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Chongchun Zeng - One of the best experts on this subject based on the ideXlab platform.

  • Unstable Manifolds of Euler Equations
    Communications on Pure and Applied Mathematics, 2013
    Co-Authors: Zhiwu Lin, Chongchun Zeng
    Abstract:

    We consider a steady state v0 of the Euler Equation in a fixed bounded domain in ℝn. Suppose the Linearized Euler Equation has an exponential dichotomy of unstable and center-stable subspaces. By rewriting the Euler Equation as an ODE on an infinite-dimensional manifold of volume-preserving maps in Wk, q (k>1+n/q) the unstable (and stable) manifolds of v0 are constructed under a certain spectral gap condition that is satisfied for both two-dimensional and three-dimensional examples. In particular, when the unstable subspace is finite dimensional, this implies the nonlinear instability of v0 in the sense that arbitrarily small Wk, q perturbations can lead to L2 growth of the nonlinear solutions. © 2013 Wiley Periodicals, Inc.

  • Unstable manifolds of Euler Equations
    arXiv: Analysis of PDEs, 2011
    Co-Authors: Zhiwu Lin, Chongchun Zeng
    Abstract:

    We consider a steady state $v_{0}$ of the Euler Equation in a fixed bounded domain in $\mathbf{R}^{n}$. Suppose the Linearized Euler Equation has an exponential dichotomy of unstable and center-stable subspaces. By rewriting the Euler Equation as an ODE on an infinite dimensional manifold of volume preserving maps in $W^{k, q}$, $(k>1+\frac{n}{q})$, the unstable (and stable) manifolds of $v_{0}$ are constructed under certain spectral gap condition which is verified for both 2D and 3D examples. In particular, when the unstable subspace is finite dimensional, this implies the nonlinear instability of $v_{0}$ in the sense that arbitrarily small $W^{k, q}$ perturbations can lead to $L^{2}$ growth of the nonlinear solutions.

  • Inviscid Dynamical Structures Near Couette Flow
    Archive for Rational Mechanics and Analysis, 2011
    Co-Authors: Zhiwu Lin, Chongchun Zeng
    Abstract:

    Consider inviscid fluids in a channel $${\{-1\leqq y\leqq1\}}$$ . For the Couette flow u _0 = ( y , 0), the vertical velocity of solutions to the Linearized Euler Equation at u _0 decays in time. Whether the same happens at the non-linear level is an open question. Here we study issues related to this problem. First, we show that in any (vorticity) $${H^{s}\left(s\frac{3}{2}\right)}$$ neighborhoods of Couette flow might be much simpler. Such contrasting dynamics in H ^ s spaces with the critical power $${s=\frac{3}{2}}$$ is a truly nonlinear phenomena, since the linear inviscid damping near Couette flow is true for any initial vorticity in L ^2.

Zhiwu Lin - One of the best experts on this subject based on the ideXlab platform.

  • Unstable Manifolds of Euler Equations
    Communications on Pure and Applied Mathematics, 2013
    Co-Authors: Zhiwu Lin, Chongchun Zeng
    Abstract:

    We consider a steady state v0 of the Euler Equation in a fixed bounded domain in ℝn. Suppose the Linearized Euler Equation has an exponential dichotomy of unstable and center-stable subspaces. By rewriting the Euler Equation as an ODE on an infinite-dimensional manifold of volume-preserving maps in Wk, q (k>1+n/q) the unstable (and stable) manifolds of v0 are constructed under a certain spectral gap condition that is satisfied for both two-dimensional and three-dimensional examples. In particular, when the unstable subspace is finite dimensional, this implies the nonlinear instability of v0 in the sense that arbitrarily small Wk, q perturbations can lead to L2 growth of the nonlinear solutions. © 2013 Wiley Periodicals, Inc.

  • Unstable manifolds of Euler Equations
    arXiv: Analysis of PDEs, 2011
    Co-Authors: Zhiwu Lin, Chongchun Zeng
    Abstract:

    We consider a steady state $v_{0}$ of the Euler Equation in a fixed bounded domain in $\mathbf{R}^{n}$. Suppose the Linearized Euler Equation has an exponential dichotomy of unstable and center-stable subspaces. By rewriting the Euler Equation as an ODE on an infinite dimensional manifold of volume preserving maps in $W^{k, q}$, $(k>1+\frac{n}{q})$, the unstable (and stable) manifolds of $v_{0}$ are constructed under certain spectral gap condition which is verified for both 2D and 3D examples. In particular, when the unstable subspace is finite dimensional, this implies the nonlinear instability of $v_{0}$ in the sense that arbitrarily small $W^{k, q}$ perturbations can lead to $L^{2}$ growth of the nonlinear solutions.

