Hydromechanics 7.1 Dimensional Analysis, Basic Estimation, and Model Testing 133
Part A | 7.1
Next breaking
wave
Runup
R
B
SWL
Fig. 7.2 Definition of the main parameters of beach runup
Example 7.2
Wave runup R is the height above the still water line
(SWL) that an ocean wave will wash up onto a beach
(Fig. 7.2). R has been found to depend on waveheight
H, deep water wavelength , and beach angle ˇ. Use
dimensional analysis to find the associated ˘ groups
and explicitly write the resulting functional dependence
of R on the other terms.
Solution 7.2
R D f .H; ;; ˇ/ ; ŒR D L ; ŒH D L ;
ŒŒ D L ; ŒˇD 1 :
4 variables, 1 dimension ! 3 ˘ -groups.
Eliminate length:
Ä R
H
D 1 ;
Ä
H
D 1 :
)
R
H
D '
Â
H
; ˇ
Ã
:
Empirical solutions for wave runup have been found
by [7.12]
R
H
D
s
2
ˇ
;
and [7.13]
R
H
D
r
H
tan ˇ :
Example 7.3
Longshore currents flow parallel to the shoreline and
are typically confined to the area between the shore and
the zone of breaking waves (Fig. 7.3). The current forms
due to the oblique approach of waves to the shore at
angle ˛ b , with the volume rate of flow depending primarily on the height of the incoming waves H b (which
break under the influence of gravity) and the slope of
Beach
Breake r zone
Longshore current
Rip
head
α b
α b
Rip
current
Oblique wave crests
Fig. 7.3 Longshore current and wave approach (after [7.14])
the beach m. Use dimensional analysis to find an expression for the functional dependence of the longshore
current speed u.
Solution 7.3
u D f .˛ b ; H b ; g; m/ ; Œu D
L
T
; Œ˛ b D 1 ;
ŒH b D L ; Œg D
L
T 2 ; Œm D 1
5 variables, 2 dimensions ! 3˘ -groups. There are already two dimensionless terms ˘ 2 D ˛ b , ˘ 3 D m.
Eliminate length:
Ä u
H b
D
1
T
;
Ä g
H b
D
1
T 2 ;
Eliminate time:
˘ 1 D
u
H b
s
H b
g
D
u
p
gH b
:
) u D
p
gH b '.˛ b ; m/ :
From theory and experiment [7.15]
u D 20:7m
p
gH b sin.˛ b / :
Example 7.4
The waterplane area is the horizontal cross section
formed by the intersection of the free surface of the water and the hull of a vessel. A catamaran (two hulls) has
Part A | 7.1
Next breaking
wave
Runup
R
B
SWL
Fig. 7.2 Definition of the main parameters of beach runup
Example 7.2
Wave runup R is the height above the still water line
(SWL) that an ocean wave will wash up onto a beach
(Fig. 7.2). R has been found to depend on waveheight
H, deep water wavelength , and beach angle ˇ. Use
dimensional analysis to find the associated ˘ groups
and explicitly write the resulting functional dependence
of R on the other terms.
Solution 7.2
R D f .H; ;; ˇ/ ; ŒR D L ; ŒH D L ;
ŒŒ D L ; ŒˇD 1 :
4 variables, 1 dimension ! 3 ˘ -groups.
Eliminate length:
Ä R
H
D 1 ;
Ä
H
D 1 :
)
R
H
D '
Â
H
; ˇ
Ã
:
Empirical solutions for wave runup have been found
by [7.12]
R
H
D
s
2
ˇ
;
and [7.13]
R
H
D
r
H
tan ˇ :
Example 7.3
Longshore currents flow parallel to the shoreline and
are typically confined to the area between the shore and
the zone of breaking waves (Fig. 7.3). The current forms
due to the oblique approach of waves to the shore at
angle ˛ b , with the volume rate of flow depending primarily on the height of the incoming waves H b (which
break under the influence of gravity) and the slope of
Beach
Breake r zone
Longshore current
Rip
head
α b
α b
Rip
current
Oblique wave crests
Fig. 7.3 Longshore current and wave approach (after [7.14])
the beach m. Use dimensional analysis to find an expression for the functional dependence of the longshore
current speed u.
Solution 7.3
u D f .˛ b ; H b ; g; m/ ; Œu D
L
T
; Œ˛ b D 1 ;
ŒH b D L ; Œg D
L
T 2 ; Œm D 1
5 variables, 2 dimensions ! 3˘ -groups. There are already two dimensionless terms ˘ 2 D ˛ b , ˘ 3 D m.
Eliminate length:
Ä u
H b
D
1
T
;
Ä g
H b
D
1
T 2 ;
Eliminate time:
˘ 1 D
u
H b
s
H b
g
D
u
p
gH b
:
) u D
p
gH b '.˛ b ; m/ :
From theory and experiment [7.15]
u D 20:7m
p
gH b sin.˛ b / :
Example 7.4
The waterplane area is the horizontal cross section
formed by the intersection of the free surface of the water and the hull of a vessel. A catamaran (two hulls) has