DOCSLIB.ORG

Thesis Dynamics and Structure of Stably

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Thesis Dynamics and Structure of Stably THESIS DYNAMICS AND STRUCTURE OF STABLY STRATIFIED TURBULENCE Submitted by Simon Schaad Department of Civil and Environmental Engineering In partial fulfillment of the requirements For the Degree of Master of Science Colorado State University Fort Collins, Colorado Fall 2012 Master’s Committee: Advisor: Subhas Karan Venayagamoorthy Pierre Y. Julien Lakshmi Prasad Dasi ABSTRACT DYNAMICS AND STRUCTURE OF STABLY STRATIFIED TURBULENCE The dynamics and turbulent structures of stably stratified turbulence are explored via direct numerical simulations (DNS). The structural features of stratified turbu- lence and its relationship to the flow dynamics has been the subject of many recent investigations. In strongly stratified turbulent flows, the formation of large-scale quasi-horizontal vortices in layers with strong vertical variability has been observed in laboratory experiments. Enstrophy isosurfaces of strongly stable flows indicate the emergence of randomly distributed ‘pancake’-like structures with near horizontal orientation at later times. The strongly stratified simulations are diffusive and dom- inated by linear internal waves. The results suggest a decoupling between horizontal and vertical dynamics as the vertical dynamics can be described using rapid-distortion theory (RDT) while horizontal dynamics continue to be dominated by non-linear ef- fects not captured by RDT. The integral flux Richardson number for decaying turbulence is the ratio of back- ground potential energy gain to turbulent kinetic energy loss. The traditional flux- based formulation converges upon this ratio only when integrations are performed over an entire event, while the irreversible formulation converges rapidly without er- ror from reversible effects. Mixing efficiency is a property of the flow for energetic flow but becomes a property of the fluid for diffusive flows and subject to Prandtl number effects. RDT models predict the flux Richardson number scales as the inverse Prandtl number at the diffusive limit when the Prandtl number is greater than unity. Mixing efficiency comparisons between DNS and physical grid-tow experiments reveal a large discrepancy for strong stratification, which is attributed in part to the low Reynolds numbers attained in both DNS and grid-tow experiments. Overturns are unstable conditions where heavier fluid resides above lighter fluid. The collapse of these local instabilities produce additional patches of turbulence and ii mixing making overturns an important mechanism in stratified turbulence. The over- turning structures in strongly stratified flow resemble the quasi-horizontal vorticity structures and were found to be correlated with increased horizontal vorticity. The Thorpe scale, a measure of overturning structure height, and the Ozmidov scale equate only at the critical condition where inertial and buoyancy effects are equal (i.e. the turbulent Froude number is unity). The error of inferred dissipation rates from equat- ing the Thorpe and Ozmidov scales was found to be up to two orders of magnitude. iii ACKNOWLEDGEMENTS This thesis would not have been possible without the guidance and teachings from my advisor Dr. Karan Venayagamoorthy. Studying fluid mechanics at CSU has been a fantastic journey and I would like to acknowledge Dr. Venayagamoorthy and his fluid mechanics research group for their support and help with the ideas presented in this work. Thanks to Ben Mater for his help, useful feedback, and collaboration on the overturning length scale ideas in Sec. (4.3). And thanks to Farid Karimpour for helping me solve those tricky calculus problems. I would like to acknowledge Dr. Pierre Julien and Dr. Prasad Dasi for generously volunteering their time to be on my committee and read my thesis. Thanks to Lobo, Argos, and Hayek for their loyal company, Caesar for warming my lap, Gaius for spilling drinks on my laptop, and Sawyer for his loud, 3-legged thumping. Of course this thesis could not have been possible without the loving support from my family (albeit far removed from Colorado). Most importantly, thanks to Emily Chryst for her love, patience, and sharing her life with me. iv DEDICATION Dedicated to my muse, my friend, and my love; Emily v TABLE OF CONTENTS 1 Introduction 1 1.1 Background.................................... 1 1.2 Objectives..................................... 2 1.3 Thesislayout ................................... 2 1.4 Research presentations and publications . ........ 3 2 Literature Review 4 2.1 Introduction.................................... 4 2.2 Equationsofmotion ............................... 5 2.2.1 Governingequations ........................... 5 2.2.1.1 Navier-Stokes equations . 5 2.2.1.2 Continuity equation . 