A Unified Model Considering Effects of Droplet Break-Up and Air Entrainment at the Initial Stage of Fuel Spray Penetration

This paper proposes a new unified single model that considers droplet break-up and air entrainment impact on three flow regimes (namely, Stokes, Allen, and Newton) in the initial stage of fuel spray penetration. Homotopy perturbation method was used to obtain semianalytical solutions of unified single model on diesel fuel’s spray penetration when the influence of air entrainment is small (up to 0.1-0.2ms after the start of injection). It is demonstrated that the applied analytical method is very straightforward in comparison with existing techniques. Furthermore, it is decidedly effectual in terms of accuracy and rapid convergence. The formulation of the problem is presented in the text as well as the analytical and numerical procedures.


Introduction
The importance of the spray penetration issue for various applications is well recognized and has been extensively studied experimentally and theoretically [1][2][3][4][5].A rigorous theory of spray penetration would be very complex if it needs to involve the modeling of the ligaments formation and breakup, droplet break-up and evaporation, the air entrainment, and the turbulence effects [6].A self-consistency of the modeling for all these processes is still a major challenge.However, this modeling is not always essential for understanding the process and engineering applications.On many occasions, it is so crucial to establish a "hierarchy" of the importance of the various processes and develop simplified models suitable for practical applications.In this manner, as mentioned in references [7,8], simplified analytical models have been developed which describes the initial stage of spray penetration in three flow regimes, namely, Stokes, Allen, and Newton without considering the air entrainment influences.
The homotopy perturbation method has also been exploited for finding the exact and approximate solutions of the linear and nonlinear integral equations and their systems [22][23][24], the integrodifferential equations [25,26], and the Volterra-Fredholm integral equation [27].
As can be seen in relevant literature, no attempt has been made so far to analytically study the effects of droplet break-up and air entrainment on the spray tip penetration.The objective of this paper is, therefore, firstly to construct a new unified model that considers the three flow regimes to consider the droplet break-up and air entrainment in the initial stage of fuel spray penetration.Secondly, an analytical result, which takes into account the flow regimes, is derived from homotopy perturbation method (HPM) employing Padé technique of the unified model.Thus, we utilize the homotopy perturbation method (HPM) as a powerful seriesbased analytical tool.

The Initial Stage
2.1.Dynamics of Droplets.The velocities of droplets injected from a nozzle exit tip are initially much greater than the velocity of the gas (air) stream but are slowed down due to the drag force, while gas is accelerated.The equation describing the dynamics of an individual droplet can be written as where   , V  , and   are droplet's mass, velocity, and crosssectional area, respectively.V  and   are gas velocity and density, respectively.In this study, only one-dimensional dynamics of gas and droplets are considered.  is the drag coefficient, which depends on the shape of the droplet and the Reynolds number defined as Re = 2  (V  − V  )  /  , with   and   being the droplet's radius and gas dynamic viscosity, respectively.
Assuming that the droplets are perfect spheres, then (1) is simplified as where   represents the droplet's density,  is the distance measured from the nozzle, and V  = /.Equation ( 2) is not easily amenable to analytical results, especially due to the drag coefficient   , which is a rather complicated function of the Reynolds number.Therefore, a number of approximations for   coefficient were suggested in the literature [7].The most convenient approximation can be found in Douglas et al. [28], which considered three ranges of Reynolds numbers: Re ≤ 0.2 (Stokes flow), 0.2 < Re ≤ 500 (Allen flow), and 500 < Re ≤ 10 5 (Newton flow).The functions   (Re) for these flows are given by the expressions:   = 24/ Re (Stokes flow),   = 18.5/Re 0.6 (Allen flow), and   = 0.44 (Newton flow).The expressions for   do not take into account the effects of droplet acceleration, internal circulation, vaporization, burning, nonspherical shape, vibrations, and heating processes.Two basic approximations were applied to (1) in relation to V  , namely, (i) V  ≪ V  and (ii) V  = √, where  = √/ 0 and  = 3 2    /(8  )  V 2 0 = constant.Here, V 0 is the initial droplet velocity,   and   are radii of the gas/air jet and the spray itself, respectively,   is the volume fraction of droplets in the spray, and   is usually slightly larger than   , but as a first approximation we will assume that they are equal.
The approximate analytical results obtained from these approximations and their physical interpretations are extensively discussed in [7,8].Now with regard to the latter approximation for V  (= √) in (2), the governing equations for Stokes, Allen, and Newton flows [7,29] are rewritten as follows: where  = 9  /2 A careful examination of (3a), (3b), and (3c) revealed that they could be combined to give the following model [30]: Here  denotes the flow regime parameter or identifier.Equation ( 4) is the unified model for the fuel spray penetration, and it has not been reported elsewhere in literature to the best of author's knowledge.The equation is nonlinear and includes all three regimes of the flow, in that the parameter  takes the values 0, 0.4, and 1, such that  0 =  (Stokes flow),  0.4 =  (Allen flow), and  1 =  (Newton flow), respectively.

