1. whereTisdiffusivity, h, hz,and hj arescalefactors,J isJacobian,and I e2 are unitbase vectors ofthe BFC coordinates. For cells such asp1. pz, p3 and p4 in Figure 1, the treatment could be similar to thc onc domain code by imaging that valuesorPI'. pz' eIc.arc available. Forcellp, however,it is not straightforward. Thefollowingpracticeisadopted in thisstudy: v whereDiand A@;isevaluatedatthe cellface,fi, and A$4, for exampleisequal to $A - $B. Thecoefficients Diwillbeaddedto thelinkcoefficientsofcellp to its childrenpi.
2. The geometry of the CFD-shaped duct is displayed in Figure 3. For such kinds of problems, a single block srmcturedgrid isvery difficulttogenerate. Even if it is possible,it will bevery expensivein termsof computer time and storage requirements since ghost cells (of blocked cells) have to be used. Unlike singledomain approach, it is very convenient for a multi-domain method to divide the problem into six rectangular blocksas shown in Figure 3 and then generatethe grid within each block. The flow has two inlets and one exit as depicted in Figure 3 and theReynolds number based on the inlet velocity and duct width is 1,000. Both the proposed implicit method and the above mentioned Schwarz method are used to calculate thc problem. Figure 4 showstheconvergencehistory from the two methods. The implicit method offers an order of magnitudeincreasein convergenceratecomparcdto the commonly used Schwarz method. This improvement is also proven by other applications in which the present method needs 2-15 times less iteration number than Schwarz method to reach the sameconvergencelevel. studied by Fletcherand McDaniel 6and numerically by manyreswrchersl*. It has been widcly used as a 3D benchmark test case. The overall geometry and flow patterns arc displayedin Figure 5. The combustortest sectionwasconfiguredwith staged,transverseinjectors locatedbehind abackwardfacingstep. Sincethehigh speed compressible gas streams injected transversely into a supersonic free stream, the flow becomes is highlythree-dimensionalanddominatedby shockwaves and shockboundary layer interactions. The boundary conditions forinletfreestreamareMachnumberof 2.07 with P= 31.4 kpa andT = 161°K. Thejet exit Mach number is 1.4 and the injector freestream dynamic pressure ratio of 1.2. Since the turbulence model is essential for this problem, a two equation k-E model was used for the simulation. The computational domainis divided intotwosubdomains: thecombustor pars and the inlet part. The grid number is 64x 30x 50and 6x30x 30respectively. 'd outlet
3. Figure 5. Sketch of Combustor Configuration and 0 2ea 6QB BQB leQB lzea Flow Condition
4. fortheSupersonicCombustor Thecomputational results have been obtained and are plotted in Figures 7-12. Figure 7 shows the velocity vectorsat theinjectorplane andthepressurecontoursat the same plane is plotted in Figure 8. As expected, there aretwo bow shocks well captured in frontthe the two transverse jets. Pressure contours at symmetry plane is also shown in Figure 9. Particle tracings at thesymmetryplane(Figure 10)showoverallfeaturesof the flowfieldwhich includes recirculationzone behind the stepand velocity change direction through shocks. Contours formasspenetration andspreadingareplotted in Figure 11. The comparison between the computation and experiment of the static pressures alongthesymmenyplane isgiven inFigure 12. d 0 ExperimentalData