The arbitrary motion of a solid sphere released in a solid-body rotating fluid is investigated theoretically in the limit of small Reynolds and Taylor numbers. The angular velocity of the fluid is assumed to be constant and under the premise that Ta1/2 ≫ Re, the simplicity of the unperturbed flow enables us to calculate analytically the force acting on a particle moving with a harmonic slip velocity (by means of matched asymptotic expansions), when both inertia and unsteady effects are taken into account. Subsequently, these single-frequency results are used in order to determine the temporal expression of the force acting on an arbitrarily moving sphere, since the problem under study is linear. This force is first determined in a co-rotating reference frame and takes the form of two convolution products involving the particle acceleration and the particle velocity. For convenience, the corresponding expression of this force is also derived in the laboratory reference frame, and the particle motion equation obtained is thereafter illustrated by dealing with two practical situations, where unsteady and inertia effects must be taken into account to predict the particle dynamics accurately.

A technique by which the effect of the random phase fluctuations or Doppler ambiguity can be removed from measurements of turbulence intensities by the laser-Doppler velocimeter is proposed. Because the ambiguity is often much broader band than the turbulence, it is proposed that a sequence of low-pass filters at frequencies above those of the turbulence be used in the measurement of turbulence intensities. The true turbulence intensity is given by the extrapolation of these measurements to zero frequency.

The simple pendulum remains one of the most fundamental systems studied in physics. It is commonly used as a model to illustrate a broad variety of mechanisms in a wide range of areas. However, in spite of this popularity, subtle behaviours still remain to be discovered and to be explored when the pendulum is strongly coupled to fluid mechanics. This is for instance illustrated in recent studies by Neill, Livelybrooks & Donnelly (Am. J. Phys., vol. 75, 2007, pp. 226–229) and Bolster, Hershberger & Donnelly (Phys. Rev. E, vol. 81, 2010, pp. 1–6) which highlight the impact on a simple spherical pendulum of vortex shedding and added mass effects. In the present work we show that the equilibrium of a pendular disk facing a flow exhibits bi-stability and hysteresis. We give a simple interpretation of this behaviour in terms of a two-potential-well description, only requiring to know the angular dependence of the normal drag coefficient of an inclined plate. We investigate the influence of turbulence on the equilibrium of the pendulum in general and on the observed bi-stability in particular. Our results have potentially important fundamental and practical consequences: (i) they extend the attractiveness of the pendulum as a model to investigate generic questions related to bi-stable stochastic processes, (ii) they highlight important fluid dynamic mechanisms, including turbulent drag enhancement and fluid–structure interaction issues.

The vibrational relaxation frequency of carbon dioxide has been determined by measuring the rate of change of thermal emission in shock waves near 4±3μ. This method of measuring the relaxation frequency depends mainly on the degree of excitation of the asymmetric stretching mode of the molecule, and the results are compared with those of earlier density measurements made in the same shock tube. The gas samples used are not optically thin, and it is shown that self-absorption can be taken into account. The results imply that the relaxation frequency of the asymmetric stretching mode is about 70% of that of the bending mode.

Observations and measurements are reported on the patterns and rates of growth in time of the double-diffusive cells that form adjacent to a cooled sidewall in a saltstratified environment. Fluid near the wall is cooled and sinks a distance $h$ where its density, increased by cooling, matches that of the salt-stratified ambient. The fluid separates from the wall, moving outwards as a cool, fresher layer beneath a warmer, more saline region. This leads to growing double-diffusive cells that advance outward at a rate, found by dimensional reasoning, to initially be proportional to $N_{0}h,$ where $N_{0}$ is the initial buoyancy frequency in the ambient and $h$ is the intrusion's vertical thickness. Near the wall at the top of each cell, the sinking colder fluid is continually replaced by selective withdrawal from the ambient ‘far field’. The fluid being withdrawn from the ambient is always the least dense in the cell, and as the experiment proceeds, the straining of the fluid in the ambient region reduces the stratification. The vertical density gradient inside the cell relaxes by continuous hydrostatic adjustment (CHA) to match the ambient and the speed of advance reduces. Measurements of the rate of advance of the cell nose were made in tanks of different lengths $L$ with a range of initial salinity gradients and temperature differences. A simple two-dimensional model is developed to describe the rate of extension of the cells and the internal density gradient as functions of time in which the tank length appears as an important variable. This effect does not seem to have been recognized previously. The rates of evolution in each run involve the time scale $\tau \,{=}\, L /( {C_H hN_0 })$, where $C_H \,{\approx}\, 10^{ - 2}$ is a heat transfer coefficient. The mean length of the cells $\skew2\bar {l}(t)$and the internal buoyancy frequency as functions of time are given by \[ \skew2\bar {l}(t) / L = t/\tau - ( t/2\tau)^2,\quad N = N_0 (1 - t / 2\tau ). \] Inversion of the first of these expressions results in $t/\tau \,{=}\, 2\,{-}\, 2\{ {1 - (\skew2\bar {l}(t) / L)}\}^{1 / 2}$ from which a time scale $\tau ^{ - 1}$ can be estimated. The measurements from individual runs when plotted in this way generally produce accurate straight lines as the model predicts, from which $C_H $ is found. This should be approximately the same for each run; the mean over all runs was found to be $9.3\,{\times}\, 10^{ - 3}$ with standard deviation 2.4$\,{\times}\, 10^{ - 3}$. The velocity scale of the intrusions at the beginning of an experiment is of order 10$^{ - 2}$ cm s$^{-1}$, for typical parameters of water at temperature 20 $^\circ$C, cooled wall temperature of 0 $^\circ$C and mean salinity of 5%.

