As reconnection begins and the enstrophy $Z$ grows for two configurations, helical trefoil knots and anti-parallel vortices, two regimes of self-similar collapse are observed. First, during trefoil reconnection a new $\sqrt{\unicode[STIX]{x1D708}}Z$ scaling, where $\unicode[STIX]{x1D708}$ is viscosity, is identified before any $\unicode[STIX]{x1D716}=\unicode[STIX]{x1D708}Z$ dissipation scaling begins. Further rescaling shows linearly decreasing $B_{\unicode[STIX]{x1D708}}(t)=(\sqrt{\unicode[STIX]{x1D708}}Z)^{-1/2}$ at configuration-dependent crossing times $t_{x}$ . Gaps in the vortex structures identify the $t_{x}$ as when reconnection ends and collapse onto $\unicode[STIX]{x1D708}$ -independent curves can be obtained using $A_{\unicode[STIX]{x1D708}}(t)=(T_{c}(\unicode[STIX]{x1D708})-t_{x})(B_{\unicode[STIX]{x1D708}}(t)-B_{\unicode[STIX]{x1D708}}(t_{x}))$ . The critical times $T_{c}(\unicode[STIX]{x1D708})$ are identified empirically by extrapolating the linear $B_{\unicode[STIX]{x1D708}}(t)$ regimes to $B_{\unicode[STIX]{x1D708}}^{{\sim}}(T_{c})=0$ , yielding an $A_{\unicode[STIX]{x1D708}}(t)$ collapse that forms early as $\unicode[STIX]{x1D708}$ varies by 256. These solutions are regular or non-singular, as shown by decreasing cubic velocity norms $\Vert u\Vert _{L_{\ell }^{3}}^{}$ . For the anti-parallel vortices, first there is an exchange of circulation, from $\unicode[STIX]{x1D6E4}_{y}(y=0)$ to $\unicode[STIX]{x1D6E4}_{z}(z=0)$ , mediated by the viscous circulation exchange integral $\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D6E4}}(t)$ , which is followed by a modified $B_{\unicode[STIX]{x1D708}}(t)$ collapse until the reconnection ends at $t_{x}$ . Singular Leray scaling and mathematical bounds for higher-order Sobolev norms are used to help explain the origins of the new scaling and why the domain size $\ell$ has to increase to maintain the collapse of $A_{\unicode[STIX]{x1D708}}(t)$ and $\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D6E4}}$ as $\unicode[STIX]{x1D708}$ decreases.

We consider the steady supercritical flow of a fluid layer. The layer is bounded above by a free surface and below by a rigid no-slip base. The base is in two parts: the downstream part of the base is stationary, while the upstream part translates in the streamwise direction with a uniform speed; there is an abrupt transition. At high Reynolds number, a boundary layer forms in the fluid above the base downstream of the transition point. The displacement due to this boundary layer creates a perturbation to the outer flow and therefore to the free surface. We show that the Blasius boundary layer solution, which applies in an infinitely deep fluid, also applies at high Froude numbers. The Blasius solution no longer applies for flows that are just supercritical, as the outer flow is strongly affected by the presence of the boundary layer. We outline possible applications of this work to depth-averaged models of gravity currents.

The stability of a hypersonic boundary layer for a $5^{\circ }$ half-angle cone at the Caltech T5 high-enthalpy flow conditions was investigated using direct numerical simulations. For the ‘linear’ stability investigations, the boundary layer was perturbed by small axisymmetric disturbances with very small amplitudes, and for the nonlinear regime, three-dimensional pulse disturbances with larger amplitudes were introduced. The surprising result from these investigations was that the 3D wave packet undergoes strong spatial modulations, which we have not observed for other experimental conditions (e.g. the Purdue BAM6QT). This modulation was found to be directly due to the synchronization between second-mode wave components and vorticity/entropy modes. Furthermore, it was found that a synchronization with slow acoustic waves leads to a sudden and strong emission of acoustic waves deep into the free stream, which was observed for both a linear wave train and a 3D nonlinear wave packet. Therefore, it can be concluded that this is a linear mechanism that is not suppressed by nonlinear effects.

