Introduction Filtering the equations More 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 9 1 / 15
Introduction Filtering the equations More LECTURE 9 LES equations 2 / 15
Introduction Filtering the equations More Questions to be answered in the present lecture Which unknowns are generated by filtering the equations? How can the residual stresses be decomposed? What does the kinetic energy balance in LES involve? 3 / 15
Introduction Filtering the equations More Roadmap The basic elements of the LES approach 1. definition of a spatial filter 2. derivation of the filtered Navier-Stokes equations 3. choice of a model for the unclosed subgrid-stress term 4. numerical solution of the closed equations 4 / 15
Introduction Continuity & momentum Filtering the equations Kinetic energy More Filtering the Navier-Stokes equations Navier-Stokes equations for incompressible flow ◮ instantaneous velocity u ( x , t ) , pressure p ( x , t ) ◮ Cartesian coordinates, index notation, u = ( u 1 , u 2 , u 3 ) T ∂ ( u j u i ) ∂ 2 u ∂u i + 1 ∂p ∂t + = ν ∂x j ρ ∂x i ∂x j ∂x j ∂u j = 0 ∂x j ◮ recall: Reynolds decomposition u ( x , t ) = � u � + u ′ u ( x , t ) = u + u ′′ ◮ here: apply spatial filtering (notation u ′′ to distinguish from statist. fluctuations) 5 / 15
Introduction Continuity & momentum Filtering the equations Kinetic energy More Filtering the Navier-Stokes equations (2) Applying a spatial filter to the equations ◮ consider a homogeneous filter → filter & derivative commute Filtered continuity equation � ∂u j � = ∂u j = 0 ∂x j ∂x j ◮ filtered field u is divergence-free ∂u ′′ ∂ j = ( u j − u j ) = 0 ∂x j ∂x j ◮ residual field u ′′ is also divergence-free → analogous to continuity in the RANS context 6 / 15
Introduction Continuity & momentum Filtering the equations Kinetic energy More Filtering the Navier-Stokes equations (3) Filtered momentum equation ∂ 2 u ∂u i ∂t + ∂u j u i + 1 ∂p = 2 ν ∂S ij = ν ∂x j ρ ∂x i ∂x j ∂x j ∂x j ◮ since u j u i � = u j u i we have: ∂u i ∂t + ∂u j u i + 1 ∂p = 2 ν ∂S ij + ∂u j u i − ∂u j u i ∂x j ρ ∂x i ∂x j ∂x j ∂x j ◮ from which: ∂τ R D u i D t + 1 ∂p = 2 ν ∂S ij ij − ρ ∂x i ∂x j ∂x j with τ R ij ≡ u j u i − u j u i 7 / 15
Introduction Continuity & momentum Filtering the equations Kinetic energy More Filtering the Navier-Stokes equations (4) Residual stress tensor τ R ij ≡ u j u i − u j u i ◮ alternatively called sub-grid scale (SGS) stress tensor ◮ analogy to RANS context: τ RANS ≡ � u j u i � − � u j �� u i � = � u ′ i u ′ j � ij Modified filtered equations k r ≡ 1 2 τ R ◮ residual kinetic energy: kk ◮ anisotropic residual stress tensor: τ r ij ≡ τ R ij − 2 3 k r δ ij ◮ define modified filtered pressure: p ≡ p + 2 ˜ 3 k r δ ij ∂τ r D u i D t + 1 ∂ ˜ p = 2 ν ∂S ij ij − ρ ∂x i ∂x j ∂x j 8 / 15
Introduction Continuity & momentum Filtering the equations Kinetic energy More Comments on the filtered equations Modified filtered equations ∂τ r D u i D t + 1 ∂ ˜ p 2 ν ∂S ij ij = − ρ ∂x i ∂x j ∂x j ∂u j = 0 ∂x j ◮ equations are unclosed ◮ residual stress tensor τ r ij needs modeling (cf. next lecture) ◮ the fields u and ˜ p are three-dimensional, unsteady, random ◮ residual stress tensor depends on type and parameters of filter 9 / 15
Introduction Continuity & momentum Filtering the equations Kinetic energy More Decomposition of the residual stresses ij + 2 u j u i = u j u i + τ r filtered convective term contains: 3 δ ij k r Leonard decomposition (1974) τ R + u j u ′′ i + u i u ′′ + u ′′ i u ′′ ij = u j u i − u j u i j j � �� � � �� � ���� ≡ L ij ≡ C ij ≡ R ij ◮ L ij are termed “Leonard stresses” ◮ C ij are the “cross stresses” ◮ R ij are the “SGS Reynolds stresses” → all terms non-zero in general case! � BUT: the stresses L ij and C ij are not Galilean invariant (when considering shifted velocity → decomposition changes) 10 / 15
