Results of experimental investigations of vibrational relaxation regions of shock waves in nitrous oxide are reported in this paper. The Mach-number range covered was from 1·5 to 7·5 and the photographs of the shocks were made using a Mach-Zehnder interferometer and a spark source. The experimental and calculated values of the over-all density ratio were found to be in good agreement, and the values of the relaxation frequencies were consistent with the data of other workers.

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 linear stability of two-dimensional surface gravity waves on fluid of finite depth is investigated for superharmonic disturbances. For this problem, Zufiria & Saffman (Stud. Appl. Maths vol. 74, 1986, p. 259) suggested that an exchange of stability occurs when the total wave energy becomes stationary as a function of wave speed for fixed ‘Bernoulli constant’. In defining the potential energy of the above total wave energy, the surface displacement was measured from the quiescent surface with the same ‘Bernoulli constant’. We have re-examined this problem both analytically and numerically, and found that the above ‘Bernoulli constant’ must be replaced by ‘mean surface height’ for the statement to be valid.

Consequences of density stratification are studied for an ideal (Euler) incompressible fluid, confined to move under gravity between rigid lids but otherwise free to move along horizontal directions. Initial conditions that generate horizontal pressure imbalances in a laterally unbounded domain are examined. The aim is to show analytically the existence of classes of initial data for which total horizontal momentum evolves in time, even though only vertical forces act on the fluid in this set-up. A simple class of such initial conditions, leading to momentum evolution, is identified by systematic asymptotic expansions of the governing inhomogeneous Euler equations in the small-density-variation limit. These results for Euler equations are compared and confirmed with long-wave asymptotic models, which can handle arbitrary density variations and provide closed-form mathematical expressions for limiting cases. In particular, the role of wave dispersion arising from the fluid inertia is captured by the long-wave models, even for short-time dynamics emanating from initial conditions outside the models’ asymptotic range of validity. These results are compared with direct numerical simulations for variable-density Euler fluids, which further validate the numerical algorithms and the analysis.

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.

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.

The characteristics of the laminar boundary layer on a continuous moving surface are described and an experiment is performed to demonstrate that such a flow is physically realizable. The hydrodynamic stability of the flow is analysed within the framework of small-perturbation stability theory. A complete stability diagram is mapped out. The critical Reynolds number is found to be substantially higher than that for the Blasius flow and, correspondingly, the critical layer lies closer to the wall. The disturbance amplitude function and its derivative are numerically evaluated, from which are derived the vector flow field of the disturbance, the resultant flow field (main flow plus disturbances), the root-mean-square distributions of the disturbance velocity components, and the distributions of the kinetic energy and the Reynolds stress. The energy criterion for stability is also investigated and is found to be consistent with the solutions of the eigenvalue problem.

The influence of inertia terms in the equations of motion on the properties of the base-flow and near-wake flow at very low Reynolds numbers is investigated by using Oseen's approximation and by comparing with the results obtained in Part 1 for Stokes flow. The general solutions of Oseen's equations of motion are derived for two-dimensional and axisymmetric flows in the half-space x > 0, for an arbitrarily given velocity field in the plane x = 0. Numerical examples are given for two-dimensional flow and compared with Stokes-flow examples.

The problem of transonic flow past an array of micro-electro-mechanical-type (MEMS-type) heating elements placed on a flat surface is investigated using the triple-deck theory. The compressible Navier–Stokes equations supplemented by the energy equation are considered for large Reynolds numbers. The triple-deck problem is formulated with the aid of the method of matched expansions. The resulting nonlinear viscous lower deck problem, coupled with the upper deck problem governed by the nonlinear Kármán–Guderley equation, is solved using a numerical method based on Chebyshev collocation and finite differences. Our results show the differences in subsonic and supersonic flow behaviour over heated elements. The results indicate the possibility of using the elements to favourably control the transonic flow field.

The statistics of a passive scalar randomly emitted from a point source is investigated analytically for the Kraichnan ensemble. Attention is focused on the two-point equal-time scalar correlation function, a statistical indicator widely used both in experiments and in numerical simulations. The only source of inhomogeneity/anisotropy is the injection mechanism, the advecting velocity being here statistically homogeneous and isotropic. The main question we address is on the possible existence of an inertial range of scales and a consequent scaling behaviour. The question arises from the observation that for a point source the injection scale is formally zero and the standard cascade mechanism cannot thus be taken for granted. We find from first principles that an intrinsic integral scale, whose value depends on the distance from the source, emerges as a result of sweeping effects. For separations smaller than this integral scale a standard forward cascade occurs. This is characterized by a Kolmogorov–Obukhov power-law behaviour as in the homogeneous case, except that the dissipation rate is also dependent on the distance from the source. Finally, we also find that the combined effect of a finite inertial-range extent and of inhomogeneities causes the emergence of subleading anisotropic corrections to the leading isotropic term, that are here quantified and discussed.

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 $N=4$.

In the energy stability theory, the critical Reynolds number is usually defined as the minimum of the first positive eigenvalue $R_{1}$ of an eigenvalue equation for all wavenumber pairs $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})$, where $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ are the streamwise and spanwise wavenumbers of the normal mode. We prove that $(\cos \unicode[STIX]{x1D703}\pm 1)R_{1}$ are decreasing functions of $\unicode[STIX]{x1D703}=\arctan (\unicode[STIX]{x1D6FD}/\unicode[STIX]{x1D6FC})$ for the parallel flows between no-slip or slip parallel plates with or without variations in temperature. Numerical results inspire us to conjecture that $R_{1}$ is also a decreasing function of $\unicode[STIX]{x1D703}$ for the parallel shear flows under the no-slip boundary condition and without variations in temperature. If the conjecture is correct, the least stable normal modes for the energy stability will be streamwise vortices for these base flows.

