Regularization modeling of turbulent mixing; sweeping the scales

Transcription

1 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

2 Turbulence sweeping scales?

3 Cyclone near Iceland

4 Cyclone near Iceland

5 Early impression of swept scales - Da Vinci

6 Practical relevance - weather dynamics

7 Vortices in the far wake - airport congestion

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

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

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

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

12 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

13 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

14 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

15 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

16 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?

17 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?

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

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

20 Some basic flow properties momentum thickness A(k) time Momentum thickness and energy spectrum Smagorinsky too dissipative, Bardina not enough dynamic models quite accurate problems, e.g., intermediate and smallest scales k

21 Instantaneous snapshots of spanwise vorticity x2 0 x x1 (a) x1 (b) x2 0 x x1 (c) x1 a: DNS, b: Bardina, c: Smagorinsky, d: dynamic Accuracy limited: regularization models better? (d)

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

23 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

24 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

25 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

26 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

27 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

28 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 a ) 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?

29 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 a ) 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?

30 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 a ) 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?

31 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 a ) 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?

32 Refining - refining

33 Refining - refining

34 Refining - refining

35 Refining - refining

36 Refining - refining

37 Refining - refining

38 Refining - refining

39 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

40 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

41 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

42 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

43 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

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

45 Momentum thickness θ as = l/16 7 θ Filtered DNS ( ) Leray-model 32 3 : dash-dotted 64 3 : solid 96 3 : dynamic model 32 3 : dashed 64 3 : dashed with t

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

47 Robustness at arbitrary Reynolds number 10 1 E k 5/ Re = : dash-dotted 96 3 : dash-dotted, Re = : dashed 96 3 : dashed, Re = : solid 96 3 : solid, k What about improvements arising from NS-α?

48 Momentum thickness θ as = l/ θ t NS-α (solid), Leray (dash), dynamic (dash-dotted), DNS approximately grid-independent at 96 3 NS-α accurate but converges slowest

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

50 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

51 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

52 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

53 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

54 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

55 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

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

57 Quality of filter-inversion Leray 10 x Inversion of Simpson top-hat filter: 0 (a) Kinetic energy (a) and momentum thickness (b) n = 1 (solid), n = 2 (dash), n 5 (b)

58 Quality of filter-inversion LANS-α Kinetic energy E and momentum thickness D 1.5 x Inversion of Simpson top-hat filter: 0 (a) n = 1 (solid), n = 2 (dash), n = 5, n 10 Conclusion: n 5 (b)

59 Quality of filter-inversion LANS-α Kinetic energy E and momentum thickness D 1.5 x Inversion of Simpson top-hat filter: 0 (a) n = 1 (solid), n = 2 (dash), n = 5, n 10 Conclusion: n 5 (b)

60 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

61 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

62 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

63 LANS-α: sensitive to discretization N = 32: Kinetic energy E and momentum thickness D 10 x (a) Second order, Fourth order, (4-2) method (b)

64 Spectral signature of numerics Leray and LANS-α spectra: (a) (b) Second order, Fourth order N = 32 (dash), N = 64 (dash-dot), N = 96 (solid)

65 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

66 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

67 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

68 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

69 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

70 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

71 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

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

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

74 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

75 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

76 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

77 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

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

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

80 Error-landscape: optimal refinement (a) (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

81 Error-landscape: optimal refinement (a) (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

82 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

83 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

84 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

85 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

86 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

87 LES with DG-FEM: dissipative numerics γ c c s N Third order DG-FEM at Re λ = 100: red symbols - optimal setting

88 Optimality of MILES? δ L δ L N (a) 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

89 Optimality of MILES? δ L δ L N (a) 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

90 Optimality of MILES? δ L δ L N (a) 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

91 Optimality of MILES? δ L δ L N (a) 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

92 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)]

93 SIPI applied to homogeneous turbulence Each iteration = separate simulation (a) (b) Re λ = 50 (a) and Re λ = 100 (b). Resolutions N = 24 (solid), N = 32 (dashed) and N = 48 (dash-dotted) Iterations: +

94 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) 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

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

96 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

97 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

98 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

99 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

100 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

101 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

102 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

103 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

104 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

105 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

106 Combustion-modulated turbulence Fuel Oxidizer

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

108 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 dx 2 4 Ω l/2

109 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 dx 2 4 Ω l/2 10 x E 9.5 δ t t Q = 1 (solid), Q = 10 (dash), Q = 100 (dash-dot)

110

111 Model: Stratified mixing layer g

112 Turbulent plumes Unstable, anisotropic, inhomogeneous: go

113 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

114 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

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

116 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?

117 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

118 Effect of Fr on surface area and wrinkling Area (f = 1) and wrinkling (f = n ) 10 4 A 10 4 W 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) t (b) 2

119 Surface-area and wrinkling: LES Localized wrinkling more difficult to capture with LES 7 2 x A5 W t (a) Fr = 2 and Sc = 10: DNS, Leray, NS-α, Dynamic t (b)

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

121 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,...)

122 Any questions?