(399e) Turbulence and Mixing - Honoring Professor Robert Stanley Brodkey

All turbulent flows are three-dimensional and time-dependent (Batchelor, 1960; Monin and Yaglom, 1965; Brodkey, 1967; Tennekes and Lumley, 1977; Pope, 2000; Davidson et al., 2011, Davidson, 2013). Reactive (and non-reactive) multiphase, multicomponent, and immiscible materials are governed by five equations-of-change: (1) mass; (2) linear momentum; (3) angular momentum; (4) energy; and, (5) entropy (Manninen et al., 1996; Churchill, 2014; Marchisio and Fox, 2013). By hypothesis, all material transport models for gases, liquids, and solids are frame indifferent (i.e., objective). If the deviatoric component of the Cauchy’s stress is not symmetric, then the intrinsic and external angular momentum can be exchanged (Dahler, 1959; Dahler and Scriven, 1961 and 1963).

The Coriolis Theorem predicts that the strain rate is objective; and, that the Reynolds “stress” is not. However, the Cauchy stress and the Reynolds “stress” are complementary, but not similar. A Universal Realizable Anisotropic PreStress (URAPS) closure predicts that the Coriolis acceleration causes an anisotropic re-distribution of the turbulent kinetic energy among the three components of the fluctuating velocity in a rotating homogeneous decay (Koppula et al., 2009; 2011; 2013). Turbulent flows are commonly decomposed into a Reynolds ensemble average and an instantaneous fluctuating field.

The URAPS closure simulates turbulent flows of interpenetrating continua. Beyond the Kolmogoroff scale, the mean flow field is spatially homogeneous and the Reynolds “stress” is spatially anisotropic. The Coriolis Theorem predicts that the material strain rate is frame insensitive (i.e., objective); and, most significantly, the Reynolds “stress” is frame sensitive (i.e., non-objective).

The Cauchy stress and the Reynolds “stress” are not similar for single phase fluids (or for multiphase fluids). However, for single phase fluids, turbulent dispersion is positive; turbulent dissipation is positive; and, turbulent “production” of kinetic energy may be either positive or negative. For a second-order, irreducible, compressible, Newtonian fluid, the Cauchy stress and the strain rate are symmetric and objective. The phenomenological coefficients for this model are objective scalar-valued functions of the local thermodynamic state and the eigenvalues of the local strain rate. For some materials, the Cauchy stress is asymmetric and objective.

The Reynolds average of the equation-of-change for linear momentum is an exact, albeit unclosed, vector-valued equation for the mean velocity field of a single-phase fluid. Unlike the Cauchy stress, the Reynolds “stress” is a dyadic-valued real, symmetric, non-negative, and non-objective operator for all inertial and non-inertial temporal frames-of-reference. The non-objectivity property of the Reynolds “stress” stems directly from the Coriolis Theorem.

Turbulent flows of multiphase fluids are encountered ubiquitously in industry. The mixture model presented herein will include equations-of-change for mass, linear momentum, angular momentum, energy, and entropy. The multiphase mixture theory is developed by using a phase average followed by a Reynolds average beyond the Kolmogoroff scale, even if the mean turbulent flow is spatially homogeneous. The purpose of this presentation is to extend this observation to a multiphase fluid.

Turbulent multiphase flows are encountered ubiquitously in the process industry. Examples include flows within hydrocyclone separators and fluidized bed reactors (see 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 turbulence closure models limit the practical utility of computational methods as an enabling technology. This presentation will extend a recently developed Reynolds stress closure for single phase fluids to a Reynolds stress closure for interpenetrating multiphase fluids. 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.

Citations

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

Brodkey, R. S., 1967, The Phenomena of Fluid Motions, Addison-Wesley Pub. Co.

Churchill, S.W., 1996, Critique of Predictive and Correlative Models for Turbulent Flow and Convection, Ind. & Eng. Chem. Res., 35, 3122.

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

Dahler, J.S., 1959, Sept. 29, Transport Phenomena in a Fluid Composed of Diatomic Molecules, The Journal of Chemical Physics, Vol. 30, No. 6, pp.1447-1475.

Dahler, J.S. and L.E. Scriven, 1961, Angular Momentum of Continua, October 7, Nature.

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, Vol. 275, No. 1363, pp.504-527.

Davidson, P.A., Y. Kaneda, K. Moffatt, K. R. Screenivasan, 2011, A Voyage Through Turbulence, Cambridge University Press

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

Koppula, K. S., A. Bénard, and C. A. Petty, 2009, “Realizable Algebraic Reynolds Stress Closure”, Chem. Eng. Sci., 64, 4611-4624.

Koppula, K. S., A. Bénard, and C. A. Petty, 2011, “Turbulent Energy Redistribution in Spanwise Rotating Channel Flows”, Ind. Eng. Chem. Res., 50 (15), 8905-8916.

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, S. Kallio, and, A. Akademi, 1996, On the Mixture Model for Multiphase Flow, Espoo, Technical Research Center of Finland, VTT Publication, 288, 67 pages.

Marchisio, D.L., and R.O. Fox, 27 May, 2013, Computational Models for Polydispered Particulate and Multiphase Systems, Physics, Cambridge University Press.

Monin, P.K., and A. M. Yaglom, 1965, Statistical Fluid Mechanics: Mechanics of Turbulence, Volume 1, The MIT Press.

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

Tennekes, H. and J.L. Lumley, 1977, A First Course in Turbulence, The MIT Press.