On the asymptotic analysis of pulsating planar Poiseuille flow instabilities

The linear stability analysis of a pulsating flow between parallel plates at finite amplitudes and different frequencies demonstrates the presence of instability with a different nature to that initially reported in earlier studies at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>W</mml:mi> <mml:mi>o</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>25</mml:mn> </mml:mrow> </mml:math> . The time-dependent stability problem is formulated using a Floquet expansion of the perturbation and solved in a matrix approach. At higher pulsation frequencies (i.e. <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>W</mml:mi> <mml:mi>o</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>30</mml:mn> </mml:mrow> </mml:math> ), pulsation amplitudes and sufficiently large Reynolds numbers, the time-dependent baseflow displays multiple inflexion points throughout the entire duration of the pulsation. A scaling analysis of the temporal growth rate reveals that the instability is of inviscid type and scales with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>R</mml:mi> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow> <mml:mo lspace="0em" rspace="0em">−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> . Freezing the time-dependent baseflow for a particular set of parameters, the growth rate is simultaneously controlled by viscosity solely acting on the perturbation, and the pulsating frequency coupling the harmonics. In the asymptotic limit ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>R</mml:mi> <mml:mi>e</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mo form="prefix" stretchy="false">+</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> ), we demonstrate that the temporal problem simplifies to a set of steady stability analyses dependent on the phase of the pulsation. In particular, we show that in this limit, Rayleigh and Fjørtoft’s criteria for instability become valid again, that sub-harmonic periodic orbits can be used to estimate the inviscid growth rate and match with the asymptotic values predicted from the viscous stability analysis.