Effect Of Pressure Gradient Research Articles

Numerical analysis and control of contaminant leakage during personnel movement across differential pressure zones

Artificial pressure differential environments play a vital role in various fields, including healthcare, industrial applications, and laboratory safety. Despite numerous studies conducted in this area, contaminant leakage due to personnel movement across pressure zones remains a significant challenge, increasing the risk of infection or contamination. This study aims to systematically analyze and control contaminant particle leakage during such movements, using numerical simulations and field experiments that incorporate the moving mesh method. This study investigates the effects of pressure gradients, door types, and temperature differences on particle migration. The results indicate that maintaining a pressure gradient can reduce particle leakage by up to 35.33 %. Sliding doors proved more effective than hinged doors, reducing leakage by up to 74.36 %. Introducing a temperature difference between areas also decreased leakage by up to 64.9 %. These findings provide practical recommendations for optimizing the design of negative-pressure isolation wards and can be extended to other environments that maintain differential pressure. This study offers valuable insights into the control of contaminant leakage, contributing to improved indoor air quality and infection control.

Thermal Performance Analysis of a Nonlinear Couple Stress Ternary Hybrid Nanofluid in a Channel: A Fractal-Fractional Approach.

Nanofluids have improved thermophysical properties compared to conventional fluids, which makes them promising successors in fluid technology. The use of nanofluids enables optimal thermal efficiency to be achieved by introducing a minimal concentration of nanoparticles that are stably suspended in conventional fluids. The use of nanofluids in technology and industry is steadily increasing due to their effective implementation. The improved thermophysical properties of nanofluids have a significant impact on their effectiveness in convection phenomena. The technology is not yet complete at this point; binary and ternary nanofluids are currently being used to improve the performance of conventional fluids. Therefore, this work aims to theoretically investigate the ternary nanofluid flow of a couple stress fluid in a vertical channel. A homogeneous suspension of alumina, cuprous oxide, and titania nanoparticles is formed by dispersing trihybridized nanoparticles in a base fluid (water). The effects of pressure gradient and viscous dissipation are also considered in the analysis. The classical ternary nanofluid model with couple stress was generalized using the fractal-fractional derivative (FFD) operator. The Crank-Nicolson technique helped to discretize the generalized model, which was then solved using computer tools. To investigate the properties of the fluid flow and the distribution of thermal energy in the fluid, numerical methods were used to calculate the solution, which was then plotted as a function of various physical factors. The graphical results show that at a volume fraction of 0.04 (corresponding to 4% of the base fluid), the heat transfer rate of the ternary nanofluid flow increases significantly compared to the binary and unary nanofluid flows.

Development of a Machine Learning Natural Ventilation Rate Model by Studying the Wind Field Inside and Around Multiple-Row Chinese Solar Greenhouses

This paper experimented with a methodology of machine learning modelling using virtual samples generated by fast CFD (Computational Fluid Dynamics) simulations in order to predict the greenhouse natural ventilation. However, the output natural ventilation rates using fast two-dimensional (2D) CFD models are not always consistent with the three-dimensional (3D) one for all the scenarios. The first contribution of this paper is a proposed comparative modelling methodology between two-dimensional and three-dimensional CFD studies, regarding its validity, especially when buildings are in rows. The results show that the error of the ventilation rate prediction could exceed 50%, if 2D models are not properly used. Subsequently, in those scenarios where the 2D and the 3D models had equal accuracy, nearly one thousand samples were generated using fast 2D CFD simulations to train a natural ventilation rate regression tree model. This model is efficient to deal with the combined effect of wind pressure and thermal gradients under various vent configurations, with only four necessary inputs. In addition, by analyzing the wind speed distribution contour of the outdoor wind field around the greenhouse rows, the optimal wind speed-measuring locations were determined to eliminate interference for predicting the natural ventilation rate.

