Solving the three-dimensional boundary layer equations carries theoretical significance and practical applications, which also poses substantial challenges due to its inherent complexity. In this paper, the laminar boundary layer equations for the symmetry plane of three-dimensional bodies are derived in an orthogonal curvilinear coordinate system associated with the principal curvatures. The derivation of the boundary layer equations is based not only on the common symmetric properties of the flow, as given by Hirschel et al. (Three-Dimensional Attached Viscous Flow, 2014, Academic Press, pp. 183–187), but also incorporates the geometric symmetry properties of the body. The derived equations are more representative and simplified. Notably, these equations can degenerate to a form consistent with or equivalent to the commonly used boundary layer equations for special bodies such as flat plates, cones and spheres. Furthermore, for hypersonic flows, the crossflow velocity gradient at the boundary layer edge on the symmetry plane is derived based on Newtonian theory. Subsequently, this parameter can provide the necessary boundary condition needed for solving the boundary layer equations using existing methods. Finally, as examples, the equations developed in this paper are solved using the difference-differential method for several typical three-dimensional blunt shapes that appeared on hypersonic vehicles. They prove to be useful in the analysis and interpretation of boundary layer flow characteristics in the symmetry plane of blunt bodies.

The injection of ${\rm CO}_2$ into depleted reservoirs carries the potential for significant Joule–Thomson cooling, when dense, supercritical ${\rm CO}_2$ is injected into a strongly under-pressured reservoir. The resulting low temperatures around the wellbore risk causing thermal fracturing of the well/near-well region or causing freezing of pore waters or formation of gas hydrates which would reduce injectivity and jeopardise well and reservoir integrity. These risks are particularly acute during injection start-up when ${\rm CO}_2$ is in the gas stability field. In this paper we present a model of non-isothermal single-phase flow in the near-wellbore region. We show that during radial injection, with fixed mass injection rate, transient Joule–Thomson cooling can be described by similarity solutions at early times. The positions of the ${\rm CO}_2$ and thermal fronts are described by self-similar scaling relations. We show that, in contrast to steady-state flow, transient flow causes slight heating of ${\rm CO}_2$ and reservoir gas either side of the thermal front, as pressure diffuses into the reservoir. The scaling analysis here identifies the parametric dependence of Joule–Thomson cooling. We present a sensitivity analysis which demonstrates that the primary controls on the degree of cooling are reservoir permeability, reservoir thickness, injection rate and Joule–Thomson coefficient. The analysis presented provides a computationally efficient approach for assessing the degree of Joule–Thomson cooling expected during injection start-up, providing a complement to complex, fully resolved numerical simulations.

The electroosmotic flow (EOF) fields in the vicinity of solids with high dielectric permittivity are studied for the case of charge-asymmetric electrolyte solutions. Corresponding solutions of the coupled Poisson–Nernst–Planck and Navier–Stokes equations are obtained analytically and numerically. When a direct-current (DC) electric field is applied to a high-permittivity uncharged sphere, a net EOF develops that translates into a non-zero electrophoretic mobility of the sphere, although it does not carry any charge. Similarly, a DC field acting on a channel in a high-permittivity material results in a net flow through the channel, although the solid is uncharged. Such phenomena are expected to occur frequently whenever high-permittivity solids are immersed in charge-asymmetric electrolyte solutions and do not rely on special scenarios such as ion crowding. Also, the net flow velocities are very significant for realistic values of the electric field strength. The derived scaling relationships even predict giant net flow velocities through nanochannels of the order of metres per second for practically relevant scenarios.

This study explores the implementation of an online control strategy based on dynamic mode decomposition in the context of flow control. The investigation is conducted mainly with a fixed Reynolds number of $Re = 100$, focusing on the flow past a circular cylinder constrained between two walls to mitigate vortex shedding. The control approach involves the activation of two synthetic jets on the cylinder through blowing and suction. Velocity fluctuations in the wake, specifically in the $x$-direction, are harnessed to ascertain the mass flow rate of the jets using the linear quadratic regulator and online dynamic mode decomposition. The study systematically assesses the control performance across various configurations, including different values of the input penalty factor ${\boldsymbol {R}}$, varying numbers of probes and distinct probe arrangement methods. The synthetic jets prove effective in stabilising the separation bubble, and their interaction with the unsteady wake leads to a notable reduction in drag force, its fluctuations and the amplitude of the lift force. Specifically, the mean and standard deviation of the drag coefficient witness reductions of $7.44\,\%$ and $96.67\,\%$, respectively, and the standard deviation of the lift coefficient experiences an impressive reduction of $85.18\,\%$. The robustness of the proposed control method has also been tested on two more complicated cases, involving unsteady incoming flows with multiple frequency components. Comparatively, the methodology employed in this paper yields results akin to those obtained through deep reinforcement learning in terms of control effectiveness. However, a noteworthy advantage lies in the substantial reduction of computational resource consumption, highlighting the efficiency of the proposed approach.

