Andreas Fiolitakis

Correlations for the laminar flame speed, adiabatic flame temperature and ignition delay time for methane, ethanol and n-decane

This paper provides a detailed database for curve fitted polynomials to deliver information about the laminar flame speed, adiabatic flame temperature and ignition delay time depending on the equivalence ratio, temperature and pressure. The fuels under investigation are methane, ethanol and n-decane. A new aspect of this paper is the envisaged broad range of validity of the correlations, covering the range of 0.6 1.5 for the equivalence ratio, 293.15 K 593.15 K for the temperature, regarding the laminar flame speed and the adiabatic flame temperature and 1400 K 1800 K, regarding the ignition delay time and 0.5 bar 6 bar for the pressure range. As basis of the mathematical expressions, curves with the highest order of four have been chosen to best describe the behavior of the quantities under examination. Evaluation of well established experimental data sets for each respective fuel generates a first rudimentary groundwork for the derivation of the polynomial. The experimental results are further enhanced by obtaining a large amount of additional points through finite rate chemistry simulations with detailed reaction mechanism. By means of a least squares fit, the coefficients for the algebraic expression are inferred. An error analysis is contained subsequent to the calculation, helping the reader to assess the reliability and quality of the fitted curves.

A Novel Progress Variable Approach for Predicting NO in Laminar Hydrogen Flames

Due to the enforcement of strict regulations on NO emissions in technical applications, the inclusion of pollutant prediction into computational fluid dynamics (CFD) is becoming more and more important in everyday engineering. In order to capture all pathways of NO formation accurately the use of detailed chemistry in CFD is mandatory. Solving a transport equation for all species of a detailed reaction mechanism however is too costly for engineering purposes. Therefore an approach is discussed in the present paper which is based on the solution of a transport equation for NO where the chemical source term is retrieved as a function of reaction progress and mixture fraction from look-up tables. These look-up tables are computed by simulating an ensemble of laminar premixed flames at dierent mixture fractions using detailed chemistry. In order to define reaction progress for a given mixture fraction, the length of the H2O-NO-trajectory is used. A transport equation for the reaction progress is then derived. This tabulation concept is validated by computing dierent laminar, counterflow, hydrogen-air diusion flames with a finite volume solver. Comparisons with experimental data (NO and temperature profiles) and with finite rate chemistry simulations prove that the new model is capable to predict NO accurately.

Transported PDF Calculations of a Turbulent, Non-Premixed, Non-Piloted, Hydrogen-Air Flame with Differential Diffusion

Di usive mass and heat uxes cause unclosed terms in transported probability density function (TPDF) methods which need to be modeled accurately, in order to account for the e ects as local extinction or stabilization of turbulent ames. The modeling of ame stabilization is addressed in this work with the example a non-piloted, non-premixed hydrogen-air ame. Experimental data suggest that di erential di usion plays a major role at the root of this ame. To account for this e ect, a di usion model recently introduced for TPDF/LES (Large-Eddy-Simulation) methods is modi ed and applied in RANS (ReynoldsAveraged-Navier Stokes) calculations. Several computations employing this model with varying particle-numbers and grid-resolutions demonstrate the capability of the di usion model to take di erential di usion into account and thus improve ame stabilization.

A New Model Approach for Convective Wall Heat Losses in DQMoM-IEM Simulations for Turbulent Reactive Flows

