We show that the Fukagata et al.'s (Phys. Fluids, vol. 14, no. 11, 2002, pp. 73–76) identity for free-stream boundary layers simplifies to the von Kármán momentum integral equation relating the skin-friction coefficient and the momentum thickness when the upper bound in the integrals used to obtain the identity is taken to be asymptotically large. If a finite upper bound is used, the terms of the identity depend spuriously on the bound itself. Differently from channel and pipe flows, the impact of the Reynolds stresses on the wall-shear stress cannot be quantified in the case of free-stream boundary layers because the Reynolds stresses disappear from the identity. The infinite number of alternative identities obtained by performing additional integrations on the streamwise momentum equation also all simplify to the von Kármán equation. Analogous identities are found for channel flows, where the relative influence of the physical terms on the wall-shear stress depends on the number of successive integrations, demonstrating that the laminar and turbulent contributions to the skin-friction coefficient are only distinguished in the original identity discovered by Fukagata et al. (Phys. Fluids, vol. 14, no. 11, 2002, pp. 73–76). In the limit of large number of integrations, these identities degenerate to the definition of skin-friction coefficient and a novel twofold-integration identity is found for channel and pipe flows. In addition, we decompose the skin-friction coefficient uniquely as the sum of the change of integral thicknesses with the streamwise direction, following the study of Renard & Deck (J. Fluid Mech., vol. 790, 2016, pp. 339–367). We utilize an energy thickness and an inertia thickness, which is composed of a thickness related to the mean-flow wall-normal convection and a thickness linked to the streamwise inhomogeneity of the mean streamwise velocity. The contributions of the different terms of the streamwise momentum equation to the friction drag are thus quantified by these integral thicknesses.

Active skin-friction reduction in a turbulent boundary layer (TBL) is experimentally studied based on time-periodic blowing through one array of streamwise slits. The control parameters investigated include the blowing amplitude A+ and frequency f+, which, expressed in wall units, range from 0 to 2 and from 0.007 to 0.56, respectively. The maximum local friction reduction downstream of the slits reaches more than 70 %; friction does not fully recover to the state of the natural TBL until 500 wall units behind the slits. A positive net power saving is possible, and 4.01 % is measured with a local friction drag reduction (DR) of 49 %. A detailed analysis based on hot-wire, particle image velocimetry and smoke-wire flow visualization data is performed to understand the physical mechanisms involved. Spectral analysis indicates weakened near-wall large-scale structures. Flow visualizations show stabilized streaky structures and a locally relaminarized flow. Two factors are identified to contribute to the DR. Firstly, the jets from the slits create streamwise vortices in the near-wall region, preventing the formation of near-wall streaks and interrupting the turbulence generation cycle. Secondly, the zero-streamwise-momentum fluid associated with the jets also accounts for the DR. A closed-loop opposing control system is developed, along with an open-loop desynchronized control scheme, to quantify the two contributions. The latter is found to account for 77 % of the DR, whereas the former is responsible for 23 %. An empirical scaling of the DR is also proposed, which provides valuable insight into the TBL control physics.

In this paper, direct numerical simulation (DNS) is presented for spatially evolving turbulent boundary layer over an isothermal flat-plate at Ma∞ = 2.25,5,6,8. When Ma∞ = 8, two cases with the ratio of wall-to-reference temperature Tω/T∞ = 1.9 and 10.03 are considered respectively. The wall temperature approaches recovery temperatures for other cases. The characteristics of compressible turbulent boundary layer (CTBL) affected by freestream Mach number and wall temperature are investigated. It focuses on assessing compressibility effects and the validity of Morkovin's hypothesis through computing and analyzing the mean velocity profile, turbulent intensity, the strong Reynolds analogy (SRA) and possibility density function of dilatation term. The results show that, when the wall temperature approaches recovery temperature, the effects of Mach number on compressibility is insignificant. As a result, the compressibility effect is very weak and the Morkovin's hypothesis is still valid for Mach number even up to 8. However, when Mach number equal to 8, the wall temperature effect on the compressibility is sensitive. In this case, when Tω/T∞ = 1.9, the Morkovin's hypothesis is not fully valid. The validity of classical SRA depends on wall temperature directly. A new modified SRA is proposed to eliminate such negative factor in near wall region. Finally the effects of Mach number and wall temperature on streaks are also studied.

A numerical investigation of the mean wall shear stress properties on a spatially developing turbulent boundary layer over a smooth flat plate was carried out by means of a zonal detached eddy simulation (ZDES) technique for the Reynolds number range $3060\leq Re_{\theta }\leq 13\, 650$. Some asymptotic trends of global parameters are suggested. Consistently with previous findings, the calculation confirms the occurrence of very large-scale motions approximately $5\delta $ to $6 \delta $ long which are meandering with a lateral amplitude of $0.3 \delta $ and which maintain a footprint in the near-wall region. It is shown that these large scales carry a significant amount of Reynolds shear stress and their influence on the skin friction, denoted $C_{f,2}$, is revisited through the FIK identity by Fukagata, Iwamoto & Kasagi (Phys. Fluids, vol. 14, 2002, p. L73). It is argued that $C_{f,2}$ is the relevant parameter to characterize the high-Reynolds-number turbulent skin friction since the term describing the spatial heterogeneity of the boundary layer also characterizes the total shear stress variations across the boundary layer. The behaviour of the latter term seems to follow some remarkable self-similarity trends towards high Reynolds numbers. A spectral analysis of the weighted Reynolds stress with respect to the distance to the wall and to the wavelength is provided for the first time to our knowledge and allows us to analyse the influence of the largest scales on the skin friction. It is shown that structures with a streamwise wavelength $\lambda _x >\delta $ contribute to more than $60\, \%$ of $C_{f,2}$, and that those larger than $\lambda _x >2\delta $ still represent approximately $45\, \%$ of $C_{f,2}$.

