Velocity Probability Density Research Articles

Experimental and numerical characterization of viscous flow and mixing in an impinging jet contactor

AbstractThe laminar flow in an impinging jet contactor is examined as a first step toward the development of new technology for fast mixing of viscous fluids. The flow, velocity, and stretching fields in an impinging jet contactor are quantified for low Reynolds number flow using three‐dimensional numerical simulations and particle image velocimetry measurements. Computational and experimental velocity fields are in close agreement, as quantified by the velocity probability density functions. Two steady‐state flow regimes are found to exist: for jet Reynolds numbers (Rej) < 10, the jets do not impinge and the velocity field scales linearly with Reynolds number; for Rej > 10, the jets begin to impinge and recirculation regions form above and below the impingement point. The magnitude of the rate‐of‐strain tensor is calculated as a function of Rej. While areas of essentially zero stretching occupy most of the flow domain, very high rates of stretching occur at specific locations in the flow. The maximum and average rates of stretching in the contactor increase roughly linearly as a function of Reynolds number. Mixing simulations show that no mixing occurs for the steady flow in a symmetric‐jet contactor. However, mixing is improved substantially by a slight modification of the impinging jet geometry that disrupts geometric symmetry.

Conditional Fluid-Particle Accelerations in Turbulence

Simple closures for average fluid-particle accelerations, conditional on fixed local fluid velocity, are considered in isotropic, homogeneous and stationary turbulence using exact probability density transport equations and are compared with direct numerical simulations (DNS). Such accelerations are common ingredients in Lagrangian stochastic models for fluid-particle trajectories in turbulence. One-particle accelerations are essentially trivial, so the focus is on two-particle relative accelerations, which are important in the relative dispersion process. The closure is simply a quadratic form in the velocity variable and this special form also defines the Eulerian velocity probability density function (pdf), and comparisons with DNS (for grids up to 5123) of both the acceleration closure and velocity pdf's are encouraging.

PDF simulation of turbulent non-premixed CH 4/H 2-air flames using automatically reduced chemical kinetics

In this paper, we present numerical simulations of a turbulent non-premixed CH4/H2-air flame based on the solution of the transport equation for the joint velocity and composition probability density function (PDF). A Monte Carlo particle-mesh approach is used to solve the modeled PDF transport equation, in which the mean pressure gradient and turbulent frequency are supplied by a finite-volume mean flow program. The finite-rate chemical kinetics of the CH4/H2-air system is simplified by the method of intrinsic low-dimensional manifolds (ILDM). Simple look-up tables guarantee an easy implementation into the turbulent PDF code. The hybrid solution algorithm is applied to calculate a recently published bluff-body stabilized turbulent non-premixed flame of CH4/H2 by Masri et al. [1]. Two different characteristic flames are presented, where one is close to chemical equilibrium and, for the other, the effects of finite-rate chemistry become significant. The mean temperature distribution and scatter plots for temperature and mass fractions of CO2 and H2O are compared with experimental data. The numerical calculations reproduce the experimentally observed phenomena, such as local extinction, quite well.

Intermittency and the slow approach to Kolmogorov scaling

From a simple path integral involving a variable volatility in the velocity differences, we obtain velocity probability density functions with exponential tails, resembling those observed in fully developed turbulence. The model yields realistic scaling exponents and structure functions satisfying extended self-similarity. But there is an additional small scale dependence for quantities in the inertial range, which is linked to a slow approach to Kolmogorov (1941) scaling occurring in the large distance limit.

Coupled map lattices simulating hydrodynamical turbulence

Cascade models driven by strongly chaotic driving forces yield a successful description of velocity probability densities and scaling exponents measured in fully developed three-dimensional turbulence. We prove that for these types of models velocity differences are bounded, in agreement with measurements in turbulence experiments. Regarding the radial component of the velocity difference as a strongly fluctuating scalar field variable, we show how a dynamics governed by coupled quadradic or cubic mappings can arise from a self-interacting momentum balance equation. The spatial coupling of the coupled map lattice at the top of the cascade is a measure for the inverse Reynolds number. Some numerical results on force patterns are presented.

