Improvement of Euler-Euler simulation of two-phase flow by particle-center-averaged method

Current Euler-Euler modeling based on phase-averages shows inconsistencies since the finite size of the bubbles is not properly accounted for. As a result, nonphysical gas concentration can appear in the center or the near wall region of a pipe if the bubble diameter is larger than the mesh size (Tomiyama, Shimada et al. 2003). In addition, mesh independent solutions may not exist in these cases.
By employing particle-center-averages (Prosperetti 1998), these inconsistencies can be remedied and mesh independent solutions are obtained. In this approach, the number density of bubble centers is the primary variable. This requires an additional wall-contact force to ensure that bubble centers cannot come arbitrarily close to walls (Lucas, Krepper et al. 2007). The gas volume fraction can be calculated from the number density by a convolution (Kitagawa, Murai et al. 2001) with a kernel that has a support corresponding to the extent of a bubble. In addition, the derivation shows explicitly that bubbles respond to pressure and stress of the continuous liquid phase such that no additional closure models for the gas phase pressure or stress are required.
In the present contribution, the convolution method is replaced by a diffusion-based method (Sun and Xiao 2015), which is much easier to implement in CFD codes using unstructured meshes like OpenFOAM. A physically motivated model for the wall-contact force is introduced. The remedy of the issues with the conventional phase-averaged two-fluid model is demonstrated using a simplified two-dimensional test case. Furthermore, comparison is made for real pipe flow cases where experimental data are available.

References
Kitagawa, A., Y. Murai and F. Yamamoto (2001). "Two-way coupling of Eulerian–Lagrangian model for dispersed multiphase flows using filtering functions." International journal of multiphase flow 27(12): 2129-2153.
Lucas, D., E. Krepper and H.-M. Prasser (2007). "Use of models for lift, wall and turbulent dispersion forces acting on bubbles for poly-disperse flows." Chemical Engineering Science 62(15): 4146-4157.
Prosperetti, A. (1998). Ensemble averaging techniques for disperse flows. Particulate Flows, Springer: 99-136.
Sun, R. and H. Xiao (2015). "Diffusion-based coarse graining in hybrid continuum–discrete solvers: Theoretical formulation and a priori tests." International Journal of Multiphase Flow 77: 142-157.
Tomiyama, A., N. Shimada and H. Asano (2003). Application of Number Density Transport Equation for the Recovery of Consistency in Multi-Field Model. ASME/JSME 2003 4th Joint Fluids Summer Engineering Conference, American Society of Mechanical Engineers.