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<p>In this paper, we give construction to a new algorithm for solving the governing equations that describes incompressible Newtonian thermal flows. This algorithm is developed based on the so-called Taylor–Galerkin/Pressure-Correction finite element method. Usually, to describe the motion and temperature of the fluid, we use the Navier–Stoke equations coupling with energy equation. The novelty in this study lies in the development of a method with high accuracy to treat a fluid flow under thermal condition. Here, new stages have been added to the Taylor–Galerkin/Pressure-Correction finite element method directly linked energy equation. The essential step in the development process involved applying the two-step Lax–Wendroff scheme on energy equation, after which, two new steps for energy equations are obtained within the (TG/BC) algorithm. To meet the analysis of new method, a start-up Poiseuille flow through axisymmetric rectangular channel for the Newtonian thermal flow is utilized as a simple test problem. In that situation, three different meshes: coarse, normal and fine are employed to study the effect of mesh refinements on the accuracy of the solution. Moreover, we have studied the effect of non-dimensional numbers such as Reynolds number (<i>Re</i>) and Prandtl number (<i>Pr</i>) for each of velocity, pressure and temperature. Influence of the thermal conductivity (<i>k</i>) and viscosity <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\((\mu )\)</EquationSource> </InlineEquation> on the solution components is presented as well. The obtained results reveal that, the developed method gives in harmony with the approved physical phenomena and other studies.</p>

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Simulation of Newtonian Axisymmetric Thermal Channel Flow by Using a Developed Taylor–Galerkin/Pressure Correction Method

  • 期刊论文

  • 发表日期：

    2025年8月18日

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  • Anas Al-Haboobi,
  • Alaa H. Al-Muslimawi

摘要

In this paper, we give construction to a new algorithm for solving the governing equations that describes incompressible Newtonian thermal flows. This algorithm is developed based on the so-called Taylor–Galerkin/Pressure-Correction finite element method. Usually, to describe the motion and temperature of the fluid, we use the Navier–Stoke equations coupling with energy equation. The novelty in this study lies in the development of a method with high accuracy to treat a fluid flow under thermal condition. Here, new stages have been added to the Taylor–Galerkin/Pressure-Correction finite element method directly linked energy equation. The essential step in the development process involved applying the two-step Lax–Wendroff scheme on energy equation, after which, two new steps for energy equations are obtained within the (TG/BC) algorithm. To meet the analysis of new method, a start-up Poiseuille flow through axisymmetric rectangular channel for the Newtonian thermal flow is utilized as a simple test problem. In that situation, three different meshes: coarse, normal and fine are employed to study the effect of mesh refinements on the accuracy of the solution. Moreover, we have studied the effect of non-dimensional numbers such as Reynolds number (Re) and Prandtl number (Pr) for each of velocity, pressure and temperature. Influence of the thermal conductivity (k) and viscosity \((\mu )\) on the solution components is presented as well. The obtained results reveal that, the developed method gives in harmony with the approved physical phenomena and other studies.