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Rollover instability due to double diffusion in a stably stratified cylindrical tank
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10.1063/1.2827488
/content/aip/journal/pof2/19/12/10.1063/1.2827488
http://aip.metastore.ingenta.com/content/aip/journal/pof2/19/12/10.1063/1.2827488

Figures

Image of FIG. 1.
FIG. 1.

Typical configuration of a cylindrical storage vessel for liquid natural gas. There are four practically constant heat fluxes at the bottom , sidewalls (, ), and top of the container . Evaporation across the vapor-liquid interface is described by an evaporation rate and an associated heat flux which includes the sensible heat flux across the interface and the latent heat of vaporization lost from the liquid. “HT” is shorthand for sensible heat transfer and represents enthalpy change by evaporation. Treating is the key challenge for a hydrodynamics-only model.

Image of FIG. 2.
FIG. 2.

Axisymmetric coordinate system with a standard mesh used for many of the simulations; 12 352 triangular elements with 106 077 degrees of freedom in the Galerkin finite element expansions for radial and axial velocities, pressure, temperature, and concentration fields. The initial concentration profile is a steep change from unit concentration in the lower layer to zero concentration in the upper layer. Kinematic arguments suggests two pseudosteady buoyant convention cells shaped like tori should evolve (see Ramsay, Ref. 7).

Image of FIG. 3.
FIG. 3.

Constant flux BC model with , , , and , showing 20 contours of temperature, gray scale concentration, and arrows for velocity vectors for times .

Image of FIG. 4.
FIG. 4.

Fixed temperature BC model with , , , and , showing 20 contours of temperature, gray scale concentration, and arrows for velocity vectors for times .

Image of FIG. 5.
FIG. 5.

Biot boundary condition simulation with , , , showing the phase averaged temperature difference, , for times .

Image of FIG. 6.
FIG. 6.

Same simulations as Fig. 5 showing the time history of the maximum point temperature difference, .

Image of FIG. 7.
FIG. 7.

Fixed temperature BC simulation showing the induced average Reynolds number , which is the domain averaged velocity magnitude.

Image of FIG. 8.
FIG. 8.

Same simulation as Figs. 5 and 6 demonstrating the Nusselt number Nu calculated as the dimensionless average heat flux, averaging Eq. (13).

Image of FIG. 9.
FIG. 9.

Time to rollover for which the volume averaged velocity magnitude is maximum: for the constant flux BC simulation with varying and fixed , , .

Image of FIG. 10.
FIG. 10.

Time to rollover for which the volume averaged velocity magnitude is maximum: for the fixed temperature BC simulation with varying and fixed , , .

Image of FIG. 11.
FIG. 11.

Constant flux BC simulation for the same conditions as Fig. 9 with three measures of velocity.

Image of FIG. 12.
FIG. 12.

Fixed temperature BC simulation for the same conditions as Fig. 10 with three measures of velocity.

Image of FIG. 13.
FIG. 13.

The decay rate vs time for the most dangerous mode for a range of . The crossing of the -axis corresponds to the most dangerous mode becoming unstable. , , for all simulations for the Dirichlet boundary condition.

Image of FIG. 14.
FIG. 14.

The most dangerous mode computed for Dirichlet boundary condition with , , , just before the neutral stability time . Apparently, the velocity field switches sense with the lower right-hand corner eddy switching from counterclockwise to clockwise rotation. However, care should be taken in recognizing that the eigenvector of the unknown is determined only up to a multiplicative constant in the linear theory. Thus, the most dangerous eigenmodes, before and after the critical time, are indistinguishable. Both temperature and concentration profiles consistently flip signs as well.

Image of FIG. 15.
FIG. 15.

Constant flux BC simulation reporting the energy stability functional for , , , .

Image of FIG. 16.
FIG. 16.

Shannon entropy evolution for , , , and for , , , with the constant flux BC model.

Tables

Generic image for table
Table I.

Time averaged temperature differences with each boundary condition: phase averaged and fixed geometry temperature difference . The averaging is taken over for and 15 000 and for and 5000.

Generic image for table
Table II.

Summary statistics for velocity (induced Reynolds number) for the constant flux boundary condition with , , and . is the time at which the simulation was ended. is the average velocity in the tank at . is the time at which the average tank velocity was achieved, which is taken as the quantity distinguishing rollover. is the maximum velocity instantaneously achieved in the tank during the entire simulation at any point.

Generic image for table
Table III.

Summary statistics for velocity (induced Reynolds number) for the fixed temperature boundary condition with , , and . is the time at which the simulation was ended. is the average velocity in the tank at . is the time at which the average tank velocity was achieved, which is taken as the quantity distinguishing rollover. is the maximum velocity instantaneously achieved in the tank during the entire simulation at any point.

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/content/aip/journal/pof2/19/12/10.1063/1.2827488
2007-12-28
2014-04-17
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Rollover instability due to double diffusion in a stably stratified cylindrical tank
http://aip.metastore.ingenta.com/content/aip/journal/pof2/19/12/10.1063/1.2827488
10.1063/1.2827488
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