Moukalled F., Mangani L., Darwish M. The finite volume method in computational fluid dynamics. An advanced introduction with OpenFOAM and Matlab (811443), страница 90
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(15.119), the wall shear stress can be approximated usingðvC vb ÞjjðvC vb Þ ½ðvC vb Þ nn¼ lbd?d8 ? 9ðuuÞðuuÞnþðvC vb Þny þ ðwC wb Þnz nx >>CbCbx< =l¼ bðvC vb Þ ðuC ub Þnx þ ðvC vb Þny þ ðwC wb Þnz nyd? > >;:ðwC wb Þ ðuC ub Þnx þ ðvC vb Þny þ ðwC wb Þnz nzswall lbð15:122Þfrom which the boundary force for a laminar flow can be obtained as8ðuC ub Þ ðuC ub Þnx þ ðvC vb Þny þ ðwC wb Þnz nx><l SbFb ¼ bðvC vb Þ ðuC ub Þnx þ ðvC vb Þny þ ðwC wb Þnz nyd? >:ðwC wb Þ ðuC ub Þnx þ ðvC vb Þny þ ðwC wb Þnz nz9>=>;ð15:123ÞUsing Eq. (15.123) the coefficients of the boundary elements for the momentumequation in the x, y and z directions are modified as follows:u-component equationauCþauC|{z}interior faces contribution0boundary face contributionauF¼bbuCþbuC|{z}lb Sb 1 n2xd?|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}interior faces contributionlb Sb ub 1 n2x þ ðvC vb Þny nx ðwC wb Þnz nx pb Sxbd?|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}boundary face contributionð15:124Þv-component equationavCavC|{z}þinterior faces contribution0bvCboundary face contributionavF¼bbvC|{z}interior faces contributionlb Sb 1 n2yd?|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}þilb Sb hðuC ub Þnx ny þ vb 1 n2y þ ðwC wb Þnz ny pb Sybd?|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}boundary face contributionð15:125Þ15.6Boundary Conditions607w-component equationawCinterior faces contribution0bwClb Sb 1 n2zd?|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}þawC|{z}boundary face contributionawF¼bþbwC|{z}interior faces contributionlb Sb ðuC ub Þnx nz þ ðvC vb Þny nz þ wb 1 n2z pb Szbd?|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}boundary face contributionð15:126ÞThe unknown boundary pressure pb is extrapolated from the interior solutionusing either a truncated Taylor series expansion around point C such that pressure isfound fromðnÞpb ¼ pC þ rpC dCbð15:127Þor the mass flow rate expressed via the Rhie-Chow interpolation asðnÞðnÞm_ b ¼ qb vb Sb qb DvC rpb rpC Sbð15:128ÞSince the mass flow rate and velocity at the wall boundary are zero, the aboveequation reduces toðnÞðnÞð15:129Þ0 ¼ 0 qb DvC rpb rpC Sbwhich can be modified intoðnÞðnÞDvC rpb Sb ¼ rpb S0 bðnÞ¼ rpC S0 bð15:130ÞExpressing S0 b as the sum of the two vector Eb and Tb, Eq.
(15.130) becomesðnÞðnÞrpb ðEb þ Tb Þ ¼ rpC S0 bð15:131ÞSince Eb is in the direction of Cb, the above equation can be modified toðnÞðnÞDC ðpb pC Þ ¼ rpC S0 b rpb Tbð15:132Þfrom which the boundary pressure is obtained aspb ¼ pC þðnÞðnÞrpC S0 b rpb TbDCð15:133Þ60815 Fluid Flow Computation: Incompressible FlowsFig. 15.20 A schematic of aslip wall boundary conditionCebvbnSbwall=0Slip Wall Boundary ðpb ¼ ?; m_ b ¼ 0; Fb ¼ 0ÞFor this boundary condition, the wall shear stress is zero (Fig. 15.20). Therefore theboundary force is zero.
The boundary pressure is computed as for the no-slip wallboundary case using Eq. (15.127) or Eq. (15.133). The coefficients of themomentum equation (in vector form) are modified as follows:avCavC|{z}interior faces contribution0bvCð15:134ÞavF¼bbvC|{z}interior faces contributionpb Sb|ffl{zffl}boundary face contribution15.6.1.2 Inlet Boundary ConditionsThree types of inlet boundary conditions are considered. (i) specified velocity;(ii) specified static pressure and velocity direction; and (iii) specified total pressureand velocity direction. All treatments of the pressure boundary conditions will bedetailed in the pressure correction boundary conditions section.15.6Boundary Conditions609Specified Velocity ðpb ¼ ?; m_ b specified; vb specified ÞFor a specified velocity boundary condition at inlet (Fig.
15.21) the convectionðm_ b vb Þ and diffusion ðFb ¼ sb Sb Þ terms at the boundary face are calculated usingthe known values of velocity vb and mass flow rate m_ b . The pressure at theboundary is extrapolated from the boundary element centroid asðnÞpb ¼ pC þ rpC dCbð15:135ÞThe terms involving the boundary velocity are treated explicitly by adding themto the source term and setting the coefficient avF¼b to zero while adding its value tothe avC coefficient.The coefficients of the boundary element are modified according toavCavCbvCbvC avF¼b vb0ð15:136ÞavF¼bFig. 15.21 A schematic ofspecified velocity boundarycondition at inletCebv b = v specifiedbnSbmb =bv b Sbpb = ?61015 Fluid Flow Computation: Incompressible FlowsSpecified 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. Thevelocity being unknown, has to be computed from the pressure gradient at theboundary. To this end, a velocity direction should be specified as part of theboundary condition.As the boundary pressure pb is known, its value is directly used in calculating thepressure gradient in the boundary element without any special treatment.
