Lecture 6. Monin-Obukhov Similarity Theory (Garratt 3.3) in This Lecture
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Lecture 7. More on BL Wind Profiles and Turbulent Eddy Structures in This
Atm S 547 Boundary Layer Meteorology Bretherton Lecture 7. More on BL wind profiles and turbulent eddy structures In this lecture… • Stability and baroclinicity effects on PBL wind and temperature profiles • Large-eddy structures and entrainment in shear-driven and convective BLs. Stability effects on overall boundary layer structure Above the surface layer, the wind profile is also affected by stability. As we mentioned previously, unstable BLs tend to have much more well-mixed wind profiles than stable BLs. Fig. 1 shows observations from the Wangara experiment on how the velocity defects and temperature profile are altered by BL stability (as measured by H/L). Within stability classes, the velocity profiles collapse when scaled with a velocity scale u* and the observed BL depth H, but there is a large difference between the stability classes. Baroclinicity We would expect baroclinicity (vertical shear of geostrophic wind) to also affect the observed wind profile. This is most easily seen for a laminar steady-state Ekman layer in a geostrophic wind with constant vertical shear ug(z) = (G + Mz, Nz), where M = -(g/fT0)∂T/∂y, N = (g/fT0)∂T/∂x. The momentum equations and BCs are: -f(v - Nz) = ν d2u/dz2 f(u - G - Mz) = ν d2v/dz2 u(0) = 0, u(z) ~ G + Mz as z → ∞. v(0) = 0, v(z) ~ Nz as z → ∞. The resultant BL velocity profile is the classical Ekman layer with the thermal wind added onto it. u(z) = G(1 - e-ζ cos ζ) + Mz, (7.1) 1/2 v(z) = G e- ζ sin ζ + Nz. -
In Classical Fluid Dynamics, a Boundary Layer Is the Layer I
Atm S 547 Boundary Layer Meteorology Bretherton Lecture 1 Scope of Boundary Layer (BL) Meteorology (Garratt, Ch. 1) In classical fluid dynamics, a boundary layer is the layer in a nearly inviscid fluid next to a surface in which frictional drag associated with that surface is significant (term introduced by Prandtl, 1905). Such boundary layers can be laminar or turbulent, and are often only mm thick. In atmospheric science, a similar definition is useful. The atmospheric boundary layer (ABL, sometimes called P[lanetary] BL) is the layer of fluid directly above the Earth’s surface in which significant fluxes of momentum, heat and/or moisture are carried by turbulent motions whose horizontal and vertical scales are on the order of the boundary layer depth, and whose circulation timescale is a few hours or less (Garratt, p. 1). A similar definition works for the ocean. The complexity of this definition is due to several complications compared to classical aerodynamics. i) Surface heat exchange can lead to thermal convection ii) Moisture and effects on convection iii) Earth’s rotation iv) Complex surface characteristics and topography. BL is assumed to encompass surface-driven dry convection. Most workers (but not all) include shallow cumulus in BL, but deep precipitating cumuli are usually excluded from scope of BLM due to longer time for most air to recirculate back from clouds into contact with surface. Air-surface exchange BLM also traditionally includes the study of fluxes of heat, moisture and momentum between the atmosphere and the underlying surface, and how to characterize surfaces so as to predict these fluxes (roughness, thermal and moisture fluxes, radiative characteristics). -
Turbulent-Prandtl-Number.Pdf
Atmospheric Research 216 (2019) 86–105 Contents lists available at ScienceDirect Atmospheric Research journal homepage: www.elsevier.com/locate/atmosres Invited review article Turbulent Prandtl number in the atmospheric boundary layer - where are we T now? ⁎ Dan Li Department of Earth and Environment, Boston University, Boston, MA 02215, USA ARTICLE INFO ABSTRACT Keywords: First-order turbulence closure schemes continue to be work-horse models for weather and climate simulations. Atmospheric boundary layer The turbulent Prandtl number, which represents the dissimilarity between turbulent transport of momentum and Cospectral budget model heat, is a key parameter in such schemes. This paper reviews recent advances in our understanding