Introduction Algebraic stress models Non-linear eddy viscosity models Modelling of turbulent flows: RANS and LES Turbulenzmodelle in der Str¨ omungsmechanik: RANS und LES Markus Uhlmann Institut f¨ ur Hydromechanik Karlsruher Institut f¨ ur Technologie www.ifh.kit.edu SS 2012 Lecture 7 1 / 16
Introduction Algebraic stress models Non-linear eddy viscosity models LECTURE 7 Algebraic stress models 2 / 16
Introduction Algebraic stress models Non-linear eddy viscosity models Questions to be answered in the present lecture How can the linear eddy viscosity assumption be avoided without the need for solving transport equations? ◮ algebraic stress models ◮ nonlinear eddy viscosity models 3 / 16
Introduction Algebraic stress models Non-linear eddy viscosity models Intermediate between RSM and Boussinesq approximation Reynolds stress transport models ◮ naturally incorporate transport effects ◮ describe stress production exactly BUT: high computational cost (equations for 6 components) Standard eddy-viscosity models (Boussinesq approximation) (A) local relation between Reynolds stress and mean strain (B) linear relation between Reynolds stress and mean strain � (A) is inevitable ⇒ (B) can be changed ⇒ non-linear Reynolds stress/mean strain relationships 4 / 16
Introduction Algebraic stress models Non-linear eddy viscosity models Algebraic stress models ¯ D � u ′ i u ′ j � = P ij + R ij − 2 Reynolds stress + ( T kij ) ,k 3 ˜ εδ ij ¯ D t equations for � �� � transport model: ≡D ij (transport) Basic idea of algebraic stress models (ASM): ◮ approximating transport terms D ij by local expressions → resulting model is free from derivatives: 6 algebraic equations relating � u ′ i u ′ j � , k , ˜ ε , � u i � ,j ⇒ approach benefits from known models for pressure-strain R ij 5 / 16
Introduction Algebraic stress models Non-linear eddy viscosity models Algebraic stress models – equilibrium assumption ¯ D � u ′ i u ′ j � = P ij + R ij − 2 Reynolds stress + ( T kij ) ,k 3 ˜ εδ ij ¯ equations for D t � �� � transport model: ≡D ij (transport) Simplest local equilibrium assumption: ◮ neglect the transport term altogether: D ij = 0 1 ⇒ implies for the turbulent energy: 2 D ll = P − ˜ ε = 0 � problem: equality P = ˜ ε not verified in general! 6 / 16
Introduction Algebraic stress models Non-linear eddy viscosity models Algebraic stress models – weak equilibrium assumption ¯ D � u ′ i u ′ j � = P ij + R ij − 2 Reynolds stress + ( T kij ) ,k 3 ˜ εδ ij ¯ equations for D t � �� � transport model: ≡D ij (transport) Weak equilibrium assumption (Rodi 1972) ◮ rewriting Reynolds stress in terms of anisotropy and TKE: j � = 2 k b ij + 2 � u ′ i u ′ 3 k δ ij ◮ neglecting transport of anisotropy: � � u ′ � ¯ D � u ′ i u ′ � u ′ i u ′ ¯ ¯ i u ′ � u ′ i u ′ ¯ j � j � j � j � D k D D k ✘ D t + ✘✘✘✘✘✘ = k ≈ ¯ ¯ ¯ ¯ k k k D t D t D t ◮ applying the approximation to the entire transport term: � u ′ i u ′ � u ′ i u ′ � u ′ i u ′ j � j � j � 1 D ij ≈ · ( transport of k ) = 2 D ll = ( P − ˜ ε ) k k k � u ′ i u ′ j � ε ) = P ij + R ij − 2 ⇒ final model: ( P − ˜ 3 ˜ εδ ij k 7 / 16
Introduction Algebraic stress models Non-linear eddy viscosity models ASM predictions for homogeneous shear flow LRR-IP pressure-strain model 0.3 b ij R ij = 0.2 b 11 εb ij − C 2 ( P ij − 2 − C R 2˜ 3 P δ ij ) ◮ corresponding ASM: 0.1 1 P ij − 2 2 (1 − C 2 ) 3 δ ij P b ij = ε · 0.0 C R − 1+ P / ˜ ε ˜ b 22 =b 33 ◮ in homogeneos shear flow: -0.1 b ij has finite limit for P b 12 ε → ∞ -0.2 ˜ b 12 (k- ε ) b 11 → 4 15 0 1 2 3 4 b 22 → − 2 P/ ε P / ˜ ε 15 b 12 → − 1 — —, ASM predictions; – – – –, k - ε model 5 (from Pope “Turbulent Flows”, 2000) ⇒ stress remains realizable 8 / 16
