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Turbulence Models and Their Applications TURBULENCE MODELS AND THEIR APPLICATIONS Presented by: T.S.D.Karthik Department of Mechanical Engineering IIT Madras Guide: Prof. Franz Durst 10th Indo German Winter Academy 2011 2 Outline 3 Turbulence models introduction Boussinesq hypothesis Eddy viscosity concept Zero equation model One equation model Two equation models Algebraic stress model Reyolds stress model Comparison Applications Developments Conclusion Turbulence Models and Their Applications Turbulence models 4 A turbulence model is a procedure to close the system of mean flow equations. For most engineering applications it is unnecessary to resolve the details of the turbulent fluctuations. Turbulence models allow the calculation of the mean flow without first calculating the full time-dependent flow field. We only need to know how turbulence affected the mean flow. In particular we need expressions for the Reynolds stresses. For a turbulence model to be useful it: must have wide applicability, be accurate, simple, and economical to run. Turbulence Models and Their Applications Common turbulence models 5 Classical models. Based on Reynolds Averaged Navier-Stokes (RANS) equations (time averaged): Zero equation model: mixing length model. One equation model Two equation models: k- style models (standard, RNG, realizable), k- model, and ASM. Seven equation model: Reynolds stress model. The number of equations denotes the number of additional PDEs that are being solved. Large eddy simulation. Based on space-filtered equations. Time dependent calculations are performed. Large eddies are explicitly calculated. For small eddies, their effect on the flow pattern is taken into account with a “sub-grid model” of which many styles are available. DNS Turbulence Models and Their Applications Classification 6 Turbulence Models and Their Applications Prediction Methods 7 Turbulence Models and Their Applications Boussinesq hypothesis 8 Many turbulence models are based upon the Boussinesq hypothesis. It was experimentally observed that turbulence decays unless there is shear in isothermal incompressible flows. Turbulence was found to increase as the mean rate of deformation increases. Boussinesq proposed in 1877 that the Reynolds stresses could be linked to the mean rate of deformation. Using the suffix notation where i, j, and k denote the x-, y-, and z-directions respectively, viscous stresses are given by: u u i j ij x j xi Similarly, link Reynolds stresses to the mean rate of deformation Turbulence Models and Their Applications Eddy Viscosity Concept 9 One of the most widely used concept Reynold’s stress tensor – U i U j T u 'u ' ij i j t x j xi A new quantity appears: the turbulent viscosity or eddy viscosity (νt ). The second term is added to make it applicable to normal turbulent stress. The turbulent viscosity depends on the flow, i.e. the state of turbulence. The turbulent viscosity is not homogeneous, i.e. it varies in space. Turbulence Models and Their Applications Eddy Viscosity Concept 10 It is, however, assumed to be isotropic. It is the same in all directions. This assumption is valid for many flows, but not for all (e.g. flows with strong separation or swirl). The turbulent viscosity may be expressed as 2 t uclc (or) lc /tc This concept assumes that Reynolds stress tensor can be characterized by a single length and time scales. Turbulence Models and Their Applications Major Drawbacks 11 Interaction among eddies is not elastic as in the case for molecular interactions in kinetic theory of gases. For many turbulent flows, the length and time scale of characteristic eddies is not small compared with the flow domain (boundary dominated flows). The eddy viscosity is a scalar quantity which may not be true for simple turbulent shear flows. It also fails to distinguish between plane shear, plane strain and rotating plane shear flows. Successful – 2D shear flows. Erroneous results for simple shear flows such as wall jets and channel flows with varying wall roughness. Turbulence Models and Their Applications Zero Equation Model - Mixing Length Model 12 On dimensional grounds one can express the kinematic turbulent viscosity as the product of a velocity scale and a length scale: 2 t (m / s) (m/ s)(m) If we then assume that the velocity scale is proportional to the length scale and the gradients in the velocity (shear rate, which has dimension 1/s): U y we can derive Prandtl’s (1925) mixing length model: U 2 t m y Algebraic expressions exist for the mixing length for simple 2-D flows, such as pipe and channel flow. Turbulence Models