Computational modeling of turbulent nanofluid heat transfer over a heated moving surface with local thermal non-equilibrium and machine-learned eddy viscosity

  • Muhammad Shoaib Arif orcid

    Department of Mathematics and Sciences, College of Sciences and Humanities, Prince Sultan University, Riyadh 11586, Saudi Arabia
  • Yasir Nawaz orcid

    Department of Mathematics, Faculty of Engineering and Computing, National University of Modern Languages (NUML), Islamabad 44000, Pakistan
Article ID: 4643
DOI: https://doi.org/10.59400/adecp4643
Keywords: turbulent nanofluid flow; local thermal non-equilibrium; adaptive modified Runge-Kutta method; k − ω turbulence model; eddy viscosity; machine learning

Abstract

This study presents an adaptive modified Runge-Kutta compact scheme for the numerical simulation of unsteady k − ω turbulent nanofluid flow over a heated moving surface under local thermal non-equilibrium conditions. The surface-interfacial model incorporates mixed convection, viscous dissipation, turbulence transport, and separate energy equations for the base fluid and nanoparticle phases, with the effective thermal conductivity described by Xue’s formulation. The proposed time-integration method is explicit and combined with a compact finite-difference discretization that provides fourth-order spatial accuracy. The temporal coefficients are selected to achieve second-order accuracy, and the method is further enhanced through adaptive time stepping based on local error control. Stability analysis for the scalar convection–diffusion problem and conditional convergence analysis for the corresponding system formulation are also established. Numerical comparisons show that the proposed adaptive scheme yields lower error than existing adaptive Euler- and Runge-Kutta-based schemes. The computed results further demonstrate that thermal buoyancy increases the mean velocity, whereas larger Prandtl numbers reduce the thermal boundary-layer thickness of the fluid and nanoparticle phases. In addition, stronger interphase coupling modifies the two-temperature fields in a manner consistent with local thermal nonequilibrium. A machine-learning model is also employed to predict eddy viscosity, and its reliability is confirmed through profile comparisons, contour analyses, sensitivity assessments, and Taylor diagram evaluations. Overall, the proposed framework provides an accurate and efficient computational tool for surface-associated turbulent nanofluid transport with interfacial thermal nonequilibrium.

Published
2026-08-28
How to Cite
Arif, M. S., & Nawaz, Y. (2026). Computational modeling of turbulent nanofluid heat transfer over a heated moving surface with local thermal non-equilibrium and machine-learned eddy viscosity. Advances in Differential Equations and Control Processes, 33(3). https://doi.org/10.59400/adecp4643
Issue
Vol. 33 No. 3 (2026): In Progress
Section
Article

Copyright (c) 2026 Muhammad Shoaib Arif, Yasir Nawaz

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

References

[1]Choi SUS, Eastman JA. Enhancing thermal conductivity of fluids with nanoparticles. In: Proceedings of the ASME International Mechanical Engineering Congress & Exposition; 12–17 November 1995; San Francisco, CA, USA.

[2]Buongiorno J. Convective transport in nanofluids. Journal of Heat Transfer. 2006; 128(3): 240–250.

[3]Varma A, Nidhul K. Thermo-hydraulic and entropy generation analysis of nanofluid flow with variable properties in various duct cross sections: A 3-D two-phase approach. Journal of Thermal Analysis and Calorimetry. 2024; 149(9): 4209–4225.

[4]Wilcox DC. Turbulence Modeling for CFD, 3rd ed. DCW Industries; 2006.

[5]Menter FR. Two-equation eddy-viscosity turbulence models for engineering applications. AIAA Journal. 1994; 32(8): 1598–1605.

[6]Roy G, Gherasim I, Nadeau F, et al. Heat transfer performance and hydrodynamic behavior of turbulent nanofluid radial flows. International Journal of Thermal Sciences. 2012; 58: 120–129.

[7]Pati S, Borah A, Boruah MP, et al. Critical review on local thermal equilibrium and local thermal non-equilibrium approaches for the analysis of forced convective flow through porous media. International Communications in Heat and Mass Transfer. 2022; 132: 105889.

