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Pressure velocity coupling

Published on: Mar 4, 2016
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Transcripts - Pressure velocity coupling

• 1. Pressure VelocityCoupling Arvind Deshpande
• 2. Semi-Implicit Method for PressureLinked Equations Patankar and Spalding - Guess and Correct procedure for calculation of pressure on staggered grid arrangement1. Initial guess for velocity and pressure field.2. Convective mass flux per unit area F is evaluated from guessed velocity components.3. Guessed pressure field is used to solve momentum equations to get velocity components.4. Values of velocity components are substituted in continuity equation to get a pressure correction equation.5. Values of pressure and velocity are updated.6. The process is iterated until convergence of pressure and velocity fields.4/11/2012 Arvind Deshpande(VJTI) 2
• 3. SIMPLE   aeue   anbunb  PP  PE Ae  be a v   a v  P  P A  b n n nb nb P N n n a u   a u  P  P A  b * e e * nb nb * P * E e e a v   a v  P  P A  b * n n * nb nb * P * N n n a u  u    a u  u   P  P   P  P A e e * e nb nb * nb P * P E * E e a v  v    a v  v   P  P   P  P A n n * n nb nb * nb P * P N * N n a u   a u  P  P A e e nb nb P E e a v   a v  P  P A n n nb nb P N n4/11/2012 Arvind Deshpande(VJTI) 3
• 4. Omit a u & a v nb nb nb nb Ae An de  & dn  ae an   ue  PP  PE d e vn  P  P d P N n ue  u  d P  P  * e e P E vn  v  d P  P  * n n P N uw  u  d P  P  * w w W P vs  v  d P  P  * s s S P Aw As dw  & ds  aw as4/11/2012 Arvind Deshpande(VJTI) 4
• 5. Continuity equationuA i 1, J   uAi , J   vAI , j 1  vAI , j  0  A u e e * e     d e PP  PE   w Aw u w  d w PW  PP *   A v n n * n  d P n P  P    A v N s s * s  ds P  P   0 S P dAe  dAw  dAn  dAs PP dAe PE  dAw PW  dAn PN  dAs PS u A  u A  v A  v A  * w * e * s * naP P  aE P  aW P  a N P  aS P  b P E W N S P 4/11/2012 Arvind Deshpande(VJTI) 5
• 6. aE  dAe aW  dAw a N  dAn aS  dAs aP  aE  aW  a N  aS      bP  u * A w  u * A e  v* A s  v* A n    PP  PP*  PP  ue  ue  d e PP  PE *  vn  vn  d n * P P P  N4/11/2012 Arvind Deshpande(VJTI) 6
• 7. Discussion of Pressure CorrectionEquation1. Omission of  a u &  a v nb nb nb nb2. Semi-Implicit3. Justification of omission4. Mass source is useful indicator of convergence5. Pressure correction equation is intermediate step to get correct pressure field4/11/2012 Arvind Deshpande(VJTI) 7
• 8. Under-relaxation Pressure correction equation is susceptible to divergence unless some under-relaxation factor is used during iterative process. αp, αu, αv,are under relaxation factors for pressure, u-velocity and v-velocity. u and v are corrected values without under relaxation and un-1 and vn-1 are values at the end of previous iteration. P new  P*   p P u new   u u  (1   u )u n 1 v new   v v  (1   v )v n 14/11/2012 Arvind Deshpande(VJTI) 8
• 9. Under-relaxation A correct choice of these factors is important for cost effective simulation. Large value of α leads to oscillatory behavior or even divergence and small value cause extremely slow convergence. There are no general rules for choosing the best value for α. Optimum values depends on nature of the problem, the number of grid points, grid spacing, and iterative procedures used. Suitable value of α can be found by experience and from exploratory computations for the given problem. Suggested values are 0.5 for α and 0.8 for αp X-momentum and Y-momentum equations are modified considering under-relaxation factors instead of applying under- relaxing velocity correction as velocity values are continuity satisfying.4/11/2012 Arvind Deshpande(VJTI) 9