7 2.2.1.3 Density transport equation . 7 2.2.1.4 Non-dimensional form of governing equations . ... 8 2.2.2 Energetics................................. 9 2.2.2.1 Turbulent kinetic energy . 9 2.2.2.2 Turbulent potential energy . 10 2.2.2.3 Totalenergy .......................... 12 2.3 Turbulentmixing................................. 13 2.3.1 Mixing efficiency and flux Richardson number . 14 2.3.2 Physicalexperiments. 16 2.4 Lengthandtimescales............................. 18 2.4.1 Kolmogorovscales ............................ 19 2.4.2 Batchelorscale .............................. 20 2.4.3 Taylormicroscale............................. 20 2.4.4 Buoyancyfrequency ........................... 20 2.4.5 Turbulent length and time scale . 22 2.4.6 Ozmidovscale .............................. 23 2.4.7 Buoyancyscale .............................. 23 2.4.8 Thorpescale ............................... 23 2.4.9 Ellisonscale................................ 24 2.5 Non-dimensionalparameters. 24 2.5.1 Initial Richardson number and Froude number . 25 vi 2.5.2 Reynolds number and Peclet number . 25 2.5.3 BuoyancyReynoldsnumber. 26 2.6 Direct Numerical Simulations . 27 2.6.1 Descriptionofcode............................ 28 2.6.2 Initialconditions . .. .. .. .. .. .. .. .. 30 2.6.3 Rapid-DistortionTheory. 30 3 Dynamics and Structure 31 3.1 Introduction.................................... 31 3.2 Validation..................................... 31 3.2.1 Resolutionissues ............................. 32 3.2.2 Comparisonwithpreviouswork. 35 3.3 Dynamicsandenergetics. 37 3.3.1 Energetics................................. 37 3.3.2 Dissipation ................................ 38 3.3.3 Taylor and turbulent Reynolds numbers . 40 3.3.4 Formulations of flux Richardson number . 41 3.4 Flowregimes ................................... 42 3.4.1 Scaling regimes of mixing efficiency . 43 3.4.2 BuoyancyReynoldsnumber. 45 3.4.3 Regimes of buoyancy Reynolds number . 47 3.5 Coherentturbulentstructures . ..... 49 3.6 Diffusivity..................................... 58 3.6.1 Prandtlnumbereffects. .. .. .. .. .. .. .. 59 3.6.2 Comparison with physical experiments . 63 3.7 Rapid-DistortionTheory. 64 3.7.1 Decoupling of horizontal and vertical dynamics . ....... 65 3.7.2 Reynoldsnumbereffects. .. .. .. .. .. .. .. 66 3.7.3 Prandtl number effects, revisited using RDT simulations....... 70 3.8 Conclusion .................................... 71 4 Overturns 76 4.1 Introduction.................................... 76 4.2 Overturningstructures. 76 4.2.1 Overturns and stratification . 77 4.2.2 Overturnsandvorticity . 77 vii 4.2.3 Investigation of an overturn . 82 4.3 Overturninglengthscales . 83 4.3.1 TheThorpescale............................. 83 4.3.2 Thorpe-Ozmidovrelation . 86 4.3.3 Inferreddissipation. 89 4.3.4 Other important length scale relations . ..... 90 4.4 Conclusion .................................... 92 5 Conclusion 95 5.1 Summaryofstudies ............................... 95 5.2 ConclusionsfromChapter3 . 95 5.3 ConclusionsfromChapter4 . 97 5.4 Recommendationsforfuturework . 98 References 100 A Description of simulations and models 106 A.1 Summaryofruns................................. 106 A.2 DescriptionofDNScode. 111 A.3 DescriptionofRDTcode ............................ 112 B Thorpe scale post-processing 114 B.1 Ensemble-averageThorpescale . 114 B.2 3DThorpescale ................................. 115 B.3 Tabulatedresults................................ 118 Glossary 119 viii LIST OF TABLES A.1 MainDNSstudy ................................. 108 A.2 ReynoldsnumberDNSstudy . 109 A.3 ReynoldsnumberRDTstudy . 109 A.4 Inviscidandnon-diffusiveRDTstudy . 110 A.5 PrandtlnumberRDTstudy. 110 A.6 SubroutinesandfilesofDNScode . 111 B.1 Thorpescaleresults .............................. 118 ix LIST OF FIGURES 2.1 Buoyancy forces in stable and unstable stratification . ........... 11 2.2 Density profile mixing in physical experiment . ........ 17 3.1 Modelspectrum ................................. 33 3.2 Energy cascades with length scales . ..... 34 3.3 Gridindependence ................................ 36 3.4 Validation with Ri and Pr variation . 36 3.5 Evolution of ensemble-average energetics . ......... 38 3.6 Horizontal and vertical kinetic energy . ....... 39 3.7 Evolution of dissipation rates . ..... 39 3.8 EvolutionofReynoldsnumbers . 41 3.9 Integrated buoyancy flux and potential energy dissipation rate . 42 3.10 Comparison of mixing efficiency formulations . ........ 43 3.11 Rehmann&Koseffscalingregimes . 45 3.12 Evolution of buoyancy Reynolds number . ...... 46 3.13 Regimes of buoyancy Reynolds number . ..... 48 3.14 Isotropicenstrophystructures . ....... 52 3.15 Pancakeenstrophystructures . ..... 53 3.16 Evolution of enstrophy structures . ....... 54 3.17 Cross-sections of enstrophy fields . ....... 55 3.18 Evolution of horizontal vorticity structures . ..........
