Chapter 3 CFD ENGINE SIMULATION 3.1 COMPUTATIONAL MODELS FOR ENGINE DEVELOPMENT

Chapter 3

CFD ENGINE SIMULATION

3.1

COMPUTATIONAL MODELS FOR ENGINE DEVELOPMENT

Current computational models play a fundamental role in engine research and development. Through their predictive capabilities they can reduce research and development time showing important information not available from the experiments. However, every model has a limited range of validity depending on the level of complexity of the model and on the fidelity of the experimental data against which it is calibrated. Within this operating range the models can also provide an acceptable level of confidence in making predictions. In addition to maintaining their good predictive capability, acceptable calculation times are also required even for detailed analysis. Sometimes, a compromise between accuracy and computational cost is necessary.

Simple kind of models like zero dimensional, quasi-dimensional or phenomenological models [45-47] can be used to predict average cylinder quantities like pressure and rate-of-heat-release with very small computational time but also with poor accuracy. However, not accounting for the complicated effects of engine geometry, turbulence, etc., on combustion and emissions, they cannot be used as a reliable design tool for real engines.

Multi-dimensional computational fluid dynamics (CFD) models solving for spatially resolved quantities are currently used for engine research and development [8,14]. A literature review of multidimensional models for engine simulations is made in this chapter, with emphasis on turbulent combustion models. Initially, the governing equations and fundamental concepts of turbulent reactive flows are reviewed then some of the computational models adopted in this study are briefly described. Finally, the application of detailed chemistry to engine combustion simulations is discussed in the last section of this chapter.

3.2. Governing Equations for Turbulent Reactive Flows

3.2.1. Navier-Stokes Equations

In turbulent reactive flows, the fluid dynamics is governed by the Navier-Stokes equations. Turbulence is considered by solving model transport equations. The interaction between the gas phase and the liquid drops is accounted for by

using exchange functions. Major assumptions include the use of the ideal gas law, Fick’s law for mass diffusion, and Fourier’s law for thermal diffusion [8].

The fundamental premise underlying turbulent combustion modeling is that it should remove the necessity of resolving all the smallest structures and fluctuations associated with turbulence. Therefore, in practice, it is both necessary and desirable to limit the dynamic range of length and time scales in the computational domain. This can be accomplished by various averaging procedures. For turbulent flows with large density changes such as occur in combustion applications, it is often convenient to use the density-weighted average, called the Favre average, to split an instantaneous quantity, for example

the flow velocity vector

u

, into an average

u

~

and a fluctuating part

u

′′

, as

u

u

u

= ~

+

′′

, where the average

u~

is defined in terms of the conventional average by

u

~

=

ρ

u

/

ρ

, and the fluctuation

u ′′

satisfies

ρ

u

′′

=

0

, where the overbar represents an averaging operator. The governing equations described in this section all follow this manipulation.

The continuity equation for species is:

k

( )

u

~

D

₁

(

k

1

,...,

K

,

t

k S c k k T k k

₊

₊

₌

⎥

⎦

⎤

⎢

⎣

⎡

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

∇

⋅

∇

=

⋅

∇

+

∂

∂

_ρ

_ρ

_δ

ρ

ρ

ρ

ρ

ρ

_&

_&

₎

(3.1)

where

ρ

is the density, subscript is the species index,

k

K

is the total number of species, is the flow velocity vector. Fick’s law diffusion with a single turbulent diffusion coefficient is assumed.

u

T

D

c

k

ρ

&

and

ρ

&

Sare source terms due to chemical reactions and spray evaporation, respectively. Species 1 represents the fuel in the spray and

δ

is the Kronecker delta

(

δ

_ij

=

1

if

i

=

j

;

δ

_ij

=

0

if

i

≠

j

)

. The summation of Eq. (3.1) over all species gives the continuity equation for the total fluid:

( )

_u

~

S

.

t

ρ

ρ

ρ

₊

_∇

_⋅

₌

_&

∂

∂

(3.2)

The momentum equation for the fluid is

(

)

~

,

3

2

~

~

~

g

F

I

k

p

u

u

t

u

_ρ

_σ

_ρ

S

_ρ

ρ

+

+

−

⋅

∇

+

−∇

=

⋅

∇

+

∂

∂

(3.3)

where

p

is the pressure,

σ

is the total viscous stress tensor (laminar plus turbulent) given by

(

)

(

~

)

,

3

2

~

~

⎥⎦

⎤

⎢⎣

⎡

_∇

₊

_∇

₋

_∇

_⋅

+

=

u

u

T

u

I

T

µ

µ

σ

(3.4) 20

µ

is the laminar dynamic viscosity,

µ

_Tis the turbulent dynamic viscosity, superscript means vector transpose, is the turbulence kinetic energy, defined by

T

k

,

2

1

~

u

u

k

=

′′

⋅

′′

(3.5)

I

is the identity tensor,

F

Sis the rate of momentum gain per unit volume due to the spray,

g

is the specific body force.

