Moukalled F., Mangani L., Darwish M. The finite volume method in computational fluid dynamics. An advanced introduction with OpenFOAM and Matlab (811443), страница 84
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Then apply under-relaxation to the discretized equation using(i) Patankar’s under-relaxation method(ii) The E-Factor relaxation method(iii) The false transient method55614Discretization of the Source Term, Relaxation, and Other DetailsExercise 4In the solution of turbulent flows (which will be the subject of Chap. 17), turbulencemodels are introduced. One such models is the well-known two-equation k eturbulence model, with k and e governed by conservation equations that have theform of the general conservation equation. The e equation in the model has thefollowing source term:ee2QeC ¼ Ce1 Pk Ce2 qkkwhere q is the fluid density, Ce1 and Ce2 are positive constants, and Pk , k, and e areall positive quantities.
Suggest a linearization for QeC .Exercise 5In solving the momentum equation in rhZ coordinates, the momentum equation inthe h direction has the following source term:QvCh ¼ qvr vhrSuggest a linearization for QvCh .Exercise 6The Fithigh-Nagumo equations, presented below, model the evolution of animalcoat pattern formation. The equations represent concentrations of two chemicalsubstances influencing skin pigmentation whose reaction-diffusion interaction resultin the formation of patterns that are reminiscent of the zebra or jaguars skins.
Thevariations in the resulting patterns depend on the constants used in the model. Themodel equations are given by@/¼ r ar/ þ / /3 u þ k@t@u¼ r bru þ / us@tSolve these equations using uFVM and OpenFOAM® for the following valuesof the constants:a ¼ 2:8 104b ¼ 5 103s ¼ 0:1k ¼ 0:005Let the computational domain be a square of dimension [4, 4] discretized with amesh of size 100 100 elements. As boundary conditions, use a zero gradient overall boundaries with a random initial conditions for the / (values within [0, 1]) and14.7Exercises557initial conditions for u ¼ 1 /. Solve the problem over 5 s with a time stepDt 104 s, using the first order Euler Implicit Scheme and the second orderCrank-Nicholson scheme and compare results. Make sure that the source term islinearized.References1.
Patankar S (1980) Numerical heat transfer and fluid flow. McGraw Hill, New York2. Van Doormaal JP, Raithby GD (1984) Enhancement to the SIMPLE method for predictingincompressible fluid flows. Numer Heat Transf 7:147–1633. Mallinson GD, de Vahl Davis G (1973) The method of the false transient for the solution ofcoupled elliptic equations.
J Comput Phys 12(4):435–4614. Raithby GD, Schneider GE (1979) Numerical solution of problems in incompressible fluidflow: treatment of the velocity-pressure coupling. Numer Heat Transf 2:417–4405. OpenFOAM (2015) Version 2.3.x. http://www.openfoam.orgPart IIIAlgorithmsChapter 15Fluid Flow Computation: IncompressibleFlowsAbstract In previous chapters the procedure for discretizing and solving thegeneral transport equation for the variable / in the presence of a known velocityfield was formulated. In general, the velocity field is not known and has to becomputed by solving the set of Navier-Stokes equations. For incompressible flowsthis task is complicated by the strong coupling that exist between pressure andvelocity and by the fact that pressure does not appear as a primary variable in eitherthe momentum or continuity equations.
The focus of this chapter is on presenting amethod that addresses these two issues, and computes the flow field for incompressible fluid flows. This is accomplished initially on a one dimensional staggeredgrid, then on a collocated one dimensional grid and finally on a collocated threedimensional unstructured grid. In addition to fully deriving the SIMPLE,SIMPLEC, PRIME and PISO algorithms, the Rhie-Chow interpolation and itsextension to transient, relaxation and body force terms are clearly formulated.Finally, the implementation details for a number of frequently encounteredboundary conditions are presented.15.1 The Main DifficultyThe general conservation equation dealt with in previous chapters can be reformedinto an equation similar to the continuity and momentum equations.
Yet thenumerical techniques presented up till now are not enough to allow for the resolution of the Navier-Stokes equations. Solving general fluid flows requires analgorithm [1] that can deal with the pressure velocity coupling. To understand thisissue, the continuity and momentum equations are reproduced below.© 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_1556156215 Fluid Flow Computation: Incompressible Flows@qþ r ðqvÞ ¼ 0@tn hio@½qv þ r fqvvg ¼ rp þ r l rv þ ðrvÞT þ f b@tð15:1Þð15:2ÞThat Eqs. (15.1) and (15.2) are nonlinear is not by itself an unsurmountabledifficulty, since such a problem is usually handled by adopting an iterativeapproach. Moreover, Eq. (15.2) is a vector equation, which when written in terms ofits components results in a system of scalar equations that can be solved sequentially.
Furthermore, the stress tensor can be reformulated into a diffusion-like termand treated implicitly, with its second part (i.e., the transpose of the velocity gradient) evaluated explicitly based on previous iteration values and added to thesource.
The main issue that cannot be addressed directly with the numerics of thegeneral scalar equation, is the unavailability of an explicit equation for computingthe pressure field that appears in the momentum equation.A review of Eqs. (15.1) and (15.2) reveals that while the velocity field can becomputed using the momentum equation, the pressure field appearing in themomentum equation cannot be computed directly from the continuity equation.This strong yet implicit coupling can be made more evident by rewriting the set ofequations in a matrix form as fbvF BTAu ¼¼:ð15:3ÞP0B 0In this form, Eq. (15.3) shows a zero diagonal block in the system, which is acharacteristic of saddle point problems, indicating that it cannot sustain the solutionof the pressure and velocity fields by any iterative mean.
