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Cold-wire measurements of a scalar, temperature, its fluctuations and the axial and radial components of the scalar dissipation between two opposed turbulent jet flows, where one jet was slightly heated, show that the residence times of the scalar in the mixing layer were short, that the scalar fluctuations and their dissipation were strongly correlated and that the probability distributions of the conditional scalar dissipation components were log-normal at values of the dissipation larger than the mean. The first finding is consistent with the fact that the scalar turbulence was ‘young’, in the sense that residence times were shorter than the large-eddy turn-over time, so that the results are likely to be representative of scalar turbulence when scalar mixing first takes place between two streams, for example close to the stabilization region of turbulent diffusion flames. The second implies that the mean scalar dissipation, conditional on the stoichiometric mixture fraction, is larger than the unconditional mean by up to an order of magnitude. Dependence of the distributions of the mean and r.m.s. conditional scalar dissipation on the shape of the scalar p.d.f. was demonstrated by relating the largest conditional dissipation values to the rarest scalar fluctuations and it was found that this dependence was also valid in other flows where scalar dissipation has been measured. The third finding implies that the use of a log-normal distribution to describe the p.d.f. of the conditional scalar dissipation, in the context of flame extinction modelling, will be in error by only 15% provided that the mean and the r.m.s. conditional scalar dissipation are accurately known.
An analysis is presented of some steady natural convection flows at large distances downstream of point heat sources on solid walls. These asymptotic self-similar flows depend only on the Prandtl number of the fluid. The flow induced by a localized source on an adiabatic wall that is vertical or facing downwards is described numerically, whereas the flow due to a localized source on a wall facing upwards separates and leads to a self-similar plume. When the wall is held at the same temperature as the ambient fluid far from the source, the flow is described by a self-similar solution of the second kind, with the algebraic decay of the temperature excess above the ambient temperature determined by a nonlinear eigenvalue problem. Numerical solutions of this problem are presented for two-dimensional and localized heat sources on a vertical wall, whereas the problem for a localized heat source under an inclined isothermal downwards-facing wall turns out to capture the Rayleigh–Taylor instability of the flow and could not be solved by the methods used in this paper. The analogous flows in fluid-saturated porous media are found to be the solutions of parameter-free problems. A conservation law similar to the one holding for a wall jet is found in the case of a two-dimensional source on an isothermal wall and numerical solutions are presented for the other cases.
The Chapman–Enskog expansion is generalized in order to derive constitutive relations for flows of inelastically colliding spheres in three dimensions – to Burnett order. To this end, the pertinent (nonlinear) Boltzmann equation is perturbatively solved by performing a (double) expansion in the Knudsen number and the degree of inelasticity. One of the results is that the normal stress differences and the ‘temperature anisotropy’, characterizing granular fluids, are Burnett effects. The constitutive relations derived in this work differ, both qualitatively and quantitatively, from those obtained in previous studies. In particular, the Navier–Stokes (order) terms have a different dependence on the degree of inelasticity and the number density than in previously derived constitutive relations; for instance, the expression for the heat flux contains a term which is proportional to ε∇ log n, where ε is a measure of the degree of inelasticity and n denotes the number density. This contribution to the heat flux is of zeroth order in the density; a similar term, i.e. one that is proportional to ε∇n, has been previously obtained by using the Enskog correction but this term is O(n) and it vanishes in the Boltzmann limit. These discrepancies are resolved by an analysis of the Chapman–Enskog and Grad expansions, pertaining to granular flows, which reveals that the quasi-microscopic rate of decay of the temperature, which has not been taken into account heretofore, provides an important scale that affects the constitutive relations. Some (minor) quantitative differences between our results and previous ones exist as well. These are due to the fact that we take into account an isotropic correction to the leading Maxwellian distribution, which has not been considered before, and also because we consider the full dependence of the corrections to the Maxwellian distribution on the (fluctuating) speed.
