Canonical Boundary Research Articles

Computing the minimal perimeter polygon for digital objects in the triangular tiling

This work presents an algorithm, together with its correctness proof, to determine the minimum perimeter polygon (MPP) for digital objects given as regular complexes in the triangular plane tiling. Such objects are edge-adjacency-connected sets of triangle tiles that have no end tiles, and the point set union of all their tiles forms a simple polygon. Nevertheless, the boundary paths of the objects are not assumed to be simple. Then the MPP is a weakly simple polygon that coincides with the relative convex hull (i.e., geodesic hull) of a set A with respect to a simple polygon B, where A⊂B, but A is not necessarily a polygon, in fact it is generally not connected. Our MPP algorithm relies on constructing and iteratively constraining cones of visibility through forthcoming boundary tiles, it uses the structure of the canonical boundary path, the MPP frontier is the shortest polygonal curve following this path. We also propose a boundary tracing algorithm to obtain such paths from the objects.

Experimental characterization and similarity scaling of smooth-body flow separation and reattachment on a two-dimensional ramp geometry

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.

Dependence of eigenvalues of fourth-order Sturm-Liouville problems on canonical boundary conditions

In this paper, we discuss the dependence of eigenvalues of fourth-order Sturm-Liouville (SL) problems on the fundamental canonical forms of separated, coupled and mixed self-adjoint boundary conditions. The eigenvalues depend not only continuously but smoothly on the boundary conditions and boundary points. Simultaneously, the derivative expression of the eigenvalues with respect to all the parameters in the boundary conditions are given, and thus its monotonicity with respect to some parameters is derived.

The connectivity-based architecture of the human piriform cortex

The anatomy of the human piriform cortex (PC) is poorly understood. We used a bimodal connectivity-based-parcellation approach to investigate subregions of the PC and its connectional differentiation from the amygdala.One hundred (55 % female) genetically unrelated subjects from the Human Connectome Project were included. A region of interest (ROI) was delineated bilaterally covering PC and amygdala, and functional and structural connectivity of this ROI with the whole gray matter was computed. Spectral clustering was performed to obtain bilateral parcellations at granularities of k = 2–10 clusters and combined bimodal parcellations were computed. Validity of parcellations was assessed via their mean individual-to-group similarity per adjusted rand index (ARI).Individual-to-group similarity was higher than chance in both modalities and in all clustering solutions. The amygdala was clearly distinguished from PC in structural parcellations, and olfactory amygdala was connectionally more similar to amygdala than to PC. At higher granularities, an anterior and ventrotemporal and a posterior frontal cluster emerged within PC, as well as an additional temporal cluster at their boundary. Functional parcellations also showed a frontal piriform cluster, and similar temporal clusters were observed with less consistency. Results from bimodal parcellations were similar to the structural parcellations. Consistent results were obtained in a validation cohort.Distinction of the human PC from the amygdala, including its olfactory subregions, is possible based on its structural connectivity alone. The canonical fronto-temporal boundary within PC was reproduced in both modalities and with consistency. All obtained parcellations are freely available.

Wall Shear Stress Characteristics Of A Turbulent Boundary Layer Obtained With Multi-Aperture Defocusing Micro-PTV And High-Speed Stereo Profile-PIV

A multi-aperture 3d 3c micro PTV system is presented which relies on single window optical access for both high-speed tracer illumination and image recording. Similar to the “defocusing” concept described by C. Willert & Gharib (1992), the wall distance of individual particles is obtained from the size of projected particle image triplets formed by a triplet of apertures on the entrance pupil of the microscope lens. The present article extends upon previously published material Klinner & Willert (2022) by applying multi-aperture micro particle tracking velocimetry (MA-micro PTV) on a canonical turbulent boundary layer (TBL) to track the near-wall motion of tracer particles with the aim of estimating the unsteady wall-shear stress from particle tracks. The technique is validated with measurements of a TBL inside the closed test section of the 1 m wind tunnel of DLR in Göttingen (1mWK) at free- stream velocities of 5.2 to 20 m/s with corresponding shear Reynolds numbers between 560 and 1630. In the viscous sublayer, the probability density distributions of stream- and span-wise wall shear stress (WSS) components could be reliably captured down to probability densities of 10E−3. As far as we know, this has not been achieved in the past, particularly not for the span-wise component. For the stream-wise component the measured Reynolds-number dependency of the root mean squared fluctuations agrees to correlations by Örlü & Schlatter (2011). Furthermore, the skewness of the stream-wise WSS agrees well to values from DNS. On the other hand, the span-wise WSS fluctuations consistently underestimate the predicted DNS values, for which it is assumed that the deviation can be explained by the rapid decrease of the span-wise velocity fluctuations with increasing wall distance. Wall-normal profiles of 3c velocity statistics were obtained by bin averaging of particle velocities. Up into the buffer layer, these profiles agree well with profiles from stereo PIV as well as from DNS and LES, indicating an accurate determination of both near-wall velocities and inner scaling.