Introduction Algebraic stress models Non-linear eddy viscosity models Stress/mean strain relation implied by ASM 0.30 LRR-IP pressure-strain model C µ ◮ define: 0.20 −� u ′ v ′ � = C µ k 2 ε � u � ,y ˜ with unknown function C µ 0.10 ◮ substituting ASM: 2 3 (1 − C 2 )( C R − 1+ C 2 P / ˜ ε ) C µ = 0.00 ε ) 2 ( C R − 1+ P / ˜ 1 2 3 4 P/ ε ⇒ C µ decreases with P / ˜ ε P / ˜ ε — —, ASM predictions (from Pope “Turbulent Flows”, 2000) 9 / 16
Introduction Algebraic stress models Non-linear eddy viscosity models Assessing the ASM approach Achievements of algebraic stress models ◮ partial differential equations reduced to algebraic equations ◮ physics of pressure-strain model is carried over Problems of the ASM approach ◮ implicit system of equations ◮ dependence is in general non-linear ◮ system can have multiple solutions ◮ numerical stiffness 10 / 16
Introduction Performance in free shear flow Algebraic stress models Performance in flow with system rotation Non-linear eddy viscosity models Explicit ASM or non-linear eddy viscosity models Explicit ASM (EASM) ◮ explicit expressions for the stress components are numerically desirable ◮ there are two routes (viewpoints) to achieve this: b ij = f i ( b ij , k 1. construct an implicit ASM (as above): ε � u i � ,j ) ˜ then derive equivalent explicit form analytically b ij = f e ( k ⇒ ε � u i � ,j ) ˜ 2. construct an explicit expression for the Reynolds stresses: b ij = f e ′ ( k ⇒ ε � u i � ,j ) ˜ ⇒ both approaches have been realized ⇒ results also known as “non-linear eddy viscosity models” 11 / 16
Introduction Performance in free shear flow Algebraic stress models Performance in flow with system rotation Non-linear eddy viscosity models Deriving explicit algebraic stress models � � S , � � ◮ ansatz: b ij = B ij Ω where normalized mean rate of strain/rotation are defined: � � S ij ≡ k Ω ij ≡ k ε ( � u i � ,j + � u j � ,i ) , ε ( � u i � ,j − � u j � ,i ) 2˜ 2˜ ◮ most general consistent expression (Pope 1975): � � = � 10 n =1 G ( n ) � T ( n ) S , � � B ij Ω ij ◮ with independent, symmetric, deviatoric functions: T (1) T (2) T (3) S 2 − 1 S 2 ) I � = � � = � S � Ω − � Ω � � = � 3 trace ( � S S ij ij ij Ω 2 − 1 Ω 2 − 2 T (4) Ω 2 ) I T (5) S 2 − � S 2 � T (6) Ω 2 � Ω 2 ) I � = � 3 trace ( � � = � Ω � � = � S + � S � 3 trace ( � S � Ω ij ij ij T (7) Ω 2 − � Ω 2 � T (8) T (9) Ω 2 � S 2 + � Ω 2 − 2 Ω 2 ) I S 2 − � S 2 � S 2 � S 2 � � = � Ω � S � S � � = � S � Ω � Ω � � = � 3 trace ( � Ω S ij ij ij Ω 2 − � T (10) S 2 � Ω 2 � S 2 � � = � Ω � Ω ij and undetermined scalar coefficients G ( n ) 12 / 16
Introduction Performance in free shear flow Algebraic stress models Performance in flow with system rotation Non-linear eddy viscosity models Examples of EASM Linear case – Boussinesq hypothesis ◮ G (1) = − C µ ; G ( n ) = 0 for n ≥ 2 b ij = − C µ � → S ij Statistically two-dimensional flow (Pope, 1975) ( G ( n ) = 0 for n ≥ 4 ) ◮ sum contains only three terms General three-dimensional flow ◮ all 10 terms are non-zero 1. ASM approach: Gatski & Speziale (1993), based on linear pressure-strain 2. direct approach: Shih, Zhu & Lumley (1995), based on realizability 13 / 16
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