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Soret-driven thermosolutal convection induced by inclined thermal and solutal gradients in a shallow horizontal layer of a porous medium
- P. A. LAKSHMI NARAYANA, P. V. S. N. MURTHY, RAMA SUBBA REDDY GORLA
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- Published online by Cambridge University Press:
- 10 October 2008, pp. 1-19
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The stability of Soret-driven thermosolutal convection in a shallow horizontal layer of a porous medium subjected to inclined thermal and solutal gradients of finite magnitude is investigated theoretically by means of a linear stability analysis. The horizontal components of these gradients induce a Hadley circulation, which becomes unstable when vertical components are sufficiently large. We employed a two-term Galerkin approximation for various modes of instability. The effect of the Soret parameter on the mechanism of instability of the thermosolutal convection is investigated. Results are presented for various values of the governing parameters of the flow. It is observed that the Soret parameter has a significant effect on convective instability and this is discussed.
Slender-body theory for steady sheared plumes in very viscous fluid
- ROBERT J. WHITTAKER, JOHN R. LISTER
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- 10 October 2008, pp. 21-44
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A simple model based on slender-body theory is developed to describe the deflection of a steady plume by shear flow in very viscous fluid of the same viscosity. The key dimensionless parameters measuring the relative strengths of the shear, diffusion and source flux are identified, which allows a number of different dynamical regimes to be distinguished. The predictions of the model show good agreement with many, but not all, observations from previous experimental studies. Possible reasons for the discrepancies are discussed.
Analysis and flamelet modelling for spray combustion
- YUYA BABA, RYOICHI KUROSE
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- Published online by Cambridge University Press:
- 10 October 2008, pp. 45-79
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The validity of a steady-flamelet model and a flamelet/progress-variable approach for gaseous and spray combustion is investigated by a two-dimensional direct numerical simulation (DNS) of gaseous and spray jet flames, and the combustion characteristics are analysed. A modified flamelet/progress-variable approach, in which total enthalpy rather than product mass fraction is chosen as a progress variable, is also examined. DNS with an Arrhenius formation, in which the chemical reaction is directly solved in the physical flow field, is performed as a reference to validate the combustion models. The results show that the diffusion flame is dominant in the gaseous diffusion jet flame, whereas diffusion and premixed flames coexist in the spray jet flame. The characteristics of the spray flame change from premixed–diffusion coexistent to diffusion-dominant downstream. Comparisons among the results from DNS with various combustion models show the modified flamelet/progress-variable approach to be superior to the other combustion models, particularly for the spray flame. Where the behaviour of the gaseous total enthalpy is strongly affected by the energy transfer (i.e. heat transfer and mass transfer) from the dispersed droplet, and this effect can be accounted for only by solving the conservation equation of the total enthalpy. However, even the DNS with the modified flamelet/progress-variable approach tends to underestimate the gaseous temperature in the central region of the spray jet flame. To increase the prediction accuracy, a combustion model for the partially premixed flame for the spray flame is necessary.
Turbulent spot flow topology and mechanisms for surface heat transfer
- D. R. SABATINO, C. R. SMITH
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- 10 October 2008, pp. 81-105
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The properties of artificially initiated turbulent spots over a heated plate were investigated in a water channel. The instantaneous velocity field and surface Stanton number were simultaneously established using a technique that combines particle image velocimetry and thermochromic liquid crystal thermography. Several characteristics of a spot are found to be similar to those of a turbulent boundary layer. The spacing of the surface heat transfer streak patterns within the middle or ‘body’ of a turbulent spot are comparable to the low-speed streak spacing within a turbulent boundary layer. Additionally, the surface shear stress in the same region of a spot is also found to be comparable to a turbulent boundary layer. However, despite these similarities, the heat transfer within the spot body is found to be markedly less than the heat transfer for a turbulent boundary layer. In fact, the highest surface heat transfer occurs at the trailing or calmed region of a turbulent spot, regardless of maturity. Using a modified set of similarity coordinates, instantaneous two-dimensional streamlines suggest that turbulent spots entrain and subsequently recirculate warm surface fluid, thereby reducing the effective heat transfer within the majority of the spot. It is proposed that energetic vortices next to the wall, near the trailing edge of the spot body, are able to generate the highest surface heat transfer because they have the nearest access to cooler free-stream fluid.
