Moukalled F., Mangani L., Darwish M. The finite volume method in computational fluid dynamics. An advanced introduction with OpenFOAM and Matlab (811443), страница 87
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15.9 Staggered Cartesian velocity components in a curvilinear grid system(a)(b)eCfFeFeCfeFig. 15.10 Curvilinear coordinates a Covariant component and b contra-variant componentsIn curvilinear grids, the use of Cartesian velocity components can lead toproblems when one or more of the surfaces become aligned with the staggeredvelocity component as shown in Fig. 15.9.Therefore a better alternative in this case is to use either covariant or contra-variantcurvilinear velocity components, as shown in Fig.
15.10a, b, respectively.An example of staggering using contra-variant velocity components is shown inFig. (15.11).Fig. 15.11 Curvilinear velocities: staggering using contra-variant velocity components58415 Fluid Flow Computation: Incompressible FlowsFig. 15.12 Cartesian components defined at each faceUnfortunately complications arise when discretizing the momentum equations incurvilinear coordinates [1, 3, 4, 7], due to the increased complexity in the treatmentof the diffusion term, and because the equations gain non-conservative terms.Another option shown in Fig.
15.12 is to stagger all Cartesian velocity components in all directions so as to have all velocity components at all faces. This woulddouble (in two dimensions) or triple (in three dimensions) the number of momentumequations to be solved. The problem is further complicated in the case of anunstructured grid. In this case there is no obvious staggering direction, and the onlyway for a staggering concept to apply is by changing the size of the cell elementsused for the pressure and velocity components, or by resorting to staggering allvelocity components along all the faces, again dramatically increasing the number ofvariables to be solved.
Finally, the geometric information stored is more than doubled, as a new unstructured grid need to be used for the velocity components.It turns out that the use of a cell-centered collocated grid system (Fig. 15.13),where all variables are stored at the same location (the cell centroid), is a moreattractive solution. It is worth noting that while the velocity components are storedat the centroids of the elements as is the case for pressure or any other variable, themass flux, a scalar value, in a collocated grid is stored at the element faces. Themass flux can actually be viewed as a contra-variant component, except that in thiscase it is computed using a custom interpolation of the discrete momentum equation, known as the Rhie-Chow interpolation, which is the subject of the nextsection.Fig.
15.13 Collocated grid arrangement15.4The Rhie-Chow Interpolation58515.4 The Rhie-Chow InterpolationThe deficiency in the original collocated formulation presented earlier was in thelinear interpolation used to calculate the velocities at the element faces. Thisinterpolation resulted in decoupling the pressure and velocity values at the cell levelgiving rise to the checkerboard problem. In 1983, Rhie and Chow [8] reported onan interpolation procedure that allowed the formulation of the SIMPLE algorithmon a collocated grid [9–16]. In their method a dissipation term, representing thedifference between two estimates of the cell face pressure gradient, is added to thelinearly interpolated cell face velocity.
As shown in Fig. 15.14 the two pressuregradient estimates are based on different grid stencils.This procedure will be shown to be equivalent to constructing a pseudomomentum equation at the element face with its coefficients linearly interpolatedfrom the coefficients of the momentum equations at the centroids of the elementsstraddling the face and its pressure gradient computed using a small grid stencil. Inthat respect, the Rhie-Chow interpolation simply mimics the small stencil pressurevelocity coupling of the staggered grid arrangement.Starting with the discretized x-momentum equations for cells C and F, which are@puC þ H C ½ u ¼@x C@puF þ HF ½u ¼ BuF DuF@x FBuCDuCð15:50ÞA uf velocity equation similar to that of Eq.
(15.50), with the pressure gradientlinked to the local neighboring pressure values, as illustrated in Fig. 15.14, willhave the following form:uf þ Hf ½u ¼ Buf DufFig. 15.14 The two pressuregradient estimates in theRhie-Chow interpolationtechnique@p@x:ð15:51ÞfInterpolated face value∂p∂xLinearinterpolationCRhie-Chowinterpolationf∂p∂xF58615 Fluid Flow Computation: Incompressible FlowsSince in a collocated grid, the coefficients of this equation cannot be directlycomputed, they are approximated by interpolation from the coefficients of theneighboring nodes.
