Towards low-order models of turbulence

Transcription

1 Towards low-order models of turbulence Turbulence in Engineering Applications Long Programme in Mathematics of Turbulence IPAM, UCLA Ati Sharma Beverley McKeon + Rashad Moarref + Paco Gomez Ashley Willis # Hugh Blackburn Murray Rudman University of Southampton + California Institute of Technology Monash # University of Sheffield 20th November 2014 Thanks to EOARD, AFOSR, ARC

2 Aim Provide a set of basis functions, derived from equations, that economically represent wall turbulence and can be used to make predictions about it. Fidelity is not required, but we would like graceful degradation. 2 / 38

3 Are low order models possible? Navier-Stokes equations are infinite-dimensional but turbulence has structure exact solutions / coherent structures should also be captured, if they continue into turbulent So turbulence should admit more formal (rather than phenomenological) reduced order models that describe structure 3 / 38

4 Overview Model formulation Sparsity of frequencies in turbulent DNS Scaling of resolvent modes, triads Representation of TW solutions 4 / 38

5 NSE in Fourier domain Fourier modes for velocity in all (three) homogeneous directions (;, θ, ) =,,ω () (+θ ω) θ,,, () :=ˆ(;,, ω), subtract out steady-state / / mean / (,, ω) = (,, ) and substitute for nonlinear term (;, θ, ) := ( ()) ( ()) (;, θ, ) :=,,ω () (+θ ω) 5 / 38

6 Steady-state equation The equation for (,, ω) = (,, ) gives the spatio-temporal mean Re stress gradients supporting mean = () () () + () time-space ave velocity requires requires just assume 6 / 38

7 Equation for fluctuations at (,, ω) ω () = () () () () + () () + () = ( ) + () 7 / 38

8 Equation for fluctuations at (,, ω) interaction between scales () = () linear resolvent The resolvent = (ω ) is the frequency-domain (so travelling waves) transfer function from nonlinear interaction between scales to velocity field 7 / 38

9 NSE as a network of resolvents (,, ω) () Mean equation () (,, ω) 8 / 38

10 () () Resolvent = ( ) ( ) ( ) ( + ) = + ( + ), states are (, θ, ) loss of translational symmetry in causes non-normality / source of energy as = ω/ and, Energetic response at critical layer becomes very important at high Re. response becomes more localised around critical layer

11 Some points on interpretation / FAQ Valid for large fluctuations Clear interpretation for linear operators formed using the mean profile (not perturbation analysis around = ) Eddy viscosity not needed ( stress model ) Haven t used scale separation arguments as in RDT Focuses discussion on forcing model/interpretation (c.f. earlier frequency-domain works by Jovanovic; Bamieh) Time (ω) on equal footing with homogeneous spatial directions (, ) Importance of critical layer is clear 10 / 38

12 () () Approximating a single resolvent SVD approximates an operator by directions of principal gain = ψ ()σ ϕ () = () χ (, ϕ ) σ ψ χ (, ϕ ) σ ψ χ (, ϕ ) σ ψ + () Each σ is a (real) gain, σ is the maximum gain. Velocity field response is ψ (). χ (, ϕ ) σ ψ χ (, ϕ ) σ ψ χ (, ϕ ) σ ψ 11 / 38

13 () () Approximation of (by gain) Since often σ σ, often reasonable to approximate by leading ψ () This gives radial form (or structure) of velocity field at Reduces NSE solution to a weighted sum of response modes (, θ,, ), χ σ ψ () (+θ ω) Using = or < is great simplification, but remains to find coefficients χ 12 / 38

14 Finding coefficients Coefficients are fixed by in a quadratic equation χ = σ χ χ, = ( ψ ψ, ϕ ) Natural truncation where σ < ε (low-order / sparsity) 13 / 38

15 Ways to proceed 1. assume forcing model over spectra 2. symmetries of ϕ, ψ ; {χ }; scaling 3. pick representative combinations of χ structure 4. solve truncated χ = σ χ χ approximate exact solutions 14 / 38

16 Localisation of response at critical layer + ( τ (, )); normalised at each + ; τ = 15 / 38

17 Turbulence as sheets of coefficients = = ω ω ω ω ω / ω ω / τ ω ω ω ω ω / ω ω / τ energy quite localised around critical layer large areas over-resolved in or ω resulting in stiffness of equations 16 / 38

18 Turbulence as sheets of coefficients = = τ τ Truncate outside τ < < ; above high,, ω; need for 4D data (DNS / experiment) 16 / 38

19 + = sparsity in ω introduced by finite-length pipe (σ,,ω, ) 1D resolvent (upper); 2D resolvent (lower) 17 / 38

20 DNS + = projection, mode amplitudes fixed by nonlinearity 2 1 (σ,,ω, ) F. Gómez, H. M. Blackburn, M. Rudman, A. S. Sharma and B. J. McKeon D M D R itz vector norms ( 5e 2) amplitude a ( 4e+2) 5,ω,1 amplification σ 5,ω,1 ( 5e 3) nonlinear forcing χ 5,ω,1 ( 1e+7) ω/2π Fig. 1 Distribution of amplitude a5,ω,1, amplif cation σ5,ω,1 and nonlinear forcing χ5,ω,1 in frequency of the f rst singular value m= 1 corresponding to the azimuthal wavenumber n = 5. Bars indicate most energetic frequencies computed via DMD of DNS data. 2D resolvent; sparsity in ω of, σ, ; amplitude peaks not exactly at σ peaks (need to understand ) 18 / 38