The turbulent dynamic viscosity

µ

_Tis related to the turbulent kinetic energy and its dissipation rate

k

ε

by

,

~

~

2

ε

ρ

µ

_T

=

c

_µ

k

(3.6)

where is a model constant that varies in different turbulence model formulations. By definition,

µ

c

T

µ

is related to the turbulent thermal diffusivity

α

_Tand mass diffusivity

D

_Tby

,

Pr

Pr

_T T T T T

_ρ

µ

υ

α

=

=

(3.7)

,

T T T T T

Sc

Sc

D

ρ

µ

υ

₌

=

(3.8)

where

υ

_Tis the turbulent kinematic viscosity, and are the turbulent Prandtl and Schmidt numbers, respectively. T

Pr

Sc

_T

Based on Eq. (3.3), the stress contribution due to turbulence, called the Reynolds stress tensor

τ

, is defined as

(

)

~

.

3

2

~

3

2

~

~

_u

_u

_I

_k

_I

u

u

u

T T

ρ

µ

ρ

τ

=

−

′′

′′

=

_⎢⎣

⎡

∇

+

∇

−

∇

⋅

⎤

_⎥⎦

−

(3.9) The internal energy transport equation is:

( )

~

~

~

~

~

~

~

,

~

S C

_Q

Q

J

u

p

I

u

t

I

₊

_∇

_⋅

₌

₋

_∇

_⋅

₋

_∇

_⋅

₊

₊

_{& +}

_&

∂

∂

ρ

_ρ

_ρ

_ε

(3.10)

where

I

is the specific internal energy, is the heat flux vector accounting for contributions due to heat conduction and enthalpy diffusion,

J

,

~

~

~

_∑

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

∇

−

∇

−

=

k k k T

h

D

T

J

ρ

ρ

ρ

λ

(3.11)

λ

is the turbulent thermal conductivity,

T

is the fluid temperature, and is the specific enthalpy of species

k

. and are source terms due to chemical heat release and spray interactions, respectively. By definition,

k

h

c

Q&

Q&

S

λ

is related to the turbulent thermal diffusivity

α

_Tand heat capacity

c

_pby

.

T p

c

α

ρ

λ

=

(3.12)

The state relations for the gas phase mixture is assumed to obey the ideal gas law,

∑

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

=

k k k u

W

T

R

p

~

ρ

,

(3.13)

where

R

_uis the universal gas constant,

W

_kis the molecular weight of species .

k

3.2.2. Turbulence Models

The in-cylinder reactive flows of IC engines are compressible and highly turbulent. Many turbulence models with different levels of complexity have been proposed for turbulent combustion modeling in engine applications. Models based on Reynolds averaged Navier-Stokes (RANS) equations, Direct Navier-Stokes (DNS) equations and large eddy simulation (LES) [48] have been adopted in the engine simulation fields in recent years.

The RANS based approach, based on a temporal average operation of the Navier-Stokes equations, focuses on the average flux properties, not considering the instantaneous variations. This allows a consistent computational resources saving, without losing much in accuracy. To close the RANS equations, modeling the Reynold stress tensor in the momentum equation, accurate turbulence models are implemented in the computational codes.

The LES approach is based on the direct calculation of the average flux in large eddies, using instead classic turbulence models to solve the physics of the small eddies. It is well known that the average flux of the large eddies strongly influences the turbulence effects, containing an high energy level. This approach is very accurate from a physical point of view, providing accurate information about the main turbulence effect but still requires large computational resources.

Currently, DNS calculation, based on the direct integration of the Navier-Stokes equations, results very expensive in terms of computational resources, since it needs very small computational domains. To completely catch the turbulence effects the average dimension of every computational cell must be in

the order of magnitude of the small eddies characteristic length, implying a very fine domain subdivision.

Although DNS and LES turbulence models are starting to be adopted for turbulence calculation in some technical applications [49], RANS-based models, will remain dominant in industrial applications in the near future due to their significantly lower demand on computational resources. This review will focus on some of most commonly used turbulence model, such as the k-ε models, which are also used in the present combustion modeling work.

