In channels with rotation (about their spanwise axis) or curvature or both, steady two-dimensional vortices develop above a critical Reynolds number Rec, owing to centrifugal or Coriolis effects. The stability of these streamwise oriented roll cells to two-dimensional, spanwise-periodic perturbations (i.e. Eckhaus stability) is examined numerically using linear stability theory and spectral methods. The results are then confirmed by nonlinear flow simulations. In channels with curvature or rotation or both, the Eckhaus stability boundary is found to be a small closed loop. Within the boundary, two-dimensional vortices are stable to spanwise perturbations. Outside the boundary, Eckhaus instability is found to cause the vortex pairs to split apart or merge together. For all channels examined, two-dimensional vortices are always unstable when Re > 1.7 Rec. Usually, the most unstable spanwise perturbations are subharmonic disturbances, which cause two pairs of vortices with small wave-numbers to be split apart by the formation of a new vortex pair, but cause two pairs of vortices with large wavenumber to merge into a single pair. Recent experimental observations of splitting and merging of vortex pairs are discussed. When Re is not too high (Re < 4.0 Rec), the wavenumbers of vortices are selected by Eckhaus instability and most experimentally observed wavenumbers are close to the ones that are least unstable to spanwise perturbations.

We perform an experimental analysis of co-current, stratified wavy pipe flow, with the aim of investigating the effect of small scale wave breaking (microscale breaking) on the airflow. Particle image velocimetry is applied simultaneously in the gas and liquid phases. Active wave breaking is identified by high levels of vorticity on the leeward side of individual waves, and the statistics of the airflow above breaking and non-breaking waves are extracted from the gas-phase velocity fields. Keeping the liquid superficial velocity constant ($U_{sl}=0.1~\text{m}~\text{s}^{-1}$), we consider two experimental cases of different gas flow rates. The lowest flow rate ($U_{sg}=1.85~\text{m}~\text{s}^{-1}$) is slightly higher than the onset of microscale breaking, while the higher flow rate ($U_{sg}=2.20~\text{m}~\text{s}^{-1}$) is within the regime where wave breaking is observed to be frequent, and the root-mean-square interface elevation $\unicode[STIX]{x1D702}_{rms}$ is independent of gas flow rate. Results show that for the lowest gas flow rate considered, active wave breaking has a stabilizing effect on the airflow above the waves, reducing the sheltered region on the leeward side of the wave and the turbulence above the wave crest compared with non-breaking waves at similar steepness. At the higher gas flow rate the effect of active wave breaking is found to be small, and the main geometrical properties of the waves are found to dominate the evolution of the separated flow region.

We use numerical simulations of the reactive Euler equations to analyse the nonlinear stability of steady-state one-dimensional solutions for gaseous detonations in the presence of both momentum and heat losses. Our results point to a possible stabilization mechanism for the low-velocity detonations in such systems. The mechanism stems from the existence of a one-parameter family of solutions found in Semenko et al. (Shock Waves, vol. 26 (2), 2016, pp. 141–160).

A non-modal linear stability analysis (NMA) of the miscible viscous fingering in a porous medium is studied for a toy model of non-monotonic viscosity variation. The onset of instability and its physical mechanism are captured in terms of the singular values of the propagator matrix corresponding to the non-autonomous linear equations. We discuss two types of non-monotonic viscosity profiles, namely, with unfavourable (when a less viscous fluid displaces a high viscous fluid) and with favourable (when a more viscous fluid displaces a less viscous fluid) endpoint viscosities. A linear stability analysis yields instabilities for such viscosity variations. Using the optimal perturbation structure, we are able to show that an initially unconditional stable state becomes unstable corresponding to the most unstable initial disturbance. In addition, we also show that to understand the spatio-temporal evolution of the perturbations it is necessary to analyse the viscosity gradient with respect to the concentration and the location of the maximum concentration $c_{m}$. For the favourable endpoint viscosities, a weak transient instability is observed when the viscosity maximum moves close to the pure invading or defending fluid. This instability is attributed to an interplay between the sharp viscosity gradient and the favourable endpoint viscosity contrast. Further, the usefulness of the non-modal analysis demonstrating the physical mechanism of the quadruple structure of the perturbations from the optimal concentration disturbances is discussed. We demonstrate the dissimilarity between the quasi-steady-state approach and NMA in finding the correct perturbation structure and the onset, for both the favourable and unfavourable viscosity profiles. The correctness of the linear perturbation structure obtained from the non-modal stability analysis is validated through nonlinear simulations. We have found that the nonlinear simulations and NMA results are in good agreement. In summary, a non-monotonic variation of the viscosity of a miscible fluid pair is seen to have a larger influence on the onset of fingering instabilities than the corresponding Arrhenius type relationship.

