First, predict the value of Fi+j by using a Taylor series where 3E/3f is evaluated at point (i, j). Denote this predicted value by Fi+j:
Fi+ij = F, j + ( ^ ) A*
In Equation (13.55), (3E/3f )jj is obtained from the continuity equation, Equation
(13.50) , using/orwa/г/ differences for the p derivatives; that is,
In Equation (13.56), all quantities on the right-hand side are known and allow the calculation of (3E/3f ), j which is, in turn, inserted into Equation (13.55). A similar procedure is used to find predicted values of G, H, and K, namely, G,+ij, Hi+ij, and Ki+ij, using forward differences in Equations (13.51) to (13.53). In turn, predicted values of the primitive variables, Pi+j, Рі+ij, etc., can be obtained from Equations (13.39a to d) and (13.44) to (13.46).
13.4.3 Corrector Step
The predicted values obtained above are used to obtain predicted values of the derivative (3E/3f );+ij, using rearward differences in Equation (13.50):
/ЭЕ) _ ( Ц^уА fi’+i. i ~ Fi+ij-1 1 (pv)i+u ~ (pv)i+ij-1
/ ~ y, dx)i+lj Art ys Ap
In turn, the results from Equations (13.56) and (13.57) allow the calculation of the average derivative
Finally, this average derivative is used in Equation (13.54a) to obtain the corrected value of Fi+j. The same process is followed to find the corrected values of G, t L/, Ні+ij, and A’^i, using rearward differences in Equations (13.51) to (13.53) and calculating the average derivatives (9G/9§)ave, etc., in the same manner as Equation
The above finite-difference procedure allows the step-by-step calculation of the flow field, marching downstream from some initial data line. In the flow given in Figure 13.8, the initial data line is the inlet, where properties are considered known. Although all the calculations are carried out in the transformed, computational plane, the flow-field results obtained at points (2, 1), (2, 2), etc., in the computational plane are the same values at points (2, 1), (2, 2), etc., in the physical plane.
There are other aspects of the finite-difference solution which have not been described above. For example, what values of At] and A§ in Equations (13.54a to d), (13.55), (13.56), and (13.57) are allowed in order to maintain numerical stability? How is the flow-tangency condition at the walls imposed on the finite-difference calculations? These are important matters, but we do not take the additional space to discuss them here. See chapter 11 of Reference 21 for details on these questions. Our purpose here has been to give you only a feeling for the nature of the finite-difference method.