Collision statistics in an isotropic particle-laden turbulent suspension. Part 1. Direct numerical simulations

Direct numerical simulations of heavy particles suspended in a turbulent fluid are performed to study the rate of inter-particle collisions as a function of the turbulence parameters and particle properties. The particle volume fractions are kept small (∼10−4) so that the system is well within the dilute limit. The fluid velocities are updated using a pseudo-spectral algorithm while the particle forces are approximated by Stokes drag. One unique aspect of the present simulations is that the particles have finite volumes (as opposed to point masses) and therefore particle collisions must be accounted for. The collision frequency is monitored over several eddy turnover times. It is found that particles with small Stokes numbers behave similarly to the prediction of Saffman & Turner (1956). On the other hand, particles with very large Stokes numbers have collision frequencies similar to kinetic theory (Abrahamson 1975). For intermediate Stokes numbers, the behaviour is complicated by two effects: (i) particles tend to collect in regions of low vorticity (high strain) due to a centrifugal effect (preferential concentration); (ii) particle pairs are less strongly correlated with each other, resulting in an increase in their relative velocity. Both effects tend to increase collision rates, however the scalings of the two effects are different, leading to the observed complex behaviour. An explanation for the entire range of Stokes numbers can be found by considering the relationship between the collision frequency and two statistical properties of the particle phase: the radial distribution function and the relative velocity probability density function. Statistical analysis of the data, in the context of this relationship, confirms the relationship and provides a quantitative description of how preferential concentration and particle decorrelation ultimately affect the collision frequency.

Scaling Properties of Tracer Trajectories in a Saturated Porous Medium

The analysis of the trajectory's scaling properties of tracer particles and preliminary valuations on their relation to experimentally determined dispersion coefficient tensor components in a saturated porous medium are presented. The matching index technique is used to assure optical access inside a three-dimensional laboratory model of a porous medium. In this way it is possible to determine the trajectories of the tracer particles. Using the methods suggested by PIV (Particle Image Velocimetry), the test section is illuminated with a light sheet and images are acquired and digitalized by means of an acquisition apparatus. A Lagrangian description of the motion is carried out and the statistical analysis performed on the experimental field allows for the estimation of velocity probability density functions, velocity components correlation functions and components of the mechanic dispersion coefficient tensor. The statistical analysis results are briefly recalled. The reconstructed tracer trajectories are analysed to verify the self-similarity or the self-affinity properties. The results show self-affine characteristics of the trajectories confirming the anisotropy of the pollutant plume. A relation between the two components of the dispersion coefficient determined by the scaling analysis of the tracer particles is derived.

On the influence of non-Gaussianity on turbulent transport

Non-Gaussianity effects, first of all the influence of the third and fourth moments of the velocity probability density function, have to be assessed for higher-order closure models of turbulence and Lagrangian modelling of turbulent dispersion in complex flows. Whereas the role and the effects of the third moments are relatively well understood as essential for the explanation of specific observed features of the fully developed convective boundary layer, there are indications that the fourth moments may also be important, but little is known about these moments. Therefore, the effects of non-Gaussianity are considered for the turbulent motion of particles in non-neutral flows without fully developed convection, where the influence of the fourth moments may be expected to be particularly essential. The transport properties of these flows can be characterized by a diffusion coefficient which reflects these effects. It is shown, for different vertical velocity distributions, that the intensity of turbulent transport may be enhanced remarkably by non-Gaussianity. The diffusion coefficient is given as a modification of the Gaussian diffusivity, and this modifying factor is found to be determined to a very good approximation by the normalized fourth moment of the vertical velocity distribution function. This provides better insight into the effect of fourth moments and explains the varying importance of third and fourth moments in different flows.