Thereforepb is used in calculating rpC .The mass flow rate at the boundary is computed from the continuity equation(see next section). Then, for a specified velocity direction given by the unit vectorev, the velocity for a specified pressure boundary condition at inlet is obtained as ¼m_ b ¼ qb vb Sb ¼ qb vb ev Sb ) vb m_ b ) vb ¼ vb e vqb ðev Sb Þð15:137ÞThe velocity at the boundary is computed at every iteration and the problem issolved as in the case of a specified velocity with the coefficients in the momentumequation modified according to Eq.
(15.136).CebbnSbpb(specified )e vb (specified )vb = ?mb = ?Fig. 15.22 A schematic of specified pressure and velocity direction boundary condition at inlet15.6Boundary Conditions611Fig. 15.23 A schematic ofspecified total pressure andvelocity direction boundarycondition at inletCebpb = ?bvb = ?mb = ?nSbptotal (specified ) = pb +bvb vb2Specified Total Pressure and Velocity Direction po;b ¼ po;specified ;m_ b ?; ev specified; vb ?ÞIn the case of a specified total pressure at inlet (Fig.
15.23) the velocity directionshould also be specified. However, the magnitude of the velocity and the pressure atthe boundary are unknown though related using the total pressure definition given byp0 ¼p|{z}static pressureþ1qv v2|fflfflffl{zfflfflffl}ð15:138Þdynamic pressurewhere p0 is the total pressure, p the static pressure, ρ the density, and v the velocityvector. The mass flow rate at the boundary is computed from the continuityequation (see next section). Knowing the mass flow rate, the velocity is computedin the same manner as for the specified pressure case using Eq. (15.137).
Thevelocity is thus treated as a known velocity condition (i.e., a Dirichlet boundarycondition) with the coefficients in the momentum equation modified according toEq. (15.136).15.6.1.3 Outlet Boundary ConditionsThree types of boundary conditions at an outlet are considered: (i) a specified staticpressure, (ii) a specified mass flow rate, and (iii) a fully developed flow.61215 Fluid Flow Computation: Incompressible FlowsSpecified Static Pressure pb ¼ pspecified ; m_ b ?; vb ?For the momentum equation, fully developed conditions are assumed at a specifiedpressure outlet (Fig.
15.24) implying a zero velocity gradient along the direction ofthe surface vector at outlet. This is also equivalent to assuming the velocity at theoutlet to be equal to that of the boundary element. Thus it is similar to a zero fluxboundary condition whose implementation is rather simple.The modifications to the coefficients are given byavCavC|{z}þboundary face contributioninterior faces contribution0m_ b|{z}ð15:139ÞavF¼bbvCbvC|{z}interior faces contributionpb Sb|ffl{zffl}boundary face contributionThis sets the contribution of the outlet velocity to zero and uses the specifiedpressure boundary value in the computation of the pressure gradient.However to ensure that the flux is zeroed in the outflow surface vector directiononly, the velocity is usually extrapolated to the outlet using the boundary flux,which is computed from the boundary element flux asrvb ¼ rvC ðrvC eb ÞebFig. 15.24 A schematic ofspecified static pressureboundary condition at outletð15:140Þvb = ?Sbmb = ?nbebCpb = pspecified15.6Boundary Conditions613This ensures that the gradient along the boundary surface vector is zero.
Then,using a Taylor series expansion, the velocity at the boundary is computed asvb ¼ vC þ rvb dCbð15:141Þwhere rvb is used instead of rvC . Therefore an additional correction is now addedto the source term and the modifications to the coefficients becomeavCþavC|{z}interior faces contribution0bvCm_ b|{z}boundary face contributionð15:142ÞavF¼bbvC|{z}m_ b ðrvb dCb Þ pb Sb|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}interior faces contributionboundary face contributionSpecified Mass Flow Rate m_ b ¼ m_ specified ; pb ?vb ?Since the flow is incompressible, a specified mass flow rate boundary condition(Fig. 15.25) is equivalent to specifying the normal component of velocity at theboundary.
The velocity is calculated by assuming its direction to be the same as thatat the main grid point, i.e., ðev Þb ¼ ðev ÞC . Thus, the velocity is expressed asvb ¼ j vb j ð e v Þ CFig. 15.25 A schematic ofspecified mass flow rateboundary condition at outletð15:143Þvb = ?Sbpb = ?nbebCmb = mspecified61415 Fluid Flow Computation: Incompressible Flowswith jvb j obtained fromm_ b ¼ qb vb Sb ¼ qb jvb jðev ÞC Sb ) jvb j ¼m_ bqb ðev ÞC Sbð15:144Þallowing vb to be calculated. Thus for momentum, a specified velocity boundarycondition is applied.
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