Transonic vane film cooling with shaped sweeping jet design

The sweeping jet (SJ) presents promising potential for film cooling due to its periodic, self-excited flow behavior. The shaped sweeping jet (SSJ) design features a compact structure and has demonstrated superior cooling performance compared to conventional holes under low-speed conditions. This study further explores the SSJ cooling performance on a transonic vane's pressure and suction surfaces. Experiments were conducted to evaluate its cooling performance and reveal the effects of blade curvature and pressure gradient under engine-relevant Mach numbers (exit Ma = 0.84). The adiabatic cooling effectiveness of the SSJ and the baseline 777 holes was measured using the fast-responding pressure-sensitive paint (Fast-PSP) technique at blowing ratios of M = 0.5–3.0. The oscillating frequency of the SSJ and its Strouhal number were determined through velocity fluctuation analysis. Experimental results consistently show enhanced cooling effectiveness on the pressure surface for the SSJ compared to 777 holes, with the maximum improvement of 85 % observed at M = 2.5. Due to stronger compressibility effects on the suction surface, the cooling effectiveness of the SSJ increases more significantly with rising blowing ratios compared to the pressure surface. Numerical simulations reveal that the SSJ generates a smaller recirculation zone than the 777 hole. The alternating vortices formed during the sweeping process enhance lateral film distribution. Additionally, the total pressure loss coefficient of the SSJ remained comparable to that of the 777 holes, with a maximum increase of 6.11 %. These results contribute to the understanding of SJ film cooling and support the potential application of SSJ in turbine film cooling design.

Investigation of the impact of transient pressure gradients on turbulent channel flow dynamics

AbstractThe effect of transient pressure gradients, or transient pumping in turbulent channel flow configuration, is investigated. Employing a cost‐efficient reduced‐order stochastic method known as one‐dimensional turbulence (ODT) modeling, simulations explore different signal shapes for the modulation of the prescribed pressure gradient forcing, including sinusoidal modulation, step‐like modulation, and piecewise sinusoidal beating. Various cycle periods and active pumping times are investigated. The simulations are conducted at a frictional Reynolds number of and a molecular Prandtl number of . The study adopts a passive scalar formulation to investigate heat transfer properties. The study quantifies the effects of transient pressure gradients on heat transfer rate, drag, and pumping power. Preliminary ODT predictions suggest that all transient cases exhibit lower heat transfer rates and a higher pumping power requirement than the constant pressure gradient case, with the step‐like modulation yields superior skin‐friction drag and heat transfer rate reductions relative to other signals.

Compact Eddy Structure of Turbulence Aft of an Inclined Slender Body

The compact eddy structure and the correlation structure of three distinct turbulent boundary layers were investigated, with a view to understanding the turbulent inflow seen by a stern-mounted propulsion tail rotor. The first is of a zero-pressure-gradient, equilibrium, flat plate boundary layer, and the second and third are on the lee and port sides of a body of revolution (BOR) inclined at a 5° angle of attack. The BOR is an axially symmetric body with an ellipsoidal nose, straight midbody, and 1.17-diam-long stern ramp with a stern-mounted tail rotor. Measurements on the BOR were taken at a body-diameter-based Reynolds number of 600,000 and a rotor advance ratio of J=1.27. Particle image velocimetry was taken at two principal locations at the stern of the BOR. Distortion of the correlation function was evident in the compact eddy structure of the BOR boundary layer on both the lee and port sides. On the port side, the turbulent modes were dominated by the embedded shear layer, whereas the lee side showed more significant pressure gradient effects in the eddy structure.

A generalized wall-pressure spectral model for non-equilibrium boundary layers

This study uses high-fidelity simulations (direct numerical simulation or large-eddy simulation) and experimental datasets to analyse the effect of non-equilibrium streamwise mean pressure gradients (adverse or favourable), including attached and separated flows, on the statistics of boundary-layer wall-pressure fluctuations. The datasets collected span a wide range of Reynolds numbers ( $Re_\theta$ from 300 to 23 400) and pressure gradients (Clauser parameter from $-0.5$ to 200). The datasets are used to identify an optimal set of variables to scale the wall-pressure spectrum: edge velocity, boundary layer thickness and the peak magnitude of Reynolds shear stress. Using the present datasets, existing semi-empirical models of the wall-pressure spectrum are shown unable to capture effects of strong, non-equilibrium adverse pressure gradients, due to inappropriate scaling of the wall pressure using the wall shear stress, calibration with limited types of flows and dependency on model parameters based on the friction velocity, which reduces to zero at the detachment point. To address these shortcomings, a generalized wall-pressure spectral model is developed with parameters that characterize the extent of the logarithmic layer and the strength of the wake. Derived from the local mean streamwise velocity profile, these two parameters inherently carry the effect of the Reynolds number, as well as those of the non-equilibrium pressure gradient and its history. Comparison with existing models shows that the proposed model behaves well and is more accurate in strong-pressure-gradient flows and in separated-flow regions.

