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Parametric and Numerical Study of Fully-Developed Flow and Heat Transfer in Rotating Rectangular Ducts

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Parametric and Numerical Study of Fully-Developed Flow and Heat Transfer in Rotating Rectangular Ducts E AMEICA SOCIEY O MECAICA EGIEES 9-G- 35 E 7 S ew Yok Y 117 e Sociey sa o e esosie o saemes o oiios aace i aes o I is- cussio a meeigs o e Sociey o o is iisios o Secios o ie i is uicaios iscussio is ie oy i e ae is uise i a ASME oua aes ae aaiae ][ om ASME o iee mos ae e meeig ie i USA Copyright © 1990 by ASME Downloaded from http://asmedigitalcollection.asme.org/GT/proceedings-pdf/GT1990/79078/V004T09A005/2399934/v004t09a005-90-gt-024.pdf by guest on 27 September 2021 aameic a umeica Suy o uy-eeoe ow a ea ase i oaig ecagua ucs IACOIES a E AUE Mecaica Egieeig eame UMIS Macese M 1 UK ASAC Ui wall shear velocity V mean velocity in y direction This work is concerned with fully-developed constant-density W mean velocity in r direction turbulent flow through rectangular straight ducts rotating in an Wb bulk velocity orthogonal mode. Ducts of both square and 2:1 aspect ratio x cross-stream flow direction normal to the axis of cross-sections have been examined. For the square duct, rotation predictions have been performed for Reynolds numbers of 33,500 Y distance from the wall to the point in question and 97,000 and for the 2:1 aspect ratio duct the computations y cross-stream flow direction parallel to the axis of were carried out for a Reynolds number of 33,500. Values of the rotation inverse Rossby number (Ro = QD/Wb) ranged from 0.005 to 0.2. y+ non-dimensional wall distance Except in the immediate vicinity of the wall, the standard z streamwise flow direction high-Reynolds-number version of the k-c model is used to account for the effects of turbulence. Across the near-wall sublayer the E turbulent kinetic energy dissipation rate damping of turbulence is modelled through a low-Reynolds-number O temperature one-equation model. v kinematic viscosity Low rotational speeds cause the formation of a pair of vt turbulent kinematic viscosity symmetric streamwise vortices. At higher rotational speeds, flow p density instabilities on the pressure side lead to transition to a more a turbulent Prandtl number complex four-vortex structure. The transition point depends on S2 angular velocity of coordinate rotation both the cross-sectional geometry and the flow Reynolds number. Moreover, over a range of Rossby number, either two- or Suscis four-vortex solutions are possible depending upon initial 0 non-rotating reference state conditions. P pressure wall The rotation leads to significant differences between the S suction wall values of friction factor and Nusselt number on the suction and T top wall essue suaces o e uc e egee o ea ase augmentation on the pressure side is found to depend on the 1. INTRODUCTION Reynolds number as well as on Rossby number. In contrast, heat-transfer attenuation on the suction side is only Rossby-number The effects of rotation on the hydrodynamic and thermal eee characteristics of internal flows have considerable practical implications for the design of turbine-blade cooling passages. OMECAUE Cooling passages have irregular cross-sections and rotate in an orthogonal mode of rotation where the axis of rotation is normal Cf friction factor to the predominant flow direction. Under such circumstances the C1 ,c 2 ,C µ turbulence modelling constants interaction between the cross-stream components of the Coriolis D hydraulic diameter force and the slow-moving near-wall fluid generates a secondary H distance from the centre of rotation flow recirculation pattern. The near-wall fluid is driven from the k turbulent kinetic energy pressure side towards the