Modelling of turbulent scalar flux in turbulent premixed flames based on DNS databases

A transport equation for scalar flux in turbulent premixed flames was modelled on the basis of DNS databases. Fully developed turbulent premixed flames were obtained for three different density ratios of flames with a single-step irreversible reaction, while the turbulent intensity was comparable to the laminar burning velocity. These DNS databases showed that the countergradient diffusion was dominant in the flame region. Analyses of the Favre-averaged transport equation for turbulent scalar flux proved that the pressure related terms and the velocity–reaction rate correlation term played important roles on the countergradient diffusion, while the mean velocity gradient term, the mean progress variable gradient term and dissipation terms suppressed it. Based on these analyses, modelling of the combustion-related terms was discussed. The mean pressure gradient term and the fluctuating pressure term were modelled by scaling, and these models were in good agreement with DNS databases. The dissipation terms and the velocity–reaction rate correlation term were also modelled, and these models mimicked DNS well.     

Using self-similar properties of turbulent premixed flames to downsize chemical tables in high-performance numerical simulations

Detailed chemical mechanisms have to be incorporated in turbulent combustion modelling to predict flame propagation, ignition, extinction or pollutant formation. Unfortunately, hundreds of species and thousands of elementary reactions are involved in hydrocarbon chemical schemes and cannot be handled in practical simulations, because of the related computational costs and the need to model the complexity of their interaction with turbulent motions. Detailed chemistry may be handled using look-up tables, where chemical parameters such as reaction rates and/or species mass fractions are determined from a reduced set of coordinates, progress variables or mixture fractions, as proposed in ILDM, FPI or FGM methods. Nevertheless, these tables may require large computer memory spaces and non-negligible access times. This issue becomes of crucial importance when running on massively parallel computers: to implement these databases in shared memories would induce a large number of data exchanges, reducing the overall code performance; on the other hand duplicating databases in every local processor memory may become impossible either for large databases or small local memories. This work proposes to take advantage of the self-similar behaviour of turbulent premixed flames to reduce the size of these chemical databases, specifically when running on massively parallel machines, under the FPI (Flame Prolongation of ILDM) framework. Several approaches to reduce the database are investigated and discussed both in terms of memory requirements and access times. A very good compromise is obtained for methane–air turbulent premixed flames, where the size of the database is decreased by a factor of 1000, while the access time is reduced by about 60%.     

Large scale effects in weakly turbulent premixed flames

In this study we numerically investigate large scale premixed flames in weakly turbulent flow fields. A large scale flame is classified as such based on a reference hydrodynamic lengthscale being larger than a neutral (cutoff) lengthscale for which the hydrodynamic or Darrieus–Landau (DL) instability is balanced by stabilizing diffusive effects. As a result, DL instability can develop for large scale flames and is inhibited otherwise. Direct numerical simulations of both large scale and small scale three-dimensional, weakly turbulent flames are performed at constant Karlovitz and turbulent Reynolds number, using two paradigmatic configurations, namely a statistically planar flame and a slot Bunsen flame. As expected from linear stability analysis, DL instability induces its characteristic cusp-like corrugation only on large scale flames. We therefore observe significant morphological and topological differences as well as DL-enhanced turbulent flame speeds in large scale flames. Furthermore, we investigate issues related to reaction rate modeling in the context of flame surface density closure. Thicker flame brushes are observed for large scale flames resulting in smaller flame surface densities and overall larger wrinkling factors.     

Large eddy simulation of turbulent premixed flames using level-set G-equation

Level-set G-equation and stationary flamelet chemistry are used in large eddy simulation of a propane/air premixed turbulent flame stabilized by a bluff body. The aim was to study the interaction between the flame front and turbulent eddies, and in particular to examine the effect of sub-grid scale (SGS) eddies on the wrinkling of the flame surface. The results indicated that the two types of turbulence eddies—the resolved large scale eddies and the unresolved SGS eddies—have different effects on the flame. The fluctuation of the flame surface, which is responsible for the broadening of the time averaged mean flame brush by turbulence, depends on the large resolved turbulence eddies. Time averaged mean flow velocity, temperature, and major species concentrations mainly depend on the large scale resolved eddies. The unresolved SGS eddies contribute to the wrinkling at the SGS level and play an important role in the enhancement of the propagation speed of the resolved flame front. In addition, the spatially filtered intermediate species, such as radicals, and the spatially filtered reaction rates strongly depend on the small SGS eddies. The asymptotic behavior of flame wrinkling by the SGS eddies, with respect to the decrease in filter size and grid size, is investigated further using a simplified level-set equation in a model shear flow. It is shown that to minimize the influence of the SGS eddies, fine grid and filter size may have to be used.     

