有限元分析 ANSYS：Modeling Turbulent Flows（PPT讲稿）Introductory FLUENT Training

∧NSY Modeling Turbulent Flows Introductory FLUENT Training e 2006 ANSYS Inc. All ANSYS, Inc. Proprietary

© 2006 ANSYS, Inc. All rights reserved. ANSYS, Inc. Proprietary Modeling Turbulent Flows Introductory FLUENT Training

Introductory FLUENT Notes Fluent User Services cente ANSYS LUENT v6. 3 December 2006 wwfluentusers. com FLUENT What is Turbulence? Unsteady, irregular(aperiodic)motion in which transported quantities (maSs, momentum, scalar species) fluctuate in time and space Identifiable swirling patterns characterize turbulent eddies Enhanced mixing(matter, momentum, energy, etc. )results fluid properties and velocity exhibit random variations Statistical averaging results in accountable, turbulence related transport mechanisms This characteristic allows for turbulence modeling Contains a wide range of turbulent eddy sizes(scales spectrum) The size/velocity of large eddies is on the order of mean flow a Large eddies derive energy from the mean flow Energy is transferred from larger eddies to smaller eddies In the smallest eddies, turbulent energy is converted to internal energy by vIscous dissipation C 2006 ANSYS. nc All ANSYS, Inc. Proprietar

© 2006 ANSYS, Inc. All rights reserved. 6-2 ANSYS, Inc. Proprietary Fluent User Services Center www.fluentusers.com Introductory FLUENT Notes FLUENT v6.3 December 2006 What is Turbulence? ◆ Unsteady, irregular (aperiodic) motion in which transported quantities (mass, momentum, scalar species) fluctuate in time and space ⚫ Identifiable swirling patterns characterize turbulent eddies. ⚫ Enhanced mixing (matter, momentum, energy, etc.) results ◆ Fluid properties and velocity exhibit random variations ⚫ Statistical averaging results in accountable, turbulence related transport mechanisms. ⚫ This characteristic allows for turbulence modeling. ◆ Contains a wide range of turbulent eddy sizes (scales spectrum). ⚫ The size/velocity of large eddies is on the order of mean flow. ◼ Large eddies derive energy from the mean flow ⚫ Energy is transferred from larger eddies to smaller eddies ◼ In the smallest eddies, turbulent energy is converted to internal energy by viscous dissipation

Introductory FLUENT Notes Fluent User Services cente ANSYS LUENT v6. 3 December 2006 wwfluentusers. com FLUENT Is the flow turbulent? External flows where Re PUL x≥5000 along a surface L L=x.d,、d,.etc Re>20000 around an obstacle Other factors such as free-stream turbulence. surface conditions and disturbances may cause transition Internal flows to turbulence at lower reynold Re,≥2,300 numbers Natural Convection Ra where Ra βgL△Tp2CD阝gD△7 is the rayleigh number P μ k a k is the Prandtl number C 2006 ANSYS. nc All ANSYS, Inc. Proprietar

© 2006 ANSYS, Inc. All rights reserved. 6-3 ANSYS, Inc. Proprietary Fluent User Services Center www.fluentusers.com Introductory FLUENT Notes FLUENT v6.3 December 2006 Is the Flow Turbulent? External Flows Internal Flows Natural Convection Rex  500,000 along a surface around an obstacle where where Other factors such as free-stream turbulence, surface conditions, and disturbances may cause transition to turbulence at lower Reynolds numbers Re  2,300 dh Re d  20,000 is the Rayleigh number   = U L Re L , , ,etc. d dh L = x k g L T Cp g L T     =    = 3 2 3 Ra 9 10 Pr Ra  k Cp =   Pr = is the Prandtl number

Introductory FLUENT Notes Fluent User Services cente ANSYS LUENT v6. 3 December 2006 wwfluentusers. com FLUENT Turbulent flow structures Small L structures structures Iniection of energy Dissipation Large-scale Flux ofenergy Ls= / Re Energy Cascade Richardson(1922) C 2006 ANSYS. nc All ANSYS, Inc. Proprietar

© 2006 ANSYS, Inc. All rights reserved. 6-4 ANSYS, Inc. Proprietary Fluent User Services Center www.fluentusers.com Introductory FLUENT Notes FLUENT v6.3 December 2006 Turbulent Flow Structures Energy Cascade Richardson (1922) Small structures Large structures

Overview of Computational Approache to Introductory FLUENT Notes Fluent User Services cente LUENT v6. 3 December 2006 wwfluentusers. com UENT Reynolds-Averaged Navier-Stokes(RANS) models Solve ensemble-averaged (or time-averaged) Navier-Stokes equations All turbulent length scales are modeled in rans The most widely used approach for calculating industrial flows Large Eddy Simulation (LES Solves the spatially averaged N-S equations. Large eddies are directly resolved, but eddies smaller than the mesh are modeled Less expensive than dns, but the amount of computational resources and efforts are still too large for most practical applications Direct Numerical Simulation (DNS Theoretically, all turbulent flows can be simulated by numerically solving the full Navier-Stokes equations Resolves the whole spectrum of scales. No modeling is required But the cost is too prohibitive! Not practical for industrial flows-DNS is not available in Fluent There is not yet a single, practical turbulence model that can reliably predict all turbulent flows with sufficient accuracy C 2006 ANSYS. nc All ANSYS, Inc. Proprietar