We present a decomposition of the temporal growth rate ${\it\omega}_{i}$ which characterises the evolution of wave-like disturbances in linear stability theory for compressible flows. The decomposition is based on the disturbance energy balance by Chu (Acta Mech., vol. 1 (3), 1965, pp. 215–234) and provides terms for production, dissipation and flux of energy as components of ${\it\omega}_{i}$. The inclusion of flux terms makes our formulation applicable to unconfined flows and flows with permeable or vibrating boundaries. The decomposition sheds light on the fundamental mechanisms determining temporal growth or decay of disturbances. The additional insights gained by the proposed approach are demonstrated by an investigation of two model flows, namely compressible Couette flow and a plane compressible jet.

We demonstrate that the critical magnetic Reynolds number $Rm_{c}$ for a turbulent non-helical dynamo in the limit of low magnetic Prandtl number $Pm$ (i.e. $Pm=Rm/Re\ll 1$ ) can be significantly reduced if the flow is subjected to global rotation. Even for moderate rotation rates the required energy injection rate can be reduced by a factor of more than $10^{3}$ . This strong decrease in the onset is attributed to the transfer of energy to the large scales, forming a large-scale condensate, and the reduction in the turbulent fluctuations that cause the flow to have a much larger cutoff length scale than in a non-rotating flow of the same Reynolds number. The dynamo thus behaves as if it is driven just by the large scales that act as a laminar flow (i.e. it behaves as a high $Pm$ dynamo) even though the actual Reynolds number is much higher than the magnetic Reynolds number (i.e. low $Pm$ ). Our finding thus points to a new paradigm for the design of new experiments on liquid metal dynamos.

A highly rarefied gas, initially in equilibrium, is compressed by an infinite plane piston. The resulting flow is considered when the gas is unbounded and also when the gas is bounded by a second stationary wall. In the first case, the density ahead of the piston is analysed for specularly and diffusely reflecting surfaces. In the second case, the average temperature of the gas is found as a function of the wall separation for specularly reflecting surfaces. This (non-equilibrium) temperature is compared with that generated in the corresponding continuum flow which involves multiple shock wave reflexion. It is shown that, for a given average density gradient, the free molecule temperatures are very much higher than the continuum temperatures.

For a particle with two or three perpendicular planes of symmetry rotating at low Reynolds number in a Couette flow field, there are three orbits in which the motion is simply periodic. The three scalars Bi involved in Bretherton's shape tensor are found experimentally from periods of rotation in these orbits. Experiments on right parallelepipeds, ranging from thin platelets through cubes to rectangular section rods, are described. Each of the Bi is found to depend on the aspect ratio of one of the particle cross-sections, with only slight influence from its third dimension. Results are expressed in terms of a relationship between this aspect ratio and that of an equivalent ellipse, incorporating a weak function of the third particle dimension. The equations of motion governing a general doubly periodic motion, and incorporating experimental Bi values, are integrated numerically and compared satisfactorily with experimental observations.

