International Journal of Rotating Machinery (C) 1998 OPA (Overseas Publishers Association) 1998, Vol. 4, No. 1, pp. 35-48 Amsterdam B.V. Published under license Reprints available directly from the publisher under the Gordon and Breach Science Photocopying permitted by license only Publishers imprint. Printed in India. Turbulent Flow and Heat Transfer in Circular Couette Flows in Concentric Annulus SHUICHI TORII a,. and WEN-JEI YANG b Department of Mechanical Engineering, Kagoshima University, 1-21-40 Korimoto, Kagoshima 890, Japan," Department of Mechanical Engineering and Applied Mechanics, University of Michigan, Ann Arbor, MI 48109, USA (Received in final form 5 September 1996) A numerical study is performed to investigate heat transfer and fluid flow in the hydro- dynamically and thermally fully-developed region of an annulus, consisting of a heated rotating inner cylinder and a stationary insulated outer cylinder. Emphasis is placed on the effect of rotation of an inner core on the flow structure and the thermal field. A Reynolds stress turbulence model is employed to determine three normal components of the Reynolds stress and its off-diagonal one. The turbulent heat flux is expressed by Boussinesq approximation in which the eddy diffusivity.for heat is given as functions of the temperature variance 7 and the dissipation rate of temperate fluctuations ct. The governing boundary- layer equations are discretized by means of a control volume finite-difference technique and numerically solved using the marching procedure. An inner core rotation causes an amplification of the three normal components of the Reynolds stress over the whole cross section, resulting in a substantial enhancement in the Nusselt number. Keywords." Circular Couette flow, Reynolds stress model, Taylor number, Two-equation heat transfer model INTRODUCTION by the centrifugal force induced by the swirl. In other words, the turbulent transfer of heat and The convective heat transfer in turbulent swirling momentum is suppressed or promoted by the flows is often encountered in chemical and mechan- interaction between turbulence and centrifugal ical mixing and separation devices, electrical and force associated with the swirl. turbo-machinery, combustion chambers, pollution Murakami and Kikuyama (1980) measured the control devices, swirl nozzles, rocketry, and fusion velocity profile and hydraulic loss in a hydrodynam- reactors. In these flow fields, the heat transport ically fully-developed flow region of a rotating phenomena in connection with the flow and the pipe. It was disclosed that both turbulence and turbulence properties are substantially influenced hydraulic loss were remarkably reduced due to pipe Corresponding author. Tel.: 099-285-8245. Fax: 099-285-8246. E-mail: [email protected]. 35 36 S. TORII AND WEN-JEI YANG rotation and that streamwise velocity profile causes an amplification of the turbulent kinetic gradually deformed into a parabolic form with an energy over the whole cross section, resulting in a increase in its speed. Kikuyama et al. (1983) substantial enhancement in the Nusselt number. analyzed the variation of streamwise velocity A k-e model, which is one of the two-equation profiles using a modified mixing length theory models of turbulence, is popular in computational proposed by Bradshaw (1969). A combined experi- analyses of turbulent flow. However, in the case of mental and theoretical study was performed by an annular duct flow, the importance of taking the Hirai and Takagi (1988) using the Reynolds stress effect of the term, -uv utOU/Or, into account is model to determine the effects of pipe rotation on recognized. This is because the time-averaged local fluid flow and heat transfer in a thermally and shear stress, -uv, does not go to zero at the radial hydrodynamically fully developed flow region. It location where the time-averaged streamwise veloc- was found that an increase in the rotation rate ity has its maximum value, i.e. a radial gradient of resulted in a decrease in heat transfer performance zero (Rehme, 1974). Furthermore, since the two- with the Nusselt number asymptotically approach- equation k-e model basically assumes isotropic ing that of a laminar pipe flow. Fluid. flow and heat turbulence structure, it cannot precisely reproduce transfer characteristics in the thermally and hydro- the anisotropy of turbulence caused by the inner dynamically developing and fully-developed regi- core rotation. In order to obtain the detailed ons of an axially rotating pipe were investigated by information pertinent to the flow structures, it is Torii and Yang (1995a,b) using the existing k-c necessary to employ the higher order closure turbulence models in which they are modified to model, i.e. a Reynolds stress turbulence model. include the swirling effect. Throughout the numerical simulations for the On the contrary, an effect of promoting the above turbulent heat transport problems in the swir- turbulent transport of heat and momentum by the