Extension of Thwaites Method to turbulent boundary layers

Thwaites developed an approximate method for predicting the momentum thickness (θ) in laminar boundary layers using a first-order expansion of the Von Karman integral equation. Such a model is lacking for turbulent flows; where the pressure gradient affects the inner and outer flow scales differently. In this work, the Thwaites method is extended for determining the momentum thickness in non-equilibrium turbulent boundary layers using boundary layer edge variables. It is shown that the method provides good estimates of the momentum thickness in comparison to recent high-fidelity simulation datasets for attached non-equilibrium boundary layers and also for boundary layers on the verge of separation in both subsonic and transonic regimes. The results indicate that the proposed linear model can capture the “history effects” associated with pressure gradient on the growth of the boundary layer.

We perform adjoint analysis using the ODE model to study the sensitivity of turbulent flow separation to upstream perturbations. Our analysis also reveals that the proposed model agrees fairly well with Alber’s method (9th Aerospace Sciences Meeting, 1971) for predicting the point of separation. Finally, using the continuity equation, and a proposed fit, the growth rate of the displacement thickness is evaluated which may be of use in practical potential flow solvers