An experimental study was made of the effect of a periodic velocity perturbation on the separation bubble downstream of the sharp-edged blunt face of a circular cylinder aligned coaxially with the free stream. Velocity fluctuations were produced with an acoustic driver located within the cylinder and a small circumferential gap located immediately downstream of the fixed separation line to allow communication with the external flow. The flow could be considerably modified when forced at frequencies lower than the initial Kelvin-Helmholtz frequencies of the free shear layer, and with associated vortex wavelengths comparable to the bubble height. Reattachment length, bubble height, pressure at separation, and average pressure on the face were all reduced. The effects on the large-scale structures were studied using flow photographs obtained by the smoke-wire technique. The forcing increased the entrainment near the leading edge. It was concluded that the final vortex of the shear layer before reattachment is an important element of the flow structure. There are two different instabilities involved: the Kelvin-Helmholtz instability of the free shear layer and the ‘shedding’-type instability of the entire bubble. The former consists of an interaction of the shear layer vorticity with itself, the latter with its images that result because of the presence of a wall. In order to determine the optimum forcing frequency, a method of frequency scaling is proposed which correlates data for a variety of bubbles and supports an analogy with Kármán vortex shedding.

At the beginning of the last century Worthington and Cole discovered that the high-speed jets ejected after the impact of an axisymmetric solid on a liquid surface are intimately related to the formation and collapse of an air cavity created in the wake of the impactor. In this paper, we combine detailed boundary-integral simulations with analytical modelling to describe the formation of such Worthington jets after the impact of a circular disk on water. We extend our earlier model in Gekle et al. (Phys. Rev. Lett., vol. 102, 2009a, 034502), valid for describing only the jet base dynamics, to describe the whole jet. We find that the flow structure inside the jet may be divided into three different regions: the axial acceleration region, where the radial momentum of the incoming liquid is converted to axial momentum; the ballistic region, where fluid particles experience no further acceleration and move constantly with the velocity obtained at the end of the acceleration region; and the jet tip region, where the jet eventually breaks into droplets. From our modelling of the ballistic region we conclude that, contrary to the case of other physical situations where high-speed jets are also ejected, the types of Worthington jets studied here cannot be described using the theory of hyperbolic jets of Longuet-Higgins (J. Fluid Mech., vol. 127, 1983, p. 103). Most importantly, we find that the velocity and the shape of the ejected jets can be well predicted at any instant in time with the only knowledge of quantities obtained before pinch-off occurs. This fact allows us to provide closed expressions for the jet velocity and the sizes of the ejected droplets as a function of the velocity and the size of the impactor. We show that our results are also applicable to Worthington jets emerging after the collapse of a bubble growing from an underwater nozzle, although this system creates thicker jets than the disk impact.

The dynamic contact angle formed when silicone oil displaces air from the surface of glass has been measured. Even though the glass was neither treated in any special way nor cleaned by an elaborate technique, the standard deviation associated with our measurements was approximately 1·5°. A sequence of experiments revealed that the dynamic contact angle depends on both the speed at which the oil spreads across the glass and the size of the characteristic length scale associated with the device within which it is measured. It is shown that the latter implies directly that: (i) the measured and the actual contact angles are not the same; (ii) the usual hydrodynamic model for fluids is inadequate when a moving contact line is present. These conclusions are consistent with recent theoretical studies.

We investigate the settling speeds and root mean square (r.m.s.) velocities of inertial particles in isotropic turbulence with gravity using experiments with water droplets in air turbulence from 32 loudspeaker jets and direct numerical simulations (DNS). The dependence on particle inertia, gravity and the scales of both the smallest and largest turbulent eddies is investigated. We isolate the mechanisms of turbulence settling modification and find that the reduced settling speeds of large particles in experiments are due to nonlinear drag effects. We demonstrate using DNS that reduced settling speeds with linear drag (e.g. see Nielsen, J. Sedim. Petrol., vol. 63, 1993, pp. 835–838) only arise in artificial flows that, by design, eliminate preferential sweeping by the eddies. Gravity and inertia both reduce the particle r.m.s. velocities and falling particles are more responsive to vertical than to horizontal fluctuations. The model by Wang & Stock (J. Atmos. Sci., vol. 50, 1993, pp. 1897–1913) captures these trends.

