An exact pressure evolution equation for the incompressible Navier-Stokes equations

隐式大涡模拟中高阶H(div)-符合有限元方法用于不可压缩纳维-斯托克斯流

Implicit LES with high-order H(div)-conforming FEM for incompressible Navier-Stokes flows

本文研究了高雷诺数下不可压缩纳维-斯托克斯流的隐式大涡模拟，采用高阶H(div)-符合有限元方法，在周期域衰减湍流和壁面受限通道流两种情况下进行验证。

The Variable Density Model for the Rayleigh-Taylor Instability and its transformation to the diffusive, inhomogeneous, incompressible Navier-Stokes equations

It is shown how the variable density model (VDM) that governs the Rayleigh-Taylor instability (RTI) for the miscible mixing of two incompressible fluids can be transformed into a diffusive version of the inhomogeneous, incompressible Navier-Stokes equations forced by gradients of the composition density $ρ$ of the mixing layer. This demonstrates how buoyancy-driven flows drive and enhance Navier-Stokes turbulence. The role of the potential vorticity $q = ω\cdot\nablaρ$ is also discussed.

Geometric Regularity Criteria for Incompressible Navier--Stokes Equations with Navier Boundary Conditions

We study the regularity criteria for weak solutions to the $3D$ incompressible Navier--Stokes equations in terms of the geometry of vortex structures, taking into account the boundary effects. A boundary regularity theorem is proved on regular domains with a class of oblique derivative boundary conditions, providing that the vorticity of the fluid is coherently aligned. In particular, we establish the boundary regularity on round balls, half-spaces and right circular cylindrical ducts, subject to the classical Navier and kinematic boundary conditions.

Possibility of Turbulence from a Post-Navier-Stokes Equation

We introduce corrections to the Navier-Stokes equation arising from the transitions between molecular states and the injection of external energy. In the simplest application of the proposed post Navier-Stokes equation, we find a multi-valued velocity field and the immediate possibility of velocity reversal, both features of turbulence.

A conservative discontinuous Galerkin scheme for the 2D incompressible Navier--Stokes equations

In this paper we consider a conservative discretization of the two-dimensional incompressible Navier--Stokes equations. We propose an extension of Arakawa's classical finite difference scheme for fluid flow in the vorticity-stream function formulation to a high order discontinuous Galerkin approximation. In addition, we show numerical simulations that demonstrate the accuracy of the scheme and verify the conservation properties, which are essential for long time integration. Furthermore, we discuss the massively parallel implementation on graphic processing units.

Stochastic Least-Action Principle for the Incompressible Navier-Stokes Equation

We formulate a stochastic least-action principle for solutions of the incompressible Navier-Stokes equation, which formally reduces to Hamilton's principle for the incompressible Euler solutions in the case of zero viscosity. We use this principle to give a new derivation of a stochastic Kelvin Theorem for the Navier-Stokes equation, recently established by Constantin and Iyer, which shows that this stochastic conservation law arises from particle-relabelling symmetry of the action. We discuss issues of irreversibility, energy dissipation, and the inviscid limit of Navier-Stokes solutions in t…

Variational methods for finding periodic orbits in the incompressible Navier-Stokes equations

Unstable periodic orbits are believed to underpin the dynamics of turbulence, but by their nature are hard to find computationally. We present a family of methods to converge such unstable periodic orbits for the incompressible Navier-Stokes equations, based on variations of an integral objective functional, and using traditional gradient-based optimisation strategies. Different approaches for handling the incompressibility condition are considered. The variational methods are applied to the specific case of periodic, two-dimensional Kolmogorov flow and compared against existing Newton iterati…

N维不可压缩Navier-Stokes方程组的解析解

Analytic Solution of the N-Dimensional Incompressible Navier-Stokes Equations

本文通过递推关系给出N维不可压缩Navier-Stokes方程的解析解，回应千禧年问题中的存在性与光滑性。

Symmetry reduction and exact solutions of the Navier-Stokes equations

Ansatzes for the Navier-Stokes field are described. These ansatzes reduce the Navier-Stokes equations to system of differential equations in three, two, and one independent variables. The large sets of exact solutions of the Navier-Stokes equations are constructed.

Quotients of Navier--Stokes equation on space curves

A Navier--Stokes system on a curve is discussed. The quotient equation for this system is found. The quotient is used to find some solutions of Navier--Stokes system. Using virial expansion of the Planck potential, we reduce the quotient equation to a series of systems of ordinary differential equations.

On the Dynamics of Navier-Stokes and Euler Equations

This is a rather comprehensive study on the 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. Besides Navier-Stokes and Euler equations, we also study two models of them.