Model Order Reduction of the Navier-Stokes Equations at High Reynolds Number Maciej Balajewicz 1 Earl Dowell 2 Bernd Noack 3 1 Aeronautics and Astronautics Stanford University 2 Mechanical Engineering Duke University 3 D´ epartment Fluides, Thermique, Combustion Institut PPRIME, Universit´ e de Poitiers December 10, 2012 Balajewicz, Dowell, Noack MOR of the Navier-Stokes December 10, 2012 1 / 13
Background and motivation Common elements in most Model Order Reduction (MOR) techniques: 1. Spectral discretization � n u ( x , t ) ≈ i =1 a i ( t ) u i ( x ) (1) 2. Projection �� n � �� v i , R i =1 a i u i Ω = 0 (2) Balajewicz, Dowell, Noack MOR of the Navier-Stokes December 10, 2012 2 / 13
Background and motivation Common elements in most Model Order Reduction (MOR) techniques: 1. Spectral discretization � n u ( x , t ) ≈ i =1 a i ( t ) u i ( x ) (1) 2. Projection �� n � �� v i , R i =1 a i u i Ω = 0 (2) Balajewicz, Dowell, Noack MOR of the Navier-Stokes December 10, 2012 2 / 13
Background and motivation Common elements in most Model Order Reduction (MOR) techniques: 1. Spectral discretization � n u ( x , t ) ≈ i =1 a i ( t ) u i ( x ) (1) 2. Projection �� n � �� v i , R i =1 a i u i Ω = 0 (2) Balajewicz, Dowell, Noack MOR of the Navier-Stokes December 10, 2012 2 / 13
Background and motivation ◮ One popular approach: first n left-singular vectors of M , i.e u i = U (: , i ) where M = U Σ V T and u ( x 1 , t 1 ) u ( x 1 , t 2 ) · · · u ( x 1 , t N t ) u ( x 2 , t 1 ) u ( x 2 , t 2 ) · · · u ( x 2 , t N t ) M = (3) . . . ... . . . . . . u ( x N x , t 1 ) u ( x N x , t 2 ) · · · u ( x N x , t N t ) ◮ These basis functions are optimal in the sense that no other basis functions capture a greater proportion of kinetic energy of the flow. Balajewicz, Dowell, Noack MOR of the Navier-Stokes December 10, 2012 3 / 13
Background and motivation ◮ One popular approach: first n left-singular vectors of M , i.e u i = U (: , i ) where M = U Σ V T and u ( x 1 , t 1 ) u ( x 1 , t 2 ) · · · u ( x 1 , t N t ) u ( x 2 , t 1 ) u ( x 2 , t 2 ) · · · u ( x 2 , t N t ) M = (3) . . . ... . . . . . . u ( x N x , t 1 ) u ( x N x , t 2 ) · · · u ( x N x , t N t ) ◮ These basis functions are optimal in the sense that no other basis functions capture a greater proportion of kinetic energy of the flow. Balajewicz, Dowell, Noack MOR of the Navier-Stokes December 10, 2012 3 / 13
Background and motivation ◮ Turbulence is a multi-scale phenomenon: large scale flow features are broken down into smaller and smaller scales until the scales are fine enough that viscous forces can dissipate their energy. ◮ Application of any POD-based MOR strategy to a turbulent flow is problematic because POD, by construction, is biased toward the large, energy containing scales of the turbulent flow. ◮ Reduced Order Models (ROMs) generated using only the first most energetic POD basis functions are, therefore, not endowed with the natural energy dissipation of the smaller, lower energy turbulent scales. Balajewicz, Dowell, Noack MOR of the Navier-Stokes December 10, 2012 4 / 13
Background and motivation ◮ Turbulence is a multi-scale phenomenon: large scale flow features are broken down into smaller and smaller scales until the scales are fine enough that viscous forces can dissipate their energy. ◮ Application of any POD-based MOR strategy to a turbulent flow is problematic because POD, by construction, is biased toward the large, energy containing scales of the turbulent flow. ◮ Reduced Order Models (ROMs) generated using only the first most energetic POD basis functions are, therefore, not endowed with the natural energy dissipation of the smaller, lower energy turbulent scales. Balajewicz, Dowell, Noack MOR of the Navier-Stokes December 10, 2012 4 / 13
Background and motivation ◮ Turbulence is a multi-scale phenomenon: large scale flow features are broken down into smaller and smaller scales until the scales are fine enough that viscous forces can dissipate their energy. ◮ Application of any POD-based MOR strategy to a turbulent flow is problematic because POD, by construction, is biased toward the large, energy containing scales of the turbulent flow. ◮ Reduced Order Models (ROMs) generated using only the first most energetic POD basis functions are, therefore, not endowed with the natural energy dissipation of the smaller, lower energy turbulent scales. Balajewicz, Dowell, Noack MOR of the Navier-Stokes December 10, 2012 4 / 13
