Optimization and Computational Fluid Dynamics - Department of ...

9 CFD-based optimization for papermaking 275
of using the 3D geometry of the headbox slice channel, the depth-averaged
Navier-Stokes equations are used. These equations are introduced next.
Let the velocity vector U =(u,v,w) and the static pressure p be a solution
of the 3D Navier-Stokes equations. The depth-averaged pressure P and the
velocity components U and V are defined as averaged values in the vertical
direction over the depth of the slice channel as follows
U (x, z)=
V (x, z)=
P (x, z)=
1
D (x, z)
1
D (x, z)
1
D (x, z)
�
D(x,z)
0
�
D(x,z)
0
�
D(x,z)
0
u (x, y, z)dy
v (x, y, z)dy
p (x, y, z)dy
(9.1)
where D =(x, z) is the depth of the slice channel depending on a position
(x, z). Then, by integrating the 3D continuity equation over the depth and by
using the definitions (9.1), the following depth-averaged continuity equation
is obtained
∇·(DV ) = 0 (9.2)
where V =(U,V ) is the reduced 2D velocity vector. The depth-averaged
momentum equation is derived similarly, and it is
where
− 1
D ∇·�2μDε (V ) � + ρC(m)V ·∇(DV )
�
4(m +1)μ
+
D2 �
V + ∇P =0
C (m)=
2m +2
2m +1
(9.3)
, m ≥ 2 (9.4)
is a coefficient associated with the velocity profile in the depth direction. For
example, m = 2 corresponds to a parabolic velocity profile, and m =7to
a turbulent plug flow profile. See also [33] for depth-averaging of the k–ε
turbulence model.