Detailed measurements of local scalar front structures in stagnation-type turbulent premixed flames

Local scalar front structures of OH mole fraction, reaction progress variable, and its three-dimensional gradient have been measured in stagnation-type turbulent premixed flames. The reaction progress variable front is observed to change with increasing turbulence from parallel iso-scalar contours but reduced progress variable gradients, called the lamella-like front, to disrupted non-parallel iso-contours that deviate substantially from those of wrinkled laminar flamelets, called the non-flamelet front. This transition is attributed to the different scales of interaction between the flame internal structure and a spectrum of turbulence extending from the integral scale to the Kolmogorov scale. The lamella-like front pattern occurs when the length scales of interaction are smaller than the laminar flame thickness but the time scales are greater than the flame residence time. The non-flamelet front pattern occurs when the length scales of interaction are greater than the laminar flame thickness but the time scales are smaller than the flame residence time. This difference corresponds to the change of combustion regime from complex-strain flame front to turbulent flame front on a revised regime diagram. A correlation is also proposed for the turbulent flame brush thickness as a function of turbulent Reynolds number and heat release parameter. The heat release parameter is considered to arise from the non-passive effects of flame-surface wrinkling.     

The compositional structure of highly turbulent piloted premixed flames issuing into a hot coflow

Simultaneous line measurements of major species and temperature by the Raman–Rayleigh technique, combined with CO two-photon laser-induced fluorescence and crossed-plane OH planar laser-induced fluorescence have been applied to a series of flames in the Piloted Premixed Jet Burner (PPJB). The PPJB is capable of stabilizing highly turbulent premixed jet flames through the use of a stoichiometric pilot and a large coflow of hot combustion products. Four flames with increasing jet velocities and constant jet equivalence ratios are examined in this paper. The characteristics of these four flames range from stable flame brushes with reaction zones that can be described as thin and “flamelet-like” to flames that have thickened reaction zones and exhibit extinction re-ignition behaviour. Radial profiles of the mean temperature are reported, indicating the mean thermal extent of the pilot and spatial location of the mean flame brush. Measurements of carbon monoxide (CO) and the hydroxyl radical (OH) reveal a gradual decrease in the conditional mean as the jet velocity is increased and the flame approaches extinction. Experimental results for the conditional mean temperature gradient show a progressive trend of reaction zone thickening with increasing jet velocities, indicating the increased interaction of turbulence with the reaction zone at higher turbulence levels. For the compositions examined, the product of CO and OH mole fractions ([CO][OH]) is shown to be a good qualitative indicator for the net rate of production of carbon dioxide. The axial variation of [CO][OH] is shown to correlate well with the mean chemi-luminescence of the flames including the extinction re-ignition regions. The experimental findings reported in this paper further support the hypothesis of an initial ignition region followed by extinction and re-ignition regions for certain PPJB flames.     

Partially premixed flamelet in LES of acetone spray flames

An effective partially premixed flamelet model for large eddy simulation (LES) of turbulent spray combustion is formulated. Different flame regimes are identified with a flame index defined by budget terms in a 2-D multi-phase flamelet formulation, and the application in LES of partially pre-vaporized spray flames shows a favorable agreement with experiments. Simulations demonstrate that, compared to the conventional single-regime flamelets, the present partially premixed flamelet formulation shows its ability in capturing the subgrid regime transitions, yielding a well prediction of peak gas temperature and the downstream flame spreading. A propagating premixed flame front is found coupled with a trailing diffusion burning through the spray evaporation, and the spray effect on regime discrimination is manifested with transport budget analysis. A two-phase regime indicator is then proposed, by which the evaporation-dictated regime is properly described. Its intended use will rely on both gas and spray flamelet structures.     

