Numerical study of Fourier-filtered rough surfaces

Rough surfaces are common in engineering applications, where a priori estimations of drag for a given flow are needed based on knowledge of the surface topography. It is likely that length-scale information is required, in addition to standard statistical quantities such as solidity or effective slope, root-mean-square height and skewness. In this work we consider a set of rough surfaces that are derived from a physical scan of a grid-blasted rough surface. Different surfaces are generated by applying Fourier band-pass filters to the surface scan, including complementary cases, for example, where a midrange of wave numbers are included or excluded. This enables comparisons as to how the roughness spectral content affects the mean flow and turbulence properties. In total, turbulent flow over five surfaces with different wave-number spectra is investigated by direct numerical simulation, with resulting variations in the roughness function of over a factor of 3. It is found that, except for the low-pass filtered surface which has very small effective slope, the roughness function scaled by the viscous proportion of total drag remains remarkably constant, while the pressure counterpart is largest for high-pass filtered surfaces. Existing correlations for the roughness function are found, at best, to reproduce only the qualitative effects, suggesting that the correlations would benefit from introducing additional parameters to account for the wave-number spectrum of the rough surfaces. Besides the friction effect, it is also of interest to determine the extent to which the turbulence in the roughness layer is influenced by the spectral characteristics of the surface. The location of the peak streamwise velocity fluctuations moves outwards in wall units as the roughness function increases, whereas wall-normal and spanwise velocity fluctuations are found to be insensitive to the surface filtering, down to a region below the maximum roughness height. A trend towards spanwise organization of the mean flow is observed for low-pass filtered surfaces, but otherwise the effect of the roughness wave-number spectrum appears to vanish rapidly above the maximum roughness elevation. Instead, significant differences are found in the profiles of the various dispersive stresses which are highly dependent on (local) topographical features of the roughness; for some quantities differences between surfaces persist well into the log layer.

Figure 1

Left: height maps; center: power spectral densities (logarithmic scale, arbitrary units); right: autocorrelation function (black and white lines indicate levels of 0.2 and 0, respectively). From top to bottom: B, L, M, H, and E.

Figure 2

Compensated angle-averaged radial spectra (left) and autocorrelation functions (right). B: $––$; L: $––\text{−}$; M: $\text{−}\text{−}\text{−}\text{−}$; H: $–.–.$; E: $–..–$.

Figure 3

Probability density (left) and cumulative distribution (right) functions for each surface. B: $––$; L: $––\text{−}$; M: $\text{−}\text{−}\text{−}\text{−}$; H: $–.–.$; E: $–..–$.

Figure 4

Viscous ($f_v$) and pressure drag ($f_p$) in proportion to the driving pressure gradient $\mathrm{\Pi }$ on the left and multiplied by $\mathrm{\Delta }{U}^{+}$ on the right.

Figure 5

Mean velocity deficit profiles with wall distance in outer units. Smooth: $––$; B: $––$; L: $––\text{−}$; M: $\text{−}\text{−}\text{−}\text{−}$; H: $–.–.$; E: $–..–$.

Figure 6

Mean streamwise velocity profiles with wall distance in inner units. The inset shows the mean velocity profiles within the roughness canopy relative to the height of the smallest and tallest element. The filled circles mark the crest of each surface. Smooth: $––$; B: $––$; L: $––\text{−}$; M: $\text{−}\text{−}\text{−}\text{−}$; H: $–.–.$; E: $–..–$.

Figure 7

Streamwise component of the dispersive stress tensor. The inset shows the same profiles with wall distance relative to the roughness height. B: $––$; L: $––\text{−}$; M: $\text{−}\text{−}\text{−}\text{−}$; H: $–.–.$; E: $–..–$.

Figure 8

Dispersive streamwise velocity contours at $z=h_{\max }$ in color scale (transparency varies linearly from $\tilde{u}=0$ up to half-way to each extreme of the scale) overlaid on the surface height map in gray scale.

