Moukalled F., Mangani L., Darwish M. The finite volume method in computational fluid dynamics. An advanced introduction with OpenFOAM and Matlab (811443), страница 91
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The coefficients of the boundary elements are modifiedaccording to Eq. (15.136).Fully Developed Outlet FlowFor a fully developed flow, the velocity gradient normal to the outlet surface isassumed to be zero. Hence the velocity at the outlet is assumed to be known andcomputed from the zero normal gradient using Eqs. (15.140) and (15.141).
As forthe pressure at the boundary, it can be extrapolated from the interior pressure fieldusingpb ¼ pC þ rpC dCbð15:145ÞThe velocity is treated as known and the coefficients of the momentum equationare modified according to Eq. (15.142).15.6.1.4 Symmetry Boundary ConditionA scalar is reflected across a symmetry boundary. Thus, a symmetry boundary condition for a scalar variable is imposed by setting its normal gradient to zero, as with aninsulated wall boundary condition.
For the velocity vector, the symmetry conditionshown in Fig. 15.26 also implies that it is reflected about the symmetry boundary withits parallel component (i.e., parallel to the symmetry boundary) retaining magnitudeand direction, while its normal component becoming zero.
This results in a zero shearstress but a non-zero normal stress along the symmetry boundary. Thus, the sameboundary condition can be used to impose a slip wall boundary condition for viscousflows.The unit vector in the direction normal to the boundary n and the normaldistance to the boundary d? are given by Eq. (15.120). Therefore the velocitycomponents normal and parallel to a symmetry boundary satisfyv? ¼ 0@vjj¼0@nð15:146Þ15.6Boundary Conditions615Fig.
15.26 A schematic of asymmetry boundary conditionCvvvbnvSbvCSymmetry PlanevvThe normal component of velocity can be written as8>< uC nx þ vC ny þ wC nz nxv? ¼ ðv nÞn ¼uC nx þ vC ny þ wC nz ny>:uC nx þ vC ny þ wC nz nz9>=ð15:147Þ>;while the parallel component is given by Eq. (15.121). The boundary force Fb canbe decomposed into a normal component F? and a parallel component Fjj .
As theshear stress along a symmetry boundary is zero, the parallel component of Fb iszero. Denoting the normal stress by r? , the force Fb is given byF b ¼ r ? Sbð15:148ÞThe normal stress component can be approximated as8uC nx þ vC ny þ wC nz nx><v?lr? ’ 2lb¼ 2 buC nx þ vC ny þ wC nz nyd?d? >:uC nx þ vC ny þ wC nz nz9>=>;ð15:149Þfrom which the boundary force is found to be8uC nx þ vC ny þ wC nz nx><l SbFb ¼ Fn ¼ 2 buC nx þ vC ny þ wC nz nyd? >:uC nx þ vC ny þ wC nz nz9>=>;ð15:150Þ61615 Fluid Flow Computation: Incompressible FlowsThe pressure gradient in the direction normal to a symmetry boundary is zero.Mathematically this is written asrpb n ¼ 0ð15:151ÞThe pressure at a symmetry boundary should be extrapolated from the interior ofthe domain.
Therefore to ensure a zero normal gradient, the pressure gradient at thesymmetry boundary is computed asrpb ¼ rpC ðrpC nÞnð15:152ÞThus, the pressure is obtained frompb ¼ pC þ rpb dCbð15:153ÞUsing the above equations, the coefficients of the boundary elements for themomentum equation in the x, y and z directions are modified as follows:u-component equationauCauC|{z}þinterior faces contribution0buC2lb Sb 2nd? x|fflfflfflffl{zfflfflfflffl}boundary face contributionauF¼b2l Sb buC bvC ny þ wC nz nx pb Sxb|{z}d?|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}interior faces contributionð15:154Þboundary face contributionv-component equationavCavC|{z}interior faces contribution0bvCþ2lb Sb 2nd? y|fflfflfflffl{zfflfflfflffl}boundary face contributionð15:155ÞavF¼b2l Sb b ½uC nx þ wC nz ny pb Sybd?|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}interior faces contributionbvC|{z}boundary face contribution15.6Boundary Conditions617w-component equationawCinterior faces contribution0bwC2lb Sb 2nd? z|fflfflfflffl{zfflfflfflffl}þawC|{z}boundary face contributionawF¼b2l Sb bwC buC nx þ vC ny nz pb Szb|{z}d?|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}interior faces contributionð15:156Þboundary face contributionWhile this is not a comprehensive list of momentum boundary conditions, itdoes cover the most common types.15.6.2 Boundary Conditions for the Pressure CorrectionEquationThe continuity equation for a boundary cell can be written asXf nbðCÞm_ f þm_ b|{z}¼0ð15:157Þboundary faceorXðm_ f þ m_ 0f Þ þ ðm_ b þ m_ 0b Þ ¼ 0|fflfflfflfflfflffl{zfflfflfflfflfflffl}f nbðCÞð15:158Þboundary facewhere m_ b is the boundary mass flux and m_ 0b is its correction.
While for an internalface the mass flux and its correction are defined by Eqs. (15.100) and (15.101), for aboundary face the definition is slightly different. Since at a boundary face only theboundary cell contributes to the average quantities, the use of Eq. (15.109) incombination with Eqs. (15.100) and (15.101) givesðnÞðnÞm_ b ¼ qb vC Sb qb DvC ðrpb rpC Þ Sbm_ 0b ¼ qb DC ðp0b p0C Þð15:159ÞIn implementing boundary conditions the values of m_ b , m_ 0b , pb, and p0b must becalculated. Based on the discussions related to the momentum equation, three typesof boundary conditions can be inferred.
