Laminar turbulent behavior in shear-thickening channel flow

We conduct direct numerical simulations (DNS) to investigate the dynamics of a non-Newtonian shear-thickening fluid in a turbulent plane channel. The primary parameter governing the non-Newtonian flow rheology is the Carreau number (<a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mtext>Cu</a:mtext></a:math>), defined as the ratio of the fluid's characteristic timescale, <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi mathvariant="normal">Λ</b:mi></b:math>, to the flow characteristic timescale, <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:mi>h</d:mi><d:mo>/</d:mo><d:msub><d:mi>u</d:mi><d:mi>τ</d:mi></d:msub></d:mrow></d:math>, where <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mi>h</e:mi></e:math> is the channel half-height and <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:msub><f:mi>u</f:mi><f:mi>τ</f:mi></f:msub></f:math> is the friction velocity at the wall. Starting from a Newtonian turbulent channel flow at <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mrow><g:msub><g:mtext>Re</g:mtext><g:mi>τ</g:mi></g:msub><g:mo>=</g:mo><g:mn>180</g:mn></g:mrow></g:math>, we examine the impact of increasing <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mtext>Cu</h:mtext></h:math> on turbulent statistics and coherent structures. The shear-dependent rheology is described using a Carreau viscosity model, with <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:mtext>Cu</i:mtext></i:math> values spanning two orders of magnitude (<j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:mrow><j:mtext>Cu</j:mtext><j:mo>=</j:mo><j:mn>0.1</j:mn><j:mo>,</j:mo><j:mn>1</j:mn><j:mo>,</j:mo><j:mn>5</j:mn><j:mo>,</j:mo><j:mn>10</j:mn></j:mrow></j:math>). While maintaining a constant mean pressure gradient (<k:math xmlns:k="http://www.w3.org/1998/Math/MathML"><k:mrow><k:mi>∇</k:mi><k:mover><k:mi>p</k:mi><k:mo>¯</k:mo></k:mover></k:mrow></k:math>) that drives the flow—ensuring a fixed shear Reynolds number based on the zero-strain-rate reference viscosity <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"><l:mrow><l:msub><l:mtext>Re</l:mtext><l:mrow><l:mi>τ</l:mi><l:mo>,</l:mo><l:mn>0</l:mn></l:mrow></l:msub><l:mo>=</l:mo><l:mn>180</l:mn></l:mrow></l:math>—we observe varying effective Reynolds numbers (<m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:msub><m:mtext>Re</m:mtext><m:mrow><m:mi>τ</m:mi><m:mo>,</m:mo><m:mi>w</m:mi></m:mrow></m:msub><m:mo>=</m:mo><m:mn>111</m:mn><m:mo>,</m:mo><m:mn>76</m:mn><m:mo>,</m:mo><m:mn>58</m:mn><m:mo>,</m:mo><m:mn>52</m:mn></m:mrow></m:math>) due to the altered viscosity distributions. Our findings reveal that shear thickening generally reduces the mean flow velocity and enhances the isotropy of velocity fluctuations compared to the Newtonian reference case. As <n:math xmlns:n="http://www.w3.org/1998/Math/MathML"><n:mtext>Cu</n:mtext></n:math> increases, the strength of vortical structures diminishes, accompanied by a decline in turbulence intensity. This behavior is primarily attributed to elevated wall viscosity, which shifts turbulent stresses towards viscous stresses. Notably, at subcritical effective shear Reynolds numbers for which a Newtonian fluid would exhibit laminar flow, the shear-thickening fluid retains unsteady behavior (turbulence) but lacks a logarithmic layer.