A weakly nonlinear theory, based on the combined amplitude–multiple timescale expansion, is developed for the flow of an arbitrary fluid governed by the low-Mach-number equations. The approach is shown to be different from the one conventionally used for Boussinesq flows. The range of validity of the applied analysis is discussed and shown to be sufficiently large. Results are presented for the natural convection flow of air inside a closed differentially heated tall vertical cavity for a range of temperature differences far beyond the region of validity of the Boussinesq approximation. The issue of possible resonances of two different types is noted. The character of bifurcations for the shear- and buoyancy-driven instabilities and their interaction is investigated in detail. Lastly, the energy transfer mechanisms are analysed in supercritical regimes.

Electroconvection in a layer of liquid subjected to unipolar injection is characterized by two stability criteria, a linear and a nonlinear one, with an associated hysteresis loop. Experimentally it is found that the velocity field fluctuates around its mean value. A temporal analysis of the measured current, which is directly related to the velocity, revealed the existence of a well-defined frequency correlated to the mean rotation time of a fluid particle in the convective cell, thus indicating that these fluctuations are not stochastic but related to the intrinsic dynamics of the system. Here a method of superparticles is used to solve the problem of the non-stationary electroconvection of the liquid. A good agreement between theoretical and experimental results is obtained.

The effects of wall blowing or suction on the stability characteristics of a laminar incompressible two-dimensional plane wall jet are investigated both experimentally and theoretically. A quantitative comparison between linear stability calculations and phase-locked experimental data, obtained when the wall jet is subjected to two-dimensional excitations, confirms the co-existence of the viscous and inviscid instability modes and the theoretically predicted effects of blowing and suction on the stability of the wall jet. According to these predicted effects, blowing stabilizes the inviscid mode while destabilizing the viscous one; suction has the opposite effect. Furthermore, blowing and suction tend to increase and decrease, respectively, the ratio between the outer and inner amplitude maxima of the streamwise velocity fluctuation. When wall blowing is applied, the instability domain is enlarged and includes higher-frequency waves. In addition, the region where both unstable modes co-exist simultaneously begins at a lower local Reynolds number. Opposite effects are caused when suction is applied. The quantitative comparison between the theory and experiment includes the cross-stream structure and the downstream growth of the streamwise velocity fluctuations. In order to accurately account for the effect of the mean flow divergence in the stability analysis, the second-order corrections to the mean flow solutions are obtained for all wall conditions. Spectral distributions, obtained when natural wall-jets are subjected to blowing and suction, support qualitatively the above results.

We present simulation results of flow-induced vibrations of an infinitely long flexible cable at Reynolds numbers Re = 100 and Re = 200, corresponding to laminar and early transitional flow states, respectively. The question as to what cable motions and flow patterns prevail is investigated in detail. Both standing wave and travelling wave responses are realized but in general the travelling wave is the preferred response. A standing wave cable response produces an interwoven pattern of vorticity, while a travelling wave cable response produces oblique vortex shedding. A sheared inflow produces a mixed standing wave/travelling wave cable response and chevron-like patterns with vortex dislocations in the wake. The lift force on the cable as well as its motion amplitudes are larger for the standing wave response. At Re = 200, the cable and wake response are no longer periodic, and the maximum amplitude of the cable is about one cylinder diameter, in agreement with experimental results.

The plane Couette system does not exhibit any secondary solutions bifurcating from the primary solution of constant shear. Since the work of Nagata (1990) it has been well known that three-dimensional steady solutions exist. Here the manifold of those steady solutions is explored in the parameter space of the problem and their instabilities are investigated. These instabilities usually lead to time-periodic solutions whose properties do not differ much from those of the steady solutions except that the amplitude varies in time. In some cases travelling wave solutions which are asymmetric with respect to the midplane of the layer are found as quaternary states of flow. Similarities with longitudinal vortices recently observed in experiments are discussed.

The modelling of anisotropies in the dissipation rate of turbulence is considered based on an analysis of the exact transport equation for the dissipation rate tensor. An algebraic model is systematically derived using integrity bases methods and tensor symmetry properties. The new model differs notably from all previously proposed models in that it depends nonlinearly on the mean velocity gradients. This gives rise to a transport equation for the scalar dissipation rate that is of the same general form as the commonly used model with one major exception: the coefficient of the production term is dependent on the invariants of both the rotational and irrotational strain rates. The relationship between the new model and other recently proposed models is examined in detail. Some basic tests and applications of the model are also provided along with a discussion of the implications for turbulence modelling.

We present a formal solution to the initial value problem for small perturbations of a straight vortex tube with constant vorticity, and show that any initial perturbation to such a tube evolves exclusively as a collection of Kelvin vortex waves. We then study in detail the evolution of the following particular initial states of the vortex tube: (i) an axisymmetric pinch in the radius of the tube, (ii) a deflection in the location of the tube, and (iii) a flattening of the tube's cross-secton. All of these initial states are localized in the direction along the tube by weighting them with a Gaussian function. In each case, the initial perturbation is decomposed into packets of Kelvin vortex waves which then propagate outward along the vortex tube. We discuss the physical mechanisms responsible for the propagation of the wave packets, and study the consequences of wave dispersion for the solution.

