A theory of time-dependent two-dimensional detonation is developed for an explosive with a finite-thickness reaction zone. A representative initial–boundary-value problem is treated that illustrates how the planar shock of an initially one-dimensional detonation becomes non-planar in response to the action of an edge rarefaction that is generated at the explosive's lateral surface. The solution of this time-dependent problem has a wave-hierarchy structure that at late times includes a weakly two-dimensional hyperbolic region and a fully two-dimensional parabolic region. The wave head of the rarefaction is carried by the hyperbolic region. We show that the shock locus is analytic at the wave head. The dynamics of the final approach to two-dimensional steady-state detonation is controlled by Burgers’ equation for the shock locus. We also present some results concerning the stability of the solutions to our problem.

The first four moments of conserved scalar probability density functions (p.d.f.'s) measured by Raman scattering in an H2 turbulent jet diffusion flame are analysed and compared with those found by Pitts & Kashiwagi (1984) in a non-reacting CH4 jet. The measurements are in good agreement, indicating that heat release and combustion have little effect on p.d.f. shapes. However, the measured p.d.f.'s are not qualitatively similar to the simple forms often assumed in combustion modelling. A three-zone model by Effelsberg & Peters was used to separate the experimental p.d.f.'s into a delta function (non-turbulent zone), a Gaussian (turbulent zone) and the remainder (interface zone). The interface zone contributed as much as 90% of the total p.d.f. in both the H2 flame and the non-reacting CH4 jet. A physical interpretation for the existence of broad interface zones in reacting and non-reacting turbulent jet flows is suggested based upon large-scale structures.

We consider plane flows of identical, smooth, nearly elastic, circular disks interacting with a boundary formed by attaching halves of similar disks at equal intervals along a flat wall. The roughness of the boundary is given in terms of the diameters of the two types of disks and the spacing of the wall disks. We suppose that the velocity distribution of the flow disks is Maxwellian and calculate the rates at which momentum and energy are supplied to the flow disks in collisions over a unit length of the boundary. At the boundary we balance these supplies with the stress and the total flux of energy in the flow and obtain boundary conditions on the shear stress, pressure, and flux of fluctuation energy. We find that the boundary can either supply fluctuation energy to the flow or absorb it, depending on the relative magnitudes of the rate of working of the boundary tractions through the slip velocity and the rate at which energy is dissipated in collisions. As an example we solve the boundary-value problem for the steady shearing flow maintained by the relative motion of parallel plates a fixed distance apart. When the dimensions and properties of the flow disks and the boundary are given, the specification of the distance between the plates and their relative velocity determines the slip velocity, the shear stress and pressure necessary to maintain the flow, and the distributions of mean velocity, fluctuation energy, and density.

The stability of a temperature–salinity front of finite width in which the density is exactly compensated for in the horizontal direction is studied by means of a linear theory. It is assumed that salt fingers are responsible for vertical transports of salinity and heat. The front is found to be always unstable even if viscosity is present. Horizontal intrusions are generated as a result of the instability, and cold/fresh water sinks while hot/salty water rises.

The stability of the front is described by three dimensionless parameters: a frontal stability parameter G = [g(1 − γ)βΔS]6/κe2a2N10, a stratification parameter $\mu = g(1-\gamma)\beta\overline{S}_z/N^2$ and a Schmidt number ε = ν/κe, where g is the acceleration due to gravity, β the salinity contraction coefficient, ΔS half of the salinity difference across the front, γ the density-flux ratio of temperature to salt due to salt fingers, N the Brunt–Väisälä frequency corresponding to the basic density stratification, a the half-width of the front, $\overline{S}_z$ the vertical gradient of the basic salinity, κe the eddy diffusivity of salt due to salt fingers, and ν is either molecular or eddy kinematic viscosity. When ε is not zero it is found that a modified frontal stability parameter R defined by R = G/ε plays an important role in determining the stability for a fixed value of μ. In particular, when ε is larger than 10, the stability is almost completely determined by R if μ is specified.