suction side of the passage and the 4 turbulent length scale faster-moving core fluid moves towards the pressure side. This Nu Nusselt number onset of secondary flow leads to an increase in the average levels P pressure of friction factor and wall heat transfer rates around the perimeter Pk turbulent kinetic energy generation rate but also to substantial circumferential variations. In the case of Pr molecular Prandtl number incompressible rotating flows, the two dimensionless parameters that Re Reynolds number determine the character of the flow are the Reynolds number and Ro inverse Rossby number the inverse Rossby number, Ro = QD/Wb. U mean velocity to x direction esee a e Gas uie a Aeoegie Cogess a Eosiio—ue 11-1 199—usses egium is ae as ee accee o uicaio i e asacios o e ASME iscussio o i wi e accee a ASME eaquaes ui Seeme 3 199 Most of the earlier experimental work on rotating duct flows a au a ( aul concentrated on flow visualization tests (Hart, 1971) and on static + c + ay v c pressure and mean velocity measurements, Wagner and Vclkoff (1972), Hill and Moon (1962) and Moore (1967). More recently Morris and Harasgama (1985) provided fully-developed heat-transfer v av + a r y aVl data for a rotating square duct. Their data appear to show a (U + Ta y( P _ - a + aU7lcff^J ( ^lcff^f relatively small perimetral variations in local Nusselt-number. Guidez (1988), on the other hand, and also Wagner, Johnson and (ow)+ (vw) _ - p^ Hajek (1989) in their studies of developing flows through rotating I22Hz - 24U + (vc f f j rectangular ducts measured substantial local wall heat flux Downloaded from http://asmedigitalcollection.asme.org/GT/proceedings-pdf/GT1990/79078/V004T09A005/2399934/v004t09a005-90-gt-024.pdf by guest on 27 September 2021 variations. Numerical studies of fully-developed flow carried out a aw y e ese auos coceig o cicua (Iacoies a + a O Launder, 1987a) and square (lacovides and Launder, 1987b) coss-secioe ucs ae aso esue i e eicio o similarly strong local wall heat flux and shear stress variations. veff = v + vt and Hx and Hz are the distances from the centre Another interesting aspect of the square-duct computations was the of rotation in the x and z directions respectively. transition at high Rossby numbers from a symmetric double-vortex structure to a symmetric four-vortex structure. This phenomenon Continuity has also been observed in corresponding laminar-flow studies by Speziale (1982) and Kheshgi and Scriven (1985). A limited au a flow-visualization investigation carried out by Hart (1971), in which the motion of a set of dylines along a line normal to the duct Enthalpy Equation symmetry plane was observed, provides evidence which tends to confirm the existence of these flow instabilities in turbulent flow. a a a v v t ae a v ae In this work we report the results of a numerical parametric (U® + (a ) _ a- {( + study of fully-developed constant-density flow through rotating ea} + +I( ®} rectangular ducts. Ducts of square cross-section and of 2:1 aspect ratio have been considered. The value of the inverse Rossby number is progressively raised from 0 to 0.2. For the square Turbulence Modelling Equations duct, two sets of predictions have been obtained: one for a The turbulent viscosity vt is obtained by the following two Reynolds-number value of 32,500 and one for 97,500. For the eessios: 2:1 aspect ratio duct the computations were confined to a Reynolds-number value of 32,500. The objective is to assess the a in the high-Reynolds-number k-E region effects of Reynolds and Rossby number on the flow. Consideration is given to both the detailed flow structure and to vt = Cµk2 /E ; parameters of engineering interest, such as local and average Nusselt numbers and friction factors. • in the low-Reynolds-number one-equation region TURBULENCE MODELLING ASPECTS 