## 1. Introduction

Optimum use of energy is a major concern in designing modern energy conversion systems. According to the second law of thermodynamics, the energy efficiency in practice is always less than that expected theoretically, because of the irreversibilities in the system. Irreversibility essentially causes degradation of available energy into internal energy and, thus, destruction of the exergy (availability) of the working fluid [

1]. This leads to a departure from thermodynamic ideality or reduction of the second-law efficiency. The rate of exergy destruction due to irreversibilities can be characterized in terms of entropy generation according to the Gouy–Stodola theorem,

I_{D} =

T_{a} S _{g} [

2,

3], where

I_{D},

T_{a} and

S _{g} denote the rate of exergy destruction (also known as lost power), ambient (dead state) temperature and the entropy generation rate, respectively. Optimizing the energy efficiency thus relies on minimizing the overall exergy destruction, which can be achieved by minimizing the rate of entropy generated within the system [

4–

9].

During the past several decades, the second-law analysis has been the subject of broad investigations. These include system-level analysis, often termed exergy analysis, to obtain the net rate of exergy destruction [

10–

23]. More detailed studies involve identification of specific processes contributing to losses by considering the local generation of entropy. Such analysis has been performed on laminar flows in many studies. Teng

et al. [

24] derived the entropy transport equation to determine the rate of local entropy generation in multicomponent laminar reacting flows. Datta and Som [

18] considered energy and exergy balance in a gas turbine combustor. Datta [

25] conducted entropy generation analysis of a laminar diffusion flame. Nishida

et al. [

26] considered premixed and diffusion flames and identified important entropy generation and exergy loss mechanisms. Datta [

27] studied the effect of gravity on the structure and generation of entropy in confined laminar diffusion flames. Shuja

et al. [

28] studied the influence of inlet velocity profile on the efficiency of heat transfer in a laminar jet. Briones

et al. [

29] studied the entropy generation processes in a partially-premixed flame. Sciacovelli and Verda [

30] used an entropy generation minimization technique for design modifications in a tubular solid oxide fuel cell. Jiang

et al. [

31] presented an analysis of entropy generation in a hydrogen/air premixedmicro-combustor with baffles, and Rana

et al. [

32] studied the exergy transfer and destruction due to premixed combustion in a heat recirculating micro-combustor.

In turbulent flows, there have been several studies involving direct numerical simulation (DNS). Okong’o and Bellan [

33–

35] performed comprehensive studies on entropy generation effects in supercritical, multicomponent shear flows; they suggested that, by containing the full extent of dissipative effects, entropy generation is useful to describe the behavior of small-scale turbulent motions. McEligot

et al. [

36] studied the entropy generation in the near wall region of a turbulent channel flow. Farran and Chakraborty [

37] conducted DNS prediction of entropy generation in a turbulent premixed flame. Ghasemi

et al. [

38,

39] used DNS and Reynolds averaged Navier–Stokes (RANS) to study the entropy generation and energy dissipation in transitional regions in wall shear flows. The prediction of entropy generation in the context of RANS has been carried out in many other studies. Stanciu

et al. [

40] performed the second-law analysis of a turbulent diffusion flame. Shuja

et al. [

41] studied local entropy generation in an impinging jet and used minimumentropy concept to evaluate various turbulence models. Adeyinka and Naterer [

42] provided a model for the entropy transport equation in turbulent flows. Kock and Herwig [

43] provided wall functions for entropy production and performed analysis of entropy generation due to fluid flow and heat transfer in the near wall region of a pipe. Yapıcı

et al. [

44] performed local entropy generation in a methane-air burner. Herwig and Kock [

45] used entropy generation as a tool for evaluating heat transfer performance in a turbulent shear flow. Stanciu

et al. [

46] studied the influence of swirl angle on the irreversibility in a turbulent diffusion flame, and Emadi and Emami [

47] studied entropy generation in a turbulent hydrogen-enriched methane/air bluff-body flame. Despite the known advantages of large eddy simulation (LES) in turbulence modeling, the extent of its usage for entropy generation analysis has been insignificant. For the most part, this is due to the challenges in subgrid-scale (SGS) modeling of the unclosed irreversibility effects. An effective strategy for modeling of SGS effects is the filtered density function (FDF) methodology [

