Wave motion of incompressible turbulent fluid

Evolution of small disturbances in a fully developed incompressible turbulent flow is considered on the base of the transport equation for the single-point probability density function (PDF) of velocity fluctuations. It is shown that at high frequencies this equation is similar to the Vlasov equation for charged plasma in a self-consistent electromagnetic field having longitudinal wave solutions for turbulent stresses similar to Langmuir waves. It is found that the longitudinal waves of turbulent stresses have a constant phase velocity and can be damped, neutral, or growing waves, depending on the type of undisturbed probability density function of velocity fluctuations. The obtained result differs from the previously published solutions to this problem using the statistical moments closures according to which the wave disturbances should be neutral or damped. The possibilities of experimental observation of longitudinal waves of turbulent stresses are analyzed.     

Eulerian transported probability density function sub-filter model for large-eddy simulations of turbulent combustion

Reactive flow simulations using large-eddy simulations (LES) require modelling of sub-filter fluctuations. Although conserved scalars like mixture fraction can be represented using a beta-function, the reactive scalar probability density function (PDF) does not follow an universal shape. A one-point one-time joint composition PDF transport equation can be used to describe the evolution of the scalar PDF. The high-dimensional nature of this PDF transport equation requires the use of a statistical ensemble of notional particles and is directly coupled to the LES flow solver. However, the large grid sizes used in LES simulations will make such Lagrangian simulations computationally intractable. Here we propose the use of a Eulerian version of the transported-PDF scheme for simulating turbulent reactive flows. The direct quadrature method of moments (DQMOM) uses scalar-type equations with appropriate source terms to evolve the sub-filter PDF in terms of a finite number of delta-functions. Each delta-peak is characterized by a location and weight that are obtained from individual transport equations. To illustrate the feasibility of the scheme, we compare the model against a particle-based Lagrangian scheme and a presumed PDF model for the evolution of the mixture fraction PDF. All these models are applied to an experimental bluff-body flame and the simulated scalar and flow fields are compared with experimental data. The DQMOM model results show good agreement with the experimental data as well as the other sub-filter models used.     

Multifractal PDF analysis for intermittent systems

The formula for probability density functions (PDFs) has been extended to include PDF for energy dissipation rates in addition to other PDFs such as for velocity fluctuations, velocity derivatives, fluid particle accelerations, energy transfer rates, etc., and it is shown that the formula actually explains various PDFs extracted from direct numerical simulations and experiments performed in a wind tunnel. It is also shown that the formula with appropriate zooming increment corresponding to experimental situation gives a new route to obtain the scaling exponents of velocity structure function, including intermittency exponent, out of PDFs of velocity fluctuations.     

A stochastic model of torques in von Karman swirling flow

A stochastic model is derived to predict the turbulent torque produced by a swirling flow. It is a simple Langevin process, with a colored noise. Using the unified colored noise approximation, we derive analytically the PDF of the fluctuations of injected power in two forcing regimes: constant angular velocity or constant applied torque. In the limit of small velocity fluctuations and vanishing inertia, we predict that the injected power fluctuates twice less in the case of constant torque than in the case of constant angular velocity forcing. The model is further tested against experimental data in a von Karman device filled with water. It is shown to allow for a parameter-free prediction of the PDF of power fluctuations in the case where the forcing is made at constant torque. A physical interpretation of our model is finally given, using a quasi-linear model of turbulence.Received: 29 January 2004, Published online: 18 June 2004PACS: 47.27.-i Turbulent flows, convection, and heat transfer - 47.27.Eq Turbulence simulation and modeling     

Rotation effect on near-wall turbulence statistics and flow structures

Turbulent flow in an axially rotating pipe, involving complicated physical mecha- nism of turbulence, is a typical problem for the study of rotating turbulent flow. The pipe rotation induces two effects on the flow. One is the stabilizing effect due to the centrifu- gal and Coriolis forces, which accounts for the relaminarization of the turbulence[1—3] and the reduction of the friction coefficient at the pipe wall. The behavior is also related to the wall streaks inclining to the azimuthal di…     

