Amirkabir University of TechnologyAmirkabir Journal of Civil Engineering2588-297X531220220220Numerical Solution of Steady Incompressible Turbulent Navier–Stokes Equations using Multiquadric Radial Basis Function (MQ-RBF) MethodNumerical Solution of Steady Incompressible Turbulent Navier–Stokes Equations using Multiquadric Radial Basis Function (MQ-RBF) Method52255356432610.22060/ceej.2021.18788.6964FAMohammad HosseinMirabiUniversity of QomEhsanJabbariUniversity of Qom0000-0002-6345-8567TaherRajaeeUniversity of QomJournal Article20200729The inconveniences of introducing and modifying the mesh grids in mesh-based numerical methods lead the researchers to meshfree methods, among which the RBF methods are probably the most interesting and powerful ones. In this research, the numerical solution of the steady-state incompressible continuity and Navier–Stokes equations, and the standard k-Ɛ turbulence model was investigated in a 2D domain. The computational domain consisting of a 0.5 m×0.5 m square lid-driven cavity was analyzed for five Reynolds numbers of 2.5×10<sup>5</sup>, 5×10<sup>5</sup>,<sub> </sub>10×10<sup>5</sup>, 2×10<sup>6</sup>, and 5.5×10<sup>6</sup>. The Multiquadric Radial Basis Function (MQ-RBF), as the most successful RBF, was employed with 36 and 121 domain computational nodes to solve the PDEs. The velocity fields in two directions, the static pressure, the turbulent kinetic energy and the turbulent energy dissipation, were computed. A try–and–error algorithm was used for solving a set of non-linear equations, and the optimal values of the shape parameter <em>c</em> and the <em>λ</em> set coefficients were evaluated and discussed for each flow field. According to the results, assuming the independence of the values of the shape parameter <em>c</em> for each flow field at different Reynolds numbers, a predictable pattern can be obtained for the <em>λ</em> set for different Reynolds’ numbers in the studied range. These patterns with the predictor functions of the flow fields were compared to existing benchmark results of the finite volume method (ANSYS Fluent). The Nash-Sutcliffe coefficients of 93-99% and RRSME of about %1 obtained from this comparison indicated the reasonable accuracy of the assumption concerning the independence of the shape parameter <em>c</em> of the Reynolds’ numbers, the repeatable patterns of the normalized <em>λ</em> set, and polynomial predictor functions in the MQRBF method for each flow field.The inconveniences of introducing and modifying the mesh grids in mesh-based numerical methods lead the researchers to meshfree methods, among which the RBF methods are probably the most interesting and powerful ones. In this research, the numerical solution of the steady-state incompressible continuity and Navier–Stokes equations, and the standard k-Ɛ turbulence model was investigated in a 2D domain. The computational domain consisting of a 0.5 m×0.5 m square lid-driven cavity was analyzed for five Reynolds numbers of 2.5×10<sup>5</sup>, 5×10<sup>5</sup>,<sub> </sub>10×10<sup>5</sup>, 2×10<sup>6</sup>, and 5.5×10<sup>6</sup>. The Multiquadric Radial Basis Function (MQ-RBF), as the most successful RBF, was employed with 36 and 121 domain computational nodes to solve the PDEs. The velocity fields in two directions, the static pressure, the turbulent kinetic energy and the turbulent energy dissipation, were computed. A try–and–error algorithm was used for solving a set of non-linear equations, and the optimal values of the shape parameter <em>c</em> and the <em>λ</em> set coefficients were evaluated and discussed for each flow field. According to the results, assuming the independence of the values of the shape parameter <em>c</em> for each flow field at different Reynolds numbers, a predictable pattern can be obtained for the <em>λ</em> set for different Reynolds’ numbers in the studied range. These patterns with the predictor functions of the flow fields were compared to existing benchmark results of the finite volume method (ANSYS Fluent). The Nash-Sutcliffe coefficients of 93-99% and RRSME of about %1 obtained from this comparison indicated the reasonable accuracy of the assumption concerning the independence of the shape parameter <em>c</em> of the Reynolds’ numbers, the repeatable patterns of the normalized <em>λ</em> set, and polynomial predictor functions in the MQRBF method for each flow field.https://ceej.aut.ac.ir/article_4326_191834802419a9104d1d11c1d6601aee.pdf