48,

49]. This methodology has been the subject of extensive previous contributions [

50–

62]. In recent works [

63,

64], a methodology based on filtered density function (FDF), termed the entropy FDF (En-FDF), has been introduced, which allows LES prediction of entropy transport and generation in turbulent reacting flows. This methodology has been presented as two formulations: the comprehensive and the marginal En-FDF. The comprehensive En-FDF contains the complete statistical information about the velocity, scalar, turbulent frequency and entropy fields and, thus, provides SGS closure for all of the unclosed moments in the filtered transport equations. The marginal En-FDF is the FDF of entropy and scalar fields and describes the unclosed entropy generation, chemical reaction and entropy-scalar statistics. This methodology is computationally more affordable and, thus, constitutes a more practical means of predicting entropy generation in complex turbulent reacting flows. However, it requires closure for all of the second order SGS moments via the conventional (non-FDF) LES models. The objective of this paper is to provide an overview of the state of progress in the application of En-FDF for LES prediction of entropy generation. Both En-FDF formulations are discussed along with their recent applications in LES of turbulent shear flows. Assessments of LES results against direct numerical simulation (DNS) and experimental data are also presented.

## 2. LES Formulation and Modeling

The primary transport variables in turbulent reacting flows, varying in space

**x** ≡

x_{i} (

i = 1, 2, 3) and time

t, are the fluid density

ρ(

**x**,

t), the velocity vector

**u** ≡

u_{i}(

**x**,

t) along the

x_{i} direction, the specific enthalpy

h(

**x**,

t), the specific entropy

s(

**x**,

t), the pressure

p(

**x**,

t) and the mass fractions of

N_{s} species,

Y_{α}(

**x**,

t) (

α = 1, 2, …,

N_{s}), respectively. Implementation in LES involves the use of spatial filtering operation [

65,

66]:

$\langle Q(\mathbf{x},t)\rangle ={\int}_{-\infty}^{+\infty}Q({\mathbf{x}}^{\prime},t)\hspace{0.17em}G({\mathbf{x}}^{\prime},\mathbf{x})\hspace{0.17em}d{\mathbf{x}}^{\prime}$ where

G denotes the filter function of width Δ and 〈

Q(

**x**,

t)〉 represents the filtered value of the transport variable

Q(

**x**,

t). In reacting flows, it is convenient to consider the Favre filtered quantity, 〈

Q(

**x**,

t)〉

_{L} =〈

ρQ〉

/〈

ρ〉. The transport variables satisfy the conservation equations of mass, momentum, energy and species mass fractions, as well as entropy transport equation. The filtered form of these equations are:

where

γ denotes the thermal and mass molecular diffusivity coefficients for all of the scalars. We assume unity Lewis number. In these equations,

R_{α},

X_{α} and

S _{α} are gas constant, mole fraction and chemical reaction source term for species

α, respectively, and

μ_{α} is the chemical potential per unit mass of species

α. Variables

T and

c_{p} denote the temperature and the specific heat capacity at constant pressure for the mixture, respectively. We use the scalar array

**φ** = [

φ_{1}, …,

φ_{Ns+1}] to represent mass fraction and enthalpy in a common form with

φ_{α} ≡

Y_{α} for

α = 1, …,

N_{s} and

φ_{Ns+1} ≡

h. We consider a Newtonian fluid and employ Fourier’s law of heat conduction and Fick’s law of mass diffusion. The viscous stress tensor

τ_{i j} is thus represented as:

${\tau}_{ij}=\mu \left({\scriptstyle \frac{\partial {u}_{i}}{\partial {x}_{j}}}+{\scriptstyle \frac{\partial {u}_{j}}{\partial {x}_{i}}}-{\scriptstyle \frac{2}{3}}{\scriptstyle \frac{\partial {u}_{k}}{\partial {x}_{k}}}{\delta}_{ij}\right)$ where

μ is the fluid dynamic viscosity. In

Equations (2)–

(4), the second order SGS moments

τ(

a,

b) = 〈

ab〉

_{L} − 〈

a〉

_{L} 〈

b〉

_{L} appear as unclosed. In addition, the filtered chemical reaction source term, the last term on the RHS of

Equation (3), and the filtered entropy generation terms, the last four terms on the RHS of

Equation (4), require SGS modeling. The modeled filtered entropy generation terms must be positive semidefinite according to the second law of thermodynamics. It is important to emphasize that filtered entropy and entropy generation cannot be obtained from other filtered variables, because of their nonlinear dependency. Moreover, it is clear that having entropy alone is not sufficient to account for the individual processes contributing to its generation.

