Baumgarten

174 4 Modeling Spray and Mixture Formation

Fig. 4.57. Influence of droplet size ratio on the coalescence/stretching separation (Eq. 4.240) and on the coalescence/reflexive separation transition criterion (Eq. 4.245)

If the two droplets permanently coalescence, the velocity u&new and the temperature Tnew of the combined drop are calculated as

u&new	m1u&1	m2u&2	,	(4.241)
	m1 m2
T	m1T1	m2T2 .		(4.242)
new	m1 m2

If stretching separation occurs, both droplets are re-formed again, and it is assumed that the temperature of the initial droplets is not altered. O’Rourke [93] derived the equations

u&new

♠m u&

m u&

m

u&

u&

≡

>B B

← 1 1

2 2

2

1

2

…

crit

(4.243)

1

>m1

m2 >1 Bcrit

and

&new

♠m u&

m u&

m

u&

u&

≡>B B

← 1 1

2 2

1

2

1

…

crit

u2

(4.244)

>m1

m2 >1

Bcrit

in order to predict the velocities of both droplets after separation. Eqs. 4.243 and 4.244 include some simplifying assumptions about the fraction of energy that is dissipated during collision.

Figure 4.58 shows the effect of the collision model on the Sauter mean radius (SMR) of a full-cone high-pressure diesel spray under evaporating and nonevaporating conditions, which is injected in pressurized air. Table 4.5 summarizes the boundary conditions used in the simulation. All curves in Fig. 4.58 show ex-

4.7 Collision and Coalescence	175

tremely large values at the beginning of injection, because the initial drops have a size equal to the nozzle hole diameter, and because the first break-up occurs after a small time delay, the break-up time. In the case of low ambient gas temperatures, the effect of evaporation on the droplets’ lifetimes is small, and the Sauter mean radius is a result of a competition between coalescence and break-up. If coalescence effects are neglected, break-up will produce an enormous number of extremely small droplets, which results in unreasonably small SMR values. On the other hand, if the collision model is used, there is even an increase in SMR over time because of the rising number of droplets per unit volume that increase the collision frequency and thus the probability of coalescence. Hence, the inclusion of an appropriate collision model is very important in the case of non-evaporating sprays, which are customarily used to validate break-up models and to investigate the effect of nozzle geometry and injection strategy on the spray formation processes. In the case of high ambient gas temperatures evaporation has a dominant effect on the droplets’ lifetimes. Especially the small droplets evaporate fast and the chance to coalesce is reduced. Thus, compared to the non-evaporating case, there are less droplets, the collision frequency is reduced, and the overall effect of the collision model on the SMR is smaller.

Table 4.5. Boundary conditions used for the calculations shown in Fig. 4.58

	Evaporating spray	Non-evaporating spray
Ambient temperature [K]	800	298
Ambient pressure [MPa]	5.7	5.7
Fuel temperature [K]	298	298
Nozzle hole diameter [µm]	200	200
Injected mass [mg]	40,7	40.7
Injection duration [ms]	4	4
Break-up model	Blob + KH/RT	Blob + KH/RT

Fig. 4.58. Effect of the collision model on the SMR of an evaporating and a nonevaporating diesel spray

176 4 Modeling Spray and Mixture Formation

In the model of O’Rourke [97, 93], only coalescence and stretching separation are considered. Reflexive separation, which is important for near head-on collisions, and shattering collisions are not included. Furthermore, the formation of satellite drops after stretching separation is ignored. For this reason, Tennison et al. [140] presented an enhanced version of the collision model, which also takes reflexive separation into account. The model extension is based on the theoretical work of Ashgriz and Poo [8], who derived the transition criterion between coalescence and reflexive separation that:

