Convective motions in rectangular boxes with one side horizontal have been studied. The critical Rayleigh numbers were determined. In most cases cell patterns with rolls parallel to the shorter side wall of the rectangular box were observed. The results have been compared with the known theoretical results of Davis (1967). In general, good agreement has been found.

A method of renormalized expansions in theory of turbulence is developed with the use of the Lagrangian position function. The introduction of this function makes it easy to express the Lagrangian development of the velocity field. A simple truncation of a set of renormalized expansions is shown to lead to an approximation which is compatible with Kolmogorov's inertial range energy spectrum.

A formal solution to the initial value problem for a plane vortex sheet in an inviscid fluid is obtained by transform methods. The eigenvalue problem is investigated and the stability criterion determined. This criterion is found to be in agreement with that obtained previously by Landau (1944), Hatanaka (1949), and Pai (1954), all of whom had included spurious eigenvalues in their analyses. It is also established that supersonic disturbances may be unstable; related investigations in hydrodynamic stability have conjectured on this possibility, but the vortex sheet appears to afford the first definite example. Finally, an asymptotic approximation is developed for the displacement of a vortex sheet following a suddenly imposed, spatially periodic velocity.

The problem of capillary-driven flow in a V-shaped surface groove is addressed. A nonlinear diffusion equation for the liquid shape is derived from mass conservation and Poiseuille flow conditions. A similarity transformation for this nonlinear equation is obtained and the resulting ordinary differential equation is solved numerically for appropriate boundary conditions. It is shown that the position of the wetting front is proportional to (Dt)½ where D is a diffusion coefficient proportional to the ratio of the liquid-vapour surface tension to viscosity and the groove depth, and a function of the contact angle and the groove angle. For flow into the groove from a sessile drop source it is shown that the groove angle must be greater than the contact angle. Certain arbitrarily shaped grooves are also addressed.

This paper describes an experimental and theoretical study of thermal convection in a sloping porous layer. The saturated layer is bounded by two parallel impermeable planes maintained at different temperatures. Several types of flows were observed: a unicellular movement and a juxtaposition of longitudinal coils or of polyhedral cells.

A theoretical analysis has been made using the standard bases of the linear theory of stability and by taking into account some assumptions suggested by experimental observations. The critical conditions for the transition between unicellular and polycellular flows has been determined. For flow in longitudinal coils or with polyhedral cells the average heat transfer depends mainly on the filtration Rayleigh number and on the slope of the layer.

The experimental study was made in a Rayleigh number range 0–800 and for various slopes (0–90°). For both the transition criterion and the heat transfer, a good fit was observed between the experimental and theoretical results. For maximum slope, i.e. 90°, a correlation which connects the Nusselt number with both the Rayleigh number and the vertical extent of the model is proposed.

Water-tunnel experiments on two cones of taper ratio 36:1 and 18:1 have shown that the frequency of vortex shedding is controlled by the local diameter and has a value which is slightly lower than that on a parallel cylinder of the same size. The shedding process is modulated by a low-frequency oscillation which has a frequency dependent only on U2/v and is independent of any physical dimension of the model. A mathematical analogue of the wake, which helps to explain these features of the flow, has been constructed. The pattern of vortices in the wake of a cone suggested by this model is in reasonable agreement with the photographic evidence.

Wind-tunnel measurements of lift, drag and wake velocity spectra were carried out under (tonal) acoustic excitation for a smooth airfoil in the chord-Reynolds-number (Rec) range of 4 × 104−1.4 × 105. The data are supported by smoke-wire flowvisualization pictures. Small-amplitude excitation in a wide, low-frequency range is found to eliminate laminar separation that otherwise degrades the airfoil performance at low Rec near the design angle of attack. Excitation at high frequencies, scaling as $U_{\infty}^{\frac{3}{2}}$, eliminates a pre-stall, periodic shedding of large-scale vortices; U∞ is the free-stream velocity. Significant improvement in lift is also achieved during post-stall, but with large-amplitude excitation. Wind-tunnel resonances strongly influence the results, especially in cases requiring large amplitudes. It is shown that large transverse velocity fluctuations, induced near the airfoil by specific cross-resonance modes, lead to the most effective separation control; resonances inducing only large-amplitude pressure fluctuations are demonstrated to be less effective.

