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

NumericalHeatTransfer（数值传热学）Chapter11ApplicationExamplesoftheGeneral Codefor2DEllipticalFF&HTProblems(2)1/98

1/98 Numerical Heat Transfer (数值传热学) Chapter 11 Application Examples of the General Code for 2D Elliptical FF & HT Problems (2)

11.12D steadyheat conductionwithout sourceterminCartesian coordinate11.2Steady heat conduction inahollow cylinder11.3Fully-developed heat transfer ina square duct11.4Fullydevelopedheat transferin annularspacewith straight fin at inner wall11.5Fluid flowand heat transfer ina2-D suddenexpansion11.6Complicatedfullydevelopedfluidflowandheattransferinsquare duct11.7Impingingflowona rotatingdisc11.8 Turbulent flow and heat transfer in duct with acentraljet2/98

11.1 2D steady heat conduction without source term in Cartesian coordinate 11.2 Steady heat conduction in a hollow cylinder 11.3 Fully-developed heat transfer in a square duct 11.4 Fully developed heat transfer in annular space with straight fin at inner wall 11.5 Fluid flow and heat transfer in a 2-D sudden expansion 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 2/98

11-6Complicatedfullydeveloped fluidflow andheattransferinasquareduct---Velocity isregardedas a @ variable11-6-1 Physicalproblemand itsmathformulationKnown: Fully developed heat transfer in a square ductshown in Fig. 1. The effect of gravitation is taken intoaccount by Boussinesg assumption. Duct top and bottomwalls are adiabatic, while left and right walls are kept atadiabatconstant and uniform temperatures: T-0, T-1; Pr-0.7,n-1.0, dp/dz=-3000, and pgβ=104XL=YL=1.0绝热Find: Cross sectional distributions of u, y, and wNatural convectiontemperature distribution andfRe.due to buoyancy3/98

3/98 11-6-1 Physical problem and its math formulation 11-6 Complicated fully developed fluid flow and heat transfer in a square duct -Velocity is regarded as a variable  Known: Fully developed heat transfer in a square duct shown in Fig. 1. The effect of gravitation is taken into account by Boussinesq assumption. Duct top and bottom walls are adiabatic, while left and right walls are kept at constant and uniform temperatures: Tc=0, Th=1; Pr=0.7, =1.0, dp/dz=-3000, and . 4  g =10 Find：Cross sectional distributions of u, v, and w, temperature distribution and fRe. Natural convection due to buoyancy

adiabaticFig.1PhysicalmodelofProblem6yy绝垫TXL = YL = 1.0X绝热Main feature: When heat transfer goes into the fully developedregion, the heat leaves the hot wall goes into the cold wall, i.e., theheattransferrateis determinedbytheflow atthe cross-section,andthe axial flow does not make any contribution to this heat transfer4/98

4/98 XL YL = =1.0 Main feature: When heat transfer goes into the fully developed region, the heat leaves the hot wall goes into the cold wall, i.e., the heat transfer rate is determined by the flow at the cross-section, and the axial flow does not make any contribution to this heat transfer. Fig. 1 Physical model of Problem 6 adiabatic

Analysis ofthegoverning eq.:> According to the fully developed conditionauouaOuauoyopp(uWnaxayOaxOzaxavOva3OvopOp(u+WnpgaxayayayOx02owowa?waopp(u17MoyazaxaOzaxThe axial flow does not make contributions to heat transfera'TaTa'TaT02aT111ayOyaxOzOx5/98

5/98 222 2 2 2 ( ) ( ) u u u p u u u u v w x y z x x y z          + + = − + + +        2 2 2 2 2 2 ( ) ( ) v v v p v v v u v w g x y z y x y z           + + = − + + + −        222 2 2 2 ( ) ( ) p z w w w w w w u v w x y z x y z    −       + + = + + +        222 2 2 2 ( ) ( ) p T T T T T T c u v w x y z x y z         + + = + +       Analysis of the governing eq.： ➢ According to the fully developed condition ➢ The axial flow does not make contributions to heat transfer: