Velocity Probability Density Function Research Articles

One-point velocity statistics in decaying homogeneous isotropic turbulence

A reasonable closure of the Monin-Lundgren hierarchy of equations for many-point probability density functions of velocity for decaying homogeneous isotropic turbulence is achieved to treat the first equation for the one-point velocity distribution. As a result, we are naturally led to a two-parameter family of solutions for the distribution, which are Gaussian.

Stochastic modeling of evaporating sprays within a consistent hybrid joint PDF framework

In the present study, a framework for modeling two-phase evaporating flow is presented, which employs an Eulerian–Lagrangian–Lagrangian approach. For the continuous phase, a joint velocity-composition probability density function (PDF) method is used. Opposed to other approaches, such PDF methods require no modeling for turbulent convection and chemical source terms. For the dispersed phase, the PDF of velocity, diameter, temperature, seen gas velocity and seen gas composition is calculated. This provides a unified formulation, which allows to consistently address the different modeling issues associated with such a system. Because of the high dimensionality, particle methods are employed to solve the PDF transport equations. To further enhance computational efficiency, a local particle time-stepping algorithm is implemented and a particle time-averaging technique is employed to reduce statistical and bias errors. In comparison to previous studies, a significantly smaller number of droplet particles per grid cell can be employed for the computations, which rely on two-way coupling between the droplet and gas phases. The framework was validated using established experimental data and a good overall agreement can be observed.

Subgrid continuum modeling of particle motion in a turbulent flow

An Eulerian continuum approach to modeling the motion of dispersed particles within the framework of the large-eddy simulation method is developed. The approach is based on a kinetic equation for the filtered probability density function for the particle velocity in a turbulent flow. Models for the subgrid turbulent stresses of the dispersed phase are presented.

Probability density functions of hydraulic head and velocity in three‐dimensional heterogeneous porous media

In this study, we assess probability density functions of hydraulic heads and specific discharges in three‐dimensional bounded heterogeneous porous media by Monte Carlo (MC) simulation. We discuss their empirical shapes and demonstrate that the intuitive use of obvious information on boundedness leads to parametric distribution functions, which fit surprisingly well. On the basis of statistical moments of hydraulic heads and velocities up to fourth order, we discuss the spatial dependence of the empirical distributions and their dependence on the variance of log conductivity. Comparison of the first and second central moment to the results from classical numerical first‐order second‐moment (FOSM) analysis reveals that FOSM predicts these moments surprisingly close for hydraulic heads. On the basis of this fact, we demonstrate that fitting the chosen parametric distributions for hydraulic heads to FOSM moments is promising for the sake of estimating exceedance probabilities. Our MC scenarios vary in variance of log conductivity (0.125 to 5.0), in the type of multivariate dependence, in correlation scale and types of boundary conditions. Our study illustrates that in contrast to the common assumption, FOSM is a reasonable choice for evaluating multivariate and univariate moments for heads, if used in conjunction with additional information on distribution shapes. In the absence of utilizable additional information, we demonstrate that second‐moment methods are mostly inadequate for assessing distributions accurately. Significant deviations from Gaussian distributions occurred for discharge components even at a variance of log conductivity as low as 0.125, and we found that the distributions of transverse discharge components are extremely fat‐tailed. The observed non‐Gaussianity questions the results of approximate approaches in solute flux and dispersion studies where velocity fields are assumed to be multi‐Gaussian and then directly represented by or generated from their covariances. The main implication is to apply more accurate schemes such as exact non‐local methods, extensive MC or higher‐order stochastic Galerkin approaches, and to include higher‐order moments, at least if no additional assumptions on the shape of distributions are available or justifiable.

Beyond eddy diffusivity: an alternative model for turbulent dispersion

Turbulent dispersion proceeds not only much faster but also in a qualitatively different manner than molecular diffusion. Yet, the majority of hydraulic, oceanic and atmospheric models rely on the concept of an eddy diffusivity. It is shown here that an alternative model can be developed to exhibit observed behavior. The new term in the diffusion equation, which is non-local, may be interpreted in terms of the probability density function (pdf) of the turbulent velocity. Different assumptions about this distribution lead to a family of models, one of which is the model proposed here and another, the classical Fickian model of diffusion. A connection is also made with models using fractional calculus.

