Flamelet Progress Variable Model Research Articles

Nonpremixed and Premixed FPV Modeling of a Turbulent CH4-H2 Bluff-Body Flame

ABSTRACT In this study, modeling of nonpremixed CH4/H2 bluff-body (HM1) flame was performed using nonpremixed and premixed-based flamelet progress variable (FPV) models. The models were developed in an OpenFOAM flow solver, and the performance of the different variants of the FPV model was evaluated in the prediction of flame structure. The premixed FPV tables were constructed by employing two different interpolation techniques, i.e. linear and cubic spline, to retrieve the reactive scalars outside the flammability range. A generalized progress variable equation was introduced in the flow solver, which can be used for both tabulation approaches. Inside the flammable range, the profiles of the reactive scalars obtained from premixed FPV were found to be similar to those of the nonpremixed FPV. However, there exists disparity in the profiles outside the flammable range, particularly in the fuel-rich side between the nonpremixed and premixed approaches. A modified k-ε model was found to be suitable for modeling the turbulence compared to standard k-ε and k-ω shear stress transport models. The predictions of mean reactive scalars by the premixed FPV model, when combined with the linear interpolation, agree very well with experimental data and with the nonpremixed FPV predictions. However, deviations were observed in the predictions of the mean reactive scalar while using the cubic spline method in premixed FPV. All three approaches exhibit almost similar variations in the predictions of minor species. The results obtained from the present approach depict that the developed solver can be extended for modeling turbulent-chemistry interactions arising in partially premixed turbulent flames.

Study on the application of artificial neural network-based flamelet/progress variable model in supersonic combustion

The flamelet model has the characteristics of high efficiency and physical intuition and has excellent application prospects in supersonic turbulent combustion simulation. Expanding the dimensions of the flamelet model is a potential direction for model development in order to improve its applicability and accuracy, but the accompanying surge in memory is a problem that must be avoided. Therefore, the idea of using the artificial neural network (ANN) model to replace the flamelet database is a feasible exploration currently and has been preliminarily applied in 2D flamelet databases based on central processing unit frameworks. Based on the 3D flamelet database of the flamelet/progress variable (FPV) model, this article studies the strategy of using ANN to replace the flamelet database of the FPV model in a graphics processing unit framework. Due to the significant influence of the progress variable source term and heat release rate on the combustion calculation and the large range of these two parameters, four data processing methods are used to train the parameters separately, and three indicators are used to evaluate the training performance. Subsequently, based on the ANN model using different data processing methods mentioned earlier, calculations are conducted on a hydrogen-fueled supersonic combustion, and the computational accuracy is evaluated. The results indicate that the strategy proposed in this study can screen out artificial neural network replacement models with the same accuracy as the traditional flamelet model.

An improved flamelet/progress variable modeling in a hydrogen-fueled scramjet

The compressible nature of the supersonic combustion flow field results in significant temperature and pressure fluctuations. In the standard Flamelet/Progress Variable (FPV) model, not only is it difficult to specify the tabulation boundary conditions for the flamelet database, but using a single fixed flamelet database to describe the entire complex flow field will likely introduce significant errors. To account for this, an improved FPV model is proposed in this study. Dynamic adjustment of the tabulation pressure in the flamelet database is achieved by scaling the benchmark flamelet database to match the local pressure. Real-time calibration of the tabulation oxidant temperature and fuel temperature is also achieved by real-time statistics and interpolation between four flamelet databases. This allows the improved FPV model to adaptively match the target supersonic combustion flow field without the difficulty of specifying tabulation pressures and temperatures as in the standard FPV model, and it avoids the limitations imposed by the single flamelet database. An additional species transport equation is also incorporated into this model to improve species prediction. Based on the simulations of a hydrogen-fueled scramjet with low-and-high total temperature, the improved model is shown to better restore wall-pressure profiles, combustion mode predictions and OH (hydroxyl radical) distributions, compared to the standard FPV model and comparable studies. This indicates that the effects of compressibility on the chemical reaction rates have been considered.

