A family of exact similarity solutions for inviscid compressible ablative flows in slab symmetry with nonlinear heat conduction is proposed for studying unsteadiness and compressibility effects on the hydrodynamic stability of ablation fronts relevant to inertial confinement fusion. Dynamical multi-domain Chebyshev spectral methods are employed for computing both the similarity solution and its time-dependent linear perturbations. This approach has been exploited to analyse the linear stability properties of two self-similar ablative configurations subjected to direct laser illumination asymmetries. Linear perturbation temporal and reduced responses are analysed, evidencing a maximum instability for illumination asymmetries of zero transverse wavenumber as well as three distinct regimes of ablation-front distortion evolution, and emphasizing the importance of the mean flow unsteadiness, compressibility and stratification.

The stability of a horizontal shear current under surface gravity waves is investigated on the basis of the Rayleigh equation. As the differential operator is non-normal, a standard modal analysis is not effective in capturing the transient growth of a perturbation. The representation of the stream function by a suitable basis of bi-orthogonal eigenfunctions allows one to determine the maximum growth rate of a perturbation. It turns out that, in the considered range of parameters, such a growth rate can be two orders of magnitude larger than the maximum eigenvalue obtained by standard modal analysis.

Fully developed two-dimensional salt-finger convection is characterized by the appearance of coherent dipolar eddies which carry relatively fresh and cold fluid upward and salty and warm fluid downward. Such structures – the double-diffusive modons – are prevalent in the regime in which density stratification is close to neutral and the salt-finger instability is extremely vigorous. The structure and translation velocities of modons are discussed in terms of the asymptotic expansion in which the background density ratio approaches unity. It is argued that the vertical salt flux is driven primarily by double-diffusive modons, which makes it possible to derive explicit expressions for the mixing rates of temperature and salinity as a function of their background gradients. Predictions of the proposed mixing model are successfully tested by direct numerical simulations.

Detailed observations have been performed on the evolution of a viscous catenary, a rope of high-viscosity fluid suspended from two points falling under gravity. Stroboscopic imaging techniques are used to obtain the position and shape of the strand as a function of time. Depending on their initial thickness and profile, the filaments are observed to evolve into either a quasi-catenary, or other, more complex shapes. A conceptually simple, energy-based theory is developed and compared with observations. It is shown to describe reasonably, except for a scaling in the time scale, the catenary-like regime.

A model is developed for turbulent natural convection in boundary layers formed next to isothermal vertical surfaces. A scaling analysis shows that the flow can be described by plume equations for an outer turbulent region coupled to a resolved near-wall laminar flow. On the laboratory scale, the inner layer is dominated by its own buoyancy and the Nusselt number scales as the one-third power of the Rayleigh number (Nu ∝ ). This gives a constant heat flux, consistent with previous experimental and theoretical studies. On larger geophysical scales the buoyancy is strongest in the outer layer and the laminar layer is driven by the shear imposed on it. The predicted heat transfer correlation then has the Nusselt number proportional to the one-half power of Rayleigh number (Nu ∝ ) so that a larger heat flux is predicted than might be expected from an extrapolation of laboratory-scale results. The criteria for transitions between flow regimes are consistent with a hierarchy of instabilities of the near-wall laminar flow, with a buoyancy-driven instability operating on the laboratory scale and a shear-driven instability operating on geophysical scales.

