(493e) Turbulence and Interpenetrating Continua, Part II

Turbulent multiphase flows are encountered ubiquitously in the process industry (Soo, 1989; Cocco et al., 2014). The potential for simulating these flows has primarily occurred because of advancements in computational hardware and software. However, longstanding deficiencies in turbulent models (Pope, 2000; Davidson, 2013) have limited the practical utility of computational methods as an enabling technology. This presentation, which complements our AIChE 2021 abstract, will define a Reynolds stress closure for interpenetrating multiphase fluids (Manninen et al., 1996; Kleinstreuer, 2003; Brenner, 2005; Marchisio & Fox, 2013).

The normalized Reynolds “stress” must be a non-negative operator for all turbulent flows of single phase and multiphase fluids (Batchelor, 1960). All of the eigenvalues of this operator must be non-negative for all turbulent flows in rotating and non-rotating frames-of-reference (Koppula et al., 2013). The mathematical requirements of the normalized Reynolds “stress” cannot be compromised if turbulence modeling is to attain its full potential in predicting 3D-flows in finite geometries.

Current CFD technology can reproduce benchmark flows. However, the ability to predict low-order statistical properties beyond a calibrating flow is not possible. The 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 is not supported by the Navier-Stokes equation and the fundamental physical principles of thermodynamics and turbulence. As noted in Part I, 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.

Batchelor, G. K., 1960, The Theory of Homogeneous Turbulence, Cambridge University Press.

Brennen, C. E., 2005, Fundamentals of Multiphase Flow, Cambridge University Press.

Cocco, R., Kari, S. B. R., Knowlton, T., 2014, “Introduction to Fluidization: Back to Basics”, CEP, AIChE

Davidson, P.A., 2013, Turbulence in Rotating Stratified and Electrically Conducting Fluids, Cambridge University Press.

Dahler, J.S. and L.E. Scriven, 1963, Theory of Structured Continua. I. General Consideration of Angular Momentum and Polarization, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Oct. 29, 1963, Vol. 275, No. 1363, pp. 504-527.

Kleinstreuer, C., 2003, Two-Phase Flow: Theory and Practice, Taylor & Francis.

Koppula, K.S., S. Muthu, A. Bénard, and C. A. Petty, 2013, “The URAPS closure for the normalized Reynolds stress”, Physica Scripta, T155.

Manninen, M., V. Taivassalo, and S. Kallio, 1996, On the Mixture Model for Multiphase Flow, Espoo, Technical Research Centre of Finland, VTT Publication 288.

Marchisio, D.L., and R.O. Fox, 2013, Computational Models for Polydispered Particulate and Multiphase Systems, Cambridge University Press.

Pope, S. B., 2000, Turbulent Flows, Cambridge University Press.

Soo, S.L., 1989, Particulates and Continuum: Multiphase Fluid Dynamics, Hemisphere.