Lagrangian Analysis of Nonreactive Pollutant Dispersion in Porous Media by Means of the Particle Image Velocimetry Technique

An experimental technique based on image analysis was used to perform a Lagrangian description of passive pollutant particle motion in a three‐dimensional saturated porous medium. To allow for optical access, the experiment was carried out with Pyrex grains as the solid matrix and glycerol as the liquid phase in order to have two phases with the same refractive index. Statistical analysis of the experimental data allowed for estimation of velocity and displacement probability density functions (pdf), velocity component correlation functions, Lagrangian integral scales, and mechanical dispersion coefficient tensor components. The results obtained suggest that the longitudinal velocity component has a log normal pdf while the transversal component has a symmetrical pdf, which is nevertheless not Gaussian for high values of the kurtosis. Furthermore, the velocity components' autocorrelation functions are well represented by exponential laws, and the integral scale is dependent on filtration velocity and grain size. As foreseen in the theory the total displacement pdf shows the tendency to reach normal distribution after many integral scales. The evaluated dispersion coefficient tensor components are dependent on travel time; the components start from zero and reach an asymptotic value after several integral scales. Furthermore, the tensor is anisotropic, with the longitudinal component greater than the transversal one by about 1 order of magnitude. Comparison with other experimental data shows agreement at least for the longitudinal dispersion component. Dagan's linear theory has been used for comparing the analytical longitudinal component of the dispersion tensor with that obtained by means of the experiments.

Electrostatic effect on the flow behavior of a dilute gas/cohesive particle flow system

AbstractCohesive (Group C) particles have been widely used in various industries. To handle and process such fine particles, a clear understanding of the flow behavior and interparticle force, is needed. To achieve that objective, a Laser Doppler Anemometer system was used to measure particle velocity, fluctuating velocity, and size and extent of agglomeration or cluster formation of particles in a dilute gas/fine oil shale particle flow system with particle density of 2,082 kg/m3, average particle volumetric concentration of 1.5%, and average particle mass flux of about 100 kg/m2·s in a controlled‐moisture environment. The flow behavior of the particles was also studied for a mixture of 99% shale particles and 1% antistatic agent (Larostat powder, a quaternary ammonium compound) to examine the role of electrostatic force in gas/cohesive particle flow behavior. The addition of Larostat powder significantly reduced the electrostatic force and, in turn, made Group C particles behave similar to Group A or in some cases to Group B particles. In addition, our experimental data showed that the Maxwellian distribution function is a reasonable assumption to describe the velocity probability density function of the shale particles with or without antistatic agents.

An equation for the particle velocity probability density function in inhomogeneous turbulent flow

The Eulerian continuum description of a gas dispersion (two-phase) turbulent flow is considered on the basis of the particle velocity probability density distribution method. A refined variant of the kinetic equation for the probability density distribution in inhomogeneous turbulent flow is presented, a chain of equations for the velocity moments of the dispersed phase is constructed, and the reverse influence of the particles on the characteristics of the carrier flow is determined.

垂直管内気液対向環状流の界面の不安定性に関する実験的研究

This paper provides an experimental understanding of liquid film characteristics to examine flooding mechanisms in the countercurrent annular two-phase flow in a vertical tube. The experiments are performed in an air-water system with a large diameter tube, 50mm or 100mm in diameter. Time variations of liquid film thicknesses at four different elevations along the tube are measured, and their power spectra are used to examine interfacial wave characteristics. At the top region of the tube periodic wave with a frequency of about 45Hz is observed at a low gas flow rate condition. Increasing gas flow rate, the periodic wave disappears and the continuous spectra indicating a chaotic aspect is observed in the power spectrum profile. At the bottom of the tube the spectra indicate the continuous feature even at a low gas flow rate condition. As the gas flow rate increases, a power of the spectra increases due to large amplitude waves produced by interaction between the gas and liquid flows. The frequency of maximum power spectra shifts slightly to the low frequency region, but the profile does not change significantly before flooding. At the just before flooding, however, the profile become more flat in the high frequency region. The probability density of interfacial velocity increases in the large velocity region at just before flooding than at low gas flow rate condition. As the gas flow rate increases, the wave number of large amplitude tends to be small.

