Grid Resolution Effects on VSFMDF/LES - Springer Link

Flow Turbulence Combust (2010) 85:677–688 DOI 10.1007/s10494-010-9272-5

Grid Resolution Effects on VSFMDF/LES Submitted for the Special Issue Dedicated to S. B. Pope Mehdi B. Nik · Server L. Yilmaz · Mohammad Reza H. Sheikhi · Peyman Givi

Received: 22 January 2010 / Accepted: 11 June 2010 / Published online: 9 July 2010 © Springer Science+Business Media B.V. 2010

Abstract The grid dependence of LES/VSFMDF is studied on a series of grids with progressively increased resolution reaching over 10 million grids for simulation of a turbulent piloted nonpremixed methane jet flame (Sandia D). In VSFMDF, the effects of the subgrid scale chemical reaction and convection appear in closed forms. The modeled transport equation for the VSFMDF is solved by a hybrid finite-difference/Monte Carlo scheme. A flamelet model is employed to relate the instantaneous composition to the mixture fraction. The simulated results are assessed via comparison with laboratory data. In addition, the dependence of predicted statistics on the grid size of the simulation is studied. The first order moments converge for the finest grid, but the higher order statistics including the PDFs are more sensitive to the grid resolution.

Keywords Large eddy simulation · Filter density function · Reacting flow

M. B. Nik (B) · P. Givi Department of Mechanical Engineering and Material Science, University of Pittsburgh, Pittsburgh, PA 15261, USA e-mail: [email protected] S. L. Yilmaz Center for Simulation and Modeling, University of Pittsburgh, Pittsburgh, PA 15213, USA M. R. H. Sheikhi Department of Mechanical and Industrial Engineering, Northeastern University, Boston, MA 02115, USA

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Flow Turbulence Combust (2010) 85:677–688

1 Introduction The filtered density function (FDF) [1–3] is now regarded as one of the most effective means of conducting large eddy simulation (LES) in turbulent combustion. In its initial form, the marginal scalar FDF (SFDF) [4]; and its mass weighted scalar filtered mass density function (SFMDF) [5] provided the first demonstration of a “transported” FDF in reacting flows. The primary advantage of SFDF (SFMDF) is that it accounts for the subgrid scale (SGS) chemical reaction in a closed form. This closure is one of the reasons for SFMDF’s popularity and its widespread recent applications [6–27]. Inclusion of the “velocity” in the FDF accounts for the effects of “convection” in a closed form as well. This is demonstrated in the velocity-FDF (VFDF) [28], the joint velocity-scalar FDF (VSFDF) [29] and its density weighted VSFMDF [30]. Currently, this is the most comprehensive form of the FDF for hydrocarbon flame simulations [31]. The objective of the present work is to assess the grid dependence of the VSFMDF for LES of turbulent flames. For that, we consider the piloted non-premixed methane jet flame as studied in the experiments at the Sandia National Laboratories [32, 33] and at the TU-Darmstadt [34]. Flame D is considered in this work; this flame has been the subject of extensive previous LES via SFMDF by several investigators [12–14, 19–24]. With utilization of progressively larger number of grid points, the effects of resolution on various statistics of the flow variables as predicted by VSFMDF are investigated.

2 Simulations Sandia Flame D consists of a main jet with a mixture of 25% methane and 75% air by volume. The nozzle is placed in a coflow of air and the flame is stabilized by a substantial pilot. The Reynolds number for the main jet is Re = 22400 based on the nozzle diameter D = 7.2mm and the bulk jet velocity 49.6m/s. Details of the VSFMDF formulation for simulation of this flame is given in Ref. [31]. In this formulation, the primary transport variables are the density ρ(x, t), the velocity vector ui (x, t) (i = 1, 2, 3), the pressure p(x, t), the enthalpy h(x, t) and the species’ mass fractions Yα (x, t) (α = 1, 2, . . . , Ns ), where x ≡ xi , i = 1, 2, 3, t and Ns denote the space coordinate, the time and the total number of species involved in the chemical reaction. Large +∞ eddy simulation involves the spatial filtering operation [35, 36] Q(x, t) = −∞ Q(x , t)G(x , x)dx , where G(x , x) denotes a filter function, and Q(x, t) is the filtered value of the transport variable Q(x, t). In variable-density flows it is convenient to use the Favre-filtered quantity Q(x, t) L = ρ Q / ρ . We consider a filter function that is spatially and tempoinvariant and localized, thus: G(x , x) ≡ G(x − x) with the properties G(x) ≥ 0, rally +∞ −∞ G(x)dx = 1. For closure of the VSFMDF transport equation, we consider the general diffusion process, [37] given by the system of stochastic differential equations (SDEs). In this context developed in Refs. [4, 28, 29, 38, 39] we utilize the simplified Langevin model (SLM) and the linear mean square estimation (LMSE) model [40].


