Objectives & contents. Turbulence: dynamics and modelling (Part 4) Motivations. I. What is LES?

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1 Turbulence: dynamics and modelling (Part 4) Pierre Sagaut D Alembert Institute Université Pierre et Marie Curie -Paris 6 Objectives & contents Part 4: advanced simulation techniques 1. What is Large-Eddy Simulation? 2. Mathematical models for the small scale removal 3. Closing the LES equations: subgrid modelling 4. The LES boundary condition problem 5. What is the real LES solution? 6. Beyond LES: multiresolution/multiscale methods 7. Bibliography pierre.sagaut@upmc.fr Motivations I. What is LES? Direct Numerical Simulation is too expensive in many cases (fully developped turbulence) Large scales carry the kinetic energy and the anisotropy Small scales are more universal at very high Reynolds number

2 LES: the key observation LES: a more complete view u: exact turbulent solution u h : discrete solution LES: discrete solution --> projection error approximate integro-differential operators --> discretization error unresolved scales --> resolution error LES: what are we really doing? LES equations: Modified PDEs Model Exact projection of the continous exact solution onto a discrete basis Accounting for errors in the discrete model Discretisation/time integration errors Approximate physical model Round-off errors? (e.g. CADNA project) Exact subgrid model numerical error Subgrid modelling error

3 Implicit non-linear couplings in LES E.g. Numerical and VMS viscosities (Ciardi & al., JOT, 2006)???? What is the optimal LES solution? Formal statistical error norm projection resolution discretization Definition of the optimal LES solution II. Mathematical models for the small scale removal Implicit LES: No physical subgrid model Ad hoc scheme Classical LES: explicit subgrid model neutral scheme

4 Statement of the problem Practical ( real world ) LES:! is not known! a mathematical model, G, is necessary for! to write LES governing equations to get some insight into properties of LES solutions This model must Account for small scale removal Be friendly Not introduce artifacts Expectation: Expected/required properties of G Uniform fields must be preserved Basic properties of the exact solution must be preserved Physical symmetries (1-parameter Lie group, asymptotic material frame indifference, ) Basic laws of thermodynamics (thermodynamic variables must remain positive, ) Governing equations must be simple G must commute with space/time derivatives Treatment of boundary conditions Remark: these choices are arbitrary E.g.: positivity of passive scalar Main existing LES models (Châtelain et al., Int. J. Numer. Meth. Fluids, 2004) Family 1: Filtering operators Projective filters Convolution filters (most popular ones) Family 2: Regularized hydrodynamic models Leray s model NS-! equations Ladyzenskaja s model Family 3: Conditional average operators McComb s Conditional Mode Elimination Yoshizawa s Partial Statistical Average

5 Example 1: convolution filters Convolution filter and BCs (a first look) Most general form (spatio-temporal convolution filter): Bounded fluid domain Cutoff length unbounded domain isotropic filter = commutation causality Symmetry preservation Solution 1: constant filter width!the exact field must be prolonged outside the domain Solution 2: decreasing filter width Bounded domains & anisotropic filters continued Spatial convolution filter Commutation error with space derivative Further results dealing with commutation errors exist (time-dependent filters, timedependent domains ) Filter kernels with reduced/controled commutation error have been proposed Explicit terms have been proposed to cancel commutation errors

6 Ideal filtered NS equations Closed LES equations Subgrid model Problems and issues: Form 1 Form 2 Resolved Subgrid Remark None is Galilean invariant Both are Galilean invariant the subgrid closure M is not given by the definition of G the filtered variable is still continuous: the projection onto the discrete basis is not taken into account for smooth invertible kernels (e.g. Gaussian filter), there is no information loss! small scales are rescaled but not removed! Artefacts: commutation errors on bounded domains no easy representation of numerical errors Example 2: Variational Projection continued General form (finite space-dependent basis) Resolved non linear term for mode i Choice of a solution space subgrid non linear term for mode i Definition of the degrees of freedom

7 continued LES appears as a specific weak formulation Main advantages Galerkin-type numerical methods are naturally described (more difficult for FD: distribution theory is needed) Main convolution filter artifacts disappear Commutation error with derivatives Boundary conditions formulation A huge amount of results dealing with weak solutions is available But some non-trivial issues may arise E.g. : P2/P1 method, Example 3: Ladyzenskaya s regularized hydrodynamic model Motivation: no existence/unicity/regularity result for arbitrary long time for the general 3D incompressible NS case (Clay prize!) Technical problem: control of Possible existence of singularities (set with null measure) Local bursts of vorticity far beyond the Kolmogorov scale The linear Newtonian model may not hold in this case! non linear correction continued continued This equation admits a regular solution for arbitrary long time if Existence of weak solutions Uniqueness of weak solutions No singularity: small scales are regularized Amplitude parameter " can be used to tune the damping effect (i.e. the cutoff scale) Regularization can be considered as an implicit filter Limits: Not a discrete model Does not account for numerical scheme Nothing about boundary condition treatment

