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The three-dimensional stability of two-dimensional natural convection flows in a heated, square enclosure inclined to the horizontal is investigated numerically. First, the computational procedure is validated by comparison of base flow solutions to results reported in literature across a range of inclinations. A bi-global linear stability analysis is then conducted to investigate the stability of these two-dimensional base flows to infinitesimal three-dimensional perturbations, and the effect that buoyancy forces (defined by a buoyancy number $R_N$) and enclosure inclination $\\theta$ have on these stability characteristics. The flow is first observed to become three-dimensionally unstable at buoyancy number $R_N = 213.8$ when $\\theta$ is $180^\\circ$; this increases to $R_N = 2.54 \\times 10^4$ at inclination $\\theta =58^\\circ$. It is found that the two-dimensional base flow is more unstable to three-dimensional perturbations with the critical $R_N$ corresponding to three-dimensional instability being significantly lower than its two-dimensional counterpart across all considered inclinations except $83^\\circ \\leq \\theta \\leq 88^\\circ$, where the most unstable mode is a two-dimensional oscillatory mode that develops in the boundary layers along the conducting walls. Eight different leading three-dimensional instability modes are identified, with inclinations $58^\\circ \\leq \\theta < 88^\\circ$ transitioning through an oscillatory mode, and inclinations $88^\\circ \\leq \\theta \\leq 180^\\circ$ transitioning through a stationary mode. The characteristics of the primary instability modes corresponding to inclinations $88^\\circ \\leq \\theta \\leq 179^\\circ$ indicate the presence of a Taylor–Görtler instability.

This work reports the application of a model-free deep reinforcement learning (DRL) based flow control strategy to suppress perturbations evolving in the one-dimensional linearised Kuramoto–Sivashinsky (KS) equation and two-dimensional boundary layer flows. The former is commonly used to model the disturbance developing in flat-plate boundary layer flows. These flow systems are convectively unstable, being able to amplify the upstream disturbance, and are thus difficult to control. The control action is implemented through a volumetric force at a fixed position, and the control performance is evaluated by the reduction of perturbation amplitude downstream. We first demonstrate the effectiveness of the DRL-based control in the KS system subjected to a random upstream noise. The amplitude of perturbation monitored downstream is reduced significantly, and the learnt policy is shown to be robust to both measurement and external noise. One of our focuses is to place sensors optimally in the DRL control using the gradient-free particle swarm optimisation algorithm. After the optimisation process for different numbers of sensors, a specific eight-sensor placement is found to yield the best control performance. The optimised sensor placement in the KS equation is applied directly to control two-dimensional Blasius boundary layer flows, and can efficiently reduce the downstream perturbation energy. Via flow analyses, the control mechanism found by DRL is the opposition control. Besides, it is found that when the flow instability information is embedded in the reward function of DRL to penalise the instability, the control performance can be further improved in this convectively unstable flow.

The results of an experimental investigation of smooth-body adverse pressure gradient (APG) turbulent boundary layer flow separation and reattachment over a two-dimensional ramp are presented. These results are part of a larger archival smooth-body flow separation data set acquired in partnership with NASA Langley Research Center and archived on the NASA Turbulence Modeling Resource website. The experimental geometry provides initial canonical turbulent boundary layer growth under nominally zero pressure gradient conditions prior to encountering a smooth, two-dimensional, backward facing ramp geometry onto which a streamwise APG that is fully adjustable is imposed. Detailed surface and off-surface flow field measurements are used to fully characterize the smooth-body APG turbulent boundary layer separation and reattachment at multiple spanwise locations over the ramp geometry. Unsteady aspects of the flow separation are characterized. It is shown that the first and second spatial derivatives of the streamwise static surface pressure profile are sufficient to determine key detachment and reattachment locations. The imposed streamwise APG gives rise to inflectional mean velocity profiles and the associated formation of an embedded shear layer, which is shown to play a dominant role in the subsequent flow development. Similarity scaling is developed for both the mean velocity and turbulent stresses that is found to provide self-similar collapse of profiles for different regions of the ramp flow. Despite the highly non-equilibrium flow environment, a new similarity scaling proved capable of providing self-similar turbulent stress profiles over the full streamwise extent of flow separation and downstream reattachment.

The effects of different geometries of two-dimensional (2-D) roughness elements in a zero pressure gradient (ZPG) turbulent boundary layer (TBL) on turbulence statistics and drag coefficient are assessed using single hot-wire anemometry. Three kinds of 2-D roughness are used: (i) circular rods with two different heights, $k= 1.6$ and 2.4 mm, and five different streamwise spacing of $s_{x}= 6k$ to $24k$, (ii) three-dimensional (3-D) printed triangular ribs with heights of $k= 1.6$ mm and spacing of $s_{x}= 8k$ and (iii) computerized numerical control (CNC) machined sinewave surfaces with two different heights, $k= 1.6$ and 2.4 mm, and spacing of $s_{x}= 8k$. These roughnesses cover a wide range of ratios of the boundary layer thickness to the roughness height ($23 < \\delta /k < 41$), where $\\delta$ is the boundary layer thickness. All roughnesses cause a downward shift on the wall-unit normalised streamwise mean velocity profile when compared with the smooth wall profiles agreeing with the literature, with a maximum downward shift observed for $s_{x}= 8k$. In the fully rough regime, the drag coefficient becomes independent of the Reynolds number. Changing the roughness height while maintaining the same spacing ratio $s_{x}/k$ exhibits little influence on the drag coefficient in the fully rough regime. On the other hand, the effective slope $(ES)$ and the height skewness $(k_{sk})$ appear to be major surface roughness parameters that affect the drag coefficient. These parameters are used in a new expression for $k_{s}$, the equivalent sand grain roughness, developed for 2-D uniformly distributed roughness in the fully rough regime.