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dX+ i

=

U i+ dt

dU i+

=

+

2γ dWi , ρ

(1a)

∂ uj L ∂ uj L 2 1 ∂ ∂ ui L 1 ∂ 2 ∂ γ − γ dt γ + ρ ∂ xj ρ ∂ xj ∂ xj ∂ xi 3 ρ ∂ xi ∂ xj

1 ∂ p + Gij U +j − u j L dt + C0 dWi + − ρ ∂ xi

dφα+ = −Cφ ω φα+ − φα L dt + Sα (φ + ) dt,

2γ ∂ ui L dWj, ρ ∂ xj

(1b) (1c)

where

1 3 Gij = −ω + C0 δij 2 4 k3/2 = C L

ω=

k

(2)

1 k = τ L (ui , ui ) . 2

HereXi+ , U i+ , φα+ are probabilistic representations of position, velocity vector, and scalar variables, respectively. W terms denote the Wiener-Lévy processes [41, 42], γ denotes the molecular diffusion coefficients, and the chemical reaction source terms Sα (φ(x, t)) are functions of compositional scalars (φ ≡ φ1 , φ2 , . . . , φ Ns +1 ). In Eq. 2, ω is the SGS mixing frequency, is the dissipation rate, k is the SGS kinetic energy, and L is the LES filter size. The SGS correlations are dented by τ L (a, b) = ab L − a L b L , and the model parameters are the same as those suggested by Sheikhi et al. [30]: C0 = 2.1, Cφ = 1.0 and C = 1.0. No attempt is made to optimize the values of these parameters. Numerical solution of the modeled VSFMDF transport equation is obtained by a hybrid finite-difference (FD)/Monte Carlo (MC) procedure. The computational domain is discretized on equally spaced finite-difference grid points and the FMDF is represented by an ensemble of statistically identical MC particles which carry information pertaining to the velocity and the scalar values. Statistical information is obtained by considering an ensemble of N E computational particles residing within an ensemble domain of volume V E centered around each of the FD grid points. To reduce the computational cost, a procedure involving the use of non-uniform weights is also considered. This procedure allows a smaller number of particles in regions where a low degree of variability is expected. Conversely, in regions of high variability, a large number of particles is allowed. The sum of weights within the ensemble domain is related to filtered fluid density [43]. The FD solver is fourth-order accurate in space and second-order accurate in time [44]. To avoids the commutation error in the filtered and VSFMDF transport equations, all of the FD operations are conducted on fixed and equally spaced grid points. The transfer of information from the FD points to the MC particles is accomplished via a linear interpolation. The inverse transfer is accomplished via

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Table 1 Grid Parameters and the corresponding filter size

Grid

Resolution (x,y,z)

Cells ×106

G1 G2 G3 G4

101 × 81 × 81 150 × 123 × 123 200 × 161 × 161 270 × 215 × 215

0.662 2.260 5.180 12.48

ensemble averaging. The FD transport equations include unclosed second order moments which are obtained from the MC. Simulations are conducted on a three-dimensional Cartesian mesh with uniform spacings in each of the three directions. With available resources, simulations are afforded within a domain spanning 20D × 16D × 16D in streamwise (x), and the two lateral (y, z) directions, respectively. The number of grid points for different test cases in the x, y and z directions are presented in Table 1. The filter size is set equal to L = 2(xyz)(1/3) where x, y and z denote the grid spacings in the corresponding directions. The MC particles are supplied in the inlet region and are free to move within the domain due to combined actions of convection and diffusion. The volume of the ensemble domain is V E = (xyz) within which N E ≈ 40. The MC simulation results are monitored to ensure the particles fully encompass and extend well beyond regions of non-zero vorticity and reaction. For efficient parallel simulations, the domain is partitioned into equally sized partitions and each partition is assigned to a different processor at the onset of the simulations. This provides an effective parallelization and is relatively easy to implement. Each CPU is assigned relatively equal number of grid points and (approximately) equal number of particles. Simulations are conducted in conjunction with MPI and the PETSC [45–47] library. The flow variables at the inflow are set the same as those in the experiments, including the inlet profiles of the velocity and the mixture fraction. The inlet condition for the velocity is presented in Fig. 1. The flow is excited by superimposing oscillating axisymmetric perturbations at the inflow. The procedure is similar to