The first type is designated by “specifiedmass flow rate” (e.g., walls or velocity specified at inlets). For this category m_ 0b ¼ 0,which is similar to a zero scalar flux boundary condition, and no modification to thepressure-correction equation is needed. The pressure however has to be computed at61815 Fluid Flow Computation: Incompressible Flowsthe boundary from the interior field. The second type of boundary conditions istermed “pressure specified” where p0b ¼ 0 and for which a Dirichlet-like conditionhas to be enforced for the pressure-correction equation.
For this condition, m_ b iscomputed from the boundary and interior pressure field. In the third type, animplicit relation exists between the pressure and the mass flow rate, as in a specifiedtotal pressure boundary condition. In this case, an explicit equation is extractedfrom the implicit relation and substituted into the pressure-correction equation.Details regarding the various types of boundary conditions and their implementation are now given.15.6.2.1 Wall Boundary Conditionðpb ¼ ?; m_ b ¼ 0; vb ¼ vwall Þ orðpb ¼ ?; m_ b ¼ 0; Fb ¼ 0ÞWhether it is a slip (Fig. 15.20) or no-slip (Fig. 15.19) wall boundary condition themass flow rate is zero.
Therefore m_ 0b ¼ 0, which is equivalent to a specified zeroflux and implying that no modification is needed for the pressure-correctionequation. However the pressure at the wall is required and is computed usingEq. (15.127) or Eq. (15.133) or a low order extrapolation profile, as shown below.pb ¼8>>>>>><>>>>>>:ðnÞpC þpC þ rpC dCbðnÞðnÞrpC S0 b rpb TbpCDCEq: ð15:127ÞEq: ð15:133Þð15:160Þloworder extrapolation15.6.2.2 Inlet Boundary ConditionsSpecified Velocity ðpb ¼ ?; m_ b specified; vb specified ÞFor a specified velocity at inlet (Fig. 15.21), the mass flux is known and its correction is set to zero, i.e., m_ 0b ¼ 0. Thus, similar to a wall boundary condition, theterm is simply dropped from the pressure-correction equation. The pressure at theboundary is extrapolated from the internal pressure field using Eq.
(15.127) orEq. (15.133) or a low order extrapolation profile as summarized in Eq. (15.160).Specified Pressure and Velocity Direction pb ¼ pspecified ; m_ b ?; ev specified; vb ?ÞIn the case of a specified static pressure at inlet (Fig. 15.22), pb is known and thusp0b is set to zero but m_ 0b 6¼ 0. The inlet is treated as a Dirichlet boundary conditionfor the pressure-correction equation. The coefficient of the p0 equation becomes15.6Boundary Conditions619X0aCp ¼qf Dfþf nbðCÞ|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}qb DC|fflffl{zfflffl}ð15:161Þboundary face contributioninterior faces contributionSpecified Total Pressure and Velocity Direction po;b ¼ po;specified ; m_ b ?;ev specified; vb ?ÞAs mentioned earlier, for a specified total pressure (Fig.
15.23), the velocitydirection should also be specified. The total pressure relation given by Eq. (15.138)is first rewritten as a function of the mass flow rate and pressure by replacing thevelocity magnitude by the mass flux. Thus,m_ b ¼ qvb Sb ¼ qjvb jev Sb ) qjvb j ¼) po;b ¼ pb þm_ bev S b1m_ 2b2qb ðev Sb Þ2ð15:162ÞUsing a Taylor expansion about pb, p0b is obtained aspb þ p0b ¼ pb þ@pb 0@pb 0ðm_ Þ ) p0b ¼m_@ m_ b b@ m_ b bð15:163ÞDifferentiating Eq. (15.162) with respect to m_ b and substituting intoEq.
(15.163), the final form of p0b is found to bep0b ¼ m_ bqb ðev Sb Þm_ 0 ¼ 2 bqb vb vb 0m_ bm_ bð15:164ÞSubstituting Eq. (15.164) in Eq. (15.159), the mass flux correction is expressed asm_ 0b ¼ qb DC p0b p0C ) m_ 0b ¼m_ bm_ b qb DCp0 DC ðqb vb qb vb Þ Cð15:165ÞReplacing m_ 0b in the expanded continuity equation (Eq. 15.158) by its expression0from Eq. (15.165), the modified apC coefficient for the boundary cell becomes0apC ¼Xqf Dff nbðCÞ|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}interior faces contributionþqb m_ b DCm_ b DC ðqb vb qb vb Þ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}boundary face contributionð15:166Þ62015 Fluid Flow Computation: Incompressible Flows15.6.2.3 Outlet Boundary ConditionsSpecified Pressure pb ¼ pspecified ; m_ b ?; vb ?For a specified pressure at outlet (Fig.
15.24) p0b is set to zero. On the other hand, m_ 0bis computed asm_ 0b ¼ qb DC ðp0b p0C Þð15:167ÞThe velocity direction being needed, it is customary to take the direction of vb to0be that of the upwind velocity vC. The expression of the apC coefficient in thepressure-correction equation becomes0apC ¼Xqf Dff nbðCÞ|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}þqb DC|fflffl{zfflffl}ð15:168Þboundary face contributioninterior faces contributionSpecified Mass Flow Rate m_ b ¼ m_ specified ; pb ?vb ?For a specified mass flow rate at outlet (Fig.
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