**3. Numerical simulation of the GDI spray dynamics**

Reducing development time, improving performances and reliability of numerical models is of crucial importance for the design of new engine components. The use of optimization methods coupled with modern CFD tools is today very effective to accomplish these tasks, especially where uncertainty exists about a number of involved constants. Numerical procedures, in fact, may be used to generate a series of progressively improved solutions to the optimization problem, starting from an initial one. The process is terminated when some convergence criterion is satisfied.

In the present section the assessment of a simulation tool reproducing the spatio-temporal dynamics of sprays issuing from new generation high pressure injectors under various operating conditions is presented. The model, developed within the AVL FireTM code environment, is conceived to exploit the previously described experimental data in part as input parameters, in part as terms of comparison for the numerical results.

In order to numerically simulate the effected tests, the spray is hypotesised to enter the top surface of a properly dimensioned computational domain of cylindrical shape, where the injector is supposed to be placed in central position. According to the discrete droplet method (DDM), the spray is considered as a train of droplets of given size, suffering various

Numerical Modelling and Optimization of the

corresponding distribution of initial droplets size.

Fig. 11. Sketch of the numerical model constants tuning procedure.

encountered in the simulation of GDI sprays from new generation injectors.

A similitude of behaviour between Injector #1 and #2 is evident, since the values of C1 are found comparable. The values relevant to Injector #3 are slightly higher, probably due to the differences in the injector geometrical characteristics, hence in the way internal perturbations affect the issuing flow. A slight increase of the variance with injection pressure may be assumed, according to a former authors' idea, physically consistent with the reduction of droplets initial diameter consequent the increase of injection pressure. A similar conjecture may apply to the trend of the Huh-Gosman constant, which slightly increases to account for the greater injection velocity. The low dispersion of the optimal values of the model constants confirms the good prediction capability of the model. The combined use of the Huh-Gosman break-up model and of a properly defined log-normal distribution for the initial droplets size allows overwhelming the problems generally

The dependence of the break-up process upon injection pressure, indeed, is a challenging issue, since it is well known that increasing injection pressure has a twofold effect on the

60000 cells are presented.

Mixture Formation Process by Multi-Hole Injectors in a GDI Engine 187

computational effort. In the following, however, results relevant to a grid made of about

Tuning of the Huh-Gosman model constant and of the distribution variance is here discussed as made by means of an optimization algorithm, instead than through a trial and error procedure. At each injection pressure, the error between the numerically computed penetration length, as averaged over the six jets, and the experimentally measured one is minimised by varying the value of both C1 and in a properly defined DOE space. A sketch of the tuning procedure, as developed within the modeFRONTIER software, is represented in Fig. 11. The Simplex algorithm is used. The results of the automatic model tuning are reported in Fig.12. The top of the figure represents the value of the C1 constant minimizing the error between numerical and experimental data as a function of the injection pressure for the three injectors. The bottom of the figure reports the value of the variance of the

concurring effects as they travel within the computational domain. Turbulent dispersion, coalescence and break-up affect the droplets diameter within a Lagrangian approach coupled with the Eulerian description of the surrounding air motion. Break-up is simulated according to the model of Huh-Gosman [Huh and Gosman, 1991], whose constant C1 (regulating the break-up time) is properly adjusted in the model tuning procedure. Initial droplets size at the nozzle exit section, is considered as not constant, but variable according to a probabilistic log-normal distribution, whose expected value is given by the following theoretical diameter:

$$D\_{\rm th} = \mathbf{C}\_d \left(\frac{2\pi\tau\_f}{\rho\_g u\_{rel}^2}\right) \mathcal{X}^\* \tag{1}$$

being f the gasoline surface tension, g the surrounding gas density, urel the relative velocity between the fuel and the gas, Cd a constant of the order of the unity (indeed taken equal to the unity), and the parameter \* deriving from the hydrodynamic stability analysis and indicating the dimensionless wavelength of the more unstable perturbation to the liquid-gas interface at the injector exit section. The variance of the distribution, , is another parameter of the model to be properly tuned.

It is worth noticing that the definition of a probabilistic distribution of initial droplets size at the nozzle exit section corresponds to specify the occurrence probability for each particle diameter entry in the particle size distribution. The sum of all elements is used to normalize the distribution. The number of particles per parcel is determined by the particle probability distribution, the number of introduced parcels per time step and the assigned mass flow rate. The number of introduced parcels per time step is fixed a priori, and the injection velocity is evaluated in such a way to fulfil the continuity equation. The single jet cone angle is set as an input parameter, according to the effected measurements.

