Showing papers on "Schmidt number published in 2005"

Two‐scale continuum model for simulation of wormholes in carbonate acidization

Abstract: A two-scale continuum model is developed to describe transport and reaction mechanisms in reactive dissolution of a porous medium, and used to study wormhole formation during acid stimulation of carbonate cores. The model accounts for pore level physics by coupling local pore-scale phenomena to macroscopic variables (Darcy velocity, pressure and reactant cup-mixing concentration) through structure-property relationships (permeability-porosity, average pore size-porosity, and so on), and the dependence of mass transfer and dispersion coefficients on evolving pore scale variables (average pore size and local Reynolds and Schmidt numbers). The gradients in concentration at the pore level caused by flow, species diffusion and chemical reaction are described using two concentration variables and a local mass-transfer coefficient. Numerical simulations of the model on a two-dimensional (2-D) domain show that the model captures the different types of dissolution patterns observed in the experiments. A qualitative criterion for wormhole formation is developed and it is given by Λ ∼ O(1), where Λ = . Here, keff is the effective volumetric dissolution rate constant, DeT is the transverse dispersion coefficient, and uo is the injection velocity. The model is used to examine the influence of the level of dispersion, the heterogeneities present in the core, reaction kinetics and mass transfer on wormhole formation. The model predictions are favorably compared to laboratory data. © 2005 American Institute of Chemical Engineers AIChE J, 2005

Dust Diffusion in Protoplanetary Disks by Magnetorotational Turbulence

Abstract: We measure the turbulent diffusion coefficient of dust grains embedded in magnetorotational turbulence in a protoplanetary disk directly from numerical simulations and compare it to the turbulent viscosity of the flow. The simulations are done in a local coordinate frame comoving with the gas in Keplerian rotation. Periodic boundary conditions are used in all directions, and vertical gravity is not applied to the gas. Using a two-fluid approach, small dust grains of various sizes (with friction times up to Ω0τf = 0.02) are allowed to move under the influence of friction with the turbulent gas. We measure the turbulent diffusion coefficient of the dust grains by applying an external sinusoidal force field acting in the vertical direction on the dust component only. This concentrates the dust around the midplane of the disk, and an equilibrium distribution of the dust density is achieved when the vertical settling is counteracted by the turbulent diffusion away from the midplane. Comparing with analytical expressions for the equilibrium concentration, we deduce the vertical turbulent diffusion coefficient. The vertical diffusion coefficient is found to be lower than the turbulent viscosity and to have an associated vertical Schmidt number (vertical diffusion Prandtl number) of about 1.5. A similar radial force field also allows us to measure the radial turbulent diffusion coefficient. We find a radial Schmidt number of about 0.85 and also find that the radial turbulent diffusion coefficient is around 70% higher than the vertical. As most angular momentum transport happens through magnetic Maxwell stresses, both the vertical and the radial diffusion coefficients are found to be significantly higher than suggested by the angular momentum transport by Reynolds stresses alone. We also find evidence for trapping of dust grains of intermediate friction time in turbulent eddies.

Mass Transport in Vegetated Shear Flows

Abstract: Submerged aquatic vegetation has the potential to greatly improve water quality through the removal of nutrients, particulates and trace metals. The efficiency of this removal depends heavily upon the rate of vertical mixing, which dictates the timescale over which these constituents remain in the canopy. Continuous dye injection experiments were conducted in a flume with model vegetation to characterize vertical mass transport in vegetated shear flows. Through the absorbance–concentration relationship of the Beer–Lambert Law, digital imaging was used to provide high-resolution concentration profiles of the dye plumes. Vertical mass transport is dominated by the coherent vortices of the vegetated shear layers. This is highlighted by the strong periodicity of the transport and its simple characterization based on properties of the shear layer. For example, the vertical turbulent diffusivity is directly proportional to the shear and thickness of the layer. The turbulent diffusivity depends upon the size of the plume, such that the rate of plume growth is lower near the source. In the far-field, mass is mixed more than twice as rapidly as momentum. Finally, plume size is dictated predominantly by X, a dimensionless distance that scales upon the number of vortex rotations experienced by the plume.

Dynamic regimes of fluids simulated by multiparticle-collision dynamics.