Dynamic Effects on the Tropical Cloud Radiative Forcing and Radiation Budget

Abstract Vertical velocity is used to isolate the effect of large-scale dynamics on the observed radiation budget and cloud properties in the tropics, using the methodology suggested by Bony et al. Cloud and radiation budget quantities in the tropics show well-defined responses to the large-scale vertical motion at 500 hPa. For the tropics as a whole, the ratio of shortwave to longwave cloud forcing (hereafter N) is about 1.2 in regions of upward motion, and increases to about 1.9 in regions of strong subsidence. If the analysis is restricted to oceanic regions with SST > 28°C, N does not increase as much for subsiding motions, because the stratocumulus regions are eliminated, and the net cloud forcing decreases linearly from about near zero for zero vertical velocity to about −15 W m−2 for strongly subsiding motion. Increasingly negative cloud forcing with increasing upward motion is mostly related to an increasing abundance of high, thick clouds. Although a consistent dynamical effect on the annual cycle of about 1 W m−2 can be identified, the effect of the probability density function (PDF) of the large-scale vertical velocity on long-term trends in the tropical mean radiation budget is very small compared to the observed variations. Observed tropical mean changes can be as large as ±3 W m−2, while the dynamical components are generally smaller than ±0.5 W m−2. For relatively small regions in the east and west Pacific, changes in the relative magnitude of longwave and shortwave cloud forcing can be related to the PDF of vertical velocity. The east Pacific in 1987 and 1998 showed large reductions of N in association with an increase in the fraction of the area in the domain with upward motion, and concomitant increases in high cloud. For the west Pacific in 1998, a large increase in N was caused not so much by a change in the mean vertical motion, but rather by a shift from top- to bottom-heavy upward motion.

Dynamics of solid particles in a tangle of superfluid vortices at low temperatures

We present results of numerical study of motion of micron-size, neutrally buoyant, solid particles in liquid helium at temperatures small enough that the normal fluid is negligible and turbulence manifests itself as a tangle of superfluid line vortices. Based on dynamically self-consistent model of interaction between a solid particle and the quantized vortex, we analyze first an influence of temperature on particle trapping on the vortex core. We find that the particle can be trapped only in the case where the particle experiences the damping force, such as the viscous drag force exerted by the normal fluid. However, at temperature below 0.7K, when the normal fluid is practically absent, the moving particle still experiences the damping force caused by the ballistic scattering of quasiparticles (phonons and rotons) off the particle surface. Using, together with our calculation of close interaction between the particle and the quantized vortex, available experimental data and theoretical results for this force, we show that trapping of micron-size, neutrally buoyant particles on quantized vortices becomes impossible at temperatures below 0.5K. At such temperatures, the particle motion in the vortex tangle can be studied based on the simpler, “one-way coupling” model ignoring the backreaction of the particle on the motion and evolution of quantized vortices. Based on such a model, we present results of numerical calculation of particle trajectories in the vortex tangle and show that, due to instability of trajectories and the mismatch of initial velocities of particles and the fluid, the motion of inertial particles does not reveal the motion of turbulent superfluid. We show that particle trajectories between vortex cores are ballistic. Interaction of particles with moving vortices leads to increase of the average particle velocity until it saturates at the value much larger than the fluid rms velocity. These results prevent the use of small tracer particles to study vortex tangles at low temperatures, but open to investigation a new and remarkably simple model of Lagrangian turbulence. We present the probability density function of the turbulent velocity and show that the particle velocity spectrum obeys a simple scaling law.

