PLIF AND IMAGE CHEMILUMINESCENCE
4.7 OH PLIF Results
4.7.2 Correction for Spatial and Temporal Variation in Excitation
There are three principal adjustments applied during the data reduction process to correct for variations in laser sheet intensity. Two of these three require the accurate determination of the location (in relation to the image frame) of the virtual focus of the plano-concave lens (L301) that expands the laser beam in the vertical direction. This information is needed in order to perform virtual ray-tracing during these corrections.
The horizontal distance between the virtual focus and burner centerline has been accurately measured by mechanical means and determined to be 1484 mm. On the other hand, the height offset is slightly more difficult to determine. With great care, mechanical measurements can be made. However, as an alternative, a real ray-tracing method was chosen. In this technique, the attenuated laser sheet is allowed to pass through the sheet forming optics and into the test section.
Just prior to entering the test section, the upper and lower edges of the laser sheet are slightly clipped by an iris (effectively knife edges on top and bottom). The profile of the laser sheet is then measured using the homemade profilometer described earlier. The corresponding end points of the two generated lines then define rays that pass back through the focal center. These end points are located algorithmically with an intensity-center-finding method which produces results down to a fraction of a pixel. Since the ‘x’ coordinate of the focal center is already known, the ‘y’ coordinate is determined by finding the intercept of one or both of these rays at the plane specified by the given
‘x’ coordinate. The accuracy and repeatability of the method is surprisingly good. Clipped laser sheet profile measurements are taken at the beginning of every experimental run in order to verify the sheet focal center location for that group of data.
With the virtual focus location in hand it is possible to proceed with the aforementioned intensity corrections. The first correction is simply for the shot-to-shot energy variation from the laser. This is applied to each collected frame after background subtraction has been performed. It is accomplished by multiplying each pixel in the image by 0
( ) I f E
where f E( ) is the relation between measured intensity versus the laser shot energy and I0 is a fixed reference intensity. The function f E( ) was determined in Figure 4-7 and found to be linear for the given OH PLIF experiments at the prescribed laser wavelength, linewidth and sheet energy fluence. This particular correction is typically applied during the phase averaging process.
After the entire dataset has been phase averaged, the remaining corrections are applied.
The first of these is to correct for the spatial (vertical) variation in the laser sheet intensity. As covered in section 4.4.4, a simple profilometer was constructed to measure the laser sheet vertical intensity distribution. Using the images collected from the profilometer experiments and by tracing virtual rays through the generated lines back toward the virtual focus of L301, a correction frame is created that provides a local laser intensity multiplier at each pixel of interest. Intrinsic to this generation method is also the correction for the (1/r) drop in the sheet intensity as the distance from L301 increases. Once this frame is generated, its reciprocal (element by element) is simply multiplied by each image frame.
Finally, the last correction is applied to adjust for the laser sheet absorption as it passes through the flame. In many cases where the absorbing species are sparse or where the absorption cross section is small, no correction of this type is required. However for the given case with the chosen OH Q-branch line, the laser absorption is not negligible and, in fact, can be a several percent over the width of the flame. To correct for this a Beer’s Law approach is taken. Beer’s Law states that
(4-5) A=
ε
c,where
ε
is the extinction coefficient, is the path length, c is the species concentration, and A is the resulting molar absorptivity. In terms of initial and final intensities, this can be written as(4-6) 10
0
log If I
ε
c
= −
.
This can now be applied to the field of view by discretizing over the image space with a length scale on the order of the pixel size. The equation becomes
(4-7)
1 10
0 0
log 1
2
n n
n i
i
I c c
I
ε
−=
= − +
∑
.Here, the subscripts correspond to the coordinate in the horizontal direction with the origin being at the left side of the frame (the side from which the laser sheet approaches). For the given problem, the unit of measure of length is taken to be one pixel. As such, for a horizontal ray, =1. For non-horizontal rays, = 1+m2 , where m is the slope of the ray. Now, equation (4-7) becomes
(4-8) 10
(
2)
12 10 0
log 1 1
2
n n
n i
i
I m c c
I
ε
−=
= − + +
∑
.Finally, based on the linear fluorescence behavior seen in section 4.7.1, it is assumed that species concentration can be related to local laser intensity and fluorescence by
(4-9) Sn ∝I cn n,
where Sn is the measured fluorescence.
The parameters of interest are now non-dimensionalized in the following way:
(4-10) * * *
0
; ;
n n n
n n n
I S c
I S c
I S c
= = = .
I0 is then chosen to be unity at the left edge of the image frame, and S and c are chosen such that (4-11) Sn* =I cn n* *.
Substituting (4-10) and (4-11) into (4-8) gives
(4-12) 10 * 10 * †
(
2)
12 * 1 *0
log log 1 1
2
n
n n n i
i
S c ε m c − c
=
− = − + +
∑
,where ε† is the modified extinction coefficient, ε† =cε. By inspection, it can be seen that if
ε
=0 then Sn*=c*n which is the degenerate result when no absorption is present. The equation can be rearranged:(4-13) 10 * †
(
2)
12 * 10 * †(
2)
12 1 *0
log 1 log 1
2
n
n n n i
i
c
ε
m c Sε
m − c=
− + = + +
∑
.In this form, the equation can be solved for all n by marching (breadth-first) from the left side of the frame toward the right, and iteratively solving for cn at each pixel location. For the algorithm developed, the ci's are linearly interpolated between pixels along rays cast back through the virtual focus of L301. For a given “input” fluorescence field, Sn*, the corrected image is output as cn*.
The only remaining unknown in equation (4-13) is the modified extinction coefficient, ε†. This is computed by taking advantage of the symmetry of the flame under examination. A line of symmetry exists down the axial centerline of the burner. The PLIF image to the left of the centerline should be the mirror image of the PLIF image to the right of the centerline after full correction. Using a group of steady-state images from the dataset being processed, an iterative approach is used to balance the resulting c s*n' within a prescribed interrogation window (and its mirror image) by adjusting ε†. Once this value is determined, the solution method given above can be used on image frames that have had the first two intensity correction methods already applied. As with the laser sheet intensity profile correction, this Beer’s Law correction is typically applied to the binned images after all phase averaging has been completed. This operation constitutes the final correction process.