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NASA NTRS2000

An Overview of the NCC Spray/Monte-Carlo-PDF Computations

This paper advances the state-of-the-art in spray computations with some of our recent contributions involving scalar Monte Carlo PDF (Probability Density Function), unstructured grids and parallel computing. It provides a complete overview of the scalar Monte Carlo PDF and Lagrangian spray computer codes developed for application with unstructured grids and parallel computing. Detailed comparisons for the case of a reacting non-swirling spray clearly highlight the important role that chemistry/turbulence interactions play in the modeling of reacting sprays. The results from the PDF and non-PDF methods were found to be markedly different and the PDF solution is closer to the reported experimental data. The PDF computations predict that some of the combustion occurs in a predominantly premixed-flame environment and the rest in a predominantly diffusion-flame environment. However, the non-PDF solution predicts wrongly for the combustion to occur in a vaporization-controlled regime. Near the premixed flame, the Monte Carlo particle temperature distribution shows two distinct peaks: one centered around the flame temperature and the other around the surrounding-gas temperature. Near the diffusion flame, the Monte Carlo particle temperature distribution shows a single peak. In both cases, the computed PDF's shape and strength are found to vary substantially depending upon the proximity to the flame surface. The results bring to the fore some of the deficiencies associated with the use of assumed-shape PDF methods in spray computations. Finally, we end the paper by demonstrating the computational viability of the present solution procedure for its use in 3D combustor calculations by summarizing the results of a 3D test case with periodic boundary conditions. For the 3D case, the parallel performance of all the three solvers (CFD, PDF, and spray) has been found to be good when the computations were performed on a 24-processor SGI Origin work-station.

Raju, M. S.↗
NASA NTRS2000

NCC: A Physics-Based Design and Analysis Tool for Combustion Systems

The National Combustion Code (NCC) is an integrated system of computer codes for physics-based design and analysis of combustion systems. It uses unstructured meshes and runs on parallel computing platforms. The NCC is composed of a set of distinct yet closely related modules. They are: (1) a gaseous flow module solving 3-D Navier-Stokes equations; (2) a turbulence module containing the non-linear k-epsilon models; (3) a chemistry module using either the conventional reduced kinetics approach of solving species equations or the Intrinsic Low Dimensional Manifold (ILDM) kinetics approach of table looking up in conjunction with solving the equations of the progressive variables; (4) a turbulence-chemistry interaction module including the option of solving the joint probability density function (PDF) for species and enthalpy; and (5) a spray module for solving the liquid phase equations. In early 1995, an industry-government team was formed to develop the NCC. In July 1998, the baseline beta version was completed and presented in two NCC sessions at the 34th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, July 1998. An overview of this baseline beta version was presented at the NASA HPCCP/CAS Workshop 98, August 1998. Since then, the effort has been focused on the streamlining, validation, and enhancement of the th baseline beta version. The progress is presented in two NCC sessions at the AIAA 38 Aerospace Sciences Meeting & Exhibit, January 2000. At this NASA HPCCP/CAS Workshop 2000, an overview of the NCC papers presented at the AIAA 38 th Aerospace Sciences Meeting & Exhibit is presented, with emphasis on the reduction of analysis time of simulating the (gaseous) reacting flows in full combustors. In addition, results of NCC simulation of a modern turbofan combustor will also be reported.

Liu, Nan-Suey↗
NASA NTRS2000

Monthly Oceanic Rainfall Derived from Space-Borne Microwave Data

A technique for estimating monthly oceanic rainfall rate using multi-channel microwave measurements has been developed. This technique is applied to the Special Sensor Microwave Imager (SSM/I) data since 1987 to infer monthly rainfall for the Global Precipitation Climatology Project (GPCP). A modified version of this algorithm is now being applied to the TRMM Microwave Imager (TMI) data. There are three prominent features of this algorithm. First, the knowledge of the form of the rainfall intensity probability density function used to augment the measurements. Second, utilizing a linear combination of the 19.35 and 22.235 GHz channels to deemphasize the effect of water vapor. Third, an objective technique has been developed to estimate the rain layer thickness from the 19.35 and 22.23 5 GHz brightness temperature histograms, TMI data with better spatial resolution and 24 hour sampling (vs. sun-synchronized sampling, which is limited to two narrow intervals of local solar time for DMSP satellites) prompt us to study the similarity and difference between these two rainfall estimates. One year of rainfall data (January to December 1998) are used in this study. The mean differences of these two rain estimates are typically less than 15%. Statistical tests are administrated to evaluate the differences between rainfall estimates from SSM/I and TMI data. Differences between TMI and SSM/I rain estimates are discussed in the context of radiative transfer calculations and rainfall estimate algorithm.

