The mixing-layer analogy is due to Raupach, Finnigan & Brunet (Boundary-Layer Meteorol., vol. 25, 1996, pp. 351–382). In the analogy, the flow in the roughness sublayer of a homogeneous deep vegetation canopy boundary layer is analogous to a plane mixing layer rather than a surface layer. Evidence for the analogy includes the inflected velocity profile, which resembles the velocity profile in a plane mixing layer, and, most notably, the following estimate as a result of the Kelvin–Helmholtz instability: $\varLambda _x=8.1L_s$, where $\varLambda _x$ is the spacing of the large-scale eddies, and $L_s$ is the shear length. The mixing-layer analogy has been very successful in vegetation canopy flow research, but has received only limited support in urban-canopy flow research. This work revisits Raupach et al.'s mixing-layer analogy, and we present the evidence for the mixing-layer analogy in urban-canopy flows: the exponential velocity profile in the canopy layer, i.e. $(U-U_c)/(U_h-U_c)=\exp (z/L_m)$, and $L_m\sim [(U_h/U_c-1)(U_h/U_c+3)]^{-1}$. Here, $z$ is the vertical coordinate, $L_m$ is the attenuation length and is a measure of the largest eddy in the canopy layer, $U_h$ is the wind speed at the canopy crest and $U_c$ is the velocity in the inactive layer. We conduct direct numerical simulations of various deep homogeneous urban-canopy flows and test the above two scalings. We also discuss why Raupach et al.'s analogy has not seen as many successes in urban-canopy flows as in vegetation canopy flows.

When manufacturing a turbine engine, the combustor annulus and the turbine annulus are created as separate parts and assembled. This leads to an inter-platform gap between the two components, which must be supplied with leakage air to prevent ingestion of the extremely hot combustion gases into the interior of the engine. The combustor and the turbine are likely to misalign because of differential thermal expansion or assembly tolerances. This paper presents a direct numerical simulation study of inter-platform misalignment with leakage flow supplied at the junction of the platforms. The geometry is two misaligned plates with a cross-flow and a leakage flow simulated as a slot jet. The misalignment of the two plates gives rise to a forward misalignment configuration and a backward misalignment configuration, and the jet/cross-flow gives rise to a windward mixing layer and a leeward mixing layer. Compared with the aligned configuration, the cooling effectiveness immediately downstream of the gap decreases in the forward misalignment configuration and increases in the backward misalignment configuration; this response amplifies as the flow rate through the gap increases. In addition to the cooling effectiveness, we report flow statistics, including the velocity, the temperature, the turbulent kinetic energy and the relevant turbulent fluxes. We find strong turbulence generation in the leeward mixing layer and high turbulence level as a result. Mixing of the thermal energy, on the other hand, occurs predominantly in the windward mixing layer. The eddy viscosity and the eddy conductivity that are critical to turbulence modelling are also reported. We find negative eddy viscosity at regions where the incoming boundary layer starts to mix with the leakage jet. The analysis shows that the negative eddy viscosity is a result of flow hysteresis: it takes time, or travel distance, before the eddies in the incoming boundary layer and the eddies in the leakage jet come to an equilibrium, thereby favouring a transport Reynolds stress model over a local eddy viscosity type model. The novelty of this paper lies in the direct numerical simulations, which provide direct access to the near-wall flow field and clarify the effects of blowing ratio and platform misalignment on heat transfer. The novelty also lies in the data analysis, which sheds light on how this flow should be modelled.

We present a derivation that begins with the Navier–Stokes equation and ends with a prediction of multiple statistically stable states identical to those observed in a spanwise rotating plane Couette flow. This derivation is able to explain the presence of multiple states in fully developed turbulence and the selection of one state over another by differently sized computational domains and different initial conditions. According to the present derivation, two and only two statistically stable states are possible in an infinitely large plane Couette flow with spanwise rotation, and that multiple states are not possible at very slow or very rapid rotation speeds. We also show the existence of limit-cycle-like behaviours near statistically stable states.

Much of our theoretical understanding of statistically stable and unstable flows is from the classical Monin–Obukhov similarity theory: the theory predicts the scaling of the mean flow well, but its prediction of the turbulent fluctuation is far from satisfactory. This study builds on Monin–Obukhov similarity theory and Townsend’s attached-eddy hypothesis. We present a model that connects the mean flow and the streamwise velocity fluctuations in both neutral and unstable boundary-layer flows at both moderate and high Reynolds numbers. The model predictions are compared to direct numerical simulations of weakly unstable boundary layers at moderate Reynolds numbers, and large-eddy simulations of unstable boundary-layer flows at high Reynolds numbers. The flow is shear dominated. The range of stability parameter considered in this work is $L/\unicode[STIX]{x1D6FF}<-0.1$, where $L$ is the Monin–Obukhov length, and $\unicode[STIX]{x1D6FF}$ is the boundary-layer height. Reasonably good prediction of velocity fluctuations based on knowledge of the mean velocity profile is obtained.

