1 Introduction
The closure of the mean reaction rate in the context of Reynolds Averaged Navier Stokes (RANS) simulations in turbulent combustion often requires the knowledge of the scalar dissipation rate of the fuel mass fraction Y
F (i.e.
\( \overset{\sim }{\varepsilon_Y}=\overline{\uprho D\nabla {Y}_F^{\prime \prime}\cdotp \nabla {Y}_F^{\prime \prime }}/\overline{\uprho} \) [
1‐
3], where
\( \overline{\mathrm{q}} \),
\( \overset{\sim }{\mathrm{q}}=\overline{\uprho \mathrm{q}}/\overline{\uprho} \) and
\( {\mathrm{q}}^{\prime \prime }=\mathrm{q}-\overset{\sim }{\mathrm{q}} \) are Reynolds average, Favre mean and Favre fluctuation of a general quantity q,
D is the mass diffusivity and ρ is the gas density). Algebraic and transport equation based closures of
\( \overset{\sim }{\varepsilon_Y} \)have previously been considered in the context of purely gaseous phase combustion where variations in equivalence ratio exist [
2‐
6]. Analyses of simulations of turbulent droplet-laden combustion have also been carried out where the scalar dissipation rate transport equation has been considered [
7]. Moreover, previous studies on droplet combustion analysed the statistical behaviour of the terms of the scalar dissipation rate transport equation [
8], but the statistical analysis of
\( \overset{\sim }{\varepsilon_Y} \) is yet to be addressed in detail. Furthermore, the validity of existing closures of
\( \overset{\sim }{\varepsilon_Y} \) and the unclosed terms of its transport equation, which were originally proposed for purely gaseous phase combustion, is yet to be assessed for turbulent spray flames. These gaps in the existing literature have been addressed here by analysing the statistical behaviours of
\( \overset{\sim }{\varepsilon_Y} \) and the terms of its transport equation using a three-dimensional compressible Direct Numerical Simulations (DNS) database [
9‐
12] of statistically planar turbulent flames propagating into droplet-laden mixtures where the fuel is supplied in the form of mono-disperse droplets ahead of the flame. The current study considers selected cases from a large database [
9‐
12] such that the effects of droplet diameter
ad and droplet equivalence ratio ϕ
d (i.e. fuel in liquid droplets to air ratio by mass, normalised by fuel to air ratio by mass under stoichiometric condition) on the statistical behaviours of
\( \overset{\sim }{\varepsilon_Y} \) and its transport can be analysed in detail. The main objectives of this study are:
(a)To analyse the statistical behaviours of \( \overset{\sim }{\varepsilon_Y} \) and the various unclosed terms of its transport equation for turbulent spray flames in the context of RANS.
(b)To assess the validity of the existing models for the unclosed terms of \( \overset{\sim }{\varepsilon_Y} \) transport equation for turbulent droplet combustion.
The rest of the paper will be organised as follows. The mathematical background and numerical implementation pertinent to this analysis are presented in the next section. This will be followed by the presentation of results and their subsequent discussion. Finally, the main findings will be summarised and conclusions will be drawn.
