Shape Of Probability Density Function Research Articles

Lagrangian intermittent modelling of turbulent lifted flames

Partially premixed combustion in turbulent non-premixed flames is studied. A possible way to describe these turbulent flames is to consider a transport equation for the probability density function (PDF) for all the fluctuating variables with a satisfactory representation of small-scale mixing phenomena influenced by chemical reactions. Here, a more tractable model based on a presumed joint PDF shape for both a mixture fraction variable and a progress variable is proposed. In this approach, the assumption of statistical independence between the two variables is avoided. On the contrary, a strong but clearly stated functional dependence between the two scalars is introduced. This allows us to express the higher probability that fluid particles have to burn for nearly stochiometric compositions. The effects of both non-infinitely fast chemistry and partially premixed combustion are taken into account by using a modified version of the MIL model (Modèle Intermittent Lagrangien) originally introduced by Borghi and Gonzalez. In the present modelling proposal, the MIL principles are essentially used as a way of distributing the mean chemical rate among a premixed and a diffusive contribution and the new formulation has the potentiality to take into account the essential features of the stabilization zone. Finally, the complete model has been used to simulate the turbulent non-premixed lifted methane–air flames studied by G. Mungal and his co-workers at Stanford University. With the proposed approach, a satisfactory agreement is obtained in terms of lift-off heights for a wide range of operating conditions.

Modelling and control of the flame temperature distribution using probability density function shaping

This paper presents three control algorithms for the output probability density function (PDF) control of the 2D and 3D flame distribution systems. For the 2D flame distribution systems, control methods for both static and dynamic flame systems are presented, where at first the temperature distribution of the gas jet flames along the cross-section is approximated. Then the flame energy distribution (FED) is obtained as the output to be controlled by using a B-spline expansion technique. The general static output PDF control algorithm is used in the 2D static flame system, where the dynamic system consists of a static temperature model of gas jet flames and a second-order actuator. This leads to a second-order closed-loop system, where a singular state space model is used to describe the dynamics with the weights of the B-spline functions as the state variables. Finally, a predictive control algorithm is designed for such an output PDF system. For the 3D flame distribution systems, all the temperature values of the flames are firstly mapped into one temperature plane, and the shape of the temperature distribution on this plane can then be controlled by the 3D flame control method proposed in this paper. Three cases are studied for the proposed control methods and desired simulation results have been obtained.

Scaling of electric field fluctuations associated with the aurora during northward IMF

We present a statistical study of scaling features of the electric field fluctuations measured by the DE2 satellite in the polar region during positive Bz IMF when the theta‐aurora was observed by the DE1 satellite. It is demonstrated that the power spectra of the fluctuations have a power‐law form at spatial scales from ∼0.5 km (the limit of resolution) to several thousands of kilometers, with a break near 100 km. The scaling properties of the field are studied by examining the generalized structure functions (GSFs) and probability density functions (PDFs) of the fluctuations. The observed PDFs have a non‐Gaussian shape with heavy tails. We also demonstrate a collapse of the re‐scaled PDFs onto a single curve. A relation of PDF shape to solar wind conditions is revealed. The same analysis is performed on the TV observations of the theta‐aurora. The scaling characteristics of the field and auroral fluctuations are compared.

Analysis of Short Latencies In Industrial Network Environments

The bibliography shows the typical shape of probability density function (PDF) that describe network latencies. The paper presents the results of the measurements of the PDF of Round Trip Time (RTT) in real network environments. The results show, that the appropriate PDFs are significantly differentiated and depend on the kind of the network, its load, peers load, logic of applications, etc. The PDF measurments is focused on industrial network control programs transmissions, where only loses and short delays are important.

Generation of Composite Surfaces With Bimodal Distribution and Contact Analysis for Optimum Tribological Performance