Dependence of ELM instability on separatrix density in EAST long-pulse H-mode plasmas

Abstract The transition from small edge-localized modes (ELMs) to large ELMs has been repetitively observed in minute-scale long-pulse high-confinement mode (H-mode) discharges in 2017 EAST campaign. The appearance of large ELMs is found to be strongly correlated with the decrease in separatrix density due to the gradual decrease in fuel recycling during the long-pulse H-mode operations (LPHOs). By numerical scan of separatrix density with a fixed temperature profile, it has been found that the dependence of ELM instability on separatrix density is related to the competition between the ion diamagnetic stabilizing effect and destabilizing effect of pressure gradient and current density in the pedestal region, shedding light on a comprehensive understanding of different roles of separatrix density in ELM instability observed in EAST experiments. With a high separatrix density, the ideal ballooning mode could be destabilized near the separatrix, which is thought to help achieve small ELMs in EAST LPHOs. In 2021 EAST campaign, the experiment of large ELMs control has been performed through actively changing fuel recycling by moving strike point location (SPL) on the lower tungsten divertor target plate. It has been demonstrated that the mitigation of large ELMs is strongly correlated with the significant increase in separatrix density, which is thought to be attributed to a higher ionization source in the scrape-off layer (SOL) region by SOLPS-ITER simulation. The high ionization source in the SOL region is believed to provide a strong fueling effect near the separatrix and thus raise the local density, which is considered as an important reason for triggering ballooning instabilities near the separatrix and achieving small ELMs.

Toroidal Alfvén mode instability driven by plasma current in low-density Ohmic plasmas of the spherical tori

Due to intrinsically low magnetic fields, at low density the drift speed of the current-carrying electrons in spherical tokamaks can exceed fraction of the Alfvén speed sufficient for the excitation of the Alfvén gap modes. A particular case of the toroidal mode, observed during minor disruptions in Ohmic shots on SUNIST [Liu et al., Phys. Plasmas 23, 120706 (2016)], is considered in the present communication. Due to the negligible effect of the electron pressure gradient, the growth rate scales linearly with the drift speed, with slope inversely proportional to electron thermal velocity. In the absence of continuum damping, the threshold value of the drift speed for TAE excitation is independent of electron temperature.

Dorsal arachnoid web of the thoracic spine associated with an arachnoid cyst and a presyrinx state as an unexplained cause of thoracic myelopathy: illustrative case.

Dorsal thoracic arachnoid web is a rare diagnosis and is not commonly seen in neurosurgical practice. Patients can present with symptoms and signs of thoracic myelopathy in the setting of an arachnoid cyst and a presyrinx state. A 57-year-old male with a 10-year history of worsening bilateral leg weakness and chronic back pain re-presented to the neurosurgery clinic after being seen by neurology and orthopedic spine surgery. Initial imaging was concerning for myelomalacia and syringomyelia, and repeat delayed computed tomography myelography findings were consistent with an evolving thoracic arachnoid web, now demonstrating spinal cord compression secondary to arachnoid cyst formation and consistent with the signs of thoracic myelopathy. Intraoperative ultrasound displayed the arachnoid web as the cause of the evolving arachnoid cyst, edematous spinal cord, and a presyrinx-like state. The patient underwent surgical decompression, which restored cerebrospinal fluid (CSF) dynamics, resulting in clinical improvement. Dorsal thoracic arachnoid web is a dynamic condition that can occur in the setting of an arachnoid cyst. There appears to be a relationship between dorsal thoracic arachnoid web formation and the presence of an arachnoid cyst resulting from a ball-valve mechanism leading to the creation of a pressure gradient effect that alters CSF fluid dynamics. https://thejns.org/doi/10.3171/CASE24313.

