# Direct numerical simulation and global stability analysis of boundary layer flows past roughness elements

2022-08

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Direct numerical simulation and global stability analysis of boundary layer flows past roughness elements

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2022-08

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Understanding wall roughness effects on flows is important in engineering applications. The objective of this dissertation is to increase understanding of roughness effects on transitional and turbulent boundary layers. The rough-wall flow is studied by performing direct numerical simulations (DNS) of the Navier-Stokes equations and global linear stability analyses. DNS of turbulent channel flow with a random-rough bottom wall is performed at friction Reynolds number $Re_{\tau}=400$ and $600$. The rough surface corresponds to the experiments of Flack et al. \cite{flack2019skin}. The computed skin friction coefficients and the roughness functions show good agreement with experimental results. The double-averaging methodology is used to investigate mean velocity, Reynolds stresses, dispersive flux, and mean momentum balance. The roll-up of the shear layer on the roughness crests is identified as a primary contributor to the wall-normal momentum transfer. The mean-square pressure fluctuations are increased within the inner layer and a phenomenological explanation is suggested by examining the dominant source terms in the pressure Poisson equation. The rapid term shows that high pressure fluctuations observed in front of and above the roughness elements are mainly due to the attached shear layer formed above the protrusions. The slow term makes a relatively smaller contribution, and is primarily increased in the troughs and in front of the roughness elements, corresponding to the occurrence of quasi-streamwise vortices and secondary vortical structures. The mean wall shear on the rough surface is highly correlated with the roughness height, and depends on the local roughness topography. Events with comparable magnitudes of the streamwise and spanwise wall-shear stress occur more frequently, corresponding to a more isotropic vorticity field in the roughness layer. The boundary layer transition due to an isolated roughness element is studied using global stability analysis and DNS. The large ratio of element height to displacement boundary layer thickness $(h/\delta^*)$ is considered to model a trip at an early location in the boundary layer. The cubical trip geometries with two aspect ratios ($\eta$) are investigated. Both steady base flows and time-averaged mean flows are able to capture the frequencies of the primary vortical structures and mode shapes. Global stability results highlight that although the varicose instability is dominant for large $h/\delta^*$, sinuous instability becomes more pronounced as $Re_h$ increases for the thin geometry ($\eta=0.5$), due to increased spanwise shear in the near-wake region. Wavemaker results indicate that $\eta$ affects the convective nature of the shear layer more than the type of instability. DNS results show that different instability mechanisms lead to different development and evolution of vortical structures in the transition process. For $\eta=1$, the varicose instability is associated with the periodic shedding of hairpin vortices, and its stronger spatial transient growth indicated by wavemaker results aids the formation of hairpin vortices farther downstream. In contrast, for $\eta=0.5$, the interplay between varicose and sinuous instabilities results in a broader-banded energy spectrum and leads to the sinuous wiggling of hairpin vortices in the near wake when $Re_h$ is sufficiently high. A sinuous mode with a lower frequency captured by dynamic mode decomposition (DMD) analysis, and associated with the ‘wiggling’ of streaks persists far downstream and promotes transition to turbulence. A new regime map is developed to classify and predict instability mechanisms based on $Re_{hh}^{1/2}$ and $d/\delta^*$ using a logistic regression model. Although the mean skin friction demonstrates different evolution for the two geometries, both of the two geometries efficiently trip the flow to turbulence at $Re_h=1100$. An earlier location of a fully-developed turbulent state is established for $\eta=1$ at $x \approx 110h$. The influence of roughness spacing on boundary layer transition over distributed roughness elements is studied using direct numerical simulation and global stability analysis, and compared to the isolated roughness element. Small spanwise spacing $\lambda_z=2.5h$ inhibits the formation of counter-rotating vortices (CVP), as a result, the hairpin vortices are not generated and the downstream shear layer is steady. For $\lambda_z=5h$, the CVP and hairpin vortices are induced by the first row of roughness, perturbing the downstream shear layer and causing transition. Although the periodicity of the primary hairpin vortices seems to be independent on the streamwise spacing, the distributed roughness leads to lower critical $Re_h$ for instability to occur and more significant breakdown of boundary layer compared to the isolated roughness. When the streamwise spacing is comparable to the region of flow separation ($\lambda_x=5h$), the high-momentum fluid hardly moves downward into the cavities and the wake flow has little impact on the following roughness elements. The leading unstable varicose mode is associated with the central low-speed streaks along the aligned roughness elements, and its frequency is close to the shedding frequency of hairpin vortices in the isolated case. For larger streamwise spacing ($\lambda_x=10h$), two distinct modes are obtained from global stability analysis. The first mode shows varicose symmetry, corresponding to the primary hairpin vortex shedding induced by the first-row roughness. The high-speed streaks formed in the longitudinal grooves are also found to be unstable and interacting with the varicose mode. The second mode is a sinuous mode with lower frequency, induced as the wake flow of the first-row roughness runs onto the second row. It extracts most energy from the spanwise shear between the high- and low-speed streaks.

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University of Minnesota Ph.D. dissertation. 2022. Major: Aerospace Engineering and Mechanics. Advisor: Krishnan Mahesh. 1 computer file (PDF); 151 pages.

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Ma, Rong. (2022). Direct numerical simulation and global stability analysis of boundary layer flows past roughness elements. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/243148.

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