▪ AbstractWe review the experimental evidence on turbulent flows over rough walls. Two parameters are important: the roughness Reynolds number ks+, which measures the effect of the roughness on the buffer layer, and the ratio of the boundary layer thickness to the roughness height, which determines whether a logarithmic layer survives. The behavior of transitionally rough surfaces with low ks+ depends a lot on their geometry. Riblets and other drag-reducing cases belong to this regime. In flows with δ/k ≲ 50, the effect of the roughness extends across the boundary layer, and is also variable. There is little left of the original wall-flow dynamics in these flows, which can perhaps be better described as flows over obstacles. We also review the evidence for the phenomenon of d-roughness. The theoretical arguments are sound, but the experimental evidence is inconclusive. Finally, we discuss some ideas on how rough walls can be modeled without the detailed computation of the flow around the roughness element...

Turbulent flows over rough walls

▪ Abstract The recent progress in three-dimensional boundary-layer stability and transition is reviewed. The material focuses on the crossflow instability that leads to transition on swept wings and rotating disks. Following a brief overview of instability mechanisms and the crossflow problem, a summary of the important findings of the 1990s is given.

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Stability and transition of three-dimensional boundary layers

This article reviews linear instability analysis of flows over or through complex two-dimensional (2D) and 3D geometries. In the three decades since it first appeared in the literature, global instability analysis, based on the solution of the multidimensional eigenvalue and/or initial value problem, is continuously broadening both in scope and in depth. To date it has dealt successfully with a wide range of applications arising in aerospace engineering, physiological flows, food processing, and nuclear-reactor safety. In recent years, nonmodal analysis has complemented the more traditional modal approach and increased knowledge of flow instability physics. Recent highlights delivered by the application of either modal or nonmodal global analysis are briefly discussed. A conscious effort is made to demystify both the tools currently utilized and the jargon employed to describe them, demonstrating the simplicity of the analysis. Hopefully this will provide new impulses for the creation of next-generation algorithms capable of coping with the main open research areas in which step-change progress can be expected by the application of the theory: instability analysis of fully inhomogeneous, 3D flows and control thereof.

Global Linear Instability

Advances in global linear instability analysis of nonparallel and three-dimensional flows

Notes on numerical fluid mechanics

Crossflow-vortex-induced laminar breakdown in a three-dimensional flat-plate boundary-layer flow is investigated in detail by means of spatial direct numerical simulations. The base flow is generic for an infinite swept wing, with decreasing favourable chordwise pressure gradient. First, the downstream growth and nonlinear saturation states initiated by a crossflow-vortex-mode packet as well as by single crossflow-vortex modes with various spanwise wavenumbers are simulated. Second, the secondary instability of the flow induced by the saturated crossflow vortices is scrutinized, clearly indicating the convective nature of the secondary instability and strengthening knowledge of the conditions for its onset. Emphasis is on the effect of crossflow-vortex-mode packets and of the spanwise vortex spacing on the secondary stability properties of the saturation states. Saturated uniform crossflow vortices initiated by single crossflow-vortex modes turn out to be less unstable than vortices initiated by a packet of vortex modes, and closely spaced saturated vortices are even stable. Third, we investigate the transition control strategy of upstream flow deformation by appropriate steady nonlinear vortex modes as applied in wind tunnel experiments at the Arizona State University. A significant transition delay is shown in the base flow considered here, and the underlying mechanisms are specified.

Mechanisms and passive control of crossflow-vortex-induced transition in a three-dimensional boundary layer

For numerical simulations of the spatially evolving laminar-turbulent transition process in boundary layers using the complete Navier-Stokes equations, the treatment of the outflow boundary requires special attention. The disturbances must pass through this boundary without causing reflections that would significantly alter the flow upstream. In this paper, we present various methods to influence the disturbed flow downstream of the region of interest, such that the disturbance level at the outflow boundary is significantly reduced, and hence the possibility of reflections is minimized. To demonstrate the effectiveness of the various techniques to alter the disturbance flow near the outflow boundary, the fundamental breakdown of a strongly decelerated boundary layer is simulated. Our results show that the most effective method is to spatially suppress the disturbance vorticity within a so-called "relaminarization zone." The suppression of the disturbance vorticity is gradually imposed within this zone by means of a weighting function. The enforced decay of the disturbance vorticity leads to a practically complete dissipation of any fluctuating component. Most importantly, this technique causes only a negligible upstream effect. The "relaminarized" boundary-layer flow then passes through the outflow boundary without significant reflections.

