SIMPLE (Semi-implicit Method for Pressure-Linked Equations), which was presented by Patankar and Spalding in 1972, is a very economical numerical calculation method for predicting three dimensional flow fields where a dominant flow direction exists. In Parts 5 and 6 of this study, the authors showed that SIMPLE was available for predicting the behaviour of warm water discharged into a rectangular open channel. However, some problems were also pointed out. One of them is that when warm water is discharged with very strong buoyancy, or is discharged near the surface of the channel water and a stable thermally stratified flow is formed, discrepancies between the computed results and experimental results appear in the reduction of the maximum temperature in the main flow direction and thermal distributions near the water surface. To solve this problem, first, it is necessary to take as fine as possible mesh intervals and diminish the influence of the apparent viscosity accompanied with upwind differencing. If fine mesh intervals are taken in the depth direction, however, the mesh intervals in the main flow direction must also be made very fine to hold the stability of calculation. Therefore, the largest merit of SIMPLE, that is, that it is economical, decreases. It is assumed that the instability of calculation is mainly caused by the following two factors: one is that the coefficients of finite differences equations for momentum equations are approximated by the values calculated with upstream velocity, temperature, and viscosity. The other is that several terms of Poisson's equation for pressure are neglected for brevity. The purpose of this paper is to improve the stability of calculation of SIMPLE and to save computation time by evaluating the coefficients in the finite differences equations correctly with downstream information and including the terms neglected in Patankar's SIMPLE. Taking these two points into account, an iterative calculation is necessary to solve the finite differences equations for the velocity and the Poisson's equation for pressure because the coefficients and the terms neglected in Patankar's SIMPLE are unknown values which are obtained with downstream information. As this point is different from Patankar's SIMPLE, the authors name the method presented in the paper "iterative SIMPLE". To discuss the performance of Patankar's SIMPLE and iterative SIMPLE, computations with both methods and an experiment on the behaviour of warm water were carried out under the condition that the warm water was discharged near the water surface of a rectangular open channel from a rectangular conduit in the center of the channel. The results of the computations and experiment clearly show that the stability of calculation and computation time of iterative SIMPLE is superior to those of Patankar's SIMPLE. A new convenient differences scheme in which we can choose at will an upwind differencing scheme (UDS), centered differences scheme (CDS), or hybrid differences scheme (HDS) by changing the value of a positive coefficient m: m=0 then UDS, m=∞ then CDS, and 0<m<∞ then HDS is also presented.
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