### Abstract

Language | English |
---|---|

Pages | 1243-1268 |

Number of pages | 25 |

Journal | Computers and Mathematics with Applications |

Volume | 52 |

Issue number | 8-9 |

DOIs | |

Publication status | Published - 2006 |

### Fingerprint

### Keywords

- Navier-Stokes equations
- free surface
- open boundary
- optimal control
- adjoint equations
- sensitivity analysis
- ocean
- waves

### Cite this

*Computers and Mathematics with Applications*,

*52*(8-9), 1243-1268. https://doi.org/10.1016/j.camwa.2006.11.004

}

*Computers and Mathematics with Applications*, vol. 52, no. 8-9, pp. 1243-1268. https://doi.org/10.1016/j.camwa.2006.11.004

**Open boundary control for Navier-Stokes equations including the free surface: adjoint sensitivity analysis.** / Gejadze, I.Y.; Copeland, G.J.M.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Open boundary control for Navier-Stokes equations including the free surface: adjoint sensitivity analysis

AU - Gejadze, I.Y.

AU - Copeland, G.J.M.

PY - 2006

Y1 - 2006

N2 - This paper develops the adjoint sensitivities to the free-surface barotropic Navier- Stokes equations in order to allow for the assimilation of measurements of currents and free-surface elevations into an unsteady flow solution by open-boundary control. To calculate a variation in a surface variable, a mapping is used in the vertical to shift the problem into a fixed domain. A variation is evaluated in the transformed space from the Jacobian matrix of the mapping. This variation is then mapped back into the original space where it completes a tangent linear model. The adjoint equations are derived using the scalar product formulas redefined for a domain with variable bounds. The method is demonstrated by application to an unsteady fluid flow in a one-dimensional open channel in which horizontal and vertical components of velocity are represented as well as the elevation of the free surface (a 2D vertical section model). This requires the proper treatment of open boundaries in both the forward and adjoint models. A particular application is to the construction of a fully three-dimensional coastal ocean model that allows assimilation of tidal elevation and current data. However, the results are general and can be applied in a wider context.

AB - This paper develops the adjoint sensitivities to the free-surface barotropic Navier- Stokes equations in order to allow for the assimilation of measurements of currents and free-surface elevations into an unsteady flow solution by open-boundary control. To calculate a variation in a surface variable, a mapping is used in the vertical to shift the problem into a fixed domain. A variation is evaluated in the transformed space from the Jacobian matrix of the mapping. This variation is then mapped back into the original space where it completes a tangent linear model. The adjoint equations are derived using the scalar product formulas redefined for a domain with variable bounds. The method is demonstrated by application to an unsteady fluid flow in a one-dimensional open channel in which horizontal and vertical components of velocity are represented as well as the elevation of the free surface (a 2D vertical section model). This requires the proper treatment of open boundaries in both the forward and adjoint models. A particular application is to the construction of a fully three-dimensional coastal ocean model that allows assimilation of tidal elevation and current data. However, the results are general and can be applied in a wider context.

KW - Navier-Stokes equations

KW - free surface

KW - open boundary

KW - optimal control

KW - adjoint equations

KW - sensitivity analysis

KW - ocean

KW - waves

UR - http://dx.doi.org/10.1016/j.camwa.2006.11.004

U2 - 10.1016/j.camwa.2006.11.004

DO - 10.1016/j.camwa.2006.11.004

M3 - Article

VL - 52

SP - 1243

EP - 1268

JO - Computers and Mathematics with Applications

T2 - Computers and Mathematics with Applications

JF - Computers and Mathematics with Applications

SN - 0898-1221

IS - 8-9

ER -