The structure of the pressure and velocity field in the air above progressive, mechanically generated water waves was investigated in order to evaluate the influence of a mobile and deformable boundary on turbulence production and the related bursting phenomena. The Reynolds stress fluctuations were measured in a transformed Eulerian wave-following frame of reference, in a wind-wave research facility at Stanford University.

The structure of the wave-coherent velocity field was found to be very sensitive to the height of the critical layer below which the waves travel faster than the wind. Because the critical-layer height changes rapidly with the ratio (c / u*) of the wave speed to the wind friction velocity, the structure of the wave-coherent velocities depends strongly on the parameter c/Uδ0, where Uδ0 is the mean free-stream wind velocity. When the critical height is large enough that most of the flow in the turbulent boundary layer is below the critical height, the structure of the wave-coherent velocities is strongly affected by the Stokes layer (in the air), which under the influence of turbulence can have thickness comparable with the wave amplitude. In contrast, when the critical height is small enough that most of the flow in the boundary layer is above the critical height, the structure of the wave-coherent velocities is strongly affected by the critical layer. The latter was found to be nonlinear and turbulently diffusive.

The dependence of the structural behaviour of the wave-coherent velocity field upon the critical and Stokes layers results in considerable modifications of the turbulence-generating mechanism during the bursting-cycle, as the dimensionless wave speed c/Uδ0 changes. Such modifications are manifested by an enhancement of the contributions to the mean Reynolds stress of the bursting events (relative to their solid-wall counterparts), and their dependence on the dimensionless wave speed. For c/Uδ0 [ges ] 0.68 (or c/u* > 20), the nonlinear critical-layer thickness is large compared to the wave amplitude (except when c/Uδ0 = 0.68), and the diffused Stokes layer stimulates the wave-associated stress production. In the water proximity, the bursting contributions remain nearly constant with dimensionless wave speed; ejections account for 90% of the mean Reynolds stress, whereas sweeps provide 77%, the excess over 100% being balanced by the outward and inward interactions. For c/Uδ0 < 0.68, the critical-layer thickness is smaller than the wave amplitude and all contributions increase gradually with c/Uao. However, the ratio of ejection to sweep contributions remains unaltered and ≈ 1.15, indicating that sweeps are nearly as energetic as ejections a t all dimensionless wave speeds. The value of c/Uδ0 ≈ 0.68 appears to separate the flow regimes of high and low critical level, respectively, where significant and weak production of the wave-associated stresses have been found. Near the water surface the height distribution of the fractional contributions of the bursting events is also sensitive to the ratio c/UJO. In the equilibrium region of the boundary layer it remains uniform and in the free stream rises sharply, independent of dimensionless wave speed.

The mean time period between ejections or sweeps depends on both the wave and wind field characteristics and does not scale with either the inner or the outer flow variables. The former can be determined from the time between the first two largest consecutive peaks of the phase-averaged Reynolds stress distribution.

In the water proximity, the height distribution of the normalized energy production is sensitive to c/Uδ0; only when c/Uδ0 [ges ] 0.68 does it show a peak of increasing magnitude with increasing dimensionless wave speed.

We investigate the transition to turbulence in a free shear layer which contains a thin central region of stable density stratification. The fluid is assumed to possess Prandtl number significantly larger than unity, and the flow may exhibit either Holmboe or Kelvin–Helmholtz (KH) instability, depending upon the intensity of the stratification. A sequence of two-dimensional nonlinear numerical simulations of flows near the KH–Holmboe transition (i.e. having bulk Richardson numbers near 1/4) clearly illustrates the structural relationship between Holmboe and Kelvin–Helmholtz waves. The time-dependent nonlinear wave states delivered by the simulations are subjected to a three-dimensional normal-mode stability analysis in order to discover the physical processes that might drive the flow towards a turbulent state. Strong secondary instability is found to persist up to large spanwise wavenumbers, with no indication of a preferred lengthscale. These results indicate that secondary instability may lead the flow directly into the turbulent state.

