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The effect of confining boundaries on gravity currents in porous media is investigated theoretically and experimentally. Similarity solutions are derived for currents when the volume increases as tα in horizontal channels of uniform cross-section with boundary height b satisfying b ~ a|y/a|n, where y is the cross-channel coordinate and a is a length scale of the channel width. Experiments were carried out in V-shaped and semicircular channels for the case of gravity currents with constant volume (α=0) and constant flux (α=1). These showed generally good agreement with the theory.
Typically, we find that the propagation of the current is well described by L ~ tc for some scalar c. We study the dependence of c on the time exponent of the volume of fluid in the current, α, and the geometry of the channel, parameterized by n. For all channel shapes, there exists a critical value of α, αc = 1/2, above which increasing n causes an increase in c and below which increasing n causes a decrease in c, where increasing n corresponds to opening up the channel boundary to the horizontal. The current height increases or decreases with respect to time depending on whether α is greater or less than αc. It is this fact, along with global mass conservation, which explains why varying the channel shape n affects the propagation rate c in different ways depending on α.
We also consider channels inclined at an angle θ to the horizontal. When the slope of the channel is much greater than the slope of the free surface of the current, the component of gravity parallel to the slope dominates, causing the current to move with a constant velocity, Vf say, regardless of channel shape n and flux parameter α, in agreement with results for a two-dimensional gravity current obtained by Huppert & Woods (1995) and some initially axisymmetric gravity currents presented by Vella & Huppert (2006). If the effect of the component of gravity perpendicular to the channel may not be neglected, i.e. if the slopes of the channel and free surface of the current are comparable, we find that, in a frame moving with speed Vf, the form of the governing equation for the height of a current in an equivalent horizontal channel is recovered. We calculate that the height of a constant flux gravity current down an inclined channel will tend to a fixed depth, which is determined by the channel shape, n, and the physical properties of the fluid and rock. Experimental and numerical results for inclined V-shaped channels agree very well with this theory.
In this study we report discovery of a new type of water bell. This is formed by impinging a vertical liquid jet on to the underside of a large horizontal flat plate. After impact, the liquid spreads radially along the plate before falling at an abrupt unspecified radius. This falling liquid may then coalesce to form a curtain which encloses a volume of air. When the flow rate of the impinging jet is altered from the value at initial formation, a pronounced hysteretic effect in the water bell shape can be observed. We present detailed observations of these new phenomena, including the size and nature of the flow underneath the plate and the shape of the liquid curtain. These observations are interpreted theoretically in a companion paper (Part 2, Button et al. vol. 649, 2010, pp. 45–68).
In a companion paper (Part 1, Jameson et al. J. Fluid Mech. vol. 649, 2010, 19–43), the discovery of a new type of water bell was reported. When a vertical liquid jet impacts on the underside of a large horizontal plate, the resulting thin film spreads radially along the plate to an unspecified abrupt departure point, from whence it falls away from the plate of its own accord. The departure radius of the fluid from the plate is seen to depend strongly on the volumetric flow rate. The falling liquid may then coalesce to form a water bell. Here we present a theoretical analysis and explanation of this phenomenon. A force balance determining the maximum radial extension of the thin film flow along the plate is considered as a mechanism for fluid departure from the plate, for which an analytical model is developed. This model gives good predictions of the measured radius of departure. When a water bell has been formed, and the flow rate is altered, many interesting shapes are produced that depend on the shapes at previous flow rates. We discuss the origin of this hysteresis, and also present a leading order theory for the bell shape under a regime of changing flow rate. The models are compared with experimental results spanning two orders of magnitude in viscosity.
The flow of compositional gravity currents past circular cylinders mounted above a wall is investigated numerically. Two- and three-dimensional Navier–Stokes simulations are employed to quantify the force load on the cylinder, along with the friction velocity at the bottom wall near the cylinder, for Reynolds numbers in the range of 2000–45 000. While two-dimensional simulations accurately capture the impact stage, they are seen to overpredict the force and friction velocity fluctuations throughout the transient stage. Comparisons between gravity current and constant-density flows past circular cylinders show that the impact and transient stages are unique to gravity current flows. During the quasi-steady stage, on the other hand, the wake structures and the values of the drag, the peak-to-peak lift, the vortex shedding frequency and the friction velocity below the cylinder are comparable.
The friction velocity below the cylinder depends chiefly on the Reynolds number formed with the front velocity and the gap width. The maximum friction velocity at impact is about 60% larger than during the quasi-steady stage or in a constant-density flow. This raises the possibility of aggressive erosion behaviour at impact, which may occur in a spanwise localized fashion because of the larger friction velocity near the lobes.
