Derivation of the conditional moment closure equations for spray combustion

Combustion and Flame 156 (2009) 62–72

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Combustion and Flame www.elsevier.com/locate/combustflame

Derivation of the conditional moment closure equations for spray combustion Mikael Mortensen ∗ , Robert W. Bilger School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, 2006 Sydney NSW, Australia

a r t i c l e

i n f o

Article history: Received 23 July 2007 Received in revised form 4 April 2008 Accepted 10 July 2008 Available online 1 September 2008 Keywords: Turbulence Mixing Combustion Spray combustion Multiphase flow Spray

a b s t r a c t In this work we derive the fundamental equations for conditional moment closure (CMC) modelling of individual phases set in a two-phase flow. The derivation is based on the instantaneous transport equations for the single phase that involve a level set/indicator function technique for accounting for interfaces. Special emphasis is put on spray combustion with the CMC equations formulated for the gas phase. The CMC equations are to be viewed as an adjunct to existing methods for the modelling of the dynamics of sprays: they provide a refinement of the modelling of chemical reactions in the gas phase. The resulting CMC equations differ significantly from those already in use in the literature. They contain, of course, unclosed terms that need to be modelled. Investigation of the unclosed terms associated with evaporation at the droplet surface is well beyond the capabilities of laboratory measurement or direct numerical simulation. It is proposed that modelling of these terms be based on the well-established ‘laws’ of similarity between heat and mass transfer: an example is detailed for one example of the general modelling of the spray dynamics. Other unclosed terms are important throughout the gas phase. Models used for these terms in single-phase flows are reviewed and it is proposed that any modifications needed for these models be investigated by DNS of suitable model problems having good resolution of the flow and mixing in the inter-droplet space. It is proposed that a spray analogue of the scalar mixing layer that has been widely studied in single-phase flows be used as the model problem for such DNS studies and also for LES and RANS modelling. © 2008 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

1. Introduction Turbulent non-premixed combustion taking place in a single gaseous phase flow has been the subject of a large body of research and several statistical models are already at a level [1] where reliable predictions can often be made. Conditional moment closure (CMC) [2] is one modelling approach that is soundly based in its derivation, is economical in computer resources and has achieved success in the prediction of auto-ignition [3], the formation of carbon monoxide and nitric oxide [4] and of soot [5]. In addition to combustion, CMC has also found application for fast, mixing sensitive liquid-phase reactions [6], commonly encountered in chemical engineering, and more recently it was used successfully as a model for laminar chaotic [7] flows. In CMC the primary dependent variables are the conditional averages of the species mass fractions and enthalpy, with the averaging being conditional on the mixture fraction having a particular value. (The mixture fraction is thus an extra independent variable.) An important modelling assumption is that the conditionally averaged reaction rates

*

Corresponding author. Current address: Norwegian Defence Research Establishment (FFI), NO-2027 Kjeller, Norway. E-mail address: mikael.mortensen@ffi.no (M. Mortensen).

may often be closed using the conditionally averaged species mass fraction and enthalpy. The underlying physical meaning of this is that fluctuations in species and temperature are assumed to closely follow fluctuations in the mixture fraction. CMC is already being applied to the combustion of two-phase flows including sprays [8–11] and solid particulates [12]. There is little agreement in these studies about the extra terms, if any, that arise from the inter-phase transfer processes. It is apparent that such terms can be important and that clarification is needed. The aim of this paper is to provide a rigorous derivation of the CMC equations for two-phase flows. This derivation is essentially mathematical. It is applicable to sprays, but also to bubbly flows. In this derivation, reactions at the inter-phase surface are not considered but extension to such flows (e.g., char particle combustion) can easily be made. It is noted that the physics involved for a particular type of flow is embodied in the closure assumptions used for the unclosed terms that result from the derivation. Only a brief outline is given here for approaches to the modelling of these terms for sprays. Faeth [13,14] gives excellent reviews of mainstream approaches to the analysis and modelling of mixing, evaporation and combustion in sprays. Kataoka [15] gives a sound mathematical formulation for analysis of sprays on a two-fluid basis. Sirignano [16] and Carrara and DesJardin [17] provide insights into the impor-

0010-2180/$ – see front matter © 2008 The Combustion Institute. Published by Elsevier Inc. All rights reserved. doi:10.1016/j.combustflame.2008.07.007

M. Mortensen, R.W. Bilger / Combustion and Flame 156 (2009) 62–72

tance of correctly volume averaging and of correctly filtering in large eddy simulation (LES). Zhu et al. [18] give a formulation in terms of the transport equation for the probability density function (PDF). Peters [19] gives an interactive unsteady flamelet approach for modelling soot and nitric oxide formation in diesel engine sprays. The mathematical derivation presented here in Section 2 closely follows that of Kataoka [15] for the ‘two-fluid’ analysis of two phase flows. In such a two-fluid analysis, equations are written separately for each phase and complications arise from the effects of transport at the inter-phase surfaces. The Kataoka analysis is accepted here as being essentially correct. Here, we extend the analysis to the derivation of the CMC equations. As a result, the terms arising in CMC from inter-phase transport are given in explicit mathematical form. The derived CMC equations are compared with the current literature in the final subsection of Section 2. Some preliminary consideration is given in Section 3 to possible methods of modelling the unclosed terms in the CMC equations for sprays. It is to be noted that CMC, in itself, does not provide a complete framework for sprays. Its strength lies in the fact that it provides an added refinement to existing methods for the prediction of mixing and primary combustion in sprays. Its aim is to provide improved predictions of secondary processes that have strong influences from chemical kinetics, such as NO, CO and soot formation. Many existing methods, that have proven to give good prediction of spray mixing and primary combustion, can be adapted to incorporate CMC predictions for such important minor species. Some assessment of the analysis and modelling used in earlier studies of CMC in two-phase flows is made. Finally, conclusions derived from this study are presented in Section 4. 2. Mathematical derivation In this section we will first elaborate on CMC for single-phase flows, which hopefully will make the transition to two-phase flows more transparent. We will briefly introduce the fundamentals of CMC and show how single-phase CMC can be derived from the local instantaneous transport equations for turbulent reacting flows and summarize the major assumptions that lead to a final model. We will then continue with the derivation for a general two-phase flow, which is the novel contribution of the current paper. Here we will emphasize the application to spray combustion, where the CMC-equations will be used for the continuous gas phase. The major assumptions used in the mathematical derivation will be discussed and a close comparison will be made to the single-phase case. Readers already familiar with the fundamentals of singlephase CMC can jump directly to Section 2.3. Note that the singlephase equations derived in Section 2.2 will be used as a starting point for the derivation of the novel two-phase equations in Section 2.4. 2.1. Fundamentals of CMC In CMC transport equations are solved for the average mass fraction Y α (x, t ) of a reactive scalar α conditional on the value of a conserved scalar or mixture fraction ξ(x, t ). Here x and t represent real space and time respectively and a boldface font is used throughout this paper to denote a vector. The conditional average or mean is defined as







Q α (η, x, t ) ≡ Y α (x, t )ξ = η ≡ Y α |η,

(1)

where the angular brackets denote an ensemble average over an ensemble of realizations of the flow and the vertical bar indicates that this average is conditional on the event ξ(x, t ) = η , where η

63

is a sample space variable for ξ . The conditional mean is related to the instantaneous mass fraction through





