Interpenetrating Mixture Theory for Fluidized Beds:Thermodynamics and Turbulence

Turbulent multiphase flows are encountered ubiquitously in the process industry (Soo, 1989; Gidaspow, 1994; Crow et al., 1998; Kolev, 2002; Kleinstreuer, 2003; Brenner, 2005; Ishii and Hibbiki, 2006). Examples include flows within hydrocyclone separators and fluidized bed reactors (Cocco et al., 2014; Yate and Lettieri, 2016). The potential for simulating these flows has primarily occurred because of advancements in computational hardware and software. However, longstanding deficiencies in turbulence closure models limit the practical utility of computational methods as an enabling technology (Davidson, 2013; Pope, 2000; Drew and Passman, 1999; Marchisio and Fox, 2013; Yeoh et al., 2014).

This presentation will extend a recently developed Reynolds stress closure for single phase fluids (Koppula et al., 2009, 2011, 2013) to a Reynolds stress closure for interpenetrating multiphase fluids (Manninen et al., 1996). The normalized Reynolds stress is a non-negative operator; therefore, it is essential that all of the eigenvalues of this operator must be non-negative for all turbulent flows in rotating and non-rotating frames-of-reference. This fundamental mathematical property cannot be compromised if turbulence modeling is to attain its full potential in predicting flows in complex geometries. Although current CFD technology can reproduce benchmark flows, the ability to predict low-order statistical properties beyond a calibrating flow is not possible. This weakness of turbulence modeling can be traced to the closure hypothesis that velocity fluctuations are objective vector fields. This assumption is embedded in the sub-grid closures associated with large-eddy simulations; the pressure/strain rate closures associated with the second-order moment equation for the Reynolds stress; and, the eddy viscosity models associated with the closure of the Reynolds averaged Navier-Stokes equation. The ad hoc assumption that the Reynolds stress is an objective operator similar to the Cauchy stress in continuum mechanics is not supported by direct numerical simulations of the Navier-Stokes equation and fundamental physical principles of thermodynamics and turbulence. Continuum scale hydrodynamic fluctuations, unlike molecular scale fluctuations, are not objective vector fields. Research at Michigan State University has identified a class of algebraic closure models for the normalized Reynolds stress that are realizable for all turbulent flows in rotating and non-rotating frames. This discovery has much potential to transform current CFD technology from an interpolating tool to a predictive tool.

References

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