A Hybrid Finite-Element/Finite-Difference Scheme for Solving the 3-D Energy Equation in Transient Nonisothermal Fluid Flow over a Staggered Tube Bank

Abstract

This article presents a hybrid finite-element/finite-difference approach. The approach solves the 3-D unsteady energy equation in nonisothermal fluid flow over a staggered tube bank with five tubes in the flow direction. The investigation used Reynolds numbers of 100 and 300, Prandtl number of 0.7, and pitch-to-diameter ratio of 1.5. An equilateral triangle (ET) tube pattern is considered for the staggered tube bank. The proposed hybrid method employs a 2-D Taylor-Galerkin finite-element method, and the energy equation perpendicular to the tube axis is discretized. On the other hand, the finite-difference technique discretizes the derivatives toward the tube axis. Weighting the 3-D, transient, convection-diffusion equation for a cube verifies the numerical results. The L2 norm of the error between numerical and exact solutions is also presented for three different hybrid meshes. A grid independence study for the energy equation preceded the final mesh. The outcome is found to be in acceptable concurrence with those from the previous studies. After the temperature field is attained, the local Nusselt number is computed for the tubes in the bundle at different times. The isotherms are also obtained at different times until a steady-state solution is reached. The numerical results converge to the exact results through refining the mesh. The implemented hybrid scheme requires less computation time compared with the conventional 3-D finite-element method, requiring less program coding.