We study natural convection driven by unstable concentration differences of sodium chloride (NaCl) across a horizontal permeable membrane at Rayleigh numbers (Ra) of 1010 to 1011 and Schmidt number (Sc)=600. A layer of brine lies over a layer of distilled water, separated by the membrane, in square-cross-section tanks. The membrane is permeable enough to allow a small flow across it at higher driving potentials. Based on the predominant mode of transport across the membrane, three regimes of convection, namely an advection regime, a diffusion regime and a combined regime, are identified. The near-membrane flow in all the regimes consists of sheet plumes formed from the unstable layers of fluid near the membrane. In the advection regime observed at higher concentration differences (ΔC) across the membrane, there is a slow overturning through-flow across the membrane; the transport across the membrane occurs mostly by advection. This phenomenology explains the observed Nub~Ra2/Sc scaling of the Nusselt number. The planforms of sheet plumes near the membrane show a dendritic structure due to the combined influence of the mean shear due to the large-scale flow and the entrainment flow of the adjacent plumes. The near-membrane dynamics show initiation, elongation and merger of plumes; a movie is available with the online version of the paper. Increase in Ra results in a larger number of closely and regularly spaced sheet plumes. The mean plume spacing in the advection regime , is larger than the mean plume spacing in Rayleigh–Bénard convection (), and shows a different Ra-dependence. The plume spacings in the advection regime (λb) show a common log-normal probability density function at all Ra. We propose a phenomenology which predicts ~ , where Zw and are, respectively, the near-wall length scales in Rayleigh–Bénard convection (RBC) and due to the advection velocity. In the combined regime, which occurs at intermediate values of ΔC, the flux scales as (ΔC/2)4/3. At lower driving potentials, in the diffusion regime, the flux scaling is similar to that in turbulent RBC.

It is known that Kolmogorov's four-fifths law for statistically homogeneous and isotropic turbulence can be generalized to anisotropic turbulence. This fundamental result for homogeneous anisotropic turbulence says that in the inertial range the divergence of the vector third-order moment 〈|δv(r)|2δv(r)〉 is constant and is equal to -4ϵ, where ϵ is the dissipation rate of the turbulence. This law can be extended to incompressible magnetohydrodyamic (MHD) turbulence where statistical isotropy is often not valid due, for example, to the presence of a large-scale magnetic field. Laws for anisotropic incompressible MHD turbulence were first derived by Politano and Pouquet. In this paper, the laws for vector third-order moments in homogeneous non-isotropic incompressible MHD turbulence are derived by a technique due to Frisch that clarifies the relationship between the energy flux in Fourier space and the vector third-order moments in physical space. This derivation is different from the original derivation of Politano and Pouquet which is based on the Kármán–Howarth equation, and provides some new physical insights. Separate laws are derived for the cascades of energy, cross-helicity and magnetic-helicity, the three ideal invariants of incompressible MHD for flows in three dimensions. These laws are of fundamental importance in the theory of MHD turbulence where non-isotropic turbulence is much more prevalent than isotropic turbulence.

The fluctuating wall pressure and its gradients in the plane of the surface were measured beneath the turbulent boundary layer that forms over the salt playa of Utah's west desert. Measurements were acquired under the condition of near-neutral thermal stability to best mimic the canonical zero-pressure-gradient boundary-layer flow. The Reynolds number (based on surface-layer thickness, δ, and the friction velocity, uτ) was estimated to be 1 × 106 ± 2 × 105. The equivalent sandgrain surface roughness was estimated to be in the range 15≤ks+≤85. Pressure measurements acquired simultaneously from an array of up to ten microphones were analysed. A compact array of four microphones was used to estimate the instantaneous streamwise and spanwise gradients of the surface pressure. Owing to the large length scales and low flow speeds, attaining accurate pressure statistics in the present flow required sensors capable of measuring unusually low frequencies. The effects of imperfect spatial and temporal resolution on the present measurements were also explored. Relative to pressure, pressure gradients exhibit an enhanced sensitivity to spatial resolution. Their accurate measurement does not, however, require fully capturing the low frequencies that are inherent and significant in the pressure itself. The present pressure spectra convincingly exhibit over three decades of approximately −1 slope. Comparisons with low-Reynolds-number data support previous predictions that the inner normalized wall pressure variance increases logarithmically with Reynolds number. The wall pressure autocorrelation exhibits its first zero-crossing at an advected length that is between one tenth and one fifth of the surface-layer thickness. Under any of the normalizations investigated, the present surface vorticity flux intensity values are difficult to reconcile with low-Reynolds-number data trends. Inner variables, however, do yield normalized flux intensity values that are of the same order of magnitude at low and high Reynolds number. Spectra reveal that even at high Reynolds number, the primary contributions to the pressure gradient intensities occur over a relatively narrow frequency range. This frequency range is shown to be consistent with the scale of the sublayer pocket motions. In accord with low-Reynolds-number data, the streamwise pressure gradient signals at high Reynolds number are also characterized by statistically significant pairings of opposing sign fluctuations.