A physical mechanism for the long-wave instability of thin liquid films is presented. We show that the many diverse systems that exhibit this instability can be classified into two large groups. Each group is studied using the model of a thin liquid film with a deformable top surface flowing down a rigid inclined plane. In the first group, the top surface has an imposed stress, while in the other, an imposed velocity. The proposed mechanism shows how the details of the energy transfer from the basic state to the disturbance are handled differently in each of these cases, and how a common growth mechanism produces the unstable motion of the disturbance.

Experimental evidence suggests that in the turbulent mixing layer the fundamental mechanism of growth is two-dimensional and little affected by the presence of vigorous three-dimensional motion. To quantify this apparent property and study the growth of streamwise vorticity, we write for the velocity field \[ {\boldmath V}(x, t) = {\boldmath U}(x, z, t) + {\boldmath u}(x, y, z, t), \] where U is two-dimensional and u is three-dimensional. In a first version of the problem U is independent of u, while in the second U is the spanwise average of V. In both cases the equation for u is linearized around U. The equations for U and u are solved simultaneously by a finite-difference calculation starting with a slightly disturbed parallel shear layer.

The solutions provide a detailed description of the growth of the three-dimensional motion. They show that its characteristics are dictated by the distribution of spanwise vorticity which results from roll-up and pairing. Pairing inhibits its growth. The solutions also demonstrate that even when the three-dimensional flow attains large amplitudes it has a negligible effect on the interaction of spanwise vortices and thus on the growth of the layer.

We compare - both analytically and numerically – two related spectral (≡ two-point) closures for the problem of the decay of temperature fluctuations convected by isotropic turbulence. The methods are the test-field model (TFM) (Kraichnan 1971; Newman & Herring 1979) and the eddy-damped quasinormal Markovian (ENQNM) approximation (Orszag 1974; Lesieur & Schertzer 1978). We show that EDQNM may be regarded as a rational approximation to, and simplification of, the TFM, except at small wavenumbers, where an additional eddy-dissipative term is needed to produce satisfactory results for the former. We consider three available methods for determining the relaxation timescales: (i) comparison with experiments, (ii) comparison with the direct-interaction approximation (DIA) in thermal equilibrium, and (iii) comparison with DIA at very small wavenumber, where it is believed to represent the dynamics properly. Comparison with both large Reynolds number and wind-tunnel Reynolds numbers is presented. For the latter, we discuss the relationship of the present theoretical results to the experiments of Warhaft & Lumley (1978) and Sreenivasan et al. (1980), and to the theoretical analysis of Corrsin (1964), Kerr & Nelkin (1980) and Antonopolos-Domis (1981).

The nonlinear development of stationary Görtler vortices leads to a highly distorted mean flow field where the streamwise velocity depends strongly not only on the wall-normal but also on the spanwise coordinates. In this paper, the inviscid instability of this flow field is analysed by solving the two-dimensional eigenvalue problem associated with the governing partial differential equation. It is found that the flow field is subject to the fundamental odd and even (with respect to the Görtler vortex) unstable modes. The odd mode, which was also found by Hall & Horseman (1991), is initially more unstable. However, there exists an even mode which has higher growth rate further downstream. It is shown that the relative significance of these two modes depends upon the Görtler vortex wavelength such that the even mode is stronger for large wavelengths while the odd mode is stronger for short wavelengths. Our analysis also shows the existence of new subharmonic (both odd and even) modes of secondary instability. The nonlinear development of the fundamental secondary instability modes is studied by solving the (viscous) partial differential equations under a parabolizing approximation. The odd mode leads to the well-known sinuous mode of break down while the even mode leads to the horseshoe-type vortex structure. This helps explain experimental observations that Görtler vortices break down sometimes by sinuous motion and sometimes by developing a horseshoe vortex structure. The details of these break down mechanisms are presented.

Nonlinear resonant wave triads composed of one finite and two infinitesimal components are unstable for sum interactions and neutrally stable for difference interactions. A similar criterion holds for tertiary interactions.

