Transonic potential flow around the leading edge of a thin two-dimensional general airfoil with a parabolic nose is analysed. Asymptotic expansions of the velocity potential function are constructed at a fixed transonic similarity parameter (K) in terms of the thickness ratio of the airfoil in an outer region around the airfoil and in an inner region near the nose. These expansions are matched asymptotically. The outer expansion consists of the transonic small-disturbance theory and it second-order problem, where the leading-edge singularity appears. The inner expansion accounts for the flow around the nose, where a stagnation point exists. Analytical expressions are given for the first terms of the inner and outer asymptotic expansions. A boundary value problem is formulated in the inner region for the solution of a uniform sonic flow about an infinite two-dimensional parabola at zero angle of attack, with a symmetric far-field approximation, and with no circulation around it. The numerical solution of the flow in the inner region results in the symmetric pressure distribution on the parabolic nose. Using the outer small-disturbance solution and the nose solution a uniformly valid pressure distribution on the entire airfoil surface can be derived. In the leading terms, the flow around the nose is symmetric and the stagnation point is located at the leading edge for every transonic Mach number of the oncoming flow and shape and small angle of attack of the airfoil. The pressure distribution on the upper and lower surfaces of the airfoil is symmetric near the edge point, and asymmetric deviations increase and become significant only when the distance from the leading edge of the airfoil increases beyond the inner region. Good agreement is found in the leading-edge region between the present solution and numerical solutions of the full potential-flow equations and the Euler equations.

We derive averaged equations for large Reynolds number laminar flows of gas–liquid dispersions accounting for slowly varying spatial and temporal fields. In particular, we obtain an exact expression for the dispersed-phase stress tensor to be used in the force balance equation for gas bubbles and illustrate its application by evaluating the stress tensor for a few special cases. It is shown that the dispersed-phase stress tensor gradient with respect to the mean relative motion or the void fraction for the uniformly random bubbly liquids under conditions of large Reynolds number laminar flows is negative and thus has a destabilizing influence on the dynamics of void fraction waves in bubbly liquids.

The paper considers the generation of Boussinesq internal waves in the framework of the Green's function method. For certain domains it is shown how to construct Green's functions using the fundamental solution of the equation. The behaviour of the solution at large times for an impulsively started monochromatic point source is studied, attention being focused on the growth rate of the oscillation amplitude on the characteristic surfaces of the steady-oscillation equation which are emitted from the point source. In addition a simple extended source is considered, for which a focusing singularity phenomenon is shown to take place.

We propose a general geometric method of derivation of invariant relations for hydrodynamic dissipationless media. New dynamic invariants are obtained. General relations between the following three types of invariants are established, valid in all models: Lagrangian invariants, frozen-in vector fields and frozen-in co-vector fields. It is shown that frozen-in integrals form a Lie algebra with respect to the commutator of the frozen fields. The relation between frozen-in integrals derived here can be considered as the Backlund transformation for hydrodynamic-type systems of equations. We derive an infinite family of integral invariants which have either dynamic or topological nature. In particular, we obtain a new type of topological invariant which arises in all hydrodynamic dissipationless models when the well-known Moffatt invariant vanishes.

The efficacy of perturbation approaches for short–long wave interactions is examined by considering a simple case of two interacting wave trains with different wavelengths. Frequency-domain solutions are derived up to third order in wave steepness using two different formulations: one employing conventional wave-mode functions only, and the other introducing a modulated wave-mode representation for the short-wavelength wave. For long-wavelength wave steepness and short-to-long wavelength ratio ε1 and ε3 respectively, the two results are shown to be identical for ε1 [Lt ] ε3 < 0.5. As ε1 approaches ε3, the conventional wave-mode approach converges slowly and eventually diverges for ε1 [Gt ] ε3. The loss of convergence is because the linear phase of conventional wave-mode functions is ineffective for modelling the modulated phase of the short wave. As expected, this difficulty can be removed by using a modulated wave-mode function for the short wave. On the other hand, for relatively large ε3 ∼O(1), the conventional wave-mode approach converges rapidly while the slowly varying interaction between the two waves cannot be accurately predicted by the present modulated wave-mode approach. These findings have important implications to (time-domain) numerical simulations of the nonlinear evolution of ocean wave fields, and suggest that a hybrid wave model employing both conventional (for large-ε3 interactions) and modulated (for small-ε3 interactions) wave-mode functions should be particularly effective.

