Gradient Parameter Research Articles

On the thermal performance of radiative stagnation-point hybrid nanofluid flow across a wedge with heat source/sink effects and sensitivity analysis

The present article aims to examine the thermal performance and the sensitivity analysis of a GO−TiO2/water hybrid nanofluid in the presence of different nanoparticle shapes along with heat absorption and thermal radiation effects over a wedge geometry. Analyzing the effects of heat generation and radiation effects is one of the key studies conducted by researchers in various nanofluid flows over some required geometries. However, a combined study of these effects has yet to be studied over a moving wedge, and that combination defines the novelty of the work. Similarity transformations are implemented to the governing equations to obtain the final set of nondimensional equations, which are solved using the bvp4c code in MATLAB. The results obtained were in close agreement with the published results. The Nusselt number decreased with an increase in the heat source parameter Q, and it increased with an increasing Hartree pressure gradient β and thermal radiation parameter Rd. The sensitivity is statistically analyzed for the variations in radiation effect, heat source, and pressure gradient parameters on the Nusselt number. The high values for R2=99.99% and Adj R2=99.96% validate the ANOVA results obtained using a Box–Behnken design (BBD) model in the response surface methodology (RSM) with 14 degrees of freedom. The input parameters Rd and β show positive sensitivity, while Q shows negative sensitivity toward the skin friction. The Nusselt number proves to be most sensitive toward the pressure gradient parameter. TiO2, graphene (Gr), and the derivative forms of graphene, are gaining much importance due to their wide applications in the oil and petroleum industries. Thus, this study contributes to lubrication purposes, emulsion stabilizers, oxalic acid removal, anti-corrosive properties, etc.

Free Vibration of Thermally Loaded FG-GPLRC Nanoplates Integrated with Magneto-Electro-Elastic Layers in Contact with Fluid

This work investigates the free vibrations of innovative thermally loaded nanoplates constructed by integrating magneto-electro-elastic (MEE) layers with functionally-graded graphene platelet-reinforced composite cores (FG-GPLRC) and accounting for viscous fluid interactions. An advanced multiphysics model is developed using the Navier–Stokes equations to capture fluid structure coupling effects, Halpin–Tsai, and the rule of mixtures micromechanics to predict the non-uniform effective properties, third-order shear deformation plates theory (TSDPT) to incorporate thickness stretching, and the nonlocal strain gradient theory (NSGT) to characterize size dependencies. The Galerkin technique is used to solve the governing equations, which are derived from the Hamilton’s principle. Parametric analyses quantify the influences of fluid depth, temperature fluctuations, temperature profiles, nonlocal and strain gradient parameters, electric and magnetic potentials, graphene distribution patterns, graphene weight fractions, and boundary conditions on the vibration response. The outcomes of this study provide design guidelines and predictive tools enabling active vibration control systems for next-generation thermally-loaded nanocomposite structures with widespread applications from aerospace vehicles to nanoelectronics.

High-resolution boundary variation diminishing scheme for two-phase compressible flow with cavitation and evaporation

We propose high-resolution numerical schemes for two-phase compressible flow simulations to reproduce dynamically created gas/vapor-liquid interfaces with phase change. In the Godunov-type finite volume framework, suppressing numerical dissipation errors in numerical schemes is crucial for capturing the discontinuous solutions. The MUSCL scheme has second-order accuracy for smooth solutions and non-oscillatory behavior near discontinuous solutions. However, the MUSCL scheme introduces excessive numerical dissipation and diffuses the discontinuities nonphysically, leading to difficulties in distinguishing gas/vapor and liquid phases, and thus causing a blur in the interfaces during the multi-phase flow simulations.The hybrid-type boundary variation diminishing (BVD) scheme in this paper combines the MUSCL scheme and the THINC scheme to reduce the numerical dissipation errors near the discontinuities. The MUSCL-THINC-BVD scheme applies the MUSCL scheme for smooth solutions and the THINC scheme for discontinuous solutions, resulting in the successful capture of the discontinuities including the dynamically created gas/vapor-liquid interfaces. The Adaptive THINC-BVD scheme, which switches two types of THINC schemes with different values of gradient parameter, also captures the discontinuities clearly. The numerical results of the benchmark tests show that the proposed BVD schemes can lucidly reproduce the vapor-liquid interfaces newly created during the dynamical process of phase change.