In order to elucidate the mechanism of cavitation erosion, the dynamics of a single laser-generated cavitation bubble in water and the resulting surface damage on a flat metal specimen are investigated in detail. The characteristic effects of bubble dynamics, in particular the formation of a high-speed liquid jet and the emission of shock waves at the moment of collapse are recorded with high-speed photography with framing rates of up to one million frames/s. Damage is observed when the bubble is generated at a distance less than twice its maximum radius from a solid boundary (γ=2, where γ=s/Rmax, s is the distance between the boundary and the bubble centre at the moment of formation and Rmax is the maximum bubble radius). The impact of the jet contributes to the damage only at small initial distances (γ[les ]0.7). In this region, the impact velocity rises to 83 m s−1, corresponding to a water hammer pressure of about 0.1 GPa, whereas at γ>1, the impact velocity is smaller than 25 m s−1. The largest erosive force is caused by the collapse of a bubble in direct contact with the boundary, where pressures of up to several GPa act on the material surface. Therefore, it is essential for the damaging effect that bubbles are accelerated towards the boundary during the collapse phases due to Bjerknes forces. The bubble touches the boundary at the moment of second collapse when γ<2 and at the moment of first collapse when γ<1. Indentations on an aluminium specimen are found at the contact locations of the collapsing bubble. In the range γ=1.7 to 2, where the bubble collapses mainly down to a single point, one pit below the bubble centre is observed. At γ[les ]1.7, the bubble shape has become toroidal, induced by the jet flow through the bubble centre. Corresponding to the decay of this bubble torus into multiple tiny bubbles each collapsing separately along the circumference of the torus, the observed damage is circular as well. Bubbles in the ranges γ[les ]0.3 and γ=1.2 to 1.4 caused the greatest damage. The overall diameter of the damaged area is found to scale with the maximum bubble radius. Owing to the possibility of generating thousands of nearly identical bubbles, the cavitation resistance of even hard steel specimens can be tested.
The deformation of a liquid capsule enclosed by an elastic membrane in an infinite simple shear flow is studied numerically at vanishing Reynolds numbers using a boundary-element method. The surface of the capsule is discretized into quadratic triangular elements that form an evolving unstructured grid. The elastic membrane tensions are expressed in terms of the surface deformation gradient, which is evaluated from the position of the grid points. Compared to an earlier formulation that uses global curvilinear coordinates, the triangular-element formulation suppresses numerical instabilities due to uneven discretization and thus enables the study of large deformations and the investigation of the effect of fluid viscosities. Computations are performed for capsules with spherical, spheroidal, and discoidal unstressed shapes over an extended range of the dimensionless shear rate and for a broad range of the ratio of the internal to surrounding fluid viscosities. Results for small deformations of spherical capsules are in quantitative agreement with the predictions of perturbation theories. Results for large deformations of spherical capsules and deformations of non-spherical capsules are in qualitative agreement with experimental observations of synthetic capsules and red blood cells. We find that initially spherical capsules deform into steady elongated shapes whose aspect ratios increase with the magnitude of the shear rate. A critical shear rate above which capsules exhibit continuous elongation is not observed for any value of the viscosity ratio. This behaviour contrasts with that of liquid drops with uniform surface tension and with that of axisymmetric capsules subject to a stagnation-point flow. When the shear rate is sufficiently high and the viscosity ratio is sufficiently low, liquid drops exhibit continuous elongation leading to breakup. Axisymmetric capsules deform into thinning needles at sufficiently high rates of elongation, independent of the fluid viscosities. In the case of capsules in shear flow, large elastic tensions develop at large deformations and prevent continued elongation, stressing the importance of the vorticity of the incident flow. The long-time behaviour of deformed capsules depends strongly on the unstressed shape. Oblate capsules exhibit unsteady motions including oscillation about a mean configuration at low viscosity ratios and continuous rotation accompanied by periodic deformation at high viscosity ratios. The viscosity ratio at which the transition from oscillations to tumbling occurs decreases with the sphericity of the unstressed shape. Results on the effective rheological properties of dilute suspensions confirm a non-Newtonian shear-thinning behaviour.
The relative dispersion framework for the non-reactive and reactive solute flux in aquifers is presented in terms of the first two statistical moments. The solute flux is described as a space–time process where time refers to the solute flux breakthrough and space refers to the transverse displacement distribution at the control plane. The statistics of the solute flux breakthrough and transversal displacement distributions are derived by analysing the motion of particle pairs. The results indicate that the relative dispersion formulation approaches the absolute dispersion results on increasing the source size (e.g. >10 heterogeneity scales). The solute flux statistics, when sampling volume is accounted for, show a flattened distribution for the solute flux variance in the space–time domain. For reactive solutes, the solute flux shows a tailing phenomenon in time while solute flux variance exhibits bi-modality in transverse distribution during the recession stage of the solute breakthrough. The solute flux correlation structure is defined as an integral measure over space and time, providing a potentially useful tool for sampling design in the subsurface.