Droplet Break-Up Models.
A number of droplet breakup models have been proposed [1,[31][32][33].One of the most popular models is according to the comparison between the eigenfrequency of droplet oscillations and the frequency of turbulent pulsation in the surrounding gas.This allows the determination of the maximal stable droplet diameter.The predictions of this model are in good agreement with some experimental observations [34].
In our case, however, the driving force of droplet breakup is not the external turbulence, but the kinetic energy of the droplets themselves is.In this case, a different approach would seem to be more appropriate.As in [7], this study is based on the model developed in [35,36] which is particularly attractive for analytical and numerical analysis and is widely used in CFD codes.
In this model, two droplet break-up regimes are identified and are shown in Figure 1.These are bag break-up assuming that We ≡   V 2    > 6. ( And stripping break-up is as given assuming that where We is the Weber number and  is the surface tension.The first regime is analogous to the Rayleigh-Taylor instability accompanied by the development of normal stresses. The second regime is analogous to the Kelvin-Helmholtz instability accompanied by the development of tangential stresses.The break-up processes described by ( 5) and ( 6) can occur simultaneously.In this case, the dominant process will be determined by the shortest lifetime as described below.Sometimes a more stringent criterion for the stripping instability ( > 0.7) was applied [1].The modifications of our analysis to meet this criterion would be straightforward.It is important not to confuse  with the Ohnesorge number, Oh ( ≡ √ We/ Re) [1].The lifetime of unstable droplets was estimated as [36] where  was considered equal to  and  = 13.As a first approximation, the creation of droplets with different radii during the break-up process was not taken into account.The radii of unstable droplets are allowed to change continuously following the given equation by where   and   were introduced by ( 7) and ( 8), respectively. () is marginally stable radii that is determined by We = 6 and  = 0.5 conditions.This approximation can be explained by the fact that the contribution of smaller droplets generated during the break-up is relatively small and can be ignored.
In addition, (9) implies that droplets preserve their spherical shape during the break-up process.The authors agree that this is a limitation of the model.Combination of solutions for (9) together with (3a), (3b), or (3c) associated with spray penetration for Stokes, Allen, and Newton flows will be investigated for the bag and stripping break-up regimes.

Effect of Droplet Bag Break-Up
From ( 5), the following is obtained: Substituting ( 7) into (9), the following equation is obtained: where Assuming that  ≫   the right hand side of ( 11) can be reduced to −1/(  √) and a definition can be written as Substituting ( 12) into (3a), (3b), and (3c), we obtain the governing equation for Stokes, Allen, and Newton flows that are achieved as follows, respectively: where A careful examination of above equations revealed that they could be combined to give the following model: Here  is defined as the flow regime parameter or identifier.Equation ( 14) is a new unified model to explain the droplet bag break-up and air entrainment impact at initial stages of fuel spray penetration, noting that it has not been reported elsewhere in literature to the best of the author's knowledge.The equation is nonlinear and embraces all three regimes of the flow in which we have  = 4/3, 1.0666, and 2/3.One has to note that the parameter  takes the values of 0, 0.4, and 1 such that  0 =   (Stokes flow),  0.4 =   (Allen flow), and  1 =   (Newton flow), respectively.This means that the specific applicable examples to (3a), (3b), and (3c) are embedded in the model equation (14), respectively.

