Nuances of theoretical and numerical aspects of the fully developed flow in a pipe
DOI:
https://doi.org/10.35819/remat2024v10i1id6667Keywords:
finite differences, circular geometry, iterative methods, Poisson equation, analytical solutionAbstract
The fully developed flow of a Newtonian fluid in a circular cross-section tube will be applied to discussions of theoretical solutions and numerical methodologies, based on the finite difference technique. The objective is to present details of obtaining the analytical solution, the numerical methodologies used and the physics involved, in order to disseminate an original didactic-scientific compilation in this area. The flow in deal is modeled by an elliptic partial differential equation subject to a Dirichlet condition at the boundary. The boundary of the tube does not coincide with the computational mesh which is uniform rectangular. Therefore, a linear extrapolation technique will be employed. The resulting linear system is solved by classical Jacobi and Gauss-Seidel methods for comparison. A mesh refinement process demonstrates that the numerical solution converges to the analytical solution with order O(h^3). The relationship between the number of iterations of the Jacobi and Gauss-Seidel methods and the condition number of the matrix of the linear system corroborates that the Jacobi method requires approximately twice as many iterations as the Gauss-Seidel method to converge. Results about the influence of the physical parameters of the model and the relationship between average and maximum velocities are investigated. It is shown, analytically, that the absolute value of the pressure gradient is directly proportional to the velocity, while the characteristics are inversely proportional to this variable, furthermore, the proportionality relationship between the average and maximum flow velocities is theoretically demonstrated and applied for other numerical verifications.
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