Flamelet modeling of thermo-diffusively unstable hydrogen-air flames

In order to reduce CO2 emissions, hydrogen combustion has become increasingly relevant for technical applications. In this context, lean H2-air flames show promising features but, among other characteristics, they tend to exhibit thermo-diffusive instabilities. The formation of cellular structures associated with these instabilities leads to an increased flame surface area which further promotes the flame propagation speed, an important reference quantity for design, control, and safe operation of technical combustors. While many studies have addressed the physical phenomena of intrinsic flame instabilities in the past, there is also a demand to predict such flame characteristics with reduced-order models to allow computationally efficient simulations. In this work, a H2-air spherical expanding flame, which exhibits thermo-diffusive instabilities, is studied with flamelet-based modeling approaches both in a-priori and a-posteriori manner. A recently proposed Flamelet/Progress Variable (FPV) model, with a manifold based on unstretched planar flames, and a novel FPV approach, which takes into account a large curvature variation in the tabulated manifold, are compared to detailed chemistry (DC) calculations. Both flamelet approaches account for differential diffusion utilizing a coupling strategy which is based on the transport of major species instead of transporting the control variables of the manifold. First, both FPV approaches are assessed in terms of an a-priori test with the DC reference dataset. Thereafter, the a-posteriori assessment contains two parts: a linear stability analysis of perturbed planar flames and the simulation of the spherical expanding flame. Both FPV models are systematically analyzed considering global and local flame properties in comparison to the DC reference data. It is shown that the new FPV model, incorporating large curvature variations in the manifold, leads to improved predictions for the microstructure of the corrugated flame front and the formation of cellular structures, while global flame properties are reasonably well reproduced by both models.

An extended flamelet/progress variable model for coal/biomass co-firing flame

Coal/biomass co-firing (CBCF) is regarded as one of the sustainable alternatives to reduce emissions from the utilization of fossil fuels. It features complex reacting stages and fuel streams caused by the asynchronous reaction behaviors of coal and biomass particles, which cannot be represented well by the traditional two-mixture-fractions (2Z) coal flamelet/progress variable (FPV) model. To address this issue, we developed an extended FPV model for the CBCF flame in the present study. Firstly, a three-dimensional (3D) point-particle direct numerical simulation (PP-DNS) was conducted to explore the combustion characteristics of the co-firing flame and served as a reference for the model development. Secondly, an extended FPV model was developed by introducing an extra parameter to distinguish the volatiles sources, and the model performance was evaluated by the apriori study as well as comparison with those of the traditional coal-/bio- 2Z-FPV models. The results showed that there were three reacting stages with four fuel streams in the CBCF flame, and their corresponding flame behaviors were obviously different from each other as demonstrated in both the one-dimensional flamelets and 3D PP-DNS solutions. The apriori results showed that the coal-/bio- 2Z-FPV models would give large deviations in the predictions of gas temperature and major species due to the lack of distinguishing the volatiles sources. In contrast, the extended FPV model could well reproduce the flame behaviors (both temperature and species profiles) for different reacting stages with complex fuel streams in the CBCF flame. This validated the extended FPV model and demonstrated its superiority against the traditional 2Z-FPV models.

Flame characterisation of gas-assisted pulverised coal combustion using FPV-LES

A multiphase flamelet/progress variable (FPV) model for the large eddy simulation (LES) of gas-assisted pulverised coal combustion (PCC) is developed. The target of the simulation is the Darmstadt turbulent gas-assisted swirling solid fuel combustion chamber. The coal particles are treated as Lagrangian point particles, the position, momentum and energy of which are tracked. The gas phase is described by the low-Mach Navier-Stokes equations alongside the Eulerian transport equations of the governing variables for the FPV model. The set of chemical states of the PCC flame is pre-tabulated in a six-dimensional flamelet table and determined by the mixing of the primary fuel stream, volatiles and char off-gases with the oxidising air, the progress of chemical reactions, the interphase heat transfer, as well as sub-grid scale variations. A presumed β-PDF approach for the total mixture fraction is applied to capture sub-grid scale effects. The discrete ordinate method (DOM) with the weighted sum of grey gases model (WSGGM) is employed to model radiation. The FPV-LES results are validated against the experimental evidence and a good agreement of the predicted mean and RMS velocities, as well as the mean gas temperature between experiments and simulations is obtained. The contributions of the pilot, volatile and char off-gas fuel streams to the coal flame are analysed. It is found that most regions of the furnace are dominated by either pilot or volatile combustion, while char conversion only occurs in the far downstream and outer furnace regions. The pilot gas dominates the near-wall region inside the quarl, whereas the volatile gas mainly released from small particles dominates a first volatile combustion zone in the interior of the internal recirculation zone. Larger particles heat up more slowly and release their volatile content further downstream, leading to a secondary volatile combustion zone.

