Moukalled F., Mangani L., Darwish M. The finite volume method in computational fluid dynamics. An advanced introduction with OpenFOAM and Matlab (811443), страница 93
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This explicit treatment is justified by thesmall contribution to the convergence of the entire flow field by the iterative sweepsof the momentum equation. On the other hand, finding the correct solution for thepressure field represents the most important factor in the overall convergence.Based on this argument, the PRIME algorithm can be summarized as follows:15.7The SIMPLE Family of Algorithms625The momentum equation is solved explicitly to obtain a new velocity field v*usingh ivC ¼ HC vðnÞ DvC rpðnÞ þBvCCð15:176ÞThis velocity field is employed to derive the pressure correction equation. Thusdefining the correction fields such that0vC ¼ vC þ v C0ðnÞpC ¼ pC þ pCð15:177Þthe corrected field would satisfyh ivv0vðnÞ0v¼H½vDðrpÞþB¼H½vþvDrpþpþ BvCCCCCCCCCð15:178Þleading to the following expression relating velocity and pressure correction: hiv0 C ¼ HC v vðnÞ þ HC ½v0 DvC rp0Cð15:179ÞSubstituting Eq.
(15.179) and its correction into the continuity equation yieldsXf nbðCÞqf Df rp0f Sf ¼ Xf nbðCÞm_ f þiiX h hqf Hf v vðnÞ þ Hf ½v0 Sff nbðCÞð15:180Þwhere the underlined terms in Eqs. (15.179) and (15.180) are neglected. The terms neglected in PRIME HC v vðnÞ þ HC ½v0 can become smallerthan the term neglected in SIMPLE ðHC ½v0 Þ if HC ½v0 and HC v vðnÞ are ofopposite signs. It is worth noting that HC ½v0 ¼ HC ½v v is a correction to satisfycontinuity, while HC v vðnÞ is a correction to satisfy momentum. Usually thecorrector added to satisfy momentum is opposite to that added to satisfy continuity and hence, the neglected term HC v vðnÞ þ HC ½v0 is smaller.
Moreover, sincethe momentum equations are explicitly solved, no under-relaxation is required. Thishas the advantage of increasing the stability of the algorithm.15.7.3 The PISO AlgorithmIn the PISO algorithm [18, 25], the HC ½v0 term is accounted for as part of acorrection procedure composed of two or more steps. The first step is similar to the62615 Fluid Flow Computation: Incompressible FlowsSIMPLE algorithm where v0 is computed from Eq. (15.83) while neglecting HC ½v0 .The continuity satisfying velocity v** and pressure p* fields are used to recalculatethe coefficients of the momentum equation and then to solve it explicitly.
The newvelocity field v*** is used to calculate the mass flow rate field m_ at the elementfaces using the Rhie-Chow interpolation. Then, HC ½v0 is partially recovered in asecond corrector step where the velocity correction is written as00v¼ vCC þv Cv 00¼ HC ½v ðDC Þ ðrp ÞC þ v C 0v ¼ HC ½v þ v ðDC Þ ðrp ÞC þ v00 C 0v 00¼ HC ½v HC ½v ðDC Þ ðrp ÞC þ v Cv v000¼ HC ½v ðDC Þ ðrp ÞC HC DC ðrp ÞC þ v C|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}ð15:181ÞvC v000 vC þ v C HC DC ðrp ÞCIn Eq. (15.181) the underlined term represents the portion of the HC ½v0 that isrecovered by the second corrector step. The second velocity correction satisfies00v 00v00 C ¼ HC ½v ðDC Þ ðrp ÞCð15:182ÞUsing Eq.
(15.181) with the Rhie-Chow interpolation between points C and F, anew pressure-correction field is obtained asXf nbðCÞqf Df rp00f Sf ¼ Xf nbðCÞm_ f þX qf Hf ½v00 Sfð15:183Þf nbðCÞwhere the underlined terms in Eqs. (15.182) and (15.183) are again neglected. Thiscorrector step may be repeated as many times as desired, each time recovering anew additional portion of HC ½v0 .By following the sequence of events, it can be easily seen that PISO may beconsidered as a combination of one SIMPLE step and one or more PRIME steps,hence combining the implicitness of the SIMPLE algorithm with the stability of thePRIME algorithm. The sequence of events in the collocated PISO algorithm can besummarized as follows:1.
