Reynolds-number-dependent turbulent inertia and onset of log region in pipe flows. (25th October 2014) - Chin, C.; Philip, J.; Klewicki, J.; Ooi, A.; Marusic, I.
Abstract A detailed analysis of the 'turbulent inertia' (TI) term (the wall-normal gradient of the Reynolds shear stress, <alternatives><inline-graphic xlink:href="ark:/27927/pgh1fhv8b6b" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$\mathrm{d} \langle -uv\rangle /\mathrm{d} y $]]> </alternatives>), in the axial mean momentum equation is presented for turbulent pipe flows at friction Reynolds numbers <alternatives><inline-graphic xlink:href="ark:/27927/pgh1fhv89g7" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$\delta ^{+} \approx 500$]]> </alternatives>, 1000 and 2000 using direct numerical simulation. Two different decompositions for TI are employed to further understand the mean structure of wall turbulence. In the first, the TI term is decomposed into the sum of two velocity–vorticity correlations (<alternatives><inline-graphic xlink:href="ark:/27927/pgh1fhv89q3" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$\langle v \omega _z \rangle + \langle - w \omega _y \rangle $]]> </alternatives>) and their co-spectra, which we interpret as an advective transport (vorticity dispersion) contribution and a change-of-scale effect (associated with the mechanism of vorticity stretching and reorientation). In the second decomposition, TI is equivalently represented as the wall-normal gradient of the Reynolds shear stress co-spectra, which serves to clarify the accelerative or decelerative effects associated with turbulent motions at different scales. The results show that the inner-normalised position, <alternatives><inline-graphic xlink:href="ark:/27927/pgh1fhv8bgr" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$y_m^{+}$]]> </alternatives>, where the TI profile crosses zero, as well as the beginning of the logarithmic region of the wall turbulent flows (where the viscous force is leading order) move outwards in unison with increasing Reynolds number as <alternatives><inline-graphic xlink:href="ark:/27927/pgh1fhv89v9" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$y_m^{+} \sim \sqrt{\delta ^{+}}$]]> </alternatives> because the eddies located close to <alternatives><inline-graphic xlink:href="ark:/27927/pgh1fhv89wv" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$y_m^{+}$]]> </alternatives> are influenced by large-scale accelerating motions of the type <alternatives><inline-graphic xlink:href="ark:/27927/pgh1fhv8bt8" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$\langle - w \omega _y \rangle $]]> </alternatives> related to the change-of-scale effect (due to vorticity stretching). These large-scale motions of <alternatives><inline-graphic xlink:href="ark:/27927/pgh1fhv88xw" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$O(\delta ^{+})$]]> </alternatives> gain a spectrum of larger length scales with increasing <alternatives><inline-graphic xlink:href="ark:/27927/pgh1fhv8823" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$\delta ^{+}$]]> </alternatives> and are related to the emergence of a secondary peak in the <alternatives><inline-graphic xlink:href="ark:/27927/pgh1fhv8bnh" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$-uv$]]> </alternatives> co-spectra. With increasing Reynolds number, the influence of the <alternatives><inline-graphic xlink:href="ark:/27927/pgh1fhv87md" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$O(\delta ^{+})$]]> </alternatives> motions promotes viscosity to act over increasingly longer times, thereby increasing the <alternatives><inline-graphic xlink:href="ark:/27927/pgh1fhv8bwc" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$y^{+}$]]> </alternatives> extent over which the mean viscous force retains leading order. Furthermore, the TI decompositions show that the <alternatives><inline-graphic xlink:href="ark:/27927/pgh1fhv897c" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$\langle v \omega _z \rangle $]]> </alternatives> motions (advective transport and/or dispersion of vorticity) are the dominant mechanism in and above the log region, whereas <alternatives><inline-graphic xlink:href="ark:/27927/pgh1fhv898x" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$\langle - w \omega _y \rangle $]]> </alternatives> motions (vorticity stretching and/or reorientation) are most significant below the log region. The motions associated with <alternatives><inline-graphic xlink:href="ark:/27927/pgh1fhv88wb" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$\langle - w \omega _y \rangle $]]> </alternatives> predominantly underlie accelerations, whereas <alternatives><inline-graphic xlink:href="ark:/27927/pgh1fhv87q2" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$\langle v \omega _z \rangle $]]> </alternatives> primarily contribute to decelerations. Finally, a description of the structure of wall turbulence deduced from the present analysis and our physical interpretation is presented, and is shown to be consistent with previous flow visualisation studies.