Hypersonic turbulent boundary layer over the windward side of a lifting body

In the present study, we performed direct numerical simulations for a hypersonic turbulent boundary layer over the windward side of a lifting body, the HyTRV model, at Mach number $6$ and attack angle 2 $^{\circ }$ to investigate the global and local turbulent features, and evaluate its difference from canonical turbulent boundary layers. By scrutinizing the instantaneous and averaged flow fields, we found that the transverse curvature on the windward side of the HyTRV model induces the transverse opposing pressure gradients that push the flow on both sides towards the windward symmetry plane, yielding significant effects of the azimuthal inhomogeneity and large-scale cross-stream circulations, moderate and azimuthal independent influences of adverse pressure gradient, and negligible impact of the mean flow three-dimensionality. Further inspecting the local turbulent statistics, we identified that the mean and fluctuating velocity become increasingly similar to the highly decelerated turbulent boundary layers over flat plates in that the mean velocity deficit is enhanced, and the outer layer Reynolds stresses are amplified as it approaches the windward symmetry plane, and prove to be self-similar under the scaling of Wei & Knopp (J. Fluid Mech., vol. 958, 2023, A9) for adverse-pressure-gradient turbulent boundary layers. Conditionally averaged Reynolds stresses based on strong sweeping and ejection events demonstrated that the Kelvin–Helmholtz instability of the strong embedded shear layer induced by the large-scale cross-stream circulations is responsible for the turbulence amplification in the outer layer. The strong Reynolds analogy that relates the mean velocity and temperature was refined to incorporate the non-canonical effects, showing considerable improvements in the accuracy of such a formula. On the other hand, the temperature fluctuations are still transported passively, as indicated by their resemblance to the velocity. The conclusions obtained in the present study provide potentially profitable information for turbulent modelling modification for the accurate predictions of skin friction and wall heat transfer.

An improved Baldwin–Lomax algebraic wall model for high-speed canonical turbulent boundary layers using established scalings

In this work, we employ well-established relations for compressible turbulent mean flows, including the velocity transformation and algebraic temperature–velocity (TV) relation, to systematically improve the algebraic Baldwin–Lomax (BL) wall model for high-speed zero-pressure-gradient air boundary layers. Any new functions or coefficients fitted by ourselves are avoided. Twelve published direct numerical simulation (DNS) datasets are employed for a priori inspiration and a posteriori examination, with Mach numbers up to 14 under adiabatic, cold and heated walls. The baseline BL model is the widely used one with semilocal scalings. Three targeted modifications are made. First, we employ a total-stress-based transformation (Griffin et al., Proc. Natl Acad. Sci. USA, vol. 118, issue 34, 2021, e2111144118) to the inner-layer eddy viscosity for improved scaling up to the logarithmic region. Second, we utilize the van Driest transformation in the outer layer based on the compressible defect velocity scaling. Third, considering the difficulty in modelling the rapidly varying and singular turbulent Prandtl number near the temperature peak in cold-wall cases, we design a two-layer strategy and use the TV relation to formulate the inner-layer temperature. Numerical results prove that the modifications take effect as designed. The prediction accuracy for mean streamwise velocity is notably improved for diabatic cases, especially in the logarithmic region. Moreover, a significant improvement in mean temperature is realized for both adiabatic and diabatic cases. The mean relative errors of temperature to DNS for all cases are down to 0.4 % in the logarithmic wall-normal coordinate and 3.4 % in the outer coordinate, around one-third of those in the baseline model.