Flamelet modeling of forced ignition and flame propagation in hydrogen-air mixtures

Modeling hydrogen-air flames is challenging for several reasons. Due to its high diffusivity and reactivity, the combustion properties of hydrogen are substantially different from conventional hydrocarbon fuels. Besides differential diffusion and flame stretch, which play an important role in hydrogen combustion, the prediction of ignition events is also relevant for the safety and control of combustors operating with hydrogen. To address these effects with a manifold-based method, a novel flamelet progress variable (FPV) model is presented which consists of a tabulated manifold and a coupling strategy. In contrast to prior work, the manifold is coupled to the CFD simulation by transporting major species and enthalpy instead of transporting the flamelet control variables only. A closure approach is developed to employ a mixture-averaged diffusion model accounting for exact diffusion coefficients. Furthermore, a novel tabulation strategy is presented which is based on a recently published composition space model (CSM). The CSM is used for computing and tabulating flamelets which are consistently blended with constant-pressure homogeneous ignition solutions at elevated temperatures. The FPV model is validated for planar and curved hydrogen-air premixed flames across a range of equivalence ratios (0.7≤ϕ≤1.4). The latter are ignited by an energy deposition followed by an outward propagation as cylindrical or spherical expanding flames. It is shown that the FPV model accurately recovers the characteristics of the forced ignition process in both quiescent mixtures and a counterflow configuration, as well as the flame structures and propagation speed of the outwardly propagating hydrogen-air flames.

Modelling extinction/re-ignition processes in fire plumes under oxygen-diluted conditions using flamelet tabulation approaches

The main objective of this article is to investigate the capability of the flamelet progress variable (FPV) model to capture the extinction processes observed in under-ventilated fire scenarios. To this end, large eddy simulation (LES) of the methane line fire plumes in oxygen-reduced environments down to global extinction, investigated experimentally at the University of Maryland (UMD), is performed. Two experimental burner configurations, that differ by the presence (anchored) or not (non-anchored) of an oxygen anchor to stabilise the flame base, are considered leading to two different extinction modes. Both the FPV and the steady laminar flamelet (SLF) model coupled with a presumed filtered density function (FDF) are considered. The Rank Correlated Full Spectrum k-distribution (RCFSK) model is used as a gas radiative property model. In both non-anchored and anchored scenarios, the FPV model reproduces with fidelity the evolution of the fire plume structure, radiative loss, and combustion efficiency with decreasing down to global extinction, without introducing any adjustable constant. The extinction in the non-anchored scenario occurs owing to flame-based detachment coupled to the generation of a buoyancy-driven vortex and is found to be very sensitive to the grid resolution in the near burner region. The present results suggest that these processes can be adequately resolved with a spatial resolution of 2.5 mm in this region. The SLF model, for its part, provides reliable predictions comparable to the FPV as long as no local extinction/re-ignition process occurs.

An improved flamelet/progress variable modeling for supersonic combustion

In order to dynamically describe the transition of the scalar dissipation rate of the mixture fraction variance from near-wall regions to mainstream in near-wall combustion, the traditional flamelet/progress variable (FPV) model, which can be used either within the Reynolds Averaged Navier Stokes (RANS) framework or the large-eddy simulation (LES) framework, is modified to form an improved FPV model within the framework of the improved delayed detached eddy simulation (IDDES). The model blends respective expressions of the scalar dissipation rate within the RANS and LES frameworks into a unified form with an employment of the IDDES concept. It also adopts an analogy between the mixture fraction variance and turbulent kinetic energy. A hydrogen-fueled near-wall combustion is simulated to validate this model. Large-scale turbulent structures, which can either locally enhance combustion or cause local extinction in the reacting zone, are reproduced. Due to the blockage to lateral extents of spanwise turbulent structures, a three-dimensional effect, that the low temperature fluid at lower corners of flow passage is forced to move away from sidewalls, is also reproduced. The results show that the improved FPV model can avoid the scalar over-mixing effect of the traditional FPV model by delaying and mitigating the development of the mixture fraction variance at the near-wall regions. Therefore, the starting point of the high temperature zone predicted by the improved FPV model is downstream of that predicted by the traditional FPV model. It is indicated that the improved FPV model tends to delay combustion. This tendency is also demonstrated by the less combustion efficiency in the upstream region predicted by the improved FPV model. As a result, it can generate a moderate growth of the mixing and reacting zone in the longitudinal direction, which therefore improves the predictions of the species concentrations. In contrast, the traditional FPV model thickens the mixing and reacting zone, which is negative to the combustion efficiency in the downstream region. It is implied that a reasonable development of the mixture fraction variance at near-wall regions is essentially required for near-wall combustion simulation using the FPV model within the IDDES framework.

