Boundary Layer Analysis Research Articles

Site-wise dynamic defects in a non-conserving exclusion process

Motivated by the significant influence of the defects in the dynamics of the natural or man-made transportation systems, we propose an open, dynamically disordered, totally asymmetric simple exclusion process featuring bulk particle attachment and detachment. The site-wise dynamic defects might randomly emerge or vanish at any lattice location, and their presence slows down the motion of the particles. Using a mean-field approach, we obtain an analytical expression for both particle and defect density and validate them using Monte Carlo simulation. The study investigates the steady-state characteristics of the system, including phase transitions, analysis of boundary layers, and phase diagrams. Our approach streamlines the defect dynamics by integrating two parameters into one called the obstruction factor, which helps in determining an effective binding constant. The impact of the obstruction factor on the phase diagram is explored across various combinations of binding constants and detachment rates. A critical value of the obstruction factor is obtained, about which an infinitesimal change results in a substantial qualitative change in the structure of the phase diagrams. Further, the effect of the detachment rate is studied, and critical values along which the system observes a quantitative transition of the stationary phases are obtained as a function of the obstruction factor. Overall, the system shows stationary phases ranging from three to seven depending upon the value of the obstruction factor, the binding constant, and the detachment rate. Moreover, we scrutinized the impact of the obstruction factor on the shock dynamics and found no finite-size effect.

Moment of momentum integral analysis of turbulent boundary layers with pressure gradient

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.

Boundary Layer Analysis of Second-Order Magnetic Nanofluid Flow with Carbon Nanotubes and Gyrotactic Microorganisms for Medical Diagnostics

Boundary Layer Analysis of Second-Order Magnetic Nanofluid Flow with Carbon Nanotubes and Gyrotactic Microorganisms for Medical Diagnostics

3D CFD analysis of pressure, boundary layer and shear stresses on a gudgeon (Gobio gobio)

The fish’s mechanosensory lateral line system detects non-acoustic hydrodynamic stimuli required for feeding, schooling, predator avoidance and underwater object detection. Biological investigations have established that flow stimuli are detected through the boundary layer as pressure gradients by canal neuromasts and as shear stresses acting on the superficial neuromasts. Previous works have also shown that the spatial distribution of neuromasts is strongly correlated with the pressure coefficient. Despite these fundamental insights, substantial knowledge gaps persist in understanding how fish body geometry influences the boundary layer, the pressure distribution and shear stresses. To address these gaps, we provide a set of numerical models based on the open-source CFD toolkit OpenFOAM which are experimentally validated using velocity measurements obtained in a laboratory fish swim tunnel. Specifically, we investigate the mid dorsal-ventral planar flow fields around a 3D fish-shaped body of gudgeon (Gobio gobio), a common freshwater bottom-dwelling fish. The contributions of this work are two-fold: First, we provide a comparison of the boundary layer thicknesses and velocity profiles at flow velocities ranging from 0.25 to 1.25 m/s. Second, we qualitatively compare the spatial distributions of the pressure coefficient, dynamic pressure and shear stresses to biological observations of the neuromast locations of adult gudgeon.

Heat transfer enhancement for electro-thermo-convection of FENE-P viscoelastic fluid in a square cavity

This study numerically investigates the heat transfer enhancement for viscoelastic electro-thermo-convection in a two-dimensional differentially heated cavity with injection from below. The flow motion is assumed to be incompressible, which is driven by the Coulomb force and the thermal buoyant force. The polymers are described by the FENE-P model which exhibits typical shear-thinning and elastic properties. Based on the extensibility parameter (L), the cases are divided into several scenarios, corresponding to weak elasticity with strong shear-thinning, moderate elasticity with moderate shear-thinning, and strong elasticity with weak shear-thinning, respectively. We find that the competition between the shear-thinning and elasticity dominates the flow state and heat transport. The shear-thinning effect tends to facilitate heat transfer, while its elastic properties tend to decrease it. In the scenario of weak elasticity with strong shear-thinning (L2=10), the polymer additives significantly improve the heat transfer enhancement (HTE) of the electric field as the polymer viscosity ratio (β) decreases or Weissenberg number (Wi) increases, where the maximum HTE reaches around 92.1%. The amount of HTE first increases rapidly with Wi but then remains almost constant once a critical Wi is exceeded. However, the HTE significantly decreases in the scenario of strong elasticity with weak shear-thinning (300≤L≤1000) since the elasticity dominates over the shear-thinning. These heat transfer performances are then corroborated with the boundary layer and kinetic energy budget analysis.

