We consider the flow of a viscous liquid along an unknown topography. A new strategy is presented to reconstruct the topography and the free surface shape from one component of the free surface velocity only. In contrast to the classical approach in inverse problems based on optimization theory we derive an ordinary differential equation which can be solved directly to obtain the inverse solution. This is achieved by averaging the Navier–Stokes equation and coupling the function parameterizing the flow domain with the free surface velocity. Even though we consider nonlinear systems including inertia and surface tension, the inverse problem can be solved analytically with a Fourier series approach. We test our method on a variety of benchmark problems and show that the analytical solution can be applied to reconstruct the flow domain from noisy input data. It is also demonstrated that the asymptotic approach agrees very well with numerical results of the Navier–Stokes equation. The results are finally confirmed with an experimental study where we measure the free surface velocity for the film flow over a trench and compare the reconstructed topography with the measured one.

An instrumental method is proposed to measure the direction of surface velocity gradients in three-dimensional boundary layers. It utilizes electrochemical techniques developed in this laboratory.

The recent study of Ren et al. (J. Fluid Mech., vol. 871, 2019, pp. 831–864) investigated the hydrodynamic linear stability of a compressible boundary layer over an insulated flat plate for a non-ideal gas (supercritical $\text{CO}_{2}$). In particular, the authors showed that in the transcritical regime (across the pseudo-critical line) the flow is strongly convectively unstable due to the co-existence of two unstable modes: Mode I, related to Tollmien–Schlichting instabilities and a new inviscid two-dimensional mode (Mode II) with a spatial growth rate one order of magnitude larger than Mode I for high Eckert numbers. In contrast to the transcritical regime, in the sub- and supercritical regimes, Mode II does not exist. Only Mode I drives the instabilities: viscous and two-dimensional for the subcritical regime and inflectional and three-dimensional for the supercritical regime.

Stuart vortices are among the few known smooth explicit solutions of the planar Euler equations with a nonlinear vorticity, and they have a counterpart for inviscid flow on the surface of a fixed sphere. By means of a perturbative approach we adapt the method used to investigate Stuart vortices on a fixed sphere to provide insight into some large-scale shallow-water flows on a rotating sphere that model the dynamics of ocean gyres.

Compressible flow varies from ideal-gas behaviour at high pressures where molecular interactions become important. It is widely accepted that density is well described through a cubic equation of state while enthalpy and sound speed are functions of both temperature and pressure, based on two parameters, $A$ and $B$, related to intermolecular attraction and repulsion, respectively. Assuming small variations from ideal-gas behaviour, a closed-form approximate solution is obtained that is valid over a wide range of conditions. An expansion in these molecular interaction parameters simplifies relations for flow variables, elucidating the role of molecular repulsion and attraction in variations from ideal-gas behaviour. Real-gas modifications in density, enthalpy and sound speed for a given pressure and temperature lead to variations in many basic compressible-flow configurations. Sometimes, the variations can be substantial in quantitative or qualitative terms. The new approach is applied to choked-nozzle flow, isentropic flow, nonlinear wave propagation and flow across a shock wave, all for a real gas. Modifications are obtained for allowable mass flow through a choked nozzle, nozzle thrust, sonic wave speed, Riemann invariants, Prandtl’s shock relation and the Rankine–Hugoniot relations. Forced acoustic oscillations can show substantial augmentation of pressure amplitudes when real-gas effects are taken into account. Shocks at higher temperatures and pressures can have larger pressure jumps with real-gas effects. Weak shocks decay to zero strength at sonic speed. The proposed framework can rely on any cubic equation of state and can be applied to multicomponent flows or to more complex flow configurations.

The wake structure of an incompressible, conducting, viscous fluid past an electrically insulating sphere affected by a transverse magnetic field is investigated numerically over flow regimes including steady and unsteady laminar flows at Reynolds numbers up to 300. For a steady axisymmetric flow affected by a transverse magnetic field, the wake structure is deemed to be a double plane symmetric state. For a periodic flow, unsteady vortex shedding is first suppressed and transitions to a steady plane symmetric state and then to a double plane symmetric pattern. Wake structures in the range $210<Re\leqslant 300$ without a magnetic field have a symmetry plane. An angle $\unicode[STIX]{x1D703}$ exists between the orientation of this symmetry plane and the imposed transverse magnetic field. For a given transverse magnetic field, the final wake structure is found to be independent of the initial flow configuration with a different angle $\unicode[STIX]{x1D703}$. However, the orientation of the symmetry plane tends to be perpendicular to the magnetic field, which implies that the transverse magnetic field can control the orientation of the wake structure of a free-moving sphere and change the direction of its horizontal motion by a field–wake–trajectory control mechanism. An interesting ‘reversion phenomenon’ is found, where the wake structure of the sphere at a higher Reynolds number and a certain magnetic interaction parameter ($N$) corresponds to a lower Reynolds number with a lower $N$ value. Furthermore, the drag coefficient is proportional to $N^{2/3}$ for weak magnetic fields or to $N^{1/2}$ for strong magnetic fields, where the threshold value between these two regimes is approximately $N=4$.