One of the commonly adopted turbulence model for CFD applications is the standard k-ε model with considerations about velocity dilatation in the ε-equation and spray-induced source terms in both k and ε equations [8, 9,14]. The Favre averaged equations, basic for the model are:

( )

(

)

~

~

~

,

Pr

~

:

~

~

3

2

~

~

~

W

k

u

u

k

k

u

t

k

k T

₋

₊

&

⎥

⎦

⎤

⎢

⎣

⎡

∇

+

⋅

∇

+

∇

+

⋅

∇

−

=

⋅

∇

+

∂

∂

ρ

_ρ

_ρ

_σ

µ

µ

_ρ

_ε

(3.14)

(

)

(

)

_⎥

⎦

⎤

⎢

⎣

⎡

∇

+

⋅

∇

+

⋅

∇

⎟

⎠

⎞

⎜

⎝

⎛

₋

−

=

⋅

∇

+

∂

∂

ρ

ε

_ρ

_ε

_ρ

_ε

µ

µ

_ε

ε ε ε

~

Pr

~

~

3

2

~

~

~

3 1

c

u

T

c

u

t

_~

(

:

~

~

~

)

,

~

2 1 S S

W

c

c

u

c

k

&

+

−

∇

+

ε

_ε

σ

_ε

ρ

ε

(3.15)

where

Pr

_k

,

Pr

_ε

,

c

_ε1

,

c

_ε2 and

c

_µare model constants, as listed in Table 3.1.

The RNG version of the k-ε model, based on the Re-Normalized Group theory, was first proposed by Yakot and Orzag [50]. The k equation in the RNG version is the same as the standard version, but the ε equation was based on rigorous mathematical derivation instead of using empirical constants. The RNG ε equation is given by

(

)

(

)

_⎥

⎦

⎤

⎢

⎣

⎡

∇

+

⋅

∇

+

⋅

∇

⎟

⎠

⎞

⎜

⎝

⎛

₊

−

=

⋅

∇

+

∂

∂

ρ

ε

_ρ

_ε

_ρ

_ε

µ

µ

_ε

ε ε ε

~

Pr

~

~

3

2

~

~

~

2 1

c

u

T

c

u

t

_~

~

(

c

₁

:

u

~

c

₂

~

c

W

~

)

R

,

k

S S

ρ

ε

ρ

σ

ε

ε ε

∇

−

+

−

+

&

(3.16)

where the R term is defined as

( ) (

)

,

~

~

1

/

1

2 3 0 3

k

c

R

ε

βη

η

η

η

µ

+

−

=

(3.17) with 23

,

~

~

ε

η

=

S

k

(3.18)

(

2

S

S

)

1/2

,

S

=

⋅

(3.19)

and S is the rate of strain tensor,

(

~

~

)

.

2

1

_u

_u

T

S

=

∇

+

∇

(3.20)

Values of the model constants and used with the RNG version are listed in Table 3.1 as well. Compared to the standard ε equation, the RNG model has one extra term R, which accounts for non-isotropic turbulence, according to Yakot and Orzag [50]. Han and Reitz modified the model to take the compressibility effect into account, and implemented the modified RNG k-ε model into the KIVA-3V code [10]. According to Han and Reitz [51],

2 1

,

,

Pr

,

Pr

_{k ε}

c

_ε

c

_ε

c

µ

(

) ( )

,

3

6

1

1

3

2

1

₂ 3

η

η µ δ ε ε

c

c

n

m

c

c

=

−

+

−

−

+

−

(3.21)

where m=0.5, n=1.4 for ideal gas, and

(

)

_,

1

/

1

3 0

βη

η

η

η

η

₊

−

=

c

(3.22) with

⎩

⎨

⎧

>

⋅

∇

<

⋅

∇

=

.

0

~

,

0

;

0

~

,

1

u

if

u

if

δ

(3.23)

Han and Reitz [51] applied also their version of the RNG k-ε model to engine simulations and obvious improvements in the results were shown compared to the standard k-ε model. Therefore, the RNG k-ε model was used in the present modeling work.

Table 3.1. Constants in the standard and RNG k-ε models [14].

µ

c

c

_ε1

c

_ε2

c

_ε3 1/

Pr

_k 1/

Pr

_ε

η

₀

β

Standard k-ε 0.09 1.44 1.92 -1.0 1.0 0.769

RNG k-ε 0.0845 1.42 1.68 Eq.(3.21) 1.39 1.39 4.38 0.012

3.3. CHEMICAL KINETICS APPLICATION

In recent years, to better understand the physics of engine combustion processes and to further improve the accuracy of computational models attention is being given to the introduction of models incorporating detailed chemical kinetic mechanisms. Instead of assuming global single-step reaction in engine combustion, the chemical mechanisms models very detailed reaction pathways with the appropriate reaction rates. Thus, this models are capable of providing more accurate and more insightful information about the real combustion process that with phenomenological combustion models would not be catched.

