热流科学与工程西步文源大学G教育部重点实验室6.1.2SourcetermsinmomentumequationsThe general governing equation is:a(pd) + div(pUp)= div(Fagradp)+ SaatComparing N-S equations in the three coordinateswith the above general governing equation, the relatedsource terms can be obtained,where bothphysical sourceterm (such as gravitation) and numerical source term areincluded;Treatment of source term is very important innumerical simulationof momentum equations.ΦCFD-NHT-EHT6/37CENTER
6/37 Comparing N-S equations in the three coordinates with the above general governing equation, the related source terms can be obtained, where both physical source term (such as gravitation) and numerical source term are included; The general governing equation is: ( ) div U div grad S ( ) ( ) t 6.1.2 Source terms in momentum equations Treatment of source term is very important in numerical simulation of momentum equations
热流科学与工程西步文源大堂E教育部重点实验室Table 6-1 (Text book)Sourcetermsof2-Dincompressibleflow(n = const. No gravitation)Coordinatesu-equationv-equationyl0Cartesian00uIVAxi--uyusymmetric02tcylindricalvupu2209au2nau_q大puynuPolar12r2ae72a6r2r中CFD-NHT-EH7/37CENTER
7/37 Source terms of 2-D incompressible flow ( const. No gravitation) v u Table 6-1 (Text book)
热流科学与工程西步文源大学G教育部重点实验室6.1.3Twokeyissuesinsolvingincompressibleflowfield1.Conventional discretization method forpressuregradient in momentum equation may lead to oscillatingpressure field.Conventionally, one grid system is used to store all kinds ofinformation. If we store pressure , velocity, temperature, etc. atthe same grids, then the discretized momentum equations cannotdetectun-reasonablepressurefieldFor example. At node ithe 1-D steady momentum equationdrdd'udu_ dpoundr?xi-2i-1ii+1i+2dxdxCFD-NHT-EHTΦ8/37CENTER
8/37 6.1.3 Two key issues in solving incompressible flow field 1. Conventional discretization method for pressure gradient in momentum equation may lead to oscillating pressure field. Conventionally, one grid system is used to store all kinds of information. If we store pressure , velocity, temperature, etc. at the same grids, then the discretized momentum equations can not detect un-reasonable pressure field. 2 2 du dp d u u dx dx dx For example. At node i the 1-D steady momentum equation
热流科学与工程西步文源大堂G教育部重点实验室canbediscretizedbvFDMasfollows:, ui+ -2u, +ui-I; O(△x2)Ui+I -ui-l --- Pi+I - Pi-l ++npu,(8x)228x28xCDCDCDDiscussion: this is the discretized momentum equationfor node i, but it does not contain the pressure at node i.while includesthepressuredifferencebetweentwonodespositioned two-steps apart, leading to following result: thediscretized momentum equation can not detect anunreasonable pressure solution! Because it is the pressuregradient rather than pressure itself that occurs in themomentumequationPressure difference over two steps is called 2- Sxpressure difference中CFD-NHT-EHT9/37CENTER
9/37 1 1 1 1 1 1 2 2 2 2 ( ) i i i i i i i i u u p p u u u u x x x ； CD CD CD Discussion：this is the discretized momentum equation for node i, but it does not contain the pressure at node i, while includes the pressure difference between two nodes positioned two-steps apart, leading to following result: the discretized momentum equation can not detect an unreasonable pressure solution！Because it is the pressure gradient rather than pressure itself that occurs in the momentum equation. Pressure difference over two steps is called pressure difference. 2 x can be discretized by FDM as follows： 2 O( ) x
热流科学与工程西步文源大堂E教育部重点实验室2i-2i-1ii+1i+2力True solutionrCFD-NHT-EHTG10/37CENTER
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