Mechanical Engineering - Research Publications
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ItemReynolds-number-dependent turbulent inertia and onset of log region in pipe flows(CAMBRIDGE UNIV PRESS, 2014-10)Abstract A detailed analysis of the ‘turbulent inertia’ (TI) term (the wall-normal gradient of the Reynolds shear stress,$\mathrm{d} \langle -uv\rangle /\mathrm{d} y $), in the axial mean momentum equation is presented for turbulent pipe flows at friction Reynolds numbers$\delta ^{+} \approx 500$, 1000 and 2000 using direct numerical simulation. Two different decompositions for TI are employed to further understand the mean structure of wall turbulence. In the first, the TI term is decomposed into the sum of two velocity–vorticity correlations ($\langle v \omega _z \rangle + \langle - w \omega _y \rangle $) and their co-spectra, which we interpret as an advective transport (vorticity dispersion) contribution and a change-of-scale effect (associated with the mechanism of vorticity stretching and reorientation). In the second decomposition, TI is equivalently represented as the wall-normal gradient of the Reynolds shear stress co-spectra, which serves to clarify the accelerative or decelerative effects associated with turbulent motions at different scales. The results show that the inner-normalised position,$y_m^{+}$, where the TI profile crosses zero, as well as the beginning of the logarithmic region of the wall turbulent flows (where the viscous force is leading order) move outwards in unison with increasing Reynolds number as$y_m^{+} \sim \sqrt{\delta ^{+}}$because the eddies located close to$y_m^{+}$are influenced by large-scale accelerating motions of the type$\langle - w \omega _y \rangle $related to the change-of-scale effect (due to vorticity stretching). These large-scale motions of$O(\delta ^{+})$gain a spectrum of larger length scales with increasing$\delta ^{+}$and are related to the emergence of a secondary peak in the$-uv$co-spectra. With increasing Reynolds number, the influence of the$O(\delta ^{+})$motions promotes viscosity to act over increasingly longer times, thereby increasing the$y^{+}$extent over which the mean viscous force retains leading order. Furthermore, the TI decompositions show that the$\langle v \omega _z \rangle $motions (advective transport and/or dispersion of vorticity) are the dominant mechanism in and above the log region, whereas$\langle - w \omega _y \rangle $motions (vorticity stretching and/or reorientation) are most significant below the log region. The motions associated with$\langle - w \omega _y \rangle $predominantly underlie accelerations, whereas$\langle v \omega _z \rangle $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.
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ItemSelf-similarity in the inertial region of wall turbulence(AMER PHYSICAL SOC, 2014-12-24)The inverse of the von Kármán constant κ is the leading coefficient in the equation describing the logarithmic mean velocity profile in wall bounded turbulent flows. Klewicki [J. Fluid Mech. 718, 596 (2013)] connects the asymptotic value of κ with an emerging condition of dynamic self-similarity on an interior inertial domain that contains a geometrically self-similar hierarchy of scaling layers. A number of properties associated with the asymptotic value of κ are revealed. This is accomplished using a framework that retains connection to invariance properties admitted by the mean statement of dynamics. The development leads toward, but terminates short of, analytically determining a value for κ. It is shown that if adjacent layers on the hierarchy (or their adjacent positions) adhere to the same self-similarity that is analytically shown to exist between any given layer and its position, then κ≡Φ(-2)=0.381966..., where Φ=(1+√5)/2 is the golden ratio. A number of measures, derived specifically from an analysis of the mean momentum equation, are subsequently used to empirically explore the veracity and implications of κ=Φ(-2). Consistent with the differential transformations underlying an invariant form admitted by the governing mean equation, it is demonstrated that the value of κ arises from two geometric features associated with the inertial turbulent motions responsible for momentum transport. One nominally pertains to the shape of the relevant motions as quantified by their area coverage in any given wall-parallel plane, and the other pertains to the changing size of these motions in the wall-normal direction. In accord with self-similar mean dynamics, these two features remain invariant across the inertial domain. Data from direct numerical simulations and higher Reynolds number experiments are presented and discussed relative to the self-similar geometric structure indicated by the analysis, and in particular the special form of self-similarity shown to correspond to κ=Φ(-2).