The stress and conformational relaxation of an initially straight flexible polymer is studied through Brownian dynamics simulations covering a broad range of time scales and polymer lengths. At short times $t\,{\ll}\, N^{-2}$, the strong stress component scales as $\sigma_{11}\,{\sim}\, N^3$ (‘1’ is the direction of the original alignment), while the weak component is $\sigma_{22} \,{\sim}\, N$ (where $N$ is the polymer length). At intermediate times $N^{-2}\,{\ll}\, t\,{\ll}\, N^{2}$, the stress decay is shown to be anisotropic: $\sigma_{11} \,{\sim}\, N^2 t^{-1/2}$ while $\sigma_{22} \,{\sim}\, N^{1/2} t^{-1/4}$. At long times $t\,{\gg}\, N^2$, both stress components show the same exponential decay associated with the reduction of the chain's length, while their magnitudes are still different; $\sigma_{11}\,{=}\, \mbox{\it O}(N)$ while $\sigma_{22} \,{=}\, \mbox{\it O}(1)$. The configuration relaxation is studied over the same extended time periods by employing scaling laws for the evolution of the eigenvalues of the gyration tensor. After the short-time free diffusion, the configuration relaxation is anisotropic at intermediate times: the chain's width grows as $R_{\perp} \,{\sim}\, N^{-1/4} t^{3/8}$ while its length is reduced as $R_{\parallel}(0)\,{-}\,R_{\parallel} \,{\sim}\, t^{1/2}$. During long times, the polymer length shows an exponential decay towards the equilibrium coil-like shape. The polymer chain remains aligned along the original direction until late in long times where the chain rotation is shown to be significant. The chain is shown to be far from equilibrium during the entire transient relaxation. By focusing on the chain's longitudinal relaxation, a quasi-steady equilibrium of the link tensions in shown to exist along the chain's length; the mechanism is identical to that found in Grassia & Hinch (1996) based on a sideways motion model. This mechanism also explains the ability of the FENE model to describe the longitudinal relaxation of the flexible bead-rod chain. A comparison with experimental findings from initially straight single tethered DNA molecules is also included.

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.

The potential flow upstream from a contraction in a rectangular flume is analysed. In order to calculate the potential function, the flow is considered as the superposition of sinks uniformly distributed in the contraction. The effect of boundaries is taken into account by introducing virtual sinks. The calculation is performed in the complex plane and provides a closed-form solution of the complex potential function. As an illustration, the effect of contraction size and position is analysed, and the solution is compared to experimental measurements and other numerical solutions for vertical sluice gates.

Abdominal aortic aneurysms are a dilatation of the aorta, localized preferentially above the bifurcation of the iliac arteries, which increases in time. Understanding their localization and growth rate remain two open questions that can have either a biological or a physical origin. In order to identify the respective role of biological and physical processes, we address in this article these questions of the localization and growth using a simplified physical experiment in which water (blood) is pumped periodically (amplitude a, pulsation ω) in an elastic membrane (aorta) (length L, cross-section A0 and elastic wave speed c0) and study the deformation of this membrane while decharging in a rigid tube (iliac artery; hydraulic loss K). We first show that this pulsed flow either leads to a homogenous deformation or inhomogenous deformation depending on the value of the non-dimensional parameter c02/(aLω2K). These different regimes can be related to the aneurysm locations. In the second part, we study the growth of aneurysms and show that they only develop above a critical flow rate which scales as A0c0/.

This paper deals with the stability of suspensions modelled as dusty gases for non-uniform profiles of mass fraction of particles. It is known that a stationary uniform fluidized bed may be unstable to small disturbances which grow until a secondary instability develops forming bubbles that rise through the bed. Interactions between particles are difficult to model and this makes it necessary to close the model with some assumptions. However, in dilute fluidized beds which are characterized by a low volume fraction of particles, interactions between particles are negligible and this motivates the study of instabilities of suspensions by means of the dusty gas equations, avoiding the problem of particular closures.

We show in this work that suspensions blown to regions of higher concentration are unstable to two-dimensional disturbances. An equation which governs this instability is obtained. A physical mechanism is proposed to explain this instability and it is related to the Rayleigh–Taylor one in the limit of long characteristic lengthscales associated with the concentration profile. Finally, the evolution of this instability is followed using a fully nonlinear numerical code showing the formation of streamers and clusters of particles.

Because the three-dimensional large-scale vortex structures dominate the dispersion of particles at intermediate Stokes numbers in shear layers, there is interest in understanding the two-way thermal coupling effect in droplet–gas flows for practical combustion applications. Using pseudo-spectral and Lagrangian approaches, three-dimensional two-way thermally coupled droplet-laden mixing layers are studied with hot air and cool water droplets. Higher air density is observed around the region of the interface of the two streams and, thus, thermal contraction occurs in this region. This thermal contraction results in an increase of the magnitude of the vorticity field, a more unstable flow, a higher droplet concentration and a lower droplet dispersion across the mixing layer.