OpenFOAMslides-02 (1185932), страница 2
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Deformation ceaseswhen the load is removed, but the plastic solid does not return to itsoriginal state.In ideal elastic solid stess vector and so stress tensor depends ondeformations:~σ~n = ~σ i ni = σ ik~ek niσ ik = λI1 (ε)g ik + 2µεikIlias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVM(23)19 ìàÿ 2016 ã.18 / 89Dierence between solids and uidsAn ideal elastic solid will deform under load and, once the load is removed,will return to its original state. Some solids are plastic. These deform underthe action of a sucient load and deformation continues as long as a loadis applied, providing the material does not rupture. Deformation ceaseswhen the load is removed, but the plastic solid does not return to itsoriginal state.In ideal elastic solid stess vector and so stress tensor depends ondeformations:~σ~n = ~σ i ni = σ ik~ek niσ ik = λI1 (ε)g ik + 2µεikE=µIlias Sibgatullin (Moscow University)3λ + 2µ,λ+µσ=(23)λ2(λ + µ)OpenFOAM course 1: theory of FVM(24)19 ìàÿ 2016 ã.18 / 89Dierentianl equations of continuum media:divergence formMass conservation:∂ρ+ ∇i (ρv i ) = 0∂t(25)∂(ρv k ) + ∇i (ρv k v i ) = ∇i σ ik + ρf k∂t(26)Momentum conservation:Scalar conservation:∂(ρϕ) + ∇i (ρϕv i ) = sϕ ρ − div ~qϕ∂tSuch a form is somtimes also calledconservationform since as we will seein FV mehod integral quantities are conserved after space discretisation(decomposition to nite volumes).So this form of equations is essential for OpenFOAMIlias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVM19 ìàÿ 2016 ã.19 / 89Divergence form of dierentianl equations in tensor notation∂ρ+ div (ρ~v ) = 0∂t∂(ρ~v ) + div (ρ~v~v ) = div σ + ρf~∂t∂(ρϕ) + div (ρϕ~v ) = sϕ ρ − div ~qϕ∂tv1 v1 v1 v2 v1 v3~v~v means ~v ⊗ ~v = [vi vk ] = v2 v1 v2 v2 v2 v3 v3 v1 v3 v2 v3 v3div (ρ~v~v )meansdiv (ρv k~v )ek;div (σ)means∇i pik ekDo you think that engeneers who uses OpenFOAM and commercialpackages for years always understand it? You're wrong!Ilias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVM19 ìàÿ 2016 ã.20 / 89Boundary condirionsSlip,No-slip boringStressfreeIlias Sibgatullin (Moscow University)StressfulOpenFOAM course 1: theory of FVM19 ìàÿ 2016 ã.21 / 89Dimensionless form of equationst∗ =t;t0x∗i =xi;L0u∗i =ui;v0p∗ =p;ρv02T∗ =T − T0.T1 − T0∂u∗i=0,∂x∗iSt(27)∂(u∗j u∗i )∂p∗∂u∗i1 ∂ 2 u∗i1−=+ 2 γi ,+∗∗∗2∗∂t∂xjRe ∂xj∂xiFr(28)∂T ∗ ∂(u∗j T ∗ )1 ∂2T ∗=.+∗∗∂t∂xjRe Pr ∂x∗2j(29)StIn theoretical books on Mechanics of Continua Media you can often seethat when they describe the problem, they add asteriscs * to all thevarables (dimensional).
And later they use usual notation for dimensionlessvariables. In CFD books this kind of notiation is less popular, since forcomplex problems and geometry dimensionless form is often useless.Ilias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVM19 ìàÿ 2016 ã.23 / 89Simplied modelsIncompressible owsdiv ~v = 0 ,(30)∂ui1+ div (ui~v ) = div (ν grad ui ) − div (p~ii ) + bi ,∂tρ(31)Inviscid (Euler) ow:∂(ρui )+ div (ρui~v ) = −div (p~ii ) + ρbi .∂tIlias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVM19 ìàÿ 2016 ã.(32)24 / 89Simplied modelsPotential owsdiv ~v = 0 ,(33)1∂ui+ div (ui~v ) = div (ν grad ui ) − div (p~ii ) + bi ,∂tρ(34)rot ~v = 0 .(35)~v = −grad ΦFor incompressible ows:div (grad Φ) = 0 .Ilias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVM19 ìàÿ 2016 ã.25 / 89Simplied modelsCreeping (Stokes) FlowWhen the ow velocity is very small, the uid is very viscous, or thegeometric dimensions are very small (i.e.