New measurements of the streamwise component of the turbulence intensity in a fully developed pipe flow at Reynolds numbers up to 145 000 indicate that the magnitude of the near-wall peak is invariant with Reynolds number in location and magnitude. The results agree with previous pipe flow data that have sufficient spatial resolution to avoid spatial filtering effects, but stand in contrast to similar results obtained in boundary layers, where the magnitude of the peak displays a prominent Reynolds number dependence, although its position is fixed at the same location as in pipe flow. This indicates that the interaction between the inner and outer regions is different in pipe flows and boundary layers.
We analyse numerically the linear stability of the fully developed flow of a liquid metal in a square duct subject to a transverse magnetic field. The walls of the duct perpendicular to the magnetic field are perfectly conducting whereas the parallel ones are insulating. In a sufficiently strong magnetic field, the flow consists of two jets at the insulating walls and a near-stagnant core. We use a vector stream function formulation and Chebyshev collocation method to solve the eigenvalue problem for small-amplitude perturbations. Due to the two-fold reflection symmetry of the base flow the disturbances with four different parity combinations over the duct cross-section decouple from each other. Magnetic field renders the flow in a square duct linearly unstable at the Hartmann number Ha ≈ 5.7 with respect to a disturbance whose vorticity component along the magnetic field is even across the field and odd along it. For this mode, the minimum of the critical Reynolds number Rec ≈ 2018, based on the maximal velocity, is attained at Ha ≈ 10. Further increase of the magnetic field stabilizes this mode with Rec growing approximately as Ha. For Ha > 40, the spanwise parity of the most dangerous disturbance reverses across the magnetic field. At Ha ≈ 46 a new pair of most dangerous disturbances appears with the parity along the magnetic field being opposite to that of the previous two modes. The critical Reynolds number, which is very close for both of these modes, attains a minimum, Rec ≈ 1130, at Ha ≈ 70 and increases as Rec ≈ 91Ha1/2 for Ha ≫ 1. The asymptotics of the critical wavenumber is kc ≈ 0.525Ha1/2 while the critical phase velocity approaches 0.475 of the maximum jet velocity.
Droplet deformation by air cushioning prior to impact is considered. A model is presented coupling the free-surface deformation of a droplet with the pressure field in the narrow air layer generated as a droplet approaches an impact. The model is based upon the density and viscosity in the air being small compared with those in the liquid. Additionally, the Reynolds number, defined using the droplet radius ℛ and approach velocity
The problem is studied numerically with a boundary-element method in the inviscid droplet coupled with a finite-difference method in the lubricating air. In normal impacts, air cushioning will be shown to deflect the free surface upwards, delaying the moment of touchdown and trapping a bubble. The volume of the bubble is found to be (μg4/3ℛ5/3/ρl4/3
We investigate theoretically the nonlinear state of ideal straight rolls in the Rayleigh–Bénard system of a fluid layer heated from below with a porous medium using a Galerkin method. Applying the Oberbeck–Boussinesq approximation, binary mixtures with positive separation ratio are studied and compared with one-component fluids. Our results for the structural properties of roll convection resemble qualitatively the situation in the Rayleigh–Bénard system without porous medium except for the fact that the streamlines of binary mixtures are deformed in the so-called Soret regime. The deformation of the streamlines is explained by means of the Darcy equation which is used to describe the transport of momentum. In addition to the properties of the rolls, their stability against arbitrary infinitesimal perturbations is investigated. We compute stability balloons for the pure fluid case as well as for a wide parameter range of Lewis numbers and separation ratios that are typical for binary gas and fluid mixtures. The stability regions of rolls are found to be restricted by a crossroll, a zigzag and a new type of oscillatory instability mechanism, which can be related to the crossroll mechanism.
We compare laboratory observations of equilibrated baroclinic waves in the rotating two-layer annulus, with numerical simulations from a quasi-geostrophic model. The laboratory experiments lie well outside the quasi-geostrophic regime: the Rossby number reaches unity; the depth-to-width aspect ratio is large; and the fluid contains ageostrophic inertia–gravity waves. Despite being formally inapplicable, the quasi-geostrophic model captures the laboratory flows reasonably well. The model displays several systematic biases, which are consequences of its treatment of boundary layers and neglect of interfacial surface tension and which may be explained without invoking the dynamical effects of the moderate Rossby number, large aspect ratio or inertia–gravity waves. We conclude that quasi-geostrophic theory appears to continue to apply well outside its formal bounds.