Numerical and experimental investigation of the supercooled large droplets icing of rotating spinner

The icing of aero-engine rotating components is more complex than traditional aircraft icing. A three-dimensional numerical method was developed based on the Eulerian method to study the supercooled large droplets icing of a rotating spinner. Typical dynamic behaviours of supercooled large droplets, such as deformation, breakup, rebound, and splashing, were considered. An icing thermodynamic model applicable to a rotating spinner was established by considering the effects of pressure gradient, air shear force, and centrifugal force on the flow of the water film. Meanwhile, an ice wind tunnel experiment on supercooled large droplets icing of a rotating spinner was carried out. The results showed that the maximum relative deviation of the simulated icing thickness at the cone tip is no more than 15% compared with the experimental result, and the icing thickness on the surface of the rotating spinner is basically consistent with the experimental result. Based on the three-dimensional numerical method developed in this study, the effects of the rotational speed, inflow velocity, and inflow temperature on the supercooled large droplets icing of a rotating spinner were investigated. As the rotational speed increases, the rebound and splashing of supercooled large droplets becomes more obvious, leading to an increase in the mass loss coefficient and a decrease in the collected mass of water droplets on the surface of the rotating spinner. Consequently, the icing thickness at the tail of the rotating spinner decreases with an increase in the rotational speed. Since the liquid water content in the environment is relatively low, water droplets freeze immediately after impacting the surface of the rotating spinner and rime ice is usually formed. When the inflow temperature is relatively high, overflow water exists and glaze ice is formed near the tip of the rotating spinner. The findings of this study can provide valuable support for the design and verification of anti-icing systems for aero-engine rotating spinner.

Fluid-Solid Interaction Analysis for Developing In-Situ Strain and Flow Sensors for Prosthetic Valve Monitoring.

Transcatheter aortic valve implantation (TAVI) was initially developed for adult patients, but there is a growing interest to expand this procedure to younger individuals with longer life expectancies. However, the gradual degradation of biological valve leaflets in transcatheter heart valves (THV) presents significant challenges for this extension. This study aimed to establish a multiphysics computational framework to analyze structural and flow measurements of TAVI and evaluate the integration of optical fiber and photoplethysmography (PPG) sensors for monitoring valve function. A two-way fluid-solid interaction (FSI) analysis was performed on an idealized aortic vessel before and after the virtual deployment of the SAPIEN 3 Ultra (S3) THV. Subsequently, an analytical analysis was conducted to estimate the PPG signal using computational flow predictions and to analyze the effect of different pressure gradients and distances between PPG sensors. Circumferential strain estimates from the embedded optical fiber in the FSI model were highest in the sinus of Valsalva; however, the optimal fiber positioning was found to be distal to the sino-tubular junction to minimize bending effects. The findings also demonstrated that positioning PPG sensors both upstream and downstream of the bioprosthesis can be used to effectively assess the pressure gradient across the valve. We concluded that computational modeling allows sensor design to quantify vessel wall strain and pressure gradients across valve leaflets, with the ultimate goal of developing low-cost monitoring systems for detecting valve deterioration.

Wall Modeling of Turbulent Flows with Varying Pressure Gradients Using Multi-Agent Reinforcement Learning

We propose a framework for developing wall models for large-eddy simulation that is able to capture pressure-gradient effects using multi-agent reinforcement learning. Within this framework, the distributed reinforcement learning agents receive off-wall environmental states, including pressure gradient and turbulence strain rate, ensuring adaptability to a wide range of flows characterized by pressure-gradient effects and separations. Based on these states, the agents determine an action to adjust the wall eddy viscosity and, consequently, the wall-shear stress. The model training is in situ with wall-modeled large-eddy simulation grid resolutions and does not rely on the instantaneous velocity fields from high-fidelity simulations. Throughout the training, the agents compute rewards from the relative error in the estimated wall-shear stress, which allows them to refine an optimal control policy that minimizes prediction errors. Employing this framework, wall models are trained for two distinct subgrid-scale models using low-Reynolds-number flow over periodic hills. These models are validated through simulations of flows over periodic hills at higher Reynolds numbers and flows over the Boeing Gaussian bump. The developed wall models successfully capture the acceleration and deceleration of wall-bounded turbulent flows under pressure gradients and outperform the equilibrium wall model in predicting skin friction.