2 Mathematical Background & Numerical Implementation
A modified single-step irreversible chemical mechanism [
13] was used to perform the present analysis:
Fuel +
s ·
Oxidiser → (1 +
s) ·
Products, where s is the oxidiser to fuel ratio by mass under stoichiometric condition. The activation energy and heat of combustion are taken to be functions of the gaseous equivalence ratio,
ϕg, so that a realistic
ϕg-dependence of unstrained laminar burning velocity
\( {S}_{b\left({\phi}_g\right)} \) can be obtained [
13]. It has been shown by Tarrazo et al. [
13] that the mechanism compares favourably with both experiments and detailed chemistry simulations for all hydrocarbon-air flames. It has been demonstrated by Swaminathan and Bray [
14] based on experimental data that the normalised laminar burning velocity
\( {S}_{b\left({\phi}_g\right)}/{\left\{{S}_{b\left({\phi}_g\right)}\right\}}_{max} \) dependence of equivalence ratio
ϕg is not sensitive to the choice of fuel for hydrocarbon-air mixtures. Moreover, it has also been found that there is no significant difference between
ϕg dependences of normalised laminar burning velocity
\( {S}_{b\left({\phi}_g\right)}/{\left\{{S}_{b\left({\phi}_g\right)}\right\}}_{max} \) and non-dimensional adiabatic flame temperature
\( {\theta}_{\left({\phi}_g\right)}=\left({T}_{ad\left({\phi}_g\right)}-{T}_0\right)/\left({T}_{ad}\left({\phi}_{g=1}\right)-{T}_0\right) \) (with
\( {T}_{ad\left({\phi}_g\right)} \) and
T0 are the adiabatic flame temperature at gaseous equivalence ratio of
\( {\phi}_g \) and unburned gas temperature respectively) obtained from the modified single-step [
13] and multi-step detailed [
15] chemical mechanisms. Furthermore, the variations of
\( {S}_{b\left({\phi}_g\right)}/{\left\{{S}_{b\left({\phi}_g\right)}\right\}}_{max} \) and
\( {\theta}_{\left({\phi}_g\right)} \) with
ϕg have been found to be in good agreement with experimental data [
16].
In this analysis, all species are taken to have unity Lewis number and are assumed to be perfect gases. Standard values have been taken for the ratio of specific heats (γ = 1.4) and Prandtl number (
Pr = 0.7) for the gaseous phase. The individual droplets are tracked in the Lagrangian sense and the quantities transported for each droplet are the position,
\( {\overrightarrow{x}}_d \), velocity,
\( {\overrightarrow{u}}_d \), diameter,
ad and temperature,
Td. The transport equations of
\( {\overrightarrow{x}}_d \),\( {\overrightarrow{u}}_d \), ad and
Td are given as [
9‐
12,
17‐
23]:
$$ \frac{d{\overrightarrow{x}}_d}{dt}={\overrightarrow{u}}_d;\frac{d{\overrightarrow{u}}_d}{dt}=\frac{\overrightarrow{u}\left({\overrightarrow{x}}_d,t\right)-{\overrightarrow{u}}_d}{\tau_d^p};\frac{d{a}_d^2}{dt}=-\frac{a_d^2}{\tau_d^u}\ \mathrm{and}\ \frac{d{T}_d}{dt}=-\frac{\hat{T}\left({\overrightarrow{x}}_d,t\right)-{T}_d-{B}_d{L}_v/{C}_P^g}{\tau_d^T} $$
(1i)
where
\( \hat{T} \) is the instantaneous dimensional temperature,
Lv is the latent heat of vaporization, and
\( {\tau}_d^p \),\( {\tau}_d^u \) and
\( {\tau}_d^T \) are relaxation/decay timescales for droplet velocity, diameter and temperature respectively, which are defined as [
17‐
22]:
$$ {\tau}_d^p=\frac{\rho_d{a}_d^2}{18{C}_u\mu };{\tau}_d^u=\frac{\rho_d{a}_d^2}{4\mu}\frac{Sc}{S{h}_c}\frac{1}{\mathit{\ln}\left(1+{B}_d\right)}\ \mathrm{and}\ {\tau}_d^T=\frac{\rho_d{a}_d^2}{6\mu}\frac{\mathit{\Pr}}{N{u}_c}\frac{B_d}{\mathit{\ln}\left(1+{B}_d\right)}\frac{C_p^L}{C_p^g} $$
(1ii)
where
ρd is the droplet density,
\( {C}_p^L \) is the specific heat for the liquid phase,
\( {C}_p^g \) is the specific heat at constant pressure for the gaseous phase,
Cu is the corrected drag coefficient and is given by [
18‐
23]:
$$ {C}_u=1+\frac{1}{6}{\mathit{\operatorname{Re}}}_d^{2/3} $$
(1iii)
Furthermore,
Red is the droplet Reynolds number,
Sc is the Schmidt number,
Bd is the Spalding mass transfer number,
Shc is the corrected Sherwood number and
Nuc is the corrected Nusselt number, which are defined as [
9‐
12,
17‐
23]:
$$ R{e}_d=\frac{\rho \left|\overrightarrow{u}\left({\overrightarrow{x}}_d,t\right)-{\overrightarrow{u}}_d\right|{a}_d}{\mu };{B}_d=\frac{Y_F^S-{Y}_F\left({\overrightarrow{x}}_d,t\right)}{1-{Y}_F^s}\ \mathrm{and}\ S{h}_c=N{u}_c=2+\frac{0.555R{e}_d Sc}{{\left(1.232+R{e}_d{Sc}^{4/3}\right)}^{1/2}} $$
(1iv)
where
\( {Y}_F^s \)is the value of
YF at the surface of the droplet. Equation (
1iv) implicitly invokes the unity Lewis number assumption. The Clausius–Clapeyron relation for the partial pressure of the fuel vapour at the droplet surface,
\( {p}_F^s \), is used to evaluate the Spalding number
Bd, which leads to:
$$ {p}_F^s={p}_{ref}\mathit{\exp}\left(\frac{L_v}{R_0}\left[\frac{1}{T_{ref}^s}-\frac{1}{T_d^s}\right]\right);{Y}_F^s={\left(1+\frac{W_{air}}{W_F}\left[\frac{p\left({\overrightarrow{x}}_d,t\right)}{p_F^s}-1\right]\right)}^{-1} $$
(1v)
where
\( {T}_{ref}^s \) is the boiling point of the fuel at pressure
pref, R0 is the gas constant,
\( {T}_d^s \) is assumed to be
Td, and
Wair and
WF are the molecular weights of air and fuel, respectively.
The droplet and gaseous phases are coupled in the gaseous transport equations [
9‐
12,
17‐
23]:
$$ \frac{\partial \rho \psi}{\partial t}+\frac{\partial \rho {u}_j\psi }{\partial {x}_j}=\frac{\partial }{\partial {x}_j}\left({\varGamma}_{\psi}\frac{\partial {\psi}_1}{\partial {x}_j}\right)+{\overset{.}{\omega}}_{\psi }+{\overset{.}{S}}_g+{\overset{.}{S}}_{\psi } $$
(1vi)
where
ψ = {1,
uj,
e,
YF,
YO} for the conservation of mass, momentum, energy and mass fractions respectively,
\( {\psi}_1=\left\{1,{u}_j,\hat{T},{Y}_F,{Y}_O\right\} \) for
ψ = {1,
uj,
e,
YF,
YO}, and
Γψ =
μ/
σψ and
λ for
ψ = {
uj,
YF,
YO} and
ψ =
e respectively, with
uj,
μ,
λ and
σψ being the velocity component in the j
th direction, dynamic viscosity, thermal conductivity and an appropriate Schmidt number for ψ, respectively. The term
\( {\overset{.}{\omega}}_{\psi } \) arises due to chemical reaction rate and
\( {\overset{.}{S}}_g \) is an appropriate gaseous phase source term. The droplet source term arising from evaporation,
\( {\overset{.}{S}}_{\psi }=-1/V{\sum}_dd\left({m}_d{\psi}_d\right)/ dt \), is interpolated from the droplet’s sub-grid position to the 8 surrounding nodes, where
V is the cell volume,
\( {m}_d={\rho}_d\left(1/6\right)\pi {a}_d^3 \) is the droplet mass and the summation is carried out over all droplets in the vicinity of each node [
9‐
12,
17‐
23].