A nonlocal theory for stress in bound, Brownian suspensions of slender, rigid fibres

A nonlocal theory for stress in bound suspensions of rigid, slender fibres is developed and used to predict the rheology of dilute, rigid polymer suspensions when confined to capillaries or fine porous media. Because the theory is nonlocal, we describe transport in a fibre suspension where the velocity and concentration fields change rapidly on the fibre's characteristic length. Such rapid changes occur in a rigidly bound domain because suspended particles are sterically excluded from configurations near the boundaries. A rigorous no-flux condition resulting from the presence of solid boundaries around the suspension is included in our nonlocal stress theory and naturally gives rise to concentration gradients that scale on the length of the particle. Brownian motion of the rigid fibres is included within the nonlocal stress through a Fokker–Planck description of the fibres’ probability density function where gradients of this function are proportional to Brownian forces and torques exerted on the suspended fibres. This governing Fokker–Planck probability density equation couples the fluid flow and the nonlocal stress resulting in a nonlinear set of integral-differential equations for fluid stress, fluid velocity and fibre probability density. Using the method of averaged equations (Hinch 1977) and slender-body theory (Batchelor 1970), the system of equations is solved for a dilute suspension of rigid fibres experiencing flow and strong Brownian motion while confined to a gap of the same order in size as the fibre's intrinsic length. The full solution of this problem, as the fluid in the gap undergoes either simple shear or pressure-driven flow, is solved self-consistently yielding average fluid velocity, shear and normal stress profiles within the gap as well as the probability density function for the fibres’ position and orientation. From these results we calculate concentration profiles, effective viscosities and slip velocities and compare them to experimental data.

Modeling of particle motion in a turbulent flow with allowance for collisions

On the basis of the kinetic equation for the colliding-particle velocity probability density distribution in a turbulent flow, a model for calculating the dispersed phase motion is constructed for a broad range of variation of the number density and particle size.

On the moments approximation method for constructing a Lagrangian Stochastic model

The exact Eulerian velocity probability density function (pdf) of a turbulent field is generally unknown, and one normally has available only partial information in the form of low order moments. We compare two alternative Lagrangian Stochastic (LS) approaches formed from this partial information, (i) the “moments approximation” approach (Kaplan and Dinar, 1993); and (ii) the well-mixed model (Thomson, 1987) that corresponds to the “maximum missing information” pdf formed from the available information. We show that the moments approximation model does not in general satisfy the well-mixed constraint, and can give an inferior prediction of dispersion.

Measurement of Cloud Droplet Size Spectra by Doppler Radar

Abstract A new technique is examined for using Doppler radars to extract information about the size spectrum of cloud droplets too small to have terminal velocities large enough to be resolvable by the radar. If the drops are very small, motions of the drops are dominated by turbulent fluctuations in the medium rather than their fall velocity. Their motion is then the convolution of the terminal velocity with the turbulent velocity probability density function, and size information about the population can be obtained only by deconvolving the spectra. Doppler radars can extract this velocity and size information, as well as cloud liquid and liquid flux, using a surprisingly simple and accurate technique assuming some functional form (e.g., gamma) for the drop number density spectrum. The method also allows Doppler radars to extract drop size information independent of up-/downdrafts in the medium in which they are embedded. Various gamma and lognormal functions are compared, and finally, a “Stokes range” ...

Velocity probability density functions in developed turbulence: A finite Reynolds theory

The probability density function of velocity differences δv i=v i( x+ r)−v i( x) in turbulent flows has a stricking evolution when r goes from large scales to small dissipative ones developing wide non gaussian wings. A formalism has been developed, generalizing the multifractal formalism, which well describes this evolution for finite Reynolds numbers. The extrapolation to infinite Reynolds number presents unexpected features.

