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

3. Layer Averaging the Governing Equations 59
zk -
Here < > is shorthand notation for the layer integral ∫
1 /2
( ) dz.
zk + 1 /2
simplified by using the fundamental theorem of calculus. Thus
The integral of a term involving a derivative of z can be
where the fractional subscripts k – 1⁄ 2 and k 1⁄ 2
z
k – 1⁄ 2
∫ F
z
k + 1⁄ 2
-----
∂F
dz
=
∂z
k – 1⁄ – F
2 k + 1⁄ , (3.11)
2
+ indicate that the variable F is evaluated at the interface between two layers.
Formally, an interface value is defined by Fk + 1⁄ = [ Fxyzt ( , , , ) ]
2
z = zk . For computational purposes, the interface values can
+ 1⁄ 2
be evaluated as the simple average of the adjacent layer-averages of the same variables, such as
or, if unequal layer heights are used, the formula
Fk + 1⁄ 2
Fk 1⁄ 2
can be applied with θ = hk ⁄ ( hk + hk + 1)
to improve the accuracy of interpolations.
3.2 Continuity Equation
Integration of the continuity equation over a layer yields
z
k – 1⁄ 2
∫z
k + 1⁄ 2
or, after substitution of equations 3.9 and 3.11,
∂Uk
∂Vk
-------- + –
∂x
∂y
1
= -- ( F
2 k + Fk + 1)
(3.12)
+ = θFk + 1 + ( 1 – θ)Fk
(3.13)
z
k – 1⁄ 2
∫z
k + 1⁄ 2
z
k – 1⁄ 2
∫z
k + 1⁄ 2
∂u
∂v
∂w
----- dz + ---- dz + ------ dz = 0
(3.14)
∂x
∂y
∂z
∂zk
– 1⁄ ∂z
2
k – 1⁄ 2
-------- uk – 1 ---------------
⁄ + v
2
k – 1 ---------------
⁄ – w
2
k – 1⁄ 2
∂x
∂y
∂zk
+ 1⁄ ∂z
2
k + 1⁄ 2
uk + 1 ---------------
⁄ + v
2
k + 1 ---------------
⁄ – w
2
k + 1⁄ 2
+ = 0 .
∂x
∂y
(3.15)
The bracketed terms in equation 3.15 are simplified by applying the boundary conditions for each layer. With the exception of the
free surface and the bottom of the estuary, the layer interfaces are horizontal; thus,
∂zk
± 1⁄ 2 --------------- = 0 . (3.16)
∂x
At the free surface and the bottom, the kinematic boundary conditions presented in Chapter 2 (eqs. 2.52 and 2.59) are applicable.
These conditions, written in the present notation, are
and
∂z1
∂z1
⁄ 2 ⁄ ∂z1
2
-------- u1 -------- ⁄ 2
+
∂t
⁄ + v1 --------
2 ∂x
⁄ – w1 2 ∂y
⁄ = 0
(3.17)
2