Y α (x, t ) = Q α ξ [x, t ], x, t + Y α (x, t ),

(2)

where Y α is the instantaneous fluctuation about the conditional

mean and Q α (ξ [x, t ], x, t ) is a nonrandom function evaluated with a random argument ξ . Note that  Q α (ξ [x, t ], x, t )|η = Q α (η, x, t ) and Y α |η = 0. The conventional average Y α  is obtained by integrating over mixture fraction space

∞ Y α  =

p (η)Y α |η dη,

(3)

−∞

where p (η) is the mixture fraction PDF. The PDF contains information about the fine-scale structure of the flow and needs to be closed in parallel with Y α |η. As indicated in the introduction CMC is motivated by the need to improve predictions of turbulence/chemistry interactions. The principal attraction of CMC is that, if the reacting and conserved scalars are strongly correlated, then the conditional average rate of chemical reactions can be very accurately predicted using only the conditional moments. (The average reaction rate can then of course be computed from the conditional average through integration, like in Eq. (3).) In other words in the exact expression

Y α Y β |η = Y α |ηY β |η + Y α Y β |η,

(4)

we assume that the conditional variance (last term) is negligible. This is first order CMC modelling. In single-phase flows it is well established that conserved and reactive scalars are strongly correlated in flames with little local extinction and reignition effects. This leads to a small conditional variance and first order CMC is thus a very good model. In two-phase flows, like spray combustion, this correlation has not yet been established with experiments nor simulations. However, in principle, the correlation should be equally strong here. 2.2. Derivation of CMC for single-phase flows The governing equations for single-phase reacting flows are the Navier–Stokes and species transport equations. To derive CMC, though, we need only consider the continuity equation

∂ρ ∂ ρ ui + =0 ∂t ∂ xi and the equation for the mass fraction of species

∂ρYα ∂ ρ Y α (u i + V α , i ) + = ρWα. ∂t ∂ xi

(5)

α (6)

Here u i (x, t ) is a component of the velocity vector, ρ (x, t ) is the density of the fluid, V α ,i represents a component of the massdiffusion velocity and W α is the rate of production of species α . For cases where Fickian diffusion can be assumed, we have V α ,i = −

Dα ∂ Yα , Y α ∂ xi

(7)

where D α is the molecular diffusion coefficient of species α . In this work we will make the common assumption that all species share the same diffusion coefficient D. If necessary, differential diffusion effects can be incorporated straightforward at a later development stage (see, e.g., Section 9.1 in [2]). Note that in Eqs. (5)–(7) and for the rest of this paper summation is implied by repeating Roman indices. Exception to this rule is given for index k, which later will be used as a phase indicator. No summation is implied for Greek letters.

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M. Mortensen, R.W. Bilger / Combustion and Flame 156 (2009) 62–72

We are now interested in averaging Eqs. (5) and (6) conditional on the event that ξ = η . In the literature there are two different approaches that can be followed, the joint PDF [20] and the decomposition method [21]. In this work we choose to follow the joint PDF method, since it is generally the less cumbersome. The fine-grained PDF is central to the joint PDF method. The fine-grained PDF Ψ is defined through the delta-function Ψ ≡ δ(ξ − η), which is zero for all ξ , except for ξ = η , where it is infinite. The delta function is a generalized function, which formally is defined by the property

∞ F (η)δ(η − η0 ) dη = F (η0 ),

(8)

−∞

where F is any good function [2]. Differentiation of the finegrained PDF is an important tool used for deriving CMC. Differentiation is performed through the following identities [2]



∂Ψ ∂ ∂ξ ≡− Ψ ∂t ∂ η ∂t





and



∂Ψ ∂ ∂ξ ≡− Ψ . ∂ xi ∂ η ∂ xi

Inserting for the identities in (9) and making use of Eqs. (5) and (6) (with ξ = Y α and W α = 0) it is simply a matter of rearrangement that leads to a transport equation for the fine-grained PDF (11)

where V ξ,i is a component of the mass-diffusion velocity of ξ . This exact equation can be further trivially manipulated by inserting for Fick’s law (7) and then by exchanging the diffusion term with an equivalent formulation based on the scalar dissipation rate. The scalar dissipation rate is defined as N (x, t ) = D

∂ξ ∂ξ ∂ xi ∂ xi

(12)

and a correlation between diffusion and dissipation is given in Eq. (42) of [2]. Inserting for this correlation and Fick’s law into (11) leads to an alternative formulation of the fine-grained PDF equation

 ∂2 ∂ρΨ ∂Ψ ∂ ρ ui Ψ ∂ − 2 (ρ N Ψ ). + = ρD ∂t ∂ xi ∂ xi ∂ xi ∂η

(13)

Equation (13) can now be averaged in the usual sense, using the fact that an average of the product of Ψ and a function of ξ can be expressed as

 Ψ F (ξ ) =  F |η p (η).



(14)

Since differentiation and regular averaging commutes the average of (13) is

∂ρ |η p ∂ρ u i |η p + ∂t ∂ xi =−

∂ 2 ρ N |η p ∂ + ∂ xi ∂ η2



 ∂ ρ D  ∂ 2 ρ D |η p η p − ,  ∂ xi ∂ xi ∂ xi

  ∂ρYαΨ ∂YαΨ ∂ ∂ Y α ∂ξ ∂ ρ ui Y α Ψ ∂ +2 + = ρD ρD Ψ ∂t ∂ xi ∂ xi ∂ xi ∂η ∂ xi ∂ xi −

(15)

which is the exact, unclosed equation for the mixture fraction PDF. Note that in high Reynolds number flows the last two terms of Eq. (15) can be neglected. The conditional fluctuations in ρ can also usually be neglected, which means that it is common (but not exact) to set ρ F |η = ρη  F |η, where ρη = ρ |η. The error in making this assumption was discussed by Klimenko and Bilger [2].

∂2 (ρ N Y α Ψ ) + ρ W α Ψ. ∂ η2

(16)

Averaging and assuming high Reynolds numbers (neglecting the first term on the right hand side) leads to the CMC equation

∂ ρη p Q α ∂ ρη p  u i Y α |η  ∂ JY + = ρη p  W α |η  + , ∂t ∂ xi ∂η where



(9)

(10)

 ∂ ρ ξ V ξ,i ∂ρΨ ∂ ρ ui Ψ ∂ Ψ , + = ∂t ∂ xi ∂η ∂ xi

can be obtained simply by multiplying Eq. (13) with Y α and rearranging. The math is straightforward and the final result reads

J Y = 2ρη D

The transport equation for the mixture fraction PDF, p (η; x, t ), can be derived starting from

∂ρΨ ∂Ψ ∂ρ ≡Ψ +ρ . ∂t ∂t ∂t

Considering (14) it is evident that a transport equation for

Y α |η can be derived from an equation for Ψ Y α . This equation

 ∂ ρη  N Y α |η  p ∂ Y α ∂ξ  η p− . ∂ xi ∂ xi  ∂η

(17)

(18)

The J Y term is unclosed. The primary closure hypothesis of CMC is obtained by neglecting correlations with Y α in modelling of J Y . Inserting for Y α = Q α and consequently ∇ Y α = ∇ Q α + ∇ξ ∂ Q α /∂ η and neglecting the term D ∇ Q ∇ξ by assuming high Reynolds numbers, we obtain the following closed expression for J Y (a more thorough analysis if the primary closure hypothesis is given in [2]) J Y = −Q α

∂ ρη  N |η  p ∂Qα + ρη  N |η  p . ∂η ∂η

(19)

