Reliability of LES in complex applications

Reliability of LES in complex applications Bernard J. Geurts Multiscale Modeling and Simulation (Twente) Anisotropic Turbulence (Eindhoven) DESIDER Symposium Corfu, June 7-8, 27

Sample of complex flow simulation DG-FEM treatment of delta-wing: Smagorinsky

Connection to Wake-Vortex Hazards Airport throughput limitations: separation up to km

LES - Tendency toward complex applications Reliability - Error-bounds? Turbulence model/numerics?? LES error-interactions? Optimal balance computational cost - accuracy?

Outline Regularization modeling 2 Role of numerics 3 Pragmatic LES 4 Concluding remarks

Energy cascading process in 3D I II III ln(e) k 5/3 k i k d ln(k) I: large-scales stirring at integral length-scale l i /k i II: inviscid nonlinear transfer inertial range E k 5/3 III: viscous dissipation dominant l d /k d

DNS and LES in a picture Classical problem: wide dynamic range capture both large and small scales: resolution problem N Re 9/4 ; W Re 3 If Re Re then N 75N and W W Coarsening/mathematical modeling instead: LES

Filtering Navier-Stokes equations Convolution-Filtering: filter-kernel G, filter-width u i = L(u i ) = G(x ξ)u(ξ) dξ Application of filter: j u j = t u i + j (u i u j ) + i p Re jju i = j (u i u j u i u j ) Turbulent stress tensor: closure problem 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)

Spatial filtering, closure problem Shorthand notation: NS(u) = NS(u) = τ(u, u) M(u) Completes basic LES formulation Find v : NS(v) = M(v) After closure system of PDE s results: dynamic range restricted to scales > Numerics does solution v resemble u? Modeling

Some explicit subgrid models Popular physics-based 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 ( ) 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 Alternative: mathematical, first principles modeling

Example: Leray regularization Alter convective fluxes: t u i + u j j u i + i p Re u i = LES template: t u i + j (u j u i ) + i p ( ) Re u i = j mij L Implied Leray model ( ) mij L = L u j L (u i ) u j u i = u j u i u j u i Uniquely coupled to filter L and its inverse L Rigorous analysis available ( 93s) Provides accurate LES model ( 3)

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

Kelvin s circulation theorem d ( ) u j dx j dt Γ(u) Re NS-α regularization Γ(u) kk u j dx j = NS eqs Filtered Kelvin theorem: yields extended Leray model u u t t 2

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

Mixing layer: testing ground for models t = 2 t = 4 t = 8 Temporal at different t Spatial at different x

Some basic flow properties 7 6 momentum thickness 5 4 3 2 A(k) 2 3 4 5 6 7 8 2 4 6 8 time 9 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 2 2 5 5 5 5 x2 x2 5 5 5 5 2 2 25 2 3 4 5 x (a) 25 2 3 4 5 x (b) 25 25 2 2 5 5 5 5 x2 x2 5 5 5 5 2 2 25 2 3 4 5 x (c) 25 2 3 4 5 x a: DNS, b: Bardina, c: Smagorinsky, d: dynamic Accuracy limited: regularization models better? (d)

Leray and NS-α predictions: Re = 5, = l/6 (DNS) (DNS) (Leray) Snapshot u 2 : red (blue) corresponds to up/down (NS-α)

Momentum thickness θ as = l/6 7 6 θ 5 4 3 2 2 3 4 5 6 7 8 9 t NS-α (solid), Leray (dash), dynamic (dash-dotted), DNS approximately grid-independent at 96 3 NS-α accurate but converges slowest

Application to complex physics Intuitive subgrid-scale modeling: collection of assumptions/parameters increases consistency becomes problem predictive power? Can first principles approach be extended? Wall-bounded flow, Combustion, Particle-laden flow,... But how much focus should be put on SGS modeling?

Outline Regularization modeling 2 Role of numerics 3 Pragmatic LES 4 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 :, ) rapidly However: computational costs N 4 : implies modest /h potentially large role of numerical method in computational dynamics because of marginal resolution

Modified closure problem high pass filter Discretization induces spatial filter: δ x f(x) = x ( f) Convective contribution: [ ] x (u 2 ) = δ x (u 2 ) + x (u 2 ) δ x (u 2 ) = δ x (u 2 ) + x (u 2 û2 ) = x (û2 ) + x (ξ) Modified mean flux Computational Turbulent Stress Tensor ξ = u 2 û2 = (u 2 u 2 ) + (u 2 û2 ) = τ + H(u 2 ) Numerically induced high-pass filter: H(f) = f f as r = /h

Dynamic importance - subgrid resolution Contributions associated with u = e ıkx : τ = A τ (k )e 2ıkx ; H(u 2 ) = A H (k, r)e 2ıkx

Dynamic importance - subgrid resolution Contributions associated with u = e ıkx : τ = A τ (k )e 2ıkx ; H(u 2 ) = A H (k, r)e 2ıkx 4 3.5 A τ A H 3 2.5 A τ : solid 2.5.5 2 3 4 5 6 k

Dynamic importance - subgrid resolution Contributions associated with u = e ıkx : τ = A τ (k )e 2ıkx ; H(u 2 ) = A H (k, r)e 2ıkx 4 3.5 A τ A H 3 2.5 2 A τ : solid A H at r = (- -) r = 2 (-.-) and r = 4 ( ).5.5 2 3 4 5 6 k

Dynamic importance - subgrid resolution Contributions associated with u = e ıkx : τ = A τ (k )e 2ıkx ; H(u 2 ) = A H (k, r)e 2ıkx 4 3.5 A τ A H 3 2.5 2.5 A τ : solid A H at r = (- -) r = 2 (-.-) and r = 4 ( ) 2nd order (thin) 4th order (thick).5 2 3 4 5 6 k

