Exact Solutions for Restricted Incompressible Navier--Stokes Equations with Dirichlet Boundary Conditions
Manuel García-Casado
备注：
This is an author-created, un-copyedited version of an article published in Journal of Physics Communications. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. This article is published under a CC BY licence. The Version of Record is available online at https://doi.org/10.1088/2399-6528/ab3837
This paper exposes how to obtain a relation that have to be hold for all free--divergence velocity fields that evolve according to Navier--Stokes equations. However, checking the violation of this relation requires a huge computational effort. To circumvent this problem it is proposed an additional antsatz to free-divergent Navier--Stokes fields. This makes available six degrees of freedom which can be tuned. When they are tuned adequately, it is possible to find finite $L^2$ norms of the velocity field for volumes of $\mathbb{R}^3$ and for $t\in[t_0,\infty)$. In particular, the kinetic energy of the system is bounded when the field components $u_i$ are class $C^3$ functions on $\mathbb{R}^3\times[t_0,\infty)$ that hold Dirichlet boundary conditions. This additional relation lets us conclude that Navier--Stokes equations with no-slip boundary conditions have not unique solution. Moreover, under a given external force the kinetic energy can be computed exactly as a funtion of time.
The Variable Density Model for the Rayleigh-Taylor Instability and its transformation to the diffusive, inhomogeneous, incompressible Navier-Stokes equations
A matrix-free high-order discontinuous Galerkin compressible Navier-Stokes solver: A performance comparison of compressible and incompressible formulations for turbulent incompressible flows
Solution to two-dimensional Incompressible Navier-Stokes Equations with SIMPLE, SIMPLER and Vorticity-Stream Function Approaches. Driven-Lid Cavity Problem: Solution and Visualization