To compute the solution at time t + Δt, use as an initial guess the solution at timet for pressure, velocity, and mass flow rate fields p(n), u(n), and m_ ðnÞ , respectively.SIMPLE Step2. Solve implicitly the momentum equation given by Eq. (15.70) to obtain a newvelocity field v*.3. Update the mass flow rate at the cell faces using the Rhie-Chow interpolationtechnique (Eq. 15.100) to obtain a momentum satisfying mass flow rate field m_ f .15.7The SIMPLE Family of Algorithms627set initial guess m(f ) ,v ( ) ,and p( ) at timennnt + t to converged values at timetassemble and solveimplicitlymomentumequation for v *compute m*f using theRhie Chow interpolationassemble and solve pressurecorrectionequation for pSIMPLEiterationcorrect m*f ,v * ,and p( ) to obtainn**f**m ,v ,and p*assemble and solveexplicitlymomentumequation for v ***repeatrepeatcompute m***using thefrepeatRhie ChowinterpolationPRIMEassemble and solve pressureiterationcorrectionequation for pcorrect m***,v *** ,and p* tofobtain m****,v **** ,and p**fnumber of corrector steps exceeded ?yessetnomf = mfv ** = v ****p* = p**set m(f ) = m****,v ( ) = v **** , p( ) = p**fnnnonconverged ?yesset solution at time t + t tobeequal tothe converged solutionadvanceintimeset t = t + tnotimeexceeded ?yesstopFig.
15.27 A flow chart of the PISO algorithm62815 Fluid Flow Computation: Incompressible Flows4. Using the new mass flow rates, assemble the pressure correction equation(Eq. 15.98) and solve it to obtain a pressure correction field p′.5. Update the pressure and velocity fields at the cell centroids and the mass flowrate at the cell faces to obtain continuity-satisfying fields using Eq. (15.101).PRIME Step(s)6.
Using the latest available velocity and pressure fields, calculate the coefficientsof the momentum equation and solve it explicitly.7. Update the mass flow rate at the cell faces using the Rhie-Chow interpolationtechnique.8. Using the new mass flow rates, assemble the pressure correction equation(Eq. 15.183) and solve it to obtain a pressure correction field.9. Update pressure, velocity, and mass flow rate fields using expressions similar tothe ones given in Eq.
(15.101).10. Go to step 6 and repeat based on the desired number of corrector steps.11. Set the initial guess for velocity, mass flow rate, and pressure as u**, m_ ,and p*.12. Go back to step 2 and repeat until convergence.13. Set the solution at time t + Δt to be equal to the converged solution and set thecurrent time t + Δt to be t.14.
Advance to the next time step.15. Go back to step 1 and repeat until the last time step is reached.A flowchart of the PISO algorithm is presented in Fig. 15.27.15.8 Optimum Under-Relaxation Factor Values for v and p0To promote convergence in the SIMPLE algorithm the momentum and continuityequations are under relaxed using the under relaxation factors λv and λp, respectively. An important task is to find the under relaxation values that will result in theoptimum convergence rate.
Recalling that the velocity correction is obtainedwithout any under relaxation fromv0 C ¼ DC ðrp0 ÞCð15:184ÞMoreover, in calculating the pressure field, the pressure correction is underrelaxed in order for the velocity correction field given by Eq.
(15.184) to satisfy theexact velocity correction equation given byv0 C ¼ HC ½v0 kp DC ðrp0 ÞCð15:185Þ15.8Optimum Under-Relaxation Factor Values for v and p0629Equating Eqs. (15.184) and (15.185), an expression for λp is obtained asHC ½v0 DC ðrp0 ÞC ¼ HC ½v0 kp DC ðrp0 ÞC ) kp ¼ 1 þ 0vCXv 0aF v F¼1þð15:186ÞF NBðCÞavC v0 CThe SIMPLEC algorithm eliminated the need to under relax pressure correctionand resulted in the optimum acceleration rate. Therefore, using the approximationintroduced in SIMPLEC, the velocity correction at C can be written as the weightedaverage of the velocity corrections at the neighboring grid points such thatXavF v0 Fv0 C F NBðCÞXð15:187ÞavFF NBðCÞFrom Eqs.
(15.70)–(15.73) the coefficient avC can be expressed as01XX1avF þavC ¼ v @avC m_ f AkFNBðCÞf nbðCÞð15:188Þwhich in the limit of a steady state solution (the case for which under relaxation isused, since for an unsteady situation the time step plays the role of the underrelaxation factor) reduces toavC ¼ 1 X vakv FNBðCÞ Fð15:189ÞSubstituting Eq. (15.189) in Eq. (15.187), the velocity correction is approximated asXXavF v0 FavF v0 Fv0 C FNBðCÞkv avC) avC v0 C FNBðCÞvkð15:190Þsubstituting Eq.
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