Moukalled F., Mangani L., Darwish M. The finite volume method in computational fluid dynamics. An advanced introduction with OpenFOAM and Matlab (811443), страница 44
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Define in the dictionary file fvSchemes the option to exclude the non orthogonalcorrection (Gauss linear uncorrected;).c. Define in the dictionary file fvSchemes the option to use a limited nonorthogonal correction (Gauss linear limited 0.8;).d. List of the possible Laplacian non orthogonal correction type available inOpenFOAM® (Hint: just mistype a scheme, i.e., banana, and launch any solveror application).References1. Patankar SV (1980) Numerical heat transfer and fluid flow. Hemisphere PublishingCorporation, McGraw-Hill, USA2. Jiang T, Przekwas AJ (1994) Implicit, pressure-based incompressible Navier-stokes equationssolver for unstructured meshes.
AIAA-94-03053. Davidson L (1996) A pressure correction method for unstructured meshes with arbitrarycontrol volumes. Int J Numer Meth Fluids 22:265–2814. Barth TJ (1992) Aspects of unstructured grids and finite-volume solvers for the Euler andNavier-stokes equations. Special course on unstructured grid methods for advection dominatedflows. AGARD Report 7875. Perez-Segarra CD, Farre C, Cadafalch J, Oliva A (2006) Analysis of different numericalschemes for the resolution of convection-diffusion equations using finite-volume methods onthree-dimensional unstructured grids, Part I: discretization schemes. Numer Heat Transfer PartB: Fundam 49(4):333–3506. Demirdzic I, Lilek Z, Peric M (1990) Finite volume method for prediction of fluid flow inarbitrary shaped domains with moving boundary. Int J Numer Meth Fluids 10:771–7907.
Demirdzic I, Lilek Z, Peric M (1993) A collocated finite volume method for predicting flowsat all speeds. Int J Numer Meth Fluids 16:1029–10508. Warsi ZUA (1993) Fluid dynamics: theoretical and computational approaches. CRC Press,Boca Raton9. Berend FD, van Wachem GM (2014) Compressive VOF method with skewness correction tocapture sharp interfaces on arbitrary meshes. J Comput Phys 279:127–14410. Jasak H (1996) Error analysis and estimation for the finite volume method with applications tofluid flow. Ph.D. thesis, Imperial College LondonReferences27111.
Balckwell BF, Hogan RE (1993) Numerical solution of axisymmetric heat conductionproblems using the finite control volume technique. J Thermophys Heat Transfer 7:462–47112. Wang S (2002) Solving convection-dominated anisotropic diffusion equations by anexponentially fitted finite volume method. Comput Math Appl 44:1249–126513. Jayantha PA, Turner IW (2003) On the use of surface interpolation techniques in generalisedfinite volume strategies for simulating transport in highly anisotropic porous media. J ComputAppl Math 152:199–21614.
Bertolazzi E, Manzini G (2006) A second-order maximum principle preserving finite volumemethod for steady convection-diffusion problems. SIAM J Numer Anal 43:2172–219915. Domelevo K, Omnes P (2005) A finite volume method for the Laplace equation on almostarbitrary two-dimensional grids. Math Model Numer Anal 39:1203–124916. Eymard E, Gallouet T, Herbin R (2004) A finite volume scheme for anisotropic diffusionproblems. Comptes Rendus Mathematiques (CR MATH) 339:299–30217.
Darwish M, Moukalled F (2009) A compact procedure for discretization of the anisotropicdiffusion operator. Numer Heat Transfer, Part B 55:339–36018. OpenFOAM, 2015 Version 2.3.x. http://www.openfoam.org19. OpenFOAM Doxygen, 2015 Version 2.3.x. http://www.openfoam.org/docs/cpp/Chapter 9Gradient ComputationAbstract As was shown in the previous chapter, the discretization of the gradientsof / at cell centroids and faces is fundamental to constructing the discretized sets ofdiffusion equations and, as will be revealed in later chapters, of equations involvingthe convection term. In addition, the evaluation of gradients is needed for theevaluation of various operators. For example, pressure derivatives are directlyneeded in the discretized momentum equations, while velocity gradients arerequired to compute the production term in turbulence models, and the strain rate innon-Newtonian viscosity models. This chapter describes several techniques forevaluating gradients on a general mesh topology.
The chapter starts with adescription of the techniques for computing the gradient on cartesian structuredgrids and proceeds with gradient evaluation on unstructured grids. The presentedmethods follow either the Green-Gauss or the least square approach. Methods tointerpolate the gradient to element faces are also presented.9.1Computing Gradients in Cartesian GridsFor the one-dimensional problem shown in Fig.
9.1 discretized using a uniformgrid, a linear profile assumption for the variation of / between cell centroids resultsin the following expression for the derivative at cell face e:d// /C¼ Edx e xE xC/ /C¼ Edxeð9:1ÞIn the same way, the derivative at the cell centroid C (Fig. 9.1) can be writtenusing the values at the two adjacent cells as© 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_92732749Gradient ComputationFig. 9.1 Computing thegradient on a uniform onedimensional gridxwxeweWxECxC d// /w¼ e¼dx C xe xw/ /W¼ E2DxC/E þ/C2/C þ/W2DxCð9:2ÞThis expression is usually referred to as the “central difference” approximationof the first derivative.For Cartesian grids in multiple dimensions, the derivatives can be computed byapplying the same principle along the respective coordinate directions.
