Initial Momentum Thickness Research Articles

Evaluating anisotropic minimum dissipation, sigma and modulated gradient subgrid-scale models in large-eddy simulation of compressible mixing layers

We evaluate three subgrid-scale (SGS) models in large eddy simulations (LES) of compressible mixing layers up to convective Mach number ( M c ) 2.0. The initial momentum-thickness based Reynolds number is 1000 to match recent high-resolution simulations of the same configuration. The simulations employ the Localised Artificial Diffusivity (LAD) shock-capturing scheme and evaluate constant-coefficient and dynamic versions of Anisotropic Minimum Dissipation (AMD), Sigma, and Modulated Gradient (MGM) SGS models. The results show that the LAD scheme is crucial for stability at higher convective Mach numbers, for M c > 1.2 . The SGS models improve the accuracy of the LES predictions, particularly for dissipation and Reynolds stresses. Dynamic versions of all three SGS models give accurate results. The AMD kernel is found to be most sensitive to the flow conditions since the dynamically obtained model coefficient varies the least with M c . Dynamic versions of the SGS models enhance the agreement with reference results compared to the constant-coefficient versions, particularly for the eddy-viscosity AMD and Sigma models, and to a lesser extent for the MGM model. The results also emphasise the importance of retaining a non-zero artificial-shear-viscosity term, even in the presence of explicit SGS models.

Double shear layer evolution on the non-uniform computational mesh

This paper considers the problem of a thin shear layer evolution at Reynolds number rmRe = 400000 using the novel Compact Accurately Boundary Adjusting high-Resolution Technique (CABARET). The study is focused on the effect of the specific mesh refinement in the high shear rate areas on the flow properties under the influence of the developing instability. The original sequence of computational meshes (2562, 5122, 10242, 20482 cells) is modified using an iterative refinement algorithm based on the hyperbolic tangent. The properties of the solutions obtained are discussed in terms of the initial momentum thickness and the initial vorticity thickness, viscous and dilatational dissipation rates and also integral enstrophy. The growth rate for the most unstable mode depending on the mesh resolution is considered. In conclusion the accuracy of calculated mesh functions is estimated via L 1, L 2, L ∞ norms.

Momentum transport process in the quasi self-similar region of free shear mixing layer

In this study, we performed a direct numerical simulation (DNS) of a spatially developing shear mixing layer covering both developing and developed regions. The aim of this study is to clarify the driving mechanism and the vortical structure of the partial counter-gradient momentum transport (CGMT) appearing in the quasi self-similar region. In the present DNS, the self-similarity is confirmed in x/L ≥ 0.67 (x/δU0 ≥ 137), where L and δU0 are the vertical length of the computational domain and the initial momentum thickness, respectively. However, the trend of CGMT is observed at around kδU = 0.075 and 0.15, where k is the wavenumber, δU is the normalized momentum thickness at x/L = 0.78 (x/δU0 = 160), and kδU = 0.075 corresponds to the distance between the vortical/stretching regions of the coherent structure. The budget analysis for the Reynolds shear stress reveals that it is caused by the pressure diffusion term at the off-central region and by −p(∂u/∂y)¯ in the pressure-strain correlation term at the central region. As the flow moves toward the downstream direction, the appearance of those terms becomes random and the unique trend of CGMT at the specific wavenumber bands disappears. Furthermore, we investigated the relationship between the CGMT and vorticity distribution in the vortex region of the mixing layer, in association with the spatial development. In the upstream location, the high-vorticity region appears in the boundary between the areas of gradient momentum transport and CGMT, although the high-vorticity region is not actively producing turbulence. The negative production area gradually spreads by flowing toward the downstream direction, and subsequently, the fluid mass with high-vorticity is transported from the forehead stretching region toward the counter-gradient direction. In this location, the velocity fluctuation in the high-vorticity region is large and turbulence is actively produced. In view of this, the trend of negative production appears in the flow where the turbulence production and non-turbulent regions mix. Then, the non-turbulent region and CGMT almost simultaneously disappear in the fully developed region.