and modeling Thermal stratification of the turbulent Prandtl number in high-Reynolds number and thermally stratified atmospheric boundary layer Turbulent Prandtl number (ABL) flows. Multiple lines of evidence suggest that there are strong linkages between the mean flowproperties such as the turbulent Prandtl number in the atmospheric surface layer (ASL) and the energy spectra in the inertial subrange governed by the Kolmogorov theory. Such linkages are formalized by a recently developed cospectral budget model, which provides a unifying framework for the turbulent Prandtl number in the ASL. The model demonstrates that the stability-dependence of the turbulent Prandtl number can be essentially captured with only two phenomenological constants. The model further explains the stability- and scale-dependences -
A Two-Layer Turbulence Model for Simulating Indoor Airflow Part I
Energy and Buildings 33 +2001) 613±625 A two-layer turbulence model for simulating indoor air¯ow Part I. Model development Weiran Xu, Qingyan Chen* Department of Architecture, Massachusetts Institute of Technology, 77 Massachusetts Avenue, MA 02139, USA Received 2 August 2000; accepted 7 October 2000 Abstract Energy ef®cient buildings should provide a thermally comfortable and healthy indoor environment. Most indoor air¯ows involve combined forced and natural convection +mixed convection). In order to simulate the ¯ows accurately and ef®ciently, this paper proposes a two-layer turbulence model for predicting forced, natural and mixed convection. The model combines a near-wall one-equation model [J. Fluid Eng. 115 +1993) 196] and a near-wall natural convection model [Int. J. Heat Mass Transfer 41 +1998) 3161] with the aid of direct numerical simulation +DNS) data [Int. J. Heat Fluid Flow 18 +1997) 88]. # 2001 Published by Elsevier Science B.V. Keywords: Two-layer turbulence model; Low-Reynolds-number +LRN); k±e Model +KEM) 1. Introduction This paper presents a new two-layer turbulence model that performs two tasks: 1.1. Objectives 1. It can accurately predict flows under various conditions, i.e. from purely forced to purely natural convection. This Energy ef®cient buildings should provide a thermally model allows building ventilation designers to use one comfortable and healthy indoor environment. The comfort single model to calculate flows instead of selecting and health parameters in indoor environment include the different turbulence models empirically from many distributions of velocity, air temperature, relative humidity, available models. and contaminant concentrations. These parameters can be 2. -
Turbulent Prandtl Number and Its Use in Prediction of Heat Transfer Coefficient for Liquids
Nahrain University, College of Engineering Journal (NUCEJ) Vol.10, No.1, 2007 pp.53-64 Basim O. Hasan Chemistry. Engineering Dept.- Nahrain University Turbulent Prandtl Number and its Use in Prediction of Heat Transfer Coefficient for Liquids Basim O. Hasan, Ph.D Abstract: eddy conductivity is unspecified in the case of heat transfer. The classical approach for obtaining the A theoretical study is performed to determine the transport mechanism for the heat transfer problem turbulent Prandtl number (Prt ) for liquids of wide follows the laminar approach, namely, the momentum range of molecular Prandtl number (Pr=1 to 600) and thermal transport mechanisms are related by a under turbulent flow conditions of Reynolds number factor, the Prandtl number, hence combining the range 10000- 100000 by analysis of experimental molecular and eddy viscosities one obtain the momentum and heat transfer data of other authors. A Boussinesq relation for shear stress: semi empirical correlation for Prt is obtained and employed to predict the heat transfer coefficient for du the investigated range of Re and molecular Prandtl ( ) 1 number (Pr). Also an expression for momentum eddy m dy diffusivity is developed. The results revealed that the Prt is less than 0.7 and is function of both Re and Pr according to the following relation: and the analogous relation for heat flux: Prt=6.374Re-0.238 Pr-0.161 q dT The capability of many previously proposed models of ( h ) 2 Prt in predicting the heat transfer coefficient is c p dy examined. Cebeci [1973] model is found to give good The turbulent Prandtl number is the ratio between the accuracy when used with the momentum eddy momentum and thermal eddy diffusivities, i.e., Prt= m/ diffusivity developed in the present analysis. -