Direct numerical simulations (DNS) are used to examine the pressure fluctuations generated by a spatially developed Mach 5.86 turbulent boundary layer. The unsteady pressure field is analysed at multiple wall-normal locations, including those at the wall, within the boundary layer (including inner layer, the log layer, and the outer layer), and in the free stream. The statistical and structural variations of pressure fluctuations as a function of wall-normal distance are highlighted. Computational predictions for mean-velocity profiles and surface pressure spectrum are in good agreement with experimental measurements, providing a first ever comparison of this type at hypersonic Mach numbers. The simulation shows that the dominant frequency of boundary-layer-induced pressure fluctuations shifts to lower frequencies as the location of interest moves away from the wall. The pressure wave propagates with a speed nearly equal to the local mean velocity within the boundary layer (except in the immediate vicinity of the wall) while the propagation speed deviates from Taylor’s hypothesis in the free stream. Compared with the surface pressure fluctuations, which are primarily vortical, the acoustic pressure fluctuations in the free stream exhibit a significantly lower dominant frequency, a greater spatial extent, and a smaller bulk propagation speed. The free-stream pressure structures are found to have similar Lagrangian time and spatial scales as the acoustic sources near the wall. As the Mach number increases, the free-stream acoustic fluctuations exhibit increased radiation intensity, enhanced energy content at high frequencies, shallower orientation of wave fronts with respect to the flow direction, and larger propagation velocity.

A systematic account is given of the derivation of the dispersion relation for helicon waves in a uniform cylindrical plasma bounded by a vacuum. By retaining finite resistivity in the equations, boundary conditions present no difficulties, since the wave magnetic field is continuous through the plasma-vacuum interface. Two unexpected results are found. First, the wave attenuation remains finite in the limit of vanishing resistivity. This is due to the energy dissipated at the interface by the surface currents required to match the plasma wave field to the vacuum wave field. Zero wave attenuation for zero resistivity is recovered if electron inertia is included. Secondly, it is found that waves with azimuthal numbers m of opposite sign propagate differently, but the sense of polarization at the axis of the cylinder is independent of the sign of m.

The argument of the dispersion function is complex and numerical results were obtained using a computer. The method of programming is described, and results are given applicable to propagation in metals at low temperatures, or in a typical gas discharge plasma for the m = 0 and m = ± 1 modes.

An example of the amplitude of the wave fields as a function of radius is given for the axisymmetric mode, and of amplitude and phase for the m = ± 1 modes.

Three-dimensional velocity distributions of a turbulent flow in the core region of a square duct at ReH = 1.2 × 105 are measured using holographic particle image velocimetry (HPIV). Spatial filtering of the 136 × 130 × 128 velocity vector maps enables calculation of subgrid-scale (SGS) stresses and parameters based on the filtered velocity gradients, such as the filtered strain-rate tensor and vorticity vector. Probability density functions (p.d.f.) of scalar parameters characterizing eigenvalue structures confirm that the most probable strain-rate topology is axisymmetric extension, and show that the most probable SGS stress state is axisymmetric contraction. Conditional sampling shows that high positive SGS dissipation occurs preferentially in regions with these preferred strain-rate and stress topologies. High negative SGS dissipation (backscatter) occurs preferentially in regions of axisymmetric contracting SGS stress topology, but is not associated with any particular strain-rate topology. The nonlinear model produces the same trends but tends to overpredict the likelihood of the preferred stress state.