The dependence of the vertical scale h of the fastest-growing mode on external parameters varies according to the value of R for a given value of μ. When R is less than 40(1 + μ)5.4 (for ε = 10−3-103) h becomes independent of R and ε and is given by h = 2.2d/(1 + μ), where d = g(1 − γ)βΔS/N2 is the scale suggested by Ruddick & Turner (1979) who performed an experiment on horizontal intrusions across a narrow front. When R > 2 × 1055 (1 + μ)4.9, on the other hand, h becomes proportional to dR−1/4, the scale suggested by Toole & Georgi (1978), who considered the stability of a temperature–salinity front of infinite width. The constant of the proportionality is a weak function of ε and varies from 2π[(2 + μ)/4(1 + μ)2]−¼ to 2π[2(1 + μ + (1 + μ)½)]½ as ε goes from zero to infinity. Thus a front can be said to be narrow if R < 40(1 + μ)5.4 and wide if R > 2 × 105(1 + μ)4.9.

The results of the theory explain those of Ruddick & Turner's experiment (1979) reasonably well.

A compact rotating shallow mass of fluid with a free boundary is released on a horizontal plane. Initially it is supposed that horizontal sections through the mass are elliptical, vertical sections are parabolic and the two velocities are linear functions of the horizontal coordinates. Thus four numbers suffice to describe the configuration of the fluid and another four the velocity field. The subsequent motion preserves this simple structure, and so the shallow-water equations reduce to ordinary differential equations in time for the eight parameters required to specify the initial condition.

This eighth-order system can be reduced to quadratures using the well-known invariants of the shallow-water equations. There are five such integrals: volume, energy, enstrophy, angular momentum, and a fifth, which lacks a familiar name. The remaining three degrees of freedom can be related to the shape (i.e. eccentricity), horizontal size (i.e. radius of gyration) and orientation of the mass. In general, all of these are periodic functions of time but with different characteristic timescales. Simplest are size changes, which occur at the inertial frequency f. Shape changes are superinertial, while orientation changes may be subinertial or superinertial.

This solution is used to discuss the unsteady motion of a non-axisymmetric Gulf Stream ring. We argue that size and shape changes excite internal gravity waves in the underlying fluid, while orientation changes generate Rossby waves. While this wave radiation decreases the energy of the ring, and may alter the angular momentum, it cannot lead to a state of no motion because the volume and enstrophy are unaffected.

The combustion of a moving liquid fuel drop has been investigated. The drop experiences a strong evaporation-induced radial velocity while undergoing slow translation. In view of the high evaporation velocity, the flow field is not in the Stokes regime. The combustion process is modelled by an indefinitely fast chemical reaction rate.

While the flow and the transport in the continuous phase and the drop internal circulation are treated as quasisteady, the drop heat-up is regarded as a transient process. The transport equations of the continuous phase require analysis by a singular perturbation technique. The transient heat-up of the drop interior is solved by a series-truncation numerical method. The solution for the total problem is obtained by coupling the results for the continuous and dispersed phases.

The enhancement in the mass burning rate and the deformation of the flame shape due to drop translation have been predicted. The initial temperature of the drop and the subsequent heating influence the temporal variations of the flamefront standoff ratio and the flame distance. The friction drag, the pressure drag and the drag due to interfacial momentum flux are individually predicted, and the total drag behaviour is discussed. The circulation inside the drop decreases with evaporation rate. A sufficiently large non-uniform evaporation velocity causes the circulation to reverse.

Approximate layer-averaged equations describing the mechanics of turbid underflows are derived. Closure of the equations describing the balance of fluid mass, sediment mass, and mean flow momentum provides for the delineation of a three-equation model. A description of sediment exchange with the bed allows for the possibility of a self-accelerating turbidity current in which sediment entrainment from the bed is linked to flow velocity. A consideration of the balance of the mean energy of the turbulence yields a constraint on physically realistic solutions to the three-equation model. It is shown that the self-acceleration predicted by the three-equation model is so strong that the energy constraint fails to be satisfied. In particular, the turbulent energy consumed in entraining new bed sediment exceeds the supply of energy to the turbulence, so that the turbulence, and thus the turbidity current, must die. The problem is rectified by the formulation of a four-equation model, in which an explicit accounting is made of the mean energy of the turbulence. Sediment entrainment from the bed is linked to the level of turbulence in the four-equation model. Self-acceleration is again predicted, although it is somewhat subdued compared with that predicted by the three-equation model. The predictions of both models are summarized over a wide range of conditions.