Consider,for example, the two dimensional grid shown in Fig. 9.2, using the central difference approximation introduced earlier, the partial derivatives in the x and y directions are obtained as( x)Fig. 9.2 Computing thegradient on a two dimensionalCartesian grideNNW( y)( x)wNESnnSeSwW( y)ECsSsySWSSE( x)Cx( y)C9.1 Computing Gradients in Cartesian Grids@/@x/ /W¼ ExE xWC275@/@y¼C/N /SxN xSð9:3ÞA similar equation holds in the z direction for three dimensional grids.When dealing with unstructured grids the computation of the gradient using theabove method becomes unpractical and leads to the use of more general methods,some of these are now presented.9.2Green-Gauss GradientThis is one of the most widely used methods for computing the gradient. It wasintroduced in the previous chapter and will not be repeated here.
Rather the finalform of the equation will be given and some additional methods to compute the facevalues will be introduced.As derived in the previous chapter, the gradient at the centroid of an elementC with volume VC ( Fig. 8.8) is computed asr/C ¼1 X/ SfVC f nbðCÞ fð9:4Þwhere f refers to a face and Sf to its surface vector. The face values /f still need tobe defined before the formula can be used. Two approaches are presented forcomputing /f . One is face-based with a generally compact stencil involving faceneighbors, and the second is vertex-based with a larger stencil involving vertexneighbors.
The number of cells in this extended stencil is about twice the compactone.The use of a compact stencil is attractive with implicit methods because it leadsto more compact Jacobian matrices. However, the enlarged stencil brings moreinformation into the reconstruction, and is therefore expected to be more accurate.Method 1: Compact Stencil [1]For the two and three dimensional grid system shown in Fig. 9.3a, b, respectively, a simple approximation for calculating /f is to use the average values of thetwo cells sharing the face.
In this case the value of /f is calculated as/f ¼ gC /C þ ð1 gC Þ/Fwhere gC is a geometric weighting factor equal toð9:5Þ2769(a)Gradient Computation(b)ySfCfrfCrfrFFeFrFzSfrCrCyx(c)x(d)ySffFCfrfrFeCrFSfzrfFrfrCrCyxxFig. 9.3 Conjunctional face in a a two dimensional and b a three dimensional configuration;non-conjunctional face in c a two dimensional and d a three dimensional configurationgC ¼k rF rf kdFf¼k rF rC k dFCð9:6Þwhere r designates the position vector and d the distance between two points. Whenthe face is situated halfway between the two cell centers, /f is found to be/f ¼/C þ /F2ð9:7ÞThis approach is simple to implement in two and three dimensional situationsand all operations involved are face-based, not requiring any additional grid connectivity.
Accuracy-wise, the above relation leads to a second order approximationof /f only when the segment [CF] and face Sf intersection point coincides with thecentroid of the face Sf , i.e., with the Gaussian integration point f. Thus a secondorder accurate representation of the gradient is generally not achieved except for theabove special case.Such a condition is not usually satisfied with a general structured non-orthogonalor unstructured grid systems as shown for the two and three dimensional gridsystems displayed in Fig. 9.3c, d, respectively. Rather, the skewness of the mesh(non-conjunctionality) results in the segment [CF] and the surface Sf intersecting at9.2 Green-Gauss Gradient277a point f 0 , different from the face centroid f.
In this case a correction is needed forthe interpolated /f 0 value to get /f . The correction equation is given by/f ¼/f 0 þ correction¼/f 0 þ ðr/Þf 0 ðrf rf 0 Þð9:8Þwhich may also be written in expanded form as/f ¼ gC /C þ ðr/ÞC rf rC þ ð1 gC Þ /F þ ðr/ÞF rf rFð9:9Þ¼ /f 0 þ gC ðr/ÞC rf rC þ ð1 gC Þðr/ÞF rf rF|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}correctionSince gC depends on f 0 , Eq.
(9.9) indicates that improved estimates of the gradientcan be obtained iteratively. At each iteration, the average value at the face iscomputed using the gradients calculated in the pervious iteration. These face valuesare then used to compute new estimates of the gradients. Doing excessive numberof iterations may cause oscillations and usually not more than two iterations areperformed.In this case, the calculation of gC requires locating the intersection point f 0between [CF] and the face Sf. For that purpose three options will be described.Option 1: In this option f 0 is taken to be the exact intersection between [CF] andthe faceSf of surface vector Sf.
With n denoting the surface unit vector (i.e.,n ¼ Sf =Sf ) and e the unit vector along CF (i.e., e ¼ CF=kCFk), the location off 0 (Fig. 9.3c, d) can be found by exploiting the orthogonality condition that existsbetween n and the segment ff 0 (i.e., n is normal to the face Sf containing thesegment ff 0 ) to writerf rf 0 n ¼ 0ð9:10ÞFurther, since f 0 is a point on CF the vector Cf 0 can be expressed in terms of e asCf 0 ¼ rf 0 rC ¼ keð9:11Þwhere k is a scalar quantity.
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