Turbulent boundary layers on immersed objects can be significantly altered by the pressure gradients imposed by the flow outside the boundary layer. The interaction of turbulence and pressure gradients can lead to complex phenomena such as relaminarization, history effects and flow separation. The angular momentum integral (AMI) equation (Elnahhas & Johnson, J. Fluid Mech., vol. 940, 2022, A36) is extended and applied to high-fidelity simulation datasets of non-zero pressure gradient turbulent boundary layers. The AMI equation provides an exact mathematical equation for quantifying how turbulence, free-stream pressure gradients and other effects alter the skin friction coefficient relative to a baseline laminar boundary layer solution. The datasets explored include flat-plate boundary layers with nearly constant adverse pressure gradients, a boundary layer over the suction surface of a two-dimensional NACA 4412 airfoil and flow over a two-dimensional Gaussian bump. Application of the AMI equation to these datasets maps out the similarities and differences in how boundary layers interact with favourable and adverse pressure gradients in various scenarios. Further, the fractional contribution of the pressure gradient to skin friction attenuation in adverse-pressure-gradient boundary layers appears in the AMI equation as a new Clauser-like parameter with some advantages for understanding similarities and differences related to upstream history effects. The results highlight the applicability of the integral-based analysis to provide quantitative, interpretable assessments of complex boundary layer physics.

By incorporating the traditionally overlooked advective term in the wall-normal momentum equation, a new momentum integral equation is developed for two-dimensional incompressible turbulent boundary layers under arbitrary pressure gradients. The classical Kármán's integral arises as a special instance of the new momentum integral equation when the pressure gradient is weak. The new momentum integral equation's validity is substantiated by direct numerical simulation data. Unlike the classical Kármán's integral, which is limited to predicting wall shear stress within mild pressure gradients, the new momentum integral equation accurately computes wall shear stress across a broad range of pressure gradients, even in the presence of strong adverse pressure gradients that lead to flow separation. Moreover, a new pressure parameter $\\beta _\\kappa$ is introduced through examining terms in the new momentum integral equation. This parameter naturally quantifies the pressure gradient's influence on turbulent boundary layers and offers guidance for applying the classical Kármán's integral. Additionally, to facilitate experimental determination of wall shear stress under strong pressure gradients, an approximate integral equation is proposed that relies solely on easily measurable variables. Validation against direct numerical simulation data demonstrates that this simplified equation provides reasonably accurate estimates of wall shear stress in turbulent boundary layers experiencing strong pressure gradients.

In this paper, we conduct a systematic study of the instability of a boundary layer over a rotating cone that is inserting into a supersonic stream with zero angle of attack. The base flow is obtained by solving the compressible boundary-layer equations using a marching scheme, whose accuracy is confirmed by comparing with the full Navier–Stokes solution. Setting the oncoming Mach number and the semi-apex angle to be 3 and 7$^\\circ$, respectively, the instability characteristics for different rotating rates ($\\bar \\varOmega$, defined as the ratio of the rotating speed of the cone to the axial velocity) and Reynolds numbers ($R$) are revealed. For a rather weak rotation, $\\bar \\varOmega \\ll 1$, only the modified Mack mode (MMM) exists, which is an extension of the supersonic Mack mode in a quasi-two-dimensional boundary layer to a rotation configuration. Further increase of $\\bar \\varOmega$ leads to the appearance of a cross-flow mode (CFM), coexisting with the MMM but in the quasi-zero frequency band. The unstable zones of the MMM and CFM merge together, and so they are referred to as the type-I instability. When $\\bar \\varOmega$ is increased to an $O(1)$ level, an additional unstable zone emerges, which is referred to as the type-II instability to be distinguished from the aforementioned type-I instability. The type-II instability appears as a centrifugal mode (CM) when $R$ is less than a certain value, but appears as a new CFM for higher Reynolds numbers. The unstable zone of the type-II CM enlarges as $\\bar \\varOmega$ increases. The vortex structures of these types of instability modes are compared, and their large-$R$ behaviours are also discussed.

