FINITE VOLUME METHOD FOR SOLVING NAVIER STOKES EQUATIONS IN FLUID DYNAMICS - Abu Arrah, Abdulraheem (An-Najah National University, 2025)
Several equations, particularly the Navier-Stokes equations, govern fluid dynamics. These equations are essential for describing fluid motion, which helps us understand many natural phenomena. The Navier-Stokes equations present significant challenges for researchers in mathematics and engineering due to their complexity and the difficulties in obtaining analytical solutions. As a result, it has become necessary to explore alternative methods for solving these equations, particularly through numerical approaches. Since numerical methods yield approximate solutions, it is vital to evaluate the effectiveness of this approach in addressing the Navier-Stokes equations. One such numerical method is the finite volume method (FVM), which provides approximate solutions to the Navier-Stokes equations. In this thesis, we conducted a thorough examination of the finite volume method using various examples of the Navier Stokes equations that have analytical solutions. We began with simpler cases and gradually increased the complexity while also comparing our numerical results with the analytical solutions to assess how closely they aligned with the exact solutions. This comparison enabled us to evaluate the effectiveness of the method. We encountered issues related to the stability and accuracy of the numerical solutions based on the specific conditions we examined while employing this method. As a result, we discussed the numerical schemes related to the Finite Volume Method (FVM) and the criteria for selecting a specific scheme, especially concerning the Peclet number. We then evaluated the effectiveness of each scheme by applying them to the same case. The results obtained from the finite volume method for solving one-dimensional steady state Navier-Stokes equations, with a suitable choice of discretization scheme, provided accurate solutions with excellent stability. However, we observed that when high Peclet numbers were used, solution instability emerged, necessitating the implementation of higher-order discretization schemes. Future research could build on this method by looking at flow situations in two or three dimensions and improving computing efficiency with adaptive mesh refinement (AMR) and better discretization schemes.
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