Moukalled F., Mangani L., Darwish M. The finite volume method in computational fluid dynamics. An advanced introduction with OpenFOAM and Matlab (811443), страница 92
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15.25), m_ 0b is zero and is simplydropped from the pressure correction equation with no modifications required forthe coefficients of the boundary elements. By setting m_ 0b to zero in Eq. (15.159), thepressure correction (or pressure) at the boundary is set equal to the pressure correction (or pressure) at the boundary cell centroid.Fully Developed Outlet FlowFor a fully developed flow, the velocity at the outlet is assumed to be known andcomputed from the zero normal gradient.
This means that m_ b at the outlet is known.Therefore no correction is needed and m_ 0b is set to zero. However, as the boundarypressure is unknown, it is extrapolated from the interior pressure field. Since thevelocity is iteratively updated, the above treatment does not guarantee overallconservation except at convergence. It is customary with incompressible flows toovercome this issue and to enforce global mass conservation at any iteration bymodifying m_ b at the boundary to satisfy overall mass conservation.This is done byPm_ in . Then based oncalculating the total mass flow rate entering the domainthe calculated mass flow rates at outlet boundaries, the total mass flow rate leaving15.7The SIMPLE Family of Algorithmsthe domainaccording toP621m_ out is computed.
The mass flow rate at an outlet is adjustedm_ outPm_ inm_ out Pm_ outð15:169ÞTo be able to apply the above treatment, the outlet should be placed far awayfrom any recirculation zone.15.6.2.4 Symmetry Boundary ConditionThe mass flow rate across a symmetry line (Fig.
15.26) is zero and as such itscorrection is set to zero, i.e., m_ 0b ¼ 0. Thus, similar to a wall boundary condition, themass flow rate correction term is simply dropped from the pressure-correctionequation. The pressure at the boundary is extrapolated from the internal pressurefield using Eq. (15.127) or Eq. (15.133) or a low order extrapolation profile assummarized in Eq. (15.160).15.6.2.5 The Relative Nature of PressureFor incompressible flow problems in which the normal velocities are prescribed onall boundaries, a difficulty arises due to the relative nature of pressure.
In suchcases, since only the pressure gradient appears in the momentum equation, there isno means for determining the absolute level of pressure, and only pressure differences have physical meaning. This indeterminacy of the pressure level leads to asingular coefficient matrix A and the direct solution of the system A/ ¼ b fails. Thesingularity is easily removed by simply setting the pressure at one point in thedomain to a prescribed value.
The remaining pressures are then calculated relativeto this value.15.7 The SIMPLE Family of AlgorithmsIn the SIMPLE algorithm [13], velocity and pressure are treated in a segregated(sequential) manner, with the pressure field computed by deriving a pressure correction equation that exploits the discrete momentum equation to replace thevelocity field in the continuity equation with a pressure term. In the derivation, avelocity correction term, Hf ½v0 , was neglected as retaining it renders the equationunmanageable.Discarding this term does not affect the final solution since its value is zero atconvergence.
Rather, it affects the path to convergence because during the initial62215 Fluid Flow Computation: Incompressible Flowsiterations its value can be significant. This large value may either cause divergenceor slow down the rate of convergence as a result of an overestimated pressurecorrection field. To counterbalance that, in SIMPLE the pressure is under relaxedby computing its value using p ¼ p þ kp p0 , where kp is the pressure underrelaxation factor. For optimum convergence, kp is usually set equal to ð1kv Þ,where kv is the momentum under relaxation factor, with more information on thisprovided later.Despite the use of under relaxation, the rate of convergence of the SIMPLEalgorithm remains problem dependent and researchers sought alternatives for further improvements.
Their effort culminated in the development of a SIMPLE-likefamily of algorithms such as SIMPLEC [17], SIMPLER [3], PISO [18], SIMPLEX[5], PRIME [19], SIMPLEM [20], and SIMPLEST [21]. Moukalled and Darwish[22] unified the formulation of these algorithms for both incompressible andcompressible flows while Darwish et al.
[23] and Jang et al. [24] assessed theirperformance. It is not the intention here to give a full account of these algorithms,rather, attention will be focussed on the two most popular variants, which are theSIMPLEC (SIMPLE Consistent) algorithm of Van Doormal and Raithby and thePISO (Pressure-Implicit Split Operator) algorithm of Issa. These two algorithmspresent two different approaches for dealing with the Hf ½v0 term. In SIMPLEC thevelocity correction at the main grid point is approximated as the weighted averageof the velocity corrections at the neighboring locations altering the term Hf ½v0 into~ f ½v0 , of smaller magnitude, which is then neglected.
In the PISOa modified one, Halgorithm, the Hf ½v0 term is accounted for as part of the split operator approach. Inall other algorithms, the Hf ½v0 term is neglected as in SIMPLE and modificationsare introduced either to the momentum equations or the Dv operator. Because thePISO algorithm is equivalent to one step of the SIMPLE algorithm and one or moresteps of the PRIME algorithm, the latter is also detailed.In the PRIME [19] algorithm the momentum equation is solved explicitly. Thisexplicit treatment of the momentum equation is justified by its small contribution tothe convergence of the entire flow field.