Abstract: We investigate the hydrodynamic properties of a fluid simulated with a mesoscopic solvent model. Two distinct regimes are identified, the ``particle regime'' in which the dynamics is gaslike and the ``collective regime'' where the dynamics is fluidlike. This behavior can be characterized by the Schmidt number, which measures the ratio between viscous and diffusive transport. Analytical expressions for the tracer diffusion coefficient, which have been derived on the basis of a molecular-chaos assumption, are found to describe the simulation data very well in the particle regime, but important deviations are found in the collective regime. These deviations are due to hydrodynamic correlations. The model is then extended in order to investigate self-diffusion in colloidal dispersions. We study first the transport properties of heavy pointlike particles in the mesoscopic solvent, as a function of their mass and number density. Second, we introduce excluded-volume interactions among the colloidal particles and determine the dependence of the diffusion coefficient on the colloidal volume fraction for different solvent mean-free paths. In the collective regime, the results are found to be in good agreement with previous theoretical predictions based on Stokes hydrodynamics and the Smoluchowski equation.

Scalar dissipation rate and dissipative anomaly in isotropic turbulence

Abstract: We examine available data from experiment and recent numerical simulations to explore the supposition that the scalar dissipation rate in turbulence becomes independent of the fluid viscosity when the viscosity is small and of scalar diffusivity when the diffusivity is small. The data are interpreted in the context of semi-empirical spectral theory of Obukhov and Corrsin when the Schmidt number, Sc, is below unity, and of Batchelor's theory when Sc is above unity. Practical limits in terms of the Taylor-microscale Reynolds number, R λ , as well as Sc, are deduced for scalar dissipation to become sensibly independent of molecular properties

Scalar dissipation rate and dissipative anomaly in isotropic turbulence

Abstract: We examine available data from experiment and recent numerical simulations to explore the supposition that the scalar dissipation rate in turbulence becomes independent of the fluid viscosity when the viscosity is small and of scalar diffusivity when the diffusivity is small. The data are interpreted in the context of semi-empirical spectral theory of Obukhov and Corrsin when the Schmidt number, Sc ,i s below unity, and of Batchelor’s theory when Sc is above unity. Practical limits in terms of the Taylor-microscale Reynolds number, Rλ ,a s well asSc, are deduced for scalar dissipation to become sensibly independent of molecular properties. In particular, we show that such an asymptotic state is reached if RλSc 1/2 � 1f orSc 1.

Plume structure in high-Rayleigh-number convection

Abstract: Near-wall structures in turbulent natural convection at Rayleigh numbers of $10^{10}$ to $10^{11}$ at A Schmidt number of 602 are visualized by a new method of driving the convection across a fine membrane using concentration differences of sodium chloride. The visualizations show the near-wall flow to consist of sheet plumes. A wide variety of large-scale flow cells, scaling with the cross-section dimension, are observed. Multiple large-scale flow cells are seen at aspect ratio (AR)= 0.65, while only a single circulation cell is detected at AR= 0.435. The cells (or the mean wind) are driven by plumes coming together to form columns of rising lighter fluid. The wind in turn aligns the sheet plumes along the direction of shear. the mean wind direction is seen to change with time. The near-wall dynamics show plumes initiated at points, which elongate to form sheets and then merge. Increase in rayleigh number results in a larger number of closely and regularly spaced plumes. The plume spacings show a common log–normal probability distribution function, independent of the rayleigh number and the aspect ratio. We propose that the near-wall structure is made of laminar natural-convection boundary layers, which become unstable to give rise to sheet plumes, and show that the predictions of a model constructed on this hypothesis match the experiments. Based on these findings, we conclude that in the presence of a mean wind, the local near-wall boundary layers associated with each sheet plume in high-rayleigh-number turbulent natural convection are likely to be laminar mixed convection type.

Stretching of material lines in shock-accelerated gaseous flows

Abstract: A Mach 1.2 planar shock wave impulsively accelerates one of five different configurations of heavy-gas (SF6) cylinders surrounded by lighter gas (air), producing one or more pairs of interacting vortex columns. The interaction of the columns is investigated with planar laser-induced fluorescence in the plane normal to the axes of the cylinders. For the first time, we experimentally measure the early time stretching rate (in the first 220μs after shock interaction before the development of secondary instabilities) of material lines in shock-accelerated gaseous flows resulting from the Richtmyer-Meshkov instability at Reynolds number ∼25000 and Schmidt number ∼1. The early time specific stretching rate exponent associated with the stretching of material lines is measured in these five configurations and compared with the numerical computations of Yang et al. [AIAA J. 31, 854 (1993)] in some similar configurations and time range. The stretching rate is found to depend on the configuration and orientation of the gaseous cylinders, as these affect the refraction of the shock and thus vorticity deposition. Integral scale measurements fail to discriminate between the various configurations over the same time range, however, suggesting that integral measures are insufficient to characterize early time mixing in these flows.

