Showing papers on "Schmidt number published in 2001"

Schmidt number witnesses and bound entanglement

Abstract: The Schmidt number of a mixed state characterizes the minimum Schmidt rank of the pure states needed to construct it. We investigate the Schmidt number of an arbitrary mixed state by studying Schmidt-number witnesses that detect it. We present a canonical form of such witnesses and provide constructive methods for their optimization. Finally, we present strong evidence that all bound entangled states with positive partial transpose in ${\mathcal{C}}^{3}\ensuremath{\bigotimes}{\mathcal{C}}^{3}$ have Schmidt number 2.

Simulating the Wave-Enhanced Layer under Breaking Surface Waves with Two-Equation Turbulence Models

Abstract: The purpose of this paper is to modify two-equation turbulence models such that they are capable of simulating dynamics in the wave-enhanced layer near the surface. A balance of diffusion of turbulent kinetic energy (TKE) and dissipation is assumed as the surface boundary condition for TKE following the suggestion of Craig and Banner. It is shown that this theory, originally developed under the assumption of a macro length scale linearly increasing down from the surface, fails for two-equation models such as the well-known k–e model. Suggestions are made how to modify such models for overcoming this deficiency. The basic idea is to insert the analytic solution of a model problem suggested by Craig into the dissipation rate equation and solve for the turbulent Schmidt number of the dissipation rate equation, which may be formulated as a function of the production/dissipation ratio. With this modification, the linear behavior of the macro length scale is properly reproduced by the k–e model. It is ...

Lagrangian characteristics of turbulence and scalar transport in direct numerical simulations

Abstract: A study of the Lagrangian statistical properties of velocity and passive scalar fields using direct numerical simulations is presented, for the case of stationary isotropic turbulence with uniform mean scalar gradients. Data at higher grid resolutions (up to 512 3 and Taylor-scale Reynolds number 234) allow an update of previous velocity results at lower Reynolds number, including intermittency and dimensionality effects on vorticity time scales. The emphasis is on Lagrangian scalar time series which are new to the literature and important for stochastic mixing models. The variance of the ‘total’ Lagrangian scalar value (ϕ˜ + , combining contributions from both mean and fluctuations) grows with time, with the velocity–scalar cross-correlation function and fluid particle displacements playing major roles. The Lagrangian increment of ϕ˜ + conditioned upon velocity and scalar fluctuations is well represented by a linear regression model whose parameters depend on both Reynolds number and Schmidt number. The Lagrangian scalar fluctuation is non-Markovian and has a longer time scale than the velocity, which is due to the strong role of advective transport, and is in contrast to results in an Eulerian frame where the scalars have shorter time scales. The scalar dissipation is highly intermittent and becomes de-correlated in time more rapidly than the energy dissipation. Differential diffusion for scalars with Schmidt numbers between 1/8 and 1 is characterized by asymmetry in the two-scalar cross-correlation function, a shorter time scale for the difference between two scalars, as well as a systematic decrease in the Lagrangian coherency spectrum up to at least the Kolmogorov frequency. These observations are consistent with recent work suggesting that differential diffusion remains important in the small scales at high Reynolds number.

Dynamics of scalar dissipation in isotropic turbulence: a numerical and modelling study

Abstract: The physical mechanisms underlying the dynamics of the dissipation of passive scalar fluctuations with a uniform mean gradient in stationary isotropic turbulence are studied using data from direct numerical simulations (DNS), at grid resolutions up to 5123. The ensemble-averaged Taylor-scale Reynolds number is up to about 240 and the Schmidt number is from ⅛ to 1. Special attention is given to statistics conditioned upon the energy dissipation rate because of their important role in the Lagrangian spectral relaxation (LSR) model of turbulent mixing. In general, the dominant physical processes are those of nonlinear amplification by strain rate fluctuations, and destruction by molecular diffusivity. Scalar dissipation tends to form elongated structures in space, with only a limited overlap with zones of intense energy dissipation. Scalar gradient fluctuations are preferentially aligned with the direction of most compressive strain rate, especially in regions of high energy dissipation. Both the nature of this alignment and the timescale of the resulting scalar gradient amplification appear to be nearly universal in regard to Reynolds and Schmidt numbers. Most of the terms appearing in the budget equation for conditional scalar dissipation show neutral behaviour at low energy dissipation but increased magnitudes at high energy dissipation. Although homogeneity requires that transport terms have a zero unconditional average, conditional molecular transport is found to be significant, especially at lower Reynolds or Schmidt numbers within the simulation data range. The physical insights obtained from DNS are used for a priori testing and development of the LSR model. In particular, based on the DNS data, improved functional forms are introduced for several model coefficients which were previously taken as constants. Similar improvements including new closure schemes for specific terms are also achieved for the modelling of conditional scalar variance.