Detailed chemical processes in combustion simulations can be described by chemical kinetic mechanisms that define the pathways of occurring chemical reaction leading to the change of species concentrations. In detailed chemical kinetic mechanisms, the reversible (or irreversible) elementary reactions involving K chemical species can be represented in the general form [52]

∑

∑

= =

′′

⇔

′

K k K k ki k k ki 1 1

χ

υ

χ

υ

(i=1,…,I). (3.24)

The production rate of the kth species in the ith reaction can be written as

(

ki ki i ki

υ

υ

)

q

ω

_&

=

′′

−

′

(k=1,…, K) , (3.25) where,

[ ]

_∏

[ ]

∏

= ′′ = ′

₋

=

K l l bi K l l fi i

k

X

li

k

X

li

q

1 1 υ υ (i=1,…,I) . (3.26)

Equation (3.26) defines the rate-of-progress variable qi for the ith elementary reaction, where [Xl] is the concentration of species l. The forward and backward

rate constants are given by the Arrhenius form

,

exp

_⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −

=

T

R

E

T

A

k

u fi fi fi fi β (3.27)

.

exp

_⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −

=

T

R

E

T

A

k

u bi bi bi bi β (3.28)

The summation of

ω

_&

_kiover all the reactions gives the chemical source term in the species continuity equation (Eq. (3.1)) as

c k

ρ

&

.

1

∑

=

=

I i ki k c k

W

ω

ρ

_&

_&

(3.29)

Correspondingly, the chemical heat release term in the energy equation is given by

(

)

( )

0

,

1 1 1 i k f I i K k ki ki I i i i c

_Q

_q

_h

_q

Q

=

−

∑

=

−

∑∑

′′

−

′

∆

= = =

υ

υ

&

(3.30)

where Qi is the heat of reaction of reaction I at absolute zero,

(

)

( )

0

,

1 f k K k ki ki i

h

Q

=

∑

′′

−

′

∆

=

υ

υ

(3.31) and

( )

k f

h

0

∆

is the heat of formation of species k at absolute zero.

A large amount of work has been done on developing and validating detailed chemical kinetic mechanisms for hydrocarbon fuel oxidation and pollutant formation [5, 6, 53]. Regarding the application to engine combustion, most previous efforts have focused on the predictability of the autoignition delay time, laminar burning velocity, and pollutant emissions of fuel/air mixtures under a wide range of engine operating conditions [54-56]. In developing detailed chemical kinetic mechanisms, experimental data measured in numerous devices are commonly used for mechanism validation, such as ignition delay time measurements in rapid compression machines and shock tubes [54,57], and laminar flame speed measurements in constant volume bombs [53, 58].

Multidimensional CFD coupled with detailed chemistry has been successfully applied to conventional and HCCI combustion simulations. For example, Kong et al. [7, 59] incorporated CHEMKIN [60] into the KIVA-3V code by direct integration of detailed chemical kinetics with CFD models, and applied the integrated model to diesel and HCCI engine simulations. Significant improvements on the prediction of ignition timing, pressure evolution, heat release rate, and pollutant emissions were achieved, compared to simulation results from the standard Shell auto-ignition model and the single turbulent time scale CTC combustion model of Kong, Han and Reitz [61].

3.4. COMBUSTION

MODELS

Although turbulent combustion is commonly encountered in many engineering devices, the insightful understanding of its physics is still very difficult. The in-cylinder turbulent combustion is a complicated process, especially due to the turbulence and chemistry interactions on tremendously different time-scale and length-scale levels. Currently there are several approaches for modeling the premixed and partially premixed combustion process occurring in engine applications. They can be classified into turbulent mixing-controlled, flamelet and probability density function (PDF) approaches.

3.4.1. MIXING-CONTROLLED MODELS

The basic concept of turbulent mixing-controlled combustion models is that the burning rate of the mixture is mainly determined by the turbulent mixing rate. Thereby, the model does not account for the influence of chemical kinetics representing only the fast chemistry. The combustion models based on this approach allows to save computational times and, by means of an accurate tuning of the model constants, provide reasonable results in terms of pressure, temperature and heat release rate for conventional engine application simulations. Eddy-Break-Up model and Characteristic Time Scale (CTC) model are briefly described below.

3.4.1.1. Eddy-Break-Up model

In the Eddy-Break-Up (EBU) model, proposed by Spalding [62], the burnt and unburned gases are assumed to be located in different eddies, and the fuel burn rate is purely controlled by eddy’s dissipation rate and fuel mass fraction. The mean fuel burn rate can be expressed as

( )

~

,

~

~

~

₂ 1/2 F EBU F

Y

k

C

′′

=

ε

ρ

ω

&

(3.32)

where is the variance of the fuel mass fraction and is the Eddy-Break-Up model constant.