According to Townsend’s attached eddy hypothesis (AEH), a boundary layer flow is comprised of wall-attached eddies, but to extract the part of the flow whose statistical behaviours are well described by the AEH is not at all straightforward. The objective of this work is to extract the part of the flow that can be described by the AEH, and study the statistical behaviours of the other part, which cannot be described by the AEH. In this process, two types of eddies are identified in addition to the Kolmogorov-scale eddies, i.e. wall-attached eddies and wall-detached eddies. The statistical behaviours of the wall-attached eddies are shown to be very well described by the AEH, i.e. the eddies are wall-attached, self-similar and, importantly, Gaussian, whereas the wall-detached eddies cannot be modelled by the AEH. Specifically, a decomposition scheme is proposed following Townsend (The Structure of Turbulent Shear Flow, Cambridge University Press, 1976). We apply our decomposition scheme to three different flows, i.e. channel, boundary layer and atmospheric surface layer flows. The results are similar with only quantitative differences, suggesting possible universality in both the wall-attached eddies and the wall-detached eddies.

The kinematics of a fully developed passive scalar is modelled using the hierarchical random additive process (HRAP) formalism. Here, ‘a fully developed passive scalar’ refers to a scalar field whose instantaneous fluctuations are statistically stationary, and the ‘HRAP formalism’ is a recently proposed interpretation of the Townsend attached eddy hypothesis. The HRAP model was previously used to model the kinematics of velocity fluctuations in wall turbulence: $u=\sum _{i=1}^{N_{z}}a_{i}$, where the instantaneous streamwise velocity fluctuation at a generic wall-normal location $z$ is modelled as a sum of additive contributions from wall-attached eddies ($a_{i}$) and the number of addends is $N_{z}\sim \log (\unicode[STIX]{x1D6FF}/z)$. The HRAP model admits generalized logarithmic scalings including $\langle \unicode[STIX]{x1D719}^{2}\rangle \sim \log (\unicode[STIX]{x1D6FF}/z)$, $\langle \unicode[STIX]{x1D719}(x)\unicode[STIX]{x1D719}(x+r_{x})\rangle \sim \log (\unicode[STIX]{x1D6FF}/r_{x})$, $\langle (\unicode[STIX]{x1D719}(x)-\unicode[STIX]{x1D719}(x+r_{x}))^{2}\rangle \sim \log (r_{x}/z)$, where $\unicode[STIX]{x1D719}$ is the streamwise velocity fluctuation, $\unicode[STIX]{x1D6FF}$ is an outer length scale, $r_{x}$ is the two-point displacement in the streamwise direction and $\langle \cdot \rangle$ denotes ensemble averaging. If the statistical behaviours of the streamwise velocity fluctuation and the fluctuation of a passive scalar are similar, we can expect first that the above mentioned scalings also exist for passive scalars (i.e. for $\unicode[STIX]{x1D719}$ being fluctuations of scalar concentration) and second that the instantaneous fluctuations of a passive scalar can be modelled using the HRAP model as well. Such expectations are confirmed using large-eddy simulations. Hence the work here presents a framework for modelling scalar turbulence in high Reynolds number wall-bounded flows.

Considering structure functions of the streamwise velocity component in a framework akin to the extended self-similarity hypothesis (ESS), de Silva et al. (J. Fluid Mech., vol. 823, 2017, pp. 498–510) observed that remarkably the large-scale (energy-containing range) statistics in canonical wall-bounded flows exhibit universal behaviour. In the present study, we extend this universality, which was seen to encompass also flows at moderate Reynolds number, to Taylor–Couette flow. In doing so, we find that also the transversal structure function of the spanwise velocity component exhibits the same universal behaviour across all flow types considered. We further demonstrate that these observations are consistent with predictions developed based on an attached-eddy hypothesis. These considerations also yield a possible explanation for the efficacy of the ESS framework by showing that it relaxes the self-similarity assumption for the attached-eddy contributions. By taking the effect of streamwise alignment into account, the attached-eddy model predicts different behaviour for structure functions in the streamwise and in the spanwise directions and that this effect cancels in the ESS framework – both consistent with the data. Moreover, it is demonstrated here that also the additive constants, which were previously believed to be flow dependent, are indeed universal at least in turbulent boundary layers and pipe flow where high Reynolds number data are currently available.