Many natural and industrial processes involve the flow of fluids made of solid particles suspended in non-Newtonian liquid matrices, which are challenging to control due to the fluid's nonlinear rheology. In the present work, a Taylor–Couette canonical system is used to investigate the flow of dilute to semi-dilute suspensions of neutrally buoyant spherical particles in highly elastic base polymer solutions. Friction measurement synchronized with direct flow visualization are combined to characterize the critical conditions for the onset of elasto-inertial instabilities, expected here as a direct transition to elasto-inertial turbulence (EIT). Adding a low particle volume fraction (${\leq }2\,\%$, dilute regime) does not affect the nature of the primary transition and reduces the critical Weissenberg number for the onset of EIT, despite a significant decrease in the apparent fluid elasticity. However, for particle volume fractions ${\geq }6\,\%$ (semi-dilute regime), EIT is no longer observed in the explored Reynolds range, suggesting an apparent relaminarization with yet not further decrease in fluid elasticity. Instead, a new regime, termed here elasto-inertial dissipative (EID), was uncovered. It originates from particle–particle interactions altering particle–polymer interactions and occurring under elasto-inertial conditions comparable to those of EIT. Increasing particle volume fraction in the semi-dilute regime and, in so, the particle contribution to the overall viscosity, delays the onset of EID similarly to what was observed previously for EIT in lower elasticity fluids. After this onset, a decrease in the pseudo-Nusselt number observed with increasing inertia and particle-to-polymer concentration ratio confirms a particle-induced alteration of energy transfer in the flow.

When a partially miscible binary mixture is quenched below its critical temperature, it transitions from its single-phase to a two-phase region, undergoing phase separation. The processes of formation and coalescence of droplets are driven by diffusive and convective phenomena, taking place isotropically in the system. The application of an external force field, which exerts a different contribution on the two species, breaks the symmetry of phase separation, leading to the segregation of two equilibrated phases separated by a single interface. This study investigates the dynamics of phase segregation under an external force. The effects of various force magnitudes, captured by the Bond number, in both high- and low-viscosity mixtures, distinguished by different fluidity numbers, are quantified via numerical simulations by using the phase field model. The intricate dynamics of formation, floating and coalescence of droplets towards complete segregation are described along with the quantification of the segregation time, revealing different patterns for high and low Bond numbers. Results show that in none of the cases, formation and floating can be regarded as strictly serial processes. A universal scaling between segregation time, Bond number, fluidity number and domain size is not possible, with a power-law dependence emerging only under the diffusion-dominated regime.

Nonlinear steady solutions of the barotropic quasi-geostrophic equation in basins, gulfs and channels on a $\beta$-plane are presented. The domains are rectangular with arbitrary aspect ratios. The two-dimensional solutions assume a linear relationship between the potential vorticity $q$ and the stream function $\psi$. The sign of the slope in the linear $q\unicode{x2013}\psi$ relationship defines two broad sets of solutions. For a positive slope, the solutions in a closed basin correspond to the inertial gyres derived by Fofonoff in 1954. The negative slope solutions consist of normal modes that can be resonant. For gulfs and channels, the conditions at the open boundaries are almost arbitrary flows entering or leaving the domain. Such conditions allow a great variety of solutions in the interior, characterised mainly by arrays of vortices with alternate signs. Several examples are presented and discussed.

We present a Mach 15 air flow over a blunt two-dimensional wedge simulated using the direct molecular simulation method. As electronically excited states are not modelled, the resulting air mixture around the wedge contains the electronic ground states only, namely ${\rm N}_2(\text {X}^1 \varSigma _g^{+})$, ${\rm O}_2(\text {X}^3 \varSigma _g^{-})$, ${\rm NO}(\text {X}^2\varPi _r)$, ${\rm N}(^4{\rm S})$ and ${\rm O}(^3{\rm P})$. All the potential energy surfaces (PESs) that are used to model the various interactions between air particles are ab initio, with two notable exceptions, namely ${\rm N}_2+{\rm NO}$ and ${\rm O}_2+{\rm NO}$. At the selected free-stream conditions, strong vibrational non-equilibrium is observed in the shock layer. The flow is characterized by significant chemical activity, with near-complete oxygen dissociation, considerable formation of NO and minimal molecular nitrogen dissociation. Complex mass diffusion kinetics, driven by composition, temperature and pressure gradients, are identified in the shock layer. All these physical phenomena are directly coupled to, and responsible for, the mechanics of the gas flow and are all solely traceable to the PESs’ inputs, without the need for any thermochemical models, mixing rules or constitutive laws for transport properties. Because the flow is entirely at near-continuum conditions, it is a gas-phase thermophysics benchmark that is useful to enhance the fidelity of continuum models used in computational fluid dynamics of hypersonic flows.

