2. What tvpe does the boundary condition of the pressurecorrection equation belong to ? Can it uniquely determine thesolution?(1)Mathematically the boundary condition of the pressurecorrection equation is Newmann condition,dp-Gresho's question (1991: A simple question to=0anSIMPLE users)(2) The adiabatic type boundary condition of the pressurecorrection equation can uniquely (唯一地)define anincompressibleflow problem, becausepressure exists intheN-S equation in terms of gradient! This formulation can uniquelyU.vU--lVp+Wdefine the flow fieldV.U=0 pThinking question: Why fortemperature not?No slip on the boundary11/43
11/43 2. What type does the boundary condition of the pressure correction equation belong to ？Can it uniquely determine the solution? (1)Mathematically the boundary condition of the pressure correction equation is Newmann condition， ' 0 p n = (2) The adiabatic type boundary condition of the pressure correction equation can uniquely (唯一地) define an incompressible flow problem, because pressure exists in the N-S equation in terms of gradient! 1 2 U U p U • = − + • = U 0 No slip on the boundary －Gresho’s question（1991：A simple question to SIMPLE users) This formulation can uniquely define the flow field. Thinking question: Why for temperature not?
(3)Theboundaryconditionofthepressurecorrectionequation makes theABEqs.being linearly dependent(线性相关），andthecoefficientmatrixissingular（奇异）；Inorder to get a unique solution the compatibility condition相容性条件）mustbesatisfied：thesumoftherightterms(右端项）oftheABEqs.shouldbezero.Mathematicallyapp,=aep,+awp+anp+asp,+bRightapp,-(aep.+awp.+anp+asp.termMass conservation of theentireZbijk=0-domain should be satisfied.Thus the requirement of mass conservation at each iterationlevel corresponds to the execution of Neumann boundarycondition for the satisfaction of compatibility requirement!12/43
12/43 (3) The boundary condition of the pressure correction equation makes the ABEqs. being linearly dependent (线性 相关)，and the coefficient matrix is singular (奇异)；In order to get a unique solution the compatibility condition (相容性条件) must be satisfied：the sum of the right terms (右端项) of the ABEqs. should be zero. ' ' ' ' ' P E W N S P E W N S a p a p a p a p a p b = + + + + ' ' ' ' ' ( ) P E W N S P E W N S a p a p a p a p a p b − + + + = , , 0 i j k b = Mass conservation of the entire domain should be satisfied. Thus the requirement of mass conservation at each iteration level corresponds to the execution of Neumann boundary condition for the satisfaction of compatibility requirement! Right term Mathematically
In our teaching program RMAX, SSUM represent bmaand b,, respectively.(4)DeterminationofabsolutepressureFor Neumann condition,1037p 'should be determined102p'levelspecifiedby computation, ratherp'level note101than specified inadvanceAspecified100After receiving theconverged solution,10-1selecting some point as a10~2reference and using the051015202530relative results as outputIterationtimes13/43
13/43 (4) Determination of absolute pressure For Neumann condition, p ’should be determined by computation , rather than specified in advance. After receiving the converged solution, selecting some point as a reference and using the relative results as output. In our teaching program RMAX，SSUM represent bmax and , respectively. i j , b
3.How to adopt the underrelaxation in solving flow fields?(1) Underrelaxation of pressure correctionp:p=p+α,pαn--pressure underrelaxation factor for non-linearity(2) Underrelaxation of velocity is organized into thesolution procedure:Iteration process is generally expressed as[Zanbgn +bapofp= +αldg]a,neweq.bofneweq,(")op=Zamlm +b+(1-α)%dThe obtained numerical results have already been underrelaxed!14/43
14/43 3.How to adopt the underrelaxation in solving flow fields？ (1) Underrelaxation of pressure correction p ’ ： * ' p p p p = + p -pressure underrelaxation factor for non-linearity (2) Underrelaxation of velocity is organized into the solution procedure： Iteration process is generally expressed as： 0 0 [ ] nb nb P P P P a b a + = + − 0 ( ) (1 ) P P P nb nb P a a a b = + + − b of new eq. of new eq. aP The obtained numerical results have already been underrelaxed!
Discussion: Can the direct underrelaxation be used forvelocity?u=u+aαuNo! No! No!Reason: The velocity correction is obtained through massconservation requirement.Its underrelaxation will violate(破坏)mass conservation condition.Thus incorporating(纳入) the underrelaxation of velocity into solutionprocedure is necessary!6.5.2Convergencecriteriaofflowfielditeration1.Two different iterations15/43
15/43 Discussion：Can the direct underrelaxation be used for velocity? * ' u u u u = + No ！No！No！ Reason：The velocity correction is obtained through mass conservation requirement. Its underrelaxation will violate (破坏) mass conservation condition. Thus incorporating (纳入) the underrelaxation of velocity into solution procedure is necessary! 6.5.2 Convergence criteria of flow field iteration 1.Two different iterations