Regularization modeling of turbulent mixing; sweeping the scales

Regularization modeling of turbulent mixing; sweeping the scales Bernard J. Geurts Multiscale Modeling and Simulation (Twente) Anisotropic Turbulence (Eindhoven) D 2 HFest, July 22-28, 2007

Turbulence sweeping scales?

Cyclone near Iceland

Early impression of swept scales - Da Vinci

Practical relevance - weather dynamics

Vortices in the far wake - airport congestion

Outline 1 Turbulence and filtering 2 Regularization modeling of small scales 3 Numerics: friend or foe? 4 Mixing: combustion and stratification 5 Concluding remarks

DNS and LES in a picture capture both large and small scales: W Re 3 N 4 Coarsening/mathematical modeling instead: LES

Filtering Navier-Stokes equations Convolution-Filtering: filter-kernel G, width u i = L(u i ) = G(x ξ)u(ξ) dξ Application of filter: t u i + j (u i u j ) + i p 1 Re jju i = j (u i u j u i u j ) Turbulent stress tensor: Properties τ: τ ij = u i u j u i u j = L(u i u j ) L(u i )L(u j ) = [L,Π ij ](u) fully specified by filter L and NS-dynamics depends on u and u closure problem

Heuristic: Eddy-viscosity modeling Obtain smoothing via increased dissipation: ( 1 ) t v i + j (v j v i ) + i P Re + ν t jj v i = 0 Damp large gradients: dimensional analysis ν t = length velocity x v Effect: Strong damping at large filter-width and/or gradients

Popular models: Some explicit subgrid models Dissipation: Eddy-viscosity models, e.g., Smagorinsky τ ij ν t S ij = (C S ) 2 S S ij ; effect Similarity: Inertial range, e.g., Bardina 1 ( 1 ) Re Re + ν t τ ij [L,Π ij ](u) = u i u j u i u j Mixed models? m ij = Bardina + C d Smagorinsky C d via dynamic Germano-Lilly procedure Are these models any good?

Mixing layer: testing ground for models Temporal model for mixing layer

Mixing layer: testing ground for models t = 20 t = 40 t = 80 Temporal at different t Spatial at different x

Some basic flow properties 7 10 1 6 10 0 10 1 momentum thickness 5 4 3 2 A(k) 10 2 10 3 10 4 10 5 10 6 1 10 7 10 8 0 0 20 40 60 80 100 time 10 9 10 0 10 1 Momentum thickness and energy spectrum Smagorinsky too dissipative, Bardina not enough dynamic models quite accurate problems, e.g., intermediate and smallest scales k

Instantaneous snapshots of spanwise vorticity 25 25 20 20 15 15 10 10 5 5 x2 0 x2 0 5 5 10 10 15 15 20 20 25 0 10 20 30 40 50 x1 (a) 0 25 10 20 30 40 50 x1 (b) 25 25 20 20 15 15 10 10 5 5 x2 0 x2 0 5 5 10 10 15 15 20 20 25 0 10 20 30 40 50 x1 (c) 0 25 10 20 30 40 50 x1 a: DNS, b: Bardina, c: Smagorinsky, d: dynamic Accuracy limited: regularization models better? (d)

Outline 1 Turbulence and filtering 2 Regularization modeling of small scales 3 Numerics: friend or foe? 4 Mixing: combustion and stratification 5 Concluding remarks

Models and connection to filter Central role of filter identified, but...... modeling rather heuristically, and...... connection with filter is only partial.. Alternative regularization models: first principles approach alter nonlinearity derive implied subgrid model achieve unique coupling to filter, e.g., Leray, LANS-α

Example: Leray regularization Alter convective fluxes: sweeping by large scales LES template: t u i + u j j u i + i p 1 Re u i = 0 t u i + j (u j u i ) + i p 1 ( ) Re u i = j mij L Implied Leray model ( ) mij L = L u j L 1 (u i ) u j u i = u j u i u j u i Rigorous analysis available ( 1930s) Uniquely coupled to filter L and its inverse L 1

Approximate inversion procedures Geometric series: repeated filtering For example: L 1 (v) = (I (I L)) 1 (v) N (I L) n (v) n=0 N = 0 : u = L 1 1 (v) = v N = 1 : u = L 1 2 (v) = v + (I L)(v) = 2v v Exact inversion of Simpson-top-hat: 3-point filters (a, 1 2a, a) n ( a 1 + 1 2a ) j L 1 vm+rj/2 (v m ) a (1 2a) 1/2 j= n Use: a = 1/3 and r = /h = 2, 4,... Convergence?