Load more

Recommended publications
  • Constraints on the Magnetic Buoyancy Instabilities of a Shear-Generated Magnetic Layer
    Submitted to The Astrophysical Journal Constraints on the magnetic buoyancy instabilities of a shear-generated magnetic layer Geoffrey M. Vasil JILA and Dept. of Atmospheric & Oceanic Sciences, University of Colorado, Boulder, CO 80309-0440 Nicholas H. Brummell Dept. of Applied Mathematics & Statistics, University of California, Santa Cruz, CA 95064 [email protected] ABSTRACT In a recent paper (Vasil & Brummell 2008), we found that the action of vertical (radial) shear on the vertical component of poloidal magnetic field could induce magnetic buoyancy instabilities that produced rising, undulating tubular magnetic structures as often envisaged arising from the solar tachocline, but only under extreme circumstances. Herein, we examine, in greater detail, the reasons underpinning the difficulties in obtaining magnetic buoyancy. Under a variety of assumptions about the maintenance of the shear, we herein discuss some analytic limits on the ability of the Ω-effect to produce toroidal magnetic field. Firstly, we consider the Ω-effect in a local time-dependent context, where an unmaintained shear is allowed to build a magnetic layer over time. In this case, we find amplitude estimates of the magnetic field made through such a process in terms of the shear-flow Richardson number. Secondly, we consider the Ω-effect in a local time-independent context, where the shear is forced, such that under certain circumstances it would be maintained. In this situation, we derive a variety of mathematical bounds on the toroidal magnetic energy that can be realized, and its gradients, in terms of the magnitude and the magnetic Prandtl number. This result implies that at low σM , unreasonably strong shear flows must be forced to produce magnetic buoyancy, as was found in our earlier paper.
    [Show full text]
  • Laboratory Studies of Turbulent Mixing J.A
    Laboratory Studies of Turbulent Mixing J.A. Whitehead Woods Hole Oceanographic Institution, Woods Hole, MA, USA Laboratory measurements are required to determine the rates of turbulent mixing and dissipation within flows of stratified fluid. Rates based on other methods (theory, numerical approaches, and direct ocean measurements) require models based on assumptions that are suggested, but not thoroughly verified in the laboratory. An exception to all this is recently provided by direct numerical simulations (DNS) mentioned near the end of this article. Experiments have been conducted with many different devices. Results span a wide range of gradient or bulk Richardson number and more limited ranges of Reynolds number and Schmidt number, which are the three most relevant dimensionless numbers. Both layered and continuously stratified flows have been investigated, and velocity profiles include smooth ones, jets, and clusters of eddies from stirrers or grids. The dependence of dissipation through almost all ranges of Richardson number is extensively documented. The buoyancy flux ratio is the turbulent kinetic energy loss from raising the potential energy of the strata to loss of kinetic energy by viscous dissipation. It rises from zero at small Richardson number to values of about 0.1 at Richardson number equal to 1, and then falls off for greater values. At Richardson number greater than 1, a wide range of power laws spanning the range from −0.5 to −1.5 are found to fit data for different experiments. Considerable layering is found at larger values, which causes flux clustering and may explain both the relatively large scatter in the data as well as the wide range of proposed power laws.
