skip to main content


This content will become publicly available on November 10, 2024

Title: Vorticity cascade and turbulent drag in wall-bounded flows: plane Poiseuille flow

Drag for wall-bounded flows is directly related to the spatial flux of spanwise vorticity outward from the wall. In turbulent flows a key contribution to this wall-normal flux arises from nonlinear advection and stretching of vorticity, interpretable as a cascade. We study this process using numerical simulation data of turbulent channel flow at friction Reynolds number$Re_\tau =1000$. The net transfer from the wall of spanwise vorticity created by downstream pressure drop is due to two large opposing fluxes, one which is ‘down-gradient’ or outward from the wall, where most vorticity concentrates, and the other which is ‘up-gradient’ or toward the wall and acting against strong viscous diffusion in the near-wall region. We present evidence that the up-gradient/down-gradient transport occurs by a mechanism of correlated inflow/outflow and spanwise vortex stretching/contraction that was proposed by Lighthill. This mechanism is essentially Lagrangian, but we explicate its relation to the Eulerian anti-symmetric vorticity flux tensor. As evidence for the mechanism, we study (i) statistical correlations of the wall-normal velocity and of wall-normal flux of spanwise vorticity, (ii) vorticity flux cospectra identifying eddies involved in nonlinear vorticity transport in the two opposing directions and (iii) visualizations of coherent vortex structures which contribute to the transport. The ‘D-type’ vortices contributing to down-gradient transport in the log layer are found to be attached, hairpin-type vortices. However, the ‘U-type’ vortices contributing to up-gradient transport are detached, wall-parallel, pancake-shaped vortices with strong spanwise vorticity, as expected by Lighthill's mechanism. We discuss modifications to the attached eddy model and implications for turbulent drag reduction.

 
more » « less
Award ID(s):
2103874
NSF-PAR ID:
10511081
Author(s) / Creator(s):
; ;
Publisher / Repository:
CUP
Date Published:
Journal Name:
Journal of Fluid Mechanics
Volume:
974
ISSN:
0022-1120
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. Two common definitions of the spatially local rate of kinetic energy cascade at some scale$\ell$in turbulent flows are (i) the cubic velocity difference term appearing in the ‘scale-integrated local Kolmogorov–Hill’ equation (structure-function approach), and (ii) the subfilter-scale energy flux term in the transport equation for subgrid-scale kinetic energy (filtering approach). We perform a comparative study of both quantities based on direct numerical simulation data of isotropic turbulence at Taylor-scale Reynolds number 1250. While in the past observations of negative subfilter-scale energy flux (backscatter) have led to debates regarding interpretation and relevance of such observations, we argue that the interpretation of the local structure-function-based cascade rate definition is unambiguous since it arises from a divergence term in scale space. Conditional averaging is used to explore the relationship between the local cascade rate and the local filtered viscous dissipation rate as well as filtered velocity gradient tensor properties such as its invariants. We find statistically robust evidence of inverse cascade when both the large-scale rotation rate is strong and the large-scale strain rate is weak. Even stronger net inverse cascading is observed in the ‘vortex compression’$R>0$,$Q>0$quadrant, where$R$and$Q$are velocity gradient invariants. Qualitatively similar but quantitatively much weaker trends are observed for the conditionally averaged subfilter-scale energy flux. Flow visualizations show consistent trends, namely that spatially, the inverse cascade events appear to be located within large-scale vortices, specifically in subregions when$R$is large.