Most contact analyses assume that the surface height distributions follow a single modal distribution. However, there are many surfaces with multi-modal roughness distributions, e.g., magnetic particulate tape, super alloys with precipitates, and hydrophobic leaves. In this study, an algorithm is developed to generate bimodal surfaces by superimposing particles with radii following a Gaussian distribution on a Gaussian rough surface. Two different cases are presented to produce composite surfaces with particles; the first case is particles sitting on a surface and the other case is particles sitting on the mean plane of a surface. Statistical analysis is carried out for the generated bimodal surfaces to study the effect of the bimodal roughness distributions on the surface’s probability density function shapes. Contact analysis is also conducted to identify optimum bimodal roughness distributions for low friction, stiction, and wear. It is assumed that particles and matrix have uniform elastic properties as it is a reasonable assumption in some applications such as magnetic tapes. Variation of fractional contact area, maximum contact pressure, and relative meniscus force as functions of relative mean radius and relative standard deviation of particles are studied for different values of particle densities. It is found that bimodal surfaces with lower particle density are beneficial to low friction and stiction, whereas those with higher particle density are beneficial to low wear. Relative mean radii of particles of 2–3 in bimodal surfaces with particles sitting on surface and 3–5 in bimodal surfaces with particles sitting on the mean plane of surface are desirable for low friction, stiction, and wear.

Analysis and Validation of the Probability Density Function of the Output Current of Optically Preamplified Receivers with Arbitrary Optical Filtering through Monte Carlo Simulation

The validation of the probability density function (PDF) of the current at the decision circuit of an optically preamplified receiver with arbitrary optical filtering is assessed, for the first time, for tight optical filtering and considering the optical filter detuning. This assessment is performed through the analytical estimation of the PDF from the moment generating function (MGF) and by comparison with estimates using Monte Carlo simulation. An excellent agreement between PDFs is found. The influence of the optical filter shape, bandwidth and detuning on the PDF of the output current of an optically preamplified receiver with tight optical filtering is also investigated. All PDFs obtained are clearly non-Gaussian. For smaller signal powers, this non-Gaussian behavior is enhanced. The absence of a polarizer at the receiver and the narrowing of the optical filter bandwidth enhance even more this non-Gaussian behavior. The smoother cut-off of the optical filter amplitude response leads to a smaller PDF asymmetry (lower Gaussian behavior). For higher signal power, the PDF shape becomes more Gaussian and the dependence of its shape on the optical filter and on the presence/absence of a polarizer is lessened. It is also shown that the optical filter detuning influence on the non-Gaussian behavior of the PDF is very small.

Modeling of Turbulent Diffusion Combustion Using a Combined Probability Density Function/Moment Method

A combined probability density function (PDF) /moment method has been developed to improve the accuracy of the Monte-Carlo method which is employed to simulate the evolution of PDFs in the PDF approach. The accuracy of Monte-Carlo method depends strongly on the number of Monte-Carlo particles and declines with the decrease in the PDF transport probability. The present method solves simultaneously a modeled PDF transport equation and moment equations of mean and variance. Both the mean and variance values obtained from the moment equations are just used as the reference values to match to the corresponding values of PDFs. In the frame of the conserved scalar concept, the usefulness of the present method is evaluated in the configuration of a turbulent jet diffusion flame, with a comparison with other methods; the PDF method and the flamelet model method. The results have showed good agreement between the present PDF method and the flamelet method, highlighting the improvement from the PDF method. Especially, the agreement in terms of PDF shape has verified that the beta function approximation in the flamelet model has high accuracy.

Scalar mixing from a concentrated source in turbulent channel flow

The mixing of a scalar (temperature) emitted from a concentrated line source in fully developed high-aspect-ratio turbulent channel flow is studied. The motivation for the work is to study the effect of the inhomogeneity on the scalar dispersion. It is most readily carried out in a flow that is inhomogeneous in only one direction, i.e. channel flow. Experiments were performed at two Reynolds numbers (). Both the mean and r.m.s. temperature profiles were found to be described well by truncated Gaussian distributions. In contrast to homogeneous flows, (i) the growth rates of the mean profile widths did not exhibit power law behaviours, (ii) the centres of the r.m.s. profiles were found to drift towards the centre of the channel for plumes emanating from off-centreline source locations and (iii) the r.m.s. profiles showed no tendency towards double peaks far downstream, as are observed in homogeneous flows. For near-wall source locations, the probability density function (PDF) of the scalar fluctuations evolved from a quasi-Gaussian distribution near the wall to a strongly positively skewed PDF (with a large spike at the cold-fluid temperature) for transverse locations away from the wall. Increasing the Reynolds number was found to improve the mixing, even though this decreases the amount of time for which the scalar can mix (owing to the more rapid advection). For the centreline source location, the PDF shape was, in general, more spiked, indicating the importance of the flapping of the plume in this case. The effect of the meandering of the plume was less significant when the plume was bounded by the wall. Second- and third-order velocity–temperature correlations were presented. The differences in their profiles for the near-wall and centreline source locations were distinct.