Chang, Alfred T. C.↗
NASA NTRS2000

Fusion of AIRSAR and TM Data for Parameter Classification and Estimation in Dense and Hilly Forests

The expanded remotely sensed data space consisting of coincident radar backscatter and optical reflectance data provides for a more complete description of the Earth surface. This is especially useful where many parameters are needed to describe a certain scene, such as in the presence of dense and complex-structured vegetation or where there is considerable underlying topography. The goal of this paper is to use a combination of radar and optical data to develop a methodology for parameter classification for dense and hilly forests, and further, class-specific parameter estimation. The area to be used in this study is the H. J. Andrews Forest in Oregon, one of the Long-Term Ecological Research (LTER) sites in the US. This area consists of various dense old-growth conifer stands, and contains significant topographic relief. The Andrews forest has been the subject of many ecological studies over several decades, resulting in an abundance of ground measurements. Recently, biomass and leaf-area index (LAI) values for approximately 30 reference stands have also become available which span a large range of those parameters. The remote sensing data types to be used are the C-, L-, and P-band polarimetric radar data from the JPL airborne SAR (AIRSAR), the C-band single-polarization data from the JPL topographic SAR (TOPSAR), and the Thematic Mapper (TM) data from Landsat, all acquired in late April 1998. The total number of useful independent data channels from the AIRSAR is 15 (three frequencies, each with three unique polarizations and amplitude and phase of the like-polarized correlation), from the TOPSAR is 2 (amplitude and phase of the interferometric correlation), and from the TM is 6 (the thermal band is not used). The range pixel spacing of the AIRSAR is 3.3m for C- and L-bands and 6.6m for P-band. The TOPSAR pixel spacing is 10m, and the TM pixel size is 30m. To achieve parameter classification, first a number of parameters are defined which are of interest to ecologists for forest process modeling. These parameters include total biomass, leaf biomass, LAI, and tree height. The remote sensing data from radar and TM are used to formulate a multivariate analysis problem given the ground measurements of the parameters. Each class of each parameter is defined by a probability density function (pdf), the spread of which defines the range of that class. High classification accuracy results from situations in which little overlap occurs between pdfs. Classification results provide the basis for the future work of class-specific parameter estimation using radar and optical data. This work was performed in part by the Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, and in part by the NASA Ames Research Center, Moffett Field, CA, both under contract from the National Aeronautics and Space Administration.

Moghaddam, Mahta↗
NASA NTRS2000

A Validation Summary of the NCC Turbulent Reacting/non-reacting Spray Computations

This pper provides a validation summary of the spray computations performed as a part of the NCC (National Combustion Code) development activity. NCC is being developed with the aim of advancing the current prediction tools used in the design of advanced technology combustors based on the multidimensional computational methods. The solution procedure combines the novelty of the application of the scalar Monte Carlo PDF (Probability Density Function) method to the modeling of turbulent spray flames with the ability to perform the computations on unstructured grids with parallel computing. The calculation procedure was applied to predict the flow properties of three different spray cases. One is a nonswirling unconfined reacting spray, the second is a nonswirling unconfined nonreacting spray, and the third is a confined swirl-stabilized spray flame. The comparisons involving both gas-phase and droplet velocities, droplet size distributions, and gas-phase temperatures show reasonable agreement with the available experimental data. The comparisons involve both the results obtained from the use of the Monte Carlo PDF method as well as those obtained from the conventional computational fluid dynamics (CFD) solution. Detailed comparisons in the case of a reacting nonswirling spray clearly highlight the importance of chemistry/turbulence interactions in the modeling of reacting sprays. The results from the PDF and non-PDF methods were found to be markedly different and the PDF solution is closer to the reported experimental data. The PDF computations predict that most of the combustion occurs in a predominantly diffusion-flame environment. However, the non-PDF solution predicts incorrectly that the combustion occurs in a predominantly vaporization-controlled regime. The Monte Carlo temperature distribution shows that the functional form of the PDF for the temperature fluctuations varies substantially from point to point. The results also bring to the fore some of the deficiencies associated with the use of assumed-shape PDF methods in spray computations.

