An analysis is presented of the shock-wave configurations which will occur when a plane shock is incident on a double wedge for which the second wedge may have a greater (concave case) or a smaller (convex case) inclination than the first wedge. It is shown that seven different reflection processes may be expected depending on the Mach number of the incidnet shock Mi and the two wedge angles θ1w and θ2w. These processes may be defined by seven regiouns in the (θ1w,θ2w)-plane, for a given value of Mi. Each of the seven processes has been verified by sequences of shadowgraph and schlieren photographs.

A shock-polar analysis of each of the seven processes has provided infomation about the pressure changes and the wave structures which develop immediately behind the main reflections along the wedge surfaces. These wave structures have been verified experimentally, and two types have been observed; one normal to the reflecting surface, and the other in the form of a regular reflection. The criteria to determine which of thses configurations will occur have not yet been established.

It is believed that the present study will be of value in predicting the loading of shock waves on structures, and may lead to a better understanding of shock reflections form concave and convex cylindrical surfaces.

The separation of pairs of particles within turbulent flow fields constructed using the ‘kinematic simulation’ method is explored. A consequence of the way the flow is constructed is that, in contrast to real turbulence, there is no ‘sweeping’ of the smaller eddies by the larger eddies. The implications of this are investigated. A simple phenoenological argument is presented which predicts that the mean-square separation of the particle pairs should grow like $t^6$ in kinematic simulation. Simulations support this result for the case where a large mean velocity is added to the flow to exaggerate the sweeping problem and the inertial subrange is sufficiently long. In the absence of a large mean velocity, the situation is more complex with the simple phenomenological argument failing in the parts of the flow where the velocity is much smaller than the r.m.s. velocity and where there is no sweeping problem. The separation process then follows $t^6$ in the bulk of the flow but follows Richardson's classical $t^3$ law in regions where the velocity is much smaller than the r.m.s. velocity. Because of the way the size of these regions varies in time, the resulting mean-square separation grows like $t^{9/2}$. Both the $t^6$ and $t^{9/2}$ behaviours contrast with the classical Richardson $t^3$ law, which is believed to hold in reality, and raise questions about the applicability of the kinematic simulation approach to the separation of pairs in real turbulent flows.

We examine various perturbations of Jeffery-Hamel flows, the exact solutions of the Navier-Stokes equations governing the steady two-dimensional motions of an incompressible viscous fluid from a line source at the intersection of two rigid plane walls. First a pitchfork bifurcation of the Jeffery-Hamel flows themselves is described by perturbation theory. This description is then used as a basis to investigate the spatial development of arbitrary small steady two-dimensional perturbations of a Jeffery-Hamel flow; both linear and weakly nonlinear perturbations are treated for plane and nearly plane walls. It is found that there is strong interaction of the disturbances up- and downstream if the angle between the planes exceeds a critical value 2α2, which depends on the value of the Reynolds number. Finally, the problem of linear temporal stability of Jeffery-Hamel flows is broached and again the importance of specifying conditions up- and downstream is revealed. All these results are used to interpret the development of flow along a channel with walls of small curvature. Fraenkel's (1962) approximation of channel flow locally by Jeffery-Hamel flows is supported with the added proviso that the angle between the two walls at each station is less than 2α2.

Gravity current and dam-break flows, resulting from the instantaneous release of fluid initially at rest behind a lock gate, are modelled theoretically using the shallow water equations. By analysing the motion in the hodograph plane, the governing equations become linear and hence it is possible to integrate them analytically from lock-release initial conditions. This approach provides many advantages: not only are numerical computations obviated, but the analysis clearly reveals how the nature of the ensuing flow depends on the Froude number, $\hbox{\it Fr}$, at the front of the current. It is also demonstrated that the motion comprises uniform and simple wave regions within which both or one of the characteristic variables are constant, respectively, in addition to complex wave regions within which both characteristic variables vary. These solutions reveal phenomena that have not previously been reported for gravity current flow. Specifically, when $\hbox{\it Fr}\,{>}\,2$, the height and velocity fields become discontinuous at late times at an interior point within the current. Conversely, when $\hbox{\it Fr}\,{<}\,2$, there is a wave-like disturbance that propagates along the length of the current, being reflected successively between the rear wall of the lock and the front of the flow.

Velocity moments have been calculated for the rapid distortion of axisymmetric turbulence in a uniform mean shear. The moments are compared with data on steady pipe flows and two-dimensional channel flows, and the turbulence structure of these flows is summarized in terms of effective distortion strain. The centre-line structure is taken to be characteristic of the undistorted state which from the data is more nearly axisymmetric rather than isotropic. A closer comparison is found than that by Townsend (1970), and in particular the differences in stress ratios $\tau/\rho\overline{u^2_1},\tau/\rho\overline{q^2} $ found between different experiments can be accounted for with the hypothesis of initially axisymmetric turbulence. Profiles for the effective strain are derived from the experiments and are shown to have the same form and to indicate the existence of a relaxation timescale for the large eddies, comparable to the energy decay timescale. An equation for the effective distortion strain is formulated that can be incorporated into a turbulence model.