The En-FDF provides an effective means of modeling the unclosed SGS effects. In its comprehensive form, the En-FDF, denoted by

$\mathcal{F}$_{en} (

**û**,

**φ̂**,

ω̂,

ŝ,

**x**;

t), contains complete statistical information about SGS variation of velocity, scalar, turbulent frequency

ω(

**x**,

t) and entropy fields. The En-FDF is defined as [

63]:

where:

is the fine-grained density [

67] and

δ denotes the Dirac delta function. In this formulation, the sample space variables

**û**,

**φ̂**,

ω̂ and

ŝ correspond to velocity vector, scalar array, turbulent frequency and entropy, respectively. The filtered value of any function

Q̃ **(****û**,

**φ̂**,

ω̂,

ŝ**)**, fully defined by velocity, scalar, frequency and/or entropy, is obtained from the En-FDF as:

The marginal En-FDF [

64], denoted by

${\mathcal{F}}_{en}^{\prime}\left(\widehat{\mathit{\phi}},\widehat{s},\mathbf{x};t\right)$, can be obtained from

Equation (5) by integrating over

**û**,

ω̂ spaces. This form of En-FDF thus only contains information on joint scalar-entropy statistics. It is thus computationally more affordable for the prediction of complex turbulent reacting flows. Both forms of En-FDF are governed by exact transport equations, which include several unclosed terms [

63,

64]. The closure is provided by a stochastic model, which consists of a system of stochastic differential equations (SDEs). The stochastic model for the comprehensive form of En-FDF consists of SDEs for position, velocity, scalars, frequency and entropy:

where

${X}_{i}^{+},{U}_{i}^{+}$,

ω^{+},

${\phi}_{\alpha}^{+}$,

T^{+},

${\mu}_{\alpha}^{+}$,

h^{+} and

s^{+} are the stochastic representations of position, velocity, frequency, scalars, temperature, chemical potential per unit mass of species

α, specific enthalpy and specific entropy, respectively. The set of SDEs include the linear mean square estimation (LMSE) [

67] and the simplified Langevin models [

68], with

${G}_{ij}=-\mathrm{\Omega}\left({\scriptstyle \frac{1}{2}}+{\scriptstyle \frac{3}{4}}{C}_{0}\right)\hspace{0.17em}{\delta}_{ij}$. The

W_{i},

${W}_{i}^{\prime}$ terms denote the Wiener–Lévy processes [

69]. In these equations,

k_{s} =

τ(

u_{i},

u_{i})

/2 denotes the SGS kinetic energy and

**Ω** is the SGS mixing frequency, modeled as:

**Ω** ≡

C_{Ω}〈

ω^{+} |

ω^{+} ≥ 〈

ω〉

_{L}〉

_{L} [

57]. The model parameters

C_{0} = 2.1,

C_{φ} = 1,

C_{f} = 1,

C_{ω} = 2 and

C_{Ω}= 0.9 are set according to previous work [

53,

56,

57]. The stochastic process corresponding to entropy (

Equation (8e)) follows the Gibbs fundamental equation, in which

ε_{t} is the total rate of turbulent dissipation, including both SGS and resolved contributions:

The Fokker–Planck equation [

70] conjugate to the set of SDEs is the modeled transport equation for the comprehensive En-FDF:

For the marginal En-FDF, the SDEs for the scalar and the entropy remain the same. However, the velocity and frequency must be obtained by other (non-FDF) means. In this case, the physical transport is modeled by [

51]:

where the mixing frequency is modeled as

**Ω** = (

γ +

γ_{t})

/〈

ρ〉Δ

^{2} [

50,

51] and

γ_{t} denotes the SGS diffusivity. The Fokker–Planck equation corresponding to the set of SDEs (

Equations (8c),

(8e) and

(11)) is the modeled transport equation for the marginal En-FDF:

In both En-FDF formulations, the filtered chemical reaction and its entropy generation effect (the last two terms on the RHS of

Equations (10) and

(12)) appear in closed forms. Integrating

Equation (10) according to

Equation (7) yields the transport equations for all SGS moments implied by the En-FDF. The first entropy moment describes the transport of filtered entropy:

where

g_{α} is the Gibbs free energy per unitmass of species

α. For themarginal En-FDF, a similar transport equation is obtained from

Equation (12), but the SGS entropy flux is modeled as

$\tau ({u}_{i},s)=-\frac{{\gamma}_{t}}{\langle \rho \rangle}\frac{\partial {\langle s\rangle}_{L}}{\partial {x}_{i}}$. Comparing

Equation (13) with

Equation (4) reveals the En-FDF implied modeling of the individual filtered entropy generation terms:

It is noted that the filtered entropy generation by chemical reaction

$\langle {S}_{{g}_{C}}\rangle =-\langle {\scriptstyle \frac{\rho}{T}}{\sum}_{\alpha =1}^{{N}_{s}}{\mu}_{\alpha}{S}_{\alpha}\rangle $ in

Equation (13) does not require modeling in the En-FDF.