The role of streamline curvature-driven favourable pressure gradients in modifying the turbulence structure of a Mach 4.9, high-Reynolds-number (${\mathit{Re}}_{\theta } = 43\hspace{0.167em} 000$) boundary layer is examined. Three pressure gradient cases ($\beta = (\mathrm{d} p/ \mathrm{d} x)({\delta }^{\ast } / {\tau }_{w} )= 0. 07, - 0. 3$ and $- 1. 0$) are characterized via particle image velocimetry. The expected stabilizing trends in the Reynolds stresses are observed, with a sign reversal in the Reynolds shear stress in the outer part of the boundary layer for the strongest favourable pressure gradient considered. The increased transverse normal strain rate and reduced principal strain rate are the primary factors. Reynolds stress quadrant events are redistributed, such that the relative differences between the quadrant magnitudes decreases. Very little preferential quadrant mode selection is observed for the strongest pressure gradient considered. Two-point correlations suggest that the turbulent structures are reoriented to lean farther away from the wall, accompanied by a slight reduction in their characteristic size, consistent with previous flow visualization studies. This reorientation is more pronounced in the outer, dilatation-dominated region of the boundary layer, whereas the alteration in structure size is more pronounced nearer the wall, where the principal strain rates are larger. In addition, integration of a simplified form of a Reynolds stress transport closure model provided a framework to assess the role of the strain-rate field on the observed Reynolds shear stresses. Given the simple geometry, the present data provide a suitable test bed for Reynolds stress transport and large-eddy model development and validation.

This paper focuses on turbulence structure in a fully developed rough-wall channel flow and its role in subgrid-scale (SGS) energy transfer. Our previous work has shown that eddies of scale comparable to the roughness elements are generated near the wall, and are lifted up rapidly by large-scale coherent structures to flood the flow field well above the roughness sublayer. Utilizing high-resolution and time-resolved particle-image-velocimetry datasets obtained in an optically index-matched facility, we decompose the turbulence into large (${\gt }\lambda $), intermediate ($3\text{{\ndash}} 6k$), roughness ($1\text{{\ndash}} 3k$) and small (${\lt }k$) scales, where $k$ and $\lambda (\lambda / k= 6. 8)$ are roughness height and wavelength, respectively. With decreasing distance from the wall, there is a marked increase in the ‘non-local’ SGS energy flux directly from large to small scales and in the fraction of turbulence dissipated by roughness-scale eddies. Conditional averaging is used to show that a small fraction of the flow volume (e.g. 5 %), which contains the most intense SGS energy transfer events, is responsible for a substantial fraction (50 %) of the energy flux from resolved to subgrid scales. In streamwise wall-normal ($x\text{{\ndash}} y$) planes, the averaged flow structure conditioned on high SGS energy flux exhibits a large inclined shear layer containing negative vorticity, bounded by an ejection below and a sweep above. Near the wall the sweep is dominant, while in the outer layer the ejection is stronger. The peaks of SGS flux and kinetic energy within the inclined layer are spatially displaced from the region of high resolved turbulent kinetic energy. Accordingly, some of the highest correlations occur between spatially displaced resolved velocity gradients and SGS stresses. In wall-parallel $x\text{{\ndash}} z$ planes, the conditional flow field exhibits two pairs of counter-rotating vortices that induce a contracting flow at the peak of SGS flux. Instantaneous realizations in the roughness sublayer show the presence of the counter-rotating vortex pairs at the intersection of two vortex trains, each containing multiple $\lambda $-spaced vortices of the same sign. In the outer layer, the SGS flux peaks within isolated vortex trains that retain the roughness signature, and the distinct pattern of two counter-rotating vortex pairs disappears. To explain the planar signatures, we propose a flow consisting of U-shaped quasi-streamwise vortices that develop as spanwise vorticity is stretched in regions of high streamwise velocity between roughness elements. Flow induced by adjacent legs of the U-shaped structures causes powerful ejections, which lift these vortices away from the wall. As a sweep is transported downstream, its interaction with the roughness generates a series of such events, leading to the formation of inclined vortex trains.

Afin d'étudier la structure fine de la turbulence pariétale en présence de variations importantes de densité, une analyse statistique des données expérimentales, dans une soufflerie à recirculation, est effectuée. Les résultats montrent que la relaxation du facteur de dissymétrie de $u'(S_{u'}$) s'effectue plus rapidement dans la couche externe que près de la paroi, aussi bien pour l'injection de l'air que pour l'injection d'hélium. $S_{u'}$ croît près de la fente d'injection de façon appréciable et cette augmentation est nettement plus accentuée pour l'injection de l'air que l'injection d'hélium. Cette croissance du facteur de dissymétrie près de la fente d'injection peut s'expliquer par l'augmentation du flux convectif longitudinal de l'énergie turbulente dans cette zone. Les résultats montrent, pour la distribution du facteur d'aplatissement $F_{u'}$, qu'il n'y a pas d'effet important du gradient de densité sur la structure intermittente de la vitesse longitudinale instantanée dans la zone développée, $x/\delta \ge 5$. L'analyse statistique menée dans cette étude montre que l'injection d'hélium, dans la couche limite, engendre des éjections plus violentes que dans le cas d'une injection d'air. Ce résultat est confirmé par la contribution importante des éjections au flux massique turbulent.