Wecoll		ª		1 '	2					1		3		2 /		3	º	' 1 '3 2					(4.245)
	3	«	4					7			'						»					,
																				6
		¬															¼		' K1		K2
where
		K		2 1 9					2	1 92			0.5			1 ,							(4.246)
			1
	K2			2 ' 9 2 '2 92 0.5													'3 ,						(4.247)
				9					crit						.								(4.248)
					0.5B						1 '

Reflexive separation occurs if the collision Weber number is above the value given by Eq. 4.245. Eq. 4.245 also includes the effect that reflexive separation is more likely to occur if the diameter ratio is close to one, see Fig. 4.57, which is in agreement with experimental observations [8]. The consideration of the reflexive separation regime in diesel sprays was found to give a slight reduction of the overall Sauter mean diameter [17].

In Eqs. 4.246 and 4.247, Κ1 and Κ2 are the fractions of the drops’ kinetic energy that participates in the reflexive separation process. The expressions are derived using a balance between kinetic energy and surface energy. It is assumed that the two combined drops form a flattened disc that quickly changes into a cylinder, which stretches out under the force of the internal flow of the fluid moving in opposite directions, Fig. 4.56. It would then seem to be logical to use a simple balance between this effective reflexive energy and the nominal surface energy. However, once the cylinder has stretched far enough, the surface energy can be reduced by forming two drops. Hence, separation can occur even if the reflexive energy is less than the surface energy. The criterion derived by Ashgriz and Poo [8] is that reflexive separation occurs if the reflexive energy is more than 75% of the nominal surface energy.

As suggested in the study of Hung and Martin [57], shattering of droplets during binary collisions can be important for Wecoll > 80. A drop-shattering collision model is has been proposed by Georjon and Reitz [40]. It is assumed that first a

large drop with radius r0 = (r13 + r23)1/3 and then a ligament (length: 2Γ, radius: r = (2r02/(3Γ))0.5) are formed during the collision of two droplets with a sufficient collision Weber number, Fig. 4.59. The ligament elongates and capillary waveinduced disturbances grow (Rayleigh linear jet break-up theory). If the break-up

4.7 Collision and Coalescence	177

Fig. 4.59. Schematic representation of ligament formation and break-up [40]

time is shorter than the time taken by the two ends of the cylindrical ligament to retract again, it disintegrates into small droplets. The drop dynamics (elongation and retraction) are formulated based on the energy equation of a half-cylinder and yield a second-order non-linear differential equation, which is solved numerically in order to get the time-dependent diameter and length of the elongating ligament. Next, the Rayleigh linear jet break-up theory is applied [117], and this finally yields the wavelength Ο of the disturbance, the break-up time of the ligament tbu, and the radius rchild of the new child drops:

	O 9.02 r ,						(4.249)
	2S		2S		Ul r3		(4.250)
tbu :			0.34		V	,
rchild			1.89 r .				(4.251)

If the break-up time is reached before the cylinder contracts again, the ligament

is assumed to break up into small droplets of size rchild. Otherwise, shattering collision does not occur. It is assumed that the shattering collision neither depends on

the impact parameter B nor on the droplet size ratio .

Georjon and Reitz [40] consider shattering collisions to be an extension of the stretching separation regime. For each pair of drops that have undergone stretching separation, it is tested whether a shattering collision is possible by solving the equation of motion of the ligament. If shattering is possible, the collision is calculated between N pairs of drops, where N is the minimum of droplets in the two colliding parcels. The remaining droplets do not change their properties and are put into a new parcel. The droplets taking part in the collision process disintegrate

178 4 Modeling Spray and Mixture Formation

into smaller droplets and undergo velocity changes that reflect momentum and energy conservation. The model has only been validated against experimental data from non-evaporating sprays, and the results show a slight overprediction of child drop sizes and velocities.