Large eddy simulation of Cambridge bluff-body coal (CCB2) flames with a flamelet progress variable model

In the present study, we report the first large eddy simulation (LES) study of the Cambridge CCB2 coal flames, one of the target flames in the Workshop on Measurement and Simulation of Coal and Biomass Conversion, with an extended flamelet progress variable (FPV) model. The extended FPV model is based on two mixture fractions considering the volatiles and char off-gases. The normalized total enthalpy is used for the interphase heat transfer modelling. Turbulence-chemistry interaction is treated with an assumed probability density function approach. The results show that the present LES can generally capture the flow field and particle distribution, while there are considerable deviations in the OH prediction due to the boundary treatment of using a mixture of volatiles and carrier gas to replace the methane-containing mixtures in the primary and pilot flow. It indicates that for such gas-assisted coal flames, the pilot fuel stream needs to be rigorously considered in the flamelet tabulation that could be resolved by extending the two-mixture-fraction model into a three-mixture-fraction model. The instantaneous Lagrangian particles histories show that the increasing of coal load has a negligible effect on devolatilization, but delays the char conversion.

Large-eddy simulation of hydrothermal flames using extended flamelet/progress variable approach

The primary purpose of this paper is to develop high-fidelity combustion models for hydrothermal flames. Based on the flamelet/progress variable (FPV) model, comprehensive real-fluid models are incorporated to treat the thermodynamic and transport properties. To account for the heat loss effects, the FPV model is extended to be non-adiabatic with the heat release damping (HRD) approach. Combined with a presumed probability density function closure, the model is then evaluated in laboratory-scale hydrothermal flames in the context of large-eddy simulation. Results show that the prediction from the FPV model is better than those from the EDM and EDC-MTS models, since it correctly predicts the flame lift-off height as well as the maximum temperature position. The wall heat loss has significant influence on the temperature and species distributions. The use of the non-adiabatic FPV approach improve considerably the predictions of the flame temperature by comparing with the experimental data.

Comparison of Finite Rate Chemistry and Flamelet/Progress-Variable Models: Sandia Flames and the Effect of Differential Diffusion

ABSTRACT In this study, large eddy simulations (LES) are conducted using both a finite-rate chemistry (FRC) model and a flamelet/progress-variable (FPV) model for a series of piloted partially premixed methane/air flames of increasing turbulence intensity (Sandia Flames D, E, and F). From Flame D to E to F, as flow velocity and strain rate increase, the flame is either pushed downstream and extended radially or weakened by enhanced local extinction. The two combustion models produce different spatial distributions of both time-averaged quantities and instantaneous flame field. The FPV model provides an overall better prediction of the time-averaged axial and radial profiles of Flame D, but a significantly worse prediction of Flame F, primarily because the FPV model significantly over predicts local extinction. In terms of the conditional statistics, in which the effects of spatial distribution of mixture fraction and subgrid-scale (SGS) modeling are largely “removed,” the FRC model provides better predictions than the FPV model for all quantities at most locations and mixture fractions in all three flames. The effect of differential diffusion on the prediction of a species depends on the molecular diffusivity of that species; the effect is typically smaller than the difference between the FRC and FPV models.