The critical-layer analysis of the nonlinear resonant-triad interaction by Goldstein & Lee (1992) is extended to include viscous effects. A generalized scaling which is valid both for the quasi-equilibrium and non-equilibrium critical-layer analyses in zero- or non-zero-pressure-gradient boundary layers is obtained. A system of partial differential equations which governs the fully coupled non-equilibrium critical-layer dynamics is obtained and it is solved by using a numerical method. Amplitude equations and their viscous limits are also presented. The parametric-resonance growth rate of the non-equilibrium critical-layer solution with finite viscosity is larger than that of the viscous-limit quasi-equilibrium solution. The viscosity delays both the onset of the fully coupled interaction and the ultimate downstream location of the singularity. The difference between the non-equilibrium critical-layer solution and the corresponding quasi-equilibrium critical-layer solution becomes smaller, at least in the parametric resonance region, as the viscosity parameter becomes large. However, the non-equilibrium solution with finite viscosity always ends in a singularity at a finite downstream position unlike the viscous-limit solution.

A solution to the hypersonic small disturbance equations is obtained for a class of two-dimensional bodies supporting logarithmic shock waves by reducing the partial differential equation to an ordinary differential equation. The body shape is calculated and shown not to be logarithmic if γ ≠ 1, where γ is the ratio of the specific heats of the gas.

A Hamiltonian formulation is used to investigate irrotational capillary wave dynamics. Dissipation is accounted for by putting the wave system in contact with a ‘heat bath’. The generation of short waves by longer waves is studied. It is found that millimetre-wavelength waves tend to be created on the forward face of a steep longer wave, while centimetre waves tend to form near the crest. Generation of capillary waves by wind waves is investigated. The results are compared with predictions of the Hasselmann transport equation. It is found that off-resonance interactions lead to significant corrections to the transport theory. The relative importance of three-wave and four-wave interactions is studied, as well as the role of triad resonances. For the capillary phenomena studied here, the four-wave terms in most cases lead to quantitative, but not qualitative, corrections to the three-wave only calculations. However, restricting interactions to the neighbourhood of triad resonances can give quite erroneous results. Use of a canonical transformation to pseudo-wave variables can greatly reduce numerical computation times.

The modulated nonlinear Schrödinger equation (Zhang & Melville 1990), describing the evolution of a weakly nonlinear short-gravity-wave train riding on a longer finite-amplitude gravity-wave train is used to study the stability of steady envelope solutions of the short-wave train. The formulation of the stability problem reduces to the solution of a pair of coupled equations for the disturbance amplitude and (relative) phase. Approximate analytical solutions and numerical solutions show that the conventional sideband (Benjamin-Feir) instability is just the first in a series of resonantly unstable regions which increase in number with increasing perturbation wavenumber. The first of these new instabilities is the result of a quintet resonance between four short waves and one long wave. Subsequent unstable regions correspond to sextet or higher-order resonances. The results presented here suggest that steady envelope solutions for unforced irrotational short waves on longer irrotational gravity waves may be unstable for a wide range of conditions.

The development of reflected waves are studied when two shocks of unequal strength collide and when a shock collides with a constant temperature wall. Both these problems are examined using the Monte Carlo technique developed by Bird (1967). Some limitations upon this technique are suggested and a modified time advance parameter used.

The steady flow of an ideal gas past a conical body is investigated by the method of matched asymptotic expansions, with particular emphasis on the flow near the singular ray occurring in linearized theory. The first-order problem governing the flow in this region is formulated, leading to the equation of Kuo, and an approximate solution is obtained in the case of compressive flow behind the main front. This solution is compared with the results of previous investigations with a view to assessing the applicability of the Lighthill-Whitham theories.

Pressure and pressure-velocity space correlations are calculated, using rapid-distortion theory, for turbulence in a uniform shear flow. It is found that pressure fluctuations remain correlated over significantly greater distances than do velocity fluctuations. When these predictions are used as a model for turbulence in free turbulent shear flows, it is found that the predicted scale of the pressure fluctuations is larger than the flow width. It is proposed that pressure fluctuations remain highly correlated right across free shear flows. Predictions from the theory are then compared with various experimental situations in which reasonable qualitative agreement is to be expected, and this is found.