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Abstract: We present a detailed study of the linear stability of planeCouette-Poiseuille flow in the presence of a cross-flow. The base flow ischaracterised by the cross flow Reynolds number, $R {inj}$ and thedimensionless wall velocity, $k$. Squire-s transformation may be applied to thelinear stability equations and we therefore consider 2D spanwise-independentperturbations. Corresponding to each dimensionless wall velocity, $k\in0,1$,two ranges of $R {inj}$ exist where unconditional stability is observed. In thelower range of $R {inj}$, for modest $k$ we have a stabilisation of longwavelengths leading to a cut-off $R {inj}$. This lower cut-off results fromskewing of the velocity profile away from a Poiseuille profile, shifting of thecritical layers and the gradual decrease of energy production. Cross-flowstabilisation and Couette stabilisation appear to act via very similarmechanisms in this range, leading to the potential for robust compensatorydesign of flow stabilisation using either mechanism. As $R {inj}$ is increased,we see first destabilisation and then stabilisation at very large $R {inj}$.The instability is again a long wavelength mechanism. Analysis of theeigenspectrum suggests the cause of instability is due to resonant interactionsof Tollmien-Schlichting waves. A linear energy analysis reveals that in thisrange the Reynolds stress becomes amplified, the critical layer is irrelevantand viscous dissipation is completely dominated by the energyproduction-negation, which approximately balances at criticality. Thestabilisation at very large $R {inj}$ appears to be due to decay in energyproduction, which diminishes like $R {inj}^{-1}$. Our study is limited to twodimensional, spanwise independent perturbations.

Author: Anirban Guha, Ian A. Frigaard

Source: https://arxiv.org/

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