The subgrid-scale (SGS) Prandtl number ($Pr$) is an important parameter in large-eddy simulation. Prior models often assume that the ‘$-5/3$’ inertial subrange scaling applies to the wavenumber range from 0 to $k_{\unicode[STIX]{x1D6E5}}$ (the wavenumber corresponding to the filter scale $\unicode[STIX]{x1D6E5}$) and yield a $Pr$ that is stability-independent and scale-invariant, which is inconsistent with experimental data and the results of dynamic models. In this study, the SGS Prandtl number is revisited by solving the co-spectral budgets of momentum and heat fluxes in an idealized but thermally stratified atmospheric surface layer. The SGS Prandtl number from the co-spectral budget model shows a strong dependence on the atmospheric stability and increases (decreases) as the atmosphere becomes stable (unstable), which is in good agreement with recent field experimental data. The dependence of $Pr$ on the filter scale is also captured by the co-spectral budget model: as the filter scale becomes smaller, the SGS Prandtl number decreases. Finally, the value of SGS Prandtl number under neutral conditions is shown to be caused by the dissimilarity between momentum and heat in the pressure decorrelation term and the flux transfer term. When the dissimilarity exists only in the flux transfer term, the fact that under neutral conditions the SGS Prandtl number is usually smaller than the turbulent Prandtl number for Reynolds-averaged Navier–Stokes simulations is an indication of a stronger spectral transfer coefficient for heat than for momentum. The model proposed in this study can be readily implemented.

We consider the flow of an electrically conducting fluid confined in a rotating spherical shell. The flow is driven by a directly imposed electromagnetic body force, created by the combination of an electric current flowing from the inner sphere to a ring-shaped electrode around the equator of the outer sphere and a separately imposed predominantly axial magnetic field. We begin by numerically computing the axisymmetric basic states, which consist of a strong zonal flow. We next compute the linear onset of non-axisymmetric instabilities, and fully three-dimensional solutions up to ten times supercritical. We demonstrate that an experimental liquid-sodium device 50 cm in diameter could achieve and exceed these parameter values.

This paper presents an extension of existing works dealing with the dynamics of a passive scalar in freely decaying isotropic turbulence, by accounting for a production mechanism of the passive scalar itself. The physically relevant case of the temperature dynamics in the presence of Joule heating via the dissipation of the turbulent kinetic energy is selected and analysed by theoretical and numerical means. In particular, the sensitivity of the temperature decay to the non-dimensional parameters Prandtl number ($\mathit{Pr}$) and Eckert number ($\mathit{Ec}$), the latter measuring the intensity of the internal energy production mechanism, is investigated. The time behaviour of the global quantities such as the temperature variance $ \overline{{\theta }^{2} } (t)$ and its destruction rate ${\varepsilon }_{\theta } (t)$ is analysed, and a detailed analysis of the temperature variance spectrum ${E}_{\theta } (k)$ is provided. In the case of a very strong heating mechanism, some important modifications of the temperature dynamics are observed. The time-decay-law exponents of the global physical quantities assume new values, which are governed only by features of the kinetic energy spectrum, while they depend on the shape of ${E}_{\theta } (k)$ in the classical free-decay case. The temperature variance spectrum ${E}_{\theta } (k)$ exhibits two new spectral ranges. One is a convective–production range such that ${E}_{\theta } (k)\propto {k}^{1/ 3} $ is observed for a finite time at all values of $\mathit{Pr}$. In the case of very diffusive fluids with $\mathit{Pr}\ll 1$, a convective–diffusive–production range with ${E}_{\theta } (k)\propto {k}^{- 7/ 3} $ is also detected.

The dynamics of multiple alternating oceanic jets has been studied in the presence of a simple bottom topography with constant slope in the zonal direction. A baroclinic quasi-geostrophic model forced with a horizontally uniform and vertically sheared background flow generates mesoscale eddies and jets that are tilted from the zonal direction and drift with constant speed. The governing dynamical equations are rewritten in a tilted frame of reference moving with the jets, and the cross-jet time-mean profiles of the linear and nonlinear stress terms are analysed. Here, the linear stress terms are present because of the zonally asymmetric topography. It is demonstrated that the linear dynamics controls the drift mechanism. Also, it is found that the drifting jets are directly forced by the imposed vertical shear, whereas the eddies oppose the jets, although this is limited to continuously forced dissipative systems. This role of the eddies is opposite to the one in the classical baroclinic model of stationary, zonally symmetric multiple jets. This is expected to be more generic in the ocean, which is zonally asymmetric nearly everywhere.

