ABSTRACT
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Marcelo J.S. de Lemos
CONTENTS Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .411 10.2 Local Instantaneous Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .411 10.3 Volume and Time Average Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 10.4 Time-Averaged Transport Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 10.5 The Double-Decomposition Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416
10.5.1 Basic Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 10.6 Turbulent Momentum Transport in Porous Media . . . . . . . . . . . . . . . . . . . 419
10.6.1 Mean Flow Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 10.6.1.1 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 10.6.1.2 Momentum-one average operator . . . . . . . . . . . . . . . 419 10.6.1.3 Momentum equation — two average operators . . 420 10.6.1.4 Inertia term — space and time (double)
decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 10.6.2 Equations for Fluctuating Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 10.6.3 Turbulent Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426
10.6.3.1 Equation for 〈k〉i = 〈u′ · u′〉i/2 . . . . . . . . . . . . . . . . . . . . . . . . 428 10.6.3.2 Comparison of macroscopic transport equations . 430
10.7 Turbulent Heat Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 10.7.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431
10.7.1.1 Time average followed by volume average . . . . . . . . 431 10.7.1.2 Volume average followed by time average . . . . . . . . 432
10.7.2 Turbulent Thermal Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 10.7.3 Local Thermal Equilibrium Hypothesis . . . . . . . . . . . . . . . . . . . . . . . 436 10.7.4 Macroscopic Buoyancy Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
10.7.4.1 Mean flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 10.7.4.2 Turbulent field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438
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10.8 Turbulent Mass Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440 10.8.1 Mean and Turbulent Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440 10.8.2 Turbulent Mass Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442
10.9 Applications in Hybrid Media. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444 10.9.1 The Stress Jump Conditions at Interface . . . . . . . . . . . . . . . . . . . . . . 444 10.9.2 Buoyant Flows in Cavities Partially Filled with Porous
Inserts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 10.9.3 FlowAround a Sinusoidal Interface in a Channel . . . . . . . . . . . 447
10.10 Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
Engineering equipment design and environmental impact analyses can benefit fromappropriatemodelingof turbulentflowinporousmedia.Accordingly, a number of natural and engineering systems can be characterized by some sort of porous structure through which a working fluid permeates. Turbulence models proposed for such flows depend on the order of application of time and volume-average operators. Two developed methodologies, following the two orders of integration, lead to different governing equations for the statistical quantities. This chapter reviews recently published methodologies to mathematically characterize turbulent transport in porous media. For hybridmedia, involving both aporous structure and a clear flow region,
difficulties arise due to the proper mathematical treatment given at the interface. This chapter also presents and discusses numerical solutions for such hybrid media, here considering a channel partially filled with a wavy porous layer throughwhich fluid flows in turbulent regime. In addition, macroscopic forms of buoyancy terms are also considered in both the mean and the turbulent fields. Cases reviewed include heat transfer in cavities partially filled with porous material. In summary, within this chapter local instantaneous governing equations
are reviewed for clear flow before volume and time-average operators are applied to them. The double-decomposition concept is presented and thoroughly discussed prior to the derivation ofmacroscopic governing equations. Equations for turbulent momentum transport in porous media follow showing detailed derivation for the mean and turbulent field quantities. The statistical k-emodel for clear domains, used tomodelmacroscopic turbulence effects, also serves as the basis for turbulent heat transport modeling. Turbulentmass transport in porousmatrices is further reviewed in the light of the double-decomposition concept.Asection on applications in hybridmedia
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covers flow over porous layers in channels and in cavities partially filledwith porous material.
