FLOW AROUND A CYLINDER - istiarto

d� t
dt � ��t �t � ��t �x
dx
dt � �� t
�y
dy
dt � ��t �z
– 5.46 –
dz
dt � 0 � �� t
�t � u ��t t
�x � v �� t
t
�y � wt (5.4)
which is known as the kinematic condition. The above equation can be solved for � t by
using a finite-volume technique on the 2D grid (xy plane) of the surface boundary. To
avoid the excessive computational time, the surface computation can be carried out at the
end of each time step. Alternatively, it can also be done after several time steps when the
pressure correction has reduced to a sufficiently small number as in the case of the
present model.
The surface function, � t , is unfortunately given for the surface (top face) center, whereas
the cell is defined by its vertices. This in effect necessitates the use of an interpolation to
get the z values of the cell vertices from this function. The simplest approach will be
using a linear interpolation (see Fig. 5.24c). Note that the usual notations PEWNS are
used to indicate the center and neighboring nodes on the 2D mesh which constitutes the
top boundary of the 3D mesh. Integrating Eq. 5.4 over the 2D cell and writing each term
in the discretized finite-volume form, one has:
�
�t
�� St, z
�� St, z
�� St, z
��S
t, z
� t dS � �t,P n�1 n
� � t,P
St,z �t
dS � �u t�
�x
n�1 n�1 n�1 n�1 n�1
�� �t �y�
� �
e � t �y�
� �
w � t �y�
� �
n � t �y�s
�
�� t
dS � �vt �
�y
n�1 n�1 n�1 n�1 n�1
�� �t �x�
� �
e � t �x�
� �
w � t �x�
� �t �x
n � �s �
��
u t
t
v t
w t dS � wt St,z (5.5)
where the face values are obtained by interpolation (see Eq. 4.19), for example for the
east face: � t,e � �1� �e�� t,P � �e � t,E . Using this interpolation to replace the face values
in Eq. 5.5, the discretized finite-volume form of Eq. 5.4 is:
n�1 n�1
aP �t,P � � anb �t,nb � b (5.6)
nb�EWNS
where the coefficients consist of the following terms:
aP � St,z �t � anb �
nb
� �ye � �yw � �yn � �y � s�u
t � �xe � �xw � �xn � �xs aE � �e��ye ut � �xe vt�, aW � �w��y w ut � �xw vt�, aN � �n��y n ut � �xn vt�, aS � �s��y s ut � �xs vt �,
b � w t S t,z � S t,z
�t .
� � v t ,