© 2006 ANSYS, Inc. All rights reserved. 6-5 ANSYS, Inc. Proprietary Fluent User Services Center www.fluentusers.com Introductory FLUENT Notes FLUENT v6.3 December 2006 Overview of Computational Approaches ◆ Reynolds-Averaged Navier-Stokes (RANS) models ⚫ Solve ensemble-averaged (or time-averaged) Navier-Stokes equations ⚫ All turbulent length scales are modeled in RANS. ⚫ The most widely used approach for calculating industrial flows. ◆ Large Eddy Simulation (LES) ⚫ Solves the spatially averaged N-S equations. Large eddies are directly resolved, but eddies smaller than the mesh are modeled. ⚫ Less expensive than DNS, but the amount of computational resources and efforts are still too large for most practical applications. ◆ Direct Numerical Simulation (DNS) ⚫ Theoretically, all turbulent flows can be simulated by numerically solving the full Navier-Stokes equations. ⚫ Resolves the whole spectrum of scales. No modeling is required. ⚫ But the cost is too prohibitive! Not practical for industrial flows - DNS is not available in Fluent. ◆ There is not yet a single, practical turbulence model that can reliably predict all turbulent flows with sufficient accuracy

Introductory FLUENT Notes Fluent User Services cente ANSYS LUENT v6. 3 December 2006 wwfluentusers. com FLUENT Turbulence models available in fluent One-Equation Models Spalart-Allmaras Two-Equation Models RANS based Standard k-e Increase in models RNG K-e Computational Realizable k-e Standard k-o Per iteration SSTK-o Reynolds stress model Detached eddy simulation arge Eddy simulation C 2006 ANSYS. nc All ANSYS, Inc. Proprietar

© 2006 ANSYS, Inc. All rights reserved. 6-6 ANSYS, Inc. Proprietary Fluent User Services Center www.fluentusers.com Introductory FLUENT Notes FLUENT v6.3 December 2006 Turbulence Models Available in FLUENT RANS based models One-Equation Models Spalart-Allmaras Two-Equation Models Standard k–ε RNG k–ε Realizable k–ε Standard k–ω SST k–ω Reynolds Stress Model Detached Eddy Simulation Large Eddy Simulation Increase in Computational Cost Per Iteration

Introductory FLUENT Notes Fluent User Services cente ANSYS LUENT v6. 3 December 2006 wwfluentusers. com FLUENT RANS Modeling- Time Averaging Ensemble(time)averaging may be used to extract the mean flow properties from the instantaneous ones n(x)=mN∑x) l(x,) (x, t) (x) l4(x,)=l(x,)+(xt) Example: Fully-Developed Ins tantaneous Time-average fluctuating Turbulent Pipe Flow component component component Velocity profile he Reynolds-averaged ntum equations are OWS 0|,Dl t ax ax. a (Reynolds stress tensor) The Reynolds stresses are additional unknowns introduced by the averaging procedure, hence they must be modeled (related to the d flow quantities) order to close the system of governing equations C 2006 ANSYS. nc All ANSYS, Inc. Proprietar

© 2006 ANSYS, Inc. All rights reserved. 6-7 ANSYS, Inc. Proprietary Fluent User Services Center www.fluentusers.com Introductory FLUENT Notes FLUENT v6.3 December 2006 RANS Modeling – Time Averaging ◆ Ensemble (time) averaging may be used to extract the mean flow properties from the instantaneous ones: ◆ The Reynolds-averaged momentum equations are as follows ⚫ The Reynolds stresses are additional unknowns introduced by the averaging procedure, hence they must be modeled (related to the averaged flow quantities) in order to close the system of governing equations. ( ) ( )  ( ) = → = N n n i N i u t N u t 1 , 1 x, lim x u ( t) u ( t) u ( t) i i i x, = x, +  x, Fluctuating component Time-average component Example: Fully-Developed Turbulent Pipe Flow Velocity Profile u ( t) i x, u ( t) i  x, Instantaneous component Rij ui uj = −   j i j j i k i j i k i x R x u x x p x u u t u   +              +   = −           +    (Reynolds stress tensor) u ( t) i x

Introductory FLUENT Notes Fluent User Services cente ANSYS LUENT v6. 3 December 2006 wwfluentusers. com FLUENT The closure problem The RaNS models can be closed in one of the following ways (1) Eddy viscosity Models(via the boussinesq hypothesis) Ou Ou,2 ai puiu 8n-=pkδ Boussinesq hypothesis Reynolds stresses are modeled using an eddy(or turbulent) viscosity, u. The hypothesis is reasonable for simple turbulent shear flows: boundary layers, round jets, mixing layers, channel flows, etc (2)Reynolds-Stress Models(via transport equations for Reynolds stresses Modeling is still required for many terms in the transport equations RSM is more advantageous in complex 3D turbulent flows with large streamline curvature and swirl, but the model is more complex, computationally intensive, more difficult to converge than eddy viscosity models C 2006 ANSYS. nc All ANSYS, Inc. Proprietar

