The deformation of a viscoelastic drop suspended in a Newtonian fluid subjected to a steady shear is investigated using a front-tracking finite-difference method. The viscoelasticity is modelled using the Oldroyd-B constitutive equation. The drop response with increasing relaxation time λ and varying polymeric to the total drop viscosity ratio β is studied and explained by examining the elastic and viscous stresses at the interface. Steady-state drop deformation was seen to decrease from its Newtonian value with increasing viscoelasticity. A slight non-monotonicity in steady-state deformation with increasing Deborah number is observed at high Capillary numbers. Transient drop deformation displays an overshoot before settling down to a lower value of deformation. The overshoot increases with increasing β. The drop shows slightly decreased alignment with the flow with increasing viscoelasticity. A simple ordinary differential equation model is developed to explain the various behaviours and the scalings observed numerically. The critical Capillary number for drop breakup is observed to increase with Deborah number owing to the inhibitive effects of viscoelasticity, the increase being linear for small Deborah number.

For the flow around a circular cylinder, the steady flow changes its topology at a Reynolds number around 6 where the flow separates and a symmetric double separation zone is created. At the bifurcation point, the flow topology is locally degenerate, and by a bifurcation analysis we find all possible streamline patterns which can occur as perturbations of this flow. We show that there is no a priori topological limitation from further assuming that the flow fulfils the steady Navier–Stokes equations or from assuming that a Hopf bifurcation occurs close to the degenerate flow.

The steady flow around a circular cylinder experiences a Hopf bifurcation for a Reynolds number about 45–49. Assuming that this Reynolds number is so close to the value where the steady separation occurs that the flow here can be considered a perturbation of the degenerate flow, the topological bifurcation diagram will contain all possible instantaneous streamline patterns in the periodic regime right after the Hopf bifurcation. On the basis of the spatial and temporal symmetry associated with the circular cylinder and the structure of the topological bifurcation diagram, two periodic scenarios of instantaneous streamline patterns are conjectured. We confirm numerically the existence of these scenarios, and find that the first scenario exists only in a narrow range after the Hopf bifurcation whereas the second one persists through the entire range of Re where the flow can be considered two-dimensional. Our results corroborate previous experimental and computational results.

We present the results of an experimental and linear stability study of the influence of viscosity on the frozen wave (FW) instability, which arises when a vessel containing stably stratified layers of immiscible liquids is oscillated horizontally. Our linear stability model consists of two superposed fluid layers of arbitrary viscosities and infinite lateral extent, subject to horizontal oscillation. The effect of the endwalls of the experimental vessel is simulated by enforcing the conservation of horizontal volume flux, so that the base flow consists of counterflowing layers.

We perform experiments with four pairs of fluids, keeping the viscosity of the lower layer (ν1) constant, and increasing the viscosity of the upper layer (ν2), so that 1.02 × 102 ≤ N1 = ν2/ν1 ≤ 1.21 × 104. We find excellent quantitative agreement between theory and experiment despite the simple model geometry, for both the critical onset parameter and wavenumber of the FW. We show that the model of lyubimov:1987 (Fluid Dyn. vol. 86, 1987, p. 849), which is valid in the limit of inviscid fluids, consistently underestimates the instability threshold for fluids of equal viscosity, but generally overestimates the threshold for fluids of unequal viscosity. We extend the experimental parameter range numerically to viscosity contrasts 1 ≤ N1 ≤ 6 × 104 and identify four regions of N1 where qualitatively different dynamics occur, which are reflected in the non-monotonic dependence of the most unstable wavenumber and the critical amplitude on N1. In particular, we find that increasing the viscosity contrast between the layers leads to destabilization over a wide range of N1, 10 ≤ N1 ≤ 8 × 103. The intricate dependence of the instability on viscosity contrast is due to considerable changes in the time-averaged perturbation vorticity distribution near the interface.

