Viscous Instabilities in Transversely Strained Channel Flows
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909232393093120 |
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| author | Abdullah, Muhammad |
| author_facet | Abdullah, Muhammad |
| contents | We investigate here linear stability in a canonical three-dimensional boundary layer generated by the superposition of a spanwise pressure gradient upon an otherwise standard channel flow. As the main result, we introduce a simple coordinate transformation that enables the complete description of modal and non-modal stability using previous results on Poiseuille flow. We leverage this insight to derive closed forms for some relevant stability metrics. In particular, the critical Reynolds number for exponential-in-time growth is found to monotonically decrease with the strength of the cross-flow. A suitably chosen re-scaling, however, shows that the stability characteristics ultimately approach those of channel flow, despite the presence of a non-zero spanwise shear. Unstable eigenmodes akin to the Tollmien-Schlichting wave are found to propagate along the direction of the net flow. From a non-modal perspective, the maximal transient (algebraic) growth increases quadratically with the spanwise pressure differential and, similar to two-dimensional flows, is fueled by the lift-up effect. In this regard, the linear energy budget highlights a dramatic increase in energy production against the spanwise shear. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_18891 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Viscous Instabilities in Transversely Strained Channel Flows Abdullah, Muhammad Fluid Dynamics Mathematical Physics We investigate here linear stability in a canonical three-dimensional boundary layer generated by the superposition of a spanwise pressure gradient upon an otherwise standard channel flow. As the main result, we introduce a simple coordinate transformation that enables the complete description of modal and non-modal stability using previous results on Poiseuille flow. We leverage this insight to derive closed forms for some relevant stability metrics. In particular, the critical Reynolds number for exponential-in-time growth is found to monotonically decrease with the strength of the cross-flow. A suitably chosen re-scaling, however, shows that the stability characteristics ultimately approach those of channel flow, despite the presence of a non-zero spanwise shear. Unstable eigenmodes akin to the Tollmien-Schlichting wave are found to propagate along the direction of the net flow. From a non-modal perspective, the maximal transient (algebraic) growth increases quadratically with the spanwise pressure differential and, similar to two-dimensional flows, is fueled by the lift-up effect. In this regard, the linear energy budget highlights a dramatic increase in energy production against the spanwise shear. |
| title | Viscous Instabilities in Transversely Strained Channel Flows |
| topic | Fluid Dynamics Mathematical Physics |
| url | https://arxiv.org/abs/2406.18891 |