Turbulence: dynamics and modelling MEC 585 (Part 4)

Transcription

1 Turbulence: dynamics and modelling MEC 585 (Part 4) Pierre Sagaut D Alembert Institute Université Pierre et Marie Curie -Paris 6 pierre.sagaut@upmc.fr

2 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. Bibliography

3 I. What is LES?

4 Motivations 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

5 LES: the key observation

6 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

7 LES: what are we really doing? 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)

8 LES equations: Modified PDEs Model Exact subgrid model numerical error Subgrid modelling error

9 Implicit non-linear couplings in LES????

10 E.g. Numerical and VMS viscosities (Ciardi & al., JOT, 2006)

11 What is the optimal LES solution? Formal statistical error norm projection resolution Definition of the optimal LES solution discretization Implicit LES: No physical subgrid model Ad hoc scheme Classical LES: explicit subgrid model neutral scheme

12 II. Mathematical models for the small scale removal

13 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:

14 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

15 (Châtelain et al., Int. J. Numer. Meth. Fluids, 2004) E.g.: positivity of passive scalar

16 Main existing LES models Family 1: Filtering operators Projective filters Convolution filters (most popular ones) Family 2: Regularized hydrodynamic models Leray s model NS-α equations Ladyzenskaja s model Family 3: Conditional average operators McComb s Conditional Mode Elimination Yoshizawa s Partial Statistical Average

17 Example 1: convolution filters Most general form (spatio-temporal convolution filter): Bounded fluid domain Cutoff length unbounded domain +isotropic filter = commutation causality Symmetry preservation

18 Convolution filter and BCs (a first look) Solution 1: constant filter width the exact field must be prolonged outside the domain Solution 2: decreasing filter width

19 Bounded domains & anisotropic filters Spatial convolution filter Commutation error with space derivative

20 continued 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

21 Ideal filtered NS equations Resolved Subgrid Remark Form 1 Form 2 None is Galilean invariant Both are Galilean invariant

22 Closed LES equations Subgrid model Problems and issues: 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

23 Example 2: Variational Projection General form (finite space-dependent basis) Choice of a solution space Definition of the degrees of freedom

24 continued Resolved non linear term for mode i subgrid non linear term for mode i

25 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,

26 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

27 continued This equation admits a regular solution for arbitrary long time if Existence of weak solutions Uniqueness of weak solutions

28 continued No singularity: small scales are regularized Amplitude parameter ε 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

29 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

30 III. Closing the LES equations: subgrid modelling

31 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

32 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

33 A famous example: Smagorinsky subgrid viscosity (1963) Subgrid tensor model: Subgrid viscosity model Arbitrary constant Cutoff length scale

34 continued Remarks: 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

35 The Smagorinsky constant Canonical analysis: Infinite inertial range Local equilibrium: production = dissipation «Universal value»:

36 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!

37 Error maps & Optimal trajectories (Meyers et al., Phys. Fluids, 2003) Framework: isotropic turbulence at finite (low) Reynolds number Control parameter: subgrid activity Smagorinsky model:

38 0.2 Optimal value regions (Meyers et al., Phys. Fluids, 2003)

39 Error with optimal Smagorinsky constant (Meyers et al., Phys. Fluids, 2003)

40 Relative error on kinetic energy (%) Coarsest grid Finest grid 0 Subgrid activity 1 (Meyers et al., Phys. Fluids, 2003)

41 Constant Subgrid activity Optimal trajectory (Meyers et al., Phys. Fluids, 2003)

42 (Meyers et al., Phys. Fluids, 2003) Relative error on Taylor microscale (%) Coarsest grid Finest grid Subgrid activity

43 Accuracy in shear flows 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)

44 More general models 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

45 Resolved scales Schematic view E(k) Large resolved scales Small resolved scales Subgrid scales Test filter LES filter k

46 Variational Multiscale Method (Hughes & al.)

47 VMS equations Neglected (or modeled) modeled Ignored (or simplified PDEs?)

48 General VMS viscosity model Subgrid viscosity Strain tensor Selected subspace Viscosity test field: X=v : Large -??? Model X=w : Small -??? Model X=v+w : All -??? Model Strain test field: Y=v:???- Large Model Y=w :???-Small Model Y= v+w :???- All Model Target subspace: Z=φ Z= ψ Z= φ +ψ

49 Embedded LES approaches 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) VMS (Hughes et al. 2000s) Spectral Vanishing Viscosity based LES (Tadmor (1989), Maday et al. (1993)) (Karniadakis,Pasquetti, 2000s)

50 VMS implementation (1) 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)

51 VMS implementation (2) v W

52 VMS implementation (3) Hyperviscosity formulation Idea: Iterated Laplacian elliptic filter as a second-order aproximation of cell-agglomeration projection All-Small Small-Small

53 Approximate Deconvolution models (Adams et al.) Key idea: find an approximate de-filtered field 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?

54 continued ADM momentum equation 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)

55 Implementation of inverse filter Approximate iterative deconvolution 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

56 Implementation (continued) Approximate differential expansions Convolution filter model Regularity hypothesis

57 Implementation (continued) Usual form: Truncated explicit Taylor series expansion 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

58 One-parameter Lie group of symmetries of incompressible 3D NS Time shift Pressure shift Rotations Generalized Galilean Transformations Rescaling (I) Rescaling (II)

59 Additionnal symmetries of incompressible NS Mirror symmetry 2D Material frame indifference (2D flow only) Time reversal (inviscid flow only)

60 Symmetry analysis (Oberläck, 1997; Razafindralandy &Hamdouni,2006) Model T-shift P-shift rotation Galilean Scaling I Scaling II Mirror 2D MFI T-reversal Smagorinsky Scale similarity Gradient Dynamic Smagorinsky

61 Generalized thermodynamics: symmetry-preserving subgrid viscosity Arbitrary scalar functions

62 Remarks Symmetry preservation invariant preservation Invariants of NS equation are non-local quantities Noether theorem must be applied to NS+adjoint NS system Asymptotic inviscid case: global kinetic energy conservation Symmetry-preserving numerical methods?

63 IV. The LES boundary condition problem

64 Main issues in LES 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)

65 Wall modelling: typical first cell

66 Wall modelling: typical first cell

67 Subgrid TKE transfer in near wall region

68 Main approaches 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

69 The turbulent inflow issue 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

70 The reconstruction issue Exact data (Druault et al. AIAA J.,2004) Random, same Reynolds stress Random, same time two-point correlations Random, same space two-point correlations Linear stochastic Estimation

71 Main approaches 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

72 VII. Bibliography

73 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

74 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

75 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

76 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