A model is presented for steady two-dimensional Stokes flow in a bifurcating channel. The geometry of the bifurcation is symmetric, but the flow rates through the two branches may be unequal. It is found that separation takes place on the outer boundary of the channel for either a smooth or a sharp flow divider, but only when the flow rates through the two branches of the channel are sufficiently different.

A two-zone model is proposed for the longitudinal dispersion of contaminants in two-dimensional turbulent flow in open channels – a fast zone in the upper region of the flow, and a slow zone nearer to the bottom. The usual one-dimensional dispersion approach (Elder 1959) is used in each zone, but with different flow speeds U1 and U2 and dispersion coefficients D1 and D2 in the fast and slow zones respectively. However, turbulent vertical mixing is allowed at the interface between the two zones, with a small vertical diffusivity ε. This leads to a pair of coupled, linear, one-dimensional dispersion equations, which are solved by Fourier transformation. The Fourier-inversion integrals are analysed using two different methods.

In the first method asymptotically valid expressions are found using the saddle-point method. The resulting cross-sectional average concentration consists of a leading Gaussian distribution followed by a trailing Gaussian distribution. The trailing Gaussian cloud disperses (longitudinally) faster than the leading one, and this gives the long tail observed in most dispersion experiments. Significantly the peak value of the average concentration is found to decay exponentially with time at a rate which is close to the rate observed by Sullivan (1971) in the early stage of the dispersion process. The solution is useful for fairly small times, and both the calculated value of D1 and the predicted bulk concentration distribution are in meaningful agreement with the experimental and simulation data of Sullivan (1971).

In the second method an exact solution is found in the form of a convolution integral for the case D1 = D2 = D0. Explicit expressions which are valid for small times and for large times from the release of contaminant are found. For small times this exact solution confirms the basic results obtained by the saddle-point method. For large times the exact solution gives a contaminant concentration which approaches a Gaussian distribution travelling with the bulk speed as predicted by the Taylor model. The overall longitudinal dispersion coefficient at large times, D(∞), consists of the diffusivity D0 plus a contribution D[ell ](∞) which depends entirely on the vertical mixing. D(∞) is in good agreement with Chatwin's (1971) interpretation of Fischer's (1966) experimental data.

We compute the axisymmetric convective motions that exist in a spherical shell heated from below with inner to outer radius ratio equal to 0.5. The boundaries are stress-free and gravity is directly proportional to radius. Accurate solutions at large Rayleigh numbers (O(105)) are made feasible by a spectral method that employs diagonal-mode truncation. By examining the stability of axisymmetric motions we conclude that the preferred form of convection varies dramatically according to the value of the Rayleigh number. While axisymmetric motions with different patterns may exist for modestly nonlinear convection, only a single motion persists at sufficiently large values of the Rayleigh number. This circulation is symmetric about the equator and has two meridional cells with rising motion at the poles. Instability of this single axisymmetric motion determines that the preferred pattern of three-dimensional convection has one azimuthal wave.

Shock fronts and fluid-particle trajectories throughout a two-dimensional shock wave flow have been measured by multiple schlieren photography in a detailed study of the Mach reflection from a 10° wedge of plane uniform shocks with Mach numbers of 1.105, 1.240 and 1.415. Correction of optical distortions throughout the field of view permitted the positions and shapes of the shock fronts and the magnitudes and directions of the particle velocities to be measured with a high degree of accuracy. No departure from self-similarity of the flow fields could be detected. The cross-sections of the reflected shocks were found to be circular and centred on a point which moved with the velocity of the flow behind the incident shock. The triple-point trajectories were linear. The velocity of the curved Mach stem shock was found to be constant at any one height above the wedge surface and to decrease monotonically with height. A deviation from perpendicularity was noticed where the Mach stems met the surface of the wedge, the shocks having a slight forward inclination of as much as 1°. The experimental results cannot be completely explained using the classical three-shock theory and an alternative model for weak Mach reflection is developed in Part 2 of this paper.

A non-stationary approach to the reflection of weak plane shocks is suggested as an alternative to the usual pseudo-stationary transformation. For regular reflection the non-stationary model produces results which are identical to those obtained using the pseudo-stationary assumption, but with simpler algebra. For weak Mach reflections, where the predictions of the pseudo-stationary model are in disagreement with experimental results, the non-stationary model predicts accurately the observed shapes and positions of the reflected and Mach stem shocks and the spatially varying flow properties behind these shocks. However, the non-stationary model predicts that the gas flows above and below the contact surface, relative to the triple point, are not quite parallel. Parallel flows could be obtained only in the limiting case of grazing incidence, when the reflected shock was sonic. The model is based on the experimental results presented in Part 1 of this paper.