Droplet evaporation leads to mixture inhomogeneities, which are characterized by the mixture fraction
: ξ = (
YF −
YO/
s +
YO∞/
s)/(
YF∞ +
YO∞/
s), where
YF∞ = 1.0 (Y
O∞ = 0.233) is the fuel (oxidiser) mass fraction in the pure fuel (air) stream. The fuel used here is n-heptane, C
7H
16, for which
s = 3.52 and the stoichiometric fuel mass/mixture fraction is:
YFst =
ξst = 0.0621. Using ξ, a reaction progress variable
c can be defined in the following manner according to several previous analyses on droplet combustion [
9‐
12,
21‐
25]:
c = [(1 −
ξ)
YO∞ −
YO]/[(1 −
ξ)
YO∞ −
max (0, [
ξst −
ξ]/
ξst)
YO∞] so that
c increases monotonically from 0 in unburned reactants to 1.0 in fully burned products. It should be noted that the mixture fraction is not a passive scalar in the strict sense as there is an extra term in the transport equation of
ξ due to evaporation. However, the Burke-Schumann relations
YOu = (1 −
ξ)
YO∞ and
YOb =
max (0, [
ξst −
ξ]/
ξst)
YO∞ (where subscripts
u and
b refer to values in unburned reactants and fully burned products, respectively) remain reasonably valid for the oxygen mass fraction
YO because the evaporation contributions in the mixture fraction transport equation remain mostly negligible in both fully unburned and fully burned gases. Accordingly, using Eq. (
1vi), one obtains the following transport equation for
\( \overset{\sim }{\varepsilon_Y} \) [
6]:
$$ {T}_2=-2D\overline{\frac{\left[\ {\overset{.}{\omega}}_F+\nabla \cdotp \left(\rho D\nabla {Y}_F\right)\right]}{\rho}\frac{\partial {Y}_F}{\partial {x}_k}\frac{\partial \rho }{\partial {x}_k}}+2\frac{\overset{\sim }{D}}{\overline{\rho}}\frac{\partial \overset{\sim }{Y_F}}{\partial {x}_k}\frac{\partial \overline{\rho}}{\partial {x}_k}\left[\overline{{\overset{.}{\omega}}_F+\nabla .\left(\rho D\nabla {Y}_F\right)}-\frac{\partial \left(\overline{\rho {u}_l^{\prime \prime }{Y}_F^{\prime \prime }}\right)}{\partial {x}_l}\right] $$
(2iii)
$$ {T}_4=2\overline{D\frac{\partial {\overset{.}{\omega}}_F}{\partial {x}_k}\frac{\partial {Y}_F}{\partial {x}_k}}-2\overset{\sim }{D}\frac{\partial \overline{{\overset{.}{\omega}}_F}}{\partial {x}_k}\frac{\partial \overset{\sim }{Y_F}}{\partial {x}_k} $$
(2v)
$$ {T}_5=2\overline{D\frac{\partial {\overset{.}{S}}_E}{\partial {x}_k}\frac{\partial {Y}_F}{\partial {x}_k}}-2\overset{\sim }{D}\frac{\partial {\overset{.}{S}}_{ER}}{\partial {x}_k}\frac{\partial \overset{\sim }{Y_F}}{\partial {x}_k} $$
(2vi)
$$ {T}_6=\overline{D\nabla {\mathrm{Y}}_F\cdotp \nabla {Y}_F\Gamma}-\overset{\sim }{D}\nabla \overset{\sim }{{\mathrm{Y}}_F}\cdotp \nabla \overset{\sim }{Y_F}\overset{\sim }{\Gamma} $$
(2vii)
$$ {D}_2=2\overline{\rho {D}^2\frac{\partial^2{Y}_F^{\prime \prime }}{\partial {x}_k\partial {x}_i}\frac{\partial^2{Y}_F^{\prime \prime }}{\partial {x}_k\partial {x}_i}} $$
(2viii)
$$ f(D)=\overline{2D\frac{\partial {Y}_F}{\partial {x}_k}\frac{\partial \left(\rho D\right)}{\partial {x}_k}\frac{\partial^2{Y}_F}{\partial {x}_j\partial {x}_j}}+\overline{2D\frac{\partial {Y}_F}{\partial {x}_k}\frac{\partial^2\left(\rho D\right)}{\partial {x}_j\partial {x}_k}\frac{\partial {Y}_F}{\partial {x}_j}}-\overline{\frac{\partial }{\partial {x}_j}\left(\rho {N}_Y\frac{\partial D}{\partial {x}_j}\right)}-\overline{2\rho D\frac{\partial D}{\partial {x}_j}\frac{\partial }{\partial {x}_j}\left(\frac{\partial {Y}_F}{\partial {x}_k}\frac{\partial {Y}_F}{\partial {x}_k}\right)}+\overline{\rho \left(\frac{\partial {Y}_F}{\partial {x}_k}\frac{\partial {Y}_F}{\partial {x}_k}\right)\left[\frac{\partial D}{\partial t}+{u}_j\frac{\partial D}{\partial {x}_j}\right]} $$