    [Show full text]
  • Applied Fluid Dynamics Handbook
    APPLIED FLUID DYNAMICS HANDBOOK ROBERT D. BLEVINS H iMhnisdia ttodisdiule Darmstadt Fachbereich Mechanik 'rw.-Nr.. [VNR1 VAN NOSTRAND REINHOLD COMPANY ' ' New York Contents Preface / v 5.4 The Bernoulli Equations / 29 5.4.1 Presentation of the Equations / 29 1. Definitions / 1 5.4.2 Example of the Bernoulli Equations / 30 5.5 Conservation of Momentum / 32 2. Symbols / 5 5.5.1 The Momentum Equations / 32 5.5.2 Examples of Conservation of 3. Units / 6 Momentum / 33 4. Dimensional Analysis / 8 4.1 Introduction / 8 6. Pipe and Duct Flow / 38 4.2 Boundary Geometry / 8 6.1 General Considerations / 38 4.3 Euler Number / 10 6.1.1 General Assumptions / 38 4.4 Reynolds Number / 10 6.1.2 Principles of Pipe Flow / 38 4.5 Mach Number / 11 6.2 Velocity Profiles for Fully Developed Pipe 4.6 Froude Number / 11 Flow / 39 4.7 Weber Number / 11 6.2.1 Inlet Length / 39 4.8 Cavitation Number / 12 6.2.2 Fully Developed Profiles / 40 4.9 Richardson Number / 12 6.3 Straight Uniform Pipes and Ducts / 42 4.10 Rayleigh Number / 12 6.3.1 General Form for Incompressible 4.11 Strouhal Number / 12 Flow / 42 4.12 Kolmogoroff Scales / 13 6.3.2 Friction Factor for Laminar Flow 4.13 Application / 13 (Re<2000) / 42 4.13.1 Then-Theorem / 13 6.3.3 Friction Factor for Transitional Flow 4.13.2 Example: Model Testing a Nuclear (2000 <Re <4000) / 48 Reactor / 14 6.3.4 Friction Factor for Turbulent Flow 4.13.3 Example: Model Testing of a (Re>4000) / 50 Surfboard / 15 6.4 Curved Pipes and Bends / 55 6.4.1 General Form ./ 55 5.
    [Show full text]
  • Global Stability of Buoyant Jets and Plumes
    This draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics 1 Global stability of buoyant jets and plumes R. V. K. Chakravarthy, L. Lesshafft and P. Huerre Laboratoire d’Hydrodynamique, CNRS/Ecole´ polytechnique, 91128 Palaiseau, France. (Received xx; revised xx; accepted xx) The linear global stability of laminar buoyant jets and plumes is investigated under the low Mach number approximation. For Richardson numbers in the range 10−4 6 Ri 6 103 and density ratios S = ρ∞/ρjet between 1.05 and 7, only axisymmetric perturbations are found to exhibit global instability, consistent with experimental observations in helium jets. By varying the Richardson number over seven decades, the effects of buoyancy on the base flow and on the instability dynamics are characterised, and distinct behaviour is observed in the low-Ri (jet) and in the high-Ri (plume) regimes. A sensitivity analysis indicates that different physical mechanisms are responsible for the global instability dynamics in both regimes. In buoyant jets at low Richardson number, the baroclinic torque enhances the basic shear instability, whereas buoyancy provides the dominant instability mechanism in plumes at high Richardson number. The onset of axisymmetric global instability in both regimes is consistent with the presence of absolute instability. While absolute instability also occurs for helical perturbations, it appears to be too weak or too localised in order to give rise to a global instability. 1. Introduction Vertical injection of light fluid into a denser ambient creates a flow that either bears the characteristics of a jet, if the injected momentum is dominant over the buoyant forces, or those of a plume, if the momentum that is generated by buoyancy is dominant over the momentum that is imparted at the orifice.