Characteristics of turbulent boundary layers generated by different tripping devices

In this study, we employ a high-fidelity flow visualisation technique to investigate the resemblance between a turbulent boundary layer generated in a lab environment with a limited development length by different tripping devices and a naturally developed canonical turbulent boundary layer. Furthermore, we aim to provide statistics of the wall-normal velocity fluctuation of the tripped boundary layers and analyse them compared with the canonical turbulent boundary layer. This is achieved by conducting two-dimensional particle image velocimetry measurements to acquire the velocity components in the stream-wise wall-normal plane downstream of the tripping location. Different criteria including integral parameters, diagnostic plot, stream-wise and wall-normal turbulence intensities, and mean velocity deficit profiles are employed to compare the boundary layer profiles generated by different trips with well-behaved turbulent boundary layers presented in the literature. The generation of Reynolds shear stress in tripped boundary layers is analysed using quadrant analysis, and suitably sized trips are shown to have contributions of sweeps and ejections in a similar manner to a well-behaved turbulent boundary layer. Furthermore, through the detection of uniform momentum zones, the presence of coherent structures in the logarithmic and outer layers of these well-behaved tripped boundary layers is shown. The findings of this study identify the blockage created by trips as the main factor affecting the turbulence statistics at a certain downstream distance. The results also suggest that the investigation of inner-scaled mean stream-wise velocity is recommended as a simple yet reliable and rapid assessment of the impact of a trip on the turbulent boundary layer development in short test sections.

Scaling high Rayleigh number natural convection boundary layer statistics: A vertical water tunnel experiment

A high Rayleigh number natural convection boundary layer adjacent to the vertical heated wall was investigated at a large-scale facility. The global Rayleigh number (Rax) measured by the temperature difference between the wall and ambient water and the distance from the bottom of the heated wall reached 1013. Experimental results confirm that the global Nusselt number Nux is scaled by power 1/3 of Rax, which is similar to the well-known asymptote of the previously achieved Rax. The velocity field obtained using particle image velocimetry (PIV) indicated that the buoyancy-dominant outer-layer scaling suggested by Wells and Worster [A geophysical-scale model of vertical natural convection boundary layers, J. Fluid Mech. 609, 111–137 (2008)] was not only applicable to scale the lower-order velocity statistics but was also valid as a reasonable measure of the spatial correlation, probability distribution, and quadrant contribution features in the outer layer. The dynamical behavior of fluid motion captured by PIV supported a robust momentum transfer to positive wall-normal direction, which was sustained by the Q1 and Q3 quadrants. In addition, merging the existing literature and current data suggested that near-wall function can be applied at a moderate Rax; the universality of this wall-function model was confirmed around z×∼0.7 (where z× represents the near-wall scaled wall-normal distance, Kiš and Herwig [The near wall physics and wall functions for turbulent natural convection, Int. J. Heat Mass Transfer 55, 2625–2635 (2012)]). At a larger buoyancy regime, it was expected to follow a canonical boundary layer flow.

Modelling the Wall Pressure Fluctuations on the VEGA-C Launcher in Supersonic Conditions

The prediction of pressure fluctuations generated over the external surface of aerospace launchers during the atmospheric flight remains a challenging task due to the complexity of the geometry and the effects of compressibility at high supersonic Mach numbers. An experimental database is here analysed to the scope of providing a procedure to model and predict the relevant statistics of the wall pressure fluctuations generated by a supersonic flow overflowing the VEGA-C Launcher Vehicle. Data have been obtained in an extensive Wind Tunnel test campaign carried out in the trisonic wind tunnel of the National Institute for Aerospace Research (INCAS) in Bucharest. Wall-mounted pressure transducers allowed for the computation of the pressure Auto- and Cross-spectra over the fourth (the payload region) and third stages of the launcher model. Coherence functions are modelled through exponential-like analytical functions following the approaches usually adopted in canonical boundary layer flows, whereas the auto-spectra models are based on polynomial fits. The approach adopted for the achievement of proper non-dimensional quantities as well as the procedure implemented for the full-scale extrapolation is presented and discussed.

Spatial evolution of the wake of discrete roughness elements in a turbulent boundary layer at a moderate Reynolds number

In this study, the effect of cylindrical discrete roughness elements on multiscale turbulent motions in a canonical turbulent boundary layer was explored. Flow fields with a considerably large field of view in streamwise—wall-normal planes—were measured by two-dimensional particle image velocimetry. It was found that large-scale motions and very-large-scale motions in the outer layer of the middle-to-far-wake regime are enhanced due to the bottom-up process of roughness-introduced disturbance, where vortical structures shedding from the roughness elements lift up to higher flow layers as they convect downstream, thus promoting the cross-layer interaction. The modulation effect was found to extend to the lower bound of the outer region and persist longer than traditionally believed. An interesting finding is that in the near-wall region of the far-wake regime, small-to-moderate-scale components of Reynolds shear stress is suppressed, which may be associated with the inhibition of the generation of near-wall prograde vortical structures. This study offers a new perspective for turbulent control by passive discrete elements.