The B-spline mapping function (BMF): representing anisotropic troposphere delays by a single self-consistent functional model

Troposphere’s asymmetry can introduce errors ranging from centimeters to decimeters at low elevation angles, which cannot be ignored in high-precision positioning technology and meteorological research. The traditional two-axis gradient model, which strongly relies on an open-sky environment of the receiver, exhibits misfits at low elevation angles due to their simplistic nature. In response, we propose a directional mapping function based on cyclic B-splines named B-spline mapping function (BMF). This model replaces the conventional approach, which is based on estimating Zenith Wet Delay and gradient parameters, by estimating only four parameters which enable a continuous characterization of the troposphere delay across any directions. A simulation test, based on a numerical weather model, was conducted to validate the superiority of cyclic B-spline functions in representing tropospheric asymmetry. Based on an extensive analysis, the performance of BMF was assessed within precise point positioning using data from 45 International GNSS Service stations across Europe and Africa. It is revealed that BMF improves the coordinate repeatability by approximately 10%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$10\\%$$\\end{document} horizontally and about 5%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$5\\%$$\\end{document} vertically. Such improvements are particularly pronounced under heavy rainfall conditions, where the improvement of 3-dimensional root mean square error reaches up to 13%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$13\\%$$\\end{document}.

Generalised Couette flow of nanoliquid among two concentric channels with particles injection

In the presence of nanoparticle injection, buoyancy power, constant pressure gradient, variable viscosity coefficient, magnetic field, mass diffusivity of thermophoresis, and Brownian motion, the hydrodynamic nanoliquid conducted with electric current flows through the gap between two concentric cylindrical channels of infinite length. The focus of this article was on generalised Couette flow that is flow in which the outer channel plate moves at a uniform velocity while still being subjected to a pressure gradient. Here we use the RKF45 numerical method and the shooting technique to solve dimensionless momentum, energy, and concentration equations. How the obtained parameters controlled on flow rate, temperature and concentration of the fluid have been investigated and appearances through respective graphs. The high magnitude of nanoliquid flow rate with outer channel wall motion and pressure gradient parameters has been observed. The nanoliquid temperature declines with an annular gap parameter while amplified thermophoresis and magnetic parameter values enrich the fluid temperature. In addition, loftier values of variable viscosity and Brownian motion parameters boost the nanoliquid concentration.

A new chromatographic response function with automatically adapting weight factor for automated method development

A critical factor for automated method development in chromatography is the maximization or minimization of an objective function describing the quality (and speed) of the separation. In chromatography, this function is commonly referred to as a chromatographic response function (CRF). Many CRFs have previously been introduced, but many have unfavourable properties such as featuring multiple optima, insufficient discriminatory power, and a too strong dependence on the weight factors needed to balance resolution and time penalty components. To overcome these problems, the present study introduces a new type of CRF wherein the relative weight of the time penalty term is a self-adaptive function of the separation quality. The ability to unambiguously identify the optimal gradient settings of this newly proposed CRF is compared to that of some of the most frequently used CRFs in a study covering 100 randomly composed in silico samples. Doing so, the new CRF is found to flawlessly lead to the correct solution (=linear gradient parameters providing the highest resolution in the shortest potential time) in 100 % of the cases, while the most frequently used literature CRFs were off-target for about 50 to 60 % of the samples, even when considering the availability of spectral peak shape data. Some slight alterations to the proposed CRF are introduced and discussed as well.

Spectral-like conjugate gradient methods with sufficient descent property for vector optimization.

Several conjugate gradient (CG) parameters resulted in promising methods for optimization problems. However, it turns out that some of these parameters, for example, 'PRP,' 'HS,' and 'DL,' do not guarantee sufficient descent of the search direction. In this work, we introduce new spectral-like CG methods that achieve sufficient descent property independently of any line search (LSE) and for arbitrary nonnegative CG parameters. We establish the global convergence of these methods for four different parameters using Wolfe LSE. Our algorithm achieves this without regular restart and assumption of convexity regarding the objective functions. The sequences generated by our algorithm identify points that satisfy the first-order necessary condition for Pareto optimality. We conduct computational experiments to showcase the implementation and effectiveness of the proposed methods. The proposed spectral-like methods, namely nonnegative SPRP, SHZ, SDL, and SHS, exhibit superior performance based on their arrangement, outperforming HZ and SP methods in terms of the number of iterations, function evaluations, and gradient evaluations.