In real engine sprays, collision phenomena are probably much more complicated than described by all of the above-mentioned models. A deeper understanding of spray physics in this regime is necessary. Some of the correlations used in the models come from experiments in entirely different regimes (rain drops, cloud physics) and need to be tested against fundamental experiments on collision of hydrocarbon droplets at high pressures. However, data about fundamental experiments under diesel engine conditions are scarce. Furthermore, it has been shown that coalescence of hydrocarbon droplets is promoted if the environment contains fuel vapor [109]. So far, no model accounts for this effect.

4.7.3 Implementation in CFD Codes

In general, two possible methods for the implementation of a collision model in CFD codes exist: the statistical and the deterministic approach. Using the deterministic approach, the exact positions and velocity vectors of all parcels are used in order to check the possibility of collision for all parcel pairs. Because this method is computationally extremely expensive, the statistical approach is usually preferred. Only collisions of parcels that lie in the same computational cell are considered. All pair combinations of parcels in a cell are checked according to the following procedure. It is assumed that the droplets of both parcels are homogeneously distributed inside the cell volume Vcell. Then, the number of collisions of one drop of parcel 1 containing the larger droplets (N1 droplets of diameter d1) with all the smaller droplets (parcel 2: N2 droplets of diameter d2) is predicted, and it is assumed that all the other large droplets of parcel 1 have identical behavior. The large droplet is called a collector droplet.

The probability that a collector collides with k droplets from parcel 2 follows a Poisson distribution [97],

	Pk	Θ12	t k	exp Θ12 t ,	(4.252)
		k !

where t is the time step of the computation and

	N2	Σ				2		&
Θ12				d1	d2			urel	E12	(4.253)
	Vcell	4

is the collision frequency, which is modeled in analogy to the kinetic theory of gases and is the product of number density N2 /Vcell, collision cross section Σ(d12 + d22)/4, see Fig. 4.60, and the magnitude of the relative velocity u&rel. The quantity E12 in Eq. 4.253 is the collision efficiency, which is shown to have a value of E12 |

4.7 Collision and Coalescence	179

1.0 in engine sprays [97]. In Eq. 4.252 the probability of no collision is P0 = exp(-

Θ12 t).

Next, the actual number n of collisions between one collector droplet of parcel 1 and the small droplets of parcel 2 must be specified. A random number [1 is sampled from a uniform distribution in the interval [0, 1], and if [1 > P0 a collision is assumed to occur (otherwise no collision occurs and the next parcel pair is checked). If [1 > P0 the corresponding value of the integrated distribution function of Pk (normalized to lie also in the interval [0, 1]),

k	n
³	Pk dk [1 ,	(4.254)
k	0

is identified and solved for n. This procedure guarantees that the possibility of sampling a discrete number of collisions is given by Eq. 4.252. Further details are given in [5], for example.

Finally, the nature of collision must be specified. In the model of O’Rourke [97], it depends on the impact parameter B (interval [0, 1]). Again, a random number [2 is sampled from a uniform distribution in the interval [0, 1], and the nondimensional off-center distance B is determined from the relation

B2 [2 .

(4.255)

If B > Bcrit (Bcrit: from Eq. 4.240 or by the following approximation from

O’Rourke [97]: B2crit = min 1.0, (2.4/Wecoll)( 3 - 2.4 2 + 2.7 )`), the collision will result in stretching separation. Otherwise permanent coalescence will occur. If fur-

ther collision regimes are included, they can be implemented as sub-regimes. For example, if reflexive separation is included, and if the O’Rourke model predicts permanent coalescence, then Eq. 4.245 is used in a second step to check whether reflexive separation occurs instead of coalescence.