Estimating statistical parameters of an elastic random medium from traveltime fluctuations of refracted waves

Traveltime fluctuations of diving-type refracted waves are studied in the framework of geometrical optics in order to estimate the statistical parameters of an elastic random medium. A stratified background medium is considered in which the velocity increases linearly with depth. Smooth and strongly anisomeric (statistically anisotropic) inhomogeneities are embedded in this medium. The covariance and the variance of traveltime fluctuations are derived and subsequently used to estimate the standard deviation of the medium fluctuations and the inhomogeneity scale lengths in horizontal and vertical directions. The theoretical estimation procedure is verified by performing numerical calculations and it is observed that, under the considered conditions, the traveltime variance decreases at large offsets. This new phenomenon has not been observed before either in acoustics and optics, or in radio wave propagation.     

Anomalous diffusion in silo drainage

The silo discharge process is studied by molecular dynamics simulations. The development of the velocity profile and the probability density function for the displacements in the horizontal and vertical axis are obtained. The PDFs obtained at the beginning of the discharge reveal non-Gaussian statistics and superdiffusive behaviors. When the stationary flow is developed, the PDFs at shorter temporal scales are non-Gaussian too. For big orifices a well-defined transition between ballistic and diffusive regime is observed. In the case of a small outlet orifice, no well-defined transition is observed. We use a nonlinear diffusion equation introduced in the framework of non-extensive thermodynamics in order to describe the movements of the grains. The solution of this equation gives a well-defined relationship (γ= 2/(3-q)) between the anomalous diffusion exponent γ and the entropic parameter q introduced by the non-extensive formalism to fit the PDF of the fluctuations.     

Statistical properties of Er/Yb co-doped random Rayleigh feedback fiber laser

In this Letter, we experimentally investigate fast temporal intensity dynamics and statistical properties of the claddingpumped Er/Yb co-doped random Rayleigh feedback fiber laser(EYRFL) for the first time, to the best of our knowledge. By using the optical spectral filtering method, strong and fast intensity fluctuations with the generation of extreme events are revealed at the output of EYRFL. The statistics of the intensity fluctuations strongly depends on the wavelength of the filtered radiation, and the intensity probability density function(PDF) with a heavy tail is observed in the far wings of the spectrum. We also find that the PDF of the intensity in the central part of the spectrum deviates from the exponential distribution and has the dependence on the laser operating regimes, which indicates some correlations among different frequency components exist in the EYRFL radiation and may play an important role in the random lasing spectrum stabilization process.     

Conditional velocity statistics in the double scalar mixing layer – A mapping closure approach

In this work we use 3D direct numerical simulations (DNS) to investigate the average velocity conditioned on a conserved scalar in a double scalar mixing layer (DSML). The DSML is a canonical multistream flow designed as a model problem for the extensively studied piloted diffusion flames. The conditional mean velocity appears as an unclosed term in advanced Eulerian models of turbulent non-premixed combustion, like the conditional moment closure and transported probability density function (PDF) methods. Here it accounts for inhomogeneous effects that have been found significant in flames with relatively low Damköhler numbers. Today there are only a few simple models available for the conditional mean velocity and these are discussed with reference to the DNS results. We find that both the linear model of Kutznetzov and the Li and Bilger model are unsuitable for multi stream flows, whereas the gradient diffusion model of Pope shows very close agreement with DNS over the whole range of the DSML. The gradient diffusion model relies on a model for the conserved scalar PDF and here we have used a presumed mapping function PDF, that is known to give an excellent representation of the DNS. A new model for the conditional mean velocity is suggested by arguing that the Gaussian reference field represents the velocity field, a statement that is evidenced by a near perfect agreement with DNS. The model still suffers from an inconsistency with the unconditional flux of conserved scalar variance, though, and a strategy for developing fully consistent models is suggested.     

