Linear Amplification in Nonequilibrium Boundary Layers

The zero pressure gradient (ZPG), turbulent boundary layer (TBL) has been studied extensively as a model for engineering flows over surfaces. In reality however, most vehicle surfaces have curvature that can decelerate the free stream flow, creating an adverse pressure gradient (APG) and significant streamwise evolution of the turbulence. Departures from the ZPGTBL are most evident in the outer region of the flow, where the flat plate APGTBL exhibits a larger wake and a secondary peak in the streamwise velocity fluctuations. This turbulence is also sensitive to how the free stream flow is decelerated, meaning that even if local conditions (Reynolds number and pressure gradient strength) are matched, the streamwise history affects the results.

Here, we study the linear amplification of turbulent structures through a resolvent operator that resolves the streamwise and wall normal directions. Nonparallel effects, such as the history of the deceleration in the free stream, are included through the operator. In particular, we find that this linear analysis predicts an increased linear amplification of large scale structures further from the wall with increased APG strength. Furthermore, the velocity fluctuations modeled using only this linear analysis predict a secondary peak for the APGTBL that is not present in the ZPGTBL. These results are corroborated with literature that observe large scale structures energized with increasing APG in simulation and experiment.

Figure: Contours of the streamwise component of the leading resolvent response modes, ψ_u,1(x, y) for a ZPG and APG TBL with spanwise wavenumber, k_zδ₉₉ = 2π, and temporal frequency, ωδ₉₉/U_e = π. Lineplots compare |σ₁ψ_u,1| at x/δ₉₉ = 10, where σ₁ denotes the linear amplification. Black dashed lines denote the boundary layer thickness, δ₉₉