A numerical investigation of the velocity, pressure and vorticity fields very near the injection of flat and thin two-dimensional gas jets or liquid sheets between two parallel high-speed gas coflows is performed. The motivation of this research is to uncover some basic physical mechanisms underlying twin-fluid atomization. Conservation equations and boundary and initial conditions are presented for both single-phase jets and two-phase liquid sheet/gas-stream systems. Both infinitely thin and thick solid walls are considered. Apart from the gas Strouhal and Reynolds numbers appearing in the dimensionless single-phase flow equations, the liquid Reynolds number, the momentum flux ratio, the gas/liquid velocity ratio and the Weber number enter the two-phase flow dimensionless formulation. The classical numerical techniques for single-phase jets are supplemented with the volume-of-fluid (VOF) method for interface tracking and the continuum surface force (CSF) method to include surface tension in two-phase flow systems. Ad hoc convection algorithms in combination with a developed version of the fractional-step scheme allows a significant reduction of the numerical diffusion, maintaining localized and sharp interfaces. The action of the surface tension is correctly found via the CSF with a smoothed scalar-field approximation.

Results for single-phase jets with thin-wall injectors indicate qualitatively correct features and trends when varying the Reynolds number and the coflow/jet ratios: thick-wall injectors significantly modify the vorticity and pressure near fields; increasing the Reynolds number leads to larger flow disturbances; larger coflow/jet velocity ratios yield more perturbed near flow fields. For single-phase jets the Strouhal number as a function of the Reynolds number follows the usual trends of flows behind a circular cylinder.

For two-phase flows, increasing the gas Reynolds number leads to larger liquid-sheet deformations and to a reduction of the breakup length; a plot of the gas Strouhal number, in the presence of a liquid sheet, as a function of the gas Reynolds number displays a monotonically decreasing curve, contrary to that for a gas jet. This observation strongly suggests that the gas vortex shedding mechanism is modified by the liquid-sheet motion. The gas vortex shedding frequency as a function of the liquid-sheet oscillation frequency follows a straight line with a slope of approximately $45^{\circ}$ for momentum flux ratios greater than roughly 0.45; for values below 0.45 the gas vortex shedding frequency remains constant while the liquid sheet varies its oscillation frequency. Increasing the surface tension leads to a larger breakup length. Thin trailing edges almost double the sheet oscillation frequency and more than halve the perturbation wavelength compared to thick trailing edges.

The flow at low Reynolds number around rectangular cylinders of varying chord-to-thickness ratios under transverse periodic forcing is studied numerically. Although of relatively low amplitude, the forcing locks the shedding from both the leading and trailing edges to the applied frequency. The base suction, and the lift and the drag on the cylinders are found to be complex functions of the forcing frequency. At low Reynolds numbers and without applied forcing, the flow is controlled by a global instability with the leading- and trailing-edge shedding locked; moreover, the reduced frequency of shedding varies in a stepwise manner with the chord-to-thickness ratio. This global instability is still evident in the flows under external forcing examined in this paper. While previous researchers have conjectured that the trailing-edge shedding plays a dominant role in the preferred frequency selection in the natural shedding case, the important role of trailing-edge shedding when the flow is forced is confirmed in the present study. In particular, the individual contributions from leading- and trailing-edge vortices on the perturbation to the leading-edge shear layer are examined. In addition, it is shown that the base suction is maximum when the forcing frequency is close to the global instability frequency observed in unforced flows, thereby strengthening the argument that the unforced, forced, and duct resonant cases are strongly influenced by the same global instability. The variations of the lift, drag and formation length with chord-to-thickness ratio are quantified.

