Relation between a singly-periodic roughness geometry and spatio-temporal turbulence characteristics

Abstract

The structure of a turbulent boundary layer over a singly-periodic roughness of large wavelength is shown to give insight into the physics of rough-wall boundary layers. To this end, a roughness consisting of a single spanwise-varying mode and a single streamwise-varying mode was 3D printed with wavelengths on the order of the boundary layer thickness. The large length scale introduced by such a roughness creates spatial inhomogeneity of the mean velocity field throughout the entire boundary layer. A hot-wire probe was used to take time series of streamwise velocity at a grid of points in the x,y, and z directions, covering the volume over a single period of roughness, and allowing Fourier transforms of field variables to isolate the spatial variations correlating to the periodic geometry.The pre-multiplied Taylor-transformed wavelength power spectrum of streamwise velocity λTΦ(y, λT, x, z) can be Fourier-transformed in space to reveal that the portion of the power spectrum which varies most strongly in the streamwise direction is the portion with Taylor-transformed wavelength λT equal to the roughness wavelength λx. The spatial variation of the power spectrum at this wavelength exhibits a systematic change in phase across the boundary layer, which can be correlated to the phase of the spatially-varying time-averaged velocity field to reveal amplitude modulation of particular wavelengths by a roughness-induced synthetic scale.In a canonical smooth-wall boundary layer, the spatial variation of the mean velocity and the power spectrum would be identically zero due to translational symmetry. The introduction of a periodic roughness introduces the spatial variation in the power spectrum, but not directly. The roughness creates a stationary time-averaged velocity mode, but this mode does not appear in the power spectrum as it does not convect. The connection to the power spectrum must therefore be through non-linear interactions. It is shown that the correlation between the mean velocity and the power spectrum can be interpreted exactly as a measure of phase organization between pairs of convecting velocity modes which are triadically consistent with the stationary roughness velocity mode, analogously to amplitude modulation in canonical flows. Implications for real-world roughness are discussed.