Direct numerical simulation of a turbulent boundary layer under spatially varying pressure gradients

Abstract

A comprehensive understanding of turbulent boundary layers (TBLs), particularly those subjected to non-equilibrium conditions such as varying streamwise pressure gradients, presents a significant challenge in fluid mechanics. Such flows are not only of academic interest but also commonly encountered, playing a critical role in numerous practical engineering applications, spanning from the external aerodynamics of aerospace and ground vehicles to internal flows within turbo-machinery components, heat exchangers, and diffusers. The dynamic behavior of these TBLs is significantly influenced by the interplay of mean shear, the accelerating (favorable pressure gradient, FPG) or decelerating (adverse pressure gradient, APG) effects of pressure forces, and often the persistent influence of upstream flow history. Conventional turbulence models, which are frequently developed and calibrated based on idealized equilibrium flow assumptions, often struggle to accurately capture the intricate physics of these non-equilibrium dynamics and the effects of such spatially varying pressure gradients. This thesis aims to deepen our understanding of TBLs developing under such spatially varying non-equilibrium conditions, specifically focusing on a TBL subjected to an APG followed by an FPG region. The core objective of this thesis is to elucidate how the interplay of local mean shear, varying pressure gradients, and flow history governs the behavior of Reynolds-shear-stress-carrying coherent structures, and thereby to characterize the evolution of the mean flow and turbulence statistics in this non-equilibrium TBL. For this purpose, a large-scale direct numerical simulation (DNS) database of a TBL subjected to an APG followed by a FPG region was thoroughly analyzed. This primary database, with $Re_\theta$ reaching up to 13,000 and shape factor ($H$) values spanning from approximately 1.4 (small-velocity defect) to regions exceeding 2.8 (large-velocity defect), serves as the main foundation for the investigations presented. DNS was employed as the primary research tool, being the only method capable of resolving the full range of turbulent scales without modeling assumptions, thus providing high-fidelity data essential for fundamental understanding of the complex flow physics. This method is uniquely capable of providing highly accurate and detailed data extending down to the wall, which is crucial for fundamental understanding of the flow and for the development and validation of turbulence models. The DNS was performed using the in-house TBLDNS code. Its strong scalability was evaluated on various high performance computer systems, demonstrating efficient scaling up to 14336 cores. To provide a broader context and to isolate specific physical effects, data from canonical zero-pressure-gradient (ZPG) TBLs and homogeneous shear turbulence (HST) databases from the literature were also employed for comparative purposes. These comparisons were particularly valuable for isolating the effects of wall-proximity, varying pressure gradients, and pure shear from the more complex interactions present in the primary APG/FPG TBL. % These flows are commonly encountered in numerous engineering applications, including aerospace vehicles to turbomachinery, and their complex dynamics are significantly affected by the interplay of mean shear, pressure gradients, and upstream flow history. Conventional turbulence models often struggle to accurately predict these non-equilibrium dynamics and pressure gradient effects. This thesis aims to deepen our understanding of turbulent boundary layers developing under spatially varying, non-equilibrium pressure gradients, specifically focusing on a TBL subjected to an adverse pressure gradient (APG) followed by a favorable pressure gradient (FPG) region, and the persistent influence of upstream flow history. The core objectives are to characterize the evolution of mean flow and turbulence statistics, to investigate the behavior of Reynolds-shear-stress carrying coherent structures, and to elucidate how local mean shear, pressure gradients, and history effects collectively govern their properties and organization. For this purpose, a large-scale direct numerical simulation (DNS) database of a TBL subjected to an APG followed by a FPG region was thoroughly analyzed. This primary database, with $Re_\theta$ reaching up to 13,000 serves as the main foundation for the investigations presented. For comparative purposes data from canonical zero-pressure-gradient (ZPG) TBLs and homogeneous shear turbulence (HST) databases from the literature were also employed to isolate the effects of wall, pressure gradients, and pure shear. The investigation into the fundamental flow physics first addressed the critical issue of defining the boundary layer edge in such complex non-equilibrium