Baroclinic geostrophic adjustment in a rotating circular basin is investigated in a laboratory study. The adjustment process consists of a linear phase before advective and dissipative effects dominate the response for longer time. This work describes in detail the hydrodynamics and energetics of the linear phase of the adjustment process of a two-layer fluid from an initial step height discontinuity in the density interface $\uDelta H$ to a final response consisting of both geostrophic and fluctuating components. For a forcing lengthscale $r_f$ equal to the basin radius $R_0$, the geostrophic component takes the form of a basin-scale double gyre while the fluctuating component is composed of baroclinic Kelvin and Poincaré waves. The Burger number $S\,{=}\,R/r_f$ ($R$ is the baroclinic Rossby radius of deformation) and the dimensionless forcing amplitude $\epsilon\,{=}\,\uDelta H/H_1$ ($H_1$ is the upper-layer depth) characterize the response of the adjustment process. In particular, comparisons between analytical solutions and laboratory measurements indicate that for time $\tau$: $1 \,{<}\, \tau \,{<}\, S^{-1}$ ($\tau$ is time scaled by the inertial period $2 \pi/f$), the basin-scale double gyre is established, followed by a period where the double gyre is sustained, given by $S^{-1} \,{<}\, \tau \,{<}\, 2\epsilon^{-1}$ for a moderate forcing and $S^{-1}\,{<}\, \tau\,{<}\,\tau_D$ for a weak forcing ($\tau_D$ is the dimensionless dissipation timescale due to Ekman damping). The analytical solution is used to calculate the energetics of the baroclinic geostrophic adjustment. The results are found to compare well with previous studies with partitioning of energy between the geostrophic and fluctuating components exhibiting a strong dependence on $S$. Finally, the outcomes of this study are considered in terms of their application to lakes influenced by the rotation of the Earth.

The so-called 2/15-law for two-point third-order velocity statistics in isotropic turbulence with helicity is computed for the first time from a direct numerical simulation of the Navier–Stokes equations in a $512^3$ periodic domain. This law is a statement of helicity conservation in the inertial range, analogous to the benchmark Kolmogorov 4/5-law for energy conservation in high-Reynolds-number turbulence. The appropriately normalized parity-breaking statistics, when measured in an arbitrary direction in the flow, disagree with the theoretical value of 2/15 predicted for isotropic turbulence. They are highly anisotropic and variable and remain so over long times. We employ a recently developed technique to average over many directions and so recover the statistically isotropic component of the flow. The angle-averaged statistics achieve the 2/15 factor to within about 7% instantaneously and about 5% on average over time. The inertial- and viscous-range behaviour of the helicity-dependent statistics and consequently the helicity flux, which appear in the 2/15-law, are shown to be more anisotropic and intermittent than the corresponding energy-dependent reflection-symmetric structure functions, and the energy flux, which appear in the 4/5-law. This suggests that the Kolmogorov assumption of local isotropy at high Reynolds numbers needs to be modified for the helicity-dependent statistics investigated here.

We consider the effects of blade mean loading on the noise generated by the interaction between convected vorticity and a blade row. The blades are treated as flat plates aligned at a non-zero incidence angle, $\delta$, to the oncoming stream, and we take harmonic components of the incident vorticity field with reduced frequency $k$, and use asymptotic analysis in the realistic limit $k\,{\gg}\, 1$, $\delta \,{\ll}\, 1$ with $k\delta=O(1)$. In a previous paper (Peake & Kerschen, J. Fluid Mech., vol. 347 (1997), pp. 315–346) we have analysed the sound radiated back upstream, but the field in the blade passages and the sound radiated downstream are also of considerable practical interest, and are considered in this paper. The flow is seen to consist of inner regions around each leading edge, in which sound is generated by the local gust–airfoil and gust–flow interactions, and an outer region in which the incident gust and the acoustic radiation interact with the non-uniform mean flow and the other blades. It is shown that the complicated multiple interactions between the blades can be represented by images in potential–streamfunction space, yielding closed-form expressions for the phase distortion experienced by sound waves propagating down the blade passages. The acoustic radiation downstream of the cascade at $O(1)$ distances is dominated by the duct-mode beams that emanate from the passages, while the far downstream field is generated by the diffraction of the duct modes by the trailing edges. The modal amplitudes of the radiation field far downstream tend to be largest when the mode direction is close to the propagation direction of the duct mode which generated it, corresponding to the way (in uniform flow) in which the radiation from a single blade passage tends to be beamed in the duct-mode directions. Although the diffraction coefficient for the scattering from a single trailing edge is singular in these directions, we show how uniformly valid expressions can be derived by combining the trailing-edge fields in an appropriate way, thereby describing the larger amplitude in the beam directions. The steady non-uniform flow downstream has the effect of tilting the directions of the beams by $O(\delta)$ angles away from the duct-mode directions, which are explicitly determined. Throughout this analysis it will be seen that the interaction with the non-uniform mean flow introduces phase corrections of size $O(k\delta)$, which, given the way in which interference effects between the multiple blades dominate unsteady cascade flow, proves to be highly significant.