A model based on the lubrication approximation is put forward for the general case of asymmetric forward roll coating of Newtonian liquids. Two more-rigorous theories are developed, one based on asymptotic expansions for small ratios of gap-to-roll diameter (H0/R), the second on Galerkin/finite-element solutions of the full Navier–Stokes equations over the relevant flow domain. The lubrication model is useful only as an approximation at high capillary numbers $(Ca\equiv \mu\overline{V}/\sigma) $. The asymptotic analysis is accurate when H0/R < 0.001 and Ca > 0.1. The ratio of the film thicknesses on the two rolls is predicted to equal the speed ratio to the 0.65 power, which is confirmed experimentally. The Galerkin/finite-element solutions give full details of the steady two-dimensional free-surface flows including complex recirculation patterns in the film-splitting region, and show how the film-splitting stagnation line becomes a static contact line in the limit as one roll surface becomes stationary.

In contrast to the steady Stokes equations for creeping motion, the time-dependent linearized Navier–Stokes equations have only been solved for very restricted geometries, the solution for the sphere being the sole solution for an isolated finite body. In the present paper, the linearized Navier–Stokes equations are further explored and a simple expression is derived which relates the force on an arbitrary axisymmetric body in oscillatory motion to the solution for the stream function in the far field. This result is applied to the case of a slightly eccentric spheroid and it is shown that the total hydrodynamic force contains four terms, three of which correspond to the classical solutions for the Stokes drag, added mass and Basset force on the perturbed sphere; the fourth term is only present when the body is non-spherical. In contrast to the three classical forces, the new term is not a simple power of the dimensionless frequency parameter iL2ω/ν, in which L is a length-scale, ω is the frequency of oscillation and ν is the kinematic viscosity of the fluid. A Laplace superposition is then used to find the force on the spheroid in an arbitrary axisymmetric motion with velocity U(t). The new memory term decays faster than the Basset force at large times and is bounded at short times.

The experimental evidence on the mechanisms of forcing of unstable vorticity waves (the Tollmien–Schlichting–Schubauer or TS waves) of circular frequency ω and wavelength λTS in wall layers by unsteady pressure gradients of amplitude A and frequency ω is reviewed and found to be confused and contradictory. It is proposed that a likely effective receptivity mechanism rests on the fact that under realistic conditions A varies with distance x along any body of finite thickness, A(x), and introduces thereby additional characteristic lengths which can match λTS. Heuristic arguments suggest that through A(x) the pressure gradient infuses vorticity at the wall and forces spatial growth of the TS mean-square vorticity $\overline{\zeta^2}$, at a rate proportional to the real part of ΔAF(κTS), the contribution of A(x) between x−½λTS and x + ½λTS to the Fourier transform AF(κ) at κ = κTS. A second input into $\overline{\zeta^2_{\rm TS}}$ growth corresponds to the conversion of the steady boundary-layer vorticity into unsteady ζTS, through the action of vf, the normal velocity of the forcing field. The rate is given by $\int_0^{\delta}\overline{v_{\rm f}\zeta_{\rm TS}}U^{\prime\prime}(y)\,{\rm d}y $ and is proportional to the imaginary part of κTSΔAF(KTS).

The proposition is consistent in all currently verifiable respects with one numerical and a series of laboratory experiments. In the laboratory experiments, various configurations of a pulsating pressure source and shielding plates located in the free stream supplied the variable-amplitude pressure gradients over the nearby flat-wall boundary layer. Three of the cases presented here demonstrate that stationary unsteady pressure fields induce Stokes-like sublayers when the boundary layer is stable and self-excited vorticity waves when it is unstable. The results of a fourth experiment suggest that unsteady pressure sources in wakes near the boundary layer can force the growth of unstable wall waves at the wake frequencies even though their propagation speeds differ. Material is also presented on key Soviet experiments and views on receptivity. Finally, these experiments and ours are examined for consistency and complementarity.