Modeling heavy-duty diesel engines using tabulated kinetics in a wide range of operating conditions

Fast and high-fidelity combustion models including detailed kinetics and turbulence chemistry interaction are necessary to support design and development of heavy-duty diesel engines. In this work, the authors intend to present and validate tabulated flamelet progress variable model based on tabulation of laminar diffusion flamelets for different scalar dissipation rate, whose predictability highly depends on the description of fuel–air mixing process in which engine mesh layout plays an important role. To this end, two grids were compared and assessed: in both grids, cells were aligned on the spray direction with such region being enlarged in the second one, where the near-nozzle and near-wall mesh resolution were also improved, which is expected to better account for both spray dynamics and flame–wall interaction dominating the combustion process in diesel engines. Flame structure, in-cylinder pressure, apparent heat release rate, and emissions for different relevant operating points were compared and analyzed to identify the most suitable mesh. Afterwards, simulations were carried out in a heavy-duty engine considering 20 operating points, allowing to comprehensively verify the validity of tabulated flamelet progress variable model. The results demonstrated that the proposed approach was capable to accurately predict in-cylinder pressure evolution and NO x formation across a wide engine map.

Analysis of flame structure using detailed chemistry and applicability of flamelet/progress variable model in the laminar counter-flow diffusion flames of pulverized coals

Pulverized coal is still found in many practical devices even though it is recognized as “dirty fuel” because of its CO2 and pollutant emissions. To overcome this problem, advanced coal utilization technologies have been developed using numerical simulations. In this study, the structures of the laminar counter-flow diffusion flames of pulverized coals were investigated by performing simulations based on detailed chemistry. The high-temperature region became narrower as the coal/air ratio increased, because of the departure from the stoichiometric mixture and local quenching by the heat transfer between the gas and solid phases. Further, the applicability of the flamelet/progress-variable (FPV) model was investigated through a priori and a posteriori tests. The a priori test confirmed that the FPV model is capable of reproducing the numerical solutions obtained using the detailed chemistry, including the mass fractions of minor species. In the a posteriori test, there was a slight difference between the FPV model and detailed chemistry results due to overestimation of the progress of the chemical reactions. Given the sufficiently high accuracy of the FPV model in various numerical conditions, it can be concluded that the extended FPV model has potential for use in turbulent coal combustion simulations.

Effects of infinitely fast chemistry on combustion behavior of coaxial diffusion flame predicted by large eddy simulation

Large eddy simulations (LES) based on turbulent combustion models aid the design and optimization of combustors. Of the various combustion models available, the eddy break up (EBU) model is widely used because it assumes an infinitely fast chemistry. However, omitting the actual chemical kinetics can cause unexpected behavior, and the characteristics of the combustion models need to be elucidated. Here, the effects of an infinitely fast chemistry on the combustion behavior of a coaxial diffusion flame as predicted by an LES were analyzed. Although the EBU model captured the overall behavior of the chemical species as well as the flow field, the gas temperature and mass fractions of the combustion products in the mixing region of the fuel and oxidizer streams were overestimated. In contrast, the flamelet/progress variable (FPV) model yielded results that were in better agreement with the experimental data, because while the EBU model assumes an infinitely fast chemistry, the look-up tables used in the FPV model are based on the actual chemical kinetics. As these models can be used for the CFD simulations of coal and spray combustion, the results of this study should be useful for efficiently simulating practical combustion systems.

A priori study of an extended flamelet/progress variable model for NO prediction in pulverized coal flames

An extended flamelet/progress variable (FPV) model is developed in this work to predict the NO formation in pulverized coal flames with a newly defined progress variable. The validity of the model is assessed through an a priori analysis, in which the thermo-chemical values predicted with the extended FPV model are compared with the corresponding reference results. It is found that the NO mass fraction in the pulverized coal flames can be well predicted with the extended model, which is sensitive to the progress variable. The prediction accuracy can be further improved by optimizing the definition of progress variable. The effects of strain rate and initial temperature on the performance of the extended FPV model are also investigated. The overall good agreements between the developed model predictions and the detailed chemistry solutions demonstrate that the extended FPV model has a better performance for NO prediction than the conventional one in pulverized coal flames. Compared with the conventional model, the deviations of maximum NO mass fraction predicted with the extended FPV model are decreased by 92.97%, 86.80% and 95.14% for three different strain rate cases as well as 86.80%, 82.81% and 87.45% for three different initial temperature cases, respectively.