flows. A comparative evaluation of four distinct methods for determining the boundary layer thickness confirmed that the local reconstruction method provided the most consistent and physically robust results across the varying pressure gradient conditions, particularly in the strong APG region where traditional methods proved less reliable. The evolution of turbulence anisotropy was then examined using Anisotropy Invariant Maps (AIM) and Barycentric Maps. These analyses revealed that while turbulence originates near the wall in a state close two-component turbulence, its trajectory through the anisotropy map is significantly affected by the local pressure gradient and, importantly, by the upstream flow history. In the large-defect APG region, a pronounced shift towards axisymmetric contraction was observed, highlighting the strong influence of the APG. However, even in the subsequent FPG region, the turbulence state did not fully recover to ZPG characteristics or achieve full isotropy in the outer layer. Instead, it retained distinct features from its prior APG development, underscoring the persistent nature of history effects. The main focus of the thesis was the detailed characterization of Reynolds-shear-stress carrying coherent structures, specifically ejections (Q2 events) and sweeps (Q4 events), identified using quadrant analysis on velocity fields. The study first quantified their contributions to the total Reynolds shear stress, showing how the balance between attached and detached structures, as well as the relative contributions of different quadrant events, shifted significantly across the investigated APG and FPG regions. For instance, detached structures and, surprisingly, Q1 (outward interaction) and Q3 (inward interaction) events gained increased importance in the near-wall region of the strong APG case, indicating a significant alteration of turbulence production mechanisms. Conversely, in the FPG region, attached structures tended to dominate throughout a larger portion of the boundary layer. The geometric properties of detached Reynolds-shear-stress carrying structures in the outer layer ($0.3 < y/\delta < 0.8$) were then analyzed using both fully-spatial and spatio-temporal data. Notably, when the characteristic size of these structures, represented by the diagonal of their circumscribing box ($d$), was normalized by the local Corrsin length scale ($L_c$), a significant trend was observed. The average aspect ratios (streamwise-to-wall-normal, $a_{xy}$, and wall-normal-to-spanwise $a_{zy}$) for the combined Q2 and Q4 structures collapsed onto nearly universal curves across all turbulent boundary layer regions examined (APG with small defect, APG with large defect, and FPG with APG history), and also aligned closely with those observed in HST flows. This robust collapse, especially for structures of moderate size (with normalized diagonal sizes, $d/L_c$, approximately between 2 and 10), indicates that the local mean shear, effectively represented by $L_c$, is the primary factor governing the shape and relative dimensions of these Reynolds-shear-stress carrying structures, largely independent of the pressure gradient condition or its upstream flow history. While the wall-normal-to-spanwise aspect ratio ($a_{zy}$) exhibited almost identical trends for both Q2 and Q4 structures across all cases, some subtle differences were observed in the streamwise-to-wall-normal aspect ratio ($a_{xy}$), particularly in the spatio-temporal analysis of the large-defect APG region. These differences may point to limitations in the application of Taylor's frozen turbulence hypothesis under conditions of strong deceleration. Finally, the investigation into the spatial organization of Q2 and Q4 structures within a wall-parallel plane, yielded highly consistent results across all analyzed flow regions. Structures of the same type (Q2-Q2 or Q4-Q4) invariably showed a preferential alignment in the upstream-downstream direction, while structures of different types (Q2-Q4 or Q4-Q2) predominantly arranged themselves side-by-side in a one-sided pair configuration. The robustness of this organizational pattern, seemingly independent of the pressure gradient or flow history within the outer layer. The overall conclusion is that while pressure gradients and flow history profoundly alter the mean flow, velocity defect, and macroscopic turbulence statistics, the fundamental geometry of detached, shear-driven coherent structures in the outer layer, when scaled by Lc, exhibits universal characteristics across different APG strengths and FPG recovery stages. Similarly, their local spatial organization appears governed by universal shear-driven mechanisms. The significant changes in macroscopic turbulence statistics are likely a consequence of how pressure gradients modify the mean shear profile, which in turn dictates the intensity, scale, and prevalence of these fundamental coherent structures.