The validity of the primary closure hypothesis has not yet been well established, but naturally it will be a good model for systems with little conditional variance, which is also where first order CMC is known to perform well. With the primary closure hypothesis in place the CMC-equations can be solved by closing the remaining unknowns p,  N |η, u i |η and u i Y α |η. Good models are here generally available for binary mixing, which is further discussed in Section 3. 2.3. Basic equations for two-fluid models of two-phase flows In Section 2.2 CMC was derived from the local instantaneous equations (5) and (6) governing single-phase flows. To derive CMC in two-phase flows, the corresponding equations for two-phase flows are required. In this section we start by giving a brief summary of the mathematical tools required for the treatment of local instantaneous two-fluid models of two phase flows. We will, in this regard, repeat some of the introduction given by Zhu et al. [18]. For more complete formulation of the approach, reference is given to Kataoka [15]. The two-fluid model (also called the separated flow model) follows the assumption that the overall flow field can be divided into non-overlapping, spatially extended regions, each region containing exclusively material from one single phase. The two different phases are separated by interfaces that are assumed to be infinitely thin and thus without mass. From this definition the interfaces are simply locations where exchange of mass, momentum or species may occur. The formulation also leads to a set of jump conditions that must be satisfied across a boundary. These jump conditions are required to conserve overall mass, momentum, energy or species. The most important tool used in derivation of the two-fluid model for two-phase flows is the introduction of a field variable β(x, t ). In phase space βˆ (the sample space of β ) the location of an interface is defined by β = β I , where the superscript I is used to describe an interface. In phase 1 we now assume that β < β I

M. Mortensen, R.W. Bilger / Combustion and Flame 156 (2009) 62–72

and in phase 2 β > β I . Tracking of an interface is achieved through the level set function

∂β + uI · ∇β = 0, ∂t

(20)

where uI is the velocity vector of the interface. Equation (20) has physical meaning only on the interfaces. In this work the level set equation will be used merely as a mathematical tool to derive statistical models and the level set equation will thus not be solved directly. Hence we do not have to give a physical interpretation of β off the interfaces. On any level surface the normal unit vector nk for phase k is defined as nk = (−1)k+1

∇β . |∇β|

(21)

Hence n1 points inwards to phase 2 and n2 points into phase 1. Note that throughout this paper there is no summation implied by repeating k’s, since k merely is used as a phase indicator. It is useful to manipulate Eq. (20) by adding and subtracting uk · ∇β to obtain a slightly different form

∂β + uk · ∇β = B k , ∂t

(22)

B k = −u¯ k |∇β|,

(23)

where the relative velocity u¯ k of phase k is defined as





u¯ k = u − uk · n1 . I

(24)

Through Eq. (22) we are now able to track an interface and for a given value of β we are able to tell whether phase 1 or 2 resides at location x at time t. However, it is useful to also define a phase indicator function θk (β; x, t ) that is unity for one phase and zero for the other. We define θ1 to be unity in phase 1 and zero in phase 2 and vice versa for θ2 . In mathematical terms θk is given as1 k−1

θk = 0

k



 I

+ (−1) H β − β .

(25)

Here H denotes the Heaviside function defined as H (a) = 0 for a < 0 and H (a) = 1 for a > 0. (The Dirac delta function, δ , introduced in Section 2.2, is the derivative of the Heaviside function.) By expanding the total derivative of θk over β the governing equation for θk can be found as [18]

∂θk + uk · ∇θk = Πk , ∂t

(26)

where

  Πk = (−1)k B k δ β − β I .

(27)

In the case of spray combustion, Πk is the volumetric rate of evaporation of fuel per unit volume. This is perhaps more evident using the local and instantaneous interfacial area concentration a (area per unit volume), which is defined as [15]





a = |∇β|δ β − β I .

Since there is only one phase present in the interior of each phase, the complementary governing equations for continuity and species transport in phase k are exactly the same as in a single-phase flow, i.e. Eqs. (5) and (6). To avoid confusion we write these equations here with a phase indicator. The continuity and species transport equations valid inside phase k are

∂ ρk ∂ ρk uk,i + =0 ∂t ∂ xi

(28)

(30)

and

∂ ρk Y k,α (uk,i + V k,α ,i ) ∂ ρk Y k,α + = ρk W k,α , ∂t ∂ xi

(31)

respectively, where ρk and Y k,α are the density and mass fraction of species α in phase k, V k,α ,i represents a component of the mass-diffusion velocity and W k,α is the rate of production of species α in phase k. The indicator function can be incorporated into Eqs. (30)–(31) directly through multiplication with θk , then rearranging and using (26), which gives transport equations that are valid throughout phase k all the way up to the interface. The extended equation set is

∂θk ρk ∂θk ρk uk,i + = ρk Πk ∂t ∂ xi

where uk is the velocity of phase k and B k is defined as

65

(32)

and

∂θk ρk Y k,α ∂θk ρk Y k,α (uk,i + V k,α ,i ) + ∂t ∂ xi = ρk Y k,α (Πk + Vˆ k,α ) + θk ρk W k,α ,

(33)

where Vˆ k,α = V k,α ,i

∂θk ∂ xi

  = (−1)k |∇β|δ β − β I Vk,α · n1 = (−1)k aVk,α · n1

(34)

is the volumetric diffusion velocity across an interface per unit volume (i.e. units are s−1 ). Here it is necessary to comment on the source terms appearing on the right hand side of (32) and (33). Through the application of a level set and indicator function the sources appear naturally as part of the transport equations. However, within phase k the surface terms (Πk and Vˆ k,α ) are zero and Eqs. (30) and (31) are regained since θk = 1. On the interface the only nonzero terms are Πk and Vˆ k,α . In the second phase both left and right hand sides are zero. Hence, Eqs. (32) and (33) are valid for all positions and times. This brief analysis reveals the modeling assumption in twofluid models that at any position and time we have either phase k, the other phase, or an interface, but never two at the same position and time. Since the interface is without mass and volume, mass cannot be generated on the interface, which leads to the jump condition [15] 2 

ρk Πk = 0.

(35)

Πk can now after trivial manipulations be written as a more infor-

k=1

mative velocity times area term

Since the interface is without mass and volume, species can neither be generated nor produced on the interface. This leads to another jump condition

k+1

Πk = (−1)

u¯ k a.

(29)

There is a great body of literature on modelling of a, that often is found from its own transport equation (cf. Morel [22], and references therein).

1

00 ≡ 1.

2 

ρk Y k,α (Πk + Vˆ k,α ) = 0.

(36)

k=1

Here we have only considered diffusion controlled volumetric reactions, i.e., no surface reaction mechanisms as found in e.g. char

66

M. Mortensen, R.W. Bilger / Combustion and Flame 156 (2009) 62–72

oxidation or mono-propellants. Equation (36) ensures that for Fickian diffusion Spalding’s [23] well known equation for interface mass transfer in sprays is obtained, i.e.

˙ (Y 1,α − Y 2,α ) = ρ2 D 2,α ∇ Y 2,α · n1 . m

(37)

˙ = ρ1 u¯ 1 = ρ2 u¯ 2 from Here we have used the mass transfer rate m (35) and assume, like Spalding, that the liquid is a pure substance. Evident from (37), the level set approach is in agreement with fundamental theories on interface mass transfer, even though the layers surrounding the spray droplets are treated as massless interfaces. However, as a consequence of Eq. (37), the species mass fractions will in general be discontinuous across an interface. Note that interface diffusion was not considered by Zhu et al. [18], who as a result cannot (without further manipulations) incorporate the balance between convection and diffusion across interfaces.

