Moukalled F., Mangani L., Darwish M. The finite volume method in computational fluid dynamics. An advanced introduction with OpenFOAM and Matlab (811443), страница 97
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15.35 One-dimensional Forchheimer model for flow in porous mediaExercise 6Compute the interface velocities ue and uw using the Rhie-Chow interpolation andcompare it to the averaged values ue and uw knowing the following data (Fig. 15.36):pWW ¼ 10; pW ¼ 12; pC ¼ 16; pE ¼ 24; pEE ¼ 40; anduW ¼ 5; uC ¼ 10; uE ¼ 40:uWWpWWuWpWuCwFig. 15.36 A one dimensional collocated gridpCuEepEuEEpEEReferences653Exercise 7In OpenFOAM® develop a SIMPLEC pressure correction algorithm by modifyingthe SIMPLE algorithm described in this chapter.
Hint: in order to find the summation of the extra diagonal coefficients use the H1() function of the fvMatrix.Exercise 8Check the pisoFoam solver located in $FOAM_SRC/../applications/solvers/incompressible/pisoFoam/pisoFoam.C and compare it with the algorithm described in this chapter, i.e., “The Collocated PISO Algorithm”.
Find out the inconsistency with the standard OpenFOAM® implementation.Exercise 9Develop a pressure correction PISO algorithm for OpenFOAM®.References1. Patankar SV (1981) A calculation procedure for two dimensional elliptic situations. NumerHeat Transfer 4(4):409–4252. Patankar SV (1980) Numerical heat transfer and fluid flow. Hemisphere, NY3. Patankar SV, Spalding DB (1972) A calculation procedure for heat, mass and momentumtransfer in three-dimensional parabolic flows. Int J Heat Mass Transf 15(10):1787–18064. Harlow FH, Welch JE (1965) Numerical calculation of time-dependent viscous incompressibleflow of fluid with free surface.
Phys Fluids 8(12):2182–21895. Van Doormaal JP, Raithby GD (1985) An evaluation of the segregated approach for predictingincompressible fluid flows. ASME Paper 85-HT-9, Presented at the national heat transferconference, Denver, Colorado6. Raithby GD, Schneider GE (1979) Numerical solution of problems in incompressible fluidflow: treatment of the velocity-pressure coupling. Numer Heat Transfer, Part A 2(4):417–4407.
Patankar SV (1975) Numerical prediction of three-dimensional flow. In Launder BE(ed) studies in convection: theory, measurement, and application, vol 1. Academic, New York,pp 1–98. Rhie CM, Chow WL (1983) Numerical study of the turbulent flow past an airfoil with trailingedge separation. AIAA J 21:1525–15329. Rhie CM (1988) A three-dimensional passage flow analysis method aimed at centrifugalimpellers. Comput Fluids 13:443–46010.
Majumdar S (1988) Role of under relaxation in momentum interpolation for calculation offlow with nonstaggered grids. Numer Heat Transfer 13:125–13211. Miller TF, Schmidt FW (1988) Use of a pressure-weighted interpolation method for thesolution of incompressible Navier-Stokes equations on a nonstaggered grid system. NumerHeat Transfer 14:213–23312. Karki KC, Patankar SV (1988) Calculation procedure for viscous incompressible flows incomplex geometries.
Numer Heat Transfer 14:295–30713. Choi SK, Nam HY, Cho M (1993) Use of the momentum interpolation method for numericalsolution of incompressible flows in complex geometries: choosing cell face velocities. NumerHeat Transfer, Part B 23:21–4114. Choi SK, Nam HY, Lee YB, Cho M (1993) An efficient three-dimensional calculationprocedure for incompressible flows in complex geometries. Numer Heat Transfer, Part B23:387–40065415 Fluid Flow Computation: Incompressible Flows15.
Choi SK, Nam HY, Cho M (1994) Use of staggered and nonstaggered grid arrangements forincompressible flow calculations on nonorthogonal grids. Numer Heat Transfer, Part B 25(2):193–20416. Choi SK, Nam HY, Cho M (1994) Systematic comparison of finite-volume calculationmethods with staggered and nonstaggered grid arrangements. Numer Heat Transfer, Part B 25(2):205–22117. Van Doormaal JP, Raithby GD (1984) Enhancement of the SIMPLE method for predictingincompressible fluid flows. Numer Heat Transfer 7:147–16318. Issa RI (1982) Solution of the implicit discretized fluid flow equations by operator splitting.Mechanical Engineering Report, FS/82/15, Imperial College, London19.
Maliska CR, Raithby GD (1983) Calculating 3-D fluid flows using non-orthogonal grid. In:Proceedings of the third international conference on numerical methods in laminar andturbulent flows, Seattle, pp 656–66620. Acharya S, Moukalled F (1989) Improvements to incompressible flow calculation on anon-staggered curvilinear grid. Numer Heat Transfer, Part B 15:131–15221. Spalding DB (1980) Mathematical modelling of fluid mechanics, heat transfer and masstransfer processes.