We study the nonlinear response of the equatorial shallow-water system at rest to a localized long-wave perturbation with small meridional to zonal aspect ratio. An asymptotic theory of such a response (adjustment) for small Rossby numbers is constructed. Possible scenarios of nonlinear adjustment are classified depending on the relation between the Rossby number and the aspect ratio. The calculations show that slow, geostrophically balanced Rossby and Kelvin waves and the fast inertia–gravity waves are dynamically split off. The fast component of motion exerts no drag on the slow one, which is proved by direct computation. Evolution equations are derived for both components confirming earlier results which were obtained by ad hoc filtering of one of the components of motion. A well-defined initialization procedure is developed for each component.

Due to the breaking of non-dispersive Kelvin waves, the asymptotic theory has obvious limits of validity. In order to go beyond these limits and to study strongly nonlinear effects during the adjustment process we undertook high-resolution shock-capturing numerical simulations based on recent progress in finite-volume numerical methods. The simulations confirm theoretical results but also reveal new effects such as fission of a strongly nonlinear Rossby-wave packet into a sequence of equatorial modons or jet formation in the wake of a breaking Kelvin wave.

An experimental study of the migration of dilute suspensions of particles in Poiseuille flow at Reynolds numbers $\hbox{\it Re}\,{=}\,67\hbox{--}1700$ was performed, with a few experiments performed at $\hbox{\it Re}$ up to 2400. The particles used in the majority of the experiments were neutrally buoyant spheres with diameters $d$ yielding a ratio of pipe to particle diameter in the range $D/d \,{=}\, 8\hbox{--}42$. The volume fraction of solids was less than 1% in all cases studied. The results of G. Segré & A. Silberberg (J. Fluid Mech.14, 136, 1962) have been extended to show that the tubular pinch effect in which particles accumulate on a narrow annulus is moved toward the wall as $\hbox{\it Re}$ increases. A careful comparison with asymptotic theory for Poiseuille flow in a channel was performed. Another inner annulus closer to the centre, and not predicted by this asymptotic theory, was observed at elevated $\hbox{\it Re}$. As $\hbox{\it Re}$ is increased, the distribution of particles over the cross-section of the tube at the measurement location, lying at a distance $L \doteq 310 D$ from the entrance, changes from one centred at the annulus predicted by the theory to one with the particles primarily on the inner annulus. The case of slightly non-neutrally buoyant particles was also investigated. A particle trajectory simulation based on asymptotic theory was performed to facilitate the comparison of theory and the experimental observations.

The flow of liquid metal in a magnetic field may become almost two-dimensional because the magnetic field inhibits velocity variations along the field lines. Two-dimensionality must break down near rigid boundaries to satisfy no-slip boundary conditions, leading to a quasi-two-dimensional flow comprising a two-dimensional core between Hartmann boundary layers. Flow in the Hartmann layers is dominated by viscosity and the Lorentz force. Pothérat, Sommeria & Moreau (J. Fluid Mech. vol. 424, 2000, p. 75, referred to herein as PSM) recently proposed a two-dimensional equation for the vertically averaged horizontal velocity to describe such flows. Their treatment extends previous work to account for inertial corrections (such as Ekman pumping) to the flow in the Hartmann layers. The inertial corrections lead to extra nonlinear terms in the vertically averaged equations, including terms with mixed spatio-temporal derivatives, in addition to the algebraic drag term found previously. The present paper shows that many of these terms coincide with a previously postulated model of two-dimensional turbulence, the anticipated vorticity method, and a subsequent modification restoring linear and angular momentum conservation that might be described as an anticipated velocity method. A fully explicit version of PSM's equation is derived, with the same formal accuracy but no spatio-temporal derivatives. This explicit equation is shown to dissipate energy, although enstrophy may increase. Numerical experiments are used to compare the effect of the various different equations (without linear drag or forcing) on both laminar and turbulent initial conditions. The mixed spatio-temporal derivatives in PSM's original equation lead to a system of differential-algebraic equations, instead of ordinary differential equations, after discretizing the spatial variables. Such systems may still be solved readily using existing software. The original and explicit versions of PSM's equation give very similar results for parameter regimes representative of laboratory experiments, and give qualitatively similar results to the anticipated velocity method. The anisotropic diffusion of vorticity along streamlines that is present in all equations studied except the Navier–Stokes equations has comparatively little effect. The additional terms in PSM's equation, and also the anticipated velocity method, that arise from Ekman pumping are much more significant. These terms lead to an outward transport of vorticity from coherent vortices, so solutions of equations with these extra terms appear much more organized and have less fine-scale structure than solutions of the Navier–Stokes equations, or even the anticipated vorticity method, with the same initial conditions. This has implications for the extent to which the self-organizing behaviour and appearance of global modes seen in laboratory experiments with thin liquid-metal layers and magnetic fields may be attributed to self-organizing properties of the unmodified two-dimensional Navier–Stokes equations.

