Mathematical Models and Measures of Mixing Part II: More Models and - PowerPoint PPT Presentation

Mathematical Models and Measures of Mixing Part II: More Models and Reconciliation Zhi George Lin 1 Katarina Bodova 1 Charles R. Doering 1 , 2 , 3 1 Department of Mathematics University of Michigan 2 Department of Physics and Michigan Center for Theoretical Physics University of Michigan 3 Center for the Study of Complex Systems University of Michigan SIAM Conference on Applications of Dynamical Systems, 2009 Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 1 / 24

Outline Recap of Part I 1 Models, Measures and Conflicts Resolution Questions Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 2 / 24

Outline Recap of Part I 1 Models, Measures and Conflicts Resolution Questions More Modeling 2 Kinetics Dispersion-Diffusion Theory Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 2 / 24

Outline Recap of Part I 1 Models, Measures and Conflicts Resolution Questions More Modeling 2 Kinetics Dispersion-Diffusion Theory Reconciliation 3 Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 2 / 24

Outline Recap of Part I 1 Models, Measures and Conflicts Resolution Questions More Modeling 2 Kinetics Dispersion-Diffusion Theory Reconciliation 3 Conjecture 4 Richardson Turbulence Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 2 / 24

Recap of Part I Models, Measures and Conflicts Recap of Part I Advection-Diffusion Equation for a Passive Scalar u = 0 , 〈 s 〉 = 0 ∂ t θ + � u ·∇ θ = κ ∆ θ + s , ∇· � Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 3 / 24

Recap of Part I Models, Measures and Conflicts Recap of Part I Advection-Diffusion Equation for a Passive Scalar u = 0 , 〈 s 〉 = 0 ∂ t θ + � u ·∇ θ = κ ∆ θ + s , ∇· � Method Measure Scaling Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 3 / 24

Recap of Part I Models, Measures and Conflicts Recap of Part I Advection-Diffusion Equation for a Passive Scalar u = 0 , 〈 s 〉 = 0 ∂ t θ + � u ·∇ θ = κ ∆ θ + s , ∇· � Method Measure Scaling Homogenization ∼ κ Pe 2 Effective Diffusivity Flux-Gradient Ansatz Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 3 / 24

Recap of Part I Models, Measures and Conflicts Recap of Part I Advection-Diffusion Equation for a Passive Scalar u = 0 , 〈 s 〉 = 0 ∂ t θ + � u ·∇ θ = κ ∆ θ + s , ∇· � Method Measure Scaling Homogenization ∼ κ Pe 2 Effective Diffusivity Flux-Gradient Ansatz Variational Methods Variance Suppression ∼ κ Pe Internal Layer Theory Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 3 / 24

Recap of Part I Models, Measures and Conflicts Recap of Part I Advection-Diffusion Equation for a Passive Scalar u = 0 , 〈 s 〉 = 0 ∂ t θ + � u ·∇ θ = κ ∆ θ + s , ∇· � Method Measure Scaling Homogenization ∼ κ Pe 2 Effective Diffusivity Flux-Gradient Ansatz �= Variational Methods Variance Suppression ∼ κ Pe Internal Layer Theory Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 3 / 24

Recap of Part I Models, Measures and Conflicts Recap of Part I Advection-Diffusion Equation for a Passive Scalar u = 0 , 〈 s 〉 = 0 ∂ t θ + � u ·∇ θ = κ ∆ θ + s , ∇· � Method Measure Scaling Homogenization ∼ κ Pe 2 Effective Diffusivity Flux-Gradient Ansatz �= Variational Methods Variance Suppression ∼ κ Pe Internal Layer Theory Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 3 / 24

Recap of Part I Models, Measures and Conflicts Recap of Part I Advection-Diffusion Equation for a Passive Scalar u = 0 , 〈 s 〉 = 0 ∂ t θ + � u ·∇ θ = κ ∆ θ + s , ∇· � Method Measure Scaling Homogenization ∼ κ Pe 2 Effective Diffusivity Flux-Gradient Ansatz �= Variational Methods Variance Suppression ∼ κ Pe Internal Layer Theory Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 3 / 24

Recap of Part I Models, Measures and Conflicts Recap of Part I Advection-Diffusion Equation for a Passive Scalar u = 0 , 〈 s 〉 = 0 ∂ t θ + � u ·∇ θ = κ ∆ θ + s , ∇· � Method Measure Scaling Homogenization ∼ κ Pe 2 Effective Diffusivity Flux-Gradient Ansatz �= Variational Methods Variance Suppression ∼ κ Pe Internal Layer Theory Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 3 / 24

Recap of Part I Models, Measures and Conflicts Recap of Part I Advection-Diffusion Equation for a Passive Scalar u = 0 , 〈 s 〉 = 0 ∂ t θ + � u ·∇ θ = κ ∆ θ + s , ∇· � Method Measure Scaling Homogenization ∼ κ Pe 2 Effective Diffusivity Flux-Gradient Ansatz �= Variational Methods Variance Suppression ∼ κ Pe Internal Layer Theory Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 3 / 24