In this paper we study some instabilities and bifurcations of two-dimensional channel flows. We use analytical, numerical and experimental methods. We start by recapitulating some basic results in linear and nonlinear stability and drawing a connection with bifurcation theory. We then examine Jeffery–Hamel flows and discover new results about the stability of such flows. Next we consider two-dimensional indented channels and their symmetric and asymmetric flows. We demonstrate that the unique symmetric flow which exists at small Reynolds number is not stable at larger Reynolds number, there being a pitchfork bifurcation so that two stable asymmetric steady flows occur. At larger Reynolds number we find as many as eight asymmetric stable steady solutions, and infer the existence of another seven unstable solutions. When the Reynolds number is sufficiently large we find time-periodic solutions and deduce the existence of a Hopf bifurcation. These results show a rich and unexpected structure to solutions of the Navier–Stokes equations at Reynolds numbers of less than a few hundred.

In this paper we develop a theory for a rheometrical device for measuring the speed of shear waves into a region at rest. The device is a Couette apparatus with a narrow gap. The outer cylinder is moved impulsively and a time of transit is measured. The linearized theory governing this apparatus is reduced to a perturbation of Stokes’ first problem between parallel planes. A method for determining an effective shear modulus from measured values of the wave speed is discussed and various cases are analysed. An experimental apparatus based on this theory, together with tabular data, is discussed in a companion paper (Part 2, Joseph, Riccius & Arney 1986).

In this paper we describe the experimental apparatus that we use to measure transit speeds. Tables of measured values of transit speeds and the corresponding values of the shear modulus are presented. The criteria we use to determine if a transit speed is a shear-wave speed are described and applied to the data. The main criteria are that transit speeds should be independent of the gap size and the corresponding value of the shear modulus should be consistent with independent rheometrical measurements. All the elastic liquids and many liquids that are usually assumed to be Newtonian satisfy our criteria for shear waves. We present evidence that the measured wave speeds are associated with slower-relaxing molecular structures in the liquids.

The response of the interface between a compliant surface and a turbulent boundary-layer flow is examined theoretically. This response is forced by transient, convected, interfacial pressure pulses that represent the footprints of turbulent flow structures in the boundary layer. Calculations are presented for coatings with a wide range of damping and densities equal to the density of the flow. For coatings with moderate damping, three regimes of response are found. When the flow speed U∞ is less than about 1.2 (non-dimensionalized by the shear wave speed of the coating), the response is stable and primarily localized under the pressure pulse. For flow speeds from 1.2 to as high as 2.8, depending on the damping, the response is also stable, but it includes a wave pattern behind the pressure pulses. For flow speeds above 2.8, the response is unstable and eventually forms a two-dimensional wavetrain moving in the flow direction. At the highest stable flow velocity, the amplitude of the surface displacements reaches 4.0% of the boundary-layer displacement thickness δ* and the energy transfer from the pressure pulse reaches 5.0 × 10−4U∞2 (δ*)2. For high damping, the coating response is again stable when the flow speed is below 2.8. However, there is no wavelike response regime; the path that is traversed by the pressure pulse is covered by a scar that heals according to the viscous relaxation properties of the material. The amplitude of the response is at most 0.01δ*.

The shock-tube problem has served as a popular test for numerical hydrodynamics codes. The development of relativistic hydrodynamics codes has created a need for a similar test problem in relativistic hydrodynamics. The analytical solution to the special relativistic shock-tube problem is presented here. The relativistic shock-jump conditions and rarefaction solution which make up the shock tube are derived. The Newtonian limit of the calculations is given throughout.

