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Two-Equation Models (The K -E Model)

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Two-Equation Models (The K -E Model) INTRODUCTION TO TURBULENCE MODELING Goodarz Ahmadi Department of Mechanical and Aeronautical Engineering Clarkson University Potsdam, NY 13699-5725 In this section, an introduction to the historical development in turbulence modeling is provided. Outline · Viscous Fluid · Turbulence · Classical Phenomenological models (Mixing Length) · One-Equation Models · Two-Equation Models (The k -e Model) · Stress Transport Models · Material-Frame Indifference · Continuum Approach · Consistency of the k -e Model · Rate-Dependent Model ME637 1 G. Ahmadi VISCOUS FLOW The conservation laws for a continuous media are: Mass ¶r + Ñ × (ru) = 0 ¶t Momentum du r = rf + Ñ × t dt Angular Momentum t T = t Energy re& = t : Ñu + Ñq + rh Entropy Inequality q rh r h& - Ñ ×( ) - > 0 T T Constitutive Equation Experimental evidence shows that for a viscous fluid, the stress is a function of velocity gradient. That is tkl = -pd kl + G kl (u i, j ) The velocity gradient term may be decomposed as ui, j = dij + wij where d ij is the deformation rate tensor and wij is the spin tensor. These are given as 1 1 d = (u + u ), w = (u - u ) kl 2 k,l l,k kl 2 k,l l,k ME637 2 G. Ahmadi The principle of Material Frame-Indifference of continuum mechanics implies that the stress is generated only by the deformation rate of media and the spin has no effect. This is because both stress and deformation rate tensors are frame-indifferent while spin is not. Thus, the general form of the constitutive equation is given as tkl = -pdkl + Fkl (dij ) For a Newtonian fluid, the constitutive equation is linear and is given as tkl = (-p + lu i,i )dkl + 2md kl The entropy inequality imposed the following restrictions on the coefficient of viscosity: 3l + 2m > 0 , m >0 , Using the constitutive equation in the balance of momentum leads to the celebrated Navier-Stokes equation. For an incompressible fluid the Navier-Stokes and the continuity equations are given as 2 ¶u i ¶u i ¶p ¶ u i r( + u j ) = - + m , ¶t ¶x j ¶x i ¶x j¶x j ¶u i = 0 ¶xi These form four equations for evaluating four unknowns u i , p. TURBULENT FLOW In turbulent flows the field properties become random functions of space and time. Thus u i = U i + u i¢ ui = Ui , u¢i = 0 p = P + p¢ p = P , p¢ = 0 Substituting the decomposition into the Navier-Stokes equation and averaging leads to the Reynolds equation. ME637 3 G. Ahmadi Reynolds Equation 2 ¢ ¢ ¶U i ¶U i 1 ¶P ¶ U i ¶u i u j + U j = - + n - ¶t ¶x j r ¶x i ¶x j x j ¶x j Here T tij = -ru¢i u¢j =Turbulent Stress Tensor First Order Modeling (Classical Phenomenology) Boussinesq Eddy Viscosity: dU tT = -ru¢v¢ = rn 21 T dy T tij ¶U i ¶U j 1 = -u¢i u¢j = n T ( + ) - u¢k u¢k dij r ¶x j ¶x i 3 Prandtl Mixing Length ¶U ¶U tT = rl2 | | 21 m ¶y ¶y 2 ¶U n T ¶T n T = lm | |, - T¢v¢ = ¶y s T ¶y Kolmogorov-Prandtl Expression Eddy Viscosity n T » cu l , u = velocity scale, l= length scale, c = const. Kinematic Viscosity n µ cl , c = speed of sound, l = mean free path Let ¶U ¶U u ~ l | |, l = l Þ n = l 2 | | m ¶y m T m ¶y For free shear flows ME637 4 G. Ahmadi l m ~ cl 0 ( l 0 = half width) Close to a wall l m = ky (y = distance from the wall) Local Equilibrium For local equilibrium Mixing length Hypothesis Production = Dissipation ó Shortcomings of the Mixing Length Model ¶U · When = 0 Þ n = 0 ¶y T · Lack of transport of scales of turbulence · Estimating the mixing length, l m . Reattachment Point ¶U At the reattachment point = 0 which leads to vanishing eddy diffusivity and ¶y thus negligible heat flux. Experiments, however, show that the heat flux is maximum at the reattachment point. ME637 5 G. Ahmadi One-Equation Models Eddy Viscosity 1 n = c k1/ 2l , k = u¢u¢ = Turbulence Kinetic Energy T m 2 i i Exact k-equation d u¢u¢ ¶ u¢u¢ P¢ ¶U ¶u¢ ¶u¢ ¶ 2 u¢u¢ i i = - u¢ ( i i + ) - u¢u¢ i - n i i + n i i dt 2 ¶x k 2 r i j ¶x ¶x ¶x ¶x x 2 14243 144k 424443 14243j 142j 43j 144j2j 443 Convective TurbulenceDiffusion Transport Production Dissipation Viscous Diffusion where d u¢i u¢i d ¶ ¶ = convective transport, = + U j dt 2 dt ¶t ¶x j ¶ u i¢u¢i P¢ u¢k ( + ) = turbulence diffusion ¶x k 2 r ¶U