LES/PDF for premixed combustion in the DNS limit

We investigated the behaviour of the composition probability density function (PDF) model equations used in a large-eddy simulation (LES) of turbulent combustion in the direct numerical simulation (DNS) limit; that is, in the limit of the LES resolution length scale Δ (and the numerical mesh spacing h) being small compared to the smallest flow length scale, so that the resolution is sufficient to perform a DNS. The correct behaviour of a PDF model in the DNS limit is that the resolved composition fields satisfy the DNS equations, and there are no residual fluctuations (i.e. the PDF is everywhere a delta function). In the DNS limit, the treatment of molecular diffusion in the PDF equations is crucial, and both the ‘random-walk’ and ‘mean-drift’ models for molecular diffusion are investigated. Two test cases are considered, both of premixed laminar flames (of thickness δ_L). We examine the solutions of the model PDF equations for these test cases as functions of Δ/δ_L and h/δ_L. Each of the two PDF models has advantages and disadvantages. The mean-drift model behaves correctly in the DNS limit, but it is more difficult to implement and computationally more expensive. The random-walk model does not have the correct behaviour in the DNS limit in that it produces non-zero residual fluctuations. However, if the specified mixing rate Ω normalised by the reaction timescale τ_c is sufficiently large (Ωτ_c ? 1), then the residual fluctuations are less than 10% and the observed flame speed and thickness are close to their laminar values. Away from the DNS limit (i.e. h/δ_L ? 1), the observed flame thickness scales with the mesh spacing h, and the flame speed scales with Ωh. For this case it is possible to construct a non-general specification of the mixing rate Ω such that the flame speed matches the laminar flame speed.     

On the numerical modelling of the turbulent layer penetration into a stably stratified fluid

The improved numerical models based on the algebraic representations of the Reynolds stresses and fluxes and the use of the differential equation for the transfer of the dispersion of fluctuations of the vertical velocity component are considered for describing the processes of a vertical turbulent exchange in a stably stratified reservoir. Numerical modelling of the penetration of a turbulent layer of a mixed fluid in a linearly stratified medium under the action of constant shear stress is carried out. Computational results agree well with known experimental data and point to a substantial influence of the anisotropy of the flow on its main characteristics.     

Approximation of joint PDFs by discrete distributions generated with Monte Carlo methods

A new method for the approximation of multivariate scalar probability density functions (PDFs) in turbulent reacting flow by means of a joint presumed discrete distribution (jPDD) is presented. The jPDDs can be generated with specified mean values and variances as well as covariances. Correlations between variables – e.g. fluctuating mixture fractions and/or reaction progress – can thereby be taken into account. In this way the new approach overcomes an important limitation of ordinary presumed PDF methods, where statistical independence between the variables is often assumed. Different methods are presented to generate discrete distributions, based either on biased random number generators or on mixing models familiar from PDF transport models.

The new approach is extensively validated on a turbulent flow configuration with simultaneous mixing and reaction. Large eddy simulation data as well as results from a transported PDF model are used for the validation of the jPDD approach. The comparison shows that in particular distributions generated with mixing models are able to predict mean reaction rates accurately. For the configuration considered, the neglect of correlations results in significant underestimation of reaction rates. Moreover it is found that higher statistical moments (e.g. the skewness) can influence reaction rates. The consequences for the generation of jPDDs are discussed.

In summary, the new jPDD model has the potential to be significantly more accurate than established presumed PDF methods, because correlations between fluctuating variables can be taken into account. At the same time, the new approach is nearly as efficient as standard presumed PDF formulations, since mean rates are computed in a pre-processing step and stored in look-up tables as a function of the first and second moments of the relevant variables.     

