FLOW AROUND A CYLINDER - istiarto

k and � equations
– 4.43 –
� set all coefficients related to the boundary node E in Eq. 4.52 to zero:
C D C D
aE � aE � a P
� � ��a E P�
� b
E D � � � b
E 1D � � � b
E 2D � � � 0
E
� extrapolate k and � at P to E: k E � k P and � E � � P
Pressure and velocity corrections
Since the discharge across the boundary is zero, the coefficients related to the
contribution from the boundary node E in the discretized pressure correction equation,
p p
� 0 and b
Eq. 4.68, are set to zero: a E
� � e � 0 .
The pressure correction gradient, �p c �x i , needed for the velocity correction, Eq. 4.57, is
computed by the finite-volume technique, Eq. 4.20, which requires the value of
p c � � � p
e c
� � . This latter is obtained by: p
E c � � � p
E c � � . The velocity at E has to be
P
corrected such that it is parallel to the symmetry plane, since the flux across the boundary
is zero. This is similar to assuming that the velocity at E is the same as the projection of
the velocity vector at P on a plane parallel to the symmetry boundary: V E � Vt,P.
4.5.6 Surface boundary
At the (water) surface (see Fig. 4.10) the velocity is parallel to the boundary, the
discharge across the (water) surface is zero, and thus there is no convective transport
across this boundary. The shear stress along the surface, in addition, is neglected. This
allows the specification of the velocity along the surface boundary the same as the
projection of the velocity at the cell center. The water surface does not create turbulence;
therefore, the kinetic energy along the surface boundary is set to zero. The energy
dissipation, �, at cell center is obtained in a similar manner as that at the wall boundary; a
correction may be given to reduce the computed value as has been reported in some
previous works (Krishnappan and Lau, 1986).
For the pressure, a hydrostatic distribution is assumed between the surface and the cell
center. The pressure at the surface is supposed to be atmospheric; if it is not the case, the
surface is moved according to the pressure defect, relative to a reference pressure, which
is prescribed at a particular cell. This reference cell is normally defined at the top-most
cell of the outflow boundary. This is similar to prescribe a constant flow-depth condition
at the outflow. The surface correction is done at the end of each time step. An underrelaxation
factor and a limitation may be imposed to avoid excessive change of the
computational domain. The procedures to handle surface boundary are described in the
following paragraphs.