Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 87
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Convective flow does not arise up to a Rayleigh number Ra, based on the layer thickness and temperature difference, of around 1700. TheBOOKCOMP, Inc. — John Wiley & Sons / Page 554 / 2nd Proofs / Heat Transfer Handbook / BejanINTERNAL NATURAL CONVECTION123456789101112131415161718192021222324252627282930313233343536373839404142434445555exact value was obtained as 1708 by several researchers, using stability analysis, asreviewed by Gebhart et al. (1988). Therefore, the Nusselt number based on thesecharacteristic quantities is 1.0 up to this value of Ra. As the Rayleigh number increases, convective flow arises and different flow regimes, including turbulent flow,and different instabilities have been studied in detail, as reviewed by Gebhart et al.(1988). Hollands et al.
(1975) presented experimental results and correlated their owndata and that from other studies for horizontal enclosures. Similarly, Hollands et al.(1976) presented correlations for inclined enclosures heated at the bottom.Natural convection in cylindrical, spherical, and annular cavities has also been ofinterest. Ostrach (1972) reviewed much of the work done on the flow inside horizontalcylindrical cavities. Kuehn and Goldstein (1976) carried out a detailed experimentaland numerical investigation of natural convection in concentric horizontal cylindricalannuli.
A thermosyphon, which is a fully or partially enclosed circulating fluid system[555], (31)driven by buoyancy, has been of interest in the cooling of gas turbines, electricalmachinery, nuclear reactors, and geothermal energy extraction. See, for instance,the review by Japikse (1973) and the detailed study by Mallinson et al. (1981).Lines: 827 to 835Partial enclosures are of importance in room ventilation, building fires, and coolingof electronic equipment. Several studies have been directed at the natural convection———flow arising in enclosures with openings and the associated thermal or mass transport0.927pt PgVar(Markatos et al., 1982; Abib and Jaluria, 1988).
Figure 7.15 shows the schematic———of the buoyancy-driven flow in an enclosure with an opening. Typical results forNormal Pageflow due to a thermal source on the left wall are shown in Fig. 7.16. The numbers in * PgEnds: Ejectthe figure indicate dimensionless stream function and temperature values. A strongstable stratification, with hot fluid overlying colder fluid, is observed. Many other such[555], (31)natural convection flows and the resulting transport processes have been investigatedin the literature.Figure 7.15 Buoyancy-driven flow due to a thermal energy source in an enclosure with anopening.BOOKCOMP, Inc.
— John Wiley & Sons / Page 555 / 2nd Proofs / Heat Transfer Handbook / Bejan123456789101112131415161718192021222324252627282930313233343536373839404142434445556BOOKCOMP, Inc. — John Wiley & Sons / Page 556 / 2nd Proofs / Heat Transfer Handbook / BejanFigure 7.16 Calculated streamlines and isotherms for the buoyancy-driven flow in an enclosure with an opening at (a) Ra = 105 and (b) Ra = 106,where Ra is based on the thermal source height and temperature difference from the ambient.
(From Abib and Jaluria, 1988.)[556], (32)Lines: 835 to 844———* 528.0pt PgVar———Normal Page* PgEnds: PageBreak[556], (32)TURBULENT FLOW1234567891011121314151617181920212223242526272829303132333435363738394041424344455577.6 TURBULENT FLOW7.6.1 Transition from Laminar Flow to Turbulent FlowOne of the most important questions to be answered in convection is whether theflow is laminar or turbulent, since the transport processes depend strongly on the flowregime. Near the leading edge of a surface, the flow is well ordered and well layered.The fluctuations and disturbances, if any, are small in magnitude compared to themean flow.
The processes can be defined in terms of the laminar governing equations,as discussed earlier. However, as the flow proceeds downstream from the leadingedge, it undergoes transition to turbulent flow, which is characterized by disturbancesof large magnitude. The flow may then be considered as a combination of a meanvelocity and a fluctuating component. Statistical methods can generally be used todescribe the flow field. In several natural convection flows of interest, particularly inindustrial applications, the flow lies in the unstable regime or in the transition regime.In a study of the transition of laminar flow to turbulence, it is necessary to determine the conditions under which a disturbance in the flow amplifies as it proceedsdownstream.
This involves a consideration of the stability of the flow, an unstablecircumstance leading to disturbance growth. These disturbances enter the flow fromvarious sources, such as building vibrations, fluctuations in heat input, and vibrations in equipment. Depending on the conditions in terms of frequency, location, andmagnitude of the input disturbances, they may grow in amplitude due to a balanceof buoyancy, pressure, and viscous forces. This form of instability, termed hydrodynamic stability, leads to disturbance growth.The disturbances gradually amplify to large enough magnitudes to cause significant nonlinear effects and secondary mean flows which distort the mean velocity andtemperature profiles. This leads to the formation of a shear layer, which fosters further amplification of the disturbances, and concentrated turbulent bursts arise.
