On the Finite-Time Blowup of a 1D Model for the 3D Incompressible Euler Equations

We study a 1D model for the 3D incompressible Euler equations in axisymmetric geometries, which can be viewed as a local approximation to the Euler equations near the solid boundary of a cylindrical domain. We prove the local well-posedness of the model in spaces of zero-mean functions, and study the potential formation of a finite-time singularity under certain convexity conditions for the velocity field. It is hoped that the results obtained on the 1D model will be useful in the analysis of the full 3D problem, whose loss of regularity in finite time has been observed in a recent numerical study (Luo and Hou, 2013).Comment: 23 page

On Finite Time Singularity and Global Regularity of an Axisymmetric Model for the 3D Euler Equations

We investigate the large time behavior of an axisymmetric model for the 3D Euler equations. In \cite{HL09}, Hou and Lei proposed a 3D model for the axisymmetric incompressible Euler and Navier-Stokes equations with swirl. This model shares many properties of the 3D incompressible Euler and Navier-Stokes equations. The main difference between the 3D model of Hou and Lei and the reformulated 3D Euler and Navier-Stokes equations is that the convection term is neglected in the 3D model. In \cite{HSW09}, the authors proved that the 3D inviscid model can develop a finite time singularity starting from smooth initial data on a rectangular domain. A global well-posedness result was also proved for a class of smooth initial data under some smallness condition. The analysis in \cite{HSW09} does not apply to the case when the domain is axisymmetric and unbounded in the radial direction. In this paper, we prove that the 3D inviscid model with an appropriate Neumann-Robin boundary condition will develop a finite time singularity starting from smooth initial data in an axisymmetric domain. Moreover, we prove that the 3D inviscid model has globally smooth solutions for a class of large smooth initial data with some appropriate boundary condition.Comment: Please read the published versio

Toward the Finite-Time Blowup of the 3D Axisymmetric Euler Equations: A Numerical Investigation

Whether the three-dimensional incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. This work attempts to provide an affirmative answer to this long-standing open question from a numerical point of view by presenting a class of potentially singular solutions to the Euler equations computed in axisymmetric geometries. The solutions satisfy a periodic boundary condition along the axial direction and a no-flow boundary condition on the solid wall. The equations are discretized in space using a hybrid 6th-order Galerkin and 6th-order finite difference method on specially designed adaptive (moving) meshes that are dynamically adjusted to the evolving solutions. With a maximum effective resolution of over (3 x 10^(12))^2 near the point of the singularity, we are able to advance the solution up to tau_2 = 0.003505 and predict a singularity time of t(s) approximate to 0.0035056, while achieving a pointwise relative error of O(10^(-4)) in the vorticity vector. and observing a (3 x 10^8)-fold increase in the maximum vorticity parallel to omega parallel to(infinity). The numerical data are checked against all major blowup/non-blowup criteria, including Beale-Kato-Majda, Constantin-Fefferman-Majda, and Deng-Hou-Yu, to confirm the validity of the singularity. A local analysis near the point of the singularity also suggests the existence of a self-similar blowup in the meridian plane

Formation of Finite-Time Singularities in the 3D Axisymmetric Euler Equations: A Numerics Guided Study

Whether the three-dimensional incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. This work attempts to provide an affirmative answer to this long-standing open question, by first describing a class of potentially singular solutions to the Euler equations numerically discovered in axisymmetric geometries, and then by presenting evidence from rigorous analysis that strongly supports the existence of such singular solutions. The initial data leading to these singular solutions possess certain special symmetry and monotonicity properties, and the subsequent flows are assumed to satisfy a periodic boundary condition along the axial direction and a no-flow, free-slip boundary condition on the solid wall. The numerical study employs a hybrid 6th-order Galerkin/finite difference discretization of the governing equations in space and a 4th-order Runge--Kutta discretization in time, where the emerging singularity is captured on specially designed adaptive (moving) meshes that are dynamically adjusted to the evolving solutions. With a maximum effective resolution of over (3 x 10¹²)² near the point of the singularity, the simulations are able to advance the solution to a point that is asymptotically close to the predicted singularity time, while achieving a pointwise relative error of O(10⁻⁴) in the vorticity vector and obtaining a 3 x 10⁸-fold increase in the maximum vorticity. The numerical data are checked against all major blowup/nonblowup criteria, including Beale-Kato-Majda, Constantin-Fefferman-Majda, and Deng-Hou-Yu, to confirm the validity of the singularity. A close scrutiny of the data near the point of the singularity also reveals a self-similar structure in the blowup, as well as a one-dimensional model which is seen to capture the essential features of the singular solutions along the solid wall, and for which existence of finite-time singularities can be established rigorously

On the Finite-Time Blowup of a 1D Model for the 3D Axisymmetric Euler Equations

In connection with the recent proposal for possible singularity formation at the boundary for solutions of 3d axi-symmetric incompressible Euler's equations (Luo and Hou, 2013), we study models for the dynamics at the boundary and show that they exhibit a finite-time blow-up from smooth data.Comment: A paragraph at the end of Section 2 and an appendix discussing kinetic energy conservation are adde

Fast Synthesis Algorithm for Uniformly Spaced Circular Array with Low Sidelobe Pattern

© 2018 ACES. In this paper, a highly efficient approach is proposed to synthesize the low sidelobe pattern of uniformly spaced circular array. The proposed approach can be generalized to deal with the pattern synthesis for the circular array with directional elements. Numerical examples are given to verify the effectiveness and advantage of this approach

Neural activity inspired asymmetric basis function TV-NARX model for the identification of time-varying dynamic systems

Inspired by the unique neuronal activities, a new time-varying nonlinear autoregressive with exogenous input (TV-NARX) model is proposed for modelling nonstationary processes. The NARX nonlinear process mimics the action potential initiation and the time-varying parameters are approximated with a series of postsynaptic current like asymmetric basis functions to mimic the ion channels of the inter-neuron propagation. In the model, the time-varying parameters of the process terms are sparsely represented as the superposition of a series of asymmetric alpha basis functions in an over-complete frame. Combining the alpha basis functions with the model process terms, the system identification of the TV-NARX model from observed input and output can equivalently be treated as the system identification of a corresponding time-invariant system. The locally regularised orthogonal forward regression (LROFR) algorithm is then employed to detect the sparse model structure and estimate the associated coefficients. The excellent performance in both numerical studies and modelling of real physiological signals showed that the TV-NARX model with asymmetric basis function is more powerful and efficient in tracking both smooth trends and capturing the abrupt changes in the time-varying parameters than its symmetric counterparts

Thermal-hydraulic modelling and analysis of hydraulic damper for impact cylinder with large flow

The hydraulic damper has a great sense for impact machine to extend life and improve the environmental performance. The objective of this paper is to provide a systematic investigation to design or evaluation of a hydraulic damper used in the impact machine. A novel hydraulic damper using guiding sleeve to enlarge buffer chamber area is designed and manufactured by ingenious tactics. The performance of a prototype hydraulic damper is acquired by the test. A nonlinear thermal-hydraulic model for the hydraulic damper is presented by analyzing the internal fluid dynamic phenomenon and heat transfer with respect to the prototype. Comparisons between test data and simulation result confirm the validity of the thermal-hydraulic model. In the meantime, evaluation of the importance of some key factors using the model for designing is discussed. It shows the influence of orifice diameter, inner diameter of buffer chamber and setting pressure of the relief valve to hydraulic damper characteristics with large flow, which gives a theoretical basis to design and optimize hydraulic damper with large flow for impact machine