FLOW AROUND A CYLINDER - istiarto

– 4.13 –
convective transport through the east face is elaborated in the following paragraphs and a
similar approach applies to the other faces.
F C n�n�1
� � � � V �S
e
��
� � e
� u e
n�n�1
n�n�1
n�n�1
��V �S�
�e e
n,� n n,� n n,� n n,��1 � ��1
� Se,x � ve Se,y � we Se,z��
e � qe �e
(4.26)
In the above equation, qe is the discharge (the mass flux per unit mass) normal to the east
�
face. For simplicity, the time index ‗n‘ is omitted and the notation
��
q e stands for the
n,� n
discharge obtained from
��
ue and Se . A linearisation has been applied to the convective
term in Eq. 4.26 by setting � as the only unknown while taking the discharge, qe,
explicitly from the previous iteration.
The unknown variable at the east face, �e, is estimated by using upwind scheme, that is
by taking its value at the upstream control volume which depends on the flow direction
(Versteeg and Malalasekera, 1995, p. 115):
��
� ��1 ��1 � ��1 ��1 �
e � �P if qe � 0, �e � �E if qe � 0 (4.27)
The convective flux across the east face, Eq. 4.26, is then:
C n�n�1
�F � �� e
�
� max�q e,0��
P
� �� E
��1 �
� max �qe,0
��1 (4.28)
Note that the discharge, qe, is a scalar product of the velocity and the surface vector and it
has a positive sign when leaving the cell. Thus the discharge across the west cell face of
cell P is equal to the opposite value of that across the east face of cell W. The same is true
for the other cell faces. The discharge across the north or top faces of cell P is equal to the
opposite value of that across the south or bottom face of cells N or T, respectively. This
property has to be kept in the calculation of the convective flux in order to maintain the
flux consistency. The convective flux leaving the cell P across the east face is equal to
that entering cell W; otherwise a discrepancy occurs between neighboring cells. The
convective flux across the west face thus reads:
C n�n�1
�F � �� w
�
� max�q w,0��
P
� �� W
��1 �
� max �qw,0
4.3.6 Discretisation of the diffusive terms
��1 (4.28a)
The discrete form of the diffusive terms reads in Eq. 4.24 for cell P:
F D � � n�n�1
� � n�n�1
� �� � �� � dS � � � �� � S
(4.29)
S
cf�ewnstb
� � cf
n�n�1