i u¢i u¢j = production ¶x j ¶u¢ ¶u¢ n i i = , = dissipation ¶x j ¶x j ¶ 2 u¢u¢ n i i = viscous diffusion ¶x jx j 2 Modeled k-equation ¶U 3/ 2 dk ¶ n T ¶k ¶Ui j ¶Ui k = ( ) + n T ( + ) - cD , dt ¶x j sk ¶x j ¶x j ¶x i ¶x j l where ¶ n T ¶ k ( ) = turbulence diffusion, sk »1(turbulence Prandlt number) ¶x j sk ¶x j ¶U ¶Ui j ¶Ui n T ( + ) = production, ¶x j ¶x i ¶x j ME637 6 G. Ahmadi k 3/ 2 c = e = dissipation. D l Note that the turbulence length scale l is given by an algebraic equation. Bradshaw’s Model Modeled k-equation dk ¶ t ¶U k 3/ 2 = (Bk Max ) + ak - c dt ¶y r ¶y D l where ¶U ak = production ¶y k 3 / 2 c = dissipation D l - u¢v¢ = ak (shear stress µ kinetic energy) tMax y y B = 2 g( ) , l = df ( ) rv 0 d d Shortcomings of One-Equation Models · Use of an algebraic equation for the length scale is too restrictive. · Transport of the length scale is not accounted for. Transport of Second Scale (Boundary Layer) The transport of a second turbulence scale, z, is given as z-Equation dz ¶ n T ¶z n T ¶U 2 k = ( ) + z[c1 ( ) - c2 ] + Sz dt ¶y sz ¶y k ¶y n T where ME637 7 G. Ahmadi ¶ n ¶z ( T ) = diffusion ¶y s z ¶y n ¶U c T ( )2 = production 1 k ¶y k c2 = destruction, n T Sz = secondary source Choices for z Turbulence Time Scale = l 2 / k Turbulence frequency Scale = k / l2 Turbulence mean-square vorticity Scale = k / l2 ¶u¢ ¶u¢ Turbulence Dissipation = e = n i i ¶x j ¶x i z = kl e -Equation (exact): de ¶ ¶u¢ ¶u¢ ¶u¢ ¶ 2u¢ ¶ 2 u¢ = - (u¢e') - 2n i i k - 2n i i dt ¶x j ¶x ¶x ¶x ¶x ¶x ¶x ¶x 14j243 144k 24l 43l 144k42l 44k 43l Diffusion Generationby Viscousdestruction vortexstretching Note that k 3/ 2 k 3/ 2 E(k) e ~ , l ~ Universal Equilibrium l e Thus , k 2 n ~ k l ~ , T e Note also that e is also the amount of energy that paths through the Inertia entire spectrum of eddies of Subrange turbulence. k Schematics of turbulence energy spectrum. ME637 8 G. Ahmadi Two-Equation Models The k - e Model c k 2 ¶U m ¶Ui j 2 n T = , - u¢i u¢j = n T ( + ) - kd ij e ¶x j ¶x i 3 k-equation dk ¶ u ¶k ¶U ¶U ¶U = ( T ) + n ( i + j ) i - e dt ¶x s ¶x T ¶x ¶x ¶x 123 14j42k443j 144j424i443j dissipation Diffusion Pr oduction e -equation ¶U 2 de ¶ u T ¶k e ¶U i j ¶U i e = ( ) + c e1n T ( + ) - ce2 , dt ¶x j se ¶x j k ¶x j ¶x i ¶x j 123k 1442443 14444244443 Distruction Diffusion Generation where cm = 0.09 , ce1 = 1.45 , c e 2 = 1.9 , s k = 1, se = 1.3, (Jones and Launder, 1973) Momentum dU i 1 ¶P ¶ = - - u i¢u¢j dt r ¶x i ¶x j Mass ¶U i = 0 ¶x i Closure: Six Equations for six unknowns, v i , P, k , e . Kolmogorov Model dk 1 2 ¶ k ¶k = 2uTSijSij - k w + A¢¢ ( ) dt 14243 2 ¶x j w ¶x j Pr oduction 14243 Dissipation 1442443 Diffusion ME637 9 G. Ahmadi dw 7 ¶ k ¶k = - w2 + 2A¢ ( ) dt 10 ¶x j w ¶x j ¶U Ak 1 ¶Ui j n T = , Sij = ( + ) w 2 ¶x j ¶x i Comparison of Model Predictions In this section comparisons of the predictions of the mixing length and one and two-equation models with the experimental data for simple turbulent shear flows are presented. Development of Plane Mixing Layer (Rodi, 1982) It is seen that the k - e model captures the features of the flow more accurately when compared with the one-equation and mixing length model. ME637 10 G. Ahmadi Turbulent Recirculating Flow (Durst and Rastogi, 1979) The k - e model predictions for turbulent flow in a channel with an obstructing block are compared with the experimental data. b) Velocity profiles Flow in a Square Cavity (Gosman and Young) The k - e model predictions for a square cavity are shown in this section. ME637 11 G. Ahmadi Free-Stream Turbulence The free stream turbulence affects the skin friction coefficient. The mixing length model can not predict such effects. The k - e model does a reasonable job in predicting the increase of skin friction coefficient with the free stream turbulence. Turbulent Channel Flow (Rodi, 1980) Distribution of mean velocity and turbulence quantities in fully developed two- dimensional channel flows was predicted by Rodi (1980) using an algebraic stress model (a modified k - e model) Closed Channel Flow ME637 12 G. Ahmadi Open Channel Flow Jets Issuing in Co-flowing Streams (Rodi, 1982) 1 ¥ For a jet, q = ò U(U - U E )dy = Excess momentum thickness U E 0 Plane Jet Round Jet ME637 13 G.
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