Refining - refining

Kelvin s circulation theorem d ( ) u dt j dx j 1 Re Γ(u) NS-α regularization Γ(u) kk u j dx j = 0 NS eqs

Kelvin s circulation theorem d ( ) u dt j dx j 1 Re Γ(u) NS-α regularization Γ(u) kk u j dx j = 0 NS eqs Filtered Kelvin theorem (Γ(u) Γ(u)) extends Leray u u t 1 t 2

NS-α regularization α-principle: Γ(u) Γ(u) Euler-Poincaré t u j + u k k u j + u k j u k + j p j ( 1 2 u ku k ) 1 Re kku j = 0 Rewrite into LES template: Implied subgrid model t u i + j (u j u i ) + i p 1 Re jju i ( ) = j u j u i u j u i 1 ) (u 2 j i u j u j i u j = j mij α Alternative expression of large-scale sweeping

Cascade-dynamics computability E M k 5/3 1/ k 13/3 k L k 3 k k NSa k DNS NS-α,Leray are dispersive Regularization alters spectrum controllable cross-over as k 1/ : steeper than 5/3

Leray and NS-α predictions: Re = 50, = l/16 (DNS) (DNS) (Leray) Snapshot u 2 : red (blue) corresponds to up/down (NS-α)

Momentum thickness θ as = l/16 7 θ 6 5 4 3 2 Filtered DNS ( ) Leray-model 32 3 : dash-dotted 64 3 : solid 96 3 : dynamic model 32 3 : dashed 64 3 : dashed with 1 0 0 10 20 30 40 50 60 70 80 90 100 t

Streamwise kinetic energy E as = l/16 E 10 1 10 0 10 1 10 2 10 3 Filtered DNS ( ) Leray-model 32 3 : dash-dotted 64 3 : solid 96 3 : dynamic model 32 3 : dashed 64 3 : dashed with 10 4 10 5 10 0 10 1 k

Robustness at arbitrary Reynolds number 10 1 E k 5/3 10 0 10 1 10 2 Re = 50 64 3 : dash-dotted 96 3 : dash-dotted, Re = 500 64 3 : dashed 96 3 : dashed, Re = 5000 64 3 : solid 96 3 : solid, 10 3 10 4 10 0 10 1 k What about improvements arising from NS-α?

Momentum thickness θ as = l/16 7 6 θ 5 4 3 2 1 0 0 10 20 30 40 50 60 70 80 90 100 t NS-α (solid), Leray (dash), dynamic (dash-dotted), DNS approximately grid-independent at 96 3 NS-α accurate but converges slowest

Outline 1 Turbulence and filtering 2 Regularization modeling of small scales 3 Numerics: friend or foe? 4 Mixing: combustion and stratification 5 Concluding remarks

Numerics in academic LES setting Goal: approximate the unique solution to system of PDE s resulting after adopting explicit closure model General requirements: Filter separates scales > from scales < Computational grid provides additional length-scale h Require /h to be sufficiently large ( /h ) Good numerics: v(x, t :, h) v(x, t :, 0) rapidly However: computational costs N 4 : implies modest /h potentially large role of numerical method in computational dynamics because of marginal resolution

Numerical aspects Study influence of: Filter inversion Spatial discretization method Also study error interactions/cancellations...