The recent theoretical discovery of families of unstable travelling-wave solutions in pipe flow at Reynolds numbers lower than the transitional range, naturally raises the question of their relevance to the turbulent transition process. Here, a series of numerical experiments are conducted in which we look for the spatial signature of these travelling waves in transitionary flows. Working within a periodic pipe of 5D (diameters) length, we find that travelling waves with low wall shear stresses (lower branch solutions) are on a surface in phase space which separates initial conditions which uneventfully relaminarize and those which lead to a turbulent evolution. This dividing surface (a separatrix if turbulence is a sustained state) is then minimally the union of the stable manifolds of all these travelling waves. Evidence for recurrent travelling-wave visits is found in both 5D and 10D long periodic pipes, but only for those travelling waves with low-to-intermediate wall shear stress and for less than about 10% of the time in turbulent flow at Re = 2400. Given this, it seems unlikely that the mean turbulent properties such as wall shear stress can be predicted as an expansion solely over the travelling waves in which their individual properties are appropriately weighted. Instead the onus is on isolating further dynamical structures such as periodic orbits and including them in any such expansion.

A long-wavelength stability analysis of two co-rotating Gaussian vertical vortices in an inviscid strongly stratified fluid is conducted for vortices separated by a large distance b compared to their radius a (b ≫ a). This analysis predicts and explains the zigzag instability found by a numerical stability analysis in a companion paper (Otheguy, Chomaz & Billant, J. Fluid. Mech. vol. 553, 2006, p. 253). The zigzag instability results from the coupling between the bending perturbations of each vortex and the external strain that one vortex induces on the other S = Γ/2 π b2, where Γ is the circulation of the vortices. The analysis predicts that the maximum growth rate of the instability is twice the strain S and that the most unstable vertical wavelength λ scales as the buoyancy length, defined by LB = Γ/πaN, multiplied by the ratio b/a, i.e. λ ∝ Fhb, where Fh = Γ/πa2N is the horizontal Froude number. The asymptotic results are in very good agreement with the numerical results.

The effects of surface roughness on a spatially developing turbulent boundary layer (TBL) are investigated by performing direct numerical simulations of TBLs over rough and smooth walls. The Reynolds number based on the momentum thickness was varied in the range Reθ = 300 ∼ 1400. The roughness elements were periodically arranged two-dimensional spanwise rods, and the roughness height was k = 1.5θin, where θin is the momentum thickness at the inlet, which corresponds to k/δ = 0.045 ∼ 0.125, δ being the boundary layer thickness. To avoid generating a rough-wall inflow, which is prohibitively difficult, a step change from smooth to rough was placed 80θin downstream from the inlet. The spatially developing characteristics of the rough-wall TBL were examined. Along the streamwise direction, the friction velocity approached a constant value, and self-preserving forms of the turbulent Reynolds stress tensors were obtained. Introduction of the roughness elements affected the turbulent stress not only in the roughness sublayer but also in the outer layer. Despite the roughness-induced increase of the turbulent Reynolds stress tensors in the outer layer, the roughness had only a relatively small effect on the anisotropic Reynolds stress tensor in the outer layer. Inspection of the triple products of the velocity fluctuations revealed that introducing the roughness elements onto the smooth wall had a marked effect on vertical turbulent transport across the whole TBL. By contrast, good surface similarity in the outer layer was obtained for the third-order moments of the velocity fluctuations.

This paper examines the widespread idea that vortex lines and tubes must either close on themselves or extend to the boundary of the fluid. A survey of the origins of this misconception, and of earlier attempts to set it right, is followed by an analysis of simple flows exhibiting vortex lines and tubes which do not fit those shapes. Two types of vortex lines are discussed: dense, which comprise open lines of infinite length but confined in a finite region, and separatrix, which comprise lines that begin or finish within the fluid, at points where the vorticity is null. The presence of these vortex lines in a vortex tube affects its topology in the following ways. Vortex tubes formed by dense vortex lines have infinite length; they self-intersect an infinite number of times but do not close on themselves. Vortex tubes formed by separatrix vortex lines (and either closed or open vortex lines) are torn apart at the points where the vorticity is null. Vortex tubes exclusively composed of separatrix vortex lines begin or finish at points or surfaces within the fluid; in this particular situation the vortex tube has zero strength.