Standard approximations expressing the drag of a sphere as a function of Reynolds number are reappraised in the light of the evident requirement that drag reverses with the direction of motion. It is thereby highlighted that the relation between the drag and the velocity of a sphere is not analytic. Another, simpler example is cited to illustrate a non-analytic relation between physical properties, which is appreciated to be a common feature of hydrodynamic models that rely on the abstract notion of an infinite incompressible fluid.

Experimental results are given for the mean and fluctuation concentration in a plume from a point source in grid-generated turbulence, which has developed so as to have a width comparable to the cylinder diameter, is disturbed by a cylinder with its axis intersecting the plume axis perpendicularly. It is shown that the disturbance effect of the cylinder appears in the upstream region of about 1.5 diameter from the stagnation point. As the stagnation point is approached, the increased travel time and the enhanced molecular diffusion cause a noticeable amplification of the dissipation of concentration variance. Also, the characteristics of some conditional statistics of the plume entrained in the wake are examined in order to investigate the intermittent structure of the plume meandering produced by the Kármán vortex street.

Equations for the steady flow of an incompressible, inviscid fluid through a collapsible tube under longitudinal tension are derived by treating the tube longitudinally as a membrane, and taking the collapsibility of the tube into account in an approximate way by replacing in the equation for an axisymmetric membrane a term representing the resistance of the tube to area change by the tube law for collapsible tubes. The flow is assumed to be uniform in a cross-section. A nonlinear differential equation is obtained for the shape of the tube for given values of total pressure p0, flow rate q, longitudinal tension τ and tube law P = P(ρ); where ρ = (A/πR2)½ is the equivalent radius of the tube (A = area of a cross-section, R = radius of the unloaded, then circular tube). The equation can be integrated and analysed in the phase plane. Equilibrium points correspond to uniform flow through cylindrical tubes; saddle points correspond to subcritical flow (S < 1), centrepoints to supercritical (S > 1) and a higher-order point to critical flow (S = 1). Here S is the speed index, the ratio of the flow speed to the speed of long waves. Near centrepoints there are solutions, that represent area-periodic tubes. For a finite tube, held open at the ends, the steady flow is formulated as a two-point boundary-value problem. On the basis of numerical calculations, and a bifurcation analysis using the method of Lyapunov–Schmidt, the existence and multiplicity of the solutions of this problem are discussed and the process of flow limitation studied. For negative total pressures two collapsed solutions are found that disappear at the flow-limitation value of the flow rate. For positive total pressures a distinction is made between subcritical, critical and supercritical total pressures. In all these cases there is a multiplicity, proportional to the ratio of the tube length to [Lscr ]1(0), the wavelength of the collapsed periodic solution for vanishing flow rate, and having maximum radius ρ = 1. For subcritical total pressures increase of the flow rate leads to a gradual loss of all solutions in higher-order flow limitations until final flow limitation occurs by the mergence of two collapsed solutions. For supercritical total pressures increase of the flow rate also leads to a gradual loss of all solutions in higher-order flow limitations in a process which now also depends upon the ratio of the tube length to the wavelength L of periodic solutions with vanishing amplitude and ρ ≡ 1.

We study the instability of finite-amplitude waves on uniform vortex layers of finite thickness bounded by a plane rigid surface. A weakly nonlinear analysis of vorticity interface perturbations, and spectral stability calculations using the full equations of motion, together show that steady progressive waves are unstable to general subharmonic pertubations in the range 0.094 < d/λ < 1.7, where d is the mean layer thickness and λ is the primary wavelength. The relevance of this instability to ultimate interface filamentation is tested by performing several numerical contourdynamical simulations of the nonlinear interface evolution for initial disturbances consisting of the finite amplitude wave plus eigenfunctions obtained from the spectral calculations. The results indicate that within the band of unstable wavelengths, small perturbations to the steady non-uniform flow given by the finite amplitude wave motion (vortex equilibrium) are able to grow in magnitude, until at a time tf, the wave extremum encounters a hyperbolic critical point of the velocity field after which filamentation occurs. Arguments are put forward based on the unsteady simulations with the purpose of identifying the preferred frame of reference for viewing the kinematical events controlling the filamentation process. An estimate for tf is then made, and the mechanism of filamentation found is discussed in relation to the recently proposed nonlinear-cascade mechanism of Dritschel (1988a).