    [Show full text]
  • CFD Analysis of Heat Transfer and Wind Shear Near Air-Water Interface Zimeng Wu Clemson University, [email protected]
    Clemson University TigerPrints All Theses Theses 5-2019 CFD Analysis of Heat Transfer and Wind Shear near Air-Water Interface Zimeng Wu Clemson University, [email protected] Follow this and additional works at: https://tigerprints.clemson.edu/all_theses Recommended Citation Wu, Zimeng, "CFD Analysis of Heat Transfer and Wind Shear near Air-Water Interface" (2019). All Theses. 3049. https://tigerprints.clemson.edu/all_theses/3049 This Thesis is brought to you for free and open access by the Theses at TigerPrints. It has been accepted for inclusion in All Theses by an authorized administrator of TigerPrints. For more information, please contact [email protected]. CFD ANALYSIS OF HEAT TRANSFER AND WIND SHEAR NEAR AIR-WATER INTERFACE A Thesis Presented to the Graduate School of Clemson University In Partial Fulfillment of the Requirements for the Degree Master of Science Civil Engineering by Zimeng Wu May 2019 Accepted by: Dr. Abdul Khan, Committee Chair Dr. Nigel Kaye, Co-committee Chair Dr. Ashok Mishra, Committee Member ABSTRACT This research was undertaken to examine a problem with respect to heat transfer and fluid dynamics, and this problem was simulated by CFD (Computational Fluid Dynamics). A tank of water with high initial temperature was exposed to an atmosphere with lower temperature than the water, which produced heat transfer at the air-water interface. The air had various magnitudes of velocity for different cases, which also generated wind shear near the interface. These two factors interacted and eventually got balanced for both thermodynamics and kinematics. The cooling plumes were observed in the initial stage when heat transfer was large due to the large difference of temperature between air and water.
    [Show full text]
  • Asymmetrical Thermal Boundary Condition Influence on the Flow
    fluids Article Asymmetrical Thermal Boundary Condition Influence on the Flow Structure and Heat Transfer Performance of Paramagnetic Fluid-Forced Convection in the Strong Magnetic Field Lukasz Pleskacz , Elzbieta Fornalik-Wajs * and Sebastian Gurgul Department of Fundamental Research in Energy Engineering, AGH University of Science and Technology, Al. Mickiewicza 30, 30-059 Kraków, Poland; [email protected] (L.P.); [email protected] (S.G.) * Correspondence: [email protected] Received: 15 November 2020; Accepted: 14 December 2020; Published: 16 December 2020 Abstract: Continuous interest in space journeys opens the research fields, which might be useful in non-terrestrial conditions. Due to the lack of the gravitational force, there will be a need to force the flow for mixing or heat transfer. Strong magnetic field offers the conditions, which can help to obtain the flow. In light of this origin, presented paper discusses the dually modified Graetz-Brinkman problem. The modifications were related to the presence of the magnetic field influencing the flow and asymmetrical thermal boundary condition. Dimensionless numerical analysis was performed, and two dimensionless numbers (magnetic Grashof number and magnetic Richardson number) were defined for paramagnetic fluid flow. The results revealed the heat transfer enhancement due to the strong magnetic field influence accompanied by possible but not essential flow structure modifications. On the other hand, the flow structure changes can be utilized to prevent the solid particles’ sedimentation. The explanation of the heat transfer enhancement including energy budget and vorticity distribution was presented. Keywords: computational fluid dynamics; strong magnetic field; thermo-magnetic convection; forced convection; Nusselt number; magnetic Richardson number 1.
    [Show full text]
  • Numerical Simulation of the Natural, Forced and Mixed Convection in a Tunnel with a Flat Track of Sinusoidal Shape and a Roof Opening
    E3S Web of Conferences 85, 02001 (2019) https://doi.org/10.1051/e3sconf/20198502001 EENVIRO 2018 Numerical simulation of the natural, forced and mixed convection in a tunnel with a flat track of sinusoidal shape and a roof opening Oumar Drame1,*, Cheikh Mbow1, Florin Bode2, Samba Dia1, and Omar Ngor Thiam1 1Cheikh Anta Diop University of Dakar, Faculty of Science and Technology Department of Physics; fluid mechanics laboratory and applications BP 5005 Dakar-Fann 2Technical University of Cluj-Napoca, Dept. of Mechanical Engineering, Bd. Muncii 103-105, 400641, Cluj-Napoca, Romani Abstract. In this work, we studied the mixed convection of the airflow in a tunnel open at both ends. The tunnel has a sinusoidal trace and the horizontal ceiling is provided with an opening in the center. The tunnel floor is uniformly heated. Although of interest for many industrial applications, the configuration of this study has been studied very little from an academic point of view. Coupled equations of Naiver-Stokes and energy are solved numerically by the finite volume method with the Boussinesq hypothesis. We analyzed the effect of the parameters that characterize heat transfer, and the flow structure. Several situations have been considered by varying the Richardson number (1.3610-3≤Ri≤2.17.104) for a Prandtl number Pr = 0.71. Nomenclature Nu Nusselt number; Pr a amplitude of the sinusoidal part (m) Prandlt number; -2 gp gravitational acceleration (ms ) PeT Peclet number; height between the ceiling and the bottom of Re Reynolds number; h the sinusoid
