A hybrid finite difference‐finite element method for solving the 3D energy equation in non‐isothermal flow past over a tube

Abstract

PurposeThis paper aims to develop a hybrid finite difference‐finite element method and apply it to solve the three‐dimensional energy equation in non‐isothermal fluid flow past over a tube.Design/methodology/approachTo implement the hybrid scheme, the tube length is partitioned into uniform segments by choosing grid points along its length, and a plane perpendicular to the tube axis is drawn at each of the points. Subsequently, the Taylor‐Galerkin finite element technique is employed to discretize the energy equation in the planes; while the derivatives along the tube are discretized using the finite difference method.FindingsTo demonstrate the validity of the proposed numerical scheme, three‐dimensional test cases have been solved using the method. The variation of L2‐norm of the error with mesh refinement shows that the numerical solution converges to the exact solution with mesh refinement. Moreover, comparison of the computational time duration shows that the proposed method is approximately three times faster than the 3D finite element method. In the non‐isothermal fluid flow around a tube for Re=250 and Pr=0.7, the results show that the Nusselt number decreases with the increase in the tube length and, for the tube length greater than six times the tube diameter, the average Nusselt number converges to the value for the two‐dimensional case.Originality/valueA hybrid finite difference‐finite element method has been developed and applied to solve the 3D transient energy equation for different test cases. The proposed method is faster, and computationally more efficient, compared with the 3D finite element method.