Vertical Dispersion of Fine and Coarse Sediments in Turbulent Open-Channel Flows

Abstract: Through using a kinetic model for particles in turbulent solid–liquid flows, underlying mechanisms of sediment vertical dispersion as well as sediment diffusion coefficient are investigated. Four hydrodynamic mechanisms, namely gravitational settling, turbulent diffusion, effect of lift force, and that of sediment stress gradient, coexist in two-dimensional (2D) uniform and steady open-channel flows. The sediment diffusion coefficient consists of two independent components: one accounts for the advective transport of sediment probability density distribution function due to sediment velocity fluctuations, and the other results from sediment–eddy interactions. Predictions of the kinetic model are in good agreement with experimental data of 2D open-channel flows. In such flows, it is shown that: (1) the parameter γ (i.e., the inverse of the turbulent Schmidt number) may be greater than unity and increases toward the bed, being close to unity for fine sediments and considerably large for coarse ones; (2) eff...

Schmidt number and near-bed boundary condition effects on a two-phase dilute sediment transport model

Abstract: [1] In this paper we investigate a concentration-dependent Schmidt number description and two near-bed boundary conditions, empirical pickup and reference concentration, for a dilute two-phase sediment transport model with a k-ɛ fluid turbulence closure. The pick-up approach adopts an empirical formula to calculate the upward sediment flux, whereas the reference concentration approach relates the upward sediment flux to the concentration at a reference location above the initially undisturbed bed. Through model-data comparisons with data measured in the U tube, we show that the variation of Schmidt number only affects the magnitude of calculated concentration and is insensitive to the predicted phase of concentration time histories. The predicted phase is found more sensitive to the near-bed sediment boundary condition. A concentration-dependent Schmidt number is then introduced, which improves the predictions of magnitude and phase of concentration. Using the concentration-dependent Schmidt number, the reference concentration approach generally predicts suspended sediment concentration better than the pick-up approach does.

Effect of Schmidt number on the velocity-scalar cospectrum in isotropic turbulence with a mean scalar gradient

Abstract: We consider transport of a passive scalar by an isotropic turbulent velocity field in the presence of a mean scalar gradient. The velocity–scalar cospectrum measures the distribution of the mean scalar flux across scales. An inequality is shown to bound the magnitude of the cospectrum in terms of the shell-summed energy and scalar spectra. At high Schmidt number, this bound limits the possible contribution of the sub-Kolmogorov scales to the scalar flux. At low Schmidt number, we derive an asymptotic result for the cospectrum in the inertial–diffusive range, with a -11/3 power law wavenumber dependence, and a comparison is made with results from large-eddy simulation. The sparse direct-interaction perturbation (SDIP) is used to calculate the cospectrum for a range of Schmidt numbers. The Lumley scaling result is recovered in the inertial–convective range and the constant of proportionality was calculated. At high Schmidt numbers, the cospectrum is found to decay exponentially in the viscous–convective range, and at low Schmidt numbers, the -11/3 power law is observed in the inertial–diffusive range. Results are reported for the cospectrum from a direct numerical simulation at a Taylor Reynolds number of 265, and a comparison is made at Schmidt number order unity between theory, simulation and experiment.

Turbulent dispersion from elevated line sources in channel and couette flow

Abstract: Turbulent dispersion of a scalar emitted from a source in homogeneous turbulence has been placed in a solid theoretical context by Taylor, Saffman, and Batchelor. However, the case of source diffusion in the near-wall region, where the turbulence is not Gaussian, and the effects of the molecular Prandtl number (Pr) on the effective dispersion have not been explored to similar depth. The present work studies the behavior of elevated sources in turbulent channel flow and in turbulent-plane Couette flow. The trajectories of heat markers are monitored in space and time as they move in a hydrodynamic field created by a direct numerical simulation. The fluids span several orders of magnitude of Pr (or Schmidt number), Pr = 0.1, 0.7, 3, 6, 10, 100, 200, 500, 1000, 2400, 7500, 15,000, and 50,000 (liquid metals, gases, liquids, lubricants, and electrochemical fluids). It is found that the molecular Pr has negligible effects in the evolution of the marker cloud for Pr ≥ 3, when the point of marker release is away from the viscous wall sublayer. However, when the markers are released close to the wall, the molecular effects on dispersion are strong. It is also found that total effective dispersion is higher in the case of plane Couette flow, where the total stress across the channel is constant. © 2005 American Institute of Chemical Engineers AIChE J, 2005