Using Probability Density Functions to Derive Consistent Closure Relationships among Higher-Order Moments

Parameterizations of turbulence often predict several lower-order moments and make closure assumptions for higher-order moments. In principle, the low- and high-order moments share the same probability density function (PDF). One closure assumption, then, is the shape of this family of PDFs. When the higher-order moments involve both velocity and thermodynamic scalars, often the PDF shape has been assumed to be a double or triple delta function. This is equivalent to assuming a mass-flux model with no subplume variability. However, PDF families other than delta functions can be assumed. This is because the assumed PDF methodology is fairly general. This paper proposes closures for several third- and fourth-order moments. To derive the closures, the moments are assumed to be consistent with a particular PDF family, namely, a mixture of two trivariate Gaussians. (This PDF is also called a double Gaussian or binormal PDF by some authors.) Separately from the PDF assumption, the paper also proposes a simplified relationship between scalar and velocity skewnesses. This PDF family and skewness relationship are simple enough to yield simple, analytic closure formulas relating the moments. If certain conditions hold, this set of moments is specifically realizable. By this it is meant that the set of moments corresponds to a real Gaussian-mixture PDF, one that is normalized and nonnegative everywhere. This paper compares the new closure formulas with both large eddy simulations (LESs) and closures based on double and triple delta PDFs. This paper does not implement the closures in a single-column model and test them interactively. Rather, the comparisons are diagnostic; that is, low-order moments are extracted from the LES and treated as givens that are input into the closures. This isolates errors in the closures from errors in a single-column model. The test cases are three atmospheric boundary layers: a trade wind cumulus layer, a stratocumulus layer, and a clear convective case. The new closures have shortcomings, but nevertheless are superior to the double or triple delta closures in most of the cases tested.

Probability Density Function of the Output Current of Cascaded Multiplexer/Demultiplexers in Transparent Optical Networks

The influence of the concatenation of arbitrary optical multiplexers/demultiplexers (MUX/DEMUXs) on the probability density function (PDF) of the output current of a transparent optical network is assessed. All PDF results obtained analytically are compared with estimates from Monte Carlo simulation and an excellent agreement is achieved. The non-Gaussian behavior of the PDFs, previously reported by other authors for square-law detectors, is significantly enhanced with the number of nodes increase due to the noise accumulation along the cascade of MUX/DEMUXs. The increase of the MUX/DEMUXs bandwidth and detuning also enhances the PDFs non-Gaussian behavior. The PDF shape variation with the detuning depends strongly on the number of nodes. Explanations for the Gaussian approximation (GA) accuracy on the assessment of the performance of a concatenation of optical MUX/DEMUXs are also provided. For infinite extinction ratio and tuned MUX/DEMUXs, the GA error probabilities are, in general, pessimistic, due to the inaccurate estimation of the error probability for both bits. For low extinction ratio, the GA is very accurate due to a balance between the error probabilities estimated for the bits “1” and “0.” With the detuning increase, the GA estimates can become optimistic.

EXIT chart analysis and performance of precoded turbo equalisation systems for long block lengths

The authors investigate the effect that the precoders' weights have on the convergence of turbo equalisation systems for long block lengths. A comparison is made between the trellis code capacities and the independent and identically distributed channel capacity. The extrinsic information transfer chart is used to analyse four precoded turbo equalisation systems. It is observed that the gradient of the transfer function at high input mutual information values, relating to the maximum a posteriori equalisation of a precoded channel, plays a greater role than the initial loss of information in determining the convergence of precoded turbo equalisation systems. Simulation results demonstrate that a low weight precoder converges at a lower signal-to-noise ratio in comparison to a high weight precoder. Further, it is noticed that the system trajectories only follow the bounds of the transfer functions on the extrinsic information transfer chart during the early iterations. The Kolmogorov–Smirnov test is used to study probability density function shapes of the extrinsic log-likelihood ratios during the iterative equalisation/decoding process. The authors also show that the extrinsic information transfer chart can be used to compute the trellis code capacity of a precoded channel.

Two-point closure strategy in the mapping closure approximation approach

A two-point closure strategy in mapping closure approximation (MCA) approach is developed for the evolution of the probability density function (PDF) of a scalar advected by stochastic velocity fields. The MCA approach is based on multipoint statistics. We formulate a MCA modeled system using the one-point PDFs and two-point correlations. The MCA models can describe both the evolution of the PDF shape and the rate at which the PDF evolves.