8 Important open issues Sensitivity of turbulence features with respect to G and! usual quantities of engineering relevance Higher-order, multi-point statistical moments Intermittency Coherent structures! What can we recover from LES?! Better understanding of turbulence needed! A fully satisfactory model is still missing III. Closing the LES equations: subgrid modelling General Explicit subgrid modelling new terms in the governing equations Structural approach: find a model for the subgrid tensor Functional approach: find a model for the subgrid force Implicit approach Find a numerical scheme whose leading error term will act as an explicit subgrid model A priori requirements for a subgrid model Consistency (1) : subgrid model should vanish in fully resolved regions Consistency (2) : some subgrid scale effects must be recovered (e.g. net kinetic energy transfer associated with kinetic energy cascade) Consistency (3) : symmetries of the exact LES equations must be preserved Subgrid models must be user friendly

9 A famous example: Smagorinsky subgrid viscosity (1963) Subgrid tensor model: Subgrid viscosity model Remarks: continued Reminiscent of von Neumann - Richtmyer artificial viscosity for shock capturing (1950) Belongs to Ladyzenskaja s class of viscosity Subgrid model constant must be tuned to enforce the desired rate of kinetic energy dissipation Arbitrary constant Cutoff length scale The Smagorinsky constant Canonical analysis: Infinite inertial range Local equilibrium: production = dissipation!«"universal value"»: Accuracy in isotropic turbulence Satisfactory recovery of E(k) at infinite Reynolds number (non-dissipative numerical method, cutoff within inertial range) Not coherent with realistic TKE spectrum, i.e. does not account for! Smagorinsky constant not constant!

10 Error maps & Optimal trajectories (Meyers et al., Phys. Fluids, 2003) Framework: isotropic turbulence at finite (low) Reynolds number Control parameter: subgrid activity Optimal value regions Smagorinsky model: 0 (Meyers et al., Phys. Fluids, 2003) Error with optimal Smagorinsky constant Relative error on kinetic energy (%) Coarsest grid Finest grid (Meyers et al., Phys. Fluids, 2003) 0 Subgrid activity 1 (Meyers et al., Phys. Fluids, 2003)

11 100 Relative error on kinetic energy (%) 10 1 Coarsest grid Finest grid Constant Subgrid activity Optimal trajectory 0 Subgrid activity 1 (Meyers et al., Phys. Fluids, 2003) (Meyers et al., Phys. Fluids, 2003) Constant Subgrid activity Optimal trajectory Relative error on Taylor microscale (%) Coarsest grid Finest grid (Meyers et al., Phys. Fluids, 2003) 0 1 Subgrid activity (Meyers et al., Phys. Fluids, 2003)

12 (Meyers et al., Phys. Fluids, 2003) Accuracy in shear flows Relative error on Taylor microscale (%) Coarsest grid Finest grid C S = 0.18 yields too high dissipation Not adapted because: It relies on asymptotic analysis in HIT Model is non-local in terms of wave number! takes into account large scale gradients New expression (Canuto & Cheng, Phys. Fluids, 1997) Subgrid activity More general models Schematic view Resolved scales Local automatic tuning of C S Germano s identity --> least square optimization Kolmogorov equation --> dissipation optimization (Shao et al.) Improved localness in terms of wave number Variational Multiscale methods Filtered models Remark: all these approaches are based on a test filter The resolved field is decomposed into spectral bands Features of turbulence are used/extrapolated to calibrate the subgrid models E(k) Large resolved scales Small resolved scales Subgrid scales Test filter LES filter k

13 Variational Multiscale Method (Hughes & al.) VMS equations Neglected (or modeled) modeled Ignored (or simplified PDEs?) General VMS viscosity model Embedded LES approaches Subgrid viscosity Strain tensor Selected subspace General multresolution LES methods (Terracol, Sagaut & Basdevant, JCP 2001, 2003) Incremental Unknowns (Jauberteau, Dubois, Temam, 1990s) 2-level LES closures (Domaradzki et al., 1990s) Wavelet-based methods CVS (Farge et al., 1990s) SCALES (Goldstein & Vasyliev, Phys. Fluids 2005) Viscosity test field: X=v : Large -??? Model Strain test field: Y=v:???- Large Model Target subspace: Z=# VMS (Hughes et al. 2000s) X=w : Small -??? Model Y=w :???-Small Model Z= $ X=vw : All -??? Model Y= vw :???- All Model Z= # $ Spectral Vanishing Viscosity based LES (Tadmor (1989), Maday et al. (1993)) (Karniadakis,Pasquetti, 2000s)