The separation point of the flow around a circular cylinder has been numerically and experimentally investigated in the regime of Reynolds number less than 280. The present results reveal that the long-existing discrepancy in the data concerning the time-averaged separation angles reported in the literature results mainly from the oscillating characteristics of the flow separation on the cylinder surface and the experimental methodologies rather than the commonly mentioned blockage-ratio effect. In the present experiment, the time-averaged separation angles are obtained by averaging the instantaneous images from a soap-film flow visualization instead of from the commonly used streakline images from finite time exposures. Excellent agreement has been achieved between the present experimental results and numerical simulations by the spectral element method. Particle-streak visualization in a towing tank has also been conducted to compare with that of the two-dimensional soap-film experiments. It reveals that the separation angle is insensitive to the three-dimensional effect. Variations of the time-averaged separation angles with Reynolds number can be represented by a four-term $\theta _{s}\hbox{--}{\it Re}^{-1/2}$ relationship in the range of $7\,{\le}\,{\it Re}\,{\le}\,200$. Moreover, if the data in the very low Reynolds number region are excluded, a simple linear $\theta_{s}\hbox{--}{\it Re}^{-1/2}$ relationship can be derived for $10\,{\le}\,{\it Re}\,{\le}\,200$. Since the dimensionless boundary layer thickness and the Strouhal–Reynolds number relationship for the circular cylinder are also known to be proportional to ${\it Re}^{-1/2}$, this linear relationship offers direct evidence that the flow characteristics of the boundary layer extend downstream along the cylinder surface to the separation point in this ${\it Re}$-range. The blockage effect on the separation angle has also been quantitatively analysed.

A weak, laminar shear flow of a monodisperse suspension of high-Reynolds-number, low-Weber-number bubbles is studied in a novel experimental configuration. Nitrogen bubbles are formed through an array of small capillaries at the base of a tall channel with a small inclination from the vertical. The bubbles generate a unidirectional shear flow, in which the denser suspension near the bottom wall falls and the lighter suspension near the top wall rises. Profiles of the bubble and liquid velocities and the bubble volume fraction are obtained using hot-film and dual impedance probes. To our knowledge, measurements of the laminar shear properties of a nearly homogeneous bubble suspension have not previously been reported.

A steady shear flow is observed in which the bubble velocity variation across the channel is typically less than 20% of the mean bubble velocity. The velocity and volume fraction gradients increase with channel inclination and exhibit little or no dependence on the mean gas volume fraction. To explain the magnitude of the volume fraction gradients, it is necessary to consider the effects of both the lift force and the effective bubble-phase diffusivity in balancing the segregating tendency of the cross-channel component of the buoyancy force. The bubble velocity gradient can be understood in terms of a balance of the component of the buoyancy force parallel to the channel walls and an effective viscosity associated with the Reynolds stresses produced by bubble-induced liquid velocity fluctuations. Theories for bubbles rising with potential-flow hydrodynamic interactions predict an instability of the homogeneous state due to a negative Maxwell pressure. However, the hydrodynamic diffusivity inferred from our experiments is large enough to mitigate the clustering effects of the Maxwell pressure. Consistent with this, a vigorous instability of the homogeneous state of the bubble suspension is only observed at volume fractions larger than 5%–20% with the critical volume fraction depending on the angle of inclination.

Two-phase systems of microstructured complex fluids are an important class of engineering materials. Their flow behaviour is interesting because of the coupling among three disparate length scales: molecular or supra-molecular conformation inside each component, mesoscopic interfacial morphology and macroscopic hydrodynamics. In this paper, we propose a diffuse-interface approach to simulating the flow of such materials. The diffuse-interface model circumvents certain numerical difficulties in tracking the interface in the classical sharp-interface description. More importantly, our energy-based variational formalism makes it possible to incorporate complex rheology easily, as long as it is due to the evolution of a microstructure describable by a free energy. Thus, complex rheology and interfacial dynamics are treated in a unified framework. An additional advantage of our model is that the energy law of the system guarantees the existence of a solution. We will outline the general approach for any two-phase complex fluids, and then present, as an example, a detailed formulation for an emulsion of nematic drops in a Newtonian matrix. Using spectral discretizations, we compute shear-induced deformation, head-on collision and coalescence of drops where the matrix and drop phases are Newtonian or viscoelastic Oldroyd-B fluids. Numerical results are compared with previous studies as a validation of the theoretical model and numerical code. Finally, we simulate the retraction of an extended nematic drop in a Newtonian matrix as a method for measuring interfacial tension.