A systematic hierarchy of partial differential equations for linear gravity waves in water of variable depth is developed through the expansion of the average Lagrangian in powers of [mid ]∇[mid ] (h=depth, ∇h=slope). The first and second members of this hierarchy, the Helmholtz and conventional mild-slope equations, are second order. The third member is fourth order but may be approximated by Chamberlain & Porter's (1995) ‘modified mild-slope’ equation, which is second order and comprises terms in ∇2h and (∇h)2 that are absent from the mild-slope equation. Approximate solutions of the mild-slope and modified mild-slope equations for topographical scattering are determined through an iterative sequence, starting from a geometrical-optics approximation (which neglects reflection), then a quasi-geometrical-optics approximation, and on to higher-order results. The resulting reflection coefficient for a ramp of uniform slope is compared with the results of numerical integrations of each of the mild-slope equation (Booij 1983), the modified mild-slope equation (Porter & Staziker 1995), and the full linear equations (Booij 1983). Also considered is a sequence of sinusoidal sandbars, for which Bragg resonance may yield rather strong reflection and for which the modified mild-slope approximation is in close agreement with Mei's (1985) asymptotic approximation.
The evolution of a vertically propagating vortex pair in stratified and sheared environments is studied with a two-dimensional numerical model. We consider a range of Froude (Fr) and Richardson (Ri) numbers, and a limited number of Reynolds numbers (Re). We find that stratification causes the formation of counter-sign vorticity around each of the original vortices through baroclinic production. At higher Fr, this wake vorticity advects the primary vortices closer together, decreasing their separation distance and increasing their vertical propagation speed, as predicted by Crow (1974) and Scorer & Davenport (1970). For these higher values of Fr, the wake vorticity also participates in an instability of the primary vortex pair, with the direction of propagation of the pair oscillating about the vertical. We term this instability the vortex head instability to distinguish it from the jet instabilities to which the wake itself is also susceptible. At lower Fr, internal gravity wave radiation dominates, and the intensity and spatial coherence of each vortex is rapidly reduced.
When a mean horizontal flow having constant shear is present in an unstratified fluid, we find that the vortices eventually rotate about one another with the same rotational sense as the background shear flow, as predicted in Lissaman et al. (1973). When stratification is also present, we find that the distribution of baroclinically generated wake vorticity is asymmetric, which sometimes leads to the emergence of a solitary vortex with the same sign as the background shear vorticity (depending on the values of Fr, Ri, and Re). Our limited survey of parameter space indicates that a solitary vortex emerges more rapidly for smaller values of Ri, smaller values of Fr, and/or larger values of Re.
It is argued that because shallow water cyclones on a β-plane drift westward at a speed equal to an available Rossby wave phase speed, they must radiate energy and cannot, therefore, be steady. The form of the Rossby wave wake accompanying a quasi-steady cyclone is calculated and the energy flux in the radiated waves determined. Further, an explicit expression for the radiation-induced northward drift of the cyclone is obtained. A general method for determining the effects of the radiation on the radius and amplitude of the vortex based on conservation of energy and potential vorticity is given. An example calculation for a cyclone with a ‘top-hat’ profile is presented, demonstrating that the primary effect of the radiation is to decrease the radius of the vortex. The dimensional timescale associated with the decay of oceanic vortices is of the order of several months to a year.
An isolated fluid mass travelling horizontally in a stratified layer is a phenomenon described alternatively as a detached gravity-current head or a strongly nonlinear solitary wave. A key feature of this flow is the transport of mass. Laboratory experiments examine the transition in time from a regime in which the flow is density driven, to one in which it is wave dominated. A simple means of creating this transitional regime, an isolated flow that exhibits both density and wave effects, is achieved by dropping a thermal into a linearly stratified layer. This transitional regime is called an ‘isolated propagating flow’. Parameters for which the transitional regime occurs are identified. Particle-tracking studies reveal the vertical flow structure. There is an upper zone that is wave dynamical, and a lower zone in which transport of mass occurs. The transported mass slowly leaks out, until the phenomenon resembles a weakly nonlinear solitary wave. The experiments mimic a thunderstorm microburst impacting a temperature inversion, which has aviation safety implications. In the ocean, cracks in the ice cap (polar leads) cause similar flows impacting the thermocline.