Mixing dynamics in flows are governed by the coupled action of diffusion and stretching by velocity gradients. This leads to the development of elongated lamellar structures in scalar fields where concentration fluctuations exist at scales set by the Batchelor scale. Because the latter is generally too small to be resolved experimentally, observation of these mechanisms remains an outstanding challenge. Here we present high-resolution experiments allowing for the precise quantification of the evolution of concentration distributions at the scale of a single lamella experiencing diffusion, stretching and aggregation with other lamellae. Quantitative agreement is found with analytical predictions for the lamella’s concentration profile, Batchelor time, Batchelor length scale, and concentration distribution for a large range of Péclet numbers and without adjustable parameter. This benchmark configuration is used to set the experimental spatial resolution required to quantify the concentration probability density functions (PDFs) of scalar mixtures in fluids. The diffusive coalescence of two nearby lamellae, the mechanism by which scalar mixtures ultimately reach uniformity, is shown to induce a complex transient evolution of the concentration PDFs.

The coupled nonlinear interaction of three-dimensional gravity–inertia internal wavepackets, in the form of beams with nearly monochromatic profile, with their induced mean flow is discussed. Unlike general three-dimensional wavepackets, such modulated nearly monochromatic beams are not susceptible to modulation instability from their inviscid, purely modulation-induced mean flow. However, streaming – the induced mean flow associated with the production of mean potential vorticity via the combined action of dissipation and nonlinearity – can cause cross-beam bending, transverse broadening and increased along-beam decay of the beam profile, in qualitative agreement with earlier laboratory experiments. For wavepackets with general three-dimensional modulations, by contrast, streaming does arise, but plays a less prominent role in the interaction dynamics.

We prove the first rigorous bound on the heat transfer for three-dimensional Rayleigh–Bénard convection of finite-Prandtl-number fluids between free-slip boundaries with an imposed heat flux. Using the auxiliary functional method with a quadratic functional, which is equivalent to the background method, we prove that the Nusselt number $\mathit{Nu}$ is bounded by $\mathit{Nu}\leqslant 0.5999\mathit{R}^{1/3}$ uniformly in the Prandtl number, where $\mathit{R}$ is the Rayleigh number based on the imposed heat flux. In terms of the Rayleigh number based on the mean vertical temperature drop, $\mathit{Ra}$, we obtain $\mathit{Nu}\leqslant 0.4646\mathit{Ra}^{1/2}$. The scaling with Rayleigh number is the same as that of bounds obtained with no-slip isothermal, free-slip isothermal and no-slip fixed-flux boundaries, and numerical optimisation of the bound suggests that it cannot be improved within our bounding framework. Contrary to the two-dimensional case, therefore, the $\mathit{Ra}$-dependence of rigorous upper bounds on the heat transfer obtained with the background method for three-dimensional Rayleigh–Bénard convection is insensitive to both the thermal and the velocity boundary conditions.

The effect of domain anisotropy on the inverse cascade occurring within the geostrophic turbulence regime of rapidly rotating Rayleigh–Bénard convection is investigated. In periodic domains with square cross-section in the horizontal, a domain-filling dipole state is present. For rectangular periodic domains, a Kolmogorov-like flow parallel to the short side and consisting of a periodic array of alternating unidirectional jets with embedded vortices is observed, together with an underlying weak meandering transverse jet. Similar transitions occurring in weakly dissipative two-dimensional flows driven by externally imposed small-amplitude noise and in classical hydrostatic geostrophic turbulence are a consequence of inviscid conservation of energy and potential enstrophy, and can be understood using statistical mechanics considerations. Rotating Rayleigh–Bénard convection represents an important three-dimensional system with only one inviscid invariant which nonetheless exhibits large-scale structures driven by intrinsically generated fluctuations.

Turbulent axisymmetric wakes under pressure gradient have received little attention in the literature, in spite of their fundamental and practical importance, for example, in the case of wind turbine wakes over topography. In this paper, we develop an analytical framework to analyse turbulent axisymmetric wakes under different pressure gradient conditions. Specifically, we develop a model to predict how an arbitrary imposed pressure gradient perturbs the evolution of the zero-pressure-gradient wake. The starting point of the model is the basic mean conservation of the streamwise momentum equation. We take advantage of the self-similarity of the wake velocity deficit and the assumption that the ratio of the maximum velocity deficit to the wake width is independent of the pressure gradient; such an assumption is supported experimentally for planar wakes, and numerically for axisymmetric wakes in this study. Furthermore, an asymptotic solution for the problem is also derived. The problem is considered for both an axisymmetric strain and a planar strain. The inputs to the model are the imposed pressure gradient and the wake width in the zero-pressure-gradient case. To validate the model results, a set of large-eddy simulations (LES) are performed. Comparing the evolution of the maximum velocity deficit and the wake width, the model results and the LES data show good agreement. Similarly to planar wakes, it is observed that the axisymmetric wake recovers faster in the favourable pressure gradient compared with the adverse one.