Onset of global instability in low-density jets

In low-density axisymmetric jets, the onset of global instability is known to depend on three control parameters, namely the jet-to-ambient density ratio $S$, the initial momentum thickness $\unicode[STIX]{x1D703}_{0}$ and the Reynolds number $Re$. For sufficiently low values of $S$ and $\unicode[STIX]{x1D703}_{0}$, these jets bifurcate from a steady state (a fixed point) to a self-excited oscillatory state (a limit cycle) when $Re$ increases above a critical value corresponding to the Hopf point, $Re_{H}$. In the literature, this Hopf bifurcation is often regarded as supercritical. In this experimental study, however, we find that under some conditions, there exists a hysteretic bistable region at $Re_{SN}<Re<Re_{H}$, where $Re_{SN}$ denotes a saddle-node point. This shows that, contrary to expectations, the Hopf bifurcation can also be subcritical, which we explore by evaluating the coefficients of a truncated Landau model. The existence of subcritical bifurcations implies the potential for triggering and the need for weakly nonlinear analyses to be performed to at least fifth order if one is to be able to predict saturation and bistability. We conclude by proposing a universal scaling for $Re_{H}$ in terms of $S$ and $\unicode[STIX]{x1D703}_{0}$. This scaling, which is insensitive to the super/subcritical nature of the bifurcations, can be used to predict the onset of self-excited oscillations, providing further evidence to support Hallberg & Strykowski’s concept (J. Fluid Mech., vol. 569, 2006, pp. 493–507) of universal global modes in low-density jets.

Numerical analysis of characteristic features of shallow and deep cavity in supersonic flow

abstractFlow past open cavities are numerically simulated at a Mach number of 1.5, and Reynolds number, based on initial momentum thickness at the front lip of cavity, of 3333 for variable depths (D) with constant length (L). The dominant frequency of oscillation shows a sudden jump when there is a transition from shallow (L/D > 1) to deep cavity (L/D < 1). The vorticity thickness displays two different growth rates along the length of cavity: (1) initial lower spreading rate, followed by (2) higher spreading rate. The lower spreading rate of shear layer is dictated by the type of cavity (either shallow or deep), while the higher spreading rate is directly related to the amplitude of oscillations. Proper orthogonal decomposition (POD) is implemented to visualise the coherent structures based on their energy content. The first two POD spatial structures in the shallow cavity represent vortex shedding, while in the deep cavity, they comprise vortex pairing interactions as in mixing layer. The higher POD modes contain coherent structures at mixed frequencies. The behaviour of coherent structures associated with a temporal frequency is further investigated using dynamic mode decomposition (DMD). The higher DMD modes confirm the dominance of mixing layer behaviour in the deep cavity.

Effect of viscosity gradients on mean velocity profile in temporal mixing layer

ABSTRACTWe present a phenomenological and statistical description of the very early stages of a temporal mixing layer (TML), in which the slow fluid is Rν = νhigh/νlow times more viscous than the rapid fluid. Direct numerical simulations (DNS) are performed for two viscosity ratios, Rν = 1 and Rν = 9, whilst the upper and lower streams are density-matched. The space–time evolution of variable-viscosity flow (VVF) is compared with that of a baseline case, which is a constant-viscosity flow (CVF, for which Rν = 1). The initial Reynolds number, based on the initial momentum thickness, δθ, 0, is for CVF, while it varies between 32 and 128 for VVF. It is shown that the mean velocity profile of VVF is affected by the viscosity variations, thus rectifying the myth that viscosity is a small-scale quantity that does not affect the large scales. The transport equation for mean velocity is derived and assessed through DNS data at the initial stages of the TML. The modification of the mean velocity profile is mainly due to the simultaneous occurrence of mean velocity and mean viscosity gradients.

Effects of Nanosecond Pulse Driven Plasma Actuators on Turbulent Shear Layers

The effects of nanosecond pulse driven dielectric barrier discharge plasma actuators on turbulent shear layers are examined experimentally on a mixing layer and backward facing step. The nanosecond pulse driven dielectric barrier discharge control mechanism is believed to be primarily thermal in contrast to most flow control actuators, including alternating current dielectric barrier discharges, which rely on periodic or pulsed momentum to excite shear-layer instabilities. Control authority using thermal perturbations has been demonstrated in various high-speed shear flows yet many questions on fundamental physics and scaling remain unanswered. This work aims to provide insight into both nanosecond pulse driven dielectric barrier discharges and thermal mechanisms in general for high-amplitude aerodynamic flow control. Previous studies suggest the efficacy of nanosecond pulse driven dielectric barrier discharges (and likely thermal perturbations in general) is strongly dependent on initial shear-layer conditions, namely some measure of the initial thickness. In an effort to support this hypothesis, boundary-layer suction is applied to a splitter plate upstream of a turbulent mixing-layer origin. This successfully reduces the initial mixing-layer momentum thickness, but does not result in substantial nanosecond pulse driven dielectric barrier discharge control authority. These results and the experimental conditions are documented in detail. Application of nanosecond pulse driven dielectric barrier discharge forcing to the turbulent shear layer downstream of a backward facing step having even smaller initial thickness produces the expected control authority for lower pulse amplitude than employed in the mixing-layer case. This supports the importance of initial shear-layer thickness (rather than state) for estimating potential control authority by nanosecond pulse driven dielectric barrier discharge and thermal perturbations in general.

