Asymptotic modeling of the wave-propagation over acoustic liners Adrien Semin Technische Universit¨ at Berlin, Institut f¨ ur Mathematik Joint work with Kersten Schmidt (TU Berlin) and B´ erang` ere Delourme (Paris 13) Research Center M ATHEON Mathematics for key technologies Analysis and Numerics of Acoustic and Electromagnetic Problems, October 17th-22nd
Background Optimized noise reduction in transportation and interior spaces Jet engines of air-planes ⊲ Absorption of the generated acoustic pressure at different frequencies (noise) Jet engines and turbo machines ⊲ Damping of acoustic instabilities in the combustion chamber at particular frequencies Courtesy of A. Th¨ ons-Zueva 2 / 34
Motivation Perforated liners absorb acoustic waves ⊲ Absorption due to viscosity and interaction with flow through holes ⊲ Measurements at realistic temperatures, pressures are extremely expensive ⊲ Direct numerical simulations not feasible (small holes, boundary layer Bias flow liner in an acoustic channel (DUCT-C) at the ≪ wave-length) institute of Institute of Propulsion Technology at DLR Berlin (courtesy of F. Bake) 3 / 34
Outline Surface homogenization for micro-structured layers with singularities 1 Extension of the macroscopic part of the solution 2 Periodic layer obstacle and transmission conditions (2 scales) End-point of the periodic layer and corner singularities Periodic layer obstacle and transmission conditions (3 scales) Numerical simulations 3 Conclusion and perspectives 4 4 / 34
Outline Surface homogenization for micro-structured layers with singularities 1 Extension of the macroscopic part of the solution 2 Periodic layer obstacle and transmission conditions (2 scales) End-point of the periodic layer and corner singularities Periodic layer obstacle and transmission conditions (3 scales) Numerical simulations 3 Conclusion and perspectives 4 5 / 34
Surface homogenization for micro-structured layers with singularities Γ Θ Θ η ( δ ) x + x + x − δ x − h ( δ ) O O O O Figure: On the left: the exact domain Ω δ,η . On the right: the limit domain Ω 0 . ⊲ Navier-Stokes equation with non-linear advection for isothermal process in Ω δ,η ∂ t v + ( v ·∇ ) v + ∇ p − ν ∆ v = f Conservation of momentum ∂ t p + c 2 div v + v ·∇ p + p div v = 0 in Ω δ,η Conservation of mass on ∂ Ω δ,η v = 0 No-slip boundary condition Here, p = p ′ /ρ 0 , where p ′ is the acoustic pressure , and ν is the kinematic viscosity . ⊲ This model contains effects of different nature: small geometrical scales ( δ , η ( δ ), h ( δ )), re-entrant corners , viscosity ( ν = ν ( δ )) , non-linearity . ⊲ In the following, we develop the surface homogenization method in which the effect of the micro-structure and the viscosity is there (taken into account through Wentzel boundary conditions) for a single frequency excitation ( f ( t , x ) = F ( x ) exp( − ıω t )). 6 / 34
Surface homogenization for micro-structured layers with singularities Γ Θ Θ η ( δ ) x + x + x − δ x − h ( δ ) O O O O Figure: On the left: the exact domain Ω δ,η . On the right: the limit domain Ω 0 . ⊲ Navier-Stokes equation with non-linear advection for isothermal process in Ω δ,η ∂ t v + ( v ·∇ ) v + ∇ p − ν ∆ v = f Conservation of momentum ∂ t p + c 2 div v + v ·∇ p + p div v = 0 in Ω δ,η Conservation of mass on ∂ Ω δ,η v = 0 No-slip boundary condition Here, p = p ′ /ρ 0 , where p ′ is the acoustic pressure , and ν is the kinematic viscosity . ⊲ This model contains effects of different nature: small geometrical scales ( δ , η ( δ ), h ( δ )), re-entrant corners , viscosity ( ν = ν ( δ )) , non-linearity . ⊲ In the following, we develop the surface homogenization method in which the effect of the micro-structure and the viscosity is there (taken into account through Wentzel boundary conditions) for a single frequency excitation ( f ( t , x ) = F ( x ) exp( − ıω t )). 6 / 34