2. Vt = C9Q µk The authors' earlier work on rotating duct flows has shown where Q = 2.4Y{l - exp-(0.016y )} that, in fully-developed constant-density cases, the use of an * algebraic second-moment closure (ASM) resulted in predictions Y similar to those obtained with a k-e effective-viscosity model, * = Iacovides and Launder (1987a and 1987b). For this reason, only k-e predictions have been carried out in this study. In the Y being the distance from the wall to the point in quesio near-wall regions, the authors' past experience in three-dimensional turbulent internal flows indicates that the use of the wall-function Turbulent Kinetic Energy (k) Transport Equation approximation is inappropriate (Choi, Iacovides and Launder, 1989). is as aso ee oe o e cocusios o i iey a a McDonald's (1985) computational investigation of developing flow a (Uk + a vt a ak u (k _ {( + 6 + } in rotating ducts. In our previous computations of rotating flows k Y {( v + 6k u Y- the use of Van Driest's mixing-length model across the wall sublayer, matched to a transport-based turbulence model outside the + Pk - E wall region has been felt to be reasonably satisfactory. Nevertheless, because of the strong near-wall secondary flows created in these flows and having regard for the sensitivity of the (aU av 2 ( 2 ( l (2 l 2 local friction factors and wall heat transfer rates on the model of where Pk = {l Y + ^^ + (TyJ + laxl + ETJY I vt turbulence in the near-wall sublayer, it was decided that Wolfshtein's (1969) low-Reynolds-number one-equation model would Where the k-E model is used, E is obtained from a separate be a more appropriate near-wall model choice. At this level aso equaio transport effects on the turbulent kinetic energy are accounted for. a a a ((I v t aE a (( V t aE a (UE + (E _ 3. EQUATIONS OF MOTION + } + O ( + 6 E I0 The equations of fluid motion are formulated for the rotating E E 2 + CE F k - C E2 Cartesian coordinate system shown in Figure 1.
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    Bibliography L. Prandtl: Selected Bibliography A. Sommerfeld. Zu L. Prandtls 60. Geburtstag am 4. Februar 1935. ZAMM, 15, 1–2, 1935. W. Tollmien. Zu L. Prandtls 70. Geburtstag. ZAMM, 24, 185–188, 1944. W. Tollmien. Seventy-Fifth Anniversary of Ludwig Prandtl. J. Aeronautical Sci., 17, 121–122, 1950. I. Fl¨ugge-Lotz, W. Fl¨ugge. Ludwig Prandtl in the Nineteen-Thirties. Ann. Rev. Fluid Mech., 5, 1–8, 1973. I. Fl¨ugge-Lotz, W. Fl¨ugge. Ged¨achtnisveranstaltung f¨ur Ludwig Prandtl aus Anlass seines 100. Geburtstags. Braunschweig, 1975. H. G¨ortler. Ludwig Prandtl - Pers¨onlichkeit und Wirken. ZFW, 23, 5, 153–162, 1975. H. Schlichting. Ludwig Prandtl und die Aerodynamische Versuchsanstalt (AVA). ZFW, 23, 5, 162–167, 1975. K. Oswatitsch, K. Wieghardt. Ludwig Prandtl and his Kaiser-Wilhelm-Institut. Ann. Rev. Fluid Mech., 19, 1–25, 1987. J. Vogel-Prandtl. Ludwig Prandtl: Ein Lebensbild, Erinnerungen, Dokumente. Uni- versit¨atsverlag G¨ottingen, 2005. L. Prandtl. Uber¨ Fl¨ussigkeitsbewegung bei sehr kleiner Reibung. Verhandlg. III. Intern. Math. Kongr. Heidelberg, 574–584. Teubner, Leipzig, 1905. Neue Untersuchungen ¨uber die str¨omende Bewegung der Gase und D¨ampfe. Physikalische Zeitschrift, 8, 23, 1907. Der Luftwiderstand von Kugeln. Nachrichten von der Gesellschaft der Wis- senschaften zu G¨ottingen, Mathematisch-Physikalische Klasse, 177–190, 1914. Tragfl¨ugeltheorie. Nachrichten von der Gesellschaft der Wissenschaften zu G¨ottingen, Mathematisch-Physikalische Klasse, 451–477, 1918. Experimentelle Pr¨ufung der Umrechnungsformeln. Ergebnisse der AVA zu G¨ottingen, 1, 50–53, 1921. Ergebnisse der Aerodynamischen Versuchsanstalt zu G¨ottingen. R. Oldenbourg, M¨unchen, Berlin, 1923. The Generation of Vortices in Fluids of Small Viscosity.
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