A viscous thread falling onto a steadily moving horizontal belt shows a surprisingly complex range of behaviour in experiments. Low belt speeds produce coiling, as might be expected from the behaviour of a thread falling onto a stationary surface. High belt speeds produce a steady thread, whose shape is predicted well by theory developed to describe a stretching viscous catenary with surface tension and inertia. Intermediate belt speeds show several novel modes of oscillation, which lay down a wide variety of patterns on the belt. The patterns include meanders, side kicks, slanted loops, braiding, figures-of-eight, Ws, and also period-doubled versions of figures-of-eight, meanders and coiling. The experimental boundary between steady and unsteady behaviour occurs at a slightly lower belt speed than the loss of the steady solution for a stretching catenary.

In this paper we present direct numerical simulations (DNS) of hypersonic turbulent boundary layers to study high-enthalpy effects. We study high- and low-enthalpy conditions, which are representative of those in hypersonic flight and ground-based facilities, respectively. We find that high-enthalpy boundary layers closely resemble those at low enthalpy. Many of the scaling relations for low-enthalpy flows, such as van-Driest transformation for the mean velocity, Morkovin’s scaling and the modified strong Reynolds analogy hold or can be generalized for high-enthalpy flows by removing the calorically perfect-gas assumption. We propose a generalized form of the modified Crocco relation, which relates the mean temperature and mean velocity across a wide range of conditions, including non-adiabatic cold walls and real gas effects. The DNS data predict Reynolds analogy factors in the range of those found in experimental data at low-enthalpy conditions. The gradient transport model approximately holds with turbulent Prandtl number and turbulent Schmidt number of order unity. Direct compressibility effects remain small and insignificant for all enthalpy cases. High-enthalpy effects have no sizable influence on turbulent kinetic energy (TKE) budgets or on the turbulence structure.

Measurements have been made of the Couette flow in the annular space between concentric cylinders with a radius ratio of 1.5, the outer cylinder being held stationary and the inner one rotated at speeds to give Taylor numbers in the range 1.0 × 104−2.3 × l06 times the critical value for first instability of the steady viscous flow. Mean velocities have been measured both with Pitot tubes and with linearized hot-wire anemometers, and turbulent intensities and stresses, frequency spectra and space-time correlations have been obtained using single hot-wire anemometers of X-form and linear arrays of eight single-wire anemometers. For Taylor-number ratios to the critical number less than 3 × l05, the most prominent feature of the flow is a system of toroidal eddies, encircling the inner cylinder and uniformly spaced in the axial direction with nearly the separation of the Taylor vortices of the viscous instability. They are superimposed on a background of irregular motion and, except within the thin wall layers, the toroidal eddies contribute more to the total intensity. With increase of rotation speed, the toroidal eddies lose their regularity, and they cannot be clearly distinguished at Taylor-number ratios beyond 5 × l05.

The change of flow type from quasi-regular toroidal to fully irregular turbulent takes place over an extensive range of Taylor-number ratio centred near 3 × l05, and it may be linked with changes in the thin wall layers that separate the flow boundaries from the central region of nearly constant circulation. For ratios over 5 × l05, an appreciable part of the wall layers is comparatively unaffected by flow curvature and has a logarithmic distribution of mean velocity similar to that found in channel flows. It is suggested that the motion in the wall layers changes from a set of Gortler vortices characteristic of curved-wall flow to the more irregular motion found on plane walls, causing the toroidal eddies to break into sections of length ranging from a considerable fraction of the flow perimeter to nearly the separation of the cylinders. Changes in the frequency spectra of the radial and azimuthal velocit'y fluctuations are consistent with such a change.

In this paper we develop a uniformly valid, second-order theory for calculating the unsteady incompressible flow that occurs when an airfoil is subjected to a convected sinusoidal gust. Explicit formulae for the airfoil response functions (i.e. fluctuating lift) are given. The theory accounts for the effect of the distortion of the gust by the steady-state potential flow around the airfoil, and this effect is found to have an important influence on the response functions. A number of results relevant to the general theory of the scattering of vorticity waves by solid objects are also presented.