The nonlinear evolution of an array of pairs of inviscid counter-rotating vortices, subjected to an applied stretching strain field, has been studied numerically using the contour-dynamics method. The array configuration is effectively the Corcos-Lin model of streamwise vortices in the braid region of a nominally two-dimensional mixing layer. For each individual vortex the simulations elucidate the strong interaction between the vortex self-induction, the vorticity amplification of the stretching strain, and the local in-plane strain applied by all other members of the array. When the initial vorticity distribution is modelled by a non-uniform piece-wise-constant vorticity field defined over a nested set of non-intersecting contours, the dynamical evolution reveals fine structure consisting of strong vortex roll-up accompanied by trailing, filament-like spiral vortex sheets, and the presence of tertiary instabilities. It is shown by a particular example that these features are largely absent in an equivalent computation in which array members are modelled by the commonly used uniform-vortex approximation.

An experimental study of the dynamic response of non-rotating, stratified reservoirs to the application of surface shear stresses is presented. The experiments were made using two-layered and linear stratifications; a moving belt was used to supply a shear stress to the fluid. Detailed measurements of the density field show that upwelling occurs at all values of the Wedderburn number, W, rather than only occurring when W < 1. Differences between the two-layer theory and the present observations are attributed to the fact that the experimental stratifications were continuous, rather than layered. Shearflow dispersion is observed to be an important mechanism for distributing the effects of localized upwelling over the entire length of the mixed layer. A model of mixed-layer deepening based on upwelling and shearflow dispersion is presented and is compared to the observations of this and other experimental studies.

The Taylor-Görtler vortex instability equations are formulated for steady and unsteady interacting boundary-layer flows. The effective Görtler number is shown to be a function of the wall shape in the boundary layer and the possibility of both steady and unsteady Taylor-Görtler modes exists. As an example the steady flow in a symmetrically constricted channel is considered and it is shown that unstable Görtler vortices exist before the boundary layers at the wall develop the Goldstein singularity discussed by Smith & Daniels (1981). As an example of an unsteady spatially varying basic state we also consider the instability of high-frequency large-amplitude two- and three-dimensional Tollmien-Schlichting waves in a curved channel. It is shown that they are unstable in the first ‘Stokes-layer stage’ of the hierarchy of nonlinear states discussed by Smith & Burggraf (1985). This instability of Tollmien-Schlichting waves in an internal flow can occur in the presence of either convex or concave curvature. Some discussion of this instability in external flows is given.

Data have been obtained for a wave-driven current system in a closed basin. Owing to interaction of the longshore current with the sidewall a strong rip current was generated. The velocity distribution over the depth of the rip current appeared to be more or less uniform. The current system has been modelled by means of a mathematical model. The effect of bottom topography, bottom friction, convection and turbulent viscosity on the current system has been investigated. The conclusions are that the rip current is dominated by convection and that the bottom topography plays a role in the convergence and divergence of the streamlines. The order of magnitude of the velocities is largely determined by the bottom-friction coefficient. The velocity field is modified by viscosity. First, turbulent viscosity entrains fluid in the longshore current and into the rip current, secondly it permits turbulent boundary layers and thirdly it is responsible for the existence of closed streamlines outside the breaker zone. Finally the model and the conclusions are extrapolated to prototype conditions.

We consider wavenumber selection for the periodic two-dimensional Rayleigh–Bénard problem on a horizontally infinite domain. The temperature difference (and hence the Rayleigh number) is assumed to be a slowly varying function of the horizontal coordinate perpendicular to the convection rolls. Under this condition Kramer et al. have shown that a unique wavenumber of convection is selected by a certain solvability condition if the domain contains a subcritical region. They performed analytic calculations for a model problem and for small-amplitude convection of an infinite-Prandtl-number fluid between stress-free boundaries. Their results are extended here to the realistic case of large-amplitude convection of a finite-Prandtl-number fluid between rigid boundaries. The temperature difference may be ‘ramped’ by changing either the temperature at the lower boundary or at the upper boundary, or both. It is shown that the choice has a significant effect on the ‘mean flow’, but no effect on the selected wavenumber.