Boundary layer analysis on magnetohydrodynamic dissipative Williamson nanofluid past over an exponentially stretched porous sheet by engaging OHAM

PurposeThis investigation delves into the rationale behind the preferential applicability of the non-Newtonian nanofluid model over alternative frameworks, particularly those incorporating porous medium considerations. The study focuses on analyzing the mass and heat transfer characteristics inherent in the Williamson nanofluid’s non-Newtonian flow over a stretched sheet, accounting for influences such as chemical reactions, viscous dissipation, magnetic field and slip velocity. Emphasis is placed on scenarios where the properties of the Williamson nanofluid, including thermal conductivity and viscosity, exhibit temperature-dependent variations.Design/methodology/approachFollowing the use of the OHAM approach, an analytical resolution to the proposed issue is provided. The findings are elucidated through the construction of graphical representations, illustrating the impact of diverse physical parameters on temperature, velocity and concentration profiles.FindingsRemarkably, it is discerned that the magnetic field, viscous dissipation phenomena and slip velocity assumption significantly influence the heat and mass transmission processes. Numerical and theoretical outcomes exhibit a noteworthy level of qualitative concurrence, underscoring the robustness and reliability of the non-Newtonian nanofluid model in capturing the intricacies of the studied phenomena.Originality/valueAvailable studies show that no work on the Williamson model is conducted by considering viscous dissipation and the MHD effect past over an exponentially stretched porous sheet. This contribution fills this gap.

Investigations into dynamic variation characteristics of near-bottom boundary flows over a long term at strait areas

Dynamic processes at the bottom boundary layers are crucial for various marine applications such as marine energy exploration and environmental protection. This paper explored hydrodynamic characteristics of near-bottom boundary currents based on the continuous measured velocity data at a site near the bottom boundary layer in the Strait of Georgia. Previous studies on the bottom boundary layer encompasses various aspects, including physical, chemical, and biological dimensions. However, the mechanism underlying the hierarchical structure of flow velocity remain poorly understood. Thus, the power spectrum and wavelet analysis were used to analyze the significant period of the averaged velocity and tidal characteristics, and to calculate the thickness of near-bottom boundary layer. The results indicate that the current velocities at the boundary layer tend to flow in south-eastern and north-western directions. There is a significant period of 15 days in August and September. At a depth of 301 m, the vertical velocities are stratified, indicating different velocity layers. The boundary layer thickness varies from 5.23 to 14.74 m, as indicated by the vertical structural characteristics. The structural characteristics of the seabed boundary layer are conducive to deepening our understanding of sediment's incipient motion, transport, and sedimentation process. The findings provide a theoretical foundation for future studies on sediment movement and the numerical simulation analysis of the bottom boundary layer. It is of great significance for the study of wave flow interaction and marine dynamic structure, and can provide scientific support for the development and utilization of the strait area.

Linear stability analysis of compressible boundary layer over an insulated wall using compound matrix method: Existence of multiple unstable modes for Mach number beyond 3

The linear spatial stability of a parallel two-dimensional (2D) compressible boundary layer on an adiabatic plate is investigated by considering both 2D and three-dimensional (3D) disturbances. The compound matrix method is employed here, for the first time, for compressible flows, which, unlike other conventional techniques, can efficiently eliminate the stiffness of the equations governing the spectral amplitudes. The method is first validated with published results in the literature corresponding to spatial and temporal instability of flows ranging from low subsonic to high supersonic Mach numbers (M), which shows a good match depending upon the proper choice of free-stream temperature and the wall dispersion relation. Subsequently, flow compressibility effects and the spanwise variation of disturbances are also investigated for M ranging from low subsonic to high supersonic cases (from M = 0.1 to 6). Mack (AGARD Report No. 709, 1984) reported the existence of two unstable modes for M > 3 from viscous calculations (the so-called “second mode”) that subsequently fuse to create only one unstable zone when M increases. Our calculations show a series of higher-order unstable modes for M > 3 in addition to the findings of Ma and Zhong [“Receptivity of a supersonic boundary layer over a flat plate. Part 1. Wave structures and interactions,” J. Fluid Mech. 488, 31–78 (2003)], where such higher-order modes for supersonic boundary layers are all noted to be spatially stable. The number and the frequency extent of the corresponding unstable zones increase with an increase in M beyond 3 while propagating downstream at a higher speed than those corresponding to incompressible, subsonic, and low supersonic (M < 2) cases.