Mechanical Engineering Department Report HTS/80/1, Imperial College ofScience, Technology and Medicine, London22. Moukalled F, Darwish M (2000) A unified formulation of the segregated class of algorithmsfor fluid flow at all speeds. Numer Heat Transfer, Part B 37:103–13923. Darwish M, Asmar D, Moukalled F (2004) A comparative assessment within a multigridenvironment of segregated pressure-based algorithms for fluid flow at all speeds. Numer HeatTransfer, Part B 45(1):49–7424. Jang DS, Jetli R, Acharya S (1986) Comparison of the PISO, SIMPLER and SIMPLECalgorithms for the treatment of the pressure-velocity coupling in steady flow problems. NumerHeat Transfer 10:209–22825. Yen RH, Liu CH (1993) Enhancement of the SIMPLE algorithm by an additional explicitcorrector step.
Numer Heat Transfer, Part B 24:127–14126. Mecinger J (2012) An alternative finite volume discretization of body force field on collocatedgrids. In: Petrova R (ed) Finite volume method-powerful means of engineering design.ISBN:978-953-51-0445-227. OpenFOAM, 2015 Version 2.3.x. http://www.openfoam.orgChapter 16Fluid Flow Computation: CompressibleFlowsAbstract The previous chapter presented the methodology for solving incompressible flow problem using pressure based algorithms. In this chapter thesealgorithms are extended to allow for the simulation of compressible flows in thevarious Mach number regimes, i.e., over the entire spectrum from subsonic tohypersonic speeds.
While incompressible flow solutions do not generally requiresolving the energy equation, compressibility effects couple hydrodynamics andthermodynamics necessitating the simultaneous solution of the continuity,momentum, and energy equations. The dependence of density on pressure andtemperature, a relation expressed via an equation of state, further complicates thevelocity-pressure coupling present in incompressible flows. The derivation of thepressure correction equation now involves a density correction that introduces tothe equation a convection-like term, in addition to the diffusion-like term introducedby the velocity correction.
Another difficulty is introduced by the complexboundary conditions that arise in compressible flow problems. Details on resolvingall these issues are presented throughout this chapter.16.1 HistoricalComputational Fluid Dynamics methods have been traditionally classified into twofamilies denoted by density-based and pressure-based. Specifically density-basedmethods have historically dominated the simulation of transonic and supersonicflows usually encountered in the aeronautics industry, and were well-establishedwhen the SIMPLE algorithm was first instigated.
SIMPLE, a pressure-basedmethod, was initially developed to address this shortcoming and was quite efficientin resolving incompressible and low Mach number flows.Early on, efforts were directed towards extending the operation of each of theseapproaches to flow regimes customarily dominated by the other. Harlow andAmsden [1, 2] were amongst the earliest to simulate fluid flow at all speeds. In theirwork, the use of pressure as a main variable in preference to density was presented an© Springer International Publishing Switzerland 2016F.
Moukalled et al., The Finite Volume Method in Computational Fluid Dynamics,Fluid Mechanics and Its Applications 113, DOI 10.1007/978-3-319-16874-6_1665565616Fluid Flow Computation: Compressible Flowsadvantage since its variations remained finite irrespective of the Mach number value.Nonetheless it was the work of Patankar [3] that provided a clear resolution to thisproblem, and allowed for SIMPLE-based methods to genuinely develop intomethods capable of resolving fluid flow at all speeds [4–13]. The critical development was the reformulation of the pressure equation to include density and velocitycorrection such that the type of the equation changed from purely elliptic forincompressible flows to hyperbolic in transonic and supersonic compressible flows[14, 15].
This allowed the SIMPLE-family of methods to seamlessly solve flowproblems across the entire Mach number spectrum, with pressure playing the dualrole of affecting density in the limit of high Mach compressible flow and velocity inthe limit of incompressible flow [16], in order to enforce mass conservation.16.2 IntroductionAn important advantage of the pressure-based approach is its ability to resolve fluidflows in the various Mach number regimes without any artificial treatment topromote convergence and stabilize computations. This ability of the pressure basedmethod is due to the dual role the pressure plays in compressible flows [17], whichcan best be described by considering the following two extreme cases:1. At very low Mach numbers, the pressure gradient needed to establish the flowfield through momentum conservation is so small that it does not significantlyinfluence the density, and the flow can be considered to be incompressible.Hence, density and pressure in addition to density and velocity are very weaklyrelated indicating that variations in density are not sensitive to variations invelocity.
In this case the continuity equation can no longer be considered as anequation for density, rather, it acts as a constraint on the velocity field.2. At hypersonic speeds, changes in velocity become relatively small as comparedto the magnitude of the velocity indicating that variations in pressure do significantly affect density. Consequently, the pressure now acts on density alonethrough the equation of state to satisfy mass conservation [16, 18] and thecontinuity equation can be viewed as the equation for density.The above limiting cases highlight the dual role played by pressure in compressible flow situations.
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