14 VMS implementation (1) VMS implementation (2) Possible implementations: hp framework h : multigrid-like cell agglomeration (Koobus & Farhat, CMAME 2004) p : hierarchical basis (Jansen, 2002) full hp : to be done! filtering framework general formulation (Vreman, Phys. Fluids, 2003) truncated pseudo-spectral basis (Hughes et al, Phys. Fluids, 2001) (Sagaut & Levasseur, Phys. Fluids, 2005) equivalent hyperviscosity: (Winckelmans et al., 2001), (Sagaut & Levasseur, Phys. Fluids, 2005) v W VMS implementation (3) Hyperviscosity formulation Approximate Deconvolution models (Adams et al.) Key idea: find an approximate de-filtered field Idea: Iterated Laplacian elliptic filter as a second-order aproximation of cell-agglomeration projection All-Small Small-Small Remarks Same grid is used --> Developed using the convolution filter model No undelying assumption on the physics Issue: how to find the adequate approximate inverse if the exact filtering operator is unknown?

15 continued ADM momentum equation Implementation of inverse filter Approximate iterative deconvolution Non linear term computed using the surrogate field: does not account for u Dissipative regularization term used to prevent energy pile-up in resolved scales (net drain associated with kinetic energy cascade) Remarks Known as the van Cittert deconvolution in signal processing l=5 in is enough in practice (Stolz & Adams) l=1 : Bardina s scale-similarity model Obviously dependent on the evaluation of G Can be replaced by other methods: conjugate gradient, Truncated expansion Implementation (continued) Approximate differential expansions Implementation (continued) Usual form: Truncated explicit Taylor series expansion Convolution filter model Regularity hypothesis Alternative forms: Padé approximant Best polynomial projection (minimax, least-square, ) Remarks l=2! Clark s gradient model (= tensor diffusivity model) Anti-diffusion in some directions

16 One-parameter Lie group of symmetries of incompressible 3D NS Time shift Pressure shift Rotations Mirror symmetry Additionnal symmetries of incompressible NS 2D Material frame indifference (2D flow only) Generalized Galilean Transformations Rescaling (I) Time reversal (inviscid flow only) Rescaling (II) Symmetry analysis (Oberläck, 1997; Razafindralandy &Hamdouni,2006) Generalized thermodynamics: symmetry-preserving subgrid viscosity Model T-shift P-shift rotation Galilean Scaling I Scaling II Mirror 2D MFI T-reversal Smagorinsky Scale similarity Gradient Arbitrary scalar functions Dynamic Smagorinsky

17 Remarks Symmetry preservation %invariant preservation Invariants of NS equation are non-local quantities Noether theorem must be applied to NSadjoint NS system Asymptotic inviscid case: global kinetic energy conservation Symmetry-preserving numerical methods? IV. The LES boundary condition problem Main issues in LES Wall modelling: typical first cell Wall modelling: if first grid cell is too large to use the no-slip BC, a specific subgrid model must be defined to account for a part of the boundary layer dynamics Turbulent inflow conditions: what kind of fluctuations should be prescribed at the inlet to mimic upstream turbulence? (reconstruction issue)

18 Wall modelling: typical first cell Subgrid TKE transfer in near wall region Main approaches The turbulent inflow issue Find an empirical explicit relation between the skin friction and the velocity at the first off-wall point (Schumann, Grötzbach, Wengle 1970s) Algebraic model Based on equilibrium boundary layer mean flow Solve a boundary layer equation within the first grid cell (Balaras et al., CTR group) Gain: pressure assumed to be constant in the wallnormal direction More general Goal: feed LES with realistic pseudo-turbulent fluctuations Question(s): What must be the characteristics of the fluctuations?! what are the important features of the real turbulent fluctuations? Correlations? Spectra? Compatibility with the numerical method

19 The reconstruction issue Main approaches (Druault et al. AIAA J.,2004) Exact data Random, same Reynolds stress Random, same time two-point correlations Random, same space two-point correlations Linear stochastic Estimation Kraichnan-type random fields Reynolds stresses & spectra:ok No correlations POD or LSE-based coherent fluctuations random noise OK for correlations, spectra and Reynolds stresses Respective weights of coherent/random TKE? Extraction/rescaling Precursor simulation Muchinsky s LES fluid model (Muchinsky, JFM, 1996) V. What is the real LES solution? Idea: LES Navier-Stokes equations = DNS equations for a non-newtonian fluid Non-linear viscosity (e.g. Smagorinsky) Mixing length Filter cutoff length C S can be considered as a free parameter