Improved Design Method for Hypersonic Intakes Using Pressure-Corrected Osculating Axisymmetric Flows Method

The efficient design of air intake is crucial for high-speed airbreathing propulsion systems. This paper presents a novel design method that integrates the recently introduced internal conical flow M (ICFM) basic flowfield (Musa et al., “New Parent Flowfield for Streamline-Traced Intakes,” AIAA Journal, Vol. 61, No. 7, 2023, pp. 2906–2921) with the pressure-corrected osculating axisymmetric flows and streamline tracing techniques. The new design method takes into account the effects of lateral pressure gradients by incorporating pressure corrections into the original design method using crossflow velocity information from the ICFM flowfield. Additionally, new entrance and throat profiles are introduced for designing two hypersonic internal waverider intakes with shape transition. One intake is designed using the new method, and the other using the original method. Viscous simulations have been performed at design (i.e., Mach 6.0) and off-design (i.e., Mach 4.0 and 3.5) conditions. The results indicate that the pressure-corrected intake exhibits superior performance in terms of total pressure recovery and reduced flow distortion. This reveals that the new design method can achieve a smooth shape and efficient aerodynamic transitions between the intake entrance and throat. The new entrance and throat profiles realized better performance than the typical ones. The startability analysis reveals that the Van Wie curve controls the considered intakes. Thus, combining the pressure-corrected osculating axisymmetric flows with the ICFM flowfield enhances the performance of hypersonic intakes, making this approach a promising avenue for future hypersonic vehicles.

Experimental investigation on the effects of complex pressure gradients on the film cooling effectiveness of a serpentine nozzle

Serpentine nozzles more easily deform and damage due to high-temperature airflow because of their multi-bend channel configuration. Therefore, film cooling technologies should be applied to serpentine nozzles. Complex pressure gradients in serpentine nozzles lead to different film cooling characteristics from combustion chambers, turbine blades, and other exhaust systems. This study experimentally investigated the effects of complex pressure gradients and film hole inclination angles on a serpentine nozzle’s film cooling effectiveness (FCE) using pressure-sensitive paint (PSP). The flow mechanisms were obtained via numerical simulations. The experimental nozzle design needs to consider the needs of PSP observations and the normal flow of serpentine nozzles. However, the complex structure of serpentine nozzles makes those increasingly difficult. The studied pressure gradients include the pressure gradient mutation (PGM), strong adverse pressure gradient (SAPG), weak adverse pressure gradient (WAPG), and favorable pressure gradient (FPG). The results show that complex pressure gradients change the coolant coverage and adhesion and may induce a recirculation zone in the SAPG region. This affects the development of the vortex construction and leads to different FCE distributions. The FCE under the effects of the PGM is the largest, followed by the SAPG, WAPG, and FPG. A larger inclination angle weakens the coolant adhesion but may enhance its downstream and lateral flow treads. This is coupled with the effects of complex pressure gradients, where larger inclination angles bring better film cooling effects in some cases. After accounting for the four blowing ratios, the area-averaged FCEs under the effects of the SAPG, WAPG, and FPG are 28.9 %, 42.8 %, and 52.3 % lower than that under the PGM, respectively. The area-averaged FCEs under the conditions α = 45° and 60° are 6.6 % and 10 % lower than that under α = 30°.

The synergistic effects of salinity and pressure gradients on sustainable energy conversion in nanofluidics

In recent years, osmotic energy conversion and electrokinetic energy conversion within nanochannels have received widespread attention. In this work, we propose four kinds of energy conversion models in nanochannels, including pressure-driven power generation, salinity-driven power generation, the positive combination of pressure gradient and salinity gradient and the negative combination. The changes in ion transport behavior and energy conversion characteristics are analyzed. The results show that the combination of salinity gradient and pressure gradient is not a simple superposition. Compared to the power of the single drive mode before optimization, the power under positive combination (PC) can reach 1.02 pW, and the power enhancement ratio is as high as 1.6. The negative combination has a certain optimization potential under a large concentration ratio and small pressure gradient condition. The higher the surface charge density, the greater the output power, but the optimization effect worsens. On the contrary, a larger channel length corresponds to a smaller output power, but the optimization effect becomes better. The reasonable choice of surface charge density and channel length needs to be weighed. These studies provide valuable guidance for the design of future nanofluidic energy conversion devices.