We present the solution of the idealized steady-state gravity current of height $h$ and density $\unicode[STIX]{x1D70C}_{1}$ that propagates into an ambient motionless fluid of height $H$ and density $\unicode[STIX]{x1D70C}_{2}$ with an upper surface open to the atmosphere (open channel) at high Reynolds number. The current propagates with speed $U$ and causes a depth decrease $\unicode[STIX]{x1D712}$ of the top surface. This is a significant extension of Benjamin’s (J. Fluid Mech., vol. 31, 1968, pp. 209–248) seminal solution for a fixed-top channel ($\unicode[STIX]{x1D712}=0$). In the present case, the determination of $\unicode[STIX]{x1D712}$ is part of the problem. The dimensionless parameters of the problem are $a=h/H$ and $r=\unicode[STIX]{x1D70C}_{2}/\unicode[STIX]{x1D70C}_{1}$. We show that a control-volume analysis determines $\tilde{\unicode[STIX]{x1D712}}=\unicode[STIX]{x1D712}/H$ and $\mathit{Fr}=U/(g^{\prime }h)^{1/2}$ as functions of $a$ and $r$, where $g^{\prime }=(r^{-1}-1)g$ is the reduced gravity. The system satisfies balance of volume and momentum (explicitly), and vorticity (implicitly). We develop two branches (models) of solution for the control-volume balances. The first assumes that the outflow velocity $u_{2}$ is $z$-independent, accompanied by a similarly homogeneous headloss; this model is exactly the counterpart of Benjamin’s solution ($z$ is the vertical coordinate). The second model assumes that $u_{2}$ increases from $0$ in a layer of finite thickness $\unicode[STIX]{x1D702}$ (say) above the $z=h$ interface, and the headloss occurs in this layer only (a vortex wake). The $\mathit{Fr},\tilde{\unicode[STIX]{x1D712}}$ results of these models are in excellent agreement, and this increases the confidence concerning the physical relevance of the analysis. (The solution is obtained numerically.) The predicted flows are in general dissipative, and thus physically energetically valid only for $a\leqslant a_{max}(r)\approx 0.5r$, where non-negative dissipation appears. In this range of validity, the open-surface $\mathit{Fr}(a,r)$ is smaller than Benjamin’s $\mathit{Fr}_{b}(a)$, but the reduction is not dramatic, typically a few per cent. The current with thickness $h=a_{max}H$ is energy conserving. For given $r$, the energy conserving $a_{max}(r)$ provides the maximum height depression $\tilde{\unicode[STIX]{x1D712}}_{max}(r)\approx 0.45r^{1/4}(1-r)$. The largest possible depression is $\tilde{\unicode[STIX]{x1D712}}_{max}\approx 0.25$ at $r\approx 0.20$, and this corresponds to the minimum of $\mathit{Fr}(a,r)/\mathit{Fr}_{b}(a)$ which is approximately 0.75. For $r<0.68$, the propagation is subcritical; for $r>0.68$, the subcritical flow condition reduces the applicability slightly to $a\leqslant a_{\mathit{crit}}\approx 0.345$. For a Boussinesq system with $r\approx 1$, we obtain $\tilde{\unicode[STIX]{x1D712}}\ll 1$, and the present $\mathit{Fr}$, dissipation and $a_{\mathit{crit}}$ results differ only slightly from Benjamin’s classical predictions, as expected.

