Mechanical Engineering - Theses
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ItemBoundary layer and bulk dynamics in vertical natural convection( 2017)Results from direct numerical simulations of vertical natural convection at Rayleigh numbers 10^5–10^9 and Prandtl number 0.709 are found to support a generalised applicability of the Grossmann–Lohse (GL) theory, which was originally developed for Rayleigh–Bénard convection. In accordance with the GL theory, we show that the normalised mean boundary- layer thicknesses of the velocity and temperature fields obey the laminar-like Prandtl– Blasius–Pohlhausen scaling, corresponding to the “classical” state. Away from the walls, the dissipation of the turbulent fluctuations, which can be interpreted as the “bulk” or “background” dissipation of the GL theory, is found to obey the Kolmogorov– Obukhov–Corrsin scaling for fully developed turbulence. The present results suggest that, similar to Rayleigh–Bénard convection, a pure power-law relationship between the Nusselt, Rayleigh and Prandtl numbers is not the best description for vertical natural convection and existing empirical relationships should be recalibrated to better reflect the underlying physics. On closer scrutiny of the boundary layers, we find evidence that the boundary layers are undergoing a transition from the classical state to the “ultimate” shear-dominated state. In particular, we observe near-wall higher-shear patches that occupy increasingly larger fractions of the wall-areas. These higher-shear patches exhibit turbulent features, for instance (i) the patches appear streaky, reminiscent of the characteristic near- wall streaks in canonical wall-bounded turbulence, (ii) the local mean temperature profile yields a logarithmic variation, in agreement with the logarithmic law of the wall for mean temperature, and (iii) the local Nusselt number follows an effective Rayleigh number power-law scaling exponent of 0.37, consistent with the logarithmically corrected 1/2 power-law scaling predicted for ultimate thermal convection. We reason that both turbulent and laminar-like boundary layers coexist in the transitional regime of vertical natural convection, consistent with the findings reported for Rayleigh–Bénard convection and Taylor–Couette flows. When the walls are instead removed and boundary layers eliminated, the new setup mimics turbulent bulk-dominated thermal convection. We refer to this new setup as homogeneous vertical natural convection. A direct application of scaling arguments to the governing equations of this new setup yields the asymptotic 1/2 power-law scaling relations for the Nusselt and Reynolds numbers, in accordance to previous theoretical predictions of turbulent bulk-dominated thermal convection. Results from direct numerical simulation of the new setup further supports the predicted 1/2 power- law relations. When employing bulk quantities for the wall-bounded setup, we too find the aforementioned 1/2 power-law scaling. This extended result suggests that the 1/2 power-law scaling relation may even be present at lower Rayleigh numbers provided the appropriate quantities in the turbulent bulk flow are employed for the definitions of the Rayleigh, Reynolds and Nusselt numbers. Lastly, we perform a straightforward assessment of the mixing efficiency in vertical natural convection. The value is predicted and found to be approximately 0.5, which suggests that the dissipation rate of kinetic energy is directly proportional to the rate at which gravitational potential energy is readily available for conversion.
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ItemDirect numerical simulation of turbulent natural convection bounded by differentially heated vertical walls( 2013)Using new, high-resolution direct numerical simulation (DNS) data, this study appraises the different scaling laws found in literature for turbulent natural convection of air in a differentially heated vertical channel. The present data is validated using past DNS studies, and covers the Rayleigh number (Ra) range between 5.4 × 10^5 to 1.0 × 10^8. This is followed by an appraisal of various scaling laws proposed by four studies: Versteegh and Nieuwstadt (77), Holling and Herwig (34), Shiri and George (63) and George and Capp (23). These scaling laws are appraised with the profiles of the mean temperature defect, mean streamwise velocity, normal velocity fluctuations, temperature fluctuations and Reynolds shear stress. Based on the arguments of an inner (near-wall) and outer (channel-centre) region, the DNS data is found to support a −1/3 power law for the mean temperature in an overlap region. Using the inner and outer temperature profiles, an implicit heat transfer equation is obtained and a correction term in the equation is shown to be not negligible for the present Ra range when compared with explicit equations found in literature. In addition, I determined that the mean streamwise velocity and normal velocity fluctuations collapse in the inner region when using the outer velocity scale. A similar collapse is noted in the profiles of temperature fluctuations with increasing Ra when normalised with inner temperature and length scale. Lastly, I show evidence of an incipient proportional relationship between friction velocity and the outer velocity scale with increasing Ra. The study is extended to the spectrum of turbulent kinetic energy and temperature fluctuations of the flow. The one-dimensional streamwise spectra collapse onto the −5/3 slope, coinciding with the standard Kolmogorov form of the power spectra reported in literature. This collapse is found to occur in the outer region of the flow in the bounds between the peaks of the mean streamwise velocity. In spectrogram form, I find evidence that the spectral peaks correspond to energetic velocity structures in the channel — the structures of streamwise velocity fluctuations appear to stretch half of the streamwise domain and occur at a quarter intervals in the spanwise direction. From 2-dimensional autocorrelations, the structures of spanwise velocity fluctuations are found to be organised in a hatched pattern in an inner location (z^× i ≈ 7) and at the channel-centre. The respective pattern angles are \theta_ i ≈ 54◦ and \theta_ o ≈ 48◦, both measured from the horizontal. For the temperature spectrum, the −5/3 collapse is also observed in the same bounds as the velocity spectrum. In pre-multiplied form, the spectral peak is found to occur at the wall-normal location which coincides with the peak temperature fluctuations in the channel. With increasing Ra, the wall-parallel isocontours of temperature are found to show standard features of turbulent pressure driven boundary layers — streaks with spanwise length of 100+ units.