Constant-density electrically conducting fluid is confined to a rapidly rotating spherical shell and is permeated by an axisymmetric potential magnetic field with dipole parity; the regions outside the shell are rigid insulators. Slow steady axisymmetric motion is driven by rotating the inner sphere at a slightly slower rate. Linear solutions of the governing magnetohydrodynamic equations are derived in the small Ekman number E-limit for values of the Elsasser number Λ less than order unity. Attention is restricted to the mainstream outside the Ekman–Hartmann layers adjacent to the inner and outer boundaries.

The low-Reynolds-number collision and rebound of two rigid spheres moving in an ideal isothermal gas is studied in the lubrication limit. The spheres are non-Brownian in nature with radii much larger than the mean-free path of the molecules. The nature of the flow in the gap between the particles depends on the relative magnitudes of the minimum gap thickness, h′o, the mean-free path of the bulk gas molecules, λo, and the gap thickness at which compressibility effects become important, hc. Both the compressible nature of the gas and the non-continuum nature of the flow in the gap are included and their effects are studied separately and in combination. The relative importance of these two effects is characterized by a dimensionless number, αo≡ (hc/λo). Incorporation of these effects in the governing equations leads to a partial differential equation for the pressure in the gap as a function of time and radial position. The dynamics of the collision depend on αo, the particle Stokes number, Sto, and the initial particle separation, h′o. While a continuum incompressible lubrication force applied at all separations would prevent particle contact, the inclusion of either non-continuum or compressible effects allows the particles to contact. The critical Stokes number for particles to make contact, St1, is determined and is found to have the form St1= 2 [ln(h′o/l) +C(αo)], where C(αo) is an O(1) quantity and l is a characteristic length scale defined by l≡ hc(1+αo)/ αo. The total energy dissipated during the approach and rebound of two particles when Sto[Gt ]St1 is also determined in the event of perfectly elastic or inelastic solid-body collisions.

Some simple but exact general expressions are derived for the viscous stresses required at the surface of irrotational capillary–gravity waves of periodic or solitary type on deep water in order to maintain them in steady motion. These expressions are applied to nonlinear capillary waves, and to capillary–gravity waves of solitary type on deep water. In the case of pure capillary waves some algebraic expressions are found for the work done by the surface stresses, from which it is possible to infer the viscous rate of decay of free, nonlinear capillary waves.

Similar calculations are carried out for capillary–gravity waves of solitary type on deep water. It is shown that the limiting rate of decay of a solitary wave at low amplitudes is just twice that for linear, periodic waves. This is due to the spreading out of the wave envelope at low wave steepnesses. At large wave steepnesses the dissipation increases by an order of magnitude, owing to the sharply increased curvature in the wave troughs. The calculated rates of decay are in agreement with recent observations.

We describe a series of laboratory experiments in which aqueous salt solutions were cooled and solidified from above. These solutions serve as model systems of metallic castings, magma chambers and sea ice. As the solutions freeze they form a matrix of ice crystals and interstitial brine, called a mushy layer. The brine initially remains confined to the mushy layer. Convection of brine from the interior of the mushy layer begins abruptly once the depth of the layer exceeds a critical value. The principal path for brine expelled from the mushy layer is through ‘brine channels’, vertical channels of essentially zero solid fraction, which are commonly observed in sea ice and metallic castings. By varying the initial and boundary conditions in the experiments, we have been able to determine the parameters controlling the critical depth of the mushy layer. The results are consistent with the hypothesis that brine expulsion is initially determined by a critical Rayleigh number for the mushy layer. The convection of salty fluid out of the mushy layer allows additional solidification within it, which increases the solid fraction. We present the first measurements of the temporal evolution of the solid fraction within a laboratory simulation of growing sea ice. We show how the additional growth of ice within the layer affects its rate of growth.

Swept-wedge flows are used to study the effects of pressure gradient and flow angle on the stability of three-dimensional laminar boundary layers. It is shown that the flow is absolutely unstable in the chordwise direction, i.e. disturbances grow in time at every chordwise location, for certain parameter combinations. However, laminar–turbulent transition may still be a convective process.

We derive a constitutive relation, relating the tangential stress, tangential velocity, thickness h, and viscosity μ, for a thin layer of Newtonian fluid on top of a fluid substrate. We find that the upper layer exerts a viscous tangential shear stress on the lower fluid, behaving as if it were a film with a two-dimensional shear viscosity equal to μh, and a dilatational viscosity 3μh.

This work is devoted to the decay of random solutions of the unforced Burgers equation in one dimension in the limit of vanishing viscosity. The initial velocity is homogeneous and Gaussian with a spectrum proportional to kn at small wavenumbers k and falling off quickly at large wavenumbers.