Low-dimensional characteristics of a transonic jet. Part 1. Proper orthogonal decomposition
- C. E. TINNEY, M. N. GLAUSER, L. S. UKEILEY
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- 10 October 2008, pp. 107-141
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An experimental investigation concerning the most energetic turbulent features of the flow exiting from an axisymmetric converging nozzle at Mach 0.85 and ambient temperature is discussed using planar optical measurement techniques. The arrangement of the particle image velocimetry (PIV) system allows for all three components of the velocity field to be captured along the (r, θ)-plane of the jet at discrete streamwise locations between x/D=3.0 and 8.0 in 0.25 diameter increments. The ensemble-averaged (time-suppressed) two-point full Reynolds stress matrix is constructed from which the integral eigenvalue problem of the proper orthogonal decomposition (POD) is applied using both scalar and vector forms of the technique. A grid sensitivity study indicates that the POD eigenvalues converge safely to within 1% of their expected value when the discretization of the spatial grid is less than 30% of the integral length scale or 10% of the shear-layer width. The first POD eigenvalue from the scalar decomposition of the streamwise component is shown to agree with previous investigations for a range of Reynolds numbers and Mach numbers with a peak in azimuthal mode 5 at x/D=3.0, and a gradual shift to azimuthal mode 2 by x/D=8.0. The eigenvalues from the scalar POD of the radial and azimuthal components are shown to be much lower-dimensional with most of their energy residing in the first few azimuthal modes, that is modes 0, 1 and 2, with little change in the relative energies along the streamwise direction. From the vector decomposition, the azimuthal eigenspectra of the first two POD modes shift from a peak in azimuthal mode 5 at x/D=3.0, followed by a gradual decay to azimuthal mode 2 at x/D=8.0, the differences in the peak energies being very subtle. The conclusion from these findings is that when the Mach number is subsonic and the Reynolds number sufficiently large, the structure of the turbulent jet behaves independently of these factors.
Vortex motion in doubly connected domains
- L. ZANNETTI, F. GALLIZIO, G. M. OTTINO
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- 10 October 2008, pp. 143-152
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The unsteady two-dimensional rotational flow past doubly connected domains is analytically addressed. By concentrating the vorticity in point vortices, the flow is modelled as a potential flow with point singularities. The dependence of the complex potential on time is defined according to the Kelvin theorem. The general case of non-null circulations around the solid bodies is discussed. Vortex shedding and time evolution of the circulation past a two-element airfoil and past a two-bladed Darrieus turbine are presented as physically coherent examples.
Trajectory and flow properties for a rod spinning in a viscous fluid. Part 1. An exact solution
- ROBERTO CAMASSA, TERRY JO LEITERMAN, RICHARD M. MCLAUGHLIN
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- 10 October 2008, pp. 153-200
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An exact mathematical solution for the low-Reynolds-number quasi-steady hydrodynamic motion induced by a rod in the form of a prolate spheroid sweeping a symmetric double cone is developed, and the influence of the ensuing fluid motion upon passive particles is studied. The resulting fluid motion is fully three-dimensional and time varying. The advected particles are observed to admit slow orbits around the rotating rods and a fast epicyclic motion roughly commensurate with the rod rotation rate. The epicycle amplitudes, vertical fluctuations, arclengths and angle travelled per rotation are mapped as functions of their initial coordinates and rod geometry. These trajectories exhibit a rich spatial structure with greatly varying trajectory properties. Laboratory frame asymmetries of these properties are explored using integer time Poincaré sections and far-field asymptotic analysis. This includes finding a small cone angle invariant in the limit of large spherical radius whereas an invariant for arbitrary cone angles is obtained in the limit of large cylindrical radius. The Eulerian and Lagrangian flow properties of the fluid flow are studied and shown to exhibit complex structures in both space and time. In particular, spatial regions of high speed and Lagrangian velocities possessing multiple extrema per rod rotation are observed. We establish the origin of these complexities via an auxiliary flow in a rotating frame, which provides a generator that defines the epicycles. Finally, an additional spin around the major spheroidal axis is included in the exact hydrodynamic solution resulting in enhanced vertical spatial fluctuation as compared to the spinless counterpart. The connection and relevance of these observations with recent developments in nano-scale fluidics is discussed, where similar epicycle behaviour has been observed. The present study is of direct use to nano-scale actuated fluidics.
