3.4 Nonhydrostatic Solver
29
Fig. 3.5 Location used to define the coefficients a t , a b , a e , and a w . The pressure part q and the
pressure correction Δq are calculated at the same grid points
3.4.4 The S.O.R. Method
The pressure equation (Eq. 3.24) can be solved by an iterative method called
Successive Over-Relaxation (or S.O.R in short) that can be formulated as:
Δq
r +1
i,k = (1 − ω) Δq
r
i,k −
ω
a o
q
∗
i,k +
+
ω
a o
a e Δq
i,k+1 + a w Δq
i,k−1 + a t Δq
i−1,k + a b Δq
i+1,k
(3.27)
where r = 0, 1, 2, · · · is the iteration index, the superscript is given by either
= r + 1 or = r dependent on whether an update of Δq already exists, and the
parameter ω determines the degree of over-relaxation. Typical values are in a range
between 1.2 and 1.4.
Start values of Δq for the S.O.R. iteration can be set to zero, but the iteration is
often faster if we use the values of the previous time step instead; that is,
Δq
r=0
i,k = Δq
n
i,k
The surface boundary value for dynamic pressure needs to be given at every step
of the S.O.R. iteration. How this is done is described in the following. First, velocity
components are updated within the S.O.R. iteration with:
u
r+1
i,k = u
∗
i,k −
Δt
ρ o Δx
Δq
r+1
i,k+1 − Δq
r +1
i,k
(3.28)
w
r+1
i,k = w
∗
i,k −
Δt
ρ o Δz
Δq
r+1
i−1,k − Δq
r +1
i,k
(3.29)
29
Fig. 3.5 Location used to define the coefficients a t , a b , a e , and a w . The pressure part q and the
pressure correction Δq are calculated at the same grid points
3.4.4 The S.O.R. Method
The pressure equation (Eq. 3.24) can be solved by an iterative method called
Successive Over-Relaxation (or S.O.R in short) that can be formulated as:
Δq
r +1
i,k = (1 − ω) Δq
r
i,k −
ω
a o
q
∗
i,k +
+
ω
a o
a e Δq
i,k+1 + a w Δq
i,k−1 + a t Δq
i−1,k + a b Δq
i+1,k
(3.27)
where r = 0, 1, 2, · · · is the iteration index, the superscript is given by either
= r + 1 or = r dependent on whether an update of Δq already exists, and the
parameter ω determines the degree of over-relaxation. Typical values are in a range
between 1.2 and 1.4.
Start values of Δq for the S.O.R. iteration can be set to zero, but the iteration is
often faster if we use the values of the previous time step instead; that is,
Δq
r=0
i,k = Δq
n
i,k
The surface boundary value for dynamic pressure needs to be given at every step
of the S.O.R. iteration. How this is done is described in the following. First, velocity
components are updated within the S.O.R. iteration with:
u
r+1
i,k = u
∗
i,k −
Δt
ρ o Δx
Δq
r+1
i,k+1 − Δq
r +1
i,k
(3.28)
w
r+1
i,k = w
∗
i,k −
Δt
ρ o Δz
Δq
r+1
i−1,k − Δq
r +1
i,k
(3.29)