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Research Papers: Flows in Complex Systems

Statistics for Assessing Mixing in a Finite Element Hydrocode

[+] Author and Article Information
M. A. Brown

AWE,
Aldermaston, Reading,
Berkshire, RG7 4PR, UK
e-mail: matthew.a.brown@awe.co.uk

C. A. Batha

AWE,
Aldermaston, Reading,
Berkshire, RG7 4PR, UK
e-mail: chris.batha@awe.co.uk

R. J. R. Williams

AWE,
Aldermaston, Reading,
Berkshire, RG7 4PR, UK
e-mail: robin.williams@awe.co.uk

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received March 22, 2013; final manuscript received May 19, 2014; published online July 9, 2014. Assoc. Editor: Oleg Schilling.

J. Fluids Eng 136(9), 091103 (Jul 09, 2014) (8 pages) Paper No: FE-13-1186; doi: 10.1115/1.4027775 History: Received March 22, 2013; Revised May 19, 2014

Direct numerical simulation of mixing processes (Rayleigh-Taylor and Richtmyer-Meshkov instabilities) is computationally expensive due to the need to resolve turbulent structures on small scales. Hence, it is common practice in both academia and industry to use phenomenological models that explicitly model the mixing processes within a host hydrodynamic code. For such schemes to be self-consistent, the mixing should be dominated by the mass introduced by the dedicated mixing model, with minimal contribution from the numerical methods of the host code. In this report, several diagnostic statistics are described that allow for the assessment of the production of mix and a determination of the quality of a mixing model. These diagnostics are implemented within an existing two-dimensional finite element hydrocode, containing an implementation of Youngs' turbulent mix model, and used to assess the mixing scheme against a number of two-fluid test problems.

Copyright © British Crown Owned Copyright 2014/AWE
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References

Figures

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Fig. 1

Initialized fluid volume fractions at t=20 ms (the turn-on time for the dynamic mix model) against tank depth (tank depth measured from the heavy fluid side). Also plotted are the initial distributions of the mix-mass, mix integral, and b-parameter. The mix-mass and mix integral have been scaled by the cell volumes.

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Fig. 2

The mix-mass in the 3:1-density two-fluid mixing problem, showing the components of the total mix-mass attributable to the dynamic mix model and the host hydrocode

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Fig. 3

For the 3:1-density two-fluid mixing problem, the width of the mixing region, defined by w = 4∫m1m2dx, where the x-axis is parallel to the density gradient. The mix region gains finite width at t = 20, when the volume fraction profiles are first initialized. The mix-width grows as t2.

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Fig. 4

The mix integral and integrated b-parameter as a function of time. Displayed are the total statistic, the contribution due to the dynamic mix model, and the contribution due to the host code. The contribution from the passive mix model is not shown, but comprises the remainder of all contributions to the total statistic. See Table 1 for explicit values of each statistic at t = 70 ms.

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Fig. 6

The contribution towards the mix-mass due to the mesh remapping algorithm of the host code, in the tilted-rig experiment at different mesh resolutions. This contribution has the largest variation with mesh size, but is a small proportion of the overall mix-mass.

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Fig. 5

The total mix-mass for the tilted-rig experiment, for different mesh resolutions

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Fig. 15

Comparison of the mix integral for the air/H2 shock tube, between moving and fixed (remapped) meshes. (a) the total mix integral. (b) the dynamic mix model and host code contributions. See Fig. 16 for further break-down of the host code contributions.

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Fig. 14

Comparison of the mix-mass for the air/H2 shock tube, between moving and fixed (remapped) meshes. (a) the total mix-mass. (b) the dynamic mix model and host code contributions. (The host code contribution is zero for the moving mesh case.)

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Fig. 13

The integrated b-parameter for the air/H2 shock tube as a function of time, for the case of purely Lagrangian mesh-movement (no mesh remapping). The b-parameter is decomposed into contributions from the passive and dynamic mix models, and the hydrodynamic host code.

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Fig. 12

The total mix integral for the air/H2 shock tube as a function of time, for the case of purely Lagrangian mesh-movement (no mesh remapping). The mix integral is also decomposed into its contributions from the passive and dynamic mix models, and the hydrodynamic host code.

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Fig. 11

The total mix-mass for the air/H2 shock tube as a function of time, for the case of purely Lagrangian mesh-movement (no mesh remapping). The mix-mass is also decomposed into its contributions from the passive and dynamic mix models; there is no contribution from the host code.

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Fig. 10

Distance–time plot for the linear shock-tube simulation; plotted are detected shock positions (gray points) and the 1% and 99% mass-fraction contours of H2. The air/H2 interface is initially at coordinate 0 cm, the shock-tube end wall is at coordinate +62 cm. Apparent are the initial shock wave moving towards the air/H2 interface from the left, and subsequent reshocks on the right.

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Fig. 9

Schematic of the shock tube problem setup. Initially shocked air is to the left; unshocked air and hydrogen is to the right. The initial position of the shock front (moving left to right) is indicated.

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Fig. 8

Contributions to the total mix integral and integrated b-parameter, in the tilted-rig experiment, comparing fixed and moving meshes (remapped to fixed coordinates, and Winslow mesh remapping, respectively) after t = 22 ms. The mesh resolution is 2.5×2.5 mm. Symbols are to distinguish the lines.

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Fig. 7

Comparison between the mix-mass for the tilted rig simulation, with fixed and moving meshes (remapped to fixed coordinates, and Winslow mesh remapping, respectively) after t = 22 ms. The mesh resolution is 2.5×2.5 mm. Symbols are to distinguish the lines.

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Fig. 16

Comparison, between fixed and moving mesh cases, of the contributions to the host code component of the mix integral. Indicated are contributions due to the initial Lagrangian phase, and the mesh-remapping phase.

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