Bubble-induced skin-friction drag reduction and the abrupt transition to air-layer drag reduction
- BRIAN R. ELBING, ERIC S. WINKEL, KEARY A. LAY, STEVEN L. CECCIO, DAVID R. DOWLING, MARC PERLIN
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- 10 October 2008, pp. 201-236
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To investigate the phenomena of skin-friction drag reduction in a turbulent boundary layer (TBL) at large scales and high Reynolds numbers, a set of experiments has been conducted at the US Navy's William B. Morgan Large Cavitation Channel (LCC). Drag reduction was achieved by injecting gas (air) from a line source through the wall of a nearly zero-pressure-gradient TBL that formed on a flat-plate test model that was either hydraulically smooth or fully rough. Two distinct drag-reduction phenomena were investigated; bubble drag reduction (BDR) and air-layer drag reduction (ALDR).
The streamwise distribution of skin-friction drag reduction was monitored with six skin-friction balances at downstream-distance-based Reynolds numbers to 220 million and at test speeds to 20.0ms−1. Near-wall bulk void fraction was measured at twelve streamwise locations with impedance probes, and near-wall (0 < Y < 5mm) bubble populations were estimated with a bubble imaging system. The instrument suite was used to investigate the scaling of BDR and the requirements necessary to achieve ALDR.
Results from the BDR experiments indicate that: significant drag reduction (>25%) is limited to the first few metres downstream of injection; marginal improvement was possible with a porous-plate versus an open-slot injector design; BDR has negligible sensitivity to surface tension; bubble size is independent of surface tension downstream of injection; BDR is insensitive to boundary-layer thickness at the injection location; and no synergetic effect is observed with compound injection. Using these data, previous BDR scaling methods are investigated, but data collapse is observed only with the ‘initial zone’ scaling, which provides little information on downstream persistence of BDR.
ALDR was investigated with a series of experiments that included a slow increase in the volumetric flux of air injected at free-stream speeds to 15.3ms−1. These results indicated that there are three distinct regions associated with drag reduction with air injection: Region I, BDR; Region II, transition between BDR and ALDR; and Region III, ALDR. In addition, once ALDR was established: friction drag reduction in excess of 80% was observed over the entire smooth model for speeds to 15.3ms−1; the critical volumetric flux of air required to achieve ALDR was observed to be approximately proportional to the square of the free-stream speed; slightly higher injection rates were required for ALDR if the surface tension was decreased; stable air layers were formed at free-stream speeds to 12.5ms−1 with the surface fully roughened (though approximately 50% greater volumetric air flux was required); and ALDR was sensitive to the inflow conditions. The sensitivity to the inflow conditions can be mitigated by employing a small faired step (10mm height in the experiment) that helps to create a fixed separation line.