In high Reynolds-number turbulence, local scalar turbulence structure parameters,$( C_{\theta }^{2}) _{r}$, local scalar variance dissipation rates, $\chi _{r}$, and local energy dissipation rates, $\varepsilon _{r}$, vary randomly in time and space. This variability, commonly referred to as intermittency, is known to increase with decreasing $r$, where $r$ is the linear dimension of the local averaging volume. Statistical relationships between $\chi _{r}$, $\varepsilon _{r}$, and $( C_{\theta }^{2}) _{r}$ are of practical interest, for example, in optical and radar remote sensing. Some of these relationships are studied here, both theoretically and on the basis of recent observations. Two models for the conditionally averaged local temperature structure parameter, $\langle( C_{\theta }^{2}) _{r}| \varepsilon _{r}\rangle $, are derived. The first model assumes that the joint probability density function (j.p.d.f.) of $\chi _{r}$ and $\varepsilon _{r}$ is bivariate lognormal and that the Obukhov–Corrsin relationship, $( C_{\theta }^{2}) _{r}=\gamma\varepsilon _{r}^{-1/3}\chi _{r}$, where $\gamma\,{=}\,1.6$, is locally valid. In the second model, small-scale intermittency is ignored and $C_{\theta }^{2}$ and $\varepsilon $ are treated traditionally, that is, as averages over many outer scale lengths, such that $C_{\theta }^{2}$ and $\varepsilon $ change only as a result of large-scale intermittency. Both models lead to power-law relationships of the form $\langle( C_{\theta }^{2}) _{r}| \varepsilon _{r}\rangle \,{=}\,c\hspace{0.03in}\varepsilon _{r}^{\delta}$, where $c$ is a constant. Both models make predictions for the value of the power-law exponent $\delta$. The first model leads to $\delta\,{=}\,\rho _{xy}\sigma _{y}/\sigma _{x}-1/3$, where $\sigma _{x}$ and $\sigma _{y}$ are the standard deviations of the {logarithms} of $\varepsilon _{r}$ and $\chi _{r}$, respectively, and $\rho _{xy}$ is the correlation coefficient of the logarithms of $\chi _{r}$ and $\varepsilon _{r}$. This model leads to $\delta\,{=}\,1/3$ if $\rho _{xy}\,{=}\,2/3$ and if $\sigma _{x}\,{=}\,\sigma _{y}$. The second model predicts $\delta\,{=}\,2/3$, regardless of whether (i) static stability and shear are statistically independent, or (ii) they are connected through a Richardson-number criterion. These theoretical predictions are compared to fine-wire measurements that were taken during the night of 20/21 October 1999, at altitudes of up to 500 m in the nocturnal boundary layer and the overlying residual layer above Kansas. The fine-wire sensors were moved up and down with the University of Colorado's Tethered Lifting System (TLS). The data were obtained during the Cooperative Atmosphere-Surface Exchange Study 1999 (CASES-99). An interesting side result is that the observed frequency spectra of the logarithms of $\varepsilon _{r}$ and $\chi _{r}$ are described well by an $f^{-1}$ law. A simple theoretical explanation is offered.

Lubrication theory is employed to examine surface-tension-dominated flows that arise during the application of thin coatings, using pressurized dies, to axially symmetric fibres. In all cases, it is assumed that the clearance between the die exit and the fibre is small compared with the fibre diameter. Previous analyses have been concerned with flows controlled by axial curvature for which the resulting solutions are unique. The present investigation examines stationary flows in which both the axial and the azimuthal curvatures are comparable. It is shown that this situation develops when the applicator volume flow is sufficiently large. Moreover, as the volume flow is increased, spatially oscillatory menisci can exist such that the solution is not always unique. These results are new, and calculations are presented that determine the maximum die clearance below which the solution remains unique. Within this regime, surface oscillations do not occur and there is a monotonic decay to the final uniform coating thickness.

