2. Flow field solution is the most important step for solvingconvective heat transfer problems.3.Numerical approaches(方法)for solution of incompressibleflow field :SimultaneouslyInsuchapproachesnospecialalgorithmis needed. The only requirement is ansolving(同时求解)extremelylargecomputerresource.Nowdifferent dependentit is called coupled methodvariables (u, y, w, p, T)Primitivevariablemethod(原始变量法，u,w,p).PressurecorrectionSegregated solutionsmethod is the most widely used one（分离式求解）ofNon-primitive variable methoddifferentdependentVortex-stream function method (涡量流函数法）wasthemostwidelyusedvariablesone (Chapter 8 of the textbook)6/39
6/39 3 . Numerical approaches(方法) for solution of incompressible flow field： 2. Flow field solution is the most important step for solving convective heat transfer problems. In such approaches no special algorithm is needed. The only requirement is an extremely large computer resource. Now it is called coupled method. Simultaneously solving (同时求解) different dependent variables (u, v, w, p, T). Segregated solutions (分离式求解）of different dependent variables Non-primitive variable method. Vortex-stream function method (涡量 流函数法) was the most widely used one (Chapter 8 of the textbook) Primitive variable method(原始变量 法, u,v,w,p). Pressure correction method is the most widely used one
6.1.2SourcetermsinmomentumequationsThe general governing equation is:a(pd) + div(pUp) = div(Fagradd) + SaatComparing N-S equations in the three coordinateswith the above general governing equation, the relatedsourceterms can be obtained,wherebothphysical sourceterm (such as gravitation) and numerical source term areincluded;Treatment of source term is very important innumerical simulation ofmomentum equations.7/39
7/39 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
Table 6-1 (Text book)Sourcetermsof2-Dincompressibleflow(n = const. No gravitation)Coordinatesu-equationv-equationyl0Cartesian00uVAxi-uyrusymmetric02acylindricalvupu2209au2nau_q大puynuPolar42r2ae72a6r2r8/39
8/39 Source terms of 2-D incompressible flow ( const. No gravitation) = v u Table 6-1 (Text book)
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 eguationdrdd'ududpoumdr?2i-2i-1ii+1i+2dxdx9/39
9/39 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
canbediscretizedbyFDMasfollows:ui++ -2u, +ui-I; O(△x?)ui -uil = - Pil- Pil + n+pu,(8x)228x28xCDCDCDDiscussion: this is the discretized momentum equationfor node i, but it does not contain the pressure at nodei.while includes the pressure difference between two nodespositioned 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 themomentumeguationPressure difference over two steps is called 2-Sxpressuredifference10/39
10/39 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