《数值传热学》研究生课程教学资源（课件讲稿）Chapter 8 Numerical Simulation for Turbulent Flow and Heat Transfer

热流科学与工程西步文源大学E教育部重点实验室Numerical HeatTransfer数值传热学）Chapter8Numerical SimulationforTurbulentFlowandHeatTransfer(Chapter9inTextbook)QInstructorTao,Wen-QuanKeyLaboratoryofThermo-FluidScience&EngineeringInt.JointResearchLaboratoryofThermalScience&EngineeringXi'an Jiaotong UniversityInnovativeHarborofWestChina,Xian2022-November-08CFD-NHT-EHTΦ1/76CENTER

1/76 Instructor Tao, Wen-Quan Key Laboratory of Thermo-Fluid Science & Engineering Int. Joint Research Laboratory of Thermal Science & Engineering Xi’an Jiaotong University Innovative Harbor of West China, Xian 2022-November-08 Numerical Heat Transfer (数值传热学) Chapter 8 Numerical Simulation for Turbulent Flow and Heat Transfer（Chapter 9 in Textbook）

热流科学与工程西步文源大堂E教育部重点实验室8.1Introductiontoturbulence8.2Time-averaged governingequation forincompressibleconvectiveheattransfer8.3Zero-equation andone-equation model8.4Two-equationmodel8.5Wallfunctionmethod8.6Low-Reynoldsnumberk-epsilonmodel8.7BriefintroductiontorecentdevelopmentsCFD-NHT-EHTG2/76CENTER

2/76 8.1 Introduction to turbulence 8.2 Time-averaged governing equation for incompressible convective heat transfer 8.3 Zero-equation and one-equation model 8.4 Two-equation model 8.5 Wall function method 8.6 Low-Reynolds number k-epsilon model 8.7 Brief introduction to recent developments

热流科学与工程西步文通大堂G教育部重点实验室8.1Introductiontoturbulence8.1.1 Present understandingof turbulence10.1.2Classificationsofturbulence simulationmethods10.1.3 Reynolds time-averages and theircharacteristicsCFD-NHT-EHTΦ3/76CENTER

3/76 8.1.1 Present understanding of turbulence 10.1.2 Classifications of turbulence simulation methods 8.1 Introduction to turbulence 10.1.3 Reynolds time-averages and their characteristics

热流科学与工程西步文源大堂E教育部重点实验室8.1lntroductiontoturbulence8.1.1 Present understanding of turbulence1. Turbulence is a highly complicated unsteady flow, withinwhich all kinds of physical quantities are randomly varying withboth time and space;2. Transient Naiver-Stokes are valid for turbulent flows;3. Turbulent flow field can be regarded as a collection of eddies(涡漩）with different geometric scales .Eddyvs.vortex(漩涡):Eddyis characterizedbyturbulentflowwith randomness, and it covers a wide range of geometric scales:Vortex is kind of flow pattern caused by a specific solid outlinecharacterized by a recirculation. Such vortex flow can be laminar orCERNHr-urbulent.ΦCENTER4/76

4/76 8.1 Introduction to turbulence 8.1.1 Present understanding of turbulence 1. Turbulence is a highly complicated unsteady flow, within which all kinds of physical quantities are randomly varying with both time and space； 2. Transient Naiver-Stokes are valid for turbulent flows； 3. Turbulent flow field can be regarded as a collection of eddies (涡漩）with different geometric scales . Eddy vs. vortex (漩涡)：Eddy is characterized by turbulent flow with randomness, and it covers a wide range of geometric scales; Vortex is kind of flow pattern caused by a specific solid outline characterized by a recirculation. Such vortex flow can be laminar or turbulent

热流科学与工程西步文源大学E教育部重点实验室8.1.2Classifications of turbulencesimulation methodsDNSDirect numerical simulationN-SLarge eddy simulationLESequationSecondmomentclosureRANSNumericalAlgebraic stress modelturbulenceReynoldsaveragedmodelTurbulent viscosity modelN-S eqs.BoltzmannMixing length theoryequationOne-equation modelTwo-equation modelNon-linear model. etc.CFD-NHT-EHTΦ5/76CENTER

5/76 N-S equation Boltzmann equation DNS LES RANS Second moment closure Algebraic stress model Turbulent viscosity model Mixing length theory One-equation model Non-linear model. etc. Two-equation model Numerical turbulence model Direct numerical simulation Large eddy simulation Reynolds averaged N-S eqs. 8.1.2 Classifications of turbulence simulation methods