Lagrangian characteristics of turbulence and scalar transport in direct numerical simulations

Abstract: A study of the Lagrangian statistical properties of velocity and passive scalar elds using direct numerical simulations is presented, for the case of stationary isotropic turbulence with uniform mean scalar gradients. Data at higher grid resolutions (up to 512 3 and Taylor-scale Reynolds number 234) allow an update of previous velocity results at lower Reynolds number, including intermittency and dimensionality eects on vorticity time scales. The emphasis is on Lagrangian scalar time series which are new to the literature and important for stochastic mixing models. The variance of the ‘total’ Lagrangian scalar value ( ~+ , combining contributions from both mean and fluctuations) grows with time, with the velocity{scalar cross-correlation function and fluid particle displacements playing major roles. The Lagrangian increment of ~+ conditioned upon velocity and scalar fluctuations is well represented by a linear regression model whose parameters depend on both Reynolds number and Schmidt number. The Lagrangian scalar fluctuation is non-Markovian and has a longer time scale than the velocity, which is due to the strong role of advective transport, and is in contrast to results in an Eulerian frame where the scalars have shorter time scales. The scalar dissipation is highly intermittent and becomes de-correlated in time more rapidly than the energy dissipation. Dierential diusion for scalars with Schmidt numbers between 1/8 and 1 is characterized by asymmetry in the two-scalar cross-correlation function, a shorter time scale for the dierence between two scalars, as well as a systematic decrease in the Lagrangian coherency spectrum up to at least the Kolmogorov frequency. These observations are consistent with recent work suggesting that dierential diusion remains important in the small scales at high Reynolds number.

BGK models for diffusion in isothermal binary fluid systems

Abstract: Two Bhatnagar–Gross–Krook (BGK) models for isothermal binary fluid systems—the classical single relaxation time model and a split collision term model—are discussed in detail, with emphasis on the diffusion process in perfectly miscible ideal gases. Fluid equations, as well as the constitutive equation for diffusion, are derived from the Boltzmann equation using the method of moments and the values of the transport coefficients (viscosity and diffusivity) are calculated. The Schmidt number is found to be equal to one for both models. The split collision term model allows the two fluid components to have different values of the viscosity, while the single relaxation time model does not have this characteristic. The value of the viscosity does not depend on the density in the split collision term model, as expected from the classical kinetic theory developed by Maxwell. Possible extension of BGK models to non-ideal gases and ideal solutions (where the Schmidt number is larger than 1) is also investigated.

Turbulent transport of a passive scalar in a round jet discharging into a co-flowing stream

Abstract: The mass transport properties of a round turbulent jet of water discharging into a low velocity co-flowing water stream, confined in a square channel, is investigated experimentally. The measurement region is the self-similar range from x/d=70 to x/d=140. Combined laser-induced fluorescence and 2D laser Doppler velocimetry are used in order to measure simultaneously, instantaneously and in the same probe volume, the molecular concentration of a passive scalar and two components of the velocity. This technique allows the determination of moments involving correlations of both velocity and concentration fields, which are necessary to validate the second-order modelling schemes. Both transport equations of Reynolds shear stress uv and turbulent mass flux vc have been considered. In both cases, advection, production and diffusion terms have been determined experimentally. The pressure-strain correlation and the pressure scrambling term are inferred with the help of the budget of Reynolds shear stress and mass turbulent transport equations. Second order closure models are evaluated in the light of the experimental data. The turbulent Schmidt number is found to be almost constant and equal to 0.62 in the center region and decreases strongly to zero in the mixing layer of the jet. The effects of the co-flow on the turbulent mixing process are also highlighted.

Linear eddy simulations of Reynolds number and Schmidt number effects on turbulent scalar mixing

Abstract: Effects of molecular diffusion on turbulent scalar mixing are studied using the linear eddy model. Unlike the energy spectrum which is determined only by the viscous and eddy time scales, the scalar spectra also depend on the diffusion time scales. The linear eddy model, being one-dimensional, offers an inexpensive way of capturing these time scale dependencies. Some of the observations made using experiments and direct simulations are verified using the model. Simulations reported here indicate that certain features of scalar mixing continue to depend on the Reynolds number (Reλ) and the Schmidt number (Sc) in the range of parameter space (32⩽Reλ⩽775 and 0.001⩽Sc⩽700) that is unattainable using the current direct simulation capabilities. However, some of the models developed for differential diffusion using direct simulations are shown to be accurate even when the difference in the Schmidt numbers of the scalars and (or) the Reynolds numbers are very high.

