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– 4.5 –<br />
The volume integrals of the convective and diffusive terms, the second and third terms on<br />
the left-hand side, can be expressed as integral over the closed surface bounding the<br />
control volume by applying Gauss divergence theorem (Versteeg and Malalasekera,<br />
1995, p. 26; Hirsch, 1988, p. 241):<br />
�<br />
�� �t<br />
���<br />
� �dV �<br />
V<br />
�� ��V� dS � �<br />
S<br />
S ��� ��dS<br />
�� � ��� R�dV (4.15)<br />
where S is the surface vector normal outward to the control volume dV .<br />
�� � ��<br />
�� ��<br />
�� Table 4.2 Terms in the general transport equation, Eq. 4.13.<br />
��� �� � t � � � � �V��<br />
��� ��� ���<br />
R� (4.13)<br />
��<br />
R �<br />
1 u �t � 1 �p �<br />
�<br />
� �x �x � ��<br />
�� t<br />
2 v �t � 1 �p �<br />
�<br />
� �y �x � ��<br />
t<br />
��<br />
��<br />
3 w �t � 1 �p �<br />
�<br />
� �z �x � ��<br />
�� t<br />
4 1 0 0<br />
5 k � t � k G-��<br />
6 �� � t � � c 1<br />
4.2 Solution strategy: the iterative method<br />
V<br />
�u��<br />
�<br />
�<br />
�x �� �y � �� �v��<br />
�<br />
�� t �<br />
�x��<br />
�z � �� �w��<br />
�� t<br />
�x �� � gx �u��<br />
�y ��<br />
��� �<br />
�y � �� �v��<br />
t<br />
��<br />
��<br />
�y��<br />
��� �<br />
�z � �� �w��<br />
t<br />
��<br />
��<br />
�y ��<br />
��� gy �u��<br />
�<br />
�<br />
�z �� �y � �� �v��<br />
�<br />
�� t �<br />
�z��<br />
�z � �� �w��<br />
�� t<br />
�z �� � gz �<br />
k G � c �<br />
2<br />
k �<br />
The integral form of the general transport equation, Eq. 4.15, is used to obtain the<br />
solution for u, v, w, k, and � by substituting these variables to the scalar variable �. The<br />
solution of the equation is sought at discrete time steps; calculations are performed at<br />
every discrete time steps and repeated until a steady-state solution is obtained. The time<br />
derivative term in Eq. 4.15 facilitates the application of the model to transient flow<br />
problems; in this case, the solution at each discrete time step must converge. When the<br />
problems concern steady case ones —as is the case in the present work— the time<br />
derivative serves as an iteration loop. In this case the solution at each time step is<br />
considered as an intermediate solution, and the end-solution (the steady-state one) is<br />
obtained when all variables �‘s converge. Since the end-solution that is sought, it is not<br />
necessary to force the intermediate solution to converge at the same degree of<br />
convergence as that of the end-solution. The complete computational procedure is<br />
depicted in the flowchart shown in Fig. 4.1. The time loop, from the initial until the
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