The phenomena of excitation-induced suppression and amplification of broadband jet noise have been experimentally investigated in an effort to understand the mechanisms, especially in relation to the near flow-field large-scale structure dynamics. Suppression is found to occur only in jets at low speeds with laminar exit boundary layers, the optimum occurring for excitation at Stθ ≈ 0.017, where Stθ is the Strouhal number based on the initial shear-layer momentum thickness. The suppression mechanism is linked to an initial-condition effect on the large-scale structure dynamics. The interaction and evolution of laminar-like structures at low jet speeds produce more (normalized) noise and turbulence, compared to asymptotically lower levels at high speeds when the initial shear layer is no longer laminar. The effect of initial condition has been demonstrated by tripped versus untripped jet data. The excitation at Stθ ≈ 0.017 results in a quick roll-up and transition of the laminar shear-layer vortices, yielding coherent structures which are similar to those at high speeds. Thus, the broadband noise and turbulence are suppressed, but at the most to the asymptotically lower levels. When at the asymptotic level, the broadband jet noise can only be amplified by the excitation; the amplification is found to be maximum for excitation in the StD range of 0.65–0.85, StD being the Strouhal number based on the jet diameter. Excitation in this StD range also produces strongest vortexpairing activity. From spectral analysis of the flow-field and the near sound-pressure field, it is inferred that the pairing process induced by the excitation is at the origin of the broadband noise amplification.

Finite-amplitude thermal convection in a horizontal layer with finite conducting boundaries is investigated. The nonlinear steady problem is solved by a perturbation technique, and the preferred mode of convection is determined by a stability analysis. Square cells are found to be the preferred form of convection in a semi-infinite three-dimensional region Ω in the (γb,γt, P)-space (γb and γt are the ratios of the thermal conductivities of the lower and upper boundaries to that of the fluid and P is the Prandtl number). Two-dimensional rolls are found to be the preferred convection pattern outside Ω. The dependence on γb, γt and P of the heat transported by convection is computed for the various solutions analysed in the paper.

A three-dimensional model of the plane mixing layer is constructed by applying digital image processing to laser fluorescence motion pictures of the layer, and displayed using computer graphic techniques. A system of streamwise counterrotating vortices is shown to exist on top of the classical spanwise eddies, and its influence in mixing is discussed briefly. Some quantitative information on their strength is also given.

The fully developed laminar mixed-convection flow in horizontal ducts of rectangular, circular and semicircular cross-sections has been studied using a numerical model of the governing equations of motion, subject to the Boussinesq approximation and an axially uniform heat-flux condition. Dual solutions with a two- and a four-vortex flow pattern have been observed in all cases. The rectangular geometry, with its aspect ratio and Grashof number as parameters, is posed as a two-parameter problem. In this parameter-space, the critical points where the transition between the two- and the four-vortex pattern occur, follow a tilted cusp. This is akin to the phenomenon in the Taylor problem which has been thoroughly investigated by Benjamin and co-workers in a general study of bifurcation phenomena for viscous flow problems. The bifurcation phenomenon in circular ducts, which is essentially a one-parameter problem, has features similar to that observed for the Dean problem, by Nandakumar and Masliyah.

Two new derivations of Darwin's theorem on the equality of the added mass for translation of a body moving in an ideal fluid of infinite extent and the drift mass are given. The first is based on the idea of time lag, used by Rayleigh (1876), Ursell (1953), and Longuet-Higgins (1953) to study fluid drift. The second is truly elementary, relying only on the concept of continuity and Newton's second law of motion. A geometrical interpretation of the result in the first derivation is given, and a few examples are provided.

An algorithm is constructed for the use of the Lagrangian kinematic specification in Newtonian fluid mechanics. The algorithm is implemented with a finite-element method, and it is demonstrated that the method accurately describes free-surface flow, including the effects of surface tension, with the use of just bilinear isoparametric elements. Moving contact lines are modelled with a small amount of slip near the contact lines. The contact angle boundary condition is included in the form of a net interfacial force specified at the contact line. Simulations of measurements in a parallel-plate geometry show that the measured apparent contact angle is not the true angle, and that the true angle is always very close to the equilibrium value.

A perturbation procedure is used to obtain first- and second-order solutions for small-amplitude internal waves in a Lagrangian coordinate system. The first-order Lagrangian equations are formally accurate to the same order as the first-order Eulerian equations; however, they are different and the Lagrangian solution gives a more realistic wave shape. First-order Lagrangian solutions for internal waves in uniformly stratified fluid have a shape similar to that found in the second-order Eulerian solution. Wave profiles in uniformly stratified fluid exhibit broad crests and narrow troughs near the surface, a sinusoidal shape at mid-depth, and narrow crests and broad troughs near the bottom. The difference between the shape of crests and troughs grows as the wave amplitude is increased. Solutions obtained in a uniformly stratified fluid with a small bottom slope yield plausible shapes for breaking waves.

The time required to pull a large object from a sandy seabed is estimated by assuming that the seabed is porous but rigid. The phenomenon of breakout (i.e., sudden release) is shown to occur without the assumption of elasticity of the soil skeleton (Foda 1982). A new case of wedge-shaped gap is also studied, and compared to a uniform gap, Laboratory experiments are shown to support the theory.

