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Research Papers: Fundamental Issues and Canonical Flows

On the Streamwise Development of Density Jumps

[+] Author and Article Information
A. Regev

Department of Civil and Environmental Engineering, Technion—Israel Institute of Technology, Haifa 32000, Israel

S. Hassid

Department of Civil and Environmental Engineering, Technion—Israel Institute of Technology, Haifa 32000, Israelcvrhasd@tx.technion.ac.il

J. Fluids Eng 132(2), 021202 (Feb 17, 2010) (9 pages) doi:10.1115/1.4000794 History: Received March 29, 2009; Revised November 25, 2009; Published February 17, 2010; Online February 17, 2010

The analysis of density jumps in two-layer channel flows of miscible fluids controlled by a downstream obstruction, in which one of the layers is infinitely deep and at rest, is extended to consider the dependence of its features on its streamwise dimension. The momentum conservation equation in the entrainment and roller regions, and the energy conservation equation after the jump are corrected to account for friction. The streamwise coordinate is related to the increase in the density layer height through a linear expression derived from CFD calculations. Three regimes are distinguished: (1) for short distances from the origin to the obstruction, only an entrainment region exists; (2) for medium distances, two regions can be distinguished, i.e., the entrainment region, and the roller region, in which no entrainment is assumed; and (3) for long distances, three regions can be distinguished—the entrainment, the roller, and the postjump regions, characterized by approximate energy conservation. It is shown that initially the dimensionless total entrainment ratio increases as the distance to the obstruction increases, until a roller region appears. A further increase in distance to the obstruction does not have a significant effect on the total entrainment, until the appearance of a postjump region, resulting in a gradual decrease in the total entrainment. These results are supported by numerical calculations using the FLUENT CFD software package, which are in good agreement with experimental results.

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Copyright © 2010 by American Society of Mechanical Engineers
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Figures

Grahic Jump Location
Figure 7

Dimensionless entrainment in the jump versus normalized streamwise coordinate for Fr1=10, Z/h1=3.33, and different values of the distance of the obstruction from the origin

Grahic Jump Location
Figure 8

The roller length divided by the downstream flow depth as a function of the Froude number at the beginning of the roller region. Comparison between data from Refs. 14,13, and Eq. 15 suggested for the density jump.

Grahic Jump Location
Figure 12

Comparison of the final entrainment as a function of the upstream Froude number from numerical simulations using FLUENT and the experiments, where Z/h1=2

Grahic Jump Location
Figure 1

Density jump in a two-layer flow controlled by a weir

Grahic Jump Location
Figure 2

Dependence of the dimensionless entrainment on the dimensionless distance of the obstruction from the origin—for Z/h1=3.33 (calculated using FLUENT )

Grahic Jump Location
Figure 3

Development of the density jump for Fr1=9 and Z/h1=3.33 for different values of L/h1 (25, 33.3, 50, 66.7, 83.33, 100, and 116.7)

Grahic Jump Location
Figure 4

Development of entrainment ε(x) in a density jump for Fr1=9 and Z/h1=3.33 for different values of L/h1 as calculated by FLUENT . (Detrainment for L/h1 higher than 50 due to including the closed roller region in the calculated entrainment.) Cases are shown in Fig. 3. Maximum entrainment calculated by 1D theory is 1.175.

Grahic Jump Location
Figure 5

Normalized height H″1 (Eq. 4) and dilution sf″ (Eq. 6) versus normalized streamwise coordinate X″

Grahic Jump Location
Figure 6

Dimensionless density layer height versus dimensionless streamwise coordinate in the entrainment region for Fr1=10, Z/h1=3.33, and different values of the distance of the obstruction from the origin

Grahic Jump Location
Figure 9

(a) Final entrainment and (b) nondimensional density layer thickness at the end of entrainment region calculated from simplified model in an obstruction-controlled density jump as a function of the distance of the obstruction from the origin for Fr1=5. If friction is neglected, the value of εf is 0.99, 0.84, 0.69, 0.510, and 0.29 for Z/h1=0, 0.5, 1, 1.5, and 2, respectively.

Grahic Jump Location
Figure 10

(a) Entrainment and (b) dimensionless maximum height calculated from simplified model in a weir-controlled density jump as a function of the distance of the obstruction from the origin, for different Froude numbers and Z/h1=3.3. If friction is neglected, εf is equal to for 0.30, 0.78, 1.46, 1.86, and 2.18 for Fr1=6.5, 8, 10, 11.5, and 13, respectively.

Grahic Jump Location
Figure 11

Schematic diagram of the flow apparatus for visualization experiment

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