Raju, M. S.↗
NASA NTRS2000

Mapping Closure Approximation to Conditional Dissipation Rate for Turbulent Scalar Mixing

A novel mapping closure approximation (MCA) technique is developed to construct a model for the conditional dissipation rate (CDR) of a scalar in homogeneous turbulence. It is shown that the CDR model from amplitude mapping closure is incorrect in asymptotic behavior for unsymmetric binary mixing. The correct asymptotic behavior can be described by the CDR model formulated by the MCA technique. The MCA approach is outlined for constructing successive approximation to probability density function (PDF) and conditional moment.

He, Gouwei↗
NASA NTRS2001

Satellite Remote Sensing of Tropical Precipitation and Ice Clouds for GCM Verification

This project, supported by the NASA New Investigator Program, has primarily been funding a graduate student, Darren McKague. Since August 1999 Darren has been working part time at Raytheon, while continuing his PhD research. Darren is planning to finish his thesis work in May 2001, thus some of the work described here is ongoing. The proposed research was to use GOES visible and infrared imager data and SSM/I microwave data to obtain joint distributions of cirrus cloud ice mass and precipitation for a study region in the Eastern Tropical Pacific. These joint distributions of cirrus cloud and rainfall were to be compared to those from the CSU general circulation model to evaluate the cloud microphysical amd cumulus parameterizations in the GCM. Existing algorithms were to be used for the retrieval of cloud ice water path from GOES (Minnis) and rainfall from SSM/I (Wilheit). A theoretical study using radiative transfer models and realistic variations in cloud and precipitation profiles was to be used to estimate the retrieval errors. Due to the unavailability of the GOES satellite cloud retrieval algorithm from Dr. Minnis (a co-PI), there was a change in the approach and emphasis of the project. The new approach was to develop a completely new type of remote sensing algorithm - one to directly retrieve joint probability density functions (pdf's) of cloud properties from multi-dimensional histograms of satellite radiances. The usual approach is to retrieve individual pixels of variables (i.e. cloud optical depth), and then aggregate the information. Only statistical information is actually needed, however, and so a more direct method is desirable. We developed forward radiative transfer models for the SSM/I and GOES channels, originally for testing the retrieval algorithms. The visible and near infrared ice scattering information is obtained from geometric ray tracing of fractal ice crystals (Andreas Macke), while the mid-infrared and microwave scattering is computed with Mie scattering. The radiative transfer is performed with the Spherical Harmonic Discrete Ordinate Method (developed by the PI), and infrared molecular absorption is included with the correlated k-distribution method. The SHDOM radiances have been validated by comparison to version 2 of DISORT (the community "standard" discrete-ordinates radiative transfer model), however we use SHDOM since it is computationally more efficient.

Evans, K. Franklin↗
NASA NTRS2000

Comparison of Fatigue Life Estimation Using Equivalent Linearization and Time Domain Simulation Methods

The Monte Carlo simulation method in conjunction with the finite element large deflection modal formulation are used to estimate fatigue life of aircraft panels subjected to stationary Gaussian band-limited white-noise excitations. Ten loading cases varying from 106 dB to 160 dB OASPL with bandwidth 1024 Hz are considered. For each load case, response statistics are obtained from an ensemble of 10 response time histories. The finite element nonlinear modal procedure yields time histories, probability density functions (PDF), power spectral densities and higher statistical moments of the maximum deflection and stress/strain. The method of moments of PSD with Dirlik's approach is employed to estimate the panel fatigue life.

Mei, Chuh↗
NASA NTRS2001

Structure and Variability of Water Vapor in the Upper Troposphere and Lower Stratosphere

