In a transonic nozzle flow in which the velocity is slightly supersonic in some neighbourhood of the nozzle throat, a shock wave may be present either very close to the throat or else somewhat further downstream. In the latter case, relatively simple series solutions in general provide an asymptotic description of the fluid motion except very close to the shock wave. These outer solutions are reviewed for symmetric two-dimensional flow, and it is shown that the shock-wave jump conditions are not satisfied. A correction is then derived in the form of an inner solution for a small region immediately behind the shock. The resulting solution exhibits the singularities in the pressure gradient, streamline curvature and shock-wave curvature which are expected to occur at the intersection of a normal shock wave and a curved wall. An extension to axisymmetric flow is also given.

Hot-wire filament vibration generated by fluctuating aerodynamic loads is found to occur in turbulent flow fields. The motion of the filament relative to the stream can cause errors in the indicated velocity fluctuations. Analysis and experiment show that these errors are probably serious only when the velocity fluctuations are confined within a narrow bandwidth spectrum such as in a Kármán vortex street. Under such conditions the filament goes into a skipping or ‘whirling’ mode of motion and the errors in the velocity perturbation reading are substantial. Errors in normally encountered boundary-layer turbulence are suspected not to be serious but this is not yet certain. A convenient experimental method for detecting these vibrations is outlined.

A study is made of the effect of filament vibration on the direct frequency-response test for hot-wire systems. These tests include shaking the probe at high frequencies in a steady stream and the Kármán vortex street method.

This paper considers the nonlinear optimal control of transition in a boundary layer flow subjected to a pair of free stream vortical perturbations using a receding horizon approach. The optimal control problem is solved using the Lagrange variational technique that results in a set of linearized adjoint equations, which are used to obtain the optimal wall actuation (blowing and suction from a control slot located in the transition region). The receding horizon approach enables the application of control action over a longer time period, and this allows the extraction of time-averaged statistics as well as investigation of the control effect downstream of the control slot. The results show that the controlled flow energy is initially reduced in the streamwise direction and then increased because transition still occurs. The distribution of the optimal control velocity responds to the flow activity above and upstream of the control slot. The control effect propagates downstream of the slot and the flow energy is reduced up to the exit of the computational domain. The mean drag reduction is $55\,\%$ and $10\,\%$ in the control region and downstream of the slot, respectively. The control mechanism is investigated by examining the second-order statistics and the two-point correlations. It is found that in the upstream (left) side of the slot, the controller counteracts the near-wall high-speed streaks and reduces the turbulent shear stress; this is akin to opposition control in channel flow, and because the time-average control velocity is positive, it is more similar to blowing-only opposition control. In the downstream (right) side of the slot, the controller reacts to the impingement of turbulent spots that have been produced upstream and inside the boundary layer (top–bottom mechanism). The control velocity is positive and increases in the streamwise direction, and the flow behaviour is similar to that of uniform blowing.

Noise intensity measurements per octave band have been carried out in the far field of a small circular nozzle, at angles 0-150° from the jet axis, with and without vortex generators. The results, interpreted in the light of recent theoretical conclusions, confirm Lighthill's original suggestion that the dominant noise radiator is the pressure-shear x-r quadrupole at all but the lowest frequencies. At very low frequencies Reynolds stress-shear x-x and x-r quadrupoles contribute comparable amounts of radiation. When the product of Strouhal number (based on nozzle diameter and jet velocity) and jet Mach number exceeds unity the geometrical acoustics approximation becomes valid, but even at such high frequencies the shear noise contribution dominates. Vortex generators, while reducing noise intensity, do not appear to modify drastically the above composition of noise radiators.