The classical lubrication-theory model of steady peristaltic transport of periodic sinusoidal waves in infinite-length tubes (Shapiro et al. 1969) is generalized to arbitrary wave shape and wavenumber in tubes of finite length. Whereas the classical model is steady in a frame of reference moving with the peristaltic waves, peristaltic transport in a finite-length tube is inherently non-steady. It may be shown, however, that pumping performance is independent of tube length if there exists an integral number of peristaltic waves in the tube. Three particularly interesting characteristics of non-steady peristalsis are described: (i) fluctuations in pressure and shear stress arise due to a non-integral number of waves in the finite-length tube; (ii) retrograde motion of fluid particles during peristaltic transport (reflux) has inherently different behaviour with single peristaltic waves as compared to multiple ‘train waves’, and (iii) finite tube length, the number of peristaltic waves and the degree of tube occlusion affect global pumping performance. We find that, whereas significant increases in pressure and shear stress result from the tube-to-wave length ratio being non-integral, global pumping performance is only slightly degraded by the existence of a non-integral number of waves in the tube during peristaltic transport. Furthermore, the extent of retrograde motion of fluid particles is much greater with single waves than with train waves. These results suggest that in the design and analysis of peristaltic pumps attention should be paid to the unsteady effects of finite tube length and to the differences between single and multiple peristaltic waves.

The interaction of linear, steady, axisymmetric deep-water gravity waves with preexisting large-scale annular currents has been investigated. Waves originating inside the annulus as well as waves approaching the annulus from the outside were studied. Exact linear ray solutions were obtained and involve two non-dimensional parameters, a radius-angle parameter and a velocity parameter. For opposing currents the linear solutions also allow the derivation of radii at which the waves are blocked, reflected at a linear caustic or stopped by the current. Various examples of rays interacting with an annular current are presented to illustrate aspects of the solutions obtained. In particular, the behaviour of the ray solutions at blocking, reflection and stopping is investigated. Linear ray theory is shown to fail at caustics and caustic solutions are briefly discussed.

This paper continues the analysis of Johnson (1990, hereinafter referred to as I) of the scattering of Kelvin waves by collections of ridges and valleys. General results, flow patterns and explicit solutions follow by restricting attention to waves whose period is long compared to the inertial period but without the additional further simplification introduced in I of approximating general features by stepped topography. A simple direct method is presented giving explicit formulae for the amplitude of the transmitted Kelvin wave and the scattered topographic long waves. A simple but accurate approximation to the solution is also given. The accuracy and usefulness of the apparently crude method of I are confirmed and a superior method presented for choosing the positions of steps in the approximation of general topography. Inviscid flows, the effects of weak dissipation and weak stratification, the form and relevance of the short-wave field over downslopes, the partition of mass and energy flux between the long-wave and short-wave fields and the size and form of higher-order effects are also discussed.

Solutions of the Poincaré equation describing equatorially trapped three-dimensional boundary travelling waves in rotating spherical systems are discussed. It is shown that the combined effects of Coriolis forces and spherical curvature enable the equatorial region to form an equatorial waveguide tube with characteristic latitudinal radius (2/m)1/2 and radial radius (1/m), where m is azimuthal wavenumber. Inertial waves with sufficiently simple structure along the axis of rotation and sufficiently small azimuthal wavelength must be trapped in the equatorial waveguide tube. The structure and frequency of the inertial waves are thus hardly affected by the presence of an inner sphere or by the condition of higher latitudes. Further calculations on rotating spherical fluid shells of finite internal viscosity and stressfree boundaries are also discussed.

The transient motion that arises in a confined rarefied gas as a container wall is rapidly heated or cooled is simulated numerically. The Knudsen number based on nominal gas density and characteristic container dimension is varied from near-continuum to highly rarefied conditions. Solutions are generated with the direct simulation Monte Carlo method. Comparisons are made with finite-difference solutions of the Navier–Stokes equations, the limiting free-molecular values, and (continuum) results based on a small perturbation analysis. The wall heating and cooling scenarios considered induce relatively large acoustic disturbances in the gas, with characteristic flow speeds on the order of 20% of the local sound speed. Steady-state conditions are reached after on the order of 5 to 10 acoustic time units, here based on the initial speed of sound in the gas and the container dimension. As rarefaction increases, the initial gas response time is decreased. For the case of a rapid increase in wall temperature, transient rarefaction effects near the wall greatly alter gas response compared to the continuum predictions, even at relatively small nominal Knudsen number. For wall cooling, the continuum solution agrees well with direct simulation at that same Knudsen number. A local Knudsen number, based on the mean free path and the scale length of the temperature gradient, is found to be a more suitable indicator of transient rarefaction effects.

