Chiodi M. An innovative 3D-CFD-Approach towards Virtual Development of Internal Combustion Engines

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126 9 3D-CFD-Modeling of the Combustion for SI-Engines

As reported in Figure 9.5 lean mixtures are particularly sensible to flame extinction. This is due

to a combination of several effects like e.g. lower laminar flame speed

and lower mixture-

specific heat-release density that make the flame propagation more sensitive to local turbulent

convection.

9.2 Flame Propagation Modeling (Weller Model)

As introduced before, a developed turbulent flame within a turbulent premixed mixture in SI-

engines is a thin reaction sheet (flame front), wrinkled and convoluted by the turbulence of the

unburned mixture. Following the one-step flame propagation model proposed by Weller [69] it is

convenient to introduce a progression variable

that permits to identify the position of the flame

within the 3D-CFD-mesh. Per definition the progress variable

has to assume the value 0 in the

unburned zone and 1 in the burned zone (see Figure 9.6).

Figure 9.6: The progress variable c and the flame propagation.

Several formulations for c have been proposed in the literature [55,59,61], but using the scalar

definition of

QuickSim the progress variable c in the 3D-CFD-cell

can be easily and directly

represented by the burned mass-fraction variable

(see Chapter 8.4):

jBj

(9.3)

The species conservation law (see Eqs. 6.38 and 6.41) as a transport equation can then be

properly used for describing the flame propagation:

Spark

Plug

Cylinder

Walls

0 c

1 c

0 c

9.2 Flame Propagation Modeling (Weller Model) 127

BBBB

TBB

rMwgradDdivvwdiv

)

()

(

)

(

Z UU

(9.4)

where

is the local reaction rate of burned gas. During the flame propagation the local

combustion rate in the cell

is then defined as follows:

,,,,, jTjTfjUjjB

SAVr 



(9.5)

Here it becomes evident that the local turbulent flame surface

jTf

within a cell is not available

numerically. Though, the Weller model describes pragmatically the surface in the cell as a

function of the gradient of the progress variable:

,,,

, jBjTjU

jjB

wgradS

r U U

(9.6)

Many formulations for

have been proposed during the years. In this work a semi-empirical

formulation of the wrinkling factor

proposed by Herweg and Maly [70] has been

implemented:













exp1

Term

















(9.7)

where

term of

) is the effect of strain on the laminar burning speed at small radiuses

(usually

1),

the integral length scale of the flow field (see Chapter 6.2.1.3),

the flame

kernel radius and



tt  the time after the spark ignition IP.

The velocity

is described as follows:



uvv



(9.8)

It represents the relevant velocity for the wrinkling factor, derived from the sum of the kinetic

energy of the main fluid, the velocity

and the turbulence

. Assuming an isotropic turbulent

field, the turbulence intensity

is given by:

2 k



(9.9)

The other terms of the formulation of

take the following effects into account:

128 9 3D-CFD-Modeling of the Combustion for SI-Engines

x 2

term (effective turbulent factor): This factor determines the behavior in the critical

region

10 



where

plays the controlling role.

and 4

term (size and time dependent integral scale): These terms taking into account

the turbulent length and time scales, respectively, influence the flame kernel development

from the very beginning. The turbulence effects become more and more important with

increasing size and life time of the flame kernel until the full spectrum of turbulence

affects the flame propagation (typical orders of magnitude:

3mm and



|

1ms).

term (fully developed turbulent combustion): This term describes the behavior of a

fully developed, freely expanding flame in an isotropic turbulent flow field.

9.3 QuickSim’s Approach: Implementation Improvement

Based on the flame propagation model approach it is then possible to calculate the variations of

all scalars that describe the properties of the working fluid in

QuickSim:



M d

jUjB ,,

(9.10)





M d

jEGRjFreshjB ,,,

(9.11)





M d

jUAirEGRjUAirjBAir ,__,_,_

(9.12)





M d

jUAirFjUFjBF ,__,_,_

(9.13)

These relations link the mass balance among the species while the energy balance refers to the

heat-release formulation introduced in Chapter 8.

9.3.1 Numerical Implementation of the Flame Propagation Model

The numerical implementation of the Weller model implies the application of a mathematical

model within discrete elements (cells) which have very often discretization lengths many times

bigger than the characteristic lengths of the investigated phenomenon. In order to ensure a low

mesh-dependent implementation of the model, first of all, investigations on the sensitivity of the

9.3 QuickSim’s Approach: Implementation Improvement 129

input variables on the results have to be performed (a model is not reliable when its input

variables are not reliable). As it will be discussed in this and also in the next chapter not rarely

local variables are affected by the local mesh structure, dimension, etc. For this reason very often

it is more convenient and numerically more “stable” to take into account global values (averaged

over the whole combustion chamber or sometimes parts of it), i.e. a loss of local information as

an input in the model is much better than meaningless final results.