A new model approach is presented in this work for including convective wall heat losses in the Direct Quadrature Method of Moments (DQMoM) approach, which is used here to solve the transport equation of the one-point, one-time joint thermochemical probability density function (PDF). This is of particular interest in the context of designing industrial combustors, where wall heat losses play a crucial role. In the present work, the novel method is derived for the first time and validated against experimental data for the thermal entrance region of a pipe. The impact of varying model-specific boundary conditions is analysed. It is then used to simulate the turbulent reacting flow of a confined methane jet flame. The simulations are carried out using the DLR in-house Computational Fluid Dynamics (CFD) code THETA. It is found that the DQMoM approach presented here agrees well with the experimental data and ratifies the use of the new convective wall heat losses model. INTRODUCTION Closure of the chemical source term poses one of the greatest challenges in modelling turbulent reactive flows. For this reason, transported probability density function (PDF) methods are an attractive modelling approach for the simulation of turbulent reactive flows, as the chemical source term appears in closed form. The Direct Quadrature Method of Moments (DQMoM) approach for solving the transported PDF equations retains the particular advantage of an already closed chemical source term, while additionally keeping computational costs relatively low compared to the traditional stochastic approaches for solving the PDF transport equation (e.g. Lagrangian Monte-Carlo methods [1] or stochastic field methods [2]). This makes the DQMoM model approach useful most notably for industrial applications, where in particular large domains are required; in addition, larger detailed reaction mechanisms become more accessible due to the lower computational expense offered by DQMoM. One particular disadvantage of the general transported PDF method is that terms involving diffusive fluxes of heat and mass are unclosed and require modelling. A variety of mixing models are available for these unknown terms, which typically provide a model for scalar dissipation. However, very few models exist which allow the inclusion of convective wall heat losses [3–5]. None of these models have been used to the authors’ knowledge in the DQMoM approach for solving the PDF transport equation. The implementation of convective wall heat losses is especially relevant in the context of industrial combustors, where for example engine cooling capabilities can play a major role in combustor design. Wall heat losses can also significantly influence the combustion process itself, and accurate prediction of this phenomenon is crucial in forecasting, for instance, cooling efficiency and its impact on noxious emissions and dynamics. Although current literature provides some insight into the capabilities of the DQMoM model in turbulent reactive flow regimes [6–10], there is little in the way of (a) wall heat losses and (b) quantification of the influence of boundary conditions. In the present work, a method for including convective wall heat c ©2018 by ASME. This manuscript version is made available under the CC-BY 4.0 license http://creativecommons.org/licenses/by/4.0/ 1 losses at isothermal walls in the DQMoM method is therefore derived for the first time; the required inclusion of molecular diffusion in the governing equations is demonstrated. This method is first validated against experimental data of a thermal entrance region of a pipe using data provided by Abbrecht and Churchill [11]. The impact of varying model-specific boundary conditions for this case is also analysed. Next, the model is verified for turbulent reacting flows with the use of a confined methane jet flame against data obtained from experiments carried out by Lammel et al. [12]. The steady Reynolds-Averaged Navier-Stokes (RANS) approach is used for all presented cases, and detailed chemical kinetics are used to model chemistry. The simulations are carried out using the in-house Computational Fluid Dynamics (CFD) code THETA [13, 14] of the German Aerospace Centre (DLR). THEORY DQMoM is an alternative approach to the traditional stochastic methods of solving the transport equations of thermochemical PDFs [15]. In this work, a joint PDF of specific enthalpy and linearly independent species mass fractions is used. For low Mach number flows the assumption of constant thermodynamic pressure is valid, and these variables are sufficient to completely determine the thermodynamic state of the chemically reacting flow. The following section first presents the Favre-PDF equation and the models used to handle unclosed terms, then the DQMoM-IEM equations used to solve the Favre-PDF equation, and finally the convective wall heat losses model for the DQMoM-IEM approach. Favre-PDF Equation The Favre-PDF P̃ is defined as: P̃(ψα ;xi, t) = ρ(ψα)P(ψα ;xi, t) 〈ρ〉(xi, t) (1) where ρ is the density, ψα the state vector corresponding to φα , which is the vector of the thermochemical variables specific enthalpy, h, and species mass fraction, Yα (which are both random variables in a turbulent flow). P is the thermochemical PDF and xi and t are the space and time coordinates respectively. The qualifier (̃·) represents a Favre-averaged quantity and 〈·〉 a Reynoldsaveraged quantity. For the DQMoM apporach, this Favre-PDF is approximated by a finite number of Dirac pulses δ as: P̃(ψα ;xi, t) = Ne ∑ n=1 pn(xi, t) Ns ∏ α=1 δ (ψα −〈φα〉n(xi, t)) (2) where Ne is the number of environments and Ns is the number of scalars in the vector φα . The probability of each environment is denoted as pn and satisfies the condition ∑e n=1 pn = 1. Assuming now that differential diffusion is negligible and Lewis number Le = 1, the unclosed form of the Favre-PDF transport equation reads (using Einstein notation, as with the rest of the paper) [16]: ∂ (〈ρ〉P̃) ∂ t + ∂ (〈ui|φγ = ψγ〉〈ρ〉P̃) ∂xi =− ∂ ∂ψα ( ∂ ∂xi 〈 D ∂φα ∂xi ∣∣∣∣φγ = ψγ〉〈ρ〉P̃) − ∂ ∂ψα ( 〈Sα |φγ = ψγ〉 〈ρ〉 ρ P̃ ) (3) where 〈·|φγ = ψγ〉 denotes a conditional expectation for given ψγ . The velocity is expressed by ui, the molecular diffusion coefficient by D and the chemical source term is Sα . The second term on the left hand side (LHS) of Eq. (3) and the first term on the right hand side (RHS) are unclosed terms, and require closure models. Closure Models The first unclosed term in Eq. (3) is treated using the Gradient Diffusion Model (GDM) [1, 17]: ∂ (〈ui|φγ = ψγ〉〈ρ〉P̃) ∂xi = ∂ ∂xi ( 〈ui|φγ = ψγ〉〈ρ〉P̃ ) = ∂ ∂xi ( 〈ũi|φγ = ψγ〉〈ρ〉P̃ ) + ∂ ∂xi ( 〈u′′ i |φγ = ψγ〉〈ρ〉P̃ ) = ∂ (ũi〈ρ〉P̃) ∂xi − ∂ ∂xi ( 〈ρ〉DT ∂ P̃ ∂xi ) (4) where (·)′′ is a quantity representing fluctuation over Favreaverage. The term DT is the turbulent diffusion coefficient. The second unclosed term in Eq. (3) is algebraically manipulated to result in the following [18]: c ©2018 by ASME. This manuscript version is made available under the CC-BY 4.0 license http://creativecommons.org/licenses/by/4.0/ 2 − ∂ ∂ψα ( ∂ ∂xi 〈 D ∂φα ∂xi ∣∣∣∣φγ = ψγ〉〈ρ〉P̃) = ∂ ∂xi ( 〈ρ〉D ∂ P̃ ∂xi ) + ∂ ∂xi ( DP̃ ∂ 〈ρ〉 ∂xi ) −〈ρ〉 ∂ 2 ∂ψα ∂ψβ (〈 D ∂φα ∂xi ∂φβ ∂xi ∣∣∣∣φγ = ψγ〉 P̃) (5) The second term on the RHS of Eq. (5), spatial density gradients, can be altogether neglected [18]. The third term represents scalar dissipation, and is closed using the Interaction by Exchange with the Mean (IEM) model [19]. Molecular Diffusion The first term on the RHS of Eq. (5) represents the molecular diffusion. This term is essential when considering convective wall heat losses as molecular diffusion plays an important role in transferring heat from the wall to the fluid, and thus needs to be included in the transport equation. It is observed that this term has the same form as the GDM term in Eq. (4). The two can therefore be combined: ∂ ∂xi ( 〈ρ〉DT ∂ P̃ ∂xi ) + ∂ ∂xi ( 〈ρ〉D ∂ P̃ ∂xi ) = ∂ ∂xi ( 〈ρ〉Deff ∂ P̃ ∂xi ) (6) providing an effective diffusivitiy Deff = DT + D. For cases where molecular diffusion does not play a pivotal role, D can simply neglected, establishing a simple on-off function for molecular diffusivity. Thus, re-writing Eq. (3) using the closure models elaborated above and the molecular diffusion term, the final form of the transport equation for the joint thermochemical Favre-PDF of specific enthalpy and species mass fractions P̃ is [16]: ∂ (〈ρ〉P̃) ∂ t + ∂ (ũi〈ρ〉P̃) ∂xi − ∂ ∂xi ( 〈ρ〉DX ∂ P̃ ∂xi ) =− ∂ ∂ψα (( Cφ 2τt (φ̃α −ψα)+ Sα ρ ) 〈ρ〉P̃ ) (7) where DX is either the turbulent diffusion coefficient DT (when molecular diffusion is neglected) or effective diffusion coefficient Deff (when molecular diffusion is accounted for), Cφ = 2.0 is the IEM mixing model constant and τt is the integral turbulent time scale. DQMoM-IEM Approach The DQMoM-IEM model approach is now used to solve Eq. (7). The method essentially involves forcing the definition of the Favre-PDF P̃, i.e. Eq. (1), to agree with values of the known (or calculable) statistical moments such that [15]: 〈φ m1 1 · · ·φ mNs Ns 〉= Ne ∑ n=1 pn Ns ∏ α=1 〈φα〉α n (8) This results in a set of transport equations for pn and 〈φα〉n. For Ne = 2 (generally sufficient1), we obtain a total of 1+2Ns equations: ∂ (〈ρ〉pn) ∂ t + ∂ (ũi〈ρ〉pn) ∂xi − ∂ ∂xi ( 〈ρ〉DX ∂ pn ∂xi ) = an (9) ∂ (〈ρ〉〈sα 〉n) ∂ t + ∂ (ũi〈ρ〉〈sα 〉n) ∂xi − ∂ ∂xi ( 〈ρ〉DX ∂ 〈sα 〉n ∂xi ) = b〈φα 〉n (10) where 〈sα〉n = pn〈φα〉n are the probability-weighted scalars. These are the final transport equations used in the DQMoM-IEM approach. The RHS of Eq. (9), an, is set to zero: this keeps the number of necessary transport equations to (Ne−1)+NsNe, and ensures that the boundedness of the conditional means is not violated [15]. The source term b〈φα 〉n on the RHS of Eq. (10) is [21]: b〈φα 〉n = (−1) n〈ρ〉 p1DX ∑i=1 ( ∂ 〈φα 〉1 ∂xi )2 + p2DX ∑i=1 ( ∂ 〈φα 〉2 ∂xi )2 〈φα 〉2−〈φα 〉1 + 〈ρ〉pn Cφ 2τt (φ̃α −〈φα 〉n)+ pn 〈ρ〉Sα ρ (11) The first term in Eq. (11) ensures correct variance. In handl