热流科学与工程亚步文源大堂G教育部重点实验室1.DNSIn DNS very small time step and space stepare needed to reveal the evolutions (演化)of eddies withdifferent scales. Required computer resource is veryhigh. Often high-performance computers (HPC) are needed.For a fully developed mixedconvectionina squareduct(L=6.4H), when Re=6400,Gr= 104 ～107 DNS is绝热conducted with 4.194x106Thnodes(=256x128x128),and8×105 time steps are needed绝热for statistical averageΦCFD-NHT-EHT6/76CENTER

6/76 In DNS very small time step and space step are needed to reveal the evolutions (演化) of eddies with different scales. Required computer resource is very high. Often high-performance computers (HPC) are needed. 1.DNS L H H x z v u w h T 绝 热 绝 热  For a fully developed mixed convection in a square duct (L=6.4H), when Re=6400, Gr= DNS is conducted with 4.194 106 nodes(=256 128 128), and time steps are needed for statistical average. 4 7 10 ~10 5 8 10    

热流科学与工程西步文源大堂E教育部重点实验室2. LESBasic idea:Turbulent fluctuations are mainlygenerated by large scale eddies, which are non-isotropic(各向异性）andvarywithflowsituation；Smallscaleeddiesdissipate(耗散) kinetic energy (from mechanic to thermalenergy),and are almost isotropic（各向同性).The N-S eqsare used to simulate the large scale eddies and the behaviorof small scale eddies is simulated by simplified modelLES requires less computer resource than that of DNS,even though still quite high, and has been used for someengineeringproblemsFor the above problem when simulated by LES only128x80x80=819200 grids are needed (compared with 4.194×106).ΦCFD-NHT-EHI7/76CENTER

7/76 Basic idea：Turbulent fluctuations are mainly generated by large scale eddies, which are non-isotropic(各 向异性) and vary with flow situation；Small scale eddies dissipate(耗散) kinetic energy (from mechanic to thermal energy), and are almost isotropic (各向同性). The N-S eqs. are used to simulate the large scale eddies and the behavior of small scale eddies is simulated by simplified model. 2. LES LES requires less computer resource than that of DNS, even though still quite high, and has been used for some engineering problems For the above problem when simulated by LES only 128 80 80=819200 grids are needed (compared with 4.194 106 ).  

热流科学与工程西步文源大堂E教育部重点实验室3.ReynoldstimeaverageN-SEqs.methodsExpressing a transient term as the sum of average term andfluctuation(脉动) term. Time average is conducted for thetransient N-S equations, and the time average terms of thefluctuations isexpressedvia somefunctionofthe averageterms8.1.3Reynoldstimeaveragesandtheircharacteristicst+A/[ p(t)dtΦ=Φ+p7At is the time step, which should be large enough relative to thefluctuation but small enough with respect to the variation period ofthe time averaged quantityΦCFD-NHT-EHT8/76CENTER

8/76 3. Reynolds time average N-S Eqs. methods Expressing a transient term as the sum of average term and fluctuation(脉动) term. Time average is conducted for the transient N-S equations, and the time average terms of the fluctuations is expressed via some function of the average terms. 8.1.3 Reynolds time averages and their characteristics '      1 ( ) t t t t dt t        t is the time step, which should be large enough relative to the fluctuation but small enough with respect to the variation period of the time averaged quantity

热流科学与工程亚步文源大堂E教育部重点实验室准稳态端流非稳态湍流AAAAN(a)Quasi-steadyUnsteadyCharacteristics of time average operations1.=0；2.=；3.+=，4.==06.g_g5.$f=(@+Φ)+f)=f+ΦOxOxadasa(f)a(gf)a(pf)8.=07.axaxaxaxaxCFD-NHT-EHTΦ9/76CENTER

9/76 Characteristics of time average operations ' 1.   0; 2.    ; 3. '      ; 4. ' '     0 5. ' ' ' '      f f f f f      ( )( ) 6. ; x x        ' ' 0 x x         7. 8. ' ' ( ) ( ) ( ) f f f x x x            非 稳 态 湍 流 Unsteady 准 稳 态 湍 流 Quasi-steady

热流科学与工程西步文源大学E教育部重点实验室8.2Time-averaged governing equationforincompre-ssibleconvectiveheattransfer8.2.1Timeaverage governing equation8.2.2Waysfor determining additional terms8.2.3 Governing equations with turbulent viscosityCFD-NHT-EHTΦ10/76CENTER

10/76 8.2.1 Time average governing equation 8.2.2 Ways for determining additional terms 8.2 Time-averaged governing equation for incompre￾ssible convective heat transfer 8.2.3 Governing equations with turbulent viscosity