Change in the Loading Behavior

We set out to observe experimentally the loading behavior change in the concentration evolution

for the equilateral triangle cross-section. Our experimental results and their comparison with

simulations are new; experiments in triangular pipe as a way to study passive tracers evolution

are not reported in the literature. Nevertheless, we are interested in pushing this study forward to

observe the predicted loading behavior change in the concentration and consequent sign change in

its skewness.

In particular, for a delta initial condition, the skewness evolution in time is reported to change

sign at non-dimensional timeτ

≈1, as shown by the black curve in figure 11.13 from [4, 7]. It should

be noted that the maximum magnitude of the negative skewness is very small (<0.05) compared to

the maximum of the positive skewness (>0.25). Here, the value of the Péclet number isP e= 10

,

which corresponds to a flow rate of0.3712cm/s, about an order of magnitude larger than those we

have been dealing with so far.

Figure 11.13: Skewness evolution in time from [4, 7]. Results from fifty simulations (black), each with

10

random walkers,

P e= 10

, and maximum time-step

10

. Initial conditionδ(x)

and spanwise

uniform. Short time (red) and long time (blue) asymptotics.

In order to observe the sign-change, we need to have a flow rate that allows for the switch to

happen before the dye reaches the end of the pipe. A similar image can be created for our flow rates,

to see how far off we are from the sign change.

Figure 11.14 (left) shows results of Monte Carlo simulations with

10

particles at flow rate

0.0732

cm/s. The blue

+indicate how far an experiment with such flow rate can be considered

reliable; that is, when the rightmost particle of the cloud reaches the right edge of the pipe. While

we are able to see the skewness decreasing in value and heading towards zero, we cannot observe the

sign change. Consequently, we have performed a numerical study to identify a flow rate value that

would allow us to see the sign change without losing any information. Figure 11.14 (right) shows the

skewness evolution in time for five flow rates. In all of these cases, the visible part of the skewness

curve (highlighted by the +’s) reaches below zero. The lower the flow rate, the further the+’s go.

10-3 10-2 10-1 100 101 log( ) -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

Cross-Sectionally Averaged Skewness

Cross-Sectionally Averaged Skewness Evolution

experiment simulation 10-3 10-2 10-1 100 101 log( ) -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 Averaged Skewness

Monte Carlo simulation of Averaged Skewness Evolution 10s-20s 10 exp 10 full 15 exp 15 full 20 exp 20 full 25 exp 25 full 30 exp 30 full

Figure 11.14: Skewness evolution in time for pipe of equilateral triangle cross-section. Left: Skewness

evolution in time for flow rate

0.0732cm/s. Simulation performed with10

random walkers and

maximum time-step10

. The red curve is the simulation result, while the blue+’s indicate how

much of the simulation would be visible in experiments with the same flow rate. Right: Skewness

evolution in time for five flow rates; top to bottom:

0.0113

cm/s,

0.0169

cm/s,

0.0225

cm/s,

0.0281cm/s, and

0.0338cm/s. The

+’s indicate how much of each simulation would be visible in

experiments with the same flow rate.

We have been working on experimental runs with these much lower flow rates to be compared

to the corresponding numerical simulations. We choose to focus on the

0.0169cm/s

flow rate and

perform longer experiments to observe the sign-change in the skewness. In particular, this much

lower flow rate requires experiments lasting

90min

(with pictures taken every1s) in order for the

dye to run along the entire length of the pipe.

Figure 11.15 shows results of such experiments compared to Monte Carlo simulations. The red

curve is obtained by merging together the two concentration curves produced by the injection and

reservoir cameras. Note that while att= 850sthe skewness value is positive, for all three of the

later times the skewness value is negative. The spikes that appear at later times are due to the

increasing amount of noise present in the data; as time passes and the solute diffuses more and

more throughout the pipe, its concentration diminishes and the amount of recorded noise increases.

Nevertheless, albeit less pretty, the results are reliable.

-10 0 10 20 30 40 50 60 70 80 90 0 0.1 0.2 t = 0 s, =0 MC Exp -10 0 10 20 30 40 50 60 70 80 90 0 0.05 0.1 t = 850 s, =0.8217, Skew = 0.033764 -10 0 10 20 30 40 50 60 70 80 90 0 0.05 t = 1550 s, =1.4984, Skew = -0.012161 -10 0 10 20 30 40 50 60 70 80 90 x/a 0 0.05 t = 2850 s, =2.7551, Skew = -0.02529 -10 0 10 20 30 40 50 60 70 80 90 x/a 0 0.05 t = 3450 s, =3.3351, Skew = -0.026212

Figure 11.15: Comparison of the concentration curve produced by experimental data (red) and

Monte Carlo simulations (blue) fora= 0.0768cm

and average flow rate0.0169cm/s. Sequence of

images at five increasing times:

t= 0,t= 850s,t= 1550s,t= 2850s, andt= 3450s. Physical

timet, corresponding non-dimensional time

τ, and full skewness value in the simulation are listed.

APPENDIX A

Peel-Off in the Parallel Plates Geometry

We report here the step-by-step calculation performed to obtain the pointwise second moment

solution in the infinite parallel plates geometry. The Peel-Off method is here shown from beginning

to end. The equations for all the intermediate steps are solved and all the residues calculated. Finally,

there are a few considerations about the complete

C

solution.