We use a newly developed set of interface conditions to revisit the problem of an evaporating thin liquid film. In particular, instead of the conventional Hertz–Knudsen–Langmuir equation for the evaporation mass flux, we impose a more general equation expressing the balance of configurational momentum. This balance, which supplements the conventional conditions enforcing the balances of mass, momentum and energy on the film surface, arises from a consideration of configurational forces within a thermodynamical framework. We study the influence of two newly introduced terms on the evolution of the liquid film. One of these terms accounts for the transport of energy within the liquid–vapour interface. The other term, which we refer to as the effective pressure, accounts for vapour recoil. Both new terms are found to be stabilizing. Furthermore, the effective pressure is found to affect a time-dependent base state of the evaporating film and to be an important factor in applications involving liquid films with thicknesses of one or two monolayers. Specifically, we demonstrate that consideration of the effective pressure makes it possible to observe the influence of the van der Waals interactions on film evolution close to the instant of rupture. Dimensional considerations indicate that one of the most significant influences of these effects occurs for molten metals.

The concept of a quasi-geostrophic singular vortex is extended to several types of two-layer model: a rigid-lid two-layer, a free-surface two-layer and a -layer model with two active and one passive layer. Generally, a singular vortex differs from a conventional point vortex in that the intrinsic vorticity of a singular vortex, in addition to delta-function, contains an exponentially decaying term. The theory developed herein occupies an intermediate position between discrete and fully continuous multilayer models, since the regular flow and its interaction with the singular vortices are also taken into account. A system of equations describing the joint evolution of the vortices and the regular field is presented, and integrals expressing the conservation of enstrophy, energy, momentum and mass are derived. Using these integrals, the initial phases of evolution of an individual singular vortex confined to one layer and of a coaxial pair of vortices positioned in different layers of a two-layer fluid on a beta-plane are described. A valuable application of the conservation integrals is related to the stability analysis of point-vortex pairs within the -layer model, -layer model, and free-surface two-layer model on the f-plane. Such vortex pairs are shown to be nonlinearly stable with respect to any small perturbation provided its regular-flow energy and enstrophy are finite.

Drift of individual β-plane vortices confined to one layer of a two-layer fluid under the rigid-lid condition is considered. For this purpose, the theory of two-layer quasi-geostrophic singular vortices is employed. On a β-plane, any non-zonal displacement of a singular vortex results in the development of a regular flow. An individual singular β-plane vortex cannot be steady on its own: the vortex moves coexisting with a regular flow, be the drift steady or not. In this paper, both kinds of drift of a singular vortex are considered. A new steady exact solution is presented, a hybrid regular–singular modon. This hybrid modon consists of a dipole component and a circularly symmetric rider. The dipole is regular, and the rider is a superposition of the singular vortex and a regular circularly symmetric field. The unsteady drift of a singular vortex residing in one of the layers is considered under the condition that, at the initial instant, the regular field is absent. The development of barotropic and baroclinic regular β-gyres is examined. Whereas the barotropic and baroclinic modes of the singular vortex are comparable in magnitudes, the baroclinic β-gyres attenuate with time, making the trajectory of the vortex close to that of a barotropic monopole on a β-plane.

A motion trapping structure can be defined as a freely floating structure under natural or externally applied restoring forces, on or below the free surface of a heavy fluid extending to infinity in at least one direction, which generates a persistent local time-harmonic oscillation of the fluid of finite energy at a particular frequency, due to its own motion at that frequency. Such an oscillation is termed a motion-trapped mode. In this paper it is shown, using accurate numerical computations, that a submerged circular cylinder making forced time-harmonic two-dimensional heave or sway motions of small amplitude in a fluid of either finite or infinite depth, can create a local flow field in which no waves radiate to infinity at particular frequencies and depths of submergence of the cylinder. By tethering such a buoyant cylinder to the bottom of a fluid of finite depth, using a vertical inelastic mooring line, it is shown, by suitable choice of buoyancy and length of tether, how the cylinder, moving freely under its mooring forces, can operate as a motion trapping structure. Such a cylinder would, if displaced from its equilibrium position and released, ultimately oscillate indefinitely at the trapped mode frequency. This simple geometry is the first example of a submerged isolated motion trapping structure free to move under its natural mooring forces.