Analysis of interfacial effect and boundary layer for forced convection on the macroscopic hydrophilic-hydrophobic hybrid surface: A large eddy simulation

Hydrophilic-hydrophobic hybrid surfaces are developed to solve the flow and heat transfer performance contradiction. However, hybrid surfaces often have micro- or nano-scale featured sizes and are used in phase change heat transfer because hydrophilic regions contribute to droplet nucleation, and hydrophobic regions contribute to bubble nucleation. In this study, large eddy simulation is used to investigate the forced convection on macroscopic hydrophilic-hydrophobic hybrid surfaces where only the surface local wettability is changed. Three hybrid surfaces with different hydrophilic-hydrophobic ratios and two homogeneous wettability surfaces are designed, and representative flow Reynolds numbers of 4000, 6000, 10 000, and 40 000 are explored to achieve different turbulent styles. The transient parameters of kinematics, vorticity, and boundary layer are analyzed to clarify the mechanism of turbulence change and eddy generation and explain the causes of variations in flow and heat transfer performances. It proves that macroscopic hydrophilic-hydrophobic hybrid surfaces are suitable for forced convection due to the drag reduction on hydrophobic regions, backflows at hydrophilic-hydrophobic interfaces, and eddies at hydrophobic-hydrophilic interfaces, which can enhance the internal disturbance and harmonize the flow and heat transfer performances. The mechanism has a profound significance in broadening the application of hydrophilic-hydrophobic hybrid surfaces and designing the arrangement of hydrophobic regions.

Use of tritium-rich fuel to improve the yield of layered deuterium/tritium inertial fusion capsules

In deuterium–tritium (DT) ice layered implosions, nearly all hot spot mass at peak burn comes from the dense fuel. Accurate prediction of the fuel mass ablation, including the enthalpy associated with mass inflow into the hot spot from the dense fuel, is essential to understanding the energetics and ignition of the hot spot in layered implosions. A recently published boundary layer analysis (Daughton et al., 2023) indicates a faster mass ablation rate than in previous analyses of layered implosions. Inclusion of this effect provides a better match to simulations and leads to a new ignition threshold where the temperature of the dense fuel plays a critical role. This analysis motivates possible new directions for improved capsule performance. Here, the authors present evidence in support of one such approach: the use of tritium-rich ice to decrease 14 MeV neutron scattering and heating of the dense fuel, resulting in less mass ablation and more robust burn of the hot spot. It is found from numerical simulations that despite a less favorable D:T ratio in the ice, the use of a 40:60 D:T ratio leads to an increase in capsule yield of 17% percent compared with that of a 50:50 D:T ratio fuel for capsules resembling those of the recent N210808 ignition experiment on the NIF (Abu-Shawareb et al., 2022) and an increase of 74% compared with that of a 60:40 D:T ratio fuel capsule. These results are potentially important for modeling all layered implosions, since some degree of DT fractionization may arise naturally during the beta layering process. In addition, this physics is important for the feasibility of high-gain capsule designs that seek to minimize tritium usage, as in some inertial fusion energy concepts.

