A review of mathematical regularization as model for small-scale turbulence Bernard J. Geurts Multiscale Modeling and Simulation Madrid, ICMAT, July 3-7
Turbulence in physical space Bernard Geurts: A review of mathematical regularization as model for small-scale turbulence
Incompressible flow Conservation of mass and momentum: ∂ j u j = 0 ∂ t u i + ∂ j ( u i u j ) + ∂ i p − 1 Re ∂ jj u i = 0 convective flux: nonlinear, destabilizing viscous flux: linear, dissipative Ratio – Reynolds number Re = convection dissipation = UL ν Turbulence requires Re ≫ 1 Bernard Geurts: A review of mathematical regularization as model for small-scale turbulence
Energy cascading process in 3D I II III ln(E) −5/3 k k i k d ln(k) I: large-scales stirring at integral length-scale ℓ i ∼ 1 / k i II: inviscid nonlinear transfer – inertial range E ∼ k − 5 / 3 III: viscous dissipation dominant ℓ d ∼ 1 / k d Bernard Geurts: A review of mathematical regularization as model for small-scale turbulence
Complexity of turbulent flow Kolmogorov: small scales are viscous, isotropic and universal Proposition: properties depend on viscosity ν and dissipation rate ε [ ν ] = length 2 / time ; [ ε ] = length 2 / time 3 Scales: make length and time � ν 3 ∼ 1 � ν � 1 / 4 � 1 / 2 η = ; τ η = ε k d ε Three dimensions: � ℓ � 3 # time-steps ∼ t end ∼ Re 9 / 4 ∼ Re 1 / 2 # dof ∼ ; η τ η Computationally intensive problem as Re ≫ 1 Bernard Geurts: A review of mathematical regularization as model for small-scale turbulence
Computational challenge Reynolds scaling of numerical resolution � ℓ � 3 ∼ Re 9 / 4 N = η Memory: if Re → Re × 10 then N → 10 9 / 4 × N ≈ 175 × N Reynolds scaling of numerical work Work ∼ Re 1 / 2 Re 9 / 4 ∼ Re 11 / 4 CPU: if Re → Re × 10 then W → 10 11 / 4 W ≈ 560 W Tough simulation problem - direct approach often impractical - capture primary features instead Bernard Geurts: A review of mathematical regularization as model for small-scale turbulence
Four themes to be mastered: Phenomenology of (coarsened) turbulence Turbulence modeling and numerical methods Error-assessment for large-eddy simulation High-performance computing
Four themes to be mastered: Phenomenology of (coarsened) turbulence Turbulence modeling and numerical methods Error-assessment for large-eddy simulation High-performance computing
Outline Filtering and closure 1 2 Regularized Navier-Stokes as turbulence model Model testing 3 Concluding Remarks 4 Bernard Geurts: A review of mathematical regularization as model for small-scale turbulence
Filtering Regularization Testing Conclusion Outline Filtering and closure 1 2 Regularized Navier-Stokes as turbulence model Model testing 3 Concluding Remarks 4 Bernard Geurts: A review of mathematical regularization as model for small-scale turbulence
Filtering Regularization Testing Conclusion DNS and LES in a picture capture both large and small scales: resolution problem Coarsening/mathematical modeling instead: LES Bernard Geurts: A review of mathematical regularization as model for small-scale turbulence
Filtering Regularization Testing Conclusion Filtering Navier-Stokes equations ; ∂ t u i + ∂ j ( u i u j ) + ∂ i p − 1 ∂ j u j = 0 Re ∂ jj u i = 0 Convolution-Filtering: filter-kernel G � u i = L ( u i ) = G ( x − ξ ) u ( ξ ) d ξ ; L ( 1 ) = 1 Large-eddy equations: ∂ j u j = 0 ∂ t u i + ∂ j ( u i u j ) + ∂ i p − 1 Re ∂ jj u i = − ∂ j ( u i u j − u i u j ) Sub-filter stress tensor τ ij = u i u j − u i u j = L (Π ij ( u )) − Π ij ( L ( u )) = [ L , Π ij ]( u ) Bernard Geurts: A review of mathematical regularization as model for small-scale turbulence
Filtering Regularization Testing Conclusion Spatial filtering, closure problem After closure - Shorthand notation: NS ( u ) = 0 ⇒ NS ( u ) = −∇ · τ ( u , u ) ⇐ −∇ · M ( u ) Basic LES formulation Find v : NS ( v ) = −∇ · M ( v ) What models M are available/reasonable? Bernard Geurts: A review of mathematical regularization as model for small-scale turbulence
Filtering Regularization Testing Conclusion Eddy-viscosity modeling Obtain smoothing via increased dissipation: � 1 � ∂ t u i + ∂ j ( u j u i ) + ∂ i p − Re + ν t ∂ jj u i = 0 Damp large gradients: dimensional analysis ν t = length × velocity ∼ ∆ × ∆ | ∂ x u | Effect: Strong damping at large filter-width and/or gradients Bernard Geurts: A review of mathematical regularization as model for small-scale turbulence
