It is shown, by direct numerical simulations, that the skin-friction drag in a fully developed channel can be sustained below that corresponding to the laminar profile when the flow is subjected to surface blowing and suction in the form of an upstream travelling wave. A key mechanism that induces the sub-laminar drag is the creation of positive (negative) Reynolds shear stress in the wall region, where normally negative (positive) Reynolds shear stress is expected given the mean shear. This mechanism is contained in the linearized Navier–Stokes equations, thus allowing linear analysis of the observed phenomena. When applied to a fully developed turbulent channel flow, skin-friction drag is also significantly reduced by an upstream travelling wave, demonstrating that the surface blowing and suction in the form of such a wave is also effective in fully developed turbulent flows. Consideration of the energy budget shows a possibility of net drag reduction in turbulent channel flows with the present open-loop control.

The response of a wall boundary layer to the motion of a convected vortex is investigated. The principal cases considered are for a rectilinear filament of strength –κ located a distance a above a plane wall and convected to the right in a uniform flow of speed U∞*. The inviscid solution predicts that such a vortex will remain at constant height a above the wall and be convected with constant speed αU∞*. Here α is termed the fractional convection rate of the vortex, and cases in the parameter range 0 [les ] α < 1 are considered. The motion is initiated at time t* = 0 and numerical calculations of the developing boundary-layer flow are carried out for α = 0, 0.2, 0.4, 0.55, 0.7, 0.75 and 0.8. For α < 0.75, a rapid lift-up of the boundary-layer streamlines and strong boundary-layer growth occurs in the region behind the vortex; in addition an unusual separation phenomenon is observed for α [les ] 0.55. For α [ges ] 0.75, the boundary-layer development is more gradual, but ultimately substantial localized boundary-layer growth also occurs. In all cases, it is argued that a strong inviscid–viscous interaction will take place in the form of an eruption of the boundary-layer flow. The generalization of these results to two-dimensional vortices with cores of finite dimension is discussed.

We consider a thin layer of a viscous fluid flowing down a uniformly heated planar wall. The heating generates a temperature distribution on the free surface which in turn induces surface tension gradients. We model this thermocapillary flow by using the Shkadov integral-boundary-layer (IBL) approximation of the Navier–Stokes/energy equations and associated free-surface boundary conditions. Our linear stability analysis of the flat-film solution is in good agreement with the Goussis & Kelly (1991) stability results from the Orr–Sommerfeld eigenvalue problem of the full Navier–Stokes/energy equations. We numerically construct nonlinear solutions of the solitary wave type for the IBL approximation and the Benney-type equation developed by Joo et al. (1991) using the usual long-wave approximation. The two approaches give similar solitary wave solutions up to an $O(1)$ Reynolds number above which the solitary wave solution branch obtained by the Joo et al. equation is unrealistic, with branch multiplicity and limit points. The IBL approximation on the other hand has no limit points and predicts the existence of solitary waves for all Reynolds numbers. Finally, in the region of small film thicknesses where the Marangoni forces dominate inertia forces, our IBL system reduces to a single equation for the film thickness that contains only one parameter. When this parameter tends to zero, both the solitary wave speed and the maximum amplitude tend to infinity.

In this study, we performed simulations of turbulent flow over rectangular ribs transversely mounted on one side of a plane in a channel, with the other side being smooth. The separation between ribs is large enough to avoid forming stable vortices in the spacing, which exhibits k-type, or sand-grain roughness. The Reynolds number Reτ of our representative direct numerical simulation case is 460 based on the smooth-wall friction velocity and the channel half-width. The roughness height h is estimated as 110 wall units based on the rough-wall friction velocity. The velocity profile and kinetic energy budget verify the presence of an equilibrium, logarithmic layer at y≳2h. In the roughness sublayer, however, a significant turbulent energy flux was observed. A high-energy region is formed by the irregular motions just above the roughness. Visualizations of vortical streaks, disrupted in all three directions in the roughness sublayer, indicate that the three-dimensional flow structure of sand-grain roughness is replicated by the two-dimensional roughness, and that this vortical structure is responsible for the high energy production. The difference in turbulence structure between smooth- and rough-wall layers can also be seen in other flow properties, such as anisotropy and turbulence length scales.