We study the flow generated by a small circular cylinder in a mixing layer. The cylinder is executing an oscillatory translation whose frequency is within the range of unstable frequencies of the shear layer. The smallness of the cylinder is measured by the ratio of its radius to the characteristic thickness of the layer. This (small) ratio serves as the expansion parameter for our theory; the flow naturally divides into inner and outer regions. The former is in the immediate vicinity of the cylinder and the latter is the far field which contains the instability waves. The solution to this problem is obtained by the method of matched asymptotic expansion. One objective is to study the dependence of this solution on various parameters such as the frequency of oscillation, velocity ratio, etc., and thus shed light on the associated receptivity. Other objectives deal with a restatement of causality and with the hydrodynamic field near the streamwise location of the cylinder. We find that receptivity is a strong function of frequency and velocity ratio and that the local hydrodynamic field may be quite large. Causality is restated in terms of the well-known exponential integral.

This paper describes experiments undertaken to study in detail the control of vortex shedding from circular cylinders at low Reynolds numbers by using feedback to stabilize the wake instability. Experiments have been performed both in a wind tunnel and in an open water channel with flow visualization. It has been found that feedback control is able to delay the onset of the wake instability, rendering the wake stable at Reynolds numbers about 20% higher than otherwise. At higher flow rates, however, it was not possible to use single-channel feedback to stabilize the wake - although, deceptively, it was possible to reduce the unsteadiness recorded by a near-wake sensor. When control is applied to a long span only the region near the control sensor is controlled. The results presented in this paper generally support the analytical results of other researchers.

The behaviour of an inviscid vortex layer of non-zero thickness near a wall is studied, both through direct numerical simulation of the two-dimensional vorticity equation at high Reynolds numbers, and using an approximate ordinary nonlinear integro-differential equation which is satisfied in the limit of a thin layer under long-wavelength perturbations. For appropriate initial conditions the layer rolls up and breaks into compact vortices which move along the wall at constant speed. Because of the effect of the wall, they correspond to equilibrium counter-rotating vortex dipoles. This breakup can be related to the disintegration of the initial conditions of the approximate nonlinear dispersive equation into solitary waves. The study is motivated by the formation of longitudinal vortices from vortex sheets in the wall region of a turbulent channel.

The dynamics of the preferred mode structure in the near field of an elliptic jet have been investigated using hot-wire measurements. A 2:1 aspect ratio jet with an initially turbulent boundary layer and a constant momentum thickness all around the nozzle exit perimeter was used for this study. Measurements were made in air at a Reynolds number ReDe (≡ UeDe/v) = 3.5 × 104. Controlled longitudinal excitation at the preferred mode frequency (StDe ≡ fDe/Ue = 0.4) induced periodic formation of structures, allowing phase-locked measurements with a local trigger hot wire. The dynamics of the organized structure are examined from educed fields of coherent vorticity and incoherent turbulence in the major and minor symmetry planes at five successive phases of evolution, and are also compared with corresponding data for a circular jet. Unlike in a circular jet, azimuthally fixed streamwise vortices (ribs) form without the aid of azimuthal forcing. The three-dimensional deformation of elliptic vortical structures and the rib formation mechanism have also been studied through direct numerical simulation. Differential self-induced motions due to non-uniform azimuthal curvature and the azimuthally fixed ribs produce greater mass entrainment in the elliptic jet than in a circular jet. The turbulence production mechanism, entrainment and mixing enhancement, and time-average measures and their modification by excitation are also discussed in terms of coherent structure dynamics and the rib-roll interaction. Various phase-dependent and time-average turbulence measures documented in this paper should serve as target data for validation of numerical simulations and turbulence modelling, and for design and control purposes in technological applications. Further details are given by Husain (1984).

We used a compound matrix method to integrate the Orr–Sommerfeld equation in an investigation of short instability waves (λ < 6 cm) on the coupled shear flow at the air–sea interface under suddenly imposed wind (a gust model). The method is robust and fast, so that the effects of external variables on growth rate could easily be explored. As expected from past theoretical studies, the growth rate proved sensitive to air and water viscosity, and to the curvature of the air velocity profile very close to the interface. Surface tension had less influence, growth rate increasing somewhat with decreasing surface tension. Maximum growth rate and minimum wave speed nearly coincided for some combinations of fluid properties, but not for others.

