Coherent vortex model for surface pressure fluctuations induced by the wall region of a turbulent boundary layer

Abstract

Exact solutions of the Navier–Stokes equations describing the interaction of streamwise vortices with a rigid surface are utilized to develop a conceptual model for the surface pressure spectrum associated with the wall region of a turbulent boundary layer. The evolution of single as well as pairs of coherent streamwise vortices, which principally govern the production of turbulence in the wall region, is considered in the presence of local straining flow induced by larger, outer-layer eddies. The surface pressure signatures of the coherent vortex motion and the associated power spectrum of the pressure are examined. Based on the results of the exact solutions, the surface pressure spectrum of an ensemble of independent coherent structures is modeled using the assumption of ergodicity in the manner described by Townsend and Lundgren for homogeneous turbulence. The free parameters in the model are estimated through comparison with available results from experiments and numerical simulations. The model, especially the one involving pairs of streamwise vortices, predicts the high frequency and high spanwise wave number range of the surface pressure spectrum quite well. Further, the probability density function of surface pressure associated with the model compares well with experimental results. Interestingly, the model also suggests that the contribution of the viscous interaction to low wave number spectral elements accounts for the discrepancy between experimental observations at such wave numbers and the prediction of the Kraichnan–Phillips theorem.