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Lift Forces in Bubbly Flows


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Figure 1 Illustration of Shear Flow around a Sphere

The dynamics of two-phase (gas/liquid) bubbly flows are complex: bubbles deform and disperse; large latent heats and heat capacity differentials influence local boiling; and relatively small changes in heated surface temperatures yield order of magnitude changes in boiling complexity.  Because the local void volume has a direct feedback effect on reactor neutron flux and fuel rod power production, prediction of local boiling rates and bulk boiling effects in nuclear reactors is key in achieving a higher fidelity prediction of the fuel system temperature distribution, local and  global power production, local fuel rod burnup, and CRUD deposition.

When the fuel rod surface temperature is high enough, steam bubbles nucleate and grow on the surface, detaching when the lift forces acting on the bubble are large     enough to overcome the suction force of the bubble on the rod.  Shear lift forces are reaction forces that result as the flowing water passes over the surface of the bubble. They evolve as the shape and size of the bubble evolve and are also influenced by asymmetries in the upstream flow field.

Figure 1 provides a general illustration of a spherical bubble in flow. The general equation describing the forces acting on the bubble can be written as [1]:

Where FI is the inertial force, FA is the added mass term, FL is the lift force, FD is the drag force, FB is the buoyancy force.  When these terms are expanded, the resolved forces that are the target of the work are more easily identified:

Where Ub is the bubble velocity, v is the fluid velocity, CM is the added mass coefficient, CL is the lift coefficient, CD is the drag coefficient, γ is the ratio of the densities (ρb/ρ), g denotes gravity and ω is vorticity.

In general, the coefficients are not constant, although their functional dependence has not been explicitly stated.  A few effects have been left out, including those due to time history, density gradients, and temperature gradients.  One of the lesser understood momentum closures is that due to the transverse forces.  These transverse forces are very critical for LWR applications as transverse motion of bubbles in the narrow and long flow channels between the fuel rods can have a huge impact on the heat transfer and neutron moderation.

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Proposed CASL Closure Form for Bubbly Flow Lift Coefficients

Many widely used two-fluid Comp utational Multiphase Fluid Dynamics (CMFD) codes, such as Ansys' CFX & Fluent, CD-adapco’s Star-CCM+,  and RPI’s NPHAS E-CMFD, implement Auton's shear lift model [5] and Antal's wall force model [7] as the two primary lift forces.  All allow the user to tweak the model lift coefficients (CL), although by default the coefficients are constants; some codes include alternative correlations for the lift coefficients. While the base form of the Auton lift force with a constant lift coefficient is an acceptable approximation at high Reynolds number (Re) and low Strouhal number (Sr) flows about roughly spherical bubbles, for different bubble shapes and orientations, the Auton closure should be modified. Additionally, Auton's model derivation assumes a Sr number of 1, and this assumption isn’t always acceptable near the wall; when Sr is not small Auton's model overpredicts the shear lift force significantly. Substitute correlations presented in the past do include some bubble deformation and alternate orientations, but they don’t appear to be sufficient for the range of PWR flow conditions.  

Thus, for CASL, the immediate need from a CMFD perspective is to include the effect of the wall on shear lift forces. To address this need, ORNL researchers Thomas Daly and Sreekanth Pannala and UT-K professor Art Ruggles collaborated in developing a new closure for lift force that includes the wall effects.  The team tested the closure using a comparison with fully developed turbulent pipe flow, and the new closure displays the correct asymptotic behavior. 

Proposed CASL Closure Form

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Table 1 Dimensionless Parameters

In order to accumulate a large body of data for use in constructing the new closure, the team first did a survey of existing work and extracted the information necessary to calculate the primary dimensionless parameters (see Table 1 for a listing; note that the characteristic length is the bubble diameter, d).  The reference data and closures were plotted against the dimensionless parameters in Figure 6.  As can be seen upon inspection of the plots, the existing closures do not capture the effect of all the different parameters and are limited in range. For example , Rastello's model [3, 4] is good at moderate to large Re and away from the wall, but is not as accurate at very low Re.

Using the referenced closures as the starting point, the research team worked to develop a new closure that could capture the empirically representative behavior of the lift coefficient versus Sr and Re. In order to better capture the influence of the wall, the proposed new closure includes an additional parameter, E, that renders the distance from the wall non-dimensional. The form was selected such that E=1 as the bubble touches the wall and approaches 0 as the bubble moves away from the wall.  Eight different variations of the closure were investigated, with the best match (based on a least-squares approach), called “DRP,” proposed for CASL adoption and future augmentation.

The new DRP lift closure was plotted with various independent parameters in Figures 6 and 7, and is shown with the reference data set.

The terms in the DRP lift closure that do not include the distance to the wall are based on Legendre & Magnaudet's model [2].  The new closure appears to capture both low and high Re behavior well. This was achieved by modifying Legendre's high Re model to allow for a local maximum near Re = 50. Like the other models it approaches the theoretical result of CL = 1/2 as Re approaches infinity. Unlike the other models discussed in this report, the proposed model also includes a dependence on distance from the wall. This allows it to remain physically reasonable as a bubble approaches the wall.

Next steps

Figure 6 DRP Lift Coefficient for Re > 10 (top) and Re < 10 (bottom) plotted with reference closures

The correlation developed represents the initial formulation of the closure.  Additional work is needed to provide a stronger theoretical footing and to include additional effects encountered in a PWR. The critical effects in increasing order of importance and difficulty that need to be accounted for in two-phase PWR conditions are:

  • Blockage (reduced area on the side of the wall for flow)
  • No-slip condition and the resulting velocity/vorticity profile
  • Turbulence (especially for small bubbles when fluctuations can influence bubble transport)  
  • Deformation (for large bubbles)
  • Swarming (high bubble density, polydispersity, break-up and coalescence
  • Boiling effects.

For more information, see CASL-I-2013-0143-000.

References

Several works are referenced in the full CASL report; those works referred to in this article include:
[1] J. Magnaudet and I. Eames. “The motion of high-Reynolds-number bubbles in inhomogeneous flows.” Annual Review of Fluid Mechanics, 32(1): 659-708, 2000.
[2] D. Legendre and J. Magnaudet. “The lift force on a spherical bubble in a viscous linear shear flow.” Journal of Fluid Mechanics, 368:81-126,1998.
[3] M. Rastello, J.L. Marie, and M. Lance. “Drag and lift forces on clean spherical and ellipsoidal bubbles in a solid-body rotating flow.” Journal of Fluid Mechanics, 682(1):434-459, 2011.
[4] M. Rastello, J.L. Marie, N. Grosjean, and M. Lance. “Drag and lift forces on interface-contaminated bubbles spinning in a rotating flow.” Journal of Fluid Mechanics, 624(1):159-178, 2009.
[5] R. Mei and JF Klausner. “Shear lift force on spherical bubbles.” International Journal of Heat and Fluid Flow, 15(1):62-65, 1994.
[6] Ernst A Van Nierop, Stefan Luther, Johanna J Bluemink, Jacques Magnaudet, Andrea Prosperetti, and Detlef Lohse. “Drag and lift forces on bubbles in a rotating flow.” Journal of Fluid Mechanics, 571(1):439-454, 2007.
[7] S. Antal, R. Lahey, and J. Flaherty.  “Analysis of phase distribution in fully developed laminar bubbly two-phase flow.” International Journal of Multiphase Flow, vol. 17, no. 5, pp 635-652, 1991.
 

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