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Research Papers: Fundamental Issues and Canonical Flows

Optical Density Measurements and Analysis for Single-Mode Initial-Condition Buoyancy-Driven Mixing

[+] Author and Article Information
Y. Doron, A. Duggleby

Exosent Engineering, College Station, TX 77845Mechanical Engineering,  TX A&M University, College Station, TX 77843

J. Fluids Eng 133(10), 101204 (Sep 27, 2011) (11 pages) doi:10.1115/1.4004943 History: Received December 16, 2010; Accepted August 17, 2011; Published September 27, 2011; Online September 27, 2011

The Texas A&M water channel experiment is modified to examine the effect of single-mode initial conditions on the development of buoyancy-driven mixing (Rayleigh-Taylor) with small density differences (low-Atwood number). Two separated stratified streams of ~5°C difference are convected and unified at the end of a splitter plate outfitted with a servo-controlled flapper. The top (cold) stream is dyed with Nigrosine and density is measured optically through the Beer-Lambert law. Quantification of the subtle differences between different initial conditions required the optical measurement uncertainties to be significantly reduced. Modifications include a near-uniform backlighting provided through quality, repeatable, professional studio flashes impinging on a white-diffusive surface. Also, a black, absorptive shroud isolates the experiment and the optical path from reflections. Furthermore, only the red channel is used in the Nikon D90 CCD camera where Nigrosine optical scatterring is lower. This new optical setup results in less than 1% uncertainty in density measurements, and 2.5% uncertainty in convective velocity. With the Atwood uncertainty reduced to 4% using a densitometer, the overall mixing height and time uncertainty was reduced to 5% and 3.5%, respectively. Initial single-mode wavelengths of 2, 3, 4, 6, and 8 cm were examined as well as the baseline case where no perturbations were imposed. All non-baseline cases commence with a constant velocity that then slows, eventually approaching the baseline case. Larger wavelengths grow faster, as well as homogenize the flow at a faster rate. The mixing width growth rates were shown to be dependent on initial conditions, slightly outside of experimental uncertainty.

Copyright © 2011 by American Society of Mechanical Engineers
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Figures

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Figure 14

Mixing height H=h(Atg/ν2)1/3 versus time τ<>=t((Atg)2/ν)1/3 for the baseline, 2, 3, 4, 6, and 8 cm perturbations. The no perturbation case has a slight quadratic dependency. Each of the initial conditions starts on a terminal velocity (straight line) and eventually deviate towards the no perturbation case. The water channel facility is not large enough to determine when or if the 6 and 8 cm perturbations would follow the same trend. All cases show the same growth rate within uncertainty, ɛ=dH/H which starts near 15% but tapers down to 5% near τ=15 as more pixels span the mixing width.

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Figure 15

Mixing height with 〈f¯1〉 at 90%/10% (top - H0.9) and 80%/20% (bottom - H0.8) for the baseline and 2, 3, 4, 6, and 8 cm perturbations. The separation between the cases is stronger, showing that the larger wavelength initial conditions homogenize the flow within the mixing layer faster.

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Figure 8

Histogram of image shows that the proper amount of light enters the camera, making use of almost all of the 14-bit CCD stored in 16-bit format (0 to 65 535)

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Figure 9

Close-up view of the first spike shown in Fig. 1 illustrates the high quality images using the Nikon D90 camera at 4288 × 2848 resolution (0.25 mm per pixel), 1/320 s exposure time, 105 mm focal length with F/16 aperture (37.1 cm depth of field), and 1/1200 s studio flash duration

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Figure 10

Parallax effects: light travels along distance L, not channel width w, and is accounted for by proportionally scaling the measured intensity. The x and y coordinate of the light is taken as the average between the entrance and exit location, with the difference of the two included as position error in the uncertainty analysis.

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Figure 11

Sample images are shown from top to bottom for the baseline, 2, 3, 4, 6, and 8 cm wavelength cases. The bubble and spike structures are clearly observed for the 8 cm case.

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Figure 12

Average density 〈f¯1〉 for the baseline, 2, 3, 4, 6, and 8 cm cases (top to bottom, left to right) as a function of y, height above the flapper, and x, downstream distance (cm)

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Figure 13

Average density uncertainty d〈f¯1〉 for the baseline, 2, 3, 4, 6, and 8 cm cases (top to bottom, left to right) cases as a function of y, height above the flapper, and x, downstream distance (cm). After the optical modifications, the relative error is now at most 1%.

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Figure 1

Rayleigh-Taylor instability: a heavy fluid accelerated into a lighter fluid (left). As the heavy fluid falls and the light fluid rises, secondary shear instabilities (Kelvin-Helmholtz) develop. The shear instability gives rise to a vorticity which drives further mixing. The vorticity exists everywhere on the interface except where the pressure gradient and density gradients are aligned. As the two fluids mix, the familiar “bubble” and “spike” structures appear and ultimately result in turbulent mixing, as shown in an instantaneous photograph of dyed cold water over clear hot water.

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Figure 2

Figure showing the water channel configuration (top left), servo motor assembly and attachment to the channel (top right), dimensions of the flapper in inches (bottom left), and close-up of the flapper and hinge (bottom right)

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Figure 3

Streakline is shown in the top section of the channel for an At=0 flow. The long coherent streakline is evidence that the flow in the channel has no free-stream turbulence. Any free-stream turbulence would quickly mix and dissipate the streak.

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Figure 4

Figure showing velocity measurements utilizing two consecutive photographs of dye blob in the flow. Notice the stop watch seconds and hundreds of a second, the interval time between both photographs is precisely one second ± 0.005 seconds.

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Figure 5

Top: If the amplitude of the oscillation is too large a non-negligible vorticity is generated at the edge of the flapper. Bottom: If the boundary layer is too large on the splitter plate, a shearing and leaning of the bubble and spikes is seen.

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Figure 6

Absorption of light as a function of width (nondimensional w/L) through a triangular wedge filled with the Nigrosine solute. Since the absorption increases linearly with the width the molar absorbtivity, ɛ is constant.

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Figure 7

Top view illustration of camera placement and flashes. The light emanating from the flashes impinges on the diffusion sheet behind the channel, travels through the experiment, and is recorded in the digital camera.

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