[8]Tayebi T, Chamkha AJ, Öztop HF, et al. Local thermal non-equilibrium (LTNE) effects on thermal-free convection in a nanofluid-saturated horizontal elliptical non-Darcian porous annulus. Mathematics and Computers in Simulation. 2022; 194: 124–140.

[9]Latha N, Shankar BM, Naveen Kumar SB, et al. Perspectives on local thermal non-equilibrium (LTNE) Darcy–Bénard convection: Variable permeability and viscosity effects. Physics of Fluids. 2024; 36(10): 104131.

[10]Nield DA, Kuznetsov AV. The effect of local thermal non-equilibrium on the onset of convection in a nanofluid. ASME Journal of Heat and Mass Transfer. 2010; 132(5): 052405.

[11]Saleh M, Kovács E, Kallur N. Adaptive step size controllers based on Runge-Kutta and linear-neighbor methods for solving the non-stationary heat conduction equation. Networks & Heterogeneous Media. 2023; 18(3): 1059–1082.

[12]Dormand JR, Prince PJ. A family of embedded Runge-Kutta formulae. Journal of Computational and Applied Mathematics. 1980; 6(1): 19–26.

[13]Roberts SB, Shirokoff D, Biswas A, et al. Runge-Kutta Methods and Stiff Order Conditions for Semilinear ODEs. arXiv preprint. 2025; arXiv:2505.15099v1.

[14]Savović S, Drljača B, Djordjevich A. A comparative study of two different finite difference methods for solving advection–diffusion reaction equation for modeling exponential traveling wave in heat and mass transfer processes. Ricerche di Matematica. 2022; 71: 245–252.

[15]Ranocha H, Winters AR, Castro HG, et al. On error-based step size control for discontinuous Galerkin methods for compressible fluid dynamics. Communications on Applied Mathematics and Computation. 2025; 7(1): 3–39.

[16]Naseem T, Mebarek-Oudina F, Vaidya H, et al. Numerical analysis of entropy generation in joule heated radiative viscous fluid flow over a permeable radially stretching disk. CMES-Computer Modeling in Engineering and Sciences. 2025; 143(1): 351–371.

[17]Mezaache A, Mebarek-Oudina F, Vaidya H, et al. Heat transfer analysis of nanofluid flow with entropy generation in a corrugated heat exchanger channel partially filled with porous medium. Heat Transfer. 2024; 53(8): 4625–4647.

[18]Shampine LF. Local extrapolation in the solution of ordinary differential equations. Mathematics of Computation. 1973; 27: 91–97.

[19]Butcher JC, Johnston PB. Estimating local truncation errors for Runge-Kutta methods. Journal of Computational and Applied Mathematics. 1993; 45: 203–212.

[20]Verner JH. Explicit Runge–Kutta methods with estimates of the local truncation error. SIAM Journal on Numerical Analysis. 1978; 15: 772–790.

[21]Chai AS. Error estimate of a fourth-order Runge-Kutta method with only one initial derivative evaluation. In: Proceedings of the April 30–May 2, 1968, Spring Joint Computer Conference; 30 April–2 May 1968; Atlantic City, NJ, USA. pp. 467–471.

[22]Merson RH. An operational methods for study of integration processes. In: Proceedings of Conference on Data Processing and Automatic Computing Machines; 3–8 June 1957; Salisbury, Australia.

[23]Alexandre dit Sandretto J. Validated B-series and Runge-Kutta pairs. Numerical Algorithms. 2024; 96(3): 1045–1062.

[24]Martens B. Error estimates for Runge–Kutta schemes of optimal control problems with index 1 DAEs. Computational Optimization and Applications. 2023; 86(3): 1299–1325.

[25]Ling J, Kurzawski A, Templeton J. Reynolds averaged turbulence modelling using deep neural networks with embedded invariance. Journal of Fluid Mechanics. 2016; 807: 155–166.

[26]Duraisamy K, Iaccarino G, Xiao H. Turbulence modeling in the age of data. Annual Review of Fluid Mechanics. 2019; 51(1): 357–377.