Introduction Continuity & momentum Filtering the equations Kinetic energy More Galilean invariant decomposition of the residual stresses Germano decomposition (1986) τ R + u j u ′′ i + u i u ′′ j − u j u ′′ i − u i u ′′ + u ′′ i u ′′ j − u ′′ i · u ′′ ij = u j u i − u j u i j j � �� � � �� � � �� � ≡ L o ≡ C o ≡ R o ij ij ij ◮ L o ij are termed “objective Leonard stresses” ◮ C o ij are the “objective cross stresses” ◮ R o ij are the “objective SGS Reynolds stresses” → terms L o ij , C o ij , R o ij are invariant under arbitrary shifts ⇒ this is the preferred decomposition 11 / 15
Introduction Continuity & momentum Filtering the equations Kinetic energy More Conservation of energy E k ( x , t ) ≡ 1 2 u · u = 1 Instantaneous kinetic energy 2 u i u i Filtered kinetic energy E k = 1 2 u i u i = 1 + 1 2 u i u i − 1 2 u i u i 2 u i u i � �� � � �� � ≡ E f ≡ k r = 1 2 τ R ii Filtered kinetic energy equation � 1 � DE f pδ ij − 2 νu i S ij + u i τ r + ρu i ˜ = −P r − ε f ij D t ,j ◮ rate of production of residual kinetic energy: P r = − τ r ij S ij ◮ viscous dissipation rate due to filtered field: ε f = 2 νS ij S ij 12 / 15
Introduction Continuity & momentum Filtering the equations Kinetic energy More Conservation of energy – comparison with RANS Reynolds decomposition: � E k � = ¯ E + k � � ∂ t ¯ � u j � ¯ E + � u i �� u ′ i u ′ j � + � u j �� p � /ρ − 2 ν � u i � ¯ E + S ij = −P − ¯ ε ,j � � � u j � k + 1 2 � u ′ i u ′ i u ′ j � + � u ′ j p ′ � /ρ − 2 ν � u ′ i S ′ ∂ t k + ij � = + P − ε ,j LES decomposition: E k = E f + k r � � u j E f + u i τ r ∂ t E f + ij + u j ˜ p/ρ − 2 νu i S ij = −P r − ε f ,j & similar equation for k r • note formal analogy between RANS and LES 13 / 15
Introduction Continuity & momentum Filtering the equations Kinetic energy More Amount of residual kinetic energy & filter width Residual kinetic energy k r as function of filter width ∆ ◮ suppose high Reynolds number, Kolmogorov spectrum ◮ choose sharp spectral filter, cut-off κ c = π/ ∆ 2 C kol ( ε ∆ /π ) 2 / 3 → � k r � = 3 Choice of filter width ∆ to resolve fraction of k r ◮ use length scale L = k 3 / 2 /ε and L ≈ ℓ EI (6 / 0 . 43) � � 3 / 2 � k r � ∆ 2 → ℓ EI = 6 π/ 0 . 43 3 C kol k ◮ e.g. with 80% resolved TKE ( � k r � = 0 . 2 ) → ∆ /ℓ EI = 1 . 16 14 / 15
Introduction Filtering the equations Output of LES More What results are provided by an LES computation? Output of LES code: filtered field u ( x , t ) Desired results 1. statistically averaged mean flow field � u � ( x , t ) � but: applying average yields � u � = � u � � = � u � ! → in practice: difference often neglected � u � ≈ � u � � u ′ i u ′ 2. Reynolds stress components j � ( x , t ) ◮ available in LES: � ( u i − � u i � )( u j − � u j � ) � = � u i u i � − � u i �� u j � ◮ defining u ∗ i = u i − � u i � one obtains: + � τ R � u ′ i u ′ j � = � u ∗ i u ∗ + � u ′′ i �� u j � + � u ′′ j �� u i � + � u ′′ i �� u ′′ j � ij � j � � �� � ���� � �� � LES result model ≈ 0 → LES result corresponds to resolved part of Reynolds stress (diagonal elements smaller than true value) 15 / 15
Conclusion Further reading References Summary of today’s lecture Derivation of the filtered flow equations ◮ analogy to RANS; physical meaning of terms different How can the residual stresses be decomposed? ◮ Leonard decompositiopn; Germano decomposition; Galilean invariance What does the kinetic energy balance in LES involve? ◮ kinetic energy of filtered field: similar equation as RANS 1 / 2
Conclusion Further reading References Further reading ◮ S. Pope, Turbulent flows , 2000 → chapter 13 ◮ J. Fr¨ ohlich, Large Eddy Simulation turbulenter Str¨ omungen , 2006 → chapter 5 ◮ P. Sagaut, Large eddy simulation for incompressible flows , 2006 → chapters 3, 9 2 / 2
Conclusion References J. Fr¨ ohlich. Large Eddy Simulation turbulenter Str¨ omungen . Teubner, 2006. M. Germano. A proposal for a redefinition of the turbulent stresses in the filtered Navier-Stokes equations. Phys. Fluids , 29:2323–2324, 1986. A. Leonard. Energy cascade in large-eddy simulations of turbulent flows. Adv. Geophys. , 18A:237–248, 1974. S.B. Pope. Turbulent flows . Cambridge University Press, 2000. P. Sagaut. Large eddy simulation for incompressible flows . Springer, third edition, 2006. 2 / 2
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