Here we have also made use of phase averaging, so that

F  ≡

2.4.1. The mixture fraction PDF The mixture fraction PDF of a single phase set in the twophase flow field can be derived starting from the fine-grained PDF Eq. (13), here with the notation used for a two-phase flow (38)

The fine-grained mixture fraction PDF of phase k is here denoted as Ψk ≡ δ(ξk − η). For simplicity the phase indicator k will be dropped in the further derivations, where it should be understood that we are deriving equations valid for a single phase of a twophase flowfield. Multiplying Eq. (38) by θk and then rearranging using Eq. (25) we obtain

(39)

where the new interface terms appear on the second line. The remaining terms are exactly equal to the single-phase equation (13), except from the appearance of the indicator function. Averaging of Eq. (39) is performed exactly as in Section 2.2, making heavy use of (14) and neglecting the first term on the right hand side by assuming high Reynolds numbers. The resulting mixture fraction PDF-equation for a single phase set in a two-phase flowfield is then

∂θρη p ∂θρη u i |η p + ∂t ∂ xi =−

∂ 2 θρη  N |η p ∂ ρη ξ Vˆ ξ |η p − + ρη Π|η p . ∂η ∂ η2

 F |η  ≡

θ F |η , θ

(40)

(41)

  ˆ ρ ΠΨ  = ρ B δ(β − β)δ(ξ − η) ˆ p ξβ (η, β) ˆ = ρ B |η, β ˆ p ˆ (η|β) ˆ p β (β), ˆ = ρ B |η, β ξ |β

The local instantaneous transport equations governing twophase flows were derived in Section 2.3 simply by incorporating the phase indicator function θk into the equations governing each individual phase. These equations can be used directly to derive further statistical models like CMC. However, since the governing equations within each phase are the same in single- and twophase flows, we can derive the mixture fraction PDF and CMC equations for the separate phases simply by incorporating θk into the equations for the the fine-grained PDF and Y α Ψ , already derived in Section 2.2. The equations derived in this section are valid for all positions and times and, unless otherwise stated, the same equations apply to both phases. In spray combustion the only interesting phase for CMC is the gas-phase, where reactions occur. Modeling specific to sprays, though, enter only through the assumption of a pure liquid substance.

∂ ∂ (θ ρ Ψ ) + (θ ρ u i Ψ ) ∂t ∂ xi  ∂ ρ ξ Vˆ ξ Ψ ∂ ∂ 2θ ρ NΨ ∂Ψ = θρD − − + ρ ΠΨ, ∂ xi ∂ xi ∂η ∂ η2

and

for any quantity F . Note that by utilizing the definition ρη p = ρ  p˜ , it is straightforward to obtain a transport equation also for the Favré PDF p˜ . The source term ρ ΠΨ  contains two delta functions, Ψ and δ(β − β I ), but in Eq. (40) we have merely used ρ ΠΨ  = ρ Π |η p. Since both β and ξ are field variables, an equivalent and perhaps more intuitive form of the source term is (keeping β unfixed in the derivation)

2.4. Derivation of CMC for separated flows

 ∂ 2 ρk N k Ψk ∂ ρk Ψk ∂Ψ ∂ ρk u i ,k Ψk ∂ + = ρk D k k − . ∂t ∂ xi ∂ xi ∂ xi ∂ η2

θ F  θ

(42)

where p ξβ is the joint PDF of ξ and β , p ξ |βˆ is the PDF of ξ conditional on β = βˆ and p β is the PDF of β . Consequently, p ξ |βˆ (η|β = β I ) is the conditional PDF of mixture fraction on the gas side of the interface and p β (βˆ = β I ) is the probability of a surface. According to (42) the source will be proportional to the mixture fraction distribution on the interface, which is perhaps more intuitive than to the gas phase PDF as it appears in Eq. (40). The conditional diffusion and convection across an interface are related through the jump conditions. If we assume the liquid contains pure fuel, as in spray combustion, the interface diffusion in Eq. (40) can be represented through (37) as

ξ Vˆ ξ = Π(1 − ξ ),

(43)

which means that the only new term that requires closure in Eq. (40) is Π |η. If the second phase does not contain pure fuel it is still possible to simplify using the jump conditions without further assumptions. Note that Eq. (40) now will be the same as obtained by [24,25], except from the (physically sound) appearance of the gas volume fraction θ in the transport equations. The mixture fraction PDF is in practical applications usually modelled with a presumed form taking the first two moments of the mixture fraction as input. These moments need to be found by individual transport equations that should be consistent with (40). Integration of Eq. (40) over η -space yields the average continuity equation

∂θρ  ∂θρ u i  + = ρ Π. ∂t ∂ xi

(44)

Integration of Eq. (40) times η and η2 yields the equations for the first and second raw moments of the mixture fraction respectively, i.e.

∂θρ ξ  ∂θρ u i ξ  + = ρ Π ∂t ∂ xi

(45)

and

  ∂θρ ξ 2  ∂θρ u i ξ 2  + = −2θρ N  + 2ρ ξ Π − ρ ξ 2 Π . (46) ∂t ∂ xi Here ρ ξ Π and ρ ξ 2 Π will depend on the modelling strategy chosen for the liquid phase. 2.4.2. The CMC equations To derive the CMC equations valid in a single phase of in a twophase flowfield, we need a transport equation for θk Y α ,k Ψk . Such an equation can be derived from the fine-grained PDF equation (39) by multiplying with Y α and then rearranging. Alternatively we can multiply Eq. (16) (with notation modified for two-phase)

M. Mortensen, R.W. Bilger / Combustion and Flame 156 (2009) 62–72

with θk and rearrange using (25). Either way we obtain after performing some exact manipulations (omitting the phase indicator k)

∂θ ρ u i Y α Ψ ∂θ ρ Y α Ψ + ∂t ∂ xi   ∂YαΨ ∂ ∂ Y α ∂ξ ∂ θρD +2 θρD = Ψ ∂ xi ∂ xi ∂η ∂ xi ∂ xi 2

∂ (θ ρ N Y α Ψ ) + θ ρ W α Ψ ∂ η2 ∂ (ρ Y α ξ Vˆ η Ψ ) + ρ Y α (Π + Vˆ α )Ψ. − ∂η −

(47)

All terms on the last line of (47) follow from the dynamic boundary conditions. The other terms are unchanged from the singlephase Eq. (16), except from the appearance of the indicator function. The above derivation is valid for any two-phase flow and the equations are equally valid for both gas and liquid phases. Averaging of Eq. (47) leads to

∂θρη p Q α ∂θρη p u i Y α |η + ∂t ∂ xi  ∂ J kY ∂  − ρη Y α ξ Vˆ ξ |η p ∂η ∂η   + ρη Y α (Π + Vˆ α )|η p ,

(48)

where we have assumed high Reynolds numbers2 and neglected the first term on the right hand side of (47). The unclosed term J kY is given as



 ∂θ ρ N Y α |η p ∂ Y α ∂ξ  η p− .  ∂ xi ∂ xi ∂η

(49)

We now proceed exactly as in single-phase flow [see Eq. (19)] and invoke the primary closure hypothesis for J kY . The closed form for J kY then becomes J kY = − Q α

∂ 2 θρη p  N |η ∂2 Q α + θρη p  N |η . 2 ∂η ∂ η2

(50)

This concludes the general derivation of CMC for a single phase set in a two-phase flowfield. For spray combustion, however, it is possible to assume the liquid is a pure substance and thus the conditional interface diffusion can be eliminated by making use of the jump conditions. Using (35) and (43) we finally get

∂θρη p Q α ∂θρη p u i Y α |η + ∂t ∂ xi = θρη p  W α |η +

 ∂ J kY ∂  − ρη (1 − η)Y α Π|η p ∂η ∂η

+ Q 1,α ρη Π|η p ,

(52)

