The mean velocity follows a logarithmic scaling in the surface layer when normalized by the friction velocity, i.e. a velocity scale derived from the wall-shear stress. The same logarithmic scaling exists for the mean temperature when one normalizes the temperature with the friction temperature, i.e. a scale derived from the wall heat flux. This temperature normalization poses challenges to adiabatic walls, for which the wall heat flux is zero, and the logarithmic temperature scaling becomes singular. This paper aims to establish a temperature transformation that applies to both isothermal walls and adiabatic walls. We show that by accounting for the diffusive flux, both the Van Driest transformation and the semi-local transformation (and other transformations alike) apply to adiabatic walls. We also show that the classic Walz equation works well for adiabatic walls because it models the diffusive flux, albeit in a rather crude way. For validation/testing, we conduct direct numerical simulations of supersonic Couette flows at Mach numbers $M=1$, 3 and 6, and various Reynolds numbers. The two walls are adiabatic, and a source term is included to cancel the aerodynamic heating in the domain. We show that the adiabatic wall data collapse onto the same incompressible logarithmic law of the wall like the isothermal wall data.

By varying the oil volume fraction, the microscopic droplet size and the macroscopic rheology of emulsions are investigated in a Taylor–Couette turbulent shear flow. Although here oil and water in the emulsions have almost the same physical properties (density and viscosity), unexpectedly, we find that oil-in-water (O/W) and water-in-oil (W/O) emulsions have very distinct hydrodynamic behaviours, i.e. the system is clearly asymmetric. By looking at the micro-scales, the average droplet diameter hardly changes with the oil volume fraction for O/W or for W/O. However, for W/O it is about $50\,\%$ larger than that of O/W. At the macro-scales, the effective viscosity of O/W is higher when compared to that of W/O. These asymmetric behaviours are expected to be caused by the presence of surface-active contaminants from the walls of the system. By introducing an oil-soluble surfactant at high concentration, remarkably, we recover the symmetry (droplet size and effective viscosity) between O/W and W/O emulsions. Based on this, we suggest a possible mechanism responsible for the initial asymmetry and reach conclusions on emulsions where interfaces are fully covered by the surfactant. Next, we discuss what sets the droplet size in turbulent emulsions. We uncover a $-6/5$ scaling dependence of the droplet size on the Reynolds number of the flow. Combining the scaling dependence and the droplet Weber number, we conclude that the droplet fragmentation, which determines the droplet size, occurs within the boundary layer and is controlled by the dynamic pressure caused by the gradient of the mean flow, as proposed by Levich (Physicochemical Hydrodynamics, Prentice-Hall, 1962), instead of the dynamic pressure due to turbulent fluctuations, as proposed by Kolmogorov (Dokl. Akad. Nauk. SSSR, vol. 66, 1949, pp. 825–828). The present findings provide an understanding of both the microscopic droplet formation and the macroscopic rheological behaviours in dynamic emulsification, and connects them.

During the high velocity impact of an object on a solid covered with a thin fluid layer, a lubricated contact exists within the short time in which the liquid is squeezed out from the contact. This is important for e.g. the grip of shoes on wet surfaces. We experimentally study the squeeze flow of such layers and find that the amount of viscous dissipation determines how much fluid remains in the contact after the kinetic energy of the impacting object is absorbed. For impacts with sufficient amount of kinetic energy, it is possible to completely drain a Newtonian or shear thinning fluid from the contact on a short time scale. Viscoelastic liquids, however, cannot be drained by increasing impact velocity, because of the fluids’ elastic tendency to retract back into the contact after rapid squeezing. This explains why the presence of polymeric fluids can lead to extreme slipperiness of surfaces. Furthermore, we show that all our experimental results agree with the predictions given by hydrodynamic theory applied to the fluid flow in the gap.

Submerged flexible vegetation occupies the core of aquatic ecosystem research. Hydrodynamics of submerged flexible vegetation and its interaction with the flow in an open channel are of great significance in studying the mass and momentum transports in the flow. In this study, a numerical model for highly flexible vegetation based on large eddy simulation and the immersed boundary method was used to simulate the flow–vegetation interaction. It is recognised that alternate vortices with opposite sense rotations appear at the flow–vegetation interface. These vortices prompt the vegetation canopy to have wave-like coherent waving motion, commonly called the monami phenomena. The spatial scale and the spreading velocity in the streamwise direction of these vortices determine the wavelength, frequency and amplitude of the vegetation coherent waving motion. In this study, the fast Fourier transform method was applied to analyse the factors affecting the characteristics of the vegetation coherent waving motion. It is revealed that, as the flow velocity increases, the wavelength of the coherent waving motion decreases, while the frequency and amplitude increase. Besides, as the vegetation spacing increases, the wavelength and amplitude of the coherent waving motion increase, but the frequency decreases. Furthermore, an increase in the relative density of vegetation magnifies the amplitude of coherent waving motion without affecting the wavelength and frequency.