Computational study of lean premixed turbulent flames using RANSPDF and LESPDF methods

A computational study is performed on a series of four piloted, lean, premixed turbulent jet flames. These flames use the Sydney Piloted Premixed Jet Burner (PPJB), and with jet velocities of 50, 100, 150 and 200 m/s are denoted PM150, PM1100, PM1150 and PM1200, respectively. Calculations are performed using the RANSPDF and LESPDF methodologies, with different treatments of molecular diffusion, with detailed chemistry and flamelet-based chemistry modelling, and using different imposed boundary conditions. The sensitivities of the calculations to these different aspects of the modelling are compared and discussed. Comparisons are made to experimental data and to previously-performed calculations. It is found that, given suitable boundary conditions and treatment of molecular diffusion, excellent agreement between the calculations and experimental measurements of the mean and variance fields can be achieved for PM150 and PM1100. The application of a recently developed implementation of molecular diffusion results in a large improvement in the computed variance fields in the LESPDF calculations. The inclusion of differential diffusion in the LESPDF calculations provides insight on the behaviour in the near-field region of the jet, but its effects are found to be confined to this region and to the species CO, OH and H₂. A major discrepancy observed in many previous calculations of these flames is an overprediction of reaction progress in PM1150 and PM1200, and this discrepancy is also observed in the LESPDF calculations; however, a parametric study of the LESPDF mixing model reveals that, with a sufficiently large mixing frequency, calculations of these two flames are capable of yielding improved reaction progress in good qualitative agreement with the mean and RMS scalar measurements up to an x/D of 30. Lastly, the merits of each computational methodology are discussed in light of their computational costs.     

Stabilization mechanism of turbulent premixed flames in strongly swirled flows

The stabilization of turbulent premixed flames in strongly swirled flows undergoing vortex breakdown is studied in the case of the ALSTOM En-Vironmental (EV) double cone burner using a simple one-dimensional boundary layer type model and computational fluid dynamics, mainly at the level of large-eddy simulation. The analysis shows that, due to flame curvature effects, the flame speed on the combustor axis is 2 D _t/R _F lower than the turbulent burning rate, where D _t is a characteristic turbulent diffusion coefficient and R _F the flame radius of curvature. Flame propagation with negative speed observed in the experiments, i.e. the flame completely embedded in the central recirculation zone on the symmetry axis, is explained with the one-dimensional model as caused by the factor 2 D _t/R _F being larger than the characteristic turbulent burning rate. A peculiar sudden displacement of the flame anchoring location deep into the burner, which takes place experimentally at a critical value of the equivalence ratio, cannot however be explained with the present one-dimensional approach due to the modelling assumptions. The mathematical analysis is supported in this case with large-eddy simulation which can accurately reproduce the flame behaviour across the full operating range. It is finally shown that steady RANS methods cannot cope with the problem due to their inability to correctly predict the velocity flowfield in this burner.     

Strain rates,flow patterns and flame surface densities in hydrodynamically unstable,weakly turbulent premixed flames

Recent numerical and experimental studies have unveiled a potentially marked difference between the laminar as well as turbulent propagation of premixed flames exhibiting Darrieus–Landau (DL) (or hydrodynamic) instabilities from flames for which instabilities are inhibited. In this study we utilize two-dimensional numerical simulations of slot burner flames as well as experimental Propane–Air Bunsen flames to analyse differences in turbulent propagation, strain rate and induced flow patterns of hydrodynamically stable and unstable flames. We also investigate the effects of hydrodynamic instability on quantities which are directly related to reaction rate closure models, such as flame surface density and stretch factor. A clear enhancement of turbulent flame speed can be observed for unstable flames, generally mitigated at higher turbulence intensity, which is attributed to a flame area increase induced by the characteristic cusp-like DL-induced corrugation, absent in stable flames, which occurs concurrently and in synergy with turbulent wrinkling. Unstable flames also exhibit, both numerically and experimentally, a different correlation between strain rate and flame curvature and are observed to give rise to a channeling of the induced flow in the fresh mixture. Conditionally averaged flame surface density is also observed to attain smaller values in unstable flames, as a result of the thicker turbulent flame brush, indicating that closure models should incorporate instability-related parameters in addition to turbulence-related parameters.     