If the outcome of the collision is permanent collision, one must check whether n·N1 > N2, because then more collisions are predicted than droplets of parcel 2 are present (each collision erases a droplet of parcel 2). In this case, n is reduced to the maximum possible number (all N2 droplets collide): n = N2 /N1. The mass of the new large droplet after n collisions is

mnew	m n m ,
1	1	2

and the new velocity and temperature are (see also Eqs. 4.241 and 4.242)

u&new	u&1m1 n u&2m2		,
1	mnew
	1
T new	T1m1 n T2m2 .
1	mnew
	1

(4.256)

(4.257)

(4.258)

The remaining droplets of parcel 2 keep their properties, but their number is reduced to

180 4 Modeling Spray and Mixture Formation

Fig. 4.60. Collision cylinder volume

N new	N	2	n N .	(4.259)
2			1

If the outcome of the collision is stretching separation, only one collision between a drop of parcel 1 with a drop of parcel 2 is considered. Temperature and droplet number of each parcel remain unchanged, and the new velocities are calculated according to Eqs. 4.243 and 4.244. If N1 Ŭ N2, the new velocity of the parcel containing more drops is calculated conserving momentum. In the case of N1 < N2 the new velocity of parcel 2 is

		N1		♣	N1	∙
	u&2,new		u2new	♦1		÷ u&2 .	(4.260)
		N2			N2
				♥		≠

Eq. 4.255 can be derived as follows. It is assumed that the probability P(X, Μ) of collision is uniformly distributed over the collision cross section A = Σ(r1 + r2), Fig. 4.60. The integrated probability function is

		1		r	X		X 2
S X					³ 2Σ r dr				,	(4.261)
	Σ r	r	2	r		r	r	2
	1	2			0	1	2

the values of which are between zero (X = 0) and one (X = r1 + r2). The value of S(X) = [2 is sampled from a uniform distribution in the interval [0, 1], and finally S(X) is solved for X using the inverse function

X r1 r2 S X r1 r2 [2 .

(4.262)

Using B = X/(r1 + r2), Eq. 4.262 can be transformed into Eq. 4.255.

4.8 Wall Impingement

Spray-wall impingement is an important process during mixture formation in direct injection small bore diesel engines as well as in direct injection and port injection gasoline engines. Usually, two main physical processes are involved. Wall-

4.8 Wall Impingement	181

spray development and wall film evolution. Both processes may strongly influence combustion efficiency and the formation of pollutants.

In a small direct injection diesel engine, the liquid penetration is sometimes longer than the distance between the nozzle tip and the piston cavity wall, especially in engines with low swirl or during cold start. In this case, the spray-wall impingement may cause a significant increase of unburned hydrocarbon and soot emissions, especially if a wall film is formed. On the other hand, if no liquid wall film is generated, it promotes combustion under hot engine conditions, because spray heating and vaporization are intensified by drop shattering, and the largescale gas vortex, which forms in the near-wall region, enhances gas entrainment.

Impingement in port injected engines causes difficulties in the transient control of the engine, because only a part of the injected fuel enters the combustion chamber during the corresponding cycle, and the rest is added to the wall film and slowly transported to the valve. At the valves, fuel separates from the film and enters the combustion chamber. However, this happens many cycles after the corresponding injection event and adds some uncontrollable amount of fuel to that of the actual injection. This effect is responsible for decreased engine response, increased fuel consumption and increased hydrocarbon emissions. In order to minimize the negative effect of a liquid film on the walls of the induction system, the fuel is sprayed directly on the back of the intake valves. In this case the interaction between valve and spray is an important source of atomization.

Hence, in diesel as well as gasoline engines, a detailed modeling of spray-wall impingement processes is necessary in order to predict their effects on engine performance and on the formation of pollutants.

4.8.1 Impingement Regimes

Figure 4.61 shows the various impingement regimes of a droplet-wall interaction. In the stick regime, a droplet with low kinetic energy adheres to the wall in nearly spherical form and continues to evaporate. In the case of spread, the droplet impacts with moderate velocity on a dry or wetted wall, spreads out and mixes with the wall film (wetted wall) or forms a wall film (dry wall). If rebound occurs, the droplet bounces off the wall (reflection) and does not break up. This regime is observed in the case of dry and hot walls, where the contact between drop and wall is prevented by a vapor cushion. Rebound also occurs in the case of a wet wall if the impact energy is low and an air film between drop and liquid film minimizes energy loss. In the boiling-induced break-up regime, the droplet disintegrates due to a rapid liquid boiling on a hot wall. The wall temperature must be near the Nakayama temperature TN, at which a droplet reaches its maximum evaporation rate. In the case of break-up, the droplet deforms into a radial film on the hot surface, which breaks up due to thermo-induced instability. The splash regime occurs at very high impact energy. A crown is formed, jets develop on the periphery of the crown, become unstable and disintegrate into many droplets.

