Computational Fluid Dynamics (CFD) is an increasingly important tool used in the analysis and design of aircraft. It provides a cost-effective way to accurately predict aerodynamic forces and moments early on in the design process. State of the art analysis for aircraft at cruise conditions utilizes the Reynolds-Averaged Navier--Stokes (RANS) equations. Unfortunately, generating computational meshes, particularly for complex geometries, is the most time intensive part of the CFD workflow. Cartesian cut-cell methods provide automatic mesh generation for arbitrarily complex geometries. Cartesian cut-cell methods have been used extensively for inviscid analysis, but struggle when applied to the RANS equations for two main reasons. The first is dealing with the irregular gradient stencils in and around cut cells. The RANS equations introduce gradient quantities into flux evaluations, something not present in inviscid flow equations. The irregular gradient stencils near cut cells produce noisy gradients that can pollute numeric solutions. The second issue is the grid resolution required to resolve the boundary layer, where states change much faster in the wall-normal direction than the wall-tangential direction. Typical boundary conforming RANS approaches stretch cells in the wall tangential direction to achieve accurate boundary layers without adding unnecessary mesh points in the wall-tangential direction. Cartesian cut-cell methods cannot stretch cells without losing the automatic meshing that makes the method desirable. The goal of this thesis is to address these issues, enabling RANS solutions on Cartesian cut-cell meshes. I start by developing a novel strategy for solving the Eikonal equation. The Eikonal equation is used to compute a distance function, necessary for solving the Spalart--Allmaras turbulence model to close the RANS equations. My approach provides an accurate and robust spatial discretization for computing solutions to the Eikonal equation on arbitrary point clouds. The approach is not limited to distance functions, and can be applied to find general solutions to the Eikonal equation. Next, I outline three techniques for solving the RANS equations on Cartesian cut-cell meshes: a slip wall treatment, an analytical wall function, and an ordinary differential equation (ODE) wall model. The slip wall treatment modifies the governing equations to produce boundary layers with more benign profiles near the wall. Accurate meshes can be coarser than their full RANS counterparts. The analytical wall function and ODE wall model approaches utilize an interior forcing point method. Data is interpolated to a point near the wall and then fit to an approximation of the boundary layer. This boundary layer approximation is used to inform cut-cell fluxes. The analytical wall function uses equilibrium assumptions to provide an approximation to the boundary layer. The ODE wall model relaxes some of these assumptions to remain accurate further into the boundary layer. Lastly, I demonstrate these three methods' effectiveness on a series of increasingly complex problems provided by the NASA turbulence modeling resource (TMR) database. The slip wall treatment required finer meshes than the two interior forcing point methods and did not have a way to deal with irregular gradient stencils around cut cells. The two forcing point methods are able to provide smooth gradients to cut cell flux evaluations because their boundary layer approximations are created using data at a fixed distance from the wall. The ODE wall model demonstrates accuracy superior to the analytical wall function on coarser meshes, particularly in the presence of strong pressure gradients.