This note has two aspects. The first concerns comparisons between the theoretical predictions of Smith (1979b) and the calculations of Fornberg (1980) for flow past a circular cylinder: there is quite good agreement overall, in such quantities as the drag, the front stagnation point pressure, the eddy pressure and the skin friction. The second aspect concerns comments and reservations on the calculated eddy lengths and discrepancies at higher Reynolds numbers.

We consider theoretically the long-time evolution of axisymmetric, high Reynolds number, Boussinesq gravity currents supplied by a constant, small-area source of mass and radial momentum in a deep, quiescent ambient. We describe the gravity currents using a shallow-water model with a Froude number closure condition to incorporate ambient form drag at the front and present numerical and asymptotic solutions. The predicted profile consists of an expanding, radially decaying, steady interior that connects via a shock to a deeper, self-similar frontal boundary layer. Controlled by the balance of interior momentum flux and frontal buoyancy across the shock, the front advances as (g′sQ/r1/4s)4/154/5, where g′s is the reduced gravity of the source fluid, Q is the total volume flux, rs is the source radius and is time. A radial momentum source has no effect on this solution below a non-zero threshold value. Above this value, the (virtual) radius over which the flow becomes critical can be used to collapse the solution onto the subthreshold one. We also use a simple parameterization to incorporate the effect of interfacial entrainment, and show that the profile can be substantially modified, although the buoyancy profile and radial extent are less significantly impacted. Our predicted profiles and extents are in reasonable agreement with existing experiments.

This paper presents a mathematical theory for premixed combustion under the influence of enthalpy fluctuations in the oncoming fresh mixture. On the basis of the assumptions of large activation energy and small Mach number, an analysis of the thermal, hydrodynamic and acoustic regions of a flame is performed to derive an interactive system that describes, on the first-principles basis, the intricate coupling between the flame and its spontaneously emitted acoustic waves. The system, in its general form, is strongly nonlinear and requires a numerical attack. In this paper, it is employed to analyse several fundamental physical processes in relatively simple cases in order to provide useful insights into the role of enthalpy fluctuations in combustion. First, the linear response of the flame to two- or three-dimensional small-amplitude enthalpy fluctuations is considered, and they are found to generate hydrodynamic motion. Secondly, enthalpy fluctuations are shown to radiate sound waves through their interaction with the flame. Thirdly, enthalpy fluctuations and the sound waves emitted by them modify the flame stability, and the analysis shows that a moderate level of enthalpy fluctuation may cause a strong subharmonic parametric instability. Finally, in the small-heat-release limit, an extended Michelson–Sivashinsky equation is derived to describe the nonlinear evolution of the flame under the influence of both the imposed enthalpy fluctuations and the induced acoustic waves. Numerical solutions suggest that the flame evolves into a time-periodic state and acquires a curved profile, which primarily vibrates in the longitudinal direction, while its overall shape remains almost unaltered.

The second-order velocity structure tensor of weakly anisotropic strong turbulence is decomposed into its SO(3) invariant amplitudes dj(r). Their scaling is derived within a scaling approximation of a variable-scale mean-field theory of the Navier–Stokes equation. In the isotropic sector j = 0 Kolmogorov scaling d0(r) ∝ r2/3 is recovered. The scaling of the higher j amplitudes (j even) depends on the type of the external forcing that maintains the turbulent flow. We consider two options: (i) for an analytic forcing and for decreasing energy input into the sectors with increasing j, the scaling of the higher sectors j > 0 can become as steep as dj(r) ∝ rj+2/3; (ii) for a non-analytic forcing we obtain dj(r) ∝ r4/3 for all non-zero and even j.

Morland, Saffman & Yuen's (1991) study of the stability of a semi-infinite, concave shear flow bounded above by a capillary–gravity wave, for which they obtained numerical solutions of Rayleigh's equation, is revisited. A variational formulation is used to construct an analytical description of the unstable modes for the exponential velocity profile U = U0 exp(y/d), −∞ < y [les ] 0. The assumption of slow waves ([mid ]c[mid ] [Lt ] U0) yields an approximation that agrees with the numerical results of Morland et al. The assumption of short waves (kd [Gt ] 1) yields Shrira's (1993) asymptotic approximation.

An analytic solution of the non-isentropic equations of gas-dynamics, for the one-dimensional motion of a non-viscous and non-conductive medium, is derived in this paper for the first time. This is a particular solution which contains only one arbitrary function. On the basis of this solution, the interaction of a centred simple wave with a shock of moderate strength is analyzed; and it is shown that, for a weak shock, this analysis is compatible with Friedrichs's theory. Furthermore, in the light of this analysis, it is explained why the empirical methods employed by the shock-expansion theory, including Whitham's rule for determining the shock path, work.