when the Reynolds number issmall), the convective (inertial) terms in the Navier-Stokes equations play aminor role and can be neglected.div (µ grad ui ) −1div (p~ii ) + bi = 0 .ρ(36)Creeping ows are found in porous media, coating technology,micro-devices etc.Ilias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVM19 ìàÿ 2016 ã.26 / 89The Algebraic Equation SystemAP φ P +XAl φl = QP ,(37)lwhere P denotes the node at which the partial dierential equation isapproximated and indexlruns over the neighbor nodes involved innite-dierence approximations. The node P and its neighbors form theso-calledcomputational moleculeNNTNNWNNENWPSEWWWSWPSEEESEWSPEBSSIlias Sibgatullin (Moscowcourse 1: theoryof FVMÐèñ.: University)Examples OpenFOAMof computationalmolecules19 ìàÿ2016 ã.in 2D and3D27 / 89The Algebraic Equation SystemThis system issparse,meaning that each equation contains only a fewunknowns. The system can be written in matrix notation as follows:~=Q~ ,AφwhereAis the square sparse coecient matrix,(38)~φis a vector (or columnmatrix) containing the variable values at the grid nodes, and~Qis thevector containing the terms on the right-hand side of Eq.
(37).The structure of matrixvector~.φAdepends on the ordering of variables in theFor structured grids, if the variables are labeled starting at acorner and traversing line after line in a regular manner (lexicographicordering),the matrix has a poly-diagonal structure.
For the case of ave-point computational molecule, all the non-zero coecients lie on themain diagonal, the two neighboring diagonals, and two other diagonalsremoved byNpositions from the main diagonal, whereNis the number ofnodes in one direction. All other coecients are zero. This structure allowsuse of ecient iterative solvers.Ilias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVM19 ìàÿ 2016 ã.28 / 89The Algebraic Equation SystemConversion of grid indices to one-dimensional storage locations for vectors or columnmatricesGrid locationCompass notationStorage locationi, j, ki − 1, j, ki, j − 1, ki, j + 1, ki + 1, j, ki, j, k − 1i, j, k + 1Pl = (k − 1)Nj Ni + (i − 1)Nj + jl − Njl−1l+1l + Njl − Ni Njl + Ni NjWSNEBTNNTNNWNNEWPESWSSENWPEWWSEEWSPEBSSIlias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVM19 ìàÿ 2016 ã.29 / 89The Algebraic Equation SystemBecause the matrixAis sparse, it does not make sense to store it as atwo-dimensional array in computer memory (this is standard practice forfull matrices).
Storing the elements of each non-zero diagonal in a separatearray of dimension1 × Ni Nj ,whereNiandNjare the numbers of grid5Ni Nj words of22storage; full array storage would require Ni Nj words of storage. In three2 2 2dimensions, the numbers are 7Ni Nj Nk and Ni Nj Nk , respectively. Thepoints in the two coordinate directions, requires onlydierence is suciently large that the diagonal-storage scheme may allowthe problem to be kept in main memory when the full-array scheme doesnot.Ilias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVM19 ìàÿ 2016 ã.30 / 89The Algebraic Equation SystemThe linearized algebraic equations in two dimensions can now be written in the form:Al,l−Nj φl−Nj + Al,l−1 φl−1 + Al,l φl + Al,l+1 φl+1 + Al,l+Nj φl+Nj = QlφWAWAS AP ANAEφS* φP = QPφNφEÐèñ.: Structure of the matrix for a ve-point computational molecule (non-zeroentries in the coecient matrix on ve diagonals are shaded; each horizontal setof boxes corresponds to one grid line)Ilias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVM19 ìàÿ 2016 ã.31 / 89The Algebraic Equation SystemMatrix storageAs noted above, it makes little sense to store the matrix as an array.