(2ix)
where Γ is the source term in the mass conservation equation due to evaporation,
\( {\overset{.}{S}}_{ER}=\overline{\Gamma}\left(1-\overset{\sim }{Y_F}\right) \) and
\( {\overset{.}{S}}_E=\Gamma \left(1-{Y}_F\right) \). The term
T1 is the turbulent transport term,
T2 arises due to density variation,
T3 originates due to the alignment of the scalar gradient with the fluid-dynamic strain rates and essentially signifies generation/destruction of the scalar gradient by the velocity gradients,
T4 arises due to the chemical reaction rate,
T5 and
T6 arise due to droplet evaporation, whereas the term
D2 arises due to molecular dissipation. In Eq. (2), the terms of
T1,
T2,
T31,
T32,
T33,
T4,
T5,
T6 and (−
D2) are unclosed terms in the context of second-moment closure and their modelling will be discussed in Section
3.
The present study uses a three-dimensional compressible DNS code SENGA [
9‐
12,
21‐
23]. High-order finite-difference (i.e. 10th central difference scheme for the internal grid points and the order of differentiation gradually reduces to a 2nd order one-sided scheme at the non-periodic boundaries) and explicit 3rd order low storage Runge-Kutta schemes are used for spatial differentiation and time advancement, respectively. A rectangular domain of size
\( 63.35{D}_0/{S}_{b\left({\phi}_g=1\right)}\times 42.17{D}_0/{S}_{b\left({\phi}_g=1\right)}\times 42.17{D}_0/{S}_{b\left({\phi}_g=1\right)} \) has been considered, where
D0 and
\( {S}_{b\left({\phi}_g=1\right)} \) are the unburned gas diffusivity and the unstrained laminar burning velocity of the stoichiometric mixture, respectively. For the present thermo-chemistry
\( {D}_0/{S}_{b\left({\phi}_g=1\right)}\approx 0.625{\updelta}_{\mathrm{th}} \) where
\( {\delta}_{th}=\left({T}_{ad\left({\phi}_g=1\right)}-{T}_0\right)/\mathit{\max}{\left(\left|\nabla \hat{T}\right|\right)}_L \) is the unstrained thermal laminar flame thickness of the stoichiometric laminar premixed flame, where subscript ‘L’ refers to the values in the unstrained stoichiometric laminar premixed flame. The simulation domain is discretised using a Cartesian grid of size 384 × 256 × 256, ensuring that both the flame thickness,
δth, and the Kolmogorov length-scale,
η, are adequately resolved. The boundaries in the mean direction of flame propagation (i.e. x-direction) are considered to be partially non-reflecting, whereas the other boundaries are taken to be periodic. The boundary conditions are specified using the well-known Navier Stokes Characteristic Boundary Conditions (NSCBC) technique [
26]. The droplets are distributed uniformly in space throughout the y- and z-directions and in the region
\( 0.0\le x{S}_{b\left({\phi}_g=1\right)}/{D}_0\le 16.53 \) ahead of the flame. The reacting flow field is initialised based on the steady laminar solution generated using COSILAB [
27] for desired values of
ad and
ϕd, as done previously by Neophytou and Mastorakos [
28] for one-dimensional laminar spray flame simulations. Initial turbulent velocity fluctuations, generated using a standard pseudo-spectral method [
29] following Batchelor-Townsend spectrum [
30], have been superimposed on top of the steady laminar spray flame solution. For the present analysis, the unburned gas temperature is taken as