Asymptotic solution of the Wang-Uhlenbeck recurrence time problem

A Langevin particle is initiated at the origin with positive velocity. Its trajectory is terminated when it returns to the origin. In 1945, Wang and Uhlenbeck posed the problem of finding the joint probability density function (PDF) of the recurrence time and velocity, naming it "the recurrence time problem". We show that the short-time asymptotics of the recurrence PDF is similar to that of the integrated Brownian motion, solved in 1963 by McKean. We recover the long-time t(-3/2) decay of the first arrival PDF of diffusion by solving asymptotically an appropriate variant of McKean's integral equation.     

High-frequency volume and boundary acoustic backscatter fluctuations in shallow water

Volume and boundary acoustic backscatter envelope fluctuations are characterized from data collected by the Toroidal Volume Search Sonar (TVSS), a 68 kHz cylindrical array capable of 360 degrees multibeam imaging in the vertical plane perpendicular to its axis. The data are processed to form acoustic backscatter images of the seafloor, sea surface, and horizontal and vertical planes in the volume, which are used to attribute nonhomogeneous spatial distributions of zooplankton, fish, bubbles and bubble clouds, and multiple boundary interactions to the observed backscatter amplitude statistics. Three component Rayleigh mixture probability distribution functions (PDFs) provided the best fit to the empirical distribution functions of seafloor acoustic backscatter. Sea surface and near-surface volume acoustic backscatter PDFs are better described by Rayleigh mixture or log-normal distributions, with the high density portion of the distributions arising from boundary reverberation, and the tails arising from nonhomogeneously distributed scatterers such as bubbles, fish, and zooplankton. PDF fits to the volume and near-surface acoustic backscatter data are poor compared to PDF fits to the boundary backscatter, suggesting that these data may be better described by mixture distributions with component densities from different parametric families. For active sonar target detection, the results demonstrate that threshold detectors which assume Rayleigh distributed envelope fluctuations will experience significantly higher false alarm rates in shallow water environments which are influenced by near-surface microbubbles, aggregations of zooplankton and fish, and boundary reverberation.     

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.     

Dispersion simulation of a wind tunnel experiment with Lagrangian particle models

Summary A particle model LAMDA suitable for dealing with the atmospheric dispersion is presented. The reliability of the model is tested comparing the results of its simulations to the wind tunnel measurements by Khurshudyanet al. (1981). Two versions of the model, both based on the Langevin equation and askewed distribution of the vertical wind velocity fluctuations, are considered. To develop the second version of the model we derived a proper scheme to produce skewed distributions of particle velocities consistent with the observed first and second moments of turbulent fluctuating velocities, with their horizontal and vertical derivatives, and their cross-correlations. The 2D geometry of the obstacle (in the wind tunnel experiment considered) allowed some simplifications. Due to the lack of some input data (such as the vertical profiles of crosswind standard deviation of wind velocity fluctuations and of the Lagrangian time scales) we looked for three different parametrizations. It was found that particular combinations of measured and parametrized data could give rise to critical vertical regions in which the derived scheme for the generation of random vertical velocity fluctuations cannot be applied. The best vertical distributions of the Lagrangian time scales (third parametrization) were estimated by fitting simple formulations for the average plume height and lateral variances to the measured data. The main results of this work were the model performance in simulating dispersion in shear flow over flat terrain, and its sensivity to the shape of the crosscorrelation term .     

Estimation of the power scintillation probability density function in free-space optical links by use of multicanonical Monte Carlo sampling

Free-space optics (FSO) can provide cost-effective, high-bandwidth, wireless connections. However, atmospheric turbulence may degrade the performance of FSO links by causing intensity and power scintillations at the receiver. Multicanonical Monte Carlo sampling is used in conjunction with the phase screen method to calculate the statistics, and particularly the probability density function (PDF), of the power fluctuations at an FSO receiver. This allows the efficient calculation of the PDF even for very small values with a limited number of iterations. The obtained PDF can be used to characterize the performance of the system in terms of the error probability.