Simulation of the blooming phenomenon in forced circular jets

The bifurcation and the blooming of jets have been numerically investigated at moderate Reynolds numbers. The study is motivated by a review article of Reynolds et al. (Annu. Rev. Fluid Mech., vol. 35, 2003, pp. 295–315) in which flow visualization images of jet blooming have been discussed, when the flow is subjected to inflow perturbations. Dual-mode perturbation, a combination of axisymmetric and helical excitations, has been used at the inflow plane to control the jet structures. In addition to the excitation frequency ratio, the effects of small-scale perturbation, excitation amplitude and initial momentum thickness have been examined. Results obtained at a Reynolds number of 2000 show that the number of branches formed in the blooming jet is strongly dependent on the excitation frequency ratio. For frequency ratios of 2, 2.5, 2.25, 2.4 and 2.22, the number of branches seen is 2, 5, 9, 12 and 20 respectively. In a blooming jet, the offset angle lies in the range 140°–180°. An equal number of branches is seen in the time-averaged flow field as well. The range of excitation frequency of the axisymmetric mode of perturbation is found to be $0.45&lt;\mathit{St}_{D}&lt;0.525$, with an excitation frequency ratio range of $2&lt;R_{f}&lt;2.6$, for which blooming jets are formed. The role of inlet shear layer thickness is less important as far as the blooming jet is concerned, while increasing excitation amplitude increases entrainment. Time-averaged data show that the blooming patterns persist in time, showing a substantial increase in spreading and entrainment.

Effects of Viscosity Variations in Temporal Mixing Layer

The objective of the present investigation is to assess the effects of viscosity variations in low-speed temporally-evolving turbulent mixing layer. The two streams are density-matched, but the slow fluid is R ν times more viscous than the rapid stream. Direct Numerical Simulations (DNS) are performed for several viscosity ratios, R ν = ν h i g h /ν l o w , varying between 1 and 9. The space-time evolution of Variable-Viscosity Flow (VVF) is compared with that of the Constant-Viscosity Flow (CVF), for which R ν = 1. The initial Reynolds number, based on the initial momentum thickness, δ 𝜃,0, is \(Re_{\delta _{\theta ,0}}=160\) for the considered cases. The study focuses on the first stages of the temporal evolution of the mixing-layer. It is shown that in VVF (with respect to CVF): (i) the velocity fluctuations occur earlier and are more enhanced for VVF. In particular, the kinetic energy peaks earlier and is up to three times larger for VVF than for CVF at the earliest stages of the flow. Over the first stages of the flow, the temporal growth rate of the fluctuations kinetic energy is exponential, in full agreement with linear stability theory. (ii) large-scale quantities, i.e. mean longitudinal velocity and momentum thickness, are affected by the viscosity variations, thus dispelling the myth that viscosity is a small-scale quantity that affects little the large scales. (iii) the transport equation for the fluctuations kinetic energy is derived and favourably compared with simulations data. The enhanced kinetic energy for VVF is mainly due to an increased production at the interface between the two fluids, in tight correlation with enlarged values of mean velocity gradient at the inflection point of the mean velocity profile.

Effects of viscosity variations in temporal mixing layer

The objective of the present investigation is to assess the effects of viscosity variations in low-speed temporally-evolving turbulent mixing layer. Direct Numerical Simulations (DNS) are performed for several viscosity ratios, Rv = vhigh/vlow, varying between 1 and 9, whereas the upper and lower streams are of equal density. The space-time evolution of Variable-Viscosity Flow (VVF) is compared with the Constant-Viscosity Flow (CVF), for which Rv = 1. The initial Reynolds number, based on the initial momentum thickness, δθ,0, is Reδθ,0 = 160 for the considered cases. The study focuses on the first stages of the temporal evolution of the mixing-layer. It is shown that in VVF (with respect to CVF): (i) the birth of turbulent fluctuations is accelerated; (ii) large-scale quantities, i.e. mean longitudinal velocity and momentum thickness, are affected by the viscosity variations, thus dispelling the myth that viscosity is a 'small-scale quantity that does not affect the large scales'; (iii) the velocity fluctuations are enhanced for VVF. In particular, the turbulent kinetic energy peaks earlier and is three times larger for VVF than CVF at the earliest stage of the mixing, and (iv) the transport equation for the turbulent kinetic energy is derived and favourably compared with simulations data.

