Numerical and experimental study of mechanisms responsible for turbulent secondary flows in boundary layer flows over spanwise heterogeneous roughness

Abstract

We study the dynamics of turbulent boundary layer flow over a heterogeneous topography composed of roughness patches exhibiting relatively high and low correlation in the streamwise and spanwise directions, respectively (i.e. the roughness appears as streamwise-aligned 'strips'). It has been reported that such roughness induces a spanwise-wall normal mean secondary flow in the form of mean streamwise vorticity associated with counter-rotating boundary-layer-scale circulations. Here, we demonstrate that this mean secondary flow is Prandtl's secondary flow of the second kind, both driven and sustained by spatial gradients in the Reynolds-stress components, which cause a subsequent imbalance between production and dissipation of turbulent kinetic energy that necessitates secondary advective velocities. In reaching this conclusion, we study (i) secondary circulations due to spatial gradients of turbulent kinetic energy, and (ii) the production budgets of mean streamwise vorticity by gradients of the Reynolds stresses. We attribute the secondary flow phenomena to extreme peaks of surface stress on the relatively high-roughness regions and associated elevated turbulence production in the fluid immediately above. An optimized state is attained by entrainment of fluid exhibiting the lowest turbulent stresses -from above -and subsequent lateral ejection in order to preserve conservation of mass.

Effects of spanwise spacing on large-scale secondary flows in rough-wall turbulent boundary layers

“…We only illustrate the case which exhibit the largest secondary flow (S/δ = 0.88), however, similar patterns were observed in the flows over each of the surfaces with coarse roughness. The patterns of the turbulent Reynolds stresses are similar to those observed in the large eddy simulations (LES) presented by Anderson et al (2015). The variance of the streamwise velocity uu is a maximum in the vicinity of the elevated roughness and particularly near the corners of the roughness elements at the interface of the small vortices and large secondary flows that occur in the low-momentum pathways.…”

“…The transverse shear stresses uw and vw are very clearly symmetric and periodic with the roughness spacing, in accordance with the directionality of the mean spanwise velocity W imposed by the large secondary flows. Anderson et al (2015) stressed the importance of the spanwise gradients of the shear stresses and the role they play in driving the secondary vortices, which were interpreted as Prantl's secondary flows of the second kind; the fact that the stresses behave similarly in the current measurements as in their results supports this interpretation. Again, we note that the main difference between these observation and those of Anderson et al (2015) is the fact that the locations of the low-momentum pathways always occur directly above the elevated roughness elements, whereas in the LES these occurred over what they referred to as 'high' roughness; however, we attempt to explain these differences in section 4.1.…”

“…These small vortices in fact appear in every case, however, for the coarse spacings, the large secondary vortices dwarf these small vortices. If you look closely, similar small vortices may also be observed in the vorticity maps by Anderson et al (2015). In this limit case with S/δ = 0.45, the larger secondary vortices still appear, flanking these smaller vortices; however, the diameter of the larger secondary vortices has nearly the same size as the small vortices that occur above the roughness element, becoming difficult to distinguish in the measurements, and having minimal effect on the mean velocity field.…”

“…The Reynolds shear stress −uv is also a maximum near the elevated roughness but exhibits high values that extend nearly to half the boundary layer thickness, following the strong updrafts that occur above the elevated surfaces. Anderson et al (2015) also observed peaks of −uv at the base of the high-momentum pathways (which occurred here in the valleys between the roughness strips); however, we do not capture this region very clearly in our measurements and it is unclear if such a region exists in figure 6b. The transverse shear stresses uw and vw are very clearly symmetric and periodic with the roughness spacing, in accordance with the directionality of the mean spanwise velocity W imposed by the large secondary flows.…”

“…The mean signed swirling strength was calculated as the ensemble average of the instantaneous swirling strength maps, multiplied by the sign of the vorticity as in Anderson et al (2015), in order to indicate both the strength and direction of rotation of the regions of swirl. For the cases with coarse spacing, large secondary vortices can be clearly observed.…”