Quality of filter-inversion Leray 10 x 104 7 9.5 6 5 9 4 8.5 3 8 2 7.5 1 7 0 10 20 30 40 50 60 70 80 90 100 Inversion of Simpson top-hat filter: 0 (a) 0 10 20 30 40 50 60 70 80 90 100 Kinetic energy (a) and momentum thickness (b) n = 1 (solid), n = 2 (dash), n 5 (b)

Quality of filter-inversion LANS-α Kinetic energy E and momentum thickness D 1.5 x 105 9 1.4 8 1.3 7 1.2 6 1.1 5 1 4 0.9 3 0.8 2 0.7 1 0.6 0 10 20 30 40 50 60 70 80 90 100 Inversion of Simpson top-hat filter: 0 (a) 0 10 20 30 40 50 60 70 80 90 100 n = 1 (solid), n = 2 (dash), n = 5, n 10 Conclusion: n 5 (b)

Spatial discretization Discretization induces spatial filter: δ x f(x) = x ( f) modified mean flux modified turbulent stress tensor Compare: Second order FV: δ x (2) (f(u)) Fourth order FV: δ x (4) (f(u)) Combination treats small scales at second order δ x (f(u)) = δ x (4) (f(u)) + δ (2) (f(u) f(u)) x

LANS-α: sensitive to discretization N = 32: Kinetic energy E and momentum thickness D 10 x 104 7 9.5 6 9 5 8.5 4 8 3 7.5 2 7 1 6.5 0 10 20 30 40 50 60 70 80 90 100 0 (a) 0 10 20 30 40 50 60 70 80 90 100 Second order, Fourth order, (4-2) method (b)

Spectral signature of numerics Leray and LANS-α spectra: 10 1 10 1 10 0 10 0 10 1 10 1 10 2 10 2 10 3 10 3 10 4 10 4 10 5 10 5 10 6 10 0 10 1 10 (a) 6 10 0 10 1 (b) Second order, Fourth order N = 32 (dash), N = 64 (dash-dot), N = 96 (solid)

Numerics or modeling or both? Observe: At marginal resolution numerics modifies the equations Likewise, a subgrid model modifies these equations Dilemma: which is to be preferred Not just best numerics/best model... LES paradoxes Governed by error-interactions: nonlinear error-accumulation

Smagorinsky fluid Homogeneous decaying turbulence at Re λ = 50, 100 Smagorinsky fluid subgrid model: m S ij = 2(C S ) 2 S S ij = 2l 2 S S S ij introduces Smagorinsky-length l S Also dynamic Smagorinsky fluid with length-scale l d

Experimental error-assessment Pragmatic: minimal total error at given computational costs Discuss: error-landscape/optimal refinement strategy LES-specific error-minimization: SIPI

Accuracy measures Monitor resolved kinetic energy E = 1 1 Ω 2 u u dx = 1 u u 2 Ω Measure relative error: top-hat filter, grid h = /r E δ E (, r) = LES (, r) E DNS (, r) E DNS (, r) with error integrated over time f 2 = 1 t 1 t 0 t1 t 0 f 2 (t)dt each simulation represented by single number concise representation facilitates comparison

Error-landscape: Definition Framework for collecting error information: l S h δ E N Each Smagorinsky LES corresponds to single point: ( N, l ) S ; error : δ h E Contours of δ E fingerprint of LES

Total error-landscape combination of central discretization and Smagorinsky optimum at C S > 0: SGS modeling is viable here

Error-landscape: optimal refinement 15 10 15 10 0.25 5 0.25 10 0.2 10 5 2.5 2.5 0.2 5 5 5 5 0.15 0.1 5 2.5 1.5 1.5 2.5 1.5 1 1.5 0.5 1 0.15 0.1 10 2.5 5 2.5 2.5 1.75 1.25 1.25 1.75 1.75 2.5 1.75 1.25 0.05 5 2.5 1 0.05 2.5 1.75 1.5 0.5 15 0 25 30 35 40 45 50 55 60 65 70 75 80 (a) 10 5 0 25 30 35 40 45 50 55 60 65 70 75 80 1.75 (b) Optimal trajectory for Re λ = 50 (a) and Re λ = 100 (b) Under-resolution leads to strong error-increase Dynamic procedure over-estimates viscosity

