Two-dimensional and three-dimensional flow models were established for fluid flow in expansion and contraction channels with symmetric structure. The analysis proved that the three-dimensional model has the same two-dimensional solution as the two-dimensional model. By numerical simulation, the velocity fields of fluid flow in two-dimensional and three-dimensional models were obtained for different Re. Based on numerical results, the nonlinear characteristics and differences of flow between two-dimensional and three-dimensional models were analyzed. The numerical results indicate that when Re is different, the flow in both the two-dimensional and three-dimensional models undergoes bifurcation of the solution, resulting in nonlinear phenomena such as symmetry breaking, self-sustained oscillation, and chaos. When Re is 120, both the two-dimensional and three-dimensional models have unique, symmetric, and steady-state velocity fields, and the solutions of the two-dimensional and three-dimensional models are exactly the same. When Re is 200, the solutions of the two-dimensional and three-dimensional models are not unique, and there is a pair of antisymmetric steady-state asymmetric solutions. The solutions of the two-dimensional and three-dimensional models are still exactly the same, and there is no three-dimensional flow. When Re is 280, a three-dimensional flow occurs in the three-dimensional model. The solutions of the two-dimensional model and the three-dimensional model are different, but both have a pair of antisymmetric steady-state asymmetric solutions. When Re is 330, both the two-dimensional and three-dimensional models have unique, symmetric, and steady-state solutions, and their solutions are exactly the same. When Re is 352, 380, and 600, the solutions of both the two-dimensional and three-dimensional models are oscillatory and non-stationary. The solutions of the three-dimensional model are different from those of the two-dimensional model, resulting in three-dimensional flow. The solutions develop from periodic oscillation, period doubling oscillation, to chaos.