Second-mode wave growth within the hypersonic boundary layer of a slender cone is investigated experimentally using high-speed schlieren visualizations. Experiments were performed in AEDC Tunnel 9 over a range of unit Reynolds number conditions at a Mach number of approximately 14. A thin lens with a known density profile placed within the field of view enables calibration of the schlieren set-up, and the relatively high camera frame rates employed allow for the reconstruction of time-resolved pixel intensities at discrete streamwise locations. The calibration in conjunction with the reconstructed signals enables integrated spatial amplification rates ($N$ factors) to be calculated for each unit Reynolds number condition and compared to $N$ factors computed from both pressure transducer measurements and linear parabolized stability equation (PSE) solutions. Good agreement is observed between $N$ factors computed from the schlieren measurements and those computed from the PSE solutions for the most-amplified second-mode frequencies. The streamwise development of $N$ factors calculated from the schlieren measurements compares favourably to that calculated from the pressure measurements with slight variations in the $N$ factor magnitudes calculated for harmonic frequencies. Finally, a bispectral analysis is carried out to identify nonlinear phase-coupled quadratic interactions present within the boundary layer. Multiple interactions are identified and revealed to be associated with the growth of disturbances at higher harmonic frequencies.

The neutral curves of the boundary layer Görtler-vortex flow generated by free-stream disturbances, i.e., curves that distinguish the perturbation flow conditions of growth and decay, are computed through a receptivity study for different Görtler numbers, wavelengths, and low frequencies of the free-stream disturbance. The perturbations are defined as Klebanoff modes or strong and weak Görtler vortices, depending on their growth rate. The critical Görtler number below which the inviscid instability due to the curvature never occurs is obtained and the conditions for which only Klebanoff modes exist are thus revealed. A streamwise-dependent receptivity coefficient is defined and we discuss the impact of the receptivity on the $N$ -factor approach for transition prediction.

Control of laminar-to-turbulent transition on a swept-wing is achieved by base-flow modification in an experimental framework, up to a chord Reynolds number of 2.5 million. This technique is based on the control strategy used in the numerical simulation by Dörr & Kloker (J. Phys. D: Appl. Phys., vol. 48, 2015b, 285205). A spanwise uniform body force is introduced using dielectric barrier discharge plasma actuators, to either force against or along the local cross-flow component of the boundary layer. The effect of forcing on the stability of the boundary layer is analysed using a simplified model proposed by Serpieri et al. (J. Fluid Mech., vol. 833, 2017, pp. 164–205). A minimal thickness plasma actuator is fabricated using spray-on techniques and positioned near the leading edge of the swept-wing, while infrared thermography is used to detect and quantify transition location. Results from both the simplified model and experiment indicate that forcing along the local cross-flow component promotes transition while forcing against successfully delays transition. This is the first experimental demonstration of swept-wing transition delay via base-flow modification using plasma actuators.

Surface tension causes the edge of a fluid sheet to retract. If the sheet is also stretched along its edge then the flow and the rate of retraction are modified. A universal similarity solution for the Stokes flow in a stretched edge shows that the scaled shape of the edge is independent of the stretching rate, and that it decays exponentially to its far-field thickness. This solution justifies the use of a stress boundary condition in long-wavelength models of stretched viscous sheets, and gives the detailed shape of the edge of such a sheet, resolving the position of the sheet edge to the order of the thickness.

The turbulent–non-turbulent interface (TNTI) of supersonic turbulent boundary layers is a fundamental but relatively unexplored physics problem. In this study, we present experimental results from fractal analysis on the TNTI of supersonic turbulent boundary layers, and test the applicability of the additive law for these flows. By applying the nanoparticle-tracer planar laser scattering (NPLS) technique in a supersonic wind tunnel, we obtain data covering nearly three decades in scale. The box-counting results indicate that the TNTI of supersonic turbulent boundary layers is a self-similar fractal with a fractal dimension of 2.31. By comparing data sets acquired from two orthogonal planes, we find that the scaling exponent does not depend on direction, consistent with the validity of the additive law for the TNTI of turbulent boundary layers in a scale range with the large-scale limit not exceeding approximately $0.05\unicode[STIX]{x1D6FF}$.

We investigate the onset of instability in a stably stratified two-component fluid in a vertical Hele-Shaw cell when the unstably stratified scalar has a settling velocity. This linear stability problem is analysed on the basis of Darcy’s law, for constant-gradient base states. The settling velocity is found to trigger a novel instability mode characterized by pairs of inclined waves. For unequal diffusivities, this new settling-driven mode competes with the traditional double-diffusive mode. Below a critical value of the settling velocity, the double-diffusive elevator mode dominates, while, above this threshold, the inclined waves associated with the settling-driven instability exhibit faster growth. The analysis yields neutral stability curves and allows for the discussion of various asymptotic limits.

Equal coaxial symmetrically located helical vortices translate and rotate steadily while preserving their shape and relative position if they move in an unbounded inviscid incompressible fluid. In this paper, the linear and angular velocities of this set of vortices ( $U$ and $\unicode[STIX]{x1D6FA}$ respectively) are computed as the sum of the mutually induced velocities found by Okulov (J. Fluid Mech., vol. 521, 2004, pp. 319–342) and the self-induced velocities found by Velasco Fuentes (J. Fluid Mech., vol. 836 2018). Numerical computations of the velocities using the Helmholtz integral and the Biot–Savart law, as well as numerical simulations of the flow evolution under the Euler equations, are used to verify that the theoretical results are accurate for $N=1,\ldots ,4$ vortices over a broad range of values of the pitch and radius of the vortices. An analysis of the flow topology in a reference system that translates with velocity $U$ and rotates with angular velocity $\unicode[STIX]{x1D6FA}$ serves to determine the capacity of the vortices to transport fluid.