Fully three-dimensional numerical simulations of concentrated suspensions of O(1000) particles in a Couette flow at zero Reynolds number are performed with the goal of determining the wall effects on concentrated suspensions of non-colloidal particles. The simulations, based on the force-coupling method, are performed for 0.2 ≤ φ ≤ 0.4 and 10 < Ly/a < 30, where φ denotes the volume fraction and Ly and a are, respectively, the channel height and the particle radius. It is shown that the suspensions can be divided into three regions depending on the microstructures; the wall region where a structured particle layering is dominant, the core region in which the suspension field is quasi-homogeneous, and the buffer region which shows the characteristics of both the particle layer and the shear structure. The width of the inhomogeneous region (wall and buffer) is a function of φ and not sensitive to Ly/a, once Ly/a is larger than a threshold. Rheological properties in the inhomogeneous and quasi-homogeneous regions are investigated. The particle stresses are compared with previous rheological models.
Aspects of large-scale organized structures in sink flow turbulent and reverse-transitional boundary layers are studied experimentally using hot-wire anemometry. Each of the present sink flow boundary layers is in a state of ‘perfect equilibrium’ or ‘exact self-preservation’ in the sense of Townsend (The Structure of Turbulent Shear Flow, 1st and 2nd edns, 1956, 1976, Cambridge University Press) and Rotta (Progr. Aeronaut. Sci., vol. 2, 1962, pp. 1–220) and conforms to the notion of ‘pure wall-flow’ (Coles, J. Aerosp. Sci., vol. 24, 1957, pp. 495–506), at least for the turbulent cases. It is found that the characteristic inclination angle of the structure undergoes a systematic decrease with the increase in strength of the streamwise favourable pressure gradient. Detectable wall-normal extent of the structure is found to be typically half of the boundary layer thickness. Streamwise extent of the structure shows marked increase as the favourable pressure gradient is made progressively severe. Proposals for the typical eddy forms in sink flow turbulent and reverse-transitional flows are presented, and the possibility of structural self-organization (i.e. individual hairpin vortices forming streamwise coherent hairpin packets) in these flows is also discussed. It is further indicated that these structural ideas may be used to explain, from a structural viewpoint, the phenomenon of soft relaminarization or reverse transition of turbulent boundary layers when subjected to strong streamwise favourable pressure gradients. Taylor's ‘frozen turbulence’ hypothesis is experimentally shown to be valid for flows in the present study even though large streamwise accelerations are involved, the flow being even reverse transitional in some cases. Possible conditions, which are required to be satisfied for the safe use of Taylor's hypothesis in pressure-gradient-driven flows, are also outlined. Measured convection velocities are found to be fairly close to the local mean velocities (typically 90% or more) suggesting that the structure gets convected downstream almost along with the mean flow.
Extended self-similarity (ESS), a procedure that remarkably extends the range of scaling for structure functions in Navier–Stokes turbulence and thus allows improved determination of intermittency exponents, has never been fully explained. We show that ESS applies to Burgers turbulence at high Reynolds numbers and we give the theoretical explanation of the numerically observed improved scaling at both the IR and UV end, in total a gain of about three quarters of a decade: there is a reduction of subdominant contributions to scaling when going from the standard structure function representation to the ESS representation. We conjecture that a similar situation holds for three-dimensional incompressible turbulence and suggest ways of capturing subdominant contributions to scaling.
It is widely accepted that both ripples and dunes form in rivers by primary linear instability; the wavelength of the former scaling on the grain size and that of the latter being controlled by the water depth. We revisit here this problem in a theoretical framework that allows to give a clear picture of the instability in terms of dynamical mechanisms. A multi-scale description of the problem is proposed, in which the details of the different mechanisms controlling sediment transport are encoded into three quantities: the saturated flux, the saturation length and the threshold shear stress. Hydrodynamics is linearized with respect to the bedform aspect ratio. We show that the phase shift of the basal shear stress with respect to the topography, responsible for the formation of bedforms, appears in an inner boundary layer where shear stress and pressure gradients balance. This phase shift is sensitive to the presence of the free surface, and the related effects can be interpreted in terms of standing gravity waves excited by topography. The basal shear stress is dominated by this finite depth effect in two ranges of wavelength: when the wavelength is large compared to the flow depth, so that the inner layer extends throughout the flow, and in the resonant conditions, when the downstream material velocity balances the upstream wave propagation. Performing the linear stability analysis of a flat sand bed, the relation between the wavelength at which ripples form and the flux saturation length is quantitatively derived. It explains the discrepancy between measured initial wavelengths and predictions that do not take this lag between flow velocity and sediment transport into account. Experimental data are used to determine the saturation length as a function of grain size and shear velocity. Taking the free surface into account, we show that the excitation of standing waves has a stabilizing effect, independent of the details of the flow and sediment transport models. Consequently, the shape of the dispersion relation obtained from the linear stability analysis of a flat sand bed is such that dunes cannot result from a primary linear instability. We present the results of field experiments performed in the natural sandy Leyre river, which show the formation of ripples by a linear instability and the formation of dunes by a nonlinear pattern coarsening limited by the free surface. Finally, we show that mega-dunes form when the sand bed presents heterogeneities such as a wide distribution of grain sizes.