The splitting of a single pressure discontinuity into a propagating two-wave system is studied for the case of saturated-liquid expansion (liquid-evaporation wave splitting) and vapour compression (vapour-condensation wave splitting). Experimental results from the Max-Planck-Institut für Strömungsforschung and from Rensselaer Polytechnic Institute show that splitting occurs in test fluids of large molar heat capacity, such as iso-octane (Cv0/R ≈ 37). Each of the two forms of splitting results in a single-phase forerunner wave carrying a pressure discontinuity followed by a phase-change wave, also with a pressure discontinuity. The thermodynamic state between the forerunner wave and the phase-change wave is metastable (supersaturated liquid or vapour). The waves are quantitatively described by systems of adiabats, e.g. shock adiabats. It appears that nucleation processes are predominantly homogeneous.

In vapour-compression shock-wave splitting, a combined wave (liquefaction shock) splits into discrete forerunner and condensation waves at a triple point, the intersection of a liquefaction shockfront, forerunner shock and condensation discontinuity: such a point occurs just at critical supersaturation (i.e. the Wilson-line state), where condensation is spontaneous and immediate. For shock waves that produce a metastable state of subcritical supersaturation, condensation is delayed, that is, the condensation discontinuity propagates more slowly; for a split-shock system, the condensation discontinuity propagates subsonically. The pressure amplitude of a real split-shock system is much larger than that predicted by an equilibrium model.

In liquid-evaporation wave splitting, the forerunner wave is an acoustic expansion wave and the second wave an evaporation wave with a propagation velocity approximately determined by the Chapman-Jouguet condition for deflagration. Such evaporation wavefronts are increasingly distinct as the temperature approaches the critical-point value. The evaporation rates across the wavefront are comparable to those found in vapour explosions.

We investigate the scattering of a plane acoustic wave by an axisymmetric vortex in two dimensions. We consider vortices with localized vorticity, arbitrary circulation and small Mach number. The wavelength of the acoustic waves is assumed to be much longer than the scale of the vortex. This enables us to define two asymptotic regions: an inner, vortical region, and an outer, wave region. The solution is then developed in the two regions using matched asymptotic expansions, with the Mach number as the expansion parameter. The leading-order scattered wave field consists of two components. One component arises from the interaction in the vortical region, and takes the form of a dipolar wave. The other component arises from the interaction in the wave region. For an incident wave with wavenumber k propagating in the positive X-direction, a steepest descents analysis shows that, in the far-field limit, the leading-order scattered field takes the form i(π−θ)eikX+½cosθcot(½θ) (2π/kR)1/2ei(kR−π/4), where θ is the usual polar angle. This expression is not valid in a parabolic region centred on the positive X-axis, where kRθ2=O(1). A different asymptotic solution is appropriate in this region. The two solutions match onto each other to give a leading-order scattering amplitude that is finite and single-valued everywhere, and that vanishes along the X-axis. The next term in the expansion in Mach number has a non-zero far-field response along the X-axis.

A detailed study into the rheology and microstructure of dense suspensions of initially spherical capsules is presented, where capsules are composed of a fluid-filled interior surrounded by an elastic membrane. This study couples a lattice-Boltzmann fluid solver to a finite-element membrane model creating a robust and scalable method for the simulation of these suspensions. A Lees–Edwards boundary condition is used to simulate periodic simple shear to obtain bulk rheological properties, and three-dimensional results are presented for capsules in the regime of negligible inertia, Brownian motion and colloidal interparticle forces. The simulation results focus on describing the suspension rheology as a function of the particle concentration and deformability, and relating these macroscopic rheological findings to changes at the particle level, i.e. the suspension microstructure. Several important findings are made: suspensions of deformable capsules are found to be shear thinning, and the initially compressive normal stresses associated with rigid spherical suspensions undergo rapid changes with moderate levels of particle deformation. These normal stress changes are particularly evident in the first normal stress difference, which undergoes a sign change at fairly minor levels of deformation, and the particle pressure, which decreases rapidly with increasing particle deformability. Changes in the microstructure as quantified by the single-body microstructure and the pair distribution function are reported. Also, results calculating particle self-diffusion are presented and related to changes in the normal stresses.

We present the effect of noise on the hysteresis characteristics of a prototypical thermoacoustic system, a horizontal Rijke tube. As we increase the noise intensity, we find that the width of the hysteresis zone decreases. However, we find that the rate of decrease in hysteresis width is constant for all the mass flow rates considered in the present study. We also show that the subcritical transition observed in the absence of noise is no longer discernible once the intensity of noise is above a threshold value and the transition appears to be continuous. We compare our experimental observations with the results obtained from a numerical model perturbed with additive Gaussian white noise and we find a qualitative agreement between the experimental and the numerical results.

