The geometrical properties of streamlines, such as the curvatures, directions and positions, are studied in steady inviscid compressible flow fields via differential geometry theories and conservation laws. The influences of the streamline geometries on the flow speeds and pressures are also identified and discussed. By transforming the streamlines to fill the domain and satisfy the boundary conditions, a unified geometry-based solver, the streamline transformation method, is proposed for both subsonic and supersonic regions. The governing equations and boundary conditions along streamlines and shock waves are also derived. This method is verified by numerical results of three typical flow fields, including the subsonic channel flow, the supersonic downstream of attached shock waves and especially the subsonic/supersonic downstream of detached bow shock waves. Both two-dimensional planar and axisymmetric flow fields are considered. Compared with the results from computational fluid dynamics, good agreements are achieved by this method, while fewer computational resources, by an order of magnitude, are consumed. Features of these flow fields are also analysed from a geometrical perspective, such as flow speeds and pressures deviated by the wall curvatures, and three-dimensional effects in the after-shock flow fields. For a hyperbolic-shaped bow shock wave, the stand-off distances and the transitions from subsonic to supersonic regions are also discussed. As indicated by the accuracy, efficiency and applicability in a wide range of flow speeds, the streamline transformation method would be a potential candidate for the theoretical analysis and inverse design of high-speed flow fields, especially where the subsonic regions exist downstream of strong shock waves.

We study the asymptotic behaviour of convective heat transfer for turbulent flows in high-porosity fluid-saturated media by two-dimensional high-resolution numerical simulation. The generalized Navier–Stokes equations for incompressible fluid flow and the heat transport equation in porous media at the representative element volume scale are solved by the lattice Boltzmann method, wherein the non-Darcian effects are taken into consideration. The asymptotic behaviour of the Nusselt number $N$ has been revealed for Rayleigh numbers $10^4\leq R\leq 10^{11}$ and Darcy numbers $10^{-6}\leq \xi \leq 10^6$: all the data for various Darcy numbers gradually collapse onto a unique line with increasing Rayleigh number. The asymptote can be well represented by $N=0.146\times R^{0.286}$ for $R>2\times 10^7$, which approaches the relationship for the Rayleigh–Bénard turbulent convection of free fluid flows. The transition can be characterized by a scaling analysis with $R^{}\xi ^{3/2}\sim 1$, below which, however, the data collapse onto the Darcy limit for porous media. The Reynolds number and the Nusselt number both increase with Darcy number above the onset of convection, whereas a premature saturation of the Nusselt number is observed in comparison with that of the Reynolds number. The counter-gradient heat transport by the large-scale flows is quantified, which compensates for the increase of the gradient heat transport with Darcy number. The heat transfer in high-porosity fluid-saturated media with a very small Darcy number $\xi \geq 10^{-6}$ can be comparable to that of free fluid flows for a sufficiently high Rayleigh number $R\geq 10^{11}$.

We consider the stability of the drawing of a long and thin viscous thread with an arbitrary number of internal holes of arbitrary shape that evolves due to axial drawing, inertia and surface tension effects. Despite the complicated geometry of the boundaries, we use asymptotic techniques to determine a particularly convenient formulation of the equations of motion that is well-suited to stability calculations. We will determine an explicit asymptotic solution for steady states with (a) large surface tension and negligible inertia, and (b) large inertia. In both cases, we will show that complicated boundary layer structures can occur. We will use linear stability analysis to show that the presence of an axisymmetric hole destabilises the flow for finite capillary number and which answers a question raised in the literature. However, our formulation allows us to go much further and consider arbitrary hole structures or non-axisymmetric shapes, and show that any structure with holes will be less stable than the case of a solid axisymmetric thread. For a solid axisymmetric thread, we will also determine a closed-form expression that delineates the unconditional instability boundary in which case the thread is unstable for all draw ratios. We will determine how the detailed effects of the microstructure affect the stability, and show that they manifest themselves only via a single function that occurs in the stability problem and hence have a surprisingly limited effect on the stability.