We solve numerically for the first time the two-fluid Hall–Vinen–Bekarevich–Khalatnikov (HVBK) equations for an He-II-like superfluid contained in a differentially rotating spherical shell, generalizing previous simulations of viscous spherical Couette flow (SCF) and superfluid Taylor–Couette flow. The simulations are conducted for Reynolds numbers in the range 1 × 102≤Re≤3 × 104, rotational shear 0.1≤ΔΩ/Ω≤0.3, and dimensionless gap widths 0.2≤δ≤0.5. The system tends towards a stationary but unsteady state, where the torque oscillates persistently, with amplitude and period determined by δ and ΔΩ/Ω. In axisymmetric superfluid SCF, the number of meridional circulation cells multiplies as Re increases, and their shapes become more complex, especially in the superfluid component, with multiple secondary cells arising for Re > 103. The torque exerted by the normal component is approximately three times greater in a superfluid with anisotropic Hall–Vinen (HV) mutual friction than in a classical viscous fluid or a superfluid with isotropic Gorter–Mellink (GM) mutual friction. HV mutual friction also tends to ‘pinch’ meridional circulation cells more than GM mutual friction. The boundary condition on the superfluid component, whether no slip or perfect slip, does not affect the large-scale structure of the flow appreciably, but it does alter the cores of the circulation cells, especially at lower Re. As Re increases, and after initial transients die away, the mutual friction force dominates the vortex tension, and the streamlines of the superfluid and normal fluid components increasingly resemble each other. In non-axisymmetric superfluid SCF, three-dimensional vortex structures are classified according to topological invariants. For misaligned spheres, the flow is focal throughout most of its volume, except for thread-like zones where it is strain-dominated near the equator (inviscid component) and poles (viscous component). A wedge-shaped isosurface of vorticity rotates around the equator at roughly the rotation period. For a freely precessing outer sphere, the flow is equally strain- and vorticity-dominated throughout its volume. Unstable focus/contracting points are slightly more common than stable node/saddle/saddle points in the viscous component, but not in the inviscid component. Isosurfaces of positive and negative vorticity form interlocking poloidal ribbons (viscous component) or toroidal tongues (inviscid component) which attach and detach at roughly the rotation period.

The stability and dynamics of an axisymmetric lifted flame are studied by means of direct numerical simulation (DNS) and linear stability analysis of the reacting low-Mach-number equations. For light fuels (such as non-premixed methane/air flames), the non-reacting premixing zone upstream of the lifted flame base contains a pocket of absolute instability supporting self-sustaining oscillations, causing flame flicker even in the absence of gravity. The liftoff heights of the unsteady flames are lower than their steady counterparts (obtained by the method of selective frequency damping (SFD)), owing to premixed flame propagation during a portion of each cycle. From local stability analysis, the lifted flame is found to have a significant stabilizing influence at and just upstream of the flame base, which can truncate the pocket of absolute instability. For sufficiently low liftoff heights, the truncated pocket of absolute instability can no longer support self-sustaining oscillations, and the flow is rendered globally stable.

In the context of geologic sequestration of carbon dioxide in saline aquifers, much interest has been focused on the process of density-driven convection resulting from dissolution of CO2 in brine in the underlying medium. Recent investigations have studied the time and length scales characteristic of the onset of convection based on the framework of linear stability theory. It is well known that the non-autonomous nature of the resulting matrix does not allow a normal mode analysis and previous researchers have either used a quasi-static approximation or solved the initial-value problem with arbitrary initial conditions. In this manuscript, we describe and use the recently developed non-modal stability theory to compute maximum amplifications possible, optimized over all possible initial perturbations. Non-modal stability theory also provides us with the structure of the most-amplified (or optimal) perturbations. We also present the details of three-dimensional spectral calculations of the governing equations. The results of the amplifications predicted by non-modal theory compare well to those obtained from the spectral calculations.

We present the results of a combined theoretical and experimental study of the stability of a uniformly stratified fluid bounded by a sidewall moving vertically with constant velocity. This arrangement is perhaps the simplest in which boundary effects can drive instability and, potentially, layering in a stratified fluid. Our investigations reveal that for a given stratification and diffusivity of the stratifying agent, the sidewall boundary-layer flow becomes linearly unstable when the wall velocity exceeds a critical value. The onset of instability is clearly observed in the experiments, and there is good quantitative agreement with some predictions of the linear stability analysis.

