Data obtained using direct numerical simulations (DNS) of pressure-driven turbulent channel flow are studied in the range 180 $\le$${Re}_{\tau }\le$ 10,000. Reynolds number effects on the mean velocity profile (MVP) and second order statistics are analyzed with a view of finding logarithmic behavior in the overlap region or even further from the wall, well in the boundary layer’s outer region. The values of the von Kármán constant for the MVPs and the Townsend–Perry constants for the streamwise and spanwise fluctuation variances are determined for the Reynolds numbers considered. A data-driven model of the MVP, proposed and validated for zero pressure-gradient flow over a flat plate, is employed for pressure-driven channel flow by appropriately adjusting Coles’ strength of the wake function parameter, Π. There is excellent agreement between the analytic model predictions of MVP and the DNS-computed MVP as well as of the Reynolds shear stress profile. The skin friction coefficient C_f is calculated analytically. The agreement between the analytical model predictions and the DNS-based computed discrete values of C_f is excellent.

Convolutional neural networks (CNN) have been widely adopted in fluid dynamics investigations over the past few years due to their ability to extract and process fluid flow field characteristics. Both in sparse-grid simulations and sensor-based experimental data, the establishment of a dense flow field that embeds all spatial and temporal flow information is an open question, especially in the case of turbulent flows. In this paper, a deep learning (DL) method based on computational CNN layers is presented, focusing on reconstructing turbulent open channel flow fields of various resolutions. Starting from couples of images with low/high resolution, we train our DL model to efficiently reconstruct the velocity field of consecutive low-resolution data, which comes from a sparse-grid Direct Numerical Simulation (DNS), and focus on obtaining the accuracy of a respective dense-grid DNS. The reconstruction is assessed on the peak signal-to-noise ratio (PSNR), which is found to be high even in cases where the ground truth input is scaled down to 25 times.

Wall-bounded turbulence modeling is fundamentally limited by the representation of the logarithmic law, yet its parameters such as the von Kármán constant, κ, and intercept, B, and its spatial bounds (ylow+, yhigh+) are often assumed a priori. Here we present a hybrid optimization – physics-informed neural network (PINN) framework that directly addresses this point. The method's novelty is a two-stage process: first, an optimization pre-processing step uses direct numerical dimulation data to calibrate the log-law constants (κ,B) and (ylow+, yhigh+). Next, these values are hard-coded into a PINN, whose loss function constrains the solution to obey both the governing Reynolds-averaged Navier–Stokes equations and the log-law within its identified region. By accurately delineating the log-layer we prevent the PINN from enforcing incorrect physics in the viscous sublayer, which is essential for obtaining physically consistent velocity profiles and their derivatives. This method suggests a more universal and accurate model across various Reynolds numbers, thereby enabling the development of data-informed, physics-grounded turbulence closures.

In this paper, we compare three mathematical models for the mean velocity and Reynolds stress profiles for fully developed pressure-driven turbulent channel flow with the aim of assessing the level of accuracy of each model. Each model is valid over the whole boundary layer thickness (0 $\le y\le$ $\delta$), and it is formulated in terms of a law of the wall and a law of the wake. To calibrate the mathematical models, we use data obtained by direct numerical simulations (DNS) of pressure-driven turbulent channel flow in the range 182 $\le {Re}_{\tau }\le$10,049. The models selected for performance evaluation are two models (Musker’s and AL84) originally developed based on high Reynolds boundary layer experimental data and Luchini’s model, which was developed when some DNS data were also available for wall-bounded turbulent flows. Differences are quantified in terms of local relative or absolute errors. Luchini’s model outperforms the other two models in the “low” and “intermediate” Reynolds number cases (${Re}_{\tau }=$ 182 to 5186). However, for the “high” Reynolds number cases (${Re}_{\tau }=$ 8016 and ${Re}_{\tau }=$ 10,049). Luchini’s model exhibits larger errors than the other two models. Both Musker’s and AL84 models exhibit comparable accuracy levels when compared with the DNS datasets, and their performance improves as the Reynolds number increases.

Data obtained by direct numerical simulations (DNS) of the Zero-Pressure-Gradient Turbulent Boundary Layer were analyzed and compared to a mathematical model of the mean velocity profile (MVP) in the range 1000 ≤ Re_θ ≤ 6500. The mathematical model is based on the superposition of an accurate description of the inner law and Coles’ wake function with appropriately chosen parameters. It is found that there is excellent agreement between the mathematical model and the DNS data in the inner layer when the Reynolds number based on momentum thickness, Re_θ, is greater than 1000. Furthermore, there is very good agreement over the entire boundary layer thickness, when Re_θ is greater than 2000. The diagnostic functions Ξ and Γ based on DNS data are examined and their characteristics are discussed in relation to the existence of a logarithmic layer or a power law behavior of the MVP. The diagnostic functions predicted by the mathematical model are also presented.

This paper employs Physics-Informed Neural Networks (PINNs) for the reconstruction and modelling of mean velocity profiles in fully developed turbulent channel flow over a high friction Reynolds number ($R{e}_{\tau }$). The network is trained with a high-fidelity Direct Numerical Simulation (DNS) dataset from channel flows, for $R{e}_{\tau }=395$–4186, and can extrapolate up to $R{e}_{\tau }$ = 10,049. The model predicts the mean velocity in terms of the inner-law variables, $u^{+}$, across Reynolds numbers using the inputs $\eta =y^{+}/R{e}_{\tau }$ and $R{e}_{\tau }$. A key novelty is the simultaneous optimisation of the network weights alongside two fundamental turbulence parameters, i.e., the von Kármán constant ($\kappa$) and the van Driest damping constant ($A^{+}$), allowing the PINN to autonomously calibrate the near-wall damping and log-law scaling directly from the physics-augmented loss function. The model performance is evaluated using profile-based metrics ($R^2$, mean square and absolute error) and integrated quantities (${\overline{V}}^{+}$, $R{e}_b$, and the skin-friction coefficient $C_f$), with comparisons against DNS-integrated friction values and classical theoretical values. The resulting hybrid framework offers a promising foundation for real-time digital twins and the acceleration of Computational Fluid Dynamics (CFD) solvers in canonical wall-bounded flows. By establishing a physically grounded connection between sparse data and structural constraints, these models enable accurate extrapolation into high Reynolds number regimes where the computational costs of traditional high-fidelity simulations are otherwise prohibitive.