Mixing due to a heated elliptic air jet

This paper presents measurements of velocity and temperature for a contoured contraction nozzle elliptic air jet with a 2:1 aspect ratio issuing into stagnant unconfined surroundings. Initial conditions at the jet nozzle exit plane were laminar. All measurements of mean and root-mean-square (RMS) velocities and temperatures were carried out using hot-wire and cold-wire anemometers. Flow development was found to be more rapid in the minor axis than major axis due to the thinner initial boundary layer momentum thickness in the minor axis plane. This is clearly demonstrated by the rapid growth of the jet width in the minor axis (compared to the major axis) in the near-field of the jet. Self-similarity in the mean velocity and temperature profiles was attained relatively early. However, RMS quantities have not become self-similar even by the streamwise distance of 38 equivalent jet nozzle diameters (De). One axis-switchover due to the initial difference in jet-width growth rates in the two axes was detected in the current jet. The rates of centerline temperature decay, temperature half-width growths and peak RMS temperatures at all measurement locations were consistently higher than their velocity counterparts. The temperature spreading rate in the major axis was found to be significantly larger vis-à-vis that of velocity, possibly due to the minima in radius of curvature of the elliptic geometry. All the results presented suggest that the use of heated elliptic jets could offer enhanced mixing performance in relevant applications.

Large eddy simulation of a circular jet: effect of inflow conditions on the near field

In the present study, the effects of the jet inflow conditions such as the initial momentum thickness (θ) and background disturbances on the downstream evolution of a circular jet are investigated using large eddy simulation (LES). We consider four different initial momentum thicknesses,D/θ = 50, 80, 120 and 180, and three different Reynolds numbers,ReD=UJD/ν = 3600, 104and 105, whereUJis the jet inflow velocity andDis the jet diameter. The present study shows that the jet characteristics significantly depend on both the initial momentum thickness and the Reynolds number. For all the Reynolds numbers considered in this study, vortex rings are generated at an earlier position with decreasing initial momentum thickness. In case of a relatively low Reynolds number ofReD= 3600, at smaller initial momentum thickness, early growth of the shear layer due to the early generation of vortex ring leads to the occurrence of large-scale coherent structures in earlier downstream locations, which results in larger mixing enhancement and more rapid increase in turbulence intensity. However, at a high Reynolds number such asReD= 105, with decreasing initial momentum thickness, rapid growth of the shear layer leads to the saturation of the shear layer and the generation of fine-scale turbulence structures, which reduces mixing and turbulence intensity. With increasingReθ(=UJθ/ν), the characteristic frequency corresponding to the shear layer mode (Stθ=fθ/UJ) gradually increases and reaches near 0.017 predicted from the inviscid instability theory. On the other hand, the frequency corresponding to the jet-preferred mode (StD=f D/UJ) varies depending onReDandD/θ. From a mode analysis, we show that, in view of the energy of the axial velocity fluctuations integrated over the radial direction, the double-helix mode (mode 2) becomes dominant past the potential core, but the axisymmetric mode (mode 0) is dominant near the jet exit. In view of the local energy, the disturbances grow along the shear layer near the jet exit: for thick shear layer, mode 0 grows much faster than other modes, but modes 0–3 grow almost simultaneously for thin shear layer. However, past the potential core, the dominant mode changes from mode 0 near the centreline to mode 1 and then to mode 2 with increasing radial direction regardless of the initial shear layer thickness.

Investigation of the shear flows over a sunroof opening based on a linear inviscid instability analysis

The large-scale shear flows over the sunroof opening of a mid-sized SUV measured using a PIV system were investigated. The shear flows were measured for five different cases of deflector protrusion (one case was the baseline test without deflector) at two different free stream flow velocities below the critical velocity where the buffeting noise level reached a maximum. The structures of the shear flows were observed to differ, apparently depending on whether the radiated buffeting noise is relatively strong or not. For strongly buffeting experimental cases, the momentum thicknesses of the shear layers were observed to grow rapidly and saturated at a station near the downstream edge of the sunroof opening, where the saturation of the transverse velocity fluctuations was also observed, and where the vortex coalescence process was presumably completed. On the other hand, no discrete large-scale vortex structures were observed for none-buffeting or weakly buffeting cases. Streamwise growth of the velocity fluctuations was found to be well predicted by a linear hydrodynamic instability analysis for the strongly buffeting cases. Numerical results obtained from a linear inviscid instability analysis using a hyperbolic tangent mean velocity profile were used to calculate the amplification factors with the initial momentum thickness and the streamwise fluctuation wavenumber. The shear flows were found to form large-scale discrete vortices when the linear inviscid amplification factors exceeded a threshold amplification factor.