This study offers an explanation of the recently observed effect of destabilization of free convective flows by weak rotation. After studying several models where flows are driven by the simultaneous action of convection and rotation, it is concluded that destabilization is observed in cases where the centrifugal force acts against the main convective circulation. At relatively low Prandtl numbers, this counter-action can split the main vortex into two counter-rotating vortices, where the interaction leads to instability. At larger Prandtl numbers, the counter-action of the centrifugal force steepens an unstable thermal stratification, which triggers the Rayleigh–Bénard instability mechanism. Both cases can be enhanced by advection of azimuthal velocity disturbances towards the axis, where they grow and excite perturbations of the radial velocity. The effect was studied by considering a combined convective and rotating flow in a cylinder with a rotating lid and a parabolic temperature profile at the sidewall. Next, explanations of the destabilization effect for rotating-magnetic-field-driven flow and melt flow in a Czochralski crystal growth model were derived.

This paper examines the emergence of the absolute instability from convectively unstable states of an inviscid compound jet. A compound jet is composed of a cylindrical jet of one fluid surrounded by a concentric annulus of a second, immiscible fluid. For all jet velocities $v$, there are two convectively unstable modes. As in the single-fluid jet, the compound jet becomes absolutely unstable below a critical dimensionless velocity or Weber number $V ({:=}\,\sqrt{v^2\,{\rho_1 R_1}/\sigma_1}$ where $\rho_{1}$, $R_{1}$ and $\sigma _{1}$ are the core density, radius and core–annular interfacial tension), which is a function of the annular/core ratios of densities $\beta$, surface tensions $\gamma$ and radii $a$. At $V\,{=}\,0$, the absolutely unstable modes and growth recover the fastest growing temporal waves. We focus specifically on the effect of $\gamma$ at $a\,{=}\,2$ and $\beta\,{=}\,1$ and find that when the outer tension is significantly less than the inner $(0.1\,{<}\,\gamma\,{<}\,0.3)$, the critical Weber number $V_{\hbox{\scriptsize{\it crit}}}$ decreases with <$\gamma$, whereas for higher ratios $(0.3\,{<}\,\gamma\,{<}\,3)$ it increases. The values (1.2–2.3) of $V_{\hbox{\scriptsize{\it crit}}}$ for the compound jet include the parameter-independent critical value of 1.77 for the single jet. Therefore, increasing the outer tension can access the absolute instability at higher dimensional velocities than for a single jet with the same radius and density as the core and a surface tension equal to the compound jet's liquid–liquid tension. We argue that this potentially facilitates distinguishing experimentally between absolute and convective instabilities because higher velocities and surface tension ratios higher than 1 extend the breakup length of the convective instability. In addition, for $0.3\,{<}\,\gamma\,{<}\,1.16$, the wavelength for the absolute instability is roughly half that of the fastest growing convectively unstable wave. Thus choosing $\gamma$ in this range exaggerates its distinction from the convective instability and further aids the potential observation of absolute instability.

We use weakly nonlinear analysis via a two-parameter expansion to study bifurcation of conduction into cellular convection of an internally heated fluid in a porous medium that forms a horizontal layer between two isothermal walls. The Darcy–Boussinesq model of convection is enhanced by including two nonlinear terms: (i) quadratic (Forchheimer) drag; and (ii) hydrodynamic dispersion enhancement of the thermal conductivity described by a weak linear relationship between effective conductivity and local amplitude of filtration velocity. The impact of the second term on the shape of the bifurcation curve for two-dimensional rolls is profound in the presence of uniform volumetric heating. The resulting bifurcation structure is unlike any pitchfork bifurcations typical of the classical Bénard problem. Although direct experimental validation of the novel bifurcation is not available, we would like to register it as an alternative or a supplement to models of small imperfections, and as an attempt to account for the scatter of observed critical values for the first bifurcation.

The Rayleigh instability of an incompressible, infinitely conducting, inviscid fluid of variable density is investigated under the influence of an horizontal magnetic field and coriolis forces. After establishing the equations of the problem when both the density and the magnetic field vary with distance in the upward direction, two special cases of density distribution are studied in detail. Both the magnetic field and coriolis forces are found to have a stabilizing influence on the configuration and further it is concluded that they may bring about stability in the configuration when it is thoroughly unstable without them.