A tiled approach to rough surface simulation is used to explore the full range of roughness Reynolds numbers, from the limiting case of hydrodynamic smoothness up to fully rough conditions. The surface is based on a scan of a standard grit-blasted comparator, subsequently low-pass filtered and made spatially periodic. High roughness Reynolds numbers are obtained by increasing the friction Reynolds number of the direct numerical simulations, whereas low roughness Reynolds numbers are obtained by scaling the surface down and tiling to maintain a constant domain size. In both cases, computational requirements on box size, resolution in wall units and resolution per minimum wavelength of the rough surface are maintained. The resulting roughness function behaviour replicates to good accuracy the experiments of Nikuradse (1933 VDI-Forschungsheft, vol. 361), suggesting that the processed grit-blasted surface can serve as a surrogate for his sand-grain roughness, the precise structure of which is undocumented. The present simulations also document a monotonic departure from hydrodynamic smooth-wall results, which is fitted with a geometric relation, the exponent of which is found to be inconsistent with both the Colebrook formula and an earlier theoretical argument based on low-Reynolds-number drag relations.

We present an investigation into the influence of upstream shear on the viscous flow around a steady two-dimensional (2-D) symmetric airfoil at zero angle of attack, and the corresponding loads. In this computational study, we consider the NACA 0012 airfoil at a chord Reynolds number $1.2\times 10^{4}$ in an approach flow with uniform positive shear with non-dimensional shear rate varying in the range 0.0–1.0. Results show that the lift force is negative, in the opposite direction to the prediction from Tsien’s inviscid theory for lift generation in the presence of positive shear. A hypothesis is presented to explain the observed sign of the lift force on the basis of the asymmetry in boundary layer development on the upper and lower surfaces of the airfoil, which creates an effective airfoil shape with negative camber. The resulting scaling of the viscous effect with shear rate and Reynolds number is provided. The location of the leading edge stagnation point moves increasingly farther back along the airfoil’s upper surface with increased shear rate, a behaviour consistent with a negatively cambered airfoil. Furthermore, the symmetry in the location of the boundary layer separation point on the airfoil’s upper and lower surfaces in uniform flow is broken under the imposed shear, and the wake vortical structures exhibit more asymmetry with increasing shear rate.

Rough surfaces have been widely used as an efficient way to enhance the heat-transfer efficiency in turbulent thermal convection. In this paper, however, we show that roughness does not always mean a heat-transfer enhancement, but in some cases it can also reduce the overall heat transport through the system. To reveal this, we carry out numerical investigations of turbulent Rayleigh–Bénard convection over rough conducting plates. Our study includes two-dimensional (2D) simulations over the Rayleigh number range $10^{7}\leqslant Ra\leqslant 10^{11}$ and three-dimensional (3D) simulations at $Ra=10^{8}$ . The Prandtl number is fixed to $Pr=0.7$ for both the 2D and the 3D cases. At a fixed Rayleigh number $Ra$ , reduction of the Nusselt number $Nu$ is observed for small roughness height $h$ , whereas heat-transport enhancement occurs for large $h$ . The crossover between the two regimes yields a critical roughness height $h_{c}$ , which is found to decrease with increasing $Ra$ as $h_{c}\sim Ra^{-0.6}$ . Through dimensional analysis, we provide a physical explanation for this dependence. The physical reason for the $Nu$ reduction is that the hot/cold fluid is trapped and accumulated inside the cavity regions between the rough elements, leading to a much thicker thermal boundary layer and thus impeding the overall heat flux through the system.

This paper deals with the motion of a single helical vortex in an unbounded inviscid incompressible fluid. The vortex is an infinite tube whose centreline is a helix and whose cross-section is a small circle where the vorticity is uniform and parallel to the centreline. Ever since Joukowsky (Trudy Otd. Fiz. Nauk Mosk. Obshch. Lyub. Estest., vol. 16, 1912, pp. 1–31) deduced that this vortex translates and rotates steadily without change of form, numerous attempts have been made to compute the velocities. Here, Hardin’s (Phys. Fluids, vol. 25, 1982, pp. 1949–1952) solution for the velocity field is used to find new expressions for the linear and angular velocities of the vortex. The theoretical results are verified by numerically computing the velocity at a single point using the Helmholtz integral and the Rosenhead–Moore approximation to the Biot–Savart law, and by numerically simulating the vortex evolution, under the Euler equations, in a triple-periodic cube. The new formulae are also shown to be more accurate than previous results over the whole range of values of the vortex pitch and cross-section.