and Their Applications Equations for mixing length 13 Wall boundary layers lm Ky (y / / K) lm (y / / K) δ = boundary layer thickness y = distance from the wall λ = 0.09 K = Von-Karman Constant Developed pipe flows l y y m 0.14 0.08(1 )2 0.06(1 )4 R R R R = radius of the pipe or the half width of the duct Turbulence Models and Their Applications Mixing Length Model Discussion 14 Advantages: Easy to implement. Fast calculation times. Good predictions for simple flows where experimental correlations for the mixing length exist. Used in higher models Disadvantages: Completely incapable of describing flows where the turbulent length scale varies: anything with separation or circulation. Only calculates mean flow properties and turbulent shear stress. Cannot switch from one type of region to another History effects of turbulence are not considered. Use: Sometimes used for simple external aero flows. Pretty much completely ignored in commercial CFD programs today. Turbulence Models and Their Applications One Equation Model 15 Different transport equation for ‘k’ is solved. L is defined algebraically, in a similar manner as mixing length. 1/ 2 uc k ν t lm k PDE for turbulent KE : Diffusion, Production and Dissipation terms k Di Ui P ε xi xi where Modeled Equation u 'u ' p'u ' j j 1 D ( i 2ν u 'u 'u ' ) i ρ x 2 j j i 3 i k k U U U k 2 U t j i j j Ui t CD P u 'u ' x x x x x x l i j x i i k i i j i c i u' u' j j ε ν x x i i Turbulence Models and Their Applications One Equation Model Discussion 16 Economical and accurate for: Attached wall-bounded flows. Flows with mild separation and recirculation. Developed for use in unstructured codes in the aerospace industry. Popular in aeronautics for computing the flow around aero plane wings, etc. Weak for: Massively separated flows. Free shear flows. Decaying turbulence. Complex internal flows. Characteristic length still experimental. Turbulence Models and Their Applications Two Equation Models – The k-ε model 17 Equations for k and ε, together with the eddy-viscosity stress- strain relationship constitute the k-ε turbulence model. ε is the dissipation rate of k. If k and ε are known, we can model the turbulent viscosity as: k 3/ 2 k 2 k1/ 2 t Turbulence Models and Their Applications The k-ε model 18 K equation: Model (simplified) equation for k after using Boussinesq assumption by which the fluctuation terms can be linked to the mean flow is as follows: 2 k k U j Ui U j t k k Ui t t xi xi x j xi xi k xi xixi k 2 with 0.09 t Turbulence Models and Their Applications Turbulent Dissipation 19 We can define the rate of dissipation per unit mass ε as: 2 eij '.eij ' Equation for ε D Dv D f D p P1 P2 P3 P4 Y Dt where v Dε Diffusive viscous transport f Dε Diffusive transportby fluctuation p Dε Diffusive transportby pressurefluctuation 1 Pε Production by deformation of mean flow field 2 Pε Production by deformation of mean flow field 3 Pε Production by gradient of mean vorticity 4 Pε Production by vortex stretching Y Viscous destruction Turbulence Models and Their Applications Turbulent Dissipation 20 u ' D 2 p' k u' Dt x x x k x x x k k k k i i v f p D D D 2 2 2 u' u' u' u' U u' U u' u' u' u' 2 i k l kl i 2 u' i i 2 k i k 2 i x x x x x k x x x x x x x x l l i k k l k l k l l k l 1 2 3 4 P P P P Y Turbulence Models and Their Applications Model Equation for ε 21 A model equation for ε is derived by multiplying the k equation by (ε/k) and introducing model constants. k 2 C t U 2 j t Cz1P Cz2 x j x j k x j K K U U U i j j P t x j xi xi Closure coefficients found emperically k 1 Cz2 1.92 Cz1 1.44 C 0.09 z 1.33 K- ε model leads to all normal stresses being equal, which is usually inaccurate. Turbulence Models and Their Applications Applications 22 Flow on a backward facing step using k-epsilon model Turbulence Models and Their Applications Single and multiple jet flows using k-epsilon models 23 Turbulence Models and Their Applications k-ε model discussion 24 Advantages: Relatively simple to implement. Leads to stable calculations that converge relatively easily. Reasonable predictions for many flows. Disadvantages: Poor predictions for: swirling and rotating flows, flows with strong separation, axis symmetric jets, certain unconfined flows, and fully developed flows in non-circular ducts. Valid only for fully turbulent flows. Requires wall function implementation. Modifications for flows with highly curved stream lines. Production of turbulence in highly strained flows is over predicted. Turbulence Models and Their Applications More two equation models 25 The k-ε model was developed in the early 1970s.
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