For numerical solution of the En-FDF, a procedure shown to be effective in FDF simulations is the hybrid Eulerian/Lagrangian Monte Carlo (MC) method [

53,

56,

57,

63,

64]. The Eulerian solver is typically a grid-based finite-difference (FD) or finite volume method. The FD solver used in these simulations is based on a fourth order compact parameter scheme [

71]. The Lagrangian solver is based on the MC method to solve the set of SDEs. In this method, an ensemble of MC particles is used to represent the FDF. These particles carry information regarding the FDF variables: position, velocity, scalars, turbulent frequency and/or entropy. From the numerical standpoint, the use of the MC method is significantly easier than solving the modeled FDF transport equations (e.g.,

Equation (10) or

(12)) directly, as shown in previous studies [

50–

53,

56,

57,

63,

64]. The filtered moments are constructed on the grid points by ensemble averaging the MC particles inside an ensemble domain around each grid point. The transfer of information from the grid points to MC particles is done by interpolation.

## 3. Simulations

In this section, we present some of the recent results pertaining to LES prediction of entropy statistics and entropy generation in turbulent shear flows. These include LES of a non-reacting temporal mixing layer and a turbulent jet flame. The former involves the comprehensive En-FDF formulation and gives assessment of this methodology against DNS data. The latter involves the marginal En-FDF and shows validation of En-FDF against laboratory data. These results demonstrate the capacity of LES/En-FDF in predicting entropy transport and generation in turbulent mixing and reacting flows.

The comprehensive form of En-FDF is applied for LES of a temporal mixing layer involving the transport of passive scalars. The objective is to assess the En-FDF prediction of entropy filtered moments and its rate of generation against DNS data. Large eddy simulation of this flow using various FDF methodologies are reported in [

52,

53,

56,

57,

63]. In the following, we present some of the latest results obtained from the En-FDF. The variables are non-dimensionalized with the corresponding reference values: the reference length

L_{o} is half of the initial vorticity thickness; the reference velocity

U_{o} is half of the velocity difference across the shear layer; and the reference temperature is

T_{o} = 298

K. The Reynolds number based on these values is

Re_{o} = 50. We assume unity Schmidt

Sc and Prandtl

Pr numbers. The computational domain spans 0 ≤

x ≤

L,

${\scriptstyle \frac{-L}{2}}\le y\le {\scriptstyle \frac{L}{2}}$ , 0 ≤

z ≤

L where

L =

L_{v}/L_{o}, and

L_{v} is specified, such that

L_{v} = 2

^{N}^{v}λ_{u}, where

N_{v} is the desired number of successive vortex pairings and

λ_{u} is the wavelength of the most unstable mode corresponding to the mean streamwise velocity profile imposed at the initial time;

x,

y and

z denote the streamwise, the cross-stream and the spanwise coordinate directions, respectively. The velocity components along these directions are denoted by

u,

v and

w, respectively. The filtered normalized streamwise velocity, density and passive scalar (mixture fraction)

φ fields are initialized using hyperbolic tangent profiles with free-stream conditions as: 〈

u〉

_{L} = 1, 〈

ρ〉 = 0.5 and 〈

φ〉

_{L} = 1 on the top and 〈

u〉

_{L} = −1, 〈

ρ〉 = 1 and 〈

φ〉

_{L} = 0 on the bottom. With the uniform initial pressure field, the initial filtered temperature is obtained from 〈

ρ〉 according to the ideal-gas equation of state. To obtain the entropy generation by mass diffusion, we consider the top and bottom streams to carry

H_{2} and

F_{2}, respectively. The flow involves pure mixing of these species; thus, mass fraction and enthalpy values are fully determined by the mixture fraction. The En-FDF predictions are assessed with DNS results. Simulations using LES and DNS are performed on 33

^{3} and 193

^{3} grid points, respectively. The LES filter size is twice as large as grid spacing in each direction. For comparison, the DNS data is filtered via a top-hat filter. The periodic boundary condition is used in the streamwise and spanwise directions, and the zero-derivative boundary condition is employed at cross-stream boundaries. The initial number of particles per grid point is 320, and the ensemble domain size is set equal to half the grid spacing in each direction. Initialization of the MC particles and their treatment at the boundaries are consistent with the FD initial and boundary conditions.