Here u i Y α |η is the turbulent conditional flux and the last term comes from Y α Π|η in the conservative CMC equation (51), where we have inserted for Y α = Q α + Y α and Π = Π|η + Π  . To summarize, we have here started from Kataoka’s two-fluid formulation as a fundamental local instantaneous description of two-phase flows. We have then averaged these equations conditional on the event that the conserved scalar ξ = η . The major assumptions are that (i) the mass-diffusion rate follows Fick’s law; (ii) the diffusion coefficients are equal for all reacting and non-reacting species; (iii) the Reynolds number is high; (iv) the conditional fluctuation in the density is negligible and (v) the primary closure hypothesis is invoked. Assumptions (i)–(iv) are made mainly to simplify the equations and can easily be relaxed by performing somewhat more elaborate mathematics. 2.5. Comparison with current literature

= θρη p  W α |η +

J kY = 2 θ ρ D

∂Qα ∂Qα +  u i |η  ∂t ∂ xi ∂θρη p u i Y α |η 1 ∂2 Q α =  N |η  +  W | η  − α θρη p ∂ xi ∂ η2 

∂ Q α Π|η + Q 1,α − Q α − (1 − η) ∂η θ ∂(1 − η)ρη p Y α Π  |η 1 − . θρη p ∂η

67

(51)

where Q 1,α is the mass fraction of species α in the liquid droplets ( Q 1,α = Y 1,α ). The equations obtained in (48) and (51) are given in conservative form. The equations can be further manipulated by using Eq. (40) to extract an equivalent ‘classical’ CMC-equation simply for Q α . By multiplying Eq. (40) with Q α and subtracting from Eq. (51) the classical form can be obtained

2 Only the gaseous flow is turbulent in spray combustion and thus the high Reynolds number assumption is valid only for the gas phase.

Mainstream approaches to modelling gas phase chemistry in sprays [13,14] usually assume that it can be treated in much the same way as in single phase systems [1]. Little attention has been given to the possible effects of strong mixture fraction variations in the neighbourhood of the evaporating droplets on the chemistry in a turbulent spray flame. Potentially, CMC has the advantage that it can handle complex chemistry and such effects of the fine scales of the mixing. As mentioned in the introduction, CMC is already being applied to the combustion of two-phase flows including sprays [8–11] and solid particulates [12]. There is considerable inconsistency between these investigations as to the additional terms that need to be included in the CMC equations. In this section we provide a brief summary of the discrepancies of the equations used in these earlier works in comparison with the rigorously derived equations presented here. The PDF transport equation is an inherently important adjunct equation in the CMC framework. It has been formulated in many works that are not specifically concerned with CMC. As noted earlier, the equation derived in many cases (e.g. [24,25]) does not include the gas volume fraction θ inside the derivatives as in our Eq. (40) and in the equations for continuity and mixture fraction moments, Eqs. (44)–(46). The contribution of this modification will be small in dilute sprays. Rogerson et al. [12] studied a bagasse-fired boiler where solid particles were treated as point sources or as a ‘pseudofluid’ in a system which was effectively gas phase only. Here both fuel and moisture were treated as separate sources to Eqs. (30) and (31). The sources (the liquid droplets) were assumed to be part of the gaseous phase, somewhat like a stochastic point signal. The major feature of the level set approach is the rigorous mathematical treatment that directly includes the volume fraction of gas/liquid into the transport equations. In the ‘pseudofluid’ approach the volume fraction of gas is set to unity. Neglecting moisture the following stationary CMC equation was reported

 ∂ ρη p  u i |η  ∂ ∂Qα  u i |η  Q α − D T = Qα + ρη p  W |η  ∂ xi ∂ xi ∂ xi

 ∂Qα ∂2 Q α + ρη p  N |η  + S (η)ρη p Q 1,α − Q α − (1 − η) , (53) ∂η ∂ η2

ρη p

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M. Mortensen, R.W. Bilger / Combustion and Flame 156 (2009) 62–72

where S (η) is the source of fuel and a gradient diffusion model has been used to close u i Y α |η. Evidently, the derived equations do not include the volume fraction of gas. Furthermore, conditional fluctuations in the source are neglected and thus the last term of Eq. (52) is not included. The last two lines are similar to the current work. The CMC equation used by Smith et al. [8] and Kim et al. [9] is similar to the one obtained by Rogerson et al. However, in this derivation the fuel source was neglected in the continuity equation. The resulting equation is the same as the single-phase CMC equation with an additional term  S |η∂ Q α /∂ η due to interface dynamics. Wright et al. [10] studied autoignition in sprays, but made no adjustments at all to the single-phase CMC nor the mixture fraction moment equations, arguing that modifications would be small. Klimenko and Bilger [2] described a higher level doubly conditioning approach for sprays with two mixture fraction variables, one for the fluctuations in the bulk gas phase and another for the inter-droplet fluctuations. Like all doubly conditioning models, the computational cost is considerably higher than for the single variable approach followed in classical CMC and in this work. The CMC equations formulated for the inter-droplet regime were not modified. 3. Closure of the CMC equation The primary objective of this paper is to rigorously derive the fundamental equations for CMC for two-phase reactive flows and in particular for evaporating and combusting sprays. It is important to realize, though, that CMC itself is merely a framework requiring further modelling of various unclosed terms. It is not the intention of the current paper to cover all aspects in relation to these unclosed terms. This would require, e.g., the discussion of all modes of spray combustion. In this section we give a general discussion of the treatment of the unclosed CMC-terms in relation to sprays. The CMC equations require closure of u i |η, u i Y α |η, Y α Π  |η, Π|η, θ and the PDF or  N |η, viz. the conditional scalar dissipation rate (CSD). If solution of Eq. (45) for the mean mixture fraction and Eq. (46) for the variance of the mixture fraction is needed in modelling the PDF then closure for ρ ξ Π and ρ ξ 2 Π in their source terms will also be needed. It is noted that it is sufficient to model either the PDF or the CSD because one can be computed from the other through Eq. (40). If the PDF and CSD are modelled separately the system is over specified and no fully consistent solution can be found. There are a wide variety of methods that are already wellestablished for the prediction of atomization, droplet dispersion, evaporation, mixing and combustion in sprays [13,14]. Problems of interest range from auto-ignition and spark ignition to soot formation, radiation and pollutant emission. In turbulent flows there is a vast number of dimensionless parameters that are of relevance. The parameter space is huge and even well-established prediction methods have a limited range in this parameter space in which they can be claimed to be valid. Important research on sprays is being devoted to improving spray modelling over the range of this parameter space and for increasing the breadth of scope of particular methods. CMC is seen as being a potential supplement to many of the existing methods for the prediction of sprays. In general, there will be no change to the existing method chosen for use in the modelling of atomization, droplet dispersion, evaporation and mixing in the spray. CMC is seen as providing a refinement on the treatment of the gas phase chemical reactions. Some feedback from the improved treatment of the chemistry will occur because of the effects on gas phase density and temperature.