Figure 9

Spanwise component of the dispersive stress tensor. The inset shows the same profiles with wall distance relative to the roughness height. B: $––$; L: $––\text{−}$; M: $\text{−}\text{−}\text{−}\text{−}$; H: $–.–.$; E: $–..–$.

Figure 10

Illustration stream lines associated with the mean velocity field for surface H in the vicinity of a roughness element. The brightness and opacity of the streamlines increases with distance to the wall.

Figure 11

Wall-normal component of the dispersive stress tensor. The inset shows the same profiles with wall distance relative to the roughness height. B: $––$; L: $––\text{−}$; M: $\text{−}\text{−}\text{−}\text{−}$; H: $–.–.$; E: $–..–$.

Figure 12

Off-diagonal ${\langle \tilde{u}\tilde{w}\rangle }^{+}$ component of the dispersive stress tensor. The inset shows the same profiles with wall distance relative to the roughness height. B: $––$; L: $––\text{−}$; M: $\text{−}\text{−}\text{−}\text{−}$; H: $–.–.$; E: $–..–$.

Figure 13

Dispersive pressure intensity. The inset shows the same profiles with wall distance relative to the roughness height. B: $––$; L: $––\text{−}$; M: $\text{−}\text{−}\text{−}\text{−}$; H: $–.–.$; E: $–..–$.

Figure 14

Streamwise component of the turbulent Reynolds stress tensor. The filled circles mark the crest of each surface. Smooth: $––$; B: $––$; L: $––\text{−}$; M: $\text{−}\text{−}\text{−}\text{−}$; H: $–.–.$; E: $–..–$.

Figure 15

Spanwise component of the turbulent Reynolds stress tensor. The filled circles mark the crest of each surface. Smooth: $––$; B: $––$; L: $––\text{−}$; M: $\text{−}\text{−}\text{−}\text{−}$; H: $–.–.$; E: $–..–$.

Figure 16

Wall-normal component of the turbulent Reynolds stress tensor. The filled circles mark the crest of each surface. Smooth: $––$; B: $––$; L: $––\text{−}$; M: $\text{−}\text{−}\text{−}\text{−}$; H: $–.–.$; E: $–..–$.

Figure 17

Off-diagonal component of the Reynolds stress tensor. The filled circles mark the crest of each surface. Smooth: $––$; B: $––$; L: $––\text{−}$; M: $\text{−}\text{−}\text{−}\text{−}$; H: $–.–.$; E: $–..–$.

Figure 18

Turbulent kinetic energy contours in color scale overlaid on the surface height map in gray scale (only half of the streamwise extent of the domain is shown).

Figure 19

Mean fluctuating pressure intensity profiles. The filled circles mark the crest of each surface. Smooth: $––$; B: $––$; L: $––\text{−}$; M: $\text{−}\text{−}\text{−}\text{−}$; H: $–.–.$; E: $–..–$.

Figure 20

$\rho (r_x,r_y=0)$ (right) and $\rho (r_x=0,r_y)$ (right) at $z^{+}=0$ ($z^{+}=0.25$ for the smooth case), from top to bottom $u$, $v$, and $w$. Smooth: $––$; B: $––$; L: $––\text{−}$; M: $\text{−}\text{−}\text{−}\text{−}$; H: $–.–.$; E: $–..–$.

Figure 21

$\rho (r_x,r_y=0)$ (right) and $\rho (r_x=0,r_y)$ (right) at $z^{+}=60$, from top to bottom $u$, $v$, and $w$. Smooth: $––$; B: $––$; L: $––\text{−}$; M: $\text{−}\text{−}\text{−}\text{−}$; H: $–.–.$; E: $–..–$.

Figure 22

Cross-correlation between $\overline{k^{\prime }}$ and $\tilde{u}$ as a function of height and streamwise separation. The horizontal dashed line indicates the height of the largest peak, the green line indicates the contour level of zero.