Figure 1 shows the formation of three-dimensional (3D) structures, visualized by the instantaneous filtered entropy fields obtained from the En-FDF and DNS. This figure shows a visual assessment of the methodology, as the large-scale coherent structures predicted by the En-FDF resemble those obtained from the DNS. Further appraisal is made by comparing the Reynolds-averaged statistics. These are constructed from the instantaneous data by spatial averaging over the homogeneous (streamwise and spanwise) directions and, hence, vary only in the cross-stream direction. The averaged quantities are denoted by an overbar.

Figure 2 shows the close agreement of the Reynolds-averaged filtered entropy predicted by the En-FDF and DNS. To illustrate the En-FDF prediction of the second order SGS moments, the SGS, the resolved and the total entropy flux in the streamwise direction are shown in

Figure 3. The resolved field is denoted by

$\overline{R(u,s)}$, with

$R(u,s)=\left({\langle u\rangle}_{L}-\overline{{\langle u\rangle}_{L}}\right)\hspace{0.17em}\left({\langle s\rangle}_{L}-\overline{{\langle s\rangle}_{L}}\right)$; the total field is

$\overline{r(u,s)}$ with

r(

u,

s) = (

u − ū) (

s − s̄). In DNS, the total component is directly available, while in LES, it is approximated by

$\overline{r(u,s)}\approx \overline{R(u,s)}+\overline{\tau (u,s)}$ [

72]. As shown, the streamwise entropy flux components are predicted well by the En-FDF. Similar agreements are obtained for other first and second order moments. The En-FDF is capable of accounting for individual filtered entropy generation effects. As shown in

Figure 4, these are predicted favorably by the En-FDF. All mean entropy generation contributions peak in the fully turbulent region in the middle of the layer where the rate of turbulent mixing is the highest. The entropy generation by heat conduction is dominant in this flow, due to the large temperature difference across the layer, followed by that of mass diffusion. The effect of viscous dissipation is slightly underpredicted, due to underprediction of turbulent dissipation by the frequency model (

Equation (8d)) [

57]. However, the contribution of viscous dissipation to overall entropy generation is much smaller than that of other effects. These simulations show the favorable agreement of En-FDF results with the DNS data.

The marginal En-FDF formulation is applied for the LES of the turbulent non-premixed piloted methane jet flame (Sandia Flame D) [

73,

74]. The main objectives are to validate the En-FDF against laboratory data and to conduct entropy generation analysis of a realistic turbulent non-premixed flame. This flame has been the subject of several previous FDF studies [

54,

60,

61,

64]. Here, we discuss some of the latest results obtained via the En-FDF. The flame configuration includes a main jet with Reynolds number of

Re_{D} = 22, 400 based on the nozzle diameter

D = 7.2

mm and the bulk jet velocity

U_{b} = 49.6

m/s. The coflow temperature is

T_{c} = 291

K. The flame is near equilibrium; thus, the methane oxidation kinetics is implemented using the flamelet concept [

75], in which the detailed kinetics of Gas Research Institute [

76] is employed in a laminar, one-dimensional counterflow (opposed jet) flame. The thermo-chemical variables are expressed as a function of the mixture fraction, which is carried as an additional passive scalar. The strain rate on the flame is assumed to be a constant value of 100

s^{−1} [

54]. The flow variables at the inflow are set similar to those in the experiment, including the inlet profiles of velocity and the mixture fraction. The molecular viscosity increases with

T^{0.7} and the molecular Schmidt (and Prandtl) number is

Sc = 0.75. Simulations are conducted on a 3D Cartesian mesh with uniform grid spacings along all coordinate directions. The computational domain spans a region of 18

D × 10

D × 10

D in the axial (

x) and the two lateral (

y,

z) directions, respectively. The number of grid points are 91 × 101 × 101 in the

x,

y and

z directions, respectively. The filter size is set equal to

$\mathrm{\Delta}=2\sqrt[3]{\mathrm{\Delta}x\mathrm{\Delta}y\mathrm{\Delta}z}$, where Δ

x, Δ

y and Δ

z denote grid spacing in the corresponding directions. The boundary conditions on FD domain boundaries are set according to the characteristic boundary conditions [