A joint SSPDF–CMC method for simulating turbulent methane–air jet combustion

A joint single scalar probability density function and conditional moment closure (SSPDF–CMC) method is proposed for modeling a turbulent methane–air jet flame. In general, the probability density function (PDF) of passive scalar (such as mixture fraction) is non-Gaussian and not fully determined by the advecting velocity field, therefore the presumed shape of PDF of mixture fraction assumed as clipped Gaussian distribution or beta function in normal conditional moment closure (CMC) method is incorrect. In SSPDF–CMC method, the PDF of mixture fraction is obtained using a Monte-Carlo method to solve a PDF transport equation. An assumption that the averaged scalar advection is approximately equal to the averaged scalar dissipation in the wake of a grid-generated turbulence flow is adopted to model the averaged scalar dissipation. The predictions using the proposed method are compared with those using the conventional CMC method and the experimental data. It is seen that the predicted Favre conditional averaged statistics and Favre unconditional averaged statistics using the proposed method are in better agreement with the measurement data than those using the conventional CMC method. The predicted conditional or unconditional mean NO even using the SSPDF model is only in fair agreement with the experiments. It shows that the first-order closure for the conditional reaction rate of NO should be improved.

Universality of probability density distributions in the overlap region in high Reynolds number turbulent boundary layers

The probability density functions (PDFs) of the streamwise mean velocity in high Reynolds number turbulent boundary layers have been studied. The hypothesis of self-similar, normalized with the root mean square velocity, PDFs has been tested using the KTH database of high Reynolds number zero pressure-gradient turbulent boundary layer flow. The self-similarity was tested using the Kullback–Leibler divergence measure and it was found that the region of self-similar PDFs extends from about 160 viscous wall units to about 0.3 boundary layer thicknesses (in δ95). This region is somewhat larger than the universal overlap region. The PDF shape in the universal overlap region is close to Gaussian allowing for a Gram–Charlier expansion approximation of the measured PDFs. A remarkable collapse was found for 57 normalized PDF distributions for different positions within the universal overlap region and Reynolds numbers based on the momentum-loss thickness between 4300 and 12 600, strongly indicating a high degree of flow universality within the universal overlap region. Within the range studied, the Gram–Charlier expansion coefficients (related to the PDF moments) show no Reynolds number trend further supporting the self-similarity hypothesis.

Examining the role of turbulence in the solar wind - magnetosphere interaction processes

In this paper we compare the intermittence properties of magnetic fluctuations (nonGaussian shape of probability density functions) observed in the solar wind (ACE) with the simultaneous occurrence of intermittence in the Earth’s plasma sheet (GEOTAIL). Intervals with different level of magnetic turbulence are investigated separately.

Probability density function method for observing reconstructed attractor structure

Probability density function (PDF) method is proposed for analysing the structure of the reconstructed attractor in computing the correlation dimensions of RR intervals of ten normal old men. PDF contains important information about the spatial distribution of the phase points in the reconstructed attractor. To the best of our knowledge, it is the first time that the PDF method is put forward for the analysis of the reconstructed attractor structure. Numerical simulations demonstrate that the cardiac systems of healthy old men are about 6-6.5 dimensional complex dynamical systems. It is found that PDF is not symmetrically distributed when time delay is small, while PDF satisfies Gaussian distribution when time delay is big enough. A cluster effect mechanism is presented to explain this phenomenon. By studying the shape of PDFs, that the roles played by time delay are more important than embedding dimension in the reconstruction is clearly indicated. Results have demonstrated that the PDF method represents a promising numerical approach for the observation of the reconstructed attractor structure and may provide more information and new diagnostic potential of the analyzed cardiac system.

