Spectrum of the curl of vorticity as a precursor to dissipation in three-dimensional Taylor-Green turbulence

Predicting when a three-dimensional turbulent flow reaches its dissipation peak is essential for both theory and adaptive algorithms in simulations and experiments. Using direct numerical simulations (DNSs) of the Taylor-Green vortex (TGV) at resolutions of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:msup> <a:mn>256</a:mn> <a:mn>3</a:mn> </a:msup> <a:mo>−</a:mo> <a:msup> <a:mn>1024</a:mn> <a:mn>3</a:mn> </a:msup> </a:mrow> </a:math> , we introduce and test a small-scale weighted diagnostic: the spectrum of <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:msup> <b:mrow> <b:mo>|</b:mo> <b:mi>∇</b:mi> <b:mo>×</b:mo> <b:mrow> <b:mi mathvariant="bold-italic">ω</b:mi> </b:mrow> <b:mo>|</b:mo> </b:mrow> <b:mn>2</b:mn> </b:msup> </b:math> (with <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mrow> <d:mrow> <d:mi mathvariant="bold-italic">ω</d:mi> </d:mrow> <d:mo>=</d:mo> <d:mi>∇</d:mi> <d:mo>×</d:mo> <d:mi mathvariant="bold">u</d:mi> </d:mrow> </d:math> ), which, for incompressible flow, is equivalent to a <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:msup> <g:mi>k</g:mi> <g:mn>4</g:mn> </g:msup> </g:math> -weighted energy spectrum. We show that the peak wavenumber of this spectrum, <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:mrow> <h:msub> <h:mi>k</h:mi> <h:mi>peak</h:mi> </h:msub> <h:msup> <h:mrow> <h:mo>[</h:mo> <h:mspace width="0.16em"/> <h:mo>|</h:mo> <h:mi>∇</h:mi> <h:mo>×</h:mo> <h:mrow> <h:mi mathvariant="bold-italic">ω</h:mi> </h:mrow> <h:mo>|</h:mo> </h:mrow> <h:mn>2</h:mn> </h:msup> <h:mrow> <h:mspace width="0.16em"/> <h:mo>]</h:mo> </h:mrow> </h:mrow> </h:math> , advances rapidly to intermediate-small scales and then levels off before the dissipation rate <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"> <l:mrow> <l:mi>ɛ</l:mi> <l:mrow> <l:mo>(</l:mo> <l:mi>t</l:mi> <l:mo>)</l:mo> </l:mrow> <l:mo>=</l:mo> <l:msub> <l:mo>∑</l:mo> <l:mi>k</l:mi> </l:msub> <l:mn>2</l:mn> <l:mi>ν</l:mi> <l:msup> <l:mi>k</l:mi> <l:mn>2</l:mn> </l:msup> <l:mi>E</l:mi> <l:mrow> <l:mo>(</l:mo> <l:mi>k</l:mi> <l:mo>)</l:mo> </l:mrow> </l:mrow> </l:math> reaches its maximum. Across all resolutions, we observe robust temporal ordering <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>t</m:mi> <m:mi>k</m:mi> </m:msub> <m:mo>&lt;</m:mo> <m:msub> <m:mi>t</m:mi> <m:mi>ɛ</m:mi> </m:msub> <m:mo>&lt;</m:mo> <m:msub> <m:mi>t</m:mi> <m:mi mathvariant="normal">Π</m:mi> </m:msub> </m:mrow> </m:math> , where <o:math xmlns:o="http://www.w3.org/1998/Math/MathML"> <o:msub> <o:mi>t</o:mi> <o:mi>k</o:mi> </o:msub> </o:math> marks the onset of the rapid rise of <p:math xmlns:p="http://www.w3.org/1998/Math/MathML"> <p:mrow> <p:msub> <p:mi>k</p:mi> <p:mi>peak</p:mi> </p:msub> <p:msup> <p:mrow> <p:mo>[</p:mo> <p:mspace width="0.16em"/> <p:mo>|</p:mo> <p:mi>∇</p:mi> <p:mo>×</p:mo> <p:mrow> <p:mi mathvariant="bold-italic">ω</p:mi> </p:mrow> <p:mo>|</p:mo> </p:mrow> <p:mn>2</p:mn> </p:msup> <p:mrow> <p:mspace width="0.16em"/> <p:mo>]</p:mo> </p:mrow> </p:mrow> <p:mo>,</p:mo> <p:mo> </p:mo> <p:msub> <p:mi>t</p:mi> <p:mi>ɛ</p:mi> </p:msub> </p:math> is the time of the maximal <t:math xmlns:t="http://www.w3.org/1998/Math/MathML"> <t:mrow> <t:mi>ɛ</t:mi> <t:mo>(</t:mo> <t:mi>t</t:mi> <t:mo>)</t:mo> </t:mrow> </t:math> , and <u:math xmlns:u="http://www.w3.org/1998/Math/MathML"> <u:msub> <u:mi>t</u:mi> <u:mi mathvariant="normal">Π</u:mi> </u:msub> </u:math> is when the cumulative flux <w:math xmlns:w="http://www.w3.org/1998/Math/MathML"> <w:mrow> <w:mo>|</w:mo> <w:mi mathvariant="normal">Π</w:mi> <w:mo>(</w:mo> <w:mi>K</w:mi> <w:mo>)</w:mo> <w:mo>|</w:mo> </w:mrow> </w:math> attains its largest peak scale. This early-warning signal correlates with the morphological transition to filament-dominated structures visible in <y:math xmlns:y="http://www.w3.org/1998/Math/MathML"> <y:mi>Q</y:mi> </y:math> -criterion isosurfaces and is consistent with integral-scale trends ( <z:math xmlns:z="http://www.w3.org/1998/Math/MathML"> <z:mrow> <z:msub> <z:mi>L</z:mi> <z:mi>int</z:mi> </z:msub> <z:mo>,</z:mo> <z:mi>λ</z:mi> <z:mo>,</z:mo> <z:mi>η</z:mi> </z:mrow> </z:math> ). The diagnostic is simple to compute from standard DNS data and highlights the incipient formation of high-curvature structures, where viscosity acts most strongly.