Simulations and experiments on the ignition probability in turbulent premixed bluff-body flames

The ignition characteristics of a premixed bluff-body burner under lean conditions were investigated experimentally and numerically with a physical model focusing on ignition probability. Visualisation of the flame with a 5 kHz OH* chemiluminescence camera confirmed that successful ignitions were those associated with the movement of the kernel upstream, consistent with previous work on non-premixed systems. Performing many separate ignition trials at the same spark position and flow conditions resulted in a quantification of the ignition probability P_ign, which was found to decrease with increasing distance downstream of the bluff body and a decrease in equivalence ratio. Flows corresponding to flames close to the blow-off limit could not be ignited, although such flames were stable if reached from a richer already ignited condition. A detailed comparison with the local Karlovitz number and the mean velocity showed that regions of high P_ign are associated with low Ka and negative bulk velocity (i.e. towards the bluff body), although a direct correlation was not possible. A modelling effort that takes convection and localised flame quenching into account by tracking stochastic virtual flame particles, previously validated for non-premixed and spray ignition, was used to estimate the ignition probability. The applicability of this approach to premixed flows was first evaluated by investigating the model's flame propagation mechanism in a uniform turbulence field, which showed that the model reproduces the bending behaviour of the S_T-versus-u′ curve. Then ignition simulations of the bluff-body burner were carried out. The ignition probability map was computed and it was found that the model reproduces all main trends found in the experimental study.     

Large eddy simulations of partially premixed ethanol dilute spray flames using the flamelet generated manifold model

The paper presents Large Eddy Simulations (LESs) for the Sydney ethanol piloted turbulent dilute spray flames ETF2, ETF6, and ETF7. The Flamelet Generated Manifold (FGM) approach is employed to predict mixing and burning of the evaporating fuel droplets. A methodology to match the experimental inflow spray profiles is presented. The spray statistical time-averaged results show reasonable agreement with mean and RMS data. The Particle Size Distribution (PSD) shows a good match downstream of the nozzle exit and up to x/D = 10. At x/D = 20 and 30 the PSD is under-predicted for droplets with mean diameter D10 > 20μm and over-predicted for the smaller size droplets. The simulations reasonably predict the reported mean flame structure and length. The effect of increasing the carrier velocity (ETF2–ETF7) or decreasing the liquid fuel injection mass flow rate (ETF2–ETF6) is found to result in a leaner, shorter flame and stronger spray–flow interactions. Higher tendency to local extinction is observed for ETF7 which is closer to blow-off compared to ETF2 and has higher scalar dissipation rates, higher range of Stokes number, and faster droplet response. The possible sources of LES-FGM deviations from the measurements are discussed and highlighted. In particular, the spray time-averaged statistical error contribution is quantified and the impact of the inflow uncertainty is studied. Sensitivity analysis to the pre-vaporized nozzle fuel mass fraction show that such small inflow perturbations (by ±?2% for the ETF2 flame) have a strong impact on the flame structure, and the droplets’ dynamics. Conditional scatter plots show that the flame exhibits wide range of mixing conditions and bimodal mixing lines particularly at upstream locations (x/D?相似文献   

A theoretical study of premixed turbulent flame development

Flame development in a statistically stationary and uniform, planar, one-dimensional turbulent flow is theoretically studied. A generalized balance equation for the mean combustion progress variable, which includes turbulent diffusion and pressure-driven transport terms, as well as the mean rate of product creation, is introduced and analyzed by invoking the sole assumption of a self-similar flame structure, well-supported by numerous experiments. The assumption offers the opportunity to simplify the problem by splitting the aforementioned partial differential equation into two ordinary differential equations, which separately model spatial variations of the progress variable and time variations of flame speed and thickness. The self-similar profile of the progress variable, obtained in numerous experiments, is theoretically predicted. Closures of the normalized pressure-driven transport term and mean rate of product creation are obtained. The closed balance equation shows that turbulent diffusion dominates during the initial stage of flame development, followed by the transition to counter-gradient transport in a sufficiently developed flame. A criterion of the transition is derived. The transition is promoted by the heat release and pressure-driven transport. Fully developed mean flame brush thickness and speed are shown to decrease when either density ratio or pressure-driven transport increases. Solutions for the development of the thickness are obtained. The development is accelerated by the pressure-driven transport and heat release.     