182 4 Modeling Spray and Mixture Formation

Fig. 4.61. Schematic illustration of different impact regimes [11]

Fig. 4.62. Droplet impingement regimes and transition conditions for a dry wall [11]

There are a number of parameters characterizing the impingement regimes such as incident drop velocity, incidence angle, liquid properties such as viscosity, temperature, surface tension, wall properties like surface roughness and temperature, wall film thickness etc. Some of these parameters can be combined to yield dimensionless parameters. The two most important numbers are the Weber number

	Υ v2d
We	l	n	,	(4.263)

ς

which represents the ratio of the droplet’s kinetic energy (vn: velocity component normal to the surface, Υl: liquid density, d: droplet diameter) and its surface energy, and the Laplace number,

	La	Υlς d	,	(4.264)
		Πl2

which measures the relative importance of surface tension and viscous forces acting on the liquid (µl: dynamic viscosity of liquid). The Laplace number is also represented by the Ohnesorge number Z = La-1/2.

Another important parameter influencing the impingement process is the wall temperature. The characteristic temperatures

	4.8 Wall Impingement 183
Tb TN Tleid	(4.265)

are the liquid boiling temperature Tb, the Nakayama temperature TN at which a droplet reaches its maximum evaporation rate, and the Leidenfrost temperature

Tleid at which a thin layer of vapor forms between the surface and the drop and evaporation is minimized. Figure 4.62 gives an overview of droplet impingement

regimes and transition conditions for a dry wall and fixed Laplace number and surface roughness [11]. In internal combustion engines the wall temperatures during injection are usually below the fuel boiling point [11]. This reduces the number of relevant impingement regimes in case of a dry wall to stick, spread and splash. In the case of a wet wall, Kolpakov et al. [66] revealed that with an increasing impact Weber number the regimes stick, rebound, spread, and splash are important.

4.8.2 Impingement Modeling

Naber and Reitz [90] developed one of the first impingement models. In their model, three regimes are considered: stick, reflection (rebound), and slide. In the slide regime a tangential motion along the surface like a jet with the same magnitude of velocity as before impact is predicted. In all regimes the size of the drops is not changed by the wall interaction. In the stick regime, droplets with low kinetic energy stick to the wall and continue to vaporize. In the case of reflection, drops rebound and the magnitude of their tangential and normal velocity components remains unchanged. However, the normal one changes its sign. This causes specular reflection and is in contrast to the experimental results of Wachters and Westerling [144], Fig. 4.63, in which the outgoing Weber number is generally smaller than the incident Weber number.

Wachters and Westerling [144] performed an experimental study of single drops falling on a hot surface in order to determine the relationship between the velocity components before and after impact. In contrast to the behavior of the tangential velocity component, the normal component is always significantly reduced. Gonzalez et al. [43] developed a numerical fit to their data, Fig. 4.63,

Weout 0.678 Wein exp 0.04415 Wein .

(4.266)

For Wein ŭ 80, the drops do not disintegrate during impact and bounce from the surface, while for Wein > 80 disintegration into small droplets on the surface occurs. In a later version of the model of Naber and Reitz [90], which was presented by Gonzalez et al. [43], a correction of the normal drop velocity (index “n”) in the rebound regime is implemented,

v	v		Weout	,	(4.267)
n,out	n,in		Wein

where Weout is determined by Eq. 4.266.