Near-critical fluids subject to simultaneous thermal quench and an imposed external acceleration field are reported to develop ‘sink zones’, where the temperature in the bulk falls below the imposed boundary value. This anomalous cooling effect persists for a long period of time corresponding to the diffusive elimination of the cold boundary layer. The sink-zone phenomenon has been captured previously in numerical simulations (Zappoli et al., J. Fluid Mech, vol. 316, 1996, pp. 53–72; Sharma et al., Phys. Rev E, vol. 96, 2017, 063102; Sharma et al., Phys. Fluids, vol. 29, 2017, 126103) and was observed experimentally (Beysens et al., Phys. Rev. E, vol. 84, 2011, 051201). The present work provides a detailed and thorough theoretical analysis and interpretation based on matched asymptotic expansions. By examining the one-dimensional transient flow and temperature with a mean-field approach, we provide further insight into this striking phenomenon, showing that the behaviour involved is unique to highly compressible near-critical van der Waals gases. We also show that the sub-cooling phenomenon cannot be predicted in perfect gases unless very extreme conditions are applied.

The stability of the two-layer film flow driven by an oscillatory plate under long-wave disturbances is studied. The influence of key factors, such as thickness ratio ($n$), viscosity ratio ($m$), density ratio ($r$), oscillatory frequency ($\beta$) and insoluble surfactants on the stability behaviours is studied systematically. Four special Floquet patterns are identified, and the corresponding growth rates are obtained by solving the eigenvalue problem of the fourth-order matrix. A small viscosity ratio ($m\le 1$) may stabilize the flow but it depends on the thickness ratio. If the viscosity ratio is not very small ($m>0.1$), in the $(\beta ,n)$-plane, stable and unstable curved stripes appear alternately. In other words, under the circumstances, if the two-layer film flow is unstable, slightly adjusting the thickness of the upper film may make it stable. In particular, if the upper film is thin enough, even under high-frequency oscillation, the flow is always stable. The influence of density ratio is similar, i.e. there are curved stable and unstable stripes in the $(\beta ,r)$-planes. Surface surfactants generally stabilize the flow of the two-layer oscillatory membrane, while interfacial surfactants may stabilize or destabilize the flow but the effect is mild. It is also found that gravity can generally stabilize the flow because it narrows the bandwidth of unstable frequencies.

We present a local stability analysis of an idealized model of the stratified vortices that appear in geophysical settings. The base flow comprises an axisymmetric vortex with background rotation and an out-of-plane stable stratification, and a radial stratification in the thermal wind balance with the out-of-plane momentum gradient. Solving the local stability equations along fluid particle trajectories in the base flow, the dependence of short-wavelength instabilities on the Schmidt number $Sc$ (ratio between momentum and mass diffusivities) is studied, in the presence of curvature effects. In the diffusion-free limit, the well-known symmetric instability is recovered. In the viscous, double-diffusive regime, instability characteristics are shown to depend on three non-dimensional parameters (including $Sc$), and two different instabilities are identified: (i) a monotonic instability (same as symmetric instability at $Sc = 1$), and (ii) an oscillatory instability (absent at $Sc = 1$). Separating the base flow and perturbation characteristics, two each of base flow and perturbation parameters (apart from $Sc$) are identified, and the entire parameter space is explored for the aforementioned instabilities. In comparison with $Sc = 1$, monotonic and oscillatory instabilities are shown to significantly expand the instability region in the space of base flow parameters as $Sc$ moves away from unity. Neutral stability boundaries on the plane of $Sc$ and a modified gradient Richardson number are then identified for both these instabilities. In the absence of curvature effects, our results are shown to be consistent with previous studies based on normal mode analysis, thus establishing that the local stability approach is well suited to capturing symmetric and double-diffusive instabilities. The paper concludes with a discussion of curvature effects, and the likelihood of monotonic and oscillatory instabilities in typical oceanic settings.

