1Department of Aerospace Engineering and Engineering Mechanics, ... The PDF approach can handle multiple combustion regimes found in spray combustion ...
7th US National Technical Meeting of the Combustion Institute Hosted by the Georgia Institute of Technology, Atlanta, GA March 20-23, 2011.
Probability Density Function Approach for Large Eddy Simulation of Turbulent Spray Combustion C. Heye, H. Koo and V. Raman1 1
Department of Aerospace Engineering and Engineering Mechanics, The University of Texas at Austin, Austin, Texas 78712, USA
Direct numerical simulation (DNS) of a piloted spray flame is used to study probability density function (PDF) based modeling of two-phase combustion. In particular, the effect of droplet Stokes number on flame evolution is analyzed. The PDF approach can handle multiple combustion regimes found in spray combustion, but the small scale mixing has to be appropriately modeled. DNS studies indicate that the mixing time scale that is used in the micromixing model is highly sensitive to the Stokes number of the droplets. A posteriori studies shows that PDF calculations with DNS-determined mixing time scale predict the flame evolution accurately.
1
Introduction
Probability density function (PDF) approach has been widely used in the modeling of single phase flames operating in non-premixed [1, 2], premixed [3, 4], and partially-premixed [5, 6] combustion regimes with minimal modifications. Since spray combustion exhibits multiple burning modes, the PDF approach appears to be an ideal candidate for describing droplet-based combustion. The goal of this work is to understand the modeling issues associated with PDF-based spray combustion using both DNS-based a priori studies and LES-PDF based a posteriori simulations of canonical spray flame configurations. The paper is organized as follows. First, the PDF model for LES-based spray combustion is described. Then, details of the DNS computations are provided. Evaluation of modeling terms in the PDF method as well as LES simulations are discussed in the final sections. 2
LES/PDF modeling of turbulent spray combustion
In the LES approach, all scales larger than a characteristic length called the filter-width (∆) are resolved by the computational grid. The Navier-Stokes equations could then be filtered to yield the large-scale equations of motion [7], which are solved on a computational grid. In spray combustion problems, spray droplet evolution needs to be described. A Lagrangian method that evolves the spray number density using a notional droplet ensemble is typically used [8–10]. Here, notional particles with weights corresponding to the mass of the droplets are injected into the computational domain. These particles carry droplet property values and evolve in physical space using a set of ordinary differential equations [8]. The spray and gas phase equations interact through the drag force on the droplet, as well as mass source due to evaporation [10, 11]. Turbulent combustion essentially takes place at the smallest scales and needs to be exclusively modeled in the context of LES. Here, the PDF approach will be used to represent the combustion
process. The PDF approach directly solves a transport equation for the the Favre-filtered PDF, which is defined in the context of LES as Z +∞ ρ (y, t) ξ [ψ, φ (y, t)] G (y − x) dy, (1) FL (ψ; x, t) = −∞
ξ [ψ, φ (y, t)] = δ [ψ − φ (y, t)] ,
(2)
where δ is an N -dimensional delta function for an N -species system and ψ is the random variable in the composition domain. The transport equation for the PDF is then given by [12–14] ∂FL ∂ g ∂ ∂ ˙ L, + (e uFL ) + u0 |ψFL = − [(M (ψ) + S(ψ) + G(ψ)) FL ] + WF (3) ∂t ∂x ∂x ∂ψ 0 |ψ is the sub-filter velocity fluctuation conditioned on the scalar, M (ψ) = ∇ · ρD∇φ|ψ where ug ˙ is the evaporation is the conditional micromixing term, S is the reaction source term, and W mass source term . The conditional velocity term in Eq. 3 is modeled using the gradient-diffusion hypothesis [15]. The term G accounts for the change in the PDF due to spray evaporation.
˙ G(ψ) = (φ∗ − ψ)W.