Fluid–structure interactions are ubiquitous in nature and technology. However, the systems are often so complex that numerical simulations or ad hoc assumptions must be used to gain insight into the details of the complex interactions between the fluid and solid mechanics. In this paper, we present experiments and theory on viscous flow in a simple bioinspired soft valve which illustrate essential features of interactions between hydrodynamic and elastic forces at low Reynolds numbers. The set-up comprises a sphere connected to a spring located inside a tapering cylindrical channel. The spring is aligned with the central axis of the channel and a pressure drop is applied across the sphere, thus forcing the liquid through the narrow gap between the sphere and the channel walls. The sphere’s equilibrium position is determined by a balance between spring and hydrodynamic forces. Since the gap thickness changes with the sphere’s position, the system has a pressure-dependent hydraulic resistance. This leads to a nonlinear relation between applied pressure and flow rate: flow initially increases with pressure, but decreases when the pressure exceeds a certain critical value as the gap closes. To rationalize these observations, we propose a mathematical model that reduced the complexity of the flow to a two-dimensional lubrication approximation. A closed-form expression for the pressure drop/flow rate is obtained which reveals that the flow rate $Q$ depends on the pressure drop $\unicode[STIX]{x0394}p$ , sphere radius $a$ , gap thickness $h_{0}$ , and viscosity $\unicode[STIX]{x1D702}$ as $Q\sim \unicode[STIX]{x1D702}^{-1}a^{1/2}h_{0}^{5/2}(1-\unicode[STIX]{x0394}p/\unicode[STIX]{x0394}p_{c})^{5/2}\unicode[STIX]{x0394}p$ , where the critical pressure $\unicode[STIX]{x0394}p_{c}$ scales with the spring constant $k$ as $\unicode[STIX]{x0394}p_{c}\sim kh_{0}a^{-2}$ . These predictions compared favourably to the results of our experiments with no free parameters.

In this paper, we investigate the ratchet mechanism of drops climbing a vibrated oblique plate based on three-dimensional direct numerical simulations, which for the first time reproduce the existing experiment (Brunet et al., Phys. Rev. Lett., vol. 99, 2007, 144501). With the help of numerical simulations, we identify an interesting and important wetting behaviour of the climbing drop; that is, the breaking of symmetry due to the inclination of the plate with respect to the acceleration leads to a hysteresis of the wetted area in one period of harmonic vibration. In particular, the average wetted area in the downhill stage is larger than that in the uphill stage, which is found to be responsible for the uphill net motion of the drop. A new hydrodynamic model is proposed to interpret the ratchet mechanism, taking account of the effects of the acceleration and contact angle hysteresis. The predictions of the theoretical analysis are in good agreement with the numerical results.

This article describes an actuator curve embedding (ACE) concept to model arbitrary lifting lines using body forces within large-eddy simulation (LES). The new method removes some inconsistencies in body-force projection of the actuator line model (ALM) commonly used to represent wind turbine blades in atmospheric boundary-layer simulations. The concept and algorithm of ACE are presented followed by selected results for various blade planform and tip shapes that signify both the predictive capability and the advantages of the ACE concept. Examples include an elliptic wing, the NREL Phase VI rotor in parked and rotating conditions, and the NREL 5-MW turbine.

Vortex reconnection under Biot–Savart evolution is investigated geometrically and numerically using a tent model consisting of vortex filaments initially in the form of two tilted hyperbolic branches; the vortices are antiparallel at their points of nearest approach. It is shown that the tips of these vortices approach each other, accelerating as they do so to form a finite-time singularity at the apex of the tent. The minimum separation of the vortices and the maximum velocity and axial strain rate exhibit nearly self-similar Leray scaling, but the exponents of the velocity and strain rate deviate slightly from their respective self-similar values of $-1/2$ and $-1$ ; this deviation is associated with the appearance of distinct minima of curvature leading to cusp structures at the tips. The writhe and twist of each vortex are both zero at all times up to the instant of reconnection. By way of validation of the model, the structure of the eigenvalues and eigenvectors of the rate-of-strain tensor is investigated: it is shown that the second eigenvalue $\unicode[STIX]{x1D706}_{2}$ has dipole structure around the vortex filaments. At the tips, it is observed that $\unicode[STIX]{x1D706}_{2}$ is positive and the corresponding eigenvector is tangent to the filament, implying persistent stretching of the vortex.

