Mechanical Engineering - Theses
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ItemThree-dimensional structure and scaling of a canonical turbulent boundary layer( 2021)Wall-bounded turbulent flows are pervasive in nature and are also encountered in many engineering applications; common examples include the flow over an airplane wing, or the atmospheric boundary layer over the Earth's surface, etc. A dominant feature present within these flows is the appearance of recurring eddies or so-called coherent structures that are highly three-dimensional (3-D) in geometry and are statistically significant over a wide range of scales. This has led to the proposal of various coherent structure-based models in the literature, with the attached eddy model (AEM) of wall-turbulence being the most popular amongst them. Their predictive capabilities, however, are still lacking due to the dearth of 3-D information on the coherent motions which they model. The present thesis reports a new and unique set of multi-point hotwire measurements conducted in a frictional Reynolds number, $Re_{\tau}$ $\sim$ $\mathcal{O}$(10$^4$) canonical turbulent boundary layer (TBL) to reconstruct the 3-D statistical picture of these energy-containing motions. The measurements are complemented by performing a similar reconstruction using published direct numerical simulation datasets at $Re_{\tau}$ $\sim$ $\mathcal{O}$(10$^3$), thereby facilitating an examination of the scaling of these structures, in flows spanning over a decade of $Re_{\tau}$. Results of these investigations provide direct empirical support towards the AEM, with the prospect of further enhancing its efficiency by defining the representative eddy geometry based on data-driven estimates. The first part of the thesis focuses on investigating characteristics of the inertially dominated wall-coherent structures (i.e. the ones extending down to the wall), which are responsible for the increased skin-friction in high-$Re_{\tau}$ TBLs. Their geometric characteristics are investigated in the wall-parallel plane by estimating, for the first time, the 2-D cross-spectrum of the streamwise velocity using the synchronous velocity fluctuations measured at a log-region ($z_{o}$) and near-wall ($z_{r}$) location. Constant energy contours of this spectrum, which are representative of the energy distribution across the range of streamwise (${\lambda}_{x}$) and spanwise (${\lambda}_{y}$) wavelengths, are found to follow the ${{\lambda}_{x}}/{z_{o}}$ $\approx$ 7(${{\lambda}_{y}}/{z_{o}}$) relationship in the large-scale range, indicative of geometric self-similarity. This suggests that a self-similar structure conforming to Townsend's attached eddy hypothesis (Townsend 1976) is ingrained in the flow, and can be conceptually modelled using the AEM framework given by Perry \& Chong (1982). The very-large-scale wall-coherent structures (i.e. the superstructures), on the other hand, do not conform to Townsend's attached eddies and are found to have a similar spanwise width as the largest motions in the self-similar hierarchy. This result, which is found via a scale-specific coherence analysis of the velocity fluctuations, also reveals the periodic organization of the superstructures along the spanwise direction. Finally, an analysis of the scale-specific phase of the coherence reveals the streamwise inclination angle of the large wall-coherent motions, which is found to be nominally 45$^{\circ}$. This fulfills the minimum geometric information required to statistically model these energetic wall-coherent motions based on the AEM. The second part of the thesis focuses on investigating the range of energy-containing structures coexisting in the log-region, which contribute significantly to the bulk turbulence production in high-$Re_{\tau}$ wall-bounded flows. Townsend (1961) hypothesized that these structures can be segregated into active and inactive motions, where the active motions are solely responsible for producing the Reynolds shear stress, the key momentum transport term in these flows. While the wall-normal component of velocity is associated exclusively with the active motions, the wall-parallel components of velocity are associated with both active and inactive motions. To test this hypothesis, the present study proposes a methodology to segregate the active and inactive components of the 2-D energy spectrum (${\Phi}_{ii}$, where $i$ denotes the velocity-component) at $z_{o}$, thereby permitting to test the self-similarity characteristics of the former which are central to theoretical models for wall-turbulence. The methodology utilizes the multi-point dataset, in conjunction with a spectral linear stochastic estimation-based procedure, to linearly decompose the total energy at $z_{o}$ (${\Phi}_{ii}$) into contributions predominantly from the active (${\Phi}^{a}_{ii}$) and inactive (${\Phi}^{ia}_{ii}$) motions. This is confirmed by ${\Phi}^{a}_{ii}$ exhibiting wall-scaling for both ${\lambda}_x$ and ${\lambda}_y$. The Reynolds shear stress cospectra, estimated solely from the active contributions, is also found to closely match the one obtained conventionally from the dataset, thereby providing direct empirical support for the concept of active and inactive motions. Both ${\Phi}^{a}_{ii}$ and ${\Phi}^{ia}_{ii}$ contours are found to depict geometric self-similarity in the log-region, suggesting that this entire region can be conceptually modelled using the AEM framework. Inactive contributions from the attached eddies also bring out the pure $k^{-1}$-scaling for the associated 1-D spectra (where $k$ is the streamwise/spanwise wavenumber), lending further empirical support to the AEM.