The stability of the variable-density Kelvin–Helmholtz billow
- JÉRÔME FONTANE, LAURENT JOLY
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- 10 October 2008, pp. 237-260
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We perform a three-dimensional stability analysis of the Kelvin–Helmholtz (KH) billow, developing in a shear layer between two fluids with different density. We begin with two-dimensional simulations of the temporally evolving mixing layer, yielding the unsteady base flow fields. The Reynolds number is 1500 while the Schmidt and Froude numbers are infinite. Then exponentially unstable modes are extracted from a linear stability analysis performed at the saturation of the primary mode kinetic energy. The spectrum of the least stable modes exhibits two main classes. The first class comprises three-dimensional core-centred and braid-centred modes already present in the homogeneous case. The baroclinic vorticity concentration in the braid lying on the light side of the KH billow turns the flow into a sharp vorticity ridge holding high shear levels. The hyperbolic modes benefit from the enhanced level of shear in the braid whereas elliptic modes remain quite insensitive to the modifications of the base flow. In the second class, we found typical two-dimensional modes resulting from a shear instability of the curved vorticity-enhanced braid. For a density contrast of 0.5, the wavelength of the two-dimensional instability is about ten times shorter than that of the primary wave. Its amplification rate competes well against those of the hyperbolic three-dimensional modes. The vorticity-enhanced braid thus becomes the preferred location for the development of secondary instabilities. This stands as the key feature of the transition of the variable-density mixing layer. We carry out a fully resolved numerical continuation of the nonlinear development of the two-dimensional braid-mode. Secondary roll-ups due to a small-scale Kelvin–Helmholtz mechanism are promoted by the underlying strain field and develop rapidly in the compression part of the braid. Originally analysed by Reinoud et al. (Phys. Fluids, vol. 12, 2000, p. 2489) from two-dimensional non-viscous numerical simulations, this instability is shown to substantially increase the mixing.
Modelling the breakup of solid aggregates in turbulent flows
- MATTHÄUS U. BÄBLER, MASSIMO MORBIDELLI, JERZY BAŁDYGA
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- 10 October 2008, pp. 261-289
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The breakup of solid aggregates suspended in a turbulent flow is considered. The aggregates are assumed to be small with respect to the Kolmogorov length scale and the flow is assumed to be homogeneous. Further, it is assumed that breakup is caused by hydrodynamic stresses acting on the aggregates, and breakup is therefore assumed to follow a first-order kinetic where KB(x) is the breakup rate function and x is the aggregate mass. To model KB(x), it is assumed that an aggregate breaks instantaneously when the surrounding flow is violent enough to create a hydrodynamic stress that exceeds a critical value required to break the aggregate. For aggregates smaller than the Kolmogorov length scale the hydrodynamic stress is determined by the viscosity and local energy dissipation rate whose fluctuations are highly intermittent. Hence, the first-order breakup kinetics are governed by the frequency with which the local energy dissipation rate exceeds a critical value (that corresponds to the critical stress). A multifractal model is adopted to describe the statistical properties of the local energy dissipation rate, and a power-law relation is used to relate the critical energy dissipation rate above which breakup occurs to the aggregate mass. The model leads to an expression for KB(x) that is zero below a limiting aggregate mass, and diverges for x → ∞. When simulating the breakup process, the former leads to an asymptotic mean aggregate size whose scaling with the mean energy dissipation rate differs by one third from the scaling expected in a non-fluctuating flow.
Endothermic and exothermic chemically reacting plumes
- DEVIN T. CONROY, STEFAN G. LLEWELLYN SMITH
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- 10 October 2008, pp. 291-310
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We develop a model for a turbulent plume in an unbounded ambient that takes into account a general exothermic or endothermic chemical reaction. These reactions can have an important effect on the plume dynamics since the entrainment rate, which scales with the vertical velocity, will be a function of the heat release or absorption. Specifically, we examine a second-order non-reversible reaction, where one species is present in the plume from a pure source and the other is in the environment. For uniform ambient density and species fields the reaction has an important effect on the deviation from pure plume behaviour as defined by the source parameter Γ. In the case of an exothermic reaction the density difference between the plume and the reference density increases and the plume is ‘lazy’, whereas for an endothermic reaction this difference decreases and the plume is more jet-like. Furthermore, for chemical and density-stratified environments, the reaction will have an important effect on the buoyancy flux because the entrainment rate will not necessarily decrease with distance from the source, as in traditional models. As a result, the maximum rise height of the plume for exothermic reactions may actually decrease with reaction rate if this occurs in a region of high ambient density. In addition, we investigate non-Boussinesq effects, which are important when the heat of reaction is large enough.