  • Inviscid Dynamical Structures Near Couette Flow
    Archive for Rational Mechanics and Analysis, 2011
    Co-Authors: Zhiwu Lin, Chongchun Zeng
    Abstract:

    Consider inviscid fluids in a channel $${\{-1\leqq y\leqq1\}}$$ . For the Couette flow u _0 = ( y , 0), the vertical velocity of solutions to the Linearized Euler Equation at u _0 decays in time. Whether the same happens at the non-linear level is an open question. Here we study issues related to this problem. First, we show that in any (vorticity) $${H^{s}\left(s\frac{3}{2}\right)}$$ neighborhoods of Couette flow might be much simpler. Such contrasting dynamics in H ^ s spaces with the critical power $${s=\frac{3}{2}}$$ is a truly nonlinear phenomena, since the linear inviscid damping near Couette flow is true for any initial vorticity in L ^2.

Kazuhiro Nakahashi - One of the best experts on this subject based on the ideXlab platform.

  • Code Development of Linearized Euler Equation on Block- Structured Cartesian Mesh for Complicated Geometries
    50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, 2012
    Co-Authors: Yuuma Fukushima, Daisuke Sasaki, Kazuhiro Nakahashi
    Abstract:

    The Linearized Euler Equation code for aeroacoustic analysis has been developed on block-structured Cartesian mesh to treat the complicated geometries robustly and accurately. In the present method, spatial derivation and time integration are conducted by high order schemes and the immersed boundary method using ghost cell approach is employed at the wall boundary. At the block boundary, higher-order data exchange is conducted by Lagrange interpolation and near the outer boundary outgoing wave is damped by absorbing boundary domain. The present code is validated for two-dimensional and three-dimensional test cases. The accuracy and effectiveness of the present method are demonstrated in two-dimensional acoustic scattering problems. In the three-dimensional cases, acoustic scattering around a sphere and noise propagation from JT15D nacelle configuration were simulated and it showed the good agreement with the analytical and experimental results. Finally, noise shielding effect of an Over-the-Wing-mounted-Nacelle (OWN) configuration was estimated as an application. The results show that far-field SPL of OWN configuration is lower than that of conventional Under-the-Wing-mounted-Nacelle configuration by about 10dB due to the shielding effect of OWN configuration.

  • Code Development of Linearized Euler Equation on Block-Structured Cartesian Mesh Combined with Immersed Boundary Method
    JOURNAL OF THE JAPAN SOCIETY FOR AERONAUTICAL AND SPACE SCIENCES, 2012
    Co-Authors: Yuuma Fukushima, Daisuke Sasaki, Kazuhiro Nakahashi
    Abstract:

    Recently, acoustic analysis using the Linearized Euler Equation (LEE) on multi-block structured or unstructured mesh is focused. However, mesh generation around complicated geometries takes time on structured mesh and cost of high order calculation gets larger on unstructured mesh. In this research, a LEE code for aeroacoustic analysis is developed on block structured Cartesian mesh of Building-Cube Method. To make an accurate calculation, the Immersed Boundary Method is implemented for wall boundary treatment and high order Lagrange interpolation is implemented at the Cube boundary for data exchange. This code is validated through acoustics scattering problems around cylinders and the calculational errors are compared. The results show good agreement with analytical solutions and the usability of these methods.

Yuuma Fukushima - One of the best experts on this subject based on the ideXlab platform.