The response of turbulent premixed flames to normal acoustic excitation

To model the thermo-acoustic excitation of flames in practical combustion systems, it is necessary to know how a turbulent flame front responds to an incident acoustic wave. This will depend partly on the way in which the burning velocity responds to the wave. In this investigation, the response of CH₄/air and CH₄/H₂/air mixtures has been observed in a novel flame stabilisation configuration, in which the premixture of fuel and air is made to decelerate under controlled conditions in a wide-angle diffuser. Control is provided by an annular wall-jet of air and by turbulence generators at the inlet. Ignition from the outlet of the diffuser allows an approximately flat flame to propagate downwards and stabilise at a height that depends on the turbulent burning velocity. When the flow is excited acoustically, the ensemble-averaged height oscillates. The fluctuations in flow velocity and flame height are monitored by phase-locked particle image velocimetry and OH-planar laser induced fluorescence, respectively. The flame stabilised against a lower incident velocity as the acoustic amplitude increased. In addition, at the lowest frequency of 52 Hz, the fluctuations in turbulent burning velocity (as represented by the displacement speed) were out-of-phase with the acoustic velocity. Thus, the rate of displacement of the flame front relative to the flow slowed as the flow accelerated, and so the flame movement was bigger than it would have been if the burning velocity had not responded to the acoustic fluctuation. With an increase in frequency to 119 Hz, the relative flame movement became even larger, although the phase-difference was reduced, so the effect on burning velocity was less dramatic. The addition of hydrogen to the methane, so as to maintain the laminar burning velocity at a lower equivalence ratio, suppressed the response at low amplitude, but at a higher amplitude, the effect was reversed.     

A DNS study of the physical mechanisms associated with density ratio influence on turbulent burning velocity in premixed flames

Data obtained in 3D direct numerical simulations of statistically planar, 1D weakly turbulent flames characterised by different density ratios σ are analysed to study the influence of thermal expansion on flame surface area and burning rate. Results show that, on the one hand, the pressure gradient induced within a flame brush owing to heat release in flamelets significantly accelerates the unburned gas that deeply intrudes into the combustion products in the form of an unburned mixture finger, thus causing large-scale oscillations of the burning rate and flame brush thickness. Under the conditions of the present simulations, the contribution of this mechanism to the creation of the flame surface area is substantial and is increased by σ, thus implying an increase in the burning rate by σ. On the other hand, the total flame surface areas simulated at σ = 7.53 and 2.5 are approximately equal. The apparent inconsistency between these results implies the existence of another thermal expansion effect that reduces the influence of σ on the flame surface area and burning rate. Investigation of the issue shows that the flow acceleration by the combustion-induced pressure gradient not only creates the flame surface area by pushing the finger tip into the products, but also mitigates wrinkling of the flame surface (the side surface of the finger) by turbulent eddies. The latter effect is attributed to the high-speed (at σ = 7.53) axial flow of the unburned gas, which is induced by the axial pressure gradient within the flame brush (and the finger). This axial flow acceleration reduces the residence time of a turbulent eddy in an unburned zone of the flame brush (e.g. within the finger). Therefore, the capability of the eddy for wrinkling the flamelet surface (e.g. the side finger surface) is weakened owing to a shorter residence time.     

Comparative analysis of methods for heat losses in turbulent premixed flames using physically-derived reduced-order manifolds

Heat losses have the potential to substantially modify turbulent combustion processes, especially the formation of pollutants such as nitrogen oxides. The chemistry governing these species is strongly temperature sensitive, making heat losses critical for an accurate prediction. To account for the effects of heat loss in large eddy simulation (LES) using a precomputed reduced-order manifold approach, thermochemical states must be precomputed not only for adiabatic conditions but also over a range of reduced enthalpy states. However, there are a number of methods for producing reduced enthalpy states, which invoke different implicit assumptions. In this work, a set of a priori and a posteriori LES studies have been performed for turbulent premixed flames considering heat losses within a precomputed reduced-order manifold approach to determine the sensitivity to the method by which reduced enthalpy states are generated. Two general approaches are explored for generating these reduced enthalpy states and are compared in detail to assess any effects on turbulent flame structure and emissions. In the first approach, the enthalpy is reduced at the boundary of the one-dimensional (1D) premixed flame solution, resulting in a single enthalpy deficit for a single premixed flame solution. In the second approach, a variable heat loss source term is introduced into the 1D flame solutions by mimicking a real heat loss to reduce the post-flame enthalpy. The two approaches are compared in methane–air piloted turbulent premixed planar jet flames with different diluents that maintain a constant adiabatic flame temperature but experience different radiation heat losses. Both a priori and a posteriori results, as well as a chemical pathway analysis, indicate that the manner by which the heat loss is accounted for in the manifold is of secondary importance compared to other model uncertainties such as the chemical mechanism, except in situations where heat loss is unphysically fast compared to the flame time scale. A new theoretical framework to explain this insensitivity is also proposed, and its validity is briefly assessed.     