A developed transition simulation model for turbulent plane jet impingement heat transfer

Jet impingement has been widely used due to its high heat transfer rate, but numerical simulation of this flow is highly challenging because of complex flow phenomena. In this paper, a turbulence model based on the physics of jet impingement is developed using the shear stress transport (SST) four-equation transition model and the SST with cross-diffusion correction (SSTCD) model. The proposed method is evaluated for plane jet impingement under various nozzle-plate spacings (2.6, 4 and 6) and different Reynolds numbers ranging from 10,200 to 20,000 in terms of heat transfer and flow fields. The results show that the developed model has the ability to capture the second peak of heat transfer and skin friction at low nozzle-plate spacing. Even at high nozzle-plate spacing, this approach also gives accurate trends for the above two features. However, without the cross-diffusion correction, the transition model provides too high energy and low velocity peaks. With the developed model in hand, the effects of pressure gradient on heat transfer and skin friction coefficient are further investigated. It is found that when the adverse pressure gradient becomes strong, the position of the secondary peak of skin friction occurs earlier than that of the secondary peak of heat transfer rate. Conversely, if the adverse pressure gradient disappears, the positions of these features coincide.

Impact of Pressure and Temperature on Charge Accumulation Characteristics of Insulators in Direct-Current Gas-Insulated Switchgear

Direct-current gas-insulated switchgear (DC GIS) is an important device for promoting the lightweight and compact design of offshore wind power platforms. Gas pressure and temperature gradients are crucial factors that must be considered during the design process of the DC GIS. In this study, the multi-physics coupling model of basin insulators considering surface charge accumulation was established, and the corresponding real-sized insulator surface charge measurement platform was constructed. The effects of gas pressure and temperature gradient on the surface charge accumulation characteristics were investigated, respectively. The results show that the effect of gas pressure on the surface charge distribution characteristics depends on the dominant mode of surface charge. When volume conduction is dominant, the effect of gas pressure on the surface charge is negligible. However, when gas conduction is dominant, the accumulation of a uniform charge pattern on the insulator surface increases with the rise in gas pressure. Furthermore, due to gas convection, the temperature of the upper part of the DC GIS is significantly higher than that of the lower part, which leads to a temperature difference between the upper and lower surfaces of the insulator. The charge density on the insulator upper surface near the central conductor rises with the increase in load current.

Investigation on formation mechanism of cavitation channel vortex in the runner of Francis turbine

Abstract Cavitation channel vortices generally cause flow state more complex in the runner, leading to the work efficiency is reduced and the cavitation erosion on flow components is enhanced. In this paper, simulation of cavitation flow is carried out for a Francis turbine model, and the evolution of vortices in the runner during cavitation development is analyzed under partial load conditions. The results show that cavitation has a great influence on the morphology and distribution of attached vortices between blades, trailing vortices at the outlet edge of the blades, and vortex near the cone in the runner. With the gradual development of cavitation, the invasion effect of vortex rope in draft tube on the runner is enhanced, and the vortex near runner cone is gradually connected as a whole. Under the effect of the downstream reverse pressure gradient in local low-pressure area, the flow separation area gradually expands to the blade and upstream, and more singular points evolve on flow separation lines, which leads to more trailing vortices at the outlet of the blade and cavitation channel vortices are formed in serious cavitation stage.

Improved Prediction of the Flow in Cylindrical Critical Flow Venturi Nozzles Using a Transitional Model

Critical flow Venturi nozzles (CFVNs) are a state-of-the-art secondary standard widely used for gas flow measurements with high precision. The flow rate correlates with the type and thickness of the boundary layer (BL) inside the nozzle throat. In the cylindrical type—one of the two standard designs of CFVNs—the nozzle throat encompasses a defined axial length in which the BL develops. This numerical study is concerned with the BL effects in a cylindrical CFVN by means of two turbulence models. Compared to experimental data, the k-ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega$$\\end{document} SST model predicts the discharge coefficient well for high and low Reynolds numbers, but not in the intermediate regime. The γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma$$\\end{document}-Reθ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Re_{\ heta }$$\\end{document} model, on the contrary, agrees well with experimental data in the entire flow range. Relevant quantities and profiles of the BL are separately investigated in the laminar, turbulent, and transitional region. The calculated laminar and turbulent BL thicknesses correspond to predictions based on integral methods for solving the BL equations. Simple representations are proposed for the Zagarola-Smits scaled laminar and turbulent deficit BL profiles removing the effects of axial position, Reynolds number, and pressure gradient. Furthermore, the shape factor is investigated as a characteristic parameter for determining the transitional region.