[27]Girimaji SS. Turbulence closure modeling with machine learning: A foundational physics perspective. New Journal of Physics. 2024; 26(7): 071201.

[28]He X, Tan J, Rigas G, et al. On the explainability of machine-learning-assisted turbulence modeling for transonic flows. International Journal of Heat and Fluid Flow. 2022; 97: 109038.

[29]Nawaz Y, Kerdid N, Arif MS, et al. AI-Assisted Hybrid Solver for Skin Friction and Sherwood Number Prediction in Eyring–Prandtl Nanofluid Flow over a Riga Plate. CMES-Computer Modeling in Engineering and Sciences. 2026; 146(2).

[30]Tao H, Aldlemy MS, Homod RZ, et al. Hybrid nanocomposites impact on heat transfer efficiency and pressure drop in turbulent flow systems: Application of numerical and machine learning insights. Scientific Reports. 2024; 14(1): 19882.

[31]Babanezhad M, Behroyan I, Nakhjiri AT, et al. Thermal prediction of turbulent forced convection of nanofluid using computational fluid dynamics coupled genetic algorithm with fuzzy interface system. Scientific Reports. 2021; 11(1): 1308.

[32]Arif MS, Abodayeh K, Nawaz Y. A Hybrid Exponential Runge-Kutta Scheme for Stochastic SIQR Disease Modeling. European Journal of Pure and Applied Mathematics. 2025; 18(3): 6176.

[33]Zhang ZJ, Duraisamy K. Machine learning methods for data-driven turbulence modeling. In: Proceedings of the 22nd AIAA Computational Fluid Dynamics Conference; 22–26 June 2015; Dallas, TX, USA.

[34]Xiao H, Wu JL, Wang JX, et al. Quantifying and reducing model-form uncertainties in Reynolds-averaged Navier–Stokes simulations: A data-driven, physics-informed Bayesian approach. Journal of Computational Physics. 2016; 324: 115–136.

[35]Wu JL, Xiao H, Paterson E. Physics-informed machine learning approach for augmenting turbulence models: A comprehensive framework. Physical Review Fluids. 2018; 3: 074602.

[36]Matai R, Durbin P. Zonal eddy viscosity models based on machine learning. Flow, Turbulence and Combustion. 2019; 103: 93–109.

[37]Maulik R, San O, Jacob JD, et al. Sub-grid scale model classification and blending through deep learning. Journal of Fluid Mechanics. 2019; 870: 784–812.

[38]Gamahara M, Hattori Y. Searching for turbulence models by artificial neural network. Physical Review Fluids. 2017; 2: 054604.

[39]Sriharan G, Mebarek-Oudina F, Harikrishnan S, et al. Experimental and machine-learning investigation of ZnO nanofluid flow in a triangular-absorber parabolic trough collector. Modern Physics Letters B. 2026; 40(15): 2650108.

[40]Vollant A, Balarac G, Corre C. Subgrid-scale scalar flux modelling based on optimal estimation theory and machine-learning procedures. Journal of Turbulence. 2017; 18: 854–878.

[41]Maulik R, San O, Rasheed A, et al. Data-driven deconvolution for large eddy simulations of kraichnan turbulence. Physics of Fluids. 2018; 30: 125109.

[42]Fayz-Al-Asad M, Mebarek-Oudina F, Vaidya H, et al. Finite element analysis for magneto-convection heat transfer performance in vertical wavy surface enclosure: fin size impact. Frontiers in Heat and Mass Transfer. 2024; 22(3): 817–837.

[43]Fukami K, Fukagata K, Taira K. Super-resolution reconstruction of turbulent flows with machine learning. Journal of Fluid Mechanics. 2019; 870: 106–120.

[44]Zhao Y, Akolekar HD, Weatheritt J, et al. Turbulence model development using cfd-driven machine learning. arXiv preprint. 2019; arXiv:1902.09075.

[45]Zhang X, Wu J, Coutier-Delgosha O, et al. Recent progress in augmenting turbulence models with physics-informed machine learning. Journal of Hydrodynamics. 2019; 31: 1153–1158.