On the relationship between manipulated inter-scale phase and energy-efficient turbulent drag reduction

We investigate the role of inter-scale interactions in the high-Reynolds-number skin-friction drag reduction strategy reported by Marusic et al. (Nat. Commun., vol. 12, 2021). The strategy involves imposing relatively low-frequency streamwise travelling waves of spanwise velocity at the wall to actuate the drag generating outer scales. This approach has proven to be more energy efficient than the conventional method of directly targeting the drag producing inner scales, which typically requires actuation at higher frequencies. Notably, it is observed that actuating the outer scales at low frequencies leads to a substantial attenuation of the major drag producing inner scales, suggesting that the actuations affect the nonlinear inner–outer coupling inherently existing in wall-bounded flows. In the present study, we find that increased drag reduction, through imposition of spanwise wall oscillations, is always associated with an increased coupling between the inner and outer scales. This enhanced coupling emerges through manipulation of the phase relationships between these triadically linked scales, with the actuation forcing the entire range of energy-containing scales, from the inner (viscous) to the outer (inertial) scales, to be more in phase. We also find that a similar enhancement of this nonlinear coupling, via manipulation of the inter-scale phase relationships, occurs with increasing Reynolds number for canonical turbulent boundary layers. This indicates improved efficacy of the energy-efficient drag reduction strategy at very high Reynolds numbers, where the energised outer scales are known to more strongly superimpose and modulate the inner scales. Leveraging the inter-scale interactions, therefore, offers a plausible mechanism for achieving energy-efficient drag reduction at high Reynolds numbers.

On wall pressure fluctuations in conical shock wave/turbulent boundary layer interaction

The structure and the frequency spectra of wall pressure fluctuations beneath a planar turbulent boundary layer interacting with a conical shock wave at Mach number $M_\infty =2.05$ and Reynolds number $\textit {Re}_\theta \approx 630$ (based on the upstream boundary layer momentum thickness) are examined to elucidate the effects of pressure gradient and flow separation on the characteristics of the wall pressure fluctuations, by exploiting a direct numerical simulation database. Upstream of the interaction, in the zero pressure gradient region, wall pressure statistics compare well with canonical compressible boundary layers in terms of fluctuation intensities and frequency spectra. Across the main interaction zone (APG1), the root-mean-square of wall pressure fluctuations becomes very large (corresponding to approximately 173.3 dB), with maximum increase approximately 12.7 dB from the incoming level. In the second adverse pressure gradient zone (APG2), the root-mean-square of wall pressure fluctuations attains a second peak (corresponding to $164.7$ dB), with an increase of 8.4 dB from the upstream level. Both the APG1 and APG2 regions feature a substantial fraction of flow reversal events, which are, however, scattered and interspersed with regions of attached flow. The wall pressure power spectral density exhibits a broadband and energetic low-frequency component associated with the global unsteadiness of the separation bubble/conical shock system. Analysis of the two-point correlations and wavenumber/frequency spectra of wall pressure fluctuations further suggests that the typical eddies become more elongated along the spanwise direction, as the flow in the separated region tends to escape the centreline, and the convection velocity is significantly reduced.

Symplectic Method-Based Analytical Solutions of Thermal Buckling for Shear Deformable Circular Functionally Graded Plates

Thermal buckling of a shear deformable circular plate made of functionally graded materials (FGM) is studied using an exact analytical method in the Hamiltonian system. Fundamental equations for the thermal buckling of the plate under transversely nonuniform temperature rising are established based on the first-order shear deformation theory. Subsequently, by introducing the symplectic method, the differential equations are converted into canonical equations in the Hamiltonian system. Buckling loads and buckling modes correspond to symplectic eigenvalues and eigen solutions, respectively. The expressions of complete buckling modes in the form of special functions are achieved by solving canonical equations and boundary conditions analytically and exactly. The symplectic eigenvalues are obtained simultaneously and the buckling temperature increments are calculated using the inverse solution. Finally, the influences of the gradient characteristics, geometric parameters, boundary conditions, and type of thermal loading on the buckling temperature increments are discussed through parameter research.