Near-infrared imaging of heat transfer behavior between gadolinium and fluid during magnetization/demagnetization process of magnetocaloric effect

This paper reports on the application of a near-infrared (NIR) imaging system for visualizing heat transfer dynamics from a bulk gadolinium (Gd) sample to the surrounding water during the magnetization/demagnetization process of the magnetocaloric effect (MCE). The suggested approach relied on the spectral variation in water absorption band at 1150 nm wavelength within the NIR spectrum. An experimental setup integrated a telecentric uniform-illumination system, a halogen lamp, and an NIR camera to enable real-time monitoring of a single magnetization and demagnetization cycle induced by an external magnetic field, which was generated by a permanent-magnet-based magnetic circuit. Two-dimensional absorbance images captured during this cycle clearly depicted the thermal energy generated by the MCE in water. Furthermore, an analysis of the thermal boundary layer and the quantification of heat transfer from Gd to water provided insights into the dynamics over time. These results indicated the potential of our NIR imaging techniques in optimizing thermal–fluid interactions within MCE systems, thereby improving the design and efficiency of magnetic refrigeration systems.

Semi-analytic PINN methods for boundary layer problems in a rectangular domain

Singularly perturbed boundary value problems pose a significant challenge for their numerical approximations because of the presence of sharp boundary layers. These sharp boundary layers are responsible for the stiffness of solutions, which leads to large computational errors, if not properly handled. It is well-known that the classical numerical methods as well as the Physics-Informed Neural Networks (PINNs) require some special treatments near the boundary, e.g., using extensive mesh refinements or finer collocation points, in order to obtain an accurate approximate solution especially inside of the stiff boundary layer. In this article, we modify the PINNs and construct our new semi-analytic SL-PINNs suitable for singularly perturbed boundary value problems. Performing the boundary layer analysis, we first find the corrector functions describing the singular behavior of the stiff solutions inside boundary layers. Then we obtain the SL-PINN approximations of the singularly perturbed problems by embedding the explicit correctors in the structure of PINNs or by training the correctors together with the PINN approximations. Our numerical experiments confirm that our new SL-PINN methods produce stable and accurate approximations for stiff solutions.

Contact area analysis of an asphalt-concrete boundary layer with X-ray computed tomography imaging

This study focusses on the bond between asphalt and concrete. The bond is a decisive factor for the transfer of traffic loads. A lack of bond between the layers can quickly reduce the load-bearing behavior and thus lead to pavement failure. In the literature, the bond between asphalt pavements and concrete substrates is usually analyzed using mechanical test methods. With mechanical test methods, the effects of interlocking, glue adhesion and friction, which have an effect at the layer boundary, are not considered individually. Based on previous findings, the aim of this study is to carry out a precise analysis of the bonding at the layer interface between concrete and asphalt using bitumen emulsion by means of measurements with a high-resolution X-ray computer tomography (CT). To determine the effect of the surface texture of the substrate, the investigations were carried out on milled and smooth surface textures. In addition to the variation of the substrate profiles, the influence of the application quantity of the bitumen emulsions C40 B5-S and C60 BP4-S was analyzed by spraying quantities of 90 g/m² and 180 g/m² bitumen after breaking. A newly developed method was used to determine the effective degree of gluing at the layer boundary, which allowed the bonding efficiency to be determined. The results of the calculated degree of gluing for the individual variants show that the highest degree of gluing of 61.94% is achieved with the smooth texture variant and high application quantity of bitumen emulsion. The lowest degree of gluing of 17.01% was achieved with the milled texture variant without the application of bitumen emulsion. Overall, the results showed that the degree of gluing was increased for both surface textures by applying the bitumen emulsion. Compared to the smooth texture, the milled textures exhibit a lower degree of gluing for both bitumen quantities, which could be explained by the distribution of the bitumen emulsions on the substrate profiles.

Singular layer physics informed neural network method for plane parallel flows

We construct in this article the semi-analytic Physics Informed Neural Networks (PINNs), called singular layer PINNs (or sl-PINNs), that are suitable to predict the stiff solutions of plane-parallel flows at a small viscosity. Recalling the boundary layer analysis, we first find the corrector for the problem which describes the singular behavior of the viscous flow inside boundary layers. Then, using the components of the corrector and its curl, we build our new sl-PINN predictions for the velocity and the vorticity by either embedding the explicit expression of the corrector (or its curl) in the structure of PINNs or by training the implicit parts of the corrector (or its curl) together with the PINN predictions. Numerical experiments confirm that our new sl-PINNs produce stable and accurate predicted solutions for the plane-parallel flows at a small viscosity.