In the present work, within the framework of thin film theory, we delineate the interaction between the interfacial dynamics of thermal Marangoni flow and non-Newtonian rheology by considering a spreading droplet over a non-isothermal substrate. The numerical simulations, performed at different equilibrium contact angles $(\theta _e)$, dimensionless thermocapillary strengths $(\beta )$ and shear-dependent viscosities $(n)$, reveal that the fluid rheology nonlinearly influences the mechanisms of disjoining pressure and Marangoni stress. Accordingly, three distinct spreading regimes for non-Newtonian drops arise. Results indicate that the Marangoni film regime, having an approximate linear drop shape, sustains at lower $\theta _e$, higher $\beta$ and $n$ ranges. Also, shear-thickening drops display an early onset of thermocapillary time scale and a steeper advancing front, while their shear-thinning counterparts retain a significant curvature for a much longer time. Contrastingly, the droplet regime is identified by fixed shape and uniform speed $(U)$ at higher $\theta _e$ and lower $(\beta$, $n)$ combinations. Here, an intricate interplay between $\beta$ and $n$ realizes a sharp increase in $U$ for shear thinning compared with its invariance for shear-thickening droplets. The transition regime appears as an intermediate regime between the other two and involves multiple ruptured droplets. In all the regimes, we observe slower (faster) spreading of shear-thinning (thickening) droplets than the Newtonian droplets. In addition, the variations in $n$ cause intense characteristic modulations to spreading attributes like droplet morphology and transient spreading behaviour, and also act as a switching mechanism between different spreading regimes. These unique results may be utilized for superior control of non-isothermal biofluid droplets in microfluidics.

Progressive capillary waves on the interface between two homogeneous fluids confined in a channel with rigid walls parallel to the undisturbed interface are investigated. This problem is formulated as a system of integrodifferential equations that can be solved numerically via a boundary integral equation method coupled with series expansions of the unknown functions. With this highly accurate scheme and numerical continuation, we explore the global bifurcation of periodic travelling waves. It is found that there are two types of limiting profile, self-intersecting and boundary-touching, which appear either along a primary branch bifurcating from infinitesimal periodic waves or on an isolated branch existing above a certain finite amplitude. For particular sets of parameters, these two types of bifurcation curves can intersect, which can be viewed as a secondary bifurcation phenomenon occurring on the primary branch. Based on asymptotic and numerical analyses of the almost limiting waves, it is found that the boundary-touching solutions feature a circular geometry, i.e. the interface is pieced together by circular arcs of the same radius. A theoretical investigation yields the necessary conditions for the existence of these extreme waves, whereby we can predict the limiting configurations for most parameter sets. The comparisons between theoretical predictions and numerical results show good agreement.

Understanding the behaviour of scalar isosurfaces in a turbulent flow is of particular interest for many problems in turbulent mixing that contain sharp interfaces between regions of the flow. Common examples include combustion, where the chemical reactions occur in thin regions within the flow, and the turbulent/non-turbulent interface in shear flows, where a thin region separates the rotational, turbulent motions from the irrotational, non-turbulent background. Recent advances in computing technology allow for in-depth analysis of these interface problems that are difficult to quantify in a laboratory setting. In this paper, the results of a direct numerical simulation of a passive scalar $\varPhi$ evolving on a turbulent, temporally developing mixing layer are described. A novel approach has been taken to calculate the surface area of individual scalar isosurfaces, $A_{iso}$, throughout the simulation, as well as the mean isosurface area density, $\varSigma$, as a function of the cross-stream direction and time. A notable finding is that the profiles of $\varSigma$ develop in a self-similar manner when scaled by the Taylor scale of the scalar field, $\lambda _\phi$. Remarkably, the scaling appears to hold for a wide range of isovalues. A rough scaling argument based on the formal definition of $\varSigma$ and properties of a temporal mixing layer is presented which also exposes a dependence on $\lambda _\phi$. Based on these results, a possible scaling for the isosurface area is presented as $A_{iso}/A_0 \sim (Re \, Sc)^{1/2}$, where $Re$ and $Sc$ are local Reynolds and Schmidt numbers, respectively.