On conditional scalar increment and joint velocity–scalar increment statistics

Conditional velocity and scalar increment statistics are usually studied in the context of Kolmogorov's refined similarity hypotheses and are considered universal (quasi-Gaussian) for inertial-range separations. In such analyses the locally averaged energy and scalar dissipation rates are used as conditioning variables. Recent studies have shown that certain local turbulence structures can be captured when the local scalar variance ⟨ϕ″2⟩r and the local kinetic energy kr are used as the conditioning variables. We study the conditional increments using these conditioning variables, which also provide the local turbulence scales. Experimental data obtained in the fully developed region of an axisymmetric turbulent jet are used to compute the statistics. The conditional scalar increment probability density function (PDF) conditional on ⟨ϕ″2⟩r is found to be close to Gaussian for ⟨ϕ″2⟩r small compared with its mean and is sub-Gaussian and bimodal for large ⟨ϕ″2⟩r, and therefore is not universal. We find that the different shapes of the conditional PDFs are related to the instantaneous degree of non-equilibrium (production larger than dissipation) of the local scalar. There is further evidence of this from the conditional PDF conditional on both ⟨ϕ″2⟩r and χr, which is largely a function of ⟨ϕ″2⟩r/χr, a measure of the degree of non-equilibrium. The velocity-scalar increment joint PDF is close to joint Gaussian and quad-modal for equilibrium and non-equilibrium local velocity and scalar, respectively. The latter shape is associated with a combination of the ramp–cliff and plane strain structures. Kolmogorov's refined similarity hypotheses also predict a dependence of the conditional PDF on the degree of non-equilibrium. Therefore, the quasi-Gaussian (joint) PDF, previously observed in the context of Kolmogorov's refined similarity hypotheses, is only one of the conditional PDF shapes of inertial range turbulence. The present study suggests that such analyses provide a connection between statistical and structural descriptions of inertial-range turbulence.

Monte Carlo ray-tracing simulation for radiative heat transfer in turbulent fluctuating media under the optically thin fluctuation approximation

The Monte Carlo ray-tracing method (MCRT) based on the concept of radiation distribution factor is extended to solve radiative heat transfer problem in turbulent fluctuating media under the optically thin fluctuation approximation. A one-dimensional non-scattering turbulent fluctuating media is considered, in which the mean temperature and absorption coefficient distribution are assumed and the shape of probability density function is given. The distribution of the time-averaged volume radiation heat source is solved by MCRT and direct integration method. It is shown that the results of MCRT based on the concept of radiation distribution factor agree with these of integration solution very well, but results of MCRT based on the concept of radiative transfer coefficient do not agree with these of integration solution. The solution of time-averaged radiative transfer equation by the concept of radiative transfer coefficient should be treated with caution.

Small-Scale and Mesoscale Variability in Cloudy Boundary Layers: Joint Probability Density Functions

The joint probability density function (PDF) of vertical velocity and conserved scalars is important for at least two reasons. First, the shape of the joint PDF determines the buoyancy flux in partly cloudy layers. Second, the PDF provides a wealth of information about subgrid variability and hence can serve as the foundation of a boundary layer cloud and turbulence parameterization. This paper analyzes PDFs of stratocumulus, cumulus, and clear boundary layers obtained from both aircraft observations and large eddy simulations. The data are used to fit five families of PDFs: a double delta function, a single Gaussian, and three PDF families based on the sum of two Gaussians. Overall, the double Gaussian, that is binormal, PDFs perform better than the single Gaussian or double delta function PDFs. In cumulus layers with low cloud fraction, the improvement occurs because typical PDFs are highly skewed, and it is crucial to accurately represent the tail of the distribution, which is where cloud occurs. Since the double delta function has been shown in prior work to be the PDF underlying mass-flux schemes, the data analysis herein hints that mass-flux simulations may be improved upon by using a parameterization built upon a more realistic PDF.

Modeling of turbulent spray combustion with application to diesel like experiment

Turbulent combustion within sprays has been studied. Experiment and theory predict that the presence of fuel droplets leads to an increase in the fluctuations of temperature and equivalence ratio, and that these fluctuations must be considered to correctly estimate the mean reaction rate. A convenient way to describe the fluctuations of the medium is to compute the probability density function of all the fluctuating variables for the gas and liquid phases. The method used to derive the probability density function (PDF) equations in a pure gaseous medium was extended to two-phase flows by using the local equations written in terms of distribution functions. The PDF for the liquid phase contained complex unclosed terms representing dispersion and atomization. Instead of modeling these terms, we have used a stochastic Lagrangian simulation of the liquid droplets which employs classical atomization models and assumes spherical droplets. This method is used to close the vaporization terms which appear in the PDF equation for the gas. The PDF equation for the gas-phase could be solved by using a Monte-Carlo method but here we have used a more tractable model based on a presumed PDF shape for the mixture fraction. The effects of finite rate chemistry were taken into account by using the MIL model. Finally, the complete model was used to simulate auto-ignition and the establishment of a flame in a diesel type experiment. The auto-ignition delay and the shape of the flame were well simulated. The importance of taking into account the influence of the droplets on the fluctuations is demonstrated by testing the model with and without the corresponding terms.