Prediction of θ will come from the modelling for spray evaporation in the general modelling framework that is adopted. No further consideration of the modelling for this term is needed. Modelling of ρ ξ Π, ρ ξ 2 Π, Π |η and Y α Π  |η requires modelling for the mixture fraction at the droplet surface and how the evaporation rate and reactive species mass fractions vary with variations in the surface mixture fraction. When approached from the perspective of investigating the effects of the details of the flow, heat and mass transfer near the droplet surface and in the bulk of the gaseous phase, this becomes a task well beyond the capabilities of present day laboratory measurements or of computer simulation. Even for spherical droplets that are not spinning due to gas phase shear and that have negligible effects of motion in the liquid phase, there are strong variations in the quantities of interest around the droplet periphery and there are significant effects of many other conditions such as the velocity of the droplet, interdroplet spacing and the bulk gaseous phase conditions. The capabilities of direct numerical simulation (DNS) are currently far from resolving conditions down to the droplet surface even for simplified problems, let alone taking into consideration droplet shape variation, spin and internal circulation. It seems, however, that acceptable models for these quantities may be obtainable from application of the principle of the similarity between heat and mass transfer. As an example of the application of the principle of similarity between heat and mass transfer to this modelling, we outline in the next subsection the use of the Spalding B number approach [23] applied to Lagrangian-type modelling [13,14] of individual droplet dynamics and evaporation in spray combustion. Essentially the same principles can be reformulated for other treatments of heat and mass transfer and for other frameworks for spray modelling. The terms u i |η and u i Y α |η are significant throughout the gas phase. Models for these terms that are used in single-phase CMC are reviewed. DNS that has good resolution of the flow and mixing in the inter-droplet space as well as providing simulation of the large-scale mixing environment should provide some insight into how these models need to be modified in sprays, if at all. Such DNS simulations will not yield valid data for the higher levels of mixture fraction associated with values relevant for the evaporating surfaces: this is due to their inability to resolve the flow right to the surfaces. These issues are addressed in Section 3.2. Modelling of the CSD and the PDF are also significant throughout the gas phase. Modelling approaches for these that are used in single-phase CMC are briefly reviewed, including the constraint imposed by the PDF transport equation which requires that only the CSD or the PDF can be independently modelled. Once again, appropriate DNS should provide insight into how these models need to be modified. These issues are addressed in Section 3.2, also. Section 3.3 gives a brief discussion of the model problems that should prove to be useful in investigating these issues. The focus is on model problems that are amenable to DNS investigations that are likely to be most productive. These model problems will also be important in LES and RANS studies. 3.1. Application of heat and mass transfer similarity Here we outline an example of the application of the principle of similarity between heat and mass transfer to the modelling of the unclosed CMC terms involving the droplet evaporation rate. We do this using the Spalding B number approach applied to Lagrangian-type modelling [13,14,23] of individual droplet dynamics and evaporation in spray combustion. Essentially the same

M. Mortensen, R.W. Bilger / Combustion and Flame 156 (2009) 62–72

principles can be reformulated for other treatments of heat and mass transfer and for other frameworks for spray modelling. The fundamentals presented here are essentially the same for Lagrangian modelling applied in Reynolds-averaged Navier Stokes (RANS) modelling of the general flow or in large eddy simulation (LES). The results may also be applicable for so-called DNS of sprays where droplets are treated as point sources and no attempt is made to resolve spatial variations in the gas phase all of the way to the droplet surface. In the RANS and LES frameworks, the effects of the unresolved gas phase turbulence are treated as a Weiner process with the parameters involved being taken from the modelling of the turbulence at the relevant scale—large scale or sub-grid scale. In the so-called DNS the droplet dynamics are determined from the resolved velocity field. Spalding’s [23] similarity law between heat and mass transfer may be stated: ln(1 + B M ) = Le2/3 ln(1 + B H ).

(54)

This derives from the effects of Prandtl number and Schmidt number on the relative rates of heat and mass transfer [11,13,14]. Here, B H is the Spalding transfer number for heat transfer and B M the Spalding transfer number for mass transfer and Le is the molecular Lewis number for the fuel. For the commonly made assumption that Le = 1, B H = B M ; but for most liquid fuels of interest, Le > 1. The Spalding transfer number for heat transfer, B H , is defined in terms of standardized enthalpy, a conserved scalar, and is assumed to be obtained from one or other of the two limiting cases defined by the presence or absence of an enveloping flame. A suitably defined critical Damköhler number can be used to discriminate between these two limiting possibilities: for droplets in which the relative velocity is too high or the droplet diameter is too small the Damköhler number would fall below the critical value that would allow for an enveloping flame. An enveloping flame is assumed for droplets with a high Damköhler number. (Most current DNS do not incorporate allowance for such consideration of envelope flames and this strategy could be an improvement. Sreedhara and Huh [11], for example, assume that their DNS is wholly within a group combustion mode according to the classification of Chiu et al. [26] and so the possibility of envelope flames is neglected.) The transfer number for mass transfer, B M , is defined in terms of a Shvab–Zeldovich conserved scalar involving species mass fractions and is also assumed to be given for one or other of the two limiting cases—with or without an envelope flame. Nominal conditions at the droplet surface for temperature and fuel mass fraction are obtained by matching the requirements of Eq. (54) and of the need for the fuel vapor pressure in the gas phase at the droplet surface to be that for equilibrium with the droplet surface temperature. The mixture fraction and the mass fractions for other species on the gas side of the droplet surface are also obtained. Details of this heat and mass transfer analysis are given in Appendix A. The evaporation rate from the droplet is that given by the framework model for the droplet dynamics. In this way, it is possible to arrive at acceptable statistics for the mixture fraction and reactive species mass fractions on the gas side of the droplet surface and their correlations with the evaporation rate of the droplets. These statistics provide closure for the terms ρ ξ Π, ρ ξ 2 Π, Π|η and Y α Π  |η without further modelling in RANS, LES and the so-called DNS frameworks for the general flow. 3.2. Gas phase terms As outlined above, we are concerned here with modelling approaches for the unclosed terms that are significant throughout the

69

gas phase. We consider first the terms u i |η, u i Y α |η. Next, we consider the modelling of the CSD and the PDF. In single-phase flows, models for u i |η include the linear model [27], the Li and Bilger model [28] and the gradient diffusion model [29]. These have been evaluated for mixing in, e.g., single [30] and double scalar mixing layers [31]. It is apparent from Ref. [31] that u i |η depends very much on the large-scale gradients of the mixture fraction of the flow, and there is no universal shape as implied by the linear model. It can be expected that such large-scale effects will also be dominant in ‘non-premixed’ sprays where there are strong large-scale gradients of the overall (liquid plus vapor) mixture fraction in the whole flow. It appears that only the gradient diffusion model can adapt to more complex shaped PDFs and also satisfy the regular criteria of a consistent model in a general flow. Since the gradient diffusion model is proportional to the gradient of the mixture fraction PDF, any improvement in modeling of the PDF will thus also indirectly affect the conditional velocity model. The term u i Y α |η is also significant throughout the gas phase. In single-phase CMC for non-premixed combustion, this term is usually modelled by a simple gradient diffusion model





u i Y α |η = − D T

∂Qα . ∂ xi

(55)

In premixed combustion the mixture fraction is fixed at a single value and this term becomes the same as the unconditional species flux. Under these conditions this flux can be counter-gradient in many problems of interest. In single-phase non-premixed problems, the simple gradient model often works well even in autoignition problems where propagation from an initial ignition center involves propagation of a premixed-like front [3]. It can be expected that this modelling will also be valid in ‘non-premixed’ sprays where there are strong large-scale gradients of the overall (liquid plus vapor) mixture fraction in the whole flow. Only at high values of η , values that are comparable with those at the inter-phase surface, can significant effects of the inter-droplet mixing field be expected to be important in such flows. The most important unclosed terms in single-phase flows are usually the CSD and the mixture fraction PDF. These terms are closely related, since one implies the other through the PDFequation (15). The most popular single-phase models include the β -PDF and the counterflow CSD, which also happens to be the model obtained using a presumed mapping function (PMF) approach [32]. There is no physical explanation known for the success of the β -PDF, but the counterflow model is easily understood from consideration of a single laminar diffusive mixing layer. In a 1D diffusive mixing layer where ξ is zero on one side and unity on the other, the conserved scalar profile is given exactly as ξ = 0.5(1 + erf( y )), where y is the local normal coordinate that spans the mixing layer. Inserting for the definition in Eq. (12), leads to the exact local and instantaneous mixing rate