Turbulence characteristics of vertical air–solid pipe flow are investigated in this paper. Direct numerical simulations of the gas phase have been performed, while the solid particles have been simulated by a Lagrangian approach, including particle collisions. The modelling of wall roughness is shown to be important to obtain agreement with experimental data. Reynolds stresses and Reynolds stress budgets are given for both phases and for a wide range of solid–air mass load ratios (mass loads), varying from 0.11 to 30. Air turbulence intensities, Reynolds shear stress, and turbulence production reduce with increasing mass load. The mean air profile does not alter for low mass loads. In this regime, a simple theory predicts that the reduction of air turbulent production relative to unladen turbulent production is approximately equal to the mass load ratio. The insight that the solids Reynolds shear stress can be significant, even for low mass loads, is essential for this explanation. It is shown that at least two mechanisms cause the turbulence reduction. In addition to the classically recognized mechanism of dissipation of turbulent fluctuations by particles, there is another suppressing mechanism in inhomogeneous flows: the non-uniform relative velocity of the phases, created because particles slip at the wall, collide, and slowly react with the continuous phase. Investigation of the air turbulent kinetic energy equation demonstrates that the relative reduction of air pressure strain is larger than the reduction of turbulent production and dissipation, and pressure strain may therefore be a cause of the reduction of the other quantities. The fluctuational dissipation induced by the drag forces from particles is small compared to the other terms, but not negligible. For intermediate and high mass loads the air turbulence remains low. The relatively small turbulence intensities are not generated by the standard turbulent mechanisms any more, but directly caused by the particle motions. The particle–fluid interaction term in the turbulent kinetic energy equation is no longer dissipative, but productive instead. On increasing the mass load, the radial and azimuthal fluctuations of the particles grow. The corresponding reduction of solids anisotropy is an effect of the inter-particle collisions, which act as a solids pressure strain term. For intermediate and high mass loads, fluctuational drag force and particle collisions appear to be the relevant dissipation mechanisms in the solids fluctuational energy equation. In contrast to the air turbulent production, the solids ‘turbulent’ production term has the same level for low and high mass loads, while it attains a clear local minimum between. With increasing mass load, large-scale coherent turbulent fluid structures weaken, and eventually disappear. Simultaneously, the fluid fluctuations at relatively small length scales are enhanced by the motion of the particles. The highest particle concentration occurs near the wall for low mass loads, but on increasing the mass load, the concentration profile becomes uniform, while for the highest mass load particles accumulate in the centre of the pipe. Two-point correlation functions indicate that the addition of a small number of small solid particles to a clean pipe flow increases the streamwise length scale of the turbulence.

The effects of polymer stresses on near-wall turbulent structures are examined by using direct numerical simulation of fully developed turbulent channel flows with and without polymer stress. The Reynolds number based on friction velocity and half-channel height is 395, and the stresses created by adding polymer are modelled by a finite extensible nonlinear elastic, dumbbell model. Both low- (18%) and high-drag reduction (61%) cases are investigated. Linear stochastic estimation is employed to compute the conditional averages of the near-wall eddies. The conditionally averaged flow fields for Reynolds-stress-maximizing Q2 events show that the near-wall vortical structures are weakened and elongated in the streamwise direction by polymer stresses in a manner similar to that found by Stone et al. (2004) for low-Reynolds-number quasi-streamwise vortices (‘exact coherent states: ECS’). The conditionally averaged fields for the events with large contribution to the polymer work are also examined. The vortical structures in drag-reduced turbulence are very similar to those for the Q2 events, i.e. counter-rotating streamwise vortices near the wall and hairpin vortices above the buffer layer. The three-dimensional distributions of conditionally averaged polymer force around these vortical structures show that the polymer force components oppose the vortical motion. More fundamentally, the torques due to polymer stress are shown to oppose the rotation of the vortices, thereby accounting for their weakening. The observations also extend concepts of the vortex retardation by viscoelastic counter-torques to the heads of hairpins above the buffer layer, and offer an explanation of the mechanism of drag reduction in the outer region of wall turbulence, as well as in the buffer layer.

The response of homogeneous and isotropic turbulence to streamwise straining action provided by planar expansion waves has been studied experimentally in the CCNY shock tube research facility at several Reynolds numbers. The reflection of a propagating shock wave at the open endwall of the shock tube generated an expansion fan travelling upstream and interacting with the induced flow behind the incident shock wave which has gone through a turbulence generating grid.

A custom-made hot-wire vorticity probe was designed and developed capable of measuring the time-dependent highly fluctuating three-dimensional velocity and vorticity vectors, and associated total temperature, in non-isothermal and inhomogeneous flows with reasonable spatial and temporal resolution. These measurements allowed the computations of the vorticity stretching/tilting terms, vorticity generation through dilatation terms, full dissipation rate of kinetic energy term and full rate-of-strain tensor. The longitudinal size of the straining zone was substantial so that measurements within it were possible. The flow accelerated from a Mach number of 0.23 to about 0.56, a value which is more than twice the initial one.