Shallow-water equations with bottom drag and viscosity are used to study the dynamics of roll waves. First, we explore the effect of bottom topography on linear stability of turbulent flow over uneven surfaces. Low-amplitude topography is found to destabilize turbulent roll waves and lower the critical Froude number required for instability. At higher amplitude, the trend reverses and topography stabilizes roll waves. At intermediate topographic amplitude, instability can be created at much lower Froude numbers due to the development of hydraulic jumps in the equilibrium flow. Second, the nonlinear dynamics of the roll waves is explored, with numerical solutions of the shallow-water equations complementing an asymptotic theory relevant near onset. We find that trains of roll waves undergo coarsening due to waves overtaking one another and merging, lengthening the scale of the pattern. Unlike previous investigations, we find that coarsening does not always continue to its ultimate conclusion (a single roll wave with the largest spatial scale). Instead, coarsening becomes interrupted at intermediate scales, creating patterns with preferred wavelengths. We quantify the coarsening dynamics in terms of linear stability of steady roll-wave trains.

In Part 1 (Kim & Yue 1989), we considered the second-order diffraction of a plane monochromatic incident wave by an axisymmetric body. A ring-source integral equation method in conjunction with a novel analytic free-surface integration in the entire local-wave-free domain was developed. To generalize the second-order theory to irregular waves, say described by a continuous spectrum, we consider in this paper the general second-order wave–body interactions in the presence of bichromatic incident waves and the resulting sum- and difference-frequency problems. For completeness, we also include the radiation problem and second-order motions of freely floating or elastically moored bodies. As in Part 1, the second-order sum- and difference-frequency potentials are obtained explicitly, revealing a number of interesting local behaviours of the second-order pressure. For illustration, the quadratic transfer functions (QTF's) for the sum- and difference-frequency wave excitation and body response obtained from the present complete theory are compared to those of existing approximation methods for a number of simple geometries. It is found that contributions from the second-order potentials, typically neglected, can dominate the total load in many cases.

A simple mathematical model for the flow in a conical cyclone is developed which allows solutions to be obtained in closed form. The flow in the main body of the cyclone is regarded as inviscid but the nature of the fluid entry to the device and the conical geometry ensure that secondary flows develop which make the flow highly rotational. The results of the theory are compared with data from two quite different experimental investigations, and good agreement is obtained.

Unsteady flow due to an oscillating sphere with a velocity U0cosωt’, in which U0 and ω are the amplitude and frequency of the oscillation and t’ is time, is investigated at finite Reynolds number. The methods used are: (i) Fourier mode expansion in the frequency domain; (ii) a time-dependent finite difference technique in the time domain; and (iii) a matched asymptotic expansion for high-frequency oscillation. The flow fields of the steady streaming component, the second and third harmonic components are obtained with the fundamental component. The dependence of the unsteady drag on ω is examined at small and finite Reynolds numbers. For large Stokes number, ε = (ωa2/2v)½ [Gt ] 1, in which a is the radius of the sphere and v is the kinematic viscosity, the numerical result for the unsteady drag agrees well with the high-frequency asymptotic solution; and the Stokes (1851) solution is valid for finite Re at ε [Gt ] 1. For small Strouhal number, St = ωa/U0 [Lt ] 1, the imaginary component of the unsteady drag (Scaled by 6πU0pfva, in which Pf is the fluid density) behaves as Dml ∼ (h0Stlog St–h1St), m = 1,3,5… This is in direct contrast to an earlier result obtained for an unsteady flow over a stationary sphere with a small-amplitude oscillation in the free-stream velocity (hereinafter referred to as the SA case) in which D1∼ –h1St (Mei, Lawrence & Adrian 1991). Computations for flow over a sphere with a free-stream velocity U0(1–α1+α1cosωt’) at Re = U02a/v = 0.2 and St [Lt ] 1 show that h0 for the first mode varies from 0 (at α1 = 0) to around 0.5 (at α1 = 1) and that the SA case is a degenerated case in which the logarithmic dependence of the drag in St is suppressed by the strong mean uniform flow.