General estimates are derived for mean velocities through and around groups or arrays of fixed and moving bodies, in unbounded and bounded domains, which lie within a defined perimeter. Robust kinematic flow concepts are introduced, namely the Eulerian spatial mean velocity $\overline{u}_E$ in the fluid volume between the bodies, the Eulerian flow outside the group, ${\bm u}_E^{(0)}$, and the Lagrangian mean velocity of material surfaces or fluid particles as they pass through the group of bodies ($\overline{u}_L^{(S)}$, $\overline{u}_L^{(P)}$). The Eulerian mean velocity is related to the momentum in the fluid domain, and is mainly influenced by fast moving regions of the flow. The Lagrangian mean velocity weights slowly moving regions of flow and is related to how material sheets deform as they are advected through groups of bodies. When the bodies are well-separated, the interstitial Eulerian and Lagrangian mean velocities ($\overline{u}_E^{(I)}$, $\overline{u}_L^{(I)}$), are defined and calculated in terms of the far-field contributions from the velocity or displacement field within the group of bodies.

In unbounded flow past well-separated bodies situated within a rectangular perimeter, the difference between the Eulerian and Lagrangian mean velocity is negligible (as the void fraction of the bodies, $\alpha\,{\rightarrow}\,0$). Within wide and short rectangular arrays, the Eulerian mean velocity is faster than the free-stream velocity $U$ because most of the incident flow passes through the array and $\overline{u}_E\,{=}\,U(1-\alpha)^{-1}$. Within long and thin rectangular arrays (and other cases where the reflux velocity is negligible), the Eulerian mean velocity, $\overline{u}_E\,{=}\,U(1-(1+C_m)\alpha)/(1-\alpha)$, is slower than the free-stream velocity, because most of the incident flow passes around the array. For a spherical or circular arrays of bodies, the particle Lagrangian mean velocity is $\overline{u}_L^{(P)}\,{=}\,U(1+C_m\alpha)^{-1}$ and differs from $\overline{u}_E$. These calculations are extended to examine the mean and interstitial flow through clouds of bodies in bounded channel flows.

The new concepts are applied to calculate the mean flow and pressure between and outside clouds of bodies, the average velocity of bubbly flows as a function of void fraction, and the tendency of clouds of bubbles to be distorted depending on their shape.

The results of experimental and numerical investigations of the onset of oscillatory convection in a sidewall heated rectangular cavity of molten gallium are reported. Detailed comparisons are made between experimental observations and calculations from numerical simulations of a three-dimensional Boussinesq model. The onset of time-dependence takes place through supercritical Hopf bifurcations and the loci of critical points in the ($\hbox{\it Gr}, \hbox{\it Pr}$)-plane are qualitatively similar with excellent agreement between the frequencies of the oscillatory motion. This provides a severe test of the control of the experiment since the mode of oscillation is extremely sensitive to imperfections.

Detailed numerical investigations reveal that there are a pair of Hopf bifurcations which exist on two asymmetric states which themselves arise at a subcritical pitchfork from the symmetric state. There is no evidence for this in the experiment and this qualitative difference is attributed to non-Boussinesq perturbations which increase with $\hbox{\it Gr}$.

However, the antisymmetric spatial structure of the oscillatory state is robust and is present in both the experiment and the numerical model. Moreover, the detailed analysis of the numerical results reveals the origins of the oscillatory instability.

The investigation involves the hydraulic behaviour of a dense layer of fluid flowing over an obstacle and subject to entrainment of mass and momentum from a dynamically inactive (but possibly moving) overlying fluid. An approach based on the use of reduced gravity, shallow-water theory with a cross-interface entrainment velocity is compared with numerical simulations based on a model with continuously varying stratification and velocity. The locations of critical flow (hydraulic control) in the continuous model are estimated by observing the direction of propagation of small-amplitude long-wave disturbances introduced into the flow field. Although some of the trends predicted by the shallow-water model are observed in the continuous model, the agreement between the interface profiles and the position of critical flow is quantitatively poor. A reformulation of the equations governing the continuous flow suggests that the reduced gravity model systematically underestimates inertia and overestimates buoyancy. These differences are quantified by shape coefficients that measure the vertical non-uniformities of the density and horizontal velocity that arise, in part, by incomplete mixing of entrained mass and momentum over the lower-layer depth. Under conditions of self-similarity (as in Wood's similarity solution) the shape coefficients are constant and the formulation determines a new criterion for and location of critical flow. This location generally lies upstream of the critical section predicted by the reduced-gravity model. Self-similarity is not observed in the numerically generated flow, but the observed critical section continues to lie upstream of the location predicted by the reduced gravity model. The factors influencing this result are explored.