Although the average value of the applied straining was only between S11 = 130 s−1 and S11 = 240 s−1 and the gradient Mach number was no more than 0.226, the amplitude of fluctuations of the strain rate S11 were of the order of 4000 s−1 before the application of straining and were reduced by about 2.5 times downstream of the interaction. This characteristic of high-amplitude bursts and the intermittent behaviour of the flow play a significant role in the dynamics of turbulence.

One of the most remarkable features of the suppression of turbulence is that this process peaks shortly after the application of the straining where the pressure gradient is substantial. It was also found that the total enthalpy variation follows very closely the temporal gradient of pressure within the straining region and peaks at the same location as the pressure gradient.

Attenuation of longitudinal velocity fluctuations has been observed in all experiments. It appears that this attenuation depends strongly on the characteristics of the incoming turbulence for a given straining strength and flow Mach number. The present results clearly show that in most of the cases, attenuation occurs at large times or distances from the turbulence generating grids where length scales of the incoming flow are high and turbulence intensities are low. Thus, large eddies with low-velocity fluctuations are affected the most by the interaction with the expansion waves. Spectral analysis has indicated that attenuation of fluctuations is not the same across all wavenumbers of the spectrum. The magnitude of attenuation appears to be higher in cases of finer mesh grids.

Helium mass flow rates in a microchannel were measured, for a wide Knudsen-number range, in isothermal steady conditions. The flow Knudsen numbers, considered here, cover the range from continuum slip regime to the near free molecular regime. We used a single-channel system involved in an experimental platform more powerful than those previously used. The experimental errors and uncertainties were accurately investigated and estimated. In the continuum slip regime, it was found that the first-order approach is pertinent for Knudsen number between 0.03 and 0.3. Moreover, the slip coefficient was deduced by comparing the experiments with the theoretical first-order slip continuum approach. For Knudsen number between 0.03 and 0.7, a polynomial second-power form is proposed for the mass flow rate expression. Otherwise, the experimental results on the mass flow rate were compared with theoretical values calculated from kinetic approaches over the 0.03–50 Knudsen number range, and an overall agreement appears through the comparison. It was also found, when the Knudsen number increased, that the wall influence on measurement occurred first through the accommodation process in the transition regime followed by the wall influence through the aspect ratio in the free molecular regime.

A computational study based on the Stokes equations is conducted to investigate the effects of gravitational forces on miscible displacements in vertical Hele-Shaw cells. Nonlinear simulations provide the quasi-steady displacement fronts in the gap of the cell, whose stability to spanwise perturbations is subsequently examined by means of a linear stability analysis. The two-dimensional simulations indicate a marked thickening (thinning) and slowing down (speeding up) of the displacement front for flows stabilized (destabilized) by gravity. For the range investigated, the tip velocity is found to vary linearly with the gravity parameter. Strongly stable density stratifications lead to the emergence of flow patterns with spreading fronts, and to the emergence of a secondary needle-shaped finger, similar to earlier observations for capillary tube flows. In order to investigate the transition between viscously driven and purely gravitational instabilities, a comparison is presented between displacement flows and gravity-driven flows without net displacements.

The linear stability analysis shows that both the growth rate and the dominant wavenumber depend only weakly on the Péclet number. The growth rate varies strongly with the gravity parameter, so that even a moderately stable density stratification can stabilize the displacement. Both the growth rate and the dominant wavelength increase with the viscosity ratio. For unstable density stratifications, the dominant wavelength is nearly independent of the gravity parameter, while it increases strongly for stable density stratifications. Finally, the kinematic wave theory of Lajeunesse et al. (J. Fluid Mech. vol. 398, 1999, p. 299) is seen to capture the stability limit quite accurately, while the Darcy analysis misses important aspects of the instability.

The linear stability of a rotating stratified inviscid horizontal plane Couette flow in a channel is studied in the limit of strong rotation and stratification. Two dimensionless parameters characterize the flow: the Rossby number ε, defined as the ratio of the shear to the Coriolis frequency and assumed small, and the ratio s of the Coriolis frequency to the buoyancy frequency, assumed to satisfy s ≤ 1. An energy argument is used to show that unstable perturbations must have large, O(ε−1), wavenumbers. This motivates the use of a WKB-approach which, in the first instance, provides an approximation for the dispersion relation of the various waves that can propagate in the flow. These are Kelvin waves, trapped near the channel walls, and inertia–gravity waves with or without turning points.