The numerical results for the unsteady drag are used to examine an approximate particle dynamic equation proposed for spherical particles with finite Reynolds number. The equation includes a quasi-steady drag, an added-mass force, and a modified history force. The approximate expression for the history force in the time domain compares very well with the numerical results of the SA case for all frequencies; it compares favourably for the PO case for moderate and high frequencies; it underestimates slightly the history force for the PO case at low frequency. For a solid sphere settling in a stagnant liquid with zero initial velocity, the velocity history is computed using the proposed particle dynamic equation. The results compare very well with experimental data of Moorman (1955) over a large range of Reynolds numbers. The present particle dynamic equation at finite Re performs consistently better than that proposed by Odar & Hamilton (1964) both qualitatively and quantitatively for three different types of spatially uniform unsteady flows.

The present investigation of aerofoil self-noise generation and propagation concerns the effects of mean flow and quadrupole sources on the broadband noise that arises from the interaction of turbulent boundary layers with the aerofoil trailing edge and the tonal noise that arises from vortex shedding generated by laminar boundary layers and trailing-edge bluntness. Compressible large-eddy simulations (LES) are conducted for a NACA0012 aerofoil with rounded trailing edge for four flow configurations with different angles of incidence, boundary layer tripping configurations and free-stream Mach numbers. The Reynolds number based on the aerofoil chord is fixed at . The acoustic predictions are performed by the Ffowcs Williams & Hawkings (FWH) acoustic analogy formulation and incorporate convective effects. Surface and volume integrations of dipole and quadrupole source terms appearing in the FWH equation are performed using a three-dimensional wideband multi-level adaptive fast multipole method (FMM) in order to accelerate the calculations of aeroacoustic integrals. In order to validate the numerical solutions, flow simulation and acoustic prediction results are compared to experimental data available in the literature and good agreement is observed in terms of both aerodynamic and aeroacoustic results. For low-Mach-number flows, quadrupole sources can be neglected in the FWH equation and mean flow effects appear only for high frequencies. However, for higher speeds, convection effects are relevant for all frequencies and quadrupole sources have a more pronounced effect for medium and high frequencies. The convective effects are most readily observed in the upstream direction.

A numerical model is proposed for the flow in a deep turbulent boundary layer over water waves. The momentum equations are closed by the use of an isotropic eddy viscosity and the turbulent energy equation. For small amplitudes the results are similar to those of Townsend's (1972) linear model, but nonlinear effects become important as the ratio of wave height to wavelength increases. With uniform surface roughness zo, the predicted fractional rate of energy input per radian advance in phase, ζ, decreases slightly with increasing amplitude and is of the same order of magnitude as in Miles’ (1957, 1959) and Townsend's linear theories. If zo is allowed to vary with position along the wave, however, the fractional rate of energy input can be significantly increased for small amplitude waves. If the variation in zo is half the mean value and the maximum wave slope zak is 0.01, we find ζ ≈ 60 (ρair/ρwater) (uo/c)2, where uo is the friction velocity and c the wave phase speed. Comparison is also made with recent laboratory and field data.

Experiments have been carried out to quantify the mixing induced by the interaction of a weak shock wave with a cylindrical volume of a gas (helium) that is lighter than its surroundings (air). In these experiments a round laminar jet was used to produce the light-gas cylinder, and planar laser-induced fluorescence (PLIF), utilizing a fluorescent tracer (biacetyl) mixed with the helium, was used to visualize the flow. These techniques provide a higher quality of flow visualization than that obtained in previous investigations. In addition, the PLIF technique could be used for the measurement of species concentration. The distortion of the helium cylinder produced by the passing shock wave was found to be similar to that displayed by images from previous experimental and computational investigations. The downstream displacement of several points on the boundary of the light-gas cylinder are measured and agree reasonably well with the results of earlier experimental and theoretical studies as well. Because the mixing process causes the helium originally contained within the cylinder to be dispersed into the surrounding air, the PLIF image area inside the contour at one half the maximum concentration of the fluorescent tracer decreases as the two gases mixed. The change in this area is used as a measure of the mixing rate, and it is found that the time rate of change of this area divided by the area of the initial jet is approximately −0.7 × 103 s−1.

