Unsteady dynamics of logarithmic wall laws under periodic mass injection: Nonlinear response and scaling behaviors

Abstract

The unsteady dynamics of logarithmic wall laws in turbulent boundary layers subjected to periodic mass injection are investigated. Using the boundary-layer approximation, the velocity and total enthalpy profiles are obtained by implicitly solving the time-dependent one-dimensional Partial Differential Equations of momentum and energy conservation. The dynamic log-law responses are analyzed by the transfer functions across a broad parametric space of blowing ratio, pulsation frequencies, and amplitudes. Periodic wall mass injection induces a generalized Stokes resonance when the Stokes penetration depth aligns with the buffer layer, yielding the maximum phase delay. The momentum and thermal transport exhibit different resonance frequencies: the momentum shows stronger nonlinear temporal persistence, with a bandpass resonance governed by the inner-scaled frequency ωres+≈6×10−3 (a band ≈(4–8)×10−3; dimensionally f≈15–30 Hz, condition-dependent), whereas the thermal field has greater temporal inertia. The velocity log-law achieves a near-universal collapse under the inner-scaled frequency (ω+); this collapse breaks down for the thermal field, which requires a (Reτ/Reref)γh correction with an empirical sign-and-order exponent [γh∼−O(1)] accounting for outer-layer thermal relaxation. The thermal relaxation disparity drives transient intra-cycle oscillations in the effective turbulent Prandtl number (Prt,eff=Ah/Au), invalidating the steady Reynolds analogy. An analytic transfer function is proposed to accurately fit the log-law dynamics, capturing the high-frequency rebound and plateau resulting from the competition between convection and diffusion. The time-averaged deviations under unsteady mass injection follow a quadratic scaling in amplitude in the weakly nonlinear regime and require a fourth-order correction at large amplitudes.