We use multiphase direct numerical simulations to identify, analyse and quantify components of wall-normal heat flux distributions in evaporative vertical falling films with surface modifications at industrially relevant conditions. Previous experiments showed a potential increase of the heat transfer rate through the film by up to 100 % using various types of modifications. We show that the modifications induce significant advective heat transport and hypothesise that four synergistic mixing mechanisms are behind the heat transfer rate improvement. Additionally, we examine how the important surface topology parameters, pitch $\hat {p}$ (distance between modifications), height $\hat {h}$ and the liquid Prandtl number $\mathit {Pr}_l$, influence the mode of heat transport and the Nusselt number $\mathit {Nu}$. We show that $\hat {p}/\hat {h} \approx 10$ maximises $\mathit {Nu}$ and that the optimal pitch is related to the recirculation zone length $L_r$ behind the modification. We find that $L_r/\hat {h} \approx 3.5$ and that $\mathit {Nu} \propto \mathit {Pr}_l^{0.42}$ in the investigated parameter ranges. We also show that all our cases on both smooth and modified surfaces have $\mathit {Pe}_l \gg 1$ and collapse well on a line $\mathit {Nu} \propto (\mathit {Pe}_l/\mathit {Re})^{0.35}$. This relation suggests that $\mathit {Nu}$ is governed by the balance of film mixing, thermal resistance and diffusivity, and that the ratio $\mathit {Pe}_l/\mathit {Re}$ can be used to estimate $\mathit {Nu}$. Our methodology and findings extend the knowledge concerning the mechanisms behind the heat transfer improvement due to surface modifications and facilitate guidelines for designing more efficient modified surfaces in industrial evaporators.

Free transverse oscillations of an elastically mounted circular cylinder with low mass-damping are studied with a focus on the effects of the cylinder end condition on structural oscillations and vortex shedding. While the top end of the cylinder pierces the free surface of a water channel, the lower end is changed to have three end conditions: an attached endplate, an endplate unattached from the cylinder at varying gaps and no endplate. All three response branches are examined, with the reduced velocity sweeping $2 \le {U^\ast } \le 15$ (corresponding to a Reynolds number range of $2000 \le Re \le 14\;700$). Although the cylinder oscillations are unaffected by the end condition in the initial and upper branches, they show significant dependency on the end condition in the lower branch. When the endplate is attached or unattached with a small gap, the upper-to-lower branch transition occurs with a sudden decrease in oscillation amplitude, after which the lower branch maintains a near-constant amplitude. For larger gaps or no endplate, with increasing reduced velocity, the oscillation amplitude decreases gradually from its peak without any discernible sign of transition between the upper and lower branches. The three-dimensional effects of the gap are the basis for these differences in oscillation. Atop the strong tip vortex, a low-magnitude streamwise velocity region develops downstream of the cylinder, which delays upper-to-lower branch transition over the cylinder span that sees this low-velocity region. With increasing gap, this low-velocity region and the delay in transition spread over larger spanwise extents, forcing the overall body to oscillate at larger amplitudes.

Scale dependence of local shearing motion is investigated experimentally in decaying homogeneous isotropic turbulence generated through multiple-jet interaction. The turbulent Reynolds number, based on the Taylor microscale, is between approximately 900 and 400. Velocity fields, measured using particle image velocimetry, are analysed through the triple decomposition of a low-pass filtered velocity gradient tensor, which quantifies the intensities of shear and rigid-body rotation at a given scale. These motions manifest predominantly as layer and tubular vortical structures, respectively. The scale dependence of the moments of velocity increments, associated with shear and rigid-body rotation, exhibits power-law behaviours. The scaling exponents for shear are in quantitative alignment with the anomalous scaling of the velocity structure functions, suggesting that velocity increments are influenced predominantly by shearing motion. In contrast, the exponents for rigid-body rotation are markedly smaller than those predicted by Kolmogorov scaling, reflecting the high intermittency of rigid-body rotation. The mean flow structure associated with shear at intermediate scales is investigated with conditional averages around locally intense shear regions in the filtered velocity field. The averaged flow field exhibits a shear layer structure with aspect ratio approximately 4.5, surrounded by rotating motion. The analysis at different scales reveals the existence of self-similar structures of shearing motion across various scales. The mean velocity jump across the shear layer increases with the layer thickness. This relationship is well predicted by Kolmogorov's second similarity hypothesis, which is useful in predicting the mean characteristics of shear layers across a wide range of scales.

