Introdução à mecânica dosfFluidos computacional

Abstract

Computational Fluid Dynamics (CFD) has consolidated itself as a fundamental tool for predicting flow behavior in engineering applications, enabling the computational solution of systems of partial differential equations whose continuous solution is, in general, unfeasible. In this context, the present work aimed to develop and validate numerical solvers for three classical incompressible and Newtonian flows: the transient Couette flow, the two-dimensional Poiseuille flow in a rectangular cross-section duct, and the flow inside a lid-driven cavity. Starting from the Navier-Stokes and mass balance equations, the partial differential equations that model each problem were simplified according to the applicable hypotheses and discretized by the Finite Difference Method. For the Couette flow, both explicit and fully implicit time-marching schemes were implemented. For this flow, a complete energetic analysis was also developed: the continuous and discrete solutions for the mechanical power delivered by the moving plate, for the viscous power converted into thermal energy through the transformation φ function, and for the total kinetic energy of the flow were derived, latter obtained through two independent calculation routes, and the mechanical energy balance relating these quantities was verified both analytically and numerically, from the impulsive start to the steady state. For the two-dimensional Poiseuille flow an implicit scheme was adopted, whose pentadiagonal linear system was solved by the iterative Gauss-Seidel method. For the cavity, projection method was employed on a staggered grid, with boundary conditions treated through ghost cells. The implementations were performed in MATLAB, C and Fortran 90, allowing a direct comparison between interpreted and compiled languages and between direct and iterative strategies for solving linear systems. The computational results were compared with the available continuous solutions for the Couette and Poiseuille flows and with the reference data from (GHIA; GHIA; SHIN, 1982) for the cavity at Reynolds numbers 100, 400 1000 and 10.000. The computational solution of the Poiseuille flow presented a relative error below 0.02% with respect to the analytical solution at steady state, and the cavity simulations faithfully reproduced the central velocity profiles, including the capture of the recirculations at the lower corners for Re = 1000 and 10.000. The performance comparison showed that the choice of algorithm had a significantly greater impact on the processing time than the choice of programming language. It is concluded that the implemented solvers are consistent, stable and accurate within the laminar regimes simulated, and that proper decisions regarding algorithm and data structure are determinant for the computational feasibility of CFD simulations.