Upper-tropospheric humidity (UTH) has been synoptically mapped via an algorithm that rejects small-scale undersampled variance, which is intrinsic to asymptotic measurements of water vapor, cloud, and other convective properties. Mapped distributions of UTH have been used, jointly with high-resolution Global Cloud Imagery (GCI), to study how the upper troposphere is humidified. The time-mean distribution of UTH is spatially correlated to the time-mean distribution of cold cloud fraction (eta)(sub c) (T < than 230 K). Regions of large UTH coincide with regions of large eta(sub c), which mark deep convection. They also coincide with regions of reduced vertical stability, in which the vertical gradient of theta is weakened by convective mixing. Coldest cloud cover is attended convective overshoots above the local tropopause, which is simultaneously coldest and highest. Together, these features reflect the upper-troposphere being ventilated by convection, which mixes in moist air from lower levels. Histograms of UTH and eta(sub c) have been applied to construct the joint probability density function, which quantifies the relationship between these properties. The expected value of UTH in convective regions is strongly correlated to the expected value of eta(sub c). In ensembles of asymptotic samples, the correlation between epsilon[UTH] and epsilon[eta(sub c)] exceeds 0.80. As these expectations reflect the most likely values, the strong correlation between epsilon[UTH] and epsilon[eta(sub c)] indicates that the large-scale organization of UTH is strongly shaped by convective pumping of moisture from lower levels. The same relationship holds for unsteady fields - even though, instantaneously, those fields are comprised almost entirely of small-scale convective structure. The spatial autocorrelation of UTH, constructed at high resolution from overpass data along ascending and descending tracks of the orbit, is limited to only a couple of degrees in the horizontal. This mirrors the spatial autocorrelation of eta(sub c), which likewise operates coherently on short scales. The short correlation scale of UTH, which reflects the scale of individual convective systems, is comparable to the spacing of retrievals from MLS. These scales are undersampled in the asynoptic measurements. Despite their prevalence, the mapping algorithm described above successfully recovers synoptic behavior operating coherently on large scales. It reveals eastward migration of anomalous UTH from the Indian ocean to the central Pacific, in association with the modulation of convection by the Madden-Julian oscillation. Additional information is contained in the original extended abstract.

Salby, Murry L.↗
NASA NTRS2001

Calculating Free Energies Using Average Force

A new, general formula that connects the derivatives of the free energy along the selected, generalized coordinates of the system with the instantaneous force acting on these coordinates is derived. The instantaneous force is defined as the force acting on the coordinate of interest so that when it is subtracted from the equations of motion the acceleration along this coordinate is zero. The formula applies to simulations in which the selected coordinates are either unconstrained or constrained to fixed values. It is shown that in the latter case the formula reduces to the expression previously derived by den Otter and Briels. If simulations are carried out without constraining the coordinates of interest, the formula leads to a new method for calculating the free energy changes along these coordinates. This method is tested in two examples - rotation around the C-C bond of 1,2-dichloroethane immersed in water and transfer of fluoromethane across the water-hexane interface. The calculated free energies are compared with those obtained by two commonly used methods. One of them relies on determining the probability density function of finding the system at different values of the selected coordinate and the other requires calculating the average force at discrete locations along this coordinate in a series of constrained simulations. The free energies calculated by these three methods are in excellent agreement. The relative advantages of each method are discussed.

Darve, Eric↗
NASA NTRS2002

Evaluation of Refractivity Profiles from CHAMP and SAC-C GPS Radio Occultation

The CHAMP and SAC-C missions are the first missions to carry a second-generation 'Blackjack' GPS receiver. One of the new features of this receiver is its ability to sense the lower troposphere closer to the surface than the proof-of-concept GPS/MET 1995 experiment. Since their launch, CHAMP and SAC-C have collected thousands of GPS radio occultations, representing a wealth of measurements available for data assimilation in Numerical Weather Prediction (NWP) models. In order to evaluate the refractivity data derived by JPL from raw radio occultation measurements, we use Data Assimilation Office (DAO) 6-hour forecasts as an independent state of the atmosphere. We compare CHAMP and SAC-C refractivity (processed by JPL) with refractivity calculated from the DAO global fields of temperature, water vapor content and humidity. We will show statistics of the differences as well as Probability Density Functions (PDFs) of the differences. Depending upon availability of AIRS data, we plan to show individual profile comparisons between GPS radio occultation and AIRS retrievals.

Poli, Paul↗
NASA NTRS2002

Evaluation of Refractivity Profiles From Champ and SAC-C GPS Radio Occultation

The CHAMP and SAC-C missions are the first missions to carry a second-generation 'Blackjack' GPS receiver. One of the new features of this receiver is its ability to sense the lower troposphere closer to the surface than the proof-of-concept GPS/MET 1995 experiment. Since their launch, CHAMP and SAC-C have collected thousands of GPS radio occultations, representing a wealth of measurements available for data assimilation in Numerical Weather Prediction (NWP) models. In order to evaluate the refractivity data derived by JPL from raw radio occultation measurements, we use Data Assimilation Office (DAO) shout forecasts as an independent state of the atmosphere. We compare CHAMP and SAC-C refractivity (processed by JPL) with refractivity calculated from the DAO global fields of temperature, water vapor content and humidity. We will show statistics of the differences as well as Probability Density Functions (PDFs) of the differences. Depending upon availability of AIRS data, we plan to show individual profile comparisons between GPS radio occultation and AIRS retrievals.