Fundamental solutions (Green’s functions) are derived for the regularised 13-moment system (R13) of rarefied gas dynamics, for small departures from equilibrium; these solutions show the presence of Knudsen layers, associated with exponential decay terms, that do not feature in the solution of lower-order systems (e.g. the Navier–Stokes–Fourier equations). Incorporation of these new fundamental solutions into a numerical framework based on the method of fundamental solutions (MFS) allows for efficient computation of three-dimensional gas microflows at remarkably low computational cost. The R13-MFS approach accurately recovers analytic solutions for low-speed flow around a stationary sphere and heat transfer from a hot sphere (for which a new analytic solution has been derived), capturing non-equilibrium flow phenomena missing from lower-order solutions. To demonstrate the potential of the new approach, the influence of kinetic effects on the hydrodynamic interaction between approaching solid microparticles is calculated. Finally, a programme of future work based on the initial steps taken in this article is outlined.

We present the results of a combined experimental and theoretical study of drop coalescence in the presence of an initial temperature difference $\unicode[STIX]{x0394}T_{0}$ between a drop and a bath of the same liquid. We characterize experimentally the dependence of the residence time before coalescence on $\unicode[STIX]{x0394}T_{0}$ for silicone oils with different viscosities. Delayed coalescence arises above a critical temperature difference $\unicode[STIX]{x0394}T_{c}$ that depends on the fluid viscosity: for $\unicode[STIX]{x0394}T_{0}>\unicode[STIX]{x0394}T_{c}$ , the delay time increases as $\unicode[STIX]{x0394}T_{0}^{2/3}$ for all liquids examined. This observed dependence is rationalized theoretically through consideration of the thermocapillary flows generated within the drop, the bath and the intervening air layer.

We explore the dynamics of starting plumes by analysis of a series of new small-scale laboratory experiments combined with a theoretical model for mass, momentum, and buoyancy conservation. We find that the head of the plume ascends with a speed which is approximately 0.6 times the characteristic speed of the fluid in the following steady plume, in accord with Turner (J. Fluid Mech., vol. 13 (03), 1962, pp. 356–368), and so the fluid released from the source eventually catches the head of the flow. On reaching the top of the plume it recirculates and mixes in the plume head. We estimate that approximately $0.61\pm 0.04$ of the total buoyancy released from the source accumulates in the plume head, with the remainder in the following steady plume. Using measurements of the volume of the head, we estimate that a fraction $0.16\pm 0.08$ of the volume of the head is entrained directly from the ambient, with the remainder of the fluid in the head being supplied by the following steady plume. These results imply that the buoyancy force exerted on the plume head plus the momentum flux supplied by the following plume exceeds the rate of change of momentum of the plume head even including the added mass of the plume head. We propose that the difference is associated with a drag force resulting from the displacement of ambient fluid around the plume head. Using our experimental data, we estimate that the drag coefficient $C_{d}$ has a value $4.2\pm 1.4$ , with the range in values associated with the uncertainty in our estimate of entrainment of fluid directly into the plume head. As a test, the proposed model is shown to provide a reasonable description of a starting plume rising through a stratified environment in the region below the maximum height of rise of the associated steady plume, although, above this point, the shape of the plume head changes and the model breaks down.

The flow in a decelerating turbulent round jet is investigated using direct numerical simulation. The simulations are initialised with a flow field from a statistically stationary turbulent jet. Upon stopping the inflow, a deceleration wave passes through the jet, behind which the velocity field evolves towards a new statistically unsteady self-similar state. Assumption of unsteady self-similar behaviour leads to analytical relations concerning the evolution of the centreline mean axial velocity and the shapes of the radial profiles of the velocity statistics. Consistency between these predictions and the simulation data supports the use of the assumption of self-similarity. The mean radial velocity is predicted to reverse in direction near to the jet centreline as the deceleration wave passes, contributing to an approximately threefold increase in the normalised mass entrainment rate. The shape of the mean axial velocity profile undergoes a relatively small change across the deceleration transient, and this observation provides direct evidence in support of previous models that have assumed that the mean axial velocity profile, and in some cases also the jet spreading angle, remain approximately constant within unsteady jets.