Background and motivation Classical benchmark: Incompressible flow inside a square, two-dimensional lid-driven cavity at Re u = 3 × 10 4 1 0 . 5 y 0 − 0 . 5 − 1 − 1 − 0 . 5 0 0 . 5 1 x Balajewicz, Dowell, Noack MOR of the Navier-Stokes December 10, 2012 5 / 13
Background and motivation POD basis functions of the lid-driven cavity (a) u 1 (b) u 2 (c) u 10 (d) u 20 (e) u 50 (f) u 200 Balajewicz, Dowell, Noack MOR of the Navier-Stokes December 10, 2012 6 / 13
Background and motivation Ω | u ( x , t ) | 2 d x � Percent of turbulent kinetic energy, e ( t ) ≡ 1/2 captured by the first n basis functions, u i of the lid-driven cavity n % 1 16.06 2 29.21 3 37.45 4 44.88 5 50.37 10 67.16 20 82.40 50 93.21 200 99.31 Balajewicz, Dowell, Noack MOR of the Navier-Stokes December 10, 2012 7 / 13
Background and motivation Standard POD-Galerkin ROMs of the lid-driven cavity ◮ ROM performance quantified using the turbulent kinetic Ω | u ( x , t ) | 2 d x � energy, e ( t ) ≡ 1/2 10 2 DNS e ( t ) 10 − 1 10 − 4 0 100 200 300 400 500 t Balajewicz, Dowell, Noack MOR of the Navier-Stokes December 10, 2012 8 / 13
Background and motivation Standard POD-Galerkin ROMs of the lid-driven cavity ◮ ROM performance quantified using the turbulent kinetic Ω | u ( x , t ) | 2 d x � energy, e ( t ) ≡ 1/2 10 2 DNS n = 5 POD-Galerkin ROM e ( t ) 10 − 1 10 − 4 0 100 200 300 400 500 t Balajewicz, Dowell, Noack MOR of the Navier-Stokes December 10, 2012 8 / 13
Background and motivation Standard POD-Galerkin ROMs of the lid-driven cavity ◮ ROM performance quantified using the turbulent kinetic Ω | u ( x , t ) | 2 d x � energy, e ( t ) ≡ 1/2 10 2 DNS n = 5 POD-Galerkin ROM n = 20 POD-Galerkin ROM e ( t ) 10 − 1 10 − 4 0 100 200 300 400 500 t Balajewicz, Dowell, Noack MOR of the Navier-Stokes December 10, 2012 8 / 13
Background and motivation Standard POD-Galerkin ROMs of the lid-driven cavity ◮ ROM performance quantified using the turbulent kinetic Ω | u ( x , t ) | 2 d x � energy, e ( t ) ≡ 1/2 10 2 DNS n = 5 POD-Galerkin ROM n = 20 POD-Galerkin ROM n = 50 POD-Galerkin ROM e ( t ) 10 − 1 10 − 4 0 100 200 300 400 500 t Balajewicz, Dowell, Noack MOR of the Navier-Stokes December 10, 2012 8 / 13
Background and motivation Standard POD-Galerkin ROMs of the lid-driven cavity ◮ ROM performance quantified using the turbulent kinetic Ω | u ( x , t ) | 2 d x � energy, e ( t ) ≡ 1/2 10 2 DNS n = 5 POD-Galerkin ROM n = 20 POD-Galerkin ROM n = 50 POD-Galerkin ROM n = 200 POD-Galerkin ROM e ( t ) 10 − 1 10 − 4 0 100 200 300 400 500 t Balajewicz, Dowell, Noack MOR of the Navier-Stokes December 10, 2012 8 / 13
Background and motivation Standard POD-Galerkin ROMs of the lid-driven cavity ◮ ROM performance quantified using the turbulent kinetic Ω | u ( x , t ) | 2 d x � energy, e ( t ) ≡ 1/2 10 2 DNS n = 5 POD-Galerkin ROM n = 20 POD-Galerkin ROM e ( t ) 10 − 1 10 − 4 0 100 200 300 400 500 t Balajewicz, Dowell, Noack MOR of the Navier-Stokes December 10, 2012 8 / 13
Proposed new approach Basic idea: Instead of using just the most energetic POD basis functions, include a few lower energy POD basis functions so that dissipative scales are resolved. ◮ For example, consider we are interested in forming a n = 3 ROM. ◮ Standard approach: use the first 3 most energetic POD basis functions, i.e. u 1 , u 2 , and u 3 ◮ Proposed Approach: u 1 , u 2 , and u 5 Balajewicz, Dowell, Noack MOR of the Navier-Stokes December 10, 2012 9 / 13
Proposed new approach Basic idea: Instead of using just the most energetic POD basis functions, include a few lower energy POD basis functions so that dissipative scales are resolved. ◮ For example, consider we are interested in forming a n = 3 ROM. ◮ Standard approach: use the first 3 most energetic POD basis functions, i.e. u 1 , u 2 , and u 3 ◮ Proposed Approach: u 1 , u 2 , and u 5 Balajewicz, Dowell, Noack MOR of the Navier-Stokes December 10, 2012 9 / 13
Proposed new approach Basic idea: Instead of using just the most energetic POD basis functions, include a few lower energy POD basis functions so that dissipative scales are resolved. ◮ For example, consider we are interested in forming a n = 3 ROM. ◮ Standard approach: use the first 3 most energetic POD basis functions, i.e. u 1 , u 2 , and u 3 ◮ Proposed Approach: u 1 , u 2 , and u 5 Balajewicz, Dowell, Noack MOR of the Navier-Stokes December 10, 2012 9 / 13
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