A Lagrangian approach to both hydrostatic non-dispersive in the short-wave range and non-hydrostatic dispersive rotating shallow-water magnetohydrodynamics is developed, and used to analyse weakly and fully nonlinear waves described by the model. Hyperbolic structure in the non-dispersive case is displayed and Riemann invariants are constructed. Characteristic equations are used to establish criteria of breaking and formation of shocks by magneto-gravity waves, and conditions of the appearance of contact discontinuities in Alfvén waves. As in the case of non-magnetic rotating shallow water, rotation cannot prevent breaking. The Lagrangian equations of the model are reduced to a single partial differential ‘master’ equation, which is used to analyse the propagation of weakly nonlinear waves of both families, with or without weak rotation, and with or without weak short-wave dispersion. Corresponding modulation equations are constructed and their main properties sketched. The same master equation is used to obtain fully nonlinear finite-amplitude wave solutions in particular cases of no short-wave dispersion or no rotation.

Over the last decade, substantial progress has been made in understanding the topology of quasi-two-dimensional (2-D) non-equilibrium fluid flows driven by ATP-powered microtubules and microorganisms. By contrast, the topology of three-dimensional (3-D) active fluid flows still poses interesting open questions. Here, we study the topology of a spherically confined active flow using 3-D direct numerical simulations of generalized Navier–Stokes (GNS) equations at the scale of typical microfluidic experiments. Consistent with earlier results for unbounded periodic domains, our simulations confirm the formation of Beltrami-like bulk flows with spontaneously broken chiral symmetry in this model. Furthermore, by leveraging fast methods to compute linking numbers, we explicitly connect this chiral symmetry breaking to the entanglement statistics of vortex lines. We observe that the mean of linking number distribution converges to the global helicity, consistent with the asymptotic result by Arnold [In Vladimir I. Arnold – Collected Works (ed. A.B. Givental, B.A. Khesin, A.N. Varchenko, V.A. Vassiliev & O.Y. Viro), pp. 357–375. Springer]. Additionally, we characterize the rate of convergence of this measure with respect to the number and length of observed vortex lines, and examine higher moments of the distribution. We find that the full distribution is well described by a k-Gamma distribution, in agreement with an entropic argument. Beyond active suspensions, the tools for the topological characterization of 3-D vector fields developed here are applicable to any solenoidal field whose curl is tangent to or cancels at the boundaries of a simply connected domain.

We study concentration-driven natural convection boundary layers on horizontal surfaces, subjected to a weak, surface normal, uniform blowing velocity $V_i$ for three orders of range of the dimensionless blowing parameter $10^{-8}\le J=Re_x^3/Gr_x\le 10^{-5}$, where $Re_x$ and $Gr_x$ are the local Reynolds and Grashof numbers at the horizontal location $x$, based respectively on $V_i$ and ${\rm \Delta} C$, the concentration difference across the boundary layer. We formulate the integral boundary layer equations, with the assumption of no concentration drop within the species boundary layer, which is valid for weak blowing into the thin species boundary layers that occur at the high Schmidt number ($Sc \simeq 600$) of concentration-driven convection. The equations are then numerically solved to show that the species boundary layer thickness $\delta _d = 1.6\,x(Re_x/Gr_x)^{1/4}$, the velocity boundary layer thickness $\delta _v=\delta _d Sc^{1/5}$, the horizontal velocity $u = V_i(Gr_x/Re_x)^{1/4}f(\eta )$, where $\eta =y/\delta _v$, and the drag coefficient based on $V_i$, $C_D = 2.32/\sqrt {J}$. We find that the vertical profile of the horizontally averaged dimensionless concentration across the boundary layer becomes, surprisingly, independent of the blowing and the species diffusion effects to follow a $Gr_y^{2/3}$ scaling, where $Gr_y$ is the Grashof number based on the vertical location $y$ within the boundary layer. We then show that the above profile matches the experimentally observed mean concentration profile within the boundary layers that form on the top surface of a membrane, when a weak flow is forced gravitationally from below the horizontal membrane that has brine above it and water below it. A similar match between the theoretical scaling of the species boundary layer thickness and its experimentally observed variation is also shown to occur.