On the other hand, finding the correctsolution for the pressure field represents the most important factor impacting theoverall convergence.In SIMPLEST [21], the coefficients in the momentum equation are separatedinto their convection and diffusion parts with the convection terms treated explicitlyand the diffusion terms implicitly, thus affecting Dv and H. The justification for theexplicit treatment of convection is based on the similarity between the propagationof disturbances at a finite speed without any change in magnitude in a pure convection situation, and the propagation of error, from a particular point to theneighboring grid points, in a single iteration of explicit iterative methods. While theimplicit treatment of diffusion is argued based on the similarity between thepropagation of disturbances in a pure diffusion situation instantaneously in alldirections with rapid decay in their amplitude and the reduction of errors throughoutthe entire solution domain, in a single iteration, by implicit solution methods.15.7The SIMPLE Family of Algorithms623In SIMPLEM (SIMPLE-Modified) [20], the pressure-correction equation issolved before the momentum equation implying that the pressure field is computedusing the old velocity field.
This results in better pressure corrections than velocitycorrections and interchange the disadvantages and advantages of the SIMPLEalgorithm.In SIMPLER (SIMPLE-Revised) [3], an additional equation is developed fromwhich the pressure is directly calculated while the SIMPLE-like pressure-correctionequation is used to update the velocity field. The reason for a separate pressureequation being that, once the velocity field is updated using the predicted pressurecorrection field, it no longer satisfies the momentum equations. Therefore, thepressure should be calculated from another equation to match the velocities, so thatthe momentum equations are also satisfied.The SIMPLEX algorithm [5] was developed with the aim of ensuring that therate of convergence will not degrade with grid refinement.
It differs from SIMPLEin the way the Dv field is computed. This is done by using an additional set ofequations, which is developed and solved based on the assumption that the influence of the spatial distribution of pressure difference changes little with gridrefinement. Therefore, if the pressure difference influence is restricted to a cell, itwould be appropriate to assume that, by extrapolation, the pressure difference at themain grid point adequately represents the pressure differences at the element faces.Though all of the above algorithms were originally derived for a segregated grid,they are applicable within a collocated grid framework.15.7.1 The SIMPLEC AlgorithmThe SIMPLEC (SIMPLE-Consistent) [17] algorithm is a modified version of theSIMPLE algorithm derived by simply assuming that the velocity correction at pointC is the weighted average of the corrections at the neighboring grid points.Mathematically this is expressed byXv0 C avF v0 FFNBðCÞXXavF v0 F v0 CXavFð15:170ÞX av v0 FX avFF v0 C) HC ½v0 v0 C HC ½1vaCavFNBðCÞFNBðCÞ Cð15:171ÞavF)FNBðCÞFNBðCÞFNBðCÞand using the H operator, Eq.
(15.170) can be written as62415 Fluid Flow Computation: Incompressible FlowsTherefore instead of neglecting the HC ½v0 term as in SIMPLE, it is replaced bythe approximate value given by the above equation. With this approximation thevelocity correction given by Eq. (15.85) becomes~ v ðrp0 Þð1 þ HC ½1Þv0 C ¼ DvC ðrp0 ÞC ) v0 C ¼ DCCð15:172ÞEquation (15.172) can then be used to derive the pressure correction equation.Theresult can be achieved by adding and subtracting the termP sameavF vC from the momentum equation obtained by combining Eqs. (15.76)F NBðCÞand (15.77), leading to the following modified equation:0@a v þCXFNBðCÞ1avF AvC þX^avF ðvF vC Þ ¼ VC ðrpÞC þ bCvð15:173ÞFNBðCÞwhich, in turn, can be written as~ C ½v vC ¼ D~ ðrpÞ þ B~vC þ HCCCvvð15:174ÞBy using Eq. (15.174), the velocity correction equation becomes~ C ½v0 v0 C D~ ðrp0 Þv0 C ¼ HCCvð15:175Þ~ C ½v0 v0 C is dropped, which is equivalent to the approximationThen the term Hgiven by Eq.
(15.171), and the modified velocity correction is used in deriving thepressure correction equation.Due to a better estimate in SIMPLEC (i.e., a smaller term is dropped), therelaxation of pressure becomes unnecessary and as compared to SIMPLE, theresulting velocity corrections will satisfy better the momentum equations.Consequently, a higher rate of convergence is obtained. Thus, with the exceptions~ C ½v0 v0 C rather than HC ½v0 and replacing Dv by D~ v , the stepsof dropping HCCinvolved in the SIMPLEC algorithm are similar to those of the SIMPLE algorithm.15.7.2 The PRIME AlgorithmIn the PRIME (PRessure Implicit Momentum Explicit) [19] algorithm, themomentum equation is solved explicitly.
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