Although the waves have real phase speeds to all algebraic orders in ε, we establish that the flow is unconditionally unstable. This is the result of linear resonances between waves with oppositely signed wave momenta. Three modes of instabilities are identified, corresponding to the resonance between (i) a pair of Kelvin waves, (ii) a Kelvin wave and an inertia–gravity wave, and (iii) a pair of inertia–gravity waves. Whilst all three modes of instability are active when the Couette flow is anticyclonic, mode (iii) is the only possible instability mechanism when the flow is cyclonic.

We derive asymptotic estimates for the instability growth rates. These are exponentially small in ε, i.e. of the form Im ω = a exp(-Ψ/ε) for some positive constants a and Ψ. For the Kelvin-wave instabilities (i), we obtain analytic expressions for a and Ψ; the maximum growth rate, in particular, corresponds to Ψ = 2. For the other types of instabilities, we make the simplifying assumption s ≪ 1 and find that the maximum growth rates correspond to Ψ=2.80 for (ii) and Ψ= π for (iii). The asymptotic results are confirmed by numerical computations. These reveal, in particular, that the instabilities (iii) have much smaller growth rates in cyclonic flows than in anticyclonic flows, even though Ψ = π in both cases.

Our results highlight the limitations of the so-called balanced models, widely used in geophysical fluid dynamics, which filter out Kelvin and inertia–gravity waves and hence predict the stability of Couette flow. They are also relevant to the stability of Taylor–Couette flows and of astrophysical accretion disks.

Amplitude thresholds for transition of localized disturbances, their breakdown to turbulence and the development of turbulent spots in the asymptotic suction boundary layer are studied using direct numerical simulations. A parametric study of the horizontal scales of the initial disturbance is performed and the disturbances that lead to the highest growth under the conditions investigated are used in the simulations. The Reynolds-number dependence of the threshold amplitude of a localized disturbance is investigated for 500≤ Re ≤ 1200, based on the free-stream velocity and the displacement thickness. It is found that the threshold amplitude scales as Re−1.5 for the considered Reynolds numbers. For Re ≤ 367, the localized disturbance does not lead to a turbulent spot and this provides an estimate of the critical Reynolds number for the onset of turbulence. When the localized disturbance breaks down to a turbulent spot, it happens through the development of hairpin and spiral vortices. The shape and spreading rate of the turbulent spot are determined for Re = 500, 800 and 1200. Flow visualizations reveal that the turbulent spot takes a bullet-shaped form that becomes more distinct for higher Reynolds numbers. Long streaks extend in front of the spot and in its wake a calm region exists. The spreading rate of the turbulent spot is found to increase with increasing Reynolds number.

We consider the behaviour of a gravity current in a porous medium when the horizontal surface along which it spreads is punctuated either by narrow fractures or by permeable regions of limited extent. We derive steady-state solutions for the current, and show that these form part of a long-time asymptotic description which may also include a self-similar ‘leakage current’ propagating beyond the fractured region with a length proportional to t1/2. We discuss the conditions under which a current can be completely trapped by a permeable region or a series of fractures.

Gravity currents formed by the release of heavy fluid from an enclosed lock on a sloping open channel were investigated experimentally. The experiments were conducted in a channel that had a running length of 13 lock depths, and could be inclined to a maximum angle of 17°. The release of heavy dyed salt solution from a lock with an aspect ratio (height to length) of 0.5, was examined using video images to determine the front velocity, and a particle-tracking technique was used to measure the two-dimensional velocity field in a vertical slice through the centre of the evolving current. The gravity current head velocity increased with time and downstream distance to a maximum at approximately 10 lock depths from the front of the lock. Flow visualization and the velocity measurements have shown that during the acceleration phase the head was being fed by a following current that increased its buoyancy as it propagated downstream. A modified version of the theory of P. Beghin, E. J. Hopfinger and R. E. Britter (J. Fluid Mech. vol. 107, 1981, p. 407) in which the measured increase in buoyancy was used, instead of the original assumption of constant buoyancy, gave results that agreed closely with the experimental velocity versus time histories.