The flow of a viscous compressible fluid in a circular tube generated by a sudden impulse at a point on the axis and directed transverse to the axis is studied on the basis of the linearized Navier–Stokes equations. A no-slip boundary condition is assumed to hold on the wall of the tube. The flow behaviour differs qualitatively from that for a point impulse in the direction of the axis in that there is no coupling to a diffusive sound mode. As a consequence, the transverse velocity autocorrelation function of a suspended Brownian particle decays at long times faster than t−3/2.
A fully three-dimensional turbulent separated flow was set up such that it had a systematic link to two-dimensional flow, as a way of investigating the more complicated nature of this flow type. The central region of the flow was fully three-dimensional, but was bounded on its sides by regions of ‘spanwise invariance’ in which the flow was invariant in the lateral direction, or very nearly so. A special case of spanwise invariance, which is statistically two-dimensional, is one in which the streamlines are also coplanar, or at least nominally so in numerous experimental studies. Another aspect of the present arrangement is that the side regions should ideally provide well-defined boundary conditions. The separation was formed downstream of a doubly swept normal flat plate, forming a ‘v’-shaped separation line, mounted on the front of a splitter plate, mounted in the centre of the wind tunnel working section. The predominantly inward flow to the central region implies a negative lateral strain rate (∂W/∂z), but all nine strain rates are non-zero. Measurements were made using pulsed-wire anemometry techniques for mean velocities, Reynolds stresses and wall shear stress. Even though the sweep angle is mild at ±10°, the effect is to increase the bubble height by over 50% in its centre to create a ‘bulge’, symmetrical about the centreline. The degree of three-dimensionality is described as moderate in that the peak inflow velocity from the side regions is less than 0.2 of the free-stream velocity, but comparable with the peak in the reverse-flow velocity. A larger sweep angle would give a larger inflow velocity. A separate study (Cao & Hancock, Eur. J. Mech. B/Fluids, vol. 23, 2004, p. 519) has shown that the bulge persists very far downstream, so that accurate physical modelling of the separated region is likely to be important in modelling the flow well downstream. An intermediate region exists between the invariant side region and the bulge, where all the stress levels are reduced, as would be expected from the effects of streamline convergence. Although overall there is a flow inward to the centre (streamline convergence), part of the overlying shear layer is subjected to diverging flow and an intensification of Reynolds stresses near the centre of the bulge.
We present an experimental study of slow laminar miscible displacement flows in vertical narrow eccentric annuli. We demonstrate that for suitable choices of viscosity ratio, density ratio and flow rate, we are able to find steady travelling wave displacements along the length of the annulus, even when strongly eccentric. Small eccentricity, increased viscosity ratio, increased density ratio and slower flow rates all appear to favour a steady displacement for Newtonian fluids. Qualitatively similar effects are found for non-Newtonian fluids, although the role of flow rate is less clear. These results are largely in line with predictions of a Hele-Shaw style of displacement model (Bittleston et al., J. Engng Math., vol. 43, 2002, pp. 229–253). The experiments also reveal interesting phenomena caused largely by secondary flows and dispersion. In the steady displacements, eccentricity drives a strong azimuthal counter-current flow above/below the advancing interface. This advects displacing fluid to the wide side of the annulus, where it focuses in the form of an advancing spike. On the narrow side we have also observed a spike, but only in Newtonian fluid displacements. For unsteady displacements, the azimuthal currents diminish as the interface elongates. With a strong enough yield stress and with a large enough eccentricity, unyielded fluid remains behind on the narrow side of the annulus.
