Discrete Kato-type theorem on inviscid limit of Navier-Stokes flows

The inviscid limit of wall bounded viscous flows is one of the unanswered central questions in theoretical fluid dynamics. Here we present a somewhat surprising result related to numerical approximation of the problem. More precisely, we show that numerical solutions of the incompressible Navier-Stokes equations converge to the exact solution of the Euler equations at vanishing viscosity and vanishing mesh size provided that small scales of the order of /U in the directions tangential to the boundary are not resolved in the scheme. Here is the kinematic viscosity of the fluid and U is the typical velocity taken to be the maximum of the shear velocity at the boundary for the inviscid flow. Such a result is somewhat counterintuitive since the convergence is ensured even in the case that small scales predicted by the conventional theory of turbulence and boundary layer are not resolved since underresolution which is allowed in our theorem in advection dominated problem usually leads to oscillation which inhibits convergence in general. The result also indicates possible difficulty in terms of numerical investigation of the vanishing viscosity problem if rigorous fidelity of the numerics is desired since we have to resolve at least small scales of the order of /U which is much smaller than any small scales predicted by the conventional theory of turbulence. On the other hand, such a result can be viewed as a discrete version of our result X. Wang, Indiana Univ. Math. J. 50, 223 2001 which generalized earlier the result of Kato in Seminar on PDE, edited by S. S. Chern Springer, NY, 1984 where the relevance of a scale proportional to the kinematic viscosity to the problem of vanishing viscosity is first discovered. © 2007 American Institute of Physics. DOI: 10.1063/1.2399752