Nonlinear evolution of the Richtmyer–Meshkov instability
- MARCUS HERRMANN, PARVIZ MOIN, SNEZHANA I. ABARZHI
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- 10 October 2008, pp. 311-338
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We report analytical and numerical results describing the dynamics of the two-dimensional coherent structure of bubbles and spikes in the Richtmyer–Meshkov instability for fluids with a finite density ratio. The theory accounts for the non-local properties of the interface evolution, and the simulations treat the interface as a discontinuity. Good agreement between the analytical and numerical results is achieved. To quantify accurately the interface dynamics in the simulations, new diagnostics and scalings are suggested. The velocity at which the interface would move if it were ideally planar is used to set the flow time scale as well as the reference point for the bubble (spike) position. The data sampling has high temporal resolution and captures the velocity oscillations caused by sound waves. The bubble velocity and curvature are both monitored, and the bubble curvature is shown to be the relevant diagnostic parameter. According to the results obtained, in the nonlinear regime of the Richtmyer–Meshkov instability the bubbles flatten and decelerate, and the flattening of the bubble front indicates the multiscale character of the coherent dynamics.
Shear-enhanced convection in a mushy layer
- JEROME A. NEUFELD, J. S. WETTLAUFER
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- 10 October 2008, pp. 339-361
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We investigate the effect of an external shear flow on the buoyant instabilities inherent in the directional solidification of a dendritic mushy layer. In the presence of an external shear flow, perturbations of the mush–liquid interface lead to perturbed flow in the bulk fluid that create pressure variations along the mush–liquid interface. These pressure variations drive flow in the mushy layer. A numerical analysis of the stability of the system provides the critical porous-medium Rayleigh number as a function of both the external flow speed and the wavenumber of the interfacial perturbations. In the limit of zero external flow we recover the so-called boundary and mushy layer modes of buoyancy-driven convection first established by Worster (J. Fluid Mech., vol. 237, 1992b, p. 649). We find that the application of an external flow can significantly reduce the stability of both the boundary and mushy layer modes. The resultant forced mushy layer mode gives rise to the formation of channels of reduced solid fraction perpendicular to the applied flow that are distinct from the planform found in the absence of an external flow. The stability of the system is examined as a function of the principal thermodynamic and dynamic parameters, and the results are applied to the solidification of sea ice in the presence of vigorous oceanic flow.
An experimental study of shear-enhanced convection in a mushy layer
- JEROME A. NEUFELD, J. S. WETTLAUFER
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- 10 October 2008, pp. 363-385
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The influence of an external shear flow on the evolution of a solidifying array of dendritic crystals, termed a mushy layer, is investigated through controlled cooling of an aqueous ammonium chloride solution in a laboratory flume. The controlled cooling produces a mushy layer that grows at a constant rate from the base of the flume over which a laminar shear flow is applied. We find a critical flow speed above which a spatiotemporal variation of the solid fraction of the layer appears with a planform transverse to the flow direction. The presence of this distinctive pattern of spanwise crevasses is compared with a simplified stability analysis in which the motion of the external fluid over the corrugated mush–liquid interface produces a pressure perturbation that drives flow and phase change within the mushy layer. This flow leads to a pattern of solidification and dissolution that is compared to the experimental results. The physical mechanism underlying the pattern formation is confirmed by the agreement between the theoretical predictions and experimental results. Finally, the comparison between theory and experiment provides a value for the mushy layer permeability, the evolution of which is of relevance to a host of geophysical, biological and engineering systems.