Inter-comparison of retrieved wind fields from large-eddy simulations and radar measurements in the convective boundary layer

Estimates of the wind retrieved from boundary layer radars are inherently averaged in space and time. Using large eddy simulation (LES) results, we are able to examine to what extent these measurements are representative of the underlying parameters of the convective boundary layer (CBL) and how they are impacted by various averaging methodologies. Comparisons between an LES with real forcing and Boundary Layer Radar (BLR) wind retrievals have been analyzed. Good agreement between the LES winds and those estimated using the BLR is observed. After removing an instrumental bias, an estimate of the probability density function (PDF) and skewness of the radar and LES vertical velocity estimates was calculated as a function of height. The PDFs of vertical velocity and the height profiles of the skewness of vertical velocity exhibit good agreement for both the LES and BLR calculations and have the expected canonical structure.

Inter-comparison of retrieved wind fields from large-eddy simulations and radar measurements in the convective boundary layer

Estimates of the wind retrieved from boundary layer radars are inherently averaged in space and time. Using large eddy simulation (LES) results, we are able to examine to what extent these measurements are representative of the underlying parameters of the convective boundary layer (CBL) and how they are impacted by various averaging methodologies. Comparisons between an LES with real forcing and Boundary Layer Radar (BLR) wind retrievals have been analyzed. Good agreement between the LES winds and those estimated using the BLR is observed. After removing an instrumental bias, an estimate of the probability density function (PDF) and skewness of the radar and LES vertical velocity estimates was calculated as a function of height. The PDFs of vertical velocity and the height profiles of the skewness of vertical velocity exhibit good agreement for both the LES and BLR calculations and have the expected canonical structure.

A non-hybrid method for the PDF equations of turbulent flows on unstructured grids

In probability density function (PDF) methods of turbulent flows, the joint PDF of several flow variables is computed by numerically integrating a system of stochastic differential equations for Lagrangian particles. A set of parallel algorithms is proposed to provide an efficient solution of the PDF transport equation modeling the joint PDF of turbulent velocity, frequency and concentration of a passive scalar in geometrically complex configurations. In the vicinity of walls the flow is resolved by an elliptic relaxation technique down to the viscous sublayer, explicitly modeling the high anisotropy and inhomogeneity of the low-Reynolds-number wall region without damping or wall-functions. An unstructured Eulerian grid is employed to extract Eulerian statistics, to solve for quantities represented at fixed locations of the domain (i.e., the mean pressure and the elliptic relaxation tensor) and to track particles. All three aspects regarding the grid make use of the finite element method employing the simplest linear shapefunctions. To model the small-scale mixing of the transported scalar, the interaction by exchange with the conditional mean (IECM) model is adopted. An adaptive algorithm to compute the velocity-conditioned scalar mean is proposed that homogenizes the statistical error over the sample space with no assumption on the shape of the underlying velocity PDF. Compared to other hybrid particle-in-cell approaches for the PDF equations, the current methodology is consistent without the need for consistency conditions. The algorithm is tested by computing the dispersion of passive scalars released from concentrated sources in two different turbulent flows: the fully developed turbulent channel flow and a street canyon (or cavity) flow. Algorithmic details on estimating conditional and unconditional statistics, particle tracking and particle-number control are presented in detail. Relevant aspects of performance and parallelism on cache-based shared memory machines are discussed.

Experimental study of higher-order moments in shear-driven boundary layers with rotation

The results of laboratory wall turbulence experiments on a shear-driven rotating boundary layer are presented. The experiments were carried out in the Turin University Laboratory rotating water tank. The flow was generated by changing the rotation speed of the platform and measured by means of particle image velocimetry. In order to analyse the influence of the rotation and of surface roughness, different cases were examined. Several rotation periods were considered. The measurements were performed both over a smooth surface and over a rough-to-smooth transition. Mean flows and the higher-order moments of the velocity probability density function are shown and discussed together with a comparison of the different experimental cases, theory and large-eddy simulations.