ESCI 485 – Air/Sea Interaction Lesson 3 – the Surface Layer References
ESCI 485 – Air/sea Interaction Lesson 3 – The Surface Layer References: Air-sea Interaction: Laws and Mechanisms , Csanady Structure of the Atmospheric Boundary Layer , Sorbjan THE PLANETARY BOUNDARY LAYER The atmospheric planetary boundary layer (PBL) is that region of the atmosphere in which turbulent fluxes are not negligible. Its depth can vary depending on the stability of the atmosphere. ο In deep convection the PBL can be of the same order as the entire troposphere. ο Usually it is of the order of 1 km or so. The PBL can be further broken down into three layers ο The mixed layer – the upper 90% or so of the PBL in which eddies have nearly mixed properties such as potential temperature, momentum, and moisture. ο The surface layer – The lowest 10% or so of the PBL in which the turbulent fluxes dominate all other terms (Coriolis and PGF) in the momentum equations. ο The viscous sub-layer – A very thin layer (order of cm or less) right near the surface where viscous effects may be important. THE SURFACE LAYER In the surface layer the turbulent momentum flux dominates all other terms in the momentum equations. The surface layer extends upwards on the order of 10 m or so. The Reynolds fluxes (stresses) in the surface layer can be assumed constant with height, and have the same magnitude as the interfacial stress (the stress between the air and water). In the surface layer, the eddy length scale is proportional to the height, z, above the surface. Based on the similarity principle , the following dimensionless group is formed from the friction velocity, eddy length scale, and the mean wind shear z dU = B , (1) u* dz where B is a dimensionless constant. -
The Atmospheric Boundary Layer (ABL Or PBL)
The Atmospheric Boundary Layer (ABL or PBL) • The layer of fluid directly above the Earth’s surface in which significant fluxes of momentum, heat and/or moisture are carried by turbulent motions whose horizontal and vertical scales are on the order of the boundary layer depth, and whose circulation timescale is a few hours or less (Garratt, p. 1). A similar definition works for the ocean. • The complexity of this definition is due to several complications compared to classical aerodynamics: i) Surface heat exchange can lead to thermal convection ii) Moisture and effects on convection iii) Earth’s rotation iv) Complex surface characteristics and topography. Atm S 547 Lecture 1, Slide 1 Sublayers of the atmospheric boundary layer Atm S 547 Lecture 1, Slide 2 Applications and Relevance of BLM i) Climate simulation and NWP ii) Air Pollution and Urban Meteorology iii) Agricultural meteorology iv) Aviation v) Remote Sensing vi) Military Atm S 547 Lecture 1, Slide 3 History of Boundary-Layer Meteorology 1900 – 1910 Development of laminar boundary layer theory for aerodynamics, starting with a seminal paper of Prandtl (1904). Ekman (1905,1906) develops his theory of laminar Ekman layer. 1910 – 1940 Taylor develops basic methods for examining and understanding turbulent mixing Mixing length theory, eddy diffusivity - von Karman, Prandtl, Lettau 1940 – 1950 Kolmogorov (1941) similarity theory of turbulence 1950 – 1960 Buoyancy effects on surface layer (Monin and Obuhkov, 1954). Early field experiments (e. g. Great Plains Expt. of 1953) capable of accurate direct turbulent flux measurements 1960 – 1970 The Golden Age of BLM. Accurate observations of a variety of boundary layer types, including convective, stable and trade- cumulus. -
Huq, P. and E.J. Stewart, 2008. Measurements and Analysis of the Turbulent Schmidt Number in Density