Design of energy-absorbing bionic workstation table for trains and research on occupant collision injury

Secondary impact is a significant contributor to occupant injuries in train crashes. To mitigate collision injuries, a workbench with energy-absorbing properties was developed, drawing inspiration from leaf-vein and honeycomb structures. A local collision model of a dummy compartment was established to evaluate the impact of different sandwich thicknesses on occupant injuries. By optimizing the wall thickness of the energy-absorbing sandwich, the optimal gradient parameters were determined. The assessment of occupant collision injury from the workstation table with a graded wall thickness sandwich was conducted using the LSTC Hybrid III 50th seating dummy. The results demonstrate that the workstation table influences the occupant's movement posture during a collision, preventing head impact with the tabletop. Apart from slightly elevated leg injuries in the dummy, which remained within permissible values, the head, neck, and chest injuries were reduced by 98.45%, 48.17%, and 81.18%, respectively.

Effects by geothermal gradient on thermal response tests for boreholes

A thermal response test (TRT) on a borehole provides information needed in the design of ground heat exchangers for ground source heat pump systems. Conventional analysis of TRT data sets uses the late-time period to estimate the ground thermal conductivity and the effective borehole resistance. During this quasi-steady-state period, the transient average fluid temperature has a logarithmic linear trend, which is consistent with conventional heat transfer models that assume a uniform undisturbed ground temperature. As the depth of the borehole increases the undisturbed ground temperature increases with depth following the geothermal gradient. This paper focuses on how the geothermal gradient affects a TRT. As the geothermal gradient increases, the required time to reach the conventional logarithmic linear trend may increase beyond any practical testing duration in coaxial boreholes. Then, conventional analysis methods based on a quasi-steady-state period incorrectly estimate the ground thermal conductivity. The magnitude of this effect is controlled through a dimensionless parameter |qratio|, which includes the geothermal gradient and other test parameters. In place of conventional models, a transient model that includes the geothermal gradient and fluid thermal storage can estimate the ground thermal conductivity and local borehole resistance within the typical TRT testing duration.

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.

Numerical analysis of novel wavy wall based control of turbulent boundary layer separation

In this work, a comprehensive insight into the new passive flow control method, i.e. a two-dimensional streamwise waviness (recently proposed by Dróżdż et al. [1]) is provided using Large Eddy Simulations. This approach was demonstrated to be effective in postponing turbulent boundary layer separation at high Re, which is out of reach for other commonly known passive flow control strategies. Although the effectiveness of the wavy wall in delaying flow separation has been confirmed, the physical mechanisms standing behind the skin friction enhancement are not fully understood which opens a space for future work. Of particular interest is an insight into the fluid flow in the near-wall region just above the waviness and this paper is aimed at doing so. To separate the effect of the global adverse pressure gradient, the results were confronted with the ones obtained under zero pressure gradient conditions for the identical wavy wall geometry and the same Reτ=2500. Also additional simulations of the turbulent boundary layer building up on the flat wall were conducted to have an additional reference for the flow fields obtained above the wavy wall under zero and adverse pressure gradients respectively. A detailed inspection of the streamwise velocity and its fluctuation profiles indicated that the contribution of the global adverse pressure gradient to the flow statistics begins when the Rotta-Clauser pressure gradient parameter β≥3. The paper confirms an enhancement of the wall shear stress τw downstream the corrugation, which is the necessary condition to postpone the separation. Interestingly, the effect of increased τw was not observed for the case with the same waviness and without pressure gradient.

A multi-band elastic metamaterial for low-frequency multi-polarization vibration absorption

The vibration of engineering structures in actual practice occurs across numerous frequency ranges and includes diverse polarization modes such as bending, torsion, and expansion. Nevertheless, most reported elastic metamaterials are designed for a single frequency range or a single elastic wave mode, thereby making it challenging to simultaneously suppress the propagation of vibrational energy across multiple frequency ranges and polarization modes. To address this limitation, we propose a two-degree-of-freedom elastic metamaterial exhibiting a multi-frequency range elastic band gap and multi-polarization mode elastic wave band gap. By incorporating periodic vibration-absorbing units, gradient parameter vibration-absorbing units, and different polarization mode vibration-absorbing units to the actual equipment support structure, we achieve respective vibration attenuation enhancement, broadband vibration absorption, and multi-polarization mode vibration absorption. We systematically verify the effectiveness of our design through kinetic theoretical analysis, full-wave field simulations, and experimental tests. This metamaterial vibration absorption device can effectively meet the application requirements of complex vibration reduction involving multi-frequency ranges and modes in practical engineering, providing novel insights into equipment vibration and noise reduction design that holds significant reference value for engineering applications.

