### Abstract

The properties of the Karhunen-Loéve (KL) expansion of the derivative u_{x}(x) of an inhomogeneous random process possessing viscous boundary-layer behavior are studied in relation to questions of efficient representation for numerical Galerkan schemes for computational simulation of turbulence. Eigenfunctions and eigenvalue spectra are calculated for the randomly forced one-dimensional Burgers' model of turbulence. Convergence of the expansion of u_{x} is much slower than convergence of the expansion of u(x), and direct expansion of u_{x} is not significantly more efficient than differentiating the expansion of u. The ordered eigenvalue spectrum of u_{x} is proportional to the square of the order parameter times the eigenvalue spectrum of u. The underlying cause of slow convergence is the earlier onset of locally sinusoidal behavior of the KL eigenfunctions when the expansion is performed over the entire domain of the solution.

Original language | English (US) |
---|---|

Pages (from-to) | 2233-2235 |

Number of pages | 3 |

Journal | Physics of Fluids |

Volume | 6 |

Issue number | 6 |

State | Published - 1994 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Condensed Matter Physics
- Physics and Astronomy(all)
- Mechanics of Materials
- Computational Mechanics
- Fluid Flow and Transfer Processes

### Cite this

*Physics of Fluids*,

*6*(6), 2233-2235.

**Karhunen-Loéve expansion of the derivative of an inhomogeneous process.** / Sung, H. J.; Adrian, Ronald.

Research output: Contribution to journal › Article

*Physics of Fluids*, vol. 6, no. 6, pp. 2233-2235.

}

TY - JOUR

T1 - Karhunen-Loéve expansion of the derivative of an inhomogeneous process

AU - Sung, H. J.

AU - Adrian, Ronald

PY - 1994

Y1 - 1994

N2 - The properties of the Karhunen-Loéve (KL) expansion of the derivative ux(x) of an inhomogeneous random process possessing viscous boundary-layer behavior are studied in relation to questions of efficient representation for numerical Galerkan schemes for computational simulation of turbulence. Eigenfunctions and eigenvalue spectra are calculated for the randomly forced one-dimensional Burgers' model of turbulence. Convergence of the expansion of ux is much slower than convergence of the expansion of u(x), and direct expansion of ux is not significantly more efficient than differentiating the expansion of u. The ordered eigenvalue spectrum of ux is proportional to the square of the order parameter times the eigenvalue spectrum of u. The underlying cause of slow convergence is the earlier onset of locally sinusoidal behavior of the KL eigenfunctions when the expansion is performed over the entire domain of the solution.

AB - The properties of the Karhunen-Loéve (KL) expansion of the derivative ux(x) of an inhomogeneous random process possessing viscous boundary-layer behavior are studied in relation to questions of efficient representation for numerical Galerkan schemes for computational simulation of turbulence. Eigenfunctions and eigenvalue spectra are calculated for the randomly forced one-dimensional Burgers' model of turbulence. Convergence of the expansion of ux is much slower than convergence of the expansion of u(x), and direct expansion of ux is not significantly more efficient than differentiating the expansion of u. The ordered eigenvalue spectrum of ux is proportional to the square of the order parameter times the eigenvalue spectrum of u. The underlying cause of slow convergence is the earlier onset of locally sinusoidal behavior of the KL eigenfunctions when the expansion is performed over the entire domain of the solution.

UR - http://www.scopus.com/inward/record.url?scp=0028011299&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0028011299&partnerID=8YFLogxK

M3 - Article

VL - 6

SP - 2233

EP - 2235

JO - Physics of Fluids

JF - Physics of Fluids

SN - 1070-6631

IS - 6

ER -