Three-dimensional eddy-resolving simulations are used to study the structure of turbulent boundary layers developing in an open channel where an array of sparse roughness elements in the form of partially burrowed mussels is placed on the smooth, flat channel bed starting at a certain streamwise location. The identical mussels are oriented with their major axis parallel to the mean flow. Their positions are randomised while making sure that the mussels are close to uniformly distributed inside the array. The turbulent flow approaching the leading edge of the array is fully developed. The protruding parts of the mussels play the role of sparse roughness elements and generate a rough-bed, internal boundary that is characterised by zero net flow exchange but non-zero local flow exchange due to active filtering. A double-averaging technique is used to obtain an equivalent, width- and time-averaged, boundary layer over a ‘flat’ rough surface containing no mussels. The paper discusses the effects of varying the mussel array density, protruding height of the mussels and filtering discharge on the spatial growth of the two-dimensional boundary layer. With proper scaling, the profiles of the (double-averaged) streamwise velocity are close to self-similar inside the inertial layer (e.g. for h $\\lt$ z $\\lt$ $\\delta$ , where h is the height of the protruding part of the mussels and $\\delta$ is the boundary layer thickness) starting some distance from the leading edge of the array. The scaled turbulent kinetic energy and concentration profiles associated with the scalar advected through the excurrent syphons are also found to be self-similar above the vertical location where the maximum value is reached. An analytical model containing three subzones is proposed for the streamwise velocity in between the bed (z $=$ 0) and z $=$ $\\delta$ . The velocity profile inside the inertial region contains a law-of-the-wall component supplemented by a law-of-the-wake component. The scaling coefficient of the law-of-the-wake component is found to be larger than typical values used to describe velocity variation in turbulent boundary layers developing in a surrounding flow with close-to-uniform free stream velocity. The equivalent roughness height for this particular type of boundary layers developing over sparse roughness elements increases monotonically with h and the mussel array density, $\\rho$ N . The paper also discusses the effect of the mussel bed density on the average refiltration fraction and the phytoplankton removal efficiency of the mussel bed.

This work presents the first experimental characterization of the flow field in the vicinity of periodically spaced discrete roughness elements (DRE) in a swept wing boundary layer. The time-averaged velocity fields are acquired in a volumetric domain by high-resolution dual-pulse tomographic particle tracking velocimetry. Investigation of the stationary flow topology indicates that the near-element flow region is dominated by high- and low-speed streaks. The boundary layer spectral content is inferred by spatial fast Fourier transform (FFT) analysis of the spanwise velocity signal, characterizing the chordwise behaviour of individual disturbance modes. The two signature features of transient growth, namely algebraic growth and exponential decay, are identified in the chordwise evolution of the disturbance energy associated with higher harmonics of the primary stationary mode. A transient decay process is instead identified in the near-wake region just aft of each DRE, similar to the wake relaxation effect previously observed in two-dimensional boundary layer flows. The transient decay regime is found to condition the onset and initial amplitude of modal crossflow instabilities. Within the critical DRE amplitude range (i.e. affecting boundary layer transition without causing flow tripping) the transient disturbances are strongly receptive to the spanwise spacing and diameter of the elements, which drive the modal energy distribution within the spatial spectra. In the super-critical amplitude forcing (i.e. causing flow tripping) the near-element stationary flow topology is dominated by the development of a high-speed and strongly fluctuating region closely aligned with the DRE wake. Therefore, elevated shears and unsteady disturbances affect the near-element flow development. Combined with the harmonic modes transient growth these instabilities initiate a laminar streak structure breakdown and a bypass transition process.

The dynamics of a shock-induced separation unit generated by a 20$^\\circ$ sharp fin placed on a cylindrical surface in a Mach 2.5 flow was investigated. Specifically, the present work investigated the mechanisms that govern the mid-frequency range of separation shock unsteadiness in the fin shock wave–boundary layer interaction (SBLI) unit. Two-dimensional pressure fields were obtained over the cylinder surface spanning the entire fin SBLI unit using high-bandwidth pressure-sensitive paint at 40 kHz imaging rate that allowed probing the low- through mid-frequency ranges of the separation shock unsteadiness. The mean pressure field showed a progressive weakening of the separation shock with downstream distance, which is an artifact of the three-dimensional relief offered by the curved mounting surface. The root-mean-square (r.m.s.) pressure field exhibited a banded structure with elevated $p_{r.m.s.}$ levels beneath the intermittent region, separation vortex and adjacent to the fin root. The power spectral density (PSD) of the surface pressure fluctuations obtained beneath the intermittent region revealed that the separation shock oscillations exhibited the mid-frequency content over the majority of its length. Interestingly, neither the PSD nor the length of the intermittent region varied noticeably with downstream distance, revealing a constant separation shock foot velocity along the entire SBLI. The pressure fluctuation PSD beneath the separation vortex also exhibited the broadband peak at the mid-frequency range of the separation shock motions over the majority of its length within the measurement domain. By contrast, the region adjacent to the fin root exhibited pressure oscillations at a substantially lower frequency compared with the separation shock and the separation vortex. Two-point coherence and cross-correlation analysis provided unique insights into the critical sources and mechanisms that drive the separation shock unsteadiness. The separation vortex and separation shock dynamics were found to be driven by a combination of convecting perturbations that originated from the vicinity of the fin leading edge and the local interactions of the separated flow with the incoming boundary layer. The boundary layer locally strengthened or weakened the convecting pressure perturbations depending on the local momentum fluctuations within the boundary layer.