A Hermite series expansion is used to describe the time evolution of the contaminant concentration at a fixed location in a flow. Exact expressions are derived for the dosage, centroid and temporal variance. A two-mode approximation is proposed which gives accurate results at moderate to large distances downstream of the discharge.

An analysis of flow visualization using small reflective flakes is introduced. This rational analysis is based on a stochastic treatment of Jeffery's (1922) solution for the motion of ellipsoidal particles in a viscous fluid, wherein thin flakes tend to align with stream surfaces. The predicted light fields are confirmed by examples of parallel flows, the flow over a rotating disk, and the spinup from rest in a cylindrical cavity. The Tollmien–Schlichting wave packet trailing a turbulent spot is taken as an example to discuss the suitability of the technique for visualizing small-amplitude waves. Attenuation of light through a suspension is described.

In stably stratified flow over a three-dimensional hill, we can define a dividing streamline that separates those streamlines that pass around the hill from those that pass over the hill. The height Hs of this dividing streamline can be estimated by Sheppard's simple energy argument; fluid parcels originating far upstream of a hill at an elevation above Hs have sufficient kinetic energy to rise over the top, whereas those below Hs must pass around the sides. This prediction provides the basis for analysing an extensive range of laboratory observations and measurements of stably stratified flow over a variety of shapes and orientations of hills and with different upwind density and velocity profiles. For symmetric hills and small upwind shear, Sheppard's expression provides a good estimate for Hs. For highly asymmetric flow and/or in the presence of strong upwind shear, the expression provides a lower limit for Hs. As the hills become more nearly two-dimensional, these experiments become less well defined because steady-state conditions take progressively longer to be established. The results of new studies are presented here of the development of the unsteady flow upwind of two-dimensional hills in a finite-length towing tank. These measurements suggest that a very long tank would be required for steady-state conditions to be established upstream of long ridges with or without small gaps and cast doubt upon the validity of previous laboratory studies.

Further experimental evidence of vortex splitting in wake flows is presented. Streakline visualizations reveal the range of conditions and the phenomenology of this novel process.

We consider the penetration of a solid medium by a foreign body which is large enough for frictional heating to melt the medium and maintain a thin liquid layer ahead of the body. This study is motivated by the possibility of the Earth's core having been formed by liquid iron diapirs melting their way through the solid, deformable mantle. Our principal results are the existence of a critical size for the body for the motion to be maintained under gravity and the ease with which an immiscible liquid body can penetrate at constant velocity compared to a solid one.

One of the possible mechanisms of forming offshore sandbars parallel to a coast is the wave-induced mass transport in the boundary layer near the sea bottom. For this mechanism to be effective, sufficient reflection must be present so that the waves are partially standing. The main part of this paper is to explain a theory that strong reflection can be induced by the sandbars themselves, once the so-called Bragg resonance condition is met. For constant mean depth and simple harmonic waves this resonance has been studied by Davies (1982), whose theory, is however, limited to weak reflection and fails at resonance. Comparison of the strong reflection theory with Heathershaw's (1982) experiments is made. Furthermore, if the incident waves are slightly detuned or slowly modulated in time, the scattering process is found to depend critically on whether the modulational frequency lies above or below a threshold frequency. The effects of mean beach slope are also studied. In addition, it is found for periodically modulated wave groups that nonlinear effects can radiate long waves over the bars far beyond the reach of the short waves themselves. Finally it is argued that the breakpoint bar of ordinary size formed by plunging breakers can provide enough reflection to initiate the first few bars, thereby setting the stage for resonant reflection for more bars.

Large-scale coherent structure in a round free jet injected into a low-speed, co-flowing stream was experimentally investigated using laser-Doppler and cold-wire techniques. Particular attention was paid to the coherent structures in the outer intermittent region of the jet in an almost self-preserving state. Velocity fluctuations u (axial) and v (radial) and temperature fluctuations θ were measured simultaneously at two positions: a reference position and a moving position. In order to clarify the pattern of coherent motion, a pattern-averaging technique was adopted and the characteristics of the turbulent fluctuations were conditionally averaged. The results show that a large-scale coherent structure exists even in the self-preserving region of a round free jet, as well as in the near field. It has a vortical structure which consists of strong outward turbulent motion from inside the jet, turbulent reverse flow and inflow in the irrotational ambient region (entrainment). In the coherent structure, the negative pattern-averaged Reynolds stress occurs at two locations: one in the irrotational ambient region outside the turbulent/irrotational interface and the other in the turbulent jet inside the interface. The former is instantaneously produced in the irrotational inflow outside the interface when the vortical motion is accelerated, and it changes even the sign of conventionally averaged Reynolds stress. The latter is instantaneously produced in the turbulent flow near the high-shear region when the turbulent motion is more strongly directed by the acceleration of the vortical motion towards the centre of the vortical structure than the averaged motion.