© 2006 ANSYS, Inc. All rights reserved. 6-8 ANSYS, Inc. Proprietary Fluent User Services Center www.fluentusers.com Introductory FLUENT Notes FLUENT v6.3 December 2006 The Closure Problem ◆ The RANS models can be closed in one of the following ways (1) Eddy Viscosity Models (via the Boussinesq hypothesis) ⚫ Boussinesq hypothesis – Reynolds stresses are modeled using an eddy (or turbulent) viscosity, μT. The hypothesis is reasonable for simple turbulent shear flows: boundary layers, round jets, mixing layers, channel flows, etc. (2) Reynolds-Stress Models (via transport equations for Reynolds stresses) ⚫ Modeling is still required for many terms in the transport equations. ⚫ RSM is more advantageous in complex 3D turbulent flows with large streamline curvature and swirl, but the model is more complex, computationally intensive, more difficult to converge than eddy viscosity models. i j i j k k i j j i i j i j k x u x u x u R u u  −     −            +   = −   =  3 2 3 2 T T

Introductory FLUENT Notes Fluent User Services cente ANSYS LUENT v6. 3 December 2006 wwfluentusers. com FLUENT Calculating turbulent Viscosity Based on dimensional analysis, Hr can be determined from a turbulence time scale(or velocity scale)and a length scale Turbulent kinetic energy [L/T2] k=u/2 Turbulence dissipation rate [L2/T3] &=v Ou /Ox, (Ou /Ox,+Ou /ax, Specific dissipation rate [1/T c/k Each turbulence model calculates uT differentl Spalart-Allmaras Solves a transport equation for a modified turbulent viscosity Standard k-8. rnG k-e Realizable k-e Solves transport equations for k and e k ● Standard k0.SSTk0 a Solves transport equations for k and o pk T C 2006 ANSYS. nc All ANSYS, Inc. Proprietar

© 2006 ANSYS, Inc. All rights reserved. 6-9 ANSYS, Inc. Proprietary Fluent User Services Center www.fluentusers.com Introductory FLUENT Notes FLUENT v6.3 December 2006 ◆ Based on dimensional analysis, μT can be determined from a turbulence time scale (or velocity scale) and a length scale. ⚫ Turbulent kinetic energy [L2 /T2 ] ⚫ Turbulence dissipation rate [L2 /T3 ] ⚫ Specific dissipation rate [1/T] ◆ Each turbulence model calculates μT differently. ⚫ Spalart-Allmaras: ◼ Solves a transport equation for a modified turbulent viscosity. ⚫ Standard k–ε, RNG k–ε, Realizable k–ε ◼ Solves transport equations for k and ε. ⚫ Standard k–ω, SST k–ω ◼ Solves transport equations for k and ω. Calculating Turbulent Viscosity 2 i i k = u  u  ( ) i j i j j i  =  u  x u  x + u  x =  k  = () ~ f T            = 2 k f T          = k f T

Introductory FLUENT Notes Fluent User Services cente ANSYS LUENT v6. 3 December 2006 wwfluentusers. com FLUENT The Spalart-Allmaras Model Spalart-Allmaras is a low-cost RANS model solving a transport equation for a modified eddy viscosity When in modified form the eddy viscosity is easy to resolve near the wall Mainly intended for aerodynamic/turbomachinery applications with mild separation, such as supersonic/transonic flows over airfoils, boundary-layer flows. etc Embodies a relatively new class of one-equation models where it is not necessary to calculate a length scale related to the local shear layer thickness Designed specifically for aerospace applications involving wall-bounded flows Has been shown to give good results for boundary layers subjected to adverse pressure gradients Gaining popularity for turbomachinery applications This model is still relatively new No claim is made regarding its applicability to all types of complex engineering fld OwS Cannot be relied upon to predict the decay of homogeneous, isotropic turbulence C 2006 ANSYS. nc All ANSYS, Inc. Proprietar

© 2006 ANSYS, Inc. All rights reserved. 6-10 ANSYS, Inc. Proprietary Fluent User Services Center www.fluentusers.com Introductory FLUENT Notes FLUENT v6.3 December 2006 The Spalart-Allmaras Model ◆ Spalart-Allmaras is a low-cost RANS model solving a transport equation for a modified eddy viscosity. ⚫ When in modified form, the eddy viscosity is easy to resolve near the wall. ◆ Mainly intended for aerodynamic/turbomachinery applications with mild separation, such as supersonic/transonic flows over airfoils, boundary-layer flows, etc. ◆ Embodies a relatively new class of one-equation models where it is not necessary to calculate a length scale related to the local shear layer thickness. ◆ Designed specifically for aerospace applications involving wall-bounded flows. ⚫ Has been shown to give good results for boundary layers subjected to adverse pressure gradients. ⚫ Gaining popularity for turbomachinery applications. ◆ This model is still relatively new. ⚫ No claim is made regarding its applicability to all types of complex engineering flows. ⚫ Cannot be relied upon to predict the decay of homogeneous, isotropic turbulence