The present study investigates streamwise ($\overline {u^2}$) energy-transfer mechanisms in the inner and outer regions of turbulent boundary layers (TBLs). Particular focus is placed on the $\overline {u^2}$ production, its inter-component and wall-normal transport as well as dissipation, all of which become statistically significant in the outer region with increasing friction Reynolds number ($Re_{\tau }$). These properties are analysed using published data sets of zero, weak and moderately strong adverse-pressure-gradient (APG) TBLs across a decade of $Re_{\tau }$, revealing similarity in energy-transfer pathways for all these TBLs. It is found that both the inner and outer peaks of $\overline {u^2}$ are always associated with local maxima in the $\overline {u^2}$ production and its inter-component transport, and the regions below/above each of these peaks are always dominated by wall-ward/away-from-wall transport of $\overline {u^2}$, thereby classifying the $\overline {u^2}$ profiles into four distinct regimes. This classification reveals existence of phenomenologically similar energy-transfer mechanisms in the ‘inner’ and ‘outer’ regions of moderately strong APG TBLs, which meet at an intermediate location coinciding with the minimum in $\overline {u^2}$ profiles. Conditional averaging suggests existence of similar phenomena even in low $Re_{\tau }$ canonical and/or weak APG TBLs, albeit with the outer-region mechanisms weaker than those in the inner region. This explains the absence of their $\overline {u^2}$ outer peak and the dominance of $\overline {u^2}$ wall-normal transport away from the wall, which potentially originates from the inner region. Given that the wall-ward/away-from-wall transport of $\overline {u^2}$ is governed by the $Q_4$(sweeps)/$Q_2$(ejections) quadrants of the Reynolds shear stress, it is argued that the emergence of the $\overline {u^2}$ outer peak corresponds with the statistical dominance of $Q_4$ events in the outer region. Besides unravelling the dynamical significance of $Q_2$ and $Q_4$ events in the outer region of TBLs, the present analysis also proposes new phenomenological arguments for testing on canonical wall-turbulence data at very high $Re_{\tau }$.

We conducted an axisymmetric numerical study of drop impact on a thin film of the same liquid in order to generate maps identifying the fluid elements in the drop and film that are transferred to the corolla during impact. We find that mass contribution from the drop comes from a surprisingly thin surface layer on the drop, and furthermore, that the shape of this layer in the drop and the film scales with film thickness, not the Weber number and Reynolds number as one might expect. The maps could be used to tailor drop composition for applications such as coatings or encapsulations.

Rotating Rayleigh–Bénard convection denotes the convection between a warm plate and a cold plate in a rotating environment. It is a classic model for understanding convective vortices in the atmosphere and ocean. The influence of background rotation on fluid inertia breaks the symmetry between cyclones and anticyclones. Such a symmetry breaking could be represented by vorticity skewness, which still lacks a systematic theory. Rapidly rotating convection with stress-free boundaries and unit Prandtl number is a convenient starting point. The investigation starts from the convective onset stage, where the vortices grow stationarily. Asymptotic analysis shows that the volumetric vorticity skewness $S$ is produced by the interaction between the $n=0,1$ and $n=1,2$ vertical eigenmodes. The $n=0$ (barotropic) mode contributes positively to $S$ mainly by stretching the vertical relative vorticity, an ageostrophic effect. The $n=2$ mode makes a minor negative contribution to $S$ by preferentially intensifying the outflow over the inflow, a non-hydrostatic effect. The theory predicts $S$ to be proportional to the global Rossby number defined with the volumetric standard deviation of vorticity, ${Ro_g}$. The proportional factor does not depend on the Rayleigh and Ekman numbers, agreeing with direct numerical simulations. Then the system enters the equilibrium stage. The stretching of vertical vorticity still contributes to $S$ dominantly. At ${Ro_g}\gtrsim 0.5$, the emergent unsteady flow significantly suppresses the asymmetry between the inflow and outflow strength, and weakens its influence on $S$.

Large-scale coherent structures in incompressible turbulent pipe flow are studied for a wide range of Reynolds numbers ($Re_\tau =180, 550, 1000, 2000$ and $5200$). Employing the Karhunen–Loève decomposition and a novel approach based on the Voronoi diagram, we identify and classify statistically coherent structures based on their location, dimensions and $Re_{\tau }$. With increasing $Re_{\tau }$, two distinct classes of structures become more energetic, namely wall-attached and detached eddies. The Voronoi methodology is shown to delineate these two classes without the need for specific criteria or thresholds. At the highest $Re_{\tau }$, the attached eddies scale linearly with the wall-normal distance with a slope of approximately $l_y\sim 1.2y/R$, while the detached eddies remain constant at the size of $l_y \approx 0.26R$, with a progressive shift towards the pipe centre. We extract these two classes of structures and describe their spatial characteristics, including radial size, helix angle and azimuthal self-similarity. The spatial distribution could help explain the differences in mean velocity between pipe and channel flows, as well as in modelling large and very-large-scale motions (LSM and VLSM). In addition, a comprehensive description is provided for both wall-attached and detached structures in terms of LSM and VLSM.