6.4.2ApproximationsinSIMPLEalgorithmSIMPLE is the dominant algorithm for solvingincompressible flows. It was proposed in 1972. Since thenmany variants (方案)were proposed to improve (overcome)following two assumptions.1.Inconsistency（不一致性）of initialfieldassumptionsIn SIMPLEu,yandp*areassumed independentlyActuallythere is someinherent（固有的）relationshipbetweenvelocityandpressure,2.Overestimating（夸大)theeffectsofpressurecorrectionof the neighboring nodes. Because ue 'is caused by boththe pressure correction and velocity corrections of its6/43
6/43 6.4.2 Approximations in SIMPLE algorithm SIMPLE is the dominant algorithm for solving incompressible flows. It was proposed in 1972. Since then many variants (方案) were proposed to improve (overcome) following two assumptions. 1.Inconsistency (不一致性) of initial field assumptions In SIMPLE u 0 ,v 0 and p * are assumed independently. Actually there is some inherent (固有的）relationship between velocity and pressure; 2.Overestimating (夸大)the effects of pressure correction of the neighboring nodes. Because ue ’ is caused by both the pressure correction and velocity corrections of its
neighboringnodes.Theneglect of velocity corrections oftheneighboringnodesattributes（归结于）thedrivingforceofutotallyto pressurecorrection,thus exaggerating（夸大)theaction ofpressurecorrection6.4.3Numericalexample[Example 6-1(Text book)]Known:Pw,Ps,ue,Vntu=0.7(pw-pp)AV,= 0.6(ps - Pp)Ar=A= 40Find: Pp,Uw,Vs7/43
7/43 6.4.3 Numerical example • • • • • • • neighboring nodes. The neglect of velocity corrections of the neighboring nodes attributes ( 归结于) the driving force of ue ’ totally to pressure correction, thus exaggerating (夸 大) the action of pressure correction. [Example 6-1(Text book)] Known: , , , W S e n p p u v 0.7( ) w W P u p p = − 0.6( ) s S P v p p = − Find: , , P w s p u v 40 S p =
The key to solve Example 6-l:how to understand :uw=0.7(pw-pp) V,=0.6 (ps- Pp)They should be regarded as followsa.u=Eanbunb +b+ A(pp-pe)Zamub+b+(pp-Pa)=i+d(pp-Pe)aeaFor thisu=0.7u,=0+0.7(pw- Pp)exampled.= 0.6Similarly,With this understanding , the problem can be solved accordingto the solution steps of SIMPLE algorithm. See Textbook8/43
8/43 The key to solve Example 6-1：how to understand : 0.7( ) w W P u p p = − They should be regarded as follows 0 0.7( ) w W P u p p = + − 0.7 w d = * * * * ( ) e e nb nb e P E a u a u b A p p = + + − * * * ( ) e e P E u d p p + − * * * * ( ) nb nb e e P E e e a u b A u p p a a + = + − = For this example * 0 w u = 0.6 ( ) s S P v p p = − Similarly， 0.6 s d = With this understanding , the problem can be solved according to the solution steps of SIMPLE algorithm. See Textbook
6.5DiscussiononSIMPLEandConvergenceCriteriaofFlowFieldIteration6.5.1DiscussiononSIMPLEalgorithm1.Can the simplification approximations affect thecomputational results?2.Mathematically what type does the boundary conditionof the pressure correction equation belong to?3.How to adopt the underrelaxation method in the flowfield iteration process?6.5.2Convergencecriteriaofflowfielditeration9/43
9/43 6.5 Discussion on SIMPLE and Convergence Criteria of Flow Field Iteration 6.5.1 Discussion on SIMPLE algorithm 1.Can the simplification approximations affect the computational results? 2.Mathematically what type does the boundary condition of the pressure correction equation belong to ？ 3.How to adopt the underrelaxation method in the flow field iteration process？ 6.5.2 Convergence criteria of flow field iteration
6.5DiscussiononSiMPLEandConvergenceCriteriaofFlowFieldIteration6.5.1DiscussiononSIMPLEalgorithm1.Can the two simplified approximations affect thecomputational results (solution accuracy) ?The approximations of SIMPLE will not affect theconverged solution, but do affect the convergence speedfor the following reasons:u°, v°, p* will be gradually(1)Theinconsistencybetweeneliminatedwiththeproceedingof iteration（随着迭代的进行）(2) The term Zun, in u will gradually approach zero(趋近于O)withtheproceedingof iterationif itconverges!10/43
10/43 6.5 Discussion on SIMPLE and Convergence Criteria of Flow Field Iteration 6.5.1 Discussion on SIMPLE algorithm 1.Can the two simplified approximations affect the computational results (solution accuracy) ? The approximations of SIMPLE will not affect the converged solution , but do affect the convergence speed for the following reasons: 0 0 * (1) The inconsistency between u v p , , will be gradually eliminated with the proceeding of iteration（随着迭代的进行）； （2）The term in will gradually approach zero (趋近于0) with the proceeding of iteration if it converges! ' e u ' nb u