77]. To account for all of the second order SGS moments in LES transport equations, the modified kinetic energy viscosity (MKEV) closure [

51] is employed. The SGS modeling of the scalar and entropy fluxes is handled using

γ_{t} =

ν_{t}/Sc_{t}, where the SGS viscosity

ν_{t} is described byMKEV, and the turbulent Schmidt (and Prandtl) number is

Sc_{t} = 0.75. The ensemble domain size is equal to the filter characteristic width, and there are approximately 48 MC particles participating in ensemble averaging at each grid point. According to extensive previous studies [

50–

53], this number of particles is sufficient to yield good statistical accuracy with minimal dispersion errors.

Figure 5 shows contours of the instantaneous filtered entropy obtained from the FD and MC solvers. The similarity of the instantaneous results indicates the consistency of the solvers in predicting the entropy field. In this figure, the fuel nozzle is located at the centerline of the

x = 0 plane, surrounded by the pilot, which exhibits the highest temperature and entropy values. The region close to the nozzle is dominated by the molecular diffusion, and the flow resembles a laminar jet. Farther downstream, the growth of perturbations causes formation of large-scale coherent structures. The overall accuracy of the En-FDF predictions is assessed by comparing various statistics with the laboratory data; the experimental data for entropy statistics are constructed using the instantaneous data corresponding to the scalars. In the following, the notations

Q̄ and

RMS (

Q) denote, respectively, the time-averaged mean and root mean square fields for a variable

Q. To show the validation of En-FDF, some of the entropy statistics are presented here. In the following, the position, velocity, temperature and entropy are normalized by

D,

U_{b},

T_{c} and

${U}_{b}^{2}/{T}_{c}$, respectively. As shown in

Figure 6, the radial (

$r=\sqrt{{z}^{2}+{y}^{2}}$) distributions of the time-averaged filtered entropy at

x = 7.5 and

x = 15 are in good agreement with the data. The validation of entropy RMS values is shown in

Figure 7. The resolved RMS is

${\overline{R(s,s)}}^{1/2}$, where

$R(s,s)=\left({\langle s\rangle}_{L}-\overline{{\langle s\rangle}_{L}}\right)\hspace{0.17em}\left({\langle s\rangle}_{L}-\overline{{\langle s\rangle}_{L}}\right)$, and the total RMS is

${\overline{r(s,s)}}^{1/2}$, where

r(

s,

s) ≈

R(

s,

s) +

τ(

s,

s). The RMS values show reasonable agreements with the experimental data. The En-FDF prediction of the instantaneous, local entropy generation effects is illustrated in

Figure 8. The entropy production by heat conduction shows local peaks in the inner (jet/pilot) shear layer near the nozzle and the fully turbulent regions downstream, where high temperature variations occur.

Figure 8b depicts the entropy generation by mass diffusion in which local large values correspond with large gradients in species concentrations characterized by the mixture fraction. The contribution of the chemical reaction is shown in

Figure 8c. As anticipated, this effect is dominant near the flame zone, identified by large temperature values. It is noted that experimental data for direct assessment of entropy generation predictions is not available for this flame; however, close agreement of filtered entropy with the data (

Figure 6) indicates accurate prediction of entropy generation terms (

Equation (4)). The mean entropy production at different axial locations (

Figure 9) shows that near the nozzle, all irreversibilities exhibit peaks in the inner shear layer, due to large velocity and scalar gradients. At downstream locations, the heat conduction effect shows increased values caused by mixing of hot combustion products with the cold jet. A secondary peak due to this effect is also observed in the outer (pilot/coflow) shear layer at

x = 15. For mass diffusion and chemical reaction, entropy generation profiles have similar shapes (

Figure 9b,c), with peaks located near

r = 1, where large concentration gradients occur because of chemical reaction. To have a quantitative comparison of these effects,

Figure 10 shows the entropy generation components along with the total entropy generation at two axial locations. Consistent with that discussed above, at

x = 5, entropy generation effects are more localized in the inner shear layer region. The contribution of heat conduction is dominant, followed by chemical reaction and mass diffusion. As the turbulent jet develops downstream, all entropy production effects show larger spread in the radial direction, and the effect of chemical reaction becomes more significant. It is noticed that the effect of mass diffusion is less important, and that of viscous dissipation (not shown) is negligible in this flame. These simulations show the validation of LES/En-FDF against experimental data and the efficacy of this methodology in analysis of local entropy generation.