Application of the Fibonacci wavelet approach to the MHD boundary layer analysis of a Casson fluid past a stretching sheet

The current work investigates the MHD boundary layer flow of Casson fluid over a stretching sheet with the help of Fibonacci wavelet approach. The stable laminar MHD flow, mass and heat transport properties of a nanofluid over a stretching sheet in the presence of a fluctuating magnetic field has been under scrutiny. By choosing relevant non-dimensional parameters, the governing boundary layer equation undergoes transformation into a dimensionless Falkner-Skan equation. The Fibonacci integral operational matrix is used to solve the achieved nonlinear equation and it has been used for the first time to solve stretching sheet problem in fluid dynamics. The nature of the boundary layer flow are analyzed graphically for a range of distinct values of physical parameters. Impact of parameters on axial and transverse velocity has been investigated. It can be observed that local skinfriction coefficient grows for larger values of Casson parameter for ϵ = 0.3 and decreases for higher values of Casson parameter and decreases for higher values of magnetic parameter, angle of inclination and pressure gradient parameter. To ensure the accuracy of the findings, local skin friction is recorded and contrasted with various approaches that are already documented in research.

Wake Formation Flow Physics and Boundary Layer Analysis on the Sides of the Isosceles Triangular Cylinder with Apex Pointing Downstream

Boundary layer interaction with downstream flow structures was numerically studied to find the region of inactivity behind an 75° isosceles triangular cylinder with apex pointing downstream at intermediate Reynolds numbers (Re = 520, 640, 840 and 1040). The Standard k-ε model in OpenFOAM was used in the study. Numerical results were validated against Particle Image Velocimetry data. Results revealed the stable region of inactivity characterized by low turbulent kinetic energy and vorticity. The onset of secondary vortex and separation point, independent of Reynolds number, was identified. The onset of the secondary vortex was located at (x = 2 mm) from the base and (y = 1.5 mm) from the apex on either side of the cylinder. The ratio of modulus of absolute primary z-component of vorticity, |ω_z^1 |, to the modulus of secondary z-component of vorticity, |ω_z^2 |, was found to be approximately equal to 1.2. This ratio is invariant of the Reynolds number of the study. These findings have practical implications. The unique properties of the inactivity region forms an ideal location that can be used for injecting fluid, placing measurement probe, active flow control and drag reduction. The research problem is formulated in the introduction. Literature is reviewed next providing the background. Details about the range of parameters, governing equations, numerical study details and software used are given in the methodology section. The results section gives the numerical results, verified by mesh refinement test and validated against experimental results. The results are finally discussed in the next section.

Effect of surface radiation on the stability of hypersonic flat plate boundary layer

Transition of hypersonic boundary layer flow is an important problem in aerodynamic research. To obtain the natural transition position using the flow stability analysis method, it is necessary to analysis the neutral curve and the perturbation growth rule. Accurate calculation of the basic flow is the basis and prerequisite for stability analysis. This paper proposes that surface radiation should be considered in the analysis of hypersonic boundary layer flow stability due to the high temperatures at the vehicle surface caused by aerodynamic heating. In order to obtain the influence of surface radiation on the flow stability, we assume that the surface satisfies the energy balance and the gas satisfies calorimetrically perfect gas properties. To simplify the analysis, gas radiation has been neglected. The boundary layer analysis of a hypersonic flat plate with and without radiation has been performed by a combination of flow stability theory and direct numerical simulation. It is found that high temperature radiation on the surface causes energy to transfer outwards. The results showed that surface temperature decreases along the streamwise direction and is no longer an adiabatic surface temperature. Instability interval decreases and the maximum growth rate increases. The range of the instability region expressed in dimensionless frequency values moves to lower frequency as the Mach number increases. Maximum growth rate shows a tendency to increase and then decrease as the Mach number increases.

