This thesis discusses the application, evaluation, and extension of dimensionally adaptive meshing to the numerical solution of velocity and pressure fields inside cylindrical reactors. Due to the high length to diameter ratios of many cylindrical reactor vessels the flow field can become axisymmetric, allowing for simplification of the governing equations and significant reduction in computational time required for solution. A fully 3D solver is developed from existing computational tools at BYU and validated against theoretical velocity profiles for pipe flow at various Reynolds numbers, as well as with experimental data for an axial-fired center jet with recirculating flow. Dimensionally adaptive meshing is then incorporated into the validated 3D solver. The boundary conditions and assumptions at the dimensional boundary are discussed. The flow information is passed across the boundary through spatial mass-weighted averaging. The 3D and axisymmetric computational domains are decoupled from one another so information can only be passed from the 3D domain downstream to the axisymmetric domain. The dimensional boundary placement must meet two main requirements, the flow must be one-way and axisymmetric. It is found that the flow becomes axisymmetric early on in the reactor (~0.3-0.4 m), but recirculation exists farther downstream (until ~0.61 m) and thus governs the placement of the dimensional boundary. The resulting computational tool capable of running simulations using dimensionally adaptive meshes is called CEDAR (Computationally Efficient Dimensionally Adaptive Recirculating flow solver). Several studies are then undertaken to examine CEDAR's ability to reproduce exit velocity profiles comparable to those produced by a fully 3D mesh, including variations in pressure, firing rate, and geometry. It is found that the flow structure inside the reactor is self-similar over a wide range of operating parameters as long as the burner jets are turbulent. This observation is supported by free and confined jet theory. These theories also provide a method for placing the dimensional boundary, which is a linear function of the confining geometry diameter only (assuming that the jet diameter is less than 1/10 the diameter of the confining geometry). All exit velocity profiles produced by CEDAR are on average within 5% of the fully 3D profiles. Timing studies reveal an average 5.16 times speedup in computational time over fully 3D computations.