    [Show full text]
  • Dimensional Analysis and Modeling
    cen72367_ch07.qxd 10/29/04 2:27 PM Page 269 CHAPTER DIMENSIONAL ANALYSIS 7 AND MODELING n this chapter, we first review the concepts of dimensions and units. We then review the fundamental principle of dimensional homogeneity, and OBJECTIVES Ishow how it is applied to equations in order to nondimensionalize them When you finish reading this chapter, you and to identify dimensionless groups. We discuss the concept of similarity should be able to between a model and a prototype. We also describe a powerful tool for engi- ■ Develop a better understanding neers and scientists called dimensional analysis, in which the combination of dimensions, units, and of dimensional variables, nondimensional variables, and dimensional con- dimensional homogeneity of equations stants into nondimensional parameters reduces the number of necessary ■ Understand the numerous independent parameters in a problem. We present a step-by-step method for benefits of dimensional analysis obtaining these nondimensional parameters, called the method of repeating ■ Know how to use the method of variables, which is based solely on the dimensions of the variables and con- repeating variables to identify stants. Finally, we apply this technique to several practical problems to illus- nondimensional parameters trate both its utility and its limitations. ■ Understand the concept of dynamic similarity and how to apply it to experimental modeling 269 cen72367_ch07.qxd 10/29/04 2:27 PM Page 270 270 FLUID MECHANICS Length 7–1 ■ DIMENSIONS AND UNITS 3.2 cm A dimension is a measure of a physical quantity (without numerical val- ues), while a unit is a way to assign a number to that dimension.
    [Show full text]
  • Numerical Convection of Viscoelastic Magnetic Flows Past a Backward Facing Step Channel
    INTERNATIONAL JOURNAL OF GEOLOGY Issue 3, Volume 4, 2010 Numerical Convection of Viscoelastic Magnetic Flows Past a Backward Facing Step Channel Cheng-Hsing Hsu, Kuang-Yung Kung, Shu-Yu Hu [email protected] magnetic visco-elastic fluid. The Maxwell model must be more Abstract—Numerical simulation presented viscoelastic magnetic than the introduction of a relaxation time, the stress tensor as flow past a backward facing step channel. The parameters of independent variables, and this model is mainly used in large Richardson Number, Reynolds Number, elastic coefficient, buoyancy magnetic visco-elastic fluid. effects and magnetic field parameters were studied. The larger The main purpose of this paper is to use numerical simulation buoyancy effect was along with a larger Richardson Number. Transient coexistence of multiple second recirculation zone appear in to explore the magnetic field fixed at the high Richardson downstream. The four symmetry vortexes appeared on the upper and Number of cases. The second model visco-elastic magnetic lower plate. The oscillation phenomena were similar to Karman fluid flow in the backward facing step is expected that the vortex. Under the same Richardson Number, Reynolds Number, and visco-elastic magnetic fluid in the post flow to the backward the elastic coefficient, the higher the magnetic field parameters facing step to have a better understanding of the characteristics. resulted to the shorter length of the main and second recirculation Backward step of the flow phenomenon has been valued by zone. everyone. The flow fields in supersonic flow first by Hama Keywords—Buoyancy effect, Elastic coefficient, Magnetic field experiment found that supersonic fluid flow through the parameter, Viscoelastic magnetic fluid.
    [Show full text]
  • Part Two Physical Processes in Oceanography
    Part Two Physical Processes in Oceanography 8 8.1 Introduction Small-Scale Forty years ago, the detailed physical mechanisms re- Mixing Processes sponsible for the mixing of heat, salt, and other prop- erties in the ocean had hardly been considered. Using profiles obtained from water-bottle measurements, and J. S. Turner their variations in time and space, it was deduced that mixing must be taking place at rates much greater than could be accounted for by molecular diffusion. It was taken for granted that the ocean (because of its large scale) must be everywhere turbulent, and this was sup- ported by the observation that the major constituents are reasonably well mixed. It seemed a natural step to define eddy viscosities and eddy conductivities, or mix- ing coefficients, to relate the deduced fluxes of mo- mentum or heat (or salt) to the mean smoothed gra- dients of corresponding properties. Extensive tables of these mixing coefficients, KM for momentum, KH for heat, and Ks for salinity, and their variation with po- sition and other parameters, were published about that time [see, e.g., Sverdrup, Johnson, and Fleming (1942, p. 482)]. Much mathematical modeling of oceanic flows on various scales was (and still is) based on simple assumptions about the eddy viscosity, which is often taken to have a constant value, chosen to give the best agreement with the observations. This approach to the theory is well summarized in Proudman (1953), and more recent extensions of the method are described in the conference proceedings edited by Nihoul 1975). Though the preoccupation with finding numerical values of these parameters was not in retrospect always helpful, certain features of those results contained the seeds of many later developments in this subject.