ling flows, the turbulent heat flux in the energy equa- centrifugal force occurs in a concentric annulus tion is modeled by using the classical Boussinesq with an inner cylinder rotating around the axis, in approximation. The unknown turbulent thermal which Taylor vortices appear (Kuzay and Scott, diffusivity Oz is obtained from the definition of the 1975; 1976). Such swirl flow is referred to as known turbulent viscosity ut and the turbulent circular Couette flow, which implies a flow with Prandtl number Prt as Ot--- ut/Prt. In this formu- one surface rotating and the other stationary (or lation, an analogy between eddy diffusivities of both surfaces rotating in the same direction at momentum and heat is implicitly assumed. different angular velocities). Hirai et al. (1987) However, shear flow measurements (Hishida et al., conducted an experimental study on an effect 1986) and direct simulation data (Kasagi et al., of inner core rotation on turbulent transport of 1992) reveal that its analogy, as represented by the momentum by using a two-color laser Doppler turbulent Prandtl nu.mber, cannot adequately velocimeter. It was disclosed that the Reynolds reflect the physical phenomenon of heat transport stresses increase due to the swirl. Torii and Yang and there are no universal values of turbulent (1994) analyzed heat transfer mechanism in annuli Prandtl number even in simple flows. If the above with an inner core rotation by means of several turbulent Prandtl number assumption is employed different two-equation k-c turbulence models. It to determine the turbulent thermal diffusivity in was found that (i) in the entrace region, the axial annuli, it yields negative along the radial direction. rotation of the inner cylinder induces a thermal This is because the turbulent viscosity, ut, becomes development and causes an increase in both the negative in the radial region between velocity Nusselt number and the turbulent kinetic energy in gradient of zero and Reynolds stress of zero, as the inner cylinder wall region, and (ii) in the fully- mentioned previously. In order to solve this developed region, an increase in the Taylor number problem and to obtain detailed information on CIRCULAR COUETTE FLOW 37 the heat transport phenomena, two-equation is placed on the mechanism of an augmentation of model for thermal field and turbulent heat flux heat transfer performance due to the inner core equation model are considered to be employed. rotation. The heat flux equation model ought to be more universal, at least in principle. This model, how- ever, is still in intensive development and is little THEORETICAL ANALYSIS used for practical applications (for example, Lai and So, 1990). The results are not as satisfactory as Governing Equations initially expected, mainly due to a few unreasonable Consider a steady turbulent flow through a hypotheses in the model (Nagano and Tagawa, concentric annulus consisting of the insulated Thus, the turbulent heat flux model should 1988). stationary outer cylinder and the slightly heated wait until the Reynolds stress models are tested and inner cylinder rotating around the axis, in which well developed, preferably with the near-wall the boundary layer is developing both thermally since the source of error in heat modeling, major and hydrodynamically. The physical configuration transfer predictions is that in calculating the and the cylindrical coordinate system are shown in velocity field (Kasagi and Myong, 1989). Fig. 1. Some approximations are deduced that: (i) This paper treats the thermal transport phenom- viscous heating is negligible; (ii) the axial conduc- ena in a concentric annulus, in which a slightly tion term in the energy equation is neglected for heated inner core rotates around the axis and an Pe > 1; and (iii) the viscous dissipation term in the insulated outer cylinder is held stationary. In order energy equation is neglected. An order-of-magni- to shed light on the mechanism of the transport tude analysis indicates all second-derivative terms phenomena, the Reynolds stress turbulence model to be negligible in the streamwise and tangential proposed by Launder and Shima is employed, (1989) directions. The simplified governing equations read because this model is able to reproduce the inherent as follows: anisotropy in the near-wall region of isothermal Continuity equation flows. The turbulent thermal diffusivity Yt is deter- mined using the two-equation model for heat tran- OU + OV + V o. sport proposed by Yousseff et al. (1992). Emphasis o-; -r r, V, v outer cylinder (stationary, insulation) ,.!iiiiiii::ii::!ii!ii::iiii::iiiii::ii::ii!::iii!!i::i,!,i.:.:,::::!! ::,:::': //l inner cylinder heatln, O, W, w rotating a Ww FIGURE A schematic of physical system and coordinate. 38 S. TORII AND WEN-JEI YANG Momentum equations." x direction." OU dP + vOU + ru rVf (2) U-Ox Or p dx r direction." OP Or2 P W2 w 2 (3) Or -P--r + -r ( + ); 0 direction." --- OW vw (7) Ux + vOW-ffF +-r W2] 0 o(w/r) u 2 r3 (4) r Or Or r2--}.