A General Strategy to Extend Turbulence Models to Rough Surfaces: Application to Smith’s $k-L$ Model

A general procedure to extend turbulence models to account for wall roughness, in the framework of the equivalent sand grain approach, is proposed. It is based on the prescription of the turbulent quantities at the wall to reproduce the shift of the logarithmic profile and hence provide the right increase in wall friction. This approach was previously applied to Spalart and Allmaras one equation (1992, “A One-Equation Turbulence Model for Aerodynamic. Flows,” 30th Aerospace Sciences Meeting and Exhibit, Reno, NV, AIAA paper No. 92-0439;1994, ibid, Rech. Aerosp. 1, pp. 5–21). Here, the strategy is detailed and applied to Smith’s two-equation $k-L$ model (1995, “Prediction of Hypersonic Shock Wave Turbulent Boundary Layer Interactions With The k-l Two Equaton Turbulence Model,” 33rd Aerospace Sciences Meeting and Exhibit, Reno, NV, Paper No. 95-0232). The final model form is given. The so-modified Spalart and Allmaras and Smith models were tested on a large variety of test cases, covering a wide range of roughness and boundary layer Reynolds numbers and compared with other models. These tests confirm the validity of the approach to extend any turbulence model to account for wall roughness. They also point out the deficiency of some models to cope with small roughness levels as well as the drawbacks of present correlations to estimate the equivalent sand grain roughness.