Robust Volume-Based Approach for the Turbulent Mixing Efficiency

[+] Author and Article Information
Roberto C. Aguirre, Jennifer C. Nathman, Philip J. Garcia

Iracletos Flow Dynamics and Turbulence Laboratories, Mechanical and Aerospace Engineering, University of California, Irvine, CA 92697

Haris J. Catrakis1

Iracletos Flow Dynamics and Turbulence Laboratories, Mechanical and Aerospace Engineering, University of California, Irvine, CA 92697catrakis@uci.edu


Corresponding author.

J. Fluids Eng 128(4), 864-873 (Nov 12, 2005) (10 pages) doi:10.1115/1.2201628 History: Received September 15, 2004; Revised November 12, 2005

This paper considers the mixture fraction which is often used to quantify the turbulent mixing efficiency in fluid engineering devices. We contrast a volume-based approach, where the mixture fraction is quantified directly using the volume bounded by the interface between mixed versus pure fluid, to a surface-based approach that requires area integrals of all mixed-fluid interfaces. Experimentally, we investigate the resolution-scale robustness of the volume-based approach compared to the small-scale sensitivity of the surface-based approach. The difference in robustness between these approaches has implications for examining, modeling, and optimizing the turbulent mixing efficiency.

Copyright © 2006 by American Society of Mechanical Engineers
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Figure 1

This experimental visualization shows an example of the sharpness of outer interface between mixed fluid and pure fluid in a gas-phase turbulent separated shear layer at a Reynolds number of Re∼106, freestream Mach number of M∼0.9, and Schmidt number of Sc∼1, recorded at a test-section pressure of ∼3atm. The image was recorded in the UC Irvine variable-pressure aero-optics facility using visible laser-induced fluorescence of acetone vapor seeded in air, excited by ultraviolet Nd:yttrium–aluminum–garnet laser illumination. The freestream is on the upper side of the image and is flowing from left to right. Ambient gas initially at rest is present below the shear layer. The sharpness of the outer interface is due, in part, to the local strain from eddies near the interface.

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Figure 2

High-resolution quantitative visualization of a turbulent outer interface derived from space-time ∼10003 measurements of the concentration field in a fully developed liquid-phase jet. The Reynolds number is Re∼20,000 and the Schmidt number is Sc∼2000 so that the flow conditions are above the mixing transition. The measurements were recorded in the UC Irvine large-scale water tank facility using visible laser-induced fluorescence of dilute disodium fluorescein seeded in water, excited by argon-ion laser illumination. This space-time visualization captures the dynamics of the outer interface with the time axis in the vertical direction. The full transverse spatial extent of the interfaces is captured including the small scales. The interface shown corresponds to a concentration threshold of c∕c1=1, cf. Figs.  1112.

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Figure 3

Schematic illustrating the local physical thickness hdc of a mixed-fluid interface associated with the differential range of concentration values {c,c+dc}, with the two surfaces shown corresponding to the c and c+dc isosurfaces. The region in between the two isosurfaces corresponds to the interfacial fluid.

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Figure 4

In this schematic, a highly wrinkled surface is shown to illustrate that small-scale features of turbulent mixed-fluid interfaces can be expected to introduce pronounced sensitivity of the surface area to those features. In contrast, the volume enclosed by such interfaces can be expected to be weakly sensitive to the small-scale features. This robustness of the volume to small-scale features, as opposed to the sensitivity of the surface area to those features, provides a useful ingredient to facilitate practical studies of the mixture fraction.

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Figure 5

The smooth surface shown in this schematic has been chosen to have a similar large-scale shape to the highly wrinkled surface in Fig. 4. Whereas the area of the smooth surface can be expected to be substantially lower than the area of the highly wrinkled surface, the volumes enclosed can be expected to be nearly the same since they are dominated by the large-scale features. In the context of mixing and mixed-fluid interfaces, this distinction can be utilized and applied to the average as well as dynamic behavior of the mixture fraction.

