案例40.流體力學(xué)numerical methods for heat,年-peakchina l

1、Lecture 30: The SIMPLE Algorithm (Contd)Last Time Looked into problem of introducing pressure into continuity equation for pressible flowsIntroduced SIMPLE algorithmDerived the pressure correction equationThis Time Look at the SIMPLE algorithm in detailExamine auxilliary issuesUnder-relaxation and c

2、onvergenceBoundary conditionsNature of pressure in pressible flowsSIMPLE AlgorithmSemi-Implicit Method for Pressure-Linked EquationsProposed by Patankar and Spalding (1972)Idea is to start with discrete continuity equationSubstitute into it the discrete u and v momentum equationsDiscrete momentum eq

3、uations contain pressure differencesHence get an equation for the discrete pressuresSIMPLE actually solves for a related quantity called the pressure correctionSIMPLE Algorithm Solve momentum equations with guessed pressure field p* - resulting velocity fields are u* and v*Do not satisfy continuity

4、because p* is wrongPropose corrections to velocities and pressure so that corrected velocities satisfy discrete continuityLet the corrected values be: Velocity correctionsPressure correctionSIMPLE Algorithm (Contd)Also require that corrected velocities and pressures satisfy momentum equations:Subtra

5、cting starred momentum equations from above:Velocity Correction EquationMake an approximation:Dropped Velocity Correction EquationsDefine:so thatandPressure Correction EquationStarred velocities do not satisfy discrete continuity equation:However, corrected velocities do:Pressure Correction Equation

6、 (Contd)Substituting for flow rate corrections:Collect terms in pressure correction p to create pressure correction equation Discrete Pressure Correction EquationScarborough criterion satisfied in the equalityb term is the amount by which the starred velocities do not satisfy continuitySIMPLE Soluti

7、on Loop1. Guess velocities and pressure p*2. Discretize and solve u momentum equation to obtain u* using p* for pressure term3. Discretize and solve v momentum equation to obtain v* using p* for pressure term4. Formulate p equation coefficients. In particular, find b term in p equation using u* and

8、v*:SIMPLE Solution Loop5. Solve p equation to obtain the pressure correction at all main control volume cell centroids6. Correct velocity and pressure:7. At this point, velocities satisfy continuity but not momentum8. Solve for other s9. Check for convergence. If converged, exit. If not, go to 2.Dis

9、cussionPressure correction equation nudges velocity and pressure fields into satisfying both continuity and momentum equations through a set of continuity-satisfying fieldsAt step 7, corrected velocities (u,v) satisfy discrete continuity exactly every single iterationHowever they dont satisfy moment

10、umNote how continuity-satisfying velocity fields are used to solve for s in step 8If this was not done, we wouldnt get bounded during iteration even with UDS! Effect of ApproximationDropping in deriving p equation does not change the final answer At convergence u and v are zero Similarly, p es a con

11、stant Can choose arbitrarily to be zero for all-velocity bcThus, approximations to primed equations cannot change the converged solutionApproximations can change rate of convergence, though Under-RelaxationIn reality, velocity correction consists of two parts:Dropping places the entire burden of vel

12、ocity correction on the pressure correctionVelocity partPressure partUnder-Relaxation (Contd)Corrected velocities always satisfy continuity, regardless of approximationHowever, large pressure correction yields poor pressure iteratesUnder-relax pressure correction in correcting p*:Do not under-relax

13、velocity correction or else corrected velocities will not satisfy continuity !Under-Relaxation (Contd)That is, do not use:Because of non-linearity, it is necessary to under-relax momentum equations:Nature of Pressure in pressible FlowsConsider domain with all-velocity bc:WallWallInflowOutflowPressur

14、e does not appear except as gradientAbsolute value of pressure does not matterOnly differences of pressure are meaningfulPressure in pressible Flows (Contd)What about pressure boundary conditions?WallWallPinPoutSay Pin =100, Pout =50.Compute velocity fieldWould velocity field change if Pin =200, Pou

15、t =150 ?Pressure in pressible FlowsWhat about mixed conditions?WallWallPoutVin Say Vin =10, Pout =10Would computed velocity be different if Vin =10, Pout =100 ?DiscussionPressure does not change densityAbsolute level of pressure does not matterOnly pressure differences matter in pressible flowsWhen

16、all bc are velocity bc, pressure level is indeterminatep and p+c are solutionsWhen at least one pressure bc is presentPressure level is fixed (not indeterminate)But only differences of pressure determine solutionChanging pressure bc while keeping pressure differences the same does not change velocity solution ClosureIn this lecture:We presented the SIMPLE solution loopFound that the