Spectral analysis of boundary layers in Rayleigh-Bénard convection

A combined experimental and numerical study of the boundary layer in a 4:1 aspect-ratio Rayleigh-Bénard cell over a four-decade range of Rayleigh numbers has been undertaken aimed at gaining a better insight into the character of the boundary layers. The experiments involved the simultaneous laser Doppler anemometry measurements of fluid velocity at two locations, i.e., in the boundary layer and far away from it in the bulk, for Rayleigh numbers varying between 1.6x10(7) and 2.4x10(9) . In parallel, direct numerical simulations have been performed for the same configuration for Rayleigh numbers between 7.0x10(4) and 7.7x10(7) . The temperature and velocity probability density functions and the power spectra of the horizontal velocity fluctuations measured in the boundary layer and in the bulk flow are found to be practically identical. Except for the smallest Rayleigh numbers, the spectra in the boundary layer and in the bulk central region are continuous and have a wide range of active scales. This indicates that both the bulk and the boundary layers are turbulent in the Ra number range considered. However, molecular effects can still be observed and the boundary layer does not behave like a classical shear-driven turbulent boundary layer.

Probability density function modeling of scalar mixing from concentrated sources in turbulent channel flow

Dispersion of a passive scalar from concentrated sources in fully developed turbulent channel flow is studied with the probability density function (PDF) method. The joint PDF of velocity, turbulent frequency and scalar concentration is represented by a large number of Lagrangian particles. A stochastic near-wall PDF model combines the generalized Langevin model of Haworth and Pope [Phys. Fluids 29, 387 (1986)] with Durbin's [J. Fluid Mech. 249, 465 (1993)] method of elliptic relaxation to provide a mathematically exact treatment of convective and viscous transport with a nonlocal representation of the near-wall Reynolds stress anisotropy. The presence of walls is incorporated through the imposition of no-slip and impermeability conditions on particles without the use of damping or wall-functions. Information on the turbulent time scale is supplied by the gamma-distribution model of van Slooten et al. [Phys. Fluids 10, 246 (1998)]. Two different micromixing models are compared that incorporate the effect of small scale mixing on the transported scalar: the widely used interaction by exchange with the mean and the interaction by exchange with the conditional mean model. Single-point velocity and concentration statistics are compared to direct numerical simulation and experimental data at Reτ=1080 based on the friction velocity and the channel half width. The joint model accurately reproduces a wide variety of conditional and unconditional statistics in both physical and composition space.

Deconvolution Technique to Separate Signal from Noise in Gravel Bedload Velocity Data

A deconvolution procedure is presented to estimate the probability density function of bedload transport velocity from noisy stationary data collected using the bottom tracking feature of acoustic Doppler current profilers (aDcps). The procedure involves the optimization of a computational summation of random variables for the instrument noise (assumed to be Gaussian with zero mean) and the spatially averaged bedload velocity within the insonified area of each acoustic beam (V) . The procedure was tested on two aDcp time series, measured in two different gravel-bed rivers (Fraser River and Norrish Creek). Models generated using either a semitheoretical compound Poisson-gamma distribution or an empirical gamma distribution for V were similar and did not differ significantly from the distribution of the original data. Optimized distributions for V were highly positively skewed. The instrument noise was comparable to instrument noise for aDcp water velocity measurements, i.e., an order of magnitude greater t...

Statistical model of collisions of bidisperse particles in an anisotropic turbulent flow

A statistical model for describing the motion and collisions of a bidisperse mixture of particles in anisotropic turbulent flows is presented. The model is based on a kinetic equation for the particle velocity probability density function (PDF). The results are compared with the data of a direct numerical simulation of the sedimentation of a bidisperse mixture of particles under the action of the gravity force.

Can Protostellar Jets Drive Supersonic Turbulence in Molecular Clouds?