Effect of Droplet Stripping Break-Up
From ( 6) the following can be achieved: The combination of ( 8), (9), and (15) yields where   = √  /  in ( 16) is valid for  >   .Otherwise  = const.Assuming that  ≫   , the right hand side of ( 16) can be simplified to −V  /  .The contribution of the term Mathematical Problems in Engineering proportional to   can be studied similarly to Section 3. In contrast to the case of bag break-up, this equation cannot be solved unless we know the time dependence of V  .
Considering the combination of (3a), (3b), and (3c) and ( 16) with  >   assumption, the governing equations for Stokes, Allen, and Newton flows are written in the following order: A careful examination of ( 17) revealed that they could be combined to give the model: Here  is defined as the flow regime parameter or identifier.Equation ( 18) is a new unified model to explain the droplet stripping break-up and air entrainment impact at initial stages of the fuel spray penetration.Note that (18) has not been reported elsewhere in the literature to the best of the author's knowledge.The equation is nonlinear and embraces all three regimes of the flow in that  = 2, 1.6, and 1 while the parameter  takes the values 0, 0.4, and 1 such that  0 =   (Stokes flow),  0.4 =   (Allen flow), and  1 =   (Newton flow), respectively.This means that the specific applicable examples to (3a), (3b), and (3c) are embedded in the model equation (18), respectively.

Basic Ideas of the HPM
To illustrate the basic idea of this method [9,10], we consider the following nonlinear differential equation: Consider the boundary given as where  is a general differential operator,  a boundary operator, () a known analytical function, and Γ the boundary of the domain Ω.The operator  can be divided into two parts of  and , where  is the linear part, whereas  is a nonlinear section in (19): therefore, it can be represented as follows: The homotopy perturbation structure is shown as where We can assume that the solution of ( 23) can be written as a power series in  as follows: Setting  = 1 results in the approximate solution as The combination of the perturbation method and the homotopy method is called the HPM, which lacks the limitations of the traditional perturbation methods, although this technique has full advantages of the traditional perturbation techniques.

Homotopy Perturbation Method Solution
In this section, we will apply the homotopy perturbation method (HPM) to nonlinear ordinary differential equation ( 14) and (18).According to this method, we can construct a homotopy as follows: where Applying the perturbation technique [9,10], we can assume that the solution of (26a) and (26b) can be expressed as a series in : where  ∈ [0, 1] is an embedding parameter.Taking  = 0, we obtain the initial guesses; in contrast having  = 1 in ( 28) presents the original one.
Tuning  = 1, we obtain an approximate solution of (26a) and (26b): One has to substitute (29) into the governing equation ( 26a) and (26b), then collects the powers of , and finally obtains a sequence of differential equations and boundary conditions.The solution for the spray penetration for Stokes, Allen, and Newton flows for droplet bag and stripping break-up regimes is given as follows.
Bag break-up for stokes flow is as follows: Bag break-up for Allen flow: Bag break-up for Newton flow: Stripping break-up for Stokes flow:  Stripping break-up for Allen flow: Stripping break-up for Newton flow: According to the above equations, we get the solution for the spray penetration field for the three flow patterns and at two break-up regimes.