• 10. Under-relaxation  aeue   anbunb  PP*  PE* Ae  be *  ue    anbunb  PP*  PE* Ae  be *  ae ue  u e   * *    anbunb  PP*  PE Ae  be * *   ue     ae   ue  u e   u  * * *    anbunb  PP*  PE Ae  be *   ue     ae   ae     a  * ue   anbunb  PP*  PE Ae  be  (1   ) e u e * *    an  * an  nb nb  P *   N  vn   a v  P  P An  bn  (1   )  v e * *   4/11/2012 Arvind Deshpande(VJTI) 10
• 11. SIMPLE algorithm1) Initial guess P*,u*,v*,φ*2) Solve discretized momentum equations and calculate u*,v* 1    * ue   anbunb  PP*  PE Ae  be   ae * * *      aeue 1    * vn   anbvnb  PP*  PN An  bn   an * * *   an u n   3) Solve pressure correction equation and calculate P’ aP PP  aW PW aE PE aS PS aN P N bP4) Correct Pressure and velocities PP  PP   P PP *  ue  u *  d e PP  PE e  vn  vn  d n * P P P  N4/11/2012 Arvind Deshpande(VJTI) 11
• 12. SIMPLE algorithm5) Solve all other discretized transport equations aPP  aW W  aEE  aSS  aNN  b6) Check for convergence. If converged, stop. Otherwise set P*  P, u*  u, v*  v,  *  7) Goto step 2 4/11/2012 Arvind Deshpande(VJTI) 12
• 13. SIMPLER (SIMPLE Revised) -Patankar Discretised continuity equation is used to derive discretised equation for pressure, instead of pressure correction equation as in simple. Pressure field is obtained without correction. Velocities are obtained through velocity corrections as in SIMPLE.4/11/2012 Arvind Deshpande(VJTI) 13
• 14. SIMPLER Algorithm ue  a u  be nb nb  Ae PP  PE   ae ae vn  a v  bn nb nb  An  PP  PN  an an u ^  a v  be nb nb e ae v ^  a v  bn nb nb n an  ue  ue^  d e PP  PE  vn  vn ^  A Pn P  PN 4/11/2012 Arvind Deshpande(VJTI) 14
• 15. Continuity equationuAe  uAw   vAn  vAs   0 A u e e ^     d e PP  PE   w Aw u w  d w PW  PP ^   A v P  P   0 e n n ^ n  dn P P  P    A v N s s ^ s  ds S PdAe  dAw  dAn  dAs PP dAe PE  dAw PW  dAn PN  dAs PS u A  u A  v A  v A  ^ w ^ e ^ s ^ naP PP  aW P  aE PE  a N PN  aS PS  bP W 4/11/2012 Arvind Deshpande(VJTI) 15
• 16. SIMPLER AlgorithmaE  dAeaW  dAwa N  dAnaS  dAsaP  aE  aW  a N  aS     bI , J  u ^ A w  u ^ A e  v ^ A s  v ^ A n  ue  ue *  d P  P  e p E vn  vn *  d P  P  n P N 4/11/2012 Arvind Deshpande(VJTI) 16
• 17. SIMPLER algorithm1) Initial guess P*,u*,v*,φ*2) Calculate pseudo velocities u^, v^ ue^  a u  be nb nb ae vn  ^ a v  be nb nb an3) Solve pressure equation and calculate Pressure at all points. aP PP  aW P  aE PE  aS PS  aN PN  bP W4) Set new value of P. P  PP * P4/11/2012 Arvind Deshpande(VJTI) 17
• 18. SIMPLER algorithm5) Solve discretized momentum equations and calculate u*,v* * * *  *  aeue   anbunb  PP  PE Ae  be an vn   anbvnb * *  P * P  P A * N n  bn6) Solve pressure correction equation and calculate P’ aP PP  aW PW aE PE aS PS aN P N bP7) Correct velocities  ue  ue  d e PP  PE *  vn  vn  d n * P P P  N4/11/2012 Arvind Deshpande(VJTI) 18
• 19. SIMPLER algorithm8) Solve all other discretized transport equations aPP  aW W  aEE  aSS  aNN  b9) Check for convergence. If converged, stop. Otherwise set P*  P, u*  u, v*  v,  *  10) Goto step 2 4/11/2012 Arvind Deshpande(VJTI) 19
• 20. SIMPLEC (SIMPLE Consistent)Algorithm Van Doormal and Raithby  ue  d e PP  PE  Ae Momentum equations de  are manipulated so that ae   anb velocity correction  vn  d n PP  PN  equations omit terms An that are less significant dn  than those omitted in an   anb SIMPLE.4/11/2012 Arvind Deshpande(VJTI) 20
• 21. PISO (Pressure Implicit with Splitingof Operators) - Issa Developed originally for non-iterative computation of unsteady compressible flows. Adapted for iterative solution of steady state problems. Involves one predictor and two corrector steps. Pressure correction equation is solved twice. Though the method implies considerable increase in computational efforts it has found to be efficient and fast. Extension of SIMPLE with a further correction step to enhance it.4/11/2012 Arvind Deshpande(VJTI) 21