A New Conceptual Picture of the Trade Wind Transition Layer

Abstract The transition layer in the trades has long been observed and simulated, but the physical processes producing its structure remain little investigated. Using extensive observations from the Elucidating the Role of Clouds–Circulation Coupling in Climate (EUREC4A) field campaign, we propose a new conceptual picture of the trade wind transition layer, occurring between the mixed-layer top (around 550 m) and subcloud-layer top (around 700 m). The theory of cloud-free convective boundary layers suggests a transition-layer structure with strong jumps at the mixed-layer top, yet such strong jumps are only observed rarely. Despite cloud-base cloud fraction measured as only 5.3% ± 3.2%, the canonical cloud-free convective boundary layer structure is infrequent and confined to large [O(200) km] cloud-free areas. We show that the majority of cloud bases form within the transition layer, instead of above it, and that the cloud-top height distribution is bimodal, with a first population of very shallow clouds (tops below 1.3 km) and a second population of deeper clouds (extending to 2–3 km depth). We then show that the life cycle of this first cloud population maintains the transition-layer structure. That is, very shallow clouds smooth vertical thermodynamic gradients in the transition layer by a condensation–evaporation mechanism, which is fully coupled to the mixed layer. Inferences from mixed-layer theory and mixing diagrams, moreover, suggest that the observed transition-layer structure does not affect the rate of entrainment mixing, but rather the properties of the air incorporated into the mixed layer, primarily to enhance its rate of moistening. Significance Statement The physical processes producing the structure of the trade wind transition layer, a thin atmospheric layer thought to be important for regulating convection, are not yet well understood. Using extensive observations from a recent field campaign, we find that the cloud-free convective boundary layer structure, with an abrupt discontinuity in thermodynamic variables, is infrequent, despite cloud-base cloud fraction being small. We show that very shallow clouds both forming and dissipating within the transition layer smooth vertical gradients compared to a jump, except in large [O(200) km] cloud-free areas. This condensation–evaporation mechanism, which is fully coupled to the mixed layer, does not appear to affect the rate of entrainment mixing, but rather the properties of air incorporated into the mixed layer.

Post-shock turbulence recovery in oblique-shock/turbulent boundary layer interaction flows

The oblique shock impinging on the supersonic turbulent boundary layer leads to a mixing layer and the emergence of large-scale coherent structures within the interaction zone which leave significant velocity defect and turbulence amplification downstream. In the present study, we investigate the turbulence recovery in the post-shock region by exploiting direct numerical simulation data of the oblique-shock/turbulent boundary layer interaction flow at the incoming Mach number of $2.28$ and the shock angle of $33.2^\circ$ , with special attention paid to the contribution of the mixing layer and large-scale structures to flow dynamics. For that purpose, we propose to split the mean velocity, Reynolds stresses and spanwise spectra into a canonical portion that is constructed according to the statistics of canonical turbulent boundary layers, and a mixing-layer-induced portion. We found that the hidden mixing layer grows with the boundary layer thickness and that the induced mean shear and Reynolds stresses decay at different rates. The mean velocity recovers to the canonical profiles at a distance of 13 boundary layer thicknesses downstream where the mixing-layer-induced mean shear ceases to have strong impacts. The recovery of Reynolds stresses requires 10 boundary layer thicknesses in the near-wall region but a much longer streamwise extent in the outer region due to the slow decay of large-scale motions. These large-scale motions superpose on the near-wall turbulence, intensifying the turbulent fluctuations, yet having a trivial impact on the skin friction, for the contribution of the mixing-layer-induced mean shear and Reynolds shear stress are balanced by the advection term. We further establish a simple physical model capable of approximately predicting the streamwise evolution of mixing-layer-induced mean shear and turbulent kinetic energy. This model suggests that the complete recovery of turbulence in the outer region requires a streamwise extent of approximately 50 boundary layer thicknesses.