An analytical treatment is given of the problem of the establishment of the flow through a dam or levee with a horizontal underdrain, when the head behind it is raised and then kept at a constant value. The essential idea employed in the analysis is to consider the unsteady flow as a time-dependent perturbation of the final steady flow. The unsteady potential ϕ(x, y, t) is expanded in a power series of e−λt, of the form $\phi (x,y,t) = \phi_0(x,y) + \phi_1(x,y)e^ {\lambda t} + O(e^{-2\lambda t})$ where ϕ0(x, y) is the known steady-state potential, ϕ1(x, y) is a perturbation potential and O(e−2λt) = ϕ2(x, y) e−2δt + ϕ3(x, y) e−3λt + …. Each of the terms ϕn(x, y)e−nλt can be thought of as being a perturbation term of its precursor in the series, and the present approach is limited to the computation of the first perturbation term ϕ1(x, y)e−λt.

It is shown that ϕ1 satisfies Laplace's equation ∇2ϕ1 = 0 in a dimensionless hodograph plane. The free-boundary condition is linear but complicated, containing the eigenvalue λ, which is fixed by a determinantal equation. The amplitude of the displacement of the free surface is left undetermined; only the mode of the motion and the eigenvalue are computed. The results of a numerical example are summarized.

A two-dimensional free-jet water tunnel developed at the St Anthony Falls Hydraulic Laboratory of the University of Minnesota is described briefly. Results of experimental measurements on a two-dimensional cup, symmetrical wedges, inclined flat plates, and a circular cylinder in the tunnel are given.

Measured force coefficients at zero cavitation number are in good agreement with theory. Shapes of the cavities were computed for one of the wedges and for one of the plates at zero cavitation number; the observed shapes are also in good agreement with the theory.

For non-zero cavitation numbers, theoretical results for force coefficients were available for comparison in only two cases. For one of these, the cup, agreement between theory and experiment was good up to a cavitation number of about 0.5. For the other, a symmetrical wedge, experimental results were compared with a linear theory with good agreement for cavitation numbers between about 0.1 and 0.3. In the case of the wedge, measured cavity lengths were somewhat shorter than predicted by the linear theory. All other comparisons with theory at non-zero cavitation number had to be made with the theory as developed for infinite fluid. The experimental force coefficients were less than predicted by infinite-fluid theory, but tended to approach the theoretical values as the cavitation number increased. A similar tendency marked the comparison between the experimental data and data taken by others in closed tunnels.

A plane turbulent wake generated by a flat plate is subjected to a uniform distortion. It is observed that nearly two-dimensional, quasi-periodic coherent structures dominate the distorted wake. Rapid distortion theory, applied to a kinematic vortex model of the coherent structures in the undistorted far wake, predicts many of the effects revealed by a hot-wire anemometry/pattern-recognition analysis of these structures. Specifically, rapid distortion theory predicts reasonably well the observed changes in the ensemble-averaged velocity patterns and the disproportionate amplification of the large-scale coherent structures relative to the smaller-scale ‘isotropic’ eddies. These results are consistent with the view that self-preservation of the distorted wake is not possible because of the selective amplification of the coherent structures, which control the development of the wake. As well, the entrainment rate in the distorted wake increases at a rate greater than that predicted by the self-preservation theory.

Turbulent boundary layers were altered with a tandem array of manipulators arranged to produce a maximum drag reduction. The Reynolds number based on the momentum thickness, Reθ, at the first manipulator position was between 1700 and 2400. Temperature was used as a passive contaminant to explore the dynamical relationship between the near-wall and outer regions of the manipulated layer. Heat was introduced by warming the wall uniformly to 15°C above the ambient temperature or with a line heater in the wake of the manipulator. Temperature and velocity measurements showed a reduction in fluctuation amplitude and a strong decrease in larger scale mixing accompanied by a reduction of the Taylor microscale and integral lengthscale. Isocorrelations indicated that the eddy size was decreased in all three directions. The net result of the manipulators was a marked decrease in the entrainment of irrotational fluid into the boundary layer. The results suggest that the manipulators do not directly affect the wall region but rather decrease the entrainment, and hence the growth of the boundary layer, leading to possible drag reduction.