(4)
The PDF approach obviates the need to presume the shape of the sub-filter PDF. Additionally, multiple reaction regimes can be handled since the reaction source term appears closed. However, the mixing model will play a crucial role in determining the accuracy of the formulation. In particular, the model must be sensitive to the local combustion regime and adapt the mixing timescales accordingly. In the following discussion, the modeling of the mixing term is analyzed using the DNS data. 3
Direct numerical simulation of spray combustion
Two different DNS configurations are considered: 1) A gas phase piloted planar jet flame with n-heptane fuel issuing in gas phase into an air coflow and 2) a piloted planar jet spray flame with the same fuel issuing in the pre-atomized liquid phase. The configurations are designed in such a way that the mass flow rates of the fuel and oxidizer are identical in both cases. In the case of the spray flame, different Stokes number droplets are introduced. Three different computations corresponding to St = 0.1, 1, and 10 are carried out. For a complete description of the DNS methodology and computational requirements, please refer to Heye, et al [14]. Figure 1 shows instantaneous temperature contours from the four DNS computations. It can be seen that although the spray flames look similar to the gaseous flame in terms of a primary flame region supported by the pilot, there are significant differences depending on the Stokes number of the droplets. To further understand the flame evolution process, a normalized flame index [16] defined below is used: ∇Yf · ∇Yo , (5) ξ= |∇Yf | |∇Yo | where ξ is bounded between -1 and 1. The flame index is defined only in regions of finite chemical reactions. As the value of ξ tends toward −1, the flame can be represented accurately as a diffusion flame, where the fuel and oxidizer are approaching the reaction zone from opposite directions. On
(a) Gaseous flame
(b) Spray St = 0.1
(c) Spray St = 1
(d) Spray St = 10
Figure 1: Instantaneous temperature contours with superimposed particle locations for spray cases
the other hand, positive indicator function values represent premixed flame regions where the fuel and oxidizer are both approaching from the same side of the reaction zone. Figure 2 shows the resulting contours for each case. In the case of the gaseous flame, a majority of the indicator function values tend to be negative, characteristic of a partially premixed flame, with a thin non-premixed interface between the fuel and coflow. In contrast, the droplets present in the two lower Stokes number cases result in a distinctly premixed inner region of the flame due to a time delay between evaporation and reaction that allows the carrier air and fuel to mix. Each has a non-premixed region similar to the gas phase flame surrounding the premixed zone, where the excess vaporized fuel penetrates the initial reaction zone and interacts with the outer coflow air. Similar observations have been made by Luo, et al[17]. The third spray flame containing droplets with Stokes number of 10 has an entirely different dynamic. The most upstream portion of the flame is primarily non-premixed because the vaporized fuel has not yet had time to mix with the surrounding carrier air flow. Due to this reaction zone, part of the fuel has been removed from the flow and results in a complex core region of partially premixed flame. Since the spray flames are
(a) Gaseous flame
(b) Spray St = 0.1
(c) Spray St = 1
(d) Spray St = 10
Figure 2: Instantaneous contours of indicator function in regions of significant chemical source term
dominated by partially-premixed and premixed flame regimes, the use of PDF method seems valid given the past success in modeling such flames using this particular combustion model.
4
Micromixing timescale and model form
In developing the PDF transport equation (Eq. 3), the most important unclosed term is the conditional diffusion term, M (ψ). In general, this micromixing term is closed based on a mixing time scale (related to the scalar dissipation rate) and a shape function [18]. For instance, the commonly used interaction by exchange with the mean (IEM) model is written as follows: M (ψ) =
Cφ e (φ − ψ) τφ
(6)
where Cφ is a model coefficient, τφ is a time scale determined based on the flow variables, and φe is the local mean of the scalar within a filter volume. Below, we use DNS data to determine the evolution of the mixing time scale and the shape function. In LES, the timescale τφ is often specified in terms of a turbulent-diffusivity based model [6, 15, 19]. Using the definition of the mixing timescale, the model coefficient for a non-reacting scalar such as mixture fraction can be defined as Cφ =
∆2 χ fφ , 00 2 D + Dt φf
(7)
where in this case ∆ is the filter to grid width ratio while D and Dt are the scalar and turbulent 00 2 diffusivity, respectively. χ fφ is the filtered scalar dissipation and φf represents the subfilter scalar variance. Consequently, the model coefficient is sensitive to the filter width, the chemical reactions that a scalar undergoes, and the distribution of length scales within the filter volume relative to the turbulence length scales. Figure 3 displays calculated values of the micromixing model coefficient for mixture fraction in each individual flowfield. A filter-width to grid ratio of four is used to obtain this data. In general, the coefficient values increase with an increase in Stokes number. It can be noticed that the coefficient relaxes to the gas phase flame values near the outer flame location, indicating that the spray flames exhibit partially-premixed behavior once the spray droplets have fully evaporated. In addition, while the vast majority of the values in the gas phase flame lie in the commonly used range between 2 and 10, there is a significant portion of the mixing zone that is well beyond these conventional values. This indicates that a constant specified value for the model coefficient is not a valid assumption, especially as larger droplets are considered in spray laden flows. The second part of the mixing model is the shape function. The IEM model (Eq. 6) makes the assumption that the scalars relax towards the mean linearly. This term was investigated using the exact conditional diffusion term evaluated for different filter widths. Figures 4a-c show the conditional diffusion term for the unity droplet Stokes number spray flame. In each plot, the vertical dashed red line indicates the mean subfilter value. In the region close to the jet exit, the large gradients in the mixture fraction value due to strong droplet evaporation leads to a wider range in the conditional diffusion values. As the gradients are slowly destroyed, this term becomes smaller downstream. Most importantly, there is a significant linear region where the IEM model is expected to be valid. However, strong curvature of this term is noticed near the end values for the range of mixture fraction present in the flow. In the case of the temperature scalar values, similar trends should be observed, leaving significant model errors at the higher values in the reaction
(a) Gaseous flame
(b) Spray St = 0.1
(c) Spray St = 1
(d) Spray St = 10
Figure 3: Contours of Cφ for gaseous fuel and various Stokes number spray-based fuel
zone. The impact of the linearity assumption will be assessed next using a posteriori comparison with DNS data.