GEOPHYSICAL RESEARCH LETTERS, VOL. 35, L23604, doi:10.1029/2008GL036056, 2008 Measurements and analysis of the turbulent Schmidt number in density stratified turbulence Pablo Huq1 and Edward J. Stewart2 Received 18 September 2008; revised 27 October 2008; accepted 3 November 2008; published 3 December 2008. [1] Results are reported of measurements of the turbulent where rw is the buoyancy flux, uw is the Reynolds shear Schmidt number Sct in a stably stratified water tunnel stress. The ratio Km/Kr is termed the turbulent Schmidt experiment. Sct values varied with two parameters: number Sct (rather than turbulent Prandtl number) as Richardson number, Ri, and ratio of time scales, T*, of salinity gradients in water are used for stratification in the eddy turnover and eddy advection from the source of experiments. Prescription of a functional dependence of Sct turbulence generation. For large values (T* 10) values on Ri = N2/S2 is a turbulence closure scheme [Kantha and of Sct approach those of neutral stratification (Sct 1). In Clayson, 2000]. Here the buoyancy frequency N is defined 2 contrast for small values (T* 1) values of Sct increase by the gradient of density r as N =(Àg/ro)(dr/dz), and g is with Ri. The variation of Sct with T* explains the large the gravitational acceleration. S =dU/dz is the velocity scatter of Sct values observed in atmospheric and oceanic shear. Analyses of field, lab and numerical (RANS, DNS data sets as a range of values of T* occur at any given Ri. and LES) data sets have led to the consensus that Sct The dependence of Sct values on advective processes increases with Ri [see Esau and Grachev, 2007, and upstream of the measurement location complicates the references therein]. -
A Review of Ocean/Sea Subsurface Water Temperature Studies from Remote Sensing and Non-Remote Sensing Methods
water Review A Review of Ocean/Sea Subsurface Water Temperature Studies from Remote Sensing and Non-Remote Sensing Methods Elahe Akbari 1,2, Seyed Kazem Alavipanah 1,*, Mehrdad Jeihouni 1, Mohammad Hajeb 1,3, Dagmar Haase 4,5 and Sadroddin Alavipanah 4 1 Department of Remote Sensing and GIS, Faculty of Geography, University of Tehran, Tehran 1417853933, Iran; [email protected] (E.A.); [email protected] (M.J.); [email protected] (M.H.) 2 Department of Climatology and Geomorphology, Faculty of Geography and Environmental Sciences, Hakim Sabzevari University, Sabzevar 9617976487, Iran 3 Department of Remote Sensing and GIS, Shahid Beheshti University, Tehran 1983963113, Iran 4 Department of Geography, Humboldt University of Berlin, Unter den Linden 6, 10099 Berlin, Germany; [email protected] (D.H.); [email protected] (S.A.) 5 Department of Computational Landscape Ecology, Helmholtz Centre for Environmental Research UFZ, 04318 Leipzig, Germany * Correspondence: [email protected]; Tel.: +98-21-6111-3536 Received: 3 October 2017; Accepted: 16 November 2017; Published: 14 December 2017 Abstract: Oceans/Seas are important components of Earth that are affected by global warming and climate change. Recent studies have indicated that the deeper oceans are responsible for climate variability by changing the Earth’s ecosystem; therefore, assessing them has become more important. Remote sensing can provide sea surface data at high spatial/temporal resolution and with large spatial coverage, which allows for remarkable discoveries in the ocean sciences. The deep layers of the ocean/sea, however, cannot be directly detected by satellite remote sensors. -
Nasa L N D-6439 Similar Solutions for Turbulent
.. .-, - , . .) NASA TECHNICAL NOTE "NASA L_N D-6439 "J c-I I SIMILAR SOLUTIONS FOR TURBULENT BOUNDARY LAYER WITH LARGE FAVORABLEPRESSURE GRADIENTS (NOZZLEFLOW WITH HEATTRANSFER) by James F. Schmidt, Don& R. Boldmun, und Curroll Todd NATIONALAERONAUTICS AND SPACE ADMINISTRATION WASHINGTON,D. C. AUGUST 1971 i i -~__.- 1. ReportNo. I 2. GovernmentAccession No. I 3. Recipient's L- NASA TN D-6439 . - .. I I 4. Title andSubtitle SIMILAR SOLUTIONS FOR TURBULENT 5. ReportDate August 1971 BOUNDARY LAYERWITH LARGE FAVORABLE PRESSURE 1 6. PerformingOrganization Code GRADIENTS (NOZZLE FLOW WITH HEAT TRANSFER) 7. Author(s) 8. Performing Organization Report No. James F. Schmidt,Donald R. Boldman, and Carroll Todd I E-6109 10. WorkUnit No. 9. Performing Organization Name and Address 120-27 Lewis Research Center 11. Contractor Grant No. National Aeronautics and Space Administration I Cleveland, Ohio 44135 13. Type of Report and Period Covered 2. SponsoringAgency Name and Address