PDF calculations of piloted premixed jet flames

A series of piloted premixed jet flames with strong finite-rate chemistry effects is studied using the joint velocity-turbulence frequency-composition PDF method. The numerical accuracy of the calculations is demonstrated, and the calculations are compared to experimental data. It is found that all calculations show good agreement with the measurements of mean and rms mixture fraction fields, while the reaction progress is overpredicted to varying degrees depending on the jet velocity. In the calculations of the flame with the lowest jet velocity, the species and temperature show reasonable agreement with the measurements, with the exception of a small region near the centerline where products and temperature are overpredicted and fuel and oxidizer are underpredicted. In the calculations of the flame with the highest jet velocity, however, the overprediction of products and temperature and underprediction of fuel and oxidizer is far more severe. An extensive set of sensitivity studies on inlet boundary conditions, turbulence model constants, mixing models and constants, radiation treatment, and chemical mechanisms is conducted to show that any parameter variation offers little improvement from the base case. To shed light on these discrepancies, diagnostic calculations are performed in which the chemical reactions are artificially slowed. These diagnostic calculations serve to validate the experimental data and to quantify the amount by which the base case calculations overpredict reaction progress. Improved calculations of this flame are achieved only through artificially slowing down the chemical reaction by a factor of about 10. The mixing model behavior in this combustion regime is identified as a likely cause for the observed discrepancy in reaction progress.     

Measurements of flame orientation and scalar dissipation in turbulent partially premixed methane flames

Three turbulent flames were studied using a new experimental facility developed at Sandia National Laboratories. Line imaging of Raman and Rayleigh scattering and CO laser-induced fluorescence (LIF) yielded information on all major species, temperature, mixture fraction, and a 1D surrogate measure of scalar dissipation. Simultaneously, crossed planar OH LIF imaging provided information on the instantaneous flame orientation, allowing estimation of the full 3D (flame-normal) scalar dissipation rate. The three flames studied were methane–air piloted jet flames (Sandia flames C, D, and E), which cover a range in Reynolds number from 13,400 to 33,600. The statistics of the instantaneous flame orientation are examined in the different flames, with the purpose of studying the prevailing kinematics of isoscalar contours. The 1D and 3D results for scalar dissipation rate are examined in detail, both in the form of conditional averages and in the form of probability density functions. The effect of overall strain and Reynolds number on flame suppression and eventual extinction is also investigated, by examining the doubly conditional statistics of temperature in the form of S-shaped curves. This latter analysis reveals that double conditioning of temperature on both mixture fraction and scalar dissipation does not collapse the data from these flames onto the same curve at low scalar dissipation rates, as might be expected from simple flamelet concepts.     

Three-dimensional direct simulations and structure of expanding turbulent methane flames

Direct numerical simulations (DNS) are ideally suited to investigate in detail turbulent reacting flows in simple geometries. For an increasing number of applications, detailed models must be employed to describe the chemical processes with sufficient accuracy. Despite the huge cost of such simulations, recent progress has allowed the direct numerical simulation of turbulent premixed flames while employing complete reaction schemes. We briefly describe our own developments in this field and use the resulting DNS code to investigate more extensively the structure of premixed methane flames expanding in a three-dimensional turbulent velocity field, initially homogeneous and isotropic. This situation typifies, for example, the initial flame development after spark ignition in a gas turbine or an internal combustion engine. First investigation steps have been carried out at low turbulence levels on this same configuration in the past Symposium, and we build on top of these former results. Here, a considerably higher Reynolds number is considered, the simulation has been repeated twice in to limit the possibility of spurious, very specific results, and several complementary post-processing steps are carried out. Characteristic features concerning the observed combustion regime are presented. We then investigate in a quantitative manner the evolution of flame surface area, global stretch-rate, flame front curvature, flame thickness, and correlation between thickness and curvature. The possibility of obtaining reliable information on flame front curvature from two-dimensional slices is checked by comparison with the exact procedure.