On the Dynamics of Navier-Stokes and Euler Equations

This is a detailed study on certain dynamics of Navier-Stokes and Euler equations via a combination of analysis and numerics. We focus upon two main aspects: (a) zero viscosity limit of the spectra of linear Navier-Stokes operator, (b) heteroclinics conjecture for Euler equation, its numerical verification, Melnikov integral, and simulation and control of chaos. Due to the difficulty of the problem for the full Navier-Stokes and Euler equations, we also propose and study two simpler models of them.     

Helical symmetric solution of 3d Navier-Stokes equations arising from geometric shape of the boundary

In this article, we investigate three-dimensional solution with helical symmetry in a gap between two concentric rotating cylinders, inside is a helicoidal surface (screw propeller) while outside is a cylindrical surface. Establish the partial differential equations and its variational formulation satisfied by a helical solution in a helical coordinate system using tensor analysis method, we provide a computational method for the power and propulsion of the screw. The existence and uniqueness of weak helical solutions are proved.     

One-dimensional viscous radiative gas with temperature dependent viscosity

This paper is concerned with the construction of global, large amplitude solutions to the Cauchy problem of the one-dimensional compressible Navier–Stokes system for a viscous radiative gas when the viscosity and heat conductivity coefficients depend on both specific volume and absolute temperature. The data are assumed to be without vacuum,mass concentrations, or vanishing temperatures, and the same is shown to be hold for the global solution constructed. The proof is based on some detailed analysis on uniform positive lower and upper bounds of the specific volume and absolute temperature.     

The effects of span-wise and chord-wise flexibility on the aerodynamic performance of micro flapping-wing

The flexibility of flapping-wing strongly affects the aerodynamic performance of Flapping-wing Micro Air Vehicle (FMAV),and the deformations in span-wise and chord-wise directions are coupled together in flight.In this study,the flexible deformation is formulated in span-wise and chord-wise separately in order to analyze its effects on aerodynamic behavior.The preconditioned Navier-Stokes equations based on chimera grid are used in the computational fluid dynamics method to study the aerodynamic effects caused by flexible deformation,and the simulation results are compared with experimental test to illustrate the capability of above method.Based on our results,it is clearly showed that the span-wise flexible deformation should be limited in a small range to achieve higher aerodynamic performance and the chord-wise deformation could enhance the aerodynamic performance.The results also suggest that FMAV designers should design the flapping-wing with high stiffness leading edge to limit the span-wise deformation,and more flexible chord ribs to keep chord-wise deformation in suitable range.     

三维Navier-Stokes方程的正则性准则

考察三维不可压Navier-Stokes方程的弱解正则性问题.基于Holder不等式和速度场的不可压缩性质,通过对速度向量的部分分量及相关导数的估计,得到了一个新的关于Leray-Hopf弱解的正则性准则的结果.在速度向量的部分分量及相关导数满足适当的条件下,三维不可压Navier-Stokes方程的弱解是整体正则的.     

Error Bounds for Uniform Asymptotic Expansions—Modified Bessel Function of Purely Imaginary Order

The authors modify a method of Olde Daalhuis and Temme for representing the remainder and coefficients in Airy-type expansions of integrals.By using a class of rational functions,they express these quantities in terms of Cauchy-type integrals;these expressions are natural generalizations of integral representations of the coe？cients and the remainders in the Taylor expansions of analytic functions.By using the new representation,a computable error bound for the remainder in the uniform asymptotic expansion of the modified Bessel function of purely imaginary order is derived.     

STRONG SOLUTIONS FOR THE INCOMPRESSIBLE FLUID MODELS OF KORTEWEG TYPE

In this article, we are concerned with the strong solutions for the incompressible fluid models of Korteweg type in a bounded domain Ω R3. We prove the existence and uniqueness of local strong solutions to the initial boundary value problem. We point out that in this article we allow the existence of initial vacuum provided initial data satisfy a compatibility condition.     

Bi-velocity hydrodynamics

Theoretical evidence derived from linear irreversible thermodynamics (LIT) jointly with Burnett’s solution of Boltzmann’s gas-kinetic equation is used to show that fluid mechanics and transport processes in both gaseous and liquid continua require the use of two independent velocities rather than one in order to correctly quantify the physics of fluid motion. This finding, reflecting the coalescence of macroscopic and molecular perspectives, undermines the current foundations of continuum fluid mechanics. Of the two required context-specific velocities, one is the mass velocity appearing in the continuity equation. The other is the volume velocity entering into the constitutive equation for the mechanical rate-of-working term appearing in the energy equation, where it serves as the multiplier of the pressure tensor . While the analysis involves only linear constitutive principles, the fundamental need for two independent velocities is noted to apply even in non-linear circumstances. A major consequence of these findings is that the Navier-Stokes-Fourier equations governing continuum fluid physics are incomplete for both single- and multi-component fluids. Our results are independently supported by the work of others based upon the use of conventional single-velocity arguments accompanied by ad hoc extensions of LIT. Our bi-velocity findings point to the existence of novel mechanodiffusive phenomena in fluid continua, entailing coupling between viscous flow and diffusion, whether referring to the diffusion of thermal energy in single-component non-isothermal fluids or of chemical species in inhomogeneous multicomponent mixtures.     

A New Boundary Problem for the Two Dimensional Navier-Stokes System

We formulate a new boundary value problem for the 2D Navier-Stokes system on the unit square. Under some suitable assumptions on the initial velocity, we obtain quantitative decay estimates of the Fourier modes of both the vorticity and the velocity. It is found that in one direction the Fourier modes decay exponentially and along the other direction their decay is only power like.     

Revisit of dilation-based shock capturing for discontinuous Galerkin methods

The idea of using velocity dilation for shock capturing is revisited in this paper, combined with the discontinuous Galerkin method. The value of artificial viscosity is determined using direct dilation instead of its higher order derivatives to reduce cost and degree of difficulty in computing derivatives. Alternative methods for estimating the element size of large aspect ratio and smooth artificial viscosity are proposed to further improve robustness and accuracy of the model. Several benchmark tests are conducted, ranging from subsonic to hypersonic flows involving strong shocks. Instead of adjusting empirical parameters to achieve optimum results for each case, all tests use a constant parameter for the model with reasonable success, indicating excellent robustness of the method. The model is only limited to third-order accuracy for smooth flows. This limitation may be relaxed by using a switch or a wall function. Overall, the model is a good candidate for compressible flows with potentials of further improvement.