The multifractal nature of plume structure in high-Rayleigh-number convection

Abstract: The geometrically different planforms of near-wall plume structure in turbulent natural convection, visualized by driving the convection using concentration differences across a membrane, are shown to have a common multifractal spectrum of singularities for Rayleigh numbers in the range $10^{10}$-$10^{11}$ at Schmidt number of 602. The scaling is seen for a length scale range of $2^5$ and is independent of the Rayleigh number, the flux, the strength and nature of the large-scale flow, and the aspect ratio. Similar scaling is observed for the plume structures obtained in the presence of a weak flow across the membrane. This common non-trivial spatial scaling is proposed to be due to the same underlying generating process for the near-wall plume structures.

Hydrodymagnetic three-dimensional flow over a stretching surface with heat and mass transfer

Abstract: A general analysis has been developed to study the combined effect of the free convective heat and mass transfer on the steady three-dimensional laminar boundary layer flow over a stretching surface. The flow is subject to a transverse magnetic field normal to the plate. The governing three-dimensional partial differential equations for the present case are transformed into ordinary differential equation using three-dimensional similarity variables. The resulting equations, are solved numerically by applying a fifth order Runge-Kutta-Fehlberg scheme with the shooting technique. The effects of the Magnetic field Parameter M, buoyancy parameter N, Prandtl number Pr and Schmidt number Sc are examined on the velocity, temperature and concentration distributions. Numerical data for the skin-friction coefficients, Nusselt and Sherwood numbers have been tabulated for various parametric conditions. The results are compared with known from the literature.

Scramjet Propulsive Flowpath Prediction Improvements Using Recent Modeling Upgrades

Abstract: Reynolds Averaged Navier Stokes simulations have been performed to examine modeling upgrades for scramjet flowpath predictions. A flush, non-reacting hydrogen fuel injector flowfield was used as the model problem, and an LES simulation of the problem was used for comparison purposes. Calculations were first performed examining the effect of Schmidt number with a constant Prandtl number. Next the effect of the compressibility correction used was examined. These findings indicated that for this injector configuration, the effect of the compressibility had a major impact on the solution, and that the average Schmidt number of about 0.45 compared closely to the LES simulation results. Next, a new scalar fluctuation model was used to obtain local values of Prandtl and Schmidt number whose values were found to vary significantly across the fuel jet mixing layer. The turbulent Prandtl number was found to vary between 0.4 to 0.9, and the turbulent Schmidt number varied from 0.6 to 1.2. Finally, a comparison was performed using an unstructured flow solver with grid adaptation. This technique is now being used to obtain grid resolved solutions in a systematic and straightforward manner in our design studies.

Turbulence Modeling for Scramjet Applications

Abstract: A variable turbulent Schmidt number model suited for reacting flows is developed. The model is used to study supersonic mixing and combustion of H2-Air mixtures. Predictions of the model are compared with the experiments of Burrows and Kurkov. Calculations were carried out in the presence and absence of turbulence chemistry interactions. A multivariate β-PDF for the mass fractions is employed. The results show that use of such a PDF is highly dissipative and thus limits the extent of mixing. As a result, use of such a PDF is not suited for combustion calculations. Better agreement with experiment is noted, when turbulence/chemistry interactions are not taken into consideration. In general, the variable Schmidt number formulation works well for both reacting and non-reacting flows.

Fluctuating thermal and mass diffusion on unsteady free convection flow past a vertical plate in slip-flow regime

Abstract: The unsteady free convective viscous incompressible flow past an infinite vertical porous flat plate with periodic heat and mass transfer in slip-flow regime is discussed. Assuming variable suction at the porous plate, approximate solutions are obtained for velocity, skin-friction, temper- ature, heat transfer and species concentration. During the course of discussion, the effects of Gr (Grashof number based on temperature), Gc (modified Grashof number based on concentration difference), Sc (Schmidt number), A (suction parameter) and ω (frequency) for Pr=0.71 (air) have been presented.