The steady simultaneous withdrawal of two inviscid fluids of different densities in a duct of finite height is considered. The flow is two-dimensional, and the fluids are removed by means of a line sink at some arbitrary position within the duct. It is assumed that the interface between the two fluids is drawn into the sink, and that the flow is uniform far upstream. A numerical method based on an integral equation formulation yields accurate solutions to the problem, and it is shown that under normal operating conditions, there is a solution for each value of the upstream interface height. Numerical solutions suggest that limiting configurations exist, in which the interface is drawn vertically into the sink. The appropriate hydraulic Froude number is derived for this situation, and it is shown that a continuum of solutions exists that are supercritical with respect to this Froude number. An isolated branch of subcritical solutions is also presented.
A pipe flow facility with a length of 32 m and a diameter of 40 mm has been designed in which a laminar flow of water can be maintained for Reynolds numbers up to 60 000. Velocity measurements taken in this facility show an asymmetric velocity profile both in the vertical as well as horizontal direction with velocities that deviate strongly from the parabolic Hagen–Poiseuille profile. The cause of this asymmetry is traced back to the influence of the Earth's rotation. This is confirmed by means of a comparison of the experimental data with the results from a perturbation solution and from a numerical computation of the full nonlinear Navier–Stokes equations. The physical background of this unforeseen result lies in the fact that a Hagen–Poiseuille flow is governed by a force equilibrium and inertia forces are everywhere negligible. This implies that the Coriolis force can be balanced only by a viscous force. So even the small Coriolis force due to the Earth's rotation causes a large velocity distortion for a case such as ours where the kinematic viscosity is small.
Individual falling balls were allowed to settle through otherwise quiescent well-mixed suspensions of non-colloidal neutrally buoyant spheres dispersed in a Newtonian liquid. Balls were tracked in three dimensions to determine the variances in their positions about a mean uniform vertical settling path. The primary experimental parameters investigated were the size of the falling ball and the volume fraction and size of the suspended particles. Unlike the horizontal variances, the vertical variances were found to be affected by short-time deterministic behaviour relating to the instantaneous local configurational arrangement of the suspended particles. For sufficiently long intervals between successive observations, the trajectories of the balls were observed to disperse about their mean settling paths in a random manner. This points to the existence of a Gaussian hydrodynamic dispersivity that characterizes the linear temporal growth of the variance in the position of a falling ball. The functional dependence of these horizontal and vertical dispersivities upon the parameters investigated was established.
The dispersivity dyadic was observed to be transversely isotropic with respect to the direction of gravity, with the vertical component at least 25 times larger than the horizontal component. The vertical dispersivity Dˆv (made dimensionless with the diameter of the suspended spheres and the mean settling velocity) was observed to decrease with increasing falling ball diameter, but to decrease less rapidly with concentration than theoretically predicted for very dilute suspensions; moreover, for falling balls equal in size to the suspended spheres, Dˆv increased linearly with increasing volume fraction ϕ of suspended solids.
In addition to the above experiments performed on suspensions of spheres, previously published settling-velocity data on the fall of balls through neutrally buoyant suspensions of rods possessing an aspect ratio of 20 were re-analysed, and vertical dispersivities calculated therefrom. (These data, taken by several of the present investigators in conjunction with other researchers, had only been grossly analysed in prior publications to extract the mean settling velocity of the ball, no attempt having been made at the time to extract dispersivity data too.) The resulting vertical dispersivities, when rendered dimensionless with the rod length and mean settling velocity, showed no statistically significant dependence upon the falling-ball diameter; moreover, all other things being equal, these dispersivities were observed to increase with increasing rod concentration.
Transition in fully developed circular pipe flow was investigated experimentally by the introduction of periodic perturbations. The simultaneous excitation of the azimuthal periodic modes m=+2 and m=−2 was chosen for detailed analysis. The experiments were carried out at three amplitudes. At the smallest amplitude the disturbances decayed in the direction of streaming. At intermediate input amplitude the disturbances amplified initially but then decayed with increasing distance downstream. Their growth was accompanied by the appearance of higher harmonics. At still higher amplitudes transition occurred. A mean velocity distortion corresponding to an azimuthal index of m=4 was observed at the intermediate and at the higher levels of forcing. When four stationary jets were introduced through the wall to emulate a similar mean velocity distortion, transition was observed at smaller amplitudes of forcing at modes ±2. Thus, weak longitudinal vortices provide an added instability needed to generate a secondary disturbance which, in turn, amplifies the steady vortical structures introduced by the jets. Such vortices may also be generated through the interaction of time-periodic helical modes.