(a) x/wj = 1
(b) x/wj = 5
(c) x/wj = 10
Figure 4: Conditional diffusion of mixture fraction in the spray flame with St = 1 at various downstream locations plotted for a range of box filter to grid width ratios
5
LES/PDF validation in spray-laden reacting flow
Our final piece of work involves a direct comparison between DNS-resolved scalar fields and the results of both LES transport of filtered scalars and moments of the subfilter scalar PDF in the spray flame with St = 1. The IEM model was used to close the mixing term. The model coefficient was set to 12, based on a series of studies designed to minimize the error in the predictions. A Lagrangian Monte-Carlo method was used to evolve the PDF transport equation [6, 14]. Figure 5 shows the temperature obtained from the DNS, LES, and LES-PDF simulations. It is seen that the LES computation with no subfilter combustion model predicts thick reaction zones. The PDF method, on the other hand, predicts the thin reaction zones supported by the pilot flame as exhibited by the DNS field. Figure 6 shows stream-normal profiles of time-averaged temperature at 10 jet widths downstream of the nozzle exit. A significant improvement is made by using the PDF transport equation instead of the LES filtered scalar transport in this spray flame. Peak temperatures for the LES scalars are
(a) DNS
(b) LES
(c) LES-PDF
Figure 5: Instantaneous temperature contours for spray flame with St = 1
seen to be almost 300 K more than the DNS predictions. As the turbulent fluctuations subside with streamwise distance, the importance of the subfilter model decreases leading to nearly consistent result for LES and LES/PDF calculations at downstream locations. It should also be noted that these results are contingent on the filter size used. For the ratio of four used here, the differences between the methods will not be significant. As the filter width becomes larger, the importance of the subfilter model increases and will potentially change the observed trends. These studies are currently being carried out.
(a) x/wj = 1
(b) x/wj = 10
Figure 6: Stream-normal profiles of time-averaged temperature for the spray flame with St = 1
6
Concluding Remarks
A LES/transported PDF method for turbulent spray combustion was formulated. The PDF formulation evolves the joint-composition PDF using the three-dimensional time-dependent LES flow field. The structure of the PDF transport equation is similar to its single-phase equivalent but includes additional terms that describe the PDF changes due to evaporation, which are accounted for using model source terms. This formulation was shown to be more appropriate than a combustion modeled tailored to a given single phase combustion regime. The commonly used IEM model closure for the additional conditional micromixing term was shown to be accurate for gaseous flames but not adaptive enough to describe the complexities of spray evolution and reaction, especially in
the presence of larger droplets. The LES/PDF method is shown to be signifcantly more accurate in predicting time averaged quantities of interest than LES filtered scalar transport equations. Acknowledgments The authors would like to thank the National Science Foundation for their generous support of this work through NSF Award No. 0747427 and the Texas Advanced Computing Center for providing the necessary computational resources. References [1] S.B. Pope. Combustion Science and Technology, 25 (1981) 159–174. [2] H. M¨obus, P. Gerlinger, and D. Br¨uggemann. Combustion and Flame, 124 (2001) 519–534. [3] M. S. Anand, S. James, and M. K. Razdan. A scalar PDF combustion model for the national combustion code. In 34th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, Cleveland, Ohio, July 1998. [4] D. C. Haworth. PROGRESS IN ENERGY AND COMBUSTION SCIENCE, 36 (2010) 168–259. [5] V. Raman, R. O. Fox, and A. D. Harvey. Combustion and Flame, 136 (2004) 327–350. [6] V. Raman and H. Pitsch. Proceedings Proceedings of the Combustion Institute, 31 (2006) 1711–1719. [7] S. B. Pope. Turbulent Flows. Cambridge University Press, 2000. [8] J. K. Dukowicz. Journal of Computational Physics, 35 (1980) 229–253. [9] N. Okongo and J. Bellan. Journal of Fluid Mechanics, 499 (2004) 1–47. [10] J. Reveillon and L. Vervisch. Journal of Fluid Mechanics, 537 (2005) 317–347. [11] N. Okongo and J. Bellan. Physics of Fluids, 12 (2000) 1573–1591. [12] M. Zhu, K. N. C. Bray, O. Rumberg, and B. Rogg. Combustion and Flame, 122 (2000) 327–338. [13] F. X. Demoulin and R. Borghi. Combustion and Flame, 129 (2000) 281–293. [14] C. Heye, H. Koo, and V. Raman. Analysis of multiple scalar large-eddy simulation/probability density function formulation for turbulent spray combustion. In 49th AIAA Aerospace Sciences Meeting and Exhibit, number AIAA-2011-782, 2011. [15] P. J. Colucci, F. A. Jaberi, and P. Givi. Physics of Fluids, 10 (1998) 499–515. [16] H. Yamashita, M. Shimada, and T. Takeno. Proceedings of the Combustion Institute 26, 1 (1996) 27–34. [17] K. Luo, H. Pitsch, M.G. Pai, and O. Desjardins. Proceedings of the Combustion Institute, In Press, Corrected Proof (2010) journal. [18] R. O. Fox. Computational Models for Turbulent Reacting Flows. Cambridge University Press, 2003. [19] F. A. Jaberi, P. J. Colucci, S. James, P. Givi, and S. B. Pope. Journal of Fluid Mechanics, 401 (1999) 85–121.