We extend the local theory of available potential energy (APE) to a general multicomponent compressible stratified fluid, accounting for the effects of diabatic sinks and sources. As for simple compressible fluids, the total potential energy density of a fluid parcel is the sum of its available elastic energy and APE density. These respectively represent the adiabatic compression/expansion work needed to bring it from its reference pressure to its actual pressure and the work against buoyancy forces required to move it from its reference state position to its actual position. Our expression for the APE density is new and is derived using only elementary manipulations of the equations of motion; it is significantly simpler than existing published expressions, while also being more transparently linked to the relevant form of APE density for the Boussinesq and hydrostatic primitive equations. Our new framework is used to clarify the links between some aspects of the energetics of Boussinesq and real fluids, as well as to shed light on the physical basis underlying the choice of reference state(s) in local APE theory.

Contained rotating flows subject to precessional forcing are well known to exhibit rapid and energetic transitions to disorder. Triadic resonance of inertial modes has been previously proposed as an instability mechanism in such flows, and that idea was developed into a successful model for predicting instability in a cylindrical container when departures from solid-body rotation are sufficiently small. Using direct numerical simulation and dynamic mode decomposition, we analyse instabilities of precessing cylinder flows whose three-dimensional basic states, steady in the gimbal frame of reference, may depart substantially from solid-body rotation. In the gimbal frame, the instability can be interpreted as resulting from a supercritical Hopf bifurcation that results in a limit-cycle flow. In the cylinder frame of reference, the basic state is a rotating wave with azimuthal wavenumber $m=1$ , and the instability satisfies triadic-resonance conditions with the instability mode maintaining a fixed orientation with respect to the basic state. Thus, we are able to demonstrate the existence of two alternative but congruent explanations for the instability. Additionally, we show that basic states may depart substantially from solid-body rotation even with modest cylinder tilt angles, and growth rates for instabilities may be sufficiently large that nonlinear saturation to disordered states can occur within approximately ten cylinder revolutions, in agreement with experimental observations.

Yawing wind turbines has emerged as an appealing method for wake deflection. However, the associated flow properties, including the magnitude of the transverse velocity associated with yawed turbines, are not fully understood. In this paper, we view a yawed turbine as a lifting surface with an elliptic distribution of transverse lift. Prandtl’s lifting line theory provides predictions for the transverse velocity and magnitude of the shed counter-rotating vortex pair known to form downstream of the yawed turbine. The streamwise velocity deficit behind the turbine can then be obtained using classical momentum theory. This new model for the near-disk inviscid region of the flow is compared to numerical simulations and found to yield more accurate predictions of the initial transverse velocity and wake skewness angle than existing models. We use these predictions as initial conditions in a wake model of the downstream evolution of the turbulent wake flow and compare predicted wake deflection with measurements from wind tunnel experiments.

The surface tension of flowing soap films is measured with respect to the film thickness and the concentration of soap solution. We perform this measurement by measuring the curvature of the nylon wires that bound the soap film channel and use the measured curvature to parametrize the relation between the surface tension and the tension of the wire. We find that the surface tension of our soap films increases when the film is relatively thin or is made of soap solution of low concentration; otherwise, it approaches an asymptotic value of $30~\text{mN}~\text{m}^{-1}$ . A simple adsorption model with only two parameters describes our observations reasonably well. With our measurements, we are also able to estimate the Gibbs elasticity of our soap film.

This study proposes an onset condition of shock-free supersonic vortex breakdown from the axial momentum variation, which applies in the presence or absence of a stagnation point. The condition is derived from a comprehensive approach to vortex breakdown. Supersonic breakdown appeared when the swirl parameter and Mach number were small. Moreover, bubble-type breakdowns with a stagnation point, which occur in subsonic conditions, could not occur under the supersonic condition in the present analysis. The predicted breakdowns under this condition were consistent with the results of the three-dimensional numerical simulations for Mach numbers ranging from 1.5 to 5.0. Supersonic vortex breakdowns were clearly captured by the helicity contours in the numerical results. The threshold of the downstream Mach number required for spiral breakdown with no stagnation point was also theoretically derived and verified in numerical results. These findings provide new insights into vortex breakdown in supersonic flows.

We use two-dimensional direct numerical simulations of Boussinesq stratified shear layers to investigate the influence of the minimum gradient Richardson number $Ri_{m}$ on the early time evolution of Kelvin–Helmholtz instability to its saturated ‘billow’ state. Even when the diffusion of the background velocity and density distributions is counterbalanced by artificial body forces to maintain the initial profiles, in the limit as $Ri_{m}\rightarrow 1/4$ , the perturbation growth rate tends to zero and the saturated perturbation energy becomes small. These results imply, at least for such canonical inflectional stratified shear flows, that ‘marginally unstable’ flows with $Ri_{m}$ only slightly less than 1/4 are highly unlikely to become ‘turbulent’, in the specific sense of being associated with significantly enhanced dissipation, irreversible mixing and non-trivial modification of the background distributions without additional externally imposed forcing.

This paper presents new asymptotic formulae for flow in a channel with one or both walls patterned with a longitudinal array of ridges and arbitrarily protruding menisci. Derived from a matched asymptotic expansion, they extend results by Crowdy (J. Fluid Mech., vol. 791, 2016, R7) for shear flow, and thus make no restriction on the protrusion into or out of the liquid. The slip length formula is compared against full numerical solutions and, despite the assumption of small ridge period in its derivation, is found to have a very large range of validity; relative errors are small even for periods large enough for the protruding menisci to degrade the flow and touch the opposing wall.

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.