A one-dimensional urban flow model with an eddy-diffusivity mass-flux (EDMF) scheme and refined turbulent transport (MLUCM v3.0)

Abstract. In recent years, urban canopy models (UCMs) have been used as fully coupled components of mesoscale atmospheric models as well as offline tools to estimate temperature and surface fluxes using atmospheric forcings. Examples include multi-layer urban canopy models (MLUCMs), where the vertical variability of turbulent fluxes is calculated by solving prognostic momentum and turbulent kinetic energy (TKE, k) using mixing length scale (l) and drag parameterizations. These parameterizations are based on the well-established 1.5-order k−l turbulence closure theory and are often informed by microscale fluid dynamics simulations. However, this approach can include simplifications such as assuming the same diffusion coefficient for momentum, TKE, and scalars. In addition, the dispersive stresses arising from spatially averaged flow properties have been parameterized together with the turbulent fluxes despite being controlled by different mechanisms. Both of these assumptions impact the quantification of the turbulent exchange of flow properties and subsequent air temperature predictions in urban canopies. To assess these assumptions and improve corresponding parameterization, we analyzed 49 large-eddy simulations (LES) for idealized urban arrays, encompassing variable building height distributions and a comprehensive range of urban densities (λp∈[0.0625,0.64]) seen in global cities. We find that the efficiency of turbulent transport (numerically described via diffusion coefficients) is similar for scalars and momentum but is 3.5 times higher for TKE. Additionally, parameterizing the dispersive momentum flux using the k−l closure was a source of error, while scaling with the pressure gradient and urban morphological parameters appears more appropriate. In response to these findings, we propose two changes to the previous version of MLUCM: (a) separate characterization for turbulent diffusion coefficient for momentum and TKE and (b) introduction of an explicit physics-based “mass-flux” term to represent the fraction of the dispersive momentum transport directly induced from buildings as an amendment to the existing “eddy-diffusivity” framework. The updated one-dimensional model, after being tuned for building height variability, is further compared against the original LES results and demonstrates improved performance in predicting vertical turbulent exchange in urban canopies.

Turbulent activity in the near-wall region of adverse pressure gradient turbulent boundary layers

Two direct numerical simulation (DNS) databases are investigated to understand the effect of the outer-layer turbulence on the inner layer’s structures and energy transfer mechanisms. The first DNS database is the non-equilibrium adverse-pressure-gradient (APG) turbulence boundary layer (TBL) of Gungor et al. [1]. Its Reynolds number and the inner-layer pressure gradient parameter reach above 8000 and 10, respectively. The shape factor spans between 1.4 and 3.3, which indicates the flow has various velocity defect situations. The second database is the same flow as the first one but the outer layer turbulence is artificially eliminated in this flow. Turbulence is removed above 0.15 local boundary layer thickness. For the analysis, we chose four streamwise positions with small, moderate, large, and very-large velocity defect. We compare the wall-normal distribution of Reynolds stresses, two-point correlations and spectral distributions of energy, production and pressure strain. The results show that the inner layer turbulence can sustain itself when the outer-layer turbulence does not exist regardless of the velocity defect or the pressure gradient. The two-point correlations of both cases show that outer large-scale structures affect the inner layer structures significantly. The streamwise extent of the correlation contours scales with pressure-viscous units. This shows the importance of the pressure gradient’s effect on the inner-layer structures. The spectral distributions demonstrate that the energy transfer mechanisms are probably the same in the inner layer regardless of the velocity defect, which suggests the near-wall cycle may exist even in very-large defect APG TBLs where the mean shear in the inner layer is considerably lower than small-defect APG TBLs.

Accurate free and forced vibration behavior prediction of functionally graded sandwich beams with variable cross-section: A finite element assessment

The current investigation proposes an enhanced finite element beam model to assess the dynamic behavior of functionally graded sandwich beams with variable cross-sections. By integrating the shear effect, this beam model integrates both C0 and C1 continuities to generate elementary matrices using an efficient three-unknown shear deformation beam theory. The model focuses on advanced sandwich beams with functionally graded material face-sheets and metallic cores, exploring various geometric arrangements. The originality of this study addresses variable cross-sections in sandwich beams, evaluating the influence of material composition and geometric configurations on free and forced vibration responses, including different boundary conditions. The methodology uses the eigenvalue method for examining free vibration and applies Newmark’s algorithm for solving forced vibration problems. The current model expands the scope of analysis compared to previous models that primarily focus on uniform cross-sections and conventional material distributions. The developed model incorporates non-uniform geometrics, material gradient parameters, and advanced finite element analysis to enhance precision and efficacy through detailed parametric investigations and comparisons with existing literature. This contribution significantly improves the mechanical modeling of composite sandwich structures, particularly those incorporating functionally graded materials and complex geometrical aspects.