Properties of single- and two-point moment generating functions (MGFs) are examined in the inertial region of wall-bounded flows. Empirical evidence for power-law scaling of the single-point MGF $\langle \text{exp}(qu^{+})\rangle$ (where $u^{+}$ is the normalized streamwise velocity fluctuation and $q$ a real parameter) with respect to the wall-normal distance is presented, based on hot-wire data from a $Re_{{\it\tau}}=13\,000$ boundary-layer experiment. The parameter $q$ serves as a ‘dial’ to emphasize different parts of the signal such as high- and low-speed regions, for positive and negative values of $q$, respectively. Power-law scaling $\langle \text{exp}(qu^{+})\rangle \sim (z/{\it\delta})^{-{\it\tau}(q)}$ can be related to the generalized logarithmic laws previously observed in higher-order moments, such as in $\langle u^{+2p}\rangle ^{1/p}$, but provide additional information not available through traditional moments if considering $q$ values away from the origin. For two-point MGFs, the scalings in $\langle \text{exp}[qu^{+}(x)+q^{\prime }u^{+}(x+r)]\rangle$ with respect to $z$ and streamwise displacement $r$ in the logarithmic region are investigated. The special case $q^{\prime }=-q$ is of particular interest, since this choice emphasizes rare events with high and low speeds at a distance $r$. Applying simple scaling arguments motivated by the attached eddy model, a ‘scaling transition’ is predicted to occur for $q=q_{cr}$ such that ${\it\tau}(q_{cr})+{\it\tau}(-q_{cr})=1$, where ${\it\tau}(q)$ is the set of scaling exponents for single-point MGFs. This scaling transition is not visible to traditional central moments, but is indeed observed based on the experimental data, illustrating the capabilities of MGFs to provide new and statistically robust insights into turbulence structure and confirming essential ingredients of the attached eddy model.

We conduct a series of large-eddy simulations (LES) to examine the mean flow behaviour within the roughness layer of turbulent boundary layer flow over rough surfaces. We consider several configurations consisting of arrays of rectangular-prism roughness elements with various spacings, aspect ratios and height distributions. The results provide clear evidence for exponential behaviour of the mean flow with respect to the wall normal distance. Such behaviour has been proposed before (see, e.g., Cionco, 1966 Tech. Rep. DTIC document), and is represented as $U(z)/U_{h}=\exp [a(z/h-1)]$, where $U(z)$ is the spatially/temporally averaged fluid velocity, $z$ is the wall normal distance, $h$ represents the height of the roughness elements and $U_{h}$ is the velocity at $z=h$. The attenuation factor $a$ depends on the density of the roughness element distribution and details of the roughness distribution on the wall. Once established, the generic velocity profile shape is used to formulate a fully analytical model for the effective drag exerted by turbulent flow on a surface covered with arrays of rectangular-prism roughness elements. The approach is based on the von Karman–Pohlhausen integral method, in which a shape function is assumed for the mean velocity profile and its parameters are determined based on momentum conservation and other fundamental constraints. In order to determine the attenuation parameter $a$, wake interactions among surface roughness elements are accounted for by using the concept of flow sheltering. The model transitions smoothly between ‘$k$’ and ‘$d$’ type roughness conditions depending on the surface coverage density and the detailed geometry of roughness elements. Comparisons between model predictions and experimental/numerical data from the existing literature as well as LES data from this study are presented. It is shown that the analytical model provides good predictions of mean velocity and drag forces for the cases considered, thus raising the hope that analytical roughness modelling based on surface geometry is possible, at least for cases when the location of flow separation over surface elements can be easily predicted, as in the case of wall-attached rectangular-prism roughness elements.

Cu-Al alloys have been recommended for application as the diffusion barriers/adhesion promoters for advanced copper based metallization schemes. This approach to barrier formation is to generate an ultra-thin interfacial layer through Cu alloying without significantly affecting the resistivity of Cu. In this paper the microstructure of the bilayers of Cu/Cu-5 at%Al and Cu-5 at%Al/Cu sputter deposited on SiO2 before and after thermal annealing is investigated by transmission electron microscopy (TEM). Interfacial layer is observed in both cases. The variation of the resistance of the Cu-Al alloy film is consistent with its microstructure. The x-ray diffraction (XRD) spectra of Cu-5 at%Al on SiO2 shows that the addition of Al into Cu intends to favor the Cu (111) texture. These results will be presented and discussed showing that films of Cu doped with Al appear to act as a suitable barrier and adhesion promoter between SiO2 and Cu.