Stochastic trajectory models for turbulent diffusion: Monte Carlo process versus Markov chains

Abstract Turbulent diffusion of passive scalars and particles is often simulated with either a Monte Carlo process or a Markov chain. Knowledge of the velocity correlation generated by either of these stochastic trajectory models is essential to their application. The velocity correlation for Monte Carlo process and Markov chain was studied analytically and numerically. A general relationship was developed between the Lagrangian velocity correlation and the probability density function for the time steps in a Monte Carlo process. The velocity correlation was found to be independent of the fluid velocity probability density function, but to be related to the time-step probability density function. For a Monte Carlo process with a constant time step, the velocity correlation is a triangle function; and the integral time scale is equal to one-half of the time-step length. When the time step was chosen randomly with an exponential pdf distribution, the resulting velocity correlation was an exponential function. Other time-step probability density functions, such as a uniform distribution and a half-Gaussian distribution, were also tested. A Markov chain, which presumes one-step memory, has a piecewise linear velocity correlation function with a finite time step. For a Markov chain with a short time step, only an exponential velocity correlation function can be realized. Thus, a Monte Carlo process with random time steps is more versatile than a Markov chain. Direct numerical calculation of the velocity correlation verified the analytical results. A new model which combines the ideas of the Monte Carlo process and the Markov chain was developed. By examining the long-time mean square dispersion, we found an exact solution for the Lagrangian integral time scale of the new model in terms of the intercorrelation parameter and the mean and the variance of the time steps. Using this new model, we can generate any velocity correlation, including one with a negative tail. Two approximate solutions that give upper and lower bounds for the Lagrangian velocity correlation are proposed.

The kinetic theory for dilute solid/liquid two-phase flow

On the basis of a brief review of the continuum theory for macroscopic descriptions and the kinetic theory for microscopic descriptions in solid/liquid two-phase flows, some suggestions are presented, i.e. the solid phase may be described by the Boltzmann equation and the liquid phase still be described by conservation laws in the continuum theory. Among them the action force on the particles by the liquid fluid is a coupling factor which connects the phases. For dilute steady solid/liquid two-phase flows, the particle velocity distribution function can be derived by analogy with the procedures in the kinetic theory of gas molecules for the equilibrium state instead of being assumed, as previous investigators did. This done, more detailed information, such as the velocity probability density distribution, mean velocity distribution and fluctuating intensity etc. can be obtained directly from the particle velocity distribution function or from its integration. Experiments have been performed for dilute solid/liquid two-phase flow in a 4 × 6 cm 2 sized circulating square pipe system by means of laser Doppler anemometry so that the theories can be examined. The comparisons show that the theories agree very well with all the measured data.

Velocity probability density functions of high Reynolds number turbulence

This paper deals with the probability density function (PDF) of velocity differences between two points separated by distance r. Measurements of PDFs were made, for r lying in the inertial range, for two different flows: in a jet with R λ = 852 and in a wind tunnel with R λ = 2720. These PDFs have a characteristic, non-Gaussian, shape with “exponential” tails. Following Kolmogorov's general ideas of log-normality, a new model for the PDF is developed which contains two parameters determined by experiments. This empirical model agrees with the experimental results that the tails of the PDF deviate from a truly exponential behaviour, in particular for small r. In addition, the model leads to the general scaling law 〈( Δ ln ε) 2〉 ∼ ( r/ r 0) - β differenr from Kolmogorov's third hypothesis 〈( Δ ln ε) 2〉 ∼ -μ ln( r/ r 0) restricted to the inertial range only ( Δ( x) is x - 〈 x〉). We develop also a formalism, based on an extremum principle, which is consistent with both the log-normality of ε and the above mentioned power law. In this formalism, β can be interpreted as the codimension of dissipative structures and asymptotically varies as β = β 1/ ln Rλ.