Under certain conditions, the free surface of a conducting liquid subject to an electric field elongates into a cone whose apex emits a thin stationary jet that carries an electric current. The structure of the flow in the cone-to-jet transition region is investigated here, assuming that the size of this region is small compared with any other length of the system where the conical meniscus is formed. The local problem depends then on three non-dimensional parameters, two of which are properties of the liquid while the third measures the flow rate injected through the meniscus. Numerical solutions are computed and the electric current is determined as a function of these parameters. A qualitative asymptotic analysis of the physically important limit of large non-dimensional flow rates gives an electric current increasing as the square root of the flow rate and independent of the dielectric constant of the liquid. When the inertia of the liquid is taken into account, the flow in this asymptotic limit is effectively inviscid in the bulk of the transition region, where the electric current is dominated by conduction in the liquid and the surface is close to an equipotential of the electric field in the gas. The effects of the viscosity of the liquid, the current transported by convection of the surface charge, and the electric shear at the surface come into play in a slender region of the jet. The limit of small non-dimensional flow rates is briefly discussed.

Direct numerical simulations (DNS) are performed to investigate the evolution of turbulence in a uniformly sheared and stably stratified flow. The spatial discretization is accomplished by a spectral collocation method, and the solution is advanced in time with a third-order Runge–Kutta scheme. The turbulence evolution is found to depend strongly on at least three parameters: the gradient Richardson number Ri, the initial value of the Taylor microscale Reynolds number Reλ, and the initial value of the shear number SK/<ε. The effect of each parameter is individually studied while the remaining parameters are kept constant. The evolution of the turbulent kinetic energy K is found to follow approximately an exponential law. The shear number SK/<ε, whose effect has not been investigated in previous studies, was found to have a strong non-monotone influence on the turbulence evolution. Larger values of the shear number do not necessarily lead to a larger value of the eventual growth rate of the turbulent kinetic energy. Variation of the Reynolds number Reλ indicated that the turbulence growth rate tends to become insensitive to Reλ at the higher end of the Reλ range studied here. The dependence of the critical Richardson number Ricr, which separates asymptotic growth of the turbulent kinetic energy K from asymptotic decay, on the initial values of the Reynolds number Reλ and the shear number SK/<ε was also obtained. It was found that the critical Richardson number varied over the range 0.04<Ricr<0.17 in our DNS due to its strong dependence on Reynolds and shear numbers.

This note presents measurements of radial and axial distributions of mean velocity, turbulent intensities and kinetic energy as well as radial distributions of the turbulent shear stress in the initial region of a turbulent air jet issuing from a long round pipe into still air. The pipe flow is transformed relatively smoothly into a jet flow. In the core subregion the mean centre-line velocity decreases slightly. The highest turbulence occurs at an axial distance of about 6d and radius of (0·7 to 0·8)d. On the axis the highest turbulent kinetic energy appears at a distance of (7·5 to 8·5)d. Normalized distributions of the turbulent quantities are in good agreement with known data on the developed region of jets issuing from short nozzles.

The plane Couette system does not exhibit any secondary solutions bifurcating from the primary solution of constant shear. Since the work of Nagata (1990) it has been well known that three-dimensional steady solutions exist. Here the manifold of those steady solutions is explored in the parameter space of the problem and their instabilities are investigated. These instabilities usually lead to time-periodic solutions whose properties do not differ much from those of the steady solutions except that the amplitude varies in time. In some cases travelling wave solutions which are asymmetric with respect to the midplane of the layer are found as quaternary states of flow. Similarities with longitudinal vortices recently observed in experiments are discussed.

Short laminar separation bubbles can develop on a flat plate due to an externally imposed pressure gradient. Here, these bubbles are computed by means of direct numerical simulations. Laminar–turbulent transition occurs in the bubble, triggered by small disturbance input with fixed frequency, but varying amplitude, to keep the bubbles short. The forcing amplitudes span a range of two orders of magnitude. All resulting bubbles differ with respect to their mean flow, linear-stability characteristics and distance between transition and mean reattachment locations. Mechanisms responsible for these differences are analysed in detail. Switching off the disturbance input or reducing it below a certain, very small threshold causes the short bubble to grow continuously. Eventually, it no longer exhibits typical characteristics of a short laminar separation bubble. Instead, it is argued that bursting has occurred and the bubble displays characteristics of a long-bubble state, even though this state was not a statistically steady state. This hypothesis is backed by a comparison of numerical results with measurements. For long bubbles, the transition to turbulence is not able to reattach the flow immediately. This effect can lead to the bursting of a short bubble, which remains short only when sufficiently large disturbances are convected into the bubble. Large-scale spanwise-oriented vortices at transition are observed for short but not for long bubbles. The failure of the transition process to reattach the flow in the long-bubble case is ascribed to this difference in transitional vortical structures.

