Oblique spray impact

(a)vN,sec versusDsec. (b) vT,sec versusDsec.

Figure 5.18: Dependence of v_N,sec and v_T,sec on the diameter D_sec for single drops of all measurements. Mean values per bin of width 5µm are also shown.

(a) Wall-normal component. (b) Wall-tangential component.

Figure 5.19: Several discrete distribution functions of secondary drops describing the velocity components per diameter bin of width 5µm. D₁₀ denotes the mean secondary diameter in the respective bin.

Figure 5.20: v_N,sec versusv_T,sec for single drops of all measurements.

Figure 5.21: Endoscopy image of the impact of a hollow cone spray in a one-cylinder test bench at BMW. The rail pressure is set to 100bar and the ambient temperature lies around70^◦C.

Courtesy Peter Helmetsberger.

are maintained for any value of the impact angle.

Secondary diameters. Presumably, film motion changes considerably with the im-pact angle. However, these changes in the film fluctuations and resulting ligaments are not known and it is therefore assumed in this work that the relation between secondary and primary mean diameters does not depend on the impact angle.

Secondary velocities. From the transmitted-light images of the reference case in Chapter 2 with an impact angle of ≈ 45^◦ and from endoscopy images taken at BMW, see Figure 5.21, it can be inferred that the direction of the secondary spray does not change significantly compared to the case of normal impact: The motion is still rather tangential to the wall.

To extrapolate the correlations from normal impact to values α₁₀ 6= 90^◦, cf. Equa-tions 5.7, a momentum consideration is applied. The absolute primary momentum is generally transferred to:

• momentum carried by secondary drops in forward and backward direction,

• momentum transferred to the wall film,

• losses.

Only the first point is of interest at this point. The associated momentum fraction is unknown and the following assumptions are made:

pN,10,sec forward = f_N,forward(α₁₀)·pabs,10,prim, pT,10,sec forward = f_T,forward(α₁₀)·pabs,10,prim, pN,10,sec backward = f_N,backward(α₁₀)·pabs,10,prim,

pT,10,sec backward = f_T,backward(α₁₀)·pabs,10,prim. (5.15)

periments for normal impact, can be considered instead of the momentum:

vN,10,sec forward = f_N,forward(α₁₀)·vabs,10,prim, vT,10,sec forward = f_T,forward(α₁₀)·vabs,10,prim, vN,10,sec backward = f_N,backward(α10)·vabs,10,prim,

vT,10,sec backward = f_T,backward(α₁₀)·vabs,10,prim. (5.17) From the experimentfN,forward/backward(90^◦)≈0.0811andfT,forward/backward(90^◦)≈0.203 result, cf. Equation 5.7.

In Figure 5.22 it is shown that the reference case with α₁₀ ≈ 45^◦ cannot be described correctly with constant functionsfN,forward/backward(α10) =fN,forward/backward(90^◦) =const.

andfT,forward/backward(α₁₀) = fT,forward/backward(90^◦) =const. : The tangential penetration is much too slow. This cannot be explained by an unaccounted change in secondary diameters with the impact angle: Assuming 0.75 · D10,prim or 1.25 ·D10,prim in the correlations does not lead to sufficient tangential penetration either. A consideration of vN/T,10,sec forward/backward relative tov_N,10,prim instead ofvabs,10,prim would even decrease the penetration.

The functions are therefore chosen intuitively as follows:

• With decreasing mean impact angle α₁₀, i.e. for flatter impact, the secondary normal momentum is expected to decrease. Assuming that especially the normal component of the primary momentum contributes to the secondary normal com-ponent, the decrease is set proportional to it with no difference between forward and backward direction:

fN,forward/backward(α₁₀) = fN,forward/backward(90^◦)·sinα₁₀. (5.18) The sensitivity of this function onα₁₀is limited as the valuefN,forward/backward(90^◦) is very small.

• The secondary tangential momentum in forward direction is expected to increase with decreasing α₁₀. Assuming that the increase is proportional to the primary tangential momentum yields:

f_T,forward(α₁₀) = fT,forward/backward(90^◦) +k_T·cosα₁₀. (5.19) The parameter k_T denotes the fraction of the primary tangential momentum which is transferred to the secondary one, additionally to the general transfor-mation of absolute momentum. It is determined in simulating the reference case and in adapting the tangential penetration to the transmitted-light im-ages. It shows that vT ,10,sec forward(45^◦)≈0.4·vabs,10,prim describes these best, see

Figure 5.22: Reference case at t = 0.60ms with constant functions fN,forward/backward(α₁₀) and fT,forward/backward(α₁₀). The sensitivity on the primary di-ameter is also presented. The results with the mean secondary velocity components considered relative to v_N,10,prim instead of vabs,10,prim are shown at the bottom.