Eigenmodes of averaged small-amplitude perturbations to a turbulent channel flow – which is one of the most fundamental canonical flows – are identified for the first time via an extensive set of high-fidelity graphics processing unit accelerated direct numerical simulations. While the system governing averaged small-amplitude perturbations to turbulent channel flow remains unknown, the fact such eigenmodes can be identified constitutes direct evidence that it is linear. Moreover, while the eigenvalue associated with the slowest-decaying anti-symmetric eigenmode mode is found to be real, the eigenvalue associated with the slowest-decaying symmetric eigenmode mode is found to be complex. This indicates that the unknown linear system governing the evolution of averaged small-amplitude perturbations cannot be self-adjoint, even for the case of a uni-directional flow. In addition to elucidating aspects of the flow physics, the findings provide guidance for development of new unsteady Reynolds-averaged Navier–Stokes turbulence models, and constitute a new and accessible benchmark problem for assessing the performance of existing models, which are used widely throughout industry.

In this paper, we investigate the receptivity of a supersonic boundary layer to impinging acoustic waves. Unlike previous studies of acoustic receptivity, where the sound waves have phase speeds comparable with or larger than the free-stream velocity $U_{\infty }$, the acoustic waves here have much slower ($O(R^{-1/8}U_{\infty })$) phase velocity, and their characteristic wavelength and frequency are of $O(R^{-3/8}L)$ and $O(R^{1/4}U_{\infty }/L)$ respectively, compatible with the triple-deck structure, where $L$ is the distance to the leading edge and $R$ the Reynolds number based on $L$ and $U_{\infty }$. A significant feature of a sound wave on the triple-deck scale is that an $O(\unicode[STIX]{x1D700}_{s})$ perturbation in the free stream generates much stronger ($O(\unicode[STIX]{x1D700}_{s}R^{1/8})$) velocity fluctuations in the boundary layer. Two receptivity mechanisms are considered. The first is new, involving the interaction of two such acoustic waves and operating in a boundary layer over a smooth wall. The second involves the interaction between an acoustic wave and the steady perturbation induced by a wavy wall. The sound–sound, or sound–roughness, interactions generate a forcing in resonance with a neutral Tollmien–Schlichting (T–S) wave. The latter is thus excited near the lower branch of the neutral curve, and subsequently undergoes exponential amplification. The excitation through sound–sound interaction may offer a possible explanation for the appearance of instability modes downstream of their neutral locations as was observed in a supersonic boundary layer over a smooth wall. The triple-deck formalism is adopted to describe impingement and reflection of the acoustic waves, and ensuing receptivity, allowing the coupling coefficient to be calculated. The two receptivity processes with the acoustic waves on the triple-deck scale are much more effective compared with those involving usual sound waves, with the coupling coefficient being greater by a factor of $O(R^{1/4})$ and $O(R^{1/8})$ in the sound–sound and sound–roughness interactions, respectively. A parametric study for both the reflection and coupling coefficients is conducted for representative T–S waves, to assess the influence of the streamwise and spanwise wavenumbers, and the phase speed (or frequency) of the acoustic wave.

The flow past a circular cylinder in a rotating frame is examined when the Rossby number Ro is O(E½), where E is the Ekman number. Previous studies of the configuration have shown that, provided the ratio Ro/E½ is less than a certain critical value, the flow around the cylinder is determined by the classical potential-flow solution. However, once Ro/E½ is greater than that critical value the E1/4 layer on the surface of the cylinder, which is rather like a boundary layer in a high-Reynolds-number non-rotating fluid, can separate from the cylinder and distort the potential flow. In this study the form of the flow once separation has occurred is examined using a method analogous to the Kirchhoff free-streamline theory in a non-rotating fluid. The results are compared with published experimental and numerical data on the flow for various values of Ro/E½.

The influence of uniform flow acceleration on the stability and the characteristics of circular-cylinder wakes over a Reynolds-number range, 20 < R < 330, was investigated. Experiments were performed to examine the temporal evolution of the wake before, during, and after the onset of the wake instability. We have demonstrated in several ways that the wake is stabilized by flow acceleration: (i) the onset of the wake instability occurs at larger Reynolds numbers than in the steady flow case, (ii) the closed wake develops to states that would be unstable in a steady flow, and (iii) once vortex shedding does occur there is a reduction in instantaneous Strouhal number. We have also examined the temporal growth rate of the wake instability and find that it is directly proportional to the applied flow acceleration. Physical mechanisms are proposed to describe the experimental observations.

Low-Rossby-number flow past a circular cylinder in a rapidly rotating frame is studied when 1 < N < 2, where N is equal to E½/Ro in terms of the Ekman number E and Rossby number Ro. For this parameter range the E¼ boundary layer contains a singularity at the rear stagantion point. The asymptotic structure of this singularity is shown to consist of three distinct asymptotic regions, one of which is viscous while the others are inviscid. New accurate numerical solutions of the boundary-layer equation confirm this singularity structure. The use of Von Mises coordinates both simplifies the analysis, and enables numerical solutions to be found closer to the critical value N = 1, beneath which the flow separates upstream of the rear stagnation point.