The dynamics of an axisymmetric liquid bridge of a fluid of density ${\it\rho}$, viscosity ${\it\mu}$ and surface tension ${\it\sigma}$ held between two co-axial disks of equal radius $e$ is studied when one disk is slowly moved with a velocity $U(t)$. The analysis is performed using a one-dimensional model for thin bridges. We consider attached boundary conditions (the contact line is fixed to the boundary of the disk), neglect gravity and limit our analysis to short bridges such that there exists a stable equilibrium shape. This equilibrium is a Delaunay curve characterized by the two geometric parameters $\ell _{0}=V_{0}/({\rm\pi}e^{3})$ and $S=\ell ({\rm\pi}e^{2})/V_{0}$, where $V_{0}$ is the volume of fluid and $\ell$ the length of the bridge. Our objective is to analyse the departure of the dynamical solution from the static Delaunay shape as a function of $\ell _{0}$, $S$, the Ohnesorge number $Oh={\it\mu}/\sqrt{{\it\rho}{\it\sigma}e}$, and the instantaneous velocity $U(t)$ and acceleration $\partial _{t}U$ (non-dimensionalized using $e$ and ${\it\sigma}/{\it\rho}$) of the disk. Using a perturbation theory for small velocity and acceleration, we show that (i) a non-homogeneous velocity field proportional to $U(t)$ and independent of $Oh$ is present within the bridge; (ii) the area correction to the equilibrium shape can be written as $U^{2}A_{i}+U\,Oh\,A_{v}+\partial _{t}UA_{a}$ where $A_{i}$, $A_{v}$ and $A_{a}$ are functions of $\ell _{0}$ and $S$ only. The characteristics of the velocity field and the shape corrections are analysed in detail. For the case of a cylinder ($S=1$), explicit expressions are derived and used to provide some insight into the break-up that the deformation would induce. The asymptotic results are validated and tested by direct numerical simulations when the velocity is constant and when it oscillates. For the constant velocity case, we demonstrate that the theory provides a very good estimate of the dynamics for a large range of parameters. However, a systematic departure is observed for very small $Oh$ due to the persistence of free eigenmodes excited during the transient. These same eigenmodes also limit the applicability of the theory to oscillating bridges with large oscillating periods. Finally, the perturbation theory is applied to the cylindrical solution of Frankel & Weihs (J. Fluid Mech., vol. 155 (1985), pp. 289–307) obtained for constant velocity $U$ when the contact lines are allowed to move. We show that it can be used to compute the correction associated with acceleration. Finally, the effect of gravity is discussed and shown to modify the equilibrium shape but not the main results obtained from the perturbation theory.

The three-dimensional instabilities of axisymmetric flow are investigated in a laterally heated vertical cylinder by linear stability analysis. Heating is confined to a central zone on the sidewall of the cylinder, while other parts of the sidewall are insulated and both ends of the cylinder are cooled. The length of the heated zone equals the radius of the cylinder. For three different aspect ratios, $A= 1.92 $, 2, 2.1 ($A=\mathrm{height}$/radius), the dependence of the critical Rayleigh number on the Prandtl number (from 0.02 to 6.7) has been studied in detail. For such a kind of laterally heated convection, some interesting stability results are obtained. A monotonous instability curve is obtained for $A= 1.92 $, while the instability curves for $A= 2 $ and $A= 2.1 $ are non-monotonous and multivalued. In particular, an instability island has been found for $A=2$. Moreover, mechanisms corresponding to different instability results are obtained when the Prandtl number changes. At small Prandtl number, the flow is oscillatory unstable, which is dominated by hydrodynamic instability. At intermediate Prandtl number, the interaction between buoyancy and shear in the base flow plays a more important role than pure hydrodynamic instability. At even higher Prandtl number, Rayleigh–Bénard instability becomes the dominant process and the flow loses stability through steady bifurcation.

In several recent papers it has been established that the addition of a shear flow to a small-scale velocity field can enhance dynamo action. Many of these studies show that the growth rate of the magnetic energy increases linearly with the shear amplitude over a range of shears. In this paper it is shown analytically that for a variety of dynamo models and very large shear amplitudes , growth rates increase no faster than under plausible conditions on the amplitude of the remaining velocity.

We investigate the bulldozing motion of a granular sandpile driven forwards by a vertical plate. The problem is set up in the laboratory by emplacing the pile on a table rotating underneath a stationary plate; the continual circulation of the bulldozed material allows the dynamics to be explored over relatively long times, and the variation of the velocity with radius permits one to explore the dependence on bulldozing speed within a single experiment. We measure the time-dependent surface shape of the dune for a range of rotation rates, initial volumes and radial positions, for four granular materials, ranging from glass spheres to irregularly shaped sand. The evolution of the dune can be separated into two phases: a rapid initial adjustment to a state of quasi-steady avalanching perpendicular to the blade, followed by a much slower phase of lateral spreading and radial migration. The quasi-steady avalanching sets up a well-defined perpendicular profile with a nearly constant slope. This profile can be scaled by the depth against the bulldozer to collapse data from different times, radial positions and experiments onto common ‘master curves’ that are characteristic of the granular material and depend on the local Froude number. The lateral profile of the dune along the face of the bulldozer varies more gradually with radial position, and evolves by slow lateral spreading. The spreading is asymmetrical, with the inward progress of the dune eventually arrested and its bulk migrating to larger radii. A one-dimensional depth-averaged model recovers the nearly linear perpendicular profile of the dune, but does not capture the finer nonlinear details of the master curves. A two-dimensional version of the model leads to an advection–diffusion equation that reproduces the lateral spreading and radial migration. Simulations using the discrete element method reproduce in more quantitative detail many of the experimental findings and furnish further insight into the flow dynamics.

This paper describes the results of a series of measurements made using a single beam schlieren system to investigate the density fluctuations present in the initial region of a supersonic axisymmetric turbulent jet with a Mach number of 1-82 in the flow a t the nozzle exit. A preheater was used to reduce the difference between the jet static temperature and that of the surrounding air to a relatively low level. The results show that significant density fluctuations are present in the potential core of the jet and that the distribution of fluctuating intensity across the shear layer differs from that obtained with a subsonic jet without preheating.