Jets and outflows from young stellar objects are proposed candidates to drive supersonic turbulence in molecular clouds. Here, we present the results from multi-dimensional jet simulations where we investigate in detail the energy and momentum deposition from jets into their surrounding environment and quantify the character of the excited turbulence with velocity probability density functions. Our study include jet--clump interaction, transient jets, and magnetised jets. We find that collimated supersonic jets do not excite supersonic motions far from the vicinity of the jet. Supersonic fluctuations are damped quickly and do not spread into the parent cloud. Instead subsonic, non-compressional modes occupy most of the excited volume. This is a generic feature which can not be fully circumvented by overdense jets or magnetic fields. Nevertheless, jets are able to leave strong imprints in their cloud structure and can disrupt dense clumps. Our results question the ability of collimated jets to sustain supersonic turbulence in molecular clouds.

Statistical analysis of small bubble dynamics in isotropic turbulence

The dynamics and dispersion of small air bubbles in isotropic turbulence are analyzed computationally. The flow field is simulated using a pseudospectral code, while the bubble dynamics are analyzed by integration of a Lagrangian equation of motion that accounts for buoyancy, added mass, pressure, drag, and lift forces. Probability density functions (pdfs) of bubble velocities, lift and drag forces, and of field velocities and vorticities along bubble trajectories are used to analyze bubble dynamics. Lagrangian bubble trajectories are also employed to determine dispersion characteristics, following the theoretical development of Cushman and Moroni [Phys. Fluids 13, 75 (2001)]. Consistent with available experimental data, bubble rise velocities are increasingly suppressed with increasing turbulence intensity. The analysis also reveals that the vertical bubble velocities are characterized by asymmetric pdfs that are positive or negative-skewed dependent upon the nondimensional turbulence intensity and the Taylor length scale. The role of the lift force in moving the bubbles to the down-flow side of turbulent eddies, and consequently retarding their rise, is consistently observed in all analyses. The dispersion of 40μm bubbles and transition to Fickian behavior is shown to be weakly affected by the turbulence level. Larger, 400μm bubbles are shown to be more sensitive to turbulence level with transition to Fickian behavior delayed in low turbulence fields.

Particle collisions in a turbulent flow

A statistical model for the continuum description of particle motion and collisions in anisotropic turbulent shear flows is presented. The model is based on a kinetic equation for the particle velocity probability density function. The results are compared with the direct numerical simulation (DNS) data for a shear layer and a plane channel.

Direct numerical simulation of turbulent heat transfer in pipe flows: Effect of Prandtl number

Direct numerical simulations of heat transfer in a fully developed turbulent pipe flow with isoflux condition imposed at the wall are performed for a Reynolds number based on pipe radius Re = 5500. Main emphasis is placed on Prandtl number effects on turbulent heat transfer in pipe flow. The scaling of mean temperature profiles is investigated in order to derive correct logarithmic law for various Pr. The rms of temperature fluctuations and turbulent heat fluxes are found to increase when increasing Prandtl number. The turbulent Prandtl number, Pr t, is almost independent of the molecular Prandtl number Pr for Pr ⩾ 0.2. The radial distributions of higher order statistics (skewness and flatness) confirm the intermittent behaviour at the close vicinity of the wall; this intermittent behaviour is more pronounced with an increase in Pr. The Nusselt number is in good agreement with the findings of the literature. Probability density functions and joint probability density functions of velocity and temperature fluctuations are used to describe the characteristics of the turbulent flow and heat transfer. The instantaneous flow and thermal fields are plotted in order to analyse the turbulent structures. To explore the impact of the wall curvature on turbulent heat transfer, predictions were compared to available results for channel flow. These comparisons show a slightly more intense temperature fluctuations in the pipe flow.

Route to non-Gaussian statistics in convective turbulence

Motivated by the work of Li and Meneveau [Phys. Rev. Lett. 95, 164502 (2005)], we propose and solve a model for the Lagrangian evolution of both longitudinal and transverse velocity and temperature increments for Boussinesq convection. From this model, the short-time evolution of an initially imposed Gaussian joint probability density function (PDF) of both velocity and temperature increments is computed analytically and the trend to non-Gaussian statistics shown in a quantitative way. Predictions for moments of the joint PDF are obtained and their behavior analyzed with respect to known experimental and numerical results. The obtained results do not depend on the model free parameters, a fact in favor of their robustness.