We investigate the Reynolds analogy over riblets, namely the analogy between the fractional increase in Stanton number $C_h$ and the fractional increase in the skin-friction coefficient $C_f$, relative to a smooth surface. We investigate the direct numerical simulation data of Endrikat et al. (Flow Turbul. Combust., vol. 107, 2021, pp. 1–29). The riblet groove shapes are isosceles triangles with tip angles $\alpha = {30}^{\circ }, {60}^{\circ }, {90}^{\circ }$, a trapezoid, a rectangle and a right triangle. The viscous-scaled riblet spacing varies between $s^+ \approx 10$ to $60$. The global Reynolds analogy is primarily influenced by Kelvin–Helmholtz rollers and secondary flows. Kelvin–Helmholtz rollers locally break the Reynolds analogy favourably, i.e. cause a locally larger fractional increase in $C_h$ than in $C_f$. These rollers induce negative wall shear stress patches which have no analogue in wall heat fluxes. Secondary flows at the riblets’ crests are associated with local unfavourable breaking of the Reynolds analogy, i.e. locally larger fractional increase in $C_f$ than in $C_h$. Only the triangular riblets with $\alpha = {30}^{\circ }$ trigger strong Kelvin–Helmholtz rollers without appreciable secondary flows. This riblet shape globally preserves the Reynolds analogy from $s^+ = 21$ to $33$. However, the other riblet shapes have weak or non-existent Kelvin–Helmholtz rollers, yet persistent secondary flows. These riblet shapes behave similarly to rough surfaces. They unfavourably break the global Reynolds analogy, and do so to a greater extent as $s^+$ increases.

To date, very few careful and direct comparisons between experiments and direct numerical simulations (DNS) have been published on free-stream turbulence (FST) induced boundary layer transition, whilst there exist numerous published works on the comparison of canonical turbulent boundary layers. The primary reason is that the former comparison is vastly more difficult to carry out simply because all known transition scenarios have large energy gradients and are extremely sensitive to surrounding conditions. This paper presents a detailed comparison between new experiments and available DNS data of the complex FST transition scenario in a flat plate boundary layer at turbulence intensity level about $Tu = 3\,\%$ and FST Reynolds number about $Re_{{fst}} = 67$. The leading edge (LE) pressure gradient distribution and the full energy spectrum at the LE are identified as the two most important parameters for a satisfying comparison. Matching the LE characteristic FST parameters is not enough as previously thought, which is illustrated by setting up two experimental FST cases with about the same FST integral parameters at the LE but with different energy spectra. Finally, an FST boundary layer penetration depth (PD) measure is defined using DNS, which suggests that the PD grows with the downstream distance and stays around 20 % of the boundary layer thickness down to transition onset. With this result, one cannot rule out the significance of the continuous FST forcing along the boundary layer edge in this transition scenario, as indicated in previous studies.

We conduct direct numerical simulations and study the evolution of a pair of counter-rotating vortices in a stratified and turbulent environment beyond the first vortex linking (which is due to the Crow instability). The initial position of the two vortices is perpendicular to the direction of thermal stratification. We vary the Froude number, the background turbulence intensity and the Reynolds number, covering strong, weak and neutral stratification; low, moderate and high background turbulence; and low and moderate Reynolds numbers. We observe a second vortex linking in a weakly stratified environment for the first time. The second vortex linking is followed by turbulence bursts that lead to the rapid decay of the residual vortices after the first vortex linking, leading to a short-living vortex pair. This challenges the conventional view that residual vortices have a long life span, which is true only in an unstratified environment. In addition to the second vortex linking, we observe the following. First, strong background turbulence quickly breaks the two vortices. Second, the two vortices induce secondary and tertiary vortices and lose energy to gravitational waves in a strongly stratified ($Fr\leqslant 0.7$) environment.

We investigate the interaction of an aqueous polymeric droplet with a tuneable continuous laser in an acoustically levitated environment. The effect of laser irradiation intensity and polymeric concentration on various spatio-temporal parameters is unearthed using high-speed shadowgraphy and theoretical scaling analysis. We observe four temporal phases: droplet evaporation, vapour bubble growth followed by membrane inflation, bubble/membrane rupture through hole nucleation and droplet breakup. During the initial droplet evaporation phase, concentration build-up at the droplet surface beyond a critical limit leads to the formation of a skin layer. It is revealed that, at a given location inside the droplet, hot spots occur, and the maximum temperature at the hot spots scales linearly with irradiation intensity until a bubble nucleates. The low-intensity laser interaction leads to symmetric membrane inflation that eventually forms holes at droplet poles and cracks on the shell surface. On the contrary, high intensity causes early bubble nucleation followed by asymmetric membrane inflation that eventually ruptures through multiple hole formation. Furthermore, the growth and rupture of the membrane is followed by a catastrophic breakup of the droplet. Two dominant atomization modes are reported at significantly high irradiation intensities: stable sheet collapse and unstable sheet breakup. The evolution of droplets into a stable/unstable sheet follows universally observed ligament and hole dynamics. A regime map is shown to describe the influence of polymer concentration and irradiation intensity on the strength and mode of droplet atomization.