Spectral analysis of a turbulent boundary layer encountering steady and unsteady complex pressure gradients

Abstract

Changes created by steady and unsteady impositions of a favorable-adverse pressure gradient (FAPG) sequence to the spectral content of a flat plate turbulent boundary layer are presented in this paper. The pressure gradients were generated in the Unsteady Pressure Gradients Facility at the University of Illinois Urbana-Champaign, which uses a ceiling mechanism that can statically or dynamically deform a false ceiling section to the shape of an inverted convex bump. The ceiling was deformed from a zero pressure gradient state (acceleration parameter, $\mathbf{K \equiv 0}$) to a strong FAPG state ($\mathbf{6 \times 10^{-6} \leq K \leq -4.8 \times 10^{-6}}$) with a speed of 1.5 m/s for the unsteady case. This speed of deformation creates an unsteady timescale that is 4.38 times faster than the convective timescale ($\mathbf{k_x}$ = 4.38) and 1.79 times faster than the large turbulent timescale ($\mathbf{k_\tau}$ = 1.79). The same range of spatial pressure gradients ($\mathbf{K \equiv 0}$ to $\mathbf{6 \times 10^{-6} \leq K \leq -4.8 \times 10^{-6}}$) were spanned by the steady cases in 6 static steps. Time-resolved planar particle image velocimetry was used to capture the spatio-temporal response of the TBL in the adverse pressure gradient (APG) region of this FAPG sequence. 1D power spectral densities of the steady and unsteady TBLs were estimated using Fourier transforms and continuous wavelet transforms on the streamwise velocity fluctuations. Effect of the pressure gradient sequence on the TBL spectrum was found to be significant. The upstream favorable pressure gradient (FPG) caused a strong suppression of spectral densities in the wake region and a shift in energy towards longer wavelengths, the extent of which both depended on the strength of the FPG. Within the APG region, an expected recovery in large scale energy happened only in $\mathbf{y < 0.3\delta}$, and was centered at $\mathbf{y = 0.06 \delta}$. This was attributed to the formation and growth of an internal boundary layer triggered by the pressure gradient sign change. In the unsteady case, similar trends were observed as the steady cases, but the response to the pressure gradients was found to be milder. The importance of studying pressure gradient TBLs with complex spatial and temporal history is demonstrated through this work.