From laminar to chaotic flow via stochastic resonance in viscoelastic channel flow

Recent research indicates that low-inertia viscoelastic channel flow experiences supercritical nonnormal mode elastic instability from laminar to sustained chaotic flow due to finite-size perturbations. The challenge of this paper is to elucidate a realization of such a pathway when the intensity of the elastic wave is too low to amplify vortex fluctuations above the instability onset. The study identifies two subregions in the transition flow regime at Weissenberg number <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mtext>Wi</a:mtext> <a:mo>&gt;</a:mo> <a:msub> <a:mtext>Wi</a:mtext> <a:mi>c</a:mi> </a:msub> </a:mrow> </a:math> , the instability onset. In the lower subregion at <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mrow> <b:msub> <b:mtext>Wi</b:mtext> <b:mi>c</b:mi> </b:msub> <b:mo>≤</b:mo> <b:mtext>Wi</b:mtext> <b:mo>≤</b:mo> <b:mn>300</b:mn> </b:mrow> </b:math> , we discover periodic spikes in the streamwise velocity time series <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mrow> <c:mi>u</c:mi> <c:mo>(</c:mo> <c:mi>t</c:mi> <c:mo>)</c:mo> </c:mrow> </c:math> that appear in the chaotic power spectrum as low-frequency, high-intensity peaks resembling stochastic resonance (SR). In contrast, the spanwise velocity power spectrum, <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:msub> <d:mi>E</d:mi> <d:mi>w</d:mi> </d:msub> </d:math> , remains flat and with low-intensity, noisy, and broad elastic wave peaks. The spikes significantly distort the probability density function of <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mi>u</e:mi> </e:math> and initiate and amplify random streaks and wall-normal vorticity fluctuations. The condition for the SR emergence is similar to that of a dynamical system where the presence of a chaotic attractor, limit cycle, and external white noise are necessary. This similarity is confirmed by presenting a phase portrait in two subregions of the transition regime. In the upper subregion at <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:mrow> <f:mtext>Wi</f:mtext> <f:mo>&gt;</f:mo> <f:mn>400</f:mn> <f:mo>,</f:mo> </f:mrow> </f:math> the periodic spikes disappear and <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:msub> <g:mi>E</g:mi> <g:mi>w</g:mi> </g:msub> </g:math> becomes chaotic with a large intensity elastic wave sufficient to self-organize and synchronize the streaks into cycles and to amplify the wall normal vorticity, according to a recently proposed mechanism of vortex-wave interactions.