Joint p.d.f.s of relative angles are used to investigate the alignments of the SGS stress eigenvectors relative to the vorticity and eigenvectors associated with eddy viscosity and similarity/nonlinear models. The results show that the most extensive SGS stress eigenvector is preferentially aligned at 32° to the most contracting strain-rate eigenvector. This alignment trend persists, with some variations in angle and peak probability, during conditional samplings based on the SGS dissipation rate, vorticity and strain-rate magnitudes. The relative alignment of the other two stress and strain-rate eigenvectors has a bimodal behaviour with the most contracting and intermediate stress eigenvectors ‘switching places’: from being aligned at 32° to the most extensive strain-rate eigenvector to being parallel to the intermediate strain-rate eigenvector. Conditional sampling shows that one of the alignment configurations occurs preferentially in regions of high vorticity magnitude, whereas the other one dominates in regions where the filtered strain-rate tensor has axisymmetric contracting topology. Analysis of DNS data for isotropic turbulence at lower Re shows similar trends.

Conversely, the measured stress eigenvectors are preferentially aligned with those of the nonlinear model. This alignment persists in various regions of the flow (high vorticity, specific flow topologies, etc). Furthermore, the alignment between the strain-rate and nonlinear model tensors also exhibits a bimodal behaviour, but the alignment angle of both configurations is 42°. Implications of alignment trends on SGS dissipation are explored and conditions for high backscatter are identified based on the orientation of the stress eigenvectors. Several dynamical and kinematical arguments are presented that may explain some of the observed preferred alignments among tensors. These arguments motivate further analysis of the mixed model, which shows good alignment properties owing to the dominance of the Leonard stress on the alignments. Nevertheless, the data also show that the mixed model produces some unrealistic features in probability distributions of SGS dissipation, and unphysical eigenvector alignments in selected subregions of the flow.

Reheat buzz is a combustion instability which occurs in the afterburners of jet aeroengines. Similar oscillations have been observed on a laboratory rig in which a confined flame is stabilized in the wake of a conical gutter. The source of energy for the instability is the unsteady heat release as the flame responds to velocity fluctuations in the approach flow. The amount of energy fed into the instability is determined primarily by the phase relationship between the unsteady heat release rate and pressure fluctuations. Two types of relationship between pressure and heat release have been observed. In the first kind, perturbations in heat release rate convect downstream from the lip of the flame stabilizer at the axial velocity of the cold reactants. In the second type, which occurs downstream of the first, the phase of the heat release rate at the buzz frequency is constant and close to the phase of the unsteady pressure. The characteristics of the resulting instability depend on which of these occupies the larger portion of the duct. Consequently two types of instability exist and the transition between them occurs sharply, with little change in mean flow conditions. The transition is associated with an abrupt change in buzz amplitude and frequency as well as with a change in the shape of the modal distributions.

This is an experimental investigation of vortex excitation of prisms with elongated rectangular, H- and [vdash ]-shaped cross-sections, where the depth parallel to the flow is much greater than the height perpendicular to the flow. Measurements are made of free oscillations in a wind tunnel and flow visualizations in a water tank. The flow around elongated bluff prisms is dominated by the impinging-shear- layer instability where the separated shear layers become unstable in the presence of different kinds of shape of the rear part of the cross-section, which may include sharp trailing edges. The two unstable shear layers interact with each other when they meet together downstream of the prism, thereby forming Kármán vortices with the same frequency of oscillation. The former impinging-shear-layer instability is largely responsible for vortex excitation of elongated bluff prisms.