Surface homogenization for micro-structured layers with singularities Γ Θ Θ η ( δ ) x + x + x − δ x − h ( δ ) O O O O Figure: On the left: the exact domain Ω δ,η . On the right: the limit domain Ω 0 . ⊲ Navier-Stokes equation with non-linear advection for isothermal process in Ω δ,η ∂ t v + ( v ·∇ ) v + ∇ p − ν ∆ v = f Conservation of momentum ∂ t p + c 2 div v + v ·∇ p + p div v = 0 in Ω δ,η Conservation of mass on ∂ Ω δ,η v = 0 No-slip boundary condition Here, p = p ′ /ρ 0 , where p ′ is the acoustic pressure , and ν is the kinematic viscosity . ⊲ This model contains effects of different nature: small geometrical scales ( δ , η ( δ ), h ( δ )), re-entrant corners , viscosity ( ν = ν ( δ )) , non-linearity . ⊲ In the following, we develop the surface homogenization method in which the effect of the micro-structure and the viscosity is there (taken into account through Wentzel boundary conditions) for a single frequency excitation ( f ( t , x ) = F ( x ) exp( − ıω t )). 6 / 34
Surface homogenization for micro-structured layers with singularities ⊲ This model is considered on a geometry that is dependent on a small parameter δ. 7 / 34
Surface homogenization for micro-structured layers with singularities ⊲ This model is considered on a geometry that is dependent on a small parameter δ. ⊲ Solving directly this problem numerically, with a mesh that resolves this exact geometry, is costly: 7 / 34
Surface homogenization for micro-structured layers with singularities ⊲ This model is considered on a geometry that is dependent on a small parameter δ. ⊲ Solving directly this problem numerically, with a mesh that resolves this exact geometry, is costly: ⊲ Then, we aim for describing an efficient model in which we don’t need to resolve the small geometrical scales. 7 / 34
Surface homogenization for micro-structured layers with singularities ⊲ Previous system can be reduced to an Helmholtz equation with Wentzel boundary conditions, with homogeneous wave number k 0 := ω/ c and viscosity parameter ν ′ ( δ ): � ∆ p δ + k 2 0 p δ = div F , ∂ n p δ + (1 + ı ) t p δ = 0 , in Ω δ,η , ν ′ ( δ ) ∂ 2 on ∂ Ω δ,η . K. Schmidt, A. Th¨ ons-Zueva, J. Joly. Asymptotic analysis for acoustics in viscous gases close to rigid walls. Math. Models Meth. Appl. Sci. , 24, 2014. ⊲ Away from the periodic layer, p δ is described by its macroscopic part p ( x ), ⊲ close to the periodic layer, the macroscopic part is corrected by a periodic boundary layer Π( x 1 , x /δ ), ⊲ close to the end-point x ± O of the periodic layer, the macroscopic part is corrected by a near field corrector P ± (( x − x ± O ) /δ ). ⊲ Goal: derive an effective macroscopic description of the solution. 8 / 34
Surface homogenization for micro-structured layers with singularities ⊲ Previous system can be reduced to an Helmholtz equation with Wentzel boundary conditions, with homogeneous wave number k 0 := ω/ c and viscosity parameter ν ′ ( δ ): � ∆ p δ + k 2 0 p δ = div F , ∂ n p δ + (1 + ı ) t p δ = 0 , in Ω δ,η , ν ′ ( δ ) ∂ 2 on ∂ Ω δ,η . K. Schmidt, A. Th¨ ons-Zueva, J. Joly. Asymptotic analysis for acoustics in viscous gases close to rigid walls. Math. Models Meth. Appl. Sci. , 24, 2014. ⊲ Away from the periodic layer, p δ is described by its macroscopic part p ( x ), ⊲ close to the periodic layer, the macroscopic part is corrected by a periodic boundary layer Π( x 1 , x /δ ), ⊲ close to the end-point x ± O of the periodic layer, the macroscopic part is corrected by a near field corrector P ± (( x − x ± O ) /δ ). ⊲ Goal: derive an effective macroscopic description of the solution. 8 / 34
Surface homogenization for micro-structured layers with singularities ⊲ Previous system can be reduced to an Helmholtz equation with Wentzel boundary conditions, with homogeneous wave number k 0 := ω/ c and viscosity parameter ν ′ ( δ ): � ∆ p δ + k 2 0 p δ = div F , ∂ n p δ + (1 + ı ) t p δ = 0 , in Ω δ,η , ν ′ ( δ ) ∂ 2 on ∂ Ω δ,η . K. Schmidt, A. Th¨ ons-Zueva, J. Joly. Asymptotic analysis for acoustics in viscous gases close to rigid walls. Math. Models Meth. Appl. Sci. , 24, 2014. ⊲ Away from the periodic layer, p δ is described by its macroscopic part p ( x ), ⊲ close to the periodic layer, the macroscopic part is corrected by a periodic boundary layer Π( x 1 , x /δ ), ⊲ close to the end-point x ± O of the periodic layer, the macroscopic part is corrected by a near field corrector P ± (( x − x ± O ) /δ ). ⊲ Goal: derive an effective macroscopic description of the solution. 8 / 34
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