Modelling of supersonic turbulent boundary layers over cubical roughness

Understanding the behaviour of compressible fluids over rough surfaces is fundamental in many aerospace applications, especially those regarding atmospheric reentry, propulsion systems, missiles and aircraft. Every surface shows a rough texture, even if it appears smooth, and it is important to distinguish the cases where the topography is relevant or not in the evolution of the fluid dynamics. The influence of roughness is well known and theorised in the incompressible regime, and this knowledge lays the foundations for further elaboration in the compressible field; here, in fact, the thermodynamic effects, such as density and viscosity variations, become substantial. On top of that, shock waves and expansion waves are typical phenomena that may come into play. For these reasons, predicting drag and heat transfer in boundary layers is a rather difficult task. Re-entry vehicles are covered with ablative shields that undergo pyrolysis when invested by an extremely hot gas. This process generates particular roughness patterns that interact with the external flow with a consequent change in the performance of the vehicle. The purpose of this thesis is to develop a wall model that is effective in describing the velocity and temperature profiles in the boundary layer, given the values at a certain distance from the wall (called the matching location) and information about the roughness topography. The main focus will be on a supersonic, zero-pressure-gradient, turbulent boundary layer at a free-stream Mach number of M=2 over cubical-shaped elements. (DNS data for comparison are provided by Cogo et al., 2025). The model presented here is an extension of the one developed by Yang et al., 2016. The present work extends and adjusts the model for supersonic applications based on the fact that, in the near-wall region, the compressibility effects are negligible due to the slowing of the flow. Outwards the roughness crest, instead, the velocity profile has a logarithmic behaviour. In order to take into account density variations, a compressible transformation (Van Driest, 1951) is implemented in the logarithmic region, while the temperature profile is obtained from the Reynolds analogy proposed by Zhang et al., 2014. The results agree with the simulation data for both the adiabatic and isothermal wall. In the future, it could be interesting to analyse more complex and realistic geometries and compare the model with simulations at hypersonic speeds. In addition, the model needs to be elaborated and tested in full scale simulation of re-entry vehicles.