Development and Application of a Transported PDF Method on Unstructured Three-Dimensional Grids for the Prediction of Nitric Oxides

The present work explores the capability of the transported PDF (probability density function) method to predict nitric oxide (NO) formation in turbulent combustion. To this end a hybrid finite-volume/Lagrangian Monte-Carlo method is implemented into the THETA code of the German Aerospace Center (DLR). In this hybrid approach the transported PDF method governs the evolution of the thermochemical variables, whereas the flow field evolution is computed with a RANS (Reynolds-Averaged Navier Stokes) method. The method is used to compute a turbulent hydrogen-air flame and a methane-air flame and computational results are compared to experimental data. In order to assess the advantages of the transported PDF method, the flame computations are repeated with the “laminar chemistry” approach as well as with an “assumed PDF” method, which are both computationally cheaper. The present study reveals that the transported PDF method provides the highest accuracy in predicting the overall flame structure and nitric oxide formation.© 2013 ASME

Transported PDF Calculations of NO in a Lifted Hydrogen Flame with Vitiated Coflow

In the present work a lifted hydrogen-air ame is calculated numerically, employing a transported probability density function (PDF) approach. This is realized by a hybrid nite volume/Lagrangian Monte-Carlo method, which is used to compute the turbulent, chemically reacting ow. Spatial pro les of temperature, major and minor species obtained with the numerical method are compared to experimental results. The quality of the nitric oxide (NO) prediction is of particular interest, because its formation is strongly in uenced by the interaction of turbulence and chemistry. The results obtained demonstrate that nitric oxide is predicted with acceptable accuracy in this ame if appropriate reaction mechanisms are used. Moreover, a strong in uence of the turbulence model on the solution is observed.