Outflow Boundary Conditions for Spatial Navier-Stokes Simulations of Transition Boundary Layers

Detailed comparison of spatial direct numerical simulations (DNS) and secondary linear stability theory (SLST) is provided for the three-dimensional crossflow- dominated boundary layer also considered at the DLR-Gfor experiments and theory. Secondary instabilities of large-amplitude steady and unsteady crossflow vortices arising from one single primary mode have been analysed. SLST results have been found to be reliable with respect to the dispersion relation and the amplitude distribution of the modal eigenfunction in the crosscut plane. However, significant deviations have been found in the amplification rates, the SLST results being strongly dependent on the necessarily simplified representation of the primary state. The secondary instability mechanisms are shown to be local, i.e. robust with respect to violations of the periodicity assumption made in the SLST for the wall-parallel directions. Perturbations associated with different local maxima of the spanwise periodic eigenfunctions develop independently from each other interacting only with the primary vortices next to them. Characteristic structures induced by different secondary instability modes have been analysed and an analogy with the Kelvin- Helmholtz instability mechanism has been highlighted.

Secondary instability of crossflow vortices: validation of the stability theory by direct numerical simulation

The laminar breakdown induced by purely travelling crossflow vortices in a three-dimensional flat-plate boundary-layer flow is investigated in detail by means of spatial direct numerical simulations. The base flow considered is generic for an infinite swept wing, with decreasing favourable chordwise pressure gradient and a sweep angle of . First, the primary downstream growth and nonlinear saturation state of a single crossflow wave are simulated. Secondly, background disturbance pulses are added, and the subsequent mechanisms triggering transition to turbulence in this scenario are identified and analysed in detail. The saturated travelling crossflow vortex is found to give rise to a co-travelling secondary instability not unlike the instability in the much investigated steady crossflow-vortex case, but with characteristic differences. An analysis method with a spanwise Galilean transformation to travel with the primary wave and a consequently adapted timewise/spanwise Fourier decomposition of the disturbance flow enables unambiguous isolation of the various secondary disturbance modes. The resulting flow structures and their dynamics in physical space are visualized.

Transition mechanisms induced by travelling crossflow vortices in a three-dimensional boundary layer

In this paper an efficient split-type Finite-Difference (FD) scheme with high modal resolu- tion - most important for the streamwise convection terms that cause wave transport and interaction - is derived for a mixed Fourier-spectral/FD method that is designed for the spatial direct numerical simulation (DNS) of boundary-layer transition and turbulence. Using a relatively simple but thor- ough and instructive modal analysis we discuss some principal trouble sources of the related FD discretization. The new scheme is based on a 6th-order compact FD discretization in streamwise and wall-normal direction and the classical 4th-order Runge-Kutta time-integration scheme with sym- metrical final corrector step. Exemplary results of a fundamental-(K-) type breakdown simulation of a strongly decelerated Falkner-Skan boundary layer (Hartree parameterH D 0.18) using 70 mega grid points in space are presented up to the early turbulent regime (Re;turb 820). The adverse pressure gradient gives rise to local separation zones during the breakdown stage and intensifies final breakdown by strong amplification of (background) disturbances thus enabling rapid transition at moderate Reynolds number. The appearance and dynamics of small-scale vortical structures in early turbulence basically similar to the large-scale structures at transition can be observed corroborating Kachanov's hypothesis on the importance of the K-regime of breakdown for coherent structures in turbulence.

Markus J. Kloker

Papers

Mechanisms and passive control of crossflow-vortex-induced transition in a three-dimensional boundary layer

Outflow Boundary Conditions for Spatial Navier-Stokes Simulations of Transition Boundary Layers

Secondary instability of crossflow vortices: validation of the stability theory by direct numerical simulation

Transition mechanisms induced by travelling crossflow vortices in a three-dimensional boundary layer

A Robust High-Resolution Split-Type Compact FD Scheme for Spatial Direct Numerical Simulation of Boundary-Layer Transition