(a)

(b)

Fig. 1 Radial distribution of the mean and RMS values of the filtered axial velocity. Ucl denotes the mean axial velocity at the centerline at the inlet, the symbols denote the experimental data. The line denotes the mean values and the thick solid line denotes the RMS values. a Mean axial velocity at x/D = 0.138, b RMS of the axial velocity at x/D = 0.138


(a)

681

(b)

Fig. 2 Radial distribution of the mean and RMS values of the filtered axial velocity. Ucl denotes the mean axial velocity at the centerline at the inlet, the symbols denote the experimental data. The line denotes the mean values and the thick solid line denotes the RMS values. a Mean axial velocity at x/D = 15, b RMS of the axial velocity at x/D = 15

that in Ref. [48], but the amplitude of forcing is set in such a way to match the experimentally measured turbulent intensity of the streamwise velocity at the inlet. Standard characteristic boundary conditions [49] are implemented in all of the FD simulations. The methane-air reaction mechanism is taken into account via the “f lamelet” model. This model considers a laminar, one-dimensional counterflow (opposed jet) flame configuration [50]. The detailed kinetics mechanism of the Gas Research Institute (GRI2.11) [51] is employed to describe combustion. The flamelet table at strain rate of a = 100 1/s is used to relate the thermo-chemical variables to the mixture fraction. The predictive capability of VSFMDF is demonstrated by comparing the flow statistics with the experimental data [32–34]. These statistics are obtained by long-time averaging of the filtered field during 6 flow through times. The notations

(a)

(b)

Fig. 3 Radial distribution of the mean and RMS values of the filtered mixture fraction. The symbols denote the experimental data. The line denotes the mean values and the thick solid line denotes the RMS values. a Mean mixture fraction at x/D = 15, b RMS of the mixture fraction at x/D = 15

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(a)

(b)

Fig. 4 Radial distribution of the mean and RMS values of the filtered temperature values. The symbols denote the experimental data. The line denotes the mean values and the thick solid line denote the RMS values. a Mean temperature (K) at x/D = 15, b RMS of the temperature (K) at x/D = 15

Q and RMS(Q) denote the time-averaged mean and root mean square values of the variable Q, respectively.

3 Results To assess the grid dependency of the VSFMDF simulations, we consider four grid configurations (G1 through G4) with a progressively increasing resolution from about 0.6 to 12.2 million grid points (Table 1). To compare with the experimental data, we consider the Reynolds-averaged moments of the thermo-chemical variables. In the presentation below, the overline denotes the Reynolds-averaged operator.

(a)

(b)

Fig. 5 Radial distribution of the mean and RMS values of filtered CH4 mass fractions. The symbols denote the experimental data. The line denotes the mean values and the thick solid line denotes the RMS values. a Mean CH4 mass fraction at x/D = 15, b RMS of CH4 mass fraction at x/D = 15


(a)

683

(b)

Fig. 6 Radial distribution of the mean and RMS values of the filtered O2 mass fractions. The symbols denote the experimental data. The line denotes the mean values and the thick solid line denotes the RMS values. a Mean O2 mass fraction at x/D = 15, b RMS of O2 mass fraction at x/D = 15

The root mean square (RMS) includes the contributions from both the resolved and the residual fields. The capability of the method in predicting the hydrodynamics field is demon strated by examining some of the (reported) radial (r = z2 + y2 ) distributions of the flow statistics in Fig. 2. The results at x/D = 15 are shown in this and all of the subsequent figures as the profiles at other streamwise location portray similar behaviors. The VSFMDF predicts the peak value of mean axial velocity profile and the spread of the jet reasonably well. The RMS values, however, is underpredicted. The radial distribution of the mixture fraction is also shown to compare well with data (Fig. 3). The mean fields of velocity and mixture fraction both converge to the results corresponding to the G4 grid. The convergence for the RMS field is slower, but the results for G4 compare best with experimental data.