This paper considers the barotropic stability of an asymmetric zonal current on a rotating earth. The current is of hyperbolic tangent form in latitude. For this stability problem the neutral wave solutions are found and estimates of the growth rates for the unstable waves are obtained from the neutral solutions as in earlier investigations by the writer. It is again found that the beta effect which is due to the earth's curvature and rotation tends to stabilize the basic flow.

The stability of the basic flow is examined for a special case. For this case the current is centered at 35 degrees latitude, has a total shear of 30 m sec−1 and a half-width of 550 km. The most unstable waves are found to be wave-numbers 6 and 7 which amplify by a factor of e in 2·9 days. In addition, wave-numbers 5, 8 and 9 are also unstable. The stability of the symmetric jet is also examined for a comparable case. It is found that a wider band of wave-numbers is unstable. The most unstable wave is wave-number 8 which amplifies by a factor of e in 2·7 days.

In conclusion it is noted that these growth rates are slower than the amplification rates for the unstable waves associated with the baroclinic stability problem.

We consider the linear temporal stability of a Couette flow of a Maxwell gas within the gap between a rotating inner cylinder and a concentric stationary outer cylinder both maintained at the same temperature. The neutral curve is obtained for arbitrary Mach (Ma) and arbitrarily small Knudsen (Kn) numbers by use of a ‘slip-flow’ continuum model and is verified via comparison to direct simulation Monte Carlo results. At subsonic rotation speeds we find, for the radial ratios considered here, that the neutral curve nearly coincides with the constant-Reynolds-number curve pertaining to the critical value for the onset of instability in the corresponding incompressible-flow problem. With increasing Mach number, transition is deferred to larger Reynolds numbers. It is remarkable that for a fixed Reynolds number, instability is always eventually suppressed beyond some supersonic rotation speed. To clarify this we examine the variation with increasing (Ma) of the reference Couette flow and analyse the narrow-gap limit of the compressible TC problem. The results of these suggest that, as in the incompressible problem, the onset of instability at supersonic speeds is still essentially determined through the balance of inertial and viscous-dissipative effects. Suppression of instability is brought about by increased rates of dissipation associated with the elevated bulk-fluid temperatures occurring at supersonic speeds. A useful approximation is obtained for the neutral curve throughout the entire range of Mach numbers by an adaptation of the familiar incompressible stability criteria with the critical Reynolds (or Taylor) numbers now based on average fluid properties. The narrow-gap analysis further indicates that the resulting approximate neutral curve obtained in the (Ma, Kn) plane consists of two branches: (i) the subsonic part corresponding to a constant ratio (Ma/Kn) (i.e. a constant critical Reynolds number) and (ii) a supersonic branch which at large Ma values corresponds to a constant product Ma Kn. Finally, our analysis helps to resolve some conflicting views in the literature regarding apparently destabilizing compressibility effects.

Viscoplastic fluids do not flow unless they are sufficiently stressed. This property can be exploited in order to produce novel flow features. One example of such flows is viscoplastically lubricated (VPL) flow, in which a viscoplastic fluid is used to stabilize the interface in a multi-layer flow, far beyond what might be expected for a typical viscous–viscous interface. Here we extend this idea by considering the encapsulation of droplets within a viscoplastic fluid, for the purpose of transportation, e.g. in pipelines. The main advantage of this method, compared to others that involve capillary forces is that significantly larger droplets may be stably encapsulated, governed by the length scale of the flow and yield stress of the encapsulating fluid. We explore this set-up both analytically and computationally. We show that sufficiently small droplets are held in the unyielded plug of a Poiseuille flow (pipe or plane channel). As the length or radius of the droplets increases, the carrier fluid eventually yields, potentially breaking the encapsulation. We study this process of breaking and give estimates for the limiting size of droplets that can be encapsulated.