Dynamic importance - subgrid resolution Contributions associated with u = e ıkx : τ = A τ (k )e 2ıkx ; H(u 2 ) = A H (k, r)e 2ıkx 4 3.5 A τ A H 3 2.5 2.5.5 2 3 4 5 6 k A τ : solid A H at r = (- -) r = 2 (-.-) and r = 4 ( ) 2nd order (thin) 4th order (thick) Strong effect r = 2; reduction as r 4

Observe: Numerics or modeling or both? At marginal resolution the numerics strongly modifies the equations that should be solved (alteration of mathematical nature) Likewise, the introduction of a subgrid model modifies these equations (alteration of flow-physics nature) Dilemma: unclear which is to be preferred Pragmatic guideline: minimal total error at given computational costs Governed by error-interactions: nonlinear error-accumulation... not just best model/best numerics

Counter-acting errors: LES-paradoxes.2.4.6.8.3.2...2.3.4.5.6 Decaying turbulence: discretization, modeling and total-error Bernard J. Geurts: Reliability of LES in complex applications

Counter-acting errors: LES-paradoxes.2.4.6.8.3.2...2.3.4.5.6 time error in kinetic energy Decaying turbulence: discretization, modeling and total-error Bernard J. Geurts: Reliability of LES in complex applications

Counter-acting errors: LES-paradoxes.2.4.6.8.3.2...2.3.4.5.6 time error in kinetic energy Decaying turbulence: discretization, modeling and total-error better model may result in worse predictions higher order discretization may result in worse predictions optimal setting and refinement? Bernard J. Geurts: Reliability of LES in complex applications

Outline Regularization modeling 2 Role of numerics 3 Pragmatic LES 4 Concluding remarks

Experimental error-assessment Pragmatic: minimal total error at given computational costs Discuss: error-landscape/optimal refinement strategy optimality of MILES in DG-FEM LES-specific error-minimization: SIPI

Smagorinsky fluid Homogeneous decaying turbulence at Re λ = 5, 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

Accuracy measures Monitor resolved kinetic energy E = Ω 2 u u dx = 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 = t t t t 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 > : SGS modeling is viable here

Error-landscape: optimal refinement 5 5.25 5.25.2 5 2.5 2.5.2 5 5 5 5.5. 5 2.5.5.5 2.5.5.5.5.5. 2.5 5 2.5 2.5.75.25.25.75.75 2.5.75.25.5 5 2.5.5 2.5.75.5.5 5 25 3 35 4 45 5 55 6 65 7 75 8 (a) 5 25 3 35 4 45 5 55 6 65 7 75 8.75 (b) Optimal trajectory for Re λ = 5 (a) and Re λ = (b) Under-resolution leads to strong error-increase Dynamic model over-estimates viscosity reduces error well with increasing resolution

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.8.6.4 γ c.2.5..5.2 c s 6 5 4 3 N 2.4.35.3.25.2.5..5. Third order DG-FEM at Re λ = : red symbols - optimal setting

Optimal refinement strategies: 2nd order From dissipative to central:.2.5.2.5.2.5.5..2.5.5..5 c s.. c s..5 c s..5.5.5.5..5..5. 2 3 4 5 6 N. 2 3 4 5 6 N. 2 3 4 5 6 N left-to-right: γ c =, γ c =., γ c =. C S = at γ c = : MILES best option γ c < implies CS : MILES sub-optimal decrease γ c implies increase CS : exchange of dissipation

Optimal refinement strategies: 3rd order From dissipative to central:.2.2.2 c s.5..5..5 c s.5..5..5 c s.5..5..5.5.5.5.5..5....5 2 3 4 5 6 N. 2 3 4 5 6 N. 2 3 4 5 6 N left-to-right: γ c =, γ c =., γ c =. for all γ c [, ] find CS : MILES sub-optimal optimal C S is less sensitive to γ c value than 2nd order

Optimality of MILES (γ c = )?.2.2.5.5 δ L2. δ L2..5.5. 2 3 4 5 6 N (a). 2 3 4 5 6 N (b) (a): 2nd order ; (b): 3rd order γ c =. (dot), γ c =. (dash) and γ c =. ( ) 2nd: MILES-error larger than with explicit SGS model 3rd: optimum requires explicit SGS model 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 (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 25 9 8 2 7 6 5 5 4 3 2 5.5..5.2.25 (a).5..5.2.25 (b) Re λ = 5 (a) and Re λ = (b). Resolutions N = 24 (solid), N = 32 (dashed) and N = 48 (dash-dotted) Iterations: +

Convergence example Re λ = 5 Re λ = n C (n) S (24) C(n) S (48) C(n) S (24) C(n) S (48) 2.747.569.874.778 3.8735.7845.937.889 4.83369.395982.2339992.868639 5.42326.442.644443.54788 6.56545.54628.6479.993576 7.46386.528373.56833.9947629 8.58422.5468589.5445586.9926872 9.62797.55489.56828.9938967 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 Regularization modeling 2 Role of numerics 3 Pragmatic LES 4 Concluding remarks

Concluding remarks Regularization modeling LES filtering approach allows grid-independent LES closure problem: intuitive or first principles modeling Leray and LANS-α: accuracy and extension to complex physics Leray robust, LANS-α (slightly) more accurate turbulent combustion: Leray not (yet) more accurate but more systematic

Concluding remarks Grid-independent LES for computational error-assessment error-decomposition: modeling, discretization effects LES-paradoxes and interacting errors: better models/numerics may not lead to better predictions error-landscape optimal refinement strategy dynamic procedure efficient error-reduction Error-interaction and a priori error-bounds hard to include: SIPI to account for modeling and numerics Thanks: Johan Meyers (Leuven), Fedderik van der Bos (Munich), Darryl Holm (London)