Influence of Reynolds Number on the Evolution of a Plane Air Jet Issuing From a Slit

Jet flows are encountered in a variety of industrial applications. Although from the points of view of manufacturing with ease and small spatial requirement it is convenient to use short slit nozzles, most of the available studies deal with turbulent jets issuing from contoured nozzles. In the present work, experiments have been conducted in the moderate Reynolds number range of 250–6250 for a slit jet. Mixing characteristics of slit jets seem to be quite different from those of jets emerging out of contoured nozzles. This is primarily due to the differences in the decay characteristics and the large scale eddy structures generated in the near field, which are functions of the initial momentum thickness. It is evident that, in the range of 250⩽Re⩽6250, the overall spreading characteristics of the slit jet flow have stronger Reynolds number dependence than those of contoured nozzle jets. In particular, the slit jets exhibit slower mean velocity decay rates and slower half-width growth rates. Normalized power spectra and probability distribution functions are used to assess the spatial evolution and the Reynolds number dependence of jet turbulence. It is seen that the fluctuating components of velocity attain isotropic conditions at a smaller axial distance from the nozzle exit than that required for mean velocity components to become self-similar.

Suppression of global modes in low-density axisymmetric jets using coflow

Experiments conducted in helium axisymmetric jets with an annular coflowing air stream yield critical values of the velocity ratio U2∕U1 needed to suppress global instability inherent in these low-density flows. Global mode suppression was achieved for coflowing velocities less than approximately 20% of the jet centerline velocity, though the critical velocity ratio displayed a nonmonotonic relationship with the initial shear layer momentum thickness. The experiments are supported by spatio-temporal inviscid stability theory, where the convective-absolute transition was tracked in an operating domain including U2∕U1 and D∕θ. For initially thick shear layers, the experimental observations are in good agreement with linear theory, but deviate considerably as the separating shear layer thickness is reduced.

An experimental investigation of the combined effects of surface curvature and streamwise pressure gradients both in laminar and turbulent flows

Flow and heat transfer characteristics over flat, concave and convex surfaces have been investigated in a low speed wind tunnel in the presence of adverse and fa- vourable pressure gradients (k), for a range of -3.6 · 10 -6 £ k £ +3.6 · 10 -6 . The laminar near zero pressure gra- dient flow, with an initial momentum thickness Reynolds number of 200, showed that concave wall boundary layer was thinner and heat transfer coefficients were almost 2 fold of flat plate values. Whereas for the same flow con- dition, thicker boundary layer and 35% less heat transfer coefficients of the convex wall were recorded with an earlier transition. Accelerating laminar flows caused also thinner boundary layers and an augmentation in heat transfer values by 28%, 35% and 16% for the flat, concave

Vortex dynamics and entrainment in rectangular free jets

Simulations of low-aspect-ratio, rectangular free jets are presented. The investigations focus on the entrainment and transitional vortex dynamics in compressible (subsonic) jets initialized with laminar conditions, a thin vortex sheet with slightly rounded-off corner regions, and uniform initial momentum thickness. A monotonically integrated large-eddy simulation approach based on the solution of the unsteady flow equations with high-resolution monotone algorithms is used. Inherent uncertainties in the jet entrainment measurement process are addressed using the database from laboratory experiments and simulations. Vorticity geometries characterizing the near flow field of low aspect-ratio (A) rectangular jets are demonstrated, involving: (i) self-deforming and (ii) splitting vortex rings; interacting ring and braid (rib) vortices including (iii) single ribs aligned with corner regions (A [ges ] 2) and (iv) rib pairs aligned with the corners (A = 1); (v) a more disorganized flow regime in the far jet downstream, where the rotational-fluid volume is occupied by a relatively weak vorticity background with strong, slender tube-like filament vortices filling a small fraction of the domain – as observed in fully developed turbulent flows. The near field entrainment properties of low-A rectangular jets are shown to be largely determined by the characteristic A-dependent coupling geometry of interacting rib and ring vortices and by vortex-ring axis-switching times.

