Geography Reference
In-Depth Information
Fig. 9.9
Flow over an obstacle for a barotropic fluid with free surface. (a) Subcritical flow (Fr < 1
everywhere). (b) Supercritical flow (Fr>1everywhere). (c) Supercritical flow on lee slope
with adjustment to subcritical flow at hydraulic jump near base of obstacle. (After Durran,
1990.)
∂x
u
h
h
M
=
∂
−
0
(9.36)
Equation (9.35) may be integrated immediately to show that the sum of kinetic
and potential energy, u
2
/2
gh, is constant following the motion. Thus, energy
conservation requires that if u increases h must decrease, and vice versa. In addi-
tion, (9.36) shows that the mass flux, u(h
+
h
M
), must also be conserved. The
direction of the exchange between kinetic and potential energy in flow over a ridge
is determined by the necessity that both (9.35) and (9.36) be satisfied.
Multiplying (9.35) by u and eliminating ∂h/∂x with the aid of (9.36) gives
−
1
Fr
2
∂u
ug
c
2
∂h
M
∂x
−
∂x
=
(9.37)