    [Show full text]
  • Enter Title Here
    CORE Metadata, citation and similar papers at core.ac.uk Provided by UM Digital Repository International Journal of Mechanical and Materials Engineering (IJMME), Vol.5 (2010), No.2, 163-170. REYNOLDS AND PRANDTL NUMBERS EFFECTS ON MHD MIXED CONVECTION IN A LID-DRIVEN CAVITY ALONG WITH JOULE HEATING AND A CENTERED HEAT CONDUCTING CIRCULAR BLOCK M. M. Rahman1, M. M. Billah2, M. A. H. Mamun 3, R. Saidur4 and M. Hasanuzzaman4 1Department of Mathematics 1Bangladesh University of Engineering and Technology (BUET), Dhaka-1000, Bangladesh 2Department of Arts and Sciences, Ahsanullah University of Science and Technology (AUST), Dhaka, Bangladesh 3The Institute of Ocean Energy, Saga University, Japan 4Department of Mechanical Engineering, University of Malaya, 50603 Kuala Lumpur, Malaysia Email: [email protected] Received 20 May 2010, Accepted 29 July 2010 ABSTRACT Nu average Nusselt number p dimensional pressure (Nm-2) The fluid flows and heat transfer induced by the combined P dimensionless pressure effects of mechanically driven lid and buoyancy force Pr Prandtl number within a rectangular cavity is investigated in this paper Ra Rayleigh number numerically. The horizontal walls of the enclosure are Re Reynolds number insulted while the right vertical wall is maintained at a Ri Richardson number uniform temperature higher than the left vertical wall. In T dimensional temperature(K) addition, it contains a heat conducting horizontal circular ∆T temperature difference (K) block in its centre. The governing equations for the problem u, v dimensional velocity components (ms-1) are first transformed into a non-dimensional form and the U, V dimensionless velocity components resulting nonlinear system of partial differential equations U0 lid velocity are solved by using the finite element formulation based on V cavity volume (m3) the Galerkin method of weighted residuals.
    [Show full text]
  • Numerical Study of Double Diffusive Convection in a Lid Driven Cavity with Linearly Salted Side Walls
    Archive of SID Journal of Applied Fluid Mechanics, Vol. 10, No. 1, pp. 69-79, 2017. Available online at www.jafmonline.net, ISSN 1735-3572, EISSN 1735-3645. DOI: 10.18869/acadpub.jafm.73.238.26231 Numerical Study of Double Diffusive Convection in a Lid Driven Cavity with Linearly Salted Side Walls N. Reddy and K. Murugesan† Mechanical and Industrial Engineering Department, Indian Institute of Technology Roorkee, Roorkee – 247 667, India †Corresponding Author Email: [email protected] (Received February 24, 2016; accepted July 26, 2016) ABSTRACT Double diffusive convection phenomenon is widely seen in process industries, where the interplay between thermal and solutal (mass) buoyancy forces play a crucial role in governing the outcome. In the current work, double diffusive convection phenomenon in a lid driven cavity model with linearly salted side walls has been studied numerically using Finite element simulations. Top and bottom walls of the cavity are assumed cold and hot respectively while other boundaries are set adiabatic to heat and mass flow. The calculations of energy and momentum transport in the cavity is done using velocity-vorticity form of Navier-Stokes equations consisting of velocity Poisson equations, vorticity transport, energy and concentration equations. Galerkin’s weighted residual method has been implemented to approximate the governing equations. Simulation results are obtained for convective heat transfer for 100<Re<500, 50<N<50 and 0.1<Ri<3.0. The average Nusselt number along the hot wall of the cavity is observed to be higher for higher Richardson number when buoyancy ratio is positive and vice versa. Maximum Nusselt number is recorded at buoyancy ratio 50 and Richardson number 3.0, on the other hand low Nusselt number is witnessed for buoyancy ratio 50.
    [Show full text]