T0 = 300K, which leads to
\( \tau =\left({T}_{ad\left({\phi}_g=1\right)}-{T}_0\right)/{T}_0=6.54 \) where
\( {T}_{ad\left({\phi}_g=1\right)} \) is the adiabatic flame temperature of the stoichiometric mixture. The fuel is supplied purely in the form of mono-disperse droplets with
ad/
δth = 0.06, 0.08, 0.10 for different values of
ϕd = 1.0, 1.7 at a distance
\( 16.53{D}_0/{S}_{b\left({\phi}_g=1\right)} \) from the point in the laminar flame at which
\( \hat{\mathrm{T}}=400\mathrm{K} \). It should be noted that the droplet diameter
ad has been non-dimensionalised using the well-defined thermal flame thickness of the stoichiometric mixture
δth, which is consistent with Buckingam’s Pi Theorem. The initial droplet number density ρ
N varies between 1.16 ≤ (
ρN)
1/3δth ≤ 2.27 in the region
\( 0.0\le x{S}_{b\left({\phi}_g=1\right)}/{D}_0\le 16.53 \) and the liquid volume fraction remains well below 0.01. Droplets are supplied at the left-hand-side boundary to maintain a constant ϕ
d ahead of the flame. Due to the high volatility of n-heptane, evaporation commences on entry and the droplet diameter decreases by at least 40%, 30% and 25% by the time it reaches the most reactive region of the flame for the initial
ad/
δth = 0.06, 0.08, 0.10 cases respectively, such that the volume of even the largest droplets remains smaller than half that of the cell volume, which validates the sub-grid point source treatment of droplets adopted for flame-droplet interactions analysed here. The droplet diameter to grid size used in the current analysis remains comparable to several previous DNS analyses [
18‐
23,
31].
The cases considered here have initial values of normalised root-mean-square (rms) turbulent velocities
\( {u}^{\prime }/{S}_{b\left({\phi}_g=1\right)}=7.5 \) and non-dimensional longitudinal integral length-scale
L11/
δth = 2.5. The ratio of droplet diameter to the Kolmogorov scale is
ad/
η ≈ 0.3,0.4,0.5 for
ad/
δth ≈ 0.06,0.08,0.1, respectively, for initial
\( {u}^{\prime }/{S}_{b\left({\phi}_g=1\right)}=7.5 \). All simulations have been carried out until t
final = max(3t
turb, 4t
chem), where t
turb =
L11/
u′ and
\( {\mathrm{t}}_{\mathrm{chem}}={D}_0/{S}_{b\left({\upphi}_g=1\right)}^2 \) are the initial eddy turnover time and chemical time, respectively. The simulation time remains either greater than or comparable to several previous analyses [
18‐
23,
32‐
35]. The volume-integrated reaction rate, flame surface area and burning rate per unit area were not changing rapidly when the statistics were extracted [
9].
The Reynolds/Favre averaged values of a quantity q (i.e. \( \overline{\mathrm{q}} \) and \( \overset{\sim }{\mathrm{q}} \)) are evaluated by ensemble-averaging q over the y-z plane at a given x-location. It should be noted that as the flames are statistically planar, the Favre averaged reaction progress variable \( \overset{\sim }{c} \) and the scalar dissipation rate \( \overset{\sim }{\varepsilon_Y} \) and the terms of its transport equation are unique functions of the x1-direction, which is aligned with the mean direction of flame propagation. The spatial distribution of the terms of the scalar dissipation rate transport equation depending on the thickness of the flame brush, which changes from one case to another. Thus, to generalise the results, all the terms and model predictions are shown as a function of \( \overset{\sim }{c}=f\left({x}_1\right) \).