Fig. 10. Results dependency on the grid size.

In order to assess the numerical results dependency on the grid cell size, preliminary tests are made. As shown in Fig. 10, the penetration length in a certain test case, as averaged on the six jets issuing from the Injector #1, is practically unchanged as computed over a grid made of 35000 cells and over a grid made of 96000 cells. The former, therefore, is found being sufficient to perform the computations with a reasonable accuracy and low

concurring effects as they travel within the computational domain. Turbulent dispersion, coalescence and break-up affect the droplets diameter within a Lagrangian approach coupled with the Eulerian description of the surrounding air motion. Break-up is simulated according to the model of Huh-Gosman [Huh and Gosman, 1991], whose constant C1 (regulating the break-up time) is properly adjusted in the model tuning procedure. Initial droplets size at the nozzle exit section, is considered as not constant, but variable according to a probabilistic log-normal distribution, whose expected value is given by the following

\*

(1)

2 2 *<sup>f</sup>*

*g rel*

*u* 

being f the gasoline surface tension, g the surrounding gas density, urel the relative velocity between the fuel and the gas, Cd a constant of the order of the unity (indeed taken equal to the unity), and the parameter \* deriving from the hydrodynamic stability analysis and indicating the dimensionless wavelength of the more unstable perturbation to the liquid-gas interface at the injector exit section. The variance of the distribution, , is another parameter

It is worth noticing that the definition of a probabilistic distribution of initial droplets size at the nozzle exit section corresponds to specify the occurrence probability for each particle diameter entry in the particle size distribution. The sum of all elements is used to normalize the distribution. The number of particles per parcel is determined by the particle probability distribution, the number of introduced parcels per time step and the assigned mass flow rate. The number of introduced parcels per time step is fixed a priori, and the injection velocity is evaluated in such a way to fulfil the continuity equation. The single jet cone angle

> 0 0.0004 0.0008 0.0012 0.0016 0.002 Time (s)

In order to assess the numerical results dependency on the grid cell size, preliminary tests are made. As shown in Fig. 10, the penetration length in a certain test case, as averaged on the six jets issuing from the Injector #1, is practically unchanged as computed over a grid made of 35000 cells and over a grid made of 96000 cells. The former, therefore, is found being sufficient to perform the computations with a reasonable accuracy and low

Grid A - 19440 number cells Grid B - 35000 number cells Grid C - 96000 number cells

 

*th d*

*D C*

is set as an input parameter, according to the effected measurements.

theoretical diameter:

of the model to be properly tuned.

0

Fig. 10. Results dependency on the grid size.

0.02

0.04

0.06

Penetration length (m)

0.08

0.1

computational effort. In the following, however, results relevant to a grid made of about 60000 cells are presented.

Tuning of the Huh-Gosman model constant and of the distribution variance is here discussed as made by means of an optimization algorithm, instead than through a trial and error procedure. At each injection pressure, the error between the numerically computed penetration length, as averaged over the six jets, and the experimentally measured one is minimised by varying the value of both C1 and in a properly defined DOE space. A sketch of the tuning procedure, as developed within the modeFRONTIER software, is represented in Fig. 11. The Simplex algorithm is used. The results of the automatic model tuning are reported in Fig.12. The top of the figure represents the value of the C1 constant minimizing the error between numerical and experimental data as a function of the injection pressure for the three injectors. The bottom of the figure reports the value of the variance of the corresponding distribution of initial droplets size.

Fig. 11. Sketch of the numerical model constants tuning procedure.

A similitude of behaviour between Injector #1 and #2 is evident, since the values of C1 are found comparable. The values relevant to Injector #3 are slightly higher, probably due to the differences in the injector geometrical characteristics, hence in the way internal perturbations affect the issuing flow. A slight increase of the variance with injection pressure may be assumed, according to a former authors' idea, physically consistent with the reduction of droplets initial diameter consequent the increase of injection pressure. A similar conjecture may apply to the trend of the Huh-Gosman constant, which slightly increases to account for the greater injection velocity. The low dispersion of the optimal values of the model constants confirms the good prediction capability of the model. The combined use of the Huh-Gosman break-up model and of a properly defined log-normal distribution for the initial droplets size allows overwhelming the problems generally encountered in the simulation of GDI sprays from new generation injectors.