The most important new finding is that, contrary to some past order of magnitude estimates made on theoretical grounds, the eigenfunctions at these short wavelengths are confined to a distance of the order of the viscous wave boundary-layer thickness from the interface. Correspondingly, the perturbation vorticity is high, the streamwise surface velocity perturbation in typical cases being five times the orbital velocity of free waves on an undisturbed water surface. The instability waves should therefore be thought of as fundamentally different flow structures from free waves: given their high vorticity, they are akin to incipient turbulent eddies. They may also be expected to break at a much lower steepness than free waves.

The integral energy method has been used to study the nonlinear interactions of the large-scale coherent structure in a spatially developing round jet. The streamwise development of a jet is obtained in terms of the mean flow shear-layer momentum thickness, the wave-mode kinetic energy and the wave-mode phase angle. With the energy method, a system of partial differential equations is reduced to a system of ordinary differential equations. The nonlinear differential equations are solved with initial conditions which are given at the nozzle exit. It is shown that the initial wave-mode energy densities as well as the initial phase angles play a significant role in the streamwise evolution of the large-scale coherent wave modes and the mean flow.

Experiment have shown that the well-known vortex pairing process may take place first in defect regions where the fundamental structure is weakest. A model is introduced here to describe this defect-induced pairing process. The model is constructed in such a way that, in a certain parameter range, a stable fundamental mode and a stable subharmonic mode may coexist. The numerical simulation demonstrates that, when initial conditions consist of a dominant fundamental with one or more defects, the subharmonic component is preferentially generated in the cores of these defects. Moreover, the results also indicate that pairing may first commence wherever the fundamental mode is weakest provided the white-noise level of the subharmonic is high enough. The numerical results are in good agreement with experimental observations.

The instability of rectangular jets is investigated using a vortex-sheet model. It is shown that such jets support four linearly independent families of instability waves. Within each family there are infinitely many modes. A way to classify these modes according to the characteristics of their mode shapes or eigenfunctions is proposed. The stability equation for jets of this geometry is non-separable so that the traditional methods of analysis are not applicable. It is demonstrated that the boundary element method can be used to calculate the dispersion relations and eigenfunctions of these instability wave modes. The method is robust and efficient. A parametric study of the instability wave characteristics has been carried out. A sample of the numerical results is reported here. It is found that the first and third modes of each instability wave family are corner modes. The pressure fluctuations associated with these instability waves are localized near the corners of the jet. The second mode, however, is a centre mode with maximum fluctuations concentrated in the central portion of the jet flow. The centre mode has the largest spatial growth rate. It is anticipated that as the instability waves propagate downstream the centre mode would emerge as the dominant instability of the jet.

Accelerations exceeding 20g in surface waves have been observed both in experiments and in numerically computed flows with a free surface. The present paper describes a family of analytic solutions which display such behaviour. They are expressible in parametric form as z = F sinh ω + iG cosh ω + γω + iH, where F, G and H are functions of the time t only, and γ is linear in t. ω is a complex parameter which is real at the free surface. The functions F(t) and G(t) satisfy two nonlinear, coupled ODEs, which can be solved numerically. Typically the solutions pass through an ‘inertial shock’, or singularity in the time, where the displacements vary as t2/3, the velocities as t1/3 and the accelerations as t-4/3. In this class of solution the free surface develops a cusp as t → ∞. In a special case, F and G vary as t4/7 and the cusp is reached in finite time. Gravity is neglected, but plays a part in setting up the initial conditions for the highly accelerated flow.

In future papers it will be shown that more general solutions exist in which the acceleration is momentarily large but bounded.

We consider the steady flow driven by turbulent mixing in a benthic boundary layer along a sloping boundary in the general case of a non-uniform background density gradient. The velocity and density fields are decomposed into barotropic and baroclinic components, and a solution is obtained by taking an expansion in the small parameter A, the aspect ratio of the boundary layer defined as the thickness divided by the alongslope length. The flow in the boundary layer is governed by a balance between alongslope baroclinic and barotropic density fluxes. A number of flow regimes can exist, and we show that in the regimes relevant to lakes and reservoirs, the barotropic flow is divergent and drives an exchange flow between the boundary layer and the interior. This leads to changes in the interior density gradient which are significant when compared to field observations.