Effect of Taylor vortices on mass transfer from a rotating cylinder

Abstract: Mass transfer from solids, which has important applications in a number of chemical and pharmaceutical industries, has been studied experimentally and semiempirically under turbulent flow conditions, and correlations are available in the literature to calculate the mass-transfer coefficients from pellets, rotating cylinders and disks etc. However, mass transfer under laminar flow has not been sufficiently addressed. One of the difficulties here is the strong Reynolds number dependence of the flow pattern, for example, due to the onset of Taylor vortices for the case of a rotating cylinder. This problem is circumvented by using a computational fluid dynamics (CFD)-based solution of the governing equations for the case of a cylinder rotating inside a stationary cylindrical outer vessel filled with liquid. The parameters cover a range of Reynolds number (based on the cylinder diameter, and the tangential speed of the cylinder), Schmidt number and the ratio of the outer to inner cylinder diameters. The results confirm that the circumferential velocity profile is a strong function of the Reynolds number and varies from a nearly Couette-type flow at very low Reynolds numbers to a boundary layer-like profile at high Reynolds numbers. The onset of Taylor vortices has a strong effect on the flow field and the mass-transfer mode. The calculations show that the Sherwood number has a linear dependence on the Reynolds number in the Couette-flow regime, and roughly square-root dependence after the onset of Taylor vortices. Correlations have been proposed to calculate the Sherwood number taking account of these effects.

Numerical study of influence of different parameters on mixing in a coaxial jet mixer using LES

Abstract: We present a comparative numerical study about the overall mixing process in a coaxial jet mixer. The two-stream mixing problem was investigated in non-reacting single phase gas and liquid mixtures using Large-Eddy Simulations with wall functions and subgrid scale models from eddy viscosity concepts. The influence of different parameters like Reynolds number, Schmidt number, Prandtl number, density ratio and flow rate ratio on the overall mixing process was investigated. Additionally two methods of control of mixing are shown to have a significant effect on the overall mixing in a coaxial jet mixer.

Numerical Modeling of Inert and Reacting Compressible Turbulent Jets

Abstract: §The problem of application of the k-e turbulence model to the numerical simulation of compressible round jets is analyzed by comparing experimental and numerical results. A limiting of the Pope correction is proposed to account for the vortex stretching and preserve the computational stability. The limited Pope correction is validated in case of an incompressible round jet in a slow coflow. Two variants of the Sarkar compressibility correction are validated for compressible plane mixing layers with different convective Mach numbers. The Sarkar and limited Pope corrections are then tested for several compressible round jets in still air and supersonic coflow. The simulations have demonstrated a good performance of the k-e model with the two corrections. The effect of the turbulent Schmidt number is investigated for compressible round jets of helium and hydrogen in supersonic air coflow. Reactive and non-reactive flows are considered for a hydrogen jet in a supersonic combustion chamber. A close agreement with the experiment has been obtained for the mixing process by adjusting the turbulent Schmidt number.

Liquid-Solid Mass Transfer at a Fixed Bed of Lessing Rings, in Relation to Electrochemical Reactor Design

Abstract: The mass-transfer behavior of a fixed bed of Lessing rings, under single- and two-phase flow, was studied by measuring the limiting current of the cathodic reduction of the ferricyanide ion. Variables studied included ring diameter (d), solution flow rate, gas flow rate, physical properties of the solution, and the effect of drag-reducing polymers. The single-phase mass-transfer data were correlated for the conditions of 1390 < Sc < 4763 (where Sc is the Schmidt number, which represents a dimensionless kinematic viscosity/molecular diffusivity coefficient), 166 < Re < 722 (where Re is the solution Reynolds number, which represents a dimensionless flow parameter), and 1 cm < d < 1.4 cm by the equation Sh = 1.57Sc0.33Re0.46 (where Sh is the Sherwood number, which represents a dimensionless mass-transfer coefficient). The two-phase mass-transfer data were correlated for the conditions 1390 < Sc < 4763, 144 < Re < 748, 60 < Reg < 818 (where Reg is the gas Reynolds number), and 1 cm < d < 1.4 cm by the equatio...

Similarity analysis in magnetohydrodynamics: Hall effects on forced convective heat and mass transfer of non–Newtonian power law fluids past a semi-infinite vertical flat plate

Abstract: The forced convective heat and mass transfer along a semi-infinite vertical flat plate is investigated for non–Newtonian power law fluids in the presence of a strong nonuniform magnetic field, and the Hall currents are taken into account. The similarity solutions are obtained using transformations group theory. These are the only symmetry transformations admitted by the field equations. The application of one-parameter groups reduces the number of independent variables by one, and consequently the system of governing partial differential equations with boundary conditions reduces to a system of ordinary differential equations with the appropriate boundary conditions. Furthermore the similarity equations are solved numerically by using a fourth-order Runge-Kutta scheme with the shooting method. Numerical results for the velocity profiles, the temperature profiles and the concentration profiles are presented graphically for various values of the power-law viscosity index n, generalized Schmidt number Sc, generalized Prandtl number Pr, the magnetic parameter M and the Hall parameter m.