FLOW CONTROL WITH NONCIRCULAR JETS

▪ Abstract Noncircular jets have been the topic of extensive research in the last fifteen years. These jets were identified as an efficient technique of passive flow control that allows significant improvements of performance in various practical systems at a relatively low cost because noncircular jets rely solely on changes in the geometry of the nozzle. The applications of noncircular jets discussed in this review include improved large- and small-scale mixing in low- and high-speed flows, and enhanced combustor performance, by improving combustion efficiency, reducing combustion instabilities and undesired emissions. Additional applications include noise suppression, heat transfer, and thrust vector control (TVC). The flow patterns associated with noncircular jets involve mechanisms of vortex evolution and interaction, flow instabilities, and fine-scale turbulence augmentation. Stability theory identified the effects of initial momentum thickness distribution, aspect ratio, and radius of curvature on the initial flow evolution. Experiments revealed complex vortex evolution and interaction related to self-induction and interaction between azimuthal and axial vortices, which lead to axis switching in the mean flow field. Numerical simulations described the details and clarified mechanisms of vorticity dynamics and effects of heat release and reaction on noncircular jet behavior. The research on noncircular jets has also led to technology transfer. A topic that started as an academic curiosity—an interesting flow phenomenon—subsequently has had various industrial applications. The investigations reviewed include experimental, theoretical, numerical, and technological aspects of the subject.

Characteristics of Self-Excited Tone of a Circular Jet Impinging upon a Coaxial Pipe.

The self-excited tone of a circular jet impinging upon a coaxial pipe can be give rise to not simply a single, however as many as well-defined frequency components in the rage of Reynoldsnumber (based on initial momentum thickness and jet velocity) 171&gne;Re&gne;271. All of these components are traceble to the nonlinear interraction frequency of two primary components : the most stable frequency of the jet shear layer, β, and a low-frequency modulating component, β/2. The modulating component β/2 arises from vortex-vortex interaction at the impingiment edge. Its upsteram storongly modulates the sensitive regime of the shear layer at the nozzle lip. The most stable component can become sudden change (i.e. frequency down) when it is coincided with resonance of the air column in pipe.

Scalar mixing and reaction in plane liquid shear layers

The scalar (concentration) field of two-dimensional liquid mixing layers was investigated at post-mixing-transition conditions. The planar laser-induced fluorescence technique was used for passive scalar measurements, and for chemical product measurements. Following the approach of Koochesfahani &amp; Dimotakis (1986), the chemical product results were used to make resolution-free estimates of mixed-fluid quantities, thus providing a check on the accuracy of the passive scalar measurements. The operating conditions were systematically varied to study the effect of various parameters (Reynolds number, speed ratio, and initial boundary-layer momentum thickness) on the structure of the layer. At conditions which are just past the mixing transition, the study essentially duplicated the results obtained by Koochesfahani &amp; Dimotakis: the chemical-product concentration profiles at high- and low-stoichio-metric ratios (flip experiments) were symmetric and the average concentration of mixed-fluid was uniform across the layer. However, when the layer was pushed to more fully developed conditions, its scalar field evolved to an asymptotic state: the two flip chemical-product concentration profiles developed modest asymmetries, and the average mixed-fluid concentration developed a small variation across the layer, but the change was less than that observed in gases. Based on the chemical reaction data, we infer that the mixture fraction probability density function (p.d.f.) for the fully-developed liquid layer evolves from a ‘non-marching’ type to a ‘tilted’ type. Despite the observed evolution, the average mixed-fluid concentration remained fixed for all the layers past the mixing transition, while the total mixed-fluid probability (the total amount of mixed fluid normalized by the layer's width) showed only a very slight increasing tendency as the layer reached fully developed conditions. The mixture fraction p.d.f., measured by the passive scalar approach, is shown and discussed for a broad range of cases. While it overpredicts the amount of mixing, it showed a qualitatively-correct ‘non-marching’ character initially, but evolved to a qualitatively-incorrect ‘marching’ character at the asymptotic state. The reasons for the poor estimation of the p.d.f. by the passive scalar approach, at fully developed conditions, are attributed to changes in the flow and lack of resolution and suggests caution when using such measures. Furthermore, the study also showed that the Reynolds number alone is inadequate to characterize the state of the layer. A different parameter (the ‘pairing parameter’), which accounts for the initial boundary layers and scales with the number of vortex mergings, was found to better explain the evolution in the structure of the scalar field.