Highly reliable measurement of temperature fluctuation in near-wall turbulence by a sophisticated cold-wire technique for considering the frequency response and spatial resolution

In turbulence measurement of the thermal field in a non-isothermal turbulent boundary layer, it is necessary to use a small-scale temperature sensor that can provide high spatial resolution. Thus, a fine-wire temperature sensor commonly called “cold-wire” is indispensable for capturing small-scale temperature fluctuations near the wall. However, the length-to-diameter ratio of the cold-wire need generally exceed 1000 to minimize the deterioration in dynamic response due mainly to the axial heat conduction of the thin-wire sensor. To meet such a conflicting requirement, the wire diameter of the cold-wire must be extremely thin. In the present study, we have tested experimentally the performance of several cold-wires of different geometrical features, and examined the effects of the temporal and spatial resolutions of the cold-wire on the temperature fluctuation measurements—root-mean-square (rms) values and instantaneous signal traces—in a canonical turbulent thermal boundary layer. As a result, it was found that the previously-reported response compensation technique for the cold-wire outputs enables us to realize sufficiently accurate temperature fluctuation measurement in the immediate vicinity of the wall while avoiding the deterioration in the spatial resolution of the cold-wire. Furthermore, to improve the spatial resolution of the cold-wire probe, we propose a novel temperature sensor consisting of a fine tungsten wire bent at its center to make an apex angle less than about 30 degrees and have named this sensor a “V-shaped” cold-wire. We have verified its measurement performance in reference to the results obtained by a standard cold-wire probe. As a result, it is shown that the V-shaped cold-wire naturally provides high spatial resolution especially for temperature fluctuation measurement near the wall and can take full advantage of the response compensation technique for the normal cold-wire probe.

Thermal capillary waves on bounded nanoscale thin films.

The effect of confining walls on the fluctuation of a nanoscale thin film's free surface is studied using stochastic thin-film equations(STFEs). Two canonical boundary conditions are employed to reveal the influence of the confinement: (1) an imposed contact angle and (2) a pinned contact line. A linear stability analysis provides the wave eigenmodes, after which thermal-capillary-wave theory predicts the wave fluctuation amplitudes. Molecular dynamics (MD) simulations are performed to test the predictions, and a Langevin diffusion model is proposed to capture oscillations of the contact lines observed in MD simulations. Good agreement between the theoretical predictions and the MD simulation results is recovered, and it is discovered that confinement can influence the entire film. Notably, a constraint on the length scale of wave modes is found to affect fluctuation amplitudes from our theoretical model, especially for 3D films. This opens up challenges and future lines of inquiry.

A comparative study of experiments with numerical simulations of free-stream turbulence transition

To date, very few careful and direct comparisons between experiments and direct numerical simulations (DNS) have been published on free-stream turbulence (FST) induced boundary layer transition, whilst there exist numerous published works on the comparison of canonical turbulent boundary layers. The primary reason is that the former comparison is vastly more difficult to carry out simply because all known transition scenarios have large energy gradients and are extremely sensitive to surrounding conditions. This paper presents a detailed comparison between new experiments and available DNS data of the complex FST transition scenario in a flat plate boundary layer at turbulence intensity level about $Tu = 3\,\%$ and FST Reynolds number about $Re_{{fst}} = 67$ . The leading edge (LE) pressure gradient distribution and the full energy spectrum at the LE are identified as the two most important parameters for a satisfying comparison. Matching the LE characteristic FST parameters is not enough as previously thought, which is illustrated by setting up two experimental FST cases with about the same FST integral parameters at the LE but with different energy spectra. Finally, an FST boundary layer penetration depth (PD) measure is defined using DNS, which suggests that the PD grows with the downstream distance and stays around 20 % of the boundary layer thickness down to transition onset. With this result, one cannot rule out the significance of the continuous FST forcing along the boundary layer edge in this transition scenario, as indicated in previous studies.

The Canon of the Pistis Sophia Books 1–3

Abstract The first three books of the Pistis Sophia demonstrate an awareness of an emerging canon in a late third-century Egyptian textual community. The canonical boundary is revealed through the scriptural practices employed by the author: texts considered to be within the canon are cited and interpreted as authoritative literature, used to authenticate the teachings of Pist. Soph., whereas texts outside of the canon are open to substantial revision and woven into the narrative without a citation formula. This paper examines the author’s use of the Hebrew Bible, the Odes of Solomon, and the Gospels as canonical literature, and asks which other texts were in his literary repertoire, suggesting that the Apocryphon of John, the Gospel of Mary, and the Ascension of Isaiah were used as inspiration for composing the new narrative and dialogue.