By making simple assumptions, an analytical theory is deduced for the mean velocity behind a two-dimensional obstacle (of height h) placed on a rigid plane over which flows a turbulent boundary layer (of thickness δ). It is assumed that h [Gt ] δ, and that the wake can be divided into three regions. The velocity deficit − u is greatest in the two regions in which the change in shear stress is important, a wall region (W) close to the wall and a mixing region (M) spreading from the top of the obstacle. Above these is the external region (E) in which the velocity field is an inviscid perturbation on the incident boundary-layer velocity, which is taken to have a power-law profile U(y) = U∞(y − y1)n/δn, where n [Gt ] 1. In (M), assuming that an eddy viscosity (= KhU(h)) can be defined for the perturbed flow in terms of the incident boundary-layer flow and that the velocity is self-preserving, it is found that u(x,y) has the form $\frac{u}{U(h)} = \frac{ C }{Kh^2U^2(h)} \frac{f(n)}{x/h},\;\;\;\; {\rm where}\;\;\;\; \eta = (y/h)/[Kx/h]^{1/(n+2)}$, and the constant which defines the strength of the wake is $C = \int^\infty_0 y^U(y)(u-u_E)dy$, where u = uE(x, y) as y → 0 in region (E).

In region (W), u(y) is proportional to In y. By considering a large control surface enclosing the obstacle it is shown that the constant of the wake flow is not simply related to the drag of the obstacle, but is equal to the sum of the couple on the obstacle and an integral of the pressure field on the surface near the body.

New wind-tunnel measurements of mean and turbulent velocities and Reynolds stresses in the wake behind a two-dimensional rectangular block on a roughened surface are presented. The turbulent boundary layer is artificially developed by well-established methods (Counihan 1969) in such a way that δ = 8h. These measurements are compared with the theory, with other wind-tunnel measurements and also with full-scale measurements of the wind behind windbreaks.

It is found that the theory describes the distribution of mean velocity reasonably well, in particular the (x/h)−1 decay law is well confirmed. The theory gives the correct self-preserving form for the distribution of Reynolds stress and the maximum increase of the mean-square turbulent velocity is found to decay downstream approximately as $ (\frac{x}{h})^{- \frac{3}{2}} $ in accordance with the theory. The theory also suggests that the velocity deficit is affected by the roughness of the terrain (as measured by the roughness length y0) in proportion to In (h/y0), and there seems to be some experimental support for this hypothesis.

An approximately homogeneous turbulent shear flow is generated in an open-return wind tunnel test-section by a plane parallel-rod grid of uniform rod diameter and non-uniform rod spacing. The grid design is based upon an analysis by Owen & Zienkiewicz (1957). Hot-wire measurements taken in this flow include mean velocities, component turbulence intensities, shear and two-point space correlations, and energy spectra. In addition, microscales, obtained both from instantaneous time derivatives of the hot-wire signal and from two-point space correlations, and integral scales, calculated both from correlations and energy spectra, are reported.

Based upon these results, it is concluded that, far enough away from the grid and the test-section wall boundary layers:

1. The turbulence intensities are maintained at uniform values by the nearly constant mean shear.

2. The turbulent shear stress approaches an asymptotic value.

3. Measured two-point space correlation coefficients and one-dimensional energy spectra attain self-preserving forms.

4. When distance downstream of the grid is measured in terms of the number of ‘local’ grid rod spacings, (see discussion of microscales obtained from time derivatives), the Taylor microscale defined by the correlation coefficient Ruu(rX, 0, 0) grows linearly with this ‘effective’ distance over most of the region measured.

5. The limited number of integral scale determinations and experimental uncertainty allow only the statements that the magnitude of the longitudinal scale is roughly one-eighth the lateral dimension of the square test-section and tends to increase slightly with ‘effective’ distance from the grid.

6. The lateral integral scales are approximately one-half the longitudinal scales and also increases with distance from the grid.

7. The integral scale which characterizes the size of the eddy primarily responsible for momentum transfer is roughly one-tenth the test-section lateral dimension (measured at one point only).