20 Extended K41 hypotheses Hyp 1: energy spectrum E(k) is of the form (in inertial range) General form: LES inertial range spectrum Extended usual form: Hyp 2: large wave number asymptotics Hyp 3: Large C S asymptotics Recovering usual form (vanishing filter width, constant mixing length) Model built-in filter (Magnient & al., Phys. Fluids, 2001) Smagorinsky s filter Remarks: Model built-in filter depends on r (i.e. on C S ) f LES depends on the subgrid model (not shown here)

21 Compensated TKE spectrum Sensitivity of compensated spectrum Kolmogorov s constant in LES Couplings between numerical errors and subgrid models Schuman nd order 4th order Smagorinsky th order Smagorinsky 120 3

22 continued Kinetic energy spectrum (Levasseur et al.,jot, 2006) Smagorinsky Dynamic Smagorinsky VMS - Small Small VMS - Large Small Dissipation Spectrum Implicit LES with GLS? Smagorinsky Dynamic Smagorinsky VMS - Small Small VMS - Large Small GLS Subgrid Model No stabilization term --> Euler equilibrium solution recovered

23 Implicit LES with GLS? Properties of computed LES solutions Question: what are the discrepancies between the filtered solution and the computed solution? Sources: Numerical errors Modelling errors GLS stabilization term only --> unphysical equilibrium solution Time correlation LES Time microscale LES DNS DNS -1 slope Time lag (He et al., Phys. Fluids, 2002) (He et al., Phys. Fluids, 2002)

24 LES microscale / DNS microscale best linear fit PDF of velocity increment Filtered experimental data Increasing filter width (He et al., Phys. Fluids, 2002) (Kang et al., JFM, 2003) PDF of velocity increment Filtered experimental data LES Experimental data Filtered data (Kang et al., JFM, 2003) (Cerutti et al., Phys. Fluids, 2000)

25 Main observations Extreme events are removed! tails of pdfs are modified LES results exhibit too high correlations loss of stochastic forcing by the subgrid scales! random acceleration may be required Might be important for prediction of aeroacoustics sources LES fields seem to be more Gaussian than filtered fields Implicit question: role of small scale events in turbulence dynamics/statistical properties? VI. Beyond LES: multiresolution/multiscale methods LES/multiresolution coupling Mathematical formulation Coarse grid domain Fine grid domain interface interface E(k) Frequency jump = discontinuity k

26 New issues Small scales generation: schematic view Discontinuous solution at the interface: Boundary conditions at the interface Reconstruction of high frequencies Postprocessing Resolution jump: Control of numerical error at the interface Discontinuous time-stepping? Jump relations for subgrid quantities Captured scales Missing scales Distance needed to recover small scales «buffer zone» Summary Exemple: Rotor/stator interaction (Courtesy: B. Raverdy, I. Mary, ONERA/DSNA) Grid refinement/small scale generation: Grid refinement enables better description of gradient/curvature of the solution! improved capturing of mean profile instabilities! small scales to be built by cascade! «"buffer zone"» (case dependent) Sudden grid clustering is not enough to recover all possible small scales Artificial triggering needed (and used!)! improved interface conditions? Re= Mach=0.1 Mesh: 6 millions points Time : 3 cycles CPU : 240 h (NEC SX 5)

27 Numerical method features Structured grid Optimized 2nd-order Finite Volume Subgrid model: Mixed scale Explicit reconstruction at the interfaces Hybridizing RANS and LES (1): Eddy-viscosity rescaling approach Most common formulation (Speziale-type) ( h! t = & ' L RANS % # $ "! RANS Very easy implementation More complex rescaling function for BL and separation (Baurle, Fasel, ) Variant: Batten s LNS (max of viscosities) Hybridizing RANS and LES (2) Limiters Idea: switch from RANS to LES by limiting some RANS scale (time, space, ) Bush-Mani limiters for k-? type models L = min( L " = max( "! = max(! RANS RANS RANS, c h) 0, c k 1, c Special case: Spalart s DES method 2 3/ 2 / h) k / h)