Filtering Regularization Testing Conclusion Some explicit subgrid models Popular models: Dissipation: Eddy-viscosity models, e.g., Smagorinsky � 1 1 � τ ij → − ν t S ij = − ( C S ∆) 2 | S | S ij ; effect Re → Re + ν t Similarity: Inertial range, e.g., Bardina 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 Bernard Geurts: A review of mathematical regularization as model for small-scale turbulence
Filtering Regularization Testing Conclusion Outline Filtering and closure 1 2 Regularized Navier-Stokes as turbulence model Model testing 3 Concluding Remarks 4 Bernard Geurts: A review of mathematical regularization as model for small-scale turbulence
Filtering Regularization Testing Conclusion Dissipation or regularization? Smagorinsky Leray Capture turbulence with eddy-viscosity or, with mathematics? Bernard Geurts: A review of mathematical regularization as model for small-scale turbulence
Filtering Regularization Testing Conclusion More regularization in LES Ciprian Foias — Darryl Holm — Edriss Titi Models with rigorously established existence, uniqueness, regularity, convergence to NS, transformation, symmetries, ... Bernard Geurts: A review of mathematical regularization as model for small-scale turbulence
Filtering Regularization Testing Conclusion Mathematical regularization as subgrid model Dynamic models popular in LES but: – expensive; ad hoc implementation features (‘clipping’) – still limited accuracy; complex flow extension difficult Regularization principle directly altering the nonlinearity Systematically obtain implied subgrid closure ‘Inherit’ rigorous mathematical properties Maintain transport structure and transformation properties of equations Consider two examples: Leray and NS- α Bernard Geurts: A review of mathematical regularization as model for small-scale turbulence
Filtering Regularization Testing Conclusion From Leray regularization to SFS model Proposal: ∂ t v i + v j ∂ j v i + ∂ i q − 1 Re ∂ jj v i = 0 Convolution filtering Leray: use ∂ j v j = 0 ∂ t v i + ∂ j ( v j v i ) + ∂ i q − 1 Re ∂ jj v i = 0 Rewrite into LES template: ∂ t v i + ∂ j ( v j v i ) + ∂ i q − 1 Re ∂ jj v i = − ∂ j ( v j v i − v j v i ) Implied Leray model: � � m L v j L − 1 ( v i ) ij = v j v i − v j v i = L − v j v i with L − 1 formally inverting L Bernard Geurts: A review of mathematical regularization as model for small-scale turbulence
Filtering Regularization Testing Conclusion Regularization requires filter-inversion Geometric series: repeated filtering N L − 1 = ( I − ( I − L )) − 1 → � ( I − L ) n n = 0 For example: L − 1 N = 0 : u = 0 ( u ) = u L − 1 N = 1 : u = 1 ( u ) = u + ( I − L ) u = 2 u − u L − 1 N = 2 : u = 2 ( u ) = u + ( I − L ) u + ( I − L )( I − L ) u = 3 u − 3 u + u Bernard Geurts: A review of mathematical regularization as model for small-scale turbulence
Filtering Regularization Testing Conclusion Regularization requires filter-inversion Bernard Geurts: A review of mathematical regularization as model for small-scale turbulence
Filtering Regularization Testing Conclusion Alternative regularizations Consider a − b models: ∂ j v j = 0 and ∂ t v i + ∂ j ( a j b i ) + ∂ i q − 1 Re ∂ jj v i = 0 Then Choose: a j = v j ; b i = v i to obtain NS Choose: a j = v j ; b i = v i to obtain Leray Choose: a j = v j ; b i = v i to obtain modified Leray Choose: a j = v j ; b i = v i to obtain modified Bardina Or, implied models: m R ij = a j b i − v j v i , i.e., m L Leray : = v j v i − v j v i ij m mL Modified Leray : = v j v i − v j v i ij m mB Modified Bardina : = v j v i − v j v i ij Bernard Geurts: A review of mathematical regularization as model for small-scale turbulence
Filtering Regularization Testing Conclusion NS- α regularization Kelvin’s circulation theorem d � � − 1 � � u j dx j ∂ kk u j dx j = 0 ⇒ NS − eqs dt Re Γ( u ) Γ( u ) Filtered Kelvin theorem ( Γ( u ) → Γ( u ) ) extends Leray u u t 1 t 2 Bernard Geurts: A review of mathematical regularization as model for small-scale turbulence
Filtering Regularization Testing Conclusion NS- α regularization Filtered Kelvin circulation theorem d � � − 1 � � u j dx j ∂ kk u j dx j = 0 dt Re Γ( u ) Γ( u ) Euler-Poincar´ e ∂ t u j + u k ∂ k u j + u k ∂ j u k + ∂ j p − ∂ j ( 1 2 u k u k ) − 1 Re ∂ kk u j = 0 Rewrite into LES template: Implied subgrid model ∂ j ( u j u i ) + ∂ i p − 1 ∂ t u i + Re ∂ jj u i − 1 � � � � = − ∂ j u j u i − u j u i u j ∂ i u j − u j ∂ i u j 2 Bernard Geurts: A review of mathematical regularization as model for small-scale turbulence
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