Modelling of turbulent flows: RANS and LES
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1 Modelling of turbulent flows: RANS and LES Turbulenzmodelle in der Strömungsmechanik: RANS und LES Markus Uhlmann Institut für Hydromechanik Karlsruher Institut für Technologie SS 2012 Lecture 9 1 / 15
2 LECTURE 9 LES equations 2 / 15
3 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
4 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
5 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 i t + (u j u i) + 1 p = ν 2 u ρ x i u j = 0 recall: Reynolds decomposition u(x, t) = u + u here: apply spatial filtering u(x, t) = u + u (notation u to distinguish from statist. fluctuations) 5 / 15
6 Filtering the Navier-Stokes equations (2) Applying a spatial filter to the equations consider a homogeneous filter filter & derivative commute Filtered continuity equation ( ) uj filtered field u is divergence-free u j = = u j = 0 (u j u j ) = 0 residual field u is also divergence-free analogous to continuity in the RANS context 6 / 15
7 Filtering the Navier-Stokes equations (3) Filtered momentum equation u i t + u ju i + 1 p = ν ρ x i since u j u i u j u i we have: 2 u = 2ν S ij from which: u i t + u ju i + 1 p = 2ν S ij ρ x i Du i Dt + 1 p = 2ν S ij ρ x i + u ju i τ R ij u ju i with τ R ij u ju i u j u i 7 / 15
8 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: τij RANS u j u i u j u i = u i u j Modified filtered equations residual kinetic energy: anisotropic residual stress tensor: define modified filtered pressure: Du i Dt + 1 p = 2ν S ij ρ x i k r 1 2 τ R kk τ r ij τ R ij 2 3 k rδ ij p p k rδ ij τ r ij 8 / 15
9 Comments on the filtered equations Modified filtered equations Du i Dt + 1 p = 2ν S ij ρ x i u j = 0 τ r ij equations are unclosed residual stress tensor τij r 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
10 Decomposition of the residual stresses filtered convective term contains: Leonard decomposition (1974) u j u i = u j u i + τ r ij δ ijk r τij R = u j u i u j u i + u j u }{{} i + uiu j }{{} L ij L ij are termed Leonard stresses C ij are the cross stresses C ij R ij are the SGS Reynolds stresses all terms non-zero in general case! + u i u j }{{} R ij BUT: the stresses L ij and C ij are not Galilean invariant (when considering shifted velocity decomposition changes) 10 / 15
11 Galilean invariant decomposition of the residual stresses Germano decomposition (1986) τij R = u j u i u j u i + u j u i }{{} + u iu j u j u i u i u j + u i }{{} u j u i u j }{{} L o ij Cij o Rij o L o ij Cij o Rij o are termed objective Leonard stresses are the objective cross stresses are the objective SGS Reynolds stresses terms L o ij, Co ij, Ro ij are invariant under arbitrary shifts this is the preferred decomposition 11 / 15
12 Conservation of energy Instantaneous kinetic energy E k (x, t) 1 2 u u = 1 2 u iu i Filtered kinetic energy E k = 1 2 u i u i = 1 2 u i u i + 1 }{{} 2 u i u i 1 2 u i u i }{{} E f k r= 1 2 τ ii R Filtered kinetic energy equation DE f Dt ( ) 1 + ρ u i pδ ij 2νu i S ij + u i τij r = P r ε f,j rate of production of residual kinetic energy: P r = τ r ij S ij viscous dissipation rate due to filtered field: ε f = 2νS ij S ij 12 / 15
13 Conservation of energy comparison with RANS Reynolds decomposition: E k = Ē + k t Ē + ( u j Ē + u i u iu j + u j p /ρ 2ν u i S ) ij ( t k + u j k + 1 ) 2 u iu iu j + u jp /ρ 2ν u is ij,j,j = P ε = +P ε LES decomposition: E k = E f + k r t E f + ( u j E f + u i τij r ) + u j p/ρ 2νu i S ij,j & similar equation for k r = P r ε f note formal analogy between RANS and LES 13 / 15
14 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/ k r = 3 2 C kol (ε /π) 2/3 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 6π/0.43 l EI = ( 2 k r 3 C kol k e.g. with 80% resolved TKE ( k r = 0.2) /l EI = / 15
15 Output of LES 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 2. Reynolds stress components u i u j (x, t) available in LES: (ui u i )(u j u j ) = u i u i u i u j defining u i = u i u i one obtains: u iu j = u i u j + τij R + u i u j + u j u i + u i u j }{{}}{{}}{{} LES result model 0 LES result corresponds to resolved part of Reynolds stress (diagonal elements smaller than true value) 15 / 15
16 Conclusion References Further reading 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
17 Conclusion References Further reading Further reading S. Pope, Turbulent flows, 2000 chapter 13 J. Fröhlich, Large Eddy Simulation turbulenter Strömungen, 2006 chapter 5 P. Sagaut, Large eddy simulation for incompressible flows, 2006 chapters 3, 9 2 / 2
18 Conclusion References J. Fröhlich. Large Eddy Simulation turbulenter Strömungen. Teubner, M. Germano. A proposal for a redefinition of the turbulent stresses in the filtered Navier-Stokes equations. Phys. Fluids, 29: , A. Leonard. Energy cascade in large-eddy simulations of turbulent flows. Adv. Geophys., 18A: , S.B. Pope. Turbulent flows. Cambridge University Press, P. Sagaut. Large eddy simulation for incompressible flows. Springer, third edition, / 2
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