As part of a long-range study of vortex rings, their dynamics, interactions with boundaries and with each other, we present the results of experiments on thin core rings generated by a piston gun in water. We characterize the dynamics of these rings by means of the traditional equations for such rings in an inviscid fluid suitably modifying them to be applicable to a viscous fluid. We develop expressions for the radius, core size, circulation and bubble dimensions of these rings. We report the direct measurement of the impulse of a vortex ring by means of a physical pendulum.

The interaction of two passive scalars (both temperature in air) emitted from concentrated line sources in fully developed high-aspect-ratio turbulent channel flow is studied. The thermal fields are measured using cold-wire thermometry in a flow with a Reynolds number (Uh/ν) of 10200.

The transverse total root-mean-square (RMS) temperature profiles are a function of the separation distance between the line sources (d/h), their average wall-normal position (ysav/h), and the downstream location (x/h), measured relative to the line sources. Similarly, profiles of the non-dimensional form of the scalar covariance, the correlation coefficient (ρ), are a function of the same parameters and quantify the mixing of the two scalars.

The transverse profiles of the correlation coefficient are generally largest at the edges of the thermal plume and smallest in its core. When the line sources are not symmetrically located about the channel centreline, the minimum in the correlation coefficient transverse profiles drifts towards the (closer) channel wall. For source locations that are equidistant from the channel centreline, the minimum correlation coefficient occurs at the centreline, due to the underlying symmetry of this geometry. The initial downstream evolution of the correlation coefficient depends significantly on d/h, similar to that in homogeneous turbulence. However, there is always a dependence on ysav/h, which increases in importance as both the downstream distance is increased and the wall is approached. Lastly, the correlation coefficient profiles tend towards positive values in the limit of large downstream distances (relative to the source separation), though further measurements farther downstream are required to confirm the exact value(s) of their asymptotic limit(s).

Spectral analysis of the cospectra and coherency spectra indicates that the large scales evolve more rapidly than the small ones. Furthermore, the fast evolution of the large scales was most evident when the sources were located close to the wall. This presumably derives from the large-scale nature of turbulence production, which is strong in the near-wall region.

The steady axisymmetric flow of viscous liquid relative to a gas bubble due to its buoyancy-driven motion in a round tube is computed by solving the nonlinear Navier–Stokes equations using a Galerkin finite-element method with a boundary-fitted mesh. When the bubble is relatively small compared with the tube size (e.g. the volume-equivalent radius of the bubble is less than a quarter of the tube radius R), the bubble exhibits similar behaviour to one moving in an extended liquid, developing a spherical-cap shape with increasing Reynolds number (Re) if the capillary number is not too small. The long-bubble (also known as a Taylor bubble) characteristics can be observed with bubbles of volume-equivalent radius greater than the tube radius, especially when the surface tension effect is relatively weak (e.g. for Weber number We greater than unity). The computed values of Froude number Fr for most cases agree well with the correlation formulae derived from experimental data for long bubbles, and even with (short) bubbles of volume-equivalent radius three-quarters of the tube radius. All of the computed surface profiles of long bubbles exhibit a prolate-like nose shape, yet various tail shapes can be obtained by adjusting the parameter values of Re and We. At large Weber number (e.g. We=10), the bubble tail forms a concave profile with a gas ‘cup’ developed at small Re and a ‘skirt’ at large Re with sharply curved rims. For We≤1, the bubble tail profile appears rounded without large local curvatures, although a slightly concave tail may develop at large Re. non-uniform annular film adjacent to the tube wall is commonly observed when Weber number is small, especially for bubbles of volume <3πR3, suggesting that the surface tension effect can play a complicated role. Nonetheless the computed value of Fr is found to be generally independent of the bubble length for bubbles of volume-equivalent radius greater than the tube radius. If the bubble length reaches about 2.5 tube radii, the value of its frontal radius becomes basically the same as that for long bubbles of much larger volume. An examination of the distribution of the z-component of traction along the bubble surface reveals the basic mechanism for long bubbles rising at a terminal velocity that is independent of bubble volume.