10th International Symposium on Turbulence and Shear Flow Phenomena (TSFP10), Chicago, USA, July, 2017
Relation Between a Singly-Periodic Roughness Geometry and
Spatio-Temporal Turbulence Characteristics
J. Morgan1∗ and B. McKeon1
1: Graduate Aerospace Laboratories, California Institute of Technology, USA.
∗ Correspondent author: [email protected]
Rough-wall turbulent boundary layers are a pervasive phenomenon throughout nature and industry, with applications to
aviation, climate, and the environment. The present work seeks to further our understanding of rough-wall boundary layers by
creating a simple, singly-periodic roughness to observe the effect on the flow of a single large roughness scale.
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. 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. Because the
roughness creates a spatial inhomogeneity in the boundary layer, we can take the spatial Fourier transform of field variables in
the periodic directions, as in Equation 1. A cosine transform is used in the z-direction to enforce symmetry.
Z λx Z λz
b k, m) =
Q(y,
0
e−ikx cos(mz)Q(x, y, z)dxdz
(1)
0
The pre-multiplied Taylor-transformed wavelength power spectrum of streamwise velocity λT Φ(y, λT , x, z) can be Fouriertransformed as in Equation 1, with a streamwise wavenumber k equal to the streamwise roughness wavenumber kx and spanwise
wavenumber m = 0. A heat map of the magnitude of this mode, shown in Figure 1, reveals 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 , denoted by a white line. The spatial variation of the power spectrum at this wavelength is plotted
spatially in Figure 2, exhibiting a systematic change in phase across the boundary layer.
In a canonical smooth-wall boundary layer, the quantities plotted in Figures 1 and 2 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 through the linearized boundary condition, 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 nonlinear interactions. It is shown that the quantity λd
T Φ(y, λT , kx , 0) 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.
Figure 1. Magnitude of streamwise-varying spatial Figure 2. Velocity power spectrum spatial Fourier mode
+
Fourier mode of the pre-multiplied Taylor wavelength λd
x Φ (y, λx , kx , 0), with white denoting a region of flow
d
power spectrum |λT Φ(y, λT , kx , 0)|. The white line cor- which is more energetic than the streamwise and spanresponds to the roughness wavelength
wise mean. The position x/λx = 0 corresponds to a
roughness peak in the streamwise direction while x/λx =
0.5 corresponds to a trough in the streamwise direction.