西安交通大学：《数值传热学》课程教学资源（课件讲稿）第十一章 通用程序在二维椭圆型流动与换热问题中的应用示例 11.6 Complicated fully developed fluid flow and heat transfer in square duct 11.7 Impinging flow on a rotating disc 11.8 Turbulent flow and heat transfer in duct with a central jet

Analysis for the computational domain:This problem looks like Problem 3 where we take 1/4 of thecross section as the computational domain. Can we still take suchpractice for this case?No!Becauseofthecross sectionalnatural convection, the entire region mustbe taken as the computational domain.adiabaticBoundary conditions:V绝热At x=0, T=T: x-XL, T=TrXL = YL =1.0XAt y-O and y=YL: adiabatic绝热At four walls: u=v=w=0.6/59

6/59 Boundary conditions： Analysis for the computational domain： No! Because of the cross sectional natural convection, the entire region must be taken as the computational domain. This problem looks like Problem 3 where we take 1/4 of the cross section as the computational domain. Can we still take such practice for this case? At x=0, T=Tc : x=XL, T=Th At y=0 and y=YL: adiabatic At four walls: u=v=w=0

Major features of the problem(1) There are three velocity components: u, y, w ; However u,v are not coupled with w ;ouOuOpuaxayaxavapOvO0axoyayowowapp(uaxayazdd(2) T is coupled with velocity u, v. The variation of pg term with Tcauses the buoyancy, driving the natural convection in cross section.aTaTaT2pc,7axayOx7/98

7/98 （1）There are three velocity components：u, v, w ; However u, v are not coupled with w ； Major features of the problem 2 2 2 2 ( ) ( ) u u p u u u v x y x x y        + = − + +      2 2 2 2 ( ) ( ) g v v p v v u v x y y x y  −        + = − + +      2 2 2 2 ( ) ( ) w w p w w x y z x u v y        + = − + +      2 2 2 2 ( ) ( ) p v T T T T c x y x u y       + = +     (2) T is coupled with velocity u, v. The variation of g term with T causes the buoyancy, driving the natural convection in cross section

11-6-2Numericalmethods(1) Boussinesq assumption is adopted for the density in thesource term of v-equation: p= Prer[1 - β(T - Trer )]Treatment of pressure gradient and gravitation termfor v-equationOvOoppg+axayoyaxapopop[1- β(T -Tref)]gPrer(1+βTref)g +gPrer βTayayoyaOPea + gPrer BT[p+ Prer(1 + βTref )gy] +gPrerβTaydyPefr~pThe source term in v-equation is a function of temperature8/98

11-6-2 Numerical methods [1 ( )] ref ref p p g g T y y    T   − − = − − −  −  Treatment of pressure gradient and gravitation term for v-equation (1 ) ref ref r f e p T g Tg y      = − − + +  [ (1 ) ] ref ref ref p T y g T y   g    =  − + + + ref eff p y g T    = − +  (1) Boussinesq assumption is adopted for the density in the source term of v-equation: [1 ( )]    = − − ref ref T T eff p p  2 2 2 2 ( ) ( ) g v v v v p v u x x y y y     − −     + = + +      The source term in v-equation is a function of temperature 8/98

(2) How to use 2-D code for solving three velocity components?OvOvopouOuap1pgBTp(o(uOxayayaxayaxaTawowawaTappocazaxaxayaxayu, y, T are coupled and should be solved simultaneously;u, y, T are not coupled with w, while w is coupled with u andy, u, y, T are solved first, then w is solved. Thus w is regardedas a scalar variable(3) The problem studied can be separated into two sub-problems:(a) Natural convection in a 2-D square cavity: u, y, T are solved:(b) Fully developed axial flow for solving w, with a pre-specifiedsource term of -dp/dz .9/98

9/98 (2) How to use 2-D code for solving three velocity components？ (3) The problem studied can be separated into two sub-problems： (a) Natural convection in a 2-D square cavity: u, v, T are solved; • u, v, T are coupled and should be solved simultaneously; (b) Fully developed axial flow for solving w, with a pre-specified source term of −dp dz. 2 2 2 2 ( ) ( ) u u p u u u v x y x x y        + = − + +      2 2 2 2 ( ) ( ) g v v p v v u v x y y x T y          + = − + + +      2 2 2 2 ( ) ( ) w w p w w x y z x u v y        + = − + +      2 2 2 2 ( ) ( ) p v T T T T c x y x u y       + = +     • u, v, T are not coupled with w, while w is coupled with u and v; u, v, T are solved first, then w is solved. Thus w is regarded as a scalar variable

Governing equations of the problem studiedOpouOuNatural convectionaxoyaxSolved firstavOvp+pgBToto get u, yaxayoyOXOand TaTaT1pc.(uX1axoySolved2ndwithowowknownu,yanddpspecifiedpressuredzaxaygradient!dp/dz (<0) can be assumed and is specified as -300010/98

10/98 2 2 2 2 ( ) ( ) eff u u u u p u v x y x x y        + = − + +      2 2 2 2 ( ) ( ) eff v v v v p u v x y T y g x y          + = − + +    +   2 2 2 2 ( ) ( ) dp d w w w w u v x y x z y       + = + +   −   2 2 2 2 ( ) ( ) p T T T T c u v x y x y       + = +     dp/dz (<0) can be assumed and is specified as -3000. Governing equations of the problem studied: Solved first to get u, v and T Solved 2nd with known u, v and specified pressure gradient! Natural convection