Results and Discussion
Spray penetration lengths are solved analytically using homotopy perturbation method (HPM) employing Padé technique.In this case, the droplet radius was 0.0001 mm, droplet density was 830 kg/m 3 .The radius of the air jet near the nozzle was 0.0001 mm, radius of the spray was 0.0001 mm, gas density was 19.7 kg/m 3 , droplet mass fraction was   = 0.001, and the gas dynamic viscosity was 3.36 × 10 −6 m 2 /s.In spite of the advantages of homotopy perturbation method, it has some drawbacks as well.By using HPM, series were acquired (in practice a truncated series solution).
The question is put forward as to how close the function can be approximated from its series expansion.This is the point where Padé approximants come into play.A possible answer is to construct a rational function whose series expansion matches the original one as much as possible.Such rational functions are called Padé approximants.Therefore, series of Padé approximants were applied to the series solution to improve the accuracy and extend the convergence domain.
Figures 2-5 are plotted in order to confirm the reliability and accuracy of the HPM that is enhanced by the Padé approximants.By using Padé approximant, one can create a fraction with some polynomials in the numerator and some polynomials in the denominator.It is of great importance to know that this solution is practical only if the orders of the numerator and the denominator are the same (diagonal approximation).In this study, Maple program was efficiently utilized in order to obtain Padé approximants with different orders [L/M].
Figure 2 elucidates the exact behaviors of spray penetration profiles that have been obtained by HPM-Padé in comparison to HPM.The results of HPM-Padé are in good agreement with numerical results.Furthermore, this figure indicates that increasing the number of HPM terms does not affect the convergence and these series diverge around infinity.
Figure 3 shows that the difference between numerical and HPM solutions depends on the order of Padé approximation.not air entrainment; both bag break-up and entrainment processes are taken into account.It is clear from these figures that spray tip penetration increases when higher injection pressure is applied.This is due to higher momentum of fuel jet.Moreover, the spray tip penetration decreases by consideration of the bag and stripping break-up effects.
All figures are presented in the validation range of the models; that is,  ≫  () .(approximated as  > 3.3  ) and  <  −1  .As can be noticed from our estimates, the range of validity for the stripping break-up model (about 0.05-0.1 ms) is noticeably narrower than that of the bag break-up model (about 0.25 ms).Hence, the plots are presented in two time ranges: 0-0.1 and 0-0.3 ms for stripping and break-up models.Since the bag break-up model has a wider range of validity, the effect of air entrainment will be more important for this model.As it can be seen from Figures 4 and 5, in all of the cases that are presented with the simplest model (no break-up and air entrainment), it clearly underestimates the observed spray penetration lengths.Air entrainment leads to the increase of the spray penetration, while the break-up process tends to decrease the spray penetration.It can be inferred that these parameters have compensating influence on the spray penetration such that a tradeoff can be established between air entrainment and break-up to obtain an optimum spray penetration.At bag break-up model, the effect of air entrainment is higher than that of break-up effect.Therefore, air entrainment causes an increase in the spray tip penetration.Meanwhile, the effect of air entrainment is negligible in the stripping break up model; the effect of breakup is only important in spray tip penetration.The model is not valid at times greater than about 0.1-0.2ms when the twophase flow approximation has to be used.illustrate the variation of spray tip penetration with nozzle-hole diameter, injection pressure, and ambient density as contour plots.As shown in Figure 6, when injection pressure increases and ambient density decreases at constant nozzle-hole diameter, spray tip penetration increases, noting that the effect of ambient density is more considerable.Also according to Figure 7, at constant injection pressure the spray tip penetration increases with the increase of nozzle-hole diameter and decrease of ambient density.
Figure 8 shows the variations of spray tip penetration with nozzle-hole diameter and injection pressure at constant ambient density.It is clear from Figure 8 that the effect of injection pressure is more considerable compared to the nozzle-hole diameter.

Conclusions
In this study, the unified model is proposed based on the droplet break-up and air entrainment influences at initial stage of the fuel spray penetration that accounts for three regimes of the flow.In the modeling equation, which is highly nonlinear, the effects of air entrainment and breakup processes on the spray penetration length were studied analytically in three flow regimes: Stokes, Allen, and Newton flows by homotopy perturbation method (HPM).The formulae for spray penetration length for three flows are presented at three cases: neither one of the break-up and air entrainment are taken into account; only breakup process (bag and stripping) is taken into account, but the air entrainment was disregarded; both break-up process (bag and stripping) and air entrainment are taken into account.The results show that the effect of air entrainment is more important in bag break-up mechanism than that of stripping break-up.The results of the analytical model are compared to the numerical method and a good agreement was accomplished.Moreover, the variations of the spray tip penetration with various parameters are 3 dimensionally plotted and commented upon.The models are expected to be valid in two time ranges: 0-0.1 and 0-0.3 ms for stripping and bag break-up models after the start of injection.

Figure 2 :
Figure 2: The effect of increasing the number of HPM terms in comparison with HPM-Padé [8/8] and numerical solution (RK4) for the spray penetration when   = 14.8 kg/m 3 ,  inj = 170 Mpa.

Figure 3 :
Figure 3: The effect of increasing the order of Padé approximation on the spray penetration when   = 14.8 kg/m 3 ,  inj = 170 Mpa.

𝑑0 ) 𝑡 6 0Figure 4 :
Figure 4: The variation of spray penetration length with time in the cases when no break-up and air entrainment are taken into account; bag break-up is taken into account; both bag break-up and air entrainment are taken into account: (a) refers to   = 3.7 kg/m 3 ,  inj = 60, and 170 MPa and (b) refers to   = 14.8 Kg/m 3 ,  inj = 60 and 170 MPa.

Figure 5 :
Figure 5: The variation of spray penetration length with time in the cases when no break-up and air entrainment are taken into account; stripping break-up is taken into account; stripping break-up and air entrainment are taken into account: (a) refers to   = 3.7 Kg/m 3 ,  = 60, and 170 MPa and (b) refers to   = 14.8 Kg/m 3 ,  = 60, and 170 MPa.

Figure 6 : 30 Figure 7 :Figure 8 :
Figure 6: Contour plot for variation of the spray tip penetration with injection pressure and ambient density at constant hole diameter of 0.1 mm in 0.3 ms.