FLOW AROUND A CYLINDER - istiarto

– 4.14 –
in which the summation extends over the six cell faces and the non-linear terms are
linearized as in the evaluation of the convection term. The diffusion across the east face is
elaborated and a similar approach applies for the other faces.
In evaluating the diffusive term across the east face, it is convenient to use a local
coordinate system attached to the east face as shown in Fig. 4.4. Across the east face,
Eq. 4.29 reads:
F
��
D
n,��1
n�n�1
n�n�1
n,������� ��
� � � � �� �S
e � � � �e e ���n��
e
n
�e n � S�e
Fig. 4.4 Evaluation of the diffusion terms across the east face
(4.30)
The evaluation of the normal gradient presents some difficulties for its y and z
components. Besides the variable at the neighbor cell E, additional ones at NE, SE, S, and
N might have to be taken into consideration. This would increase the number of
unknowns. To overcome this problem, the so called deferred-correction approach
(Ferziger and Peric, 1997) is selected in the present model, where only the immediate
neighbor cell needs to be considered. In this approach the normal gradient term is
evaluated implicitly by a simple approximation and a correction is added. The correction
is taken as the difference between the correct and approximate gradients; both are
explicitly obtained from the previous iteration. This correction is put in the source terms
at the right-hand side. The diffusion term evaluated with this approach reads (Ferziger
and Peric, 1997, pp. 218-222):
F D n�n�1 � � � � �
e
e Se
��
������
��
��
����
�� e
��1
�
������
�
���n��
e
��
��
�
�� ��
��
��
��
��
����
��
��
��
��
��
e
�� ������ �� �� �������� �� ��
�
� �e Se
correction, explicit
(4.31)
In the above expression, the time index n is omitted for simplicity and a term without any
index refers to the initial solution of the time step n � n �1 (for example Se is constant