**3.5 Development of the empirical correlations for spray penetration into the cross flow**

Literature sources suggest correlations for the spray outer border *x/d=f(z/d)* in several different forms that definitely include power function of the momentum flux ratio *qn*. Correlations may or may not include power function of Weber number. Shape of the spray pattern is typically described using logarithmic or power function. In spite of the fact that the accuracy of correlation can be improved by increasing number of empiric constants, current study seeks to simplify correlations. This was achieved by using self explained proportionality of droplets penetration into the cross flow to their velocity at the point of discharge (i.e. *x/d~Ul~q0.5*) and reducing number of the empiric constants by one (i,e *qn* = *q0.5* ). This significant simplification was proved experimentally on both tested injectors in a wide range of momentum ratios between *q=5* and *q=100.*

Another simplification of correlation function was attained by limitation of the Weber number range between *We=400* and *We=1600*. This in turn limited number of possible mechanisms of the jet disintegration to only one mode of liquid jet breakup; i.e., shear breakup excluding column break up. Independence of spray penetration upon the Weber number in the investigated range allowed an exclusion of the Weber number from correlations.

As a result spray penetration for both injectors was correlated using only one empiric coefficient (*a1*) that depends only upon the shape of the injector internal surface by the following formula:

76 Advanced Fluid Dynamics

Measurements of the spray border obtained in the current study using high speed imaging technique were compared with the spray border data obtained using Phase Doppler method. For this purpose the data rate measured with the PDPA is used as a metric to locate the edge of the spray. The edge of the spray is assumed to be around a region showing 10% of the maximum data rate as shown in Fig. 23-b. Figure 23-a demonstrates a good agreement between the spray trajectories obtained using statistically relevant high speed imaging technique and borders of the spray measured by the processing of the PDPA data rate. It is clearly seen that the maximum spray penetration determined as *X\*=Xmean+ 2.8RMS* is equal to the border determined at the level of 10%

**Data rate, cps**

Fig. 23. Comparison of the maximum spray penetration (i,e *X\*=Xmean+ 2.8RMS*) at *q=*20

**3.5 Development of the empirical correlations for spray penetration into the cross** 

Literature sources suggest correlations for the spray outer border *x/d=f(z/d)* in several different forms that definitely include power function of the momentum flux ratio *qn*. Correlations may or may not include power function of Weber number. Shape of the spray pattern is typically described using logarithmic or power function. In spite of the fact that the accuracy of correlation can be improved by increasing number of empiric constants, current study seeks to simplify correlations. This was achieved by using self explained proportionality of droplets penetration into the cross flow to their velocity at the point of discharge (i.e. *x/d~Ul~q0.5*) and reducing number of the empiric constants by one (i,e *qn* = *q0.5* ). This significant simplification was proved experimentally on both tested injectors in a wide

Another simplification of correlation function was attained by limitation of the Weber number range between *We=400* and *We=1600*. This in turn limited number of possible mechanisms of the jet disintegration to only one mode of liquid jet breakup; i.e., shear breakup excluding column break up. Independence of spray penetration upon the Weber number in the investigated range allowed an exclusion of the Weber number from

As a result spray penetration for both injectors was correlated using only one empiric coefficient (*a1*) that depends only upon the shape of the injector internal surface by the

**Distance in direction of injection X, mm**

b) Spray border determination using PDPA data rate curve

0 5 10 15

**10%**

threshold of the PDPA data rate curve maximum.

**We = 406 We = 1014 We = 1600 PDPA / We = 500 PDPA / We = 1000 PDPA / We = 1500**

measured by the high speed (HS) imaging technique and by the PDPA

**0 10 20 30 40 50**

range of momentum ratios between *q=5* and *q=100.*

**Axial distance from the orifice, mm**

(a) Maximum spray penetration

**flow** 

correlations.

following formula:

**X\*, mm**

$$\frac{\mathbf{x}}{\mathbf{d}} = a\_1 \sqrt{\mathbf{q}} \left( \frac{1}{(1 + a\_2 \frac{\mathbf{z}}{\mathbf{d}})} + \ln(1 + a\_2 \frac{\mathbf{z}}{\mathbf{d}}) \right) \tag{3}$$

The other coefficient (*a2*) only shaped the spray border described by the logarithmic function and was independent of the injector design. Thus average and maximum spray penetrations were correlated using coefficients *a1* and *a2* presented in the table 1.


Table 1. Empirical correlation coefficients for the average and maximum spray penetration into the cross flow.

Comparison of the experimentally measured and correlated spray penetrations *X* are presented on the Fig. 24 for the average and maximum penetration of the spray created by the sharp edged injector.

Fig. 24. Comparison between the correlated and experimentally measured values of spray penetration *X*

Fuel Jet in Cross Flow – Experimental Study of Spray Characteristics 79

The authors would like to thank General Electric–Aviation (Dr. Nayan Patel, contract monitor) for the fund that allowed conduct this study and for additional technical guidance.

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**5. Acknowledgment** 

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498, pp. 73-111

179-197.

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**6. References** 