Jet noise is a significant contributor to aircraft noise, and on modern aircraft it is considerably enhanced at low frequencies by a closely installed wing. Recent research has shown that this noise increase is due to the scattering of jet instability waves by the trailing edge of the wing. Experimentalists have recently shown that noise can be reduced by using wings with swept trailing edges. To understand this mechanism, in this paper, we develop an analytical model to predict the installed jet noise due to the scattering of instability waves by a swept wing. The model is based on the Schwarzschild method and Amiet’s approach is used to obtain the far-field sound. The model can correctly predict both the reduction in installed jet noise and the change to directivity patterns observed in experiments due to the use of swept wings. The agreement between the model and experiment is very good, especially for the directivity at large azimuthal angles. It is found that the principal physical mechanism of sound reduction is due to destructive interference. It is concluded that in order to obtain an effective noise reduction, both the span and the sweep angle of the wing have to be large. Such a model can greatly aid in the design of quieter swept wings and the physical mechanism identified can provide significant insight into developing other innovative noise-reduction strategies.

By extracting unstable invariant solutions directly from body-forced three-dimensional turbulence, we study the dynamical processes at play when the forcing is large scale and unidirectional in either the momentum or the vorticity equations. In the former case, the dynamical processes familiar from recent work on linearly stable shear flows – variously called the self-sustaining process (Waleffe, Phys. Fluids, vol. 9 (4), 1997, pp. 883–900) or vortex–wave interaction (Hall & Smith, J. Fluid Mech., vol. 227, 1991, pp. 641–666; Hall & Sherwin, J. Fluid Mech., vol. 661, 2010, pp. 178–205) – are important even when the base flow is linearly unstable. In the latter case, where the forcing drives Taylor–Green vortices, a number of mechanisms are observed from the various types of periodic orbits isolated. In particular, two different transient growth mechanisms are discussed to explain the more complex states found.

A nonlinear theory for thermoacoustic waves in a gas-filled, narrow channel and pore subject to an axial temperature gradient is developed based on the fluid dynamical equations for an ideal gas. Under the narrow-tube approximation, three small parameters are introduced as asymptotic parameters, one being the ratio of a span length to a typical thickness of the thermoviscous diffusion layer, another the ratio of the typical propagation speed of thermoacoustic waves to an adiabatic sound speed and the final parameter is the ratio of the typical magnitude of a pressure disturbance to uniform pressure in a quiescent state. No thermal interaction between the gas and the solid wall is taken into account on assuming that the wall has a large heat capacity. Using the three small parameters, the fluid dynamical equations are approximated asymptotically to be reduced to a single nonlinear diffusion wave (advection) equation for an excess pressure. All field variables are determined consistently in terms of the excess pressure so as to satisfy the boundary conditions on the wall. Supposing a time-periodic solution to the equation derived, the mean value of the excess pressure over one period is examined. It is shown that while the mean vanishes in the linear theory, it decreases monotonically due to nonlinearity. It is also shown that mean values of the shear stress and the heat flux at the wall, as well as those of the vector fields of the mass and energy fluxes representing, respectively, acoustic and thermoacoustic streaming, are expressed in terms of the mean values of the products of the spatial and/or temporal pressure gradients, which are reduced to the spatial derivatives of the mean pressure.

The interactions between homogeneous turbulence and a planar shock wave are analytically investigated using rapid distortion theory (RDT). Analytical solutions in the solenoidal modes are obtained. Qualitative answers to unsolved questions in a report by Andreopoulos et al. (Annu. Rev. Fluid Mech., vol. 524, 2000, pp. 309–345) are provided within the linear theoretical framework. The results show that the turbulence kinetic energy (TKE) is increased after interaction with a shock wave and that the contributions to the amplification can be interpreted primarily as the combined effect of shock-induced compression, which is a direct consequence of the Rankine–Hugoniot relation, and the nonlinear effect, which is an indirect consequence of the Rankine–Hugoniot relation via the perturbation manner. For initial homogeneous axisymmetric turbulence, the amplification of the TKE depends on the initial degree of anisotropy. Furthermore, the increase in energy at high wavenumbers is confirmed by the one-dimensional spectra. The enstrophy is also increased; its increase is more significant than that of the TKE because of the significant increase in enstrophy at high wavenumbers. The vorticity components perpendicular to the shock-induced compressed direction are amplified more than the parallel vorticity component. These results strongly suggest that a high resolution is needed to obtain accurate results for the turbulence–shock-wave interaction. The integral length scales ($L$) and the Taylor microscales ($\unicode[STIX]{x1D706}$) are decreased for most cases after the interaction. However, $L_{22,3}(=\,L_{33,2})$ and $\unicode[STIX]{x1D706}_{22,3}(=\,\unicode[STIX]{x1D706}_{33,2})$ are amplified. Here, the subscripts 2 and 3 indicate the perpendicular components relative to the shock-induced compressed direction. The dissipation length and TKE dissipation rate are amplified.