Neural network architecture to optimize the nanoscale thermal transport of ternary magnetized Carreau nanofluid over 3D wedge

SignificanceIncorporation of nanoparticles in base fluid water is significant for analysis of thermal behavior of nanofluid mixtures, which has various applications in materials science and thermal engineering, and supervised neural scheme predicts the thermal behavior by solving Carreau nanofluid model. MotiveThis article brings the investigation related to prediction of thermal transport of a ternary magnetized hybrid nanofluid [(Al2O3, CuO, TiO2)/H2O] with a three-dimensional Carreau nanofluid model over a wedge. Three nanoparticles dispersed in water (H2O). Inclined magnetic field is considered for judgement of velocity profile and thermal radiation is utilized to scrutinize the temperature distribution of nanofluid. The Carreau mathematical model is chosen to depict the rheological characteristics of non-Newtonian fluids at very high and very low shear rate. MethodologyPhysical assumptions creates the system of Partial differential equations (PDEs) and these are converted into ordinary differential equations (ODEs) by similarity tool. Further ODEs are dealt with bvp4c scheme and further prediction of solution is made by Levenberg-Marquardt neural network (LM-NN) supervised neural scheme. FindingsIncreased volume friction coefficients of nanoparticles increases the transport of heat. High inclined magnetic effect, thermal radiation, pressure gradient and shear strain parameter predict higher thermal transport.

Transformer fault diagnosis method based on SMOTE and NGO-GBDT

In order to improve the accuracy of transformer fault diagnosis and improve the influence of unbalanced samples on the low accuracy of model identification caused by insufficient model training, this paper proposes a transformer fault diagnosis method based on SMOTE and NGO-GBDT. Firstly, the Synthetic Minority Over-sampling Technique (SMOTE) was used to expand the minority samples. Secondly, the non-coding ratio method was used to construct multi-dimensional feature parameters, and the Light Gradient Boosting Machine (LightGBM) feature optimization strategy was introduced to screen the optimal feature subset. Finally, Northern Goshawk Optimization (NGO) algorithm was used to optimize the parameters of Gradient Boosting Decision Tree (GBDT), and then the transformer fault diagnosis was realized. The results show that the proposed method can reduce the misjudgment of minority samples. Compared with other integrated models, the proposed method has high fault identification accuracy, low misjudgment rate and stable performance.

Frontal slope: A new measure of ocean fronts

Ocean fronts, as typical marine phenomena, significantly impact the marine environment and the distribution of organisms. However, the research on their tilted structure is limited. This study considered the frontal slope as a key parameter for describing the tilting characteristics of ocean fronts. A novel method, based on high-resolution in situ observations and reanalysis data, was proposed to extract global frontal slopes using contour lines and surfaces. To validate this approach, a mathematical model was developed for the oceanic front slope and verified through in situ observations. The frontal slope equation produced reliable results, indicating that the front tilting primarily arises from shear flow and density differences. Furthermore, a modified oceanic front gradient parameter was proposed based on the frontal slope, and its significance in understanding the impacts of ocean fronts on underwater acoustic propagation was validated through numerical simulation and statistical analysis. The distribution characteristics of frontal slope for ocean fronts on a global scale were analyzed for the first time. Marginal ice zone fronts had the highest slope (>1.5 m/km), followed by tidal mixing fronts, western boundary current fronts, and water mass convergence fronts (1–1.5 m/km). In contrast, upwelling fronts and estuarine plume fronts generally had slopes below 1 m/km. This parameter provides a new perspective on ocean front systems and their influence on dynamic structures and ecological resource distributions. This study helps in a comprehensive understanding of the tilting characteristics of the ocean front and their implications.

Federated aggregation method based on cosine similarity approximation Shapley value method contribution degree

When calculating participants’ contribution to federated learning, addressing issues such as the inability to collect complete test data and the impact of malicious and dishonest participants on the global model is necessary. This article proposes a federated aggregation method based on cosine similarity approximation Shapley value method contribution degree. Firstly, a participant contribution calculation model combining cosine similarity and the approximate Shapley value method was designed to obtain the contribution values of the participants. Then, based on the calculation model of participant contribution, a federated aggregation algorithm is proposed, and the aggregation weights of each participant in the federated aggregation process are calculated by their contribution values. Finally, the gradient parameters of the global model were determined and propagated to all participants to update the local model. Experiments were conducted under different privacy protection parameters, data noise parameters, and the proportion of malicious participants. The results showed that the accuracy of the algorithm model can be maintained at 90% and 65% on the MNIST and CIFAR-10 datasets, respectively. This method can reasonably and accurately calculate the contribution of participants without a complete test dataset, reducing computational costs to a certain extent and can resist the influence of the aforementioned participants.

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.