Poli, Paul↗
NASA NTRS2002

Visualizing 2D Probability Distributions from Satellite Image-Derived Data

Creating maps of biophysical and geophysical variables using Earth Observing System (EOS) satellite image data is an important component of Earth science. These 2D maps have a single value at every location and standard techniques are used to visualize them. Current tools fall short, however, when it is necessary to describe a distribution of values at each location. Distributions may represent a frequency of occurrence over time, frequency of occurrence from multiple runs of an ensemble forecast or possible values from an uncertainty model. 'Distribution data sets' are described, then a case study is presented to visualize such 2D distributions. Distribution data sets are different from multivariate data sets in the sense that the values are for a single variable instead of multiple variables. Our case study data consists of multiple realizations of percent forest cover, generated using a geostatistical technique that combines ground measurements and satellite imagery to model uncertainty about forest cover. We present several approaches for analyzing and visualizing such data sets. The first is a pixel-wise analysis of the probability density functions for the 2D image while the second is an analysis of features identified within the image. Such pixel-wise and feature-wise views will give Earth scientists a more complete understanding of distribution data sets.

Kao, David↗
NASA NTRS2003

Transonic Unsteady Aerodynamics of the F/A-18E at Conditions Promoting Abrupt Wing Stall

A transonic wind tunnel test of an 8% F/A-18E model was conducted in the NASA Langley Research Center (LaRC) 16-Foot Transonic Tunnel (16-Ft TT) to investigate the Abrupt Wing Stall (AWS) characteristics of this aircraft. During this test, both steady and unsteady measurements of balance loads, wing surface pressures, wing root bending moments, and outer wing accelerations were performed. The test was conducted with a wide range of model configurations and test conditions in an attempt to reproduce behavior indicative of the AWS phenomenon experienced on full-scale aircraft during flight tests. This paper focuses on the analysis of the unsteady data acquired during this test. Though the test apparatus was designed to be effectively rigid. model motions due to sting and balance flexibility were observed during the testing, particularly when the model was operating in the AWS flight regime. Correlation between observed aerodynamic frequencies and model structural frequencies are analyzed and presented. Significant shock motion and separated flow is observed as the aircraft pitches through the AWS region. A shock tracking strategy has been formulated to observe this phenomenon. Using this technique, the range of shock motion is readily determined as the aircraft encounters AWS conditions. Spectral analysis of the shock motion shows the frequencies at which the shock oscillates in the AWS region, and probability density function analysis of the shock location shows the propensity of the shock to take on a bi-stable and even tri-stable character in the AWS flight regime.

Schuster, David M.↗
NASA NTRS2002

Simultaneous Laser Raman-rayleigh-lif Measurements and Numerical Modeling Results of a Lifted Turbulent H2/N2 Jet Flame in a Vitiated Coflow

An experiment and numerical investigation is presented of a lifted turbulent H2/N2 jet flame in a coflow of hot, vitiated gases. The vitiated coflow burner emulates the coupling of turbulent mixing and chemical kinetics exemplary of the reacting flow in the recirculation region of advanced combustors. It also simplifies numerical investigation of this coupled problem by removing the complexity of recirculating flow. Scalar measurements are reported for a lifted turbulent jet flame of H2/N2 (Re = 23,600, H/d = 10) in a coflow of hot combustion products from a lean H2/Air flame ((empty set) = 0.25, T = 1,045 K). The combination of Rayleigh scattering, Raman scattering, and laser-induced fluorescence is used to obtain simultaneous measurements of temperature and concentrations of the major species, OH, and NO. The data attest to the success of the experimental design in providing a uniform vitiated coflow throughout the entire test region. Two combustion models (PDF: joint scalar Probability Density Function and EDC: Eddy Dissipation Concept) are used in conjunction with various turbulence models to predict the lift-off height (H(sub PDF)/d = 7,H(sub EDC)/d = 8.5). Kalghatgi's classic phenomenological theory, which is based on scaling arguments, yields a reasonably accurate prediction (H(sub K)/d = 11.4) of the lift-off height for the present flame. The vitiated coflow admits the possibility of auto-ignition of mixed fluid, and the success of the present parabolic implementation of the PDF model in predicting a stable lifted flame is attributable to such ignition. The measurements indicate a thickened turbulent reaction zone at the flame base. Experimental results and numerical investigations support the plausibility of turbulent premixed flame propagation by small scale (on the order of the flame thickness) recirculation and mixing of hot products into reactants and subsequent rapid ignition of the mixture.