(a)

(b)

Fig. 7 Radial distribution of the mean and RMS values of the filtered CO mass fractions. The symbols denote experimental data. The line denotes the mean values and the thick solid line denotes the RMS values. a Mean CO mass fraction at x/D = 15, b RMS of CO mass fraction at x/D = 15

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(a)

(b)

Fig. 8 Radial distribution of the mean and RMS values of the filtered CO2 mass fractions. The symbols denote the experimental data. The line denotes the mean values and the thick solid line denotes the RMS values. a Mean CO2 mass fraction at x/D = 15, b RMS of CO2 mass fraction at x/D = 15

The radial distribution of the mean temperature and its corresponding RMS values are presented in Fig. 4. Similar to hydrodynamic quantities, the mean profiles show a relatively good agreement with measured data while the RMS values are underpredicted at the outerlayer. The statistics of the mass fractions (denoted by Y) of several of the species at different streamwise locations are compared with data in Figs. 5, 6, 7, 8 and 9. The mean profiles of the species show a close agreements with measurements. The RMS values show close agreements with measured data at the inner layer, but not as good at the outer layer. These disagreements can be attributed, in part, to the shortcoming of the flamelet model in relating the thermo-chemical variables to the mixture fraction. The mean fields of temperature and species mass fractions both converge to the results corresponding to the G4 grid.

(a)

(b)

Fig. 9 Radial distribution of the mean and RMS values of the filtered H2 O mass fractions. The symbols denote the experimental data. The line denotes the mean values and the thick solid line denotes the RMS values. a Mean H2 O mass fraction at x/D = 15, b RMS values of H2 O mass fraction at x/D = 15


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In Fig. 10, the PDFs of the resolved mixture fraction as predicted by the VSFMDF are compared with those measured experimentally at several locations throughout the domain. In general, both the peak and the spreads of the PDFs are predicted

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 10 PDF of the resolved filtered mixture fraction at x/D = 15 and different radial locations. The symbols and the thick lines denote the experimental data and LES predictions, respectively. a r = 2mm, b r = 4mm, c r = 6mm, d r = 8mm, e r = 10mm, f r = 12mm

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well. The PDF’s converge to the results corresponding to the G4 grid; in which the best agreement with experimental data is observed.

4 Summary and Conclusions Since its original development, the SFMDF [4, 5] has experiences widespread applications for LES of a variety of reacting flows [6–8, 10, 11, 13–26, 29]. The extended VSFMDF methodology [30, 31] is significantly more powerful as it also accounts for the effects of SGS convection in an exact manner. In this work, we asses the grid dependence of VSFMDF in prediction of turbulent flyames. For that, we consider the piloted, nonpremixed, turbulent, methane jet flame (Sandia D). For this flame, the thermo-chemical variables are related to the mixture fraction via a flamelet table. A modeled transport equation for the mass weighted joint FDF of the velocity and the mixture fraction is considered [30]. This equation is solved by a hybrid finite-difference/Monte Carlo method. The predictive capability of the overall scheme is assessed by comparison of the ensemble (long time Reynolds-averaged) values of the thermo-chemical variables. The grid dependency of the predicted results is assessed via consideration of progressively larger number of grid points. In the case with the finest resolution, there are over 10 million grid points. The results of this grid resolution assessment indicate that the first order moments converges quickly; but the rate of convergence for the second order moments and the PDFs of the resolved field is slower. All of the mean quantities are, generally, in good comparison with experimental data. For the RMS values, the predicted values agree closely with the experimental data in the inner layer, but not as good in the outer layer. This discrepancy is attributed, in part, to the use of the flamelet model in relating thermo-chemical variables to the mixture fraction. Acknowledgements This work is sponsored by NASA under Grant NNX08AB36A. The computations support is provided by the National Science Foundation through TeraGrid resources at the Pittsburgh Supercomputing Center.

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