Effect of radiation and slip parameter on MHD mixed convective Casson fluid flow along a vertical stretching sheet

The flow behavior of Casson fluids along a vertical stretching sheet is important for a wide range of industrial processes where such fluids are encountered. For example, knowledge of this study might be used in the design of coating processes or in biomedical applications involving blood flow. Aims of this study are the optimization of engineering processes, ensuring efficiency and quality in various applications and analysis of boundary layer along stretching sheet. Mathematical formulation of the problem consists parial differential equations and later on converted to a set of ordinary differential equations (ODEs) with boundary conditions and solved using “Bvp4c solver” in MATLAB. The Bvp4c solver is a numerical technique used to solve boundary value problems (BVPs) for ODEs. For various values of physical parameters the velocity, temperature, and concentration distribution are presented through figures. Skin friction coefficient, Nusselt number, and Sherwood number are tabulated and analyzed through bar graphs. It is noted that for assisting flow [Formula: see text], velocity profile decreases near the sheet while it increases away from the sheet for growing inputs of Casson fluid parameter ([Formula: see text]), while opposite seniro has been observed for Hartmann number ([Formula: see text]). Coefficient of surface drag is lowered with growing inputs of Casson fluid parameter and heat transfer rate is enhanced with increased inputs of Hartmann number in both assisting and opposing flow. The temperature profile enhances for raising inputs of radiation parameter ([Formula: see text]) in both assisting and opposing flow. The concentration profiles also fall, when chemical reaction parameter ([Formula: see text]) and Schmidt number ([Formula: see text]) are enhanced.

Eddy-viscosity-improved resolvent analysis of compressible turbulent boundary layers

An improved resolvent analysis is proposed in the regime of compressible turbulent boundary layers. To better model nonlinear processes in the input, the resolvent framework is augmented by adding eddy viscosity. To this end, we propose two eddy-viscosity models: a modified Cess eddy-viscosity model coupling the compressibility transformation and outer-layer correction, and a new eddy-viscosity model based on an empirical relationship and mixing-length theory. Both are incorporated into the resolvent operator to examine the performance of the eddy-viscosity-improved resolvent-based reduced-order modelling. Results of the augmented resolvent analysis are compared qualitatively and quantitatively with the first leading mode of spectral proper orthogonal decomposition, by checking the profiles and cross-spectral densities of velocities, density and temperature in two hypersonic turbulent boundary layers under different wall conditions. Higher accuracy of the turbulence prediction is achieved by adding the proposed eddy-viscosity models, particularly for the energetic cycle in the outer-layer region where strong nonlinear energy transfer exists.

Consistent outer scaling and analysis of adverse pressure gradient turbulent boundary layers

Under adverse pressure gradient (APG) conditions, the outer regions of turbulent boundary layers (TBLs) are characterized by an increased velocity defect $U_{e}-U$ , an outwards shift of the peak value of the Reynolds shear stress $-\langle uv\rangle$ and an appearance of the outer peak value of the Reynolds normal stress $\langle uu\rangle$ . Here $U_{e}$ is the TBL edge velocity. Scaling APG TBLs is challenging due to the non-equilibrium effects caused by changes in the APG. To address this, the response distance of TBLs to non-equilibrium conditions is utilized to extend the Zagarola–Smits scaling $U_{zs} = U_{e}({\delta ^{*} }/{\delta })$ and ensure that the original properties of the Zagarola–Smits scaling are maintained as $Re \to \infty$ . Here $\delta ^{*}$ is the displacement thickness and $\delta$ is the boundary layer thickness. Based on the established correlation between $U_{e}-U$ and $-\langle uv\rangle$ , the scaling is extended to $-\langle uv\rangle$ . Furthermore, considering the coupling relationship between Reynolds stress components, the scaling is extended to encompass each Reynolds stress component. The proposed consistent scaling is verified using five non-equilibrium databases and five near-equilibrium databases, successfully collapsing the data of the TBL outer region. The pressure gradient parameter $\beta =({\delta ^{*} }/{\rho u_{\tau }^{2} }) ({\mathrm {d} P_{e} }/{\mathrm {d}\kern0.7pt x})$ of these databases spans two orders of magnitude. Here $P_{e}$ is the boundary layer edge pressure, $u_{\tau }$ is the friction velocity and $\rho$ is the density. Finally, the influence of the APG on the inner and outer regions of TBLs is analysed using the mean momentum balance equation. The analysis suggests that the shift of the $-\langle uv\rangle$ peak to the outer region under APG conditions is due to an insufficient inertia term near the inner region to balance the APG. It is observed that the APG promotes interaction between the inner and outer regions of TBLs, but the inner and outer regions still retain distinctive properties.