A Semi-Implicit, Three-Dimensional Model for Estuarine ... - USGS

τ xx
τ yy
1
--
2
τ ∂ xx
– -------- Δ x
∂x
y
1
--
2
τ ∂ yy
+ -------- Δ y
∂y
τ yx
1
--
2
τ ∂ yx
– -------- Δ x
∂x
2. Governing Equations and Boundary Conditions 35
volume approach zero. The application of the equation of motion then reduces to a stress balance on the control volume. Summing
the forces (stress × area) in the x-direction from figure 2.4 gives
τ xz
1
τ ΔxΔy τ --
xs xx 2
∂τ ⎛ --------- xx
+ Δx⎞
1
Δz --
⎝ ∂x ⎠ 2
∂ζ ⎛ + -----Δx ⎞ 1
Δy τxx --
⎝ ∂x ⎠
2
∂τ ⎛ – ----------- xx
Δx⎞
1
Δz --
⎝ ∂x ⎠ 2
∂ζ
+
–
⎛ – -----Δx ⎞Δy ⎝ ∂x ⎠
1
τ --
xy 2
∂τ ⎛ + ---------- xyΔy⎞
1
Δz --
⎝ ∂y ⎠ 2
∂ζ ⎛ + -----Δy ⎞ 1
Δx τxy --
⎝ ∂y ⎠
2
∂τ ⎛ – --------- xyΔy⎞
1
Δz --
⎝ ∂y ⎠ 2
∂ζ
+
–
⎛ – -----Δy ⎞Δx ⎝ ∂y ⎠
1
τ --
xz 2
∂τxz – ⎛ – ---------Δz ⎞ΔxΔy= 0.
⎝ ∂z ⎠
Solving this equation for τ xs , then cancelling terms and eliminating the terms which vanish as Δz → 0, yields
For the y-direction, the corresponding relation is
z
Datum
1
--
2
τ ∂ xz
– -------- Δz
∂z
τ xs
τ ys
x
τ xy
τ ys
τ yz
1
--
2
τ ∂ xy
+ -------- Δ y
∂y
1
--
2
τ ∂ yz
– -------- Δz
∂z
τxs
τ xy
τ yy
τ yx
1
--
2
τ ∂ xy
– -------- Δ y
∂y
Control
volume
1
--
2
τ ∂ yx
+ -------- Δ x
∂x
(2.53)
∂ζ ∂ζ
= – τ ----- – τ ----- + τ . (2.54)
xx ∂x xy ∂y xz
∂ζ ∂ζ
= – τ ----- – τ ----- + τ . (2.55)
yx ∂x yy ∂y yz
τ xx
1
--
2
τ ∂ yy
– -------- Δ y
∂y
1
--
2
τ ∂ xx
+ -------- Δ x
∂x
Figure 2.4. Viscous stresses on a surface layer of a water body affected by a wind stress with components
τxs and τys .
1
Expressions in the form τ --
xy
represent the viscous stresses on each vertical control volume face.
2
∂τ xy
+
----------Δy
∂y