The initial development of the viscous flow in the vicinity of the sharp trailing edge of a symmetrical body, impulsively set into motion is studied within the framework of boundary-layer theory. For a blunt trailing edge there exists a similarity solution for the inviscid outer flow as has been pointed out by Proudman & Johnson (1961). The present work shows that for sharp trailing edges, however, no such solution exists.

For small or moderate trailing-edge angles, a moving singularity occurs in the solution fairly early in the flow development. The flow in the vicinity of this singularity exhibits the characteristics of unsteady separation. For large trailing-edge angles, the boundary layer becomes so thick that solutions must be terminated before there is any evidence of a singularity. For a cusped trailing edge, the solution is arbitrarily terminated to avoid large computational times and with the realization that the trailing edge flow must eventually be influenced by the leading edge at which time the present solutions cease to have meaning.

Scalar image velocimetry (SIV) is the technique to extract velocity vectors from scalar field measurements. The usual technique involves minimising a cost functional, that penalises the deviation from the scalar conservation equation. This approach requires the measured scalar field to be sufficiently resolved and relatively noise free, such that space and time derivatives of the measured scalar field can be accurately evaluated. We quantify these requirements for a synthetic two-dimensional (2-D) turbulent flow field by evaluating the velocity reconstruction accuracy as a function of the temporal and spatial resolution and the noise level. We propose an improved SIV scheme, that reconstructs not only the velocity field but also the scalar field, which does not require approximating the space and time derivatives of the measured scalar field. Improved velocity reconstruction is demonstrated for the 2-D synthetic field. We furthermore apply the scheme to interferograms of the thickness field of a falling soap film, where 2-D turbulence is generated by an array of cylindrical obstacles. The statistics of the reconstructed velocity field are within 10 % of laser Doppler velocimetry measurements.

We propose a fully spatio-temporal approach for identifying spatially varying modes of oscillation in fluid dynamics simulation output by means of multitaper frequency–wavenumber spectral analysis. One-dimensional spectrum estimation has proven to be a valuable tool in the analysis of turbulence data applied spatially to determine the rate of energy transport between spatial scales, or temporally to determine frequencies of oscillatory flows. It also allows for the quantitative comparison of flow characteristics between two scenarios using a standard basis. It has the limitation, however, that it neglects coupling between spatial and temporal structures. Two-dimensional frequency–wavenumber spectral analysis allows one to decompose waveforms into standing or travelling variety. The extended higher-dimensional multitaper method proposed here is shown to have improved statistical properties over conventional non-parametric spectral estimators, and is accompanied by confidence intervals which estimate their uncertainty. Multitaper frequency–wavenumber analysis is applied to a canonical benchmark problem, namely, a direct numerical simulation of von Kármán vortex shedding off a square wall-mounted cylinder with two inflow scenarios with matching momentum-thickness Reynolds numbers $Re_{\unicode[STIX]{x1D703}}\approx 1000$ at the obstacle. Frequency–wavenumber analysis of a two-dimensional section of these data reveals that although both the laminar and turbulent inflow scenarios show a turbulent $-5/3$ cascade in wavenumber ($\unicode[STIX]{x1D708}$) and frequency ($f$), the flow characteristics differ in that there is a significantly more prominent discrete harmonic oscillation near $(f,\unicode[STIX]{x1D708})=(0.2,0.21)$ in wavenumber and frequency in the laminar inflow scenario than the turbulent scenario. This frequency–wavenumber pair corresponds to a travelling wave with velocity near one near the centre path of the vortex street.