Technical Note National Aeronautics and Space Administration 14. SponsoringAgency Code C. 20546Washington, D. C. I 5. SupplementaryNotes " ~~ 6. Abstract In order to provide a relatively simple heat-transfer prediction along a nozzle, a differential (similar-solution) analysis for the turbulent boundary layer is developed. This analysis along with a new correlation for the turbulent Prandtl number gives good agreement of the predicted with the measured heat transfer in the throat and supersonic regionof the nozzle. Also, the boundary-layer variables (heat transfer, etc. ) can be calculated at any arbitrary location in the throat or supersonic region of the nozzle in less than a half minute of computing time (Lewis DCS 7094-7044). -
Ocean Near-Surface Boundary Layer: Processes and Turbulence Measurements
REPORTS IN METEOROLOGY AND OCEANOGRAPHY UNIVERSITY OF BERGEN, 1 - 2010 Ocean near-surface boundary layer: processes and turbulence measurements MOSTAFA BAKHODAY PASKYABI and ILKER FER Geophysical Institute University of Bergen December, 2010 «REPORTS IN METEOROLOGY AND OCEANOGRAPHY» utgis av Geofysisk Institutt ved Universitetet I Bergen. Formålet med rapportserien er å publisere arbeider av personer som er tilknyttet avdelingen. Redaksjonsutvalg: Peter M. Haugan, Frank Cleveland, Arvid Skartveit og Endre Skaar. Redaksjonens adresse er : «Reports in Meteorology and Oceanography», Geophysical Institute. Allégaten 70 N-5007 Bergen, Norway RAPPORT NR: 1- 2010 ISBN 82-8116-016-0 2 CONTENT 1. INTRODUCTION ................................................................................................2 2. NEAR SURFACE BOUNDARY LAYER AND TURBULENCE MIXING .............3 2.1. Structure of Upper Ocean Turbulence........................................................................................ 5 2.2. Governing Equations......................................................................................................................... 6 2.2.1. Turbulent Kinetic Energy and Temperature Variance .................................................................. 6 2.2.2. Relevant Length Scales..................................................................................................................... 8 2.2.3. Relevant non-dimensional numbers............................................................................................... -
Stability and Instability of Hydromagnetic Taylor-Couette Flows
Physics reports Physics reports (2018) 1–110 DRAFT Stability and instability of hydromagnetic Taylor-Couette flows Gunther¨ Rudiger¨ a;∗, Marcus Gellerta, Rainer Hollerbachb, Manfred Schultza, Frank Stefanic aLeibniz-Institut f¨urAstrophysik Potsdam (AIP), An der Sternwarte 16, D-14482 Potsdam, Germany bDepartment of Applied Mathematics, University of Leeds, Leeds, LS2 9JT, United Kingdom cHelmholtz-Zentrum Dresden-Rossendorf, Bautzner Landstr. 400, D-01328 Dresden, Germany Abstract Decades ago S. Lundquist, S. Chandrasekhar, P. H. Roberts and R. J. Tayler first posed questions about the stability of Taylor- Couette flows of conducting material under the influence of large-scale magnetic fields. These and many new questions can now be answered numerically where the nonlinear simulations even provide the instability-induced values of several transport coefficients. The cylindrical containers are axially unbounded and penetrated by magnetic background fields with axial and/or azimuthal components. The influence of the magnetic Prandtl number Pm on the onset of the instabilities is shown to be substantial. The potential flow subject to axial fieldspbecomes unstable against axisymmetric perturbations for a certain supercritical value of the averaged Reynolds number Rm = Re · Rm (with Re the Reynolds number of rotation, Rm its magnetic Reynolds number). Rotation profiles as flat as the quasi-Keplerian rotation law scale similarly but only for Pm 1 while for Pm 1 the instability instead sets in for supercritical Rm at an optimal value of the magnetic field. Among the considered instabilities of azimuthal fields, those of the Chandrasekhar-type, where the background field and the background flow have identical radial profiles, are particularly interesting.