21 Comparison with DMD structure in DNS Comparisons between resolvent modes (left) and DMD modes (right) at same frequencies ω/2π = (top) ω/2π = (middle) ω/2π = (bottom) at n = 2. Colored isosurfaces indicate ±1/3 of maximum streamwise f uctuating velocity. Gomez et al PoF / 38

22 Spectra: direct from resolvent gains for unit white noise forcing on first two modes only at + = vs DNS λ + λ + 20 / 38

23 Scaling of modes from symmetries in the resolvent mean profile scaling regions induce symmetries in resolvent reveals scalings of response modes 21 / 38

24 -scaling of, λ streamwise energy spectra inner self-similar (analytical) outer (; ) = + = (;, ) + (;, ) + = + (;, ) / τ λ + λ λ/τ + + τ =,, 22 / 38

25 -scaling of, λ streamwise energy spectra inner self-similar (analytical) outer (; ) = + = (;, ) + (;, ) + = + (;, ) / τ τ = λ / 38

26 -scaling of, λ streamwise energy spectra inner self-similar (analytical) outer (; ) = + = (;, ) + (;, ) + = + (;, ) / τ τ = λ / 38

27 -scaling of, λ streamwise energy spectra inner self-similar (analytical) outer (; ) = + = (;, ) + (;, ) + = + (;, ) / τ τ = λ / 38

28 Fitting the {χ } Introducing () fitted at τ =, using mode scalings, and assuming scalings for (), generate SW energy intensity for high τ, Note logarithmic scaling in overlap region τ = (DNS) τ =, (DNS) τ =, τ =, τ =, 23 / 38

29 Fitting the {χ } Can fix {χ } by projection on DNS Norm choice is biased towards at these You can do better ( etc) with more modes 24 / 38

30 Geometric self-similarity; hierarchies of modes Under assumptions: 1. modes local in 2. critical layer term scales geometrically () = (/ ) 3. (() ) term balances with τ term log region is necessary to obtain invariant (and mode scalings). = + ( + ) ( + ) Similar arguments derive classical inner and outer Re scalings for resolvent modes 25 / 38

31 Scaling of the interaction coefficient tensor Forcing is given by (uu) of triadically-consistent modes, σj Njab in model. Scalings for self-similar modes (log region) have been found. For a given triad, decays exponentially with cu c. We get hierarchies of triads; an interacting triad at one yc determines triads at all y in a region. Moarref et al, in review; triad from Sharma & McKeon JFM / 38

32 Self-exciting processes in the model Notice that for a mode combination to self-excite, we just require resulting nonlinear forcing terms not to be orthogonal to the resolvent forcing modes. This is true for the example plotted. We have still not yet solved for χ we need a toy problem. 27 / 38

33 The exact travelling wave solutions These provide a nice testbed, since single In high turbulence, if model modes represent structure well, and if exact solutions also do, then modes pick out regions of state space more densely crossed due to nearby solutions with only low unstable dimension So, in neighbourhood of a solution, the modes should compactly describe that solution 28 / 38

34 Projection of solutions onto model modes 15 pipe solutions provided by A Willis of Sheffield, generated by continuation to = Also channel solutions provided by M Graham of Wisonsin-Madison (presented at APS) S and N solutions presented, upper and lower branch modes generated using of solution 29 / 38

35 S1 solution close to laminar; well represented with one mode per actual solution =... = fraction of energy.... fraction of solution energy, keeping singular values per Fourier mode 30 / 38

36 N3L solution lower branch; close to laminar; well represented actual solution =... = fraction of energy.... fraction of solution energy, keeping singular values per Fourier mode 31 / 38

37 N2U solution more turbulent ; less well represented actual solution =... = fraction of energy.... fraction of solution energy, keeping singular values per Fourier mode 32 / 38

38 N4U solution more turbulent ; less well represented; not visited in turbulent DNS actual solution =... = fraction of solution energy, keeping singular values per Fourier mode fraction of energy / 38

39 N2U solution more turbulent ; but well represented?! actual solution =... = fraction of energy.... fraction of solution energy, keeping singular values per Fourier mode 34 / 38

40 What is missing in the model? Modes capture S-type and N-type (lower branch) extremely well Some upper branch solutions captured, others not; don t know why First step to demonstrating χ solution for exact solutions Methods may extend to finding new exact solutions (using modes as seeds ) 35 / 38

41 Conclusions Derived a gain-optimal basis from NSE; sparse Captures turbulent structure and TW solns quite nicely Self-exciting mode combinations exist (amplitude dependent) Scalings of modes (; geometric); interaction coeffs & triads (log region) now known 36 / 38

42 For the future Investigating solutions for coefficients using TW solutions Use approximate solutions in χ-space to seed exact solution solvers Truncated models may lead to reduced/toy DNS 37 / 38

43 Pipe mode code available at Feel free to play. 38 / 38