2 

N (ξ ) = D /π exp −2 erf−1 (2ξ − 1)

,

(56)

which, when conditionally averaged and scaled, is exactly the counterflow and PMF mixing models. A remarkable feature of (56) is that the models shape is independent of mixing rate. In homogeneous mixing the counterflow CSD leads to the PDF of the mapping closure [33] through the exact PDF equation. Note that the shape predicted by Eq. (56) also is very close to the shape obtained using an assuming β -PDF and then integrating the PDF-equation to obtain  N |η [34]. The single-phase models are not likely to perform well in sprays since the laminar mixing layer scenario outlined above does not occur at any scale in the spray configuration. The high values of mixture fraction are confined to the near droplet region and in three dimensions with modest droplet loading this implies a very

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low probability, and not a secondary peak like in purely nonpremixed systems (see, e.g., Demoulin and Borghi [24]). In fact, the fuel will enter the gas-phase with an unknown premixed composition, a quantity that can be described by the mixture fraction distribution on the gas-side of the droplet surfaces, i.e. p ξ |βˆ (η|β = β I ) in Eq. (42). The single-phase models could provide adequate results in sprays for low values of η and modifications are then required only for the high end. This observation is also plausible for the other unclosed terms discussed above. The adequacy of the single-phase flow models at lower values of η can be investigated by appropriately designed DNS. Such DNS will need to provide sufficient resolution of the inter-droplet gaseous field while at the same time simulating the large-scale mixing field. 3.3. Model problems for study As indicated above, DNS studies have the potential of being helpful in giving insight into the modelling of the terms that are significant in the gas phase in spray mixing and combustion problems. The design of such DNS studies needs to take into account the wide range of issues that are associated with the many length and time scales that are associated with the droplets, their spacing and their large-scale distribution; and also with the length and time scales associated with the turbulent velocity field. It is noted that, here, we seek DNS that resolves the flow and mixing in the inter-droplet space while still simulating the overall mixing environment of the spray: resolution down to the droplet surface is not sought. Specification of the overall characteristics for model problems suitable for DNS studies is discussed. The suitability of these model problems for RANS and LES study is also noted. In existing DNS of sprays such as Réveillon and Vervisch [35], Smith et al. [8] and Sreedhara and Huh [11], the resolution is not sufficient to resolve the mixture fraction structure in the interdroplet space. Better resolution of inter-droplet flow and mixing is obtained in studies of mixing and combustion of droplet ‘clouds’ involving only a few dozen droplets [16]. It is difficult, however, to obtain statistical quantities of interest from such droplet cloud simulations as they are essentially 4-dimensional (space and time) in nature and only involve a relatively small number of droplets. In investigations of single-phase problems in turbulence, most progress has been made by the study of problems of lower dimensionality such as in temporally-evolving scalar mixing layers in decaying isotropic turbulence [36]. It seems that DNS investigations of spray analogues of such problems should have high potential for elucidating many of the issues in the modelling of mixing and combustion of sprays if they have appropriate largescale simulation and sufficient resolution at the inter-droplet scale. Combustion science has traditionally recognized the limiting cases of ‘premixed combustion’ and of ‘non-premixed combustion’ for which the underlying fundamental science is completely different. In single-phase combustion problems, theoretical analysis and modelling approaches reflect this major difference in the basic science [1]. It is apparent that investigations of spray combustion would also be well served by recognition of such differences in the nature of the limiting case that is most appropriate to the problem under study. The limiting case in which the spatial variations of a spatially filtered overall (liquid plus gaseous) mixture fraction are small could be designated as a ‘premixed spray’ problem. In these situations, such as in explosions, the auto-ignition and flame propagation in such sprays are problems of major interest. Definition of the ‘non-premixed’ limiting case is clouded by the perceived need to define the unmixed fuel stream as a two-phase fuel mixture uncontaminated by the presence of air. While this is theoretically possible for an unmixed fuel stream consisting of just liquid fuel and its pure vapor, it is not very practical as in most

problems of interest the evaporation of the fuel involves heat and mass and momentum transfer from the air stream. Extensive work in single-phase flows involving mixing and combustion of fuel streams that are partially premixed with air, have indicated that these problems can be treated as essentially non-premixed problems if the fuel stream equivalence ratio is well-beyond the rich flammability limit [37,38]. It is apparent that useful simulations for ‘non-premixed’ spray problems can be for an unmixed ‘fuel’ stream that is partially premixed with air, provide that the unmixed ‘fuel stream’ is well beyond the limits for flammable front propagation in such a mixture. In many practical situations of interest in engineering combustor design and analysis involving sprays, the basic ‘nonpremixed’ nature of the process is likely to be dominant. This will be associated with the fact that mixing processes are associated with the large scales of the turbulence and with largescale variations in the spray distribution. In such problems the mixing characteristics associated with these large-scale features of the flow are likely to be of at least equal significance to those associated with the inter-droplet spacing and the near droplet surrounds. ‘Premixed’ treatments of the physics of the local combustion in the spray are thus likely to be questionable. It is apparent that the simplest model problem suitable for the study of such ‘non-premixed’ spray problems will be a suitable analogue of the single-phase scalar mixing layer in decaying isotropic turbulence. This problem is essentially one of onedimension in space and one-dimension in time. It is proposed that the ‘fuel’ stream in such studies be a partially premixed mixture of the liquid fuel of interest and of the air stream into which it is mixing. This will allow the definition of a single overall (liquid plus vapor) mixture fraction to define the mixing. Furthermore it is proposed that the unmixed (but partially premixed) ‘fuel’ stream be in heat and mass transfer equilibrium. This is so that heat and mass transfer processes do not need to be followed in this unmixed region of the fuel stream. This simplifies the scalar boundary condition in this part of the flow. This general modelling scenario is applicable for auto-ignition problems and for problems that are directed at modelling of pollutant formation and extinction and re-ignition. It appears that no studies of this niche in parameter space of such simple spray mixing problems have so far been carried out. Model problems of such spray scalar mixing layers in decaying isotropic turbulence with combustion conditions appropriate for auto-ignition or for finite rate chemistry effects near to the fast chemistry limit will be also be of interest in LES and RANS studies. This arises from the reduced overall dimensionality and from the simplicity of the spatial boundary conditions. It is concluded that the study of the spray analogue of the scalar mixing layer is an important model problem for nonpremixed spray studies. 4. Conclusions In this work we provide a sound derivation of the CMC equations for two-phase flows. The local instant two-fluid formulation of Kataoka [15] has been used to derive the unclosed fundamental equations for the conditional moment closure [2], applicable to a single fluid phase of a general two-phase flow. The resulting equations have significant differences from the equations used in earlier CMC studies. It is noted that CMC only seeks to provide a refinement of the treatment of gas-phase reactions in sprays and that it can be used as an adjunct to many existing approaches for the modelling of atomization, droplet dispersion, mixing and evaporation in sprays.

