A Design Tool for Aerodynamic Lens Systems - Department of ...

Figure 5 shows the flow field through two lenses separated
by a spacer. The flow separates from the wall when it passes
through the lens and a recirculation zone forms downstream
of the orifice. After a distance lr (reattachment length), the flow
reattaches to the wall, and becomes fully developed at a distance
of l f (redevelopment length) downstream of the orifice. The
existence of the downstream lens is felt by the flow la (approach
length) upstream of the next lens where the flow starts to curve
toward the centerline. Ideally, the minimum length of spacers
(ls) should be ls1 = lr + la, and more safely, ls2 = l f + la.
If the spacer length is less than ls1, the recirculation zone will
likely fill the whole length of the spacer and less focusing will
occur. Furthermore, particles are more likely to be entrained in
the recirculation zones and be lost to the wall. Therefore, when
particle diffusion is negligible and the overall length of the lens
system is not a concern, we suggest using spacer lengths at
least ls2. When diffusion is important, ls1 < ls ≤ ls2 can be
used.
To estimate the values of lr, l f , and la,wehave carried out numerical
simulations of flow through single lenses in the Reynolds
number range of 0.1–200. Calculated values for lr are shown in
Figure 6, together with data provided by other researchers (Back
and Roschke 1972; Oliveira et al. 1998). Note that our numerical
results are very close to those by Oliveria (1998), which apply
to similar Reynolds numbers and expansion ratios. The results
by Back et al. differ somewhat from ours and those of Oliveria,
probably due to the difference in ER. Since we did not carry out
simulations for Re > 200, we will use the data by Back et al.
to estimate lr in this Re range, noting that applying Back’s data
(ER = 2.6) to larger ER values might lead to some error. Also
shown in Figure 6 is a fitted curve to the whole Reynolds number
range, which can be described as follows
lr
h =
⎧
−1.016 × 10
⎪⎨
⎪⎩
−6 Re3 + 3.150 × 10−4 Re2 + 8.407
× 10−2 Re + 0.3689
�
for Re ≤ 200
9.096 + 3860.168 × exp − log10 (Re)
�
.[13]
�
� 0.431
× sin − 4.226 for Re > 200.
2π log 10 (Re)
1.272
A DESIGN TOOL FOR AERODYNAMIC LENS SYSTEMS 325
FIG. 5. Streamlines through two lenses separated by a spacer.
The redevelopment length l f can be estimated by l f = lr + le
where le is given as (Young et al. 2000)
le
dt
�
0.06Res for 0 ≤ Res < 2300
=
. [14]
for Res ≥ 2300
4.4Re 1
6
s
The approach length la is found in our simulation to be independent
of Reynolds number (Re < 150) with
la/h ≈ 1.625. [15]
Since la is relatively shorter than lr and l f ,weuse the above
equation to estimate la for the whole Reynolds number range.
Flow Limiting Orifice, Relaxation Chamber,
and Accelerating Nozzle
The flow limiting orifice, relaxation chamber, and the accelerating
nozzle are important components of the aerodynamic
FIG. 6. Reattachment length of flow downstream of an axisymmetric
expansion.