Pareto-efficient combustion modeling for improved CO-emission prediction in LES of a piloted turbulent dimethyl ether jet flame

This study extends the Pareto-efficient combustion (PEC) framework to adaptive LES combustion simulations of turbulent flames. With the focus on improving predictions of CO emissions, PEC is employed to augment a flamelet/progress variable (FPV) model through local sub-model assignment of a finite-rate chemistry (FRC) model. A series of LES-PEC calculations are performed on a piloted partially-premixed dimethyl ether flame (DME-D), using a combination of FPV and FRC models. The drift term is utilized in the PEC framework to estimate the model error for quantities of interest. The PEC approach is demonstrated to be capable of significantly improving the prediction of CO emissions compared to a monolithic FPV simulation. The improved accuracy is achieved by enriching the FPV model with FRC in regions where the low-order model is determined insufficient through the evaluation of the drift term. The computational cost is reduced by a factor of two in comparison to the full finite-rate calculation, while maintaining the same level of accuracy for CO predictions.

Evaluation of a flamelet/progress variable approach for pulverized coal combustion in a turbulent mixing layer

A steady flamelet/progress variable (FPV) approach for pulverized coal flames is employed to simulate coal particle burning in a turbulent shear and mixing layer. The configuration consists of a carrier-gas stream of air laden with coal particles that mixes with an oxidizer stream of hot products from lean combustion. Carrier-phase DNS (CP-DNS) are performed, where the turbulent flow field is fully resolved, whereas the coal is represented by Lagrangian point particles. CP-DNS with direct chemistry integration is performed first and provides state-of-the-art validation data for FPV modeling. In a second step the control variables for FPV are extracted from the CP-DNS and used to test if the tabulated manifold can correctly describe the reacting flow (a priorianalysis). Finally a fully coupled a posteriori FPV simulation is performed, where only the FPV control variables are transported, and the chemical state is retrieved from the table and fed back to the flow solver. The a priori results show that the FPV approach is suitable for modeling the complex reacting multiphase flow considered here. The a posteriori data is similarly in good agreement with the reference CP-DNS, although stronger deviations than a priori can be observed. These discrepancies mainly appear in the upper flame (of the present DNS), where premixing and highly unsteady extinction and re-ignition effects play a role, which are difficult to capture by steady non-premixed FPV modeling. However, the present FPV model accurately captures the lower, more stable flame that burns in non-premixed mode.

Analysis of transient blow-out dynamics in a swirl-stabilized combustor using large-eddy simulations

In the present work, the flame structure and dynamics of a swirl-stabilized methane/air flame near and at blow-out conditions are investigated using large-eddy simulations (LES). To this end, simulations at stable conditions and during a transient blow-out sequence are performed employing a flamelet/progress variable (FPV) model and a thickened flame (TFLES) approach with finite-rate chemistry. Good agreement is obtained for velocities, OH mass fraction profiles, and statistics between LES and measurements at conditions near blow-out. Lift-off height comparisons reveal that the TFLES model predicts a significantly less stable flame compared with the FPV model, which is consistent with the stability limits observed for the two models. Subsequently, transient blow-out simulations are performed and examined. This analysis is guided by employing early warning indicators from dynamical system analysis to identify critical transition points and precursors that trigger the onset of blow-out. It is found that the variance of the integrated heat release is a sensible quantity as an early warning signal in detecting blow-out. Detailed comparisons are then performed between the two models to examine the difference in terms of the blow-out mechanism. The comparison reveals that the different phases of the transient blow-out sequence are qualitatively similar for the two models. However, local extinction phenomena are more significant for the TFLES model, reducing the stabilizing feedback by hot combustion products and leading to a total blow-off time, which is three times shorter compared to FPV.

Evaluation of real-fluid flamelet/progress variable model for laminar hydrothermal flames

In this article, hydrothermal flames were studied and an extended flamelet model was developed accounting for the real-fluid effects. The newly formulated real-fluid flamelet/progress variable (FPV) model was evaluated in a laminar counterflow configuration through both a priori and a posteriori analyses. At supercritical conditions, the ideal-gas model results showed a higher flame temperature and intermediate species mass fractions compared to the real-fluid model results in both the detailed chemistry (DC) and FPV approaches. To thoroughly evaluate the real-fluid flamelet library, the a priori tests were carried out under atmospheric and supercritical conditions with different strain rates. The comparisons of tabulated quantities and reference values from DC simulations showed that the real-fluid FPV model can correctly predict the thermal properties and species mass fractions in hydrothermal flames. Finally, the a posteriori analysis showed overall good agreement between the FPV and DC results, verifying the newly proposed real-fluid FPV approach.