The interaction of an array of voids collapsing after passage of a stress wave is studied as a model problem relevant to porous materials, for example, to energy localization leading to hotspot formation in energetic materials. Dynamic experiments are designed to illuminate the hydrodynamic processes of collapsing void interactions for eventual input into device-scale initiation models. We examine a stress wave loading representative of accidental mechanical insult, for which the wave passage length scale is comparable with the void and inter-void length scales. A single void, two-void linear array, and a four-void staggered array are studied. Diagnostic techniques include high-speed imaging of cylindrical void collapse and the first particle image velocimetry measurements in the surrounding material. Voids exhibit an asymmetrical collapse process, with the formation of a high-speed internal jet. Volume and diameter versus time data for single void collapse under stress wave loading are compared with literature results for single voids under shock-wave loading. The internal volume history does not fall on a straight line and is in agreement with simulations, but in contrast to existing linear experimental data fits. The velocity field induced in the surrounding material is measured to quantify a region of influence at selected stages of single void collapse. In the case of multiple voids, the stress wave diffracts in response to the presence of the upstream void, affecting the loading condition on the downstream voids. Both collapse-inhibiting (shielding) and collapse-triggering effects are observed.
In this study, we represent transient and unsteady dynamics of a cylinder undergoing vortex-induced vibration, by employing measurements of the fluid forces for a body controlled to vibrate sinusoidally, transverse to a free stream. We generate very high-resolution contour plots of fluid force in the plane of normalized amplitude and wavelength of controlled oscillation. These contours have been used with an equation of motion to predict the steady-state response of an elastically mounted body. The principal motivation with the present study is to extend this approach to the case where a freely vibrating cylinder exhibits transient or unsteady vibration, through the use of a simple quasi-steady model. In the model, we use equations which define how the amplitude and frequency will change in time, although the instantaneous forces are taken to be those measured under steady-state conditions (the quasi-steady approximation), employing our high-resolution contour plots.
The resolution of our force contours has enabled us to define mode regime boundaries with precision, in the amplitude–wavelength plane. Across these mode boundaries, there are discontinuous changes in the fluid force measurements. Predictions of free vibration on either side of the boundaries yield distinct response branches. Using the quasi-steady model, we are able to characterize the nature of the transition which occurs between the upper and lower amplitude response branches. This regime of vibration is of practical significance as it represents conditions under which peak resonant response is found in these systems. For higher mass ratios (m* > 10), our approach predicts that there will be an intermittent switching between branches, as the vortex-formation mode switches between the classical 2P mode and a ‘2POVERLAP’ mode. Interestingly, for low mass ratios (m* ~ 1), there exists a whole regime of normalized flow velocities, where steady-state vibration cannot occur. However, if one employs the quasi-steady model, we discover that the cylinder can indeed oscillate, but only with non-periodic fluctuations in amplitude and frequency. The character of the amplitude response from the model is close to what is found in free vibration experiments. For very low mass ratios (m* < 0.36 in this study), this regime of unsteady vibration response will extend all the way to infinite normalized velocity.
A linear spatio-temporal stability analysis is conducted for the ice growth under a falling water film along an inclined ice plane. The full system of linear stability equations is solved by using the Chebyshev collocation method. By plotting the boundary curve between the linear absolute and convective instabilities (AI/CI) of the ice mode in the parameter plane of the Reynolds number and incline angle, it is found that the linear absolute instability exists and occurs above a minimum Reynolds number and below a maximum inclined angle. Furthermore, by plotting the critical Reynolds number curves with respect to the inclined angle for the downstream and upstream branches, the convectively unstable region is determined and divided into three parts, one of which has both downstream and upstream convectively unstable wavepackets and the other two have only downstream or upstream convectively unstable wavepacket. Finally, the effect of the Stefan number and the thickness of the ice layer on the AI/CI boundary curve is investigated.
In this paper, the early development of turbulent vortex rings at two Reynolds numbers is studied using two-dimensional and stereoscopic particle image velocimetry (PIV). In the late 1980s, a similarity theory of turbulent vortex rings was proposed and this theory was tested primarily using laser Doppler velocimetry (LDV). However, because of limitations of the experimental technique, the tests were inconclusive and important assumptions could not be checked. Because single-point measurements were used, vortex ring structures could only be inferred using a complex signal analysis technique. In this study, the PIV technique provides spatial measurements of the full field of the cross-section of a ring from which a more rigorous investigation of the similarity theory is possible. Because the region over which the similarity theory appears to hold starts at about 2.5 orifice diameters downstream, this study focusses on the early development region from this point to 8 diameters downstream. Finally, Reynolds stresses and turbulence production contours are presented. The effects of ring dispersion on the measurements is also studied and quantified.