Recap of Part I Models, Measures and Conflicts Recap of Part I Advection-Diffusion Equation for a Passive Scalar u = 0 , 〈 s 〉 = 0 ∂ t θ + � u ·∇ θ = κ ∆ θ + s , ∇· � Method Measure Scaling Homogenization ∼ κ Pe 2 Effective Diffusivity Flux-Gradient Ansatz �= Variational Methods Variance Suppression ∼ κ Pe Internal Layer Theory Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 3 / 24

Recap of Part I Models, Measures and Conflicts Recap of Part I Advection-Diffusion Equation for a Passive Scalar u = 0 , 〈 s 〉 = 0 ∂ t θ + � u ·∇ θ = κ ∆ θ + s , ∇· � Method Measure Scaling Homogenization ∼ κ Pe 2 Effective Diffusivity Flux-Gradient Ansatz �= Variational Methods Variance Suppression ∼ κ Pe Internal Layer Theory Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 3 / 24

Recap of Part I Models, Measures and Conflicts Recap of Part I Advection-Diffusion Equation for a Passive Scalar u = 0 , 〈 s 〉 = 0 ∂ t θ + � u ·∇ θ = κ ∆ θ + s , ∇· � Method Measure Scaling Homogenization ∼ κ Pe 2 Effective Diffusivity Flux-Gradient Ansatz �= Variational Methods Variance Suppression ∼ κ Pe Internal Layer Theory Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 3 / 24

Recap of Part I Models, Measures and Conflicts Recap of Part I Advection-Diffusion Equation for a Passive Scalar u = 0 , 〈 s 〉 = 0 ∂ t θ + � u ·∇ θ = κ ∆ θ + s , ∇· � Method Measure Scaling Homogenization ∼ κ Pe 2 Effective Diffusivity Flux-Gradient Ansatz �= Variational Methods Variance Suppression ∼ κ Pe Internal Layer Theory Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 3 / 24

Recap of Part I Models, Measures and Conflicts Recap of Part I Advection-Diffusion Equation for a Passive Scalar u = 0 , 〈 s 〉 = 0 ∂ t θ + � u ·∇ θ = κ ∆ θ + s , ∇· � Method Measure Scaling Homogenization ∼ κ Pe 2 Effective Diffusivity Flux-Gradient Ansatz �= Variational Methods Variance Suppression ∼ κ Pe Internal Layer Theory Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 3 / 24

Recap of Part I Models, Measures and Conflicts Recap of Part I Advection-Diffusion Equation for a Passive Scalar u = 0 , 〈 s 〉 = 0 ∂ t θ + � u ·∇ θ = κ ∆ θ + s , ∇· � Method Measure Scaling Homogenization ∼ κ Pe 2 Effective Diffusivity Flux-Gradient Ansatz �= Variational Methods Variance Suppression ∼ κ Pe Internal Layer Theory Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 3 / 24

Recap of Part I Models, Measures and Conflicts Recap of Part I Advection-Diffusion Equation for a Passive Scalar u = 0 , 〈 s 〉 = 0 ∂ t θ + � u ·∇ θ = κ ∆ θ + s , ∇· � Method Measure Scaling Homogenization ∼ κ Pe 2 Effective Diffusivity Flux-Gradient Ansatz �= Variational Methods Variance Suppression ∼ κ Pe Internal Layer Theory Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 3 / 24

Recap of Part I Resolution Resolution Exact HT: 1+Pe 2 10 10 〈� θ 0 � 2 〉 / 〈� θ � 2 〉 ILT: r 7/6 Pe 5/6 10 8 10 6 2 6 5 = k / s k u = 10 4 r � E 0 = 10 2 10 0 10 0 10 2 10 4 Pe = U κ k u Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 4 / 24

Recap of Part I Resolution Resolution r ↑ Exact HT: 1+Pe 2 10 10 〈� θ 0 � 2 〉 / 〈� θ � 2 〉 HT: r � 1 ILT: r 7/6 Pe 5/6 r > Pe r � 1 10 8 10 6 ILT: Pe � 1 10 4 Pe > r � e E 0 = P r � 1 10 2 = r 10 0 → 10 0 10 2 10 4 Pe Pe � 1 Pe � 1 Pe = U κ k u Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 4 / 24

Recap of Part I Questions Questions HT fails to predict the scalar variance sustained by steady sources and 1 sinks when Pe > r . Why? Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 5 / 24

Recap of Part I Questions Questions HT fails to predict the scalar variance sustained by steady sources and 1 sinks when Pe > r . Why? How can information about particle dispersion predict variance 2 suppression at high P ´ eclet numbers without assuming the scale separation? Lin, Bodova, Doering (U. Michigan) Mixing Models and Measures SIAM DS09 5 / 24