Using a linear interpolation profile, these coefficients arecomputed as1Hf ½u ¼ ðHC ½u þ HF ½uÞ ¼ Hf ½u21Buf ¼ BuC þ BuF ¼ Buf21Duf ¼ DuC þ DuF ¼ Duf2ð15:52ÞEmploying the values given in Eq. (15.52), the pseudo-momentum equation atthe element face becomesuf þ H f ½ u ¼BufDuf@p@xð15:53ÞfThis is in all practical sense the momentum equation on a “staggered” grid,which is reconstructed using the collocated grid momentum coefficients.In all above equations and for later use, values with an over bar are obtained bylinear interpolation between the values at points C and F according tohf ¼ gC hC þ gF hFð15:54Þwhere gC and gF are geometric interpolation factors related to the position of theelement face f with respect to the nodes C and F, as explained in previous chapters.Using Eq.
(15.50), Hf is rewritten as 1uu @puu @puC þ BC DCHf ½u ¼uF þ BF DF2@x C@x F @p¼ uf Dufþ Buf@x fð15:55Þwhere the coefficient approximation can be shown to be second order accurate, i.e.,Duf@p@x Duff@p@x 1@p@pDuCþ DuF2@x C@x F 1 u1@p@pþ DC þ DuF 22@x C@x F 1 u@p@p1 u@p@pþ DF¼ DC4@x C@x F4@x F@x C¼f OðDx2 Þð15:56Þ15.4The Rhie-Chow Interpolation587Substituting Eq. (15.55) into Eq. (15.53), the velocity at the element face usingthe Rhie-Chow interpolation method is obtained asuf ¼ Hf ½u þBufDuf@p@x !@p@p@x f@x f|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}f¼ uf|{z} Dufaveragevelocityð15:57Þcorrection termFor a multi dimensional situation, similar interpolation formulae can be derivedfor the y and z velocity components and are given byvf ¼ vf Dvf !@p@p@y f@y fwf ¼ wf Dwf@p@z !@p@z ffð15:58Þð15:59ÞEquations (15.57)–(15.59) can be written in a vector form, more suitable forderiving the multi-dimensional pressure correction equation, asvf ¼ vf Dvf rpf rpfð15:60Þwhere2Duf6 0vDf ¼ 400Dvf0300 75Dwfð15:61Þand where ∇pf is computed as per Sect.
9.4 usingrpf ¼ rpf þpF pC rpf eCF eCFdCF|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}ð15:62ÞCorrection to interpolated face gradientand yielding a stencil in the CF direction formed only from the adjacent cell valuespF and pC aspF pC rpf eCF eCF eCFrpf eCF ¼ rpf eCF þdCFð15:63ÞpF pC¼dCFWith the face velocities closely linked to the pressure of adjacent cells, checkerboard fields are inadmissible rendering solutions on collocated grids viable.58815 Fluid Flow Computation: Incompressible Flows15.5 General DerivationBefore proceeding with the development of the multidimensional collocatedpressure correction equation, the discretized multidimensional momentum equationis first presented.15.5.1 The Discretized Momentum EquationThe momentum equation given by Eq. (15.2) is slightly modified and written as@½qv þ r fqvvg ¼ rp þ r flrvg þ r flðrvÞT g þ f b@tð15:64ÞThe discretized form of Eq.
(15.64) in the time interval [t − Δt/2, t + Δt/2] issought over element C shown in Fig. 15.15.In Eq. (15.64), the three underlined expressions represent, from left to right, theunsteady, convection, and diffusion term, respectively. The discretization of theseterms proceeds as presented in previous chapters. The remaining terms are evaluated explicitly and treated as sources. The volume integral of the second part of theshear stress term is transformed into a surface integral using the divergence theoremand then into a summation of surface fluxes asFig. 15.15 Element C in ageneral unstructured gridsystemF1F2Convectionf2Diffusionf1TransientCf6f3F3Source/Sinkf4f5F4F5F615.5General DerivationZ589Z nnooXTr lðrvÞ dV ¼lðrvÞT dS ¼lðrvÞTf Sf@VCVCð15:65Þf nbðCÞwhere the expanded form of ðrvÞTf Sf in a three dimensional coordinate system isgiven by3@u x @u y @u zSSSþþ6 @x f @y f @z f 7766 @v@v y @v z 776TxS þ S þ S 7ðrvÞf Sf ¼ 66 @x f @y f @z f 7764 @w@w y @w z 5xS þS þS@x f@y f@z f2ð15:66ÞThe volume integral of the pressure gradient is also treated as a source term andevaluated explicitly asZrp dV ¼ ðrpÞC VCð15:67ÞVCor transformed into a surface integral according to Eq.
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