2

~

F

Y ′′

C

_EBU

3.4.1.2. Shell Auto-ignition model

The SHELL ignition model [63] considered uses a simplified reaction mechanism to simulate the autoignition of hydrocarbon fuels. The mechanism consists of five generic species and eight generic reactions representing the initiation, propagation, branching and termination steps. The five generic species include fuel, oxygen, radicals, intermediate species and branching agents. These reactions are based on the degenerate branching characteristics of hydrocarbon fuels. The premise is that degenerative branching controls the two-stage ignition and cool flame phenomena seen during hydrocarbon autoignition. A chain propagation cycle is formulated to describe the history of the branching agent together with one initiation and two termination reactions. Details of the model can be found in Kong and Reitz [64]. The standard model constants were tuned to match the ignition delay (ID) and the apparent heat release rate for gasoline and diesel fuels.

This ignition model was used for the low temperature chemistry. After the ignition process (i.e., when the local gas temperature is greater than 1100 K), usually a more accurate combustion model takes place for the subsequent high temperature combustion calculation.

3.4.1.3. Characteristic Time Scale Combustion Model

As an extension of the Eddy-Break-Up concept, Abraham et al. [65] proposed a characteristic time scale combustion (CTC) model assuming that the species densities approach their local thermodynamic equilibrium (indicated by superscript ∗) with a characteristic time scale

τ

_C, which is a combination of a turbulent mixing time

τ

_T

∝

k

/

ε

and a laminar chemical time scale

τ

_L. The conversion rate of species m can be written as [61]

,

* * T L m m C m m m

f

Y

Y

Y

Y

dt

dY

τ

τ

τ

+

−

−

=

−

−

=

(3.33)

where is the concentration of species m, is the local and instantaneous thermodynamic equilibrium concentration and is a delay coefficient simulating the influence of turbulence on combustion [61].

m

Y

*

m

Y

f

The laminar timescale

τ

_Lis derived from a one-step kinetic reaction rate and is modeled as [61]

[

_{2 2}

] [ ]

0.75 ₂ 1.5

exp

,

1

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

=

− + −

T

R

E

O

H

C

A

u n n L

τ

(3.34)

where A is a model constant, E is the activation energy 18,479 cal/mol, Ru is the

universal gas constant and T is the local gas temperature.

The turbulent time scale

τ

_Tis based on the Eddy-Break-Up concept and is modeled as [61]

ε

τ

_T

=

C

_m2

k

(3.35)

where Cm2 is a model constant to account for the mixing characteristics in the engine, k is the turbulent kinetic energy and

ε

is its dissipation rate.

For conventional engine applications the CTC model is usually coupled with the Shell auto-ignition model described above.

3.4.2 FLAMELET MODELS

Flamelet models are based on tracking a combustion zone interface either defined by a combustion progress variable, c, or by a non-reacting scalar, G, that divides the flow into burned and unburned portions [66]. One of the most widely used model, the G-Equation model, is described below.

3.4.2.1. G-equation Model

Recently, Flamelet models for premixed flame propagation based on the level set method are receiving growing attention and interest. The level set method is a powerful numerical tool for tracking interface evolution [67]. With its application to combustion, Williams [68] first suggested a transport equation of a non-reactive scalar, G(x, t), for laminar flame propagation, and this approach is known as the “G-equation” method in the combustion literature.

Figure 3.1 Application of G-equation to premixed turbulent flames [66]. The G-equation method is based on the postulate that the flame propagation is driven by the bulk fluid velocity u of the unburnt mixture ahead of the flame front, and the laminar flame speed SL along the normal direction n. The flame front is

defined by a G(x, t) = G0 iso-surface with G0 taking an arbitrary but fixed value. The

flame propagation velocity can be written as

,

L f

_u

_nS

dt

dx

+

=

(3.36)

where the normal vector n is defined as

( )

[

]

_1/2

.

2

G

G

G

G

n

∇

∇

−

=

∇

∇

−

=

The transport equation of G can be derived by differentiating G(x, t) = G0

with respect to t,

(

S

)

G

.

G

u

t

G

L

∇

=

⎟

⎠

⎞

⎜

⎝

⎛

₊

_⋅

_∇

∂

∂

_ρ

ρ

(3.37) 29

As shown in Fig. 3.1, a turbulent flame front represented by the G(x, t) = G0

iso-surface divides the unburnt region, where G<G0, from a burnt region, where

G>G0. The turbulent flame front is regarded as an ensamble of local laminar

flamelets with considerations of flame stretch effects on the flame speed SL. The

stretched laminar flame speed SL can be written as

S

S

S

S

_L

=

_L0

−

_L0

ςκ

−

ς

_(3.38)

where

S

_L0 is the unstretched planar laminar flame speed,

ς

the Markstein length,

κ

the flame curvature, and S the strain rate. The flame curvature

κ

is defined based on G values as

(

)

_.