Short circuits in the Corrsin–Obukhov cascade

Abstract: Experiments show that a blob of scalar released in the inertial range of scales of a turbulent medium is rapidly converted into a set of disjointed sheets whose spectrum exhibits a k−1 shape for wave numbers larger than the injection wave number 1/d. The sheets diffusive uniformization onsets at a time ts=(d/u)f(Sc) with f(Sc)∼ln(Sc) function of the injection time of the blob in the medium d/u, of the Schmidt number Sc, independently of the Reynolds number. This time is appreciably smaller than the time needed to cross the Kolmogorov cascade, which is “bypassed” by a strong and constant stretching rate acting at the injection scale.

Heat and mass transfer effects on flow past an impulsively started vertical plate

Abstract: An exact solution to the problem of lfow past an impulsively started infinite vertical plate in the presence of uniform heat and mass flux at the plate is presented by the Laplace-transform technique. The velocity, the temperature and the concentration profiles are shown graphically. The rate of heat transfer, the skin-friction, and the Sherwood number are also shown on graphs. The effect of different parameters like Grashof number, mass Grashof number, Prandtl number, and Schmidt number are discussed.

Fluid flow and mass transfer characteristics in a sinusoidal wavy-walled tube at moderate Reynolds numbers for steady flow

Abstract: Flow and mass transfer characteristics in an axisymmetric sinusoidal wavy-walled tube are investigated experimentally in the Reynolds number range from 50 to 1000. The flow patterns are visualized by the aluminum dust method. The measurements of the wall shear stress and the mass transfer rate for high Schmidt number are performed by the electrochemical method. The attention is focused on their characteristics, including the transitional flow. It was observed that steady flow changes into unsteady flow when the Reynolds number exceeds about 160. The wall shear stress in the minimum circular cross section of the tube and the mass transfer rate for one wavelength are shown to depend upon the Reynolds number. In the laminar flow regime they increase with the slope of 1 and 1/3, respectively, whereas in the turbulent flow regime they increase with the slope of 3/2 and 3/5, respectively. In the transitional flow regime both of them increase significantly with further larger slope. It is found that the laminar-like motion and turbulent-like motion occur alternatively, with different time intervals, indicating intermittent flow behavior. This is quite different from the flow instability of the wavy-walled channel in which Tollmien-Schlichting waves keeping a time-periodic flow appear.

The difference in turbulent diffusion between active and passive scalars in stable thermal stratification

Abstract: The difference in turbulent diffusion between active scalar (heat) and passive scalar (mass) in a stable thermally stratified flow is investigated both experimentally and numerically. The experiments are conducted in an unsheared thermally stratified water flow downstream of a turbulence-generating grid. Passive mass is released into the stable thermally stratified flow from a point source located 60 mm downstream from the grid. Instantaneous streamwise and vertical velocities, the temperature of the active scalar and the concentration of the passive scalar are simultaneously measured using a combined technique with a two-component laser-Doppler velocimeter (LDV), a resistance thermometer and a laser-induced fluorescence (LIF) method. From the measurements, turbulent heat and mass fluxes and eddy diffusivities for both active heat and passive mass are estimated. To investigate the Prandtl or Schmidt number effects on the difference in turbulent diffusion between active heat and passive mass, a three-dimensional direct numerical simulation (DNS) based on a finite difference method is applied to stable thermally stratified flows of both water and air behind the turbulence grid. The Schmidt number of passive mass in the DNS is set to the same value as the Prandtl number of active heat.The results show that stable stratification causes a large difference in eddy diffusivities between active heat and passive mass. The numerical predictions by the DNS are in qualitative agreement with the measurements despite the assumption of the same molecular diffusivity for active heat and passive mass. The difference suggests that the assumption of identical eddy diffusivity for active heat and passive mass, used in conventional turbulence models, gives significant errors in estimating heat and mass transfer in a plume under stably stratified conditions.

Modelling Differential Diffusion in Nonpremixed Reacting Turbulent Flow: Model Development

Abstract: The modelling of differential diffusion in nonpremixed reacting turbulent flows is considered. The present approach is based on the conditional moment closure (CMC) method. As in the nonreacting case the terms involving the deviations from the conditional averages, the eyterms, cannot be neglected and need modelling. The development of new closures for the eyterms and their validation is aided by DNS studies. Non-unity Schmidt number effects of reacting scalars in homogeneous iso-tropic decaying turbulence are investigated. Chemical kinetics are approximated by irreversible one-step temperature independent and temperature dependent reactions. It is found that the ey-terms and their modelling in reacting flows strongly resemble ey-terms and -models in nonreacting flows (Kronenburg and Bilger, 1997). However, a new definition of a suitable differential diffusion variable is necessary for accurate enclosure in reacting flows. The dependence of the ey-modelling on reaction rate and temperature effects is anal...