• 22. PISO P  P  P ** * u **  u *  u v  v  v ** *  ue*  ue  d e PP  PE * *  v  v  dn ** n * n P P P N 4/11/2012 Arvind Deshpande(VJTI) 22
• 23. PISO  aeue*   anbunb  PP**  PE** Ae  be * *a v   a v  P  P A  b ** n n * nb nb ** P ** N n na u   a u  P  P A  b *** e e ** nb nb *** P *** E e ea v   a v  P  P A  b *** n n ** nb nb *** P *** N n na u  u    a u  u   P  P   P e *** e ** e nb ** nb * nb *** P ** P *** E  PE * Ae * a v  v    a v  v   P  P   P n *** n ** n nb ** nb * nb *** P ** P *** N  PN* * A nu u  ***  a u  u   d P  P  ** nb ** nb * nb e e e P E aev *** v  **   anb vnb  vnb ** *   d P  PN  n n n P an4/11/2012 Arvind Deshpande(VJTI) 23
• 24. PISO aP PP  aE PE  aW PW  a N PN  aS PS  bP aE  dAe aW  dAw a N  dAn aS  dAs aP  aE  aW  a N  aS  A   A    a    anb unb  unb   ** *    anb unb  unb   ** *   bP   w  a e   A   A    **    anb vnb  vnb   *    anb vnb  vnb  ** *    a  s  a n 4/11/2012 Arvind Deshpande(VJTI) 24
• 25. PISO algorithm1) Initial guess P*,u*,v*,φ*2) Solve discretized momentum equations and calculate u*,v* * * *  *  aeue   anbunb  PP  PE Ae  be an vn   anbvnb * *  P * P  P A * N n  bn3) Solve pressure correction equation and calculate P’ aP PP  aW PW aE PE aS PS aN P N bP4) Correct Pressure and velocities PP *  PP  PP * *  ue*  ue  d e PP  PE * *  vn*  vn * *  d P n P P  N4/11/2012 Arvind Deshpande(VJTI) 25
• 26. PISO algorithm5) Solve second pressure correction equation and calculate P’’ aP P P  aW P W aE P E aS P S aN P N b P6) Correct Pressure and velocities again. PP***  PP  PP  PP * u ***   u  de P  P  *   **  anb unb  unb *   d P  PE  e e P E e P ae v *** *   v  d e P  PN   a v nb ** nb  vnb *   d P  PN  n n P n P an7) Set P = P***, u = u***, v = v***4/11/2012 Arvind Deshpande(VJTI) 26
• 27. PISO algorithm8) Solve all other discretized transport equations aI , J  I , J  aI 1, J  I 1, J aI 1, J  I 1, J aI , J 1 I , J 1 aI , J 1 I , J 1 bI , J9) Check for convergence. If converged, stop. Otherwise set P*  P, u*  u, v*  v,  *   10) Goto step 2 4/11/2012 Arvind Deshpande(VJTI) 27
• 28. General Comments Performance of each algorithm depends on flow conditions, the degree of coupling between the momentum equation and scalar equations, amount of under relaxation and sometimes even on the details of the numerical techniques used for solving the algebraic equations. SIMPLE algorithm is straightforward and has been successfully implemented in numerous CFD procedures. In SIMPLE, pressure correction P’ is satisfactory for correcting velocities, but not so good for correcting pressure. SIMPLER uses pressure correction for calculating velocity correction only. A separate pressure equation is solved to calculate the pressure field. Since no terms are omitted to derive the discretised pressure equation in SIMPLER, the resulting pressure field corresponds to velocity field. The method is effective in calculating the pressure field correctly. This has significant advantages when solving the momentum equations.4/11/2012 Arvind Deshpande(VJTI) 28
• 29. General Comments Although calculations are more in SIMPLER, convergence is faster and effectively computer time reduces. SIMPLEC and PISO have proved to be as efficient as SIMPLER in certain types of flows. When momentum equations are not coupled to a scalar variable, PISO algorithm showed robust convergence and required less computational efforts than SIMPLER and SIMPLEC. When scalar variables were closely linked to velocities, PISO had no significant advantage over other methods. Iterative methods using SIMPLER and SIMPLEC have robust convergence behavior in strongly coupled problems. It is still unclear which of the SIMPLE variant is the best for general purpose computation.4/11/2012 Arvind Deshpande(VJTI) 29