{
  "id": "1407.3776",
  "version": "v3",
  "published": "2014-07-14T19:39:21.000Z",
  "updated": "2015-11-08T21:20:26.000Z",
  "title": "Navier-Stokes solver using Green's functions II: spectral integration of channel flow and plane Couette flow",
  "authors": [
    "Divakar Viswanath"
  ],
  "categories": [
    "physics.flu-dyn"
  ],
  "abstract": "The Kleiser-Schumann algorithm has been widely used for the direct numerical simulation of turbulence in rectangular geometries. At the heart of the algorithm is the solution of linear systems which are tridiagonal except for one row. This note shows how to solve the Kleiser-Schumann problem using perfectly triangular matrices. An advantage is the ability to use functions in the LAPACK library. The method is used to simulate turbulence in channel flow at $Re=80,000$ (and $Re_{\\tau}=2400$) using $10^{9}$ grid points. An assessment of the length of time necessary to eliminate transient effects in the initial state is included.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-07-16T17:41:30.000Z",
      "abstract": "The Kleiser-Schumann (1980) and Kim-Moin-Moser (1987) algorithms for solving the incompressible Navier-Stokes equations have been used to simulate a great variety of turbulence phenomena in plane channels and in plane Couette flow. In this article, we derive new versions of either algorithm which completely eliminate numerical differentiation in the wall-normal direction. We argue that the new versions should be able to reach higher Reynolds number than currently possible, thanks to their greater accuracy. Using $10^{9}$ grid points and only $10$ compute nodes, they reach a frictional Reynolds number ($Re_{\\tau}$) of $2380$ (with bulk flow $Re=80,000$), which may be compared with the simulation of Hoyas and Jim\\'enez (2006, 2008) which reached $Re_{\\tau}=2003$\\ . The methods derived here appear capable of going well beyond $Re_{\\tau}=2380$.",
      "comment": "minor fixes",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-11-08T21:20:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "plane couette flow",
      "navier-stokes solver",
      "greens functions",
      "spectral integration",
      "channel flow"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.3776V"
    }
  }
}