Passive scalar measurements in a planar mixing layer by PLIF of acetone

Abstract: Quantitative passive scalar measurements were performed in an incompressible planar mixing layer at Re δ up to 104 using planar laser-induced fluorescence of acetone seeded into one side of the layer. Probability density functions compiled from sets of images showed a preferred mixture composition, favoring the high-speed fluid, which extended across the layer. This preferred composition produced non-marching PDFs and an inflection in the average mixture fraction profile. The spatial resolution of the experiment was found to be sufficient to accurately measure the fraction of mixed fluid within the layer. The mixed fluid fraction was found to increase to an asymptotic value of 0.5 by Re δ ≈ 5,000, the approximate location of turbulent transition, in contrast to high Schmidt number experiments which show minimal mixing before the transition point.

Mass transfer characteristics of artificial lungs.

Abstract: An artificial lung is used during cardiopulmonary bypass to oxygenate blood and control blood temperature. The oxygen transfer rate-flow rate characteristics of three hollow fiber membrane artificial lungs (Sarns Turbo 440, Cobe Optima, Dideco Compactflo) were determined in vitro to characterize design features. Results are presented as a unique dimensionless relationship between Sherwood number, NSh (ratio of convective to diffusive mass transfer), Schmidt number, NSc (ratio of momentum to diffusive transport), and Reynolds number, NRe (ratio of inertial to viscous forces). This relationship is a function of device porosity, epsilon, and characteristic device length, xi, defined as the ratio of the mean blood path and manifold length: Nsh/NSc(1/3) x xi(1/2) = phi x (epsilon(1/m) x NRe)(m) where phi = 0.26 and m = 1.00 for NPe 3,200 where NPe is the dimensionless Peclet number defined as NRe x NSc. We found good correspondence between the model predictions and in vitro blood oxygen transfer rates. We conclude that this dimensionless approach allows us (1) to compare artificial lungs independently, (2) to relate water tests to blood, and (3) to predict the oxygen transfer rate of a new artificial lung design.

Measurement of the Coefficient of Transverse Dispersion in Flow Through Packed Beds for a Wide Range of Values of the Schmidt Number

Abstract: Experimental values of the coefficient of transverse dispersion (DT) were measured with the system 2-naphthol/water, over a range of temperatures between 293K and 373K, which corresponds to a range of values of viscosity (μ) between 2.83×10−4 Ns/m2 and 1.01×10−3 Ns/m2 and of molecular diffusion coefficient (Dm) between 1.03×10−9 m2/s and 5.49×10−9 m2/s. Since the density (ρ) of water is close to 103 kg/m3, the corresponding variation of the Schmidt number (Sc=μ/ρDm) was in the range 1000 – 50. More than 200 experimental values of the transverse dispersion coefficient were obtained using beds of silica sand with average particle sizes (d) of 0.297 and 0.496mm, operated over a range of interstitial liquid velocities (u) between 0.1mm/s and 14mm/s and this gave a variation of the Reynolds number (Re=ρdue/μ) between 0.01 and 3.5.

Direct numerical simulation of reacting scalar mixing layers

Abstract: Understanding the passive reaction of two chemical species in shear-free turbulence with order unity Schmidt number is important in atmospheric and turbulent combustion research. The canonical configuration considered here is the reacting scalar mixing layer; in this problem two initially separated species mix and react downstream of a turbulence generating grid in a wind tunnel. A conserved scalar in this flow is, with some restrictions, analogous to temperature in a thermal mixing layer, and considerable laboratory data are available on the latter. In this paper, results are reported from high resolution, direct numerical simulations in which the evolution of the conserved scalar field accurately matches that of the temperature field in existing laboratory experiments. Superimposed on the flow are passive, single-step reactions with a wide range of activation energies and stoichiometric ratios (r). The resulting data include species concentrations as a function of three spatial dimensions plus time, and...