Small perturbations of a choked flow through a thin annular nozzle are investigated. Two cases are considered, corresponding to a ‘choked outlet’ and a ‘choked inlet’ respectively. For the first case, either an acoustic or entropy or vorticity wave is assumed to be travelling downstream towards the nozzle contraction. An asymptotic analysis for low frequency is used to find the reflected acoustic wave that is created. The boundary condition found by Marble & Candel (1977) for a compact choked nozzle is shown to apply to first order, even for circumferentially varying waves. The next-order correction can be expressed as an ‘effective length’ dependent on the mean flow (and hence the particular geometry of the nozzle) in a quantifiable way.

For the second case, an acoustic wave propagates upstream and is reflected from a convergent–divergent nozzle. A normal shock is assumed to be present. By considering the interaction of the shock's position and flow perturbations, the reflected propagating waves are found for a compact nozzle. It is shown that a significant entropy disturbance is produced even when the shock is weak, and that for circumferential modes a vorticity wave is also present. Numerical calculations are conducted using a sample geometry and good agreement with the analysis is found at low frequency in both cases, and the range of validity of the asymptotic theory is determined.

Material flow in partially filled rotating cylinders (rotary kilns) is encountered in many practical applications of material processing, for example incineration, calcination, grain drying, etc. The flow behaviour in the cross-section is important to other transport mechanisms such as mixing and energy distribution within the bed material. The paper describes an experimental study which was carried out with the objective of understanding and improving our predictive capabilities of the rheological behaviour of granular materials in rotary cylinders. Measurement techniques similar to that used in chute flows have been employed to measure flow characteristics, e.g. particle velocities, granular temperature, and solid concentration (in the shear layer developed between the free surface and the bulk of the bed) for different materials having a wide range of coefficients of restitution. The results of the experiments provide the necessary assumptions, constraints, and data for granular flows in partially filled rotating cylinders.

A two-dimensional direct numerical simulation of the natural convection flow of air in a differentially heated square cavity was performed for a Rayleigh number of 1010. The simulation was commenced from isothermal and quiescent conditions and was allowed to proceed to a statistical steady state. Two-dimensional turbulence resulted without the introduction of random forcing. Good agreement of mean quantities of the statistically steady flow is obtained with available experimental results. In addition, the previously proposed (George & Capp 1979) −$\frac{1}{3}$ and $+\frac{1}{3}$ temperature and velocity variations in the buoyant sublayer are confirmed. Other statistics of the flow are consistent with available experimental data. Selected frames from a movie generated from the computational results show very clearly turbulence production via the sequence from initial instability, proceeding through transition, and eventually reaching statistical steady state. Prominent large-scale structures are seen to persist at steady state.

The transition and the development of turbulence in the near wake of a circular cylinder are investigated using hot-wire anemometry and flow visualization. The formation zone of the large regular vortices is studied in the subcritical regime (2000 < U0D/v < 60000). with and without the introduction of a splitter plate. Two different regimes are identified in the interaction between the von Kármán vortices and those of the shear layer emerging from the separated boundary layer. Experimental evidence is given in support of the strong coupling at low Reynolds numbers characterized by phase modulations between the two types of structures. The interaction is weaker at high Reynolds numbers where the small-scale vortices are disconnected from the regular vortex shedding, giving rise to an intermittent pattern. Spectral properties are used to describe the different stages of the interaction between the shear-layer vortices and the alternating ones. Physical properties of the interaction are examined separately in a numerical simulation using a pressure-velocity formulation. Both unexcited and excited two-dimensional plane mixing layers are studied using streakline maps and time traces of the dynamical properties. The main features of the simulated vortex development are in agreement with the experimental results.