Geography Reference
In-Depth Information
and the mass inflow per unit volume is just
(ρ
U
), which must equal the rate
of mass increase per unit volume. Now the increase of mass per unit volume is just
−
∇·
the local density change ∂ρ
∂t.Therefore,
∂ρ
∂t
+
∇·
(ρ
U
)
=
0
(2.30)
Equation (2.30) is the mass divergence form of the continuity equation.
An alternative form of the continuity equation is obtained by applying the vector
identity
∇·
(ρ
U
)
≡
ρ
∇·
U
+
U
·∇
ρ
and the relationship
D
Dt
≡
∂
∂t
+
U
·∇
to get
1
ρ
Dρ
Dt
+
∇·
U
=
0
(2.31)
Equation (2.31) is the velocity divergence form of the continuity equation. It states
that the fractional rate of increase of the density
following the motion
of an air parcel
is equal to minus the velocity divergence. This should be clearly distinguished from
(2.30), which states that the
local
rate of change of density is equal to minus the
mass divergence.
2.5.2 A Lagrangian Derivation
The physical meaning of divergence can be illustrated by the following alternative
derivation of (2.31). Consider a control volume of fixed mass δM that moves with
the fluid. Letting δV
=
δx δyδz be the volume, we find that because δM
=
ρδV
=
ρδxδyδz is conserved following the motion, we can write
1
δM
D
Dt
(δM)
1
ρδV
D
Dt
(ρδV )
1
ρ
Dρ
Dt
+
1
δV
D
Dt
(δV )
=
=
=
0
(2.32)
but
D
Dt
(δz)
Referring to Fig. 2.6, we see that the faces of the control volume in the y, z
plane (designated A and B) are advected with the flow in the x direction at speeds
u
A
1
δV
D
Dt
(δV )
1
δx
D
Dt
(δx)
1
δy
D
Dt
(δy)
1
δz
=
+
+
=
=
+
Dx/Dt and u
B
D(x
δx)/Dt, respectively. Thus, the difference
in speeds of the two faces is δu
=
u
B
−
u
A
=
D(x
+
δx)/Dt
−
Dx/Dt or
δu
=
D(δx) /Dt. Similarly, δv
=
D(δy)/Dt and δw
=
D(δz)/Dt. Therefore,
1
δV
Dt
(δV )
D
∂u
∂x
+
∂v
∂y
+
∂w
∂z
=∇
·
lim
δx,δy,δz
=
U
→
0
