Investigation of transitional and turbulent heat and momentum transport in a rotating cavity

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Abstract

The paper gives the results of the DNS/LES which was performed to investigate the transitional and turbulent non-isothermal flows within a rotor/stator cavity. Computations were performed for the cavity of aspect ratio L = 2–35, Rm = 1.8 and for rotational Reynolds numbers up to 290000. The main purpose of the investigations was to analyze the influence of aspect ratio and Reynolds number on the flow structure and heat transfer. The numerical solution is based on a pseudo-spectral Chebyshev–Fourier–Galerkin collocation approximation. The time scheme is semi-implicit second-order accurate, which combines an implicit treatment of the diffusive terms and an explicit Adams–Bashforth extrapolation for the non-linear convective terms. In the paper we analyze distributions of the Reynolds stress tensor components, the turbulent heat flux tensor components, Nusselt number distributions and the turbulent Prandtl number and other structural parameters, which can be useful for modeling purposes. Selected results are compared with the experimental data obtained for single heated rotating disk by Elkins and Eaton (2000).

Highlights

► The turbulence is concentrated at the junction between outer cylinder and stator. ► Selected structural parameters are more universal than turbulent heat flux components. ► Agreement with 2D boundary layer results near the wall.

Introduction

The instability structures of the flow in the rotor/stator and rotor/rotor cavity have been investigated since the 1960s of the last century, mostly with reference to the applications in turbomachinery (Owen and Rogers, 1989, Owen and Rogers, 1995). The flow between the rotor and stator is also an interesting fundamental problem, the study of which allows to investigate the influence of mean flow parameters on the strongly 3D boundary layers. The flow in rotor/stator cavity was investigated experimentally and numerically among others by Schouveiler et al., 2001, Serre and Pulicani, 2001, Serre et al., 2004, Lygren and Andersson, 2004, Poncet and Schiestel, 2007, Séverac et al., 2007, Moisy et al., 2004, Séverac and Serre, 2007. Séverac and Serre (2007) used Spectral Vanishing Viscosity (SVV) method and compared the results with their LDV experimental data; SVV technique turned out to be very effective numerical tool for investigation of the turbulent flow in the rotating cavity. The non-isothermal flow conditions were also taken into consideration in some investigations (Randriamampianina et al., 1987, Poncet and Schiestel, 2007, Tuliszka-Sznitko et al., 2009a, Tuliszka-Sznitko et al., 2009b). These showed that the rotation-induced buoyancy influences the stability characteristics and the critical conditions. Tuliszka-Sznitko et al., 2009a, Tuliszka-Sznitko et al., 2009b, Tuliszka-Sznitko and Majchrowski, 2010 performed computations of the non-isothermal flow in the rotor/stator cavity, delivering distributions of the local Nusselt numbers along the stator and rotor for different configurations and Reynolds numbers. Pellé and Harmand (2007) performed measurements over the rotor (in the rotor/stator configuration), using a technique based on infrared thermography. A very detailed experimental investigation of the turbulent flow around a single heated rotating disk was performed by Elkins and Eaton (2000) and Littell and Eaton (1991).

The rotor/stator cavity has been also used in investigations of the Rayleigh–Bénard convection with superimposed moderate rotation. The flow in the cavity between two disks heated from below (the Rayleigh–Bénard convection) with superimposed moderate rotation is mostly used as a model problem for predicting geophysical phenomena (solar and giant planetary convection, deep oceanic convection). The Rayleigh–Bénard convection with superimposed rotations has been studied, among others, by Kunnen et al. (2009) and Julien et al. (1996).

In the present paper, we investigate the flow with heat transfer in the rotor/stator annular cavity of the aspect ratio between 2 and 35 and curvature parameters Rm = 1.8, for a wide range of Reynolds numbers. The main motivation of our work is to investigate the influence of aspect ratio and Reynolds number on axial profiles of the Reynolds stress tensor components, the turbulent heat flux components, turbulent Prandtl numbers and other structural parameters. Obtained data can be useful for modeling purposes. The mathematical model and numerical approach are described in Sections 2 Mathematical and geometrical model, 3 Numerical approach. Mean velocity and temperature profiles are presented in Section 4. Section 5 Turbulent velocity characteristics, 6 Profiles of structural parameters is devoted to the turbulent velocity characteristics and the turbulent heat flux components. The profiles of the structural parameters are discussed in Section 6. The local Nusselt number distributions are presented in Section 7. In Section 8 selected results obtained for the Rayleigh–Bénard convection with superimposed rotations (performed in the rotor/stator cavity) are presented and discussed. Conclusions are given in Section 9.