The fundamental resonance (FR) in the nonlinear phase of the boundary-layer transition to turbulence appears when a dominant planar instability mode reaches a finite amplitude and the low-amplitude oblique travelling modes with the same frequency as the dominant mode, together with the stationary streak modes, undergo the strongest amplification among all the Fourier components. This regime may be the most efficient means to trigger the natural transition in hypersonic boundary layers. In this paper, we aim to reveal the intrinsic mechanism of the FR in the weakly nonlinear framework based on the large-Reynolds-number asymptotic technique. It is found that the FR is, in principle, a triad resonance among a dominant planar fundamental mode, a streak mode and an oblique mode. In the major part of the boundary layer, the nonlinear interaction of the fundamental mode and the streak mode seeds the growth of the oblique mode, whereas the interaction of the oblique mode and the fundamental mode drives the roll components (transverse and lateral velocity) of the streak mode, which leads to a stronger amplification of the streamwise component of the streak mode due to the lift-up mechanism. This asymptotic analysis clearly shows that the dimensionless growth rates of the streak and oblique modes are the same order of magnitude as the dimensionless amplitude of the fundamental mode $(\bar {\epsilon }_{10})$, and the amplitude of the streak mode is $O(\bar {\epsilon }_{10}^{-1})$ greater than that of the oblique mode. The main-layer solution of the streamwise velocity, spanwise velocity and temperature of both the streak and the oblique modes become singular as the wall is approached, and so a viscous wall layer appears underneath. The wall layer produces an outflux velocity to the main-layer solution, inclusion of which leads to an improved asymptotic theory whose accuracy is confirmed by comparing with the calculations of the nonlinear parabolised stability equations (NPSEs) at moderate Reynolds numbers and the secondary instability analysis (SIA) at sufficiently high Reynolds numbers.

This study reveals the competitive evolutionary process of the main driving factors in the early, middle and late stages of sandstorms, as shear turbulence becomes dominant and is then suppressed by enhanced thermal stability, based on quadrant analysis of the sand-laden turbulent wind field acquired from field observations over the entire sandstorm process. Moreover, the self-organized state of multiscale structures in the energy-containing region of the sand-laden turbulence is found to change significantly as the sandstorm develops. The logarithmic scaling law that governs the cumulative turbulent kinetic energy for the non-stationary flow in the early and late stages of the sandstorm is different from the existing theoretical formula. The corresponding rate of increase in the cumulative kinetic energy with increasing scale is much higher in these stages than in the middle stage of the sandstorm with steady flow. The change in self-organized state of turbulence is responsible for the flow acceleration and the thermal superimposed effect, rather than the addition of sand particles.

The development of a bubble plume from a vertical gas-evolving electrode is driven by buoyancy and hydrodynamic bubble dispersion. This canonical fluid mechanics problem is relevant for both thermal and electrochemical processes. We adopt a mixture model formulation for the two-phase flow, considering variable density (beyond Boussinesq), viscosity and hydrodynamic bubble dispersion. Introducing a new change of coordinates, inspired by the Lees–Dorodnitsyn transformation, we obtain a new self-similar solution for the laminar boundary layer equations. The results predict a wall gas fraction and gas plume thickness that increase with height to the power of 1/5 before asymptotically reaching unity and scaling with height to the power 2/5, respectively. The vertical velocity scales with height to the power of 3/5. Our analysis shows that self-similarity is only possible if gas conservation is entirely formulated in terms of the gas specific volume instead of the gas fraction.

A model is formulated of a two-dimensional migrating, or swimming, inviscid bubble in a viscous fluid whose unsteady displacement is caused by the spreading over its surface of an initial distribution of insoluble surfactant. Assuming small capillary and Reynolds numbers, and a linear equation of state giving the surface tension as a function of surfactant concentration, the quasi-steady Stokes flow around the bubble is found analytically and explicit formulas are determined for the time-dependent bubble speed and its final overall displacement. At infinite surface Péclet number this is done using a complex version of the method of characteristics to solve a complex partial differential equation of Burgers type. For a finite non-zero surface Péclet number, the problem is shown to be linearizable by a complex variant of the classical Cole–Hopf transformation. The formulation allows general statements to be made on the bubble speed and its total net displacement in terms of the initial surfactant distribution. A weak finite-time singularity in the surface activity associated with an isolated clean point on the bubble surface is also identified and studied in detail.