Reynolds-averaged Navier–Stokes (RANS) simulations with turbulence closure models continue to play important roles in industrial flow simulations. However, the commonly used linear eddy-viscosity models are intrinsically unable to handle flows with non-equilibrium turbulence (e.g. flows with massive separation). Reynolds stress models, on the other hand, are plagued by their lack of robustness. Recent studies in plane channel flows found that even substituting Reynolds stresses with errors below 0.5 % from direct numerical simulation databases into RANS equations leads to velocities with large errors (up to 35 %). While such an observation may have only marginal relevance to traditional Reynolds stress models, it is disturbing for the recently emerging data-driven models that treat the Reynolds stress as an explicit source term in the RANS equations, as it suggests that the RANS equations with such models can be ill-conditioned. So far, a rigorous analysis of the condition of such models is still lacking. As such, in this work we propose a metric based on local condition number function for a priori evaluation of the conditioning of the RANS equations. We further show that the ill-conditioning cannot be explained by the global matrix condition number of the discretized RANS equations. Comprehensive numerical tests are performed on turbulent channel flows at various Reynolds numbers and additionally on two complex flows, i.e. flow over periodic hills, and flow in a square duct. Results suggest that the proposed metric can adequately explain observations in previous studies, i.e. deteriorated model conditioning with increasing Reynolds number and better conditioning of the implicit treatment of the Reynolds stress compared to the explicit treatment. This metric can play critical roles in the future development of data-driven turbulence models by enforcing the conditioning as a requirement on these models.

A spatially developing supersonic boundary layer at Mach 2 is analysed by means of direct numerical simulation of the compressible Navier--Stokes equations, with the objective of quantitatively characterizing the coherent vortical structures. The study shows structural similarities with the incompressible case. In particular, the inner layer is mainly populated by quasi-streamwise vortices, while in the outer layer we observe a large variety of structures, including hairpin vortices and hairpin packets. The characteristic properties of the educed structures are found to be nearly uniform throughout the outer layer, and to be weakly affected by the local vortex orientation. In the outer layer, typical core radii vary in the range of 5–6 dissipative length scales, and the associated circulation is approximately constant, and of the order of 180 wall units. The statistical properties of the vortical structures in the outer layer are similar to those of an ensemble of non-interacting closed-loop vortices with a nearly planar head inclined at an angle of approximately 20° with respect to the wall, and with an overall size of approximately 30 dissipative length scales.

Dissolution of carbon dioxide (CO2) injected into saline aquifers causes an unstable high-density diffusive front. Understanding how instability fingers develop has received much attention because they accelerate dissolution trapping, which favours long-term sequestration. The time for the onset of convection as the dominant transport mechanism has been traditionally studied by neglecting dispersion and treating the CO2–brine interface as a prescribed concentration boundary by analogy to a thermal convection problem. This work explores the effect of these simplifications. Results show that accounting for the CO2 mass flux across the prescribed concentration boundary has little effect on the onset of convection. However, accounting for dispersion causes a reduction of up to two orders of magnitude on the onset time. This implies that CO2 dissolution can be accelerated by activating dispersion as a transport mechanism, which can be achieved adopting a fluctuating injection regime.

The stability of parallel shear flow of an inviscid compressible fluid is investigated by a linear analysis. The extension of the Rayleigh stability criterion and Howard's semi-circle theorem to compressible flows, obtained by Lees & Lin (1946) and Eckart (1963) respectively, are each rederived by a different approach. It is then shown that a subsonic neutral solution of the stability equation may be found when the basic flow is represented by the hyperbolic-tangent velocity profile. With the aid of this solution, the unstable eigenvalues, eigenfunctions and Reynolds stress are determined by numerical methods. A brief discussion of the results follows.

Micro-cavitation bubbles generated by ultrasound have wide and important applications in medical ultrasonics and sonochemistry. An approximate theory is developed for nonlinear and non-spherical bubbles in a compressible liquid by using the method of matched asymptotic expansions. The perturbation is performed to the second order in terms of a small parameter, the bubble-wall Mach number. The inner flow near the bubble can be approximated as incompressible at the first and second orders, leading to the use of Laplace's equation, whereas the outer flow far away from the bubble can be described by the linear wave equation, also for the first and second orders. Matching between the two expansions provides the model for the non-spherical bubble behaviour in a compressible fluid. A numerical model using the mixed Eulerian–Lagrangian method and a modified boundary integral method is used to obtain the evolving bubble shapes. The primary advantage of this method is its computational efficiency over using the wave equation throughout the fluid domain. The numerical model is validated against the Keller–Herring equation for spherical bubbles in weakly compressible liquids with excellent agreement being obtained for the bubble radius evolution up to the fourth oscillation. Numerical analyses are further performed for non-spherical oscillating acoustic bubbles. Bubble evolution and jet formation are simulated. Outputs also include the bubble volume, bubble displacement, Kelvin impulse and liquid jet tip velocity. Bubble behaviour is studied in terms of the wave frequency and amplitude. Particular attention is paid to the conditions if/when the bubble jet is formed and when the bubble becomes multiply connected, often forming a toroidal bubble. When subjected to a weak acoustic wave, bubble jets may develop at the two poles of the bubble surface after several cycles of oscillations. A resonant phenomenon occurs when the wave frequency is equal to the natural oscillation frequency of the bubble. When subjected to a strong acoustic wave, a vigorous liquid jet develops along the direction of wave propagation in only a few cycles of the acoustic wave.