Figures 5.23 and 5.24. This givesk_T ≈0.28. Note that the reference case with an injection pressure of200bar and an oblique impact angle of45^◦ differs noteworthy from the experimental conditions.

The backward direction, i.e. the reflection towards the inside of the spray cone cannot be adjusted considering transmitted-light images and no additional in-formation is available. Therefore, no contribution from the primary tangential momentum is accounted for:

fT,backward(α10) = fT,forward/backward(90^◦). (5.20) The presented functions are used for modelling oblique impacts in the following. A quantitative validation is not possible due to a lack of data. Yet, in Chapter 6 a case is presented where mean impact angles in an approximate range of 30^◦ < α₁₀ < 60^◦ are considered. Together withα₁₀ ≈90^◦ from the experiment, it can be presumed that the correlations cover also the angle range in-between with 60^◦ < α₁₀<90^◦ and that they can therefore be used in an estimated range of 30^◦ < α₁₀ ≤90^◦ in engine calculations.

Figure 5.23: Reference case for different values of f_T,forward(45^◦) at t = 0.60ms, fN,forward/backward(α₁₀) = fN,forward/backward(90^◦)·sinα₁₀ and f_T,backward(α₁₀) = fT,forward/backward(90^◦).

This meets the conditions occuring in in-cylinder calculations rather well: Except very rarely on the spark plug, impacts under even smaller angles do not occur.

Secondary mass. In case of normal impact, half the total secondary mass is reflected on either side of the impacting spray which can be locally approximated by an impacting liquid sheet, see above. For an extrapolation to oblique angles, a simple analogy to the stationary impact of a liquid volume flux onto a plane is considered, see Figure 5.25.

Applying Bernoulli’s equation one gets, see e.g. [19]:

qm,sec forward

q_m,prim = 1

2· qm,sec total

q_m,prim ·(1 + cosα₁₀) , (5.21)

qm,sec backward

q_m,prim = 1

2· qm,sec total

q_m,prim ·(1−cosα₁₀) , (5.22)

where α₁₀ is defined relative to the wall tangent⁶.

6The analogy cannot be used to give correlations for the secondary velocity as the latter is main-tained in the secondary sheets (viscosity is neglected), which does not hold for the measured data.

Figure 5.24: Reference case for different values of f_T,forward(45^◦) at t = 0.80ms, fN,forward/backward(α₁₀) = fN,forward/backward(90^◦)·sinα₁₀ and f_T,backward(α₁₀) = fT,forward/backward(90^◦).

(a) Normal impact. (b) Oblique impact.

Figure 5.25: Impact of a liquid sheet onto a plane. Gravity and viscosity are neglected.

D_10,sec

D_10,prim ≈ 0.673, pdf_gev

D_sec D_10,sec

≈ 1

0.367 ·exp −z^−1/0.111

·z^{−1−1/0.111}

with z= 1 + 0.111·

Dsec

D10,sec −0.737

0.367 . (5.23)

• Wall-normal velocity components:

vN,10,sec

vabs,10,prim

≈ 0.0811·sinα₁₀,

pdf_Weibull

v_N,sec v_N,10,sec

≈ 1.416

1.100 ·z^0.416·exp −z^1.416 with z = v_N,sec

v_N,10,sec/1.100. (5.24)

• Wall-tangential velocity components:

vT,10,sec forward

vabs,10,prim

≈ +1·(0.203 + 0.42·cosα₁₀), vT,10,sec backward

v_T,10,prim ≈ −1·0.203, pdf_Weibull

v_T,sec v_T,10,sec

≈ 1.302

1.076 ·z^0.302·exp −z^1.302 with z = v_T,sec

v_T,10,sec/1.076. (5.25)

• Mass ratios:

qm,sec forward

q_m,prim = 0.399·(1 + cosα₁₀), (5.26)

qm,sec backward

q_m,prim = 0.399·(1−cosα₁₀). (5.27)

The correlations are empirical and not universally valid. If they are applied to other spray nozzles, the probability density functions, i.e. the pdfs, should be adapted if known. It is vital to gather further quantitative data for different volume fluxes, Reynolds numbers and impact angles in future work in order to extend and confirm

the correlations, especially the extrapolation to oblique impacts.

Yet, the empirical correlations represent the first successful attempt to describe the impact of the considered spray type, which can furthermore be easily implemented in numerical code, see the following section. The correlations describe the experiment and the reference case very well, where it has to be emphasized that the conditions of both cases differ strongly, e.g. a different injector is used and the injection pressure differs by a factor of four. Another case in Chapter 6 leads to the conclusion that the correlations are presumably applicable in a range of 30^◦ < α₁₀<90^◦.

Im Dokument Modelling wall interactions of a high-pressure, hollow cone spray (Seite 146-155)