1、. . sNATIONALADVISORYCOMMITTEEFOR AERONAUTICSTECHNICAL NOTE 4323NATURAL CONVECTION INSIDE A FMT ROTATINGBy Simon Ostrach and Willis H. BraunLewis Flight Propulsion LaboratoryCleveland, OhioCONTAINERWashingtonSeptember 1958Provided by IHSNot for ResaleNo reproduction or networking permitted without l
2、icense from IHS-,-,-.#- -NACA TECET?ICALNOTE 4323By Simon Ostrach and Willis H. Braun8eptember 1958Page 6, eqmtion (10): The second term on the right side should befthat is, the heating in the stagnationregion imposes a negative temperature gradient pars31el to the retarda-tion force. In such a conf
3、igurationthe fluid remains at rest until acrical value of the Rayleigh number (PrGr) is attained (see ref. 3).Thus, even for appreciable retardation forues, the internal fluid is in-effective as a coolant. Therefore, a large body force (such as a cen-trifugal force) transverse to the temperature gra
4、dientmight increase theeffectivenessof the heat sink by starting the motion sooner. Also, therotationaleffects must be evaluatedbecause rotation of the vehicle maybe used for aerodynamic stability and, therefore,be inherent to thesystem.Specific considerationis given in this report to the flow and h
5、eattransfer of a fluid subject to an axial body force inside a rotating-right circular cylinder of small height (see fig. 3). In this wqy, therotational effects can be evaluated, and internal conditions in the vi-cinity of the external stagnationregion of blunt bodies are simulated.In order to avoid
6、 the complications introducedby unsteady effects,weshsU assmne that the retardation force is constant *Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-.lVACATN 4323 3and (2) where the deceleration force predominates, that is, where a flowis generated
7、by the sxial body force with heating irrespectiveof whetherthe container is rotating.The governing equationsANALYSISfor the coolsat are those ressing theconservation of mass, momentum, smd energy in cylindrical coordinatesfor compressible,viscous, amd heat-conducting uids stiject to a bodyforce. Wit
8、h fluid properties constant, these equations are, respectively:Continuity:(1)Radial momentum:Azimuthal momentum:av ( avvavp- m=. (m;zrotation of the fluid when there is no heatingin e, that is, the perturbations from solid-(7)body rotation due to heating, equations (1) to 5) arein fact, from the def
9、initions ofthe Reynolds and Grashof nmbers based on -Q, the reciprocal of the Froudenumber is just e. Therefore, the dominant inertia terms are the buoyancy and Coriolis terms, that is, the last two terms on the left side, respec-tively, of eqpation (8) and the last one on the left in equation (9).
10、Inthe energy eqution the convection and frictionalheating terms are neg-ligible with respect to the conduction term so that the heat transfer isnot affected by the motion and is merely due to conduction. All this im-plies, of course, that large velocities csrmot be obtainedby applying a - , _tempera
11、ture gradient transverse to the centrifugalbody force in such arotating configuration. To understand this seemingly unusual result, letus solve the equations that result by omitting the negligible terms andassming h/Rcases.Equationresctively,c 0 IEquations (Ill)are appropriate for a flow which incre
12、aseswith externalheating =d reduces to solid-body rotation when there is no heating.Substituting into the governing equations (1) to (5), letting h/R 0,and neglecting compressibilityand dissipation terms yielde(u + ) - e2n g + er - #2v = -Gzpr + d%elzP+n(uvr+Wvz + uv/r) + em2u = #R&z1(B2)Pz=oIn the
13、first of equations (B2), no term canbe of greater order of mag-nitude than the driving force er. Hence, m = n = 2 = 1. Then equations(Bl)become identicsl to (6) and equations (B2)becomeC(U% + WUz) + rr- 2V = -pr + Rlze(uvr + wvz + uv/r) + 2U = Re%zz I (B3)If smy value other than 1 is chosen for the
14、indices m and n, theCoriolis term in either one or the other of equations (B3)will dominate,and a null solutionwill result. me energy equation becomesa-.(c/T(UTr + WTZ) = (I%Re&zz (B4)Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN 4323 19Fro
15、m the definitionsOh2ReQ . , a?14GrQ = it follows thatThis accounts for tie form of equations (8) ta (U).1. Schmidt, Ernst H. W.: Heat Trsnsmission by Natursl Convection ata High Centrifugal Accelezation in Water-Cooled Gas-Turbine Blades.8P Proc. Gen. Discussion on Heat Transfer, H, Sept. 11-13, 195
16、1.i!i 2. Allen, H. Julian, and Eggers, A. J., Jr.: A Study of the Motion andAerodynamic Heating of Missiles Entering the Earths Atmosphere atHigh Sqersonic Speeds. NACA TN”4047, 1957. (SupersedesNACARMA53D28.)3. Rayleigh: On Convection Currents in a Horizontal Layer of Fluid,when the Higher Temperat
17、ure is on the Under Side. Phil.Mag. andJour. Sci., ser. 6, vol. 32, no. 192, Dec. 1916, pp. 529-547.4. Davies, T. V.: The Flow of a Liquid Which is Heated from Below. Sci.Rep. 2, Dept. Meteorolo, M.I.T., Apr. 1, 1952.5. Chandrasekhar, S.: The Instability of a Layer of Fluid Heated Belowsmd Subject t
18、o Coriolis Forces. Rroc. Roy. Soc. (Lendon), ser. A,vol. 217, May 7, 1953, pp. 306-327.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-20 NACA TN 4323 .1.2IH/ A/&.8/.4 iAO* 8 6 4 2 0Dhensionless altitudeFigure 1. - Heat flux and decelerationas functi
19、onof altitudefor a representativevehicle.II/Externalstagnationregionreentry-type.1+$a.a4+f Retaxdingforceon externalshell1If Apparentbodyforce on internalmetal shield.Figure 2. - Cooling shieldin a deceleratingnose cone.Provided by IHSNot for ResaleNo reproduction or networking permitted without lic
20、ense from IHS-,-,-, ,z = 1/2 ,hz = -1/2+,Figure 3. - schematic(a)Flat circ- cyltiaer.sketchof configurationfor floww in rotatingcontainers. NJPProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-.TO - (Tl -Il-J&J&(b) Sector or a rotatm cyl-r.z a 1/2z = -
21、1/2Ffgure 3. - Ccmcl. Schematic sketch of configumtion for flmm inI-Otating cOntafiera.a71b .II ,IProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-.U(z).10.0 40010.06.020,.02.04.06.08.105 4 -.3 -.2 -.1 0 .1 .2 .3 .4 .5-%(a)&_e&B.Figure4. - Velocitypro
22、filesinrotatingcylfir.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-24 NACA TN 4323v(z).24”.16.080.08.16.245-. -. 4 -. 3 -. 2 -. 10 .1 .2 .3 .4 .5z.(b) Azimuthal component.Figure 4. - Continued. Velocity profiles in rotating cylinder.Provided by IH
23、SNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-a71a1525T?(z).05.04.03.02/.01 /o-.5 -.4 3 2 -.1 (-zRea400 .1 .2 .3 .4 .(c) Axial component.Figure 4. - Concluded. Velocity profiles in rotating cylinder.Provided by IHSNot for ResaleNo reproduction or networking perm
24、itted without license from IHS-,-,-.-. %To- (T1 -To).“” Tz = 1/21 z = -1/2F-e 5. - Coo.limtiflow loop hv .I.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN 4323 27Vector velocityCoriolis forceFigure 6. - Coriolis force in upper half of rotatingcylinder.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