Convective heat and mass transfer in a visco‐elastic fluid flow through a porous medium over a stretching sheet

Abstract: Presents a numerical solution of the two‐dimensional laminar boundary layer problem on free and forced convection of an incompressible visco‐elastic fluid immersed in a porous medium over a stretching sheet. Here, the driving force for the flow is provided by an impermeable sheet stretched with a velocity proportional to the distance from a slit and buoyancy effects due to both temperature and concentration gradients. The resultant governing boundary layer equations are highly non‐linear and coupled form of partial differential equations, and they have been solved by employing a numerical shooting technique with fourth order Runge‐Kutta integration scheme. Numerical computations are carried out for the non‐dimensional physical parameters. The results are analyzed for the effect of different physical parameters like visco‐elasticity, permeability of the porous medium, Grashof number, Schmidt number and Prandtl number on the flow, heat and mass transfer characteristics. One of the several important observations is that the combined effect of thermal diffusion and diffusion of species is to increase the horizontal velocity profile and to decrease the temperature and concentration profiles in the boundary layer flow field.

Effective diffusion coefficient in the low temperature process of silica aerogel production

Abstract: In low-temperature aerogel production, replacement of an original solvent with liquid or supercritical carbon dioxide is the crucial stage, and it is one fully controlled by a diffusional mechanism. The diffusion of ethanol as the primary pore fluid through an alcogel structure to the surrounding near-critical CO 2 was investigated in carefully controlled experiments with cylindrical alcogel samples. Changes of the alcohol concentration in carbon dioxide leaving the autoclave were followed with an online gas chromatographic analysis. The experiments covered the range of temperatures from 292 to 315 K and pressure from 6.9 to 9.1 MPa. On the basis of the concentration histories during the drying, the diffusion coefficient in the ethanol–CO 2 mixture inside the silica gel was identified for each experiment. A comparison of measured concentrations with values derived from a mass transfer model is presented and discussed. The method for identifying an effective diffusion coefficient is described and the values of ethanol–CO 2 effective diffusivity in a silica network are presented. The correlation of effective diffusion coefficient in the porous structure of silica gel with the Schmidt number ratio covering the entire investigated range of temperature and pressure is proposed.

Application of a submerged impinging jet for corrosion studies : Development of models for the impedance response

Abstract: A model was established for the impedance response associated with convective diffusion to a disk electrode subjected to a submerged impinging jet. The model accounted for a finite Schmidt number by employing a series expansion in terms of Sc -1/3 The model was incorporated into a quantitative analysis of impedance data for the corrosion of a steel disk in brine saturated with CO 2 . This work showed that the impedance response was influenced by diffusion through a stagnant layer as well as through a convective region. The Schmidt number obtained was very large, suggesting that the thick layer, observed by in situ video microscopy, is a colloidal gel.

Streamwise evolution of a high-Schmidt-number passive scalar in a turbulent plane wake

Abstract: The concentration fluctuation c of diluted fluorescein dye, a high-Schmidt-number passive scalar (Sc=ν/D ≈ 2000, ν and D are the fluid momentum and dye diffusivities, respectively), is measured in the wake of a circular cylinder using a single-point laser-induced fluorescence (SPLIF) technique. The streamwise decay rate of the mean and rms values of c is slow in comparison to that of θ, the temperature fluctuation for which the molecular Prandtl number Pr=ν/κ is about 0.7 (κ is the thermal diffusivity). The comparison between mean and rms distributions of c and θ highlights the combined role the Reynolds and Schmidt numbers play in terms of dispersing the scalar. The streamwise evolution of the probability density functions (pdfs) of c and θ suggest that while p(θ) is approximately Gaussian in the intermediate wake (x/d ≈ 80), p(c) is strongly non-Gaussian, and depends on both x/d and Re. The skewness of c is larger than that of θ along the wake centreline. Arguably, the asymmetry of p(c) reflects the relatively strong organisation of the large-scale motion in the far-wake.

The Effect of Schmidt Number on Air-Water Interface Mass Transfer

Abstract: The mass transfer across an air-water interface, specifically, the exchange of carbon dioxide between the atmosphere and the sea surface, was investigated by direct numerical simulation (DNS). As the Schmidt number of carbon dioxide in the seawater is known to be very high (~700), we have employed a Lagrangian approach, which has been successfully applied to the passive scalar transfer in DNS of turbulent channel flow. The result obtained by this method shows good agreement with the Eulerian DNS data at a low Schmidt number (Sc = 1.0) and also with experimental data at high Schmidt numbers (Sc = 100, 500, 1000). Although the eddy diffusivity near a wall is decreased as increasing the Schmidt number, it does not largely depend on the Schmidt number at a free surface. This suggests that the correlation between velocity and concentration fields near a free surface is kept unchanged even at a very high Schmidt number at a free surface