The problem of viscous fingering in a Hele-Shaw cell with moving contact lines is considered. In contrast to the usual situation where the displaced fluid coats the solid surface in the form of thin films, here, both the displacing and the displaced fluids make direct contact with the solid. The principal differences between these two situations are in the ranges of attainable values of the gapwise component of the interfacial curvature (the component due to the bending of the fluid interface across the small gap of the Hele-Shaw cell), and in the introduction of two additional parameters for the case with moving contact lines. These parameters are the receding contact angle, and the sensivity of the dynamic angle to the speed of the contact line. Our objective is the prediction of the shape and widths of the fingers in the limit of small capillary number, Uμ/σ. Here, U denotes the finger speed, μ denotes the dynamic viscosity of the more viscous displaced fluid, and σ denotes the surface tension of the fluid interface. As might be expected, there are similarities and differences between the two problems. Despite the fact that different equations arise, we find that they can be analysed using the techniques introduced by McLean & Saffman and Vanden-Broeck for the thin-film case. The nature of the multiplicity of solutions also appears to be similar for the two problems. Our results indicate that when contact lines are present, the finger shapes are sensitive to the value of the contact angle only in the vicinity of its nose, reminiscent of experiments where bubbles or wires are placed at the nose of viscous fingers when thin films are present. On the other hand, in the present problem at least two distinct velocity scales emerge with well-defined asymptotic limits, each of these two cases being distinguished by the relative importance played by the two components of the curvature of the fluid interface. It is found that the widths of fingers can be significantly smaller than half the width of the cell.

A transformation has been found which reduces steady non-homentropic flows of a compressible fluid to homentropic flows, provided diffusive and gravity effect are negligible. With this transformation, the equation governing steady two-dimensional flows of a compressible fluid with variable entropy in a gravitational field is derived, which is then applied to the study of atmospheric waves in the lee of mountains. The corresponding equation governing swirling axisymmetric flows is also given.

Equilibrium shapes and stability of annular fluid menisci held together by surface tension are analysed by applying asymptotic and computer-aided techniques from bifurcation theory. The shapes and locations of the menisci are governed by the Young–Laplace equation. These shapes are grouped together into families of like symmetry that branch from the basic family of annular shapes at specific values of the aspect ratio, α. Multiple equilibrium shapes exist over certain values of α. The inner, outer or both the inner and outer interfaces may possess either a cylindrical or sinusoidal equilibrium shape. Changes in applied pressure, fluid volume, or gravitational Bond number break families of the same symmetry which now develop limit points. Numerical calculations rely on a finite-element representation of the interfaces and the results compare very well with asymptotic analysis which is valid for small deformations. The results are important for the blow moulding process and are invaluable in understanding its dynamics. These dynamics are expected to be considerably different from the dynamics of a liquid jet first analysed by Rayleigh.

A numerical and experimental study has been made of the three-dimensional flow and heat transfer by natural convection in a closed, rectangular thermosyphon. At low Rayleigh numbers, the flows in the two halves of the cavity remain separate, with heat transfer across the mid-height plane occurring only by conduction. At increasing Rayleigh numbers, an exchange process of increasing complexity occurs. The numerical solutions were used to explore this process and to predict flow patterns which were found to resemble closely those observed during previous investigations of cylindrical thermosyphons. The results were verified by a flow visualization study. Heat-transfer rates are presented and augment previous data for higher values of the governing parameters.

The parabolic approximation which has recently been found to be useful in other physical contexts, is extended to head-sea diffraction of short waves by a slender raft on deep water. In particular, it is a much more direct way of getting the inner approximation of the outer solution in a scheme of matched asymptotics than the original method of Faltinsen (1971). The present results are compared with a more involved integral equation method and are found to be remarkably accurate even when the raft length is comparable to the wavelength. Finally, the asymptotic method is modified for a compliant raft which absorbs wave power by suitably controlled impedance. Optimum efficiency and other performance characteristics are predicted.