Direct numerical simulations of the Taylor–Couette (TC) problem, the flow between two coaxial and independently rotating cylinders, have been performed. The study focuses on TC flow with mild curvature (small gap) with a radius ratio of ${\it\eta}=r_{i}/r_{o}=0.909$, an aspect ratio of ${\it\Gamma}=L/d=2{\rm\pi}/3$, and a stationary outer cylinder. Three inner cylinder Reynolds numbers of $1\times 10^{5}$, $2\times 10^{5}$ and $3\times 10^{5}$ were simulated, corresponding to frictional Reynolds numbers between $Re_{{\it\tau}}\approx 1400$ and $Re_{{\it\tau}}\approx 4000$. An additional case with a large gap, ${\it\eta}=0.5$ and driving of $Re=2\times 10^{5}$ was also investigated. Small-gap TC was found to be dominated by spatially fixed large-scale structures, known as Taylor rolls (TRs). TRs are attached to the boundary layer, and are active, i.e. they transport angular velocity through Reynolds stresses. An additional simulation was also conducted with inner cylinder Reynolds number of $Re=1\times 10^{5}$ and fixed outer cylinder with an externally imposed axial flow of comparable strength to the wind of the TRs. The axial flow was found to convect the TRs without any weakening effect. For small-gap TC flow, evidence was found for the existence of logarithmic velocity fluctuations, and of an overlap layer, in which the velocity fluctuations collapse in outer units. Profiles consistent with a logarithmic dependence were also found for the angular velocity in large-gap TC flow, albeit in a very reduced range of scales. Finally, the behaviour of both small- and large-gap TC flow was compared to other canonical flows. Small-gap TC flow has similar behaviour in the near-wall region to other canonical flows, while large-gap TC flow displays very different behaviour.

A detailed study of the intermittency in the outer region of a flat-plate turbulent boundary layer has been carried out using digital sampling and processing techniques. Conditional averages are used to generate mean and fluctuating components for the turbulent and non-turbulent zones of fluid. More particularly, point averages of these variables, taken with reference to the instantaneous position of the turbulent/non-turbulent interface, have been made to show the distribution of various quantities through the turbulent front. The results indicate that significant differences exist at leading and trailing edges of the turbulent bursts and a more complete picture of the motion of an average large eddy is deduced.

Complex bioconvection patterns are observed when a suspension of the oxytactic bacterium Bacillus subtilis is placed in a chamber with its upper surface open to the atmosphere. The patterns form because the bacteria are denser than water and swim upwards (up an oxygen gradient) on average. This results in an unstable density distribution and an overturning instability. The pattern formation is dependent on depth and experiments in a tilted chamber have shown that as the depth increases the first patterns formed are hexagons in which the fluid flows down in the centre.

The linear stability of this system was analysed by Hillesdon & Pedley (1996) who found that the system is unstable if the Rayleigh number Γ exceeds a critical value, which depends on the wavenumber k of the disturbance as well as on the values of other parameters. Hillesdon & Pedley found that the critical wavenumber kc could be either zero or non-zero, depending on the parameter values.

In this paper we carry out a weakly nonlinear analysis to determine the relative stability of hexagon and roll patterns formed at the onset of bioconvection. The analysis is different in the two cases kc≠0 and kc=0. For the kc≠0 case (which appears to be more relevant experimentally) the model does predict down hexagons, but only for a certain range of parameter values. Hence the analysis allows us to refine previous parameter estimates. For the kc=0 case we carry out a two-dimensional analysis and derive an equation describing the evolution of the horizontal planform function.

A nonlinear variable-coefficient representation for the rapid-pressure covariance appearing in the Reynolds stress and heat-flux equations, consistent with the Taylor–Proudman theorem, is presented. The representation ensures that the modelled second-order equations are frame indifferent with respect to rotation in a number of different flows for which such an invariance is required. The model coefficients are functions of the state of the turbulence; they are valid for all states of a mechanical turbulence, attaining their limiting values only when the limit state is achieved. This is accomplished by a special ansatz that is used to obtain – analytically – the coefficients valid away from the realizability limit. Unlike other rapid-pressure representations in which extreme states are used to set model constants, here the coefficients are variable functions asymptotically consisted with – not fixed by – the limit states of the turbulence field. The mathematical principles invoked do not specify all the coefficients in the model; undetermined coefficients appear as free parameters which are used to ensure that the representation is asymptotically consistent with an experimentally determined equilibrium state of homogeneous sheared turbulence. This is done by ensuring that the modelled evolution equations have the same fixed points as those obtained from numerical and laboratory experiments for the homogeneous shear. Results of computations of homogeneous shear, with rotation and with curvature, are shown. Results are better, in a wide class of planar flows for which the model has not been calibrated, than those of other nonlinear models.

