Modelling of turbulent flows: RANS and LES Turbulenzmodelle in der - PowerPoint PPT Presentation

Introduction Constructing Reynolds-stress models Performance 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 5 1 / 28

Introduction Constructing Reynolds-stress models Performance LECTURE 5 Reynolds-stress transport models 2 / 28

Introduction Constructing Reynolds-stress models Performance Questions to be answered in the present lecture How can the equations be closed at the second-moment level? ◮ why resort to Reynolds-stress models? ◮ how to derive the � u ′ i u ′ j � transport equation? ◮ how to model the principal unknown terms? How do Reynolds-stress models perform? 3 / 28

Introduction Constructing Reynolds-stress models Performance Why use Reynolds-stress transport models? Fundamental deficiency of turbulent viscosity models: ◮ Reynolds stress is assumed local function of mean strain-rate → transport/history effects are neglected (e.g. failure in relaxation from mean strain – cf. lecture 3) Attractive features of Reynolds-stress transport models: ◮ avoid any turbulent viscosity hypothesis ◮ transport & production terms are in closed form → transport effects “built-in” → stress production model-free (important in complex strain) 4 / 28

Introduction Deriving the stress transport equation Constructing Reynolds-stress models Modelling the pressure-strain correlation Performance Inhomogeneous flows Deriving the transport equation for the Reynolds stress Steps in deriving the exact equation from Navier-Stokes note that ∂ t � u ′ i u ′ j � = � u ′ j ∂ t u ′ i + u ′ i ∂ t u ′ j � 1. write transport equation for fluctuating velocity u ′ 2. multiply i th-component with u ′ j 3. multiply j th-component with u ′ i 4. add results from 2. and 3. 5. take average of result from 4. 5 / 28

Introduction Deriving the stress transport equation Constructing Reynolds-stress models Modelling the pressure-strain correlation Performance Inhomogeneous flows The exact transport equation for the Reynolds stress     ¯ D � u ′ i u ′ j �  k � + 1 j � δ ik + 1  � u ′ i u ′ j u ′ ρ � p ′ u ′ ρ � p ′ u ′ i � δ jk − ν � u ′ i u ′   + j � ,k =   ¯ D t     � �� � turbulent transport T kij ,k + 1 � � −� u i � ,k � u ′ k u ′ j � − � u j � ,k � u ′ k u ′ � p ′ u ′ j,i � + � p ′ u ′ 2 ν � u ′ i,k u ′ i � i,j � − j,k � ρ � �� � � �� � � �� � production P ij dissipation tensor ε ij pressure-strain R ij ◮ pressure–rate-of-strain R ij and dissipation ε ij are unclosed ◮ first three terms of turbulent transport T kij are unclosed ◮ half the trace of this equation yields TKE equation ◮ pressure–rate-of-strain is absent in TKE equation 6 / 28

Introduction Deriving the stress transport equation Constructing Reynolds-stress models Modelling the pressure-strain correlation Performance Inhomogeneous flows Importance of the terms in a boundary layer flow ¯ D � u ′ i u ′ j � −� u ′ i u ′ j u ′ + ν � u ′ i u ′ 0 = − k � ,k j � ,kk + P ij +Π ij − ε ij ¯ D t (1) (2) (3) (4) (5) (6) Budget of streamwise normal stress � u ′ u ′ � ◮ principal production: P 11 ≈ − 2 � u ′ v ′ �� u � ,y ◮ mainly balanced by: dissipation & pressure–rate-of-strain note: Π ij ≡ R ij − � � 1 � p ′ u ′ j � δ ik + � p ′ u ′ i � δ jk ρ ,k (DNS Spalart 1988, Re θ = 1410 ) 7 / 28

Introduction Deriving the stress transport equation Constructing Reynolds-stress models Modelling the pressure-strain correlation Performance Inhomogeneous flows Importance of the terms in a boundary layer flow ¯ D � u ′ i u ′ j � −� u ′ i u ′ j u ′ + ν � u ′ i u ′ 0 = − k � ,k j � ,kk + P ij +Π ij − ε ij ¯ D t (1) (2) (3) (4) (5) (6) Budget of wall-normal stress � v ′ v ′ � ◮ no production: P 22 ≈ 0 ◮ gain from pressure–rate-of-strain ◮ approximately balanced by dissipation (DNS Spalart 1988, Re θ = 1410 ) 8 / 28

Introduction Deriving the stress transport equation Constructing Reynolds-stress models Modelling the pressure-strain correlation Performance Inhomogeneous flows Nature of the pressure–rate-of-strain correlations Observation from flow data: ◮ pressure terms are of significant magnitude ◮ pressure–rate-of-strain correlation has redistributive character ◮ due to incompressibility, term has zero trace: � � R ij ≡ 1 � p ′ u ′ j,i � + � p ′ u ′ i,j � → R ii = 0 ρ ⇒ no contribution to turbulent kinetic energy Pressure–rate-of-strain is main challenge for modelling! 9 / 28