Observation: MILES philosophy practical LES implies marginal resolution which implies large role of specific numerical discretization next to dynamics due to subgrid model and leads to strong interactions and complex error-accumulation Proposal: obtain smoothing via appropriate numerical method alone accept that there is no grid-independent solution, other than DNS accept that predictions become discretization dependent Is no-model/just numerics option optimal/viable? Consider example: DG-FEM and homogeneous turbulence

DG-FEM of homogeneous turbulence Discretization: Approximate Riemann solver F = F central + γf dissipative ; HLLC flux c ILES plane N ELES plane c s Three-dimensional accuracy charts

LES with DG-FEM: dissipative numerics 1 0.8 0.6 0.4 γ c 0.2 0 0 0.05 0.1 0.15 0.2 c s 60 50 40 30 N 20 10 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 Third order DG-FEM at Re λ = 100: red symbols - optimal setting

Optimality of MILES? 0.20 0.20 0.15 0.15 δ L2 0.10 δ L2 0.10 0.05 0.05 0.00 10 20 30 40 50 60 N (a) 0.00 10 20 30 40 50 60 N (b) (a): 2nd order ; (b): 3rd order γ c = 1.00 (dot), γ c = 0.10 (dash) and γ c = 0.01 ( ) MILES-error larger than with explicit SGS model Exchange of dissipation in optimum... Option: directly optimize total simulation error - SIPI

SIPI - basic algorithm Goal: minimize total error at given N δe C S a b d c Initial triplet: no-model, dynamic and half-way New iterand CS d = b 1 (b a) 2 [δ E (b) δ E (c)] (b c) 2 [δ E (b) δ E (a)] 2 (b a)[δ E (b) δ E (c)] (b c)[δ E (b) δ E (a)]

SIPI applied to homogeneous turbulence Each iteration = separate simulation 10 25 9 8 20 7 6 15 5 4 10 3 2 5 1 0 0 0.05 0.1 0.15 0.2 0.25 0 (a) 0 0.05 0.1 0.15 0.2 0.25 (b) Re λ = 50 (a) and Re λ = 100 (b). Resolutions N = 24 (solid), N = 32 (dashed) and N = 48 (dash-dotted) Iterations: +

Convergence example Re λ = 50 Re λ = 100 n C (n) S (24) C(n) S (48) C(n) S (24) C(n) S (48) 2 0.17470000 0.15690000 0.18740000 0.17780000 3 0.08735000 0.07845000 0.09370000 0.08890000 4 0.08330619 0.03959082 0.23399921 0.08686309 5 0.11142326 0.04010420 0.16404443 0.11154788 6 0.11156545 0.05141628 0.16400079 0.09935076 7 0.10461386 0.05283713 0.15068033 0.09947629 8 0.10508422 0.05468589 0.15445586 0.09926872 9 0.10602797 0.05504089 0.15600828 0.09938967 Computational overhead SIPI: CPU-time T N 4 implies approximate optimization at N can be (almost) completed within cost of one simulation at 3N/2

Outline 1 Turbulence and filtering 2 Regularization modeling of small scales 3 Numerics: friend or foe? 4 Mixing: combustion and stratification 5 Concluding remarks

Application to complex physics Intuitive subgrid-scale modeling: collection of assumptions/parameters increases consistency becomes problem predictive power? Can first principles approach be extended? Consider combustion and stratified mixing

Example: Turbulent combustion Computational model: t ρ + j (ρu j ) = 0 t (ρu i ) + j (ρu i u j ) + i p j σ ij = 0 t e + j ((e + p)u j ) j (σ ij u i ) + j q j h k ω k = 0 t (ρc k ) + j (ρc k u j ) j (π kj ) ω k = 0 F-O-P flame: Arrhenius law F + O P ω F = (ρc F )(ρc O ) Da exp( Ze W O T ) ω O = αω F ; ω P = (1 + α)ω F Damköhler: Da and Zeldovich: Ze weight-ratio α = W O /W F Energy equation source term: h j ω j = Qω F ; k = 1,...,N s