Cabra, R.↗
NASA NTRS2003

Modeling and Visualizing Uncertainty in Continuous Variables Predicted using Remotely Sensed Data

The use of remotely sensed images to map continuous biophysical variables, such as those related to terrestrial vegetation amount, sea surface temperature, and many other targets of NASA s Earth Observing System (EOS), includes variable, parametric, positional, spatial support and structural sources of uncertainty. A complete description of uncertainty will lead to a probability distribution at each location, allowing the exploration of the spatial dimension of uncertainty, that is, where the field is not well quantified. To achieve this purpose, convenient visualization tools are required. We have produced such a tool, called PDFVis, that facilitates the display of probability density functions (pdfs) on a per-grid-cell basis. The density estimate from Monte-Carlo generated realizations is interactively displayed as well as parametric and non-parametric summaries of the pdf field (such as mean, median, quartiles, standard deviation, number of modes, and locations of modes). Shaded surface renderings of pdfs along a transect can also be projected onto a plane. This tool will become more useful as richer descriptions of spatial uncertainty become available.

Dungan, Jennifer L.↗
NASA NTRS2003

Simultaneous Ocean Wave Measurements by the Jason and Topex Satellites, With Buoy and Model Comparisons

The verification phase of the Jason-1 satellite altimeter mission presents a unique opportunity for comparing near-simultaneous, independent satellite measurements. We here examine simultaneous significant wave height measurements by the Jason-1 and Topex/Poseidon altimeters. These data are also compared with in-situ measurements from deep-ocean buoys and with predicted wave heights from the Wave Watch 111 operational model. The rms difference between Jason and Topex wave heights is 21 cm, and this can be further lowered by application of median filters to reduce high-frequency noise. This noise is slightly larger in the Jason dataset, amounting to about 7 cm rms for frequencies above 0.05 Hz, which is the frequency at which the coherence between Topex and Jason measurements drops to zero. The probability density function for Jason shows a dearth of small waves relative to Topex. Buoy comparisons confirm that this problem lies with the Jason measurements. The buoy comparisons confirm previous reports that Topex wave heights are roughly 5% smaller than buoy measurements for waves between 2 and 5m; Jason heights in general are 2.7% smaller than Topex. Spurious dips in the Topex density function for 3- and 6-meter waves, a problem that has existed since the beginning of the mission, can be solved by waveform retracking..

Ray, Richard D.↗
NASA NTRS2003

A Self-Contained Mapping Closure Approximation for Scalar Mixing

Scalar turbulence exhibits interplays of coherent structures and random fluctuations over a broad range of spatial and temporal scales. This feature necessitates a probabilistic description of the scalar dynamics, which can be achieved comprehensively by using probability density functions (PDFs). Therefore, the challenge is to obtain the scalar PDFs (Lundgren 1967; Dopazo 1979). Generally, the evolution of a scalar is governed by three dynamical processes: advection, diffusion and reaction. In a PDF approach (Pope 1985), the advection and reaction can be treated exactly but the effect of molecular diffusion has to be modeled. It has been shown (Pope 1985) that the effect of molecular diffusion can be expressed as conditional dissipation rates or conditional diffusions. The currently used models for the conditional dissipation rates and conditional diffusions (Pope 1991) have resisted deduction from the fundamental equations and are unable to yield satisfactory results for the basic test cases of decaying scalars in isotropic turbulence, although they have achieved some success in a variety of individual cases. The recently developed mapping closure approach (Pope 1991; Chen, Chen & Kraichnan 1989; Kraichnan 1990; Klimenko & Pope 2003) provides a deductive method for conditional dissipation rates and conditional di usions, and the models obtained can successfully describe the shape relaxation of the scalar PDF from an initial double delta distribution to a Gaussian one. However, the mapping closure approach is not able to provide the rate at which the scalar evolves. The evolution rate has to be modeled. Therefore, the mapping closure approach is not closed. In this letter, we will address this problem.

He, Guo-Wei↗