2

G

G

n

n

G

G

G

n

∇

∇

⋅

∇

⋅

−

∇

−

=

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∇

∇

−

⋅

∇

=

⋅

∇

=

κ

(3.39)

The strain rate S due to velocity gradients is defined as

.

n

u

n

S

=

−

⋅

∇

⋅

(3.40)

Based on Eq. (3.38), the G-equation is finally written as

( )

_S

0

_G

( )

_D

_G

( )

_S

_G

,

G

u

t

G

L

∇

−

∇

−

∇

=

∇

⋅

+

∂

∂

_ρ

_ρ

_ρ

_κ

_ρς

ρ

ς (3.41) where

D

_ς = 0

ς

. L

S

The turbulent G-equation concept has been successfully applied to the KIVA 3 [10] code by Tan and Reitz [11, 13, 69].

3.4.3 PDF APPROACH

Models based on PDF transport equations are usually formulated considering one-point statistics implemented by Lagrangian Monte Carlo particle based techniques [70]. In PDF-based turbulent combustion models, a central role is played by the joint PDF of the velocity u (x, t) and of the composition variables

φ(x, t) that are required to specify the thermo-chemical state of the mixture [70].

The mean value of a scalar quantity

Q

, denoted that

Q

can be defined based on the velocity-composition joint PDF function, i.e.,

( )

=

_∫∫

(

) (

)

=

Q

x

,

t

Q

V

,

ψ

f

,ϕ

V

,

ψ

;

x

,

t

dVd

ψ

,

Q

u (3.42)

where

f

u,ϕ

(

V

,

ψ

;

x

,

t

)

is the velocity-composition joint PDF of the event

( )

( )

{

u

x

,

t

=

V

,

ϕ

x

,

t

=

ψ

}

. Based on Eq. (3.42), joint PDF transport equations for the velocity and the reactive scalars can be derived. Thus in PDF combustion models, the emphasis shifts from the modeling of the chemical source terms to the modeling of molecular mixing processes.

The PDF formulation can represent a very general statistical description of turbulent reacting flows, and its predictive capability for turbulent combustion depends on the quality of the models that can be constructed for the unclosed mixing terms. Detailed assessment and discussion of the application of PDF methods can be found in Pope and Anand [71].

3.5. OTHER SUBMODELS IN ENGINE MODELING

3.5.1. SPRAY MODEL

Fuel injection and breakup process play a key role in the in-cylinder mixture formation process. Accurate modeling of the fuel spray dynamics and its interactions with flowing multi-component gases is fundamental to successful combustion modeling in DI engines. The spray model should include sub-models for spray atomization and breakup, droplet collision and coalescence, vaporization, and wall impingement. A comprehensive review of the spray modeling theory has been given by Reitz [72].

In gasoline direct injection, pressure-swirl atomizers are widely used to create a hollow-cone spray. Apposite internal swirl vanes, produce rotational motion in the liquid fuel, are present inside the injector tip. The liquid forms a film along the inside walls of the injector becoming a free sheet when exiting the injector. For the mass conservation, the sheet becomes thinner as it progresses, and subsequently disintegrates into droplets, forming a hollow cone spray.

In diesel high-pressure direct injection, multi-hole atomizers are usually adopted to create several solid-cone jets progressing along the centerline of every hole. In this injection strategy, due to the high injection pressure, the primary breakup process is very fast and occurs at the boundary of the spray cloud.

Once droplets are formed, their behavior is governed by secondary breakup, drag, collision, coalescence and vaporization. Some sub-models commonly adopted in engine simulation for sheet atomization, secondary breakup, droplets and wall film behaviors are described below.

3.5.1.1. Linearized Instability Sheet Atomization (LISA) Model

The transition from internal injector flow to a fully developed hollow-cone spray is modeled using the so-called linearized instability sheet atomization (LISA) model [73]. In the LISA model, the process is divided into three stages: film formation, sheet breakup and atomization. The model formulation is briefly introduced here, more detailed discussions were given by Schmidt et al. [73].

Film Formation - The centrifugal motion of the liquid within the injector creates a

liquid film surrounding an air core. The thickness of the film, tf , is related to the

mass flow rate,

m&

, by

(

_{0 f}

)

,

f l

ut

d

t

m

=

πρ

−

(3.43)

where

ρ

_lis the liquid density, u the axial component of velocity at the injector exit,

d0 the injector hole diameter. u is related to the total velocity U by

( )

,

cos

θ

U

u

=

(3.44)

where the cone half-angle θ is assumed to be known for a specific injector. The total velocity U is related to the pressure drop across the injector exit by

,

2

l v

p

k

U

ρ

∆

=

(3.45)

where kv represent a discharge coefficient usually assumed to be in the range of

0.6 - 0.8 in engine simulations.