Spectral properties and universal behaviour of advecting–diffusing scalar fields in finite-length channels
- M. GIONA, S. CERBELLI, F. CRETA
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- 10 October 2008, pp. 387-406
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This paper analyses the relaxation towards the steady state of an advecting–diffusing field in a finite-length channel. The dominant eigenvalue, −-ΛF, of the advection–diffusion operator provides the slowest relaxation time scale for achieving steady state in open flow devices. We focus on parallel flows and analyse how ΛF depends on the velocity profile and the molecular diffusivity. As a result of the universal localization features of the eigenfunction associated with ΛF, we find that ΛF can be predicted analytically based on the local behaviour of the velocity profile near the stagnation points. Microfluidic applications of the theory are also addressed.
Finite-wavelength scattering of incident vorticity and acoustic waves at a shrouded-jet exit
- ARNAB SAMANTA, JONATHAN B. FREUND
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- 10 October 2008, pp. 407-438
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As the vortical disturbances of a shrouded jet pass the sharp edge of the shroud exit some of the energy is scattered into acoustic waves. Scattering into upstream-propagating acoustic modes is a potential mechanism for closing the resonance loop in the ‘howling’ resonances that have been observed in various shrouded jet configurations over the years. A model is developed for this interaction at the shroud exit. The jet is represented as a uniform flow separated by a cylindrical vortex sheet from a concentric co-flow within the cylindrical shroud. A second vortex sheet separates the co-flow from an ambient flow outside the shroud, downstream of its exit. The Wiener–Hopf technique is used to compute reflectivities at the shroud exit. For some conditions it appears that the reflection of finite-wavelength hydrodynamic vorticity modes on the vortex sheet defining the jet could be sufficient to reinforce the shroud acoustic modes to facilitate resonance. The analysis also gives the reflectivities for the shroud acoustic modes, which would also be important in establishing resonance conditions. Interestingly, it is also predicted that the shroud exit can be ‘transparent’ for ranges of Mach numbers, with no reflection into any upstream-propagating acoustic mode. This is phenomenologically consistent with observations that indicate a peculiar sensitivity of resonances of this kind to, say, jet Mach number.
Differential diffusion of high-Schmidt-number passive scalars in a turbulent jet
- T. M. LAVERTU, L. MYDLARSKI, S. J. GASKIN
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- 10 October 2008, pp. 439-475
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The separate evolution, or differential diffusion, of high-Schmidt-number passive scalars in a turbulent jet is studied experimentally. The two scalars under consideration are disodium fluorescein (Sc ≡ ν/D = 2000) and sulforhodamine 101 (Sc = 5000). The objectives of the research are twofold: to determine (i) the Reynolds-number-dependence, and (ii) the radial distribution of differential diffusion effects in the self-similar region of the jet. Punctual laser-induced fluorescence (LIF) measurements were obtained 50 jet diameters downstream of the nozzle exit for five Reynolds numbers (Re ≡ uod/ν = 900, 2100, 4300, 6700 and 10600, where u0 is the jet exit velocity, d is the jet diameter, and ν is the kinematic viscosity) and for radial positions extending from the centreline to the edges of the jet cross-section (0 ≤ r/d ≤ 7.5). Statistics of the normalized concentration difference, Z, were used to quantify the differential diffusion. The latter were found to decay slowly with increasing Reynolds number, with the root mean square of Z scaling as Zrms ≡ 〈Z2〉1/2 ∝ Re−0.1, (or alternatively 〈Z2〉 ∝ Re−0.2). Regardless of Reynolds number, differential diffusion effects were found to increase away from the centreline. The increase in differential diffusion effects with radial position, along with their increase with decreasing Reynolds number, support the hypothesis of increased differential diffusion at interfaces between the jet and ambient fluids. Power spectral densities of Z were also studied. These spectra decreased with increasing wavenumber – an observation attributed to the decay of the scalar fluctuations in a turbulent jet. Furthermore, these spectra showed that significant differential diffusion effects persist at scales larger than the Kolmogorov scale, even for moderately high Reynolds numbers.