In this chapter two meshes of different engines are used for presenting few results (see Figures

9.7 and 9.8):

One cylinder of a mass-production passenger car engine with two valves (

575 cm

bore 95 mm, asymmetric bowl shape with big squish area, coarse discretization with

Cells

=15,000 cells for the combustion chamber alone).

The cylinder of a mass-production bike engine with four valves (

650 cm

bore 100 mm, symmetric pent-roof chamber with squish areas, coarse discretization with

Cells

=25,000 cells for the combustion chamber alone).

Figure 9.7: Calculation mesh of a passenger

car engine (MB M102 E23 – 2 valves).

Figure 9.8: Calculation mesh of a bike engine

(Rotax 650 – 4 valves).

Investigations on different approaches in the implementation of the chosen combustion model

require first of all analyses of the processes under the same “thermodynamic conditions”, so that

the influence of each input variable can be better isolated from other factors. This has been

achieved by imposing burn profiles to the 3D-CFD-simulations (see e.g. Figure 9.9), which have

been determined by the real working-process analysis calibrated with measured pressure profiles.

With the setting of target burn rate profiles approximated by Wiebe formulations (see

Chapter 4.4.3) consequently target heat releases are imposed to the 3D-CFD-simulations, i.e. as

Car engine

MB M102-E23

2 valves

Bike engine

ROTAX 650

4valves

130 9 3D-CFD-Modeling of the Combustion for SI-Engines

assumed in this approach for reliable investigations on the model implementation the operating

cycles between experiments and 3D-CFD-simulations are thermodynamically equivalent.

Figure 9.9: Burn mass fraction profile at different operating conditions

(MB M102 E23).

Target

A target heat release



.Exp

ddQ M

is performed in the 3D-CFD-code QuickSim using its

evaluation tool for the determination first of



CFD

ddw M

(see Eq. 9.14) and then



CFD

ddQ M

in a “closed loop” for a comparison with experimental data as inputs (see

Figure 9.10).



Cells

jjB

CFD

(9.14)



CFD

Exp

(9.15)

,,,,, jBjTjUHRcorrjB

wgradScr U

(9.16)

At any time-step the target heat release is automatically compared with the simulated one and if

they differ a global heat-release correction factor

)(M

is calculated and provided to the 3D-

CFD-simulation as a feedback (see Eq. 9.15). This factor is then used iteratively for adjusting the

1500 rpm

Low Load

imep = 3.5 bar

3000 rpm - WOT

imep = 10.8 bar

Burn mass fraction w

, kg/kg

0.0

0.2

0.4

0.6

0.8

1.0

Crank an

,de

700 FTDC 740 760

MB M102-E23

9.3 QuickSim’s Approach: Implementation Improvement 131

calculation of the local reaction rate of the burned gas (see Eq. 9.16), so that at the end the target

profile is matched with high accuracy.

Investigations on the Implementation Phase

Similarly to the heat-transfer investigations (see Chapter 10) in the development phase of a

combustion model (or in this case the procedure for an improved implementation of an existing

one towards less mesh-dependence) the variation of

)(M

during the combustion should

become very small, so that

)(M

can be implemented as a constant in the combustion model

formulation.

Figure 9.10: The “internal coupling” between a 3D-CFD-code and experimental data for

burn rate control during the implementation phase of the combustion model.

Starting with the investigation of input variable reliability on the model results, it becomes clear

that in the 3D-CFD-simulation few of these variables are per definition not available at all. The

problem, as discussed below, resides in the numerical discretization of the flame front.

9.3.2 Numerical Inconsistencies at the Flame Front

When a combustion process takes place in the cylinder, i.e. a flame front is present or self-

ignition processes occur, high gradients of temperature are generated due to the local heat

Evaluation

Tool

3D-CFD-

Simulation

Comparison

)(M







,,,,

xvxT





CFD

ddQ M

Experiments



.Exp

ddQ M

132 9 3D-CFD-Modeling of the Combustion for SI-Engines

release. The oxidation energy rises the temperature of the already burned mixture up to a

temperature difference of ca. 2000 K to the unburned zone; while heat transfer through the

border of the burned zone slowly warms the unburned zone. A cell

of the 3D-CFD-mesh

involved in the oxidation process at the flame front (SI-engines), i.e. where the burned mass

fraction 0



< 1, shows a cell temperature



M,xT

(per definition in the central node of the

cell P – see Figure 9.11) with a value in-between the unburned and burned zone, respectively.

Because the cell temperature



M,xT

and most of the other variables at the central node (e.g.

the density) are representative neither for the unburned zone nor for the burned zone, it is a

remarkable cause of inaccuracy to use them in 3D-CFD-models which, e.g. expressively require

information about the unburned zone (laminar flame speed, diesel-self-ignition, local burn rate,

etc.) or about the burned zone (e.g. modeling of the burned gas properties). The example

reported in Figure 9.11 helps understanding the influence of the cell dimension, the structure or

even the orientation on the flame propagation.