The second variation of a linear combination of energy and angular momentum is used to investigate the formal stability of circular vortices. The analysis proceeds entirely in terms of Lagrangian displacements to overcome problems that otherwise arise when one attempts to use Arnol'd's Eulerian formalism. Specific attention is paid to the simplest possible model of an isolated vortex consisting of a core of constant vorticity surrounded by a ring of oppositely signed vorticity. We prove that the linear stability regime for this vortex coincides with the formal stability regime. The fact that there are formally stable isolated vortices could imply that there are provable nonlinearly stable isolated vortices. The method can be applied to more complicated vortices consisting of many nested rings of piecewise-constant vorticity. The equivalent expressions for continuous vorticity distributions are also derived.

Using stationary rakes of hot-wire anemometer probes, a series of temperature measurements have been conducted within rising buoyant thermals created by detonation of large-diameter gas-filled balloons. These atmospheric detonation simulation tests were performed at the SANDIA Laboratories Aerial Cable Test Facility, Kirtland AFB, during the late fall of 1973. All measurements were performed after pressure equilibrium (t > 1 s) but prior to completion of the toroid formation process and within the time required (t > 4 s) for the fireball to rise one diameter.

Based on an extensive processing of measured data for two detonation events, unique results describing the mean flow field and turbulent structure statistics for the early-time large-scale buoyant thermal have been determined. Final results include: mean-velocity and temperature data; temperature and temperature gradient intensities; space, auto- and ‘eddy-lifetime’ correlation scales; power-spectral densities; probability density distributions and approximate microscale data.

The present temperature data illustrate that the early-time rising fireball is characterized by a highly intermittent irregular top or cap, a constant (high-) temperature core and a thermal wake exhibiting a linear decay in mean temperature. Mean convection velocities for the fireball wake were measured to be approximately twice as large as fireball cap velocities (13.1 m/s). Maximum fireball temperatures (1100 °C) were in good agreement with peak brightness-temperature data determined from an independent IR measurements experiment. Core leading-edge mean-temperature gradients compare favourably with predictions (10 °C/cm) whereas trailing-edge gradients (0.4 °C/cm) are smaller than predicted results (3 °C/cm).

Normalized temperature intensities at the fireball top and within the thermal wake were substantial (≈ 0.6) in contrast with quiescent fireball core levels (0.1). Moving-frame autocorrelation times (‘eddy lifetimes’, 0.17 + 0.04 s) were approximately four times larger than measured autocorrelation times. This result illustrates that the early-time fireball eddy structure is highly persistent, a finding which tends to substantiate those radar models for large-scale fireballs which postulate relatively slow ‘smearing’ of early-time electron density gradients. Wake integral scales, determined from vertical space correlation data, were approximately 5% of the fireball mean radius. Probability density distributions for the fireball thermal wake were ‘spiky’ in nature, with some evidence of bimodal behaviour, and slightly skewed to the negative side of their normalized values. Normalized spectra data, in general, follow a − &5/3; falloff behaviour for several decades in spectral power. The noted large ‘spread’ in the inertial subrange is consistent with measured turbulence Reynolds numbers (2000) and with the large-scale fully developed turbulent structure for the GEST fireballs (ReD = 3 × 107). Wake microscale data, estimated from temperature and temperature-gradient intensity data and the isotropic turbulence assumption, varied from 2 to 4 ern and were in favourable agreement with predicted microscale magnitudes.

The formation of patterns of granules on the bottom of a differentially rotating tank, partially filled with a fluid, is described. The pattern formation is initiated when the tank is spun up from an initial constant rotation rate by a sufficiently large increment to another constant higher rotation rate. The granules are observed to be set into motion, slide across the bottom of the tank and organize themselves into two distinct, superposed geometrical patterns. The first pattern constitutes a system of tightly wound equi-angular spirals and is identified as being simply a visualization of the well-known stationary Class B waves of the unstable boundary layer. This pattern is not considered further here. The second pattern, which is considered here, is shown not to be a visualization of some other known stationary wave mode. It consists of a number of large-scale spiral arms extending outward towards the wall of the tank and originating at a patch of salt in the tank's centre. The presence of Class B waves is apparently a necessary condition for the spiral arm formation. The experiments show that the radius r0 of the inner salt patch is inversely proportional to the increment Δω of the rotation rate for the larger values of Δω. The angle ε(r) between the direction of the spiral arms and the tangential direction is observed to decrease with the distance r from the centre as rb where b = −0.64 ± 0.1. The relationship between the number n of spiral arms and the associated Reynolds number Re and Rossby number Ro is found from the experimental observations and dimensional considerations to be \[n^2 = Re \, Ro. \] Various aspects related to the mechanism involved in the pattern formation are discussed. However, no conclusive model for the spiral arm formation can at present be suggested.