Section snippets

Mathematical and geometrical model

We investigate the non-isothermal flow in the cavity between stationary and rotating disks of the inner and outer radius R0 and R1, respectively. The interdisks spacing is denoted by 2 h (Fig. 1). The rotor rotates at uniform angular velocity Ω = Ωez, where ez is the unit vector on the axis. The flow is described by the Navier–Stokes, continuity and energy equations written in a cylindrical coordinate system (r, ϕ, z). To take into account the buoyancy effects induced by the involved body forces,

Numerical approach

The numerical solution is based on a pseudo-spectral Chebyshev–Fourier–Galerkin collocation approximation. In the time approximation we use a second-order semi-implicit scheme, which combines an implicit treatment of the diffusive terms and an explicit Adams–Bashforth extrapolation for the non-linear convective terms. In the non-homogeneous radial and axial directions we use Chebyshev polynomials with the Gauss–Lobatto distributionsri=cos(πi/NPR)0iNPRzj=cos(πj/NPZ)0jNPZto ensure high

Mean velocity and temperature

In all considered configurations (L = 2–35) the flow is of Batchelor type. That means that the flow consists of two boundary layers on each disk, separated by an inviscid rotating core in which the velocity gradient is weak. The structure of the flow is visible in Fig. 3 where the axial profiles of the azimuthal velocity component obtained for different aspect ratios and different Reynolds numbers Re (or Reh) are presented; profiles are obtained in the middle section of cavity. We can see, that

Turbulent velocity characteristics

The axial profiles of the three Reynolds stress tensor components vv¯1/2/uσ, uu¯1/2/uσ and ww¯1/2/uσ versus the wall coordinate, obtained for different Re (or Reh) and L, are presented in Fig. 6 (middle section of cavity). From Fig. 6 we see that vv¯1/2/uσ overestimates uu¯1/2/uσ about 2.2 times. We see that the maximum values of azimuthal component vv¯1/2/uσ obtained for configurations with different L occur in the range from z+ = 13.2 to z+ = 16 and the maximum values of radial

Profiles of structural parameters

In this section we aim to show that axial profiles of different structural parameters are more uniform than profiles of the Reynolds stress tensor components and the turbulent heat flux components. Let us start our analysis from the turbulent Prandtl number, which is defined as the ratio of the eddy diffusivity for momentum to the eddy diffusivity for heat:Prt=-wv¯v¯/zwT¯T¯/zMany 2D boundary layers investigations show that the turbulent Prandtl number equals 1 in the area near the wall

Local Nusselt number distribution

Fig. 14 shows the distributions of the local Nusselt numbers Nu=αr/λ versus dimensionless radius r obtained for different aspect ratios: L = 9, 11, 25 and 35 (where α denotes the heat transfer coefficient and λ is the thermal conductivity coefficient). For laminar flow α is independent from the radius but increases with azimuthal speed; consequently Nu varies linearly with radius. For turbulent flow α increases with speed and radius. In the present investigations the local Nusselt number is

Rayleigh–Bénard convection with superimposed rotation

The algorithm shortly described in Sections 2 Mathematical and geometrical model, 3 Numerical approach was also used by us for investigating the Rayleigh–Bénard convection with superimposed moderate rotation (the classic Rayleigh–Bénard convection is defined as an unbounded horizontal layer of fluid heated from below). In this flow case the acceleration due to gravity plays vital role and cannot be ignored in the Navier–Stokes (1a–1c). In numerical simulation of Rayleigh–Bénard convection is

Conclusions

We have performed DNS/LES computations of the flow with heat transfer in rotor/stator cavity. Computations have been performed for configurations of a wide range of the aspect ratio L, L = 2–35, and different Reynolds numbers. All considered flow cases are of the Batchelor type: the flow consists of two boundary layers separated by a rotating inviscid core. For the considered Reynolds numbers the stator boundary layer was turbulent whereas the rotor boundary layer was transitional. In the

Acknowledgment

We are grateful to The Poznan Supercomputing and Networking Center, where the computations have been performed.

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