  • Code Development of Linearized Euler Equation on Block- Structured Cartesian Mesh for Complicated Geometries
    50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, 2012
    Co-Authors: Yuuma Fukushima, Daisuke Sasaki, Kazuhiro Nakahashi
    Abstract:

    The Linearized Euler Equation code for aeroacoustic analysis has been developed on block-structured Cartesian mesh to treat the complicated geometries robustly and accurately. In the present method, spatial derivation and time integration are conducted by high order schemes and the immersed boundary method using ghost cell approach is employed at the wall boundary. At the block boundary, higher-order data exchange is conducted by Lagrange interpolation and near the outer boundary outgoing wave is damped by absorbing boundary domain. The present code is validated for two-dimensional and three-dimensional test cases. The accuracy and effectiveness of the present method are demonstrated in two-dimensional acoustic scattering problems. In the three-dimensional cases, acoustic scattering around a sphere and noise propagation from JT15D nacelle configuration were simulated and it showed the good agreement with the analytical and experimental results. Finally, noise shielding effect of an Over-the-Wing-mounted-Nacelle (OWN) configuration was estimated as an application. The results show that far-field SPL of OWN configuration is lower than that of conventional Under-the-Wing-mounted-Nacelle configuration by about 10dB due to the shielding effect of OWN configuration.

  • Code Development of Linearized Euler Equation on Block-Structured Cartesian Mesh Combined with Immersed Boundary Method
    JOURNAL OF THE JAPAN SOCIETY FOR AERONAUTICAL AND SPACE SCIENCES, 2012
    Co-Authors: Yuuma Fukushima, Daisuke Sasaki, Kazuhiro Nakahashi
    Abstract:

    Recently, acoustic analysis using the Linearized Euler Equation (LEE) on multi-block structured or unstructured mesh is focused. However, mesh generation around complicated geometries takes time on structured mesh and cost of high order calculation gets larger on unstructured mesh. In this research, a LEE code for aeroacoustic analysis is developed on block structured Cartesian mesh of Building-Cube Method. To make an accurate calculation, the Immersed Boundary Method is implemented for wall boundary treatment and high order Lagrange interpolation is implemented at the Cube boundary for data exchange. This code is validated through acoustics scattering problems around cylinders and the calculational errors are compared. The results show good agreement with analytical solutions and the usability of these methods.

Daisuke Sasaki - One of the best experts on this subject based on the ideXlab platform.

  • Code Development of Linearized Euler Equation on Block- Structured Cartesian Mesh for Complicated Geometries
    50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, 2012
    Co-Authors: Yuuma Fukushima, Daisuke Sasaki, Kazuhiro Nakahashi
    Abstract:

    The Linearized Euler Equation code for aeroacoustic analysis has been developed on block-structured Cartesian mesh to treat the complicated geometries robustly and accurately. In the present method, spatial derivation and time integration are conducted by high order schemes and the immersed boundary method using ghost cell approach is employed at the wall boundary. At the block boundary, higher-order data exchange is conducted by Lagrange interpolation and near the outer boundary outgoing wave is damped by absorbing boundary domain. The present code is validated for two-dimensional and three-dimensional test cases. The accuracy and effectiveness of the present method are demonstrated in two-dimensional acoustic scattering problems. In the three-dimensional cases, acoustic scattering around a sphere and noise propagation from JT15D nacelle configuration were simulated and it showed the good agreement with the analytical and experimental results. Finally, noise shielding effect of an Over-the-Wing-mounted-Nacelle (OWN) configuration was estimated as an application. The results show that far-field SPL of OWN configuration is lower than that of conventional Under-the-Wing-mounted-Nacelle configuration by about 10dB due to the shielding effect of OWN configuration.

  • Code Development of Linearized Euler Equation on Block-Structured Cartesian Mesh Combined with Immersed Boundary Method
    JOURNAL OF THE JAPAN SOCIETY FOR AERONAUTICAL AND SPACE SCIENCES, 2012
    Co-Authors: Yuuma Fukushima, Daisuke Sasaki, Kazuhiro Nakahashi
    Abstract:

    Recently, acoustic analysis using the Linearized Euler Equation (LEE) on multi-block structured or unstructured mesh is focused. However, mesh generation around complicated geometries takes time on structured mesh and cost of high order calculation gets larger on unstructured mesh. In this research, a LEE code for aeroacoustic analysis is developed on block structured Cartesian mesh of Building-Cube Method. To make an accurate calculation, the Immersed Boundary Method is implemented for wall boundary treatment and high order Lagrange interpolation is implemented at the Cube boundary for data exchange. This code is validated through acoustics scattering problems around cylinders and the calculational errors are compared. The results show good agreement with analytical solutions and the usability of these methods.