Introduction Deriving the stress transport equation Constructing Reynolds-stress models Modelling the pressure-strain correlation Performance Inhomogeneous flows Dissipation tensor components in boundary layer flow 2.5 Observations: 2.0 ε ij ◮ for high Reynolds number: ~ ε 11 2 ε 3 1.5 dissipation tensor is ε 33 1.0 approximately isotropic ε 22 ◮ low Reynolds in DNS: 0.5 → some residual anisotropy 0.0 ε 12 but: significant anisotropy near -0.5 0.0 0.2 0.4 0.6 0.8 1.0 wall (cf. lecture 6) y/ δ (DNS Spalart 1988, Re θ = 1410 ) ⇒ dissipation tensor often modelled as isotropic: ε ij = 2 3 ˜ ε δ ij 10 / 28

Introduction Deriving the stress transport equation Constructing Reynolds-stress models Modelling the pressure-strain correlation Performance Inhomogeneous flows Unclosed terms in the Reynolds-stress transport equation Reynolds-stress equation, assuming isotropic dissipation:       ¯ D � u ′ i u ′ j �   + 1 j � δ ik + 1 � u ′ i u ′ j u ′ ρ � p ′ u ′ ρ � p ′ u ′ = P ij + R ij − 2  + ν � u ′ i u ′  + k � i � δ jk j � ,k 3 ˜ εδ ij ¯   D t   � �� �   � �� �  T ( u )  T ( p ) kij kij ,k Models need to be prescribed for the following terms: ◮ triple correlation T ( u ) kij and pressure transport T ( p ) kij ◮ the scalar (pseudo) dissipation rate ˜ ε → similar to k - ε model ◮ the pressure–rate-of-strain correlation R ij 11 / 28

Introduction Deriving the stress transport equation Constructing Reynolds-stress models Modelling the pressure-strain correlation Performance Inhomogeneous flows The Poisson equation for pressure fluctuations Fluctuating pressure: given by linear equation 1 ρ ∇ 2 p ′ = − 2 � u i � ,j u ′ � � u ′ i u ′ j − � u ′ i u ′ j,i − j � ,ij ◮ pressure can be decomposed into 3 contributions p ′ = p ( h ) + p ( r ) + p ( s ) ∇ 2 p ( h ) = 0 ◮ homogeneous pressure p ( h ) : ∇ 2 p ( r ) = − 2 ρ � u i � ,j u ′ ◮ rapid pressure p ( r ) : j,i � � ∇ 2 p ( s ) = − ρ ◮ slow pressure p ( s ) : u ′ i u ′ j − � u ′ i u ′ j � ,ij ⇒ 3 different contributions to pressure-strain correlation 12 / 28

Introduction Deriving the stress transport equation Constructing Reynolds-stress models Modelling the pressure-strain correlation Performance Inhomogeneous flows Contributions to pressure-strain correlation R ( h ) ij ≡ � p ( h ) ( u ′ i,j + u ′ Homogeneous pressure → j,i ) � /ρ ◮ influenced by boundary conditions only ◮ R ( h ) vanishes in homogeneous turbulence ij ◮ contribution important near walls (details in lecture 6) Rapid pressure ◮ reacts instantly to mean velocity gradients ◮ dominant contribution for large strain rate S k/ε Slow pressure ◮ determined by self-interaction of turbulent field ◮ principal mechanism for return to isotropy without strain 13 / 28

Introduction Deriving the stress transport equation Constructing Reynolds-stress models Modelling the pressure-strain correlation Performance Inhomogeneous flows Modelling the slow part of pressure-strain Homogeneous turbulence without mean velocity gradients: j � = R ij − 2 ∂ t � u ′ i u ′ 3 ˜ εδ ij ◮ no mean velocity gradients → R ij = R ( s ) ij R ( s ) ε F ( s ) ◮ modelling ansatz: ij = ˜ ij ( b ij ) ◮ the most general tensor function is: � � ij − 1 F ( s ) b 2 3 b 2 = C 1 b ij + C 2 kk δ ij ij ◮ C 1 , C 2 are scalar functions ◮ Rotta’s linear model: C 1 = − 2 C R , C 2 = 0 ⇒ linear return-to-isotropy: d t b ij = − ( C R − 1) ˜ ε k b ij 14 / 28

M = 50 : 8 mm 1 : 5 m 5 : 13 m Introduction Deriving the stress transport equation Constructing Reynolds-stress models Modelling the pressure-strain correlation Performance Inhomogeneous flows Slow pressure-strain models: comparison with experiment Return-to-isotropy after distorting duct evolution of b ij in straight duct section b = u u =u u � 1 = 3 � b 33 ij i j k k ij I I = b b I I I = b b b ij ij ik k j j i I I I I I I I I > 0 I I I < 0 b 22 ◮ mean strain is imposed in a I I I < 0 distorting duct b 11 � 1 � 1 U = 6 : 06 ms I I I > 0 7 : 2 ms o I I I < 0 ◮ then: in straight section, distance x turbulence relaxes to isotropy ( × , ◦ , • experiment LePenven et al. 1985) ◮ simple linear model works in (— — Rotta model, C R = 1 . 5 ) this case 15 / 28