The dependence of the break-up process upon injection pressure, indeed, is a challenging issue, since it is well known that increasing injection pressure has a twofold effect on the

Numerical Modelling and Optimization of the

0 1.5 3 Time (ms)

0 1.5 3 Time (ms)

0 1.5 3 Time (ms)

lengths for Injector #1 (top), #2 (centre), #3 (bottom).

pinj = 6 MPa

pinj = 6 MPa

pinj = 6 MPa

0

0

0

0.04

0.08

Penetration length (m)

0.12

0.04

0.08

Penetration length (m)

0.12

0.04

0.08

Penetration length (m)

0.12

Mixture Formation Process by Multi-Hole Injectors in a GDI Engine 189

opposite consequences on the overall spray length, that should be reduced by the presence of smaller droplets, but should be increased by the greater velocities. The last effect is indeed dominant, as confirmed by the experiments, but may be overestimated in the phase of spray modeling. Malaguti *et al.* [Malaguti *et al.*, 2010], say, faced the problem by resorting to what authors call an "artificial" introduction of atomized droplets made at a given distance from the injector tip: a Rosin-Ramler distribution whose average diameter was computed to match the experimentally measured penetration length and droplets size was used. Present idea is believed to better follow the actual physics of the phenomenon, since the expected value of the initial droplets distribution, inserted at the nozzle exit section, is a theoretical diameter linked to the injection pressure through the value of the relative velocity between the liquid and the air, that moves towards lower values as injection pressure is raised. The distribution variance may be maintained almost constant or slightly increasing with injection pressure, as confirmed by the here presented optimization procedure. Fig. 13, as an example, reports the distributions used for the four considered injection pressure for Injector #1. Analogous shapes are relevant to the other two injectors.

All the distributions are cut at the value corresponding to the nozzle hole diameter.

0

0

0

0.04

0.08

0.12

0.04

0.08

0.12

0.04

0.08

0.12

0 1.5 3 Time (ms)

0 1.5 3 Time (ms)

0 1.5 3 Time (ms)

Fig. 14. Numerical (continuous line) and experimental (dashed line with dots) penetration

pinj = 10 MPa

pinj = 10 MPa

pinj = 10 MPa

0

0

0

0.04

0.08

0.12

0.04

0.08

0.12

0.04

0.08

0.12

0 1.5 3 Time (ms)

0 1.5 3 Time (ms)

0 1.5 3 Time (ms)

pinj = 20 MPa

pinj = 20 MPa

pinj = 20 MPa

Fig. 12. Results of the tuning procedure of the two model constants.

Fig. 13. Log-normal distribution of the initial droplets size for Injector #1 as a function of the injection pressure.

spray behavior. From one hand, the spray fragmentation is enhanced and the droplets diameter reduced, from the other, injection velocity is increased. The two effects have

inj 1 - 6 holes inj 2 - 7 holes inj 3 - 6 holes

Fig. 12. Results of the tuning procedure of the two model constants.

0

injection pressure.

0.2

0.4

Probability density

0.6

0.8

inj 1 - 6 holes inj 2 - 7 holes inj 3 - 6 holes

4 6 8 10 12 14 16 18 20 22 Injection pressure (MPa)

> pinj = 6 MPa - Dth = 4.99e-5 m - = 0.56 pinj = 10 MPa - Dth = 2.54e-5 m - = 0.55 pinj = 15 MPa - Dth = 1.67e-5 m - = 0.58 pinj = 20 MPa - Dth = 1.27e-5 m - = 0.6

0E+000 4E-005 8E-005 1E-004 2E-004 2E-004 Diameter (m)

Fig. 13. Log-normal distribution of the initial droplets size for Injector #1 as a function of the

spray behavior. From one hand, the spray fragmentation is enhanced and the droplets diameter reduced, from the other, injection velocity is increased. The two effects have

0.3

0.4

0.5

0.6

0.7

0.8

4

8

12

C1

16

opposite consequences on the overall spray length, that should be reduced by the presence of smaller droplets, but should be increased by the greater velocities. The last effect is indeed dominant, as confirmed by the experiments, but may be overestimated in the phase of spray modeling. Malaguti *et al.* [Malaguti *et al.*, 2010], say, faced the problem by resorting to what authors call an "artificial" introduction of atomized droplets made at a given distance from the injector tip: a Rosin-Ramler distribution whose average diameter was computed to match the experimentally measured penetration length and droplets size was used. Present idea is believed to better follow the actual physics of the phenomenon, since the expected value of the initial droplets distribution, inserted at the nozzle exit section, is a theoretical diameter linked to the injection pressure through the value of the relative velocity between the liquid and the air, that moves towards lower values as injection pressure is raised. The distribution variance may be maintained almost constant or slightly increasing with injection pressure, as confirmed by the here presented optimization procedure. Fig. 13, as an example, reports the distributions used for the four considered injection pressure for Injector #1. Analogous shapes are relevant to the other two injectors. All the distributions are cut at the value corresponding to the nozzle hole diameter.