Topological chaos may be used to generate highly effective laminar mixing in a simple batch stirring device. Boyland, Aref & Stremler (2000) have computed a material stretch rate that holds in a chaotic flow, provided it has appropriate topological properties, irrespective of the details of the flow. Their theoretical approach, while widely applicable, cannot predict the size of the region in which this stretch rate is achieved. Here, we present numerical simulations to support the observation of Boyland et al. that the region of high stretch is comparable with that through which the stirring elements move during operation of the device. We describe a fast technique for computing the velocity field for either inviscid, irrotational or highly viscous flow, which enables accurate numerical simulation of dye advection. We calculate material stretch rates, and find close agreement with those of Boyland et al., irrespective of whether the fluid is modelled as inviscid or viscous, even though there are significant differences between the flow fields generated in the two cases.

The temporal evolution of small three-dimensional disturbances on viscous flows between parallel walls is studied. The initial-value problem is formally solved by using Fourier–Laplace transform techniques. The streamwise velocity component is obtained as the solution of a forced problem. As a consequence of the three-dimensionality, a resonant response is possible, leading to algebraic growth for small times. It occurs when the eigenvalues of the Orr–Sommerfeld equation coincide with the eigenvalues of the homogeneous operator for the streamwise velocity component. The resonance has been investigated numerically for plane Couette flow. The phase speed of the resonant waves equals the average mean velocity. The wavenumber combination that leads to the largest amplitude corresponds to structures highly elongated in the streamwise direction. The maximum amplitude, and the time to reach this maximum, scale with the Reynolds number. The aspect ratio of the most rapidly growing wave increases with the Reynolds number, with its spanwise wavelength approaching a constant value of about 3 channel heights.

By tracking small particles in the bulk of an intensely turbulent laboratory flow, we study the effect of long-chain polymers on the Eulerian structure functions. We find that the structure functions are modified over a wide range of length scales even for very small polymer concentrations. Their behaviour can be captured by defining a length scale that depends on the solvent viscosity, the polymer relaxation time and the Weissenberg number. This result is not captured by current models. Additionally, the effects we observe depend strongly on the concentration. While the dissipation-range statistics change smoothly as a function of polymer concentration, we find that the inertial-range values of the structure functions are modified only when the concentration exceeds a threshold of approximately 5 parts per million (p.p.m.) by weight for the 18 × 106 atomic mass unit (a.m.u.) molecular weight polyacrylamide used in the experiment.

We investigate source mechanisms for subsonic jet noise using experimentally obtained datasets of high-Reynolds-number Mach 0.4 and 0.6 turbulent jets. The focus is on the axisymmetric mode which dominates downstream sound radiation for low polar angles and the frequency range at which peak noise occurs. A linearized Euler equation (LEE) solver with an inflow boundary condition is used to generate single-frequency hydrodynamic instability waves, and the resulting near-field fluctuations and far-field acoustics are compared with those from experiments and linear parabolized stability equation (LPSE) computations. It is found that the near-field velocity fluctuations closely agree with experiments and LPSE computations up to the end of the potential core, downstream of which deviations occur, but the LEE results match experiments better than the LPSE results. Both the near-field wavepackets and the sound field are observed directly from LEE computations, but the far-field sound pressure levels (SPLs) obtained are more than an order of magnitude lower than experimental values despite close statistical agreement of the near hydrodynamic field up to the potential core region. We explore the possibility that this discrepancy is due to the mismatch between the decay of two-point coherence with increasing distance in experimental flow fluctuations and the perfect coherence in linear models. To match the near-field coherence, experimentally obtained coherence profiles are imposed on the two-point cross-spectral density (CSD) at cylindrical and conical surfaces that enclose near-field structures generated with LEEs. The surface pressure is propagated to the far field using boundary value formulations based on the linear wave equation. Coherence matching yields far-field SPLs which show improved agreement with experimental results, indicating that coherence decay is the main missing component in linear models. The CSD on the enclosing surfaces reveals that the application of a decaying coherence profile spreads the hydrodynamic component of the linear wavepacket source on to acoustic wavenumbers, resulting in a more efficient acoustic source.

The results of an experimental investigation of separation in oscillating laminar boundary layers is reported. Instantaneous velocity profiles obtained with multiple hot-wire anemometer arrays reveal that the onset of wake formation is preceded by the initial vanishing of shear at the wall, or reverse flow, throughout the entire cycle of oscillation. Correlation of the experimental data indicates that the frequency, Reynolds number and dynamic history of the boundary layer are the dominant parameters and oscillation amplitude has a negligible effect on separation-point displacement.