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Numerical solvers of the incompressible Navier-Stokes equations havereproduced turbulence phenomena such as the law of the wall, the dependence ofturbulence intensities on the Reynolds number, and experimentally observedproperties of turbulence energy production. In this article, we begin asequence of investigations whose eventual aim is to derive and implementnumerical solvers that can reach higher Reynolds numbers than is currentlypossible. Every time step of a Navier-Stokes solver in effect solves a linearboundary value problem. The use of Green's functions leads to numerical solverswhich are highly accurate in resolving the boundary layer, which is a source ofdelicate but exceedingly important physical effects at high Reynolds numbers.The use of Green's functions brings with it a need for careful quadrature rulesand a reconsideration of time steppers. We derive and implement Green'sfunction based solvers for the channel flow and plane Couette flow geometries.The solvers are validated by reproducing turbulent signals which are in goodqualitative and quantitative agreement with experiment.

Navier-Stokes solver using Green's functions I: channel flow and plane Couette flow