Thesebursts then increase in magnitude and in the fraction of time they occur, eventuallycrowding out the remaining laminar flow and giving rise to a completely turbulentflow. The general mechanisms underlying transition are shown in Fig. 7.17 from thework of Jaluria and Gebhart (1974). Several books on flow stability, such as those byChandrasekhar (1961) and Drazin and Reid (1981), are available and may be consulted for further information on the underlying mechanisms.7.6.2 TurbulenceMost natural convection flows of interest in nature and in technology are turbulent.The velocity, pressure, and temperature at a given point in these flows do not remainconstant with time but vary irregularly at relatively high frequency. There is considerable amount of bulk mixing, with fluid packets moving around chaotically, givingrise to the fluctuations observed in the velocity and temperature fields rather thanthe well-ordered and well-layered flow characteristic of the laminar regime.
Due tothe importance of turbulent natural convection flows, a considerable amount of effort, both experimental and analytical, has been directed at understanding the basicBOOKCOMP, Inc. — John Wiley & Sons / Page 557 / 2nd Proofs / Heat Transfer Handbook / Bejan[557], (33)Lines: 844 to 866———5.7pt PgVar———Normal PagePgEnds: TEX[557], (33)558123456789101112131415161718192021222324252627282930313233343536373839404142434445NATURAL CONVECTION[558], (34)Lines: 866 to 878———0.42099pt PgVar———Normal PagePgEnds: TEX[558], (34)Figure 7.17 Growth of the boundary layer and the sequence of events during transition innatural convection flow over a vertical surface in water, Pr = 6.7.
(From Jaluria and Gebhart,1974.)mechanisms and determining the transport rates. The work done in forced flows hasbeen more extensive, and in fact, much of our understanding of turbulent flows innatural convection is based on this work. The transport mechanisms in turbulent floware very different from those in laminar flow, and some of the basic considerationsare given here. For further information, books on turbulent transport, such as thoseby Hinze (1975) and Tennekes and Lumley (1972), may be consulted.In describing a turbulent flow, the fluctuating or eddy motion is superimposed ona mean motion. The flow may, therefore, be described in terms of the time-averagedvalues of the velocity components (denoted as ū, v̄, and w̄) and the disturbance orfluctuating quantities (u , v , and w ).
The instantaneous value of each of the velocitycomponents is then given asu = ū + uv = v̄ + v BOOKCOMP, Inc. — John Wiley & Sons / Page 558 / 2nd Proofs / Heat Transfer Handbook / Bejanw = w̄ + w (7.63)TURBULENT FLOW123456789101112131415161718192021222324252627282930313233343536373839404142434445559Similarly, the pressure and temperature in the flow may be written asp = p̄ + p T = T̄ + T (7.64)The time averages are found by integrating the local instantaneous value of theparticular quantity at a given point over a time interval that is long compared to thetime period of the fluctuations.
For steady turbulence, the time-averaged quantitiesdo not vary with time. Then, from the definition of the averaging process, the timeaverages of the fluctuating quantities are zero. For unsteady turbulence, the timeaveraged quantities are time dependent. Here we consider only the case of steadyturbulence, so that the average quantities are independent of time and allow the flowand the transport processes to be represented in terms of time-independent variables.If the instantaneous quantities defined by eqs. (63) and (64) are inserted into thegoverning continuity, momentum, and energy equations and a time average taken,additional transport terms, due to the turbulent eddies, arise.
An important conceptemployed for treating these additional transport components is that of eddy viscosityεM and diffusivity εH . Momentum and heat transfer processes may then be expressedas a combination of a molecular component and an eddy component.
This gives forū(y) and T (y)d ūτ= (ν + εM )ρdy(7.65)qd T̄= (α + εH )ρcpdy(7.66)where τ is the total shear stress and q is the total heat flux. For isotropic turbulence,εM and εH are independent of direction and are of the form −u v /(∂ ū/∂y) andv T /(∂ T̄ /∂y), respectively.If the preceding relationships for τ and q are introduced into the governing equations for the mean flow, obtained by time averaging the equations written for the totalinstantaneous flow in boundary layer form, we obtain∂ ū ∂ v̄+=0∂x∂y∂ ū∂∂ ū∂ ūū+ v̄= gβ(T̄ − T∞ ) +(ν + εM )∂x∂y∂y∂y∂ T̄∂ T̄∂ T̄∂ū+ v̄=(α + εH )∂x∂y∂y∂y(7.67)(7.68)(7.69)where the viscous dissipation and energy source terms are neglected. A replacementof ν and α in the laminar flow equations by ν + εM and α + εH , respectively, yields thegoverning equations for turbulent flow.
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