28 Fully overlapping zonal approach Example 1: T106 turbine blade (Labourasse & Sagaut, JCP, 2002 ) Non-Linear Disturbance Equations (Morris, Labourasse, ) Idea: Mean field from RANS on coarse grid Fluctuations via LES on embedded finer grid!multidomain/multiresolution/multimodel! Extension of LES/multiresolution Advantages/drawbacks: Usual RANS simulation LES in subdomains only Pb: cycling, information transfer, interfaces RANS simulation 2D, steady S-A Model 41 equations NLDE simulation 3D, unsteady MSM model 5 0 equations Global dynamics Results Bubble oscillation 1900 Hz Mixing Layer K-H 3500 Hz Wake 2500 Hz Aeroacoustic loop 500Hz Upstream edge Of separation bubble Middle Of separation bubble Downstream edge Of separation bubble

29 Example 2: High lift wing (Labourasse et al., ONERA/DSNA) Results: mean field correction RANS mean field New full mean field: RANStime average of NLDE Vortical structures in a plane Acoustic waves in a plane

30 State of the art Zonal RANS/LES without overlap Significant reduction of CPU cost Local enrichment in frequency/wave number Correction of the RANS solution: If LES performed on a fine grid If non-equilibrium due to large structures Need for an accurate LES solution: RANS domain (coarse grid) interface LES domain (fine grid) Controled numerical error Boundary conditions Subgrid scale model E(k) Frequency jump = discontinuity k Main features of these methods Choice of RANS model and LES model Coupling strategy between subdomains: Explicit reconstruction of frequency jump Fully general approach, but open problem! Discontinuous solution E.g.: Quéméré et al., Sergent et al. No reconstruction Continuous solution (no jump!) Progressive reconstruction thanks to instabilities! efficient for strongly unstable flows (free shear flows) But existence of a case-dependent «buffer layer» Numerical methods and grids a priori different on each side Example: Detached Eddy Simulation (Spalart et al.) Ad hoc scheme in each domain (Strelets) Smooth solution (no jump) Modelling: RANS domain: usual turbulence model LES domain: RANS model with limited scale(s) " advanced wall model for LES RANS LES

31 DES example: S-A model Main features of the numerical method Structured/unstructured grids RANS: Stabilized 2nd-order FV method LES: Stabilized 2nd- or 4th-order FV method d = min ( d RANS, d LES ) Flow around a sphere (Courtesy: D. Squires et al.) continued

32 continued continued Drag Lateral torque continued DES of complex flows (D. Squires & al.) F-15

33 DES sensitivity DES: grid dependency F-15 F-15 More zonal methods Fully general methods (Quéméré et al., Sergent et al.): Jump reconstruction at the interface Arbitrary turbulence/subgrid models TBLE wall model (Balaras et al., Wang et al.): Boundary layer equations in the RANS zone No explicit jump treatment Continuous eddy-viscosity (Wang et al.) Arbitrary turbulence/subgrid models Zonal methods: state of the art DES efficient for massively separeted flows Case-dependent buffer layer Attached boundary layer: Non-DES methods better suited «fine» grids are required Interface treatment required

34 VII. Bibliography P. Sagaut «"Large-Eddy simulation for incompressible flows"- 3rd edition», Springer, 2005 "a general introduction to all LES issues/models/approaches incuding RANS/LES and the scalar case, fundamentals of numerical methods not discussed B.J. Geurts «"Elements of Direct and Large Eddy Simulation"», Edwards, 2003 "an introduction to DNS and LES, fundamentals of numerical methods are recalled M. Lesieur, O. Métais, P. Comte «"Large-Eddy Simulations of turbulence"», Cambridge University Press, 2005 "an overview of the LES works of the authors. Nicely illustrated, with short introduction to LES for compressible flows L.C. Berselli, T. Iliescu, W.J. Layton «"Mathematics of large-eddy simulation of turbulent flows"», Springer, 2006 "a general presentation of mathematical results dealing with numerical analysis of LES V. John «"Large eddy simulation of turbulent incompressible flows"», Springer, 2004 (Lecture notes in computational science and engineering Vol. 34) "an overview of mathematical results dealing with LES obtained by the author for a class of subgrid models P. Sagaut, S. Deck, M. Terracol «"Multiscale and multiresolution approaches in turbulence"», Imperial College Press, 2006 "an general presentation including multiscale RANS, LES, hybrid RANS/LES, adaptive basis methods C. Wagner, T. Hüttl, P. Sagaut (eds.) «"LES for Acoustics"», Cambridge University Press (available in jan. 2007) "an general presentation of LES for aeroacoustics: fundamental of LES and aeroacoustics, acoustic analogies, numerical methods, boundary conditions

35 F. Grinstein, B. Rider, L. Margolin (eds.) «"Implicit LES: computing turbulence dynamics"», Cambridge University Press, 2006 "an extensive presentation of the ILES approach. Both theoretical analyss of ad hoc numerical schemes and results are provided