4.3. LONG-WAVE HYDRODYNAMIC MODELS
137
X
~ y
Z
;
h
x=X’
v = x’
Z=z'
h=z
(4101>
where
V - characteristic horizontal velocity
X - characteristic horizontal length
Z - characteristic vertical length
T - characteristic time
P - characteristic pressure
Different characteristic lengths were chosen for the horizontal and vertical directions to make them distinguishable in a geometrically distorted
hydrodynamic model.
Making these substitutions into Eqn. 4.97 and multiplying by XfV
yields the nondimensional long-wave continuity equation:
d ù(h + 7/)
qü(Â + d
/ x \ dû
= - (vrU <4102’
Similarly, substitution of the nondimensional parameters into the equations of motion and multiplying by X/V2 gives the nondimensional form of
the long-wave equations of motion.
x-Direction
dû
_du ~dû
( gZ\ dr)
di + U dx + V dy
\ V2 ) dx
d2û
ô2û\
dx2 + dy2 )
(A + rj) dz -h
1
dû
(4.103)
y-Direction
VT
dv _dv
~dv
(gZ\ dfj
— +
+ v— = — ( 777 I -7T +
dt
dx
dy
\V2J dy
d2v
d2v\
dx2 + dy2 J
Ch + fi) dz
1
di)
(4.104)