Modified Runge–Kutta Framework for k–ω Turbulent Nanofluid Flow with Machine Learning-Based Eddy Viscosity Prediction and Sensitivity Analysis

This study develops modified second-order Runge–Kutta (RK) time integration schemes that feature a significantly extended stability region compared with classical RK approaches. To ensure superior numerical accuracy, first- and second-order spatial derivatives are approximated by a fourth-order compact finite difference scheme. Furthermore, we rigorously analyze the scheme’s stability for scalar convection–diffusion problems and its convergence for systems of equations, demonstrating its reliability and effectiveness for complex flow problems. The developed numerical framework is then applied to simulate heat and mass transfer in k–ω turbulent nanofluid boundary-layer flow over a flat plate. The proposed scheme solves the governing equations for velocity, temperature, nanoparticle concentration, and turbulent quantities to examine the flow’s transport characteristics. Furthermore, an artificial neural network-based machine learning model predicts eddy viscosity using input features comprising the cross-stream variable, velocity, temperature, and nanoparticle concentration profiles, with multiple strategies for training and testing the network. The results show that the proposed numerical method provides stable, accurate solutions with a larger stability region than the existing Runge–Kutta method. At the same time, the machine learning model captures the nonlinear relationship between flow variables and eddy viscosity, offering an efficient data-driven approach to turbulence prediction. In addition, analysis of variance is used to identify the dominant parameters and their interaction effects on eddy viscosity.