     
    more » « less
  2. Researchers have hypothesized that the post-stall lift benefit of bird’s alular feathers, or alula, stems from the maintenance of an attached leading-edge vortex (LEV) over their thin-profiled, outer hand wing. Here, we investigate the connection between the alula and LEV attachment via flow measurements in a wind tunnel. We show that a model alula, whose wetted area is 1 % that of the wing, stabilizes a recirculatory aft-tilted LEV on a steadily translating unswept wing at post-stall angles of attack. The attached vortex is the result of the alula’s ability to smoothly merge otherwise separate leading- and side-edge vortical flows. We identify two key processes that facilitate this merging: (i) the steering of spanwise vorticity generated at the wing’s leading edge back to the wing plane and (ii) an aft-located wall jet of high-magnitude root-to-tip spanwise flow ( ${>}80\,\%$ that of the free-stream velocity). The former feature induces LEV roll-up while the latter tilts LEV vorticity aft and evacuates this flow toward the wing tip via an outboard vorticity flux. We identify the alula’s streamwise position (relative to the leading edge of the thin wing) as important for vortex steering and the alula’s cant angle as important for high-magnitude spanwise flow generation. These findings advance our understanding of the likely ways birds leverage LEVs to augment slow flight. 
    more » « less
  3. null (Ed.)
    We use an online database of a turbulent channel-flow simulation at $Re_\tau =1000$ (Graham et al. J. Turbul. , vol. 17, issue 2, 2016, pp. 181–215) to determine the origin of vorticity in the near-wall buffer layer. Following an experimental study of Sheng et al. ( J. Fluid Mech. , vol. 633, 2009, pp.17–60), we identify typical ‘ejection’ and ‘sweep’ events in the buffer layer by local minima/maxima of the wall stress. In contrast to their conjecture, however, we find that vortex lifting from the wall is not a discrete event requiring $\sim$ 1 viscous time and $\sim$ 10 wall units, but is instead a distributed process over a space–time region at least $1\sim 2$ orders of magnitude larger in extent. To reach this conclusion, we exploit a rigorous mathematical theory of vorticity dynamics for Navier–Stokes solutions, in terms of stochastic Lagrangian flows and stochastic Cauchy invariants, conserved on average backward in time. This theory yields exact expressions for vorticity inside the flow domain in terms of vorticity at the wall, as transported by viscous diffusion and by nonlinear advection, stretching and rotation. We show that Lagrangian chaos observed in the buffer layer can be reconciled with saturated vorticity magnitude by ‘virtual reconnection’: although the Eulerian vorticity field in the viscous sublayer has a single sign of spanwise component, opposite signs of Lagrangian vorticity evolve by rotation and cancel by viscous destruction. Our analysis reveals many unifying features of classical fluids and quantum superfluids. We argue that ‘bundles’ of quantized vortices in superfluid turbulence will also exhibit stochastic Lagrangian dynamics and satisfy stochastic conservation laws resulting from particle relabelling symmetry. 
    more » « less
  4. Large-eddy simulation was used to model turbulent atmospheric surface layer (ASL) flow over canopies composed of streamwise-aligned rows of synthetic trees of height,$h$, and systematically arranged to quantify the response to variable streamwise spacing,$\delta _1$, and spanwise spacing,$\delta _2$, between adjacent trees. The response to spanwise and streamwise heterogeneity has, indeed, been the topic of a sustained research effort: the former resulting in formation of Reynolds-averaged counter-rotating secondary cells, the latter associated with the$k$- and$d$-type response. No study has addressed the confluence of both, and results herein show secondary flow polarity reversal across ‘critical’ values of$\delta _1$and$\delta _2$. For$\delta _2/\delta \lesssim 1$and$\gtrsim 2$, where$\delta$is the flow depth, the counter-rotating secondary cells are aligned such that upwelling and downwelling, respectively, occurs above the elements. The streamwise spacing$\delta _1$regulates this transition, with secondary cell reversal occurring first for the largest$k$-type cases, as elevated turbulence production within the canopy necessitates entrainment of fluid from aloft. The results are interpreted through the lens of a benchmark prognostic closure for effective aerodynamic roughness,$z_{0,{Eff.}} = \alpha \sigma _h$, where$\alpha$is a proportionality constant and$\sigma _h$is height root mean square. We report$\alpha \approx 10^{-1}$, the value reported over many decades for a broad range of rough surfaces, for$k$-type cases at small$\delta _2$, whereas the transition to$d$-type arrangements necessitates larger$\delta _2$. Though preliminary, results highlight the non-trivial response to variation of streamwise and spanwise spacing.

     
    more » « less
  5. The statistical properties of uniform momentum zones (UMZs) are extracted from laboratory and field measurements in rough wall turbulent boundary layers to formulate a set of stochastic models for the simulation of instantaneous velocity profiles. A spatiotemporally resolved velocity dataset, covering a field of view of$8 \times 9\,{\rm m}^2$, was obtained in the atmospheric surface layer using super-large-scale particle image velocimetry (SLPIV), as part of the Grand-scale Atmospheric Imaging Apparatus (GAIA). Wind tunnel data from a previous study are included for comparison (Heiselet al.,J. Fluid Mech., vol. 887, 2020, R1). The probability density function of UMZ attributes such as their thickness, modal velocity and averaged vertical velocity are built at varying elevations and modelled using log-normal and Gaussian distributions. Inverse transform sampling of the distributions is used to generate synthetic step-like velocity profiles that are spatially and temporally uncorrelated. Results show that in the wide range of wall-normal distances and$Re_\tau$up to$\sim O(10^6)$investigated here, shear velocity scaling is manifested in the velocity jump across shear interfaces between adjacent UMZs, and attached eddy behaviour is observed in the linear proportionality between UMZ thickness and their wall normal location. These very same characteristics are recovered in the generated instantaneous profiles, using both fully stochastic and data-driven hybrid stochastic (DHS) models, which address, in different ways, the coupling between modal velocities and UMZ thickness. Our method provides a stochastic approach for generating an ensemble of instantaneous velocity profiles, consistent with the structural organisation of UMZs, where the ensemble reproduces the logarithmic mean velocity profile and recovers significant portions of the Reynolds stresses and, thus, of the streamwise and vertical velocity variability.

     
    more » « less