热流科学与工程西步文源大学G教育部重点实验室(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+ap+asp,+bRightapp,-(aep,+awpw+anp+asp)termMass conservation of the entirebijk=0 -domain should be satisfiedThus the reguirement of mass conservation at each iterationlevel corresponds to the execution of Neumann boundarycondition for the satisfaction of compatibility requirement中CFD-NHT-EH11/41CENTER
11/41 (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
热流科学与工程西步文源大堂G教育部重点实验室In our teaching program RMAX, SSUM represent bnanand bi., respectively.(4)DeterminationofabsolutepressureFor Neumann condition,103p 'should be determined102p'level specifiedby computation, ratherp'level note101than specified inadvanceAspecified100After receiving theconverged solution,10-1selecting some point as a10~2reference and using the051015202530relative results as outputIterationtimesΦCFD-NHT-EHT12/41CENTER
12/41 (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
热流科学与工程西步文源大堂G教育部重点实验室3.How to adopt the underrelaxation in solving flow fields?(1) Underrelaxation of pressure correction p':p=p+α,pαn--pressureunderrelaxationfactor(2) Underrelaxation of velocity is organized into thesolution procedure:Iterationprocess isgenerally expressed as:Zambgn +bapofp= +αdga,neweq.bofneweq(),=Zam中 +b+(1-α)%dThe obtained numerical results have already be underrelaxed!ΦCFD-NHT-EH13/41CENTER
13/41 3.How to adopt the underrelaxation in solving flow fields？ (1) Underrelaxation of pressure correction p ’ ： * ' p p p p -pressure underrelaxation factor p (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 be underrelaxed!
热流科学与工程西步文源大堂E教育部重点实验室Discussion: Can the direct underrelaxation be used forvelocity?No! No! No!u=u+αuReason: 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 iterationsΦCFD-NHT-EHT14/41CENTER
14/41 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