The flow around a finite circular cylinder with a height-to-diameter ratio of 6 was studied using large eddy simulation (LES). The cylinder was mounted on a ground plane where a thin boundary layer with a thickness of only 7% of the cylinder's diameter was formed. The flow resulting from the present LES was used to present a detailed picture of both the instantaneous and the time-averaged flow. Among the results of the present study is the explanation for the formation of two time-averaged tip vortices from two pairs of instantaneous vortices. The main recirculation region on the cylinder's top was found to be one vortex with both legs attached to the top surface, in agreement with some previous investigations. The mean flow in the near wake region was found to contain two horseshoe vortices, one large arch vortex stretching from the cylinder's top and another vortex resulting from the flow moving down below the rear stagnation point on the cylinder. The instantaneous horseshoe vortex near the ground was found to be unsteady and changed shape over time. The time-averaged flow showed three complete horseshoe vortices, in agreement with previous knowledge. The downwash process above the free end into the near wake occurs at a rather constant angle with respect to the plane normal to the streamwise direction. The same angle was preserved in the inclination of the arch vortex in the time-averaged flow.

An experimental investigation is described which has as its objectives the extension of the technical data base pertaining to roughness-induced transition and the advancement of the understanding of the physical processes by which three-dimensional roughness elements induce transition from laminar to turbulent flow in boundary layers. The investigation was carried out primarily with single hemispherical roughness elements surface mounted in a well-characterized zero-pressure-gradient laminar boundary layer on a flat plate. The critical roughness Reynolds number at which turbulence is regarded as originating at the roughness was determined for the roughness elements herein considered and evaluated in the context of data existing in the literature. The effect of a steady and oscillatory free-stream velocity on eddy shedding was also investigated. The Strouhal behaviour of the ‘hairpin’ eddies shed by the roughness and role they play in the evolution of a fully developed turbulent boundary layer, as well as whether their generation is governed by an inflexional instability, are examined. Distributions of mean velocity and intensity of the u-fluctuation demonstrating the evolution toward such distributions for a fully developed turbulent boundary layer were measured on the centreline at Reynolds numbers below and above the critical Reynolds number of transition. A two-region model is postulated for the evolutionary change toward a fully developed turbulent boundary layer: an inner region where the turbulence is generated by the complex interaction of the hairpin eddies with the pre-existing stationary vortices that lie near the surface and are inherent to a flow about a three-dimensional obstacle in a laminar boundary layer; and an outer region where the hairpin eddies deform and generate turbulent vortex rings. The structure of the resulting fully developed turbulent boundary layer is discussed in the light of the proposed model for the evolutionary process.

A boundary layer flows over a flat plate which has on it a small hump situated downstream of the leading edge. The description of the boundary-layer flow, based upon a triple-deck structure, shows how the presence of the hump generates an interaction between the inviscid region just outside the layer and the viscous region near the hump. The pressure force dominant in the boundary layer and the connexion of the local flow with the main stream develop together and are self-perpetuating, and both remain of primary significance for a wide range of hump sizes, even for a hump buried well inside the boundary layer. By consideration of the limiting cases of very small and very large humps, a consistent account of the nature of the disturbances due to the various sizes of hump is produced. The forces and couples on the hump are also evaluated.