Sheet Breakup - The model assumes that a two-dimensional, viscous,

incompressible liquid sheet of thickness, 2h, moves with velocity, U, through a quiescent, incompressible gas. A spectrum of infinitesimal disturbances is imposed on the initially steady motion and produces fluctuating velocities and pressures for both the liquid and the gas. The most unstable disturbance, denoted by , is assumed to be responsible for sheet breakup. Once the unstable waves on the sheet surface grow to a critical amplitude, ligaments are formed due to the sheet breakup. The breakup time

Ω

b

τ

for this process can be formulated based on an analogy with the breakup length of cylindrical liquid jets, i.e.,

,

ln

1

0

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

Ω

=

η

η

τ

b b (3.46)

where

η

_b is the critical amplitude at breakup. The corresponding breakup length L can be estimated by

,

ln

0

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

Ω

=

=

η

η

τ

b b

U

U

L

(3.47)

where the quantity ln(ηb/η0) is given a constant value 12. From a mass balance the

drop size dD results

,

3

2 3 L L D

K

d

d

=

π

(3.48)

where KL is the most unstable wavelength.

Atomization - As a consequence of the sheet breakup process described above,

fuel droplets are introduced into the computational domain with certain initial conditions. Subsequently, the droplets are subject to secondary breakup, collision and coalescence, aerodynamic drag, vaporization, and wall impingement. The models used to describe the atomization physics are described in the following sections.

3.5.1.2. Kelvin-Helmholtz Rayleigh-Taylor (KH-RT) Model

To predict diesel spray characteristics, Kelvin-Helmholtz and Rayleigh-Taylor hybrid model proposed by Patterson et al. [74] is widely used in the computational codes. In this droplet breakup model, it is assumed that the primary breakup takes place mainly by KH instability while secondary breakup is caused by RT instability. First the droplets are detached by the intact liquid core by KH instability. Then, secondary breakup occurs by the competition of KH and RT instability. The initial sheet velocity is written as

,

2

2 / 1

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

−

=

f amb inj v

P

P

K

V

ρ

(3.49)

where the Kv coefficient is derived from the nozzle dimension and the spray cone

angle [75, 76].

KH Breakup – According to Reitz [77], KH breakup is governed by the maximum

growth rate. During the breakup, the droplet radius reduces to radius rc with

uniform rate. The new droplet radius rn is calculated using the following equation:

,

KH c n

r

r

dt

r

r

τ

−

=

−

(3.50)

where

τ

_KH is the time to reach the critical radium rc.

RT Breakup – Similarly to the KH breakup part, also the RT model provide

expressions for the wavelength and the frequency of a fast-growing wave. In the drag calculation the TAB model (described below) is used to consider the drop oscillation and distortion. It is assumed that the droplet diameter is larger than the maximum growing rate while the RT waves are growing. If the waves are growing

longer than the breakup time,

τ

_RT, the droplet is split into small droplets with radius rc. In the RT breakup,

τ

_RT and rc are defined by

,

RT RT

C

Ω

=

τ

τ

(3.51)

,

RT RT c

K

C

r

=

π

(3.52)

where

C

_τand

C

_RTare model constants,

Ω

_RTis the maximum growing rate and is related to the maximum growing rate wavelength

RT

K

Λ

_RTby

.

2

RT RT RT

C

K

Λ

=

π

(3.53)

3.5.1.3. Taylor-Analog-Breakup (TAB) Model

The Taylor-Analog-Breakup (TAB) model proposed by O’Rourke and Amsden [78] is used to model the secondary breakup of the droplets. The TAB model utilizes the analogy between a distorted droplet and an oscillating spring-mass-system. The external forces acting on the mass, the restoring force of the spring, and the damping force are analogous to the gas aerodynamic force, the liquid surface tension force, and the liquid viscosity force, respectively. The force balance on the droplet gives

0

3

2

8

5

2 2 3 2 2 2

=

−

+

+

r

U

y

r

dt

dy

r

dt

y

d

l g l l l

ρ

ρ

ρ

σ

ρ

µ

(3.54)

where t is time, y is the normalized drop distortion parameter, σ is the surface tension coefficient, and the subscripts g and l denote the gas and liquid phase, respectively. According to O’Rourke and Amsden [78], it is assumed that breakup occurs if and only if y > 1. When this condition is satisfied, the droplet breaks up into smaller children droplets with sizes determined by an energy balance taken before (subscript 1) and after (subscript 2) the breakup as

.

