary we can state -
The rough turbulent boundary layer formed beneath
waves in short-wave models will be in similitude provided the bottom roughness scale is the same as the
geometric length scale.
Unidirectional Current Shear Stress (Tidal Currents)
When the relative roughness, a.b/ks, gets very large (as in long-wave motion or tidal currents), the friction factor relationship given by Eqn. 4.71 is
no longer valid (Kamphuis 1973). Therefore, a different shear stress scaling requirement is needed to represent unidirectional currents9 and tidal
motions.
9 The term unidirectional is this context means fluid flow that tends to move from one
point to another in a reasonably smooth path and without an oscillatory component. It
does not mean the flow is limited strictly to one vectorial direction.
Yalin (1971) gave the following expression relating maximum bottom
shear stress and bottom roughness for unidirectional flow.
(^o)mar
7W
(4-77)
where Uc is the maximum value of average current velocity and ho is the
vertical width of the boundary layer (which is considered to be extended
up to the free surface).
Yalin approximated the logarithmic profile over the range
10 < ^ < 105
ks
as a power curve so that
This power curve approximation simplifies the shear stress in Eqn. 4.77 to
P (W
,(h-V114
= const —
k «5 7
(4.78)