M. Mortensen, R.W. Bilger / Combustion and Flame 156 (2009) 62–72

The unclosed terms in the CMC equations that are associated with conditions at the droplet surface are recognized as being well beyond the scope of assessment by even the most advanced laboratory measurements or DNS. It is proposed that the use of the well-established similarity between heat and mass transfer is the way ahead for the modelling of these terms: an outline is given for how this approach can be applied in detail for one example of a general approach to spray modelling. Other unclosed terms in the CMC equations are significant throughout the gas phase in sprays. Experience in the modelling of these terms in single-phase CMC studies is reviewed. It is suggested that investigation of any modification of such modelling for the effects of evaporation from the liquid phase can best be studied by appropriate DNS that has good resolution of the flow and mixing in the inter-droplet space: resolution of the field close to the droplet surface need not be necessary. The spray analogue of the classical turbulent mixing problem of a scalar mixing layer in decaying isotropic turbulence is proposed as the model problem for DNS studies. It is noted that this model problem will also be relevant for LES and RANS modelling studies.

If there is significant reaction near the droplet, then for a onestep simple chemical reaction that is assumed fast enough so that no oxygen is present on the gas side of the droplet surface (the S state), the Spalding [23] result is BH =

BM =

In this appendix we provide further detail of the application of heat and mass transfer similarity outlined in Section 3.1. The Spalding transfer number for heat transfer, B H , is assumed to be obtained from one or other of the two limiting cases defined by the presence or absence of an enveloping flame. A suitably defined critical Damköhler number can be used to discriminate between these two limiting possibilities: for droplets in which the relative velocity is too high or the droplet diameter is too small the Damköhler number would fall below the critical value that would allow for an enveloping flame. (Most current DNS do not incorporate allowance for such consideration of envelope flames and this strategy could be an improvement. Sreedhara and Huh [11], for example, assume that their DNS is wholly within a group combustion mode according to the classification of Chiu et al. [26] and so the possibility of envelope flames is neglected.) At such low Damköhler number, B H will be that for chemically frozen transfer to the droplet surface: BH =

C P (T G − T S ) Q

(A.1)

where C P is the gas specific heat at constant pressure, Q is the latent heat of evaporation of the droplet liquid, corrected for any contributions from radiation heat transfer and droplet heat up, T G is the bulk gas temperature remote from the droplet and T S the temperature on the gas side of the interface and determined from the balance between heat and mass transfer and physical equilibrium between the liquid and gas phases. Here, and in what follows, property values at the gas-side of the surface are assumed to be an appropriate circumferential average. G-state values are also assumed to be suitably averaged. (Any corrections that may arise from detailed definitions of these averages are neglected here.) Fluid properties, such as C P , are assumed to be evaluated at a suitable ‘film’ [23] temperature and interface properties such as Q at the droplet surface temperature and pressure.

Q

.

(A.2)

Y F,S − Y F,G 1 − Y F,S

(A.3)

,

where Y F is the fuel mass fraction. For the case with an envelope flame with fast chemistry, it is given by BM =

Appendix A. Heat and mass transfer analysis

C P ( T G − T S ) + Y O2 , G Q c / s

Here, Y O2 ,G is the mass fraction of oxygen in the bulk gas remote from the droplet, Q c is the heat of combustion per unit mass of fuel and s is the stoichiometric ratio of oxygen to fuel by mass. (In DNS with significant resolution in the interdroplet space, some care may be needed in defining the G state. Conceptually, the situation is similar to defining the bulk gas-phase properties at a particular location in a porous bed.) The transfer number for mass transfer, B M , is given for the case without an envelope flame by

Acknowledgments This work is supported by the Australian Research Council. The first author gratefully acknowledges this support during his time in Sydney as a postdoctoral fellow where most of his work on this project was completed. He also acknowledges partial support from the Norwegian Research Council through the Center of Excellence grant to the Center for Biomedical Computing.

71

Y F,S + Y O2,G /s 1 − Y F,S

.

(A.4)

Equilibrium between gas phase and the liquid at the interface requires that Y F,S = Y Fe ( T S ),

(A.5)

Y Fe ( T S )

can be computed from the vapor pressure vs temwhere perature relationship for the fuel and the system pressure of interest by making a suitable assumption for the mean molecular weight at the S state. The set of Eqs. (A.1)–(A.5) is sufficient to determine the S state temperature, T S , and fuel mass fraction Y F,S for any Lagrangian droplet, covering both cases—i.e. with or without an envelope flame. It is assumed that this refinement is also carried through into the expressions used for the heat and mass transfer rates. Of particular interest is the mixture fraction at the droplet surface. This is given by

ξS =

ξG + B , 1+ B

(A.6)

where B lies somewhere between B H and B M . Equation (A.6) is valid for both limits of whether there is an envelope flame or not. It can be derived from consideration of the mixture fraction being defined in terms of a conserved scalar such as Y F − Y O2 /s and with limits of unity in the pure liquid fuel and zero in the unmixed oxidant (air). At this stage it is not clear whether the B in (A.6) should be taken as closer to B H or to B M , and whether the choice should depend on the presence, or not, of an envelope flame. It is noted that for Le = 1 there is no problem as B H = B M . The gradient of the mixture fraction at the droplet surface can be estimated from the diffusive balance at the droplet surface. The circumferential mean magnitude will be given by

|∇ξ |S =

˙ (1 − ξS ) m (ρ D )S

.

(A.7)

˙ is the mass transfer rate per unit area of the droplet surHere, m face and is obtainable from the Sherwood number for the droplet, Sh: Sh ≡

˙ S 2mr . (ρ D )S ln(1 + B )

(A.8)

In DNS, Sh is usually assumed to be a simple function of Reynolds number and Schmidt number and is used to evaluate the droplet evaporation rate. In RANS and LES additional corrections are often

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M. Mortensen, R.W. Bilger / Combustion and Flame 156 (2009) 62–72

applied for the effects of the unresolved turbulence. Here, rS is the radius of the droplet. From (A.7) and (A.8) we obtain

|∇ξ |S =

Sh ln(1 + B )(1 − ξS ) 2rS

.

(A.9)

In (A.8) and (A.9) B will lie somewhere between B H and B M as noted above in determining ξS . Such estimates of the mean magnitude of the mixture fraction gradient at the droplet surface provide some measure of the upper limit on the conditional scalar dissipation for the mixture fraction which is proportional to the square of this gradient. In reality, there are strong circumferential variations of this gradient as there are, to a lesser extent, for the value of the mixture fraction at the droplet surface. In the context of the present analysis, it seems that the effects of such circumferential variations will be secondary to those arising from the fluctuations in ambient velocity, mixture fraction and reactedness. Other reactive species mass fractions on the gas side of the droplet surface, can be obtained from the usual frozen or fast chemistry functions [39] of mixture fraction for a simple reacting system. References [1] R.W. Bilger, S.B. Pope, K.N.C. Bray, J.F. Driscoll, Proc. Combust. Inst. 30 (2005) 21–42. [2] A.Y. Klimenko, R.W. Bilger, Prog. Energy Combust. Sci. 25 (1999) 595–687. [3] E. Mastorakos, R.W. Bilger, Phys. Fluids 10 (6) (1998) 1246–1248. [4] M.R. Roomina, R.W. Bilger, Combust. Flame 125 (2001) 1176–1195. [5] A. Kronenburg, R.W. Bilger, J.H. Kent, Combust. Flame 121 (2000) 24–40. [6] M. Mortensen, Chem. Eng. Sci. 59 (2004) 5709–5723. [7] A. Vikhansky, S.M. Cox, AIChE J. 53 (1) (2007) 19–27.

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