The sedimentation of small clusters of uniform spheres, falling freely through a viscous liquid, has been studied with Reynolds numbers (based on diameter of the sphere and its velocity of free fall in the unbounded fluid) of individual spheres ranging from 10−4 to 10. The fall velocity of a cluster is, in all cases, greater than that of individual spheres, the more so when the spheres are closer together. Two spheres falling side-by-side rotate inwards and separate as they fall if Re > 0·05, but no rotation nor separation is observed for Re < 0·03. When equal-sized spheres of Re > 1 fall vertically one behind the other, the rear sphere is accelerated into the wake of the leader, rotates, round it and separates from it when the line of centres is horizontal. If two spheres of unequal size but the same individual terminal velocity fall together, the smaller always travels faster than the larger. When three similar equally spaced spheres are dropped in a horizontal line, they interchange positions but do not separate when 0·06 < Re < 0·16. But, if 0·16 < Re < 3, one sphere is always left behind; which sphere depends critically upon the initial spacing. If three to six equal spheres, of 0·06 < Re < 7, start falling as a compact cluster, they eventually draw level and arrange themselves in the same horizontal plane at the vertices of a regular polygon. The polygon expands at a decreasing rate during fall. When three spheres are arranged initially in a horizontal isosceles triangle, the spheres oscillate about their equilibrium positions but eventually the spheres form a stable equil triagnle. If Re > 7, or the cluster contains 7 or more equal spheres, it shows no tendency to form a regular polygon but breaks up into two or more groups. A regular heptagon, and a hexagon with an additional sphere at its centre, are also unstable.

The interaction of eddies with variations in topography, together with a tendency for large-scale wave propagation, generates a systematic stress which acts upon large-scale mean flows. This stress resists the midlatitude tropospheric westerlies, resists the oceanic Antarctic Circumpolar Current, and may be a dominant mechanism in driving coastal undercurrents. Associated secondary circulation provides a systematic upwelling in coastal oceans, pumping deeper water onto continental shelf areas. The derivation rests in turbulence closure theory and is supported by numerical experiments.

The instability of small gas bubbles moving uniformly in various liquids is investigated experimentally and theoretically.

The experiments consist of the measurement of the size and terminal velocity of bubbles at the threshold of instability in various liquids, together with the physical properties of the liquids. The results of the experiments indicate the existence of a universal stability curve. The nature of this curve strongly suggests that there are two separate criteria for predicting the onset of instability, namely, a critical Reynolds number (202) and a critical Weber number (1.26). The former criterion appears to be valid for bubbles moving uniformly in liquids containing impurities and in the somewhat more viscous liquids, whereas the latter criterion is for bubbles moving in pure, relatively inviscid liquids.

The theoretical analysis is directed towards an investigation of the possibility of the interaction of surface tension and hydrodynamic pressure leading to unstable motions of the bubble, i.e. the existence of a critical Weber number. Accordingly, the theoretical model assumes the form of a general perturbation in the shape of a deformable sphere moving with uniform velocity in an inviscid, incompressible fluid medium of infinite extent. The calculations lead to divergent solutions above a certain Weber number, indicating, at least qualitatively, that the interaction of surface tension and hydrodynamic pressure can result in instabilities of the bubble motion.

A subsequent investigation of the time-independent equations, however, shows the presence of large deformations in shape of the bubble prior to the onset of unstable motion, which is not compatible with the approximation of perturbing an essentially spherical bubble. This deformation and its possible effects on the stability calculation are therefore determined by approximate methods. From this it is concluded that the deformation of the bubble serves to introduce quantitative, but not qualitative, changes in the stability calculation.

The objective of this study is to develop a method of controlling vortex shedding behind a bluff body using control theory. A suboptimal feedback control procedure for local sensing and local actuation is developed and applied to the flow behind a circular cylinder. The location of sensors for feedback is limited to the cylinder surface and the control input from actuators is the blowing and suction on the cylinder surface. Three different cost functionals to be minimized (J1 and J2) or maximized (J3) are investigated: J1 is proportional to the pressure drag of the cylinder, J2 is the square of the difference between the target pressure (inviscid flow pressure) and real flow pressure on the cylinder surface, and J3 is the square of the pressure gradient on the cylinder surface, respectively. Given the cost functionals, the flow variable to be measured by the sensors and the control input from the actuators are determined from the suboptimal feedback control procedure. Several cases for each cost functional have been numerically simulated at Re = 100 and 160 to investigate the performance of the control algorithm. For all actuations, vortex shedding becomes weak or disappears, and the mean drag and drag/lift fluctuations significantly decrease. For a given magnitude of the blowing/suction, reducing J2 provides the largest drag reduction among the three cost functionals.