Consequently, an equationfor pressure is required and should be derived.One approach is to simply reformulate the system of momentum and continuityequations by decomposing matrix A into a lower (L) and an upper (U) triangularmatrices as F0F BTI F1 BTA¼¼ LUð15:4Þ¼B BF1 BTB 00Iwhere the term −BF−1BT is the Schur complement matrix.This is in essence the approach that needs to be followed in order to iterativelysolve the Navier-Stokes equations. This technique is embodied in the classicalsegregated SIMPLE (Semi Implicit Method for Pressure Linked Equations) algorithm of Patankar and Spalding [1–3].The solution procedure is based on reformulating the Navier-Stokes equations interms of a momentum and a pressure equation, which are then discretized andsolved sequentially. The pressure equation is constructed by combining the semidiscretized momentum and continuity equations (approximation of the Schurcomplement matrix).15.2A Preliminary Derivation563The algorithm is driven by a Picard type iterative procedure during which themomentum equation is solved using the pressure field of the previous iteration.
Theresulting velocity field conserves momentum but not necessarily mass. Thisvelocity field is then used to construct the pressure equation whose solution is usedto correct both the pressure and velocity fields so as to enforce mass conservation.A new iteration is then started and the sequence is repeated until the velocity andpressure fields satisfy both mass and momentum conservation.This algorithm can be described in matrix form asD1 BTII0 vv¼ppð15:5Þfollowed by an update to the velocity field usingFB0BD1 BTvp¼fb0ð15:6Þwhere in Eqs. (15.5) and (15.6) F−1 is approximated by its inverse diagonal, D−1,and the superscript (*) refers to intermediate values at the current iteration.The steps required are summarized as follows:••••Solve: Fv* = fbSolve: −BD−1BTp* = −Bv*Update: v = v* − D−1BTp*Update: p = p*This kind of splitting is similar to that used in the SIMPLE family of algorithms,which is the subject of this chapter.15.2 A Preliminary DerivationThe difficulties faced in developing a solution algorithm for incompressible flowproblems will be highlighted by performing the discretization in a one dimensionalspace over the uniform grid displayed in Fig.
15.1. For simplicity, the flow isassumed to be steady. The simplified continuity and momentum equations (writtenin conservative form) are given by@ ðquÞ¼0@x@ ðquuÞ@@u@p¼l@x@x@x@xð15:7Þð15:8Þ56415 Fluid Flow Computation: Incompressible FlowsxCuWWuWWWuCwWuEeCuEEEEEyCxexwFig. 15.1 One dimensional domain15.2.1 Discretization of the Momentum EquationThe discretization of the momentum equation starts by integrating Eq. (15.8) overelement C shown in Fig. 15.1 to yieldZVC@ ðquuÞdV¼@xZVCZ@@u@pldVdV @x@x@xð15:9ÞVCThe volume integrals of the convection and diffusion terms in Eq.
(15.9) are thentransformed into surface integrals by invoking the divergence theorem to giveZZðquuÞdy ¼@VC@VC@ul dy @xZVC@pdV@xð15:10ÞRepresenting the surface integrals by summation of fluxes over the faces of theelement, and using a single Gaussian point for the face integrals, thesemi-discretized forms of the left and right hand sides of Eq. (15.8) becomeðquDyÞ ue þ ðquDyÞw uw ¼|fflfflfflffl{zfflfflfflffl}e|fflfflfflfflfflffl{zfflfflfflfflfflffl}m_ em_ w Z@u@u@pdVl Dy l Dy @x@x@xewð15:11ÞVCwhich can be rewritten asm_ e ue þ m_ w uw |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}Convection Z@u@u@pdVl Dy l Dy¼@x@x@xew|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}VCDiffusionð15:12Þ15.2A Preliminary Derivation565The convection and diffusion terms can be discretized using any of the techniques described in previous chapters to yield an algebraic equation of the formauC uC þX ZauF uF ¼ buC FNBðCÞVC@pdV@xð15:13ÞThe discretization of the pressure term is deferred till after the discretization ofthe continuity equation.15.2.2 Discretization of the Continuity EquationThe discretized form of the continuity equation is obtained by integrating Eq.
(15.7)over element C displayed in Fig. 15.1 to giveZVC@ ðquÞdV ¼ 0@xð15:14ÞAgain making use of the divergence theorem to transform the volume integralinto a surface integral and then into summation of fluxes over the faces of theelement, the discrete form of the continuity equation is obtained asXðquDyÞf ¼ ðquDyÞe ðquDyÞw ¼ 0ð15:15Þf nbðCÞorXm_ f ¼ m_ e þ m_ w ¼ 0ð15:16Þf nbðCÞ15.2.3 The Checkerboard ProblemThe discretization of the pressure term may be accomplished by adopting either ofthe following two approaches. In the first approach, the volume integral is computed via a single Gaussian integration point resulting inZVC@pdV ¼@x @pVC@x Cð15:17Þ56615 Fluid Flow Computation: Incompressible FlowsUsing a central difference scheme, the discretized form of Eq. (15.17) is obtainedasZ@ppE pWdV ¼VC@x2DxVCð15:18ÞIn the second approach, the volume integral of the pressure gradient term istransformed into a surface integral such thatZVC@pdV ¼@xZð15:19Þp dy@VCRewriting the surface integral as a summation of fluxes over the faces of theelement, Eq.
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