### Abstract

A three-dimensional wave packet generated by a local disturbance in a two-dimensional hypersonic boundary layer flow is studied with the aid of the previously solved initial-value problem. The solution to this problem can be expanded in a biorthogonal eigenfunction system as a sum of modes consisting of continuous and discrete spectra of temporal stability theory. A specific disturbance consisting of an initial temperature spot is considered, and the receptivity to this initial temperature spot is computed for both the two-dimensional and three-dimensional cases. Using previous analysis of the discrete and continuous spectrum, the inverse Fourier transform is computed numerically. The two-dimensional inverse Fourier transform is calculated for two discrete modes: Mode F and Mode S. The Mode S result is compared with an asymptotic approximation of the Fourier integral, which is obtained using the Gaussian model as well as the method of steepest descent. It is shown that the method of steepest descent provides an excellent approximation to the more computationally intensive numerical evaluation of the inverse Fourier transform. Additionally, the three-dimensional inverse Fourier transform is found using an asymptotic approximation of the Fourier integral. A main feature of the resulting three-dimensional wave packet is its two-dimensional nature, which arises from an association of Mode S with Mack's second mode.

Original language | English |
---|---|

Article number | 104103 |

Journal | Physics of Fluids |

Volume | 18 |

Issue number | 10 |

DOIs | |

State | Published - 1 Oct 2006 |

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### Keywords

- Boundary layer turbulence
- Compressible flow
- Eigenvalues and eigenfunctions
- Flow instability
- Fourier transforms
- Hypersonic flow
- Initial value problems
- Laminar flow
- Laminar to turbulent transitions
- Waves

### Cite this

*Physics of Fluids*,

*18*(10), [104103]. https://doi.org/10.1063/1.2359003

}

*Physics of Fluids*, vol. 18, no. 10, 104103. https://doi.org/10.1063/1.2359003

**Three-dimensional wave packets in a compressible boundary layer.** / Forgoston, Eric; Tumin, Anatoli.

Research output: Contribution to journal › Article › Research › peer-review

TY - JOUR

T1 - Three-dimensional wave packets in a compressible boundary layer

AU - Forgoston, Eric

AU - Tumin, Anatoli

PY - 2006/10/1

Y1 - 2006/10/1

N2 - A three-dimensional wave packet generated by a local disturbance in a two-dimensional hypersonic boundary layer flow is studied with the aid of the previously solved initial-value problem. The solution to this problem can be expanded in a biorthogonal eigenfunction system as a sum of modes consisting of continuous and discrete spectra of temporal stability theory. A specific disturbance consisting of an initial temperature spot is considered, and the receptivity to this initial temperature spot is computed for both the two-dimensional and three-dimensional cases. Using previous analysis of the discrete and continuous spectrum, the inverse Fourier transform is computed numerically. The two-dimensional inverse Fourier transform is calculated for two discrete modes: Mode F and Mode S. The Mode S result is compared with an asymptotic approximation of the Fourier integral, which is obtained using the Gaussian model as well as the method of steepest descent. It is shown that the method of steepest descent provides an excellent approximation to the more computationally intensive numerical evaluation of the inverse Fourier transform. Additionally, the three-dimensional inverse Fourier transform is found using an asymptotic approximation of the Fourier integral. A main feature of the resulting three-dimensional wave packet is its two-dimensional nature, which arises from an association of Mode S with Mack's second mode.

AB - A three-dimensional wave packet generated by a local disturbance in a two-dimensional hypersonic boundary layer flow is studied with the aid of the previously solved initial-value problem. The solution to this problem can be expanded in a biorthogonal eigenfunction system as a sum of modes consisting of continuous and discrete spectra of temporal stability theory. A specific disturbance consisting of an initial temperature spot is considered, and the receptivity to this initial temperature spot is computed for both the two-dimensional and three-dimensional cases. Using previous analysis of the discrete and continuous spectrum, the inverse Fourier transform is computed numerically. The two-dimensional inverse Fourier transform is calculated for two discrete modes: Mode F and Mode S. The Mode S result is compared with an asymptotic approximation of the Fourier integral, which is obtained using the Gaussian model as well as the method of steepest descent. It is shown that the method of steepest descent provides an excellent approximation to the more computationally intensive numerical evaluation of the inverse Fourier transform. Additionally, the three-dimensional inverse Fourier transform is found using an asymptotic approximation of the Fourier integral. A main feature of the resulting three-dimensional wave packet is its two-dimensional nature, which arises from an association of Mode S with Mack's second mode.

KW - Boundary layer turbulence

KW - Compressible flow

KW - Eigenvalues and eigenfunctions

KW - Flow instability

KW - Fourier transforms

KW - Hypersonic flow

KW - Initial value problems

KW - Laminar flow

KW - Laminar to turbulent transitions

KW - Waves

UR - http://www.scopus.com/inward/record.url?scp=33750576987&partnerID=8YFLogxK

U2 - 10.1063/1.2359003

DO - 10.1063/1.2359003

M3 - Article

VL - 18

JO - Physics of Fluids

JF - Physics of Fluids

SN - 1070-6631

IS - 10

M1 - 104103

ER -