If,instead, the diagonals are kept in separate arrays, it is better to give eachdiagonal a separate name. Since each diagonal represents the connection tothe variable at a node that lies in a particular direction with respect to theAW , AS , AP , AN and AE ; their locationsin the matrix for a grid with 5 × 5 internal nodes are shown in Fig. 3. Withthis ordering of points, each node is identied with an index l, which is alsocentral node, we shall call themthe relative storage location. In this notation the equation can be writtenAW φ W + AS φ S + AP φ P + AN φ N + AE φ E = Q P ,(39)where the index l, which indicated rows, is understood, and the indexindicating column or location in the vector has been replaced by thecorresponding letter.
We shall use this shorthand notation from now on.When necessary for clarity, the index will be inserted. A similar treatmentapplies to three-dimensional problems.Ilias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVM19 ìàÿ 2016 ã.32 / 89Ilias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVM19 ìàÿ 2016 ã.33 / 89<++>Lecture 2. Finite Volume ApproximationsIlias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVM19 ìàÿ 2016 ã.33 / 89Finite Volume MethodsZZρφ~v · ~n dS =SZΓ grad φ · ~n dS +Sqφ dΩ .(40)ΩThe solution domain is subdivided into a nite number of small controlvolumes (CVs) by a grid which, in contrast to the nite dierence (FD)method, denes the control volume boundaries, not the computationalnodes.Surface integrals over inner CV faces cancel out.Ilias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVM19 ìàÿ 2016 ã.34 / 89The net ux through the CV boundary is the sum of integrals over the four(in 2D) or six (in 3D) CV faces:Zf dS =SIlias Sibgatullin (Moscow University)XZkf dS ,(41)SkOpenFOAM course 1: theory of FVM19 ìàÿ 2016 ã.35 / 89yj+1NNWyjWWWnwnwPNEnee neEEE∆ysswyj-1SWyseSES∆xjixxi-1xix i+1Ðèñ.: A typical CV and the notation used for a Cartesian 2D gridIlias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVM19 ìàÿ 2016 ã.36 / 89TntN∆znWewnEPsnbSz∆yyk jBix∆xÐèñ.: A typical CV and the notation used for a Cartesian 3D gridIlias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVM19 ìàÿ 2016 ã.37 / 89To calculate the surface integral exactly, one would need to know theintegrandfeverywhere on the surfaceSe .
This information is notφ are calculated so anavailable,as only the nodal (CV center) values ofapproximation must be introduced. This is best done using two levels ofapproximation:the integral is approximated in terms of the variable values at one ormore locations on the cell face;the cell-face values are approximated in terms of the nodal (CVcenter) values.Ilias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVM19 ìàÿ 2016 ã.38 / 89The simplest approximation to the integral is the midpoint rule: theintegral is approximated as a product of the integrand at the cell-facecenter (which is itself an approximation to the mean value over the surface)and the cell-face area:Zf dS = f e Se ≈ fe Se .Fe =(42)SeThis approximation of the integral provided the value offat location `e'is known is of second-order accuracy.Ilias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVM19 ìàÿ 2016 ã.39 / 89Another second-order approximation of the surface integral in 2D is thetrapezoid rule, which leads to:Zf dS ≈Fe =SeSe(fne + fse ) .2(43)In this case we need to evaluate the ux at the CV corners.Ilias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVM19 ìàÿ 2016 ã.40 / 89Approximation of Volume IntegralsThe simplest second-order accurate approximation is to replace the volumeintegral by the product of the mean value of the integrand and the CVvolume and approximate the former as the value at the CV center:Zq dΩ = q ∆Ω ≈ qP ∆Ω ,QP =(44)ΩwhereqPstands for the value ofIlias Sibgatullin (Moscow University)qat the CV center.OpenFOAM course 1: theory of FVM19 ìàÿ 2016 ã.41 / 89Interpolation and Dierentiation PracticesThe approximations to the integrals require the values of variables atlocations other than computational nodes (CV centers).f c = ρφ~v · ~n for the convective uxf d = Γgrad φ · ~n for the diusive uxWe assume that the velocity eld and the uid propertiesρandΓareknown at all locations.To calculate the convective and diusive uxes, the value ofφand itsgradient normal to the cell face at one or more locations on the CV surfaceare needed.
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