Spatial filtering and subgrid closure Filtered model equations: (ρ, ũ j, p and c k ) t ρ + j (ρũ j ) = 0 t (ρũ i ) + j (ρũ i ũ j ) + i p j σ ij = (ρτ ij ) + j (σ ij σ ij ) t ĕ + j ((ĕ + p)ũ j ) j ( σ ij ũ i ) + j q j = h k ω k + A t (ρ c k ) + j (ρ c k ũ j ) j ( π kj ) = j (ρζ jk ) + ω k + j (π kj π kj ) Terms requiring closure: Turbulent stress tensor: τ ij Velocity-species stress tensor: ζ jk Chemical source terms Many more contributions - systematic approach needed Investigate extended Leray approach

Combustion-modulated turbulence Fuel Oxidizer

Surface c F = c O Stoichiometric surface t = 35 t = 55 t = 75 DNS at 256 3 resolution modest heat-release Q = 1

Turbulence combustion interaction Leray predictions: 32 3 and = l/16 1 E(t) = 2 u iu i dx ; δ = 1 l/2 ( )( ) 1 u 1 u 1 + 1 dx 2 4 Ω l/2

Turbulence combustion interaction Leray predictions: 32 3 and = l/16 1 E(t) = 2 u iu i dx ; δ = 1 l/2 ( )( ) 1 u 1 u 1 + 1 dx 2 4 Ω l/2 10 x 104 7 E 9.5 δ 6 5 9 4 8.5 3 2 8 1 7.5 0 10 20 30 40 50 60 70 80 90 100 0 0 10 20 30 40 50 60 70 80 90 100 t t Q = 1 (solid), Q = 10 (dash), Q = 100 (dash-dot)

Model: Stratified mixing layer g

Turbulent plumes Unstable, anisotropic, inhomogeneous: go

Stratification: DNS and LES Stratification: t u + u u + p 1 Re 2 u = 1 Fr 2(ce g) Advection-diffusion for scalar c t c + u c 1 Re Sc 2 c = 0 Characteristic numbers: (Re, Fr, Sc) LES: subgrid modeling in momentum and scalar equations

Unstable stratification at Fr = 2, Re = 50 mixing of heavy fluid on top of lighter fluid: t = 30,..., 55

LES capturing of plumes DNS (t = 40) Leray LES (t = 40) Case: Fr = 2 and Sc = 10 at = l/16 What about prediction mixing efficiency?

Level-set: surface-area and wrinkling Measure mixing in terms of level-set properties ( ) I = da f(x, t) = dx δ c(x, t) a c(x, t) f(x, t) S(a,t) V where S(a, t) = {x R 3 c(x, t) = a} V Laplace transform

Effect of Fr on surface area and wrinkling Area (f = 1) and wrinkling (f = n ) 10 4 A 10 4 W 10 3 10 2 10 0 10 1 10 2 t DNS: Sc = 10 and Fr = 1, 1.5, 2, 4, 8 W : Initially t then from t 3 to t 4 global features captured well with LES 10 (a) 1 10 0 10 1 10 t (b) 2

Surface-area and wrinkling: LES Localized wrinkling more difficult to capture with LES 7 2 x 104 6 A5 W 1.8 1.6 1.4 4 1.2 1 3 0.8 2 0.6 1 0 0 5 10 15 20 25 30 35 40 45 50 t 0.4 0.2 0 (a) 0 10 20 30 40 50 60 Fr = 2 and Sc = 10: DNS, Leray, NS-α, Dynamic t (b)

Outline 1 Turbulence and filtering 2 Regularization modeling of small scales 3 Numerics: friend or foe? 4 Mixing: combustion and stratification 5 Concluding remarks

Concluding Remarks Discussed: turbulence and coarsening closure problem heuristic modeling eddy-viscosity regularization approach large-scale sweeping role of spatial discretization Leray approach robust and generally accurate under-resolved LANS-α exaggerates small scales extended to complex flows - combustion, stratification (separation, rotation, particles, electromagnetism,...)

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