Figure 9.11: Flame propagation through cells with different orientation.

),( MxT

Flame Front

),( MxT

Flame Front

),( MxT

Flame Front

),( MxT

Flame Front

Case 1

Case 2

9.3 QuickSim’s Approach: Implementation Improvement 133

Case 1

In case 1 the heat release of the flame “heats” the temperature



M,xT

of the cell very slowly

from the start value (

 

M M ,, xTxT

). As soon as the temperature



M,xT

rises, the

laminar flame speed

“irrationally” increases and the flame propagation velocity due to this

numerical problem accelerates up to its inconsistent maximum when the flame leaves the cell (

 

M M ,, xTxT

Case 2

Here the cell has the same dimension as in case 1 but a different orientation. In this case the

flame heat release increases the cell temperature



M,xT

with higher gradients. The resulting

flame propagation speed

is “irrationally” many times higher than in case 1. Within a short

time the flame leaves the cell while in case 1, during the same period, it has propagated through

a smaller distance at the beginning.

Conclusion

Concluding, this means that the calculation of the flame speed using the only available

temperature in the cell



M,xT

is formally wrong and cannot be adjusted with a corrector

factor, because numerically all depends on the “heating speed” of each cell, which is a result of

complex non linear equations influenced by the position of the cell vertices, the flow field, the

flame propagation direction, the gradients of the progress variables, etc. In particular considering

the combustion chamber mesh of an internal combustion engine this has, in order to reproduce

the complex geometry and the piston motion, moving cells with a great spectrum of dimensions,

internal angles, etc., which actually can be taken into account only within a statistical approach.

9.3.2.1 Expedients for the Numerical Inconsistencies at the Flame Front

Since the local thermodynamic variables within a cell of the mesh are not reliable for a correct

implementation of the flame propagation model an expedient that allows to handle the problem

has to be found, e.g. referring to the temperature in the unburned zone at the flame front the

following approaches may be an easy and reasonable solution of the problem:

the average temperature of the whole unburned zone is taken as reference temperature

(this approach misses local details)

the average temperature of the neighbor unburned cells is taken as reference temperature

(This approach is geometrically very complex and can be inconsistent in many cases.)

134 9 3D-CFD-Modeling of the Combustion for SI-Engines

During the years and several investigations a more comprehensive and numerical convenient

approach has been chosen. This approach is presented here below.

9.3.3 Local Two-Zones Model

In order to avoid the inaccuracy due to a wrong detection of temperature gradients described in

Chapter 9.3.2, several years ago [66] a so-called “Local CFD-Two-Zones-Model” has

been developed and implemented into

QuickSim. This model permits a continuously

thermodynamic splitting of each cell

involved in a combustion process into a burned

and unburned zone with a similar procedure like in the real working-process analysis

for the whole combustion chamber (see Figure 9.12). Following this approach also the

local volumes of the two zones can be detected so that a more reliable evaluation of the

flame front can be performed. Using the conservation equations for the mass and the

energy within the cell

and a simple assumption for the relation between the unburned

jU,

and burned density

jB,

, the equation system can be closed and a reliable

thermodynamic splitting of all the cells involved in the oxidation process can be

achieved.

Figure 9.12: Scheme of the “thermal splitting” of a flame front cell

using a local two-zones model.

Cell j

at the flame front

Flame propagation



Flame front



Local

Two-Zones-Model

9.3 QuickSim’s Approach: Implementation Improvement 135

The procedure starts with the following relations:

jBjUj

VVV



(9.17)

jBjUj

mmm



(9.18)

jBjBjUjUjj

VVV

,,,,

UU U

(9.19)

jBjBjUjUjj

hmhmhm

,,,,

 

(9.20)

jBjBpjBjUjUpjUjjpj

TcmTcmTcm

,,,,,,,,,

 

(9.21)

where the mean specific heat

, defined as follows, can be directly calculated from the thermal

enthalpy available in

QuickSim (see Chapter 8.4.2):



refref

pTh

dTpTc

pTc

ref



O

),,(

(9.22)

In a 3D-CFD-simulation it is more convenient to calculate with specific variables, i.e. the

progress variable

can be used in these equations:

(9.23)

so that the equations for the local unburned

and burned temperature

are given by:

jUpjB

jBjBpjBjjp

TcwTc

,,,

,,,,,

)1( 



(9.24)

)1(

,,,

,,,,,

jBpjB

jUjUpjBjjp

TcwTc





(9.25)

Considering the average densities over the entire combustion chamber and the global two-zones

(e.g. in Figure 9.13 and 9.14 from

QuickSim’s evaluation tool) the density profiles show a

relatively constant ratio during the whole combustion duration over widespread applications.

From this analysis and extending these considerations from the “global” zones to the local cell

it is possible to introduce an assumption that links the local unburned and burned density,

respectively:

jBjBjU

,,,

3)( U#U U

(9.26)