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Figure 6

Schematic of the large-scale water tank facility at UC Irvine, in which various flow configurations can be examined. The flow geometry examined in the present work is the round turbulent jet, as indicated in the schematic. A liquid-phase jet is issued vertically from the top and grows downwards. Imaging of the concentration field of the jet is conducted in the similarity plane normal to the jet axis, at a downstream location of ∼500 nozzle diameters. This is in the far field of the jet and corresponds to a physical location approximately halfway down the tank in order to avoid end-wall and sidewall effects.

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Figure 7

Photograph of the water tank facility at UC Irvine. The tank is octagonally shaped with an internal height of 2.74m(6ft) and a diameter of 1.83m(9ft). The capacity of the tank is 8tons or 2000gallons of water. The tank has extensive optical access by way of eight vertical 0.61m by 2.13m (2ft by 7ft) windows and one horizontal 0.61m(2ft) diameter window at the base. The large size of the tank facilitates large-scale high-resolution imaging of the concentration field while at the same time enabling flow conditions above the mixing transition.

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Figure 8

Coarse-grained quantitative visualization of an outer interface derived from three-dimensional space-time measurements of the concentration field in a liquid-phase fully developed turbulent jet. This image corresponds to coarse graining of the full-resolution ∼10003 data to a reduced resolution of ∼323. Time varies horizontally and increases from left to right. The flow conditions are above the mixing transition with a Reynolds number of Re∼20,000 and a Schmidt number of Sc∼2000.

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Figure 9

Coarse-grained visualization of a turbulent outer interface at a resolution of ∼2563, derived from the same full-resolution ∼10003 data as for Fig. 8. Additional smaller-scale features are evident, as expected, compared to the ∼323 coarse-grained visualization in Fig. 8.

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Figure 10

Full-resolution ∼10003 quantitative visualization of a turbulent outer interface corresponding to the coarse-grained ∼2563 and ∼323 visualizations in Figs.  89. The full transverse spatial extent of the interface is captured including the small scales.

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Figure 11

Comparison between the probability density function of the concentration field (solid curve with diamonds) and the surface area of the mixed-fluid interfaces (dashed curve with crosses). Both curves correspond to the high-resolution scalar measurements of the turbulent jet. Both quantities plotted are normalized by their respective peak values, excluding the peak value of the probability density of pure ambient fluid. The deviation between the two curves, at large scalar thresholds, is attributable to insufficient resolution of the internal interfaces corresponding to those thresholds.

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Figure 12

Dependence of the interfacial thickness as a function of concentration threshold for the high-resolution measurements of the turbulent jet. The high resolution (∼10003) is adequate to capture the uniformity of the interfacial thickness for the lower scalar thresholds or internal interfaces near the outer regions of the jet. However, the ∼10003 resolution is inadequate at the higher scalar thresholds with deviations as large as 30%, illustrating the sensitivity of the interfacial thickness to the resolution.

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Figure 13

Resolution-scale effects on the mixing efficiency evaluated directly in terms of the volume fraction of mixed fluid. Robustness is evident for several reductions in resolution scale. For example, for a resolution reduction of 1:10−1.5∼30:1 per dimension the mixing efficiency is captured by ∼10−0.014∼97%. In other words, there is only a ∼3% difference in the mixing efficiency even though there is a substantial ∼303:1=27,000:1 reduction in three-dimensional concentration field information.

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Figure 14

Evaluation of resolution-scale effects on the probability of finding pure fluid as a function of distance from the jet centerline. The solid curve corresponds to the ∼10003 resolution. The dashed curve corresponds to a resolution reduced by a factor of 10 per dimension, i.e., at a coarse-grained resolution of ∼1003. The robustness of the probability of finding pure fluid to resolution is evident by the very weak effect of the coarse graining. As the area above each curve corresponds to the mixing efficiency, this robustness directly applies to the mixing efficiency as well and is evaluated further in Fig. 1.

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Figure 15

Dynamical behavior of the mixing efficiency and demonstration of the robustness to resolution-scale effects. Solid curve: mixing efficiency for full-resolution (∼10003) data. Dashed curve: mixing efficiency for data coarse grained at ten times lower resolution per dimension, i.e., at a reduced resolution of ∼1003. The robustness to coarse graining is evident dynamically. Even though large excursions of the instantaneous value of the mixing efficiency are observed, the robustness to resolution-scale effects persists dynamically.




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