《数值传热学》研究生课程教学资源（课件讲稿）Chapter 6 Primitive Variable Methods for Elliptic Flow and Heat Transfer（2/3，6.4-6.6）

热流科学与工程西步文源大学E教育部重点实验室Numerical HeatTransfer（数值传热学）Chapter6PrimitiveVariableMethodsforEllipticFlowandHeatTransfer(2)raInstructorTao,Wen-QuanKeyLaboratoryofThermo-FluidScience&EngineeringInt.JointResearchLaboratoryofThermalScience&EngineeringXi'anJiaotongUniversityInnovativeHarborofWestChina,Xian2022-October-25CFD-NHT-EHTΦ1/41CENTER

1/41 Instructor Tao, Wen-Quan Key Laboratory of Thermo-Fluid Science & Engineering Int. Joint Research Laboratory of Thermal Science & Engineering Xi’an Jiaotong University Innovative Harbor of West China, Xian 2022-October-25 Numerical Heat Transfer (数值传热学) Chapter 6 Primitive Variable Methods for Elliptic Flow and Heat Transfer（2）

热流科学与工程西步文通大堂E教育部重点实验室6.4ApproximationsinSIMPLEalgorithm6.4.1CalculationprocedureofSiMPLEalgorithm6.4.2ApproximationsinSIMPLEalgorithm1.Inconsistenacy（不一致性）ofinitialfieldassumptions2.Overestimating（夸大）theeffectsofpressurecorrection of neighboring nodes6.4.3NumericalexampleΦCFD-NHT-EHT2/41CENTER

2/41 6.4.1 Calculation procedure of SIMPLE algorithm 6.4.2 Approximations in SIMPLE algorithm 6.4.3 Numerical example 1.Inconsistenacy (不一致性) of initial field assumptions 2.Overestimating (夸大) the effects of pressure correction of neighboring nodes 6.4 Approximations in SIMPLE algorithm

热流科学与工程西步文源大堂E教育部重点实验室6.4ApproximationsinSIMPLEAlgorithm6.4.1CalculationprocedureofSIMPLEalgorithm1. Assuming initial velocity fields, u o and v o, to determinethe coefficients of the discretized momentum equations;2. Assuming an initial pressure field, p *;3. Solving the discretized momentum equation based on p *obtaining u *,y *;4. Solving pressure correction equation, obtaining p ';5. Revising pressure and velocities by p ': p =p *+α,pV=V,+V,=V,+d,Ap,u=u。+u=u.+dApCFD-NHT-EHTΦ3/41CENTER

3/41 6.4 Approximations in SIMPLE Algorithm 6.4.1 Calculation procedure of SIMPLE algorithm 1. Assuming initial velocity fields, u 0 and v 0 , to determine the coefficients of the discretized momentum equations； 2. Assuming an initial pressure field, p *； 3. Solving the discretized momentum equation based on p * , obtaining u *,v *; 4. Solving pressure correction equation, obtaining p ’； 5. Revising pressure and velocities by p ’ ：p = p *+ p ’  p * ' * ' e e e e e u u u u d p      * ' * ' n n n n n v v v v d p     

热流科学与工程西步文源大学G教育部重点实验空6a. Solving other scalar variables coupled with velocity ;6b. Starting next iteration with u = u。 + ue, v = v, + v,and p= p' +α,p as the solutions of the flow fieldat present iteration level.In the following discussion focus will be paid on the solutionof flow field, and Step 6a will be ignored. The entire solutionprocedure is composed of six stepsSIMPLE=Semi-implicit method for pressure-linkedequations(求解压力耦合问题的半隐方法)where“semi-implicit""refers to the neglect of velocity correctioneffects of the neighboring grids.ΦCFD-NHT-EHT4/41CENTER

4/41 6a. Solving other scalar variables coupled with velocity； In the following discussion focus will be paid on the solution of flow field, and Step 6a will be ignored. The entire solution procedure is composed of six steps. 6b. Starting next iteration with and as the solutions of the flow field at present iteration level. * ' e e u u u   ， * ' p p p p   * ' n n v v v   SIMPLE=Semi-implicit method for pressure-linked equations(求解压力耦合问题的半隐方法)- where “semi-implicit” refers to the neglect of velocity correction effects of the neighboring grids

热流科学与工程西步文源大学E教育部重点实验室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）ofinitialfieldassumptionsIn SIMPLEu,andp* are assumed independentlyActuallythereissomeinherent（固有的）relationshipbetweenvelocityandpressure,2.Overestimating（大)theeffectsofpressurecorrectionof the neighboring nodes. Because ue 'is caused by boththe pressure correction and velocity corrections of itsCFD-NHT-EHTΦ5/41CENTER

5/41 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

热流科学与工程西步文源大堂E教育部重点实验室neighboringnodes.The neglectofvelocity corrections oftheneighboringnodesattributes（归结于）thedrivingforceof utotallyto pressure correction,thus exaggerating（夸大)theaction of pressurecorrection6.4.3Numericalexample[Example 6-1(Text book))Known:Pw, Ps,ue, Vnuu.=0.7(pw-Pp)EV, = 0.6(ps - Pp)Ar=A=400Find: Pp,uw,V's中FD-NHT-EH6/41CENTER

6/41 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 

热流科学与工程西步文源大堂E教育部重点实验室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 followsau =anbunb +b+ A(pp - pe)Eanun +b+PA(p-pe)=u +d.(pp-pe)aedFor 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 Textbook中CFD-NHT-EHT7/41CENTER

7/41 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

热流科学与工程西步文源大学G教育部重点实验室6.5DiscussiononSIMPLEandConvergenceCriteriaofFlowFieldIteration6.5.1DiscussiononSIMPLEalgorithm1.Can the simplification approximations affect thecomputational results?2.Mathematically what type does the boundary conditionof thepressure correctionequation belong to?3.How to adopt the underrelaxation method in the flowfield iteration process?6.5.2ConvergencecriteriaofflowfielditerationCFD-NHT-EHTΦ8/41CENTER

8/41 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

热流科学与工程西步文源大堂E教育部重点实验室6.5DiscussiononSiMPLEandConvergenceCriteriaofFlowFieldIteration6.5.1DiscussiononSIMPLEalgorithm1.Can the two simplification 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 un, inu。 will gradually approach zero(趋近于O)withtheproceedingof iterationif itconverges!CFD-NHT-EHT中9/41CENTER

9/41 6.5 Discussion on SIMPLE and Convergence Criteria of Flow Field Iteration 6.5.1 Discussion on SIMPLE algorithm 1.Can the two simplification 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

热流科学与工程西步文源大学E教育部重点实验室2. What type 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,ap-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 anincompressible flow problem, becausepressure exists in theN-S equation in terms of gradient! This formulation can uniquelyU.u--lVp+Wdefine the flow fieldV.u=0 pThinking question: Why fortemperature not?No slip on the boundarySCFD-NHT-EI10/41CENTER

10/41 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?