Fig. 14. Numerical (continuous line) and experimental (dashed line with dots) penetration lengths for Injector #1 (top), #2 (centre), #3 (bottom).

Numerical Modelling and Optimization of the

Mixture Formation Process by Multi-Hole Injectors in a GDI Engine 191

Pinj= 10 MPa

Pinj= 15 MPa

Fig. 16. Numerically computed (top) and experimentally collected (bottom) images of the spray issuing from the Injector #2 under injection pressures equal to 10 and 15 MPa.

The prediction capability of the model is demonstrated in Fig. 14, where a comparison between the computed and the measured penetration lengths relevant to the three injectors are reported. The agreement is satisfactory under all the injection pressures considered.

Fig. 15. Experimentally collected images (top) and numerically computed sprays (bottom) for Injector #1 under injection pressures of 20 MPa. Frontal and lateral views.

The spray structure is represented in Figs. 15 and 16. Fig. 15 is a sequence of images of the evolution of the spray issuing from Injector #1, in two different views, as experimentally collected and as numerically simulated for an injection pressure Pinj=20 MPa and an injected total mass mf=50 mg. The frontal view of the sprays allows appreciating the jet propagation. The regularity appears destroyed for the 500 and 700 μs images, where the interference between the single jets is evident and the single jet evolution cannot be longer followed. The fuel has to be considered as a single, large and composite spray. Fig. 16 shows experimental images and numerically computed sprays from Injector #2 at various instants of time from the SOI at the injection pressure of 10 and 15 MPa. The greater penetration length at the higher pressure is evident. Slight differences appear in the spray tip, that is sharper in the experiments. This discrepancy can be reduced by further adjusting the far field break-up characteristic time of droplets, although it is to be considered that, at the distance from the nozzle where the differences become appreciable, the effect of evaporation should be also taken into account under real engine working conditions.

The prediction capability of the model is demonstrated in Fig. 14, where a comparison between the computed and the measured penetration lengths relevant to the three injectors are reported. The agreement is satisfactory under all the injection pressures considered.

Fig. 15. Experimentally collected images (top) and numerically computed sprays (bottom)

The spray structure is represented in Figs. 15 and 16. Fig. 15 is a sequence of images of the evolution of the spray issuing from Injector #1, in two different views, as experimentally collected and as numerically simulated for an injection pressure Pinj=20 MPa and an injected total mass mf=50 mg. The frontal view of the sprays allows appreciating the jet propagation. The regularity appears destroyed for the 500 and 700 μs images, where the interference between the single jets is evident and the single jet evolution cannot be longer followed. The fuel has to be considered as a single, large and composite spray. Fig. 16 shows experimental images and numerically computed sprays from Injector #2 at various instants of time from the SOI at the injection pressure of 10 and 15 MPa. The greater penetration length at the higher pressure is evident. Slight differences appear in the spray tip, that is sharper in the experiments. This discrepancy can be reduced by further adjusting the far field break-up characteristic time of droplets, although it is to be considered that, at the distance from the nozzle where the differences become appreciable, the effect of evaporation should be also

for Injector #1 under injection pressures of 20 MPa. Frontal and lateral views.

taken into account under real engine working conditions.

Fig. 16. Numerically computed (top) and experimentally collected (bottom) images of the spray issuing from the Injector #2 under injection pressures equal to 10 and 15 MPa.

Numerical Modelling and Optimization of the

Fig. 17. Top view of the complete engine surface.

characteristic data.

numerical errors due to reflections from the outlet surface.

Mixture Formation Process by Multi-Hole Injectors in a GDI Engine 193

cylinder axis. Fig. 18 represents one of the grids used for the range of valves overlap, particularly at 380°, together with a table reporting its geometrical characteristics. Note the thick part on the right of the exhaust ducts, which is a part added properly to avoid

> Number of tetrahedrical cells 7698 Number of Hexahedrical cells 613785 Number of piramidal cells 67811 Number of prismatic cells 133259 Total number of cells 822553

Fig. 18. Computational grid (cells on surface) corresponding to a crank angle of 380° and