8

1

3

7

3 ₁ 2 1 2 1

_⎟

⎠

⎞

⎜

⎝

⎛

+

=

dt

dy

r

r

r

_l

σ

ρ

(3.55) 34

3.5.1.4. Droplet Collision and Coalescence Model

Droplet collisions are very important physical phenomena in dense sprays. The model of O’Rourke [79] has been widely used for computing drop collisions in engine spray modeling studies [72]. Three collision regimes were identified according to O’Rourke [69], that is, permanent coalescence, “grazing” (drops separate after collision, possibly forming satellite droplets), and shattering, depending on forces acting on the coalesed pair of drops. At low Weber numbers, surface forces dominate over liquid inertia forces, and the drops coalesce permanently. At higher Weber numbers, the liquid inertia forces become more important and the grazing collisions occur. With further increase of Weber number, the dominant liquid inertia forces cause shattering of the colliding drops, forming a group of small droplets. Reitz and Diwakar [80] studied the effect of drop collision and coalescence in sprays. They concluded that the drop size is the outcome of a competition between drop breakup and drop coalescence in non-evaporating sprays. Drop breakup was found to dominate in hollow-cone sprays because the possibility of coalescence is minimized due to the expanding spray geometry. 3.5.1.5. Droplet Vaporization Model

The droplet vaporization model of Spalding [81] is used in the present study. This model assumes uniform temperature within the droplet and single-component fuel composition. The rate of change of the fuel droplet mass due to vaporization is given as

BSh

D

r

dt

dm

g d 0

2

π

ρ

−

=

(3.56)

where md is the droplet mass, ρg the ambient gas density, D0 the laminar mass

diffusivity of the fuel vapor in the air, B the Spalding mass transfer number and Sh is the Sherwood number. The temperature change of the droplet is determined from an energy balance between the latent heat of vaporization and the heat transfer to the droplet,

,

4

d d d l d

rQ

dt

dm

L

dt

dT

c

m

−

=

π

(3.57)

where cl is the specific heat of the liquid fuel, Td the droplet temperature, L the

latent heat of fuel vaporization, Q the heat transfer rate to the droplet that is given by

(

)

,

2

r

Nu

T

T

k

Q

g d g d

−

=

(3.58)

where Tg is the gas temperature, and Nu is the Nusselt number.

3.5.1.6. Wall Impingement Model

Spray wall impingement is an important phenomenon found in PFI and DI engines. In PFI engines, the buildup of fuel wall film in the intake ports can cause an undesirable fuel delivery delay and an associated fuel metering error, while in some DI engines, the fuel is directly injected into the specially designed piston bowl and impingement is used to help create an optimal stratified mixture [82]. Therefore, an accurate spray/wall interaction model also plays important role in DI engine simulations. A model formulated by Naber and Reitz [83] considers two impingement regimes, i.e., rebounding and sliding. If the Weber number of the incoming droplet is less than 80, the droplet is assumed to rebound from the wall after a characteristic residence time, proportional to the drop’s natural vibration frequency. For a droplet with Weber number more than 80, the droplet is assumed to slide along the wall and the resulting droplet motion is calculated by analogy with a liquid jet flow [83].

3.5.2.

HEAT TRANFER MODEL

Heat transfer between the working fluid and the cylinder wall in an engine can significantly affect engine performance, efficiency and emissions. Therefore, accurate modeling of the heat transfer process is important to successful engine simulations. The heat flux through the chamber walls is mainly due to gas phase convection, fuel film conduction and radiation. In current multidimensional computations, since the boundary layer of an engine in-cylinder flow is thin relative to the practical computational grid size, velocity and temperature wall functions are used to solve for the near-wall shear stress and heat transfer. One of the most widely used model for calculating the gas phase wall heat transfer is the model proposed by Han and Reitz [84]. By considering the gas density variation and the increase of the turbulent Prandtl number in the boundary layer, the temperature profile equation (wall function) is formulated as

( )

2

.

1

33

.

4

2

.

5

,

ln

1

.

2

+

+

+

=

+ + + + +

_y

_G

_y

_G

T

(3.59)

with y+ and G+ defined as

,

;

_* *

u

q

Q

G

y

u

y

w c

υ

υ

=

=

+ + _(3.60)

where u∗ _{is the friction velocity, y is the normal distance to the wall, q}

w is the heat

flux through the wall,

Q

_c is the average volumetrical chemical heat release, and

υ

is the laminar kinematic viscosity. The corresponding formulation for wall heat flux is given as

(

)

(

)

( )

,

5

.

2

ln

1

.

2

/

4

.

33

1

.

2

ln

* *

+

+

−

=

₊ +

y

u

Q

y

T

T

T

u

c

q

p w c w

υ

ρ

(3.61)

where T and Tw are the gas temperature and wall temperature, respectively.